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in classical logic , _ material implication _ or _ material conditional _ is defined by negation and disjunction . specifically , `` if _ p _ then _ q _ '' [ ole_link3][ole_link4]or `` _ _ p _ _ implies _ q _ '' is defined as `` not _ p _ or _ , i.e. , _ p _ @xmath0 _ q _ @xmath1 @xmath2__p _ _ @xmath3 _ q_. it is well known that this definition is problematic in that it leads to `` paradoxes '' such as \1 ) @xmath2__p _ _ @xmath0 ( _ p _ @xmath0 _ q _ ) ( a false proposition implies any proposition ) , \2 ) _ p _ @xmath0 ( _ q _ @xmath0 _ p _ ) ( a true proposition is implied by any proposition ) , and \3 ) ( _ p _ @xmath0 _ q _ ) @xmath3 ( _ q _ @xmath0 _ p _ ) ( for any two propositions , at least one implies the other ) . these formulas can be easily proved as tautologies in classical logic with the _ rule of replacement _ for material implication , i.e. , replacing _ @xmath0 _ q _ with @xmath2__p _ _ @xmath3 _ q_. but these tautologies are counterintuitive so they are called `` paradoxes '' of material implication in classical logic . there are many such paradoxes , see , e.g. , ( bronstein , 1936 ) , and ( lojko , 2012 ) where lists sixteen `` paradoxes '' of material implication . it should be noted that some paradoxes of material implication are even worse than `` paradoxical '' because they are actually wrong . let us take ( _ p _ @xmath0 _ q _ ) @xmath3 ( _ q _ @xmath0 _ p _ ) as an example . using the rule of replacement _ p _ @xmath0 _ q _ = @xmath2__p _ _ @xmath3 _ q _ ( where the equals sign denotes logical equivalence , see 1.2 ) , we have ( _ p _ @xmath0 _ q _ ) @xmath3 ( _ q _ @xmath0 _ p _ ) = ( @xmath2__p _ _ @xmath3 _ q _ ) @xmath3 ( @xmath2__q _ _ @xmath3 _ p _ ) = @xmath2__p _ _ @xmath3 ( _ q _ @xmath3 @xmath2__q _ _ ) @xmath3 _ p _ = @xmath2__p _ _ @xmath3 @xmath4 @xmath3 _ p _ @xmath5 @xmath4 . thus , ( _ p _ @xmath0 _ q _ ) @xmath3 ( _ q _ @xmath0 _ p _ ) is a tautology , meaning that for any propositions _ p _ and _ q _ , it must be that `` _ _ p _ _ implies _ q _ '' or `` _ _ q _ _ implies _ p _ '' ( or both ) . this is absurd since it is possible that it is not the case even if _ p _ and _ q _ are `` relevant '' . for instance , suppose _ n _ is an integer , let _ `` _ _ n _ _ = 1 '' and _ q _ @xmath1 `` _ _ n _ _ = 0 '' , then , since ( _ p _ @xmath0 _ q _ ) @xmath3 ( _ q _ @xmath0 _ p _ ) is a tautology , it must be that `` if _ n _ = 1 then _ n _ = 0 '' is true or `` if _ n _ = 0 then _ n _ = 1 '' is true ( or both are true ) , and this is an absurdity . many efforts has been made to resolve this problem such as _ relevance logic _ which requires that antecedent and consequent be relevant , _ modal logic _ which uses concept of strict implication , _ intuitionistic logic _ which rejects the law of excluded middle , and _ inquisitive logic _ ( see , e.g. , ( ciardelli & roelofsen , 2011 ) ) which considers inquisitive semantics of sentences rather than just their descriptive aspects . these developments of non - classical logic are important to modern logics and other disciplines__. _ _ however , classical logic exists on its own reason , and we are reluctant to discard it , see e.g. ( fulda , 1989 ) . indeed , classical logic , with its natural principles such as the law of excluded middle , is not only fundamental , but also simple and useful . classical logic is an important part of logic education , so it has been an essential part of most logic textbooks . on the other hand , _ implication is a kernel concept in logic_. so it is not satisfying that the both simple and useful classical logic has an unnatural and even wrong definition of material implication appeared in many textbooks . therefore , the motivation of this work is to improve the definition of implication to replace that of the material implication in classical logic , so that 1 ) it is `` natural '' and `` correct '' , 2 ) it keeps the system still `` simple '' , and 3 ) it keeps the classical logic still as `` useful '' . in order to satisfy the second requirement , this work uses only concepts that already exist in classical logic rather than use those as specifically introduced in non - classical logics like relevance logic , modal logic , intuitionistic logic , many - valued logic , probabilistic logic , etc . the third requirement is to prevent developing `` too narrow '' a definition of implication , as bronstein ( 1936 ) commented on e. j. nelson s `` intensional logic '' : `` although his system does avoid the ` paradoxes ' , it does so only by unduly narrowing his conception of implication . '' although classical logics include at least propositional and first - order logic , this work concentrates on classical _ propositional logic _ , as the relevant concepts are the same . there may exist different notations for one thing . for example , @xmath6 _ @xmath7 _ @xmath8 _ @xmath9 _ , @xmath6 _ @xmath7 _ @xmath10 _ @xmath9 _ , _ @xmath7 _ @xmath10 _ @xmath9 _ , _ @xmath7 _ = _ @xmath9 _ , and _ @xmath7 _ @xmath5 _ @xmath9 _ may all mean _ logical equivalence _ between _ @xmath7 _ and _ @xmath9 _ ( i.e. , they have the same truth - value in every model ) , which can also be denoted with the unicode symbol `` left and right double turnstile '' ( u+27da ) . on the other hand , a notation may denotes different things . for instance , the equals sign ` = ' has a variety of usages in different contexts . to prevent ambiguities , the usage of some important notions in this work is explained as follows . @xmath7 _ and _ @xmath9 _ are _ well - formed formulas _ ( hereinafter just referred to as formulas ) in a formal language . the _ equals sign _ ` = ' is used to denote logical equivalence in a logical equation like the form _ @xmath7 _ = _ @xmath9 _ that need not to be a tautology ( it may be just a condition or is conditional ) as in a derivation . the _ identical to _ symbol ` @xmath5 ' is used to denote logical equivalence in a tautology like the form _ @xmath7 _ @xmath5 _ @xmath9 _ , or used in derivations to emphasize that _ @xmath7 _ = _ @xmath9 _ is a tautology . the symbols ` @xmath11 ' and ` @xmath12 ' are used to denote the complementary cases of ` = ' and ` @xmath5 ' , respectively . for example , if _ @xmath7 _ = @xmath2__@xmath9 _ _ , then @xmath2__@xmath7 _ _ = @xmath2@xmath2_@xmath9 _ @xmath5 _ @xmath9_. the equals sign ` = ' is of special importance in this work to denote , e.g. , `` conditional '' logical equivalence as explained . the symbol ` @xmath0 ' is used to denote both traditional material implication and the implication relation defined in this work . the other symbols are used conventionally and unambiguously without explanation . for instance , ` @xmath6[ole_link140][ole_link141 ] ' and ` @xmath13 ' are used to denote `` logically implies ( is logical consequence of ) '' and `` not logically imply ( is not logical consequence of ) '' , respectively ; ` @xmath4 ' denotes any tautology and ` @xmath14 ' denotes any contradiction , etc . the definition of material implication in classical logic is based on that , for propositions _ p _ and _ q _ , `` if _ p _ then _ q _ '' or `` _ _ p _ _ implies _ q _ '' is logically equivalent to `` it is false that _ p _ and not _ q _ '' , and the latter again is logically equivalent to `` not _ p _ or it is this `` logical equivalence '' that leads to the definition _ @xmath0 _ q _ @xmath1 @xmath2__p _ _ @xmath3 _ q _ and hence the _ rule of replacement _ _ p _ @xmath0 _ q _ = @xmath2__p _ _ @xmath3 _ q _ for material implication . this definition makes the material implication a _ truth - functional _ connective , i.e. , the truth - value of the compound proposition _ @xmath0 _ q _ is a function of the truth - values of its sub - propositions . this means that the truth - value of `` if _ p _ then _ q _ '' is determined solely by the combination of truth - values of _ p _ and _ q_. this is unnatural as shown in the following . \1 ) when we know that _ p _ is true and _ q _ is true , can we decide that `` if _ p _ then _ q _ '' is true ( or false ) ? no , not sure . \2 ) when we know that _ p _ is true and _ q _ is false , can we decide that `` if _ p _ then _ q _ '' is true ( or false ) ? yes , we can decide that it must be false . \3 ) when we know that _ p _ is false and _ q _ is true , can we decide that `` if _ p _ then _ q _ '' is true ( or false ) ? no , not sure . \4 ) when we know that _ p _ is false and _ q _ is false , can we decide that `` if _ p _ then _ q _ '' is true ( or false ) ? no , not sure . in only one case , namely the second one , the truth - value of the compound proposition `` if _ p _ then _ q _ '' can be determined by the truth - value combination of _ p _ and _ q_. this indicates that `` if _ p _ then _ q _ '' is not logically equivalent to `` not _ p _ or _ q _ '' of which the truth - value is solely determined by the truth - value combination of _ p _ and _ q_. on the other hand , since `` if _ p _ then _ q _ '' is surely false when _ p _ is true and _ q _ is false , it suggests that when `` if _ p _ then _ q _ '' is true it must not be the case that _ p _ is true and _ q _ is false ( which is equivalent to `` not _ p _ or _ q _ '' ) . thus , it should be that `` if _ p _ then _ q _ '' logically implies `` not _ p _ or _ q _ '' but not vice versa , so that _ p _ @xmath0 _ q _ @xmath12 @xmath2__p _ _ @xmath3 _ q _ since _ p _ @xmath0 _ q _ @xmath6 @xmath2__p _ _ @xmath3 _ q _ but @xmath2__p _ _ @xmath3 _ q_. @xmath13 _ p _ @xmath0 _ q_. some researchers have already pointed out , or addressed the problem , although they might have different motivations or explained it in different ways , such as maccoll ( 1880 ) , bronstein ( 1936 ) , woods ( 1967 ) , dale ( 1974 ) , and lojko ( 2012 ) . therefore , the definition of material implication in classical logic is not only unnatural but also , even more severely , not correct . this is why it leads to some unacceptable results as exemplified in introduction . the use of the ( mistaken ) equivalence _ p _ @xmath0 _ q _ = @xmath2__p _ _ @xmath3 _ q _ makes the classical logic unfortunately defective . in classical logic , `` not _ p _ '' , `` _ _ p _ _ and _ q _ '' , `` _ _ p _ _ or _ q _ '' , and `` if _ p _ then _ q _ '' are all viewed as the same kind of compound sentences formed with operations or functions , called `` logical connectives '' or `` logical operators '' . however , `` if _ p _ then _ q _ '' is actually different . essentially , `` implication '' should not be viewed as an operation but a relation . in mathematics , 1 + 2 is an expression formed by an operation while `` 1 @xmath15 2 '' is a sentence formed by a relation . we can not say that a mathematical expression like 1 + 2 is `` true '' or `` false '' , while we can say that a mathematical sentence like 1 @xmath15 2 is `` true '' . so , in mathematics , usually a function expression has no truth - value while a relation expression has . in propositional logic , a function expression such as @xmath2__p _ _ , _ p _ @xmath16 _ q _ or _ p _ @xmath3 _ q _ , unlike their mathematical counterpart , does have a truth - value , but this is because the output of such a function happens to be a proposition that has a truth - value itself . in contrast , when we say that _ p _ @xmath0 _ q _ is `` true '' or `` false '' , we concern about the implication `` @xmath0 '' itself being `` true '' or `` false '' . this intuition indicates that implication is a relation rather than a function . so , a relation expression is a `` higher level '' sentence than a function expression which if it happens to be a sentence . therefore , it is natural and important to view implication as a relation rather than an operation or function , and of course , the relation to represent the implication is not truth - functional ( woods , 1967 ) . let us now analyze possible relations between any two propositions _ p _ and _ q_. there are three distinct cases as shown using venn diagram in figure 2.2.1 , where the square ` @xmath4 ' represents `` set of all interpretations '' , the circle ` _ _ p _ _ ' represents `` set of interpretations that make _ p _ true '' , and the circle ` _ _ q _ _ ' represents `` set of interpretations that make _ q _ true '' . thus , we have case 1 `` disjoint '' that is characterized by the equation _ p _ @xmath16 _ q _ = @xmath14 ; case 2 `` joint '' that is characterized by the equations @xmath16 _ q _ @xmath11 @xmath14 , _ p _ @xmath16 _ q _ @xmath11 _ p _ , and _ @xmath16 _ q _ @xmath11 _ q _ ; and case 3 `` inclusion '' that is characterized by the equation _ @xmath16 _ q _ = _ p_. the three cases are mutually exclusive except for trivial circumstances such as that _ p _ or _ q _ equals to @xmath14 or @xmath4 . let us focus on case 3 characterized by _ @xmath16 _ q _ = _ p_. in this case , whenever _ p _ is true _ q _ must be true , so it is just the case that `` if _ p _ then _ q _ '' . therefore , it indicates that we can use _ @xmath16 _ q _ = _ p _ to define the implication , noting that _ r _ @xmath1 \{(_p _ , _ q _ ) @xmath17 _ p _ @xmath16 _ q _ = _ p _ } is a binary relation on the set of propositions . this is formalized in section 3 as follows . consider a propositional language with the set of logical connectives \{@xmath4 , @xmath14 , @xmath2 , @xmath16 , @xmath3 , @xmath0}. the semantics of the logical connectives is the same as in standard classical logic except for the implication symbol ` @xmath0 ' that is to be defined . * definition 3.1.1 . * ( propositional language ) let _ p _ be a finite set of [ ole_link7][ole_link8]propositional letters and _ o _ @xmath1 \{@xmath4 , @xmath14 , @xmath2 , @xmath16 , @xmath3 , @xmath0 } be the set of logical connectives . the propositional language _ @xmath18 _ = _ @xmath18_(_p _ , _ o _ ) is the set of formulas built from letters in _ p _ using logical connectives in _ o_. the _ valuation functions _ for the logical connectives , except for the implication symbol ` @xmath0 ' , are defined the same as in standard classical logic . * definition 3.1.2 . * ( semantics of implication ) for any formulas _ @xmath7 _ and _ @xmath9 _ in _ @xmath18 _ , the _ valuation function _ _ @xmath19 _ of the implication formula _ @xmath7 _ @xmath0 _ @xmath9 _ is defined as @xmath20 * proposition 3.1.1 . * ( criterion ) for any formulas _ @xmath7 _ and _ @xmath9 _ in _ @xmath18 _ , it holds that _ @xmath7 _ @xmath0 _ @xmath9 _ iff _ @xmath7 _ @xmath16 _ @xmath9 _ = _ @xmath7 _ iff _ @xmath2__@xmath9 _ _ = @xmath14 iff @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 . * by the semantics given in definition 3.1.2 , _ @xmath7 _ @xmath0 _ @xmath9 _ iff _ @xmath16 _ @xmath9 _ = _ @xmath7_. on the other hand , if _ @xmath7 _ @xmath16 _ @xmath9 _ = _ @xmath7 _ then _ @xmath7 _ @xmath2__@xmath9 _ _ = ( _ @xmath7 _ @xmath16 _ @xmath9 _ ) @xmath16 @xmath2__@xmath9 _ _ @xmath5 @xmath14 , and if _ @xmath7 _ @xmath2__@xmath9 _ _ = @xmath14 then , _ @xmath7 _ @xmath16 _ @xmath9 _ @xmath5 ( _ @xmath7 _ @xmath16 _ @xmath9 _ ) @xmath3 @xmath14 @xmath5 ( _ @xmath7 _ @xmath16 _ @xmath9 _ ) @xmath3 ( _ @xmath7 _ @xmath16 @xmath2__@xmath9 _ _ ) @xmath5 _ @xmath7 _ @xmath16 ( _ @xmath9 _ @xmath3 @xmath2__@xmath9 _ _ ) @xmath5 _ @xmath7 _ @xmath16 @xmath4 @xmath5__@xmath7__. so , we have _ @xmath7 _ @xmath16 _ @xmath9 _ = _ @xmath7 _ iff _ @xmath2__@xmath9 _ _ = @xmath14 , and the latter is equivalent to @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 by _ de morgan s laws_. @xmath21 * remark 3.1.1*. it should be noted that _ @xmath7 _ @xmath0 _ @xmath9 _ iff @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 in this work , while _ @xmath7 _ @xmath0 _ @xmath9 _ = @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ in traditional definition . the implication ` @xmath0 ' is thus a binary relation over _ @xmath18 _ , the set of formulas . this binary relation is determined by a logical equation on _ @xmath18_. by definition 3.1.2 , the truth - value of an implication statement is not determined by the combination of the truth - values of its antecedent and consequent . in other words , the so defined implication relation is non - truth - functional . * proposition 3.1.2 . * ( association to logical implication ) for any formulas _ @xmath7 _ and _ @xmath9 _ in _ @xmath18 _ , _ @xmath7 _ @xmath6 _ @xmath9 _ iff @xmath6 _ @xmath7 _ @xmath0 _ @xmath9 _ * proof . * _ [ ole_link54][ole_link55]@xmath7 _ @xmath6 _ @xmath9 _ means that the set of the interpretations that make _ @xmath7 _ true is a subset of the set of the interpretations that make _ @xmath9 _ true . in other words , there is no interpretation that make _ @xmath7 _ true and make _ @xmath9 _ false , i.e. _ @xmath7 _ @xmath16 @xmath2__@xmath9 _ _ is false in all interpretations , this means _ @xmath2__@xmath9 _ _ @xmath14 , or @xmath6 _ @xmath7 _ @xmath2__@xmath9 _ _ = @xmath14 , i.e. @xmath6 _ @xmath7 _ @xmath0 _ @xmath9 _ by proposition 3.1.1 . @xmath21 * proposition 3.1.3 . * ( equivalence ) for any formulas _ @xmath7 _ and _ @xmath9 _ in _ @xmath18 _ , ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) @xmath16 ( _ @xmath9 _ @xmath0 _ @xmath7 _ ) iff _ @xmath7 _ = _ @xmath9_. * proof . * if _ @xmath7 _ = _ @xmath9 _ , then _ @xmath7 _ @xmath16 _ @xmath9 _ = _ _ @xmath7 _ _ @xmath16 _ @xmath7 _ = _ @xmath7 _ and _ @xmath9 _ @xmath16 _ @xmath7 _ = _ @xmath9 _ @xmath16 _ @xmath9 _ = _ @xmath9 _ , so ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) @xmath16 ( _ @xmath9 _ @xmath0 _ @xmath7 _ ) ; if ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) @xmath16 ( _ @xmath9 _ @xmath0 _ @xmath7 _ ) , then _ @xmath7 _ = _ @xmath7 _ @xmath16 _ @xmath9 _ = _ @xmath9_. @xmath21 * remark 3.1.2*. the definition of implication relation in this work is clearly based on logical equivalence ` = ' , so there is no need to introduce another equivalence symbol such as ` @xmath8 ' . the implication relation defined in 3.1 has some important properties as listed in the following . * proposition 3.2.1 . * ( properties of the implication relation ) let _ @xmath7 _ , _ @xmath9 _ , and _ @xmath22 _ are any formulas in _ @xmath18 _ , the implication relation ` @xmath0 ' given by definition 3.1.2 has the following properties as a binary relation : \1 ) _ @xmath7 _ @xmath0 _ @xmath7 _ ( reflexivity ) ; \2 ) if _ @xmath7 _ @xmath0 _ @xmath9 _ and _ @xmath9 _ @xmath0 _ @xmath7 _ , then _ @xmath7 _ = _ @xmath9 _ ( anti - symmetry ) ; \3 ) if _ @xmath7 _ @xmath9 _ and _ @xmath9 _ @xmath0 _ @xmath22 _ , then _ @xmath7 _ @xmath0 _ @xmath22 _ ( transitivity ) ; \4 ) _ @xmath7 _ @xmath16 _ @xmath9 _ @xmath0 _ @xmath7 _ ( meet ) ; \5 ) _ @xmath7 _ @xmath0 _ @xmath7 _ @xmath3 _ @xmath9 _ ( join ) ; \6 ) @xmath14 @xmath0 _ @xmath7 _ ( bottom ) ; \7 ) _ @xmath7 _ @xmath0 @xmath4 ( top ) . * proof . * from results in 3.1 : \1 ) _ @xmath7 _ @xmath16 _ @xmath7 _ @xmath5 _ @xmath7 _ , so _ @xmath7 _ @xmath0 _ @xmath7 _ ; \2 ) it follows immediately from proposition 3.1.3 ; \3 ) _ @xmath7 _ @xmath0 _ @xmath9 _ and _ @xmath9 _ @xmath0 _ @xmath22 _ , so _ @xmath7 _ @xmath16 _ @xmath9 _ = _ @xmath7 _ and _ @xmath9 _ @xmath16 _ _ @xmath22__= _ @xmath9 _ , thus _ @xmath7 _ @xmath16 _ @xmath22 _ = ( _ @xmath7 _ @xmath16 _ @xmath9 _ ) @xmath16 _ @xmath22 _ = _ @xmath7 _ @xmath16 ( _ @xmath9 _ @xmath16 _ @xmath22 _ ) = _ @xmath7 _ @xmath16 _ @xmath9 _ = _ _ @xmath7 _ _ , therefore _ @xmath7 _ @xmath16 _ @xmath22 _ = _ _ @xmath7 _ _ , so _ _ @xmath7__@xmath0 _ @xmath22 _ ; \4 ) ( _ @xmath7 _ @xmath16 _ @xmath9 _ ) @xmath16 _ @xmath7 _ @xmath5 _ @xmath7 _ @xmath16 _ @xmath9 _ , so _ @xmath7 _ @xmath16 _ @xmath9 _ @xmath0 _ @xmath7 _ ; \5 ) _ @xmath7 _ @xmath16 ( _ @xmath7 _ @xmath3 _ @xmath9 _ ) @xmath5 _ @xmath7 _ @xmath3__@xmath7 _ _ @xmath16__@xmath9 _ _ @xmath5 _ @xmath7 _ , so _ @xmath7 _ @xmath0 _ @xmath7 _ @xmath3 _ @xmath9 _ ; \6 ) @xmath14 @xmath16 _ @xmath7 _ @xmath5@xmath14 , so @xmath14 @xmath0 _ @xmath7 _ ; \7 ) _ @xmath7 _ @xmath16 @xmath4 @xmath5 _ @xmath7 _ , so _ @xmath7 _ @xmath0 @xmath4 . * remark 3.2.1 . * the implication relation ` @xmath0 ' is a _ partial order _ over _ @xmath18 _ by properties 1 ) * * 3 ) , and the _ partial ordered set _ ( _ @xmath18 _ , @xmath0 ) is a _ bounded lattice _ with properties 4 ) * * 7 ) . * proposition 3.2.2 . * ( rule of replacement and inference ) let _ @xmath7 _ and _ @xmath9 _ are any formulas in _ @xmath18 _ , for the implication relation ` @xmath0 ' given by definition 3.1.2 , we have \1 ) _ @xmath7 _ @xmath0 _ @xmath9 _ @xmath6 @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ , \2 ) @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ @xmath6 _ @xmath7 _ @xmath0 _ @xmath9 _ iff @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 iff _ @xmath7 _ @xmath9_. * proof . * it is based on propositions 3.1.1 and 3.1.2 . @xmath22 _ @xmath1 ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) @xmath16 @xmath2(@xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ ) @xmath5 ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) @xmath16 ( _ @xmath7 _ @xmath16 @xmath2 _ @xmath9 _ ) . if _ @xmath7 _ @xmath9 _ is false , then _ @xmath22 _ is false ; if _ @xmath7 _ @xmath9 _ is true , then _ @xmath2 _ @xmath9 _ = @xmath14 , so _ @xmath22 _ is also false . thus _ @xmath22 _ is false in any interpretation , that is _ @xmath22 _ @xmath5 @xmath14 , or @xmath6 _ @xmath22 _ = @xmath14 , or @xmath6 ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) @xmath16 @xmath2(@xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ ) = @xmath14 , i.e. @xmath6 ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) @xmath0 ( @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ ) , that is _ @xmath7 _ @xmath2__@xmath7 _ _ @xmath3 _ @xmath9_. \2 ) if @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 , then _ @xmath7 _ @xmath0 _ @xmath9 _ is always true , so @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ @xmath6 _ @xmath7 _ @xmath0 _ @xmath9 _ ; if @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ @xmath11 @xmath4 , then @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ @xmath6 _ @xmath7 _ @xmath0 _ @xmath9 _ can not hold , since this means that whenever @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ is true then _ @xmath7 _ @xmath0 _ @xmath9 _ must be true so that @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 must be also true . @xmath21 * remark 3.2.2*. this means : 1 ) the traditional _ rule of replacement _ _ @xmath7 _ @xmath0 _ @xmath9 _ = @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ can not be used unless it is known that @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 or _ @xmath0 _ @xmath9 _ is indeed true ; 2 ) the _ rule of inference _ _ @xmath7 _ @xmath0 _ @xmath9 _ @xmath23 @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ can be used in any cases safely . * remark 3.2.3*. since the rule of replacement _ @xmath7 _ @xmath0 _ @xmath9 _ = @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ can not be universally used , defining logical connectives @xmath2 , @xmath16 , or @xmath3 by @xmath0 is not appropriate . consider the most common `` paradoxes '' of traditional material implication mentioned in 1.1 and re - listed here for convenience . \1 ) @xmath2__p _ _ @xmath0 ( _ p _ @xmath0 _ q _ ) ( a false proposition implies any proposition ) , \2 ) _ p _ @xmath0 ( _ q _ @xmath0 _ p _ ) ( a true proposition is implied by any proposition ) , and \3 ) ( _ p _ @xmath0 _ q _ ) @xmath3 ( _ q _ @xmath0 _ p _ ) ( for any two propositions , at least one implies the other ) . now let us check these `` paradoxes '' under the implication relation defined in 3.1 . * proposition 3.3.1 . @xmath7 _ and _ @xmath9 _ are formulas in _ @xmath18 _ , the implication relation ` @xmath0 ' is given by definition 3.1.2 , then we have that @xmath2__@xmath7 _ _ @xmath0 ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) , _ @xmath7 _ @xmath0 ( _ @xmath9 _ @xmath0 _ @xmath7 _ ) , and ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) @xmath3 ( _ @xmath9 _ @xmath0 _ @xmath7 _ ) are not tautologies . * proof . * consider the case that _ @xmath11 @xmath14 , _ @xmath7 _ @xmath11 @xmath4 , _ @xmath9 _ @xmath11 @xmath14 , _ @xmath9 _ @xmath11 @xmath4 , and _ @xmath7 _ @xmath16 _ @xmath9 _ = @xmath14 . \1 ) in this case , _ @xmath7 _ @xmath16 _ @xmath9 _ @xmath11 _ @xmath7 _ , so _ @xmath7 _ @xmath0 _ @xmath9 _ is false , thus @xmath2(@xmath2__@xmath7 _ _ ) @xmath3 ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) = _ @xmath7 _ @xmath11 @xmath4 , therefore , @xmath2__@xmath7 _ _ @xmath0 ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) is false . \2 ) in this case , _ @xmath7 _ @xmath16 _ @xmath9 _ @xmath11 _ @xmath9 _ , so _ @xmath9 _ @xmath0 _ @xmath7 _ is false , thus @xmath2__@xmath7 _ _ @xmath3 ( _ @xmath9 _ @xmath0 _ @xmath7 _ ) = @xmath2__@xmath7 _ _ @xmath11 @xmath4 , therefore , _ @xmath7 _ @xmath0 ( _ @xmath9 _ @xmath0 _ @xmath7 _ ) is false . \3 ) in this case , _ @xmath7 _ @xmath9 _ @xmath11 _ @xmath7 _ and _ @xmath9 _ @xmath16 _ @xmath7 _ @xmath11 _ @xmath9 _ , so both _ @xmath7 _ @xmath0 _ @xmath9 _ and _ @xmath9 _ @xmath0 _ @xmath7 _ are false , thus ( _ @xmath7 _ @xmath0 _ @xmath9 _ ) @xmath3 ( _ @xmath9 _ @xmath0 _ @xmath7 _ ) is false . @xmath21 this means that common `` paradoxes '' of traditional material implication do not exist under the implication relation defined in 3.1 . classical logic such as standard propositional logic is simple and useful except that the problematic definition of material implication making it unfortunately defective . this work defines an implication relation to replace the traditional material implication based on logical equivalence . specifically , _ @xmath7 _ @xmath0 _ @xmath9 _ is defined by the equation _ @xmath7 _ @xmath16 _ @xmath9 _ = _ @xmath7 _ that is equivalent to _ @xmath7 _ @xmath2__@xmath9 _ _ = @xmath14 or @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 . this prevents it from common `` paradoxes '' of traditional material implication , while keeps the system still simple and useful . it becomes clear that the rule of replacement _ @xmath7 _ @xmath0 _ @xmath9 _ = @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ can only be used safely in the case that _ @xmath7 _ @xmath9 _ and @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 are known being true . the definition of the implication relation of this work is very natural so it is much easier to understand , thus it is also beneficial to logic education . several more points are noted as follows about the implication relation defined in this work . \1 ) it should distinguish truth - values \{true , false } from tautology and contradiction symbols \{@xmath4 , @xmath14 } , since the latter are not truth - values but ( special ) propositions . so , @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ is true ( or its truth - value equals `` true '' ) is not the same thing as @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 . \2 ) defining _ @xmath7 _ @xmath0 _ @xmath9 _ by @xmath2__@xmath7 _ _ @xmath3 _ @xmath9 _ = @xmath4 is rational , since `` an alternative sentence expresses not only 1@xmath24 our knowledge of the fact that one at least of the alternants is true and 2@xmath24 our ignorance as to which of them is true , but moreover 3@xmath24 our readiness to infer one alternant from the negation of the other . '' ( ajdukiewicz , 1978 ) . \3 ) @xmath14 @xmath0 _ @xmath7 _ and _ @xmath7 _ @xmath0 @xmath4 are not paradoxical . they can be interpreted to intuition like `` if an always - false - thing is true , then anything is true '' and `` an always - true - thing is true without any premises '' . the similar is also mentioned , e.g. , in ( ceniza , 1988 ) . the corresponding formal system and the soundness and completeness of the system under the implication relation defined in 3.1 , are not investigated in this work . : : k. ( 1978 ) . conditional statement and material implication ( 1956 ) . in k. ajdukiewicz , _ the scientific world - perspective and other essays , 19311963 _ ( j. giedymin , trans . , pp . 222238 ) . dordrecht , holland : d. reidel publishing caompany . : : d. j. ( 1936 , apr . ) . the meaning of implication . _ mind , new series , 178 _ , pp . 157180 . : : c. r. ( 1988 ) . material implication and entailment . _ notre dame journal of formal logic , 29_(4 ) , pp . 510519 . : : i. , & roelofsen , f. ( 2011 ) . inquisitive logic . _ journal of philosophical logic , 40 _ , pp . doi:10.1007/s10992 - 010 - 9142 - 6 : : a. j. ( 1974 , jan . ) . a defence of material implication . _ analysis , 34_(3 ) , pp . : : v. ( 2003 ) . which notion of implication is the right one ? from logical considerations to a didactic perspective . _ educational studies in mathematics , 53_(1 ) , pp . 534 . : : j. s. ( 1989 , mar . ) . material implication revisited . _ the american mathematical monthly , 96_(3 ) , pp . : : p. ( 2012 ) . paradoxes of material implication and non - classical logics . in p. lojko , _ inquisitive semantics and the paradoxes of material implication . master s thesis _ ( pp . 3050 ) . universiteit van amsterdam , amsterdam . : : h. ( 1880 , jan . ) . symbolical reasoning . _ mind , 5_(17 ) , pp . : : j. ( 1967 , jul . ) . is there a relation of intensional conjunction ? _ mind , new series , 76_(303 ) , pp .

simple and useful classical logic is unfortunately defective with its problematic definition of material implication . this paper presents an implication relation defined by a simple equation to replace the traditional material implication in classical logic . common `` paradoxes '' of material implication are avoided while simplicity and usefulness of the system are reserved with this implication relation . * keywords . * implication ; material implication ; conditional ; relation ; classical logic ; propositional logic ; paradox * defining implication relation for classical logic * fu li school of software engineering , chongqing university , chongqing , china fuli@cqu.edu.cn

cataclysmic variables ( cvs ) are short period ( p@xmath2 typically @xmath3 d ) , semi - detached binary systems in which matter is transferred through the @xmath4 point from a roche - lobe - filling , lower main sequence dwarf ( the secondary star ) , into an accretion disk around the white dwarf ( wd ) primary star . in most nonmagnetic cvs , energy output from the disk dominates the optical x - ray luminosity ; hence , the properties of the disk ( which are primarily governed by the mass transfer rate @xmath5 in the case of an optically thick disk ) determine most of the observed properties of the entire system . uu aquarii ( @xmath6 ) was identified as a variable star early in this century ( beljawsky 1926 ) . as observations of uu aqr have been accumulated , its classification has evolved from a long - period semi - regular variable ( payne - gaposchkin 1952 ) , to a low @xmath5 , optically thin disk , dwarf nova type cv ( volkov , shugarov , & seregina 1986 ) , to a high @xmath5 , optically thick disk , novalike cv ( diaz & steiner 1991 , henceforth ds91 ; baptista , steiner , & cieslinski 1994 , henceforth bsc94 ) . volkov et al . ( 1986 ) and goldader & garnavich ( 1989 ) observed deep ( @xmath7 mag ) eclipses in uu aqr , and established an orbital period of p@xmath8 d ( @xmath9 h ) . this places uu aqr in the 34 hr period range just above the 23 hr cv period gap ( thought to be due to the transition from orbital angular momentum loss primarily via magnetic braking for p@xmath10 hr to loss due to gravitational radiation for p@xmath11 hr ; king 1988 ) . among novalike cvs , the 34 hr period range is occupied almost solely by the group of systems known as the _ sw sextantis stars _ ( e.g. , thorstensen et al . 1991 ) , which are characterized by high inclination , single - peaked optical emission lines ( in contrast to the double - peaked lines expected in high inclination disk systems ) , orbital - phase - dependent absorption in the balmer lines , and a phase offset in the emission line radial velocity curves ( implying non - uniform emission from the disk ) . as a result of their multiwavelength eclipse mapping study , baptista , steiner , & horne ( 1996 ; henceforth bsh96 ) conclude that uu aqr is most likely an sw sex star ( a conclusion that we share based on the results presented here see [ s - disc ] ) . bsc94 obtained photometric data of uu aqr on 28 nights during 19881992 . they discovered that this system , like many other novalike cvs ( dhillon 1996 ) , exhibits occasional transitions from high to low brightness states . the amplitude of the state change in uu aqr , however , is only @xmath12@xmath13 mag , which is small compared to other novalike cvs ( e.g. , bh lyncis : @xmath14mag@xmath7@xmath15 , hoard & szkody 1997 ; dw ursae majoris : @xmath14mag@xmath16@xmath17 , dhillon , jones , & marsh 1994 ) . the shape of the eclipse profile in uu aqr does change significantly from the high to low states , and bsc94 concluded that the main difference between the two states is the presence of a bright spot on the outer edge of the disk in the high state . both haefner ( 1989 ) and ds91 determined radial velocity curves for the h@xmath18 line in uu aqr ; both noted the strongly asymmetric , phase - dependent profile of the line . haefner ( 1989 ) found a velocity semi - amplitude of 160 km s@xmath19 from the wings of the h@xmath18 line , while ds91 obtained 120 km s@xmath19 using the same method . ds91 determined system parameters of @xmath20 ( m@xmath21/m@xmath22 ) @xmath23 with m@xmath24m@xmath25 . more recently , the detailed eclipse - mapping study of uu aqr by bsc94 and bsh96 provided a photometric model that requires @xmath26 km s@xmath19 , substantially smaller than estimates made from the h@xmath18 emission line . ( this is not surprising as low excitation emission lines , which form primarily in the outer parts of the disk , can not be expected to accurately map the orbital motion of the wd ; e.g. , still , dhillon , & jones 1995 , shafter , hessman , & zhang 1988 . ) bsc94 derive system parameters of @xmath27 with m@xmath28m@xmath25 and an inclination @xmath29 . we present here the results of an extensive spectroscopic monitoring campaign on uu aqr . previously published time - resolved spectra ( ds91 ) were obtained during a low state , while our spectra explore the high state of the system ( see [ s - ew ] ) . in addition , the spectra of ds91 covered only 200 around h@xmath18 , whereas our spectra span h@xmath30 to h@xmath31 on all nights , with additional coverage from h@xmath18 to @xmath32 on one night . our results include a line profile simulation that reproduces the gross characteristics of the orbital - phase - resolved emission line behavior of uu aqr in terms of several discrete emission and absorption components ( [ s - linesim ] ) , doppler tomography resolved over half an orbit of the cv ( [ s - halfdt ] ) , and a qualitative model for the system ( [ s - model ] ) . we observed uu aqr using the 1.9 m radcliffe telescope at the south african astronomical observatory ( saao ) on 1995 august 18 ut , 1995 august 1820 ut , and 1995 september 1926 ut . the observations were made using the two - channel reticon photon counting system ( rpcs ; menzies & glass 1996 ) with a 1200 line mm@xmath19 grating ( giving a resolution of 1.8 ) . exposures were 90 s with a total time per spectrum of @xmath33 s ; a total of 1492 spectra were obtained . sky was recorded in one channel of the rpcs and sky+object in the other channel . following normalization via bias - subtraction and division by a tungsten lamp flat - field , the spectra were extracted by simply subtracting the sky channel from the sky+object channel . the spectra were wavelength - calibrated by comparison to spectra of a cu - ar arc lamp standard ; the usable wavelength range was 42505000 . they were flux - calibrated via observations of the standard stars l1788 and l7379 ( hamuy et al . 1994 ) . in order to increase the signal - to - noise , the reduced rpcs spectra were averaged into phase bins of width @xmath34 in blocks of 2 consecutive nights ( see table [ t - log ] ) , using the photometric ephemeris of bsc94 ( we will refer to these as the `` binned rpcs data sets '' ) . we also constructed sets of phase - binned ( @xmath34 ) spectra using @xmath35 of the total set of rpcs spectra ( some of the rpcs spectra were rejected because they were extremely weak , possibly due to slit losses ; we will refer to this as the `` combined rpcs data set '' ) . we also obtained spectra of uu aqr on 1995 october 1213 ut using the double imaging spectrograph ( dis ) on the apache point observatory 3.5 m telescope ( e.g. , see hoard & szkody 1997 ) . exposure times were 300 s with a total time of @xmath36 s per spectrum ; a total of 65 spectra were obtained . the `` blue '' side of dis covered a wavelength range of 42005000 and the `` red '' side covered 5800 - 6800 , with an overall resolution of @xmath7@xmath37 . the spectra were extracted and calibrated using standard iraf routines ( massey , valdes , & barnes 1992 ) ; they were wavelength - calibrated via comparison with a he - ne arc lamp standard . fine corrections to the wavelength calibration were made by examining the positions of night sky lines in the spectra . both the blue and red spectra were flux - calibrated using spectra of the standard stars feige 110 and bd + 28@xmath38 4211 ( massey et al . 1988 ) . the spectroscopic observations of uu aqr are summarized in table [ t - log ] . [ t - log ] the average spectrum of uu aqr , outside orbital phases 0.9 to 0.1 ( the eclipse ) and uncorrected for any orbital motion , is shown in figure [ f - avsp ] . each blue spectrum from the binned rpcs data sets listed in table [ t - log ] is plotted in the figure . the spectrum shows prominent balmer emission lines and several weaker emission lines ( @xmath39 , @xmath40 , @xmath41 , @xmath32 , @xmath42 ) . the @xmath43 emission line is weak and blended with a / emission complex of similar strength . all of the emission lines are single - peaked ( this is true for the individual spectra as well for example , see fig . [ f - simprof ] so this is not just an effect of smearing of the line profiles in the average spectrum ) . the blue spectrum is quite constant over the @xmath44 month span of the observations . the continuum from 4200 to 6800 follows a power law , @xmath45 with @xmath46 . this is somewhat smaller than the power law index @xmath47 expected from both theoretical calculations in which all disk elements radiate as blackbodies ( pringle 1981 ) and observations of dwarf novae at outburst ( szkody 1985 ) . a relatively flat spectral energy distribution is considered typical of novalike cvs ( oke & wade 1982 ) , and may be related to departures of the disk structure from simple theoretical models and/or self - eclipse of the inner disk in high inclination systems ( e.g. , rutten , van paradijs , & tinbergen 1992 ) . radial velocities were measured from the wings of the balmer emission lines in all of the binned rpcs data sets , and from the balmer and helium lines in each of the unbinned dis spectra , using the double gaussian fitting technique described by shafter , szkody , & thorstensen ( 1986 ) and schneider & young ( 1980 ) . before applying this technique to each line , a section of the spectrum around the wavelength of the line center ( @xmath48 km s@xmath19 ) was converted to a uniform ( i.e. , linear ) velocity scale with a step - size of the local velocity dispersion , @xmath49 where @xmath50 is the speed of light ( km s@xmath19 ) , @xmath51 is the wavelength of the line center ( ) , and @xmath52 is the wavelength dispersion ( @xmath530.8 for the rpcs spectra , @xmath531.3 for the red dis spectra , and @xmath531.6 for the blue dis spectra ) . for the balmer and @xmath32 lines , the fwhm of the gaussians was set at twice the local velocity resolution of the spectra for each line ( fwhm@xmath53 160 km s@xmath19 at h@xmath30 to 180 km s@xmath19 at h@xmath31 in the rpcs spectra ; fwhm@xmath53 120 km s@xmath19 at h@xmath18 to 230 km s@xmath19 at h@xmath31 in the dis spectra ) . the gaussian separation @xmath54 was varied in steps of 100 km s@xmath19 from 500 km s@xmath19 to 2500 km s@xmath19 . in order to obtain usable velocities from the weaker @xmath39 and @xmath43 lines ( in the dis spectra ) , the fwhm of the gaussians was increased to 400 km s@xmath19 . the equation @xmath55\ ] ] was fitted to the velocity data obtained from the optimum value of the gaussian separation for each line . because we are comparing spectra obtained at different times , the velocities from each data set were shifted by the appropriate heliocentric correction prior to fitting . representative samples of the diagnostic diagrams ( from the dis data ) used to determine the optimum gaussian separation are shown in figure [ f - diags ] . for the balmer lines , the diagrams are well - behaved and we chose @xmath56 km s@xmath19 . the diagrams for the helium lines , on the other hand , are rather more uncertain . all three he lines display a trough in the fractional uncertainty of the velocity semi - amplitude , @xmath57 , between @xmath58 km s@xmath19 and @xmath59 km s@xmath19 , indicating that this range of gaussian separations best samples the line wings away from the highly variable line cores but before the adjacent continuum contaminates the wings significantly . however , the values of the system parameters ( especially @xmath60 ) still fluctuate significantly over this range of @xmath54 . the @xmath32 line is the best - behaved of the he lines , and we have used @xmath56 km s@xmath19 for this line also , based on the overall shapes of its @xmath57 , @xmath60 , and @xmath61 curves , which are similar to those of the balmer lines . the values ( and @xmath62 uncertainies determined from monte carlo simulations ) of the uu aqr velocity parameters for the balmer and @xmath32 lines are listed in table [ t - vels ] for each of the data sets . [ t - vels ] for @xmath39 and , the velocity parameter values between @xmath58 km s@xmath19 and @xmath59 km s@xmath19 ( where @xmath57 is smallest ) oscillate around a roughly flat mean trend , which makes it difficult to pick an optimum gaussian separation . instead , we have computed a weighted average of the parameter values in this range of @xmath54 using @xmath63 where @xmath64 is the uncertainty in parameter @xmath65 at a given @xmath54 as determined from a monte carlo simulation . this provides a representative velocity solution , rather than relying on any one of the individual solutions ( which might lie above or below the general trend ) . the average parameters for these two lines are also listed in table [ t - vels ] . the balmer , @xmath39 and @xmath32 , and @xmath43 radial velocity curves obtained from the dis spectra are shown in figure [ f - rvel ] , along with the best fit sine functions . for the @xmath39 and lines , we have plotted the radial velocities for @xmath56 km s@xmath19 with the sine function produced by the average velocity parameters . the rpcs radial velocity curves for h@xmath30 and h@xmath31 do not differ significantly from the dis curves , so are not plotted . the he lines in the rpcs data sets were generally too weak to yield reliable wing velocity curves . all of the lines have positive phase offsets ( @xmath66 ) corresponding to a delay of the red - to - blue velocity crossing of the emission lines relative to photometric phase 0.0 . the phase offsets range from @xmath67@xmath68 in the balmer and lines , to @xmath69 in the line ( again , however , we can not rule out the possibility of a skewed result owing to blending of the line ) . this implies that the main source of emission is off - center in the disk . if we consider the systemic velocities ( @xmath31 ) as a function of time , then the h@xmath30 and h@xmath31 lines display similar behavior . in both lines , the gamma velocity decreases from data set # 1 to # 2 . in h@xmath30 this decrease continues through data set # 3 , then @xmath31 has its maximum value in data set # 4 ; in h@xmath31 the maximum value is reached in data set # 3 . the value of @xmath31 then decreases following the maximum for each line , until a minimum value ( @xmath70 for h@xmath30 and several of the other lines ) is reached in data set # 7 . this pattern does not appear to correlate with the smaller ( @xmath71 ) , somewhat more random fluctuations of the other velocity parameters ( @xmath60 and @xmath66 ) . this suggests that the systemic velocity is influenced by a separate emission region than the velocity semi - amplitude and phase offset . for example , the presence of an emission component from a ( non - rotating or only slightly rotating ) disk wind could systematically skew the zero level of the radial velocities without overtly disrupting the rotational behavior of underlying emission from the disk . variability of the flux of emitting material in this wind might then account for the changing @xmath31 values in uu aqr . the semi - amplitude of the h@xmath30 radial velocity ranges from @xmath72@xmath73 km s@xmath19 , with the best - determined value ( @xmath74 ) from the dis data , @xmath75 km s@xmath19 . the variability of the @xmath60 value may be related to small changes in the location and/or structure of the off - center emitting region that dominates the balmer lines . for the h@xmath18 line , the dis spectra give @xmath76 km s@xmath19 . this is slightly smaller than the h@xmath18 wing velocity @xmath77 km s@xmath19 measured by haefner ( 1989 ) , but larger than that found by ds91 , @xmath78 km s@xmath19 . in general , the balmer lines and ( especially ) the line have velocity semi - amplitudes larger than the nominal wd orbital velocity predicted by the photometric model of bsc94 , @xmath79 km s@xmath19 , although many of the h@xmath30 and h@xmath31 ( but not h@xmath18 or ) velocities are within @xmath80 of the bsc94 value . the line is often thought to originate in the hotter region close to the wd and , hence , to be a good indicator of the wd orbital velocity . however , both its large difference from the independently determined value of bsc94 and its large phase offset argue against adopting our value for @xmath81 as @xmath82 . thus , we will refrain from computing additional system parameters and will instead accept those determined for uu aqr by bsc94 from their multicolor eclipse mapping investigation ( i.e. , @xmath83 , m@xmath84m@xmath25 , m@xmath85m@xmath25 , and @xmath86 ) . we note for the sake of completeness that the effect on the derived system parameters of adopting a larger wd orbital velocity would be to increase @xmath87 and decrease @xmath88 ( e.g. , garnavich et al . 1990 ) . we measured equivalent widths ( ews ) for all 65 of the dis spectra and for all of the spectra from the binned rpcs data sets . the ew curves for the dis spectra are shown in figure [ f - ew ] ; the curves for the rpcs spectra do not differ in any significant way from the dis curves , so are not shown . the mean ews and @xmath62 uncertainties outside of eclipse @xmath89 for the h@xmath30 , h@xmath31 , @xmath39 , and @xmath43 emission lines of uu aqr are listed in table [ t - ew ] for each of the binned data sets . [ t - ew ] note that the line is blended with the adjacent / emission complex and the ew measures the flux from both . the mean ews of all of these lines were essentially constant from 1995 august to 1995 october . we note , however , that data set # 7 ( which has the smallest emission line systemic velocities see table [ t - vels ] ) has the smallest mean balmer and ews and the largest mean ew . this might be explained if the balmer and emission contains a variable wind emission component that produces a positive systemic velocity when present ( as discussed in [ s - rvel ] ) , but is absent or reduced in amplitude in data set # 7 . outside of eclipse , the ews go through roughly sinusoidal modulations on the orbital period , with minima at @xmath900.70.8 . this modulation is most apparent in the h@xmath18 line , where it has a full amplitude of @xmath91 ( see top right panel of fig . [ f - ew ] ) . this modulation is not seen in broadband ( i.e. , continuum ) light curves of uu aqr ( see fig . [ f - lc ] and text below ) , indicating that it is due to variability in the emission lines . in the novalike cvs bh lyn ( hoard & szkody 1997 ) and pg 0859 + 415 ( hoard & szkody 1996a ) , similar modulation of the ews was interpreted as due to absorption in a vertically extended `` bulge '' on the edge of the disk . the presence of such bulges in the disk edge is predicted at phases of 0.2 , 0.5 , and/or 0.8 by numerical simulations of accretion stream - disk interaction ( e.g. , hirose , osaki , & mineshige 1991 ; meglicki , wickramasinghe , & bicknell 1993 ) . absorption by a bulge at @xmath92 could explain the uu aqr ew modulation . the most obvious feature of the ew curves is the presence of a large peak around @xmath93 in all of them . the height of this peak varies from curve to curve ; among the balmer series in particular , its amplitude is largest for h@xmath18 ( @xmath5355 ) and smallest for h@xmath31 ( @xmath5320 ) . the eclipse peaks reach maximum amplitude at @xmath94 but are asymmetrically shaped , with a steep rise before @xmath94 ( @xmath95@xmath96 ) and a longer decline ( @xmath97@xmath98 ) . ds91 reported no change in the emission line ews during eclipse , but their spectra were obtained in 1988 october , when bsc94 determined that uu aqr was in a low accretion state . yet , the narrowband ( 50 ) h@xmath18 light curve of uu aqr obtained by bsh96 during another low state ( in 1992 ) is uneclipsed relative to light curves in both the broadband @xmath99 and adjacent narrowband @xmath100 filters . this would produce a peak in the h@xmath18 ew as seen in our high state data . the lack of eclipse in the emission lines implies that there is an extended source of emission that remains partially visible during eclipse ; however , the presence of such a region does not appear to be strongly tied to the brightness state in uu aqr . we acquired ccd photometry of uu aqr at saao on the nights of 1995 august 0102 and 0607 ut , concurrent with binned rpcs data sets # 1 and # 2 ( see table [ t - log ] ) . a total of 570 measurements were obtained , with typical time resolution of @xmath101 s , and calibrated to strmgren b magnitudes via comparison to observations of e - region standards ( e.g. , cousins 1987 ) obtained under photometric conditions . the eclipse profile in the resultant light curve of uu aqr ( shown in fig . [ f - lc ] ) is most similar in shape to that of the high state light curves of bsc94 ( compare to their fig . 5 ) . the small differences in the profile from orbit to orbit immediately before and after eclipse are consistent with those seen by bsc94 . the asymmetry of the ew peaks is reflected in the shape of the high state eclipse light curve , which shows a steep , smooth ingress to the eclipse minimum over @xmath102 , followed by a longer egress , @xmath103 , back to the quiescent level . the eclipse egress has a sharp shoulder @xmath102 after the minimum . bsc94 interpreted the asymmetry of the eclipse as an effect of the bright spot , which must have a large azimuthal extent on the edge of the disk . the projected surface area of the spot along the line - of - sight to the system decreases as the orbital phase approaches @xmath94 , leading to a rapid spot eclipse ingress . after the eclipse minimum , the projected spot area along the line - of - sight increases as @xmath104 increases , leading to a longer egress from the eclipse of the spot ( also see rutten et al . 1992 ) . in addition , the light curve of uu aqr during 1995 from the indiana automated ccd photometric telescope ( `` roboscope ; '' e.g. , honeycutt & turner 1992 , honeycutt et al . 1994 ) shows that the system was bright ( @xmath105 ) during our spectroscopic observations ( robertson 1997 ) . as noted in [ s - intro ] , the difference in brightness between high and low states of uu aqr is small , but the eclipse profile shapes in the two states show significant differences ; hence , the latter is a more precise diagnostic of the accretion state in uu aqr than the former . based on the comparison of the system brightness , ew behavior , and eclipse profile to previously published observations , we conclude that uu aqr was in a high state during our observations . we used the fourier - filtered back - projection algorithm described by horne ( 1991 ) and others ( e.g. , marsh & horne 1988 ; kaitchuck et al . 1994 ) to produce doppler tomograms of the velocity distribution of emitting material in uu aqr . figures [ f - hbdt]a f show the h@xmath30 tomograms for the six binned rpcs data sets ; figure [ f - hbdt]g is the h@xmath30 tomogram constructed from the dis data ( binned to @xmath34 ) ; figure [ f - hbdt]h is the h@xmath30 tomogram for the combined set of rpcs spectra . the tomogram for 1995 sep 1920 ut ( fig . [ f - hbdt]d ) is of poorer quality than the others because the binned data set used to construct it ( # 4 see table [ t - log ] ) contains substantially fewer total spectra than the other rpcs data sets . as might be expected following the similar behavior of the average spectra ( see [ s - avsp ] ) and emission line equivalent widths ( [ s - ew ] ) , the h@xmath30 tomogram does not change significantly from 1995 august to 1995 october this implies that the changes in the radial velocity parameters of the emission lines must be due to small changes in the structure of a discrete emitting region rather than the whole disk . there are two main features in the tomograms : ( 1 ) a roughly circular region of diffuse emission , centered around the velocity origin and extending to a radius of @xmath53600800 km s@xmath19 ; and ( 2 ) a localized region of strong emission centered at @xmath106 to @xmath107 km s@xmath19 and @xmath108 to @xmath109 km s@xmath19 . the latter region is suggestive of the bright spot at the impact site of the accretion stream with the edge of the disk , but is somewhat inconsistent with the expected velocity position . such emission would normally lie further inside the @xmath110 quadrant of the tomogram ( see , for example , fig . 11 of kaitchuck et al . 1994 ) . in figures [ f - hbdt]g and [ f - hbdt]h , we have plotted the positions of the secondary star roche lobe and the accretion stream trajectory , using the system parameters determined by bsc94 ( see [ s - intro ] ) . the strong emission region does not coincide with the stream trajectory as would be expected for normal bright spot emission . by arbitrarily reducing the wd orbital velocity from the value of @xmath111 km s@xmath19 calculated by bsc94 to @xmath112 km s@xmath19 , we can force the accretion stream to pass roughly through the center of the emission region in the tomograms . however , this requires the masses of the stellar components to be unreasonably small ( @xmath113 , @xmath114 ) and , consequently , does not provide a useful explanation for the discrepancy between stream and emission locations . the strong emission region also appears to be elongated towards the @xmath115 quadrant ; several of the tomograms ( notably a , b , g , and h ) show a `` tail '' of emission which extends from this region to the @xmath116 axis , intersecting at @xmath117 km s@xmath19 . this velocity is comparable to an estimate of the orbital velocity of the @xmath4 point around the center - of - mass assuming the mass ratio calculated by bsc94 . indeed , the velocity at the end of the emission tail on the @xmath116-axis coincides with that of the tip of the secondary star roche lobe ( i.e. , the @xmath4 point ) as plotted in figures [ f - hbdt]g and [ f - hbdt]h ; the tail itself follows the expected trajectory of the accretion stream . none of the h@xmath30 tomograms contains a distinct ring of emission ( the signature of an accretion disk ; e.g. , see fig . 24 of kaitchuck et al . 1994 ) , although there is some indication of such a structure in figures [ f - hbdt]c and [ f - hbdt]f . figure [ f - hbdt]g contains the most obvious hint of an emission ring . the lack of a prominent disk signature in the tomograms is not surprising given the single- rather than double - peaked emission lines in the spectrum of uu aqr ; this may be consistent with only a small disk being present , or only weak emission from a larger disk , or emission from a non - disk source ( e.g. , a wind ) that masks the disk emission . the h@xmath31 and , for the dis spectra , h@xmath18 tomograms are very similar to the h@xmath30 tomograms ( the wavelength coverage of the rpcs spectra does not extend to h@xmath18 ) . representative examples of the tomograms of these lines , constructed from the dis data , are shown in figure [ f - disdt]a b . the trailed spectrum used to construct the h@xmath31 tomogram is shown in figure [ f - trails ] ( the trailed spectra for h@xmath18 and h@xmath30 are shown in fig . [ f - linesim ] ) . there is a somewhat more distinct disk - ring of emission in the h@xmath31 tomogram than in the other balmer tomograms ; the disk may be more optically thin to the more energetic balmer radiation ( i.e. , at hotter temperatures ) . the h@xmath31 ring emission is strongest along an arc starting at the strong emission region in the @xmath110 quadrant and trailing counter - clockwise in the tomogram into the @xmath118 quadrant , which suggests that the disk edge is nonuniform ( in temperature , or density , or local turbelent velocity , etc . ) . the spectra binned at a given phase from the dis data set have somewhat higher signal - to - noise than the corresponding binned rpcs spectra . thus , we also attempted to construct doppler tomograms for the weaker @xmath39 and @xmath43 lines in the dis spectra . these are shown in figure [ f - disdt]c d , and the trailed spectra for these lines are shown in figure [ f - trails ] . the high velocity regions of the tomogram are severely contaminated by noise in the continuum adjacent to the line . however , there are still two features of note in the tomogram : ( 1 ) a region of enhanced emission on the @xmath119 axis , consistent with the location of the one seen in the balmer tomograms ; and ( 2 ) a centralized lack of emission and slightly enhanced arc of emission spanning the @xmath120 quadrants that are suggestive of the ring signature of disk emission . the tomogram is practically indistinguishable from noise , although there may be a slight enhancement of the emission level in the @xmath121 quadrant ( corresponding to the location of the strong emission region in the other tomograms ) . the poor quality of the tomogram is probably the result of both the weakness of the line and its severe blending with the adjacent / complex . the only other published tomogram for uu aqr is that of the h@xmath30 line shown in the cv tomography atlas of kaitchuck et al . ( 1994 ) , and it is significantly different from our h@xmath30 tomograms . the strong emission region in the kaitchuck et al . tomogram is displaced upward to @xmath122 to @xmath123 km s@xmath19 and there is a much more prominent ring of disk emission ( centered at @xmath124 km s@xmath19 ) than in our tomograms . overall , the kaitchuck et al . tomogram of uu aqr is much more similar to that of a dwarf nova type cv with a prominent , optically thin disk and a discrete bright spot at the expected site of the accretion stream impact with the outer edge of the disk . the kaitchuck et al . tomogram was constructed using data obtained in 1988 october , during one of the photometric low states classified by bsc94 . thus , we may attribute the differences in the tomograms to differences in the accretion state of uu aqr when the spectra were obtained : low for kaitchuck et al . , high for our data . the eclipse mapping of bsc94 shows that the disk in uu aqr has a larger radius in the high state than in the low ; however , the increased mass transfer in the high state may produce a strong wind emission component that masks the disk emission . the consistency of our h@xmath30 tomogram from 1995 august to 1995 october implies that the physical conditions in the uu aqr disk were very stable during the @xmath44 month span of our observations . as a simple model , cv emission line profiles can be simulated as the sum of several component profiles with the appropriate radial velocity vs. orbital phase behavior expected for various emission regions in the system ( e.g. , accretion stream , bright spot , wd , etc . ) . in this manner , hellier & robinson ( 1994 ) and hellier ( 1996 ) have had some success at reproducing the general line profile behavior of sw sex stars ( specifically , px andromedae and v1315 aquilae ) . we have applied a similar technique to simulating the balmer emission lines in uu aqr . figure [ f - linesim ] shows trailed spectrograms and doppler tomograms for the observed h@xmath18 ( dis data ) and h@xmath30 ( combined rpcs data ) emission lines of uu aqr , and a simulated balmer line . the velocity and phase resolution of the simulated data was made comparable to that of the observed data . our simulated line profiles are made up of five components , described here : * double - peaked keplerian disk emission : * this is the only one of the components that is _ not _ represented by a gaussian profile ; instead , the disk profile is calculated as in robinson , marsh , & smak ( 1993 ) , using an emissivity index @xmath125 ( small changes in @xmath30 around this value do not have a significant effect on the simulation owing to the relatively small amplitude of the disk profile and the qualitative nature of the fitting process see below ) . we used the system parameters and wd orbital velocity for uu aqr derived by bsc94 to define the shape and orbital phase behavior of the disk profile . we initially assumed inner and outer disk radii of 0.015r@xmath25 and 0.4r@xmath25 ( bsc94 ; harrop - allin & warner 1996 ) . subsequent comparison with the observed profiles ( see fig . [ f - simprof ] ) caused us to pick 0.045r@xmath25 and 0.45r@xmath25 as final inner and outer disk radii ( the latter value is @xmath126% of the distance from the wd to the @xmath4 point ) . these disk radii are used throughout the simulation . the amplitude of the disk profile in the simulation is 0.40 above the continuum level ( which has been normalized to a value of 1.00 in the observed spectra ) . * single - peaked wd / disk wind emission : * this is represented by a gaussian function with fwhm of 1000 km s@xmath19 centered at the midpoint of the disk profile . it is assumed to follow the orbital motion of the wd . the amplitude of the wind component is 0.80 above the continuum . the amplitudes of both the disk and wind components were artificially enhanced by a factor of 2.25 during phases 0.980.05 to emulate the general appearance of the observed balmer emission lines during eclipse . we have not explicitly calculated an eclipse profile for the simulated line ( which would require determination of the precise 3-d locations of emission and continuum regions in the system ) . hence , the treatment of the eclipse phases shown in the simulated trailed spectrogram in figure [ f - linesim ] is , at best , only an approximation of the actual eclipse behavior . ( we note , however , that even this simple approximation yields line profiles that are similar to the observed profiles during eclipse see fig . [ f - simprof ] although it is not clear whether or not this is merely coincidental . ) since the data from the eclipse phases are excluded from the creation of a doppler tomogram , any incorrect treatment of the simulated eclipse behavior has no effect on the simulated tomogram . * accretion stream emission : * the free - fall velocities along the accretion stream trajectory from the l@xmath127 point were calculated as in lubow & shu ( 1975 ) . gaussian profiles with fwhm of 25% of the local stream velocity were summed along a specified length of this trajectory to simulate emission from the material in the stream . in addition , the keplerian rotational velocities of the disk material underlying the stream trajectory were calculated , to provide the option of simulating disk material excited to emission through interaction with a stream that overflows the disk from its initial impact site . in the case of uu aqr , we have assumed that the stream emits only between the l@xmath127 point and its impact with the edge of the disk this reproduces the `` tail '' seen in the balmer tomograms ( see [ s - dt ] ) . the peak amplitude of the stream component is 0.25 above the continuum . * bright spot emission : * in the model proposed by hellier & robinson ( 1994 ) and hellier ( 1996 ) for the sw sex stars px and and v1315 aql , the accretion stream continues coherently past its initial impact with the edge of the disk , absorbing the underlying disk emission along its trajectory , to a secondary impact site in the inner disk ( thereby creating a secondary bright spot whose velocity behavior is determined by the dynamics of the inner disk ) . the eclipse maps of uu aqr in a high state ( bsh96 ) , however , show evidence for a bright spot only at the expected phase of the initial stream impact with the disk ( @xmath128 ) . further , we find that the velocity of a secondary impact site near the radius expected for re - impact ( lubow 1989 ) must be arbitrarily reduced by a very large amount ( @xmath129% ) to match the observed velocity offset of the strong emission region in the balmer tomograms of uu aqr . consequently , we have simulated the line profiles using a bright spot ( gaussian fwhm = 50% of the average of the local stream+disk rotation velocity ) that originates at the intersection of the stream with the disk edge , and expands to lower velocities ( this is consistent with , for example , emission from impacting stream material that does not deeply penetrate the disk and is primarily reflected away from the disk edge in the general direction of the secondary star ) . this scenario is suggested by recent numerical simulations of the stream - disk interaction ( armitage & livio 1996 , 1997 ) , in which the impact of the stream with the disk , in the presence of inefficient cooling ( appropriate for high @xmath5 systems such as novalike cvs ) , does _ not _ produce coherent stream flow over the disk , but instead causes an `` explosion '' of material away from the impact point in all directions ( except those blocked by the disk ) . thus , the stream component in the line simulation does not contribute beyond the outer edge of the disk . the peak amplitude of the bright spot profile is 1.20 above the continuum . * non - axisymmetric absorption : * hellier ( 1997 ) has recently proposed accretion stream overflow onto an axisymmetrically flared disk to explain the transient absorption features in the emission lines of the sw sex stars . unfortunately , this mechanism appears only to be able to reproduce the absorption seen around @xmath130 in some of the sw sex stars . in order to account for absorption at both 0.5 and other phases , there must be a non - axisymmetric absorbing structure located at different positions in the disk in different systems , as suggested by hoard & szkody ( 1996a , 1997 ) . in order to match the observed behavior of the balmer emission lines in uu aqr , we utilized non - axisymmetric absorption represented by an inverted ( i.e. , negative amplitude ) gaussian function . the absorbing structure can be visualized as a vertically- and azimuthally - extended bulge or wall along the disk edge , which starts ( with zero absorption ) at @xmath131 , ramps up linearly to peak absorption ( amplitude of @xmath132 relative to the continuum ) between phases 0.7 and 0.8 , then ramps back down to zero absorption at @xmath133 . the formation of such a structure could result from stream material `` exploding '' around the impact site ( near @xmath92 ) to form the highest section of the wall , with additional material carried along the disk edge by the rotation of the disk to form the long , declining tail of the wall . this is consistent with the large azimuthal extent of the bright spot expected from the eclipse profiles of uu aqr ( see [ s - ew ] ) . the fwhm of the absorption is 300 km s@xmath19 ; this value was initially set at the rotational velocity of the outer edge of the disk ( @xmath134 km s@xmath19 ) and was refined by comparing the simulated absorption width with that seen between phases 0.5 and 1.0 in the observed spectra . in the simulation shown in figure [ f - linesim ] , the velocity behavior of the absorption was set to follow that of the bright spot , in essence producing a self - absorption component of the bright spot ( as suggested by dickinson et al . 1997 to explain the line profile behavior of the probable sw sex star v795 herculis ) . we also experimented with somewhat more physical forms of velocity behavior for the absorption , such as following either the rotational velocity of the edge of the disk or the continuation of the stream trajectory over the disk , and found these to produce qualitatively similar results . considering the simplicity of our simulation , we elected to present here the case involving the fewest unknown parameters ( i.e. , self - absorption ) . in general , this technique of line profile simulation can produce reasonable - looking results . yet , the large number of input parameters and use of simplifying approximations beg the question : how _ realistic _ are these results ? we have calculated the @xmath135 values of the simulated profiles using different input parameters compared to the observed profiles , in a number of phase bins ; the comparison for the simulation presented here is shown in figure [ f - simprof ] . this provided a means of comparing the relative `` goodness '' of simulations for a particular data set . however , the parameters were adjusted based only on a visual inspection of the simulated vs. real profiles ; for example , the blue shoulder visible in the @xmath136 profile in figure [ f - simprof ] provided a measure of the correct amplitudes for the disk and wind components , while the height of the central peak provided an estimate of the correct amplitude for the bright spot emission . this manual fitting can produce somewhat erratic results . further , like doppler tomography , the line profile simulation process essentially collapses the velocity field of the emission into two dimensions , which can lead to some ambiguity if significant sources of emission have large velocity components in the third dimension ( i.e. , vertical relative to the plane of the disk ) . the presence of a non - axisymmetric absorbing structure in the disks of sw sex stars appears to be quite successful at explaining the transient absorption features in the emission lines of these cvs . unfortunately , it poses a potential problem for the technique of doppler tomography , which is widely used in the analysis of both the sw sex stars and cvs in general . a basic tenet of the tomography process is that the emission region(s ) being mapped should be equally visible at all orbital phases of the system ; this is why eclipse phase data are neither used in the tomography process nor accurately reproduced in forward - projections of the velocity maps . in order to circumvent the violation of this tenet in uu aqr , we have constructed two additional sets of balmer line tomograms . in the first set ( shown in the top panels of fig . [ f - halfdt ] ) , we used only the spectra from orbital phases 0.0 to 0.5 the stronger , `` unabsorbed '' spectra ( see top panels of fig . [ f - linesim ] ) . the second set uses the remaining , `` absorbed '' spectra , from @xmath137@xmath138 ( bottom panels of fig . [ f - halfdt ] ) . thus , each of these half - orbit time - resolved tomograms should show the emission from only one side of the absorbing structure described in [ s - linesim ] . in the @xmath94@xmath139 tomograms , we are effectively looking across most of the width of the disk to the absorbing structure on the far edge . the strong emission region lies on the @xmath119 axis in the same location as in the full - orbit tomograms . this emission originates at the explosive stream impact with the disk edge , and is viewed on the inner disk side of the absorbing structure . little or no emission is visible along the accretion stream trajectory itself ; the stream is likely obscured at most of these phases by the absorbing bulge , accretion disk , and/or secondary star . in the @xmath137@xmath138 tomograms , we are looking only at the stream and the small amount of disk material at radii outside the absorbing structure . there is prominent emission along the accretion stream trajectory ; however , there is little or no emission on the @xmath119 axis . presumably , most of the stream impact emission is obscured by the absorbing structure at these phases . it was noted in [ s - dt ] that the h@xmath31 tomogram showed the most prominent suggestion of the ring indicative of an accretion disk . the half - orbit tomograms of this line show that the strongest part of this ring emission is visible only at @xmath137@xmath138 , in the form of enhanced emission that trails out along the disk - ring from the bright spot at the stream impact site . this corresponds to material carried along the outer edge of the disk by the disk rotation ( e.g. , armitage & livio 1997 ) ; it is hidden behind the absorbing structure and/or secondary star for @xmath93@xmath139 , so is not seen in the @xmath94@xmath139 tomograms . a similar trail of emission is seen in tomograms of the archetype sw sex ( dhillon , marsh , & jones 1997 ) and the non - sw sex system wz sagittae ( spruit & rutten 1997 ) ; spruit & rutten suggest that this emission is due to recombination in material that is cooling down from the shock of the stream - disk impact as it is swept `` downstream '' by the disk rotation . figure [ f - diag ] is a schematic diagram of a cv showing the features of the model for uu aqr suggested by our line profile simulations and half - orbit tomography . the high @xmath5 accretion stream strikes the edge of the disk and forms a roughly spherical `` explosion '' of emitting material ( region a in fig . [ f - diag ] ) . this emitting material is flung primarily in directions of least resistance , away ( and above ) the disk , but some also disperses across the inner disk , producing diffuse emission in the tomograms ( region b ) . an optically thick absorbing wall is built up by large amounts of the impacting stream material ( region d ) , either carried along the disk edge or roughly following the continuation of the stream trajectory ( although there is no coherent overflow of the stream ) . optically thin emitting gas heated by the shock of the stream impact is also carried along the very outer edge of the disk by the disk rotation , outside the absorbing structure ( relative to the wd ) , forming the trailing arc of emission seen most prominently in the h@xmath31 tomograms ( region c ) . ( this situation is likely to be further complicated by the probable presence of a wind from the disk that is mentioned , but not explicitly addressed , in this work . ) the definition of the sw sex stars as a distinct class of cv is somewhat ambiguous and considerably more phenomenological than that of other cv classes ( e.g. , u geminorum stars , polars , etc . ) the sw sex stars tend to be high inclination systems , but this is almost certainly a selection effect . they have similar orbital periods , @xmath140@xmath141 h , which suggests a possible evolutionary link to the onset of sw sex behavior . in general , they display single - peaked emission lines , orbital - phase - dependent absorption in the balmer lines , and phase offsets in the emission line radial velocity curves . uu aqr has all of these general qualities . early work on the sw sex stars ( e.g. , szkody & pich 1990 ; thorstensen et al . 1991 ) specified that the absorption occurred only at @xmath130 , whereas it occurs at @xmath92 in uu aqr ( and in pg 0859 + 415 ) . yet , this variation in behavior can be accommodated if the sw sex `` class '' contains systems comprising a continuum of observational characteristics determined by the relative values of fundamental parameters ( such as the mass transfer rate ) , rather than many examples of a fixed system morphology . we note that strong @xmath43 emission ( relative to the balmer emission lines , for example ) is also typically mentioned as a distinguishing property of the sw sex stars , but uu aqr has weak . the strength of , however , does vary among the members of this class ( hoard & szkody 1996b ) , from comparable to h@xmath30 ( e.g. , dw uma ; shafter et al . 1988 ) to much weaker than h@xmath30 ( e.g. , wx arietis ; beuermann et al . thus , although the reason for the weak emission in uu aqr is not entirely clear , just the fact that it is weak is not so troubling . it is reasonable to suggest that the strength of is related to inclination ; for example , wx ari ( @xmath142 ; beuermann et al . 1992 ) and pg 0859 + 415 ( @xmath143 ; hoard & szkody 1996a ) , which are both moderate inclination systems , have weak emission . however , this correlation does not appear to extend to other systems : bh lyn ( @xmath144 ; hoard & szkody 1997 ) and sw sex ( @xmath144 ; penning et al . 1984 ) have inclinations similar to that of uu aqr but have strong emission , while px and ( @xmath145 ; thorstensen et al . 1991 ) has a smaller inclination than uu aqr , but stronger emission . the presence of optically thick parts of the outer disk that obscure the hot inner regions where is expected to form might reduce the emission strength ; for example , pg 0859 + 415 and uu aqr are known or suspected of having thick disks and both display weak emission . the dependency of the amount of stream overflow on cooling efficiency noted by armitage & livio ( 1997 ) provides at least a partial explanation for the range of behavior among the sw sex stars . at high mass transfer rates , cooling is inefficient and the stream impact produces a forceful explosion , resulting in no coherent stream overflow and the presence of line absorption at the initial impact site ( @xmath92 ) . at low mass transfer rates , cooling is efficient and there is substantial , coherent overflow of the stream . this allows the formation of a bright spot in the inner disk at the site of the secondary stream impact site , resulting in line absorption around @xmath130 . in the sw sex stars uu aqr and pg0859 + 415 ( hoard & szkody 1996a ) , the line absorption is deepest near @xmath146 and there is evidence for a substantial bright spot on the edge of the disk . in bh lyn , the absorption occurs around @xmath137 but there is also evidence for the presence of a bright spot on the edge of the disk from time to time ( hoard & szkody 1997 ) . in dw uma ( shafter et al . 1988 ) , px and ( thorstensen et al . 1991 ) , and v1315 aql ( dhillon , marsh , & jones 1991 ) , the absorption occurs around @xmath137 and there is little or no evidence for a bright spot on the disk edge . we can understand this range of behavior in the context of the numerical simulations of armitage & livio ( 1997 ) if the mass transfer rate is largest for uu aqr and pg 0859 + 415 , @xmath5 is smaller for bh lyn , and still smaller for dw uma , px and , and v1315 aql . this produces primarily explosive impact of the stream with the disk in the first two systems , a mix of explosive impact and stream overflow in bh lyn , and primarily stream overflow in the last three systems . mass transfer rates in cvs are notoriously difficult to estimate , but we note that if their @xmath5 s do compare as described here , then the orbital periods of these six systems ( uu aqr = 3.9 h ; pg 0859 + 415 = 3.7 h ; bh lyn = 3.7 h ; px and = 3.5 h ; v1315 aql = 3.3 h ; dw uma = 3.3 h ) roughly follow a trend where mass transfer rate decreases with decreasing orbital period . some evidence for such a trend has been seen among the novalike cvs ( dhillon 1996 ) . we wish to thank keith horne for making his doppler tomography software available to us , and stefanie wachter for reading a draft of this paper . the research of ps and dwh was supported by nasa grant nag - w-3158 and nsf grant ast9217911 . mds was supported by the particle physics and astronomy research council grant k46019 . the work of rcs was partially supported by pparc grant gr / k45555 . horne , k. 1991 , in fundamental properties of cataclysmic variable stars : proceedings of the 12th north american workshop on cataclysmic variables and low mass x - ray binaries , ed . a. w. shafter ( san diego : san diego state university ) , 23 ccclcccc 1 & 2449932 & rpcs & 1995 aug 0102 & 21:24 & 02:04 & 0.511.70 & 161 + & & rpcs & 1995 aug 0203 & 22:16 & 02:28 & 0.841.91 & 146 + + 2 & 2449937 & rpcs & 1995 aug 0607 & 21:33 & 02:59 & 0.111.50 & 187 + & & rpcs & 1995 aug 0708 & 21:58 & 03:00 & 0.331.61 & 172 + + 3 & 2449949 & rpcs & 1995 aug 1819 & 22:44 & 02:40 & 0.771.78 & 145 + & & rpcs & 1995 aug 1920 & 22:43 & 02:18 & 0.881.80 & 122 + + 4 & 2449980 & rpcs & 1995 sep 1920 & 23:20 & 01:21 & 0.551.06 & 69 + & & rpcs & 1995 sep 20 & 20:05 & 22:58 & 0.831.57 & 91 + + 5 & 2449982 & rpcs & 1995 sep 2122 & 20:16 & 00:37 & 0.992.10 & 146 + & & rpcs & 1995 sep 2223 & 23:13 & 00:56 & 0.861.29 & 58 + + 6 & 2449984 & rpcs & 1995 sep 2425 & 21:43 & 01:05 & 0.701.56 & 111 + & & rpcs & 1995 sep 2526 & 22:40 & 00:58 & 0.061.64 & 84 + + 7 & 2450003 & dis & 1995 oct 12 & 04:45 & 08:18 & 0.411.31 & 35 + & & dis & 1995 oct 13 & 05:06 & 08:07 & 0.611.38 & 30 ccccccccccc @xmath1471 & 36(2 ) & @xmath148117(12 ) & 0.15(2 ) & 0.10 & @xmath14843 & 27(4 ) & 119(8 ) & 0.15(2 ) & 0.07 & @xmath14836 + @xmath1472 & 8(1 ) & @xmath148113(18 ) & 0.14(2 ) & 0.16 & @xmath14859 & 15(3 ) & 107(15 ) & 0.11(2 ) & 0.14 & @xmath14848 + @xmath1473 & 4(4 ) & @xmath148114(12 ) & 0.16(2 ) & 0.11 & @xmath14850 & 81(5 ) & 97(11 ) & 0.18(2 ) & 0.11 & @xmath14845 + @xmath1474 & 86(5 ) & @xmath14895(11 ) & 0.13(3 ) & 0.12 & @xmath14845 & 65(10 ) & 117(22 ) & 0.12(4 ) & 0.19 & @xmath14878 + @xmath1475 & 8(6 ) & @xmath14877(12 ) & 0.10(3 ) & 0.16 & @xmath14849 & 45(4 ) & 121(19 ) & 0.15(3 ) & 0.16 & @xmath14867 + @xmath1476 & 31(4 ) & @xmath148113(16 ) & 0.09(2 ) & 0.14 & @xmath14851 & 31(8 ) & 104(12 ) & 0.08(3 ) & 0.11 & @xmath14851 + @xmath1477 & @xmath14919(1 ) & @xmath148115(6 ) & 0.13(1 ) & 0.05 & @xmath14834 & 18(2 ) & 103(8 ) & 0.10(1 ) & 0.08 & @xmath14843 + + & & + @xmath1477 & @xmath1494(1 ) & @xmath148151(8 ) & 0.21(1 ) & 0.05 & @xmath14838 & @xmath14958(1 ) & 61(7 ) & 0.16(2 ) & 0.12 & @xmath14837 + + & & + @xmath1477 & 33 & @xmath148156 & 0.13 & 0.16 & @xmath148135 & @xmath14987 & 226 & 0.27 & 0.15 & @xmath148165 clcccc 1 & 1995 aug 0103 & 18.2(1.1 ) & 12.1(2.0 ) & 1.4(0.7 ) & 9.8(1.6 ) + 2 & 1995 aug 0608 & 18.3(2.3 ) & 12.7(1.4 ) & 2.1(0.8 ) & 9.7(1.4 ) + 3 & 1995 aug 1820 & 21.1(3.4 ) & 11.8(2.5 ) & 2.1(1.4 ) & 6.7(1.8 ) + 4 & 1995 sep 1920 & 19.0(3.0 ) & 13.3(7.5 ) & 2.0(1.9 ) & 8.2(5.7 ) + 5 & 1995 sep 2123 & 21.9(3.6 ) & 13.9(2.2 ) & 2.6(1.2 ) & 4.6(5.5 ) + 6 & 1995 sep 2426 & 17.6(2.1 ) & 12.1(3.2 ) & 2.7(1.2 ) & 8.5(1.8 ) + 7 & 1995 oct 1213 & 17.2(2.8 ) & 9.2(2.5 ) & 0.0(1.0 ) & 10.2(1.9 ) +

we present 14 nights of medium resolution ( 1 - 2 ) spectroscopy of the eclipsing cataclysmic variable uu aquarii obtained during a high accretion state in 1995 august october . uu aqr appears to be an sw sextantis star , as noted by baptista , steiner , & horne ( 1996 ) , and we discuss its spectroscopic behavior in the context of the sw sex phenomenon . emission line equivalent width curves , doppler tomography , and line profile simulation provide evidence for the presence of a bright spot at the impact site of the accretion stream with the edge of the disk , and a non - axisymmetric , vertically- and azimuthally - extended absorbing structure in the disk . the absorption has maximum depth in the emission lines around orbital phase 0.8 , but is present from @xmath0 to @xmath1 . an origin is explored for this absorbing structure ( as well as the other spectroscopic behavior of uu aqr ) in terms of the explosive impact of the accretion stream with the disk .

theoretical studies of light hadron spectroscopy have led to the widespread belief that gluonic excitations are present in the spectrum of hadrons , so more resonances should be observed than are predicted by the conventional @xmath0 and @xmath9 quark model . the two general categories of gluonic mesons expected are glueballs ( dominated by pure glue basis states ) and hybrids ( dominated by basis states in which a @xmath0 is combined with a gluonic excitation ) . some of these novel states , notably the light hybrids , are predicted to have exotic quantum numbers ( forbidden to @xmath0 ) , such as j@xmath10 . the confirmation of such a resonance would be proof of the existence of exotic non-@xmath0 states , and would be a crucial step towards establishing the spectrum of gluonic states . there are detailed theoretical predictions for the decays of these exotic hybrids @xcite , which have motivated several experimental studies of purportedly favored hybrid channels such as @xmath11 and @xmath12 . although one would prefer to find these unambiguously non-@xmath0 j@xmath13-exotics , glueballs and hybrids with non - exotic quantum numbers are also expected . for example , in the flux tube model the lowest hybrid multiplet , expected at @xmath14-@xmath15 gev @xcite , contains the non - exotics j@xmath16 and @xmath17 in addition to the exotics @xmath18 , @xmath19 and @xmath20 . to identify these non - exotic states one needs to distinguish them from the `` background '' of radial and orbital @xmath0 excitations in the mass region @xmath21-@xmath22 gev , where the first few gluonic levels are anticipated@xcite . our point of departure is to calculate the two - body decay modes of all radial and orbital excitations of @xmath1 states ( @xmath23 ) anticipated up to 2.1 gev . this includes 2s , 3s , 2p , 1d and 1f multiplets , a total of 32 resonances in the @xmath1 sector . we also summarize the experimental status and important decays of candidate members of these multiplets , and compare the predictions for decay rates with experiment . we start by briefly reviewing the established 1s and 1p states that confirm that p0 pair creation dominates most hadronic decays . sho wavefunctions are employed for convenience ; these lead to analytic results for decay amplitudes and are known to give reasonable empirical approximations . this is sufficient for our main purpose , which is to emphasize selection rules and to isolate major modes to aid in the identification of states . in addition to the 1s and 1p states we also find reasonable agreement between the model and decays of 1d , 2p and 1f states where data exist ; this confirms the extended utility of the model and adds confidence to its applications to unknown states . examples of new results include the following . @xmath24 the radial 2@xmath4p@xmath5 @xmath25 is strongly suppressed in s - wave , and dominant in d - wave . this contrasts with the expectation for a hybrid @xmath26 . the model s prediction of a dominant d - wave has been dramatically confirmed for the @xmath27@xcite and thereby establishes 1.7 gev as the approximate mass of the @xmath1 members of the 2p nonets . this includes the @xmath28 nonet whose i=0 members share the quantum numbers of the scalar glueball . @xmath24 in the scalar glueball sector , we find that the decays of the @xmath29 and the @xmath30 are inconsistent with radially excited quarkonia . @xmath24 we identify the 2s @xmath31 nonet . the @xmath32 members are predicted to have narrow widths relative to the @xmath33 counterpart . this is consistent with the broad @xmath34 and the narrower candidates @xmath35 and @xmath36 . @xmath24 the vector states @xmath37 and @xmath38 are interesting in that the decay branching fractions appear to show anomalous features requiring a hybrid component . we identify the experimental signatures needed to settle this question . @xmath24 the @xmath39 has been cited as a likely hybrid candidate@xcite on the strength of its decay fractions . the 3s @xmath31 @xmath0 @xmath33 is also anticipated in this region . we find that the decays of the hybrid and 3s @xmath31 have characteristic differences which enable them to be distinguished . we identify modes that may enable the separation of these two configurations . our other results for the many @xmath1 states predicted up to 2.1 gev should be useful in the identification of these higher quarkonia , and in confirming that non - exotic gluonic or molecular states are indeed inconsistent with quarkonium assignments . the order of discussion is 1s and 1p ( section 2 ) ; 2s and @xmath40 ( section 3 ) ; 3s ( section 4 ) ; 2p ( section 5 ) ; 1d ( section 6 ) ; 1f ( section 7 ) . a summary and an outline for experimental strategy is in section 8 . first we will use the well known decays of light 1s and 1p @xmath1 states to motivate and constrain the p0 decay model . ackleh , barnes and swanson@xcite have carried out a systematic study of @xmath41 decays in the p0 and related pair creation decay models : in that work a p0-type amplitude was established as dominant in most light @xmath42 decays . ( for other discussions of @xmath43 decays in the p0 model see ref.@xcite ) . fig.1 , from ref.@xcite , shows p0 model predictions for the decay widths . large widths are indeed predicted to be large and smaller widths are found to be correspondingly small . if we choose the pair creation strength @xmath44 ( eq . a3 ) to set an approximately correct overall width scale , then @xmath45 and @xmath46 are both @xmath47-@xmath48 gev ; @xmath49 , @xmath50 and @xmath51 are all @xmath52-@xmath53 gev , and @xmath54 is smallest , @xmath55 gev ; all are reasonably close to the observed widths . @xmath56 figure 1 . partial widths of light 1s and 1p @xmath0 mesons in the p0 model . the model parameters shown are @xmath57-@xmath58 gev ( with @xmath59 gev preferred ) and @xmath44 . the optimum parameter values found in a fit to the partial widths of fig.1@xcite are @xmath60 gev ( which is actually the length scale most commonly used in light @xmath0 decays ) and @xmath61 ; with these values the rms relative error for these six decays is @xmath62 . in this work we have actually found that the pair production amplitude @xmath44 is somewhat large for higher - l @xmath0 states , so in our discussions of higher quarkonia we will instead use @xmath63 . in constrained-@xmath64 fits we find that using @xmath63 only moderately decreases the accuracy of the fit to the light 1s and 1p decays , to @xmath65 , with an optimum @xmath66 gev . a more sensitive test of the p0 model involves amplitude ratios in the decays @xmath67 and @xmath68 . in these decays both s- and d - wave final states are allowed , and the ratio of these decay amplitudes is known to be d / s = @xmath69 for the @xmath70 and @xmath71 for the @xmath26 @xcite . this ratio is quite sensitive to the quantum numbers of the produced pair ; with p0 quantum numbers and the usual @xmath72 we find reasonable agreement in sign and magnitude , whereas a oge pair production mechanism gives the wrong sign for d / s @xcite . this ratio test for @xmath67 was historically very important in establishing the p0 decay model @xcite . these successes of the p0 model motivate its use in predicting decays of the less familiar radial and orbital excitations of light quarkonia . we first consider the decays of the low - lying radially - excited pseudoscalar and vector states . our general approach will be to review recent data on the state in question and compare these data to predictions for candidate @xmath0 and ( where appropriate ) hybrid states . in each case we will attempt to identify decay modes that distinguish between competing assignments most clearly . @xmath24 @xmath34 the @xmath34 was first reported by bellini _ et al._@xcite in 1982 but remains rather poorly known . it is seen in @xmath75 , @xmath76 and @xmath77 , with a width of 200 - 600 mev ; there is however no accurate measurement of the branching fractions @xcite . recently higher statistics have been obtained for the @xmath34 by ves@xcite and by e852 at bnl@xcite . the ves data shows a clear @xmath34 peak in @xmath78 , with a width of @xmath79-@xmath80 mev in both @xmath81 and @xmath82 ; the latter is particularly strong and dominates this channel below 2 gev . it should be noted , however , that the size of the deck background in @xmath83 is uncertain , and it is not clear whether the @xmath34 reported in @xmath83 is actually due to the resonance . fig.1c of ref.@xcite suggests that the deck mechanism could cause _ all _ of the @xmath84 enhancement in fig.4a of that reference . we will assume that this is essentially correct , and that the @xmath34 resonance decays dominantly to @xmath85 . in the p0 decay model we expect @xmath85 to be the dominant mode of a 2s @xmath0 @xmath34 , since this is the only open two - body channel . ( we assume that the @xmath86 and @xmath87 are dominantly k@xmath88 , so the mode @xmath89 is a more complicated three body or virtual two - body decay . ) with our parameter set @xmath63 and @xmath90 gev we predict a partial width of @xmath91 this rate is given in table b2 of appendix b. ( app.b is a tabulation of all our numerical results for partial widths in the p0 model . ) in fig.2 we show the dependence of this prediction on the wavefunction length scale @xmath72 . evidently the prediction of a large width , comparable to observation , follows from any plausible choice for @xmath72 . thus the observed @xmath34 is consistent with expectations for a 2@xmath6s@xmath7 @xmath0 state . @xmath92 figure 2 . the @xmath85 partial width of a 2s @xmath34 , with p0 model parameters @xmath57-@xmath58 gev and @xmath63 . although the mode @xmath93 is nominally closed by phase space , the @xmath94 is a very broad state , so one might anticipate a significant @xmath95 mode through the low - mass tail of the @xmath94 . this possibility may be tested by varying @xmath96 ; the resulting @xmath97 does not exceed @xmath98 mev over the range @xmath99-@xmath100 mev . thus , the population of a @xmath81 mode by @xmath34 decays through an intermediate @xmath101 state is predicted to be a small effect . if there actually is a large @xmath102 mode , rather than a nonresonant deck effect , this would be in disagreement with the p0 model . thus it would be very interesting to establish the branching fraction for @xmath102 accurately in future work . @xmath24 @xmath35 this state has a width of @xmath103 mev@xcite , much narrower than its i=1 2@xmath6s@xmath7 partner @xmath34 . it has been reported in @xmath104 and @xmath105 . this small width is natural if the @xmath34 does indeed decay dominantly to @xmath106 , since g - parity forbids the analogous processes @xmath107 and @xmath108 ; to the extent that the @xmath87 and @xmath86 are dominantly @xmath109 there are no quasi - two - body @xmath43 modes open to the @xmath35 . consequently the decays must proceed through the weaker direct three - body and virtual two - body channels such as @xmath110 and @xmath111 . it is interesting to note the rle that the 2s initial wavefunction has played in our discussion . suppose for illustration that we had instead used 1s wavefunctions for the @xmath34 and @xmath35 ; we would then have predicted partial widths of several hundred mev into the low - energy tails of the modes @xmath101 and @xmath112 , with consequent broad widths for the @xmath34 _ and _ the @xmath35 , in contradiction with experiment . @xmath24 @xmath36 these successes raise provocative questions regarding the @xmath36 state(s ) . this is a purportedly complicated region which may contain more than one resonance@xcite . the pdg width of the @xmath36 is only @xmath113 mev , with signals reported in @xmath114 , @xmath104 , @xmath115 and @xmath116 . except for @xmath116 these modes are not inconsistent with a dominantly @xmath117 state . the only two - body strong channel open for a 2@xmath6s@xmath7 @xmath117 @xmath36 is @xmath114 , but this could rescatter from kk@xmath33 into the other reported modes @xmath118 and @xmath119 . the p0 model prediction for the partial width @xmath120 versus the wavefunction length scale @xmath72 is shown in fig.3 . @xmath121 figure 3 . the k@xmath122 + h.c . partial width of a 2@xmath6s@xmath7 @xmath117 @xmath36 in the p0 model . other two - body modes are excluded by phase space . evidently the predicted @xmath114 partial width is comparable to the observed width , so a 2@xmath6s@xmath7 @xmath117 assignment appears possible for this state . of course the @xmath116 mode is not expected from @xmath117 , and if confirmed may imply large @xmath123 mixing in this sector as is observed in the 1s i=0 pseudoscalars . this can be parameterized as @xmath124 a remeasurement of @xmath125 , which should be possible at bepc and tcf in @xmath126 , would be very useful in clarifying the nature of this state . ideally we would like to know the invariant mass distributions of @xmath116 , @xmath127 and @xmath128 final states , since these are flavor - tagging modes that allow investigation of possible flavor mixing in the parent resonances . similarly , an accurate measurement of the branching fractions in the flavor - tagging @xmath129 and @xmath130 hadronic decays , with @xmath131 , would be useful for the determination of the @xmath132-@xmath117 mixing angle . in summary , from the total widths alone it is possible to describe the @xmath35 and @xmath36 as unmixed @xmath1 and @xmath117 2@xmath6s@xmath7 radial excitations . the report of a large @xmath133 radiative mode however suggests flavor mixing between these states , and should be remeasured with greater sensitivity together with other @xmath134 modes . this mixing could also account for the large @xmath36 signal seen in @xmath135 by gams @xcite . @xmath24*@xmath37 , @xmath139 * if one accepts that the @xmath34 and @xmath35 belong to a @xmath73s@xmath7 @xmath0 nonet , it is then natural to assign the @xmath37 and the @xmath38 @xcite to @xmath140s@xmath5 states . indeed , one expects the contact hyperfine interaction to raise the mass of the vector nonet with respect to the pseudoscalar nonet by approximately this amount @xcite . it is unlikely that the vectors near 1.4 - 1.5 gev are dominantly d - waves , since the @xmath4d@xmath5 @xmath1 states should lie close to the other 1d candidates such as the @xmath141 , @xmath142 and @xmath143 . in the godfrey - isgur potential model a mass of 1660 mev was predicted for the @xmath4d@xmath5 state , whereas they expect the 2@xmath4s@xmath5 radial excitation at 1450 mev @xcite . the @xmath37 also lies well below flux - tube model expectations of m@xmath144-@xmath15 gev@xcite for vector hybrids , so although the possibility of light vector hybrids has been discussed @xcite , these do not appear likely unless the flux tube model for hybrids is misleading . the experimental branching fractions of these @xmath136 states are somewhat obscure , because there are at least two broad , overlapping resonances in each flavor sector in this mass region . the status of these vector states as seen in @xmath145 annihilation was reviewed recently by clegg and donnachie @xcite . in the @xmath137 sector they find that at least two states are present . the lighter state is assigned a mass of m @xmath146 gev and a width of @xmath147 gev ; it couples strongly to @xmath148 states ( including @xmath149 but not @xmath150 ) and @xmath151 , and less strongly to @xmath152 . the higher state has m @xmath153 gev , @xmath154 gev , couples most strongly to @xmath148 ( @xmath149 and @xmath150 are not separated ) and perhaps @xmath155 ; @xmath152 is also important , but the @xmath151 width is found to be small . these states have also been reported recently by crystal barrel @xcite in @xmath156 states in @xmath157 ; both vectors appear in @xmath156 , with masses and widths of m @xmath158 gev , @xmath159 gev , and m @xmath160 gev , @xmath161 gev , quite similar to the @xmath145 results . the p0 model predictions for pure 2@xmath4s@xmath5 and @xmath4d@xmath5 @xmath137 states at 1.465 gev and 1.700 gev are given in table i ( see also tables b1 , b8 ) , together with flux tube model predictions for a hypothetical 1.5 gev vector hybrid . very characteristic differences between the states are evident in their couplings to @xmath148 final states ; 2s couples very weakly to these , 1d couples strongly to both @xmath149 and @xmath150 , and the hybrid couples strongly to @xmath149 but not to @xmath150 . both quarkonium states have moderately large couplings to @xmath152 and @xmath151 , whereas the hybrid couples strongly only to @xmath149 . . partial widths of 2s , 1d and hybrid @xmath137 states . [ cols="<,^,^,^,^,^,^,^,^,^",options="header " , ] in table vi we compare the decay modes expected for a hybrid at 1875 mev with p0 model predictions for a hypothetical @xmath6d@xmath8 @xmath162 quarkonium . both assignments lead to a significant @xmath163 signal , and both predict a much larger @xmath164 mode . the most characteristic modes are @xmath165 and @xmath166 , which should be very weak for a hybrid but large for a 1d quarkonium . similar results follow for @xmath167 and @xmath149 . clearly searches for @xmath164 , @xmath165 and @xmath166 would be most useful . the large predicted coupling to @xmath168 for the @xmath169 encourages a search in @xmath170 for this state . here we consider only the @xmath4d@xmath172 and @xmath4d@xmath8 states since the @xmath4d@xmath5 vectors were previously discussed with the 2@xmath4s@xmath5 states . the @xmath173 states @xmath142 and @xmath143 are well established @xmath4d@xmath172 @xmath1 quarkonia , with masses as expected for 1d states and widths of about 200 mev . the @xmath174 ( table b7 ) is expected to decay mainly to @xmath165 @xmath175 and @xmath152 @xmath176 , with a somewhat weaker @xmath151 mode @xmath177 . experimentally the decays to @xmath148 are about @xmath178 , of which @xmath179 is @xmath151 . the @xmath152 branching fraction is observed to be @xmath180 . there are also kk and @xmath167 modes of a few percent , roughly as predicted . the total width is predicted to be 174 mev with these parameters , consistent with observation . thus the @xmath142 appears to decay approximately as predicted by the p0 model , which supports the application of the model to decays of high - l states . its isoscalar partner @xmath143 is a more interesting case . since few modes are open and the couplings are rather weak , we predict a total width of only 69 mev . although this appears inconsistent with the pdg width of 168(10 ) mev , this observed value is presumably broadened by the hadronic width of the @xmath137 and @xmath70 in the two - body modes @xmath85 and @xmath11 . the reported modes are @xmath85 and @xmath181 ; we expect @xmath85 to be dominant , with @xmath182 branches to @xmath11 ( the source of @xmath181 ? ) and kk . the kk mode affords an opportunity to measure the actual width of the @xmath183 , which may be much smaller than it appears in @xmath85 and @xmath11 modes . our results for the @xmath4d@xmath8 @xmath184 states @xmath185 and @xmath186 are especially interesting because these are missing mesons " in the quark model . we find that these are rather broad states , with total widths of about 300 - 400 mev . the @xmath187 is predicted to have a large branching fraction of @xmath188 to @xmath164 , so it should be observable in this final state or in the secondary modes @xmath151 or k@xmath189k . the @xmath190 is predicted to have an even larger branching fraction of @xmath191 to @xmath85 . it too couples significantly to k@xmath189k , and may also be observable in @xmath192 . the 1f states provide us with an opportunity to test the accuracy of the p0 decay model predictions for higher quarkonium states , since the @xmath193 and @xmath194 states expected near 2.05 gev do not have competing assignments as glueballs or hybrids . at present only two of these states are reasonably well established , the @xmath195 and @xmath196 @xcite . there is also some evidence for an @xmath197 @xcite . we do not yet have experimental branching fractions for the i=1 1f states . the @xmath196 is seen in kk and @xmath78 , and the @xmath197 is reported in @xmath78 and @xmath198 , with @xmath199 dominant . the branching fractions of the @xmath195 are known with more accuracy ; @xmath166 and @xmath152 are important modes , @xmath200 and @xmath201 . kk and @xmath202 modes are both known , with reported branching fractions of about @xmath203 and @xmath204 respectively . p0 predictions for the decays of these @xmath4f@xmath171 states are given in tables b11 and b12 . the @xmath205 is indeed expected to appear in @xmath78 ( mainly @xmath85 ) , and the dominant mode is predicted to be @xmath206 . this state is predicted to be rather narrower than reported . the @xmath197 is predicted to decay dominantly to @xmath199 , as is observed . the @xmath78 mode is also predicted to be large , and to arise from both @xmath85 and @xmath207 . the @xmath195 p0 model predictions are also in qualitative agreement with experiment , in that @xmath152 and @xmath166 are expected to be important modes , as observed . the @xmath208 partial widths to pseudoscalar pairs are uniformly too large , for example @xmath209 mev but @xmath210 mev . this decay however is g - wave , so the rate has a prefactor of @xmath211 ; this extreme sensitivity means that a small increase of @xmath72 by @xmath212 , halves the decay rate and gives agreement with experiment . thus this disagreement is quite sensitive to parameters and is probably not significant . the predictions for branching fractions of the five missing i=0,1 1f states suggest that several of them may easily be found by reconstructing the appropriate final states . the total widths of all except the @xmath4f@xmath8 states are predicted to be @xmath213 mev , so they should be observable experimentally . the @xmath214 is predicted to couple dominantly to @xmath164 . in the spin - singlet @xmath6f@xmath172 sector , the @xmath215 should appear in @xmath85 and @xmath198 , just as we found for the @xmath197 . the @xmath216 should be evident in @xmath164 , and less strongly in @xmath217 , @xmath151 and @xmath165 . modes such as @xmath164 are preferable because the two - body mesons are not excessively broad and they are far from threshold , so a resonance can be distinguished from a threshold effect . in some cases the amplitude structure of these final states is also characteristic ; these can be determined from the results quoted in app.a . the missing @xmath4f@xmath8 states may be more difficult to identify , as we predict large total widths of @xmath218 mev for these states . the @xmath219 couples most strongly to @xmath11 ; @xmath220 and @xmath221 are other important modes . its i=0 partner @xmath222 should be evident in @xmath223 and will also populate @xmath221 final states . identification of these 1f states and determination of their branching fractions and decay amplitudes will be a very useful contribution to the study of resonances , as it will allow detailed tests of the usefulness of the p0 model as a means for identifying quarkonium states in this crucial 2 gev region . we have established that the @xmath27 is very likely a 2p radial excitation . this follows from the weak s - wave and strong d - wave in @xmath82 . this also establishes the natural mass scale for the 2p multiplets as @xmath224 gev . we have been unable to identify radial scalars . these are predicted to be broad , and so their non - appearance is not surprising . conversely it raises interest in the ( relatively narrow ) @xmath29 and possible scalar @xmath225 . we do identify some ( more speculative ) potential candidates for @xmath226 2p members . we note that @xmath170 production may help identify these radial 2p states and also clarify the nature of @xmath29 and @xmath225@xcite . the @xmath34 and @xmath35 appear to be convincing 2s states . this conclusion is based on their relative widths ; the large @xmath82 mode of the @xmath34 has no analog for its @xmath32 counterparts . the status of the @xmath36 remains open ; the mass and width suggest a dominantly @xmath117 state , but the @xmath227 mode argues against it . studies of @xmath228 and @xmath229 may identify the flavor content of these @xmath32 states . the @xmath37 and @xmath38 have masses that are consistent with radial 2s but their decays show characteristics of hybrids , as noted previously @xcite . we suggest that these states may be 2s - hybrid mixtures analogous to the 3s - hybrid mixing suggested for the @xmath230@xcite . this can be tested by accurate measurement of the partial widths of these states and their vector partners at 1.6 - 1.7 gev to @xmath152 , @xmath151 , and especially @xmath150 and @xmath149 . the 3s @xmath33 is expected in the @xmath231 mev mass region as is a @xmath232 hybrid . we find that the decay patterns of these states are very different . a strong @xmath233 from the hybrid contrasted with a large @xmath234 mode from the 3s quarkonium is the sharpest discriminant . the ves state @xmath39 clearly exhibits this hybrid signature . it is now necessary to establish the presence of @xmath31 in the @xmath234 channel , and to see if any resonant state is present that is distinct from the @xmath39 seen in @xmath233 . it is possible that there are two @xmath235 states , @xmath0 and hybrid , whose production mechanisms and decay fractions differ sufficiently so that they can be separated . we suggest that the possibility of two such @xmath235 states be allowed for in data analyses . in the immediate future there are opportunities for @xmath170 physics at lep2 and at b factories . possible strategies for isolating some of these higher quarkonia include : @xmath24 @xmath236 contains ( i ) @xmath234 which may access the radial @xmath237 and @xmath238 near 1700 mev and a possible @xmath239 . ( ii ) @xmath240 which can isolate the @xmath237 if the helicity selection rule@xcite is used to suppress the @xmath238 . @xmath24 @xmath241 may access the radial @xmath242 near 1700 mev through its decay into @xmath168 . the @xmath243 channel may also be searched for the @xmath29 since this state is known to have a significant branching fraction to @xmath244 but should have a suppressed @xmath170 coupling if it is a glueball @xcite . @xmath24 @xmath245 may be searched for @xmath17 states in order to verify whether the established @xmath141 is accompanied by a higher @xmath246 in @xmath247 and @xmath248 . this @xmath78 system may also be studied for evidence of one or more @xmath39 states . @xmath24 @xmath249 may access the isoscalar partners of these @xmath250 states . in the near future it will be possible to study @xmath251 annihilation up to @xmath252 gev at dafne . the channels @xmath253 should be measured and @xmath254 and @xmath255 states separated in order to carry out the analysis of hybrid and radial vector components in section 3b . the isoscalar partners of the vectors also need confirmation , and final states with kaons are needed to investigate possible @xmath138-@xmath256 mixing ; a potential weakness of the present data analyses is that such flavor mixing is assumed to be unimportant . in the next century there will be new opportunities at the compass facility at cern . this will enable further studies of central production and also of diffractive excitation . for the latter one may anticipate improved studies of the @xmath33 excitations ( such as the @xmath34 and @xmath39 states ) , possibly including primakoff excitation . judicious studies of specific final states as discussed above may help separate 3s and hybrid states . the use of k beams will allow analogous studies of the strange counterparts of these states and may help to clarify the spectrum of quarkonia , glueballs and hybrids . experiments with @xmath33 beams can access the following interesting channels . @xmath24 @xmath257 , to confirm the d - wave dominance of @xmath258 and to seek its partner @xmath238 . @xmath24 @xmath259 can access both @xmath260 and @xmath261 . these can be separated in @xmath11 ; the singlet selection rule forbids this mode for @xmath260 but allows it for @xmath262 . @xmath263 can also separate @xmath260 from @xmath262 ; @xmath264 is the dominant mode whereas @xmath262 is much suppressed into s+s hadrons . @xmath24 @xmath265 , @xmath266 and @xmath267 are important in the interpretation of the vectors between 1.4 and 1.7 gev , which may contain large hybrid components . @xmath24 @xmath268 and @xmath269 can all be searched for evidence of @xmath39 states . @xmath24 @xmath270 or @xmath271 access respectively @xmath272 and @xmath273 . finally , many two - body channels are predicted to couple strongly to specific 2p , 1d and 1f states , as shown in appendix b. these include missing mesons " such as the @xmath4f@xmath8 and most 2p states , and studies of these two - body final states may reveal the missing resonances . the modes @xmath164 , @xmath165 and @xmath11 are important for many of these missing states and merit careful investigation . we reiterate that it is in general a good strategy to study decays into both s+s and s+p meson modes , as the relative couplings of these modes are usually quite distinct for hybrid versus quarkonium assignments . we would like to acknowledge useful communications with c.amsler , d.v.bugg , s.u.chung , g.condo , k.danyo , a.dzierba , s.godfrey , i.kachaev , y.khokhlov , a.kirk , d.ryabchikov and a.zaitsev . this work was supported in part by the united states department of energy under contracts de - fg02 - 96er40944 at north carolina state university and de - ac05 - 96or22464 managed by lockheed martin energy systems inc . at oak ridge national laboratory . fec is supported in part by european community human capital mobility programme eurodafne , contract chrx - ct92 - 0026 . n.isgur , r.kokoski and j.paton , phys . 54 , 869 ( 1985 ) . f.e.close and p.r.page , nucl . b443 , 233 ( 1995 ) ; 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( the cancelations are due to flavor symmetries such as g - parity . ) thus for a physical decay such as @xmath291 one should multiply the unit - flavor amplitude @xmath288 in a3 by @xmath292 before computing the decay width using ( a2 ) . some states populate several decay channels , for example @xmath293 as well as @xmath294 ; to sum over all channels one should multiply the width by the flavor multiplicity factor @xmath295 in the table . in these flavor weights the pairs @xmath296 and @xmath297 are equivalent , up to factors due to identical particles in the final state . @xmath298 figure a1 . @xmath0 meson decay diagrams in the @xmath4p@xmath7 decay model . @xmath299 generic decay @xmath299 & @xmath299 subprocess @xmath299 & @xmath300 & @xmath301 & @xmath302 + @xmath303 & @xmath304 & @xmath292 & @xmath305 & @xmath306 + @xmath307 & @xmath308 & @xmath305 & @xmath305 & @xmath309 + @xmath310 & @xmath311 & @xmath312 & @xmath305 & @xmath313 + @xmath314 & @xmath315 & @xmath292 & @xmath305 & @xmath313 + @xmath316 & @xmath317 & @xmath312 & @xmath318 & @xmath306 + @xmath319 & @xmath320 & @xmath292 & @xmath292 & @xmath306 + @xmath321 & @xmath322 & @xmath305 & @xmath305 & @xmath323 + @xmath324 & @xmath325 & @xmath292 & @xmath312 & @xmath323 + @xmath326 & @xmath327 & @xmath328 & @xmath312 & @xmath313 + we take all spatial wavefunctions to be sho forms with the same width parameter @xmath72 ; as a result the @xmath329 decay amplitudes are proportional to an overall gaussian in @xmath330 times a channel - dependent polynomial @xmath331 , @xmath332 where @xmath64 is the p0 pair production coupling constant @xcite . to specify these amplitudes it suffices to quote the polynomial @xmath331 for each decay channel . the complete set of p0 decay amplitudes for all @xmath0 resonances with `` excitation level '' @xmath333n@xmath334l@xmath335 decaying into final states with @xmath336 and c @xmath337s@xmath7 ( and c = @xmath338s@xmath5 in most cases ) is given below . for the relatively obscure transitions 3s @xmath280 1d + c , 1f @xmath280 1p + c , 1f @xmath280 2p + c and 1f @xmath280 1d + c we restrict c to @xmath339s@xmath7 ; this does not exclude any decays allowed by phase space . we include a few additional amplitudes in this list . some of these are of interest as couplings to virtual two - body states , although phase space nominally forbids the decay . ' '' '' .1 cm ' '' '' .5 cm @xmath340 @xmath341 .5 cm ' '' '' .1 cm ' '' '' .5 cm ( _ see 1s @xmath280 1s + 1s for channel coefficients . ) _ .5 cm @xmath342 .5 cm ' '' '' .5 cm @xmath343 @xmath344 .5 cm ' '' '' .1 cm ' '' '' .5 cm ( _ see 1s @xmath280 1s + 1s for channel coefficients . ) _ .5 cm @xmath345 .5 cm ' '' '' .5 cm @xmath346 @xmath347 .5 cm ' '' '' .5 cm ( _ see 2s @xmath280 1p + 1s for channel coefficients . ) _ .5 cm @xmath348 .5 cm ' '' '' .5 cm ( _ see 2s @xmath280 1p + 1s for channel coefficients . ) _ .5 cm @xmath349 .5 cm ' '' '' .5 cm @xmath350 @xmath351 .5 cm ' '' '' .1 cm ' '' '' .5 cm @xmath352 @xmath353 .5 cm ' '' '' .1 cm ' '' '' .5 cm ( _ see 1p @xmath280 1s + 1s for channel coefficients . ) _ .5 cm @xmath354 .5 cm ' '' '' .5 cm @xmath355 @xmath356 .5 cm ' '' '' .5 cm @xmath357 .5 cm ' '' '' .1 cm ' '' '' .5 cm @xmath358 @xmath359 .5 cm ' '' '' .5 cm @xmath360 @xmath361 @xmath362 .5 cm ' '' '' .5 cm @xmath363 .5 cm ' '' '' .1 cm ' '' '' .5 cm @xmath364 @xmath365 .5 cm ' '' '' .5 cm @xmath366 @xmath367 .5 cm ' '' '' .5 cm ( _ see 1f @xmath280 2s + 1s for channel coefficients . ) _ .5 cm @xmath368 .5 cm ' '' '' .5 cm @xmath369 .5 cm ' '' '' .5 cm @xmath370 .5 cm ' '' '' .5 cm @xmath371 in this appendix we quote numerical values for partial widths predicted by the p0 model . the masses used are experimental values of well established candidates , usually taken from the 1996 pdg , otherwise we used an approximate multiplet mass . these are 1700 mev ( 2p ) , 1670 mev ( 1d ) , 2050 mev ( 1f ) , and 1900 mev and 1800 mev respectively for the 3@xmath372s@xmath5 and 3@xmath373s@xmath7 . the lighter meson masses assumed are @xmath374 mev , @xmath375 mev , @xmath376 mev , @xmath377 mev and @xmath378 mev . for other states we used the 1996 pdg masses except for the broad @xmath379 , which we left at 1300 mev . although we found optimum parameters near @xmath380 and @xmath381 gev in a fit to light 1s and 1p decays , these parameters lead to moderate overestimates of the widths of the well established higher - l states @xmath141 and @xmath195 ; with this @xmath72 a value closer to @xmath382 is preferred . consequently we quote widths for all these higher quarkonia with the parameters @xmath383 the tables are largely self explanatory . except in a few cases the states are specified uniquely by their labels . the exceptions include the @xmath384 and @xmath385 , which we take to be the usual @xmath386 combinations of @xmath387 and @xmath388 basis states . we assume that the @xmath389 and @xmath390 are pure @xmath387 states . the strange mesons k@xmath391 and k@xmath392 are taken to be the linear combinations and @xmath394 this gives a zero s - wave @xmath395 coupling ; experimentally d / s @xmath396 , and the small partial width implies a small s - wave amplitude . the orthogonal state @xmath397 ( b3 ) is predicted to have a d / s ratio of @xmath398 in @xmath399 , quite close to the experimental d / s @xmath400 . the large @xmath401 mode is not predicted and is possibly due to a virtual intermediate state such as k@xmath402 followed by a final - state interaction . the tables give partial widths for all nonstrange 2s , 3s , 2p , 1d and 1f quarkonia to all two - body modes allowed by phase space , rounded to the nearest mev . the predictions of the dominant modes of the `` missing states '' in the quark model , such as the @xmath184 states and most of the 1f states , are especially interesting . if the p0 model has even moderate accuracy these tables should be very useful in searches for these states . mode & .5 cm @xmath37 .5 cm & mode & .5 cm @xmath38 .5 cm + + @xmath403 & 74 . & & + @xmath404 & 122 . & @xmath405 & 328 . + @xmath406 & 25 . & @xmath407 & 12 . + + @xmath408 & 0 . & & + + @xmath409 & 1 . & @xmath410 & 1 . + @xmath411 & 3 . & & + @xmath412 & 0 . & & + + @xmath413 & 35 . & & 31 . + @xmath414 & 19 . & & 5 . + + @xmath415 & 279 . & & 378 . + @xmath416 & 310(60 ) & & 174(59 ) + mode & .5 cm @xmath418 .5 cm & mode & .5 cm @xmath419 .5 cm + + @xmath403 & 1 . & & + @xmath404 & 5 . & @xmath405 & 14 . + @xmath406 & 8 . & @xmath407 & 8 . + @xmath420 & 11 . & @xmath421 & 10 . + @xmath422 & 92 . & & + + @xmath408 & 70 . & & + @xmath423 & 50 . & @xmath424 & 121 . + + @xmath409 & 32 . & @xmath410 & 75 . + @xmath425 & 4 . & @xmath426 & 6 . + @xmath411 & 26 . & & + @xmath412 & 46 . & & + + @xmath427 & 0 . & @xmath428 & 0 . + @xmath429 & 0 . & & + @xmath430 & 0 . & & + + @xmath431 & 0 . & & + @xmath432 & 0 . & @xmath433 & 0 . + @xmath434 & 0 . & @xmath435 & 0 . + @xmath436 & 0 . & @xmath437 & 0 . + mode & .5 cm @xmath418 .5 cm & mode & .5 cm @xmath419 .5 cm + + @xmath413 & 1 . & & 1 . + @xmath414 & 21 . & & 21 . + @xmath438 & 27 . & & 27 . + + @xmath439 & 5 . & & 5 . + @xmath440 & 4 . & & 4 . + + @xmath415 & 403 . & & 292 . + mode & .5 cm @xmath39 .5 cm & mode & .5 cm @xmath441 .5 cm + + @xmath417 & 31 . & & + @xmath442 & 73 . & @xmath443 & 112 . + & & @xmath444 & 36 . + + @xmath424 & 53 . & & + + @xmath445 & 7 . & @xmath446 & 30 . + @xmath447 & 28 . & @xmath448 & 61 . + + @xmath414 & 36 . & & 36 . + + @xmath415 & 228 . & & 275 . + @xmath416 & 212(37 ) & & + mode & .5 cm @xmath449 .5 cm & .5 cm @xmath27 .5 cm & .5 cm @xmath450 .5 cm + + @xmath451 & 23 . & & 5 . + @xmath452 & 10 . & & 5 . + @xmath453 & 104 . & 58 . & + @xmath454 & 109 . & 15 . & 46 . + + @xmath455 & 3 . & & 43 . + @xmath424 & 0 . & 41 . & + + @xmath410 & 28 . & 41 . & 165 . + @xmath456 & & 2 . & + @xmath457 & 4 . & 18 . & 30 . + @xmath458 & 20 . & 39 . & + + @xmath413 & 20 . & & 0 . + @xmath414 & 17 . & 33 . & + + @xmath415 & 336 . & 246 . & 293 . + mode & .5 cm @xmath459 .5 cm & .5 cm @xmath460 .5 cm & .5 cm @xmath461 .5 cm + + @xmath403 & 81 . + @xmath462 & 4 . & & 0 . + @xmath463 & 1 . & & 16 . + @xmath464 & 159 . & 27 . & 72 . + @xmath465 & 56 . & 6 . & 22 . + + @xmath408 & 8 . & & 130 . + + @xmath466 & & 1 . & + @xmath411 & 16 . & 70 . & 122 . + @xmath412 & 43 . & 86 . & + + @xmath413 & 20 . & & 0 . + @xmath414 & 17 . & 33 . & + + @xmath415 & 405 . & 224 . & 409 . + mode & .5 cm @xmath467 .5 cm & mode & .5 cm @xmath468 .5 cm + + @xmath469 & 56 . & @xmath453 & 173 . + @xmath406 & 18 . & @xmath407 & 17 . + @xmath422 & 60 . & & + + @xmath423 & 13 . & @xmath424 & 31 . + + @xmath409 & 0 . & @xmath410 & 0 . + @xmath466 & 2 . & & + @xmath411 & 10 . & & + @xmath412 & 67 . & & + + @xmath414 & 30 . & & 30 . + + @xmath415 & 257 . + mode & .5 cm @xmath142 .5 cm & .5 cm @xmath185 .5 cm & .5 cm @xmath470 .5 cm + + @xmath403 & 59 . + @xmath471 & 19 . & 73 . & 35 . + @xmath472 & 2 . & 28 . & 16 . + @xmath464 & 71 . & 15 . & 14 . + + @xmath408 & 0 . & & 0 . + @xmath423 & 0 . & 0 . & 0 . + + @xmath409 & 6 . & 5 . & 124 . + @xmath466 & & 0 . & + @xmath411 & 1 . & 3 . & 134 . + @xmath412 & 4 . & 201 . & + + @xmath413 & 9 . & & 36 . + @xmath414 & 2 . & 44 . & 26 . + + @xmath415 & 174 . & 369 . & 435 . + @xmath416 & 215(20 ) & & 235(50 ) + mode & .5 cm @xmath143 .5 cm & .5 cm @xmath186 .5 cm & .5 cm @xmath473 .5 cm + + @xmath453 & 50 . & 221 . & 101 . + @xmath474 & 2 . & . + + @xmath424 & 0 . & 0 . & 0 . + + @xmath410 & 7 . & 8 . + + @xmath413 & 8 . & & 35 . + @xmath414 & 2 . & 44 . & 21 . + + @xmath415 & 69 . & 300 . & + @xmath416 & 168(10 ) & & 220(35 ) + mode & .5 cm @xmath141 .5 cm & mode & .5 cm @xmath475 .5 cm + + @xmath453 & 118 . & @xmath476 & 33 . + @xmath454 & 41 . & @xmath444 & 8 . + + @xmath424 & 0 . & & + + @xmath410 & 0 . & & + @xmath456 & 0 . & @xmath466 & 0 . + @xmath457 & 1 . & @xmath411 & 5 . + @xmath458 & 75 . & @xmath412 & 189 . + + @xmath414 & 30 . + + @xmath415 & 250 . & & 261 . + @xmath416 & 258(18 ) & & 180 @xmath477(25 ) + mode & .5 cm @xmath196 .5 cm & .5 cm @xmath197 .5 cm & .5 cm @xmath219 .5 cm + + @xmath478 & 12 . + @xmath479 & 3 . & & 13 . + @xmath480 & 33 . & 86 . & 37 . + @xmath481 & 54 . & 28 . & 19 . + + @xmath455 & 1 . + @xmath482 & 0 . & & 0 . + @xmath424 & 0 . & 1 . + + @xmath410 & 20 . & 12 . & 140 . + @xmath456 & & 4 . & + @xmath457 & 2 . & 6 . & 36 . + @xmath458 & 10 . & 67 . & + @xmath483 & & 0 . & + @xmath484 & 0 . & 1 . & 16 . + @xmath485 & 0 . & 24 . & + @xmath486 & 0 . & 40 . & 21 . + @xmath487 & 0 . & . + + @xmath428 & 0 . & 0 . & 2 . + @xmath488 & & 0 . & + @xmath489 & 0 . & 0 . & 0 . + @xmath490 & 0 . & + mode & .5 cm @xmath196 .5 cm & .5 cm @xmath197 .5 cm & .5 cm @xmath219 .5 cm + + @xmath491 & 0 . & + @xmath433 & 0 . & 1 . & 1 . + @xmath435 & 0 . & 1 . & 89 . + @xmath437 & 2 . & 127 . & 1 . + + @xmath413 & 8 . & & 14 . + @xmath414 & 4 . & 28 . & 15 . + @xmath438 & 9 . & + + @xmath492 & & 0 . & + @xmath439 & 0 . & + @xmath440 & 0 . & 0 . & 0 . + @xmath493 & 0 . & 31 . & + + @xmath415 & 161 . & 483 . & + @xmath416 & 427(120 ) & 340(80 ) & + mode & .5 cm @xmath195 .5 cm & .5 cm @xmath214 .5 cm & .5 cm @xmath222 .5 cm + + @xmath403 & 62 . & & 34 . + @xmath462 & 2 . & & 4 . + @xmath463 & 0 . + @xmath494 & 0 . & & 0 . + @xmath464 & 86 . & 37 . & 31 . + @xmath465 & 27 . & + + @xmath408 & 2 . & & 1 . + + @xmath495 & 0 . & & 0 . + + @xmath466 & & 2 . & + @xmath411 & 9 . & 20 . & 113 . + @xmath412 & 22 . & 192 . & 40 . + @xmath496 & & 0 . & + @xmath497 & 0 . & 0 . & 13 . + @xmath498 & 1 . & 25 . & 5 . + + @xmath499 & & 0 . & + @xmath429 & 0 . & 0 . & 1 . + @xmath430 & 0 . & + mode & .5 cm @xmath195 .5 cm & .5 cm @xmath214 .5 cm & .5 cm @xmath222 .5 cm + + @xmath431 & 1 . & 4 . + + @xmath413 & 9 . & & 14 . + @xmath414 & 5 . & 26 . & 15 . + @xmath438 & 10 . & + + @xmath492 & & 0 . & + @xmath439 & 0 . & 2 . & 91 . + @xmath440 & 0 . & 0 . & 0 . + @xmath493 & 0 . & 23 . & + + @xmath415 & 237 . & 350 . & 579 . + @xmath416 & 208(13 ) & & + mode & .5 cm @xmath216 .5 cm & mode & .5 cm @xmath215 .5 cm + + @xmath469 & 37 . & @xmath453 & 115 . + @xmath406 & 13 . & @xmath407 & 13 . + @xmath420 & 4 . & @xmath421 & 4 . + @xmath422 & 33 . & & + + @xmath423 & 1 . & @xmath424 & 1 . + @xmath500 & 0 . & @xmath501 & 0 . + + @xmath409 & 0 . & @xmath410 & 0 . + @xmath425 & 0 . & @xmath426 & 0 . + @xmath466 & 1 . & & + @xmath411 & 14 . & & + @xmath412 & 107 . & & + @xmath502 & 3 . & @xmath503 & 12 . + + @xmath427 & 0 . & @xmath428 & 0 . + @xmath499 & 0 . & & + @xmath429 & 0 . & & + @xmath430 & 1 . & & + mode & .5 cm @xmath216 .5 cm & mode & .5 cm @xmath215 .5 cm + + @xmath504 & 0 . & & + @xmath505 & 0 . & @xmath433 & 0 . + @xmath506 & 1 . & @xmath435 & 2 . + @xmath507 & 48 . & @xmath437 & 138 . + + @xmath414 & 22 . & & 22 . + @xmath508 & 5 . & & 5 . + + @xmath492 & 0 . & & 0 . + @xmath439 & 0 . & & 0 . + @xmath440 & 0 . & & 0 . + @xmath493 & 17 . & & 17 . + + @xmath415 & 308 . & & 330 .

we discriminate gluonic hadrons from conventional @xmath0 states by surveying radial and orbital excitations of all i=0 and i=1 @xmath1 systems anticipated up to 2.1 gev . we give detailed predictions of their quasi - two - body branching fractions and identify characteristic decay modes that can isolate quarkonia . several of the `` missing mesons '' with l@xmath2 and l@xmath3 are predicted to decay dominantly into certain s+p and s+d modes , and should appear in experimental searches for hybrids in the same mass region . we also consider the topical issues of whether some of the recently discovered or controversial meson resonances , including glueball and hybrid candidates , can be accommodated as quarkonia . s1@xmath4s@xmath5 s0@xmath6s@xmath7 p2@xmath4p@xmath8 p1@xmath4p@xmath5 p0@xmath4p@xmath7 p1@xmath6p@xmath5 * abstract *

epidemic , one of the most important issues related to our real lives , such as computer virus on internet and venereal disease on sexual contact networks , attracts a lot of attention . among all the models on the process of the epidemic , susceptible - infected ( si ) model @xcite , susceptible - infected - susceptible ( sis ) model @xcite , and susceptible - infected - removed ( sir ) model @xcite , are considered as the theoretical templates since they can , at least , capture some key features of real epidemics . after some classical conclusions have been achieved on regular and random networks , recent studies on small - world ( sw ) networks @xcite and scale - free ( sf ) networks @xcite introduce fresh air into this long standing area ( see the reviews @xcite and the references therein ) . the most striking result is that in the sis and sir model , the critical threshold vanishes in the limit of infinite - size sf networks . it is also a possible explanation why some diseases are able to survive for a long time with very low spreading rate . in this paper , we focus on the sis model . although it has achieved a big success , the standard sis style might contain some unexpected assumption while being introduced to the sf networks directly , that is , each node s potential infection - activity ( infectivity ) , measured by its possibly maximal contribution to the propagation process within one time step , is strictly equal to its degree . as a result , in the sf networks the nodes with large degree , named _ hubs _ , will take the greater possession of the infectivity , so - called _ super - spreader_. this assumption may fail to mimic some cases in the real world where the relation between degree and infectivity is not simply equal @xcite . the first example is that , in most of the existing peer - to - peer distributed systems , although their long - term communicating connectivity shows the scale - free characteristic @xcite , all peers have identical capabilities and responsibilities to communicate at a short term , such as the gnutella networks @xcite . second , in sexual contact networks , even the hub node has many acquaintances ; he / she has limited capability to contact with others during limited periods @xcite . third , the referral of a product to potential consumers costs money and time in network marketing processes ( e.g. a salesman has to make phone calls to persuade his social surrounding to buy the product ) . therefore , the salesman will not make referrals to all his acquaintances @xcite . the last one , in some email service systems , such as the gmail system schemed out by google @xcite , the clients are assigned by limited capability to invite others to become gmail - user after being invited by an e - mail from another gmail - user . similar phenomena are common in our daily lives , thus need a further investigation . in the epidemic contact network , node presents individual and link denotes the potential contacts along which infections can spread . each individual can be in two discrete states , whether susceptible ( s ) or infected ( i ) . at each time step , the susceptible node which is connected to the infected one will be infected with rate @xmath2 . meanwhile , infected nodes will be cured to be again susceptible with rate @xmath3 , defining the effective spreading rate as @xmath4 . without losing of generality , we set @xmath5 . individuals run stochastically through the cycle susceptible - infected - susceptible , which is also the origin of the name , sis . denote @xmath6 and @xmath7 the density of the susceptible and infected population at the time step @xmath8 , respectively . then @xmath9 in the standard sis model , each individual will contact all its neighbors once at each time step , thus the infectivity of each node is equal to its degree . in the present model , we assume that every individual has the same infectivity @xmath0 . that is to say , at each time step , each infected individual will generate @xmath0 contacts where @xmath0 is a constant . multiple contacts to one neighbor are allowed , and the contacts to the infected ones , although without any effect on the epidemic dynamics , are also counted . in this paper , with half nodes infected initially , we run the spreading process for sufficiently long time , and calculate the fraction of infected nodes averaging over the last 1000 steps as the density of infected nodes in the steady stage ( denoted by @xmath10 ) . all of our simulation results are obtained from averaging over @xmath11 different network realizations , and for each @xmath12 independent runs with different initial configurations . [ 0.7 ] ( color online ) average value of @xmath10 as a function of the effective spreading rate @xmath13 on a ba network with average degree @xmath14 and network size @xmath15 . the black points represent the case of standard sis model , and the red , green and blue points correspond to the present model with @xmath16 , 3 and 2 , respectively . the arrows point at the critical points gained from the simulation . the insert shows the threshold @xmath17 scaling with @xmath1 , with solid line representing the analytical results.,title="fig : " ] let @xmath18 denote the fraction of vertices of degree @xmath19 that are infected at time @xmath8 . then using the mean - field approximation , the rate equation for the partial densities @xmath18 in a network characterized by a degree distribution @xmath20 can be written as : @xmath21\sum_{k'}\frac{p(k'|k ) i_{k'}(t)a}{k'},\ ] ] where @xmath22 denotes the conditional probability that a vertex of degree @xmath19 is connected to a vertex of degree @xmath23 . considered the uncorrelated networks , where @xmath24 , the rate equation takes the form : @xmath25i(t)a.\ ] ] using @xmath26 to denote the value of @xmath18 in the steady stage with sufficiently large @xmath8 , then @xmath27 which yields the nonzero solutions @xmath28 where @xmath29 is the infected density at the network level in the steady stage . then , one obtains @xmath30 to the end , for the critical point where @xmath31 , we get @xmath32 this equation defines the epidemic threshold @xmath33 below which the epidemic prevalence is null , and above which it attains a finite value . the previous works about epidemic spreading in sf networks present us with a completely new scenario that a highly heterogeneous structure will lead to the absence of any epidemic threshold @xcite , while now , in the present model , it is @xmath1 instead . as shown in fig . 1 , the analytical result agrees very well with the simulations . furthermore , it is also clear that the larger infectivity @xmath0 will lead to the higher prevalence @xmath10 . [ 0.65](color online ) average value of @xmath10 as a function of the effective spreading rate @xmath13 on ( a ) the random and ba networks ; ( b ) the sf configuration networks for different values of @xmath34 ; ( c ) the ba networks with different average degree . in ( a ) and ( b ) , the average degree is with @xmath35 , and for all the simulations , @xmath15 and @xmath36 are fixed.,title="fig : " ] [ 0.65](color online ) average value of @xmath10 as a function of the effective spreading rate @xmath13 on ( a ) the random and ba networks ; ( b ) the sf configuration networks for different values of @xmath34 ; ( c ) the ba networks with different average degree . in ( a ) and ( b ) , the average degree is with @xmath35 , and for all the simulations , @xmath15 and @xmath36 are fixed.,title="fig : " ] [ 0.65](color online ) average value of @xmath10 as a function of the effective spreading rate @xmath13 on ( a ) the random and ba networks ; ( b ) the sf configuration networks for different values of @xmath34 ; ( c ) the ba networks with different average degree . in ( a ) and ( b ) , the average degree is with @xmath35 , and for all the simulations , @xmath15 and @xmath36 are fixed.,title="fig : " ] from the analytical result of the threshold value , @xmath37 , we can also acquire that the critical behavior is independent of the topology of networks which are valid for the mean - field approximation @xcite . to demonstrate this proposition , we implement the present model on various networks ; these include the random networks , the scale - free configuration model @xcite with different power - law exponent @xmath34 , and the ba networks with different average degree . as shown in fig . 2 , under a given @xmath0 , the critical value are the same , which strongly support the valid of eq . furthermore , there is no distinct finite - size effect as shown in fig . 3 . in the original sis model , the node s infectivity relies strictly on its degree @xmath19 and the threshold is @xmath38 . since the variance of degrees gets divergent with the increase of @xmath39 , the epidemic propagation on scale - free networks has an obvious size effect @xcite . however , in the current model , each infected node is just able to contact the same number of neighbors , @xmath0 , rather than its degree . thus the threshold value and the infected density beyond the threshold are both independent of the size @xmath39 . [ 0.7](color online ) average value of @xmath10 as a function of the effective spreading rate @xmath13 on the different sizes of ba networks with @xmath35 and @xmath36.,title="fig : " ] for further understanding of the epidemic dynamics of the proposed model , we study the time behavior of the epidemic propagation . first of all , manipulating the operator @xmath40 on both sides of eq . ( 3 ) , and neglecting the terms of order @xmath41 , we obtain @xmath42 thus the evolution of @xmath7 follows an exponential growing as @xmath43 where @xmath44 . [ 0.7](color online ) average value of @xmath7 in normal plots as time @xmath8 ( a ) and @xmath45 in single - log plots as rescaled time @xmath46 ( b ) for different spreading rate @xmath13 . the numerical simulations are implemented based on ba networks of size @xmath15 , @xmath47 , and @xmath36.,title="fig : " ] [ 0.7](color online ) average value of @xmath7 in normal plots as time @xmath8 ( a ) and @xmath45 in single - log plots as rescaled time @xmath46 ( b ) for different spreading rate @xmath13 . the numerical simulations are implemented based on ba networks of size @xmath15 , @xmath47 , and @xmath36.,title="fig : " ] in fig . 4 , we report the simulation results of the present model for different spreading rates ranging from 0.7 to 0.9 . the rescaled curves @xmath45 ( fig . 4(b ) ) can be well fitted by a straight line in single - log plot for small @xmath8 and the curves corresponding to different @xmath13 will collapse to one curve with rescaling time @xmath46 , which strongly supports the analytical result eq . ( 10 ) . furthermore , a more precise characterization of the epidemic diffusion through the network can be achieved by studying some convenient quantities in numerical experiments . first , we measure the average degree of newly infected nodes at time @xmath8 as @xmath48 then , we present the inverse participation ratio @xmath49 to indicate the detailed information on the infection propagation , which is defined as @xcite : @xmath50 where the weight of recovered individuals in each @xmath19-degree class ( here @xmath19-degree class means the set of all the nodes with degree @xmath19 ) is defined by @xmath51 . from this definition , one can acquire that if @xmath52 is small , the infected are homogeneously distributed among all degree classes ; on the contrary , if @xmath52 is relatively larger then , the infection is localized on some specific degree classes . [ 0.7](color online ) time behavior of the average degree of the newly infected nodes ( top ) and inverse participation ratio @xmath52 ( bottom ) in ba networks of size @xmath15 , @xmath53 for different values of @xmath0 ( with @xmath54 fixed ) ( a ) and @xmath55 ( with @xmath36 fixed ) ( b).,title="fig : " ] [ 0.7](color online ) time behavior of the average degree of the newly infected nodes ( top ) and inverse participation ratio @xmath52 ( bottom ) in ba networks of size @xmath15 , @xmath53 for different values of @xmath0 ( with @xmath54 fixed ) ( a ) and @xmath55 ( with @xmath36 fixed ) ( b).,title="fig : " ] in fig . 5 , we exhibit the time behaviors of these quantities for ba networks and find a hierarchical dynamics , that is , all those curves show an initial plateau , which denotes that the infection takes control of the large degree nodes firstly . once the highly connected hubs are reached , the infection pervades almost the whole network via a hierarchical cascade across smaller degree classes . thus , @xmath56 decreases to the next plateau , which approximates the average degree @xmath57 . immunity , relating to the people s strategies to struggle with the disease epidemics , shows great importance in practice @xcite . since the current model , which can mimic some real cases more accurately , shows different characters with the standard sis model , it requires some in - depth and detailed investigation about the immunity on this model . as we know , immunized nodes can not become infected and , therefore , will not transmit the infection to their neighbors . the simplest immunization strategy is to select immunization population completely randomly , so - called _ random immunization _ @xcite . however , this strategy is inefficient for heterogenous networks . similar to the preferential attachment mechanism introduced by ba model @xcite , dezs and barabsi proposed the _ proportional immunization _ strategy @xcite , in which the immunizing probability of each node is proportional to its degree . this preferential selection strategy can remarkable enhance the immunization efficiency in scale - free networks . the extreme strategy for immunization in heterogenous networks is the so - called _ targeted immunization _ @xcite , where the most highly connected nodes are chosen to be immunized . compared with the random immunization and proportional immunization , the targeted immunization is demonstrated as the most efficient one for various networks @xcite , and several different but relative dynamics @xcite . [ 0.7](color online ) reduced prevalence @xmath58 from numerical simulations of the present model in the random ( square point ) and ba ( circle point ) network with random ( black line ) , proportional ( red line ) and targeted immunizations ( blue line ) . in the simulations , the parameter @xmath53 , @xmath36 , @xmath35 and @xmath15 are fixed.,title="fig : " ] in fig . 6 , we report the simulation results about the three mentioned immunization strategies on the current model . the @xmath59-axis , @xmath60 , denotes the fraction of immunized population , and the @xmath61-axis , @xmath58 , represents the performance , where @xmath62 is the prevalence of infected nodes without immunization and @xmath63 the one after immunization . from the simulation results , one can find that the epidemic thresholds under random , proportional and targeted immunizations of random networks are @xmath64 , @xmath65 and @xmath66 , respectively . and those of ba networks are @xmath64 , @xmath67 and @xmath68 . it is clear from the simulation results , even in the current model where the infectivities of large - degree nodes are greatly suppressed , the targeted immunization performs best . combine with the hierarchical behavior observed in sec . iv , it strongly indicates that the heterogeneities of degree and infectivity could both contribute to the violent spreading of disease . hence even for the current model with identical infectivity , the hub nodes play much more important roles in determining the dynamical property . note that , in this model , for heterogenous networks , the random immunization is more efficient than the standard case , and the threshold @xmath69 is the same for ba and random networks . actually , the random immunization is implemented by randomly selecting and immunizing @xmath70 nodes on a network of fixed size @xmath39 . at the mean - field level , the presence of uniform immunity will effectively reduce the spreading rate @xmath13 by a factor @xmath71 . according to eq . ( 8) , the immunization threshold is given by @xmath72 as shown in fig . 6 , the simulated result ( @xmath73 ) agrees with the analytical result ( @xmath74 ) well . to compare , the random immunization threshold of standard sis model is given by @xmath75 @xcite . namely , to control the spreading , one have to immunize all the population as @xmath76 in the thermodynamic limit @xmath77 . [ 0.7](color online ) average value of @xmath7 as time @xmath8 for no immunization ( black ) , random immunization ( red ) , proportional immunization ( green ) , and targeted immunization ( blue ) at @xmath53 and @xmath78 . the numerical simulations are implemented based on ba networks of size @xmath15 , @xmath35 , and @xmath36 . the arrows indicate the time that the whole spreading process comes to the steady stage.,title="fig : " ] for further understanding the effects of those different immunization strategies , we study the time behaviors as shown in fig . 7 . in accordance with the above results , the spreading velocity under target immunization is the lowest . note that , different from the standard sis model , the random immunization can obviously slow down the spreading in the early stage even with a tiny population @xmath79 . in this paper , we investigated the behaviors of sis epidemics with the identical infectivity @xmath0 . by comparing the dynamical behaviors of the present model of different values of @xmath0 with the standard one on ba networks , we found the existence of epidemic spreading threshold . the analytical result of the threshold @xmath1 is provided , which agrees with numerical simulation very well . the critical value is independent of the topology of underlying networks , just depends on the dynamical parameter @xmath0 and the whole spreading process does not have the distinct finite - size effect . for sf networks , the infected population grows in an exponential form in the early stage , and then follows a hierarchical dynamics . in addition , the time scale is also independent of the underlying topology . the last but not the least , the numerical results of random , proportional , and targeted immunization are presented . we found that the targeted immunization performs best , while the random immunization is much more efficient in heterogenous networks than the standard case . bhwang acknowledges the support of 973 project under grant no . 2006cb705500 , the special research founds for theoretical physics frontier problems under grant no . a0524701 , the specialized program under the presidential funds of the chinese academy of science , and the national natural science foundation of china under grant no . 10472116 . tzhou acknowledges the support of the national natural science foundation of china under grant nos . 70471033 and 10635040 . m. barthlemy , a. barrat , r. pastor - satorras , and a. vespihnani , phys . lett . * 92 * , 178701 ( 2004 ) ; m. barthlemy , a. barrat , r. pastor - satorras , and a. vespihnani , j. theor . biol . * 235 * , 275 ( 2005 ) . r. pastor - satorras , and a. vespignani , _ epidemics and immunization in scale - free networks_. in : s. bornholdt , and h. g. schuster ( eds . ) _ handbook of graph and networks _ , wiley - vch , berlin , 2003 ; t. zhou , z. -q . fu , and b. -h . wang , prog . * 16 * , 452 ( 2006 ) ; s. boccaletti , v. latora , y. moreno , m. chavez , and d. -u . hwang , phys . rep . * 424 * , 175 ( 2006 ) . j. joo , and j. l. leboitz , phys . e * 69 * , 066105 ( 2004 ) ; r. olinky , and l. stone , phys . e * 70 * , 030902 ( 2004 ) ; t. zhou , j .- liu , w .- j . chen , and b .- h . wang , phys . e * 74 * , 056109 ( 2006 ) ; r. yang , b .- h . wang , j. ren , w .- j . bai , z .- w . shi , w .- x . wang , and t. zhou , phys . a * 364 * , 189 ( 2007 ) . j. mller , siam j. appl . math . * 59 * , 222 ( 1998 ) ; d. s. callway , m. e. j. newman , s. h. strogatz , and d. j. watts , phys . rev . lett . * 85 * , 5468 ( 2000 ) ; r. cohen , k. erez , d. ben - avraham , and s. havlin , phys . rev . lett . * 85 * , 4626 ( 2000 ) . z. h. liu , y. c. lai , and n. ye , phys . e * 67 * , 031911 ( 2003 ) ; d. h. zanette , and m. kuperman , physica a * 309 * , 445 ( 2002 ) ; y. c. lai , z. h. liu , and n. ye , int . j. mod . b * 17 * , 4045 ( 2003 ) ; h. zhang , z. h. liu , and w. c. ma , chin . * 23 * , 1050 ( 2006 ) . n. madar , t. kalisky , r. cohen , d. ben - avraham , and s. havlin , eur . j. b * 38 * , 269 ( 2004 ) ; t. zhou , and b. -h . wang , chin . * 22 * , 1072 ( 2005 ) ; f. takeuchi , and k. yamamoto , lect . notes comput . * 3514 * , 956 ( 2005 ) ; w. -j . bai , t. zhou , and b. -h . wang , arxiv : physics/0610138 .

in this paper , a susceptible - infected - susceptible ( sis ) model with identical infectivity , where each node is assigned with the same capability of active contacts , @xmath0 , at each time step , is presented . we found that on scale - free networks , the density of the infected nodes shows the existence of threshold , whose value equals @xmath1 , both demonstrated by analysis and numerical simulation . the infected population grows in an exponential form and follows hierarchical dynamics , indicating that once the highly connected hubs are reached , the infection pervades almost the whole network in a progressive cascade . in addition , the effects of random , proportional , and targeted immunization for this model are investigated . based on the current model and for heterogenous networks , the targeted strategy performs best , while the random strategy is much more efficient than in the standard sis model . the present results could be of practical importance in the setup of dynamic control strategies .

the alkali atoms have a simple ground state electronic structure , with only one valence electron in an _ s_-state . on a level of accuracy , where the relativistic corrections to the spectrum can be ignored , the bound state spectrum of the excited valence electron can be well described by the spherically symmetric effective single - particle potential of marinescu _ et al_. @xcite : @xmath15 where @xmath16\,.\label{effective charge}\ ] ] this is actually a nonlocal potential , because it depends for each proton number @xmath17 of the alkali atom under consideration parametrically on the orbital angular momentum @xmath18 of the valence electron . at small distance @xmath19 to the atomic nucleus this effective interaction potential mutates into a coulomb potential , describing the interaction of @xmath17 protons with the outermost electron , and an additional ( large ) constant ; that is @xcite , @xmath20\;\;\text{for\;\ } r\ll1\,.\label{effective potential very close to the origin-1}\ ] ] conversely , far outside the ionic core region the potential converts into a superposition of a long - ranged coulomb term , describing the interaction between a net positive charge @xmath21 and the valence electron ( like in hydrogen atom ) , and a short - ranged core polarization term ; that is @xcite , @xmath22 in the region around the ionic core , comprising @xmath23 strongly bound electrons filling the inner electron shells of the atom , the two parameters @xmath24 and @xmath25 represent the effects of the polarizability of the latter , while the parameters @xmath26 , @xmath27 , @xmath28 , and @xmath29 shape the spatial dependence of the effective charge @xmath30 , as it alters as a function of @xmath19 from unity to a value @xmath17 . for rubidium @xmath31 , for cesium @xmath32 , and for francium @xmath33 . recently , a phenomenological modification of the potential for @xmath34 has been suggested in terms of a cutoff @xmath35 in the core region , which successfully predicts for all principal quantum numbers @xmath36 and total angular momentum @xmath37 the fine splittings of the rydberg levels @xcite : @xmath38 where @xmath39 denotes the spin - orbit potential . new precise spectroscopic data of @xmath13 indeed comply for all principal quantum numbers @xmath40 very well with the ( semi ) analytical results obtained from quasiclassical wkb calculations , cf . tables i and ii in ref . @xcite . in what follows , sec . [ section ii ] , we first check the accuracy of our recent quasiclassical calculations of the spectrum of the highly excited valence electron in @xmath13 @xcite with the potential ( [ modified potential ] ) , employing for the solution of the radial schrdinger eigenvalue problem a modern numerical collocation method based on the barycentric chebyshev interpolation @xcite . the results of these numerical calculations indeed agree very well with our recent quasiclassical calculations of the quantum defects for orbital angular momentum @xmath41 and also @xmath42 , but for @xmath2 we spot for the heavy - metal alkali atoms rubidium and cesium a discrepancy . in sec . [ section iii ] we then provide an explanation for this discrepancy bringing out for @xmath2 a hitherto unnoticed feature of the reputable potential of marinescu _ et al_. ( [ marinescu et al . ] ) . in sec . [ section iv ] we show how to construct for the radial eigenfunctions of the rydberg states carrying an arbitrary orbital angular momentum @xmath43 two complementary uniform quasiclassical approximations . the first is the uniform wkb approximation of langer @xcite , where we determine the normalization constant by the procedure described by bender and orszag @xcite . the obtained analytical formula for the radial eigenfunctions of the rydberg states for @xmath1 in fact agrees remarkably well with the numerical calculations almost everywhere with exception of a small region around the origin at @xmath10 . close to the origin , however , the langer approximation becomes invalid . we thus patch in the region well below the remote turning point the quasiclassical approximation of langer with an ansatz for the radial wave function in terms of a bessel function first proposed by fock @xcite , that is asymptotically exact for @xmath44 , thus enabling us , for example for @xmath7 , to analytically determine at the origin @xmath10 the value of the radial wave function for the highly excited _ s_-states . in sec . [ section v ] , finally , we use these results to present a simple elementary proof for the semi - empirical fermi - segr formula @xcite determining the hyperfine splittings of the highly excited _ s_-states of the alkali atoms . to verify the accuracy of the quasiclassical calculations presented in ref . @xcite a numerically accurate method ( see supplementary material @xcite ) is required , that solves the radial schrdinger eigenvalue problem for the radial eigenfunctions @xmath45 with the modified potential ( [ modified potential ] ) reliably and accurately also for large principal quantum numbers @xmath0 @xcite : @xmath46u_{n , j , l}(r)=0\,.\label{schroedinger eigenvalue problem}\ ] ] to achieve this goal we use here a spectral collocation method @xcite on a grid consisting of @xmath47 chebyshev grid points obtained by projecting equally spaced points on the unit circle down to the interval @xmath48 $ ] . trivial scaling and shift leads then to the not - equally spaced point set @xmath49 which clusters near @xmath10 and near @xmath50 . in sharp contrast to a traditional finite difference method that controls the error of numerical discretization by the choice of grid spacing , the accuracy of a _ spectral _ collocation method ( a well - known concept in modern numerical mathematics ) is only limited by the smoothness of the function being approximated @xcite . implementing now spectral chebyshev collocation the seeked wave function @xmath51 solving the radial schrdinger eigenvalue problem ( [ schroedinger eigenvalue problem ] ) is represented in terms of a finite vector @xmath52 of its values at the chebyshev grid points @xmath53 , thus defining implicitely a stable and accurate lagrange polynomial interpolant of degree @xmath54 . of particular value and simplicity is the numerically robust barycentric representation of this interpolant due to salzer @xcite : @xmath55 where @xmath56 as a matter of fact , @xmath57 is a polynomial of degree @xmath54 , coinciding with the function values @xmath52 at the grid points @xmath53 . well - known accuracy and stability concerns regarding convergence of high order polynomial interpolants do not apply to a chebyshev grid with its not - equispaced points clustering around the corner points of the grid @xcite . replacing the function @xmath58 by such a polynomial interpolant @xmath57 of degree @xmath54 implies that derivative operations on those functions are replaced by the same operations applied to their interpolant . thus , the first derivative @xmath59 is now represented by a matrix @xmath60 of size @xmath61 acting on the vector of function values @xmath52 at @xmath47 grid points @xmath53 @xcite , likewise the second - order derivative @xmath62 is represented by a matrix @xmath63 . this approach converts the radial schrdinger eigenvalue problem ( [ schroedinger eigenvalue problem ] ) into a standard matrix eigenvalue problem . a crucial point here is that in the calculations of the spectrum of the highly excited bound valence electron the grid should be fine enough to resolve the oscillations of the wave functions @xmath58 under consideration also in the coarsest part of the grid in accordance with the sampling theorem @xcite . moreover , the largest grid point @xmath64 should be located in the region well beyond the remote classical turning point @xmath65 , say , @xmath66 . in effect , one then requires dirichlet boundary conditions for the eigenfunction @xmath58 at both ends of the grid : @xmath67 these boundary conditions imply that the first and the last columns as well as the first and the last row of the matrix @xmath68 can be stripped off @xcite , thus leading to a @xmath69 matrix eigenvalue problem to be solved for the @xmath70 unknown function values @xmath71 at the inner points of the grid . it should be noted that only eigenvectors with associated eigenvalue @xmath72 need to be searched @xcite . moreover , because only eigenvectors with components @xmath71 becoming exponentially small for @xmath53 well beyond the remote classical turning point @xmath73 are meaningfull , all other solutions of the discrete matrix eigenvalue problem being physically meaningless . for a detailed discussion and demonstration of the accuracy of the spectral collocation method on a chebyshev grid , we refer to our supplementary material @xcite , where we present a comparison with the well - known analytical eigenfunctions of the hydrogen atom . vs. scaled distance @xmath74 for orbital angular momentum @xmath2 and total angular momentum @xmath75 of the excited bound valence electron @xmath76 for rubidium ( red ) and cesium ( dashed blue ) atoms , calculated with the effective potential of marinescu _ et al_. ( [ marinescu et al . ] ) . there exists a tiny second classical region located deep inside the atom core close to the origin , where the quasiclassical momentum acquires again real values , well below the positions of the inner turning points @xmath77 and @xmath78 for rubidium and cesium , respectively , representing the lower boundary of their respective outer classical regions extending up to their remote turning point @xmath3 . + ] , cf . ( [ action integral ] ) , as calculated from a barycentric polynomial interpolant @xmath79 on a chebyshev grid , for the excited bound valence electron of @xmath13 with principal quantum number @xmath80 , orbital angular momentum @xmath7 , and total angular momentum @xmath81 . the bound state spectrum of the valence electron in @xmath13 , as calculated by the aforementioned spectral collocation method @xcite , indeed agrees for almost all orbital angular momenta @xmath82 , as well with the spectroscopic data @xcite as with the quasiclassical calculations @xcite , with the exception of the @xmath2 rydberg states @xcite , where a small systematic discrepancy is discernible between the results obtained by the quasiclassical and the full numerical calculations , cf . table [ table 1 ] . we offer here a simple explanation for this anomaly , that applies only to the heavy alkali atoms rubidium and cesium ( and most likely also to francium ) , and which to the best of our knowledge has not been reported before . [ cols="<,^,^,^,^,^,^",options="header " , ] [ table 3 ] for the @xmath7 , @xmath81 states of the valence electron in the alkali atoms the fine structure splitting due to spin - orbit coupling ( assuming exact spherical symmetry of the effective potential ) is zero . neglecting the electric quadrupole moment of the nucleus a detectable shift in the spectrum can now be attributed to the hyperfine interaction of the magnetic moment of the valence electron with the nuclear magnetic moment @xcite . within the range of validity of the fermi - contact - interaction model , the size of the spectral splitting is then determined by the magnetic dipole interaction ( hyperfine splitting ) constant @xcite @xmath83 here @xmath84 is the vacuum permeability , @xmath85 denotes the bohr magneton , and the _ g_-factors of electron and nucleus are @xmath86 and @xmath87 , respectively . for @xmath13 it is found that @xmath88 , and for @xmath12 , @xmath89 @xcite . it should be noted that in our system of units , see @xcite , the particle density distribution @xmath90 is being measured as the number of particles per unit volume @xmath91 . the value of the wave function of the rydberg _ s_-states @xmath92 at the origin @xmath10 can be calculated analytically using the asymptotics of the action integral ( [ s fock ] ) for small @xmath19 : @xmath93 insertion of ( [ eq : fock action for small r ] ) into ( [ u fock ] ) leads then together with the analytical result ( [ eq : normalization constant fock ii ] ) for the normalization constant to the exact result @xmath94 this formula connects the value of the _ @xmath95_-state wave function at the origin to the derivative @xmath96 of the bound state spectrum in a radial schrdinger eigenvalue problem . in the literature it is is often referred to as the semi - empirical fermi - segr formula @xcite . for a rigorous derivation for differential equations of the type ( [ schroedinger eigenvalue problem ] ) , based on an identity for the wronski determinant , see ref . @xcite . equation ( [ hyperfine constant ] ) engenders that the magnetic dipole interaction ( hyperfine splitting ) constant @xmath11 for the highly excited valence electron of the alkali atoms @xmath97 indeed should obey to the scaling relation @xmath98 in experiment the hyperfine level shift depends on nuclear spin @xmath99 , total angular momentum of the valence electron @xmath100 , and on total angular momentum @xmath101 assuming values in the interval @xmath102 . if only the magnetic dipole interaction was considered , then for @xmath7 , @xmath81 a level @xmath103 would split as a result of the magnetic hyperfine interaction for the special case of nuclear spin @xmath104 into a doublet structure with quantum numbers @xmath105 @xcite : @xmath106 table [ table 3 ] compares the theoretical values of the magnetic dipole interaction ( hyperfine splitting ) constant @xmath11 obtained from ( [ eq : hyperfine scaling relation ] ) for @xmath12 and @xmath13 atoms with spectroscopic data @xcite . overall , a very good agreement between theory and experiment can be observed . using a numerically accurate and easy to implement modern numerical method , namely , spectral collocation on a chebyshev grid @xcite based on the barycentric interpolation formula of salzer ( [ salzer formula ] ) , we solved the radial schrdinger eigenvalue problem and determined the excitation spectrum of the bound valence electron in the alkali atoms , thus confirming the high accuracy of recent quasiclasscial calculations of the quantum defect for the rydberg states carrying orbital angular momentum @xmath41 or @xmath107 , with exception of the @xmath2 rydberg states of rubidium and cesium atoms . as a reason for this anomaly we identified as a feature of the potential of marinescu _ et al_. @xcite , existing only for orbital angular momentum @xmath2 , a tiny second classical region located deep inside the atom core around the nucleus of alkali atoms with proton number @xmath108 , cf . ( [ fig 1 . quasiclassical momentum for l 3 ] ) , thus invalidating for the heavy alkali atoms , rubidium and cesium ( and possibly also francium ) , a standard wkb calculation with only two widely spaced turning points . also , we found that the uniform wkb approximation of langer for the radial wave function of the valence electron for @xmath109 indeed represents almost everywhere a remarkably accurate approximation to the exact solution of the radial schrdinger eigenvalue problem ( [ schroedinger eigenvalue problem ] ) , omitting a tiny interval near to the lower turning point of the classically accessible region . in the region around the origin , where the unifrom wkb approximation of langer for the _ @xmath95_-states ceases to be valid , we then showed using an ansatz of fock @xcite , that a complementary uniform quasiclassical solution in terms of a bessel function can be constructed , that coincides with the exact solution of the radial wavefunction for @xmath44 . the uniform quasiclassical approximation of fock for the rydberg the _ s_-states was found to approximate the exact radial eigenfunction almost everywhere in the classically accessible region remarkably well , with exception of a small interval around the remote turning point . a substantial reduction of computer time was achieved in the evaluations of the quasiclassical wave functions ( [ uniform langer - wkb ] ) and ( [ u fock ] ) , when we replaced the action integral @xmath110 by a corresponding ( high - order ) barycentric interpolation polynomials @xmath111 in the interval @xmath112 . upon patching the wave function of langer and fock inside the classically accessible region and making use of an exact result for the normalization integral of the langer wave function , due to bender and orszag @xcite , we finally derived an analytical result determining the quantum defect for @xmath7 and also the value of the radial _ s_-wave eigenfunctions at the origin , thus providing a very simple and short proof of the fermi - segr formula . also , within the range of validity of the fermi - contact model an analytic scaling relation for the constant @xmath11 describing the size of the hyperfine shifts and splittings of the rydberg _ s_-states of the valence electron in alkali atoms was found , cf . ( [ eq : hyperfine scaling relation ] ) , that apparently is for all principal quantum numbers @xmath0 in good agreement with precise spectroscopic data of @xmath12 and @xmath13 .

we present numerically accurate calculations of the bound state spectrum of the highly excited valence electron in the heavy - metal alkali atoms solving the radial schrdinger eigenvalue problem with a modern spectral collocation method that applies also for a large principal quantum number @xmath0 . as an effective single - particle potential we favor the reputable potential of marinescu _ et al_. , [ phys . rev . a * 49 * , 982 ( 1994 ) ] . recent quasiclassical calculations of the quantum defect of the valence electron agree for orbital angular momentum @xmath1 overall remarkably well with the results of the numerical calculations , but for the rydberg states of rubidium and also cesium with @xmath2 this agreement is less fair . the reason for this anomaly is that in rubidium and cesium the potential acquires for @xmath2 deep inside the ionic core a second classical region , thus invalidating a standard wkb calculation with two widely spaced turning points . comparing then our numerical solutions of the radial schrdinger eigenvalue problem with the uniform analytic wkb approximation of langer constructed around the remote turning point @xmath3 we observe everywhere a remarkable agreement , apart from a tiny region around the inner turning point @xmath4 . for _ s_-states the centrifugal barrier is absent and no inner turning point exists , @xmath5 . with the help of an ansatz proposed by fock we obtain for the _ s_-states a second uniform analytic approximation to the radial wave function complementary to the wkb approximation of langer , which is exact for @xmath6 . from the patching condition , that for @xmath7 the langer- and fock solutions should agree in the intermediate region @xmath8 , not only an equation determining the quasiclassical quantum defect @xmath9 but also the value of the radial _ s_-wave function at @xmath10 is analytically found , thus validating the fermi - segr formula for the hyperfine splitting constant @xmath11 . as an application we consider recent spectroscopic data for the hyperfine splittings of the isotopes @xmath12 and @xmath13 and find a remarkable agreement with the predicted scaling relation @xmath14 pacs numbers : : 31.10.+z , 31.15.-p , 31.30.gs , 32.80.ee

the data on rare decays from cleo , babar , belle and tevatron hold promise for deepening our understanding of the interplay between flavour - changing and qcd dynamics . in particular , the surprise regarding the @xmath0 amplitude , existing already for considerable time @xcite , calls for an explanation . namely , the branching ratio for the decay mode @xmath11 is measured to be almost six times bigger than the one for @xmath12 , although the same basic @xmath13 penguin mechanism is expected to drive both processes . apparently , this mechanism may have a different appearance when instead of the flavour octet pion there is an ( almost ) flavour singlet @xmath3 particle involved . it is well known that some extraordinary properties of the @xmath3 particle are related to the qcd anomaly . therefore , a suggestion to explain the enhancement of the @xmath0 amplitude by the qcd anomaly at first sight looks very intriguing @xcite . our preceding investigation in this direction found that the @xmath14 amplitude in the hard gluon regime represents a well defined short distance ( sd ) mechanism @xcite , but of minor numerical importance for the process at hand . the inability of this sd mechanism to account for the measured amplitude invites us to explore here the complementary long - distance ( ld ) mechanism . there are conclusions in the literature @xcite that the `` singlet penguin '' amplitude ( in the language of su(3 ) diagrammatic approach @xcite ) contributes substantially to the @xmath0 enhancement . one of the purposes of this note is to investigate this from another angle than done previously , i. e. to identify the singlet penguin contribution and estimate it within the well - defined microscopic dynamical framework . in order to deal with the low energy properties of some of the involved gluons and to calculate the physical @xmath0 amplitude , we have to introduce an appropriate low - energy description . we will employ ideas based on the chiral quark model ( @xmath15qm)@xcite which has been used to describe @xmath16 decays @xcite . here we will rely on the extension of such models , namely the heavy - light chiral quark model ( hl@xmath15qm ) @xcite which has , as the @xmath15qm , turned out to be a convenient calculational tool for addressing soft - gluon contributions , that can be expressed in terms of gluon condensate effects @xcite . one should note that when the quarks and soft gluons in the hl@xmath15qm are integrated out , one obtains standard heavy light chiral perturbation theory ( hl@xmath15pt ) @xcite . the hl@xmath15qm has been applied to @xmath17 mixing @xcite and to decays of the type @xmath18 @xcite . it has also been applied to the decay mode @xmath19 @xcite which has similar aspects as the mode we consider in this paper . in principle , the hl@xmath15qm naturally accounts only for soft kaons in the final state . however , as we show in the next section , assuming the general form of the relevant form factor enables one to perform extrapolation from the soft to the hard kaon case . the ld mechanism that we propose for the @xmath0 decay is shown in figure [ bketagg ] . it accounts for the contribution obtained when a soft gluon ( @xmath21 ) is emitted from the @xmath22 transition together with the virtual gluon ( @xmath23 ) associated with the penguin @xmath24 transition . in addition , one of the gluons ( @xmath25 ) in the @xmath26 vertex in figure [ bketagg ] is also assumed to be soft , so that two soft gluons form a vacuum condensate . the remaining off - shell gluon from the @xmath27 penguin is propagating into the @xmath3 . a mechanism for @xmath28 due to soft gluons . the crosses correspond to the gluon condensate . ] now , the @xmath29 vertex denoted by a large circle in figure [ bketagg ] can be calculated in the soft @xmath30 limit within the hl@xmath15qm , as displayed in figure [ bkggst ] . to begin with it , let us recall that the involved vector current form factors in heavy quark physics are defined by @xmath31 & = f_{b}(q^2 ) m_{b } v^{\mu } + f_{k}(q^2 ) p_{k}^\mu \ ; , \end{split}\ ] ] where @xmath32 in the @xmath33 limit , @xmath34 dominates , as it can be seen from the scaling properties @xmath35 and @xmath36 @xcite . in addition , considering the soft @xmath30 limit within heavy - light chiral perturbation theory ( hl@xmath15pt ) @xcite , @xmath34 is dominated by the @xmath37 pole , and is given by @xmath38 where @xmath39 , and @xmath40 , @xmath41 are the coefficients determined by qcd renormalization of the weak heavy - light current @xcite . the soft @xmath30 limit is of course unphysical in our case , and to overcome this we employ a double pole structure @xmath42 such a structure , proposed in @xcite , seems to fit very well the existing data and the theoretical requirements on the heavy - light vector form factor , for , say , @xmath43 , and some parameter @xmath44 fitting @xmath45 . to determine @xmath44 we observe that ( [ doublepole ] ) in our limits reads @xmath46 so that a comparison with ( [ bpolefk ] ) gives @xmath47 which is general within hl@xmath15pt . knowing the value for @xmath48 , @xmath44 can be determined . we will use the result of the qcd sum rules on the light - cone analysis , @xmath49 @xcite , implying @xmath50 , in agreement with lattice fits @xcite . thus , extrapolating from the soft @xmath30 to the general case we obtain the substitution rule @xmath51 below we will assume that the form factor for the @xmath29 vertex also has the dipole form ( [ doublepole ] ) , because of @xmath37 pole dominance in both cases . this assumption will hold within hl@xmath15pt in the region where it is valid . therefore we will adopt the rule ( [ extrapolation ] ) also for the @xmath29 form factor . for this case , the position of the second pole might be somewhat different , which means that @xmath44 and @xmath52 in ( [ lambdafromfk ] ) should be replaced by @xmath53 and @xmath54 , respectively . the @xmath55 penguin operator at the quark level is @xmath56 where @xmath57 means a covariant derivative , and @xmath58 for the quantity @xmath59 we take the one loop result @xcite , @xmath60 . this result might be slightly changed by perturbative qcd effects like in @xcite for @xmath61 , but we do not enter such details here . @xmath62 in the soft kaon limit . the black box denotes the @xmath27 penguin transition . ] note that the contribution of the _ dipole _ penguin operator @xmath63 is suppressed by the small form factor @xmath64 , and we neglect it . the bosonization of the coloured quark current in ( [ peng ] ) with emission of an additional soft gluon is known @xcite , @xmath65\bigg\ } \ ; , \label{btog}\ ] ] where @xmath66 is the heavy meson `` superfield '' @xcite and @xmath67 , with @xmath68 being the standard goldstone boson @xmath69 matrix field . furthermore , @xmath70 is the meson - quark coupling given by @xmath71 , where @xmath72 gev is the constituent light quark mass , @xmath73 is a hadronic parameter @xcite and @xmath74 is the axial coupling of goldstone bosons to heavy mesons . moreover , within the hl@xmath15qm we find @xmath75 and @xmath76 is a logarithmically divergent loop integral which is expressed in terms of @xmath77 and the gluon condensate @xcite , as follows @xmath78 it should be noted that the structure in ( [ btog ] ) should be rather general , while @xmath70 and the explicit expressions for @xmath79 are model dependent . taking the vector ( @xmath80 ) part of the @xmath81 and connecting with the @xmath37 propagator in figure [ bkggst ] , we obtain in the soft @xmath30 limit the amplitude @xmath82 \quad\times \big\ { -(a_1+a_2)\epsilon^{\mu\nu\alpha\beta } + 2 a_2 v^\mu \epsilon^{\lambda\nu\alpha\beta}v_\lambda \big\ } \;. \label{softamp}\end{gathered}\ ] ] in order to obtain the general amplitude for @xmath29 from this equation , we perform the substitution ( [ extrapolation ] ) for @xmath83 in the denominator above with @xmath84 replaced by @xmath85 . concerning the @xmath86 interaction , it has the general form already used in ref . @xcite , @xmath87 and several groups @xcite calculated the form - factor @xmath88 in the perturbative qcd approach . since this approach becomes unreliable for gluon momenta of @xmath89 gev , we adopt a formula from @xcite which interpolates between the perturbative qcd region and the anomaly value for zero momentum . this formula gives @xmath90 gev@xmath91 , where an error of 25 % has been allowed . accordingly , together with the value of the gluon condensate , this is the major source of uncertainty in our result below . taking now the vacuum expectation value of the two soft gluons @xmath92 we obtain the final amplitude @xmath93 where @xmath94 . numerically , with @xmath95 , @xmath96 gev , and @xmath97 given by the numbers below eq . ( [ lambdafromfk ] ) , we get @xmath98 it should be noted , that even if the uncertainty of @xmath44 is increased to 50 % when replaced by @xmath53 , it has no significant impact on the form factor @xmath85 at the physical point @xmath99 , and thereby not on the final result ( [ eight ] ) . this means that our assumption for the form factor @xmath85 should be rather sound . our result ( [ eight ] ) should be compared to the experimental amplitude @xmath100 gev thus , according to our analysis , only of order 10@xmath101 of the @xmath28 rate enhancement can be ascribed to this gluonic creation of @xmath3 . some additional mechanisms are necessary , such as constructive interference of amplitudes for creating @xmath3 in @xmath102 and @xmath103 state @xcite . however , the gluonic mechanism studied here , with its distinctive flavour - singlet nature , seems to go in the direction of the result of the su(3 ) symmetry analysis in @xcite that shows a substantial singlet penguin contribution to @xmath28 amplitude . let us comment here on how our result fits into the existing accounts of joining the two - gluon configurations arising from the @xmath1 and @xmath2 transitions with those inherent to the @xmath3 particle , in particular on those relying on properties of the @xmath3 particle that are related to the qcd anomaly . the attempts to explain some puzzling hadronic weak decays by qcd anomalies are well known : the @xmath104 enhancement in @xmath105 by the trace anomaly @xcite and the enhancement of @xmath106 decay rates by the axial anomaly @xcite . we entered such study by employing the fact that anomaly permeates all distance scales , that enables one to study the role of two - gluon anomalous configurations from an extreme sd to a truly ld regime . our recent study @xcite shows that there is merely a remnant , the anomaly tail , in the extreme sd case . it should be noted that this contribution is obtained as a two quark operator for @xmath107 and is very different from the ld contribution presented in detail in the previous section . in addition , we have also identified some other contributions which we have found to be negligible . for example , fritzsch @xcite has suggested that an effective interaction of the form @xmath108 might contribute significantly . we will describe elsewhere @xcite different perturbative and nonperturbative contributions to such an effective interaction stemming from anomalous two - gluon configurations . note that already a rotation of an appropriate term from simma and wyler s paper @xcite to fritzsch s form enables one to read off a perturbative contribution to coefficient @xmath109 above , @xmath110 , which gives an amplitude 34 times smaller than ( [ eight ] ) . after observing a systematic suppression of the anomalous two - gluon contributions through all the distance scales @xcite , we are focusing in the present paper to ld nonperturbative gluon configurations that may be phenomenologically more relevant . thereby we are considering the low energy contribution where a gluon condensate accounts for the emission of soft gluons . a priori , our calculation would only be valid in the unphysical case where the outgoing kaon is soft . however , one can extrapolate to the physical point by introducing a dipole form factor for the @xmath111 transition ( as for the standard @xmath112 transition current ) . as a result we find a more significant contribution from this mechanism , which can account for @xmath113 of the measured amplitude . this result has to be compared with findings of @xcite that flavour singlet contributions to @xmath28 may be marginal . however , due to quite large uncertainties in both amplitudes these two results are actually not inconsistent . note that the major portion of the singlet penguin amplitude in @xcite comes from the operators corresponding to singlet quark configurations forming @xmath3 particle . recently another analysis within scet appeared @xcite . this analysis concludes that the `` singlet penguin '' contribution is essential to understand the process @xmath114 . unfortunately , a direct comparison between our treatment of one soft ( @xmath115 ) and one semi - hard ( @xmath116 ) gluon and the one by scet is difficult to perform . anyway , our contribution corresponding to the gluonic configurations forming the gluon condensate is a novel one , and may significantly increase the role of the singlet penguin mechanism in the direction of the result based on the su(3 ) symmetry analysis @xcite . a number of authors already used the surprising @xmath28 enhancement to infer on the contributions beyond the standard model . however , at this stage we need first to consider possible contributions from the specific mechanisms within the standard model , like the one presented here . this mechanism seems to provide an additional contribution to the singlet penguin topology , the understanding of which may be of importance for explaining the data on cp asymmetries in penguin dominated modes @xcite . k. k. and i. p. gratefully acknowledge the support of the norwegian research council and the hospitality of the department of physics in oslo , as well as support of croatian ministry of science , education and sport under the contract no . 0119261 . j. o. e. is supported in part by the norwegian research council and by the european union rtn network , contract no . hprn - 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intriguing @xmath0 decays provide a unique opportunity to study a joining of two - gluon configurations arising from the penguin @xmath1 and @xmath2 transitions , with those inherent to the @xmath3 particle . we employ the heavy - light chiral quark model , applied previously to a somewhat related @xmath4 decay , as a calculational tool accounting for the nonperturbative soft gluon contributions to the amplitude at hand . thereby we arrive at a novel contribution to the singlet penguin amplitude , which within our model accounts for @xmath5 of the measured @xmath0 amplitude . oslo - tp-2 - 05 + zagreb - ztf-05 - 01 + = 2 * * * * + = 0.5 jan o. eeg@xmath6 , kreimir kumeriki@xmath7 , and ivica picek@xmath8 + _ @xmath9department of physics , university of oslo , p.o.b . 1048 blindern , n-0316 oslo , norway + @xmath10department of physics , faculty of science , university of zagreb , p.o.b . 331 , hr-10002 zagreb , croatia + _ + _ pacs _ : 12.15.ji ; 12.39.-x ; 12.39.fe ; 12.39.hg + _ keywords _ : b mesons , rare decays , heavy quarks , chiral lagrangians

the sunyaev - zeldovich effect ( sze ) is a weak distortion of the cosmic microwave background ( cmb ) spectrum introduced as cmb photons propagate through foreground galaxy clusters @xcite . because this signal does not suffer cosmological dimming and is expected to closely track cluster mass , the sze is a potentially powerful tool for producing large , mass - limited galaxy cluster samples that can be used to constrain dark energy , under the proviso that the relationship between cluster mass and the observed signal is well calibrated @xcite . the relationship between the sze observable , the integrated compton @xmath1-parameter , @xmath2 , and cluster mass is difficult to calibrate because of the difficulty of measuring mass directly . to date all examinations of the mass-@xmath2 relationship have assumed hydrostatic equilibrium ( hse ) when converting sze or x - ray measurements of the intracluster gas into estimates of the total mass (; e.g. * ? ? ? * ; * ? ? ? these methods measure scaling relations between closely related quantities derived from the same data ; deviations in one parameter will therefore inevitably be correlated with the other , and the true intrinsic scatter will likely be underestimated . furthermore , both simulations ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and observations @xcite suggest that the hse assumption is incorrect ; non - thermal sources of pressure support may bias or increase the scatter with @xmath2 . the scatter in mass at fixed @xmath2 is an important quantity to determine in preparation for the sze cluster surveys designed to constrain the dark energy equation of state , as scatter in the mass - observable relation is generally degenerate with the cosmological parameters of interest ( e.g. * ? ? ? * ; * ? ? ? these surveys will rely on self - calibration techniques to build cluster mass functions from sze - selected cluster catalogs , however these techniques can be rendered ineffective if the scatter is much larger than the few percent predicted by simulations @xcite . at present there are only weak observational limits , but independent measurements of the mass-@xmath2 scatter can be used to refine the self - calibration @xcite . the local cluster substructure survey ( locuss ; g. p. smith et al . 2009 , in preparation ) is a morphologically unbiased survey of x - ray luminous galaxy clusters in a narrow redshift range ( @xmath6 ) . the survey unites a variety of probes of total mass , intracluster gas , and cluster member galaxies ( e.g. , @xcite , hereafter ; @xcite ; @xcite ) . one of the main goals of the survey is to measure the shape , normalization and scatter of cluster mass - observable relations as an input to cosmological cluster studies . here we use a pilot sample of clusters , assembled from , @xcite , and early locuss observations with the sunyaev - zeldovich array ( sza ) , to compare the sze signal with cluster masses derived from gravitational lensing ( ) and thus to construct the first mass-@xmath2 relation that is independent of the assumption of hse . we summarize the observations and modeling in section [ s : obs ] . the @xmath3relationship , scatter , and its implications are discussed in section [ s : res ] . we assume = 0.3 , = 0.7 , and @xmath7=0.7 ; the median cluster redshift is @xmath8 , at which 1 corresponds to a physical scale of 3.6 kpc . lccccccc a68 & 0.255 & 0.55@xmath90.08 & 3.67@xmath90.52 & 1 & 3.48@xmath90.07 & 2,3 & disturbed + a209 & 0.206 & 0.94@xmath90.14 & 4.57@xmath90.67 & 4 & 1.23@xmath90.39 & 2 & disturbed + a267 & 0.230 & 0.53@xmath90.06 & 3.08@xmath90.34 & 1 & 2.20@xmath90.34 & 2 & disturbed + a383 & 0.188 & 0.39@xmath90.04 & 1.61@xmath90.16 & 4 & 3.71@xmath90.82 & 2 & undisturbed + a611 & 0.288 & 0.39@xmath90.04 & 3.13@xmath90.34 & 1 & 2.12@xmath90.05 & 5 & undisturbed + a773 & 0.217 & 1.03@xmath90.11 & 5.40@xmath90.57 & 1 & 4.03@xmath90.12 & 2,5 & disturbed + z2701 & 0.214 & 0.28@xmath90.03 & 1.46@xmath90.16 & 4 & 1.92@xmath90.07 & 5 & disturbed + a1413 & 0.143 & 1.83@xmath90.26 & 4.90@xmath90.70 & 1 & 2.59@xmath90.50 & 5 & undisturbed + a1689 & 0.181 & 1.86@xmath90.15 & 7.51@xmath90.60 & 1 & 7.44@xmath90.05 & 6 & disturbed + a1763 & 0.288 & 0.56@xmath90.06 & 3.10@xmath90.32 & 4 & 1.42@xmath90.54 & 2 & disturbed + a1835 & 0.253 & 1.03@xmath90.07 & 6.82@xmath90.48 & 1 & 3.35@xmath90.06 & 2,5 & undisturbed + a2218 & 0.171 & 1.12@xmath90.10 & 4.23@xmath90.38 & 1 & 4.23@xmath90.09 & 2 & disturbed + a2219 & 0.228 & 1.12@xmath90.05 & 6.27@xmath90.26 & 4 & 3.48@xmath90.07 & 2 & disturbed + a2537 & 0.297 & 0.42@xmath90.03 & 3.47@xmath90.24 & 4 & 1.75@xmath90.90 & 7 & disturbed + we analyze sze and gravitational lensing observations of 14 clusters , listed in table [ t : cls ] . sze results for eight of these come from @xcite and @xcite , the observational details being found in @xcite . new sze measurements of a209 , a383 , a1763 , a2219 , a2537 , and z2701 were obtained at 31 ghz using the sza , which has baselines of 350@xmath31300@xmath10 for sze sensitivity ( angular scales of 110 - 7 ) and 2 - 7.5 k@xmath10 for radio source removal . a detailed description of sza observations and analysis methods can be found in @xcite . typical integration times were @xmath11 hr per cluster , and the rms noise per beam @xmath12mjy . in six of 14 clusters we detect radio sources within an arcminute of the cluster center , these sources were presented in @xcite . our interferometric observations do not measure directly the total sze flux of the cluster within an aperture , this must be derived from a model cluster profile . as in previous works , we have modeled these clusters with an isothermal @xmath13-model , which has been shown to provide @xmath2 measurements indistinguishable from those from a simulation - motivated pressure profile for radii smaller than @xmath14 @xcite . the shape parameters for the @xmath13-model and an isothermal x - ray temperature were jointly derived from the sze data and from _ chandra _ x - ray images after applying a 100 kpc core cut @xcite . best - fit cluster parameters were obtained using the markov chain monte carlo method described in @xcite . the fluxes of detected radio sources and central sources present in the 1.4 ghz nvss catalog @xcite but undetected at 31 ghz are included as parameters in the markov chains and marginalized over . details of the gravitational lensing mass measurements are described by and @xcite ; the source of each measurement is listed in table [ t : cls ] . briefly , the mass measurements are based on _ hubble space telescope _ ( _ hst _ ) imaging of the cluster cores , ground - based spectroscopy of multiply imaged galaxies identified in these data , and by weakly sheared background galaxies . these data constrain parameterized models of the projected mass distribution in the cluster cores comprising one or more cluster - scale mass components ( to describe the dark matter and intracluster gas ) plus typically @xmath15 galaxy - scale mass components per cluster . measurements of the projected cluster mass and the uncertainty on the mass are then obtained by integrating the mass distributions of the family of models within the chosen confidence interval in parameter space ( in this letter we quote errors at 68% confidence ) out to the chosen radius . ( see also * ? ? ? * ) used the structure of the cluster mass distributions inferred from these lens models the substructure fraction , @xmath16 in conjunction with _ chandra _ observations to classify the cluster cores as `` disturbed '' or `` undisturbed '' . the most straightforward criterion in this classification was the offset between the peaks of the x - ray emission and the lensing - based mass map , in the sense that a statistically significant offset indicates that the cluster core is disturbed in some way , probably due to a cluster - cluster merger ( see also * ? ? ? * ) . we adopt the classification of @xcite in this letter and apply it to the five additional clusters in our sample ( table [ t : cls ] ) . we first define the aperture within which and are measured . after some experimentation we chose to retain the aperture used by , a clustercentric radius of 250@xmath17 kpc , or 350 kpc in our adopted cosmology . this fixed physical aperture is well - matched to the _ hst_/wfpc2 and advanced camera for surveys ( acs ) fields of view , and is somewhat larger than the typical resolution of our sze data . the shape of our @xmath13-model fit is jointly determined from both sza and high - resolution x - ray data and thus is well resolved . and values measured at this radius are listed in table [ t : cls ] and plotted in figure [ f : m_sz ] . we fit our projected mass ( ) and data points in the log(@xmath18)-log ( ) plane , where a power law scaling relation takes the form @xmath19 . starting with the self - similar scaling @xcite , it can be shown that the relation between mass and @xmath2 within a fixed physical radius has a slope @xmath20 with no dependence on the redshift evolution of the hubble parameter . we seek to measure the slope and normalization of the scaling relation , as well as the intrinsic scatter between these parameters , and therefore follow the prescription of @xcite for linear regression in the presence of intrinsic scatter . this bayesian method includes the intrinsic scatter in the @xmath1-coordinate ( @xmath21 ) as a parameter of the fit and maximizes the probability of the model ( @xmath22 , @xmath13 , @xmath21 ) given the data and errors . the likelihood takes the form ( equation a3 of @xcite ) @xmath23 ^ 2}{\beta^2e_{xi}^2+e_{yi}^2+\sigma_y^2},\ ] ] where the @xmath24 are the @xmath25 or @xmath1 errors in the individual data points . we perform the regression in two different ways : first , with both the normalization ( @xmath22 ) and slope ( @xmath13 ) of the fit free we obtain @xmath26 and @xmath27 , with an intrinsic scatter in mass at fixed @xmath2 of @xmath28% . with the slope fixed to the self - similar value ( @xmath20 ) we obtain @xmath29 , again with @xmath28% . despite large uncertainties , the @xmath3 slope is consistent with self - similarity . for comparison , @xcite and @xcite found that the @xmath3 scaling relation is consistent with self - similarity at ( the radius where the average interior density is @xmath30=2500 times the critical density of the universe , typically 30@xmath3100% larger than the radius used here ) . in contrast , were unable to constrain the slope of the @xmath31 relation , suggesting that the intrinsic scatter in -@xmath2 is smaller than in -@xmath4 . we now compare the intrinsic scatter in the @xmath3 relation with that in @xmath3@xmath4 from . we first use our regression method to re - fit their data , taking the opportunity to adopt the theoretical slope ( @xmath32 , @xmath33 , in their nomenclature ) appropriate for this scaling relation within a fixed aperture . with their 10 clusters we find a scatter in mass of @xmath34% , which is 28% larger ( 1.5@xmath35 significance ) than @xmath36 . indeed , both @xmath36 and @xmath37 are large compared to the scatter predicted from numerical simulations ( e.g. * ? ? ? * ) , likely due to a combination of modeling uncertainties , astrophysical processes in cluster cores , and projection effects . the effects of modeling uncertainties are hard to quantify but should not be ignored , as both and are derived from parameterized models . to the extent that these models inadequately represent the cluster or underestimate the uncertainty in the derived parameters , our inferred scatter will be artificially increased . thus our measured @xmath36 is an upper limit to the true scatter between mass and @xmath2 in cluster cores . in general , cluster mass measurements based on gravitational lensing and sze observations are more susceptible to projection effects than those based on x - ray data because the density - squared dependence of the x - ray emissivity limits contributions from material along the line of sight . nevertheless , simulations @xcite suggest that the contribution to the sze signal from projection is unimportant for clusters with @xmath38 , with the projection effects diminishing further at the small radii used here . gravitational lensing is sensitive to all mass along the line of sight through the cluster , which will likely increase the scatter in our scaling relation , however the contribution of correlated large - scale structure to lensing - based mass measurements should decrease at smaller radii in a manner similar to the impact on the sze signal . we also emphasize that our sample is x - ray selected with no consideration paid to the presence / absence of strong lensing arcs , so it should not suffer the halo orientation bias that boosts the core mass in strong - lensing - selected samples ( e.g. * ? ? ? * ) . indeed , 5/14 clusters contain no obvious strong - lensing signal . projection - induced scatter therefore appears insignificant for and may be somewhat more important for , although this affects both @xmath36 and @xmath37 . given that the mass measurement methods and 9/14 of the clusters are identical between this letter and , we expect the difference between the scatter in these two relations to be dominated by the relative sensitivity of @xmath2 and @xmath4 to the gas physics of cluster cores . the intracluster medium ( icm ) in cluster cores is often disturbed by active galactic nucleus ( agn ) activity ( e.g. * ? ? ? * ) , and cluster - cluster mergers @xcite . for this reason , cluster cores are often excluded when deriving x - ray observables for use as mass proxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? we therefore ascribe the difference between @xmath36 and @xmath37 to the difference in the sze and x - ray emissivity dependencies on icm density and temperature . the sze is unaffected by isobaric disturbances ( excluding relativistic components such as cosmic rays ) , while the energy - integrated x - ray emissivity varies approximately as the square of the density , which may be significantly perturbed in the core . @xcite found potentially important changes in the sze and x - ray signals from cool cores in simulations incorporating cosmic rays , but the effect and therefore the induced scatter was found to be larger for x - ray emission . in larger measurement apertures , such as the frequently used and ( 350 kpc corresponds to @xmath30=4000@xmath38000 for our sample ) , core - removed x - ray mass proxies are found to correlate more tightly with mass ( e.g. * ? ? ? we expect that @xmath36 will also decrease when measured on these scales . found that the normalization of the @xmath31 relation for disturbed clusters was @xmath39 hotter than for undisturbed clusters at @xmath40 ( after excising the central region of the x - ray data ) . we re - fit the - relation to the undisturbed and disturbed sub - samples ( table [ t : cls ] ) finding no significant difference in normalization between undisturbed and disturbed clusters ( @xmath41 , @xmath13 fixed to 0.5 ) . again , this suggests that the integrated sze signal , even in cluster cores , is less sensitive to the icm physics than the x - ray temperature , although the significance of this result is low given the small samples used here and in . as discussed in sections [ s : intro ] and [ s : intsc ] , the icm may be far from hse ; we search for this effect in our data by combining our results with those of @xcite who studied the - relation . our lensing measurements are sensitive to the projected mass ( @xmath42 ) ; we therefore convert them to the spherical masses ( @xmath43 ) required for comparison with the hydrostatic analysis . we do this by assuming that the halo mass distributions are described by the navarro - frenk - white ( nfw ) profile @xcite , which is specified by mass and concentration , @xmath44 and @xmath45 ; given and @xmath45 we can than derive an approximate conversion factor @xmath46 at 350 kpc . this conversion factor depends sensitively on @xmath45 , but there is considerable theoretical uncertainty about the mean value and scatter in concentration at fixed mass , and its variation with redshift and cluster mass . without direct measurements of @xmath45 for most of our clusters we marginalize over the range of values observed by @xcite for 19 clusters from the locuss sample , which have very similar masses and redshifts to our sample . we randomly choose a concentration from the 19 in @xcite for each of our clusters , calculate the corresponding value of = @xmath47 , and fit a scaling relation between the estimated and . we repeat this process thousands of times , and derive a mean normalization of the @xmath3@xmath2 relation of @xmath48 for @xmath20 . this value can be directly compared to the normalization of the @xmath3@xmath2 relation from @xcite : @xmath49 . the statistically insignificant difference between these values implies a mass ratio of /=@xmath5 . using hydrostatic masses from x - ray observations and weak lensing masses from the literature , @xcite and @xcite estimate the weak lensing to hydrostatic mass ratio to be @xmath50 and @xmath51 , respectively , measured at . @xcite , however , demonstrate a radial trend in this ratio between and , suggestive of a 20% deficit in hydrostatic masses at that can not be accounted for by the projection of unrelated structures along the line of sight . our result , interior to the innermost radius examined by @xcite , is consistent with their non - detection of a bias in the hydrostatic mass at small radii . the precision of our measurement is limited by the intrinsic scatter in our relation , with a smaller contribution coming from the unknown halo concentration values required to convert between and . future combinations of weak and strong lensing measurements , specifically joint strong@xmath52weak lens modeling of the clusters combining the _ subaru _ and _ hst _ data ( g. p. smith et al . 2009 , in preparation ) should allow us to evaluate the bias as a function of radius . we have presented the first calibration of the mass-@xmath2 relation based on gravitational - lensing measurements of cluster mass , based largely on previously published sze data and gravitational lens models . we construct the relation at a fixed physical radius of 350kpc to minimize uncertainties arising from extrapolation of cluster models based on both datasets . in contrast to the mass-@xmath4 relation within the same aperture , we succeed in fitting a model with both free slope and normalization . the best - fit slope is consistent with the self - similar prediction , and the intrinsic scatter in mass at fixed @xmath2 is 32% , in contrast to the 41% scatter in mass at fixed @xmath4 . we also fit the relation to sub - samples of disturbed and undisturbed cluster cores but find that the best - fit normalizations for these sub - samples are consistent within the errors . finally , we combine our results with those of @xcite to test whether the cluster cores are in hse . cluster core masses estimated from lensing and sze data ( assuming hse ) are , on average , consistent within the errors , suggesting that departures from equilibrium are modest in cluster cores . we conclude that the difference between the mass-@xmath2 and mass-@xmath4 relations is mainly attributable to the relative insensitivity of the sz effect to the physics of the icm in cluster cores . future articles will explore the mass-@xmath2 relation at larger radii through weak lensing and in larger samples . we thank our colleagues in the locuss collaborations for much support , encouragement and help . gps acknowledges support from the royal society and stfc , and thanks the kavli insitute of cosmological physics at the university of chicago for their hospitality whilst working on this letter . gps thanks alain blanchard for helpful comments . we gratefully acknowledge the james s. mcdonnell foundation , the national science foundation and the university of chicago for funding to construct the sza . the operation of the sza is supported by nsf division of astronomical sciences through grant ast-0604982 . partial support is provided by nsf physics frontier center grant phy-0114422 to the kavli institute of cosmological physics at the university of chicago , and by nsf grants ast-0507545 and ast-05 - 07161 to columbia university .

we present the first measurement of the relationship between the sunyaev - zeldovich effect ( sze ) signal and the mass of galaxy clusters that uses gravitational lensing to measure cluster mass , based on 14 x - ray luminous clusters at @xmath0 from the local cluster substructure survey . we measure the integrated compton @xmath1-parameter , @xmath2 , and total projected mass of the clusters ( ) within a projected clustercentric radius of 350 kpc , corresponding to mean overdensities of 4000@xmath38000 relative to the critical density . we find self - similar scaling between and @xmath2 , with a scatter in mass at fixed @xmath2 of 32% . this scatter exceeds that predicted from numerical cluster simulations , however , it is smaller than comparable measurements of the scatter in mass at fixed @xmath4 . we also find no evidence of segregation in between disturbed and undisturbed clusters , as had been seen with @xmath4 on the same physical scales . we compare our scaling relation to the @xcite relation based on mass measurements that assume hydrostatic equilibrium , finding no evidence for a hydrostatic mass bias in cluster cores ( = @xmath5 ) , consistent with both predictions from numerical simulations and lensing / x - ray - based measurements of mass - observable scaling relations at larger radii . overall our results suggest that the sze may be less sensitive than x - ray observations to the details of cluster physics in cluster cores .

in this paper , we present a simple proof of perelman s collapsing theorem for @xmath0-manifolds ( cf . theorem 7.4 of @xcite ) , which perelman used to verify thurston s geometrization conjecture on the classification of @xmath0-manifolds . [ thm0.1 ] let @xmath9 be a sequence of compact oriented riemannian @xmath0-manifolds , closed or with convex incompressible toral boundary , and let @xmath10 be a sequence of positive numbers with @xmath11 . suppose that \(1 ) for each @xmath12 there exists a radius @xmath13 , @xmath14 , not exceeding the diameter of the manifold , such that the metric ball @xmath15 in the metric @xmath16 has volume at most @xmath17 and sectional curvatures of @xmath18 at least @xmath19 ; \(2 ) each component of toral boundary of @xmath20 has diameter at most @xmath21 , and has a topologically trivial collar of length one and the sectional curvatures of @xmath20 are between @xmath22 and @xmath23 . then , for sufficiently large @xmath24 , @xmath25 is diffeomorphic to a graph - manifold . if a @xmath0-manifold @xmath26 admits an almost free circle action , then we say that @xmath26 admits a _ seifert fibration _ structure or _ seifert fibred_. a _ graph - manifold _ is a compact @xmath0-manifold that is a connected sum of manifolds each of which is either diffeomorphic to the solid torus or can be cut apart along a finite collection of incompressible tori into seifert fibred @xmath0-manifolds . perelman indeed listed an extra assumption ( 3 ) in above . however , perelman ( cf . page 20 of @xcite ) also pointed out that , if the proof of his stability theorem ( cf . @xcite ) is available , then his extra smoothness assumption ( 3 ) is in fact redundant . the conclusion of fails if the assumption of toral boundary is removed . for instance , the product @xmath0-manifold @xmath27 $ ] of a @xmath28-sphere and an interval can be collapsed while keeping curvatures non - negative . however , @xmath29 $ ] is not a graph - manifold . under an assumption on both upper and lower bound of curvatures @xmath30 , the collapsed manifold @xmath20 above admits an @xmath31-structure of positive rank , by the cheeger - gromov collapsing theory ( cf . it is well - known that a @xmath0-manifold @xmath20 admits an @xmath31-structure of positive rank if and only if @xmath20 is a graph - manifold , ( cf . @xcite ) . on page 153 of @xcite , shioya and yamaguchi stated a version of above for the case of closed manifolds , but their proof works for case of manifolds with incompressible convex toral boundary as well . morgan and tian @xcite presented a proof of without assumptions on diameters but with perelman s extra smoothness assumption ( 3 ) which we discuss below . it turned out that the diameter assumption is related to the study of diameter - collapsed @xmath0-manifolds with curvature bounded from below . to see this relation , we state a local re - scaled version of perelman s collapsing . * theorem 0.1. * ( re - scaled version of ) _ let @xmath32 be and let @xmath10 be a sequence of positive numbers as in above , @xmath33 and @xmath34 . suppose that there exists a re - scaled family of pointed riemannian manifolds @xmath35 satisfying the following conditions : _ 1 . the re - scaled riemannian manifold @xmath36 has @xmath37 on the ball @xmath38 ; 2 . the diameters of the re - scaled manifolds @xmath39 are uniformly bounded from below by @xmath40 ; i.e. ; @xmath41 3 . the volumes of unit metric balls collapse to zero , i.e. : @xmath42 \le \omega_\alpha \to 0,\ ] ] as @xmath43 . then , for sufficiently large @xmath24 , the collapsing @xmath0-manifold @xmath20 is a graph - manifold . without inequality above , the volume - collapsing @xmath0-manifolds could be diameter - collapsing . perelman s condition ensures that the normalized family @xmath44 can _ not _ collapse to a point uniformly . by _ collapsing to a point uniformly _ , we mean that there is an additional family of scaling constants @xmath45 such that the sequence @xmath46 is convergent to a @xmath0-dimensional ( possibly singular ) manifold @xmath47 with non - negative curvature . professor karsten grove kindly pointed out that the study of diameter - collapsing theory for @xmath0-manifolds might be related to a weak version of the poincar conjecture . for this reason , shioya and yamaguchi made the following conjecture . [ conj0.2 ] suppose that @xmath48 is a @xmath0-dimensional compact , simply - connected , non - negatively curved alexandrov space without boundary and that @xmath47 is a topological manifold . then @xmath47 is homeomorphic to a sphere . shioya and yamaguchi ( @xcite , page 4 ) commented that _ if is true then the study of collapsed @xmath0-manifolds with curvature bounded from below would be completely understood_. they also observed that is true for a special case when the closed ( possibly singular ) manifold @xmath47 above is a _ smooth _ riemannian manifold with non - negative curvature ; this is due to hamilton s work on @xmath0-manifolds with non - negative ricci curvature ( cf . @xcite ) . coincidentally , perelman added the extra smoothness assumption ( 3 ) in his collapsing theorem . [ asum0.3 ] for every @xmath49 there exist @xmath50 and constants @xmath51 for @xmath52 , such that for all @xmath24 sufficiently large , and any @xmath53 , if the ball @xmath54 has volume at least @xmath55 and sectional curvatures at least @xmath56 , then the curvature and its @xmath57-th order covariant derivatives at @xmath58 are bounded by @xmath59 and @xmath60 for @xmath61 respectively . let us explain how perelman s smoothness is related to the smooth case of and let @xmath62 be as in theorem 0.1. if we choose the new scaling factor @xmath63 such that @xmath64 as @xmath43 , then the newly re - scaled metric @xmath65 will have sectional curvature @xmath66 as @xmath67 . suppose that @xmath68 is a pointed gromov - hausdorff limit of a subsequence of @xmath69 . then the limiting metric space @xmath70 will have non - negative curvature and a possibly singular metric . when @xmath71 = 3 $ ] , by perelman s smoothness , the limiting metric space @xmath72 is indeed a _ smooth _ riemannian manifold of non - negative curvature . in this smooth case , is known to be true , ( see @xcite ) . for simplicity , we let @xmath73 be as in theorem 0.1. by gromov s compactness theorem , there is a subsequence of a pointed riemannian manifolds @xmath74 convergent to a lower dimensional pointed space @xmath75 of dimension either @xmath40 or @xmath28 , i.e. : @xmath76 \le 2\ ] ] using . to establish , it is important to establish that , for sufficiently large @xmath24 , the collapsed manifold @xmath20 has a decomposition @xmath77 such that each @xmath78 admits an almost - free circle action : @xmath79 we also need to show that these almost - free circle actions are compatible ( almost commute ) on possible overlaps . let us first recall how perelman s collapsing theorem for @xmath0-manifolds plays an important role in his solution to thurston s geometrization conjecture on the classification of @xmath0-manifolds . in 2002 - 2003 , perelman posted online three important but densely written preprints on ricci flows with surgery on compact @xmath0-manifolds ( @xcite , @xcite and @xcite ) , in order to solve both the poincar conjecture and thurston s conjecture on geometrization of @xmath0-dimensional manifolds . thurston s geometrization conjecture states that _ for any closed , oriented and connected @xmath0-manifold @xmath26 , there is a decomposition @xmath80 = n^3_1 \cup n^3_2 \cdots \cup n^3_{m}$ ] such that each @xmath81 admits a locally homogeneous metric with possible incompressible boundaries @xmath82 , where @xmath82 is homeomorphic to a quotient of a @xmath28-sphere or a @xmath28-torus " . _ there are exactly 8 homogeneous spaces in dimension 3 . the list of @xmath0-dimensional homogeneous spaces includes 8 geometries : @xmath83 , @xmath84 , @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 and @xmath90 . thurston s geometrization conjecture suggests the existence of especially nice metrics on @xmath0-manifolds and consequently , a more analytic approach to the problem of classifying @xmath0-manifolds . richard hamilton formalized one such approach by introducing the ricci flow equation on the space of riemannian metrics : @xmath91 where @xmath92 is the ricci curvature tensor of the metric @xmath93 . beginning with any riemannian manifold @xmath94 , there is a solution @xmath93 of this ricci flow on @xmath95 for @xmath96 in some interval such that @xmath97 . in dimension @xmath0 , the fixed points ( up to re - scaling ) of this equation include the riemannian metrics of constant ricci curvature . for instance , they are quotients of @xmath83 , @xmath84 and @xmath85 up to scaling factors . it is easy to see that , on compact quotients of @xmath98 or @xmath84 , the solution to ricci flow equation is either stable or expanding . thus , on compact quotients of @xmath98 and @xmath84 , the solution to ricci flow equation exists for all time @xmath99 . however , on quotients of @xmath87 or @xmath85 , the solution @xmath100 to ricci flow equation exists only for finite time @xmath101 . hence , one knows that in general the ricci flow will develop singularities in finite time , and thus a method for analyzing these singularities and continuing the flow past them must be found . these singularities may occur along proper subsets of the manifold , not the entire manifold . thus , perelman introduced a more general evolution process called ricci flow with surgery ( cf . @xcite and @xcite ) . in fact , a similar process was first introduced by hamilton in the context of four - manifolds . this evolution process is still parameterized by an interval in time , so that for each @xmath96 in the interval of definition there is a compact riemannian @xmath0-manifold @xmath102 . however , there is a discrete set of times at which the manifolds and metrics undergo topological and metric discontinuities ( surgeries ) . perelman did surgery along @xmath28-spheres rather than surfaces of higher genus , so that the change in topology for @xmath103 turns out to be completely understood . more precisely , perelman s surgery on @xmath0-manifolds is _ the reverse process of taking connected sums : _ cut a @xmath0-manifold along a @xmath28-sphere and then cap - off by two disjoint @xmath0-balls . perelman s surgery processes produced exactly the topological operations needed to cut the manifold into pieces on which the ricci flow can produce the metrics sufficiently controlled so that the topology can be recognized . it was expected that each connected components of resulting new manifold @xmath104 is either a graph - manifold or a quotient of one of homogeneous spaces listed above . it is well - known that any graph - manifold is a union of quotients of 7 ( out of the 8 possible ) homogeneous spaces described above . more precisely , perelman presented a very densely written proof of the following result . [ thm0.4 ] let @xmath105 be a closed and oriented riemannian @xmath0-manifold . then there is a ricci flow with surgery , say @xmath106 , defined for all @xmath107 with initial metric @xmath94 . the set of discontinuity times for this ricci flow with surgery is a discrete subset of @xmath108 . the topological change in the @xmath0-manifold as one crosses a surgery time is a connected sum decomposition together with removal of connected components , each of which is diffeomorphic to one of @xmath109 or @xmath110 . furthermore , there are two possibilities : \(1 ) either the ricci flow with surgery terminates at a finite time @xmath111 . in this case , @xmath112 is diffeomorphic to the connect sum of @xmath109 and @xmath110 . in particular , if @xmath112 is simply - connected , then @xmath112 is diffeomorphic to @xmath113 . \(2 ) or the ricci flow with surgery exists for all time , i.e. , @xmath114 . the detailed proof of perelman s theorem above can be found in @xcite , @xcite and @xcite . in fact , if @xmath26 is a simply - connected closed manifold , then using a theorem of hurwicz , one can find that @xmath115 . hence , there is a map @xmath116 of degree @xmath40 . one can view @xmath113 as a two - point suspension of @xmath117 , i.e. , @xmath118/\{0 , 1\}$ ] . thus , for such a manifold @xmath26 with a metric @xmath119 , one can define the @xmath28-dimensional width @xmath120 \}$ ] , ( compare with @xcite ) . colding and minicozzi ( cf . @xcite , @xcite , @xcite ) established that @xmath121\ ] ] for perelman s solutions @xmath122 to ricci flow with surgery , where @xmath123 are positive constants independent of @xmath124 . therefore , for a simply - connected closed manifold @xmath112 , it follows from that the ricci flow with surgery must end at a finite time @xmath111 . thus , by , the conclusion of the poincar conjecture holds for such a simply - connected closed @xmath0-manifold @xmath112 . other proofs of perelman s finite time extinction theorem can be found in @xcite and @xcite . it remains to discuss the long time behavior of ricci curvature flow with surgery . we will perform the so - called margulis thick - thin decomposition for a complete riemannian manifold @xmath125 . the thick part is the _ non - collapsing part _ of @xmath125 , while the thin part is the _ collapsing _ portion of @xmath125 . let @xmath126 denote the radius @xmath127 of the metric ball @xmath128 , where we may choose @xmath126 so that @xmath129 and @xmath130 for all @xmath131 . the re - scaled thin part of @xmath102 can be defined as @xmath132 < \omega \rho^3(x , t ) \}\ ] ] and its complement is denoted by @xmath133 , which is called the thick part for a fixed positive number @xmath134 . when there is no surgery occurring for all time @xmath99 , hamilton successfully classified the thick part @xmath133 . [ thm0.5 ] let @xmath135 , @xmath136 and @xmath133 be as above . suppose that @xmath104 is diffeomorphic to @xmath112 for all @xmath99 . then there are only two possibilities : \(1 ) if there is no thin part ( i.e. , @xmath137 ) , then either @xmath138 is convergent to a flat @xmath0-manifold or @xmath139 is convergent to a compact quotient of hyperbolic space @xmath140 ; \(2 ) if both @xmath133 and @xmath136 are non - empty , then the thick part @xmath133 is diffeomorphic to a disjoint union of quotients of real hyperbolic space @xmath141 with finite volume and with cuspidal ends removed . + + perelman ( cf . @xcite ) asserted that the conclusion of holds if we replace the classical ricci flow by _ the ricci flow with surgeries _ , ( cf . detailed proof of this assertion of perelman can be found in @xcite , @xcite and @xcite . suppose that @xmath142 is a complete but non - compact hyperbolic @xmath0-manifold with finite volume . the cuspidal ends of @xmath142 are exactly the thin parts of @xmath143 . each cuspidal end of @xmath142 is diffeomorphic to a product of a torus and half - line ( i.e. , @xmath144 ) . hence , each cusp is a graph - manifold . it should be pointed out that possibly infinitely many surgeries took place _ only _ on thick parts of manifolds @xmath145 after appropriate re - scalings , due to the celebrated perelman s @xmath146-non - collapsing theory , ( see @xcite ) . moreover , perelman ( cf . @xcite ) pointed out that the study of the thin part @xmath136 has nothing to do with ricci flow , but is related to _ his version _ of critical point theory for distance functions . we now outline our simple proof of using perelman s version of critical point theory in next sub - section . in order to illustrate main strategy in the proof of perelman s collapsing theorem for @xmath0-manifolds , we make some general remarks . roughly speaking , perelman s collapsing theorem can be viewed as a generalization of the implicit function theorem . suppose that @xmath147 is a sequence of collapsing @xmath0-manifolds with curvature @xmath148 and that @xmath147 is not a diameter - collapsing sequence . to verify that @xmath20 is a graph - manifold for sufficiently large @xmath24 , it is sufficient to construct a decomposition @xmath149 and a collection of _ regular _ functions ( or maps ) @xmath150 , where @xmath151 or @xmath28 . we require that the collection of locally defined functions ( or maps ) @xmath152 satisfy two conditions : 1 . each function ( or map ) @xmath153 is _ regular _ enough so that perelman s version of implicit function theorem ( cf . below ) is applicable ; 2 . the collection of locally defined _ regular _ functions ( or maps ) are compatible on any possible overlaps in the sense of cheeger - gromov ( cf . @xcite @xcite ) . more precisely , if @xmath154 and if @xmath155 \neq \varnothing $ ] with @xmath156 \le \dim [ f_{\alpha , j}^{-1}(z ) ] $ ] , then we require that either @xmath157 or the union @xmath158 $ ] is contained in a @xmath28-dimensional orbit of an almost - free torus action . if the above two conditions are met , with additional efforts , we can construct a _ compatible _ family of the locally defined seifert fibration structures ( which is equivalent to an @xmath31-structure @xmath159 of positive rank in the sense of cheeger - gromov ) on a sufficiently collapsed @xmath0-manifold @xmath20 . it follows that @xmath20 is a graph - manifold for sufficiently large @xmath24 , ( cf . @xcite ) . perelman s choices of locally defined _ regular _ functions ( or maps ) are related to distance functions @xmath160 from appropriate subsets @xmath161 . we briefly illustrate the main strategy of our proof for the following two cases . * * case 1.**_the metric balls @xmath162 collapse to an open interval_. we will show that @xmath163 is homeomorphic to a slim cylinder @xmath164 with shrinking spherical or toral factor @xmath165 for sufficiently large @xmath24 . when a sequence of the pointed @xmath0-manifolds @xmath166 with curvature @xmath148 are convergent to an @xmath40-dimensional space @xmath167 and @xmath168 is an interior point , perelman - yamaguchi fibration theory is applicable . thus , we are led to consider the fibration @xmath169 + we now discuss the topological type of the fiber @xmath170 . let @xmath171 and @xmath172 be the diameter of @xmath173 in @xmath20 . we further consider the limiting space @xmath174 of re - scaled spaces @xmath175 as @xmath176 . there are two sub - cases : @xmath177 or @xmath178 . let us consider the subcase of @xmath179 : @xmath180 where both @xmath181 and @xmath47 are manifolds with possibly singular metrics of non - negative curvature . to classify singular surfaces @xmath181 with non - negative curvature , we use a splitting theorem and the distance non - increasing property of perelman - sharafutdinov retraction on the universal cover @xmath182 , when @xmath181 has non - zero genus . with some extra efforts , we will conclude that @xmath181 must be homeomorphic to a quotient of 2-sphere or 2-torus , ( see below ) . it now follows from a version of perelman s stability theorem that the fiber @xmath170 is homeomorphic to @xmath181 , for sufficiently large @xmath24 . hence , @xmath170 is a quotient of 2-sphere or 2-torus as well , for sufficiently large @xmath24 . our new proof of perelman s collapsing theorem for this subcase is much simpler than the approach of shioya - yamaguchi presented in @xcite . the sub - case of @xmath183 is related to the following case : * case 2*. _ the metric balls @xmath184 collapse to an open disk_. we will show that @xmath163 is homeomorphic to a fat solid torus @xmath185 with shrinking core @xmath186 for sufficiently large @xmath24 . + in this case , our strategy can be illustrated in the following diagram b_m^3_(x_,)&^f_&^2 + _ & & + b_x^2(x_,)&^f_&^2 where the sequence of metric balls @xmath187 are convergent to the metric disk @xmath188 for @xmath189 . we will construct an admissible map @xmath190 which is regular at the punctured disk , using perelman s multiple conic singularity theory , ( see below ) . among other things , we will use the following result of perelman to construct the desired map @xmath191 . [ thm0.6 ] let @xmath192 be an alexandrov space of dimension @xmath193 , @xmath37 and @xmath194 be an interior point of @xmath192 . then the distance function @xmath195 has _ no _ critical points on @xmath196 $ ] for sufficiently small @xmath197 depending on @xmath58 . we will use and perelman s semi - flow orbit stability theorem ( cf . below ) to conclude that the lifting maps @xmath198 is regular on the annular region @xmath199 $ ] . with extra efforts , one can construct a local seifert fibration structure : @xmath200 in summary , perelman s collapsing theorem for 3-manifolds can be viewed an extension of the implicit function theorem . our proof of perelman s collapsing theorem benefited from _ his version _ of critical point theory for distance functions , including his conic singularity theory and fibration theory . perelman s multiple conic singularity theory and his fibration theory are designed for possibly singular alexandrov spaces @xmath201 . therefore , the smoothness of metrics on @xmath201 does _ not _ play a big role in the applications of perelman s critical point theory , unless we run into the so - called essential singularities ( or extremal subsets ) . when essential singularities do occur on surfaces , we use the mcs theory ( e.g. ) and the multiple - step perelman - sharafutdinov flows to handle them , ( see below ) . without using perelman s version of critical point theory , shioya - yamaguchi s proof of the collapsing theorem for @xmath0-manifolds was lengthy and involved . for instance , they use their singular version of gauss - bonnet theorem to classify surfaces of non - negative curvature , ( see chapter 14 of @xcite ) . the proof of the singular version of the gauss - bonnet theorem was non - trivial . in addition , shioya - yamaguchi extended the cheeger - gromoll soul theory to 3-dimensional singular spaces with non - negative curvature , which was rather technical and occupied the half of their first paper @xcite . using perelman s version of critical point theory , we will provide alternative approaches to classify non - negatively curved surfaces and open 3-manifolds with possibly singular metrics , ( e.g. , the 3-dimensional soul theory ) . our arguments inspired by perelman are considerably shorter than shioya - yamaguchi s proof for the @xmath0-dimensional soul theory , ( see below ) . for the readers who prefer a traditional proof of the collapsing theorem without using perelman s version of critical point theory , we recommend the important papers of morgan - tian @xcite and shioya - yamaguchi @xcite , @xcite . finally , we should also mention the recent related work of gerard besson et al , ( cf . another proof of perelman s collapsing theorem for 3-manifolds has been announced by kleiner and lott ( cf . @xcite ) . we refer the organization of this paper to the table of contents at the beginning . in -[section4 ] below , we mostly discuss interior points of alexandrov spaces , unless otherwise specified . in 1 - 2 , we will discuss our proof of theorem 0.1 for a special case . in this special case , we assume that the sequence of metric balls @xmath202 is convergent to a metric ball @xmath203 , where @xmath168 is an interior point of @xmath204 . using several known results of perelman , we will show that there is a ( possibly singular ) circle fibration : @xmath205 for some @xmath206 . in other words , we shall show that @xmath207 looks like a _ fat _ solid tori with a shrinking core , i.e. , @xmath208 \sim [ ( d^2 \times ( \mathbb r/ \varepsilon \mathbb z)]$ ] , ( see above and below ) . in fact , using the conic lemma ( above ) , kapovitch @xcite already established a circle - fibration structure over the annular region @xmath209 . let @xmath210 denote the space of unit directions of an alexandrov space @xmath192 of curvature @xmath148 at point @xmath58 . when @xmath211 , it is known ( cf . @xcite ) that @xmath204 must be a @xmath28-dimensional topological manifold . thus , @xmath212 is a circle , and hence an @xmath40-dimensional manifold . [ prop1.1 ] suppose that @xmath213 , where @xmath214 is a sequence of @xmath215-dimensional riemannian manifolds with sectional curvature @xmath148 . suppose that there exists @xmath216 such that @xmath217 is a closed riemannian manifold . then there exists @xmath218 such that for any @xmath219 we have : for any sufficiently large @xmath24 , and @xmath220 , there exists a topological fiber bundle @xmath221 such that 1 . @xmath222 and @xmath223 are topological manifolds ; 2 . both @xmath222 and @xmath223 are connected ; 3 . the fundamental group @xmath224 of the fiber is almost nilpotent . we will use perelman s fibration theorem and an multiple conic singularity theory to establish the desired circle fibration over the annular region @xmath225 for @xmath226 . our strategy can be illustrated in the following diagram b_m^3_(x_,r)&^f_&^2 + _ & & + b_x^2(x_,r)&^f_&^2 where the sequence of metric balls @xmath227 are convergent to the metric disk @xmath203 . if @xmath228 were a _ topological submersion _ " to its image , then we would be able to obtain the desired topological fibration . for this purpose , we will recall perelman s fibration theorem for non - smooth maps . we postpone the definition of admissible maps to . in , we will also recall the notion of _ regular points _ for a sufficiently wide class of _ admissible mappings _ " from an alexandrov space @xmath229 to euclidean space @xmath230 . let @xmath231 be an admissible map . the points of an alexandrov space @xmath192 that are not regular are said to be critical points , and their images in @xmath230 are said to be critical values of @xmath232 . all other points of @xmath230 are called regular values . [ thm1.2 ] 1 . an admissible mapping is open and admits a trivialization in a neighborhood of each of its regular points . 2 . if an admissible mapping has no critical points and is proper in some domain , then its restriction to this domain is the projection of a locally trivial fiber bundle . there are several equivalent definitions of alexandrov spaces of @xmath233 . roughly speaking , a length space @xmath192 is said to have curvature @xmath234 if and only if , for any geodesic triangle @xmath235 in @xmath192 , the corresponding triangle @xmath236 of the same side - lengths in @xmath237 is thinner than @xmath235 . more precisely , let @xmath238 be a simply connected complete surface of constant sectional curvature @xmath193 . a triangle in a length space @xmath192 consists of three vertices , say @xmath239 and three length - minimizing geodesic segments @xmath240 . let @xmath241 be the length of @xmath242 . given a real number @xmath193 , a comparison triangle @xmath243 is a triangle in @xmath238 with the same side lengths . its angles are called the comparison angles and denoted by @xmath244 , etc . a comparison triangle exists and is unique whenever @xmath245 or @xmath246 and @xmath247 . [ def1.3 ] a length space @xmath192 is called an alexandrov space of curvature @xmath248 if any @xmath194 has a neighborhood @xmath249 for any @xmath250 , the following inequality @xmath251 alexandrov spaces with @xmath252 have several nice properties , ( cf . @xcite ) . for instance , the dimension of an alexandrov space @xmath192 is either an integer or infinite . moreover , for any @xmath194 , there is a well defined tangent cone @xmath253 along with an inner product " on @xmath253 . in fact , if @xmath192 is an alexandrov space with the metric @xmath254 , then we denote by @xmath255 the space @xmath256 . let @xmath257 be the canonical map . the gromov - hausdorff limit of pointed spaces @xmath258 for @xmath259 is the tangent cone @xmath253 at @xmath58 , ( see @xmath2607.8.1 of @xcite ) . for any function @xmath261 , the function @xmath262 such that @xmath263 is called the differential of @xmath264 at @xmath58 . let us now recall the notion of regular points for distance functions . [ def1.4 ] let @xmath265 be a closed subset of an alexandrov space @xmath192 and @xmath266 be the corresponding distance function from @xmath267 . a point @xmath268 is said to be a regular point of @xmath269 if there exists a non - zero direction @xmath270 such that @xmath271 it is well - known that if @xmath192 has @xmath272 then @xmath273 ^ 2 $ ] has the property that @xmath274 , ( see @xcite ) . to explain such an inequality , we recall the notion of semi - concave functions . [ def1.5 ] a function @xmath261 is said to be @xmath275-concave in an open domain @xmath276 if for any length - minimizing geodesic segment @xmath277\to u$ ] of unit speed , the function @xmath278 is concave . when @xmath264 is 1-concave , we say that @xmath279 . it is clear that if @xmath280 is a semi - concave function , then @xmath281 is a concave function . in order to introduce the notion of semi - gradient vector for a semi - concave function @xmath264 , we need to recall the notion of _ inner product _ " on @xmath253 . for any pair of vectors @xmath282 and @xmath283 in @xmath253 , we define @xmath284 is the angle between @xmath285 and @xmath283 , @xmath286 , @xmath287 and @xmath288 denotes the origin of the tangent cone . [ def1.6 ] for any given semi - concave function @xmath264 on @xmath192 , a vector @xmath289 is called a gradient of @xmath264 at @xmath58 ( in short @xmath290 ) if 1 . @xmath291 for any @xmath292 ; 2 . @xmath293 . it is easy to see that any semi - concave function has a uniquely defined gradient vector field . moreover , if @xmath294 for all @xmath295 , then @xmath296 . in this case , @xmath58 is called a critical point of @xmath264 . otherwise , we set @xmath297 where @xmath298 is the ( necessarily unique ) unit vector for which @xmath299 attains its positive maximum on @xmath210 , where @xmath210 is the space of direction of @xmath192 at @xmath58 . [ prop1.7 ] let @xmath229 be a metric space with curvature @xmath148 and @xmath300 be an interior point of @xmath229 . then there exists a strictly concave function @xmath301 $ ] such that ( 1 ) @xmath302 and @xmath303 ) \subset b(\hat x , \lambda s ) $ ] for @xmath304 ; ( 2 ) the distance function @xmath305 has no critical point in punctured ball @xmath306 $ ] , for some @xmath307 depending on @xmath308 . \(1 ) the construction of the strictly concave function @xmath309 described above is available in literature ( see @xcite , @xcite ) . in fact , let @xmath310 be defined as on page 129 of @xcite for @xmath311 . we choose @xmath312 . kapovitch showed that the inequality @xmath313 holds for @xmath314 and @xmath315 , ( see page 132 of @xcite ) . thus , there exists a @xmath275 such that @xmath303 ) \subset b(\hat x , \lambda s ) $ ] for @xmath304 . \(2 ) for the convenience of readers , we add the following alternative proof of the second assertion . let us recall that the tangent cone @xmath316 is the gromov - hausdorff limit of the pointed re - scaled spaces @xmath317 as @xmath318 , i.e. @xmath319 as @xmath318 , where @xmath320 is the apex of the tangent cone @xmath321 . let @xmath322 and @xmath323 . we consider @xmath324 . by an equivalent definition of curvature @xmath148 , @xmath325 and @xmath326 are semi - concave functions . lemma 1.3.4 of @xcite implies that if @xmath327 as @xmath328 then @xmath329 . let @xmath330 $ ] be an annular region . our energy function @xmath331 has property @xmath332 on @xmath333 . it follows from lemma 1.3.4 of @xcite that , for sufficiently large @xmath334 , the function @xmath335 has no critical point on the annual region @xmath336 since we have @xmath337 = \cup_{\lambda\ge \lambda_0}a_{x } ( \hat x , \frac{1}{2\lambda},\frac { 1}{\lambda}),\ ] ] we conclude that the radial distance function has no critical point on the punctured ball @xmath306 $ ] . in this subsection , we recall explicit definitions of admissible mappings and their regular points introduced by perelman . [ def1.9 ] ( 1 ) let @xmath229 be a complete alexandrov space of dimension @xmath215 and @xmath338 and @xmath339 . a function @xmath340 is called admissible if @xmath341 where @xmath342 is a closed subset and @xmath343 is continuous . \(2 ) a map @xmath344 is said to be admissible in a domain @xmath345 if it can be represented as @xmath346 , where @xmath347 is bi - lipschitz homeomorphism and each component @xmath348 of @xmath349 is admissible . the definition of regular points for admissible maps @xmath350 on general alexandrov spaces is rather technical . for the purpose of this paper , we only need to consider two lower dimensional cases of @xmath229 : either @xmath351 is a smooth riemannian @xmath0-manifold or @xmath204 is a surface with curvature @xmath352 . [ def1.10 ] suppose that @xmath353 is an admissible map from a smooth riemannian @xmath0-manifold @xmath26 to @xmath237 on a domain @xmath345 and @xmath346 , where @xmath354 is bi - lipschitz homeomorphism and each component @xmath348 of @xmath355 is admissible . if @xmath356 are linearly independent at @xmath357 , then @xmath58 is said to be a regular point of @xmath358 . \(2 ) ( @xcite page 210 ) . suppose that @xmath359 is an admissible map from an alexandrov surface @xmath204 of curvature @xmath360 to @xmath237 on a domain @xmath361 and @xmath346 , where @xmath354 is bi - lipschitz homeomorphism and each component @xmath348 of @xmath355 is admissible . suppose that @xmath362 and @xmath363 satisfy the following conditions : 1 . @xmath364 ; 2 . there exits @xmath365 such that @xmath366 . then @xmath367 is called a regular point of @xmath368 . it is clear that perelman s condition ( 2.a ) implies that @xmath369 . this together with ( 2.b ) implies that @xmath370 conversely , we would like to point out that if @xmath369 , then there exists an admissible map @xmath371 satisfying perelman s condition ( 2.a ) and ( 2.b ) mentioned above , where @xmath372 is a small neighborhood of @xmath373 in @xmath204 . we need to single out _ bad points _ " ( i.e. , essential singularities ) for which the condition @xmath374 fails . these bad points are related to the so - called extremal subsets ( or essential singularities ) of and alexandrov space with curvature @xmath352 . [ def1.12 ] let @xmath204 be an alexandrov surface and @xmath375 be an interior point of @xmath204 . if the diameter of space of unit tangent directions @xmath376 has diameter less than or equal to @xmath377 , i.e. @xmath378 then @xmath379 is called an extremal point of the alexandrov surface @xmath204 . if @xmath380 , then we say that @xmath379 is a regular point of @xmath204 . a direct consequence of ( i.e. , ) is the regularity of sufficiently small punctured disk in an alexandrov surface . [ cor1.12 ] let @xmath204 be an alexandrov space of curvature @xmath148 and @xmath197 be as in . then each point @xmath381 $ ] in punctured disk is regular . we recall the perelman - sharafutdinov gradient semi - flows for semi - concave functions . [ def1.13 ] a curve @xmath382\to x$ ] is called an @xmath264-gradient curve if for any @xmath383 $ ] @xmath384 it is known that if @xmath385 is a semi - concave function then there exists a unique @xmath264-gradient curve @xmath386 with a given initial point @xmath387 , ( cf . prop 2.3.3 of @xcite ) . we will frequently use the following result of perelman ( cf . @xcite ) and perelman - petrunin ( cf . @xcite ) . [ prop1.14 ] let @xmath388 be a sequence of alexandrov space of curvature @xmath148 which converges to an alexandrov space @xmath389 . suppose that @xmath390 where @xmath391 is a sequence of @xmath275-concave functions and @xmath392 . assume that @xmath393 is a sequence of @xmath394-gradient curves with @xmath395 and @xmath396 be the @xmath264-gradient curve with @xmath397 . then the following is true \(1 ) for each @xmath99 , we have @xmath398 as @xmath43 ; \(2 ) @xmath399 . consequently , if @xmath400 is a bounded sequence of critical points of @xmath401 , then @xmath402 has a subsequence converging to a critical point @xmath403 of @xmath404 . as we pointed out earlier , the pointed spaces @xmath405 converge to the tangent cone of @xmath192 at @xmath58 , i.e. , @xmath406 as @xmath407 , where @xmath408 is the origin of tangent cone . when @xmath204 is an alexandrov surface of curvature @xmath148 , it is known that @xmath204 is a @xmath28-dimensional manifold . moreover we have the following observation . [ prop1.15 ] let @xmath192 be an alexandrov space of curvature @xmath148 . suppose that @xmath308 is an interior point of @xmath192 . then @xmath409 is homeomorphic to @xmath410 , where @xmath411 is given by . furthermore , there exits an admissible map @xmath412 such that @xmath413 is bi - lipschitz homeomorphism and @xmath413 is regular at @xmath414 . this is an established result of perelman , ( cf . @xcite , @xcite ) . we provide a short proof here only for the convenience of readers . let us first prove that @xmath409 is homeomorphic to @xmath410 , where @xmath411 is given by . recall that @xmath415 is convergent to @xmath416 , as @xmath417 . by perelman s stability theorem ( cf . theorem 7.11 of @xcite ) , @xmath418 is homeomorphic to @xmath419 for sufficiently small @xmath197 . thus , @xmath420 is homeomorphic to @xmath421 . by , the function @xmath422 has no critical point in punctured ball @xmath423 $ ] . thus , we can apply perelman s fibration theorem to the following diagram : @xmath424 consequently , we see that @xmath425 is homeomorphic to a cylinder @xmath426 . furthermore , the metric sphere @xmath427 is homeomorphic to @xmath428 . it follows that the metric ball @xmath429 $ ] is homeomorphic to @xmath410 , where @xmath411 is given by . it remains to construct the desired map @xmath413 . if @xmath430 , then @xmath431 . let us choose six vectors @xmath432 such that @xmath433 for @xmath434 or @xmath435 . it is easy to construct affine map @xmath436 from an euclidean sector of angle @xmath437 to an euclidean sector of angle @xmath438 . in fact we can first isometrically embed an euclidean sector as @xmath439 . we could choose @xmath440 , ( see ) . each affine map @xmath441 has height functions ( distance functions from axes ) up to scaling factors as its components . we can arrange the euclidean sectors in an appropriate order so that @xmath436 is admissible . for instance , we could change the role of @xmath442 and @xmath443 for adjunct sector such that , by gluing six euclidean sectors together , we can recover @xmath444 and construct an admissible map @xmath445 . + recall that @xmath446 . by lifting the admissible map @xmath447 to @xmath448 , we have the following result . [ cor1.16 ] let @xmath204 be an alexandrov surface of curvature @xmath148 and @xmath449 be an interior point . then there exist sufficiently small @xmath450 and admissible maps @xmath451 such that @xmath452 is regular on @xmath453 for @xmath454 , where @xmath455 is the lift of the euclidean sector bounded by @xmath456 from @xmath457 to @xmath204 . let @xmath458 and @xmath459 be as in the proof of . we choose an 1-parameter family of length - minimizing geodesic segments @xmath460 \to \lambda x^2 $ ] such that @xmath461 are converging to @xmath462 in @xmath463 , as @xmath259 . we also choose another length - minimizing geodesic segment @xmath464 \to \lambda x^2 $ ] outside the geodesic hinge @xmath465 such that @xmath466 . ( see above ) . we consider lifting distance functions as follows : @xmath467 and @xmath468 it follows from that two functions @xmath469 are regular on @xmath470 for sufficiently large @xmath275 . choosing a sufficiently large @xmath471 , we consider the map @xmath472 it follows from that @xmath452 is regular on the curved trapezoid - like region @xmath473 , ( see ) . recall that there is a sequence @xmath474 convergent to @xmath475 . we conclude by the following circle - fibration theorem . [ thm1.17 ] suppose that a sequence of pointed @xmath0-manifolds @xmath476 is convergent to a @xmath28-dimensional alexandrov surface @xmath477 such that @xmath478 is an interior point of @xmath479 . suppose that @xmath480 is a regular map as above . then there exist maps @xmath481 such that @xmath228 is regular for sufficiently large @xmath24 . moreover , the map @xmath198 gives rise to a circle fibration : @xmath482 for sufficiently large @xmath24 . we will use the same notations as in proofs of and . for each @xmath452 constructed in , we consider the map @xmath483 up to re - scaling factors , which is regular on @xmath484 . since @xmath485 as @xmath43 , we can choose geodesic segment @xmath486 in @xmath25 with @xmath487 . then we can define a map of @xmath488 by @xmath489 we further choose @xmath490 such that @xmath491 as @xmath492 , where @xmath493 is an euclidean sector of @xmath237 described in the proof of and , ( see and ) . + it follows from that @xmath494 is also regular in @xmath495 , for sufficiently large @xmath24 . using perelman s fibration theorem ( above ) , we obtain that @xmath153 defines a @xmath496 fibration on @xmath497 . to get a global @xmath496-fibration on @xmath498 , we need to glue these local fibration structures together . we will discuss the detail of the gluing procedure in below . we begin with an example of collapsing manifolds with exceptional orbits of a circle action on a solid tori . [ ex2.0 ] let @xmath499 be an infinite long @xmath0-dimensional cylinder . we consider an isometry @xmath500 given by @xmath501 where @xmath502 is a fixed integer @xmath503 . let @xmath504 be an sub - group generated by @xmath505 . it is clear that the following equation @xmath506 holds . the quotient space @xmath507 is a solid torus . let @xmath508 be the corresponding quotient map . the orbit @xmath509 is an exceptional orbit in @xmath510 . it is clear that such an exceptional orbit @xmath511 has non - zero euler number . let @xmath407 , the solid tori @xmath510 is convergent to @xmath512 , where @xmath513 is a subgroup generated by @xmath514 let us now return to the diagram constructed in the previous section : a^3_(m^3_,g^)(x _ , , ) & ^f_&^2 + _ & & + a_x^2(x _ , , ) & ^f_&^2 among other things , we shall derive the following theorem . [ thm2.1 ] let @xmath198 , @xmath191 , @xmath515 and @xmath516 as in and the diagram above . then there is a @xmath517 such that @xmath518 is homeomorphic to a solid torus for sufficiently large @xmath519 . moreover , a finite normal cover of @xmath520 admits a free circle action . we will establish by using the cheeger - gromoll - perelman s soul theory for _ singular metrics _ on open @xmath0-dimensional manifolds with non - negative curvature , ( comparing with @xcite ) . it will take several steps . we start with the following observation . [ prop2.2 ] let @xmath521 be a sequence of metric balls convergent to @xmath522 as above . then , there is another sequence of points @xmath523 such that @xmath524 as @xmath525 and for sufficiently large @xmath24 , the following is true : 1 . @xmath526 is homeomorphic to a quotient of torus @xmath527 ; 2 . there exists @xmath528 such that there is no critical point of @xmath529 for @xmath530 ; 3 . there is the furthest critical point @xmath531 of the distance function @xmath532 in @xmath533 with @xmath534 as @xmath67 . this result can be found in shioya - yamaguch s paper @xcite . for convenience of readers , we reproduce a proof inspired by perelman and yamaguchi ( cf . @xcite , @xcite ) with appropriate modifications . our choices of the desired points @xmath535 are related to certain averaging distance functions @xmath536 described below . let us first construct the limit function @xmath537 of the sequence @xmath536 . similar constructions related to @xmath537 can be found in the work of perelman , grove and others ( see @xcite page 211 , @xcite page 223 , @xcite page 210 , @xcite , @xcite ) . let @xmath538 denote the set of directions of geodesics from @xmath373 to @xmath267 in @xmath539 . it is clear that if we choose @xmath540 and if @xmath541 is the apex of @xmath542 , then @xmath543 for any @xmath544 . recall that @xmath545 as @xmath259 . suppose that @xmath204 has curvature @xmath148 . applying , we see that , for any @xmath546 , there is a sufficiently small @xmath547 such that ( 2.2.4 ) _ the minimal set @xmath548 is @xmath197-dense in @xmath549 . _ we now choose @xmath550 and @xmath551 . let @xmath552 be a maximal @xmath553-separated subset in @xmath554 and @xmath555 be a @xmath556-net in @xmath557 . yamaguchi ( @xcite ) considered @xmath558 for @xmath559 . our choice of @xmath537 is given by @xmath560 we now verify that @xmath537 has a unique maximal point @xmath561 . it is sufficient to establish @xmath562 $ ] for all @xmath563 . this can be done as follows . for each @xmath564 , we choose @xmath565 such that @xmath566 for @xmath567 . suppose that @xmath568 \to x^2 $ ] is a length - minimizing geodesic segment of unit speed from @xmath168 to @xmath58 . by comparison triangle comparison theorems , one can show that if @xmath569 then @xmath570 for @xmath571 . it follows from the first variational formula that @xmath572 for @xmath567 . in fact , kapovitch @xcite observed that @xmath573 $ ] , ( see @xcite page 129 or @xcite ) . it follows that @xmath574 has the unique maximum point @xmath168 with @xmath575 , because @xmath576 $ ] for @xmath577 . since @xmath578 as @xmath43 , we can construct a @xmath579-approximation of @xmath580 of @xmath581 with @xmath582 . let @xmath583 let @xmath584 be a local maximum set of @xmath585 and @xmath586 . applying to the sequence @xmath587 , we see that @xmath588 and @xmath589 , as @xmath590 and @xmath43 . let @xmath591 . using again , we can show that there exists a sequence @xmath592 such that neither the function @xmath593 nor @xmath594 has any critical points in the annual region @xmath595 , as @xmath596 . it follows from that the boundary @xmath597 is homeomorphic to @xmath527 or klein bottle @xmath598 for @xmath599 . however , @xmath600 is a riemannian @xmath0-manifold @xmath601 is homeomorphic to a @xmath28-sphere for @xmath602 less than the injectivity radius @xmath603 at @xmath604 . thus , we let @xmath605 and @xmath606 clearly , we have @xmath607 $ ] . as @xmath608 , we have @xmath609 . this completes the proof of . in what follows , we re - choose @xmath610 as in the proof of for each @xmath24 . we now would like to study the sequence of re - scaled metrics @xmath611 clearly , the curvature of @xmath612 satisfies @xmath613 , as @xmath43 . by passing to a subsequence , we may assume that the pointed riemannian @xmath0-manifolds @xmath614 converge to a pointed alexandrov space @xmath615 with non - negative curvature . [ prop2.3 ] let @xmath616 , @xmath204 , @xmath617 and @xmath618 be as above . then @xmath70 is a complete , non - compact alexandrov space of non - negative curvature . furthermore , we have 1 . @xmath619 ; 2 . @xmath70 has no boundary . this is an established result of @xcite and @xcite . we outline a proof here only for convenience of readers , using our proofs of above and and below . the metric @xmath620 defined above has curvature @xmath621 , as @xmath622 . by our construction , the diameter of @xmath70 is infinite . moreover , our alexandrov space @xmath70 has no finite boundary . it remains to show that @xmath623 suppose contrary , @xmath178 . then , for each @xmath624 , the subset @xmath625 has at most two elements . we will find a vector @xmath626 such that @xmath627 has at least @xmath628 elements for some @xmath565 , a contradiction to @xmath629 . our choice of @xmath630 will be related to a tangent vector to the minimum set @xmath267 of a convex function @xmath631 , which we now describe . we will retain the same notations as in the proof of above . let @xmath632 , @xmath633 be a critical point of @xmath634 be as above and @xmath635 be its image in the scaled manifold @xmath636 . suppose that @xmath403 is the limit point of a subsequence of @xmath637 . it follows from that the limiting point @xmath403 must be a critical point of the distance function @xmath638 with @xmath639 , where @xmath640 . in addition , there exist @xmath641-many geodesic segments @xmath642 in @xmath25 from @xmath643 to @xmath644 , where @xmath645 as @xmath646 with @xmath43 . let @xmath647 \to \frac{1}{\lambda_\alpha } m^3_\alpha$ ] be the re - scaled geodesic with starting point @xmath648 in the re - scaled manifold , for @xmath649 it can be shown that @xmath650 as @xmath651 , after passing to appropriate subsequences of @xmath652 . therefore , we have @xmath653-many distinct geodesic rays starting from @xmath654 in @xmath72 . let us now consider limiting busemann functions : @xmath655 \quad \text{and } \quad \hat { h}_i ( y ) = \frac{1}{n_i}\sum_{j= 1}^{n_i}\tilde{h}_{i , j}(y).\ ] ] since @xmath72 has non - negative curvature , each busemann function @xmath656 is a convex function , ( see @xcite , @xcite or below ) . if @xmath657 then @xmath264 is convex . choose @xmath658 $ ] , @xmath659 and @xmath660 defined on @xmath661 . it follows that @xmath662 . because @xmath648 is a maximum point of @xmath663 with @xmath664 and @xmath665 as @xmath43 , the point @xmath654 is a critical point of the limiting function @xmath666 with @xmath667 . thus , @xmath668 is a critical value of the _ convex _ function @xmath669 with @xmath670 . there are two cases for @xmath671 . if @xmath672 , then it is known ( cf . below ) that the distance function @xmath638 does not have any critical point in @xmath673 $ ] , which contracts to the existence of critical points @xmath403 of @xmath638 mentioned above . thus , this case can not happen . @xmath674 . in this case , our proof becomes more involved . if @xmath675 , then @xmath676 . using the proof of below , we see that @xmath403 can _ not _ be a critical point of @xmath677 either , a contradiction . thus , @xmath678 holds . if , for any quasi - geodesic segment @xmath679 \to y_\infty$ ] with ending points @xmath680 , the inclusion relation @xmath681 ) \subset \omega$ ] holds , then @xmath682 is called a totally convex subset of @xmath72 . it follows from the proof of below that the sub - level set @xmath683 ) $ ] is totally convex . let us choose @xmath684 , where @xmath685 denotes the set of directions of geodesics from @xmath373 to @xmath58 in @xmath539 . because @xmath686 are contained in the totally convex minimal set @xmath687 of a convex function @xmath264 , one has @xmath688 holds for all @xmath689 by our construction of @xmath690 , because @xmath691 is a support vector of @xmath692 , ( see the proof of below ) . let @xmath693 \to y$ ] be a geodesic segment from @xmath654 to @xmath403 . since @xmath267 is totally convex and @xmath264 is convex , we have @xmath694 ) \subset a$ ] and @xmath695 for all @xmath696 $ ] . hence , @xmath697 for all @xmath698 $ ] . recall that @xmath699 . we choose @xmath700 such that @xmath701 . because @xmath702 is a concave function of @xmath96 with @xmath703 and @xmath704 , one concludes that @xmath705 for all @xmath696 $ ] . since @xmath705 for all @xmath696 $ ] , choosing @xmath706 one has @xmath707,\ ] ] this together with inequalities @xmath708 implies that @xmath709 holds for @xmath710 hence , we conclude that @xmath711 , for @xmath712 , where @xmath713 . therefore , we demonstrated that @xmath714 . this contradicts to @xmath715 when @xmath716 . this completes the proof of the assertion @xmath717 . in what follows , if @xmath192 is an open alexandrov space of non - negative curvature , then we let @xmath718 be the boundary ( or called the ideal boundary ) of @xmath192 at infinity . for more information about the ideal boundary @xmath718 , one can consult with work of shioya , ( cf . @xcite ) . in this sub - section , we briefly review the soul theory for non - negatively curved space @xmath719 of dimension @xmath720 . the soul theory and the splitting theorem are two important tools in the study of low dimensional collapsing manifolds . let @xmath192 be an @xmath215-dimensional non - negatively curved alexandrov space . suppose that @xmath192 is a non - compact complete space and that @xmath192 has no boundary . fix a point @xmath721 , we consider the cheeger - gromoll type function @xmath722.\ ] ] let us consider the sub - level sets @xmath723)$ ] . we will show that @xmath724 is a totally convex subset for any @xmath725 in below . h.wu @xcite and z. shen @xcite further observed that @xmath726 where @xmath727 and @xmath728 is a busemann function associated with a ray @xmath729 by @xmath730.\ ] ] since @xmath723)$ ] is convex , by we see that @xmath724 contains no geodesic ray starting from @xmath731 . choose @xmath732 . since @xmath733 is totally convex and contains no geodesic rays , @xmath733 must be compact . it follows that @xmath734 is compact as well . thus the cheeger - gromoll function @xmath735 has a lower bounded @xmath736 if @xmath737 is a space without boundary , @xmath738 is called a soul of @xmath192 . otherwise , @xmath739 we further consider @xmath740 when @xmath192 is a smooth riemannian manifold of non - negative curvature , cheeger - gromoll @xcite showed that @xmath741 remains to be convex . for more general case when @xmath192 is an alexandrov space of non - negative curvature , perelman @xcite also showed that @xmath741 remains to be convex , ( see @xcite and @xcite as well ) . let @xmath742 and @xmath743 . if @xmath744 has no boundary , then we call @xmath745 a soul of @xmath192 . otherwise , we repeat above procedure by setting @xmath746 for @xmath747 . observe that @xmath748 because @xmath192 has finite dimension , after finitely many steps we will eventually get a sequence @xmath749 such that @xmath750 for @xmath751 and @xmath752 . moreover , @xmath753 is a convex subset without boundary , which is called a soul of @xmath192 . a subset @xmath682 is said to be _ totally convex _ in @xmath192 if for any quasi - geodesic segment @xmath754\to x$ ] with endpoints @xmath755 , we must have @xmath756)\subset \omega$ ] . the definition of quasi - geodesic can be found in @xcite . [ thm2.4 ] let @xmath192 be an @xmath215-dimensional open complete alexandrov space of curvature @xmath234 , @xmath757 $ ] , @xmath758 and @xmath759 be as above . then the following is true . \(1 ) for each @xmath760 , @xmath761)$ ] is a totally convex and compact subset of @xmath192 ; \(2 ) if @xmath762 , then @xmath763 remains to be totally convex ; \(3 ) the soul @xmath764 is a deformation retract of @xmath192 via multiple - step perelman - sharafutdinov semi - flows , which are distance non - increasing . \(1 ) for @xmath760 , we would like to show that @xmath761)$ ] is totally convex . suppose contrary , there were a quasi - geodesic @xmath754\to x$ ] with @xmath765 and @xmath725 with @xmath766 and @xmath767 + for each integer @xmath768 , we choose @xmath769 such that @xmath770 let @xmath771 and @xmath772 . since @xmath192 has non - negative curvature and @xmath754\to x$ ] is a quasi - geodesic , it is well - known ( @xcite ) that @xmath773 it follows that @xmath774 after passing a sub - sequence and re - indexing , we may assume that @xmath775 for all @xmath776 . by law of cosine , we have @xmath777 ^ 2\le[{{\rm d}}(\sigma(c),y_{i_j})]^2+|b - c|^2.\ ] ] therefore , we have @xmath778\\ & \ge \lim_{j\to+\infty}[i_j- \sqrt{[{{\rm d}}(\sigma(c),y_{i_j})]^2+|b - c|^2}]\\ & = \lim_{j\to+\infty}\frac{i_j^2-[{{\rm d}}(\sigma(c),y_{i_j})]^2 -|b - c|^2}{i_j+ \sqrt{[{{\rm d}}(\sigma(c),y_{i_j})]^2+|b - c|^2}}\\ & = \lim_{j\to+\infty } [ i_j - { { \rm d } } ( \sigma(c),y_{i_j } ) ] + 0 \\ & = \lim_{j\to+\infty } [ i_j - { { \rm d } } ( \sigma(c ) , \partial b(\hat x , i_j ) ) ] \\ & = f(\sigma(c))\end{aligned}\ ] ] which is contracting to @xmath779 hence , @xmath724 is a totally convex subset of @xmath192 . \(2 ) perelman @xcite showed that if @xmath724 is a convex subset of @xmath192 with non - empty boundary , then the distance function @xmath780 is concave for @xmath781 , ( see @xcite and @xcite as well ) . \(3 ) because our function @xmath782 and @xmath783 are concave in @xmath784 , the corresponding semi - flows are distance non - increasing , ( see chapter 6 of @xcite , section 2 of @xcite or @xcite ) . using the perelman - sharafutdinov flow @xmath785 , perelman ( cf . @xcite ) showed that @xmath192 is contractible to @xmath738 . let @xmath786 be the distance function @xmath787 if @xmath788 for @xmath789 . for the same reasons , @xmath738 is contractible to @xmath745 via the perelman - sharafutdinov flow @xmath790 is a deformation retract of @xmath192 . [ prop2.5 ] let @xmath791 be a _ function on @xmath792 with @xmath793 and @xmath794 as in the proofs of and above . suppose that @xmath792 is an open and complete alexandrov space with non - negative curvature and @xmath795 is a closed subset of @xmath267 . then the distance function @xmath796 from @xmath797 has no critical points in the complement @xmath798 $ ] of @xmath267 . for each @xmath799 and @xmath800 , we observe that @xmath801 . let @xmath802 \to y$ ] be a length - minimizing geodesic segment of unit speed from @xmath797 to @xmath803 with @xmath804 and @xmath805 . since @xmath792 has no boundary , any geodesic @xmath729 can be extended to a longer quasi - geodesic of unit speed @xmath806 \to y$ ] , ( see @xcite ) . since @xmath264 is convex , the composition of function @xmath807 remains convex for any quasi - geodesics @xmath808 ( see @xcite ) . it follows that @xmath809 let us consider a minimum direction @xmath810 of @xmath811 and @xmath812 \vec \xi_{min } $ ] , where we used the fact that @xmath813 \ge \frac{d^+ ( f \circ \tilde \sigma)}{dt } ( \ell ) > 0.\ ] ] hence we have @xmath812 \vec\xi_{min } \neq 0 $ ] . the vector @xmath814 is called a support vector of @xmath815 . for any support vector @xmath814 , one has ( cf . @xcite page 143 ) that inequality @xmath816 holds for all @xmath817 , where @xmath818 is a semi - concave function . let @xmath819)$ ] . when @xmath264 is convex , one has @xmath820 for any quasi - geodesic @xmath821 \to y$ ] and @xmath822 $ ] . thus , @xmath724 is totally convex . it follows that , for any direction @xmath823 , we have @xmath824 . moreover , we have @xmath825 for all @xmath823 , since @xmath818 is a concave function . because @xmath814 is a support vector of @xmath815 , one also has @xmath826 holds for @xmath823 , ( cf . @xcite page 143 ) . thus , we have @xmath827 for all @xmath823 . it follows that @xmath828 and @xmath829 is not a critical point of the distance function @xmath830 , where @xmath831 , ( compare with @xcite ) . this completes the proof of . + using soul theory and splitting theorem , we can classify non - negatively curved surfaces with possibly singular metrics . [ thm2.6 ] let @xmath204 be an oriented , complete and open surface of non - negative curvature . then @xmath204 is either homeomorphic to @xmath237 or isometric to a flat cylinder . it is known that @xmath204 is a manifold . let @xmath832 be a soul of @xmath204 . if the soul @xmath832 is a single point , then @xmath204 is homeomorphic to @xmath237 . when @xmath833 has dimension @xmath40 , then @xmath834 is isometric to embedded closed geodesic @xmath835 , ( i.e. , @xmath836 ) . let @xmath837 be the universal cover of @xmath204 with lifted metric and @xmath838 be a lift of @xmath834 in @xmath204 . we observe that x^2 & ^p & + & & + x^2&^p & n^1 suppose that @xmath839 is the perelman - sharafutdinov distance non - increasing projection from open space @xmath204 to its soul @xmath840 . such a distance non - increasing map @xmath839 can be lifted to a distance non - increasing map @xmath841 . thus @xmath842 is a line in an open surface @xmath837 of non - negative curvature . applying the splitting theorem , we see that @xmath837 is isometric to @xmath843 . it follows that @xmath204 is a flat cylinder . let us now turn our attention to closed surfaces of curvature . let @xmath204 be a closed @xmath28-dimensional alexandrov space of non - negative curvature . then the following holds : \(1 ) if the fundamental group @xmath844 is finite , then @xmath204 is homeomorphic to @xmath117 or @xmath845 . \(2 ) if the fundamental group @xmath844 is an infinite group , then @xmath204 is isometric to a flat tours or flat klein bottle . after passing through to its double when needed , we may assume that @xmath204 is oriented . when @xmath846 , @xmath204 is covered by @xmath117 . when @xmath847 and @xmath204 is oriented , for a non - trivial free homotopy class of a closed curve @xmath848\ne 0 $ ] in @xmath844 with @xmath849\ne 0 $ ] for all @xmath850 , we choose a length minimizing closed geodesic @xmath851 . suppose that @xmath837 is a universal cover of @xmath204 and @xmath852 is a lift of @xmath729 in @xmath204 . then we can check that @xmath853 is a geodesic line of @xmath854 . thus , @xmath837 is isometric to @xmath237 . it follows that @xmath204 is isometric to a flat torus , whenever @xmath204 is oriented with @xmath847 . [ ex2.9 ] when @xmath204 is an open surface of non - negative curvature , it might happen that @xmath738 is an interval . for instance , let @xmath855\times [ 0,+\infty)$ ] be a flat half - strip in @xmath237 . if we take two copies of @xmath856 and glue them along the boundary , the resulting surface @xmath857 is homeomorphic to @xmath237 . a result of petrunin implies that @xmath857 still have non - negative curvature ( e.g. , @xcite or @xcite ) . in this case , we have @xmath738 is an interval . of course , the soul @xmath858 of @xmath204 is a single point . we now say a few words for non - negatively curved surfaces @xmath204 with non - empty convex boundary . by definition of surface @xmath204 with curvature @xmath859 , its possibly non - empty boundary @xmath860 must be convex . [ cor2.9 ] let @xmath204 be a surface with non - negative curvature and non - empty boundary . then \(1 ) if @xmath204 is compact , then @xmath204 is either homeomorphic to @xmath861 or isometric to @xmath862 $ ] or a flat mbius band ; \(2 ) if @xmath204 is non - compact and oriented , then @xmath204 is either homeomorphic to @xmath863 or isometric to one of three types : @xmath864 , a half flat strip or @xmath865\times ( -\infty,+\infty)$ ] . if we take two copies of @xmath204 and glue them together along their boundaries , the resulting surface @xmath866 still has curvature @xmath234 , due to a result of petrunin @xcite . clearly , @xmath866 has no boundary . \(1 ) when @xmath866 is compact and oriented , then @xmath867 is homeomorphic to the unit @xmath28-sphere or is isometric to a flat strip . hence , @xmath204 is either homeomorphic to @xmath861 or isometric to @xmath862 $ ] or a flat mbius band . \(2 ) when @xmath866 is non - compact , then @xmath866 is homeomorphic to @xmath237 or isometric to @xmath868 or @xmath204 is isometric to @xmath869 \times [ 0 , \infty)$ ] . to verify this assertion , we consider the soul @xmath870 of @xmath867 . if @xmath870 is a circle , then @xmath866 is isometric to an infinite flat cylinder : @xmath871 . if the soul @xmath870 is a point , then @xmath866 is homeomorphic to @xmath237 . there is a special case which we need to single out : @xmath204 is isometric to @xmath869 \times [ 0 , \infty)$ ] . we will elaborate this special case in below . [ remark2.11 ] in below , we will estimate the number of extremal points , i.e. essential singularities , on surfaces with non - negative curvature , using multi - step perelman - sharafutdinov flows associated with the cheeger - gromoll convex exhaustion . finally , we would like to classify all non - negatively curved open @xmath0-manifolds with possibly singular metrics . [ thm2.11 ] let @xmath872 be an open complete @xmath0-manifold with a possibly singular metric of non - negative curvature . suppose that @xmath872 is oriented and @xmath870 is a soul of @xmath47 . then the following is true . 1 . when @xmath873 , then the soul of @xmath872 is isometric to a circle . moreover , its universal cover @xmath874 is isometric to @xmath875 , where @xmath837 is homeomorphic to @xmath876 ; 2 . when @xmath877 , then the soul of @xmath872 is homeomorphic to @xmath878 or @xmath879 . furthermore , @xmath872 is isometric to one of four spaces : @xmath880 , @xmath881 , @xmath882 or @xmath883 , where @xmath884 is the flat klein bottle and @xmath885 is homeomorphic to @xmath886 $ ] ; 3 . when @xmath887 , then the soul of @xmath872 is a single point and @xmath872 must be homeomorphic to @xmath888 . this theorem is entirely due to shioya - yamaguchi @xcite . a special case of for smooth open @xmath0-manifold with non - negative curvature was stated as theorem 8.1 in cheeger - gromoll s paper @xcite . shioya - yamaguchi s proof is quiet technical , which occupied the half of their paper @xcite . for convenience of readers , we present an alternative shorter proof of shioya - yamaguchi s soul theorem for 3-manifolds with possible singular metrics . * case 1 . * when the soul @xmath840 of @xmath47 is a closed geodesic @xmath889 , there are distance non - increasing multi - step perelman - sharafutdinov retractions from @xmath47 to @xmath889 . thus , @xmath889 is length - minimizing in its free homotopy class . it follows that the lifting geodesic @xmath890 is a geodesic line in the universal covering space @xmath891 of @xmath47 . using the splitting theorem ( cf . @xcite ) for non - negatively curved space @xmath892 , we see that @xmath892 is isometric to @xmath893 , where @xmath837 is a contractible surface with non - negative curvature . hence , @xmath894 is homeomorphic to @xmath876 . * when the soul @xmath895 of @xmath47 is a surface @xmath204 , we observe that @xmath896 is a convex subspace of @xmath47 , where @xmath897 $ ] and @xmath898 . since @xmath264 is convex and @xmath899 has non - negative curvature , @xmath204 has non - negative curvature as well . by , we see that @xmath204 is either homeomorphic to a quotient of @xmath117 or isometric to a quotient of a flat torus . for this case , our strategy goes as follows . we will show that there is a _ normal line bundle " _ over the soul @xmath204 . after passing its double cover if needed , we may assume that such a _ normal line bundle " _ is topologically trivial in @xmath47 . in this case , with some extra efforts , one can show that there is a geodesic line @xmath900 orthogonal to @xmath204 in @xmath47 . thus , the space @xmath47 ( or its double cover ) splits isometrically to @xmath901 . here is the detail of our _ normal line bundle " _ argument . for each point @xmath58 in the soul @xmath204 , its unit tangent space @xmath902 is homeomorphic to @xmath496 . recall that the space of unit tangent directions @xmath903 of @xmath47 at @xmath803 is homeomorphic to the sphere @xmath117 , because @xmath47 is a @xmath0-manifold . observe that @xmath902 is a convex subset of @xmath903 . moreover we see that @xmath902 divides @xmath903 into exactly two parts : @xmath904 = \omega^2_{x , + } \cup \omega^2_{x , + } \ ] ] since the curvature of @xmath905 is greater than or equal to 1 , using theorem 6.1 of @xcite ( cf . @xcite ) , we obtain that there is a unique unit vector @xmath906 such that @xmath907 we claim that @xmath908 . suppose contrary , @xmath909 were true . we derive a contradiction as follows . let @xmath910 \to \sigma^2_{y_\infty}(y^3_\infty)$ ] be a length - minimizing geodesic segment of unit speed from @xmath902 of length @xmath911 with @xmath912 and @xmath913 . we now choose another geodesic segment @xmath914 \to \sigma_x^1(x^2)$ ] be a geodesic segment of unit speed with @xmath915 . since @xmath916 , applying the triangle comparison theorem ( cf . @xcite ) to the our geodesic hinge at @xmath917 in @xmath903 , we see that @xmath918 . thus , for the point @xmath919 , there are at least two points @xmath920 with angular distance @xmath921 . in another words , there were at least two distinct length - minimizing geodesic segments from @xmath919 to @xmath902 . hence , @xmath922 } $ ] is no longer length - minimizing for any @xmath923 , a contradiction . it follows that @xmath908 . moreover , the equality @xmath924 holds if and only if @xmath903 is isometric to the two point spherical suspension of @xmath902 . in this case , @xmath925 is isometric to @xmath926 . recall that @xmath896 is a level set of the busemann function . we can write @xmath927 $ ] for @xmath928)$ ] . by the first variational formula ( cf . @xcite page 125 ) , we see that @xmath929 combining our earlier inequality @xmath908 , we see that @xmath924 . therefore , we conclude that @xmath930 is isometric to @xmath926 . hence , there is a _ normal line bundle " _ over the soul @xmath204 . after passing its double cover if necessary , such a _ normal line bundle " _ of @xmath204 in @xmath47 is topologically trivial . thus , we assume that @xmath931 = \omega^3_+ \cup \omega^3_-$ ] has exactly two ends , where we replace @xmath47 by its double cover @xmath932 if needed . for each end and each @xmath933 , there exists a ray @xmath934 with starting point @xmath58 . one can verify that @xmath935 is a geodesic line in @xmath47 ( or in its double cover ) . by the splitting theorem , we conclude that @xmath47 ( or its double cover ) is isometric to @xmath936 . \(3 ) when the soul @xmath937 of @xmath47 is a single point @xmath938 , our proof becomes more involved . let @xmath939 $ ] and @xmath940 be as above . there are three possibilities for @xmath941 = 0 , 1 , 2 $ ] . * subcase 3.0 . * _ @xmath941 = 0 $ ] and @xmath942_. in this subcase , the space of unit tangent directions @xmath943 at @xmath803 is homeomorphic to the sphere @xmath117 and its tangent cone @xmath944 is homeomorphic to @xmath888 . recall that the pointed spaces @xmath945 is convergent to the tangent cone @xmath946 as @xmath259 , where @xmath320 is the origin of @xmath944 . by the pointed version of perelman s stability theorem ( cf . theorem 7.11 of @xcite ) , we see that for sufficiently small @xmath411 , @xmath947 is homeomorphic to @xmath948 . it follows that @xmath949 is homeomorphic to the unit ball @xmath950 for sufficiently small @xmath951 , because @xmath47 is a @xmath0-manifold . we now use perelman s fibration theorem to complete our proof for this subcase . it follows from that @xmath952 has no critical value in @xmath953 . perelman s fibration theorem ( our above ) implies that there is a fibration structure @xmath954 \stackrel{r_a}{\longrightarrow } ( \frac{\varepsilon}{2 } , \infty)\ ] ] it follows that @xmath955 $ ] is homeomorphic to @xmath956 and that @xmath48 is homeomorphic to @xmath957 $ ] . thus , @xmath48 is homeomorphic to @xmath888 , for this subcase . * subcase 3.1 . * _ @xmath941 = 1 $ ] and @xmath958 is a geodesic segment . _ it follows from that @xmath952 has no critical value in @xmath953 . for the same reasons as above , it remains important to verify that @xmath959 is homeomorphic to @xmath950 . let @xmath693 \to y^3 $ ] be as above and @xmath960 ) $ ] be the minimal set of @xmath264 . we denote a @xmath411-neighborhood of @xmath267 by @xmath959 . let @xmath961)$ ] for some @xmath962 . we observe that @xmath963 for @xmath964 . for the same reason as in subcase 3.0 above , both @xmath965 and @xmath966 are homeomorphic to @xmath950 , because @xmath47 is a @xmath0-manifold . it is sufficient to show that @xmath967 is homeomorphic to a finite cylinder @xmath968 \times d^2 $ ] for sufficient small @xmath602 . let @xmath969 . we consider the distance function @xmath970 . we observe that the distance function has no critical point on geodesic sub - segment @xmath971)$ ] . a result of petrunin ( cf . @xcite page 142 ) asserts that if @xmath972 as @xmath973 , then @xmath974 . hence , there exists a sufficiently small @xmath951 such that @xmath975 has no critical point in @xmath976 . for the same reason as in subcase 3.0 , we can apply perelman s fibration theorem to our case : @xmath977 where we used the fact that @xmath978 \cap u_{\varepsilon}(a _ { s/2 } ) $ ] is homeomorphic to @xmath861 . it follows that @xmath967 is homeomorphic to a finite cylinder @xmath979 . therefore , @xmath980 is homeomorphic to @xmath950 . it follows that @xmath981 $ ] is homeomorphic to @xmath888 . * subcase 3.2 . * _ @xmath941 = 2 $ ] and @xmath982 is a totally convex surface with boundary . _ for the same reason as the two subcases above , it is sufficient to establish that @xmath983 is homeomorphic to the unit @xmath0-ball @xmath950 : @xmath984 let @xmath985 . by our discussion in case 2 above , we see that for each interior point @xmath986 , there is a unique _ normal line _ orthogonal to @xmath987 at @xmath58 . thus , the interior @xmath988 has a normal line bundle @xmath989 . because @xmath988 is contractible to a soul point @xmath990 , any line bundle over @xmath988 is topologically trivial . in this subcase , our technical goals are to show the following : ( 3.2a ) @xmath991 is homeomorphic to @xmath992 ; ( 3.2b ) @xmath993 is homeomorphic to a solid tori @xmath994 . to establish ( 3.2a ) , we use a theorem of perelman ( cf . theorem 6.1 of @xcite ) to show that there is a product metric on a subset @xmath991 of @xmath47 . inspired by perelman , we consider the distance function @xmath995 . since @xmath47 has non - negative curvature and @xmath996 is _ weakly concave _ towards its complement @xmath997 $ ] , perelman observed that @xmath998 is _ concave _ on @xmath999 $ ] , ( see the proof of theorem 6.1 in @xcite , @xcite or @xcite ) . we already showed that for each interior point @xmath1000 , there is a unique _ normal line _ orthogonal to @xmath1001 at @xmath58 . with extra efforts , we can show that , for each interior point @xmath1002 and each unit normal direction @xmath1003 , there is a unique ray @xmath1004 with @xmath1005 and @xmath1006 . moreover , we have @xmath1007 therefore each @xmath1008 $ ] with @xmath1009 , we have @xmath1010 hence , our busemann function @xmath264 is both convex and concave on the subset @xmath999 $ ] . thus , for any geodesic segment @xmath1011 \to [ u_{s/4}(a_s ) - a_s]$ ] , the function @xmath1012 is a linear function in @xmath96 . using the fact that @xmath1012 is a linear function in @xmath96 and the sharp version of triangle comparison theorem ( cf . @xcite ) , we can show that there is a sub - domain @xmath1013 of @xmath47 such that the metric of @xmath47 on @xmath1013 splits isometrically as @xmath1014 since @xmath1015 is homeomorphic to @xmath861 , we conclude that @xmath991 is homeomorphic to @xmath1016 ( compare with the proof of below ) . hence , our assertion ( 3.2a ) holds . it remains to verify ( 3.2b ) . we consider the doubling surface @xmath1017 . it follows from a result of petrunin that @xmath1018 has non - negative curvature . by , we see that the essential singularities ( extremal points ) in @xmath1018 are isolated . thus , there are only finitely many points @xmath1019 on @xmath1020 such that @xmath1021 \le \frac{\pi}{2}\ ] ] for @xmath1022 . we can divide our boundary curve @xmath1020 into @xmath193-many arcs , say @xmath1023 = \cup \gamma_j$ ] . using a similar argument as in subcase 3.1 , we can show that , for each @xmath1024 , its @xmath411-neighborhood @xmath1025 is homeomorphic to a finite cylinder @xmath1026 $ ] . since @xmath47 is a @xmath0-manifold , by the proof of , we know that @xmath1027 is homeomorphic to @xmath950 . consequently , we have @xmath1028 \bigcup [ \cup b_{y^3_\infty}(x_j , \varepsilon)],\ ] ] which is homeomorphic to a solid tori @xmath1029 . this completes our proof of the assertion that @xmath1030 \ } $ ] is homeomorphic to @xmath950 . therefore , @xmath1031 $ ] is homeomorphic to @xmath1032 . we now finished the proof of our soul theorem for all cases . using , we can complete the proof of . let @xmath1033 and @xmath1034 be defined by . we may assume that @xmath1035 is convergent to a pointed alexandrov space @xmath1036 of non - negative curvature , by replacing @xmath1037 with @xmath643 in if needed . by , we see that the limiting space @xmath174 is a non - compact and complete space of @xmath1038 . furthermore , @xmath47 has no boundary . by perelman s stability theorem ( cf . @xcite ) , we see that the limit space @xmath47 is a topological @xmath0-manifold . by , we see that @xmath1039 is homeomorphic to a quotient of the @xmath28-torus @xmath527 . the notion of ideal boundary @xmath1040 of @xmath47 can be found in @xcite . in our case , the ideal boundary @xmath1040 at infinity of @xmath47 is homeomorphic to a circle @xmath1041 . we will verify that @xmath1042 as follows . let @xmath1043 be a @xmath411-tubular neighborhood of the soul @xmath937 in @xmath1044 . by perelman s stability theorem and our assumption that @xmath1045 is oriented , we observe that , for sufficiently large @xmath24 , the boundary @xmath1046 is homoeomorphic to @xmath1047 . thus , the soul @xmath937 of @xmath899 must be a circle @xmath496 . in this case , it follows from that the metric on @xmath899 ( or on its universal cover ) splits . therefore , it follows from perelman s stability theorem ( cf . @xcite ) that @xmath1048 is homeomorphic to a solid tori , which is foliated by orbits of a free circle action for sufficiently large @xmath24 . we discuss more about gluing and perturbing our local circle actions in . we will use -[thm2.11 ] to derive more refined results for collapsing @xmath0-manifolds with curvature @xmath148 in upcoming sections . let @xmath1049 be a round sphere of constant curvature @xmath1050 . it is clear that @xmath1051 $ ] is convergent to @xmath1052 $ ] with non - negative curvature as @xmath407 . the product space @xmath1053 $ ] is not a graph manifold . however , if @xmath1054 is contained in the interior of collapsed @xmath0-manifold @xmath25 with boundary , then for topological reasons , @xmath1055 still has a chance to become a part of graph - manifold @xmath25 . let us now use the language of cheeger - gromov s @xmath31-structure theory to describe @xmath0-dimensional graph - manifold . it is known that a @xmath0-manifold @xmath26 is a graph - manifold if and only if @xmath26 admits an @xmath31-structure of positive rank , which we now describe . an f - structure , @xmath1056 , is a topological structure which extends the notion of torus action on a manifold , ( see @xcite and @xcite ) . in fact , the more significant concept is that of atlas ( of charts ) for an f - structure . an atlas for an f - structure on a manifold @xmath1057 is defined by a collection of triples @xmath1058 , called charts , where @xmath1059 is an open cover of @xmath1057 and the torus , @xmath1060 , acts effectively on a finite normal covering , @xmath1061 , such that the following conditions hold : 1 . there is a homomorphism , @xmath1062 , such that the action of @xmath1060 extends to an action of the semi - direct product @xmath1063 , where @xmath1064 is the fundamental group of @xmath276 ; 2 . if @xmath1065 , then @xmath1066 is connected . if @xmath1067 , then on a suitable finite covering of @xmath1068 , their lifted tori - actions commute after appropriate re - parametrization . the compatibility condition ( 3.2 ) on lifted actions implies that @xmath1057 decomposes as a disjoint union of orbits , @xmath1069 , each of which carries a natural flat affine structure . the orbit containing @xmath1070 is denoted by @xmath1071 . the dimension of an orbit of minimal dimension is called the rank of the structure . [ prop3.1 ] a @xmath0-dimensional manifold @xmath26 with possible non - empty boundary is a graph - manifold if and only if @xmath26 admits an f - structure of positive rank . for @xmath0-dimensional manifolds , we will see that 7 out of 8 geometries admits f - structure . therefore , there seven types of locally homogeneous spaces are graph - manifolds . [ ex3.2 ] let @xmath26 be a closed locally homogeneous space of dimension 3 , such that its universal covering spaces @xmath1072 is isometric to seven geometries : @xmath1073 and @xmath90 . then @xmath26 admits an f - structure and hence it is a graph - manifold . let us elaborate this issue in detail as follows . 1 . if @xmath1074 is a flat @xmath0-manifold , then it is covered by @xmath0-dimensional torus . hence it is a graph - manifold . if @xmath1075 is a lens space , then its universal cover @xmath113 admits the classical hopf fibration : @xmath1076 it follows that @xmath26 is a graph - manifold . if @xmath1077 is a closed @xmath0-manifold , then a theorem of eberlein implies that a finite normal cover @xmath1078 of @xmath26 is diffeomorphic to @xmath1079 , where @xmath895 is a closed surface of genus @xmath1080 , ( see proposition 5.11 of @xcite , @xcite ) . @xmath1081 , then a finite cover is isometric to @xmath1082 . clearly , @xmath26 is a graph - manifold . we should point out that a quotient space @xmath1083 may be homeomorphic to @xmath1084 . 5 . if @xmath1085 , then a finite cover @xmath1086 of @xmath26 is diffeomorphic to the unit tangent bundle of a closed surface @xmath1087 of genus @xmath1088 . thus , we may assume that @xmath1089 it follows that @xmath26 is a graph - manifold . 6 . if @xmath1090 , then the universal cover @xmath1091 + is a @xmath0-dimensional heisenberg group . let + @xmath1092 be the integer lattice group of @xmath89 . a finite cover @xmath1078 of @xmath26 is a circle bundle over a @xmath28-torus . therefore @xmath26 is a graph - manifold , which can be a diameter - collapsing manifold . 7 . if @xmath1093 , then @xmath26 is foliated by tori , mbius bands or klein bottles , which is a graph - manifold . let us consider a graph - manifold which is not a compact quotient of a homogeneous space . let @xmath1094 be a surface of genus @xmath503 and with a boundary circle , for @xmath1095 . clearly , @xmath1096 . we glue @xmath1097 to @xmath1098 along their boundaries with @xmath496-factor switched . the resulting manifold @xmath1099 does not admit a global circle fibration , but @xmath26 is a graph - manifold . as we pointed out above , @xmath1100 $ ] can be collapsed to an interval @xmath1052 $ ] with non - negative curvature . suppose that @xmath1101 is a portion of collapsed @xmath0-manifold @xmath26 such that @xmath1101 is diffeomorphic to @xmath1102 $ ] . we need to glue extra solid handles to @xmath1101 so that our collapsed @xmath0-manifold under consideration becomes a graph - manifold . for this purpose , we divided @xmath1100 $ ] into three parts . in fact , @xmath117 with two disks removed , @xmath1103 , is diffeomorphic to an annulus @xmath267 . thus , @xmath1100 $ ] has a decomposition @xmath1104=\big(d^2_1\times [ a , b]\big ) \sqcup ( d^2_2\times [ a , b ] ) \sqcup ( a\times [ a , b]).\ ] ] the product space @xmath1105 $ ] clearly admits a free circle action , and hence is a graph - manifold . for solid cylinder part @xmath1106 $ ] , if one can glue two solid cylinders together , then one might end up with a solid torus @xmath1107 which is again a graph - manifold . we will decompose a collapsed @xmath0-manifold @xmath25 with curvature @xmath148 into four major parts according to the dimension @xmath193 of limiting set @xmath201 : @xmath1108 where @xmath1109 denotes the interior of the space @xmath1110 . the portion @xmath1111 of @xmath25 consists of union of closed , connected components of @xmath25 which admit riemannian metric of non - negative sectional curvature . [ prop3.4 ] let @xmath1111 be a union of one of following : 1 . a spherical @xmath0-dimensional space form ; 2 . a manifold double covered by @xmath1082 ; 3 . a closed flat @xmath0-manifold . then @xmath1111 can be collapsed to a @xmath408-dimensional manifold with non - negative curvature . moreover , @xmath1111 is a graph - manifold . we denote the regular part of @xmath204 by @xmath1112 . let us now recall a reduction for the proof of perelman s collapsing theorem due to morgan - tian . earlier related work on 3-dimensional collapsing theory was done by xiaochun rong in his thesis , ( cf . @xcite ) . [ thm3.5 ] let @xmath474 be a sequence of compact @xmath0-manifolds satisfying the hypothesis of theorem 0.1 and @xmath1111 be as in above . if , for sufficiently large @xmath24 , there exist compact , co - dimension 0 submanifolds @xmath1113 and @xmath1114 with @xmath1115 satisfying six conditions listed below , then holds , where six conditions are : 1 . each connected component of @xmath1116 is diffeomorphic to the following . 1 . a @xmath527-bundle over @xmath496 or a union of two twisted i - bundle over the klein bottle along their common boundary ; 2 . @xmath1117 or @xmath1118 , where @xmath1119 $ ] is a closed interval ; 3 . a compact @xmath0-ball or the complement of an open @xmath0-ball in @xmath1120 which is homeomorphic to @xmath1121 $ ] ; 4 . a twisted i - bundle over the klein bottle , or a solid torus . + in particular , every boundary component of @xmath1116 is either a @xmath28-sphere or a @xmath28-torus . 2 . @xmath1122 ; 3 . if @xmath1123 is a @xmath28-torus component of @xmath1124 , then @xmath1125 if and only if @xmath1126 is not boundary of @xmath1127 ; 4 . if @xmath1123 is a @xmath28-sphere component of @xmath1124 , then @xmath1128 is diffeomorphic to an annulus ; 5 . @xmath1129 is the total space of a locally trivial @xmath496-bundle and the intersection @xmath1130 is saturated under this fibration ; 6 . the complement @xmath1131 $ ] is a disjoint union of solid tori and solid cylinders . the boundary of each solid torus is a boundary component of @xmath1129 , and each solid cylinder @xmath1132 in @xmath1133 meets @xmath1116 exactly in @xmath1134 . ( @xcite ) the proof of is purely topological , which has noting to do with the collapsing theory . morgan and tian @xcite first verified for special cases under additional assumption on @xmath1116 : a. @xmath1116 has no closed components ; b. each @xmath28-sphere component of @xmath1135 bounds a @xmath0-ball component of @xmath1116 ; c. each @xmath28-torus component of @xmath1135 that is compressible in @xmath25 bounds a solid torus component of @xmath1116 . the general case can be reduced to a special case by a purely topological argument . [ def3.6 ] if a collapsed @xmath0-manifold @xmath25 has a decomposition @xmath1136 satisfying six properties listed in and if @xmath1111 is a union of closed @xmath0-manifolds which admit smooth riemannian metrics of non - negative sectional curvature , then such a decomposition is called an admissible decomposition of @xmath25 . in -[section2 ] , we already discussed the part @xmath1137 and a portion of @xmath1055 . in next section , we discuss the collapsing part @xmath1138 with spherical or toral fibers for our @xmath0-manifold @xmath25 , where @xmath1109 is the interior of @xmath1110 . in this section , we discuss the case when a sequence of metric balls @xmath1139 collapse to @xmath40-dimensional space @xmath1140 . there are only two choices of @xmath1141 , either diffeomorphic to a circle or an interval @xmath865 $ ] . by perelman s fibration theorem @xcite or yamaguchi s fibration theorem , we can find an open neighborhood @xmath1142 of @xmath1143 such that , for sufficiently large @xmath24 , there is a fibration @xmath1144 where @xmath1145 is isometric to a circle @xmath496 or an open interval @xmath1146 . we will use the soul theory ( e.g. , ) to verify that a finite cover of the collapsing fiber @xmath170 must be homeomorphic to either a @xmath28-sphere @xmath117 or a @xmath28-dimensional torus @xmath527 , ( see in 0 ) let us begin with two examples of collapsing @xmath0-manifold with toral fibers [ ex4.1 ] let @xmath1147 be an oriented and non - compact quotient of @xmath140 such that @xmath26 has finite volume and @xmath26 has exactly one end . suppose that @xmath1148 be a geodesic ray . we consider the corresponding busemann function @xmath1149 $ ] . for sufficiently large @xmath725 , the sup - level subset @xmath1150 has special properties . it is well known that , in this case , the cusp end @xmath1151 is diffeomorphic to @xmath1152 . of course , the component @xmath1153 admits a collapsing family of metric @xmath1154 , such that @xmath1155 is convergent to half line @xmath1156 . we would like to point out that @xmath28-dimensional collapsing fibers can be collapsed at two different speeds . let @xmath1157 $ ] be the product of rectangle torus @xmath1158 and an interval . let us fix a parametrization of @xmath1159 , s , t\in \mathbb r\}$ ] and @xmath1160 . then the re - scaled pointed spaces @xmath1161 are convergent to the limiting space @xmath1162 , where @xmath1163 is isometric to @xmath1164 . similarly , when the collapsing fiber is homeomorphic to a 2-sphere @xmath117 , the collapsing speeds could be different along longitudes and latitudes . we may assume that latitudes shrink at speed @xmath1165 and longitudes shrink at speed @xmath411 . for the same reason , after re - scaling , the limit space @xmath1166 could be isometric to @xmath1167 \times ( -\infty , + \infty)$ ] . thus , the non - compact limiting space @xmath1168 could have boundary . let us return to the proof of perelman s collapsing theorem . according to the second condition of , we consider a boundary component @xmath1169 , where the diameter of @xmath1170 is at most @xmath1171 as @xmath43 . moreover , there exists a topologically trivial collar @xmath1172 of length one and sectional curvatures of @xmath25 are between @xmath1173 and @xmath1174 . in this case , we have a trivial fibration : @xmath1175\ ] ] such that the diameter of each fiber @xmath1176 is at most @xmath1177 $ ] by standard comparison theorem . as @xmath622 , the sequence @xmath1178 converge to an interval @xmath1179 $ ] . we are ready to work on the main result of this subsection . [ thm4.3 ] let @xmath1180 be as in theorem 0.1. suppose that an @xmath40-dimensional space @xmath1141 is contained in the @xmath40-dimensional limiting space and @xmath1181 is an interior point of @xmath1141 . then , for sufficiently large @xmath24 , there exists a sequence of subsets @xmath1182 such that @xmath1172 is fibering over @xmath1145 with spherical or toral fibers . @xmath1183 where @xmath170 is homeomorphic to a quotient of a @xmath28-sphere @xmath117 or a @xmath28-torus @xmath527 . when @xmath20 is oriented , @xmath170 is either @xmath117 or @xmath527 . as we pointed out above , since @xmath1184 is a @xmath40-dimensional space , there exists a fibration @xmath1183 for sufficiently large @xmath24 . it remains to verify that the fiber is homeomorphic to @xmath1185 or klein bottle @xmath1186 . for this purpose , we use soul theory for possibly singular space @xmath719 with non - negative curvature . by our discussion , @xmath1187 is homeomorphic to @xmath1188 , where @xmath170 is a closed @xmath28-dimensional manifold . thus , the distance function @xmath1189 has at least one critical point @xmath1190 in @xmath1191 , because @xmath1192 is not contractible . let @xmath1033 be @xmath1193 we claim that @xmath1194 and @xmath1195 as @xmath1196 . to verify this assertion , we observe that the distance functions @xmath1197 are convergent to @xmath1198 . by perelman s convergent theorem , the trajectory of gradient semi - flow @xmath1199 is convergent to the trajectory in the limit space @xmath1141 : @xmath1200 ( see @xcite or @xcite ) . clearly , @xmath1201 has no critical value in @xmath1202 for @xmath1203 . thus , for sufficiently large @xmath24 , the distance function @xmath1204 has no critical value in @xmath1205 , where @xmath1206 and @xmath1207 as @xmath1196 . let us now re - scale our metrics @xmath1208 by @xmath1209 again . suppose @xmath1210 . then a subsequence of the sequence of the pointed spaces @xmath1211 will converge to @xmath1212 . the curvature of @xmath719 is greater than or equal to @xmath408 , because @xmath1213 as @xmath1214 . since the distance @xmath1215 has a critical point @xmath1216 with @xmath1217 . thus @xmath1218 . when @xmath1219 , we observe that @xmath872 has exactly two ends in our case . thus @xmath872 admits a line and hence its metric splits . a soul @xmath181 of @xmath872 must be of @xmath28-dimensional . thus @xmath872 is isometric to @xmath1220 . it follows that the soul @xmath181 of @xmath872 has non - negative curvature . by perelman s stability theorem , @xmath170 is homeomorphic to @xmath181 for sufficiently large @xmath24 . a closed possibly singular surface @xmath181 of non - negative curvature has been classified in : @xmath1185 or klein bottle @xmath1186 . let us consider the case of @xmath1221 . we may assume that the limiting space @xmath1163 has exactly two ends . when @xmath1166 has no boundary , then the limit space is isometric to @xmath868 , because @xmath1163 has exactly two ends . by our discussion in 1 - 2 , we have fibration structure @xmath1222 for sufficiently large @xmath24 . when @xmath1166 has non - empty boundary ( i.e. , @xmath1223 ) , there are two subcases . if @xmath1224 , by our discussion in , we still have @xmath1225 . if @xmath1226 , using below , we see that @xmath1227 \sim s^2 $ ] . this completes the proof of our theorem for all cases . in next section , we will discuss the end points or @xmath1228 when @xmath1229 $ ] is an interval . in addition , we also discuss the @xmath28-dimensional boundary @xmath860 when @xmath204 is a surface with convex boundary . suppose that a sequence of @xmath0-manifolds is collapsing to a lower dimensional space @xmath1110 . in previous sections we showed that there is ( possibly singular ) fibration @xmath1230 in this section we will consider the points on the boundary of @xmath1110 , we will divided our discussion into two cases : namely ( 1 ) when @xmath1231 and @xmath1141 is a closed interval ; and ( 2 ) when @xmath1232 and @xmath204 is a surface with boundary . since @xmath204 is a topological manifold with boundary , without loss of generality , we can assume that @xmath1233 . first we provide two examples to demonstrate that the collapsing could occur in many different ways . [ ex 5.1 ] let @xmath1234 be a solid torus , where @xmath1235 on the disk so that @xmath1236 is converging to the interval @xmath1237 . it follows that if @xmath1238 then the sequence of @xmath0-spaces is converging to a finite cylinder : i.e. , @xmath1239 as @xmath1240 . for this purpose , we let @xmath1241,z\ge 0 , x^2 + y^2 + z^2 < 1 \}\ ] ] we further make a smooth perturbation around the vertex @xmath1242 , while keeping curvature non - negative . clearly , as @xmath1243 , the family of conic surfaces @xmath1244 collapses to an interval @xmath1237 . let @xmath1245 be the limiting space . as @xmath1240 , our @xmath0-manifolds @xmath1246 collapsed to @xmath204 with @xmath1247 . in this example , for every point @xmath1248 , the space of direction @xmath1249 is a closed half circle with diameter @xmath1250 . [ ex5.2 ] in this example , we will construct the limiting surface with boundary corner points . let us consider the unit circle @xmath1251 and let @xmath1252 be an involution given by @xmath1253 . then its quotient @xmath1254 is isometric to @xmath1255 $ ] . our target limiting surface will be @xmath1256 $ ] . clearly , @xmath204 has boundary corner point with total angle @xmath377 . to see that @xmath204 is a limit of smooth riemannian @xmath0-manifolds @xmath1257 while keeping curvatures non - negative , we proceed as follows . we first viewed a 2-sheet cover @xmath26 as an orbit space of @xmath1258 by a circle action . let @xmath1259 and let @xmath1260 be a parametrization of @xmath1261 and hence for its quotient @xmath1258 . there is a circle action @xmath1262 given by @xmath1263 for each @xmath1264 . we also define an involution @xmath1265 by @xmath1266 . it follows that @xmath1267 . let @xmath1268 be subgroup generated by @xmath1269 . we introduce a family of metrics : @xmath1270 the transformations @xmath1271 remain isometries for riemannian manifolds @xmath1272 . thus , @xmath1273 is a smooth riemannian @xmath0-manifold with non - negative curvature . it is easy to see that @xmath1274 is homeomorphic to a solid torus . as @xmath1240 , our riemannian manifolds @xmath1275 collapse to a lower dimensional space @xmath1276 = \ { ( r , s ) | 0 \le r < 1 , 0 \le s \le \frac 12 \}$ ] . the surface @xmath204 has a corner point with total angle @xmath377 . in above two examples , if we set @xmath1277 , then there exist an open subset @xmath1278 and a continue map @xmath1279 such that @xmath1280 is homeomorphic to a solid torus , where @xmath267 is an annular neighborhood of @xmath860 in @xmath204 . let us recall an observation of perelman on the distance function @xmath1281 from the boundary @xmath860 . [ lem5.3 ] let @xmath204 be a compact alexandrov surface of curvature @xmath352 and with non - empty boundary @xmath860 , @xmath1282 and @xmath1283 $ ] . then , for sufficiently small @xmath411 , the distance function @xmath1284 from the boundary has no critical point in @xmath1285 . we will use a calculation of perelman and a result of petrunin to complete the proof . by the definition , if @xmath204 has curvature @xmath1286 , then @xmath860 must be convex . for any @xmath1287 , we let @xmath1288 $ ] be the total tangent angle of the convex domain @xmath204 . it follows from the convexity @xmath1289 . we consider the distance function @xmath1290 for all @xmath1291 . perelman ( cf . @xcite page 33 , line 1 ) calculated that @xmath1292 for all @xmath1293 . ( perelman stated his formula for spaces with non - negative curvature , but his proof using the first variational formula is applicable to our surface @xmath204 with curvature @xmath148 ) . corollary 1.3.5 of @xcite asserts that if there is a converging sequence @xmath1294 with @xmath1295 as @xmath973 then latexmath:[\[\liminf_{n \to \infty}|\nabla f ( x_n ) |\ge the above discussion that there is an @xmath951 such that @xmath1297 for @xmath1298 . we now recall a theorem of shioya - yamaguchi with our own proof . the proof of shioya - yamaguchi used a version of margulis lemma , which we will use our in 1 instead . [ thm5.4 ] let @xmath1299 be a sequence of collapsing @xmath0-manifolds as in theorem 0.1. suppose that @xmath1300 with @xmath1301 and @xmath1302 is homeomorphic to @xmath496 . then there is @xmath1303 such that @xmath1304 is homeomorphic to @xmath1305 for all sufficiently large @xmath24 . moreover , there exist an @xmath1306 and a sequence of closed curves @xmath1307 with @xmath1308 as @xmath1309 such that a @xmath411-tubular neighborhood @xmath1310 $ ] of @xmath1311 in @xmath20 is homeomorphic to a solid tori @xmath1029 for sufficiently large @xmath24 . by conic lemma ( cf . ) , the number of points @xmath1248 with @xmath1312 is finite , denoted by @xmath1313 . since @xmath1314 is compact , it follows from that there is a common @xmath1315 such that ( 1 ) the distance function @xmath1316 has not critical point in @xmath1317 $ ] ; and ( 2 ) @xmath1318 is homeomorphic to the upper half disk @xmath1319 , for all @xmath1320 . since @xmath1302 is homeomorphic to @xmath496 , we can approximate @xmath860 by a sequence of closed broken geodesics @xmath1321 with vertices @xmath1322 . we may assume that @xmath1313 is a subset of @xmath1323 for all @xmath1324 . we also require that the distance between two consecutive vertices is less than @xmath1325 ( i.e. , @xmath1326 ) , for sufficiently large @xmath1327 . we now choose a sequence of finite sets @xmath1328 such that @xmath1329 as @xmath43 and @xmath1330 span an embedded broken geodesic @xmath1331 in @xmath20 . it follows that @xmath1332 as @xmath43 . therefor , we may assume that there is a sequence of smooth embedded curves @xmath1333 such that @xmath1334 as @xmath1335 . thus , when @xmath1336 , the corresponding closed curves @xmath1337 in the gromov - hausdorff topology , as @xmath1338 . we now choose a sufficiently large @xmath1339 and divide our closed curve @xmath1340 into @xmath1341-many arcs of constant speed : @xmath1342 \to m^3_\alpha\ ] ] for @xmath1343 , where @xmath1344 . for a fixed @xmath8 , we let @xmath1345)\ ] ] and @xmath1346 . we will show that there is an @xmath1347 such that @xmath1348 is homeomorphic to the unit @xmath0-ball @xmath950 . our theorem will follow from . it remains to establish . we will show that each of @xmath1349 is homeomorphic to @xmath950 . for @xmath1350 , we first observe that @xmath860 is convex . thus , the boundary arc @xmath1351 ) \subset \partial x^2 $ ] is a perelman - sharafutdinov semi - gradient curve of the distance function @xmath1352 . we already choose @xmath1353 sufficiently small and @xmath1341 sufficiently large so that @xmath1354 has no critical point on @xmath1355 $ ] . by our construction , we have @xmath1356 , as @xmath43 . it follows from that the distance function has no critical points on @xmath1357 . using perelman s fibration theorem , we obtain a fibration @xmath1358 it follows that @xmath1359 $ ] . we will first use and its proof to show that @xmath1360 . let @xmath1361 be given by and @xmath1362 . by the proof of , the map @xmath1363 is regular on @xmath1364 $ ] . there is a circle fibration @xmath1365 . this proves that @xmath1366 . it follows that @xmath1367 - b_{m^3_\alpha}(q_{\alpha , i } , \varepsilon_i ) - b_{m^3_\alpha}(q_{\alpha , i+1 } , \varepsilon_i ) \ } $ ] is homeomorphic to a cylinder @xmath1368 . using two points suspension of the cylinder @xmath1369 , we can further show that @xmath1370 \sim s^2\ ] ] it remains to show that @xmath1371 for sufficiently small @xmath1372 . suppose contrary , we argue as follows . using , we see that @xmath594 has no critical points in @xmath1373 $ ] . let @xmath1033 be the largest critical value of @xmath1374 in @xmath1375 . by our assumption , @xmath1376 as @xmath43 . we now consider a sequence of re - scaled spaces @xmath1377 . its sub - sequence converges to a limiting space @xmath1378 . recall that @xmath1379 . for the same reason as in the proof of , we can show that @xmath1380 and that @xmath47 has no boundary . let @xmath1381 be the soul of @xmath47 . there are three possibilities . \(1 ) if the soul @xmath1381 is a point , then by we obtain that @xmath1382 . it follows that @xmath1383 by perelman s stability theorem , we are done . \(2 ) if the soul @xmath1381 is a circle , then @xmath47 ( or its double cover ) is isometric to @xmath1384 , where @xmath1385 is homeomorphic to @xmath237 . let @xmath1386 be the limit curve in the re - scaled limit space @xmath47 . since @xmath47 ( or its double cover ) is isometric to @xmath1387 , we have @xmath1388))$ ] is homeomorphic to @xmath1389 . by perelman s stability theorem , we would have @xmath1390\sim \partial [ s^1\times d^2 ] \sim t^2 $ ] , which contradicts to the assertion @xmath1391\sim s^2 $ ] . \(3 ) if the soul @xmath1381 of @xmath47 has dimension @xmath28 , then it follows from that the infinity of @xmath47 would have at most two points . however , since @xmath1392 , the infinity of @xmath47 has an arc , a contradiction . this completes the proof of @xmath1383 for sufficiently large @xmath24 . with extra efforts , we can also show that @xmath1393 for sufficiently large @xmath24 . hence , @xmath1394 \sim \cup_i d^3_i \sim [ d^2 \times s^1]$ ] . this completes our proof . since all our discussions in this sub - section are semi - local , we may have the following setup : @xmath1395 in the pointed gromov - hausdorff distance , and @xmath1396 $ ] is an interval , @xmath1397 is an endpoint of @xmath1398 . we will study the topology of @xmath1399 for a given small @xmath547 . we begin with four examples to illustrate how smooth riemannian @xmath0-manifolds @xmath20 collapse to an interval @xmath869 $ ] with curvature bounded from below . collapsing manifolds in these manifolds are homeomorphic to one of the following : @xmath1400 , s^1 \times d^2 , k^2 \ltimes [ 0,\frac 12]\}$ ] , where @xmath1401 and @xmath884 is the klein bottle . [ ex5.5 ] ( @xmath1402 is homeomorphic to @xmath950 ) . for each @xmath1403 , we consider a convex hypersurface in @xmath1404 as follows . we glue a lower half of the @xmath0-sphere @xmath1405 to a finite cylinder @xmath1406 = \{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb r^4 |\varepsilon \le x_4 \le 1 , x_1 ^ 2 + x_2 ^ 2 + x_3 ^ 2 = \varepsilon^2 \}.\ ] ] our @xmath0-manifold @xmath1407 \big)$ ] collapse to the unit interval as @xmath1240 . for other cases , we consider the following example . [ ex5.6 ] ( @xmath1402 homeomorphic to @xmath1408 ) . let us glue a lower half of @xmath28-sphere @xmath1409 to a finite cylinder @xmath1410 $ ] . the resulting disk @xmath1411 \big)\ ] ] is converge to unit interval , as @xmath1240 . we could choose @xmath1412 . it is clear that @xmath1413 $ ] as @xmath1240 . we now would like to consider the remaining cases . of course , two un - oriented surfaces @xmath1414 and @xmath1415 would converge to a point , as @xmath1240 . however , the _ twisted _ @xmath1398-bundle over @xmath1416 ( or @xmath884 ) is homeomorphic to an oriented manifold @xmath1417 = [ \mathbb { rp}^3 -d^3]$ ] ( or @xmath1418 = m$]@xmath1419 ) , where @xmath95 is the mbius band . [ ex5.7](@xmath1402 homeomorphic to @xmath1420 $ ] or @xmath95@xmath1421 ) . let us first consider round sphere @xmath1422 . there is an orientation preserving involution @xmath1423 $ ] given by @xmath1424 . suppose that @xmath1425 is the subgroup generated by @xmath1426 . thus , the quotient of @xmath1427 $ ] is an orientable manifold @xmath1420 = [ \mathbb { rp}^3 -d^3]$ ] . similarly , we can consider the case of @xmath1428 \to [ 0 , \frac 12]$ ] , where @xmath1429 is a klein bottle . shioya and yamaguchi showed that the above examples exhausted all cases up to homeomorphisms . [ thm5.8 ] suppose that @xmath1430 with curvature @xmath148 and @xmath1396 $ ] . then @xmath20 is homeomorphic to a gluing of @xmath1431 and @xmath1432 to @xmath1433 , where @xmath1431 and @xmath1432 are homeomorphic to one of @xmath1434 , s^1 \times d^2 , k^2 \ltimes [ 0 , \frac 12]\}$ ] and @xmath895 is a quotient of @xmath527 or @xmath117 . for the proof of , we need to establish two preliminary results ( see - [ thm5.10 ] below ) . let us consider possible exceptional orbits in the seifert fibration @xmath1435 for sufficiently large @xmath24 . we emphasize that the topological structure of @xmath20 depends on the number of extremal points ( or called essential singularities ) of @xmath1436 in this case . moreover , the topological structure of @xmath20 also depends on the type of essential singularity of @xmath1436 , when we glue a pair of solid tori together , ( see below ) . therefore , we need the following theorem with a new proof . [ thm5.9 ] let @xmath1436 be a connected , non - compact and complete surface with non - negative curvature and with possible boundary . then the following is true . 1 . if @xmath1436 has no boundary , then @xmath1436 has at most two extremal points ( or essential singularities ) . when @xmath1436 has exactly two extremal points , @xmath1436 is isometric to the double @xmath1437 \times [ 0 , \infty))$ ] of the flat half strip . if @xmath1436 has non - empty boundary @xmath1438 , then @xmath1436 has at most one interior essential singularity . for ( i ) , we will use the multi - step perelman - sharafutdinov semi - flows to carry out the proof . for the assertion ( i ) , we consider the cheeger - gromoll type busemann function @xmath1439.\ ] ] in , we already showed that @xmath1440)$ ] is compact for any finite @xmath725 . let @xmath1441 . then the level set @xmath1442 has dimension at most @xmath40 . recall that @xmath738 is convex by the soul theory . thus , @xmath738 is either a point or isometric to a length - minimizing geodesic segment : @xmath1443 \to y^2 $ ] . _ case a. _ if @xmath1444 is a point soul of @xmath1436 , by an observation of grove @xcite we see that the distance function @xmath1445 has no critical point in @xmath1446 $ ] . it follows that @xmath1447 is only possible extremal point of @xmath1436 . _ case b. _ if @xmath1448)$ ] , then for @xmath1449 petrunin @xcite ] showed that @xmath1450 . hence , two endpoints @xmath1451 and @xmath1452 are only two possible extremal points of @xmath1436 . suppose that @xmath1436 has exactly two extremal points @xmath1451 and @xmath1452 . we choose two geodesic rays @xmath1453 and @xmath1454 from two extremal points @xmath1451 and @xmath1452 respectively . the broken geodesic @xmath1455 divides our space @xmath1436 into two connected components : @xmath1456 = \omega_- \cup \omega_+.\ ] ] we now consider the distance function @xmath1457 let us consider the perelman - sharafutdinov semi - gradient flow @xmath1458 for busemann function @xmath1459 . recall that the semi - flow is distance non - increasing , since the curvature is non - negative . hence , we see that @xmath1460 is a non - increasing function of @xmath1461 $ ] . it follows that @xmath1462 for @xmath1463 . on other hand , we could use multi - step geodesic triangle comparison theorem as in @xcite to verify that @xmath1464 with equality holds if and only if four points @xmath1465 span a flat rectangle in @xmath1466 . therefore , @xmath1467 is isometric to @xmath869 \times [ 0 , \infty)$ ] . it follows that @xmath1436 is isometric to the double @xmath1437 \times [ 0 , \infty))$ ] . the second assertion ( ii ) follows from ( i ) by using @xmath1468 . for compact surfaces @xmath204 of non - negative curvature , we use an observation of perelman ( @xcite page 31 ) together with multi - step perelman - sharafutdinov semi - flows , in order to estimate the number of extremal points in @xmath204 . [ thm5.10 ] ( 1 ) suppose that @xmath204 is a compact and oriented surface with non - negative curvature and with non - empty boundary @xmath1469 . then @xmath204 has at most two interior extremal points . when @xmath204 has two interior extremal points , then @xmath204 is isometric to a gluing of two copies of flat rectangle along their three corresponding sides . \(2 ) suppose that @xmath204 is a closed and oriented surface with non - negative curvature . then @xmath204 has at most four extremal points . \(1 ) we consider the double @xmath866 of @xmath204 . if @xmath867 is not simply - connected and if it is an oriented surface with non - negative curvature , then we already showed that @xmath867 is a flat torus . we may assume that @xmath204 is homeomorphic to a disk : @xmath1470 and @xmath1233 . let @xmath1471 . perelman @xcite already showed that @xmath1472 remains to be convex , ( see also @xcite ) . if @xmath1473 , then @xmath1474 is either a geodesic segment or a single point set . thus , @xmath204 has at most two interior extremal points . the rest of the proof is the same as that of with minor modifications . \(2 ) suppose that @xmath204 has 2 distinct extremal points @xmath373 and @xmath367 . let @xmath1475 be a geodesic segment connecting @xmath373 and @xmath367 . we need to show for the function @xmath1476 is concave for all @xmath1477 . clearly for @xmath1477 there exists @xmath1478 such that @xmath1479 . there are two possibilities : a. @xmath1480 is in the interior of @xmath1475 , then proof of the concavity of @xmath264 is exactly same as the one of theorem 6.1 in @xcite ( see also @xcite p156 and @xcite ) . b. @xmath1480 is one of the endpoints , say @xmath373 , then by first variational formula we have @xmath1481 where @xmath685 denotes the set of directions of geodesics from @xmath373 to @xmath58 in @xmath539 . on the other hand , by our assumption @xmath373 is an interior extremal point so @xmath1482 combine and we have @xmath1483 the rest of the proof is same as the one of theorem 6.1 in @xcite ( or @xcite , @xcite ) . + hence , we have shown that @xmath264 is concave function on @xmath1484 . let @xmath267 be the maximum set . then @xmath1485 is either a geodesic segment or one point . thus , @xmath1486 contains at most 2 extremal points by our proof of ( 1 ) . therefore , the number of total extremal points is at most 4 . the case of exactly 4 extremal points on a topological @xmath28-sphere can be illustrated in . let us now complete the proof of . * proof of :* the proof is due to shioya - yamaguchi @xcite . our new contribution is the simplified proof of , which will be used in the study of subcases ( 1.b ) and ( 2.b ) below . for the convenience of readers , we provide a detailed argument here . recall that in , we already constructed a fibration @xmath1487 with shrinking fiber @xmath170 is homeomorphic to either @xmath117 or @xmath527 for sufficiently large @xmath24 . when @xmath1488 , since the fibers @xmath1489 are shrinking , we may assume that @xmath1490 for sufficiently large @xmath24 , where @xmath1491 $ ] and @xmath1492 . if there exists an @xmath1361 such that @xmath1493 has no critical points on the punctured ball @xmath1494 $ ] for sufficiently large @xmath24 , then , using the proof of , we can show that @xmath1495 for sufficiently large @xmath24 . thus , conclusion of holds for this case . otherwise , there exists a subsequence @xmath1496 such that @xmath1497 has a critical point @xmath1498 with @xmath1499 . it follows that @xmath1500 . using a similar argument as in the proof of , we can show that the limiting space @xmath1501 has non - negative curvature and @xmath1502 , where @xmath1503 is the image of @xmath1504 under the re - scaling . there are two cases described in above . @xmath1505 . if @xmath1506 is homeomorphic to @xmath1507 , then no further verification is needed . otherwise , we may assume @xmath1508 and @xmath1509 . under these two assumptions , we would like to verify that @xmath1506 is homeomorphic to @xmath1510 . there are two sub - cases : _ subcase 1.a . _ if @xmath1511 , then @xmath899 is non - negatively curved , open and complete . let @xmath937 be a soul of @xmath899 . using perelman s stability theorem , we claim that the soul @xmath937 can not be @xmath496 . otherwise the boundary of @xmath1506 would be @xmath527 , a contradiction . for the same reason , the soul @xmath937 can not be @xmath527 or @xmath884 . otherwise the boundary @xmath1512 would be homeomorphic to @xmath527 , contradicts to our assumption . because the boundary of @xmath1506 only consists of one component , the soul of @xmath937 of @xmath792 can not be @xmath117 ; otherwise we would have @xmath1513 and @xmath1514 , a contradiction . therefore , we have demonstrated that the soul @xmath937 must be either a point or @xmath1416 . it follows that either @xmath1515 by soul theorem . hence , we conclude that @xmath1516 for sufficiently large @xmath1517 . using perelman s stability theorem , we conclude that @xmath1518 is homeomorphic to @xmath1519 for this sub - case . _ subcase 1.b . _ if @xmath1520 , i.e. @xmath792 is a surface with possibly non - empty boundary . first we claim @xmath1521 . for any fixed @xmath1522 , we have @xmath1523 by our assumption that the regular fiber is @xmath117 . suppose contrary , @xmath1524 . we would have @xmath1525 ( or a mbius band ) by , because @xmath1436 has one end . thus , for sufficiently large @xmath1517 we have @xmath1526 . applying fibration theorem for the collapsing to the surface case , we would further have @xmath1527 which contradicts to our boundary condition . hence , @xmath792 has non - empty boundary : @xmath1528 . since @xmath1436 has one end and @xmath1528 , by we know that @xmath1436 is homeomorphic to either upper - half plane @xmath1529 or isometric a half cylinder . by previous argument , @xmath1436 can not be isometric to a half cylinder . hence , we have @xmath1530 and that @xmath1531 is a non - compact set . we further observe that if @xmath1532 is a boundary point of @xmath1436 , then , by , we would have @xmath1533 \cong d^3\ ] ] where @xmath1319 is closed half disk , a contradiction . thus we may assume that @xmath1532 is an interior point of @xmath1436 . let us consider the seifert fiber projection @xmath1534 for some large @xmath1517 . if @xmath1535 contains no interior extremal point , then by the proof of one would have @xmath1536 , a contradiction . therefore , there exists an extremal point inside @xmath1537 . without loss of generality , we may assume this extremal point is @xmath320 . by and the fact that @xmath1538 , we observe that @xmath1436 has at most one interior extremal point . in our case , @xmath1436 is isometric to the following flat surface with a singularity . let @xmath1539 \times [ 0 , \infty)$ ] be a half flat strip and @xmath1540 . our singular flat surface @xmath1436 is isometric to a gluing two copies of @xmath682 along the curve @xmath1541 . the picture of @xmath1542 for large @xmath1517 will look like , where the bold line denotes @xmath1543 . in @xmath1436 for large @xmath1517,title="fig : " ] + by our assumption , we have @xmath1544 . thus we can glue a @xmath0-ball @xmath950 to @xmath1545 along @xmath1546 to get a new closed @xmath0-manifold @xmath1547 . recall that we have a ( possibly singular ) fibration @xmath1548 by , we have @xmath1549 with some efforts , we can further show that @xmath1550 \cong s^1 \times d^2 . $ ] the exceptional orbit @xmath1551 is corresponding to the case of @xmath1552 in . finally we conclude that @xmath1553 is homeomorphic to real projective @xmath0-space : @xmath1554 therefore , by our construction , we have @xmath1555 \cong [ \mathbb{rp}^3\setminus d^3 ] \cong \mathbb{rp}^2\ltimes i.\ ] ] this completes the first part of our proof for the case of @xmath1556 . if @xmath1557 , our discussion will be similar to the previous case with some modifications . in fact , there are still two sub - cases : _ subcase 2.a . _ if @xmath1511 , then the @xmath792 has a soul @xmath937 . we assert that @xmath937 can not be a point nor @xmath117 nor @xmath1416 because the boundary @xmath1558 . in addition , we observe that @xmath937 can not be @xmath527 since the boundary of @xmath1506 consists only one component . therefore , there are two remaining cases : @xmath937 is homeomorphic to either @xmath496 or klein bottle @xmath884 . if the soul is @xmath496 then @xmath1559 . similarly , if soul is @xmath884 , then @xmath1560 . _ subcase 2.b . _ @xmath1520 . it is clear that our limiting space @xmath1436 is non - compact . if @xmath1524 , we proceed as follows . ( i ) when the soul of @xmath1436 is @xmath496 , by the connectedness of @xmath1561 we know that @xmath792 is a open mbius band . therefore , @xmath1506 is homeomorphic to product of mbius band and @xmath496 , i.e. a twist @xmath1398-bundle over @xmath884 . ( ii ) when the soul of @xmath1436 is a single point , @xmath1562 , which is non - compact . by , we see that the number @xmath193 of extremal points in @xmath1436 is at most @xmath28 . recall that there is ( possibly singular ) fibration : @xmath1563 . if @xmath1564 , we have @xmath1565 . if @xmath1566 , by , we can further show that @xmath1560 . let us now handle the remaining subcase when @xmath1538 is not empty . by a similar argument as in subcase 1.b above , we conclude that @xmath1506 is homeomorphic to either @xmath1567 or @xmath950 , which contradicts to our assumption @xmath1568 . this completes the proof of for all cases . in this section , we complete the proof of perelman s collapsing theorem for @xmath0-manifolds ( theorem 0.1 ) . in previous five sections , we made progress in decomposing each collapsing @xmath0-manifold @xmath25 into several parts : @xmath1569 where @xmath1111 is a union of closed smooth @xmath0-manifolds of non - negative sectional curvature , @xmath1116 is a union of fibrations over @xmath40-dimensional spaces with spherical or toral fibers and @xmath1137 admits locally defined almost free circle actions . extra cares are needed to specify the definition of @xmath1570 for each @xmath24 . for example , we need to choose specific parameters for collapsing @xmath0-manifolds @xmath1571 so that the decomposition in becomes well - defined . the choices of parameters can be made in a similar way as in of @xcite . [ thm6.0 ] suppose that @xmath1572 be as in and that @xmath20 has no connected components which admit metrics of non - negative curvature . then there are two small constants @xmath1573 such that @xmath1574)\le \varepsilon_1 \}\ ] ] and @xmath1575 ^ 3 \right\}\ ] ] which are described as in above and @xmath1576 satisfy inequality below . if the inequality holds , then the unit metric ball is very thin and can be covered by at most @xmath1590 many small metric balls @xmath1591 . by bishop volume comparison theorem , we have a volume estimate : @xmath1592 \le \frac{4\ell}{\varepsilon_1 } c_0 [ \sinh ( 2\varepsilon_1)]^3 \le c_0^ * \ell \varepsilon_1 ^ 2\ ] ] in this subcase , one can choose constant @xmath1596 such that if a metric disk in @xmath204 with radius @xmath547 satisfies @xmath1597 and area @xmath1598 then by comparison theorems one can prove @xmath1599)\le \varepsilon_1/3\ ] ] for some @xmath1600 . it follows that @xmath1601 ) & \le { { \rm d}}_{gh}(b_{(m^3_\alpha , \rho^{-2}_\alpha g_{ij}^\alpha)}(x_\alpha , 1),b_{x^2}(x_\infty , 1 ) ) \\ & \quad+ { { \rm d}}_{gh}(b_{x^2}(x_\infty , 1 ) , [ 0,\ell])\\ & \le\frac{\varepsilon_1}{3}+\frac{\varepsilon_1}{3}<\varepsilon_1 \end{split}\ ] ] in this subcase , we may assume that the length of metric circle @xmath1041 is 100 times greater than @xmath1606 for @xmath1607 $ ] . hence , the length of collapsing fiber @xmath1608 for @xmath1609 . thus , the metric spheres @xmath1610 collapse in only one direction . because @xmath1611 is an almost metric submersion due to perelman s semi - flow convergence theorem , ( cf . above ) , volumes of metric balls collapse at an order @xmath1612 : @xmath1613 \le c_1 \varepsilon_1 r^3\ ] ] for @xmath1607 $ ] . moreover , the free homotopy class of collapsing fibers is _ unique _ in the annular region @xmath1614 . by the discussion above , we have the following decomposition of the manifold @xmath20 for @xmath24 large . more precisely , for sufficiently large @xmath24 , @xmath20 has decomposition @xmath1615 where @xmath1133 contains collapsing parts near @xmath1228 and @xmath860 described in above . this completes the proof . [ prop6.1 ] let @xmath1618 and @xmath1619 be as in theorem 0.1. suppose that none of connected components of @xmath25 admits a smooth riemannian metric of non - negative sectional curvature . then , by changing @xmath1620 by a factor @xmath1621 independent of @xmath24 , we can choose @xmath1622 so that @xmath1623 and @xmath1624 for all @xmath1625 let us choose @xmath1626 for each connected component @xmath1627 of @xmath25 , the riemannian sectional curvature of @xmath1627 can not be everywhere non - negative . thus , for each @xmath1628 , there is a maximum @xmath1629 such that sectional curvature of @xmath1630 on @xmath1631 is greater than or equal to @xmath1632 . consequently , curvature of @xmath1633 on @xmath1634 is @xmath148 . by bishop - gromov relative comparison theorem and our assumption @xmath1635 , we have @xmath1636 \le r^3\frac{v_{\rm hyp}(1)}{v_{\rm hyp}(\frac{\rho_\alpha}{r } ) } { { \rm vol}}(b_{\frac{1}{r^2}g_{\alpha}}(x,\frac{\rho_\alpha}{r } ) ) \\ & = \frac{v_{\rm hyp}(1)}{v_{\rm hyp}(\frac{\rho_\alpha}{r } ) } { { \rm vol}}(b_{g_{\alpha}}(x , \rho_\alpha ) ) \le \frac{v_{\rm hyp}(1)}{v_{\rm hyp}(\frac{\rho_\alpha}{r } ) } w_{\alpha}\rho_{\alpha}^3 \\ & = \frac{v_{\rm hyp}(1)}{v_{\rm hyp}(\frac{\rho_\alpha}{r } ) } ( \frac{\rho_{\alpha}}{r})^3\frac{1}{(\frac{\rho_{\alpha}}{r})^3 } w_{\alpha}\rho_{\alpha}^3 \le c_0 v_{\rm hyp}(1 ) w_{\alpha}r^3 \end{split}\ ] ] where @xmath1637 is the volume of the ball @xmath1638 of radius @xmath602 in @xmath0-dimensional hyperbolic space with constant negative curvature @xmath1588 . let @xmath1639 be a constant number independent of @xmath24 , and @xmath1640 be the sectional curvature of the metric @xmath119 . we now replace @xmath1641 by @xmath1642 our new choice @xmath1643 clearly satisfies our next goal in this section is to show that we can perturb our decomposition above along their boundaries so that the new decomposition admits an f - structure in the sense of cheeger - gromov , and hence @xmath25 is a graph manifold for sufficiently large @xmath24 . let us begin with a special case . in this subsection , we prove theorem 0.1 for a special case when perelman s fibrations are assumed to be circle fibrations or toral fibrations . in next sub - section , we reduce the general case to the special case , ( i.e. the case when no spherical fibration occurred ) . as we pointed out earlier , for the proof of it is sufficient to verify that @xmath25 admits an f - structure of positive rank in the sense of cheeger - gromov for sufficiently large @xmath24 , ( cf . @xcite , @xcite , @xcite ) . recall that an f - structure on a @xmath0-manifold @xmath26 is a collection of charts @xmath1647 such that @xmath1060 acts on a finite normal cover @xmath1648 of @xmath1649 and the @xmath1060-action on @xmath1648 commutes with deck transformation on @xmath1648 . moreover the tori - group actions satisfy a compatibility condition on any possible overlaps . [ def6.2 ] @xmath1650 + let @xmath1647 be a collection of charts as above . if , for any two charts @xmath1651 and @xmath1652 with non - empty intersection @xmath1653 and with @xmath1654 , the @xmath1060 actions commutes with the @xmath1655-actions on a finite normal cover of @xmath1656 after re - parametrization if needed , then the collection @xmath1657 is said to satisfy cheeger - gromov s compatibility condition . for the purpose of this paper , since our manifolds under consideration are @xmath0-dimension , the choice of free tori @xmath1060 actions must be either circle action or @xmath28-dimensional torus action . thus we only have to consider following three possibilities : a. both overlapping charts @xmath1658 and @xmath1659 admit almost free circle actions ; b. both overlapping charts @xmath1660 and @xmath1661 admit almost free torus actions ; c. there are a circle action @xmath1662 and a torus - action @xmath1663 with non - empty intersection @xmath1664 . [ prop6.3 ] let @xmath1665 and @xmath1666 be two overlapping open subsets in @xmath1667 with toral fibers @xmath1668 described in . suppose that @xmath1669 . then we can modify charts @xmath1670 so that the perturbed torus - actions on modified charts satisfy the cheeger - gromov s compatibility condition . without loss of generality , we may assume that @xmath1671 and @xmath1672 with non - empty intersection @xmath1673 , where @xmath1674 . let @xmath1675 be a partition of unity of @xmath1676 $ ] corresponding to the open cover @xmath1677 . after choosing @xmath1678 sign and @xmath1679 carefully , we may assume that @xmath1680 will not have any critical point @xmath1681 $ ] . we also require that @xmath1682 is a _ regular _ function in the sense of perelman ( i.e. @xmath1683 is a composition of distance function given by -[def1.10 ] ) . thus , we can lift the admissible function @xmath1683 to a function @xmath1684 such that @xmath1685 for @xmath1686 $ ] , @xmath1687 on @xmath1688 $ ] and @xmath1689 on @xmath1690 $ ] , where @xmath1691 is a perturbation of @xmath1692 and @xmath1693 is a perturbation of @xmath1694 . thus there is a new perturbed torus fibration . @xmath1695 this completes the proof . [ prop6.4 ] let @xmath1665 and @xmath1666 be two open sets contained in @xmath1696 corresponding to a decomposition of @xmath20 described in . suppose that two charts @xmath1697 have non - empty overlap @xmath1698 . then the union @xmath1699 admits a global circle action after some modifications when needed . we may assume that , in the limiting processes of @xmath1700 and @xmath1701 , both limiting surfaces @xmath1702 and @xmath1703 are fat ( having relatively large area growth ) . for the remaining cases when either @xmath1702 and @xmath1703 are relatively thin , by the proof of we can view either @xmath1704 or @xmath1705 is a portion of @xmath1706 instead . these remaining situations can be handled by either above or below respectively . since both limiting surfaces @xmath1702 and @xmath1703 are very _ fat _ , the lengths of metric circles @xmath1708 is at least 100 times larger than @xmath1606 , which is great than the length of collapsing fibers , where @xmath1709 $ ] . hence , on the overlapping region @xmath1710 , two collapsing fibres are freely homotopic each other in the shell - type region @xmath1711 . [ prop6.5 ] let @xmath1728 and @xmath1729 , where @xmath1730 , @xmath1731 is a decomposition of @xmath25 as in -[section5 ] . suppose that @xmath1732 and @xmath1733 have an interface @xmath1734 . then , after a perturbation if need , the circle orbits are contained in torus orbits on the overlap . let @xmath1735 be the regular function which induces the chart @xmath1736 . suppose that @xmath1737 is the corresponding regular map . since any component of a regular map must be regular , after necessary modifications , we may assume that @xmath1738 is equal to @xmath1739 in the modified regular map @xmath1740 on the overlap @xmath1741 . it follows that the modified @xmath496-orbits are contained in the newly perturbed @xmath527-orbits . thus , the modified charts @xmath1742 and @xmath1743 satisfy the cheeger - gromov s compatibility condition . [ thm6.6 ] suppose that @xmath1744 satisfies all conditions stated in . suppose that all perelman fibrations are either ( possible singular ) circle fibrations or toral fibrations . then @xmath20 must admits a f - structure of positive rank in the sense of cheeger - gromov for sufficiently large @xmath24 . consequently , @xmath20 is a graph manifold for sufficiently large @xmath24 . it remains to verify that our collection @xmath1749 satisfies the cheeger - gromov compatibility condition , after some modifications . since exceptional circle orbits with non - zero euler number are isolated , we may assume that on any possible overlap @xmath1750 there is no exceptional circular orbits . applying -[prop6.5 ] when needed , we can perturb our charts so that the modified collection of charts @xmath1751 satisfy the cheeger - gromov compatibility condition . it follows that @xmath20 admits an f - structure @xmath1752 of positive rank . therefore , @xmath20 is a graph manifold for sufficiently large @xmath24 . [ prop6.7 ] let @xmath1744 be a sequence of riemannian @xmath0-manifolds as in . if there is a perelman fibration @xmath1754 with spherical fiber @xmath1755 , then @xmath1667 must be contained in the interior of @xmath20 when @xmath20 has non - empty boundary @xmath1756 . according to condition ( 2 ) of , for each boundary component @xmath1757 , there is a topologically trivial collar @xmath1758 of length one such that @xmath1758 is diffeomorphic to @xmath1759 . thus , we have @xmath1760 \(3 ) if @xmath1765 is a finite cylinder as in ( 2.b ) , then it must be contained in a chain @xmath1769 of finite solid cylinders such that their union @xmath1770 is homeomorphic to a solid torus @xmath1107 . the first two assertions are trivial . it remains to verify the third assertion . by our construction , if @xmath1765 is a finite cylinder homeomorphic to @xmath1132 , then @xmath1765 meets @xmath1116 exactly in @xmath1134 . moreover , such a finite solid cylinder @xmath1765 is contained in a chain @xmath1771 of solid cylinders . it is easy to see that the union @xmath1772 is homotopic to its core @xmath496 . i.e. @xmath1773 is homeomorphic to a solid torus @xmath1107 . by our discussion above , for sufficiently large @xmath24 , our @xmath0-manifold @xmath20 admits a collection of ( possibly singular ) perelman fibration . thus @xmath20 has a decomposition @xmath1776 for each chart in @xmath1777 , it admits a seifert fibration . however , remaining charts could be homeomorphic to @xmath1118 , @xmath1778 $ ] or a solid cylinder @xmath1132 . it follows from ( 3 ) that @xmath20 has a more refined decomposition @xmath1779 such that a. either a finite normal cover @xmath1758 of @xmath1765 admits a free @xmath1655-action with @xmath1780 ; b. or @xmath1765 is homeomorphic to a solid torus @xmath1781 , which is obtained by a chain of solid cylinders . it remains to verify that our collection of charts @xmath1784 satisfy cheeger - gromov compatibility condition . by observations on exceptional orbits and cores of solid cylinders , we may assume that on possibly overlap @xmath1785 there is no exceptional orbits nor cores of solid cylinders . we require that @xmath1137 meets @xmath117-factors as annular type @xmath584 . thus , if @xmath1786 , we can introduce @xmath527-actions on @xmath1787 which are compatible with @xmath496-actions on @xmath584 . after modifying our charts as in proofs of -[prop6.5 ] , we can obtain a new collection of charts @xmath1788 satisfying the cheeger - gromov s compatibility condition . therefore , @xmath20 admits an f - structure @xmath1789 of positive rank . it follows that @xmath20 is a @xmath0-dimensional graph manifold , ( cf . @xcite ) . * acknowledgement : * both authors are grateful to professor karsten grove for teaching us the modern version of critical point theory for distance functions . professor takashi shioya carefully proofread every sub - step of our simple proof in the entire paper , pointing out several inaccurate statements in an earlier version . he also generously provided us a corrected proof of . we are very much indebted to professor xiaochun rong for supplying and its proof . hao fang and christina sormani made useful comments on an earlier version . finally , we also appreciate professor brian smyth s generous help the exposition in our paper . we thank the referee for his ( or her ) suggestions , which led many improvements . yu burago ; 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we will simplify earlier proofs of perelman s collapsing theorem for @xmath0-manifolds given by shioya - yamaguchi @xcite-@xcite and morgan - tian @xcite . a version of perelman s collapsing theorem states : _ let @xmath1 be a sequence of compact riemannian @xmath0-manifolds with curvature bounded from below by @xmath2 and @xmath3 . suppose that all unit metric balls in @xmath4 have very small volume at most @xmath5 as @xmath6 and suppose that either @xmath4 is closed or has possibly convex incompressible toral boundary . then @xmath7 must be a graph - manifold for sufficiently large @xmath8"_. this result can be viewed as an extension of the implicit function theorem . among other things , we apply perelman s critical point theory ( e.g. , multiple conic singularity theory and his fibration theory ) to alexandrov spaces to construct the desired local seifert fibration structure on collapsed @xmath0-manifolds . the verification of perelman s collapsing theorem is the last step of perelman s proof of thurston s geometrization conjecture on the classification of @xmath0-manifolds . a version of geometrization conjecture asserts that any closed 3-manifold admits a _ piecewise locally homogeneous metric_. our proof of perelman s collapsing theorem is accessible to advanced graduate students and non - experts .

the phenomena which led to the formulation of the ozi rule @xcite have had a definitive impact on our understanding of strong interactions . the fact that aces " ( i.e. , quarks ) led to a simple interpretation of the properties of the @xmath5 meson was clearly a very important clue for zweig @xcite since it was natural for the @xmath5 to be pure @xmath6 and for certain @xmath5 production cross sections to be small so long as hairpin graphs " were dynamically suppressed ( see fig . [ fig : ozireaction ] ) . the dynamics behind the suppression of hairpin graphs in qcd has remained unexplained . the _ phenomenology _ of meson mixing angles in qcd - based quark models was described in the mid-1970 s in a number of papers @xcite . in such models , processes with the quark line topology of the double - hairpin graphs of fig . [ fig : ozi](b ) ( but with arbitrary time orderings ) modify the quark - antiquark transition amplitudes from the totally flavor diagonal form associated with the scattering " quark line topology of fig . [ fig : ozi](a ) , namely @xmath7 ~~,\ ] ] ( for illustrative purposes we have suppressed all space - time labels and specialized to the case of @xmath8 flavor where the matrix spans the basis @xmath9 , @xmath10 , @xmath11 , @xmath12 ) by the addition of the annihilation amplitudes @xmath13 @xmath14 ~~.\ ] ] using this framework @xcite , it was noted that the ozi mixing amplitude @xmath13 characterizing fig . [ fig : ozi](b ) was of order 10 mev in the established meson nonets , with the sole exception of the ground state pseudoscalar meson nonet , where @xmath13 is an order of magnitude larger . these observations were consistent with the pattern one would expect for heavy quarkonia where the ground state pseudoscalar double - hairpin is larger than the vector double - hairpin by one factor of @xmath15 , and excited state double - hairpins are suppressed by having vanishing wave functions at @xmath16 . however , an explanation for this pattern in light quark systems was lacking . ( 120,130)(0,0 ) ( 20,20)(0 , 1)90 ( 30,20)(0 , 1)90 ( 35,65)(1 , 0)1 ( 40,65)(1 , 0)1 ( 45,65)(1 , 0)1 ( 35,66)(1 , 0)1 ( 40,66)(1 , 0)1 ( 45,66)(1 , 0)1 ( 35,65)(0 , 1)1 ( 40,65)(0 , 1)1 ( 45,65)(0 , 1)1 ( 36,65)(0 , 1)1 ( 41,65)(0 , 1)1 ( 46,65)(0 , 1)1 ( 50,20)(0 , 1)90 ( 60,20)(0 , 1)90 ( 80,110)(10,80)[bl ] ( 80,110)(10,80)[br ] ( 74,113)@xmath17 ( 84,113)@xmath18 the large size of the ground state pseudoscalar double - hairpin is a manifestation of the @xmath3 problem " @xcite : the equations of motion of qcd , taken naively , would imply that spontaneous chiral symmetry breaking leads to _ nine _ and not just eight goldstone bosons @xcite , but the large mass of the @xmath19 seems to disqualify it from the role of the flavor singlet goldstone boson . however , the @xmath20 current is anomalous , and by the late 1970 s it was understood through the study of instantons @xcite that the anomaly leads to a nonconservation of the @xmath20 charge and thereby to the evasion of goldstone s theorem in the flavor singlet channel when chiral symmetry is spontaneously broken . the connection between the quark model picture of double - hairpins and instantons was discussed by witten @xcite , veneziano @xcite , and others , who explored more generally the conflict between instantons and the large @xmath0 expansion @xcite . ( 120,130)(0,0 ) ( 20,20)(0 , 1)90 ( 30,20)(0 , 1)90 ( 100,20)(10,80)[tl ] ( 100,20)(10,80)[tr ] ( 100,110)(10,80)[bl ] ( 100,110)(10,80)[br ] ( 94,14)@xmath21 ( 104,14)@xmath22 ( 94,113)@xmath17 ( 104,113)@xmath18 ( 22,4)(a ) ( 96,4)(b ) ( the reader familiar with instanton lore may be puzzled by the connection between the annihilation amplitudes @xmath23 of eq . ( 2 ) in the pseudoscalar mesons and instanton - induced effects in the pseudoscalar mesons . the latter effects are associated with the t hooft interaction @xcite which ( in our illustrative @xmath8 flavor case ) leads to @xmath24 and @xmath25 but _ not _ the diagonal entries in eq . ( 2 ) for @xmath26 corresponding to @xmath27 or @xmath28 transitions . recall , however , that the t hooft interaction also has @xmath29 and @xmath30 interactions , i.e. , the @xmath31-like amplitudes of eq . thus the instanton - induced interactions also admit the decomposition of eqs . ( 1 ) and ( 2 ) with @xmath32 . we will elaborate upon this point below . ) .ozi - violating amplitudes in meson nonets . these amplitudes are defined to be the contribution of the @xmath33 double - hairpin to the nonet mass matrix . [ cols="^,^,^,^,^ " , ] [ tab : latticeozi ] @xmath34 for the conversion from @xmath35 extracted from the lattice via eq . ( [ eq : fit ] ) to @xmath36 for comparison to the amplitudes quoted in table [ tab : ozi ] based on mass matrices , see ref . @xcite . even without any detailed analysis , the overall empirical ozi pattern of table [ tab : ozi ] is strikingly confirmed by the lattice results . this is easily seen from the size of the various double - hairpin correlators . in figs . [ fig : psprop]-[fig : scprop ] , we have plotted the double - hairpin correlators for the pseudoscalar , vector , axial vector , and scalar sources . all plots have the same scale for comparison . the calculations have been done for 9 different choices of quark mass . the data shown in the figures are from one of the lightest quark masses , for which the pion mass is about 300 mev ( @xmath37 ) . the results quoted in table [ tab : latticeozi ] are chirally extrapolated to the physical pion mass . the errors in figs . 6 - 9 and in table [ tab : latticeozi ] are statistical only . by far the largest and longest - range correlator is the pseudoscalar correlator of fig . [ fig : psprop ] . this is expected for two reasons : the anomaly introduces a large double - hairpin vertex responsible for the large @xmath4 mass , and , as explained above , in the quenched approximation the external @xmath38 meson propagators on either side of the double - hairpin vertex are light goldstone bosons . the results extracted from fig . [ fig : psprop ] have been reported in ref @xcite . compared to the very strong pseudoscalar double - hairpin , the vector and axial vector double - hairpins of figs . [ fig : vecprop ] and [ fig : avecprop ] are dramatically suppressed , consistent with the empirical observations described in section [ sec : background ] . since quenched lattice qcd gives reasonable values for the three - point functions associated with the meson virtual loop processes depicted in fig . [ fig : oziloop ] , these results provide not only a first derivation of the ozi rule from qcd , but also a dramatic example of the evasion in qcd of the second order paradox " described in section [ sec : background ] and a confirmation of the fact that in a complete calculation a conspiracy of the type described in section [ sec : resolution ] must occur . ( of course the results reported here include not only the meson loop contributions but also the other time orderings of the double - hairpin graphs of fig . [ fig : ozi](b ) . ) we in fact see no significant signals in the vector and axial vector channels and so report in table [ tab : latticeozi ] only one standard deviation upper bounds . as described in section [ sec : resolution ] and illustrated in fig . [ fig : closure ] , if the conspiratorial cancellation amongst meson loops is associated with @xmath1 pair creation , one would expect @xmath39 to be very large . [ fig : scprop ] shows this behaviour : after taking into account the heavier mass of the scalar meson ( about 1.3 in lattice units @xcite ) , we find that the scalar ozi amplitude is comparable in size to the pseudoscalar amplitude but of the opposite sign ( see table [ tab : latticeozi ] ) . a full amplitude @xmath40 in general has glueball , instantaneous , and loop contributions , and in a given amplitude , any or all of these components might be important . ( recall , for example , that while the loop contribution to @xmath41 is believed to be small @xcite , the full @xmath41 is large . ) that the measured @xmath39 is actually consistent in sign and magnitude with the hadronic loop contribution predicted by the quark model has interesting implications which we will discuss below . a large and negative @xmath42 has been previously reported in ref . @xcite . to obtain the quantitative results for the ozi mixing amplitudes quoted in table [ tab : latticeozi ] , we carried out an analysis similar to that used to obtain the @xmath19 mass from the pseudoscalar double - hairpin @xcite . for that case , the time - dependence of the pseudoscalar double - hairpin correlator corresponding to fig . [ fig : ozi](b ) was found to be quite well described by a `` double - pole '' form consisting of a @xmath43-independent double - hairpin insertion between a pair of meson propagators ( see also ref . @xcite ) . in momentum space @xmath44 where @xmath45 is the vacuum - to - one - particle matrix element @xmath46 and @xmath47 is the ( mass)@xmath48 version of the @xmath49 defined previously @xcite ( called @xmath50 in refs . @xcite ) . this gives a time - dependent double - hairpin correlator at zero 3-momentum of the form @xmath51 to be compared to the usual valence quark ( e.g. , isovector ) correlator corresponding to fig . [ fig : ozi](a ) @xmath52 which gives @xmath53 ( the relative sign of eqs . ( [ eq : hairpinfit ] ) and ( [ eq : valencefit ] ) is tricky ; with our convention a positive @xmath47 makes a positive contribution to the ( mass)@xmath48 of a state . ) since the values of @xmath45 and @xmath54 can be -0.5 cm = 4.2 in -0.5 cm = 4.2 in -0.5 cm = 4.2 in -0.5 cm = 4.2 in separately determined from fitting eq . ( [ eq : valencefit ] ) to the valence quark correlator , the double - hairpin vertex insertion @xmath55 can be determined by a one - parameter fit of ( [ eq : fit ] ) to the overall size of the double - hairpin correlator . a similar analysis of the scalar double - hairpin led to the result quoted in table [ tab : latticeozi ] , while for the other channels such analyses provided the quoted upper bounds for the very tiny mixing amplitudes in these channels . the most straightforward conclusions of this work are that qcd can explain the ozi rule in channels where it is observed and that it predicts that @xmath39 is large and negative @xcite . this supports the quark model s explanation of the dynamical suppression of the typical scale of hadron - loop - induced ozi violation below @xmath56 expectations , and in so doing provides further evidence for the standard @xmath1 pair creation amplitude , since this is the critical feature which produces this result . while the precise consequences are unclear , the implications for phenomenology are serious . with @xmath39 large , the lightest scalar meson nonet ( the @xmath57 states ) will be close to the @xmath58 limit . we may therefore expect an octet of scalar mesons in the 1400 mev range with the other @xmath57 states while the nearly singlet scalar state will be substantially lower in mass . thus the usual assumption of phenomenological analyses that this region will contain the unmixed nonet of @xmath57 states and the scalar glueball is incorrect . for example , this region might well contain the isoscalar state of the @xmath59 nonet . in addition , since @xmath39 is comparable to the @xmath60 splitting , there is no reason to assume that either the @xmath57 or @xmath59 singlet s properties can be related by nonet symmetry to those of its octet . the net effect is that the definitive extraction of the glueball state from the scalar meson spectrum may be quite difficult . given the importance of this task , it is certainly worthwhile to study the scalar mesons more carefully in the light of this result @xcite . on the lattice it might be possible to obtain the matrix of ozi - violating amplitudes connecting the @xmath61-like and @xmath5-like @xmath57 and @xmath59 states ; in models the low - lying scalar meson spectrum can be studied including the effects of a strong annihilation channel . perhaps most critical is to use quenched lattice calculations of the mixed propagators from quarkonia to glueballs @xcite to help resolve the scalar meson ozi violation reported here into the contributions of @xmath62 intermediate states , purely gluonic intermediate states associated with true double - hairpin " graphs , and instantaneous contributions . ultimately , quenched lattice calculations of three - point functions could directly check the predicted negative loop contributions to @xmath39 by measuring the vertex functions which are the raw ingredients " of the quark model calculation . in particular , in other than the @xmath2 channel , one should see the required magnitudes and _ opposite signs _ of the virtual @xmath63-wave decays to two @xmath64 mesons and the @xmath31-wave decays to one @xmath65 and one @xmath64 meson required to build up the near cancellation that is at the heart of the quark model mechanism . in contrast , for @xmath2 mesons these channels should have the same sign . the results described here clearly have serious implications for the spectroscopy of @xmath2 states , and define the series of investigations described above required to clarify the physics behind @xmath39 . such investigations are not only important for their impact on phenomenology , however . they are also important because our results highlight other more fundamental questions raised long ago by witten @xcite , on the apparent conflict between the instanton solution of the @xmath19 mass ( i.e. , @xmath3 ) problem and the large @xmath0 limit . the quark model mechanism for the loop contributions to @xmath39 is based on large @xmath0 . while our discussion of the @xmath1 model has focused on its prescription for the quantum numbers of the created @xmath66 pair , it is also an essential ingredient of the model that this pair creates @xmath67 and not @xmath68 mesons , i.e. , that it respects the ozi rule at tree level . the physical picture behind this feature of the model is that pair creation ( at order @xmath56 ) occurs by the breaking of the color flux tube connecting @xmath69 and @xmath70 . more generally , as mentioned above , this limit provides the only known field - theoretic basis for the success of not only the valence quark model , but also of regge phenomenology , the narrow resonance approximation , and many of the systematics of hadronic spectra and matrix elements @xcite . in contrast , it is widely believed that the @xmath3 problem is solved through instanton contributions to the axial anomaly . however , as emphasized by witten , instantons vanish like @xmath71 and so do not appear in the large @xmath0 expansion . insofar as [ instantons play ] a significant role in the strong interactions , the large @xmath0 expansion must be bad . it is necessary to choose between the two . " @xcite note that these arguments draw an important distinction between semiclassically calculated instanton effects , which vanish like @xmath71 , and more general topological gauge fluctuations , which _ can _ contribute at order @xmath56 to @xmath72 . the real issue is not whether there are large fluctuations of @xmath73 in the qcd vacuum , but whether these fluctuations arise as local semiclassical lumps with quantized winding numbers or simply as a result of the generically large gauge fluctuations of a confining vacuum . to place this conflict in context , recall eqs . ( 1 ) and ( 2 ) . from section [ sec : lattice ] it is apparent that the amplitude for any of @xmath74 massless @xmath66 pairs to annihilate to any other pair is the same , i.e. , that @xmath26 does indeed have the form of the @xmath75 matrix shown in eq . as explained earlier , this is consistent with the t hooft instanton interaction since the scattering " amplitude @xmath31 in eq . ( 1 ) contains a contribution @xmath76 from instantons . thus to leading order in @xmath13 the decomposition of eqs . ( 1 ) and ( 2 ) is general and the analyses of ozi violation in refs . @xcite - - - including that in the pseudoscalar sector - - - are valid . it follows that from a purely phenomenological perspective it is irrelevant whether or not there is an instanton contribution to hadronic physics : a phenomenology with @xmath77 is legal " in any case , since the anomaly allows a resolution of the @xmath3 problem with @xcite or without @xcite instantons . what remains unclear is the physics behind the annihilation amplitudes . since a lattice simulation sums over all paths , it contains the instantons as tunnelling events between classical vacua , but the feynman diagrams of qcd , which represent the quantum corrections around these vacua , are incapable of representing instanton physics . thus if instantons are important in qcd , feynman diagrams would have to be supplemented by effective interactions ( like the t hooft interaction ) . as noted by witten @xcite , the foremost victim of the failure of feynman diagrams implied if instantons are important would be the large @xmath0 expansion , since it assumes that all - orders properties of the qcd feynman diagrammatic expansion are properties of qcd . the observations reported in this paper on @xmath39 add one more item to a growing and closely linked set of issues where the physics of instantons and the physics of large @xmath0 confront each other . assuming that confinement and the nambu - goldstone mechanism @xcite are properties of the all - orders feynman diagrammatic expansion of qcd , the large @xmath0 expansion provides a consistent framework embracing all strong interaction phenomena . among these phenomena are the hadron spectrum for all flavors of hadrons ( including the @xmath56-suppressed hadronic widths which seem to be critical to @xmath39 ) , the ozi rule ( now including @xmath39 ) , and the @xmath66 condensate . as witten argued long ago @xcite , given the @xmath3 anomaly and confinement , the large @xmath0 limit is also capable of explaining the @xmath19 mass at order @xmath56 _ without _ instantons . while its limited range of applicability makes it somewhat less attractive for phenomenology ( instantons offer a competing explanation only for the properties of the lightest @xmath78 hadrons),the instanton picture @xcite has received strong support from recent lattice results @xcite . measurements of the topological charge @xcite of cooled " gauge configurations show that in such circumstances this charge is quantized and localized as expected for instantons . moreover , the zero - modes of the dirac operator associated with the solution of the @xmath3 problem and the near - zero - modes associated with the @xmath66 condensate are also localized and in cooled " configurations can be associated with these same instantons . the lattice results on these and other hadronic properties are consistent with the instanton liquid model @xcite . since , as argued by witten , confinement can replace instantons as the source of the @xmath3 anomaly and since confinement can also produce a space - time localization of the origin of the @xmath19 mass and of the @xmath66 condensate , in our view the true origin of these effects remains unsettled . the results of this paper may help to resolve this situation since for @xmath39 the two competing pictures lead to mechanisms that are very distinct . flux - tube - breaking pair creation , a prototypical large @xmath0 phenomenon , led to the prediction that the hadron loop contribution to @xmath39 is large and negative as found here . moreover , as stated in the beginning of this paper , quark models , with their confined constituent quarks , naturally generate a large positive @xmath23 @xcite . in this case the loop contribution should be typically small @xcite , and the large positive quark model amplitude is associated with an instantaneous interaction . instantons , through the instantaneous t hooft interaction , would lead to a superficially similar pattern of ozi violation : a large positive @xmath23 and a large negative @xmath39 . however , the origins of the large negative @xmath39 are very different in the two cases : the instanton @xmath39 is associated with an instantaneous contribution while the quantitative similarity between the quark model prediction and our measured @xmath39 suggests that this amplitude is associated instead with the meson loop contributions . our result thus favors the large @xmath0 and not the instanton interpretation of the solution to the @xmath4 mass problem . nevertheless , while suggestive , the quark model prediction is not of sufficient quantitative accuracy for this conclusion to be reliable . fortunately , with recent advances in lattice methods and in computing power , we believe that the results we have described here can not only be improved but also understood more deeply . in particular , through the program we described of decomposing the ozi - violating amplitudes into their component parts , it should be possible to define the mechanism driving @xmath39 . we also believe it will be particularly fruitful to define and test confinement - based interpretations of the lattice results on such quantities as the topological susceptibility , the localization of zero modes , the correlation function of the topological charge operator , and the space - time association of the @xmath66 condensate with the topological charge . through such studies , the conflict between large @xmath0 and instanton physics can at last be resolved . we are grateful to stephen sharpe and thomas schaefer and to chris michael for alerting us to a serious sign error in the first version of this paper and to important references which had escaped our attention . this work was supported by doe contract de - ac05 - 84er40150 under which the southeastern universities research association ( sura ) operates the thomas jefferson national accelerator facility . the work of h.b . thacker was supported in part by the department of energy under grant de - fg02 - 97er41027 . g. zweig , cern report no . 8419 th 412 , 1964 ( unpublished ) ; reprinted in _ developments in the quark theory of hadrons _ , edited by d. b. lichtenberg and s. p. rosen ( hadronic press , massachusetts , 1980 ) . the history of the discovery of the quark model ( or aces " ) as seen by zweig is related in baryon 1980 " , proceedings of the ivth international conference on baryon resonances , ed . n. isgur ( toronto , 1980 ) , p. 439 . there is an interesting subtlety associated with this decomposition . [ fig : ozi](b ) includes time - orderings in which the pair creation occurs before the annihilation ( see fig . [ fig : oziloop ] ) corresponding to meson loop processes . however , the complete set of meson loop processes produce not only fig . [ fig : ozi](b ) but also z - graphs of fig . [ fig : ozi](a ) and graphs with closed @xmath66 loops inserted into fig . [ fig : ozi](a ) . thus a consistent treatment actually requires that @xmath31 be not just the simple valence graph shown , but also include graphs with internal @xmath66 loops @xcite . a.m. polyakov , phys . * 59b * , 82 ( 1975 ) ; nucl . phys . * b121 * , 429 ( 1977 ) ; a.a . belavin , a.m. polyakov , a. schwartz , and y. tyupkin , phys . lett . * 59b * , 85 ( 1975 ) ; c. callan , r. dashen , and d.j . gross , phys . lett . * 63b * , 334 ( 1976 ) ; r. jackiw and c. rebbi , phys . 37 * , 172 ( 1976 ) . the extraction of the ozi amplitudes from the data can be complex . if @xmath49 is small , @xmath79-like mixing is small as is mixing to excited nonets and glueballs , so @xmath49 may readily be extracted from the relation @xmath80 . note that since this extraction is in the @xmath81 sector , issues of @xmath58-breaking do not arise . also note that the relation between @xmath49 and @xmath47 defined by the analogous ( mass)@xmath48 formula @xmath82 is simply @xmath83 at the light quark mass scale . when @xmath49 is large , mixing and @xmath58 breaking can become important . in such circumstances , no simple comparison to the data can be made without further assumptions . since excited nonets will normally be 0.5 to 1.0 gev away , when @xmath49 is as large as it is in the pseudoscalars and scalars , the validity of ignoring internonet mixing is certainly questionable . if one nevertheless makes this assumption , then in the @xmath58 limit one would have @xmath84 or alternatively @xmath85 where @xmath86 is the mass of the @xmath58 singlet meson , leading to the relation @xmath87 $ ] . for weak @xmath58 breaking these formulas simply become @xmath88 and @xmath89 where now @xmath90 and @xmath91 are the masses of the mainly singlet and mainly octet mesons ( e.g. , the @xmath4 and the @xmath92 , respectively ) . it is this latter formula that is commonly used in defining @xmath93 in the witten - veneziano formula @xcite and which leads to @xmath94(0.85 gev)@xmath48 . these broken @xmath58 relations lead to @xmath95 { \bf a}_{ozi}$ ] . the inaccuracy of these formulas from both internonet mixing and higher order corrections in @xmath58-breaking are not obviously small . thus , given the dynamical assumptions required to relate a large value of @xmath49 to observed masses , the empirical value of @xmath96 ( or @xmath97 ) is quite uncertain ( as is @xmath98 and @xmath99 ) and therefore only semiquantitative statements about these amplitudes can be made at this time . to convert the lattice propagator data for @xmath47 to @xmath49 in table [ tab : latticeozi ] for comparison to the @xmath49 quoted in table [ tab : ozi ] , we note that in the single flavor case considered , if we once again make the assumption of small mixing with other states , the mass and ( mass)@xmath48 formulas are @xmath100 and @xmath101 . ( note that we have identified the mass without the annihilation amplitude as being the @xmath102 mass that would be found for two or more flavors . ) it follows that @xmath103 . for small @xmath49 this relation gives @xmath104 and ( as in our extraction of these quantities from the data ) in this case our assumption of small mixing is justified . for large @xmath49 such an assumption is probably not good ; however , the lattice relation can be systematically improved by measuring additional elements of the mass and ( mass)@xmath48 matrices should such an improvement be justified by an improvement in the relevant experimental data . l. micu , nucl . phys . * b10 * , 521 ( 1969 ) ; a. le yaouanc , l. oliver , o. pene , and j .- c . raynal , phys . , 2233 ( 1973 ) ; phys . lett . * b71 * , 397 ( 1977 ) ; _ ibid . _ * b72 * , 57 ( 1977 ) ; w. roberts and b. silvestre - brac , few body syst . * 11 * , 171 ( 1992 ) ; p. geiger and e.s . swanson , phys . d * 50 * , 6855 ( 1994 ) ; r. kokoski and n. isgur , phys . d * 35 * , 907 ( 1987 ) ; fl . stancu and p. stassart , phys . d * 38 * , 233 ( 1988 ) ; * 39 * , 343 ( 1989 ) ; * 41 * , 916 ( 1990 ) ; * 42 * , 1521 ( 1990 ) ; s. capstick and w. roberts , phys . rev . d * 47 * , 1994 ( 1993 ) ; phys . d * 49 * , 4570 ( 1994 ) ; e.s . ackleh , t. barnes , and e.s . swanson , phys . d * 54 * , 6811 ( 1996 ) ; and for a recent and detailed description of the implications of the @xmath1 model for mesons , see t. barnes , f.e . close , p.r . page , and e.s . swanson , phys . d * 55 * , 4157 ( 1997 ) . unfortunately , ref . @xcite did not check the loop contribution to @xmath23 since the @xmath3 anomaly guaranteed that ozi violation in this sector would be large so that there was no second order paradox to evade . however , given the identified mechanism of the suppression of the loop diagrams , there is every reason to suppose that the loop contribution to @xmath23 is as small as in any other nonet with @xmath105 . we have determined the @xmath106 , @xmath107 , and @xmath108 masses from chirally extrapolated fits to our data on the valence ( i.e. , isovector ) propagators associated with fig . [ fig : ozi](a ) , which give @xmath109 gev , @xmath110 gev , and @xmath111 gev . the latter two masses seem to be too high . it is quite possible that they are especially sensitive to quenching since both the @xmath108 and the @xmath107 have ( or are predicted to have ) very strong s - wave couplings to nearby thresholds . ( see , for example , r. kokoski and n. isgur , phys . d * 35 * , 907 ( 1987 ) ; n. isgur , phys . d * 57 * , 4041 ( 1998 ) . ) we have found no other recent lattice measurements of these masses close to the chiral limit ( see , however , ref . @xcite ) , or of other spectroscopic properties of the p - wave mesons , and consequently hope to report on their spectroscopy in a forthcoming publication . in any event , at the level of accuracy relevant to the results reported here , these @xmath112 effects are not significant . w. lee and d. weingarten , nucl . b ( proc . suppl . ) * 63 * , 194 ( 1998 ) ; nucl . phys . b ( proc . suppl . ) * 73 * , 249 ( 1999 ) ; phys . rev . d * 61*:014015 ( 2000 ) . see also ref . @xcite on @xmath1-glueball mixing . the dual topological expansion is very closely related to the large @xmath0 expansion . chew and c. rosenzweig , nucl . phys . * b*104 , 290 ( 1976 ) ; g. veneziano , proc . @xmath113 ecole det de physique des particules , gif - sur - yvette , 1977 ( 1978 ) , vol . 2 , p. 23 .

the ozi rule is prominent in hadronic phenomena only because ozi violation is typically an order of magnitude smaller than expected from large @xmath0 arguments . with its standard @xmath1 pair creation operator for hadronic decays by flux tube breaking , the quark model respects the ozi rule at tree level and exhibits the cancellations between ozi - violating meson loop diagrams required for this dramatic suppression . however , if the quark model explanation for these cancellations is correct , then ozi violation would be expected to be large in the nonet with the same quantum numbers as the pair creation operator : the @xmath2 mesons . experiment is currently unable to identify these mesons , but we report here on a lattice qcd calculation which confirms that the ozi rule arises from qcd in the vector and axial vector mesons as observed , and finds a large violation of the rule in the scalar mesons as anticipated by the quark model . in view of this result , we make some remarks on possible connections between the @xmath1 pair creation model , scalar mesons , and the @xmath3 anomaly responsible for the large ozi violation which drives the @xmath4 mass . in particular , we note that our result favors the large @xmath0 and not the instanton interpretation of the solution to the @xmath4 mass problem . epsf 0.25 cm

blazars are radio - loud active galactic nuclei ( agn ) with relativistic jets viewed at small angles to the earth s line of sight . their broad - band spectral energy distributions ( seds ) are typically dominated by two non - thermal emission components widely believed to be due to synchrotron and inverse - compton emission from a single population of ultra - relativistic electrons accelerated within the innermost parts of the jets ( e.g. , * ? ? ? the exact particle acceleration processes at work , as well as the location and structure of the dominant energy dissipation zone in blazar sources ( hereafter ` the blazar zone ' ) , are still under debate . blazars are typically sub - divided into flat spectrum radio quasars ( fsrqs ) and bl lacertae objects ( bl lacs ) based on the equivalent widths of the emission lines in their optical spectra ( e.g. , * ? ? ? bl lacs are characterized by much weaker emission lines than fsrqs , or even featureless optical continua , that can be understood in terms of distinct accretion rates in an otherwise homogeneous population of sources ( e.g. , * ? ? ? * ) . since the launch of the _ fermi _ satellite in 2008 , high - redshift blazars have been established as very - high - energy ( vhe ; @xmath9gev ) emitters @xcite , with the highest - redshift case reported being a 126 gev photon from b2 0912 + 29 , a z = 1.521 bl lac , though the redshift is still uncertain ( see * ? ? ? although collectively detected , individually these associations are based on single photons and the chance coincidence probabilities for the vhe events to be detected around the sources are not low enough to claim discovery of a single source . detection of sub - tev vhe photons from @xmath10 blazars is in principle reconcilable with most of the recently refined ebl models @xcite . here , we report the discovery of vhe @xmath1-ray emission from the blazar pks 0426@xmath3380 , classified early on as a bl lac ( e.g. * ? ? ? * ) , and only recently recognized as a fsrq based on a new classification scheme advocated by @xcite and @xcite . these authors proposed that the dividing line between the two types of blazars corresponds to the critical blr luminosity in the eddington units , @xmath11 . in the case of pks 0426@xmath3380 , the precise characterization of the optical spectrum by very large telescope ( vlt ) observations revealed a broad [ mgii ] line with an equivalent width of 5.7 and a relatively high luminosity of @xmath12ergs@xmath13 @xcite . this leads to @xmath14 for a black hole mass @xmath15 ( see section 4 ) , and therefore the fsrq classification according to the proposed scheme @xcite . the vlt detection of the broad [ mgii ] @xmath162798 emission line mentioned above by @xcite , together with ciii ] and [ oii ] @xmath163727 , enabled the determination of a redshift , @xmath17 , for the source ( @xmath18gpc , assuming standard cosmology with @xmath19kms@xmath13mpc@xmath13 , @xmath20 , and @xmath21 ) . independently , @xcite also derived @xmath22 based on the single [ mgii ] @xmath162798 emission line ; the non - detection of a host galaxy in the _ hubble _ space telescope image @xcite is consistent with the high redshift . given all these findings , we conclude that the redshift determination for pks 0426@xmath3380 is robust . the _ fermi_-lat detection of a vhe event from near the direction of pks 0426@xmath3380 in january 2010 was previously reported @xcite . however , the probability that the vhe event originated from other sources ( including background / foreground diffuse emissions ) was relatively large , and the significance did not reach @xmath23 . in this letter , we report the _ fermi_-lat detection of a second vhe event from the directional vicinity of pks 0426@xmath3380 in january 2013 , and claim convincingly that it is now the most distant vhe emitter currently detected . cccccc + energy@xmath24 & met & r.a . ( j2000 ) & dec . ( j2000 ) & angular separation@xmath25 & gtsrcprob@xmath26 + @xmath27gev ] & ( ut ) & [ deg ] & [ deg ] & [ deg ] & probability + + + 134 & 285043901.724 & 67.182 & @xmath337.930 & 0.013 & 0.9999763 + & ( 2010 jan 13 02:51:39.724 ) & & & & + 122 & 380539944.325 & 67.194 & @xmath337.943 & 0.021 & 0.9999720 + & ( 2013 jan 22 09:32:21.325 ) & & & & + + @xmath24 the energy resolution is of the order of 10% @xcite . @xmath25 angular separation is calculated from the radio position of pks 0426@xmath3380 , r.a.=67.1684342@xmath28 and dec.=@xmath337.9387719@xmath28 ( j2000 ) @xcite . @xmath26 the probability that the event belongs to pks 0426@xmath3380 , which is calculated by using gtsrcprob . the _ fermi_-lat pass7 event and spacecraft data ( ft1 and ft2 files , respectively ) were downloaded from the lat data server at the _ fermi _ science support center webpage . we took the ft1 photon event data file spanning mission elapsed time ( met , measured in seconds from 2001 jan . 1 ) 239557417 ( 2008 august 4 15:43:36 ut ) to 389039485 ( 2013 april 30 18:31:23 ut ) and chose a 10@xmath28 radius for the region of interest ( roi ) centered at the radio position of pks 0426@xmath3380 . the event selection and data analysis were performed in a standard manner using version v9r27p1 of the _ fermi science tools_. only the clean class events from 100mev to 300gev were selected . the maximum zenith angle was set to 100@xmath28 to avoid contamination from earth limb @xmath1 rays . the good time interval was generated by applying a recommended filter expression of ( data_qual==1)&&(lat_config==1)&&abs(rock _ angle)@xmath2952 and roi - based zenith angle cut ( roicut = yes ) . we used the p7clean_v6 instrument response functions ( irfs ) . we first performed unbinned maximum likelihood ( gtlike ) analysis for the 4.7-year lat data by using an xml source model in which the spectral parameters of all the sources included in the second _ fermi_-lat catalog ( 2fgl ; * ? ? ? * ) within 10@xmath28 radius were set free , while those within an annulus from 10@xmath28 to 15@xmath28 were fixed to their 2fgl values . for pks 0426@xmath3380 we assumed a log - parabola spectral shape , as measured in the 2fgl catalog . the template files gal_2yearp7v6_v0.fits and iso_p7v6clean.txt were used to represent the galactic and isotropic diffuse emission components , respectively . to allow for potential small errors in the flux and spectrum of the galactic diffuse emission model we multiplied it by a power law in energy whose normalization and index were free during the fit . using the output xml file obtained after running gtlike , we ran the gtsrcprob tool to calculate a probability that each detected vhe event originates from the direction of pks 0426@xmath3380 . for the same output xml file , we freed only the normalizations of the surrounding 2fgl sources and the galactic and extragalactic diffuse emission components and modeled pks 0426@xmath3380 with a single power - law with both the normalization and the photon index allowed to vary . then , we ran gtlike and generated a weekly ( 7-day binned ) light curve for pks 0426@xmath3380 . finally , when calculating daily fluxes and power - law indices for pks 0426@xmath3380 , only the source normalization and power - law slope were left free , along with the normalizations of the two diffuse emission components , while all the other parameters were fixed to the values obtained with gtlike for the entire 4.7-year dataset to avoid unreasonably large errors . to construct the @xmath1-ray spectrum of the source , we first selected energy intervals chosen as 11 octaves ranging from 0.1 to 204.8gev ( namely , 0.10.2 , 0.20.4 , ... , 102.4204.8gev ) . in the xml source model , the normalizations of the surrounding sources within 10@xmath28 and of the galactic and extragalactic diffuse emission components were set free , while all the other spectral parameters were fixed to the values derived for the lat data accumulated for the selected period . pks 0426@xmath3380 was modeled using a broken power - law model with the photon indices and break energy fixed to the values that we derived for the entire dataset , but with the normalization set free . using this xml file , we ran gtlike for each energy bin and generated the spectrum of the source . in figure 1 we present the lat count map of ultraclean events ( a subset of the clean class with the highest probability of being @xmath1 rays ) with 5300gev energies around pks 0426@xmath3380 . one can clearly see that the two vhe photons , with energies 134gev and 122gev , coincide with the position of the blazar ( angular separations from the blazar of @xmath30 ) and are well inside the point spread function ( psf ) of _ fermi_-lat . a detailed summary of the vhe events , including their precise localizations and arrival times , is given in table 1 . based on the gtsrcprob results ( see table 1 ) , we calculated the null hypothesis probability that both events originate from foreground / background ( galactic / extragalactic ) diffuse emission components , or other surrounding 2fgl sources rather than pks 0426@xmath3380 , using fisher s method and obtained @xmath31 . we can therefore reject the null hypothesis at the @xmath4 confidence level ( cl ) , and claim robustly that at least one of the two detected vhe @xmath1 rays originates from pks 0426@xmath3380 , making it the most distant vhe emitter known to date . although the two vhe events are classified as ultraclean ones , we further visually inspected the tracks in the lat tracker subsystem after the point of pair conversion . the first event converted at the top layer of a tower and the long straight paths of the converted electron - positron pairs were nicely tracked . the second event converted at the third layer of the tracker and similarly showed no unusual signatures . the showers in the calorimeters for both events were also well - behaved . in conclusion , we did not find any problematic features in the case of the analyzed vhe detections . figure 2 presents the weekly ( 7-day binned ) _ fermi_-lat light curve of pks 0426@xmath3380 within the energy range @xmath32gev , along with the arrival times of @xmath1 rays with energies above 50 gev . it is clear that the two vhe events were detected during high states of the source . interestingly however , the vhe detections did not coincide with the flux maxima of the flaring epochs . complex structures of the source light curve , with multiple peaks within each flaring state , preclude us however from speculating if the vhe photons preceded or followed the flux maxima at lower photon energies . to further investigate the temporal and spectral variations within @xmath33 days of the two vhe events , in figure 3 we show the daily changes of the flux and power - law index for the source . in neither case did the @xmath32gev flux dramatically increase on the day of the vhe detection . on the other hand , a significant spectral hardening can be noticed on the day of the first vhe detection , but not on the day of the second one . figure 4 shows the @xmath1-ray spectrum of pks 0426@xmath3380 during the most energetic flare , derived from the accumulated _ fermi_-lat data between met 280000000 and 302000000 ( see figure 2 ) . no high - energy cutoff expected from the ebl - related attenuation of the @xmath1-ray continuum seems to be present up to several tens of gev , although the highest energy ( @xmath34gev ) datapoint is a 95% cl upper limit due to the limited photon statistics ( corresponding to a single net photon ) . we found that a broken power - law model of @xmath35 , @xmath36 , and @xmath37gev maximized the likelihood , and this is also drawn in figure 4 . we emphasize that the presented spectrum was derived from the _ fermi_-lat data accumulated over @xmath8 months . hence , keeping in mind the large - amplitude variability of the source in @xmath1 rays , one has to be very careful in interpreting spectral features apparent in the figure , like for example the discontinuity around 10gev ( see * ? ? ? * for other examples ) . based on the observed sed , we generated the ebl - corrected spectra of the source ( figure 4 ) using two different ebl models @xcite . the de - absorbed spectra seem to reveal an additional high - energy flat - spectrum component above several tens of gev ( see also * ? ? ? . however , the significance of this feature is marginal and , more importantly , model dependent . nonetheless , the results obtained do suggest that pks 0426@xmath3380 is a very promising target for future follow - up studies with imaging atmospheric cherenkov telescopes ( iacts ) such as h.e.s.s . ii and cta @xcite . three blazars classified as fsrqs have previously been detected in the vhe range by iacts , 3c 279 ( @xmath38 ; * ? ? ? * ) , pks 1510@xmath3089 ( @xmath39 ; * ? ? ? * ) , and 4c + 21.35 ( @xmath40 ; * ? ? ? vhe emissions from more distant objects such as kuv 00311@xmath31938 at @xmath41 ( * ? ? ? * though the redshift is still tentative ) and pks 1424 + 240 at @xmath42 @xcite have been recently detected . the observational results presented in this letter therefore establish pks 0426@xmath3380 , which is located at @xmath43 , as the most distant vhe emitter observed to date . we note that the redshift of pks 0426@xmath3380 is just around the horizon for @xmath6gev @xmath1 rays ( namely , ebl - related optical depth of the universe @xmath44 ) as recently determined by _ fermi_-lat ( * ? ? ? * and figure 1 therein ) , hence the vhe detection from the @xmath43 blazar is not unreasonable . as mentioned previously , the observed luminosity of the broad [ mgii ] line in the optical spectrum of pks 0426@xmath3380 is @xmath45ergs@xmath13 @xcite , implying a blr luminosity @xmath46ergs@xmath13 using the scaling relation @xmath47 @xcite . spectral modeling of the accretion - related continuum in the source @xcite yields @xmath48 . based on the [ mgii ] line fwhm of @xmath49kms@xmath13 , and the observed @xmath50-band magnitude of 18.6 ( assuming negligible starlight contamination ) , we derived a slightly larger value of @xmath51 , using the scaling relations from @xcite and @xcite . this corresponds to the eddington luminosity @xmath52ergs@xmath13 , the ratio @xmath14 , and the accretion rate at the level of @xmath53 ( assuming the standard bolometric correction factor @xmath54 for the @xmath55-band source luminosity @xmath56ergs@xmath13 ) . the derived high accretion rate is consistent with pks 0426@xmath3380 being a fsrq . the intense @xmath1-ray emission of fsrqs is widely thought to arise due to inverse - compton up - scattering of low - energy photons generated outside of a jet by ultra - relativistic electrons accelerated within the innermost parts of the relativistic outflows @xcite . if this blazar emission zone is located at sub - parsec distances from the central black hole , as is often anticipated in the literature , then the abundant circum - nuclear photon fields provided by the blr and/or hot dust are expected to attenuate the vhe blazar emission substantially due to the photon - photon pair production , leading to the formation of breaks and cut - offs in the @xmath1-ray continua of fsrqs ( see in this context the discussion in * ? ? ? * ; * ? ? ? the observed sharp break ( @xmath57 ) at @xmath8gev would be understood by a scenario with the blazar emission zone deep within the blr . however , we note again that the @xmath1-ray spectrum when corrected for the cosmological absorption showed a flattened shape at energies above 10 gev . this flat component , if connected to the sub - tev range , should come from another emission region outside the blr to avoid @xmath58 attenuation . care must be taken not to over - interpret such high - energy features ( flattening ) in unfolded blazar spectra constructed using _ fermi_-lat data accumulated over longer periods of time , since those may simply arise due to averaging over different activity states characterized by different spectral properties ( see the related discussion and analysis in * ? ? ? * concerning the well - known bl lac object mrk 501 ) . still , the results presented here for pks 0426@xmath3380 are in principle consistent with the emergence of an additional very high - energy flat - spectrum component during the flaring states of the source . one possibility for the production of such a component could be electron pile - up at the highest energies due to the efficient and continuous acceleration processes limited only by the radiative losses @xcite . another possibility could be an additional hadronic emission component dominating occasionally the source spectrum in the vhe range ( e.g. , * ? ? ? * ; * ? ? ? in particular , assuming that pks 0426@xmath3380 generates cosmic rays with energies above 10eev , or @xmath1 rays above 30tev , the induced intergalactic cascade emission may provide a non - negligible contribution within the vhe range of the source spectrum @xcite . the time scale of the expected delay between the production of the primary ultra - high energy cosmic rays/@xmath1-ray photons and the observed re - processed vhe signal is @xmath59days and @xmath60days for the photon and the cosmic - ray induced cascade emission at @xmath43 , respectively , where @xmath55 is the intergalactic magnetic field strength , @xmath61 is the mean free path of the pair creation , and @xmath62 is the mean free path of the bethe - heitler process @xcite . rather speculatively , if there are axion - like particles , photon and axion mixing in the intergalactic medium may in addition enhance the ebl - absorbed photon flux ( e.g. , * ? ? ? lorentz invariance violation , on the other hand , could inhibit pair production ( e.g. , * ? ? ? * ) . in summary , with _ we have detected two vhe @xmath1 rays from close to the direction of a high - redshift fsrq , pks 0426@xmath3380 at @xmath43 . both of the events were detected during high @xmath1-ray states of the source , although the specific vhe arrival times did not coincide with any particularly large @xmath32gev source flux . only in the first case was the observed lat spectrum harder than usual . the very complex @xmath1-ray variability patterns revealed by the _ fermi_-lat data for pks 0426@xmath3380 calls for sensitive follow - up studies with simultaneous vhe coverage provided by iacts such as h.e.s.s . ii or future cta . we appreciate the referee s critical reading and valuable comments . is supported by kakenhi 24840031 . work by c.c.c . at nrl is supported in part by nasa dpr s-15633-y . .s . was supported by polish nsc grant dec-2012/04/a / st9/00083 . the _ fermi_-latcollaboration acknowledges support from a number of agencies and institutes for both development and the operation of the lat as well as scientific data analysis . these include nasa and doe in the united states , cea / irfu and in2p3/cnrs in france , asi and infn in italy , mext , kek , and jaxa in japan , and the k. a. wallenberg foundation , the swedish research council and the national space board in sweden . additional support from inaf in italy and cnes in france for science analysis during the operations phase is also gratefully acknowledged . , y. , boisson , c. , cerruti , m. , & h. e. s. s. collaboration . 2012 , in american institute of physics conference series , vol . 1505 , american institute of physics conference series , ed . f. a. aharonian , w. hofmann , & f. m. rieger , 490493

we report the _ fermi _ large area telescope ( lat ) detection of two very - high - energy ( vhe , @xmath0gev ) @xmath1-ray photons from the directional vicinity of the distant ( redshift , @xmath2 ) blazar pks 0426@xmath3380 . the null hypothesis that both the 134 and 122 gev photons originate from unrelated sources can be rejected at the @xmath4 confidence level . we therefore claim that at least one of the two vhe photons is securely associated with pks 0426@xmath3380 , making it the most distant vhe emitter known to date . the results are in agreement with recent _ fermi_-lat constraints on the extragalactic background light ( ebl ) intensity , which imply a @xmath5 horizon for @xmath6gev photons . the lat detection of the two vhe @xmath1-rays coincided roughly with flaring states of the source , although we did not find an exact correspondence between the vhe photon arrival times and the flux maxima at lower @xmath1-ray energies . modeling the @xmath1-ray continuum of pks 0426@xmath3380 with daily bins revealed a significant spectral hardening around the time of the first vhe event detection ( lat photon index @xmath7 ) but on the other hand no pronounced spectral changes near the detection time of the second one . this combination implies a rather complex variability pattern of the source in @xmath1 rays during the flaring epochs . an additional flat component is possibly present above several tens of gev in the ebl - corrected _ fermi_-lat spectrum accumulated over the @xmath8-month high state .

a simplicial complex is called @xmath0-neighborly if every subset of vertices of size at most @xmath0 is the set of vertices of one of its faces . neighborly complexes , especially neighborly polytopes and spheres , are interesting objects to study . in the seminal work of mcmullen @xcite and stanley @xcite , it was shown that in the class of polytopes and simplicial spheres , of a fixed dimension and with a fixed number of vertices , the cyclic polytope simultaneously maximizes all the face numbers . the @xmath1-dimensional cyclic polytope is @xmath2-neighborly . since then , many other classes of neighborly polytopes have been discovered . we refer to @xcite , @xcite and @xcite for examples and constructions of neighborly polytopes . meanwhile , the notion of neighborliness was extended to other classes of objects : for instance , neighborly cubical polytopes were defined and studied in @xcite , @xcite , and @xcite , and neighborly centrally symmetric polytopes and spheres were studied in @xcite , @xcite , @xcite , and @xcite . one goal of this manuscript is to discuss a similar notion for balanced simplicial complexes . balanced complexes were defined by stanley in @xcite , where they were called completely balanced . a @xmath3-dimensional simplicial complex is called balanced if its graph is @xmath1-colorable . we say that a balanced simplicial complex is _ balanced @xmath0-neighborly _ if every set of @xmath0 or fewer vertices with _ distinct _ colors forms a face . it is natural to ask whether apart from cross - polytopes , balanced @xmath0-neighborly polytopes or balanced @xmath0-neighborly spheres exist . to the best of our knowledge , no examples of such objects appear in the current literature , even for @xmath4 , and there is not even a plausible sharp upper bound conjecture for balanced spheres . in this manuscript , we provide two constructions of balanced 2-neighborly 3-spheres with 16 vertices . in @xcite , pfeifle , pilaud and santos , in answering the question of when a given graph is the graph of a polytope , studied the polytopality of cartesian products of certain classes of non - polytopal graphs . as a consequence , our constructions show that the complete 4-partite graph on 16 vertices with parts of equal size is the graph of a 3-sphere . in a different direction , it is also interesting to ask whether every rank - selected subcomplex of a balanced simplicial polytope or sphere has a convex ear decomposition . this statement , if true , would imply that rank - selected subcomplexes of balanced simplicial polytopes possess certain weak lefschetz properties , see theorem 3.9 in @xcite . as a consequence , it would also provide an alternative proof of the balanced generalized lower bound theorem , see theorem 3.3 and remark 3.4 in @xcite . in this manuscript , we present examples giving a negative answer to this question for 3-dimensional spheres . the structure of this manuscript is as follows . in section 2 , after reviewing basic definitions , we establish basic properties of balanced neighborly spheres ; in particular , we prove that for some values of @xmath5 , such spheres can not exist . in section 3 , we present our first construction of a balanced 2-neighborly 3-sphere with 16 vertices . we also show that several of its rank - selected subcomplexes do not have an ear decomposition , and find two balanced non - shellable 3-balls as its subcomplexes . in section 4 , we provide our second construction of a balanced 2-neighborly 3-sphere with 16 vertices . it is different from the first one since all of its rank - selected subcomplexes have ear decompositions . several other examples are provided in section 5 . a _ simplicial complex _ @xmath6 with vertex set @xmath7 is a collection of subsets @xmath8 , called _ faces _ , that is closed under inclusion , and such that for every @xmath9 , @xmath10 . for @xmath11 , let @xmath12 and define the _ dimension _ of @xmath6 , @xmath13 , as the maximum dimension of the faces of @xmath6 . we say that a simplicial complex @xmath6 is _ pure _ if all of its facets have the same dimension . if @xmath6 is a simplicial complex and @xmath14 is a face of @xmath6 , the _ star _ of @xmath14 in @xmath6 is @xmath15 . we also define the _ link _ of @xmath14 in @xmath6 as @xmath16 , and the _ deletion _ of a subset of vertices @xmath17 from @xmath6 as @xmath18 . if @xmath19 and @xmath20 are simplicial complexes on disjoint vertex sets , then the join of @xmath19 and @xmath20 , denoted @xmath21 , is the simplicial complex with vertex set @xmath22 whose faces are @xmath23 . a simplicial complex @xmath6 is a _ simplicial sphere _ _ simplicial ball _ ) if the geometric realization of @xmath6 is homeomorphic to a sphere ( resp . a simplicial sphere is called _ polytopal _ if it is the boundary complex of a convex polytope . for instance , the boundary complex of an octahedron is a polytopal sphere ; we will refer to it as an octahedral sphere . if @xmath6 is a pure @xmath3-dimensional complex such that every @xmath24-dimensional face of @xmath6 is contained in at most 2 facets , then the boundary complex of @xmath6 consists of all @xmath24-dimensional faces that are contained in exactly one facet , as well as their subsets . the boundary complex of a simplicial @xmath1-ball is a simplicial @xmath3-sphere . for a fixed field @xmath25 , we say that @xmath6 is a @xmath3-dimensional _ @xmath25-homology sphere _ if @xmath26 for every face @xmath27 ( including the empty face ) and @xmath28 . a _ homology @xmath1-ball _ ( over a field @xmath25 ) is a @xmath1-dimensional simplicial complex @xmath6 such that ( i ) @xmath6 has the same homology as the @xmath1-dimensional ball , ( ii ) for every face @xmath29 , the link of @xmath29 has the same homology as the @xmath30-dimensional ball or sphere , and ( iii ) the boundary complex , @xmath31 , is a homology @xmath3-sphere . the classes of simplicial @xmath3-spheres and homology @xmath3-spheres coincide when @xmath32 . from now on we fix @xmath25 and omit it from our notation . next we define a special structure that exists in some pure simplicial complexes . an _ ear decomposition _ of a pure @xmath3-dimensional simplicial complex @xmath6 is an ordered sequence @xmath33 of pure @xmath3-dimensional subcomplexes of @xmath6 such that : 1 . @xmath19 is a simplicial @xmath3-sphere , and for each @xmath34 , @xmath35 is a simplicial @xmath3-ball . 2 . for @xmath36 , @xmath37 . 3 . @xmath38 we call @xmath19 the _ initial complex _ , and each @xmath35 , @xmath39 , an _ ear of this decompostion_. notice that this definition is more general than chari s original definition of a _ convex ear decomposition _ , see ( * ? ? ? * section 3.2 ) , where the @xmath40 s are required to be subcomplexes of the boundary complexes of polytopes . in particular , if a complex has no ear decomposition , then it has no convex ear decomposition . however , by the steinitz theorem , all simplicial 2-spheres are polytopal , and hence also all simplicial 2-balls can be realized as subcomplexes of the boundary complexes of 3-dimensional polytopes . so for 2-dimensional simplicial complexes , the notion of an ear decomposition coincides with that of a convex ear decomposition . a @xmath3-dimensional simplicial complex @xmath6 is called _ balanced _ if the graph of @xmath6 is @xmath1-colorable , or equivalently , there is a coloring map @xmath41 $ ] such that @xmath42 for any edge @xmath43 . here @xmath44=\{1,2,\cdots , d\}$ ] denotes the set of colors . a balanced simplicial complex is called _ balanced @xmath0-neighborly _ if every set of @xmath0 or fewer vertices with distinct colors forms a face . for @xmath45 $ ] , the subcomplex @xmath46 is called the _ rank - selected subcomplex _ of @xmath6 . we also define the _ flag @xmath47-vector _ @xmath48)$ ] and the _ flag @xmath49-vector _ @xmath50)$ ] of @xmath6 , respectively , by letting @xmath51 , where @xmath52 , and @xmath53 . the usual @xmath47-numbers and @xmath49-numbers can be recovered from the relations @xmath54 and @xmath55 . the following symmetry of flag @xmath49-vectors of balanced spheres is well - known , see @xcite . [ lemma1 ] if @xmath6 is a balanced homology @xmath3-sphere , then @xmath56\backslash s}(\delta)$ ] for all @xmath45 $ ] . in the reminder of this section , we establish some restrictions on the possible size of color sets of balanced neighborly spheres . in the following , @xmath57 always denotes the set of vertices of color @xmath58 . [ prop : d=2k k - neighborly ] let @xmath6 be a balanced @xmath0-neighborly homology @xmath59-sphere . then @xmath6 has the same number of vertices of each color . let @xmath17 be an arbitrary subset of the set of colors , @xmath60 $ ] , of size @xmath0 . since @xmath6 is balanced @xmath0-neighborly , @xmath61 is also balanced @xmath0-neighborly , and hence @xmath61 is the join of @xmath0 color sets of colors in @xmath17 , each considered as a 0-dimensional complex . by the fact that @xmath62 and by lemma [ lemma1 ] , @xmath63\backslash w}h_1(\delta_{\{i\}})=\prod_{i\in[2k]\backslash w}(|v_{i}|-1).\ ] ] since @xmath64\backslash w|=|w|=k$ ] and since @xmath65 $ ] can be chosen arbitrarily , it follows that each color set in @xmath6 must have the same size . [ lm : intersection of links is sphere ] let @xmath66 . if @xmath6 is a balanced homology @xmath3-sphere and @xmath67 is the set of vertices of color @xmath1 , then @xmath68 is a homology @xmath24-ball for any @xmath69 , and @xmath70 is a homology @xmath71-sphere . let @xmath72 and @xmath73 . every facet @xmath14 of @xmath74 is a @xmath71-face whose link in @xmath6 is a 6-cycle that contains the vertices @xmath75 , and hence @xmath14 is contained in exactly one facet @xmath76 of @xmath77 , where @xmath78 is the unique adjacent vertex to both @xmath79 in @xmath80 . we conclude that @xmath74 is the boudary complex of @xmath77 . we prove that @xmath77 and @xmath74 have the same homology as a @xmath24-ball and @xmath71-sphere respectively . since each @xmath24-face of @xmath6 is contained in exactly 2 facets , it follows that @xmath81}$ ] . by the mayer - vietoris sequence , for any @xmath82 , @xmath83})\to h_n(\sigma)\to h_n(\operatorname{\mathrm{lk}}_\delta v_i)\oplus h_n(\operatorname{\mathrm{lk}}_\delta v_j)\to h_{n}(\delta_{[d-1]})\to\cdots.\ ] ] note that @xmath84}$ ] is a deformation retract of @xmath6 minus three points , hence @xmath85})=2 $ ] and @xmath86})=0 $ ] for @xmath87 . we conclude from ( [ eq:1 ] ) that @xmath88 for all @xmath89 . again by the mayer - vietoris sequence and the fact that @xmath90-\{i , j\}}\cup \sigma=\delta_{[d-1]}$ ] , we obtain @xmath91})\to h_n(\gamma)\to h_n(\operatorname{\mathrm{lk}}_\delta v_{[3]-\{i , j\}})\oplus h_n(\sigma)\to h_{n}(\delta_{[d-1 ] } ) \to\cdots.\ ] ] hence @xmath92 and @xmath93 for @xmath94 . next , for any @xmath95 , we have @xmath96 and @xmath97 . since @xmath98 is a balanced homology @xmath99-sphere , using the same argument as above , we may show that @xmath100 and @xmath101 has the same homology as a @xmath102-ball and @xmath103-sphere respectively . therefore @xmath74 is a homology @xmath71-sphere . finally , for any interior face @xmath14 of @xmath77 , @xmath104 , and hence @xmath105 is a homology sphere . by definition we conclude that @xmath77 is a homology @xmath24-ball . [ rm : 1 ] the complex @xmath74 in lemma [ lm : intersection of links is sphere ] is not balanced , since @xmath74 is @xmath3-colorable instead of being @xmath24-colorable . by lemma [ prop : d=2k k - neighborly ] , if balanced @xmath0-neighborly homology @xmath59-spheres exist , then the number of vertices must be @xmath106 for some @xmath107 . however , as the following proposition shows , we can not hope for the existence of such spheres for all values of @xmath108 and @xmath107 . [ prop : 2-nbly 3-sphere 12 ver ] no balanced 2-neighborly homology 3-spheres with 12 vertices exist . for @xmath109 , where @xmath110 , @xmath111 and @xmath112 are the three other color sets . right : the missing edges between vertices of different color of @xmath113.,title="fig : " ] for @xmath109 , where @xmath110 , @xmath111 and @xmath112 are the three other color sets . right : the missing edges between vertices of different color of @xmath113.,title="fig : " ] assume that @xmath6 is such a sphere . by lemma [ prop : d=2k k - neighborly ] , each color set of @xmath6 has three vertices . we let @xmath114 be the set of vertices of color 4 . since each @xmath113 is a 2-sphere with 9 vertices , its @xmath47-vector is ( 1,9,21,14 ) . furthermore , the balancedness implies that every vertex @xmath115 has @xmath116 or 6 . if @xmath117 is the number of vertices of degree 6 in @xmath113 , then @xmath118 and hence @xmath119 . a balanced 2-sphere with 9 vertices , 3 of which have degree 6 , is unique up to an isomorphism , as shown in figure [ figure : triangulation of the vertex link ] . it is immediate that the missing edges between vertices of different colors in this sphere form a 6-cycle . on the other hand , @xmath120 is a triangulated 2-ball by lemma [ lm : intersection of links is sphere ] . if we delete all of the boundary edges from @xmath77 , the resulting complex @xmath121 is still contractible . however , @xmath77 does not have interior vertices . ( an interior vertex of @xmath77 would not be in @xmath122 , which would contradict the 2-neighborliness of @xmath6 . ) hence the missing edges of @xmath123 that form a 6-cycle are the only interior edges of @xmath77 , i.e. , @xmath121 is a 6-cycle . this contradicts that @xmath121 is contractible , so no such sphere exists . in fact , a stronger result holds . [ lm : 3 triangulations of balanced 3-sphere ] up to an isomorphism , there are only three triangulations of balanced 3-spheres with each color set of size 3 . let @xmath6 be such a sphere and let @xmath114 . each vertex link of @xmath6 is a balanced 2-sphere with at most 9 vertices , hence it is either the octahedral sphere , the suspension of a 6-cycle , or the connected sum of two octahedral spheres . we denote these three 2-spheres as @xmath124 , @xmath125 and @xmath126 respectively . by lemma [ lm : intersection of links is sphere ] , @xmath127}$ ] is the union of three triangulated 2-balls @xmath128 , where @xmath129 $ ] , glued along their common boundary complex @xmath130 . assume that @xmath131 when @xmath132 . an easy counting leads to @xmath133})=f_0(c)+\sum_{i=1}^{3 } f_0(b_i\backslash c)=9 , \quad f_0(\operatorname{\mathrm{lk}}_\delta z_i)=f_0(c)+f_0(b_j\backslash c)+f_0(b_k\backslash c)\in \{6,8,9\}.\ ] ] in what follows we enumerate all possible values of the triple @xmath134 : 1 . @xmath135 or @xmath136 . since @xmath137 is combinatorially equivalent to @xmath124 , it follows that @xmath138 is obtained from @xmath139 by a cross flip ( see @xcite for a reference ) . so either @xmath140 , @xmath141 , and the cross flip replaces a 2-face of @xmath139 with its complement in the octahedral sphere ; or @xmath142 , @xmath141 , and the cross flip replaces the union of three 2-faces of @xmath139 with its complement in the octahedral sphere . 2 . @xmath143 or @xmath144 . in the former case , @xmath130 is a 6-cycle and @xmath127}\backslash c$ ] consists of three disjoint vertices . it is easy to see that at least one of these vertices has degree 6 . then the other two vertices must be of degree 6 as well , and hence @xmath127}$ ] is the join of @xmath130 and @xmath127}\backslash c$ ] . in the latter case , since the vertices of degree 6 in @xmath123 form a 3-cycle , the two disjoint vertices in @xmath127}\backslash c$ ] can not both have degree 6 or 4 . however , if one vertex of @xmath127}\backslash c$ ] is of degree 6 , then since @xmath137 and @xmath145 are combinatorially equivalent to @xmath125 and @xmath130 is a 7-cycle , @xmath146 must be the join of one vertex @xmath147 and a path of length 6 . then @xmath147 is not connected to any vertex of @xmath127}-c$ ] , a contradiction . 3 . @xmath148 . in this case , @xmath149 is a triangulated 2-ball with 9 vertices on the boundary . there is only one balanced 2-sphere with 9 vertices that contains @xmath149 ( it is isomorphic to @xmath126 ) , and hence @xmath150 , a contradiction . in sum , we obtain three balanced 3-spheres with 12 vertices : @xmath151 , the connected sum of two octahedral 3-spheres ; @xmath152 , the join of two 6-cycles , and @xmath153 , with @xmath154 for @xmath155 . [ prop : 2-nbly 4-sphere 15 ver ] no balanced 2-neighborly homology @xmath156-spheres with each color set of size 3 exist . let @xmath6 be such a sphere and let @xmath75 be the vertices of color 5 . by alexander duality , @xmath157})$ ] . in particular , since @xmath158 is balanced 2-neighborly , @xmath159})=\beta_1(\delta_{\{4,5\}})=4 $ ] and @xmath160})=0 $ ] . hence @xmath161})=(f_1-f_0+\chi)(\delta_{[3]})=\frac{9\cdot 6}{2}-9 + 5=23.\ ] ] by double counting , @xmath162 } f_2(\delta_{w})=\binom{4}{2}f_2(\delta_{[3]})=138 $ ] . since @xmath163 , @xmath164 and @xmath165 , it follows that either @xmath166 and @xmath167 , or @xmath168 for all @xmath58 . consider the first case above . it can be checked that for any @xmath169 , @xmath170 and @xmath171 or 9 , depending on whether @xmath172 is a 6-cycle or not . hence @xmath173 , a contradiction . as for the second case , since @xmath174 is a homology 3-ball with 12 vertices on the boundary , by lemma [ lm : 3 triangulations of balanced 3-sphere ] there is a unique balanced 3-sphere combinatorially equivalent to @xmath153 that contains @xmath174 as a subcomplex . it follows that @xmath175 , a contradiction . hence no balanced 2-neighborly homology 4-spheres with 15 vertices exist . [ cor : 3-nbly 5-sphere 18 ver ] the only balanced 3-neighborly homology 5-sphere with @xmath176 vertices is the octahedral 5-sphere . let @xmath6 be such a sphere . the vertex links of @xmath6 are balanced 2-neighborly 4-spheres with @xmath177 vertices . by proposition [ prop : 2-nbly 4-sphere 15 ver ] , it must be the suspension of a balanced 2-neighborly 3-sphere with @xmath178 vertices . then the result follows from lemma [ prop : d=2k k - neighborly ] and proposition [ prop : 2-nbly 3-sphere 12 ver ] . we propose the following conjecture motivated by proposition [ prop : 2-nbly 3-sphere 12 ver ] and corollary [ cor : 3-nbly 5-sphere 18 ver ] . for an arbitrary @xmath108 , there does not exist a balanced @xmath0-neighborly homology @xmath59-sphere with 6k vertices . in this section we present our first construction of a balanced 2-neighborly 3-sphere with 16 vertices . we denote it by @xmath6 . by proposition [ prop : d=2k k - neighborly ] , each color set of @xmath6 has four vertices . [ figure : four links ] + [ first example ] we denote the color sets of @xmath6 by @xmath179 , @xmath180 , @xmath181 and @xmath182 . in figure 2 we illustrate the construction of the vertex links @xmath113 for @xmath183 . all these links are realized as cylinders . two links @xmath137 and @xmath145 share the same top and bottom , which are triangulated hexagons spanned by vertices @xmath184 and @xmath185 , respectively . to construct @xmath123 from @xmath137 , we switch the positions of vertices @xmath186 with vertices @xmath187 respectively and form a new cylinder . the new top and bottom hexagons contain the 2-faces @xmath188 and @xmath189 . similarly , we construct the link @xmath190 from @xmath145 by switching the positions of vertices @xmath186 with vertices @xmath187 and letting @xmath188 and @xmath189 be the 2-faces that appear in the triangulation of the top and bottom hexagons . it follows that @xmath123 and @xmath190 also share the same top and bottom . now since @xmath6 is balanced 2-neighborly , by our construction , it only remains to show that @xmath6 is a 3-sphere . the geometric realizations of @xmath191 and @xmath192 are filled cylinders that share top and bottom . so their union is a filled torus ( that is , a genus-1 handlebody ) ; so is the union of @xmath193 and @xmath194 . note that these two genus-1 handlebodies have identical boundary complexes , thus they provide a genus-1 heegaard splitting of a 3-sphere , which is our @xmath6 . [ rm : property of construction 1 ] our construction @xmath6 has the following properties : 1 . one can check that all vertex links are combinatorially equivalent . furthermore , all 2-dimensional rank - selected subcomplexes of @xmath6 are isomorphic . the intersection of two vertex links , where the vertices are of the same color , always has at least two connected components . by the construction of @xmath6 , @xmath195 has two connected components when @xmath196 or @xmath197 ( they are the top and bottom hexagons as shown in figure 2 ) ; and it has three connected components when @xmath198 and @xmath199 ( each component is the union of two facets along the side of the cylinders ) . 3 . there are at least three group actions on the vertices of @xmath6 : 1 . fix the subscript and rotate the corresponding vertices of color 1 , 2 and 3 respectively . the generator is given by @xmath200 . 2 . rotate vertices of the same color . the generator is @xmath201 3 . exchange @xmath137 and @xmath145 , @xmath123 and @xmath190 , by exchanging @xmath202 and @xmath203 ( or @xmath204 and @xmath203 , @xmath204 and @xmath202 ) for all @xmath205 $ ] . the generators are @xmath206 , @xmath207 and @xmath208 . + hence the automorphism group of @xmath6 is of size at least 96 . [ prop : no convex ear decomposition ] there exist balanced 3-spheres whose 2-dimensional rank - selected subcomplexes do not have an ear decomposition . consider the complex @xmath6 in construction [ first example ] and its rank - selected subcomplex @xmath127}$ ] . suppose @xmath127}$ ] has an ear decomposition @xmath209 . since @xmath127}$ ] is the deformation retract of the sphere @xmath6 minus 4 points , @xmath159})=3 $ ] and so @xmath0 must be 3 . note that @xmath127}$ ] is the union of three 3-balls @xmath210 ( @xmath155 ) with three vertices @xmath211 removed . the only subcomplexes of @xmath127}$ ] that can be realized as the boundary complexes of 3-balls must be the boundary complex of either @xmath210 , @xmath212 or @xmath213 . using the second property of @xmath6 in remark [ rm : property of construction 1 ] , we conclude that @xmath214 for some @xmath205 $ ] . furthermore , @xmath215 divides @xmath216 into two 2-balls @xmath149 and @xmath217 . hence @xmath218 is also a 2-sphere . then the above argument yields that @xmath219 for some @xmath220 . this , however , leads to a contradiction , since @xmath195 has at least two connected components . hence @xmath127}$ ] does not have an ear decomposition . in the following we describe two balanced non - shellable 3-balls which are subcomplexes of @xmath6 . recall that a simplicial @xmath3-ball @xmath221 is called _ shellable _ if its facets can be ordered @xmath222 in such a way that the intersection of a facet with the union of previous facets is pure @xmath24-dimensional . such an ordering of facets is called a _ shelling_. a facet of @xmath221 is defined to be _ free _ if its intersection with the boundary of @xmath221 is a @xmath24-ball . a simplicial ball @xmath221 is called _ strongly non - shellable _ if it has no free facet . observe that if @xmath221 is a shellable ball with more than one facet , and @xmath223 is a shelling of @xmath221 , then @xmath224 must be a free facet . hence if @xmath221 is a strongly non - shellable ball , then @xmath221 is non - shellable . for more discussion on shellability , see ziegler s book @xcite . [ first ball example ] our construction begins with the balanced 3-sphere @xmath6 in construction [ first example ] . remove the vertex @xmath225 from @xmath6 and denote the resulting complex by @xmath149 . then @xmath149 is a balanced 3-ball with 15 vertices ( @xmath226 and @xmath227 are the only interior vertices ) and 60 facets . the boundary of @xmath149 is exactly @xmath190 . in particular , six 2-faces that form three connected components are from @xmath228 , another six 2-faces are from @xmath229 and the remaining eight 2-faces that form two connected components are from @xmath230 ( see part 2 of remark [ rm : property of construction 1 ] ) . we denote by @xmath231 the union of two adjacent faces @xmath232 and @xmath233 in @xmath228 , by @xmath234 the union of @xmath235 and @xmath236 in @xmath229 , and by @xmath237 the top triangulated hexagon in @xmath238 , as shown in figure 2(c ) . from @xmath149 we construct another 3-ball @xmath217 by deleting the eight 3-faces in @xmath239 . the resulting complex , denoted as @xmath217 , is a balanced 3-ball with 15 vertices ( all of them are on the boundary of @xmath217 ) and 52 facets , see figure [ figure : front and back view ] for a view of the boundary of @xmath217 . [ figure : front and back view ] [ non - shellability of b2 ] the balanced 3-ball @xmath217 in construction [ first ball example ] is strongly non - shellable . since every vertex of @xmath217 is on the boundary , the intersection of any facet @xmath14 and @xmath240 must contain four vertices of different colors . furthermore , if @xmath241 is pure 2-dimensional , then it contains at least two adjacent facets of @xmath240 . it can be checked that there are 18 pairs of adjacent 2-faces of @xmath240 such that all four vertices are of different colors , but none of them can be realized as a subcomplex of a facet of @xmath217 , a contradiction . hence @xmath217 has no free facet , i.e. , it is strongly non - shellable . we proceed to show that @xmath149 is also non - shellable . [ non - shellability of b1 ] the balanced 3-ball @xmath149 in construction [ first ball example ] is non - shellable . assume that @xmath149 has a shelling @xmath242 , and for @xmath243 , @xmath244 is the last facet of @xmath245 that appears in the shelling ; assume further that @xmath246 belongs to a connected component , denoted as @xmath247 , in @xmath248 , where @xmath6 is the balanced 3-sphere in construction [ first example ] . note that @xmath249 is a facet of @xmath250 . now assume that @xmath251 are facets of @xmath74 for some @xmath252 . if @xmath253 , then by our assumption @xmath254 for any @xmath243 , and so @xmath255 is a 3-ball with no interior vertices . moreover , the intersection @xmath256 satisfies the following two conditions : 1 ) it contains all four vertices of @xmath257 , and 2 ) it is a subcomplex of @xmath258 . however , a similar argument as in the proof of proposition [ non - shellability of b2 ] yields that @xmath258 can not be pure 2-dimensional . hence @xmath259 . by induction , the eight facets of @xmath74 are @xmath260 . this implies @xmath261 is shellable . however , the proof of proposition [ non - shellability of b2 ] shows that @xmath262 is strongly non - shellable , a contradiction . the 3-sphere @xmath6 in construction [ first example ] is non - polytopal . if @xmath6 is polytopal , then there exists a shelling of @xmath6 that ends with the facets of @xmath263 , see ( * ? ? ? * lemma 8.10 and theorem 8.12 ) . hence @xmath264 is shellable , which contradicts proposition [ non - shellability of b1 ] . at present , we do not know if the complex @xmath6 is shellable or not . in section 3 we constructed a balanced 2-neighborly 3-sphere all of whose 2-dimensional rank - selected subcomplexes have no ear decompositions . now we provide a second construction that is not combinatorially equivalent to construction [ first example ] . [ second example ] r r=1.9 cm ( 0:r ) cycle ( 360:r ) node[right ] @xmath265 cycle ( 30:r ) node[above right ] @xmath266 cycle ( 60:r ) node[above right ] @xmath267 cycle ( 90:r ) node[above ] @xmath268 cycle ( 120:r ) node[above left ] @xmath269 cycle ( 150:r ) node[above left ] @xmath270 cycle ( 180:r ) node[left ] @xmath271 cycle ( 210:r ) node[below left ] @xmath272 cycle ( 240:r ) node[below left ] @xmath273 cycle ( 270:r ) node[below ] @xmath274 cycle ( 300:r ) node[below right ] @xmath275 cycle ( 330:r ) node[below right ] @xmath276 ; ( 330:r ) ( 30:r ) ( 90:r)cycle ; ( 330:r ) ( 120:r ) ( 300:r)(150:r ) ; ( 150:r ) ( 270:r ) ( 210:r)cycle ; ( 0:r)(30:r ) (60:r) ( 90:r)(120:r)(150:r)(180:r)(210:r ) (240:r)(270:r)(300:r)(330:r)(0:r ) ; r r=1.9 cm ( 0:r ) cycle ( 360:r ) node[right ] @xmath265 cycle ( 30:r ) node[above right ] @xmath266 cycle ( 60:r ) node[above right ] @xmath267 cycle ( 90:r ) node[above ] @xmath268 cycle ( 120:r ) node[above left ] @xmath269 cycle ( 150:r ) node[above left ] @xmath270 cycle ( 180:r ) node[left ] @xmath271 cycle ( 210:r ) node[below left ] @xmath272 cycle ( 240:r ) node[below left ] @xmath273 cycle ( 270:r ) node[below ] @xmath274 cycle ( 300:r ) node[below right ] @xmath275 cycle ( 330:r ) node[below right ] @xmath276 ; ( 0:r ) ( 60:r ) ( 120:r)cycle ; ( 150:r ) ( 0:r ) ( 180:r)(330:r ) ; ( 180:r ) ( 300:r ) ( 240:r)cycle ; ( 0:r)(30:r ) (60:r) ( 90:r)(120:r)(150:r)(180:r)(210:r ) (240:r)(270:r)(300:r)(330:r)(0:r ) ; r r=1.9 cm ( 0:r ) cycle ( 360:r ) node[right ] @xmath265 cycle ( 30:r ) node[above right ] @xmath266 cycle ( 60:r ) node[above right ] @xmath267 cycle ( 90:r ) node[above ] @xmath268 cycle ( 120:r ) node[above left ] @xmath269 cycle ( 150:r ) node[above left ] @xmath270 cycle ( 180:r ) node[left ] @xmath271 cycle ( 210:r ) node[below left ] @xmath272 cycle ( 240:r ) node[below left ] @xmath273 cycle ( 270:r ) node[below ] @xmath274 cycle ( 300:r ) node[below right ] @xmath275 cycle ( 330:r ) node[below right ] @xmath276 ; ( 150:r ) ( 90:r ) ( 180:r)(60:r ) ( 210:r ) ( 30:r)(240:r)(0:r ) ( 270:r ) ( 330:r ) ; ( 0:r)(30:r ) (60:r) ( 90:r)(120:r)(150:r)(180:r)(210:r ) (240:r)(270:r)(300:r)(330:r)(0:r ) ; r r=1.9 cm ( 0:r ) cycle ( 360:r ) node[right ] @xmath265 cycle ( 30:r ) node[above right ] @xmath266 cycle ( 60:r ) node[above right ] @xmath267 cycle ( 90:r ) node[above ] @xmath268 cycle ( 120:r ) node[above left ] @xmath269 cycle ( 150:r ) node[above left ] @xmath270 cycle ( 180:r ) node[left ] @xmath271 cycle ( 210:r ) node[below left ] @xmath272 cycle ( 240:r ) node[below left ] @xmath273 cycle ( 270:r ) node[below ] @xmath274 cycle ( 300:r ) node[below right ] @xmath275 cycle ( 330:r ) node[below right ] @xmath276 ; ( 210:r ) ( 120:r ) ( 240:r)(90:r ) ( 270:r ) ( 60:r)(300:r) ( 30:r ) ; ( 150:r)(30:r ) ; ( 210:r)(330:r ) ; ( 0:r)(150:r)(300:r ) ; ( 120:r)(330:r)(180:r ) ; ( 0:r)(30:r ) (60:r) ( 90:r)(120:r)(150:r)(180:r)(210:r ) (240:r)(270:r)(300:r)(330:r)(0:r ) ; r r=1.9 cm ( 0:r ) cycle ( 360:r ) node[right ] @xmath265 cycle ( 30:r ) node[above right ] @xmath266 cycle ( 60:r ) node[above right ] @xmath267 cycle ( 90:r ) node[above ] @xmath268 cycle ( 120:r ) node[above left ] @xmath269 cycle ( 150:r ) node[above left ] @xmath276 cycle ( 180:r ) node[left ] @xmath271 cycle ( 210:r ) node[below left ] @xmath272 cycle ( 240:r ) node[below left ] @xmath273 cycle ( 270:r ) node[below ] @xmath274 cycle ( 300:r ) node[below right ] @xmath275 cycle ( 330:r ) node[below right ] @xmath270 ; ( 150:r)(210:r ) ( 120:r ) ( 240:r)(90:r ) ( 270:r ) ( 60:r)(300:r) ( 30:r)(330:r ) ; ( 0:r)(30:r ) (60:r) ( 90:r)(120:r)(150:r)(180:r)(210:r ) (240:r)(270:r)(300:r)(330:r)(0:r ) ; r r=1.9 cm ( 0:r ) cycle ( 360:r ) node[right ] @xmath265 cycle ( 30:r ) node[above right ] @xmath266 cycle ( 60:r ) node[above right ] @xmath267 cycle ( 90:r ) node[above ] @xmath268 cycle ( 120:r ) node[above left ] @xmath269 cycle ( 150:r ) node[above left ] @xmath276 cycle ( 180:r ) node[left ] @xmath271 cycle ( 210:r ) node[below left ] @xmath272 cycle ( 240:r ) node[below left ] @xmath273 cycle ( 270:r ) node[below ] @xmath274 cycle ( 300:r ) node[below right ] @xmath275 cycle ( 330:r ) node[below right ] @xmath270 ; ( 30:r)(90:r)(150:r)cycle ; ( 210:r)(270:r)(330:r)cycle ; ( 150:r)(0:r)(180:r)(330:r ) ; ( 0:r)(30:r ) (60:r) ( 90:r)(120:r)(150:r)(180:r)(210:r ) (240:r)(270:r)(300:r)(330:r)(0:r ) ; r r=1.9 cm ( 0:r ) cycle ( 360:r ) node[right ] @xmath265 cycle ( 30:r ) node[above right ] @xmath266 cycle ( 60:r ) node[above right ] @xmath267 cycle ( 90:r ) node[above ] @xmath268 cycle ( 120:r ) node[above left ] @xmath269 cycle ( 150:r ) node[above left ] @xmath276 cycle ( 180:r ) node[left ] @xmath271 cycle ( 210:r ) node[below left ] @xmath272 cycle ( 240:r ) node[below left ] @xmath273 cycle ( 270:r ) node[below ] @xmath274 cycle ( 300:r ) node[below right ] @xmath275 cycle ( 330:r ) node[below right ] @xmath270 ; ( 0:r)(60:r)(120:r)cycle ; ( 180:r)(240:r)(300:r)cycle ; ( 330:r)(120:r)(300:r)(150:r ) ; ( 0:r)(30:r ) (60:r) ( 90:r)(120:r)(150:r)(180:r)(210:r ) (240:r)(270:r)(300:r)(330:r)(0:r ) ; [ figure : a and b ] assume that @xmath277 , @xmath180 , @xmath181 and @xmath182 are the four color sets of a balanced 3-sphere @xmath74 . we let @xmath278 and @xmath279 , where @xmath280 , @xmath221 and @xmath281 are triangulated 2-balls sharing the same boundary as shown in figure [ fig : patches a , b , c ] . all possible edges that do not appear in @xmath280 , @xmath221 and @xmath281 are shown in figure [ fig : patch d ] as solid red edges in disc @xmath282 . notice that the dashed edges in @xmath282 are edges in discs @xmath280 and @xmath221 , so we may rearrange the boundary of @xmath283 by switching the positions of vertices @xmath270 and @xmath276 , and then replacing the edges containing @xmath270 or @xmath276 in @xmath284 by the dashed edges . in this way , we obtain a triangulation of a 12-gon @xmath283 as shown in figure [ fig : patch d ] . furthermore , @xmath285 , and @xmath286 divides the sphere @xmath287 into two discs @xmath288 and @xmath289 as shown in figure 6 . we let @xmath290 and @xmath291 . it is not hard to see that the complex @xmath74 with the given four links is indeed a balanced 2-neighborly 3-sphere . here we list some properties of @xmath74 in construction [ second example ] . first , @xmath292 is an ear decomposition of @xmath293}$ ] . since property 1 of remark [ rm : property of construction 1 ] and proposition [ prop : no convex ear decomposition ] implies that all 2-dimensional rank - selected subcomplexes of the 3-sphere in construction [ first example ] do not have an ear decomposition , we conclude that @xmath74 is not combinatorially equivalent to construction [ first example ] . alternatively , we may prove non - equivalence by inspecting the vertex links . in construction [ first example ] , all vertex links are combinatorially equivalent ; however , there are two types of vertex links of @xmath74 : @xmath294 and @xmath295 for @xmath296 are combinatorially equivalent to the triangulated spheres in figure 2 ( to see this , note that the light red triangles that appear in discs @xmath280 and @xmath288 correspond to the middle triangles in the top and bottom hexagons of the cylinders in figure 2 ) ; the other 12 vertex links are combinatorially equivalent to the balanced 2-sphere with 12 vertices such that exactly two of its vertices have degree 8 . second , the automorphism group of @xmath74 has at least two generators @xmath297 ( the second generator is given by switching vertices of color 1 and 3 , and color 2 and 4 , but with the same subscript . ) hence @xmath298 has at least 8 elements . the construction [ second example ] is shellable . for @xmath299 , there exist two shellings @xmath300 and @xmath301 such that for any @xmath302 , @xmath303 are facets from @xmath281 and @xmath304 are facets from @xmath280 . similarly , there exist two shellings @xmath305 and @xmath306 for @xmath307 , where @xmath308 are facets from @xmath221 . then @xmath309 gives a shelling of @xmath310 . we may extend this shelling into a complete shelling of @xmath74 by constructing two similar shellings of @xmath311 and @xmath312 . however , we have not been able to check whether @xmath74 is polytopal or not . it is easy to see that if @xmath19 is a balanced 2-neighborly @xmath313-sphere and @xmath20 is a balanced 2-neighborly @xmath314-sphere , then @xmath21 is a balanced 2-neighborly @xmath315-sphere . hence by taking joins , we find balanced 2-neighborly @xmath316-spheres with @xmath317 vertices for any @xmath318 . let @xmath66 and @xmath319 be arbitrary integers . is there a balanced 2-neighborly simplicial @xmath3-sphere all of whose color sets have the same size @xmath320 ? is there a polytopal sphere with these properties ? in general , there are many balanced 3-spheres , not necessarily 2-neighborly , such that some of their rank - selected subcomplexes have no ear decompositions . here we present an example different from construction [ first example ] . its rank - selected subcomplex can be embedded in @xmath321 . the strategy is that we want to construct a balanced 3-sphere @xmath6 so that for a fixed color set @xmath322 , the intersection of any two vertex links @xmath68 always has at least two connected components . in figure [ fig1:links ] we show the links @xmath323 . every label represents the color of the vertex . also each connected component of @xmath174 is colored in green , @xmath324 is colored in blue for @xmath296 , and @xmath325 is colored in pink for @xmath326 . immediately we check that all these intersections of vertex links have 2 or 3 connected components . figure [ fig2:union of links ] shows how @xmath327 is formed from these links . first we glue @xmath328 and @xmath139 along two green triangles . the resulting complex @xmath329 is shown in figure [ link12 ] . then we place @xmath138 on top of @xmath330 . as we see from figure [ link123 ] , the boundary complex of @xmath331 is a triangulated torus . finally , we place @xmath332 on top of @xmath333 so that @xmath334 covers the 1-dimensional hole " in @xmath331 , see figure [ link1234 ] . we denote the subspace of @xmath321 enclosed by @xmath335 as @xmath336 for @xmath337 , and let @xmath338 . from our construction it follows that the boundary complex of @xmath339 is a 2-sphere ; we let it be @xmath340 . since each @xmath68 has at least two connected components for @xmath341 , the mayer - vietoris sequence implies that @xmath342 is not contractible for all @xmath341 . a similar inspection of @xmath343 also implies that the boundary complexes of @xmath344 s can not be triangulated 2-spheres for distinct @xmath345 . now we prove that @xmath327 does not have an ear decomposition by using a similar argument to the one in the proof of proposition [ prop : no convex ear decomposition ] . in the following we denote the union of interior faces of a complex @xmath346 by @xmath347 . suppose @xmath327 has an ear decomposition @xmath209 . since @xmath348 and @xmath349 , @xmath0 must be 4 . notice first that @xmath350 divides @xmath321 into five subspaces , namely , @xmath351 and the complement of @xmath339 , each having @xmath335 as the boundary complex for @xmath352 respectively . since @xmath353 must be a triangulated 2-sphere , by the jordan theorem , it separates @xmath321 into two connected components , hence the bounded component must be either @xmath342 or @xmath344 for some @xmath345 . ( we may assume that it is not @xmath336 , since otherwise we may consider the 2-sphere @xmath354 instead of @xmath353 , where the subset enclosed by this sphere in @xmath321 can not be @xmath336 anymore . ) this contradicts the fact that the boundaries of @xmath342 or @xmath344 are not 2-spheres . one can think of all the figures illustrated above as projections of a subcomplex of @xmath355 onto @xmath321 . however , we do not know whether the complex provided in this section can be realized as the boundary of a 4-polytope . the author was partially supported by a graduate fellowship from nsf grant dms-1361423 . i thank moritz firsching for pointing out the automorphism groups of the constructions in section 3 and 4 and running some computational tests to decide whether the constructions are polytopal .

we find the first two examples of non - octahedral balanced 2-neighborly spheres . each construction is of dimension 3 and with 16 vertices . along the way , we show that the rank - selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition .

quasielastic electron scattering from nuclei has long been used to study various aspects of nuclear structure . the strong quenching of the measured coulomb response relative to fermi gas estimates is one prominent example@xcite . such studies are facilitated by ( 1 ) the ( reasonably ) well - understood nature of the fundamental @xmath5 interaction and ( 2 ) by the relative weakness of that interaction . however , this same interaction suffers from some important shortcomings in that it probes the nucleus in a very restricted fashion . hence only two independent observables exist ( without making polarization measurements ) , namely the coulomb ( or longitudinal ) and transverse responses . these limitations have motivated a number of quasielastic scattering experiments employing hadronic probes@xcite . the more complicated interaction of the probe with nucleons means that in principle new responses which are inaccessible with electrons can be studied . one important example of such investigations is found in the @xmath6 ( refs . @xcite ) and @xmath7 ( refs . @xcite ) experiments performed ( in part ) to extract the _ longitudinal _ spin response to complement information on the transverse spin response determined by electron scattering . analysis of such hadronic measurements is , however , hampered by the strength and complexity of the projectile- nucleon interaction . the former causes strong distortions in the incoming and outgoing projectile wavefunctions and localizes the scattering process in the region of the nuclear surface . the latter typically implies significant modification of the interaction in the nuclear medium . both sets of effects greatly complicate the theoretical description of the scattering process and the extraction of nuclear structure information . these considerations have led to much interest in @xmath0-nucleus scattering . the relative weakness of the @xmath8 interaction relative to , _ e.g. _ , @xmath9 and @xmath10 interactions suggests that distortions and medium effects may be simpler to handle and that the scattering process may be more sensitive to the nuclear interior . estimates of the mean - free - paths ( mfp s ) of the various probes support the latter assertion . for example , at a laboratory momentum of 700 mev / c , the proton and the @xmath11 have mfp s of about 2 fm while the @xmath0 mfp is roughly twice that value . while @xmath0-nucleus scattering experiments are still in their infancy , their promise has been some what diminished by early results . in particular , there is a persistant discrepancy between measured @xmath0-nucleus elastic and total cross sections and theoretical results based on multiple scattering models which are expected to be accurate due again to the weakness of the @xmath8 interaction@xcite . data for @xmath0-nucleus quasielastic scattering from c , ca and pb at a laboratory @xmath0 momentum of 705 mev / c have recently become available@xcite . a preliminary theoretical analysis of the @xmath12= 300 and 500 mev / c data for @xmath13c and @xmath3ca has already appeared along with the data@xcite . in the present paper , we extend those preliminary calculations and focus on comparison with @xmath14 data at @xmath4 mev / c . in so doing , we make the first attempt to realize the promise of @xmath0 scattering to provide nuclear structure information complementary to that extracted from electron data . the remainder of this paper is organized as follows : in section ii , we derive well known results for @xmath2 quasielastic scattering so as to establish a framework for treating @xmath0 scattering which is addressed in section iii . section iv touches upon problems associated with ambiguities in the forms of both the @xmath5 and @xmath8 on - shell elastic scattering amplitudes . past efforts to resolve the @xmath5 ambiguity are summarized and a resolution for @xmath8 is proposed . the theoretical treatment of nuclear structure is outlined in section v. specific calculations of electron and @xmath0 cross sections and comparisons with data appear in section vi where it is shown that , due to details of the structure of the @xmath0 quasielastic cross section , the strong quenching observed for the @xmath2 coulomb response does _ not _ appear for the @xmath0 process even though the latter is dominated by a contribution which is superficially very similar to the coulomb response . section vii contains a summary of the present work and our conclusions . the discussion of electron scattering from nucleons and nuclei presented in this section is considerably more detailed than would seem warranted given that no new results appear . however , this detail is supplied so as to provide a familiar context for developing our treatment of @xmath0 scattering which will be outlined in section iii . the unpolarized @xmath1 scattering differential cross section in plane wave born approximation is @xmath15 where @xmath16 and @xmath17 , @xmath18 are the initial and final electron 4-momenta , respectively , and the electron electromagnetic tensor is @xmath19 \nonumber\\ & = & { 1\over{\epsilon_f\epsilon_i}}\ \bigl[p_\mu p_\nu + { 1\over 4 } ( q^2 g_{\mu\nu}-q_\mu q_\nu)\bigr ] \label{bb}\end{aligned}\ ] ] where @xmath20 and , _ e.g. _ , @xmath21 where @xmath22 is the electron mass . also , @xmath23 is the electromagnetic response of the scatterer . for nuclear scattering , @xmath24 where the polarization insertion @xmath25 is defined via @xmath26|i > \nonumber\\ & = & \int{{d\omega}\over{2\pi}}\ e^{-i\omega(y_0-{y_0}')}\ \pi^{\mu\nu}_{em}(\bfy,\bfy';\omega ) . \label{bl}\end{aligned}\ ] ] in these expressions , the nucleon electromagnetic current operator is @xmath27 where @xmath28 is the nucleon mass and @xmath29 ( @xmath30 ) is the initial ( final ) nucleon momentum . we now consider @xmath31-nucleon scattering . it is useful to recall the lehmann representation of the polarization insertion which implies , _ e.g. _ , @xmath32 where @xmath33 for a single nucleon , @xmath34 is readily evaluated and we find @xmath35 where the kroenecker @xmath36 comes from the box normalization of the free nucleon wavefunction and where the _ nucleon _ electromagnetic tensor is @xmath37 \nonumber\\ & \rightarrow&{1\over{e_f e_i}}\biggl [ { { g^2_e+\tau g^2_m}\over{1+\tau}}\ p^\mu p^\nu + g^2_m\ { 1\over 4}\bigr(q^2 g^{\mu\nu}-q^\mu q^\nu\bigl)\biggr ] \label{bo}\end{aligned}\ ] ] with @xmath38 , @xmath39 and sach s form factors , @xmath40 and @xmath41 . then , in the initial nucleon rest frame , using @xmath42 , @xmath43 evaluating the integral _ at fixed scattering angle _ yields @xmath44 where we now understand that @xmath45 , @xmath46 and @xmath47 take on their on - shell values in the final state . for an infinitely massive structureless proton , @xmath48 , @xmath49 , @xmath50 , @xmath51 and @xmath52 , @xmath53 which implies @xmath54 , @xmath55 . in this limit , @xmath56 and @xmath57 we then have @xmath58 which is the mott cross section . for ultrarelativistic electrons , this expression becomes @xmath59 for physical nucleons ; _ i.e. _ , when no approximation to @xmath60 is made , we find , for ultrarelativistic electrons , the familiar result @xmath61 . \label{bbb}\ ] ] next , we examine @xmath31-nucleus scattering . using @xmath62 , eq . [ ba ] yields , in the nuclear rest frame , @xmath63 using the fact that @xmath64 due to current conservation , we find @xmath65 \nonumber\\ & \qquad&\qquad\qquad -\biggl(1+{{q^2}\over{4m^2}}\biggr ) w^{11}_{em } \nonumber\\ & \qquad&-{{q^2}\over{4m^2}}\ { { q^2}\over{\bfq^2}}\ w^{00}_{em } + { { q^2}\over{2m^2}}\ w^{11}_{em } \ \ \biggr\ } . \label{bc}\end{aligned}\ ] ] for ultrarelativistic electrons this reduces to the familiar expression@xcite @xmath66 \label{bd}\ ] ] in which we identify the coulomb and transverse responses , namely , @xmath67 and @xmath68 , respectively . we now observe that the expression for the ultrarelativistic @xmath5 cross section in the initial nucleon rest frame ( eq . [ bbb ] ) can be expressed as @xmath69 where @xmath70 .\end{aligned}\ ] ] comparison with eq . [ bd ] allows us to determine the _ single nucleon _ electromagnetic responses @xmath71 these expressions also follow directly from eq . [ bi ] when applied to a free nucleon . this section presents a treatment of @xmath8 and @xmath0-nucleus scattering which emphasizes the relation to the formulation of @xmath1 scattering appearing in the previous section . we begin by considering the ( fictitious ) scattering of a @xmath0 which interacts only electromagnetically with the scatterer . ( alternatively , we can think of the scattering of a `` spinless electron '' . ) using arguments analogous to those which led to eq . [ ba ] in section ii , we find that the @xmath0 _ electromagnetic _ cross section is given in plane wave born approximation by @xmath72 where now @xmath73 where @xmath74 and @xmath75 are the initial and final @xmath0 momenta and @xmath76 is the @xmath0 electromagnetic tensor given by @xmath77 this expression is to be compared with the electron electromagnetic tensor , eq . differences are clearly due to the presence of `` spin currents '' in the electron case . we now treat @xmath8 electromagnetic scattering . it is simple to show that , in analogy with eq . [ bg ] , @xmath78 for a massive structureless proton , we have @xmath79 which should be compared with the mott cross section appearing in eq . [ bh ] . ( since the limit of ultrarelativistic @xmath0 s is irrelevant for the measurements to be discussed below , we do not present the corresponding formulae . ) the @xmath0 electromagnetic analogue to the @xmath31-nucleus double differential cross section appearing in eq . [ bc ] is @xmath80 \nonumber\\ & \qquad&\qquad\qquad -\biggl(1+{{q^2}\over{4m^2}}\biggr ) w^{11}_{em } \ \ \biggr\ } \label{cc}\end{aligned}\ ] ] where now @xmath22 is the @xmath0 mass . our next task is to generalize this treatment to handle _ hadronic _ interactions . for _ electromagnetic _ @xmath0 scattering , the @xmath81-matrix which leads to eq . [ ca ] is explicitly @xmath82 where @xmath83 is the box volume arising from box normalizing the free @xmath0 wavefunctions and where the electromagnetic transition current @xmath34 is defined in eq . [ bi ] . we next write @xmath84 clearly , the dynamics specific to electromagnetic scattering appear only in the quantity @xmath85 ; the remaining factors are of a form appropriate to a general 4-vector current - current interaction . we propose that for @xmath0 _ hadronic _ scattering , we should let @xmath86 where now a scalar interaction appears in addition to the vector term . in this expression , @xmath87 and @xmath88 are complex numbers related to the @xmath8 elastic scattering amplitude in a manner to be established below and where the exact definitions of @xmath89 and @xmath34 will be given shortly . it is plausible that an interaction of this form while , as will be discussed below , not unique is sufficiently general since we know that an on - shell spin 0-spin @xmath90 elastic scattering amplitude can be completely specified at a given kinematic point ( including an irrelevant overall phase ) by two complex numbers . it is now convenient to introduce the notation @xmath91 where @xmath92 and @xmath93 with @xmath94 with these definitions , it is straightforward to show that the _ hadronic _ @xmath0 cross section analogous to the electron and @xmath0 electromagnetic cross sections of eqs . [ ba ] and [ ca ] , respectively , is @xmath95 where the @xmath0 hadronic tensor @xmath96 is defined as @xmath97 and the hadronic response is defined via ( compare with eqs . [ bj ] through [ bl ] ) @xmath98 where @xmath99 is specified by @xmath100|i > \nonumber\\ & = & \int { { d\omega}\over{2\pi}}\ e^{-i\omega(y^0-{y^0}')}\ \pi^{ab}(\bfy,\bfy';\omega)\end{aligned}\ ] ] and @xmath101 . we now specifically consider @xmath8 scattering . in eq . [ bg ] for electron scattering and eq . [ cb ] for @xmath8 electromagnetic scattering , we presented expressions for the differential cross sections in the initial nucleon rest frame . it is straight forward to derive a similar expression for the @xmath8 _ hadronic _ cross section . a slight generalization yields this cross section in a reference frame in which the @xmath8 relative motion is colinear but otherwise arbitrary . it will be convenient to have a formula for the @xmath8 _ hadronic _ cross section _ in the center - of - momentum frame _ , namely @xmath102 where @xmath103 and where @xmath104 and @xmath105 are the @xmath0 and @xmath106 energies , respectively , in the center - of - momentum frame . it is straightforward to show @xmath107u(p_i s_i ) \nonumber\\ & = & \chi^\dagger(s_f)\ \biggl [ f(q ) + i\sigma_n g(q ) \biggr]\ \chi(s_i ) \label{cf}\end{aligned}\ ] ] where @xmath108 and @xmath109 , @xmath110 as well as @xmath111 \nonumber\\ & + & { { e^2-m^2}\over{2me}}{\cal f}_v \biggl(1+\cos\theta\biggr)\end{aligned}\ ] ] and @xmath112 -{{e^2-m^2}\over{2me}}{\cal f}_v \biggr\}\ \sin\theta .\ ] ] we recognize @xmath113 and @xmath114 as the wolfenstein amplitudes for @xmath0-nucleon scattering and take them from arndt s sp88@xcite @xmath0-proton and @xmath0-neutron phase shift solutions . the above relations may readily be inverted to find @xmath115 and @xmath116 or , equivalently , @xmath87 and @xmath88 . we finally investigate @xmath0-nucleus scattering . in analogy with eq . [ bn ] for @xmath31-nucleus scattering , the @xmath0-nucleus cross section in the nuclear rest frame is @xmath117 using @xmath118 which follows from the assumption that the baryon current is conserved and also using @xmath119 , we can evaluate @xmath120 to obtain in analogy with eq . [ bc ] for @xmath31-nucleus scattering and eq . [ cc ] for @xmath0-nucleus electromagnetic scattering the following @xmath0-nucleus hadronic cross section : @xmath121 \nonumber\\ & \qquad&\qquad\qquad -\biggl(1+{{q^2}\over{4m^2}}\biggr ) w^{11 } \biggr\ } \nonumber\\ & + & 2{\rm re}({\cal f}_s ' { \cal f}_v'^*)\ \biggl({{\epsilon_f+\epsilon_i}\over{2m}}\biggr)\ { { q^2}\over{\qs^2 } } w^{0s}\ \ \biggr ] . \label{ce}\end{aligned}\ ] ] this is the main result of section iii . we note that two new responses namely @xmath122 and @xmath123 which were not present for the electromagnetic processes enter in the hadronic cross section . we finally observe that , for a single nucleon , the hadronic responses @xmath124 and @xmath125 are given by the corresponding single nucleon electromagnetic responses of eqs . [ be ] and [ bff ] in the limit that @xmath52 and @xmath53 or , equivalently , @xmath126 , @xmath127 . furthermore , it is easy to show that , in the initial nucleon rest frame , @xmath128 for a single nucleon . the expression for the electromagnetic current of a _ free _ nucleon ( eq . [ bi ] ) contains the matrix element of the nucleon electromagnetic current operator , @xmath129 ( eq . [ bm ] ) , evaluated between _ free _ nucleon dirac spinors . using the gordon decomposition , we may write @xmath130 where @xmath131 with @xmath132 and where @xmath133 is _ arbitrary_. in reference to the _ dirac _ character of the operators in the three terms ( as opposed to their _ lorentz _ character which is of course vectorial on all three cases ) , we have identified the first through third terms as `` scalar '' , `` vector '' and `` tensor '' , respectively . we may then transform to three distinct on - shell equivalent forms of @xmath129 each containing only two terms according to @xmath134 where , _ e.g. _ , `` vt '' means that only vector and tensor terms are present . ( note that , in evaluating the free nucleon electromagnetic tensor , @xmath135 of eq . [ bo ] , it is most convenient to employ the vs ( or @xmath136 ) form of @xmath137 . ) the impulse approximation used in section ii to evaluate the _ nuclear _ @xmath2 cross section does not in and of itself prescribe which representation of @xmath137 to use . this well - known ambiguity is problematic since as soon as the nucleon wave functions are modified by the nuclear medium , the various forms of @xmath137 are no longer equivalent . put another way , if we define the generalized electromagnetic response , @xmath138 , just as in eqs . [ bj ] through [ bl ] , except that @xmath137 is employed in eq . [ bl ] , we find that @xmath138 is not generally independent of @xmath133 . this represents an important qualification of the oft - repeated statement that the nuclear electromagnetic interaction is well - understood . ( there are , of course , any number of other effects which can modify the in - medium electromagnetic interaction ; we are at present restricting our attention to the ambiguities inherent in the _ impulse approximation_. ) it is worth noting that the @xmath133-dependence of electromagnetic observables is especially strong in the relativistic model of nuclear structure we will employ and which is outlined in section v. a definitive resolution of the @xmath133 ambiguity awaits a dynamical description of nucleon structure and how it is affected by the nuclear environment . we do not , of course , propose to solve that problem here . indeed , we simply adopt the vt or @xmath139 form of @xmath137 according to `` conventional wisdom''@xcite . since the treatment of @xmath0-nucleus scattering presented in section iii is also based on the impulse approximation applied to the underlying @xmath8 interaction , it is plagued by its own on - shell ambiguities . in the expression for the center - of - momentum frame @xmath8 cross section ( eq . [ cd ] ) , we find the free nucleon matrix element @xmath140u(p_i s_i)\nonumber\\ & = & \bar u(p_f s_f)\biggl\ { { \cal f}_s(\xi)\ { \bf 1}+{{p_\mu}\over m } \biggl[{\cal f}_v(\xi)\gamma^\mu+i{\cal f}_t(\xi ) { { \sigma^{\mu\nu}(p_f - p_i)_\nu}\over{2m}}\biggr]\biggl\ } u(p_i s_i ) \label{dc}\end{aligned}\ ] ] where @xmath141 and @xmath133 is again arbitrary . we have once more identified dirac scalar , vector and tensor terms and may also define scalar , vector and tensor forms of the interaction according to @xmath142 ( we note that , by choosing the vt form of the @xmath8 interaction , @xmath0 hadronic scattering becomes in plane wave born approximation _ formally _ identical to the ( fictitious ) @xmath0 electromagnetic process discussed in the beginning of section iii . ) as will be discussed in detail below , @xmath0-nucleus cross sections show some sensitivity to the form of the @xmath8 interaction . since there is no `` conventional wisdom '' to guide our choice as there was for electron scattering , we must appeal to a dynamical model of @xmath8 scattering for help in resolving the on - shell ambiguity . we turn specifically to the meson exchange model of @xmath8 scattering developed by the bonn group@xcite along the lines of their model of the @xmath9 interaction . this model has as its input parameters meson masses , @xmath0-meson and @xmath106-meson coupling constants and form factors . certain of the interactions implied by these couplings are iterated to all orders and the resulting phase shifts are in good agreement with those determined by experiment . for our purposes , we focus on the @xmath8 interactions mediated by the exchange of @xmath143 , @xmath144 and ( fictitious ) @xmath145 mesons . to proceed , we identify the t=0 ( isoscalar ) and t=1 ( isovector ) combinations of the @xmath8 elastic scattering amplitude . in terms of , _ e.g. _ , the wolfenstein amplitudes of eq . [ cf ] , we define @xmath146 where @xmath147 ( @xmath148 ) is the @xmath0-proton ( @xmath0-neutron ) amplitude . the isoscalar and isovector pieces of @xmath87 and @xmath88 then follow as described at the end of section iii . for @xmath149 targets , the @xmath0-nucleus cross section is simply the sum of t=0 and t=1 contributions given by eq . [ ce ] with different @xmath150 s and @xmath151 s for each isospin . in born approximation , @xmath143 exchange contributes to the isovector amplitudes while @xmath144 and @xmath145 exchange contribute to the isoscalar amplitudes . since the @xmath152 coupling has a vector - tensor dirac character ( with the tensor component dominant ) , the _ born _ isovector amplitude generated by @xmath143 exchange has a vt ( or , referring to eq . [ db ] , a @xmath153 ) form . numerically , the @xmath143 exchange born amplitude predicted by the bonn model at the kinematic point appropriate to the @xmath154 mev / c @xmath0-nucleus data , namely @xmath155 mev / c and @xmath156 degrees@xcite , is qualitatively consistent with the empirical amplitude@xcite we employ . ( specifically , dominance of @xmath157 over @xmath116 is observed . ) similarly , since the @xmath158 coupling is assumed to be purely vector , the born isoscalar amplitude due to @xmath145 and @xmath144 exchange has a vs ( or @xmath159 ) form . again , the born amplitudes from the bonn model are qualitatively consistent with the relevant empirical results . specifically , it is observed that @xmath115 and @xmath116 are of opposite signs ( the scalar interaction is attractive ) and are nearly equal in magnitude . the arguments presented above are clearly only suggestive . while we have argued that the form for the @xmath8 interaction employing the vs representation for the isoscalar amplitudes and the vt representation for the isovector amplitudes is physically most plausible , we will also show in the analysis to follow calculations which utilize the vs representation for both the isoscalar and isovector amplitudes . in this section we describe our nuclear structure model . throughout this paper we assume the impulse approximation to be valid . hence , apart from the ambiguities addressed in the previous section , the in - medium @xmath160 and @xmath161 couplings are entirely determined from on - shell data . moreover , we assume that both sets of couplings are small so that distortions and multi - step processes ( _ e.g. _ , multi - photon exchanges in the case of @xmath2 ) may be ignored . in this framework , all relevant nuclear structure information is contained in the responses of the nuclear target introduced in sections ii and iii to be discussed further in this section . the response of the nuclear target will be calculated in a relativistic random - phase - approximation to the walecka model . in the walecka model nucleons interact via the exchange of isoscalar scalar ( @xmath145 ) and vector ( @xmath144 ) fields . the dynamics of the system is , thus , described in terms of the following lagrangian density @xmath162 where @xmath163(@xmath164 ) is the scalar(vector ) coupling constant , @xmath28 , @xmath165 , and @xmath166 are the nucleon , @xmath145-meson , and @xmath144-meson masses , respectively , and @xmath167 , @xmath168 , and @xmath169 are the corresponding field operators . the term @xmath170 contains renormalization counterterms and the antisymmetric field - strength tensor , @xmath171 , has been defined by @xmath172 the nuclear ground state will be obtained in a relativistic mean field approximation to the walecka model . in this case , the meson - field operators are replaced by their ground - state expectation values . this approximation yields a set of dirac single - particle states that are determined self - consistently from the equations of motion . the one - body response of the nuclear ground state to an external probe is fully contained in the polarization tensor . the polarization tensor is a fundamental many - body operator that can be systematically computed using well - known many - body techniques ( e.g. , feynman diagrams ) . to illustrate these techniques we concentrate on the vector polarization , for simplicity this is defined as the ground - state expectation value of a time - ordered product of nuclear ( vector ) currents @xmath173 | i \rangle \;. \label{pimunu}\ ] ] in a mean - field approximation to the ground state the polarization insertion can be written , exclusively , in terms of the single - nucleon propagator @xmath174 @xmath175 \;. \label{pixy}\ ] ] the nucleon propagator contains information about the interaction of the nucleon with the average mean field provided by the nuclear medium . note that even if the interactions are ignored , such as in a fermi - gas description , the propagator would still be different than its free - space value because of the existence of a filled fermi sea . this fact suggests the following decomposition of the nucleon propagator @xmath176 the feynman part of the propagator , @xmath177 , has the same analytic structure as the free propagator , namely , antiparticle poles above the real axis , particle poles below the real axis , and residues proportional to the single - particle wave functions @xmath178 \;. \label{gfeyn}\ ] ] the density - dependent part of the propagator , @xmath179 , corrects @xmath177 for the presence of a filled fermi surface . formally , one effects this correction by shifting the position of the pole of every occupied state from below to above the real axis @xmath180 \\ & = & 2 \pi i \sum_{\alpha < { \rm f } } \delta\big(\omega - e_{\alpha}^{(+)}\big ) { u}_{\alpha}({\bf x } ) \overline{u}_{\alpha}({\bf y } ) \;. \label{gdens}\end{aligned}\ ] ] the decomposition of the nucleon propagator into feynman and density - dependent contributions suggests an equivalent decomposition for the polarization insertion @xmath181 the feynman part of the polarization , or vacuum polarization , @xmath182 , describes the excitation of nucleon - antinucleon ( @xmath183 ) pairs @xmath184 \;. \label{pifeynman}\ ] ] note that this contribution diverges and must be renormalized . a lowest order calculation of the response , however , requires of only the imaginary part of the polarization insertion . in infinite nuclear matter , the threshold for pair production is at @xmath185 ( @xmath186 is the effective nucleon mass in the medium ) . this timelike threshold lies far away from the spacelike region accessible in electron and kaon scattering . thus , a lowest - order description of the process is not sensitive to vacuum polarization . however , @xmath183 excitations can be virtually produced . hence , in a more sophisticated treatment of the response ( e.g. , rpa ) the effective coupling of the nucleon to the probe can be modified by vacuum polarization . indeed , it has been suggested that virtual @xmath183 pairs play an important role in the quenching of the coulomb sum@xcite . the density - dependent part of the polarization , @xmath187 , is finite and can be organized in terms of three distinct contributions @xmath188 with each one of them of at least linear in @xmath179 @xmath189 \big [ \overline{u}_{\alpha'}({\bf y})\gamma^{\nu}u_{\alpha}({\bf y } ) \big ] \delta\big(\omega+e_{\alpha}^{(+)}-e_{\alpha'}^{(+)}\big ) \;. \label{pidd}\end{aligned}\ ] ] the density - dependent part of the polarization describes the traditional excitation of particle - hole pairs . a spectral decomposition of the feynman propagator , for @xmath190 , is useful when discussing the spectral content of the polarization ( the @xmath191 term , with the opposite time ordering , contains the same physical information as @xmath190 ) @xmath192 \;. \label{pifdspect}\ ] ] the first term in the sum represents the formation of a particle - hole pair after the system has absorbed ( e.g. , a photon carrying ) energy @xmath144 . note , however , that some of these particle - hole transitions should be pauli - blocked since the feynman part of the propagator includes an unrestricted ( @xmath193 ) sum over all single - particle states . the role of @xmath194 in the present formalism is , precisely , to enforce the pauli principle . note that since the feynman part of the propagator will be evaluated nonspectrally , these pauli - forbidden transition can not be simply removed by hand . the density - dependent part of the polarization contains , in addition to particle - hole pairs , a contribution that has no nonrelativistic counterpart . this contribution is contained in the second term of the sum and represents the pauli blocking of @xmath183 excitations . recall that the feynman part of the polarization insertion represents the unconstrained excitation of @xmath183 pairs . at finite density , however , some of these excitations should be pauli - blocked . it is important to note that the inclusion of antinucleon degrees of freedom is not an unnecessary complication . if one is satisfied with computing the lowest order , or uncorrelated , nuclear response , then a `` nucleons - only '' approximation is certainly justified . if , however , one wishes to examine the role of correlations by means of an rpa response , then one is forced to include antinucleon degrees of freedom in order to satisfy fundamental physical principles such as gauge invariance . traditionally , relativistic calculations of the nuclear response have been carried out using two mean - field approximations to the walecka model . in the mean - field theory ( mft ) the feynman contribution to the single - particle propagator is neglected from the calculation of the nucleon self - energy . in contrast , one incorporates the effect from the ( full ) dirac sea in the relativistic hartree approximation ( rha ) . for a mean - field ground state obtained in the mft approximation , it has been shown that the consistent linear response of the mean - field ground state is obtained by neglecting the feynman part of the polarization insertion . this consistency is reflected , for example , in the proper treatment of spurious excitations associated with an overall translation of the center of mass . notice , however , that in the mft approximation one retains the pauli blocking of an ( @xmath195 ) excitation that has not been included from the outset . it has recently been shown that this approximation leads to severe inconsistencies in the description of the effective @xmath144-meson mass in the nuclear medium@xcite . thus , in this work we favor an rha treatment of the scattering process in which vacuum loops are included in both the description of the ground state as well as in the linear response of the system . we start with a discussion of the density - dependent contribution to the polarization . this contribution is finite and can be evaluated exactly in the finite system . according to eq . ( [ pidall ] ) we must compute self - consistently all occupied single - particle states and the feynman part of the propagator . the calculation proceeds by , first , calculating a set of occupied single - particle states satisfying the following dirac equation @xmath196u_{\alpha}({\bf x } ) = 0 \;. \label{speq}\ ] ] a hartree calculation of the mean - field ground state yields , in addition to the single - particle spectrum , the self - consistent ( scalar and vector ) mean - fields used to generate the spectrum @xmath197 knowledge of the self - consistent mean fields now enables one to compute nonspectrally the feynman part of the nucleon propagator by solving the equation @xmath198g_{f}({\bf x},{\bf y};\omega ) = \delta({\bf x}-{\bf y } ) \ ; , \label{greenseq}\ ] ] with the appropriate boundary conditions . the evaluation of the polarization insertion , although still highly nontrivial , gets simplified for the case of a spherically symmetric ground state . in this case , one can classify the single - particle states according to a generalized angular momentum @xmath199 @xmath200 where @xmath201 and we have introduced the spin - spherical harmonics defined by @xmath202 the feynman part of the propagator can be , similarly , written as a sum over partial waves @xmath203 once the bound - state orbitals and the feynman propagator have been determined , the evaluation of the polarization tensor in momentum space @xmath204 becomes straightforward . the angular integrals are done analytically leaving two radial integrals to be performed numerically . we stress that the procedure outlined above enables one to calculate the density - dependent part of the polarization exactly in the finite system . the feynman part of the polarization , however , must be calculated in a local - density approximation . to our knowledge , the renormalization of the divergent integrals has never been carried out in the finite system . thus , in the present work we adopt the following form for the feynman contribution to the response @xmath205 where @xmath206 is the renormalized vacuum polarization calculated in nuclear matter at an average momentum @xmath207 , and at a local value of the effective nucleon mass @xmath208 the nuclear response will be calculated in a variety of models and approximations . the most sophisticated calculation that we will present involves calculating the nuclear response in a relativistic random phase approximation ( rpa ) to the walecka model . in rpa one incorporates many - body correlations through an infinite summation of the lowest order ( or uncorrelated ) polarization . due to scalar - vector mixing the rpa equations form a set of @xmath209 coupled integral equations @xmath210 where we have introduced latin indices @xmath211 that run over scalar and vector lorentz structures , and a residual interaction @xmath212 given by @xmath213 note that the free vector and scalar propagators have been defined , respectively , by @xmath214 the rpa equations are solved for every spin and parity @xmath215 by , first , performing the radial ( @xmath216)-integral using a gauss quadrature scheme and then solving the resulting matrix equation using standard matrix - inversion techniques . we conclude this section with a brief discussion of the response of infinite nuclear matter . due to the translational - invariant character of nuclear matter the previous discussion simplifies considerably . in a mean - field approximation to the walecka model the meson - field operators are replaced by their classical ground - state expectation values which are constants in nuclear matter @xmath217 the ground - state of the system is , thus , characterized by a filled fermi ( and dirac ) sea of nucleons with an effective mass @xmath186 determined self - consistently from the equations of motion @xmath218 and effective nucleon and antinucleon energies which are shifted by the presence of a constant vector field @xmath219 in particular , this implies that the nucleon propagator is , formally , indistinguishable from the free nucleon propagator . the feynman and density - dependent propagators , which are the basic building blocks for the response , are thus given , respectively , by @xmath220 \ ; , \\ \label{gpf } g_{d}(k ) & = & \big ( { \rlap/\bar{k } } + m^ { * } \big ) \left [ { i\pi \over e^{*}_{\bf k } } \delta\big(\bar{k}^{0}-e^{*}_{\bf k}\big ) \theta\big({\rm k}_{\rm f}-|{\bf k}|\big ) \right ] \ ; , \label{gpd}\end{aligned}\ ] ] where @xmath221 is the fermi momentum and we have defined @xmath222 from the nucleon propagator it is simple to construct the lowest - order nuclear response @xmath223 \;. \label{pinm}\ ] ] as before , the polarization contains a divergent feynman component that must be renormalized , and a finite density - dependent contribution that describes particle - hole excitations and the pauli blocking of @xmath183 pairs . from this lowest order polarization one computes the correlated response by solving the rpa equation which becomes , in nuclear matter , a simple algebraic equation . the only ingredients that remain to be specified are the effective number of nucleons and for nuclear - matter calculations the average density at which the scattering occurs . these quantities are determined from eikonal formulae that read : @xmath224 where @xmath145 is the elementary projectile - nucleon total cross section and @xmath225 is the nuclear - thickness function defined by @xmath226 from the effective density a self - consistent nucleon mass @xmath186 is determined @xcite which then serves as input for the calculation of the various nuclear responses per nucleon . these responses are subsequently scaled by the effective number of nucleons @xmath227 and then compared to experiment . in the case of @xmath228-nucleus scattering we ignore the small electromagnetic distortions ( i.e. , assume @xmath229 ) and compute for @xmath230ca : @xmath231 , @xmath232 @xmath233 , and @xmath234 . the equivalent expressions for @xmath235-nucleus scattering ( where the isospin - averaged total cross section is @xmath236 mb ) become : @xmath237 , @xmath238 @xmath233 , and @xmath239 . ( note that we use _ experimentally _ determined@xcite values of @xmath227 to normalize the @xmath0 quasielastic calculations to be presented below . the reason for the discrepancy between our eikonal estimates and the experimental value of @xmath240 is not understood at present . ) we begin our comparison of @xmath1 and @xmath0 quasielastic scattering by presenting calculations of the longitudinal ( or coulomb ) and transverse @xmath2 responses for @xmath241 at @xmath242 mev / c in figures [ figa ] and [ figb ] , respectively . the data are from reference @xcite . all calculations are based on the relativistic hartree approximation ( rha ) to qhd as described in the previous section and thus include effects due to polarization of the nucleon sea at the one - loop level . we show both nuclear matter ( nm ) and finite nucleus ( fn ) calculations without ( har ) and with ( rpa ) rpa correlations . figure [ figa ] clearly shows the dramatic quenching of the longitudinal response due to the rpa which results in reasonable agreement with experiment . as explained in , _ e.g. _ , ref . @xcite , this quenching is most readily interpreted as a screening of the nucleon charge due to polarization of the nucleon sea . the magnitude of the quenching is comparable for nm and fn calculations but an acceptable description of the shape of the measured response evidently requires inclusion of finite nucleus effects . figure [ figb ] compares our calculations with the measured transverse response . since , as for all calculations reported here , we have included isoscalar correlations only , the differences between har and rpa results are small for this predominantly isovector response . good agreement with experiment is found for the low - energy side of the quasielastic peak , especially for the finite nucleus calculations . the underestimation of transverse strength on the high - energy side of the peak , believed to be dominated by isobar formation and meson - exchange currents , is a common shortcoming of most `` one - nucleon '' models such as ours . comparable calculations are displayed with the @xmath0 data@xcite in figures [ figc ] and [ figd ] where we employ the vs and mixed vs ( for t=0 ) and vt ( for t=1 ) representations , respectively , for the @xmath8 @xmath81-matrix as discussed in section iv . in figure [ figc ] the calculations are normalized using @xmath243 which is the mean value extracted from experiment@xcite . the calculations appearing in figure [ figd ] use @xmath244 which is the experimental upper limit . the agreement between our most complete calculations , namely those labelled fn(rpa ) , and the data is excellent although the vs calculations reproduce the overall magnitude of the measured cross sections slightly better than the mixed vs - vt results which slightly underestimate the measurements . we note that , while the simple mass @xmath28 fermi - gas result reproduces the observed _ peak _ @xmath0 quasielastic cross section of @xmath245 mb / sr / mev using the eikonal value of @xmath246 , its accounting of the _ shape _ of the cross section is poor . in particular , the mass @xmath28 fermi - gas cross section peaks at @xmath247 mev and drops too rapidly at high @xmath144 , vanishing at @xmath248 mev . for this reason , the mass @xmath28 fermi - gas calculations must be considered inadequate regardless of normalization . returning to the full calculations , we see that two differences relative to the @xmath2 calculations are immediately apparent . first , nm and fn calculations are much more alike for @xmath0 than for @xmath2 . second and more striking is the fact that rpa effects are relatively small for the @xmath0 cross sections in contrast to the large quenching they generate for the @xmath2 longitudinal response . this difference is even more surprizing since the @xmath0 cross section is _ dominated _ by `` longitudinal '' contributions ( _ i.e. _ , by terms in eq . [ ce ] not involving @xmath125 ) which are closely related to the @xmath2 longitudinal response . we will show that this difference arises from a subtle interplay of kinematic and relativistic nuclear structure effects . in the course of this discussion , we will see that the lack of strong quenching in @xmath0 quasielastic scattering is quite remarkable since the additional `` longitudinal '' responses appearing in eq . [ ce ] , namely @xmath122 and @xmath123 , are quenched _ even more _ than @xmath124 to which the @xmath2 longitudinal response is most closely related . to understand the important physical contributions which determine the @xmath2 longitudinal response and the @xmath0 quasielastic cross section , we focus on eq . [ ce ] , our plane - wave expression for the latter . with this formula as our starting point , we can define @xmath249 \label{ffa}\ ] ] where @xmath250 , @xmath251 , @xmath252 and @xmath253 the `` @xmath254 '' subscript indicates we have dropped all transverse ( _ i.e. _ , @xmath255 ) contributions , retaining only `` longitudinal '' terms . we furthermore consider only isoscalar ( @xmath256 ) contributions since it is here that strong rpa correlations appear . these restrictions are intended to simplify the analysis by focussing exclusively on the physics of greatest interest . we recall that the isoscalar coulomb response is roughly half of the full response . to assess the relative importance of the longitudinal isoscalar contribution to the @xmath0 cross sections , we observe that , for @xmath8 scattering from a free nucleon , the longitudinal @xmath256 and @xmath257 contributions are @xmath258 and @xmath259 mb / sr , respectively , while the corresponding transverse contributions are @xmath260 and @xmath261 mb / sr . this shows that the longitudinal isoscalar contribution accounts for about two thirds of the total . thus the isoscalar longitudinal contributions to which we temporarily restrict our attention for the sake of simplicity are very significant components of the measured @xmath1 and @xmath0 quantities . let us now consider the specific case of @xmath0 scattering from a free nucleon of mass @xmath28 originally at rest . then eq . [ ffa ] can be expressed as @xmath262 where we have used the fact ( eq . [ be ] ) that @xmath263 which also implies @xmath264 . in this expression we have also defined @xmath265 and @xmath266 and have redefined @xmath267 , all at the particular kinematics of this specific case . we now modify this situation by letting @xmath268 . since the kinematics change , we define @xmath269 and @xmath270 where now the `` starred '' quantities refer to the _ new _ kinematics of the @xmath268 case while the `` zero '' designation is for the mass @xmath28 kinematics . we still have @xmath264 and can write @xmath271 next we let @xmath272 and write @xmath273 and @xmath274 then , because @xmath275 is an excellent approximation , we have @xmath276 where @xmath277 is now equal to the integrated uncorrelated nuclear matter cross section per nucleon obtained from the cross sections designated in figures [ figc ] and [ figd ] by nm ( har ) . finally we consider the effects of the rpa . with @xmath278 and @xmath279 we find @xmath280 where @xmath281 is equal to the integrated uncorrelated nuclear matter cross section per nucleon designated in figures [ figc ] and [ figd ] by nm ( rpa ) . we also observe that the uncorrelated integrated longitudinal @xmath2 response per nucleon ; _ i.e. _ , the uncorrelated coulomb sum per nucleon , is given by @xmath282 its correlated counterpart is @xmath283 the point of this formulation is that , by examining the numerical values of @xmath284 , @xmath285 and @xmath286 as well as the behavior of @xmath287 , we can understand the physical relationships between the @xmath0 cross section and the longitudinal @xmath2 response . in pursuit of this understanding we first determine that @xmath288 for @xmath289 mev / c and @xmath290 which is the self - consistent nuclear matter value for the average density of @xmath3ca . ( note that , because of @xmath0 absorption in the nuclear medium , we should use the slightly larger value of @xmath291 appropriate to the lower average density at which the @xmath8 interaction occurs for quasielastic scattering . however , differences are small and we use the overall average density of @xmath3ca , which is appropriate for electron scattering , in both cases so as to facilitate the following comparisons . ) it remains to establish the behavior of the @xmath286 and the @xmath285 or , more relevantly , the values of @xmath292 as we include the effects of ( _ i _ ) @xmath293 , ( _ ii _ ) @xmath294 and ( _ iii _ ) rpa correlations . these behaviors are summarized in figure [ fige ] where we plot @xmath286 , @xmath292 and @xmath295 as they evolve from case ( _ i _ ) through case ( _ iii _ ) . we see that @xmath293 causes only a very small change in @xmath193 . hence , the ratio of the corresponding coulomb sums , @xmath296 , is nearly unity which implies that changing @xmath297 from @xmath28 , by itself , has very little consequence for electron scattering . the effect on the integrated @xmath0 quasielastic cross section is appreciable , however ; we find @xmath298 . this large effect is traceable to the delicate cancellation between the first two and the third terms of eq . [ ce ] and the changes in the kinematic factors multiplying these terms brought about by @xmath293 . overall , taking into account the effect of @xmath284 , the @xmath0 cross section is reduced by @xmath299 simply by letting @xmath300 . we next examine the effect of @xmath294 . the @xmath193 factor does not change as the coulomb strength is merely redistributed . however , now @xmath301 due to the lorentz contraction of the scalar density and in consequence the delicate cancellation alluded to above is somewhat altered . we find @xmath302 and the uncorrelated @xmath0 cross section goes up slightly so that @xmath303 . we finally consider the influence of the rpa correlations . as shown in figure [ fige ] , they cause @xmath193 to drop dramatically to @xmath304 ! this just reflects the strong rpa quenching of the coulomb response which , as mentioned above , is a well - known feature of the rha rpa@xcite . we further observe that @xmath305 goes from @xmath306 without correlations to @xmath307 with them which means that the summed responses @xmath308 and @xmath309 are quenched even more strongly by the rpa than is @xmath310 ! indeed , @xmath309 is reduced by a factor of @xmath311 . this would seem to imply a _ strong _ quenching of the @xmath0 cross section . however , as is evident in figure [ fige ] , such is not the case because @xmath312 , an increase which offsets the rpa quenching to the degree that @xmath313 drops only from @xmath314 to @xmath315 ( _ i.e. _ , by a factor of 0.91 ) when rpa effects are included . the large increase in @xmath287 in this case is due to the large reduction in @xmath305 , that is , due to _ differential _ quenching of time - like vector and scalar contributions in the rpa . if no such differential quenching were present , @xmath313 would necessarily decrease like @xmath193;_i.e . _ , like the coulomb sum . as the distinction between scalar and vector contributions is purely relativistic , it is hard to see how non - relativistic models of nuclear structure could simultaneously account for the strong quenching of the coulomb sum and the absence of quenching in the @xmath0 quasielastic scattering cross section in a manner as natural as for the present relativistic model . the preceeding analysis is based on a number of simplifying assumptions which can be tested , _ e.g. _ , by comparing with the results of more complete nuclear matter calculations which as we have already established are quite consistent with the full finite nucleus results . figure [ fige ] compares the ratio of the integrated nuclear matter cross sections labelled `` nm '' with the @xmath316 ratio discussed above . the agreement is good enough to inspire confidence in the simplified analysis . overall , we find that the _ full _ integrated rha - rpa @xmath0 quasielastic cross section per ( effective ) nucleon which includes isoscalar transverse as well as isovector contributions _ is _ reduced from the ( isospin averaged ) @xmath8 cross section . for the vs representation of the @xmath8 amplitude , the reduction factor is 0.94 while for the mixed representation the factor is 0.82 . these differences are clearly due to the different behavior of the isovector contributions , namely that they are appreciably enhanced by @xmath268 in the vs representation but little changed in the vt representation . by comparison , the full integrated rha - rpa coulomb response per nucleon is reduced from the isospin averaged single nucleon value by a factor of 0.76 . clearly , while the @xmath0 reductions are less than for the coulomb , they are not dramatically different , especially when the mixed representation is used for the former . however , it is important to observe that the reduction of the @xmath0 cross section is not due just to the quenching of the underlying responses but depends also on the interference effects discussed above . most importantly , if it were not for the _ differential quenching _ of scalar versus time - like vector contributions which emerges naturally in our relativistic model of nuclear structure , the reduction factor for the @xmath0 cross section would be _ much _ smaller and therefore inconsistent with the @xmath0 quasielastic data . as it is , we have a gratifyingly accurate and consistent description of both the @xmath0 and @xmath2 data . we have formulated a treatment of @xmath0-nucleus quasielastic scattering in a manner which parallels as closely as possible more - or - less standard treatments of @xmath1-nucleus quasielastic scattering . the latter depends in a straightforward way on the coulomb ( or longitudinal ) and transverse nuclear responses which in turn are of great importance in understanding essentials of nuclear structure . we have shown that in the present formulation @xmath0 quasielastic scattering depends on these same dirac vector and tensor nuclear responses as well as additional ones containing dirac _ scalar _ contributions . thus , in principle , @xmath0 quasielastic data can supplement and extend structure information extracted from the electron data and perhaps shed light on important issues such as the strong quenching of the coulomb sum@xcite . our treatment of the underlying @xmath8 interaction relies on the impulse approximation and we have been careful to spell out the connection between the @xmath8 amplitudes appearing in our expression for the @xmath0-nucleus quasielastic cross section , eq . [ ce ] , and @xmath8 phase shift solutions@xcite . we have also briefly summarized problems associated with on - shell ambiguities in the form of the @xmath317 amplitude @xcite and have indicated how these problems carry over to the form of the @xmath8 amplitude . we rely on a meson - exchange model of the @xmath8 interaction@xcite to justify a specific form of this interaction expressed as a `` mixture '' of dirac vector and scalar invariants for the isoscalar channel and vector and tensor invariants for the isovector channel . our nuclear structure model is based on quantum hadrodynamics , a successful relativistic phenomenology of nuclear dynamics . we specifically focus on the relativistic hartree approximation@xcite and the rpa based on it@xcite . this treatment takes into account the polarization of the nucleon sea in one - loop approximation and in so doing provides a unique mechanism for quenching the coulomb response which is found to be in reasonable accord with experiment . we have given a thorough discussion of both full finite nucleus and nuclear matter calculations of the nuclear responses in the rha - rpa . we also have indicated how to fix the _ effective _ nuclear densities and @xmath291 values for the nuclear matter treatment of @xmath0 quasielastic scattering which is complicated by the absorption of the @xmath0 scattering waves . we have compared our calculations with the coulomb and transverse responses for @xmath3ca at @xmath4 mev / c@xcite . we reproduce the low-@xmath144 side of the transverse response quite well , but , as is typical of `` one - nucleon '' models such as ours , we underestimate the response on the high-@xmath144 side , presumably due to the omission of meson - exchange - current and @xmath318-isobar effects . rpa effects strongly quench the coulomb response relative to the uncorrelated results and bring about reasonable agreement with the data . this quenching can be interpreted as a screening of the nucleon charge due to polarization of the nucleon sea . finite nucleus effects appear to be important in reproducing the details of the shape of the measured coulomb response . we have also shown similar calculations for the new @xmath319 data at the same momentum transfer @xcite . here the quenching due to the rpa is much less than for the coulomb response and differences between finite nucleus and nuclear matter calculations are smaller . agreement of the full rha - rpa calculations with the measured cross sections is very good although the calculations slightly underestimate the data when the `` mixed '' representation of the @xmath8 amplitude is employed . we have gone on to explain why the rpa quenching of the @xmath0 cross section is so much less than what is observed for the coulomb response , a phenomenon which is all the more surprizing given the dominanance of the `` longitudinal isoscalar '' contribution in the former which is where rpa effects occur in our model . the situation becomes even more puzzling upon observing that the _ new _ responses containing scalar contributions which arise in the case of @xmath0 scattering ( see eq . [ ce ] ) are even more strongly quenched than the coulomb response . however , careful analysis shows that this _ differential quenching _ of responses alters a sensitive cancellation in the expression for the @xmath0 cross section in such a way that the cross section is only slightly reduced . the situation is qualitatively unchanged for the full @xmath0 quasielastic cross section which also includes transverse isoscalar and isovector components . we note that the phenomenon of differential quenching is purely relativistic in origin and that , without it , the calculated @xmath0 cross sections would be much smaller and in strong disagreement with experiment . we have concluded that our relativistic model of nuclear structure provides a gratifyingly accurate and consistent description of both @xmath0-nucleus and @xmath1-nucleus quasielastic scattering . 0.75 true in the authors gratefully acknowledge helpful comments by s. pollock and v. dmitrasinovi ' c. this work supported in part by the u.s.d.o.e .

we formulate @xmath0-nucleus quasielastic scattering in a manner which closely parallels standard treatments of @xmath1-nucleus quasielastic scattering . for @xmath0 scattering , new responses involving scalar contributions appear in addition to the coulomb ( or longitudinal ) and transverse @xmath2 responses which are of vector character . we compute these responses using both nuclear matter and finite nucleus versions of the relativistic hartree approximation to quantum hadrodynamics including rpa correlations . overall agreement with measured @xmath2 responses and new @xmath0 quasielastic scattering data for @xmath3ca at @xmath4 mev / c is good . strong rpa quenching is essential for agreement with the coulomb response . this quenching is notably less for the @xmath0 cross section even though the new scalar contributions are even more strongly quenched than the vector contributions . we show that this `` differential quenching '' alters sensitive cancellations in the expression for the @xmath0 cross section so that it is reduced much less than the individual responses . we emphasize the role of the purely relativistic distinction between vector and scalar contributions in obtaining an accurate and consistent description of the @xmath2 and @xmath0 data within the framework of our nuclear structure model . = 10000 = 10000 # 1#2#1#2 # 1to 0pt-.6em#1 comparison of @xmath0 and @xmath1 quasielastic scattering supercomputer computations research institute , florida state university , tallahassee fl 32306 department of physics , university of colorado , boulder , co 80309 - 0446

this article is concerned with the approximation of the distribution of markov processes conditioned to not hit a given absorbing state . let @xmath0 be a discrete time markov process evolving in a state space @xmath1 , where @xmath2 is an absorbing state , which means that @xmath3 where @xmath4 . our first aim is to provide an approximation method based on an interacting particle system for the conditional distribution @xmath5 where @xmath6 denotes the law of @xmath0 with initial distribution @xmath7 on @xmath8 . our only assumption to achieve our aim will be that survival during a given finite time is possible from any state @xmath9 , which means that @xmath10 our second aim is to provide a general condition ensuring that the approximation method is uniform in time . the main assumption will be that there exist positive constants @xmath11 and @xmath12 such that , for any initial distributions @xmath13 and @xmath14 , @xmath15 this property has been extensively studied in @xcite . in particular , it is known to imply the existence of a unique quasi - stationary distribution for the process @xmath0 on @xmath8 . another main result of our paper is that , under mild assumptions , the approximation method can be used to estimate this quasi - stationary distribution . the nave monte - carlo approach to approximate such distributions would be to consider @xmath16 independent interacting particles @xmath17 evolving following the law of @xmath0 under @xmath6 and to use the following asymptotic relation @xmath18 and then @xmath19 however , the number of particles remaining in @xmath8 typically decreases exponentially fast , so that , at any time @xmath20 , the actual number of particles that are used to approximate @xmath21 is of order @xmath22 for some @xmath23 . as a consequence , the variance of the right hand term typically grows exponentially fast and then the precision of the monte - carlo method worsens dramatically over time . in fact , for a finite number of particles @xmath24 , the number of particles @xmath25 belonging to @xmath8 eventually vanishes in finite time with probability one . thus the right hand term in the above equation eventually becomes undefined . since we re typically interested in the long time behavior of or in methods that need to evolve without interruption for a long time , the nave monte carlo method is definitely not well suited to fulfill our objective . in order to overcome this difficulty , modified monte - carlo methods have been introduced in the recent past years by del moral for discrete time markov processes ( see for instance @xcite or the well documented web page @xcite , with many applications of such modified monte - carlo method ) . the main idea is to consider independent particles @xmath26 evolving in @xmath8 following the law of @xmath0 , but such that , at each time @xmath27 , any absorbed particle is re - introduced to the position of one other particle , chosen uniformly among those remaining in @xmath8 ; then the particles evolve independently from each others and so on . while this method is powerful , one drawback is that , at some random time @xmath28 , all the particles will eventually be absorbed simultaneously . at this time , the interacting particle system is stopped and there is no natural way to reintroduce all the particles at time @xmath29 . when the number of particles is large and the probability of absorption is uniformly bounded away from zero , the time @xmath28 is typically very large and this explain the great success of this method . however , many situations does not enter the scope of these assumptions , such as diffusion processes picked at discrete times or the neutron transport approximation ( see section [ sec : example_neutron ] ) . our method is non - failable in these situations . moreover the uniform convergence theorem provided in section [ sec : main2 ] also holds in these cases , under suitable assumptions . when the underlying process is a continuous time process , one alternative to the methods of @xcite has been introduced recently . the idea is to consider a continuous time @xmath24-particles system , where the particles evolve independently until one ( and only one ) of them is absorbed . at this time , the unique absorbed particle is re - introduced to the position of one other particle , chosen uniformly among those remaining in @xmath8 . this continuous time system , introduced by burdzy , holyst , ingermann and march ( see for instance @xcite ) , can be used to approximate the distribution of diffusion processes conditioned not to hit a boundary . unfortunately , it yields two new difficulties . the first one is that it only works if the number of jumps does not explode in finite time almost surely ( which is not always the case even in non - trivial situations , see for instance @xcite ) . the second one is that , when it is implemented numerically , one has to compute the exact absorption time of each particles , which can be cumbersome for diffusion processes and complicated boundaries . note that , when this difficulties are overcome , the empirical distribution of the process is known to converge to the conditional distribution ( see for instance the general result @xcite and the particular cases handled in @xcite ) . finally , it appears that both methods are not applicable in the generality we aim to achieve in the present paper and , in some cases , both method will fail ( as in the case of the neutron transport example of section [ sec : example_neutron ] ) . let us now describe the original algorithm studied in the present paper . fix @xmath30 . the particle system that we introduce is a discrete time markov process @xmath31 evolving in @xmath32 . we describe its dynamic between two successive times @xmath33 and @xmath34 , knowing @xmath35 , by considering the following random algorithm which act on any @xmath24-uplet of the form @xmath36 * algorithm 1 . * initiate @xmath37 by setting @xmath38 and @xmath39 for all @xmath40 and repeat the following steps until @xmath41 for all @xmath40 . 1 . choose randomly an index @xmath42 uniformly among @xmath43 2 . choose randomly a position @xmath44 according to @xmath45 . then * if @xmath46 , chose an index @xmath47 among @xmath48 and replace @xmath49 by @xmath50 in @xmath37 . * if @xmath51 , replace @xmath49 by @xmath52 in @xmath37 . after a ( random ) finite number of iterations , the @xmath24-uplet @xmath37 will satisfy @xmath41 for all @xmath40 . when this is achieved , we set @xmath53 . our first main result , stated in section [ sec : main1 ] , is that , for all @xmath20 , the empirical distribution of the particle system evolving following the above dynamic actually converges to the conditional distribution of the original process @xmath0 at time @xmath33 . we prove this result by building a continuous time markov process @xmath54 such that @xmath55 is distributed as @xmath56 for all entire time @xmath27 , and such that the general convergence result of @xcite applies . our second main result , stated in section [ sec : main2 ] , shows that , if the conditional distribution of the process @xmath0 is exponentially mixing ( in the sense of @xcite or @xcite for the time - inhomogeneous setting ) and under a non - degeneracy condition that is usually satisfied , then the approximation method converges uniformly in time . in section [ sec : example_neutron ] , we illustrate our method by proving that it applies to neutron transport process absorbed at the boundary of an open set @xmath57 . in this section , we consider the particle system defined by algorithm 1 . we state and prove our main result in a general setting . [ thm : intro - main ] assume that @xmath58 converges in law to a probability measure @xmath59 on @xmath8 . then , for any @xmath60 and any bounded continuous function @xmath61 , @xmath62{law } \mathbb{e}_{\mu_0}(f(x_n ) | n<\tau_\partial).\ ] ] moreover , @xmath63 we emphasize that our result applies to any process @xmath0 satisfying , overcoming the limitations of all previously cited particle approximation methods , as illustrated by the application to a neutron transport process in section [ sec : example_neutron ] . the proof is divided in two steps . first , we provide an implementation of algorithm 1 as the discrete time included chain of a continuous time fleming - viot type particle system . in particular , this step provides a mathematically tractable implementation of algorithm 1 . in a second step , we use existing results on fleming - viot type particle systems to deduce that the empirical distribution of the particle system converges to the conditional distribution . _ step 1 : algorithm 1 as a fleming - viot type process _ + let us introduce the continuous time process @xmath54 defined , for any @xmath64 , by @xmath65}1_{t < u_{[t ] } } + x_{[t]+1}1_{t \geq u_{[t]}}\ ] ] where @xmath66 $ ] denotes the integer part and @xmath67 is a family of independent random variables such that , for all @xmath60 , @xmath68 follows a uniform law on @xmath69 $ ] . with this definition , @xmath70 is a non - markovian continuous time process such that @xmath55 and @xmath56 have the same law for all @xmath27 . now , we define the continuous time process @xmath71 by @xmath72 by construction , the continuous - time process @xmath73 defined by @xmath74 is a strong markov process evolving in @xmath75 , with absorbing set @xmath76 ( see figure [ fig : graph ] for an illustration when @xmath77 and @xmath78 ) . and the thin lines are the trajectory of @xmath79 , which jumps from @xmath56 to @xmath80 at time @xmath68 . at any time @xmath81 , @xmath82 is equal to @xmath83 if the process has already jumped during any interval @xmath84 $ ] and is equal to @xmath85 otherwise . ] let us now define a fleming - viot type system whose particles evolve as independent copies of @xmath86 between their absorption times . more precisely , fix @xmath30 and consider the following continuous time fleming - viot type particle system , denoted by @xmath87 , starting from @xmath88 and evolving as follows . * the @xmath24 particles evolve as @xmath24 independent copies of @xmath86 until one of them reaches @xmath89 . note that it is clear from the definition of @xmath86 that only one particle jumps at this time . * then the unique killed particle is taken from the absorbing point @xmath89 and is instantaneously placed at the position of an other particle chosen uniformly between the @xmath90 remaining ones ; in this situation we say that the particle undergoes a _ rebirth_. * then the particles evolve as independent copies of @xmath86 until one of them reaches @xmath89 and so on . for all @xmath91 , we denote by @xmath92 the number of rebirths of the @xmath93 particle occurring before time @xmath94 and by @xmath95 the total number of rebirths before the time @xmath94 . clearly , @xmath96 also , for all @xmath91 , we set @xmath97 , where @xmath98 and @xmath99 are the marginal component of @xmath100 in @xmath8 and @xmath101 respectively . one can easily check that this fleming - viot system ( considered at discrete times ) is a particular implementation of the informal description of algorithm 1 in the introduction . indeed , at any time @xmath20 , the fleming - viot system is defined so that @xmath102 then , at each time @xmath81 , the index of the next moving particle @xmath103 belongs to the set of particles @xmath104 such that @xmath105 . moreover , conditionally to @xmath106 , the jumping times of these particles are independent and identically distributed ( uniformly on @xmath107 ) . as a consequence @xmath108 is chosen uniformly among these indexes ( this is the first step of algorithm 1 ) . then , at the jumping time @xmath109 , the position @xmath110 of the particle @xmath108 is chosen according to @xmath111 and @xmath112 is set to @xmath83 . if the position at time @xmath109 is @xmath113 , then the particle @xmath108 undergoes a rebirth and hence @xmath114 is replaced by @xmath115 , where @xmath116 is chosen uniformly among @xmath117 . hence the second step of algorithm 1 is completed . finally , the procedure is repeated until all the marginals @xmath118 are equal to @xmath83 , as in algorithm 1 . in particular , for any @xmath20 , the random variable @xmath119 obtained from algorithm 1 and the variable @xmath120 obtained from the fleming - viot type algorithm have the same law . _ step 2 : convergence of the empirical system . _ in this step , we consider a sequence of initial positions @xmath121 such that @xmath122 converges in law to a probability measure @xmath59 on @xmath8 . our aim is to prove that , for any @xmath60 and any bounded continuous function @xmath61 , @xmath123{law } \mathbb{e}_{\mu_0}(f(y_n ) | n<\tau_\partial).\ ] ] note that , since @xmath56 and @xmath55 share the same law , this immediately implies the first part of theorem [ thm : intro - main ] . since @xmath124 is a fleming - viot type process without simultaneous killings , ( * ? ? ? * theorem 2.2 ) implies that it is sufficient to prove that , for all @xmath30 and almost surely , @xmath125 where we recall that @xmath126 is number of rebirths undergone by the fleming - viot type system with @xmath24 particles before time @xmath94 . first , let us remark that @xmath127 + 1}^n = \infty).\ ] ] moreover , @xmath128 + 1}^n = \infty ) & = \mathbb{p}\left(\bigcup_{i=0}^{[t]}\{a^n_{i+1}-a_i^n = \infty\}\right ) \leqslant \sum_{i=0}^{[t]}\mathbb{p}\left(a_{i+1}^n - a_i^n = \infty\right).\end{aligned}\ ] ] using the weak markov property at time @xmath108 , it is sufficient to prove that @xmath129 for any initial distribution of the fleming - viot type process @xmath130 in order to conclude that @xmath131 and hence that holds true . but @xmath132 if and only if there exists at least one particle for which there is an infinity of rebirths , hence @xmath133 now , when a particle undergoes a rebirth , it jumps on the position of one of the @xmath90 remaining particles . as a consequence , at any time @xmath134 , the position @xmath98 of the particles @xmath40 such that @xmath135 are included in the set @xmath136 . in particular , the probability that such a particle undergoes a rebirth during its next move is bounded above by @xmath137 hence , a classical renewal argument shows that the probability that a particle undergoes @xmath33 rebirths is bounded above by @xmath138 . this implies that the probability that a particle undergoes an infinity of rebirths is zero . this , together with implies , which concludes the proof of the first part of theorem [ thm : intro - main ] . in order to conclude the proof , let us simply remark that , for a deterministic value of @xmath139 , the inequality of theorem [ thm : intro - main ] is directly provided by ( * ? ? ? * theorem 2.2 ) . now , if @xmath139 is a random measure , the inequality is obtained by integrating the deterministic case inequality with respect to the law of @xmath139 . this concludes the proof of theorem [ thm : intro - main ] . in a recent paper @xcite , necessary and sufficient conditions on an absorbed markov process @xmath0 were obtained to ensure that a process satisfies @xmath140 where @xmath12 and @xmath11 are positive constants . in particular this implies the existence of a unique quasi - stationary distribution @xmath141 for @xmath0 , that is a unique probability measure on @xmath8 such that @xmath142 , for all @xmath20 . general and classical results on quasi - stationary distributions ( see for instance @xcite ) implies that there exists @xmath143 such that @xmath144 the exponential convergence property [ eq : expo - cv ] holds for a large class of processes , including birth and death processes with catastrophe , branching brownian particles , neutron transport approximations processes ( see @xcite ) , one dimensional diffusion with or without killing ( see @xcite ) , multi - dimensional birth and death processes ( see @xcite ) and multi - dimensional diffusion processes ( see @xcite ) . also , similar properties can be proved for time - inhomogeneous processes , as stressed in the recent paper @xcite , with applications to time - inhomogeneous diffusion processes and birth and death processes in a quenched random environment . in this section , we state and prove our second main result , which states that , if holds and if , for any @xmath145 , there exists @xmath146 such that @xmath147 then the convergence of the empirical distribution of the particle system described in algorithm 1 converges uniformly in time to the conditional distribution of the process @xmath0 . we emphasize that the additional assumption is true for many processes satisfying , for instance in the case of one - dimensional diffusion processes , multidimensional diffusion processes or piecewise deterministic markov processes ( this the detailed examples of section [ sec : example_neutron ] ) . as far as we know , none of this processes were covered in this generality by previous methods . in particular , this is the first method that allows the approximation of the conditional distribution of the neutron transport approximation process ( see section [ sec : example_neutron ] ) , since in this case it easy to check that , with probability one , all the particles will eventually hit the boundary at the same time when using previous algorithms . the methods also allows to handle the case of the diffusion process on @xmath148 $ ] killed at @xmath85 , reflected at @xmath149 and solution to the following stochastic differential equation @xmath150,\ \beta>2.\end{aligned}\ ] ] in this case , the continuous time fleming - viot approximation method introduced in @xcite explodes in finite time almost surely , as proved in @xcite . as a matter of fact , it is not known if the fleming - viot type particle system is well defined as soon as the diffusion coefficient is degenerated or not regular toward the boundary @xmath85 . on the contrary , our assumption holds true for fairly general one dimensional diffusion processes , thanks to the study provided in @xcite . hence our approximation method is valid and , using the next results , converges uniformly in time for both neutron transport processes and degenerate diffusion processes . for any @xmath27 , we define the empirical distribution of the process at time @xmath33 as @xmath151 , and for any bounded measurable function @xmath152 on @xmath8 , we set @xmath153 [ thm : main - two ] assume that and holds true . then there exist two constants @xmath154 and @xmath155 , such that , for all @xmath156 and all measurable function @xmath157 bounded by @xmath83 , @xmath158 with @xmath159 in the case where the initial position of the particle system are drawn as independent random variables distributed following the same law @xmath7 , then , choosing @xmath156 small enough so that @xmath160 , basic concentration inequalities and the equality @xmath161 imply that @xmath162 for some @xmath163 . this implies the following corollary . assume that @xmath164 are independent and identically distributed following a given law @xmath7 on @xmath8 . if and hold true , then there exist two constants @xmath165 and @xmath155 such that , and all measurable function @xmath157 bounded by @xmath83 , @xmath166 where @xmath167 is the same constant as in theorem [ thm : main - two ] . we emphasize that the above results and their proofs can be adapted to the time - inhomogeneous setting of @xcite , with appropriate modifications of assumption . the following result is specific to the time - homogeneous setting and is proved at the end of this section . [ thm : invariant - quasi - stationary - distribution ] under the assumptions of theorem [ thm : main - two ] , the particle system @xmath168 is exponentially ergodic , which means that it admits a stationary distribution @xmath169 ( which is a probability measure on @xmath32 ) and that there exists positive constants @xmath170 and @xmath171 such that @xmath172 moreover , there exists a positive constant @xmath154 such that , for all measurable function @xmath157 bounded by @xmath83 , @xmath173 where @xmath167 is the same as in theorem [ thm : main - two ] and @xmath164 is distributed following @xmath169 . using the exponential convergence assumption , we deduce that , for any function @xmath157 such that @xmath174 and all @xmath175 , @xmath176 denoting by @xmath177 the natural filtration of the particle system @xmath31 , we deduce from theorem [ thm : intro - main ] that , almost surely , @xmath178 hence , for all @xmath179 , @xmath180 but ( * ? ? ? * theorem 2.1 ) entails the existence of a measure @xmath181 on @xmath8 and positive constants @xmath182 and @xmath183 such that for any @xmath60 and @xmath184 , @xmath185 note that , from now on , @xmath186 is a fixed constant . this entails @xmath187 where @xmath143 is the constant of . we deduce that @xmath188 with @xmath189 . our aim is now to control @xmath190 , uniformly in @xmath20 and for all @xmath191 . in order to do so , we make use of the following lemma , proved at the end of this subsection . [ lem : controle_pos_part ] there exists @xmath192 and @xmath156 such that , for any value of @xmath139 , @xmath193 moreover , if @xmath194 for some @xmath195 , then @xmath196 from this lemma ( where we assume without loss of generality that @xmath197 , from the markov property applied to the particle system and since @xmath198 implies @xmath194 , we deduce that , for any @xmath199 , @xmath200 where the last line is obtained by iteration over @xmath33 . this and equation imply that , for any @xmath201 , @xmath202 taking @xmath203 and assuming , without loss of generality , that @xmath204 , we obtain @xmath205 finally , using inequality and taking @xmath206 straightforward computations implies the existence of a constant @xmath154 such that , for all @xmath207 , @xmath208 with @xmath209 now , for @xmath210 , we have @xmath211 using the same computations as above , this concludes the proof of theorem [ thm : main - two ] . assume that @xmath194 . we obtain from theorem [ thm : intro - main ] that @xmath212 markov s inequality thus implies that , for all @xmath156 , @xmath213 and hence that @xmath214 but , by assumption , we have @xmath215 , so that @xmath216 . we deduce that @xmath217 choosing @xmath218 , we finally obtained @xmath196 in the general case ( when one does not have a good control on @xmath219 ) , the above strategy is bound to fail since we do not have a good control on the distance between the conditioned semi - group and the empirical distribution of the particle system . as a consequence , we need to take a closer look at algorithm 1 . as explained in the description of this algorithm , the position of the system at time @xmath83 is computed from the position of the system at time @xmath85 through several steps , each step being composed of two stages . we denote by @xmath220 the number of steps needed to compute the position of the system at time @xmath83 . for any step @xmath221 , we denote by @xmath222 the position of the @xmath93 particle at the beginning of step @xmath116 , by @xmath223 the state of the @xmath93 particle at the beginning of step @xmath116 , and by @xmath224 the index of the particle chosen during the first stage of step @xmath116 . with this notation , the process @xmath225 is a markov chain . in what follows , we denote by @xmath226 the natural filtration of this markov chain . we also introduce the quantities @xmath227 of course , we have @xmath228 and , at the beginning of the first step , one has @xmath229 for any @xmath221 , conditionally to @xmath230 and on the event @xmath231 , the position of @xmath232 is chosen with respect to @xmath233 hence , conditionally to @xmath230 and on the event @xmath231 , @xmath234 is equal to @xmath235 from assumption , we deduce that , conditionally to @xmath230 and on the event @xmath231 , @xmath234 is equal to @xmath236 let us denote by @xmath237 the successive step numbers during which the sequence @xmath238 jumps , that is @xmath239 it is clear that , for all @xmath240 , @xmath241 is a stopping time with respect to the filtration @xmath242 . we are interested in the sequence of random variables @xmath243 , defined by @xmath244 conditionally to @xmath245 ( the filtration @xmath226 before the stopping time @xmath241 ) , we deduce from that , for all @xmath9 , @xmath246 since @xmath247 almost surely . we deduce that @xmath248 moreover , assumption entails that @xmath249 . as a consequence , there exists a coupling between @xmath250 and the markov chain @xmath251 with initial law @xmath252 and transition probabilities @xmath253 such that @xmath254 , for all @xmath255 . the process @xmath256 is a positive super - martingale ( and a martingale if @xmath257 ) and hence it converges to a random variable @xmath258 almost surely as @xmath259 . let us now prove that @xmath258 is not equal to zero almost surely . consider a plya urn starting with @xmath186 balls with one white one , that is a markov chain @xmath260 in @xmath261 such that @xmath262 and @xmath263 it is well known that @xmath264 is a positive and bounded martingale which converges almost surely to a random variable @xmath265 distributed following a beta distribution with parameters @xmath266 . in particular , this implies that the event @xmath267 has a positive probability and that , conditionally to this event , @xmath268 converges to a positive random variable : @xmath269{a.s . } \mathbf{1}_{w_k / k\leq c_0,\,\forall k\geq 0}\,s_\infty.\end{aligned}\ ] ] since @xmath270 and @xmath260 have the same transition probabilities at time @xmath33 from states @xmath271 such that @xmath272 , there exists a coupling such that @xmath273 and hence such that @xmath274 since the right hand side is positive with positive probability , we deduce that @xmath258 is positive with positive probability . but @xmath275 implies that @xmath276 , thus there exists @xmath156 such that @xmath277 because of the relation between @xmath278 and @xmath279 , we deduce that @xmath280 by definition of @xmath281 , @xmath282 implies @xmath283 and hence @xmath193 this concludes the proof of lemma [ lem : controle_pos_part ] . we first prove the exponential ergodicity of the particle system and then deduce . using lemma [ lem : controle_pos_part ] , we know that , for any initial distribution of @xmath164 , @xmath284 on the event @xmath285 , there exists at least one particle satisfying @xmath286 . let us denote by @xmath287 the set of indexes of such particles and by @xmath288 the set of indexes @xmath289 such that @xmath290 . the probability that the @xmath291 first steps of algorithm 1 concern the indexes of @xmath291 in strictly increasing order is strictly lowered by @xmath292 and hence by @xmath293 . for each of this step , the probability that the chosen particle with index in @xmath294 is killed and then is sent to the position of a particle with index in @xmath295 is lowered by @xmath296 and hence by @xmath297 . overall , the probability that , after the @xmath291 first steps of algorithm 1 , all the particles with index in @xmath294 have jumped on a particle with index in @xmath295 is bounded below by @xmath298 and hence by @xmath299 . on this event , the probability that , for each next step in the algorithm up to time @xmath300 , the chosen particle jumps without being absorbed is bounded below by @xmath301 . but , using ( * ? ? ? * theorem 2.1 ) , we know that , under assumption [ eq : expo - cv ] , there exist a probability measure @xmath181 on @xmath8 and a constant @xmath302 such that @xmath303 . since the particles are independent on the event where none of them is killed , we finally deduce that the distribution of the particle system at time @xmath300 satisfies @xmath304 classical coupling criteria ( see for instance @xcite ) entails the exponential ergodicity of the particle system . let us now prove that holds . consider @xmath9 and @xmath305 such that @xmath306 . then , applying theorem [ thm : main - two ] to the particle system with initial position @xmath307 , we deduce that , for all @xmath20 , @xmath308 using the exponential ergodicity of the particle system and the exponential convergence , we deduce that @xmath309 letting @xmath33 tend toward infinity implies and concludes the proof of theorem [ thm : invariant - quasi - stationary - distribution ] . the propagation of neutrons in fissible media is typically modeled by neutron transport systems , where the trajectory of the particle is composed of straight exponential paths between random changes of directions @xcite . the behavior of a neutron before its absorption by a medium is related to the behavior of neutron tranport before extinction , where extinction corresponds to the exit of a neutron from a bounded set @xmath57 . we recall the setting of the neutron transport process studied in @xcite . let @xmath57 be an open connected bounded domain of @xmath310 , let @xmath311 be the unit sphere of @xmath310 and @xmath312 be the uniform probability measure on @xmath311 . we consider the markov process @xmath313 in @xmath314 constructed as follows : @xmath315 and the velocity @xmath316 is a pure jump markov process , with constant jump rate @xmath23 and uniform jump probability distribution @xmath317 . in other words , @xmath318 jumps to i.i.d . uniform values in @xmath311 at the jump times of a poisson process . at the first time where @xmath319 , the process immediately jumps to the cemetery point @xmath113 , meaning that the process is absorbed at the boundary of @xmath57 . an example of path of the process @xmath320 is shown in fig . [ fig : sample - path ] . for all @xmath321 and @xmath322 , we denote by @xmath323 ( resp . @xmath324 ) the distribution of @xmath320 conditionned on @xmath325 ( resp . the expectation with respect to @xmath323 ) . we also assume the following condition on the boundary of the bounded open set @xmath57 . this is an interior cone type condition satisfied for example by convex open sets of @xmath328 and by open sets with @xmath329 boundaries . * @xmath330 is non - empty and connected ; * there exists @xmath331 and @xmath332 such that , for all @xmath333 , there exists @xmath334 measurable such that @xmath335 and for all @xmath336 , @xmath337 for all @xmath338 $ ] and @xmath339 for all @xmath340 $ ] . by ( * ( 4.3 ) ) , for any @xmath145 , there exists a constant @xmath341 such that @xmath342 in particular , we deduce that , for all @xmath343 , @xmath344 now , fix @xmath345 and consider the first ( deterministic ) time @xmath346 when the ray starting from @xmath347 with direction @xmath348 hits @xmath349 or @xmath350 , defined by @xmath351 let us first assume that @xmath352 . then @xmath353 for some @xmath354 and hence @xmath355 where @xmath356 denotes the successive jump times of the process @xmath327 . using the markov property , we deduce from that @xmath357 this implies that , for all @xmath145 and all @xmath358 such that @xmath352 , @xmath359 let us now assume that @xmath360 . using assumption ( h ) , we obtain @xmath361 using the strong markov property and , we deduce that @xmath362 if @xmath363 , then @xmath364 implies that the process jumps at least one time before reaching @xmath350 , that is before time @xmath365 , so that @xmath366 . using the markov property , we deduce that , for @xmath363 or @xmath367 , @xmath368 this , equations and together imply that assumption is fulfilled for all @xmath369 . this concludes the prood of proposition [ prop : neutron ] .

we consider a general method for the approximation of the distribution of a process conditioned to not hit a given set . existing methods are based on particle system that are failable , in the sense that , in many situations , they are not well defined after a given random time . we present a method based on a new particle system which is always well define . moreover , we provide sufficient conditions ensuring that the particle method converges uniformly in time . we also show that this method provides an approximation method for the quasi - stationary distribution of markov processes . our results are illustrated by their application to a neutron transport model . _ _ keywords : _ _ particle system ; process with absorption ; approximation method for degenerate processes _ 2010 mathematics subject classification . _ primary : 37a25 ; 60b10 ; 60f99 . secondary : 60j80

this paper describes several arithmetic properties of the bc - system , showing new and interesting connections with the theory of witt vectors over the algebraic closure of finite fields and with p - adic analysis . the bc - system is a system of quantum statistical mechanics defined by a noncommutative hecke algebra of double classes in @xmath12 with respect to the subgroup @xmath13 , where @xmath14 is the @xmath15 " algebraic group ( _ cf . _ the complex hecke algebra @xmath16 of the system has a highly non - trivial structure since its regular representation , in the hilbert space of one sided classes , generates a factor of type iii@xmath17 and a canonical time evolution " @xmath18 . the study of the kms - equilibrium states at different temperatures has revealed the arithmetic nature of this dynamical system in view of the following facts @xmath19 the partition function of the system is the riemann zeta function @xmath19 there is a phase transition with spontaneous symmetry breaking at the pole of zeta function @xmath19 the zero temperature vacuum states implement the global class field isomorphism for @xmath10 . the study of the bc - system inaugurated the interplay between number - theory and noncommutative geometry . it is exactly the noncommutativity of the hecke algebra of the system which generates its non - trivial dynamics . moreover , on the noncommutative space of adles classes @xmath20 , which is naturally associated to the type ii dual of the bc - system , one obtains the spectral realization of zeros of @xmath7-functions and the trace formula interpretation of the riemann - weil explicit formulas ( _ cf . _ @xcite ) . further study ( _ cf . _ @xcite ) has shown that the integral hecke algebra @xmath21\rtimes\n$ ] supplies an integral model to the bc - system . the endomorphisms @xmath22 , @xmath23 act on the canonical generators @xmath24 $ ] , for @xmath25 and have natural linear quasi - inverses @xmath26 \to \z[\q/\z]\ , , \ \ \tilde\rho_n(e(\gamma))= \sum_{n\gamma'=\gamma}e(\gamma'),\ ] ] which are used in the construction of the crossed product and in the presentation of the algebra . in this paper we establish , for each prime @xmath0 , a strong relation connecting the integral bc - system and the universal witt ring @xmath27 of an algebraic closure of a prime field . the witt construction is in fact considered in the following three different forms @xmath28 as a @xmath29-theory endofunctor @xmath30 , in the category of commutative rings ( with unit ) @xmath28 as the big witt ring @xmath31 @xmath28 as the functor @xmath32 , for @xmath33 of characteristic @xmath0 . in the first two cases , the key structures are provided by the following operators @xmath28 the multiplicative lift @xmath34 @xmath28 the frobenius endomorphisms @xmath35 , @xmath36 @xmath28 the verschiebung additive functorial maps @xmath37 , @xmath36 @xmath28 the @xmath38-th ghost components @xmath39 , @xmath36 . these basic operators extend from the universal ring @xmath40 to its completion @xmath31 whose elements are expressed by witt vectors , in terms of which all the algebraic operations can be defined in terms of polynomials with integral coefficients . this integrality property encodes a rich and deep arithmetical information . moreover , the ring structure restricts to divisor stable subsets of @xmath41 yielding , for the set of powers of a prime @xmath0 , the functor @xmath42 . in proposition [ bcrelatenew ] and theorem [ bcrelate1 ] we prove that the @xmath0-primary structure of the integral bc - system is completely encoded by the universal ring @xmath27 , with a precise dictionary expressing the key operators @xmath43 and @xmath44 of the bc - system as respectively the frobenius @xmath45 and verschiebung @xmath46 on @xmath27 . the isomorphism connecting these algebraic structures depends upon the choice of a group isomorphism of the multiplicative group of @xmath4 with the group of _ complex _ roots of unity of order prime to @xmath0 : the ambiguity inherent to this choice is the same as that pertaining to the construction of brauer lift of characters . the completion process associated to the inclusion @xmath47 with dense image , is then used in theorem [ bcrelate2 ] to obtain , when @xmath48 and for each injective group homomorphism @xmath49 , a @xmath0-adic _ indecomposable _ representation @xmath3 of the integral bc - system as additive endomorphisms of the big witt ring @xmath50 . the construction uses the identification proven in theorem [ bcrelate1 ] and the implementation of the artin hasse exponentials . these representations are the @xmath0-adic analogues of the complex , extremal kms@xmath5 states of the bc - system . in section [ sectl ] this analogy is pursued much further . by implementing the theory of @xmath0-adic @xmath7-functions , we construct an analogue , in the @xmath0-adic case , of the partition function and of the kms@xmath51 states . in particular , we show that the division relations for the @xmath0-adic polylogarithms at roots of unity correspond to the kms condition . in [ sectexten ] we prove that the definition of the functionals satisfying such condition extends from the standard `` extended s - disk '' to the natural multiplicative group covering of @xmath8 . these results are the @xmath0-adic counterparts of the statements proven in @xcite for function fields . however , we also recognize an important difference with respect to the complex case , namely the presence of an added symmetry at non - zero temperature , due to the invariance of the states under the natural involution of @xmath52 which replaces each root of unity by its inverse . this added symmetry is a consequence of the vanishing of the @xmath0-adic @xmath7-functions associated to odd dirichlet characters . for @xmath0 a prime number , the set @xmath53 of all injective group homomorphisms @xmath54 is the parameter space for the @xmath0-adic representations of the integral bc - system . in section [ sectval ] , we relate this set with the space @xmath55 of extensions of the @xmath0-adic valuation to the maximal abelian field extension @xmath52 of @xmath10 . we view @xmath52 as an abstract field defined as the quotient of the group ring @xmath56 $ ] by the cyclotomic ideal ( _ cf . _ definition [ acf ] ) . let @xmath57 be the subgroup of @xmath58 of fractions with denominator prime to @xmath0 and let @xmath59 be the subfield ( _ i.e. _ the inertia subfield ) of @xmath52 generated by the group @xmath60 of roots of unity of order prime to @xmath0 . we describe canonical isomorphisms of @xmath55 with each of the following spaces @xmath61 the space of sequences of irreducible polynomials @xmath62 $ ] , @xmath36 , fulfilling the basic conditions of the conway polynomials ( _ cf . _ theorem [ conway0 ] ) . @xmath63 the space @xmath64 of bijections of the monoid @xmath65 commuting with their conjugates , as in definition [ defnsp ] ( _ cf . _ proposition [ hope ] ) . @xmath66 the space @xmath67 of field homomorphisms , where @xmath68 is the decomposition subfield , _ i.e. _ the fixed field under the frobenius automorphism ( _ cf . _ proposition [ homfield ] ) . @xmath69 the quotient of the space @xmath53 by the action of @xmath70 ( _ cf . _ proposition [ try ] ) . @xmath71 the algebraic spectrum of the quotient algebra @xmath72/j_p$ ] , where @xmath73 is the reduction modulo @xmath0 of the cyclotomic ideal ( _ cf . _ definition [ acf ] and proposition [ cpcp1 ] ) . for a global field @xmath74 of positive characteristic ( _ i.e. _ a function field associated to a projective , non - singular curve over a finite field @xmath75 ) it is a well known fact that the space of valuations of the maximal abelian extension @xmath76 of @xmath74 has a geometric meaning . in fact , for each finite extension @xmath77 of @xmath78 , the space @xmath79 of ( discrete ) valuations of @xmath77 is an algebraic , one - dimensional scheme whose non - empty open sets are the complements of the finite subsets @xmath80 . the structure sheaf is locally defined by the intersection @xmath81 of the valuation rings inside @xmath77 . then the space @xmath82 is the projective limit of the schemes @xmath79 , @xmath83 . for the global field @xmath84 of rational numbers , one can consider its maximal abelian extension @xmath52 as an abstract field and try to follow a similar idea . in section [ curve ] , we show however that the space @xmath85 provides only a rough analogue , in characteristic zero , of @xmath82 . our approach to this problem is guided and motivated by the following three results contained in our previous work @xmath86 the adelic interpretation of the loop groupoid @xmath87 of the abelian cover of the algebraic curve @xmath88 associated to a function field ( _ cf . _ @xcite and [ subsectprel ] ) @xmath89 the determination of the counting function @xmath90 ( a distribution on @xmath91 ) which replaces , for @xmath84 , the classical weil counting function for a function field ( _ cf . _ @xcite and [ subsectwhync ] ) @xmath92 the interpretation of the counting function @xmath90 as an intersection number , using the action of the idle class group on the class space ( _ cf . _ @xcite ) . by applying these results we find that the sought for geometric fiber over a non - archimedean , rational prime @xmath0 is the total space of a principal bundle , with base @xmath55 and structure group given by a connected , compact solenoid @xmath93 whose definition is given in proposition [ solenoid ] . then , in proposition [ fibration ] we derive a natural construction for the fiber as the mapping torus @xmath94 of the action of the frobenius on the space @xmath53 . in section [ subsectarch ] , we consider the fiber @xmath95 over the archimedean prime , with the implementation of the theory of multiplicative norms . the interpretation given in @xmath92 for the counting function as intersection number shows that the fibers @xmath94 should not be considered in isolation , but as being part of an ambient noncommutative space which is responsible for the transversality factors due to the archimedean contribution to the explicit formulas . this interpretation is explained in details in section [ subsectwhync ] . in section [ subsectendo ] , we show that the integral bc - system gives , for each @xmath0 ( including the archimedean prime ) , a natural embedding of the fiber @xmath94 into a noncommutative space constructed using the set @xmath96 of the @xmath8-rational points of the affine group scheme @xmath97 which defines the abelian part of the system ( _ cf . _ @xcite ) . here @xmath8 denotes the @xmath0-adic completion of an algebraic closure of @xmath9 . this result shows that the space @xmath98 matches , for any rational prime @xmath0 including @xmath99 , the definition of the class space . in proposition [ groupscheme ] we show , using the fact that @xmath97 is a group scheme , that @xmath100 is a free module of rank one over the hyperring @xmath101 of the adle classes . the problem of a correct interpretation of the connected factor @xmath102 in remains open . it is a general principle that in our constructions the noncommutative spaces arise as @xmath103 for a commutative ring @xmath33 ( _ cf . _ ) , while the classical subspaces of @xmath103 are defined as the support of the cyclotomic ideal ( in the affine scheme @xmath104)$ ] ) . we end the paper by showing in section [ sectlens ] the relevance of the recent work of b. de smit and h. lenstra ( _ cf . _ @xcite ) on the standard model " for the algebraic closure of a finite field . when @xmath74 is a function field , the intermediate extension @xmath105 plays an important geometric role , namely the extension of scalars to an algebraically closed field , for the algebraic curve associated to @xmath74 . when @xmath84 , we show that the intermediate extension @xmath106 used by de smit and lenstra , comes very close to fulfill the expected properties for a similar intermediate extension @xmath107 . their construction provides a conceptual construction of the subfield of @xmath4 union of all extensions whose degree is prime to @xmath0 . in the very last part of the paper we recall one of the first applications provided by e. witt of his functor , which is a conceptual construction of the missing piece @xmath108 , using the simple equation @xmath109 in witt vectors in this section we recall the definition and the main properties of the universal ring @xmath40 , where @xmath33 is any commutative ring with unit . we refer to @xcite to read more details . the second part of the section describes @xmath111 , for an algebraically closed field @xmath112 . one lets @xmath113 ( or @xmath114 ) be the category of endomorphisms of projective @xmath33-modules of finite rank . the objects are pairs @xmath115 where @xmath77 is a finite , projective @xmath33-module and @xmath116 . the morphisms in this category are required to commute with the endomorphisms @xmath117 . the following operations of direct sum and tensor product @xmath118 turn the grothendieck group @xmath119 into a ( commutative ) ring . the pairs of the form @xmath120 generate the ideal @xmath121 . we denote the quotient ring by @xmath40 @xmath122 by construction , @xmath110 is an endofunctor of the category @xmath123 of commutative rings with unit . several key operators and maps act on @xmath110 , the following are the most relevant ones for our applications @xmath61 the lift @xmath34 which is a multiplicative map . @xmath63 for @xmath36 , the frobenius ring endomorphisms @xmath124 . @xmath66 for @xmath36 , the verschiebung ( shift ) additive functorial maps @xmath37 . @xmath69 for @xmath36 , the @xmath38-th ghost component homomorphisms @xmath39 . we shortly recall their definitions . @xmath61 the lift @xmath125 : a \to \w_0(a)$ ] is defined as @xmath126=(a , f)$ ] . @xmath63 for @xmath23 , the operations in @xmath113 of raising an endomorphism @xmath117 to the @xmath38-th power induce the frobenius ring endomorphims in @xmath40 @xmath127 @xmath66 for @xmath23 , the verschiebung maps are defined by the following operations on matrices @xmath128 @xmath69 for @xmath23 , the ghost components are given by @xmath129 let @xmath130 $ ] be the multiplicative abelian group of formal power series with constant term @xmath131 . the ( inverse of the ) characteristic polynomial defines a homomorphism of abelian groups @xmath132 where @xmath133 is the matrix associated to @xmath134 ( _ i.e. _ @xmath135 , @xmath136 , @xmath137 ) . by a fundamental result of g. almkvist ( @xcite theorem 6.4 or @xcite main theorem ) , one has [ almthm ] the map @xmath7 is injective and its image is the subgroup of @xmath138 @xmath139 note in particular that for @xmath77 a finite , projective @xmath33-module and @xmath140 one has @xmath141 one also has @xmath142 the following proposition collects together several standard equations connecting these operators [ rabi10 ] let @xmath33 be a commutative ring and @xmath143 . the following hold @xmath61 @xmath144 . @xmath63 @xmath145 . @xmath66 if @xmath146 , @xmath147 . @xmath69 for @xmath36 , @xmath148 . @xmath71 for @xmath36 , @xmath149 . @xmath150 for @xmath151 , @xmath152 . @xmath153 @xmath154 all proofs are straightforward , we just check @xmath69 as an example . for @xmath155 , the action of @xmath156 on vectors @xmath157 is given by @xmath158 similar formulas hold for @xmath159 , for @xmath160 . by definition , @xmath161 corresponds to @xmath162 . this endomorphism decomposes as the direct sum of @xmath38 endomorphisms of @xmath163 , each of these is of the form @xmath164 by applying , one checks that each of the above endomorphisms is equivalent to @xmath165 . the equality @xmath148 follows . we shall apply the following proposition to the case @xmath166 an algebraic closure of @xmath2 . [ wofield ] let @xmath112 be an algebraically closed field . then the map which associates to @xmath167 the divisor @xmath168 of non - zero eigenvalues of @xmath117 ( with multiplicity taken into account ) extends to a ring isomorphism @xmath169.\ ] ] under the above isomorphism , the frobenius @xmath45 on @xmath111 is given on @xmath170 $ ] by the natural linearization of the group endomorphism @xmath171 . by applying theorem [ almthm ] , the characteristic polynomial extends to a complete invariant on @xmath172 and to an isomorphism of @xmath172 with the ring of quotients of monic polynomials in @xmath173 $ ] . moding out this ring by @xmath174 means that one removes the powers of the variable . thus the divisor of non - zero eigenvalues of @xmath117 extends to define a bijection of sets @xmath175 $ ] . it remains to check that this bijection preserves the ring operations . for addition , the set underlying the divisor @xmath176 is the disjoint union of the two sets of roots of @xmath177 and hence @xmath178 . for the product , it is enough and easy to check that the tensor product of two rank one elements @xmath179 is given by @xmath180 for non - zero elements of @xmath112 . the statement about @xmath45 is checked in the same way using on elements @xmath181 . we recall the following formula for @xmath182 in terms of the divisor @xmath183\in \z[k^\times]$ ] @xmath184 [ noncan ] for any given isomorphism @xmath185 of the multiplicative group of the algebraic closure @xmath186 with the subgroup @xmath187 of fractions with denominator prime to @xmath0 , one derives an isomorphism @xmath188.\ ] ] under the isomorphism @xmath189 , the frobenius @xmath45 of @xmath190 is given on @xmath191 $ ] by the natural linearization of the group endomorphism @xmath192 , @xmath193 ( _ i.e. _ @xmath194 in additive notation ) . for each @xmath23 , one defines group ring endomorphisms @xmath195 \to \z[\q/\z],\qquad \sigma_n(e(\gamma))=e(n\gamma)\ ] ] and the following additive maps @xmath196 \to \z[\q/\z ] , \qquad \tilde\rho_n(e(\gamma))= \sum_{n\gamma'=\gamma}e(\gamma').\ ] ] we recall from @xcite , proposition 4.4 , the following result [ mapstilderho ] the endomorphisms @xmath43 and the maps @xmath197 fulfill the following relations @xmath198 @xmath199\ ] ] @xmath200 where @xmath201 . note that taking @xmath202 in gives @xmath203.\ ] ] on the contrary , if we take @xmath204 and @xmath205 to be relatively prime we get @xmath206 we recall from @xcite ( definition 4.7 and 4.2 ) the following facts . the integral @xmath207-algebra is the algebra @xmath208\rtimes_{\tilde\rho}\n$ ] generated by the group ring @xmath209 $ ] , and by the elements @xmath210 and @xmath211 , with @xmath23 , which satisfy the relations @xmath212 \mu_n^ * x = \sigma_n(x ) \mu_n^ * \\[3 mm ] x \tilde\mu_n = \tilde\mu_n \sigma_n(x ) , \end{array}\ ] ] where @xmath197 , @xmath213 is defined in , as well as the relations @xmath214 \mu_{nm}^ * = \mu_{n}^*\mu_{m}^ * \qqq n , m\\[3 mm ] \mu_n^ * \tilde\mu_n = n \\[3 mm ] \tilde\mu_n\mu_m^ * = \mu_m^*\tilde\mu_n \ \ \ \ ( n , m ) = 1 . \end{array}\ ] ] after tensoring by @xmath10 , the hecke algebra @xmath215 has a simpler explicit presentation with generators @xmath216 , @xmath211 , @xmath36 and @xmath217 , for @xmath25 , satisfying the relations @xmath19 @xmath218 , @xmath219 , @xmath19 @xmath220 , @xmath221 , @xmath222 , @xmath19 @xmath223 , if @xmath224 , @xmath19 @xmath225 , @xmath226 , and @xmath227 , @xmath228 , @xmath19 for all @xmath36 and all @xmath25 @xmath229 after tensoring by @xmath6 and completion one gets a @xmath230-algebra with a natural time evolution @xmath231 ( @xcite , @xcite chapter iii ) . the extremal kms states below critical temperature vanish on the monomials @xmath232 for @xmath233 and @xmath234 $ ] and their value on @xmath56 $ ] is given by @xmath235 where @xmath236 determines an embedding in @xmath6 of the cyclotomic field @xmath52 generated by the abstract roots of unity . in @xcite quillen makes use of the choice of an embedding @xmath237 in the study of the algebraic k - theory of the general linear group over a finite field . in this section we compare the description of the universal witt ring @xmath190 , endowed with the structure given by the frobenius endomorphisms @xmath45 and the verschiebung maps @xmath46 with the integral bc - algebra @xmath238 . by a simple comparison process we notice that the relations , , holding on @xmath238 are the same as those fulfilled by the frobenius endomorphisms @xmath45 and the verschiebung maps @xmath46 on @xmath27 . more precisely , under the correspondences @xmath239 , @xmath240 the two relations of correspond to , and the first three relations of proposition [ rabi10 ] correspond respectively to , and . these results evidently point out to the existence of a strong relation between the ( @xmath241)-ring @xmath190 and the group ring @xmath209 $ ] endowed with the aforementioned operators . next , we compare the two groups rings : @xmath191 $ ] and @xmath209 $ ] which arise in the description of @xmath27 and in the construction of the bc - algebra respectively . one has a surjective group homomorphism : @xmath242 induced by the canonical factorization of the groups @xmath243 where @xmath244 is the group of fractions whose denominator is a power of @xmath0 . thus one obtains a corresponding factorization of the rings @xmath245=\z[\mup]\otimes_\z \z[\mupin].\ ] ] by using the trivial representation of @xmath244 ( _ i.e. _ the augmentation @xmath246 of @xmath247 $ ] ) , one gets a retraction @xmath248 producing the splitting @xmath249\stackrel{j_p}{\longrightarrow } \z[\q/\z]\stackrel{id \otimes \epsilon}{\longrightarrow}\z[\mup].\ ] ] notice that @xmath57 is preserved by the action of the map @xmath194 , @xmath250 ) . this implies that the endomorphisms @xmath43 acting on the bc - algebra restrict naturally to determine endomorphisms @xmath251\to \z[\mup]$ ] . let us denote by @xmath252 the set of integers which are prime to @xmath0 . the following lemma describes the projection of the operators @xmath44 of the bc - algebra on the group ring @xmath191 $ ] [ rrhon ] let @xmath253 , with @xmath254 . for @xmath255 , we write modulo @xmath131 @xmath256 then , with @xmath44 as in we have @xmath257 where @xmath258 , @xmath259 , is the unique solution in @xmath58 , with denominator prime to @xmath0 of the equation @xmath260 the existence and uniqueness of the decomposition derives from the factorization . for @xmath261 , the endomorphism of @xmath58 : @xmath262 restricts to an automorphism on the subgroup @xmath263 . for @xmath264 , this fact shows the existence and uniqueness of the solution @xmath258 of . one has @xmath265 for some integer @xmath266 , thus @xmath267 by applying , one also has a decomposition of the form @xmath268 one has @xmath269 modulo @xmath131 , @xmath270 , thus the solutions of the equation @xmath271 in @xmath58 which enter in are of the form @xmath272 by using one derives @xmath273 for the projection @xmath274 $ ] one thus gets that @xmath275 which is repeated with multiplicity @xmath276 . the equation follows . [ corproj ] one has @xmath277\ , , \ n\in \n.\ ] ] and @xmath278,~k\in\n\,.\ ] ] the two statements follow from . [ defnxp ] for @xmath0 a prime number , we denote by @xmath53 the space of all injective group homomorphisms @xmath54 . the relation between @xmath190 and the abelian part @xmath209 $ ] of the integral bc - algebra @xmath238 is described by the following lemma [ bcrelatenew ] let @xmath279 and let @xmath189 be the associated ring isomorphism @xmath280\subset \z[\q/\z].\ ] ] then the frobenius @xmath45 and verschiebung maps @xmath46 on @xmath190 are obtained by restriction of the ring endomorphisms @xmath43 and the maps @xmath44 on @xmath209 $ ] by the formulas @xmath281 in section [ sectwzero ] we recalled ( _ cf . _ @xcite for details ) that the frobenius @xmath45 on @xmath40 is given by @xmath282 . at the level of the divisor of the eigenvalues of @xmath117 ( it is a divisor in the virtual case ) , _ i.e. _ at the level of the associated element in @xmath170 $ ] , @xmath166 , the frobenius @xmath45 corresponds to the group homomorphism @xmath193 ( _ cf . _ proposition [ wofield ] ) . the verschiebung maps @xmath46 are described by the operation on matrices . the maps @xmath46 are additive and hence determined by the elements @xmath283)$ ] where @xmath284 . they correspond to the @xmath38 eigenvalues of the following matrix @xmath285 since the @xmath38-th power of the above matrix is the multiplication by @xmath286 , all its eigenvalues fulfill the equation @xmath287 . in fact the characteristic polynomial of the above matrix is @xmath288 . let @xmath253 , where @xmath289 is prime to @xmath0 . since @xmath4 is a perfect field , the root @xmath290 of @xmath291 is unique and it admits @xmath289 distinct roots of order @xmath289 : @xmath292 , which are the @xmath289 roots of @xmath293 . they take the form @xmath294 , with @xmath295 . thus the corresponding divisor is @xmath296.\ ] ] we now compare the above description of the divisor associated to @xmath297 with @xmath298 , where @xmath299 . the elements @xmath300 are the @xmath289 distinct roots of the equation @xmath301 . similarly , with the notations of , the elements @xmath302 are the @xmath289 solutions in @xmath57 of the equation @xmath303 . one thus gets @xmath304 thus shows that @xmath305))=\tilde\sigma(\delta)=r\circ \tilde\rho_n(e(\gamma))= r\circ\tilde\rho_n\circ \tilde\sigma([\alpha]).\ ] ] [ bcrelate1 ] let @xmath279 . the following formulas define a representation @xmath3 of the integral bc - system @xmath238 as additive endomorphisms of @xmath190 @xmath306 for all @xmath307 , @xmath308 $ ] and @xmath36 . by construction @xmath309 is a homomorphism of the group ring @xmath209 $ ] to @xmath190 and hence , by composition with the left regular representation , @xmath3 gives a representation of @xmath209 $ ] . the @xmath45 and @xmath46 are additive . it remains to check the relations and . the latter ones follow from for the first two , and from @xmath61 and @xmath66 of proposition [ rabi10 ] for the last two . to check the first relation of one needs to show that @xmath310 one has @xmath311 for all @xmath308 $ ] . thus , by applying , one can replace @xmath312 by @xmath313 without changing both sides of the equation . thus we can assume that @xmath314 for some @xmath315 . then @xmath316 is just the multiplication by @xmath317 . one has by @xmath318 thus @xmath319 is the multiplication by @xmath320 and follows from @xmath321 which is statement @xmath63 of proposition [ rabi10 ] . let us check the other two relations of . the second one means @xmath322 and since @xmath323 we can assume as before that @xmath314 , for some @xmath315 . then @xmath316 is the multiplication by @xmath317 and , by , @xmath324 is the multiplication by @xmath325 . the required equality then follows since @xmath45 is multiplicative . the last relation of means @xmath326 and assuming @xmath314 it reduces to @xmath327 which in turn follows from statement @xmath63 of proposition [ rabi10 ] . in this section we provide a short overview on the construction of the universal witt scheme in the form that is most suitable to the applications contained in this paper , for more details we refer to @xcite . in the second part of the section we connect the universal ring @xmath40 with @xmath31 . the construction of the ring of big witt vectors ( or generalized witt vectors ) is described by a covariant endofunctor @xmath328 in the category of commutative rings ( with unit ) . for @xmath329 , and as a functor to the category of sets , one defines @xmath330 to a truncation set @xmath331 ( _ i.e. _ a subset of @xmath41 which contains every positive divisor of each of its elements ) , one associates the truncated functor @xmath332 as a functor to the category of sets , @xmath333 is left represented by the polynomial ring @xmath334 $ ] . then it follows that the big witt vectors functor @xmath335 is left represented by the symmetric algebra @xmath336 $ ] @xmath337 as an endofunctor in the category of commutative rings @xmath338 is _ uniquely _ determined by requiring that for any commutative ring @xmath33 and for any @xmath339 , the following map , called the @xmath38-th ghost component is a ring homomorphism @xmath340 for @xmath341 a variable , the functorial bijection of sets @xmath342,\qquad x = ( x_n)_{n\in\n}\mapsto f_x(t ) = \prod_{n\in\n}(1-x_nt^n)^{-1}\ ] ] transports the ring structure from @xmath31 to the multiplicative abelian group @xmath138 of power series over @xmath33 with constant term @xmath131 , under the usual multiplication of power series ( the power series @xmath131 acts as the identity element ) . in other words one has @xmath343 to make the description of the corresponding product @xmath344 on @xmath138 more explicit one introduces first the @xmath38-ghost components @xmath345 , @xmath23 , which are defined by the formula @xmath346 for example , the first three ghost components are given by the universal formulas @xmath347 for products of the form @xmath348 this means that @xmath349 thus the ghost components are given by the power sums in the @xmath350 s . then , the product @xmath344 on @xmath138 is _ uniquely _ determined by requiring that these ghost components are ( functorial ) ring homomorphisms . in fact , distributivity and functoriality together force the multiplication of power series in @xmath138 be expressed by the following rule @xmath351 where @xmath352 it follows that multiplication according to translates into component - wise multiplication for the ghost components on @xmath138 . it is expressed by explicit polynomials with integral coefficients of the form @xmath353 the ghost components @xmath354 of a witt vector @xmath355 become the ghost components of @xmath356 , _ i.e. _ @xmath357 it follows that the bijection @xmath358 becomes a ring isomorphism . note moreover that the homomorphism of abelian groups @xmath359 of , preserves the product , _ i.e. _ @xmath360 so that it defines an injective ring homomorphism . two witt vectors @xmath361 are added and multiplied by means of universal polynomials with integer coefficients @xmath362 the polynomials @xmath363 are recursively computed using the ghost components by the formulas @xmath364 @xmath365 notice that the polynomials @xmath354 depend only on the @xmath366 for @xmath367 a divisor of @xmath38 , hence the @xmath38-th addition and multiplication polynomials @xmath368 , @xmath369 are polynomials that only involve the @xmath366 and @xmath370 with @xmath367 a divisor of @xmath38 . thus , for a truncation set @xmath331 , the polynomial ring @xmath334 $ ] is a sub hopf algebra and a sub co - ring object of @xmath371 , this means that it defines a quotient functor , which coincides with @xmath333 . this result applies in particular to the truncation set @xmath372 , where @xmath0 is a prime number . thus the p - adic witt vectors @xmath373 can be interpreted as a functorial quotient of the big witt vectors ( similarly one obtains @xmath374 as the p - adic witt vectors of length @xmath375 ) . the teichmller representative is a multiplicative map which defines a section to the ghost map @xmath376 . if @xmath377 is a truncation set , the teichmller representative is defined as @xmath378_n : a \to \w_n(a),\quad a \mapsto [ a]_n = ( [ a]_n)_{n\in n},\quad [ a]_{n , n}=\begin{cases } a&\text{if $ n=1$},\\0&\text{if $ n>1$}.\end{cases}\ ] ] one has @xmath379_n)=a^n$ ] for all @xmath339 . on the functorial ring @xmath31 one can introduce several functorial operations which derive from ( the large number of ) ring endomorphisms of @xmath371 and by applying the representability property . for instance , the verschiebung ( shift ) additive functorial endomorphisms on @xmath380 and its quotients , arise from the ring endomorphism @xmath381 which corresponds to the map @xmath382 in @xmath138 . for @xmath377 a truncation set , the shift is the additive map given by @xmath383 where @xmath384 . this means that the composite with the ghost components is given by @xmath385 the n - th frobenius is the ( unique ) natural ring homomorphism @xmath386 which is defined on the ghost components by the formula @xmath387 . thus by definition the n - th frobenius map makes the following diagram commute @xmath388 where @xmath389 takes a sequence @xmath390 to the sequence whose d - th component is @xmath391 . at the level of the components @xmath392 of a witt vector @xmath393 , the frobenius @xmath45 is given by polynomials with integral coefficients . for instance , the following are the first @xmath394 components of @xmath395 @xmath396 note that when @xmath0 is a rational prime one has ( _ cf . _ @xcite proposition 5.12 ) @xmath397 one also has ( _ cf . _ @xcite proposition 5.9 ) @xmath398 where for the maps @xmath45 one assumes @xmath399 and @xmath400 . proposition [ rabi10 ] extends without change , ( _ cf . _ @xcite proposition 5.10 ) . [ rabi10bis ] let @xmath377 be a truncation set , and @xmath339 with @xmath399 . let @xmath33 be a commutative ring and @xmath401 . then @xmath61 @xmath144 . @xmath63 @xmath145 . @xmath66 if @xmath289 is prime to @xmath38 , one has @xmath147 . @xmath69 one has @xmath148 . we refer to @xcite proposition 5.10 . the statement @xmath69 differs slightly from this reference , it can be checked directly using proposition [ rabi10 ] . it implies that when @xmath38 is invertible in @xmath402 then @xmath403 defines a ring endomorphism . it is important to see how the description of the universal ring @xmath40 fits with the definition of @xmath31 . there is a canonical ring monomorphism @xmath404 which is given as the composite of the injective ring homomorphism @xmath359 as in and of the ring isomorphism @xmath405 ( _ cf . _ ) @xmath406 in the case @xmath407 the characteristic polynomial @xmath408 factorizes as a product of terms @xmath409 of degree one , where the @xmath410 are the eigenvalues of @xmath117 ( _ cf . _ ) . [ embedlem ] let @xmath411 : \bar \f_p\to \w(\bar \f_p),~ x\mapsto \tau(x):=[x]$ ] be the lift and let @xmath412 $ ] be the isomorphism of . then the canonical map is given explicitly as @xmath413\ni \sum n_j \alpha_j\mapsto \sum n_j \tau(\alpha_j)\in \w(\bar \f_p).\ ] ] this lemma together with theorem [ almthm ] shows that the subring @xmath414 is just the group ring @xmath415 $ ] and is freely generated over @xmath416 by the lifts . in this section we shall implement the results of @xcite to describe the ring @xmath50 , then using the embedding with dense image @xmath417 , we will extend the representation @xmath3 of @xmath238 on @xmath27 ( theorem [ bcrelate1 ] ) to a representation of the integral bc - system on @xmath50 . such representation is the @xmath0-adic analogue of the irreducible complex representation . we begin by recalling the definition of the isomorphism @xmath418 where @xmath419 is the set of positive integers which are prime to @xmath0 and @xmath420 is the set of integer powers of @xmath0 . at the conceptual level , this isomorphism is a special case of the general functorial isomorphism holding for any commutative ring @xmath33 with unit ( @xcite theorem 1 ) @xmath421 when @xmath33 is an @xmath2-algebra , every element of @xmath422 is invertible in @xmath423 , thus one derives a canonical isomorphism @xmath424 which is defined in terms of the ghost components . let @xmath425 be the ring @xmath416 localized at the prime ideal @xmath426 so that every element of @xmath422 is invertible in @xmath425 . a central role , in the ring @xmath427 , is played by the artin - hasse exponential , this is the power series @xmath428 the following properties are well known ( _ cf . _ @xcite ) [ proj ] @xmath61 @xmath429 is an idempotent of @xmath427 . @xmath63 for @xmath430 , the series @xmath431 determine an idempotent . as @xmath38 varies in @xmath422 , the @xmath432 form a partition of unity by idempotents . @xmath66 for @xmath433 , @xmath434 and @xmath435 , @xmath436 . to check @xmath61 directly , one shows that there exists a unique sequence @xmath437 such that @xmath19 @xmath438 @xmath19 @xmath439 for all @xmath440 @xmath19 @xmath441 for all @xmath254 and @xmath442 . this follows by noticing that the coefficient of @xmath443 in @xmath444 is @xmath254 which is invertible in @xmath425 , so that one determines the @xmath445 inductively . one then checks that the ghost components of @xmath437 are the same as those of @xmath429 , _ i.e. _ @xmath446 is equal to @xmath131 if @xmath447 and is zero otherwise . note that any @xmath430 is invertible in @xmath427 . division by @xmath38 corresponds to the extraction of the @xmath38-th root of the power series @xmath448 . formally , this is given by the binomial formula @xmath449 the @xmath0-adic valuation of the rational coefficient of @xmath450 is positive because @xmath451 , thus this coefficient can be approximated arbitrarily by a binomial coefficient . it follows from proposition [ rabi10 ] , @xmath69 that @xmath452 is an endomorphism of @xmath453 and also a right inverse of @xmath45 . one easily derives from @xcite the following result [ proj1 ] let @xmath33 be an @xmath2-algebra . @xmath86 the map @xmath454 is an isomorphism onto the reduced ring @xmath455 . @xmath89 for @xmath430 , the composite @xmath456 is an isomorphism of the reduced algebra @xmath457 with @xmath458 . @xmath92 the composite @xmath459 is a canonical isomorphism @xmath460 . @xmath461 the composite isomorphism @xmath462 is given explicitly on the components by @xmath463 the first three statements follow from @xcite 3.b , @xcite , thm . 1 and prop . 1 , @xcite thm we prove @xmath461 . since the frobenius @xmath45 is an endomorphism and @xmath464 , one can rewrite as @xmath465 thus , to show it is enough to prove it for @xmath466 . one needs to check that for all @xmath467 , one has @xmath468 indeed , this follows from distributivity and the identity @xmath469 the above identity can be checked directly knowing that @xmath470 and by applying the equality @xmath471 together with proposition [ proj ] @xmath66 and the equality @xmath472 which holds for any element @xmath473 . in particular , for the lift @xmath474 $ ] of an element @xmath475 one gets @xmath476 where , on the right hand side , @xmath477 denotes the original lift @xmath478 . indeed one has @xmath479 . [ fixedfr ] let @xmath33 be an @xmath2-algebra . then , the common fixed points of the endomorphisms @xmath480 for @xmath430 , are the elements of the form @xmath481 one also has @xmath482 let @xmath467 with @xmath483 for all @xmath430 . then , it follows from and that all the components @xmath484 are equal , so that for some @xmath485 one has @xmath486 and @xmath312 is of the required form . conversely , by proposition [ proj ] @xmath66 , one has @xmath487 for all @xmath488 . thus when one applies @xmath489 to @xmath490 , one gets @xmath491 unless @xmath492 using proposition [ rabi10bis ] @xmath63 , @xmath66 . when @xmath492 one obtains @xmath493 , with @xmath494 . thus the elements of the form are fixed under all @xmath489 . we now apply these results to the case @xmath48 . we identify @xmath495 with a subring of @xmath8 ( the @xmath0-adic completion of an algebraic closure of @xmath9 ) . let @xmath496 be the completion of the maximal unramified extension of @xmath9 . then one knows that @xmath497 is the ring of integers of @xmath498 . with @xmath499 the isomorphism of , we have @xmath500 thus @xmath499 makes @xmath50 a module over @xmath501 . to the frobenius automorphism of @xmath4 corresponds , by functoriality , a canonical automorphism @xmath502 of @xmath501 which extends to a continuous automorphism @xmath503 we can now describe the @xmath0-adic analogues of the complex irreducible representations of the bc - system ( _ cf . _ ) . we recall that @xmath53 denotes the space of all injective group homomorphisms @xmath504 . the choice of @xmath279 determines an embedding @xmath505 of the cyclotomic field generated by the abstract roots of unity of order prime to @xmath0 inside @xmath8 . in the following we shall use the simplified notation @xmath506 . for @xmath254 , we let @xmath507 be the vector in @xmath50 with only one non - zero component : @xmath508 . [ bcrelate2 ] let @xmath279 . the representation @xmath3 as in theorem [ bcrelate1 ] extends by continuity to a representation of the integral bc - algebra @xmath238 on @xmath509 . for @xmath430 and for @xmath308 $ ] , @xmath510 , @xmath316 and @xmath511 are @xmath512-linear operators on @xmath50 @xmath513 @xmath514 one has @xmath311 for all @xmath308 $ ] ( @xmath515\to \z[\mup]$ ] the retraction as in ) and @xmath516 where @xmath502 is the frobenius automorphism , acting componentwise as a skew - linear operator . theorem [ bcrelate1 ] and the density of @xmath27 in @xmath50 ( _ cf . _ _ e.g. _ @xcite , 1.8 ) show that @xmath3 extends by continuity to a representation of the integral bc - algebra @xmath238 on @xmath509 . in view of the invertibility of the elements @xmath430 in @xmath50 , the description of the representation @xmath3 is simplified by using the elements @xmath517 , to stress the analogy with the complex case . it follows from corollary [ fixedfr ] that the subring @xmath512 of @xmath509 is the fixed subring for the action of the operators @xmath45 , for all @xmath430 . for @xmath430 , the operators @xmath45 are @xmath512-linear likewise the @xmath46 ( _ cf . _ proposition [ rabi10 ] , @xmath63 ) which correspond to the @xmath210 by means of the representation @xmath3 . thus we obtain the first equality in . the operators @xmath518 are the multiplication operators ( _ cf . _ corollary [ noncan ] ) by @xmath519 , thus they are @xmath512-linear and the second equation in follows from . by applying one has @xmath520 for all @xmath38 . taking @xmath521 , one gets that @xmath522 which coincides with @xmath502 acting componentwise , as it follows from the commutation @xmath523 for @xmath430 and . since @xmath524 and @xmath502 is invertible one gets . [ theideal ] we denote by @xmath525 the two sided ideal generated by the elements @xmath526 [ ideal ] one has @xmath527 ( _ cf . _ ) and the intersection @xmath209\cap \cj_p$ ] is the ideal @xmath528 of @xmath209 $ ] generated by the elements as in . the sequence of commutative algebras @xmath529\stackrel{r}{\longrightarrow}\z[\mup]\to 0\ ] ] is exact . let @xmath530 } \otimes \epsilon:\z[\q/\z]\to \z[\mup]$ ] be the retraction map introduced in . by construction , one has @xmath531 since @xmath518 only depends upon @xmath532 it follows that @xmath533 . one knows ( _ cf . _ @xcite , lemma 4.8 ) that any element of the algebra @xmath238 can be written as a finite sum of monomials of the form @xmath534.\ ] ] we show that for any finite sum @xmath88 as in we have @xmath535 it is enough to prove that @xmath536 , @xmath537 and since @xmath191 $ ] is torsion free it suffices to show that @xmath538 , @xmath537 . define @xmath539 $ ] , @xmath540 , then @xmath541 has finite support . for any group homomorphism @xmath542 there is a unique ring homomorphism @xmath543 with @xmath544\to \co,\qquad h_\chi(e(\gamma))=\chi(\gamma)\qqq \gamma\in \mup.\ ] ] this applies in particular , for any integer @xmath545 , to @xmath546 where we view @xmath505 as a group homomorphism @xmath547 . one has @xmath548 since an injective character of a finite cyclic group generates the dual group . let @xmath549 be relatively prime . then one has for any @xmath550 and @xmath551 @xmath552 thus if @xmath553 one has for all @xmath545 and @xmath554 as above @xmath555 for @xmath317 a root of unity one has @xmath556 , thus the polynomial @xmath557 vanishes , for @xmath38 large enough , on all roots of unity thus it is identically zero , hence all its coefficients must vanish _ @xmath558 it then follows from that @xmath559 , hence holds and the proof that any element of @xmath560 is in @xmath561 is complete . finally , if @xmath308 $ ] belongs to @xmath560 one has @xmath562 by and thus the intersection @xmath209\cap \cj_p$ ] is the ideal @xmath528 . [ defnhp ] we denote by @xmath563 the quotient by @xmath561 of the subalgebra of @xmath238 generated by @xmath564 $ ] , @xmath210 , @xmath211 , for @xmath565 . the algebra @xmath563 is generated by @xmath191 $ ] the operators @xmath210 and @xmath211 , for @xmath430 and its presentation is similar to the presentation of @xmath238 . the relations are @xmath566 \mu_n^ * \tilde\mu_n = n\qqq n\in i(p ) \\[3 mm ] \tilde\mu_n\mu_m^ * = \mu_m^*\tilde\mu_n \qqq n , m\in i(p)\quad\text{with}~ ( n , m ) = 1 \end{array}\ ] ] as well as the relations @xmath567 where @xmath44 , @xmath430 is defined by @xmath568 given an algebra @xmath569 , an automorphism @xmath570 and an integer @xmath0 we let @xmath571 be the subalgebra of the algebraic cross product @xmath572 determined by the condition @xmath573 if we let @xmath574 and @xmath575 , then it is easy to see that @xmath571 is generated by @xmath576 with the relations @xmath577 [ crossfrob ] there exists a unique automorphism @xmath578 such that @xmath579 one derives an isomorphism @xmath580 the map @xmath581 defines an automorphism of @xmath57 . its linearization @xmath502 acts on @xmath191 $ ] and commutes with the endomorphisms @xmath43 and @xmath582 . in fact by applying the isomorphism of proposition [ bcrelatenew ] , @xmath502 corresponds to the frobenius automorphism of @xmath4 . thus it extends to an automorphism @xmath578 . the second statement follows by comparing the presentation of @xmath583 with that of the crossed product @xmath584 as in . [ bcrelate3]let @xmath279 . @xmath61 the restriction @xmath585 of the representation @xmath3 ( as in theorem [ bcrelate2 ] ) to @xmath563 is @xmath512-linear and indecomposable over @xmath512 . @xmath63 the representations @xmath585 are pairwise inequivalent . @xmath66 the representation @xmath3 is linear and indecomposable over @xmath586 . @xmath69 two representations @xmath3 and @xmath587 are equivalent over @xmath586 if and only if there exists @xmath588 such that @xmath589 . @xmath61 the @xmath512-linearity property is checked directly on the generators using theorem [ bcrelate2 ] . it follows from that the vector @xmath590 is cyclic for @xmath563 , _ @xmath591 is dense in @xmath592 . one has @xmath593 for any @xmath512-linear continuous operator @xmath594 in the commutant of @xmath563 one has @xmath595 , @xmath596 and by there exists @xmath597 such that @xmath598 . thus since @xmath590 is cyclic , @xmath594 is given by the module action of @xmath597 . @xmath63 by , the action of @xmath599 for @xmath600 on the subspace is given by the multiplication by @xmath601 . thus @xmath602 is an invariant of the representation . @xmath66 any element of the commutant of the action of @xmath238 is given by the module action of @xmath597 , where @xmath241 is fixed for the action of the frobenius on @xmath512 , _ i.e. _ @xmath603 . this shows that @xmath3 is indecomposable . @xmath69 we show first that if there exists @xmath588 such that @xmath604 , the representations @xmath3 and @xmath587 are equivalent over @xmath586 . let @xmath605 and define @xmath606 . one has @xmath607 for all @xmath430 and if @xmath594 is an @xmath512-linear operator so is @xmath608 . it thus follows from and that @xmath609 and @xmath610 . for @xmath308 $ ] , @xmath611 only depends on @xmath313 and for @xmath612 , @xmath613 , one has @xmath614 . moreover since @xmath615 commutes with @xmath502 , it follows from that @xmath616 and @xmath617 . thus one gets the required equivalence . conversely , assume that two representations @xmath3 and @xmath587 are equivalent over @xmath586 . by the @xmath586-linear representation @xmath3 ( and similarly @xmath587 ) determines uniquely the following representation of @xmath191 $ ] in the @xmath586-module @xmath512 @xmath618 in turns this determines an extension of the @xmath0-adic valuation to the subfield @xmath619 generated over @xmath10 by @xmath620 . indeed the formula @xmath621\ ] ] only depends on the class of @xmath312 in @xmath59 and extends uniquely to a valuation on @xmath59 . the conclusion then follows from proposition [ try ] . in @xcite it was shown that the extremal , complex kms states below critical temperature of the bc - system ( _ cf . _ ) are of the form @xmath622 where @xmath623 is the hamiltonian operator of multiplication by @xmath624 in the canonical basis @xmath625 of the hilbert space @xmath626 and @xmath627 is the irreducible representation of the algebra @xmath628 given by @xmath629 where @xmath236 determines an embedding in @xmath6 of the cyclotomic field @xmath52 generated by the abstract roots of unity . thus the extremal kms states @xmath630 are directly computable using the representation @xmath627 and the explicit description of the hamiltonian . in section [ sectcompl ] , we have described the @xmath0-adic analogue of the representation @xmath627 . in this section , our goal is to obtain the @xmath0-adic analogue of the kms states @xmath630 . the guiding equation is provided by the general algebraic formulation of the kms condition which is described by the equality @xmath631 where @xmath632 is a linear form on an algebra @xmath569 endowed with an automorphism @xmath633 . in our case the algebra is @xmath634 in [ sectkmsauto ] we introduce , using the iwasawa logarithm as a substitute for the above complex hamiltonian @xmath623 , the automorphisms @xmath635 . these automorphisms are defined for @xmath636 in the extended @xmath637-disk " @xmath638 ( _ cf . _ below ) . in [ sectexten ] we shall show how to extend their definition from the domain @xmath638 to a covering @xmath639 of @xmath8 . the construction of the kms states is based on the classical construction of the @xmath0-adic l - functions and @xmath0-adic polylogarithm and many properties that we obtain rely on the simplifications which occur when @xmath640 ( @xmath641 ) . in [ sectcycloid ] we prove the identities in the cyclotomic field , involving bernoulli polynomials , which are behind the verification of the kms condition . in [ sectlinform ] we provide the construction of the linear forms @xmath630 using some of the results from @xcite ( _ cf . _ chapter v ) . in [ sectkmscond ] we prove that the functionals @xmath630 fulfill the kms condition with respect to the automorphism @xmath635 . unlike the complex case , this construction exhibits the ( new ) phenomenon of the invariance of the linear forms @xmath630 under the symmetry of @xmath642 given by the automorphism @xmath643 . throughout this section we fix a finite , rational prime @xmath0 and an algebraic closure @xmath644 whose completion is denoted @xmath8 . we also use the following notation @xmath645 and @xmath646 we consider the extended @xmath637-disk " @xmath647 and first develop the theory for @xmath648 . in [ sectexten ] we shall explain how the iwasawa construction of @xmath0-adic @xmath7-functions allows one to extend the whole theory from the domain @xmath638 to the covering of @xmath8 given by the multiplicative group @xmath639 which is the open disk of radius one and center @xmath131 in @xmath8 . let @xmath649 be the multiplicative group of rational fractions whose numerator and denominator are prime to @xmath0 . [ uniqueext ] let @xmath650 . there exists a unique analytic function @xmath651 such that @xmath652 we recall that the iwasawa logarithm @xmath653 is the unique extension of the function defined in the open unit disk centered at @xmath131 by @xmath654 to a map @xmath655 such that @xmath656 one has @xmath657 since @xmath658 is a root of unity , and @xmath659 moreover the exponential function is defined by the series @xmath660 we define @xmath661 this is a well - defined , analytic function of @xmath648 since @xmath662 and thus @xmath663 by . we show that holds . this follows from the equality @xmath664 which holds for @xmath665 since @xmath666 is even . in general , follows from the formula @xmath667 as shown in @xcite ( chapter 5 , p. 52 ) , where the notation @xmath668 is introduced . the uniqueness follows from the discreteness of the set of zeros of analytic functions . [ multext ] let @xmath669 , then @xmath670 is a group homomorphism . moreover , for @xmath650 @xmath671 this follows from and the equality ( _ cf . _ @xcite ) @xmath672 the standard notation for @xmath673 is @xmath674 : it is the unique @xmath666 root of unity which is congruent to @xmath675 modulo @xmath676 . in particular one has @xmath677 [ kmsauto ] @xmath61 for @xmath648 there exists a unique automorphism @xmath635 such that @xmath678 @xmath63 one has @xmath679 and @xmath680 is an automorphism of order @xmath666 . it suffices to check that @xmath681 preserves the presentation given by the relations and . this follows from the multiplicativity shown in lemma [ multext ] . similarly follows from . the last statement follows from . we recall that the bernoulli polynomials @xmath682 are defined inductively as follows @xmath683 equivalently , these polynomials can be introduced using the generating function @xmath684 the first few are @xmath685 these polynomials fulfill the equation @xmath686 . the bernoulli numbers are @xmath687 . using , one checks the identity ( _ cf . _ @xcite , chapter 4 , proposition 4.1 ) @xmath688 we also introduce inductively the rational fractions @xmath689 for @xmath690 , as follows @xmath691 for @xmath692 we denote by @xmath693 the class of @xmath694 $ ] modulo the cyclotomic ideal ( _ cf . _ definition [ acf ] ) . it is a root of unity whose order is the denominator of @xmath286 . [ indeplem ] let @xmath695 , @xmath696 . then @xmath697 the equality for @xmath698 follows from . thus we can assume that @xmath699 . the taylor expansion at @xmath700 of @xmath701 is given by @xmath702 since @xmath703 and @xmath704 agrees with @xmath705 . then for @xmath706 and @xmath341 such that @xmath707 one has @xmath708 since @xmath709 , one derives @xmath710 since @xmath711 , taking the taylor expansion at @xmath700 using , gives the equality @xmath712 [ alternat1 ] let @xmath713 , @xmath696 . @xmath61 the following sum only depends upon @xmath38 and @xmath714 @xmath715 @xmath63 one has @xmath716 @xmath66 for @xmath717 , @xmath718 one has @xmath719 @xmath61 follows from . to obtain @xmath63 , note that @xmath720 @xmath66 one checks as an identity between rational fractions by induction on @xmath36 . it holds for @xmath466 by applying the operation @xmath721 to both sides of the identity @xmath722 to obtain for @xmath38 assuming it for @xmath723 one applies the operation @xmath705 to both sides of the identity for @xmath723 . combining with we obtain , using when @xmath724 @xmath725 in this section we shall provide a meaning to expressions of the form @xmath726 where @xmath727 , @xmath613 is an integer prime to @xmath0 and @xmath728 . note that as a function of @xmath254 , @xmath729 only depends on the residue of @xmath289 modulo @xmath730 . we let @xmath731 and decompose the sum according to the residue @xmath286 of @xmath289 modulo @xmath117 . one has @xmath732 . the elements of @xmath422 are characterized by the fact that their residues mod . @xmath117 are given by pairs @xmath733 , with @xmath734 . for @xmath735 , we let @xmath736 be the smallest integer with residue modulo @xmath117 equal to @xmath286 . then , the sum can be written as @xmath737 notice that the first sum ( over @xmath286 ) in only involves finitely many terms . each infinite sum in is of the form ( with @xmath738 ) @xmath739 and it is well known that this expression retains a meaning in the @xmath0-adic context ( _ cf . _ @xcite chapter v ) . more precisely , the asymptotic expansion in the complex case , for @xmath740 ( this process goes back to euler s computation of @xmath741 ) @xmath742 motivates the following precise formula , where we prefer to leave some freedom in the choice of the multiple @xmath117 of @xmath743 . [ indlem ] with @xmath676 as in , and @xmath744 , @xmath745 , a multiple of @xmath743 , the expression @xmath746 defines a meromorphic function of @xmath648 with a single pole at @xmath747 . it follows from @xcite ( proposition 5.8 ) and the inequality ( _ cf . _ @xcite theorem 5.10 ) @xmath748 that the series @xmath749 converges for @xmath750 . [ indep ] for @xmath636 a negative odd integer of the form @xmath751 , and @xmath744 , @xmath745 , @xmath117 a multiple of @xmath743 , one has , with @xmath752 defined by @xmath753 for @xmath754 , @xmath755 , one has @xmath756 . the binomial coefficients @xmath757 in all vanish for @xmath758 and the sum defining @xmath759 is therefore finite . one has @xmath760 moreover for any integer @xmath761 , the bernoulli polynomials fulfill the equation @xmath762 for @xmath754 , @xmath755 , one thus gets , taking @xmath763 @xmath764 one defines for any @xmath765 @xmath766 one has @xmath767 since @xmath730 divides @xmath117 , one derives @xmath768 while , since @xmath730 divides @xmath769 one gets @xmath770 the equality follows . [ indcoro ] the function @xmath771 is independent of the choice of @xmath772 , @xmath745 . for two choices @xmath773 the analytic function of @xmath648 @xmath774 vanishes at all negative integers @xmath775 by the equality , thus it is identically @xmath776 . [ defz ] the following equation defines a linear form @xmath630 on @xmath563 for any @xmath648 @xmath777 for @xmath549 relatively prime . the next lemma will play an important role in the proof ( _ cf . _ next section ) that @xmath630 fulfills the kms condition . [ tilderhon ] for any @xmath430 and @xmath648 , @xmath778 , one has @xmath779\ ] ] ( _ cf . _ for the definition of @xmath44 ) . after multiplication by @xmath780 , both sides of are analytic functions of @xmath648 . thus it is enough to show that holds for @xmath781 . in this case one has @xmath782 and , from one gets @xmath783 to prove the equality we can assume that @xmath784 for @xmath785 . one has @xmath786 so that @xmath787 then follows from . since @xmath0 is prime to @xmath38 and the rational numbers @xmath788 form the same subset as the set made by the @xmath789 , we derive @xmath790 the main result of this section is the following [ thmkms ] for any @xmath648 , @xmath778 and @xmath791 , the linear form @xmath630 fulfills the kms@xmath792 condition : @xmath793 moreover the partition function is the @xmath0-adic @xmath7-function @xmath794 which does not vanish for @xmath648 . we fix @xmath795 , then after multiplication by @xmath780 , both sides of are analytic functions of @xmath648 . we first assume that @xmath778 ; we shall consider the case @xmath747 separately later . since any element of the algebra @xmath642 can be written as a finite linear combination of @xmath796 , for @xmath797 $ ] , we may assume that @xmath798 where @xmath549 , @xmath799 , @xmath800 , @xmath801 and @xmath802 $ ] . then , we use the presentation of @xmath642 to compute @xmath803 let @xmath804 be the gcd of @xmath805 and @xmath806 . one has @xmath807 @xmath808 let @xmath809 be the gcd of @xmath810 and @xmath811 . one has @xmath812 , @xmath813 @xmath814 we obtain @xmath815 it follows that unless @xmath816 and @xmath817 one has @xmath818 and @xmath819 thus we can assume that @xmath816 and @xmath817 . then we have @xmath820 so that , by one derives @xmath821 similarly one has , by applying again @xmath822 thus follows from the equality @xmath823 which in turn derives from and . now , we turn to the normalization factor ( _ i.e. _ partition function ) in which is given by @xmath824 this is the @xmath0-adic @xmath7-function for the character @xmath825 ( _ cf . _ @xcite , chapter 5 , theorem 5.11 ) @xmath826 moreover , notice that the iwasawa construction of @xmath7-functions ( _ cf . _ @xcite , chapter 7 , theorem 7.10 ) yields a formal power series @xmath827^\times$ ] such that ( with @xmath676 as in ) the following equality holds @xmath828 since @xmath827^\times $ ] is invertible ( _ cf . _ @xcite lemma 7.12 ) , this gives the required result . note that @xmath829 has a pole at @xmath747 , with residue given by @xmath830 [ critval ] when @xmath831 one has @xmath832 assume first that @xmath833 . then @xmath834 is a non - trivial root of unity , whose order @xmath835 divides @xmath730 which is prime to @xmath0 and hence prime to @xmath676 . thus using the decomposition @xmath836 we get @xmath837 if @xmath838 the result follows from the above discussion . notice in particular that the limit of the functional values @xmath839 as @xmath831 is independent of values of @xmath602 ( _ i.e. _ independent of the choice of @xmath279 ) . in the complex case , the functional values for @xmath840 , are given by the formula . in that case , we shall now check directly that for @xmath841 , @xmath842 , the functional values determine @xmath843 as an embedding of the abstract cyclotomic field @xmath52 in @xmath6 . [ dense ] @xmath61 let @xmath844 , @xmath845 . then the graph of the multiplication by @xmath241 in @xmath58 is a dense subset of @xmath846 . @xmath63 let @xmath847 . assume that @xmath848 . then the graph of @xmath849 is dense in @xmath846 . @xmath61 the set @xmath850 is a subgroup of @xmath846 and so is its closure @xmath851 . if @xmath852 were not dense , then there would exist a non - trivial character @xmath853 of the compact group @xmath846 whose kernel contains @xmath851 . thus there would exist a non - zero pair @xmath854 such that @xmath855 , for all @xmath692 . this would imply that the multiplication by @xmath844 in the group @xmath856 ( @xmath857 are the finite adles ) ought to fulfill @xmath858 . this implies @xmath859 and hence @xmath860 for all primes @xmath0 . if @xmath861 , this contradicts the fact that @xmath844 _ i.e. _ @xmath862 for all @xmath0 . @xmath63 by lemma [ gaut ] the group @xmath863 is the group of automorphisms of the group @xmath57 viewed as the additive group @xmath864 . let @xmath865 represent @xmath847 . then the same proof as in @xmath61 shows that if the graph of @xmath849 is not dense , there exists a non - zero pair @xmath854 such that @xmath866 for all primes @xmath867 . it follows that @xmath868 and @xmath869 . from lemma [ dense ] we derive that , if @xmath870 is a continuous non - constant function , and @xmath871 , are injective , an equality of the form @xmath872 necessarily implies that @xmath873 or @xmath874 . in the latter case one also gets @xmath875 by uniqueness of the fourier decomposition however , this case can not occur if @xmath876 , for @xmath877 . next , we fix an integer @xmath751 , @xmath440 , and we investigate the dependence on @xmath602 in the expressions . for a chosen pair of embeddings @xmath878 , assume that @xmath879 holds for all @xmath880 , _ i.e. _ the equality holds for all fractions with denominator @xmath730 prime to @xmath0 . it follows from that one has ( with @xmath502 the frobenius automorphism of @xmath498 ) @xmath881 thus we get @xmath882 since both @xmath602 and @xmath883 are isomorphisms of the group of roots of unity in @xmath59 with the group of roots of unity in @xmath8 of order prime to @xmath0 , there exists an automorphism @xmath847 such that @xmath884 for all @xmath880 . one has @xmath885 by uniqueness of the fourier transform for the finite group @xmath886 , yields the equality @xmath887 [ symbreak ] let @xmath888 and let @xmath847 . if @xmath889 one has @xmath890 if @xmath891 and @xmath751 , @xmath440 , then the functionals @xmath892 and @xmath893 are distinct . to prove we can assume that @xmath894 _ i.e. _ that @xmath895 for all @xmath880 . then we have , with @xmath896 : @xmath897 . let first @xmath751 . one has @xmath898 since @xmath899 is even , the bernoulli polynomial @xmath900 fulfills the equality @xmath901 thus follows for all values @xmath751 . since these values admit @xmath776 as an accumulation point , one derives the equality of the analytic functions on their domain @xmath638 . now , we assume that @xmath848 . then it follows from lemma [ dense ] that the graph of @xmath849 is dense in @xmath846 . thus implies that @xmath902 is constant which is a contradiction . it remains to show that for non - zero powers @xmath903 of @xmath0 one can not have an equality of the form @xmath904)\qqq x\in [ 0,1]\ ] ] where @xmath905 $ ] is the integral part of @xmath906 . in fact , this would imply that @xmath907 has infinitely many zeros , thus @xmath908 which is a contradiction . in this section we show that the construction of the kms@xmath51 states @xmath630 , for @xmath648 , extends naturally to the covering of @xmath8 defined by the following group homomorphism @xmath909 where @xmath910 is the open unit disk in @xmath8 with radius @xmath131 , viewed as a multiplicative group . up to the normalization factor @xmath911 , this group homomorphism coincides with the definition of the iwasawa logarithm , it is surjective with kernel the subgroup of roots of unity of order a @xmath0-power ( _ cf . _ @xcite , theorem p. 257 ) and it defines by restriction a bijection @xmath912 whose inverse is given by the map @xmath913 by construction , this local section is a group homomorphism which allows one to view the additive group @xmath638 as a subgroup of @xmath639 . we start by extending the definition of the functions @xmath914 as in which were implemented in the construction of the automorphisms @xmath915 ( _ cf . _ proposition [ kmsauto ] ) . for @xmath650 the equality @xmath916 defines a group homomorphism from @xmath917 to the additive group @xmath586 . [ agreelem ] for @xmath648 , @xmath650 and @xmath918 one has @xmath919 one has @xmath920 . thus @xmath921 and @xmath922 the second equality follows from the definition .proposition [ kmsauto ] and its proof thus extend from @xmath638 to @xmath639 . this means that for @xmath923 there exists a unique automorphism @xmath924 \in \aut(\ch_{\c_p}^{(p)})$ ] such that @xmath925 ( \tilde\mu_a e(\gamma)\mu_b^ * ) = \omega(b / a)\lambda^{i_p(b / a ) } \tilde\mu_a e(\gamma)\mu_b^*\qqq a , b\in i(p),\ \gamma\in \mup.\ ] ] next , we extend the construction of the linear forms @xmath630 given in [ sectlinform ] . it is sufficient to extend the definition of the functions of lemma [ indlem ] ( which we proved to be independent of the choice of @xmath745 multiple of @xmath743 ) @xmath926 to define the sought for extension it is convenient to express the above function in terms of the @xmath0-adic @xmath7-functions @xmath927 associated to even dirichlet characters of conductor @xmath928 prime to @xmath0 . by definition , a dirichlet character @xmath853 is a character of the multiplicative group @xmath929 and its conductor @xmath928 is the integer such that the kernel of @xmath853 is the kernel of the projection @xmath930 . the definition of @xmath927 is similar to precisely as follows @xmath931 where @xmath117 is any multiple of @xmath932 and where @xmath853 has been extended to a periodic function of period @xmath928 vanishing outside @xmath933 . we recall that the @xmath7-function @xmath927 is identically zero when the character @xmath853 is odd , _ i.e. _ when @xmath934 ( _ cf . _ @xcite remarks p. 57 ) . moreover when @xmath853 is even , non - trivial , and its conductor is prime to @xmath0 , there exists an analytic function @xmath935 on @xmath639 such that ( _ cf . _ @xcite theorem 7.10 ) @xmath936 the extension of the functions @xmath937 to @xmath639 is a consequence of the following [ comblem ] for any @xmath880 there exists coefficients @xmath938 such that @xmath939 where @xmath367 varies among the divisors of @xmath730 , and , for fixed @xmath367 , @xmath853 varies among the set of dirichlet characters whose conductor @xmath928 divides @xmath940 . the integers @xmath941 are the primes which divide @xmath942 but not @xmath928 . let @xmath730 be an integer prime to @xmath0 , and @xmath943 . the expression @xmath944 is independent of the choice of the multiple @xmath745 of @xmath743 . let @xmath853 be a dirichlet character ( with values in @xmath8 ) with conductor @xmath928 and let @xmath289 be a multiple of @xmath928 . then the following defines a multiplicative map from @xmath945 to @xmath8 @xmath946 if @xmath289 divides @xmath730 and one replaces @xmath853 with @xmath947 in one obtains instead of @xmath927 the function @xmath948 where the integers @xmath941 are the primes which divide @xmath942 without dividing @xmath928 . next , define for any divisor @xmath367 of @xmath730 and any function @xmath949 , @xmath940 , @xmath950 one then gets @xmath951 thus using and it is enough to prove that for any function @xmath943 there exists coefficients @xmath938 such that @xmath952 it is in fact enough to check this for @xmath953 where @xmath954 . let then @xmath367 be the gcd of @xmath955 and @xmath730 . one has @xmath956 where @xmath957 is prime to @xmath940 . moreover for any element @xmath958 one has @xmath959 which gives the required equality.we thus obtain the following extension of theorem [ thmkms ] [ thethmkms ] there exists an analytic family of functionals @xmath960 , @xmath961 , on @xmath563 such that @xmath19 @xmath962 @xmath19 @xmath960 fulfills the kms condition @xmath963(y))=\psi_{\lambda,\rho}(y\,x)\qqq x , y\in \ch_{\c_p}^{(p)}.\ ] ] @xmath19 for @xmath648 and @xmath918 one has @xmath964 it follows from that there exists an analytic function @xmath965 of @xmath961 such that @xmath966 by applying , lemma [ agreelem ] and lemma [ comblem ] , we see that there exists , for @xmath613 and @xmath967 , an analytic function @xmath968 of @xmath961 such that @xmath969 this proves the existence of the analytic family of functionals @xmath960 fulfilling the required conditions . for a global field @xmath74 of positive characteristic ( _ i.e. _ a function field associated to a projective , non - singular curve @xmath970 over a finite field @xmath75 ) it is a well known fact that the space of valuations of the maximal abelian extension @xmath76 of @xmath74 has a geometric meaning . in fact , for each finite extension @xmath77 of @xmath78 the space @xmath79 of ( discrete ) valuations of @xmath77 is turned into an algebraic , one - dimensional scheme whose non - empty open sets are the complements of finite subsets @xmath80 . the structure sheaf is locally defined by the intersection @xmath81 of the valuation rings inside @xmath77 . then the space @xmath82 is the projective limit of the schemes @xmath79 , @xmath83 . for the global field @xmath84 of rational numbers , one can consider its maximal abelian extension @xmath52 as an abstract field ( _ cf . _ definition [ acf ] ) and try to follow a similar idea . in section [ curve ] , we will see however that the space @xmath85 provides only a rough analogue , in characteristic zero , of @xmath82 . this section develops the preliminary step of presenting 5 different but equivalent descriptions of the space @xmath55 of extensions of the @xmath0-adic valuation of @xmath10 to the abstract cyclotomic field @xmath52 . the field @xmath52 is the composite of the field generated by roots of unity of order a @xmath0-power and the field @xmath59 generated by the roots of unity of order prime to @xmath0 . we describe canonical isomorphisms of @xmath55 with : @xmath61 the space of sequences of irreducible polynomials @xmath62 $ ] , @xmath36 , fulfilling the basic conditions of the conway polynomials ( _ cf . _ theorem [ conway0 ] ) . @xmath63 the space @xmath64 of bijections of the monoid @xmath65 of roots of unity of order prime to @xmath0 which commute with their conjugates , as in definition [ defnsp ] ( _ cf . _ proposition [ hope ] ) . @xmath66 the space @xmath67 of field homomorphisms , where @xmath68 is the fixed field under the frobenius automorphism ( _ cf . _ proposition [ homfield ] ) . @xmath69 the quotient of the space @xmath53 of definition [ defnxp ] by the action of @xmath70 ( _ cf . _ proposition [ try ] ) . @xmath71 the algebraic spectrum of the quotient algebra @xmath72/j_p$ ] , where @xmath73 is the reduction modulo @xmath0 of the cyclotomic ideal ( _ cf . _ definition [ acf ] and proposition [ cpcp1 ] ) . incidentally , we notice that @xmath61 describes the link between @xmath55 and the explicit construction of an algebraic closure @xmath4 of @xmath2 , by means of a sequence of irreducible polynomials over @xmath2 , fulfilling the basic conditions of the conway polynomials . theorem [ conway0 ] states that the map which associates to a valuation @xmath971 the sequence @xmath972 of characteristic polynomials for the action ( by multiplication ) of the primitive root @xmath973 on the residue field of the restriction of @xmath809 to @xmath59 , determines a bijection between @xmath55 and sequences of polynomials in @xmath974 $ ] fulfilling the basic conditions of the conway polynomials . [ acf ] the abstract cyclotomic field @xmath52 is the quotient of the group ring @xmath56 $ ] by the ideal @xmath975 generated by the idempotents @xmath976 in general , if we let @xmath977 then one knows that the @xmath38-th cyclotomic polynomial @xmath978 is the gcd of the polynomials @xmath979 , for @xmath835 , @xmath980 and @xmath981 . for @xmath982 , and @xmath983 one has @xmath984 thus @xmath985 . it follows that the homomorphism @xmath986/j\to \c\ , , \ \ \rho_0(e(\gamma))=e^{2\pi i\gamma}\ ] ] induces an isomorphism of @xmath52 with the subfield of @xmath6 generated by roots of unity . using the identification @xmath987 the group @xmath929 acts by automorphisms of @xmath58 and hence by automorphisms of the group ring @xmath56 $ ] . this action preserves globally the @xmath38-torsion in @xmath58 and hence fixes each of the projection @xmath988 . it follows that it leaves the ideal @xmath975 globally invariant and hence it induces an action on the quotient field @xmath52 . this action gives the galois group @xmath989 which acts on roots of unity as it acts on @xmath58 . for each prime @xmath0 , one has ( @xmath990 rational prime ) @xmath991 one lifts @xmath992 to the subgroup @xmath993 , with all components equal to @xmath131 except at @xmath0 . this subgroup acts trivially on @xmath57 . its fixed subfield @xmath619 is the subfield of @xmath52 generated over @xmath10 by the group @xmath994 of roots of unity of order prime to @xmath0 . it coincides with the _ inertia subfield _ @xmath995 for any extension @xmath971 of the @xmath0-adic valuation to @xmath52 . more precisely let @xmath996 be the completion of @xmath52 for the valuation @xmath809 . then one knows that the composite subfield @xmath997 is the maximal abelian extension @xmath998 of @xmath9 . this extension is the composite ( _ cf . _ @xcite ) @xmath999 where @xmath1000 denotes the maximal unramified extension of @xmath9 and @xmath1001 is obtained by adjoining to @xmath9 all roots of unity of order a @xmath0-power . the translation theorem of galois theory gives a canonical isomorphism ( by restriction ) of galois groups @xmath1002 the _ decomposition subfield _ : @xmath1003 is independent of the choice of the valuation @xmath971 since @xmath852 is abelian and acts transitively on @xmath55 , more precisely one has the following classical result [ ggalois ] @xmath61 the group @xmath863 is the group of automorphisms of the group @xmath57 . @xmath63 the inertia subfield @xmath59 is the fixed subfield of @xmath1004 and its galois group is canonically isomorphic to @xmath1005 acting on @xmath1006 as it acts on @xmath57 . @xmath66 let @xmath1007 be the element of @xmath863 with all components equal to @xmath0 . then the associated automorphism @xmath1008 is the unique automorphism which acts by @xmath1009 on the multiplicative group @xmath1010 . @xmath69 the fixed subfield @xmath68 of @xmath502 is the decomposition subfield @xmath1003 . @xmath71 the group @xmath1011 acts transitively on @xmath55 with isotropy @xmath1012 , where @xmath1013 is the closure of @xmath1014 . @xmath61 let @xmath1015 viewed as a discrete group . the pontrjagin dual @xmath1016 is the product @xmath1017 . we claim that the group of automorphisms of @xmath1018 is @xmath1019 indeed , one has @xmath864 , so that the dual of @xmath1018 is @xmath1020 . this is a compact ring which contains @xmath416 as a dense subring . thus an automorphism @xmath849 of the additive group is characterized by the assignment @xmath1021 and is given by multiplication by @xmath955 . invertibility shows that @xmath1022 . this proves . @xmath63 under the isomorphism the galois group @xmath1023 becomes the subgroup @xmath993 . the fixed subfield of this subgroup is @xmath619 and is the inertia subfield of @xmath52 . the quotient @xmath1024 is canonically isomorphic to @xmath1005 . @xmath66 under the isomorphism @xmath1025 the action of @xmath502 on @xmath620 corresponds to the multiplication by @xmath0 in @xmath57 . @xmath69 the galois group @xmath1026 is topologically generated by the frobenius automorphism @xmath1027 whose action on the roots of unity of order prime to @xmath0 is given by @xmath1028 . under the isomorphism this automorphism restricts to the automorphism @xmath1008 . notice that the fields @xmath59 and @xmath9 are linearly disjoint over their intersection @xmath1029 then , the translation theorem in galois theory shows that , by restriction to @xmath59 , one has an isomorphism @xmath1030 this shows that @xmath29 is the fixed subfield @xmath68 of @xmath502 . @xmath71 it is well known that the galois group acts transitively on extensions of a valuation . moreover the isotropy subgroup is the subgroup of the galois group corresponding to the decomposition subfield and is hence given by @xmath1012 . [ restrict ] the natural map @xmath1031 given by restriction of valuations is equivariant and bijective . the restriction map is equivariant for the action of @xmath852 on both spaces , these actions are transitive and have the same isotropy group so the restriction map is bijective . in fact it is worth giving explicitly the unique extension of a valuation @xmath1032 to @xmath52 . the latter field is obtained by adjoining to @xmath59 primitive roots of unity of order a power of @xmath0 , _ i.e. _ a solution @xmath317 of an equation of the form @xmath1033 one writes @xmath1034 and finds that the equation fulfilled by @xmath1035 is of eisenstein type , the constant term being equal to @xmath0 , and reduces to @xmath1036 , modulo @xmath0 . this shows that @xmath1037 then the valuation @xmath809 , normalized so that @xmath1038 , extends uniquely to elements of the extension @xmath1039 $ ] by setting @xmath1040 [ infinite ] the decomposition subfield @xmath1041 is an infinite extension of @xmath10 which contains for instance @xmath1042 for @xmath38 a quadratic residue modulo @xmath0 . its galois group @xmath1043 is the quotient of @xmath1005 by the closure of the group of powers of @xmath1044 and is a compact group which contains for each prime @xmath867 the cyclic group of order @xmath1045 coming from the torsion part of @xmath1046 . [ defnsp ] let @xmath1047 be the monoid obtained by adjoining a zero element to the multiplicative group @xmath620 . we denote by @xmath64 the set of bijections @xmath1048 which commute with all their conjugates @xmath1049 under rotations @xmath1050 by elements of @xmath620 , and fulfill the relations : @xmath1051 , @xmath1052 . the maps @xmath637 encode the addition of @xmath131 on @xmath1053 , when one enriches the multiplicative structure of the monoid @xmath1053 with an additive structure turning it to a field of characteristic @xmath0 ( _ i.e. _ an algebraic closure of @xmath2 ) . notice that using distributivity the addition of @xmath131 encodes the full additive structure ( _ cf . _ @xcite ) . [ gaut ] the group @xmath863 acts transitively on @xmath64 with isotropy @xmath1013 . we check that @xmath1005 acts transitively on @xmath64 . let @xmath1054 , for @xmath1055 and let @xmath1056 be the two corresponding field structures on @xmath1053 . then the two fields @xmath1056 are algebraic closures of @xmath2 and hence they are isomorphic . we let @xmath1057 be such an isomorphism . by construction @xmath849 is an automorphism of the multiplicative group @xmath620 and it transports the operation @xmath1058 of addition of @xmath131 in @xmath1059 into the operation @xmath1060 of addition of @xmath131 in @xmath1061 . since the galois group of @xmath4 is topologically generated by the frobenius @xmath1009 one gets , using galois theory , that the isotropy of any @xmath1062 is the closure of the group of powers of @xmath1044 , _ i.e. _ the subgroup @xmath1013 . we are now ready to state the main result of this section [ conway0 ] an element @xmath1063 is entirely characterized by a sequence of polynomials @xmath62 $ ] of degree @xmath1064 , such that @xmath19 each @xmath1065 is monic and irreducible . @xmath19 @xmath1066/(p_n(t))$ ] is a generator of the multiplicative group of the quotient field . @xmath19 for any integer @xmath1067 and for @xmath1068 , @xmath1069 is a multiple of @xmath1065 . the first step in the proof is to construct a natural map @xmath1070 . we know that @xmath1071 and that @xmath1072 , thus we consider the valuation ring @xmath1073 of @xmath1000 . it contains @xmath586 and @xmath620 . note that the ring generated by @xmath416 and @xmath620 is the ring of integers of the subfield @xmath619 generated over @xmath10 by @xmath620 . one has the diagram of inclusions @xmath1074 & \\ \bar\f_p & \z_p^{\rm ur } \ar[u]\ar[l ] _ { \epsilon } & \z_p^{\rm ur}\cap \qcy\ar[u]^ { } \ar[l]^ { } & \emup\cup\{0\ } \ar[l]^ { } \\ \f_p \ar[u]&\z_p \ar[l ] _ { \epsilon } \ar[u]^- { } & \z_p\cap \qcy\ar[u]^ { } \ar[l ] _ { } & \tau(\f_p ) \ar[u]^ { } \ar[l]^ { } \\ } \hspace{140pt}\end{gathered}\ ] ] where @xmath1075 is the lift . note that @xmath1076 since this lift is formed of roots of unity ( of order @xmath1077 ) . in the middle line of the above diagram , the composite map @xmath246 from @xmath1078 to @xmath4 is an isomorphism of multiplicative monoids . indeed , the lift @xmath1079 gives the inverse map . since @xmath4 is a field one can transport its additive structure using @xmath246 and one obtains a unique element @xmath1080 . [ hope ] the map @xmath1070 is a bijection and is equivariant for the action of @xmath1081 . the action of @xmath1005 on the subset @xmath620 is the one described in lemma [ gaut ] . this shows that the map @xmath1082 is equivariant . since both spaces @xmath55 and @xmath64 are homogeneous spaces over @xmath1005 with the same isotropy groups @xmath1083 as follows from lemmas [ ggalois ] and [ gaut ] , the map @xmath1082 is bijective . we can produce a concrete construction of the valuation @xmath809 associated to the map @xmath1084 . one first determines @xmath809 on the subfield @xmath619 . it is enough to determine the valuation @xmath809 on elements of the form @xmath1085 let @xmath1086 be the algebraic closure of @xmath2 obtained by endowing the multiplicative monoid @xmath1078 with the addition associated to @xmath1084 . one then has @xmath1087 where @xmath1088 is the @xmath0-adic valuation in the witt ring @xmath1089 and @xmath477 the lift . finally since the field @xmath52 is the composite of the subfields @xmath59 and the fixed field of the action of @xmath1090 which is generated by roots of unity of order a @xmath0-power , one can use to extend the valuation @xmath809 uniquely to @xmath52 . and roots of unity [ conway ] ] we are now ready to complete the proof of theorem [ conway0 ] , _ i.e. _ we prove that : [ hope1 ] an element @xmath1062 is entirely characterized by a sequence @xmath1065 of polynomials of @xmath974 $ ] fulfilling the conway conditions as in theorem [ conway0 ] . let @xmath1062 . for each @xmath36 , let @xmath1091 be the corresponding field structure on the union @xmath1092 , where @xmath1093 is the group of roots of unity of order @xmath1094 in @xmath52 generated by @xmath1095 . the @xmath2 vector space @xmath1091 is of dimension @xmath38 since its cardinality is @xmath1096 . the canonical generator @xmath1097 of @xmath1093 acts on the @xmath2 vector space @xmath1091 by the multiplication @xmath1098 . we let @xmath1065 be its characteristic polynomial _ i.e. _ the determinant @xmath1099 . it is a monic polynomial of degree @xmath38 with coefficients in @xmath2 . in the field @xmath1091 one has @xmath1100 , since @xmath1098 fulfills its characteristic equation . thus we derive a homomorphism of algebras @xmath1101/(p_n(t))\to\k_n(s)$ ] which sends @xmath1102 . it is surjective since any non - zero element of @xmath1091 is a power of @xmath1097 . since @xmath1065 has degree @xmath38 , the two algebras have the same dimension over @xmath2 and thus @xmath602 is an isomorphism . it follows that @xmath1065 is irreducible over @xmath2 . the second property of @xmath1065 also follows , since @xmath1097 is a generator of the multiplicative group . now let @xmath1067 be a divisor of @xmath38 . then @xmath1103 divides @xmath1104 and the group @xmath1105 is a subgroup of @xmath1093 . thus one has a field inclusion @xmath1106 , where the canonical generator @xmath1107 of @xmath1108 is sent to @xmath1109 , where @xmath1110 is the canonical generator @xmath1111 of @xmath1091 and @xmath1068 . one has @xmath1112 and hence @xmath1113 so that , using the above isomorphism @xmath602 , it follows that the polynomial @xmath1069 is a multiple of @xmath1065 . conversely , given a sequence @xmath1065 of polynomials fulfilling the conditions of the theorem , one constructs an algebraic closure @xmath4 and an isomorphism @xmath1114 as follows . one lets for each @xmath38 , @xmath1115/(p_n(t))$ ] and one gets an inductive system using for @xmath1067 the field homomorphism which sends the generator @xmath1116 of @xmath1117 to @xmath1118 , @xmath1068 . the inductive limit @xmath1119 is an algebraic closure @xmath4 of @xmath2 and the map @xmath1120 , @xmath1104 , defines an isomorphism @xmath545 of @xmath1121 with @xmath620 . note that this construction makes sense also for @xmath466 and that the first polynomial is of degree one and thus picks a specific generator of the multiplicative group of @xmath2 . one checks that the sequence of polynomials associated to the pair @xmath1122 is the sequence @xmath1065 . thus there is a complete equivalence between elements @xmath1123 and sequences of polynomials fulfilling the conway conditions of the theorem . to make the above map from @xmath64 to sequences of polynomials more explicit we introduce the trace invariant " of an element @xmath1124 . we continue to denote by @xmath1091 the field structure on the union @xmath1092 , where @xmath1093 is the group of roots of unity generated by @xmath1095 . in particular , @xmath1125 is a field uniquely isomorphic to @xmath2 . let @xmath1126 , then the orbit @xmath1127 of the map @xmath1128 is a finite set , let @xmath1129 be its cardinality . then the following sum @xmath1130 computed in any @xmath1091 , for @xmath1131 is the same and determines an element of @xmath1132 . [ tracemap ] let @xmath1133 be the space of orbits of the map @xmath1128 acting on @xmath620 . let @xmath1062 . we call the map @xmath1134 the trace invariant of @xmath637 . the trace invariant characterizes @xmath637 as shown by the next proposition . [ tracemapprop ] let @xmath1062 . then for each @xmath23 the polynomial @xmath1135 $ ] associated to @xmath637 by lemma [ hope1 ] is given by @xmath1136 for @xmath1137 where @xmath1138 is the set of fractions @xmath1139 where @xmath1140 and the digits of @xmath955 in base @xmath0 are all zeros except for @xmath112 of them which are equal to @xmath131 . in the field @xmath1091 the @xmath38 roots of the polynomial @xmath1065 are the elements @xmath1141 , for @xmath1142 . for each @xmath1143 , the set of products of @xmath112 distinct roots is the set of elements of the form @xmath1144 one thus gets that the @xmath112-th symmetric function @xmath1145 of the roots of @xmath1065 is given by the sum , over orbits @xmath1146 satisfying the prescribed condition @xmath1147 . we now give a third equivalent description of the space @xmath55 . we recall that the decomposition subfield @xmath1148 is independent of the choice of @xmath1063 and is equal to @xmath1149 . [ homfield ] the map @xmath1150 where the fields inclusion @xmath1151 derives from , determines a canonical and equivariant isomorphism of sets . notice that the inclusion @xmath1152 depends upon the choice of the valuation @xmath809 . one has @xmath1153 and the map @xmath1154 is equivariant for the action of @xmath1155 on @xmath55 and on the space @xmath67 by @xmath1156 since for both spaces the action of @xmath1155 is free and transitive , it follows that the map @xmath636 is bijective . we let @xmath191 $ ] be the group ring of @xmath57 and let @xmath1157)$ ] be the frobenius automorphism given by the natural linearization of the group automorphism @xmath192 , of multiplication by @xmath0 ( _ cf . _ corollary [ noncan ] ) . the natural ring homomorphism @xmath1158\to \qcp\ ] ] is equivariant for the action of @xmath502 , its image is the subring of integers of @xmath59 while the kernel is described by the intersection @xmath191\cap j$ ] , where @xmath975 is the ideal of definition [ acf ] . the @xmath2-algebra @xmath1159\otimes_\z\f_p=\f_p[\mup]\ ] ] is perfect since the group @xmath57 is uniquely @xmath0-divisible . by restriction to the fixed points of @xmath502 and composition with the residue map @xmath1160 , one obtains the map @xmath1161^\fr,\f_p),\quad \alpha\mapsto { \rm res}(\alpha)=\epsilon\circ\alpha\circ \delta.\ ] ] note that elements of @xmath1162^\fr,\f_p)$ ] are finitely supported maps from @xmath1133 to @xmath2 , thus they can be lifted to elements of @xmath191^\fr$ ] . one derives @xmath1163^\fr=\z[\mup]^\fr\otimes_\z\f_p.\ ] ] next , we show that the map res as in is injective . [ homfield1 ] let @xmath971 . we denote by @xmath1080 and @xmath1164 the corresponding elements as in lemma [ hope ] and proposition [ homfield ] . then the trace invariant map of @xmath1084 has the following description @xmath1165 the map @xmath1166 as in is injective . the additive structure @xmath1084 on @xmath1167 is the same as that of the residue field of the completion @xmath59 for the restriction of @xmath809 . it follows that on each orbit @xmath1146 of the action of @xmath502 on @xmath57 , the sum @xmath1168 coincides with the residue @xmath1169 since @xmath1170 where @xmath1171^\fr$ ] one gets . then , it follows from proposition [ tracemapprop ] that the map @xmath1166 is injective . we now briefly explain how one can reconstruct @xmath1172 from its residue @xmath1173 , using the witt functor @xmath42 . given @xmath1174^\fr,\f_p)$ ] , the witt functor @xmath42 yields a homomorphism @xmath1175^\fr),\z_p).\ ] ] if @xmath1176 , one can reconstruct @xmath286 directly using @xmath1177 . this gives a direct proof of the injectivity of the map @xmath1166 . indeed , for an orbit @xmath1146 of the action of @xmath502 on @xmath57 , the element ( @xmath1178 lift ) @xmath1179)\ ] ] is fixed by the frobenius , _ i.e. _ @xmath1180^\fr)$ ] . one then sees that @xmath1181 we end this section by giving the relation between @xmath1182 and the space @xmath53 of all injective group homomorphisms @xmath1183 ( _ cf . _ definition [ defnxp ] ) . we recall that the galois group @xmath1184 is the closure @xmath1185 of the group generated by the frobenius @xmath1044 . [ try ] let @xmath4 be a fixed algebraic closure of @xmath2 . then @xmath61 @xmath1005 acts freely and transitively on @xmath53 . @xmath63 the quotient of @xmath53 by @xmath1185 is isomorphic to @xmath1182 . let @xmath279 . the range of @xmath1 is the group @xmath620 of all roots of unity in @xmath52 of order prime to @xmath0 . thus for a pair @xmath1186 , @xmath1055 , one has @xmath1187 . this proves the first statement . for any isomorphism @xmath1188 of the multiplicative group of the algebraic closure @xmath186 with the group @xmath57 , the following defines an element @xmath1062 , @xmath1189 all elements of @xmath64 arise this way . two pairs @xmath1190 , @xmath1055 whose associated @xmath1191 are the same are easily seen to be related by an automorphism @xmath1192 _ i.e. _ @xmath1193 . the second statement thus follows . proposition [ homfield ] suggests a more appropriate equivalent description of @xmath53 using a chosen algebraic closure @xmath644 of the @xmath0-adic field and its completion @xmath8 . [ cpcp ] the map @xmath1194 where @xmath1195 is composed with the lift to determine a field homomorphism from @xmath59 to @xmath8 , is a bijection of sets . the canonical surjection @xmath1196 of proposition [ try ] @xmath63 is the restriction map @xmath1197 let @xmath279 , then @xmath1198 composed with the lift @xmath1199 extends to a unique homomorphism @xmath1200 . the map @xmath1201 is equivariant for the action of @xmath1005 on @xmath53 as in proposition [ try ] and on @xmath1202 by composition with elements of @xmath1081 . since both actions are free and transitive , @xmath1201 is bijective . for any @xmath1203 , the range of @xmath1204 is the subfield of the maximal unramified extension @xmath1205 generated over @xmath10 by roots of unity of order prime to @xmath0 . one has by construction @xmath1206 . thus the image @xmath1207 is contained in the fixed subfield @xmath9 for the action of @xmath1027 on @xmath1000 this shows that the restriction map is well defined . for @xmath1055 , let @xmath1208 , then @xmath1209 and this automorphism fixes @xmath1210 pointwise if and only the restrictions @xmath1211 are equal . since @xmath1212 is topologically generated by @xmath502 , this happens if and only if the @xmath1213 are the same in the quotient of @xmath53 by @xmath1185 . we implement the homomorphism @xmath1214\to \qcp$ ] of to associate to an element @xmath1215 its residue @xmath1216,\bar\f_p).\ ] ] the image of @xmath1217 is the ring of integers of @xmath59 , thus the image of @xmath1218 in @xmath8 is contained in @xmath501 and the composite @xmath1219 is well defined . moreover , since @xmath1220 $ ] , it follows that @xmath1221 contains the ideal @xmath73 reduction of @xmath1222 modulo @xmath0 . [ cpcp1 ] let @xmath1223 be the quotient algebra @xmath72/j_p$ ] . then @xmath61 the map @xmath1224 is a bijection of sets . @xmath63 the algebraic spectrum @xmath1225 is in canonical bijection with the set @xmath64 . @xmath66 the canonical surjection @xmath1196 of proposition [ try ] @xmath63 corresponds to the natural map @xmath1226 @xmath61 for any integer @xmath289 prime to @xmath0 , the ideal @xmath73 contains the projection ( _ cf . _ definition [ acf ] ) @xmath1227 . thus an element @xmath1228 is given by a group homomorphism @xmath1229 such that ( for @xmath835 prime to @xmath0 ) @xmath1230 . notice that this equality holds if and only if @xmath602 is injective and hence , by restriction to the finite level subgroups in the projective limit @xmath57 , if and only if it is bijective . thus @xmath61 follows from the first statement of corollary [ cpcp ] . @xmath63 consider the finite field @xmath1231 . two generators of the multiplicative group @xmath1232 have the same characteristic polynomial if and only if they are conjugate under the action of the galois group @xmath1233 . this shows that the cardinality of the set @xmath1234 of irreducible primitive polynomials of degree @xmath38 over @xmath2 is @xmath1235 , where @xmath632 is the euler totient function . each of these polynomials @xmath293 divides the reduction modulo @xmath0 of the cyclotomic polynomial @xmath1236 , thus one derives , modulo @xmath0 , the following equality @xmath1237 since the degrees of the polynomials are the same and the right hand side divides the left one . moreover one also has @xmath1238 this determines a canonical isomorphism @xmath1239/(j_p\cap \f_p[\emup(n)])\to\prod_{i_n}\f_{p^n}\ ] ] and thus a canonical bijection of sets @xmath1240 . since @xmath1223 is the inductive limit of the @xmath1241 , @xmath1225 is the projective limit of the @xmath1234 , _ i.e. _ the space of sequences of conway polynomials as in theorem [ conway0 ] . this space is in canonical bijection with @xmath64 . @xmath66 follows from the proof of @xmath63 . the restriction to the fixed points of the frobenius automorphism @xmath1242 does not change the algebraic spectrum as a set , thus we derive a canonical bijection of sets @xmath1243 finally , we characterize the image of the map @xmath1166 as in . [ imageres ] let @xmath1174^\fr,\f_p)$ ] . then @xmath1244 belongs to the image of the map @xmath1166 as in if and only if @xmath1245 contains @xmath72^\fr\cap j_p$ ] . by , and proposition [ cpcp1 ] , @xmath63 , one has natural bijections of sets @xmath1246 then , the statement follows by noticing that the elements of @xmath1247 are the elements of @xmath1162^\fr,\f_p)$ ] whose kernel contains @xmath72^\fr\cap j_p$ ] . in this section we compare the space @xmath55 of extensions of the @xmath0-adic valuation to @xmath52 ( studied at length in section [ sectval ] ) , with the fiber over a prime @xmath0 of a space @xmath1248 which represents , in this set - up , the analogue of the curve that , for function fields , plays a fundamental role in a. weil s proof of the riemann hypothesis . our results show that for each place @xmath1249 , there is a natural model @xmath1250 for the fiber over @xmath809 and an embedding of this model in a noncommutative space @xmath1251 which is a @xmath809-adic avatar of the adle class space @xmath1252 . we shall denote by @xmath74 a global field . to motivate our constructions we first recall a few relevant facts holding for function fields . in this subsection we assume that @xmath74 is a function field . we let @xmath1253 be the field of constants . let @xmath1254 be a fixed separable closure of @xmath74 and let @xmath1255 be the maximal abelian extension of @xmath74 . we denote by @xmath1256 the algebraic closure of the finite field @xmath75 inside @xmath76 . a main result holding for function fields is that for each finite field extension @xmath77 of @xmath1257 the space of ( discrete ) valuations @xmath79 inherits the structure of an algebraic , one - dimensional scheme @xmath1258 whose non - empty open sets are the complements of the finite subsets and whose structure sheaf is defined by considering the intersection of the valuation rings inside @xmath77 . more precisely , @xmath79 coincides with the set of ( closed ) points of the unique projective , nonsingular algebraic curve @xmath1258 with function field @xmath77 . we recall ( _ cf . _ @xcite corollary 6.12 ) that the category of nonsingular , projective algebraic curves and dominant morphisms is equivalent to the category of function fields of dimension one over @xmath1256 . thus , one associates ( uniquely ) to @xmath1259 the projective limit @xmath1260 which is the abelian cover @xmath1261 of the non singular projective curve @xmath88 over @xmath75 with function field @xmath74 . by restricting valuations , one also derives a natural projection map @xmath1262 onto the space @xmath1263 of valuations of @xmath74 . thus , in the function field case one derives a geometric interpretation for the natural fibration associated to the space of valuations of the field extension @xmath1264 . in @xcite we have given an adelic description of the loop groupoid @xmath87 of the abelian cover @xmath1261 . we recall that the class space @xmath1265 of any global field @xmath74 has a natural structure of hyperring @xmath1266 ( _ cf . _ @xcite ) and that the prime elements @xmath1267 of this hyperring determine a groupoid . the units of this groupoid form the set @xmath1263 of places of @xmath74 and the source and range maps coincide with the map @xmath1268 which associates to a prime element of @xmath1266 the principal prime ideal of @xmath1266 it generates ( and thus the associated place ) . when @xmath74 is a function field , the groupoid @xmath1267 is canonically isomorphic to the loop groupoid @xmath87 of the abelian cover @xmath1261 , and the isomorphism is equivariant for the respective actions of the abelianized weil group @xmath1269 ( _ i.e. _ the subgroup of elements of @xmath1270 whose restriction to @xmath1256 is an integral power of the frobenius ) , and of the idle class group @xmath1271 . it follows that , as a group action on a set , the action of @xmath1269 on @xmath82 is isomorphic to the action of the idle class group @xmath1272 on @xmath1267 . in other words , by choosing a set theoretic section @xmath1097 of the projection @xmath1273 one obtains an equivariant set theoretic bijection @xmath1274 which depends though , in a crucial manner , on the choice of the base point @xmath1275 , for each place @xmath1276 . this dependence prevents one from transporting the algebraic geometric structure of @xmath1277 onto @xmath1267 , and it also shows that the adelic space @xmath1267 carries only the information on the curve @xmath1277 given in terms of a set with a group action . now , we turn to the global field @xmath84 . a natural starting point for the construction of a replacement of the covering @xmath1277 in this number field case is to consider the maximal abelian extension of @xmath10 , _ i.e. _ the cyclotomic field @xmath52 as analogue of @xmath76 . then , the space @xmath55 of extensions of the @xmath0-adic valuation to @xmath52 appears as the first candidate for the analogue of the fiber , over a finite place , of the abelian cover @xmath1261 . thus , the first step is evidently that to compare @xmath55 with the fiber @xmath1278 of the fibration @xmath1279 over a rational prime @xmath1280 . at the level of sets with group actions , this process shows that @xmath55 is not yet the correct fiber . the following discussion indicates that one should consider instead the total space of a principal bundle , with base @xmath55 and structure group a connected compact solenoid @xmath93 whose definition is given in proposition [ solenoid ] . then , a natural construction of the fiber is provided by the mapping torus @xmath94 of the action of the frobenius on the space @xmath53 of definition [ defnxp ] . [ compare ] let @xmath1278 be the fiber of the groupoid @xmath1281 over a non - archimedean , rational prime @xmath1282 . then , the following results hold . @xmath61 the idle class group @xmath1283 acts transitively on @xmath1278 . the isotropy group of any element of @xmath1278 is the cocompact subgroup @xmath1284 of classes of idles @xmath1285 such that @xmath1286 , @xmath1287 . @xmath63 under the class field theory isomorphism @xmath1288 @xmath1289 acts transitively on @xmath55 and the isotropy group of any element of @xmath55 is @xmath1290 @xmath1291 is the closed subgroup @xmath1292 generated by @xmath0 in @xmath863 . @xmath61 follows from theorem 7.10 of @xcite . @xmath63 follows from lemma [ ggalois ] . notice that if @xmath74 is a function field and @xmath809 is a valuation of @xmath76 extending the valuation @xmath1293 of @xmath74 , any @xmath1294 such that @xmath1295 , belongs to the local weil group @xmath1296 . this is due to the fact that the restriction of @xmath1297 to an automorphism of @xmath1298 is an integral power of the frobenius . when @xmath84 , the isotropy group of the valuation @xmath809 is instead larger than the local weil group @xmath1299 . the difference is determined by the presence of the quotient @xmath1300 of the isotropy group @xmath1301 by the local weil group @xmath1302 . here , @xmath1303 is represented by the idle all of whose components are @xmath131 except at the place @xmath0 where it is equal to @xmath1304 . by multiplying with the principal idle @xmath0 , one gets the same class as the element of @xmath1305 which is equal to @xmath0 everywhere except at the place @xmath0 where it is equal to @xmath131 . thus , its image in @xmath863 is @xmath0 . the quotient group @xmath1306 is a compact connected solenoid which is described in the following proposition [ solenoid ] . the presence of the connected piece @xmath93 is due to the fact that the connected component of the identity in the idle class group acts trivially , at the galois level , on @xmath52 . [ solenoid ] the group @xmath93 is compact and connected and is canonically isomorphic to the projective limit of the compact groups @xmath1307 , under divisibility of the labels @xmath38 . we consider first the factor @xmath1308 of the projective limit @xmath93 , where @xmath416 acts diagonally , _ i.e. _ by the element @xmath1309 , on @xmath1310 . one has a natural map @xmath1311 given by @xmath1312 where one views @xmath1313 as a subgroup of @xmath1307 . the map @xmath1314 is an isomorphism of groups . when @xmath38 divides @xmath289 , the subgroup @xmath1315 is contained in @xmath1316 and this inclusion corresponds to the projection @xmath1317 . under the isomorphisms @xmath1314 , this corresponds to the projection @xmath1318 . thus the projective system defining @xmath93 is isomorphic to the projective system of the projections @xmath1318 and the projective limits are isomorphic . next , we describe a general construction of mapping torus which yields , when applied to the groups @xmath1319 the fiber @xmath1278 of the groupoid @xmath1281 over a finite , rational prime @xmath1320 . [ fibration ] let @xmath863 be the group of automorphisms of the multiplicative group @xmath620 of roots of unity in @xmath52 of order prime to @xmath0 and let @xmath1007 be the element @xmath1321 . let @xmath1005 act freely and transitively on a compact space @xmath88 . let @xmath1248 be the quotient space @xmath1322 where @xmath1323 acts on the product @xmath1324 as follows @xmath1325 then , the following results hold . @xmath61 the space @xmath1248 is compact and is an @xmath93-principal bundle over the quotient @xmath1326 of @xmath88 by @xmath1013 , where @xmath93 is the solenoid group of proposition [ solenoid ] . @xmath63 let @xmath88 and @xmath1326 be as in , then @xmath1248 is canonically isomorphic to the fiber @xmath1278 . @xmath61 we first look at the action of @xmath416 on the open interval @xmath1327 given by @xmath1328 . we consider the map @xmath1329 given by @xmath1330 one has @xmath1331 which shows that the action of @xmath416 on @xmath1327 given by @xmath1328 is isomorphic to the action of @xmath416 on @xmath1332 given by translation by @xmath1333 . by construction @xmath863 is a compact , totally disconnected group next , we show that the map which associates to @xmath1334 the element @xmath1335 extends to a bijection of @xmath11 with the closed subgroup of @xmath1005 generated by @xmath1044 . in fact , the isomorphism follows from the isomorphism between @xmath1005 and @xmath1336 , with @xmath1044 being the frobenius . the result follows by applying _ e.g. _ @xcite ( chapitre v , appendice ii , exercice 5 ) . this gives a natural inclusion @xmath1337 , @xmath1338 , as a closed subgroup . we now consider the action of the product group @xmath1339 on @xmath1324 given by @xmath1340 by construction the element @xmath1341 acts as @xmath1 ( _ cf . _ ) . the quotient group @xmath1342 is isomorphic to the solenoid @xmath93 , by using the isomorphism of the group @xmath1343 with @xmath1332 given by the logarithm in base @xmath0 . to see that @xmath1248 is a principal bundle over @xmath93 one uses the map @xmath1344 of to check that @xmath93 acts freely on @xmath1248 . the quotient of @xmath1248 by the action of @xmath93 is the quotient of @xmath88 by the action of @xmath11 . @xmath63 the fiber @xmath1278 has a canonical base point given by the idempotent @xmath1345 . hence by applying proposition [ compare ] , this fiber is canonically isomorphic to the quotient @xmath1346 . by identifying @xmath1289 with @xmath1347 , this quotient coincides with the quotient of @xmath1348 by the powers of the element @xmath1349 . under the bijection @xmath1350 from @xmath1327 to @xmath1343 , one obtains the same action as in and hence the desired isomorphism . in order to obtain the analogue , for the global field @xmath84 , of the fiber of the algebraic curve @xmath1277 , we should apply the construction of proposition [ fibration ] to a compact space @xmath53 so that the following requirements are satisfied @xmath61 @xmath1005 acts freely and transitively on @xmath53 @xmath63 the quotient of @xmath53 by @xmath1185 is canonically isomorphic to @xmath55 . proposition [ try ] provides a natural candidate for @xmath53 . moreover , equation shows that one can equivalently describe @xmath53 as the space @xmath1202 and that the canonical identification of @xmath1351 with @xmath55 is given by the restriction map to the fixed points of @xmath502 as in . we derive the definition of the following model for the fiber @xmath94 over a finite prime @xmath0 @xmath1352 we move now to the discussion of the analogues of the spaces @xmath55 , @xmath53 and @xmath94 , when @xmath0 is the archimedean prime @xmath99 ( _ i.e. _ the archimedean valuation ) . the space @xmath1353 is the space of multiplicative norms on @xmath52 whose restriction to @xmath10 is the usual absolute value . for @xmath1354 , the field completion @xmath996 is isomorphic to @xmath6 , thus one derives @xmath1355 where @xmath1356 corresponds to complex conjugation . it follows that for @xmath99 the space @xmath53 is simply @xmath1357 on the other hand , the fiber @xmath1358 is the quotient @xmath1359 , where @xmath1360 is the cocompact subgroup of @xmath1289 given by classes of idles , whose components are all @xmath131 except at the archimedean place . then , we derive that @xmath1361 this discussion shows that at @xmath99 there is no need for a mapping torus , and that the expected fiber is simply given by @xmath1362 the model for the fiber over a rational prime @xmath0 is only a preliminary step toward the global construction of the `` curve '' which we expect to replace , when @xmath84 , the geometric cover @xmath1277 . in fact , one still needs to suitably combine these models into a noncommutative space to account for the presence of transversality factors in the explicit formulas . we explain why in some details below . in @xcite we showed how to determine the counting function @xmath90 ( a distribution on @xmath91 ) which replaces , for @xmath84 , the classical weil counting function for a field @xmath74 of functions of an algebraic curve @xmath1248 over @xmath2 ( _ cf . _ @xcite ) . the weil counting function determines the number of rational points on the curve @xmath1248 defined over field extensions @xmath75 of @xmath2 @xmath1363 the numbers @xmath286 s are the complex roots of the characteristic polynomial of the frobenius endomorphism acting on the tale cohomology @xmath1364 , for @xmath867 . in @xcite we have shown that the distribution @xmath90 associated to the ( complete ) riemann zeta function is described by the similar formula @xmath1365 where @xmath1326 is the set of non trivial zeros of the riemann zeta function . this distribution is positive on @xmath1366 and fulfills all the expected properties of a counting function . in particular , it takes the correct value @xmath1367 in agreement with the ( expected ) value of the euler characteristic . in @xcite we pushed these ideas further and we explained how to implement the trace formula understanding of the explicit formulas in number - theory , to express the distribution @xmath90 as an _ intersection number _ involving the scaling action of the idle class group on the class space . this development involves a lefschetz formula whose geometric side corresponds to the following expression of the counting distribution @xmath1368 @xmath1369 here , @xmath1370 is the von - mangoldt function taking the value @xmath1371 at prime powers @xmath1372 and zero otherwise and @xmath1373 is the distribution defined , for any test function @xmath117 , as @xmath1374 where @xmath1375 is the euler constant . the distribution @xmath1373 is positive on @xmath1366 and in this domain it is equal to the function @xmath1376 . the contribution in the counting distribution @xmath1368 coming from the term @xmath1377 in can be understood geometrically as arising from a counting process performed on the fibers @xmath94 ( each of them accounting for the delta functions located on the powers of @xmath0 ) . the value @xmath1333 coming from the von - mangoldt function @xmath1370 corresponds to the length of the orbit in the mapping torus ( _ cf . _ @xcite , 2.2 ) . on the other hand , as explained in @xcite , the contribution of the archimedean place can not be understood in a naive manner as a simple counting process of points and its expression involves a transversality factor measuring the transversality of the action of the idle class group with respect to periodic orbits . this shows that the periodic orbits can not be considered in isolation and must be thought as ( suitably ) embedded in the ambient class space . this development supplies a precious hint toward the final construction of the `` curve '' and shows that the role of ergodic theory and noncommutative geometry is indispensable . next , we shall explain how the bc - system over @xmath416 gives , for each @xmath0 , a natural embedding of the fiber @xmath94 ( _ cf . _ ) into a noncommutative space constructed using the set @xmath96 of the @xmath8-rational points of the affine group scheme @xmath97 which describes the abelian part of the system ( _ cf . _ @xcite ) . since the fields @xmath8 are abstractly pairwise isomorphic the obtained spaces are also abstractly isomorphic , but in a non canonical manner . in @xcite , following a proposal of c. soul for the meaning of the ring @xmath1379 , we noted that the inductive limit @xmath1380\ ] ] coincides with the abelian part of the algebra defining the integral bc - system . the description given in that paper of the bc - system as an affine pro - group scheme @xmath97 over @xmath416 together with the dynamic of the action of a semigroup of endomorphisms , allows one to consider its rational points over any ring @xmath33 @xmath1381,a)\,.\ ] ] then , one can implement , for each rational prime @xmath0 , the canonical inclusion @xmath1382,\c_p)=\ce(\c_p)\,.\ ] ] the next result shows that the space @xmath1383 matches , for any @xmath0 including @xmath99 , the definition of the class space @xmath101 . the action of @xmath1384 ( in the semigroup @xmath1385 ) is the product of the linearization of the action @xmath1386 on the ( @xmath8-rational points of the ) scheme @xmath97 , with the action on @xmath102 given by the map @xmath1387 . [ casec ] @xmath61 the space @xmath1388 is canonically isomorphic to the class space @xmath101 . @xmath63 the subspace of the class space made by classes whose archimedean component vanishes corresponds to the quotient @xmath1389 @xmath61 the space @xmath1390 is the space of complex characters of the abelian group @xmath58 and is canonically isomorphic to @xmath11 . we use the map @xmath1350 to map the interval @xmath102 to @xmath1332 . under this map the transformation @xmath1387 becomes the multiplication by @xmath289 . the action @xmath1386 on the scheme @xmath97 corresponds to the multiplication by @xmath289 in @xmath11 . since any class is equivalent to an element of @xmath1391 , gives , for @xmath99 @xmath1392 @xmath63 follows from the identification . note that by using the inclusion @xmath1393 , one derives a natural inclusion @xmath1394 for @xmath99 one has the natural inclusion @xmath1395 which is obtained by using the canonical inclusion for @xmath99 and the fixed point @xmath1396 . the group ring @xmath209 $ ] is a hopf algebra for the coproduct @xmath1397 and the antipode @xmath643 , thus @xmath97 is a group scheme . [ groupscheme ] let @xmath33 be a commutative ring . @xmath61 the abelian group @xmath1398 is torsion free . @xmath63 the space @xmath1399 is a module over the hyperring @xmath101 . @xmath66 for any rational prime @xmath0 , @xmath100 is a free module of rank one over @xmath101 . @xmath61 one has @xmath1400,a)=\hom(\q/\z , a^\times)\ ] ] where the second @xmath1401 is taken in the category of abelian groups . since the group @xmath58 is divisible the group @xmath1402 has no torsion , for any abelian group @xmath623 . @xmath63 we first show that @xmath103 is a hypergroup and in fact a vector space over the krasner hyperfield @xmath1403 ( _ cf . _ @xcite ) . the two abelian groups @xmath1398 and @xmath102 are both torsion free , thus one gets @xmath1404 which is a projective space , hence a vector space over @xmath1405 ( _ cf . _ @xcite ) . next we show that @xmath103 is a module over @xmath101 . we use the canonical ring isomorphism @xmath1406 to define the following ring homomorphism from @xmath11 to the ring @xmath1407 @xmath1408 the map @xmath1409 is a ring homomorphism , thus @xmath1410 defines a ring homomorphism from @xmath1391 to the endomorphisms of the abelian group @xmath1411 . for any @xmath1412 , one has @xmath1413 thus the restriction of @xmath1410 to the monoid of non - zero elements of @xmath416 gives the equivalence relation which defines @xmath103 as in . it follows an action of the hyperring @xmath1414 on the hypergroup . @xmath66 it is easy to see that , once one fixes an embedding @xmath1415 and an @xmath1416 and a real number @xmath1417 , the element @xmath1418 is a generator of @xmath100 as a free module over @xmath101 . the next result displays some interesting arithmetic - geometric properties of the scheme @xmath97 . [ ramified ] @xmath61 let @xmath1419 be the maximal abelian extension of @xmath9 . then the natural map @xmath1420 is a bijection of sets . @xmath63 let @xmath1421 be the maximal unramified extension of @xmath9 and @xmath1422 the valuation ring of the @xmath0-adic valuation . then the natural map @xmath1423 is a bijection of sets . @xmath66 let @xmath1424 be the residue homomorphism . then the associated map @xmath1425 is a bijection . @xmath69 the @xmath101-module @xmath1426 is described as @xmath1427 where @xmath1428 is the prime ideal of adle classes whose @xmath0-component vanishes . @xmath61 let @xmath1429,\c_p)$ ] . then the image of @xmath602 is contained in the subfield of @xmath8 generated over @xmath10 by roots of unity , which is contained in @xmath998 . @xmath63 let @xmath1430,\q_p^{\rm ur})$ ] . then the image of @xmath602 is contained in the subring of @xmath1000 generated over @xmath416 by roots of unity ( of order prime to @xmath0 ) which is contained in @xmath1431 . @xmath66 let @xmath1432,\z_p^{\rm ur})$ ] . then @xmath602 is entirely characterized by the group homomorphism @xmath1433 where @xmath852 is the group of roots of unity in @xmath1431 , which is non canonically isomorphic to the group @xmath620 of abstract roots of unity of order prime to @xmath0 . similarly an element of @xmath1434,\bar\f_p)$ ] is entirely characterized by the associated group homomorphism from @xmath58 to @xmath1435 . since the residue morphism @xmath246 gives an isomorphism @xmath1436 one obtains the conclusion . @xmath69 one has @xmath1437 . let , as above , @xmath1438 be the subgroup of elements of denominator prime to @xmath0 . then the subset @xmath1439 is given by @xmath1440 which corresponds to the prime , principal ideal @xmath1441 of the hyperring structure @xmath101 inherent to the class space ( _ cf . _ @xcite ) . as shown in section [ sectval ] , the space @xmath55 is intimately related to the space of sequences of irreducible polynomials @xmath62 $ ] , @xmath36 , fulfilling the basic conditions of the conway polynomials ( _ cf . _ theorem [ conway0 ] ) and hence to the explicit construction of an algebraic closure of @xmath2 . the normalization condition using the lexicographic ordering just specifies a particular element @xmath1442 of @xmath55 . since the explicit computation of the sequence @xmath62 $ ] , @xmath36 , associated to @xmath1442 has been proven to be completely untractable , b. de smit and h. lenstra have recently devised a more efficient construction of @xmath4 ( _ cf . _ @xcite ) . our goal in this section is to make explicit the relation between their construction , the bc - system and the sought for curve " . when @xmath74 is a global field of positive characteristic _ i.e. _ the function field of an algebraic curve over a finite field @xmath75 , the intermediate extension @xmath1443 plays an important geometric role since it corresponds to working over an algebraically closed field . for @xmath84 , it is therefore natural to ask for an intermediate extension @xmath107 playing a similar role . one feature of the former extension is that the residue fields are algebraically closed . in their construction , de smit and lenstra use the intermediate extension @xmath106 which comes very close to fulfill the expected properties . for each prime @xmath941 , let us denote by @xmath1444 the torsion subgroup . for @xmath1445 one has @xmath1446 , while for @xmath1447 one gets @xmath1448 , where @xmath1449 is the lift . the product @xmath1450 is a compact group , and a subgroup of the galois group @xmath1451 . by galois theory , one can thus associate to @xmath1452 a ( fixed ) field extension @xmath1453 notice that one derives a subsystem of the bc - system given by the fixed points of the action of @xmath1452 . at the rational level and by implementing the cyclotomic ideal @xmath975 of definition [ acf ] , one obtains the exact sequence of algebras @xmath1454 the image of the restriction to @xmath1455 of the homomorphism @xmath676 is contained in the integers of @xmath1456 and one has @xmath1457 the space @xmath55 is the total space of a principal bundle whose base is the space @xmath1458 of valuations on @xmath1456 extending the @xmath0-adic valuation . the group of the principal bundle is the quotient of @xmath1452 by its intersection @xmath1459 with the isotropy group of elements of @xmath55 . the projection @xmath1460 is given by restriction of valuations from @xmath52 to @xmath1456 . for @xmath1461 , the isotropy group @xmath1462 of @xmath1293 for the action of @xmath1463 is the image of the isotropy group of @xmath809 in @xmath1464 for any extension @xmath809 of @xmath1293 to @xmath52 . it follows from lemma [ ggalois ] that the isotropy subgroup of @xmath809 is @xmath1465 , thus one gets @xmath1466 [ subopen ] for each prime @xmath941 the group @xmath1467 is canonically isomorphic to the additive group @xmath1468 . moreover , for each prime @xmath1469 the closed subgroup of @xmath1467 generated by @xmath0 is open and of finite index @xmath1470 where @xmath1471 for each prime @xmath941 there is a canonical isomorphism of groups @xmath1472 where the group @xmath1468 is viewed as an additive group . for @xmath941 odd , @xmath1473 is the unique @xmath1045 root of unity which is congruent to @xmath312 modulo @xmath941 and @xmath1474 , as in , is the ratio @xmath1475 . for @xmath1445 , @xmath1476 is congruent to @xmath312 modulo @xmath1477 and @xmath1478 . the first statement thus follows . the second statement follows since one has @xmath1479 and the closed subgroup of @xmath1468 generated by @xmath1480 is @xmath1481 . under the isomorphisms @xmath1482 one gets , by the chinese remainder theorem , that @xmath1483 notice the independence of the places @xmath941 in the above formula which makes the group @xmath1462 a cartesian product and allows one to express @xmath1458 as an infinite product of finite sets . to label concretely these finite sets consider , for each prime @xmath941 the @xmath1468-extension @xmath1484 of @xmath10 . one has @xmath1485 where , for @xmath1486 , the finite extension @xmath1487 of @xmath10 is associated to @xmath1488 viewed as a character of @xmath1489 . for @xmath941 odd , @xmath1487 is the fixed subfield for the action of @xmath1490 on the extension of @xmath10 generated by a primitive root of unity of order @xmath1491 . for @xmath1445 one uses a primitive root of unity of order @xmath1492 . we denote by @xmath1493 : this is a cyclic extension of @xmath10 of degree @xmath1470 . the artin reciprocity law shows that , for @xmath0 a prime @xmath1469 , the reduction modulo @xmath0 of the integers of @xmath1494 , decomposes as a product of @xmath1470 copies of @xmath2 , parameterized by the set @xmath1495 of extensions of the @xmath0-adic valuation to @xmath1494 , which is a finite set of cardinality @xmath1470 . @xmath66 by extending @xmath809 to an element of @xmath55 one gets that the restriction to @xmath59 and hence to @xmath1499 is unramified . moreover the residue field is determined by the topology on the closure set of the action of the frobenius , _ i.e. _ on @xmath1020 . the result follows . next , we shall explain the link with the notations used by de smit and lenstra and their construction . first , we recall that the additive group @xmath58 is the direct sum of its @xmath941-torsion components @xmath1505 thus the group ring @xmath209 $ ] can be written as a tensor product @xmath1506=\bigotimes_{\ell\,\rm prime } \z[h_\ell].\ ] ] the natural action of @xmath929 on @xmath209 $ ] by automorphisms of the group @xmath58 factorizes in the individual actions of @xmath1507 . one lets @xmath1508 be the ring @xmath1509 $ ] modulo the ideal generated by @xmath1510 thus one has @xmath1511 in @xmath1508 and @xmath1512 for all @xmath442 . the algebra @xmath1513 of de smit and lenstra is defined as @xmath1514 . the next lemma shows that the algebra @xmath1513 is intimately related to the fixed point algebra @xmath1515^{\delta_\ell}$ ] . one has the trace map @xmath1520 and natural ring homomorphisms @xmath1521 . de smit and lenstra ( _ cf . _ @xcite ) lift the natural generator of @xmath1522 as an extension of @xmath1523 , and the galois conjugates under @xmath1524 as the following elements of @xmath1513 @xmath1525 when @xmath1445 , one has simply @xmath1526 and in this case the above list of elements reduces to @xmath1527 the two authors show that the prime ideals @xmath1528 of @xmath1513 which contain @xmath0 , are uniquely specified by a finite system of elements @xmath1529 , @xmath1530 . more precisely , @xmath1528 is generated by @xmath0 and by the @xmath1531 for @xmath1532 and @xmath1533 . theorem [ bcsl ] does not yield the full algebraic closure of @xmath2 but only the subfield @xmath1537 thus it remains to understand how to produce naturally the missing part @xmath1538 in such a way that the tensor product over @xmath2 yields @xmath4 . de smit and lenstra construction of @xmath1539 is performed using the following artin schreier equations @xmath1540 which have the advantage of simplicity . e. witt gave in @xcite a conceptual construction of @xmath1541 based on the witt functor @xmath42 and its finite truncations @xmath1542 . the addition of two witt vectors @xmath1543 and @xmath1544 is a vector whose components @xmath1545 were proven by witt to be polynomials with integer coefficients . note also that for @xmath1546 the witt components of @xmath1547 ( the additive inverse of @xmath312 ) are simply @xmath1548 , but this result does not hold for @xmath1549 . recall also that in terms of the witt vectors , the frobenius @xmath1550 is given in characteristic @xmath0 , by @xmath1551 . [ wittextension ] let @xmath36 . let @xmath1552 $ ] be the ring of polynomials in @xmath38 variables and @xmath1553 the ideal generated by the components of the witt vector @xmath1554 , where @xmath1555 is the witt vector with components @xmath392 . then @xmath1556 is a prime ideal and the quotient field of the integral ring @xmath1557 defines the field extension @xmath1558 . one derives , for instance , that the first extensions for @xmath1549 are given by the equations with coefficients in @xmath1562 @xmath1563 for @xmath1564 one gets the following equations with coefficients in @xmath1565 @xmath1566 n. bourbaki _ algebra ii . chapters 47_. translated from the 1981 french edition by p. m. cohn and j. howie . reprint of the 1990 english edition . elements of mathematics ( berlin ) . springer - verlag , berlin , 2003 . a. connes , c. consani _ characteristic @xmath131 , entropy and the absolute point _ , in proceedings of the 21st jami conference `` noncommutative geometry , arithmetic and related topics '' , baltimore 2009 , jhup ( 2011 ) ; ( in press ) . arxiv:0911.3537v1 [ mathag ] . e. witt , _ zyklische krper und algebren der charakteristik @xmath0 vom grad @xmath1096 . struktur diskret bewerteter perfekter krper mit vollkommenem restklassen - krper der charakteristik @xmath0_. j. reine angew . , 176 ( 1937 ) , 126140 .

for each prime @xmath0 and each embedding @xmath1 of the multiplicative group of an algebraic closure of @xmath2 as complex roots of unity , we construct a @xmath0-adic indecomposable representation @xmath3 of the integral bc - system as additive endomorphisms of the big witt ring of @xmath4 . the obtained representations are the @xmath0-adic analogues of the complex , extremal kms@xmath5 states of the bc - system . the role of the riemann zeta function , as partition function of the bc - system over @xmath6 is replaced , in the @xmath0-adic case , by the @xmath0-adic @xmath7-functions and the polylogarithms whose values at roots of unity encode the kms states . we use iwasawa theory to extend the kms theory to a covering of the completion @xmath8 of an algebraic closure of @xmath9 . we show that our previous work on the hyperring structure of the class space , combines with @xmath0-adic analysis to refine the space of valuations on the cyclotomic extension of @xmath10 as a noncommutative space intimately related to the integral bc - system and whose arithmetic geometry comes close to fulfill the expectations of the arithmetic site " . finally , we explain how the integral bc - system appears naturally also in de smit and lenstra construction of the standard model of @xmath4 which singles out the subsystem associated to the @xmath11-extension of @xmath10 .

various cosmological observations , including the type ia supernova @xcite , the cosmic microwave background radiation @xcite and the large scale structure @xcite , et al . , have revealed that the universe is undergoing an accelerating expansion and it entered this accelerating phase only in the near past . this unexpected observed phenomenon poses one of the most puzzling problems in cosmology today . usually , it is assumed that there exists , in our universe , an exotic energy component with negative pressure , named dark energy , which dominates the universe and drives it to an accelerating expansion at recent times . many candidates of dark energy have been proposed , such as the cosmological constant , quintessence , phantom , quintom as well as the ( generalized ) chaplygin gas , and so on . however , alternatively , one can take this observed accelerating expansion as a signal of the breakdown of our understanding to the laws of gravitation and , thus , a modification of the gravity theory is needed . one of the most popular modified gravity models is obtained by generalizing the spacetime curvature scalar @xmath12 in the einstein - hilbert action in general relativity to a general function of @xmath12 . the theory so obtained is called as the @xmath13 theory ( see @xcite for recent review ) . recently , a new modified gravity by extending the teleparallel gravity @xcite is proposed to account for the present accelerating expansion @xcite . differing from general relativity using the levi - civita connection , in teleparallel gravity , the weitzenbck connection is used . as a result , the spacetime has only torsion and thus is curvature - free . similar to general relativity where the action is a curvature scalar , the action of teleparallel gravity is a torsion scalar @xmath1 . in analogy to the @xmath13 theory , bengochea and ferraro suggested , in ref . @xcite , a new model , named @xmath0 theory , by generalizing the action of teleparallel gravity , and found that it can explain the observed acceleration of the universe . let us also note here that models based on modified teleparallel gravity may also provide an alternative to inflation @xcite . another advantage the generalized @xmath0 torsion theory has is that its field equations are second order as opposed to the fourth order equations of f(r ) theory . more recently , linder proposed two new @xmath0 models to explain the present cosmic accelerating expansion @xcite and found that the @xmath0 theory can unify a number of interesting extensions of gravity beyond general relativity . in this letter , we plan to first perform a statefinder analysis and an @xmath14 diagnostic to these models and then discuss the constraints on them from the latest observational data , including the type ia supernovae released by the supernova cosmology project collaboration , the baryonic acoustic oscillation from the spectroscopic sloan digital sky survey , and the cosmic microwave background radiation from wilkinson microwave anisotropy probe seven year observation . we find that for both models the crossing of the @xmath15 line is impossible . this is consistent with what obtained in ref . @xcite , but in conflict with the result obtained in ref . @xcite where a crossing is found for the exponential model . in this section , following refs . @xcite , we briefly review the @xmath0 theory . we start with teleparallel gravity where the action is the torsion scalar @xmath1 defined as @xmath16 where @xmath17 is the torsion tensor @xmath18 and @xmath19 here @xmath20 is the orthonormal tetrad component , where @xmath21 is an index running over @xmath22 for the tangent space of the manifold , while @xmath23 , also running over @xmath22 , is the coordinate index on the manifold . the spacetime metric is related to @xmath20 through @xmath24 and @xmath25 is the contorsion tensor given by @xmath26 by assuming a flat homogeneous and isotropic friedmann - robertson - walker universe which is described by the metric @xmath27 where @xmath28 is the scale factor , one has , from eq . ( [ st ] ) , @xmath29 with @xmath30 being the hubble parameter . in order to explain the late time cosmic accelerating expansion without the need of dark energy , linder , following ref . @xcite , generalized the lagrangian density in teleparallel gravity by promoting @xmath1 to be @xmath31 . the modified friedmann equation then becomes @xmath32 @xmath33 where a prime denotes a derivative with respect to @xmath34 , @xmath35 is energy density and @xmath36 is the pressure . here we assume that the energy component in the universe is only matter with radiation neglected , thus @xmath37 . from eqs . ( [ mf ] , [ mf2 ] ) , we can define an effective dark energy , whose energy density and the equation of state can be expressed , respectively , as @xmath38 @xmath39 some models are proposed in refs . @xcite to explain the present cosmic accelerating expansion , which satisfy the usual condition @xmath40 at the high redshift in order to be consistent with the primordial nucleosynthesis and cosmic microwave background constraints . here we consider two models proposed by linder @xcite : @xmath41 model 1 @xmath42 here @xmath43 and @xmath44 are two model parameters . using the modified friedmann equation , one can obtain @xmath45 where @xmath46 is the dimensionless matter density today . substituting above expression into the modified friedmann equation and defining @xmath47 , one has @xmath48 let us note that this model has the same background evolution equation as some phenomenological models @xcite and it reduces to the @xmath10cdm model when @xmath49 , and to the dgp model @xcite when @xmath50 . when @xmath51 , the friedmann equation ( eq . ( [ mf ] ) ) can be rewritten as @xmath52 , which is the same as that of a standard cold dark matter ( scdm ) model if we rescale the newton s constant as @xmath53 . therefore , in order to obtain an accelerating expansion , it is required that @xmath54 . @xmath41 model 2 @xmath55 which is similar to a @xmath13 model where an exponential dependence on the curvature scalar is proposed @xcite . using the modified friedmann equation again , we have @xmath56 and @xmath57 it is easy to see that @xmath58 corresponds to the case of @xmath10cdm . in order to discriminate different dark energy models from each other , sanhi et al . proposed a geometrical diagnostic method by adding higher derivatives of the scale factor @xcite . in this method , two parameters @xmath59 , named statefinder parameters , are used , which are defined , respectively , as @xmath60 @xmath61 where @xmath62 is the decelerating parameter . apparently , @xmath10cdm model corresponds to a point @xmath63 in @xmath59 phase space . the statefinder diagnostic can discriminate different models . for example , it can distinguish quintom from other dark energy models @xcite . the @xmath2 is a new diagnostic of dark energy proposed by sahni et al . it is defined as @xmath64 apparently , this diagnostic only depends on the first derivative of the luminosity @xmath65 ( see eq . ( [ dl ] ) ) . thus , its advantage , as opposed to the equation of state of dark energy , is that it is less sensitive to the observational errors and the present matter energy density @xmath66 . one can use this diagnostic to discriminate different dark energy models by examining the slope of @xmath2 even if the value of @xmath66 is not exactly known , since the positive , null , or negative slopes correspond to @xmath67 or @xmath68 , respectively . the evolutionary curves of statefinder pair @xmath59 ( left ) , pair @xmath69 ( middle ) and @xmath2 ( right ) for model 1 with @xmath70 . , title="fig:",width=188 ] the evolutionary curves of statefinder pair @xmath59 ( left ) , pair @xmath69 ( middle ) and @xmath2 ( right ) for model 1 with @xmath70 . , title="fig:",width=188 ] the evolutionary curves of statefinder pair @xmath59 ( left ) , pair @xmath69 ( middle ) and @xmath2 ( right ) for model 1 with @xmath70 . , title="fig:",width=188 ] the evolutionary curves of statefinder pair @xmath59 ( left ) , pair @xmath69 ( middle ) and @xmath2 ( right ) for model 2 with @xmath70 . , title="fig:",width=188 ] the evolutionary curves of statefinder pair @xmath59 ( left ) , pair @xmath69 ( middle ) and @xmath2 ( right ) for model 2 with @xmath70 . , title="fig:",width=188 ] the evolutionary curves of statefinder pair @xmath59 ( left ) , pair @xmath69 ( middle ) and @xmath2 ( right ) for model 2 with @xmath70 . , title="fig:",width=188 ] here , we perform the statefinder and @xmath14 diagnostics to two @xmath0 models , i.e. , model 1 and model 2 given in the previous section . in figs . ( [ figs1 ] ) and ( [ figs2 ] ) , we show the diagnostic results with @xmath70 , which is the best fit value obtained from sne ia and bao with a model independent method @xcite and is also consistent with the result in the next section of the present letter . the left panels show the evolutionary curves of statefinder pair @xmath71 , the middle panels are the evolutionary curves of pair @xmath69 , and the right panels are the @xmath2 diagnostic . although , both model 1 and model 2 evolve from the scdm to a de sitter ( ds ) phase as one can see from the middle panels of these figures , the effective dark energy for model 2 with @xmath72 is similar to a cosmological constant both in the high redshift regimes and in the future , while for model 1 with @xmath73 this similarity occurs only in the future . as demonstrated in ref . @xcite , for a simple power law evolution of the scale factor @xmath74 , one has @xmath75 and @xmath76 . thus , a phantom - like dark energy corresponds to @xmath77 , a quintessence - like dark energy to @xmath78 , and an evolution from phantom to quintessence or inverse is given by a crossing of the point @xmath79 in @xmath59 phase plane . a crossing of phantom divide line is also represented by a crossing of the red solid line ( @xmath10cdm ) in middle panels ( @xmath69 plane ) of figs . ( [ figs1 ] , [ figs2 ] ) . therefore , we find , from the left and middle panels of figs . ( [ figs1 ] , [ figs2 ] ) , that @xmath80 ( model 1 ) or @xmath81 ( model 2 ) @xmath0 corresponds to a quintessence - like dark energy model , while @xmath82 ( model 1 ) or @xmath83 ( model 2 ) corresponds to a phantom - like one . a crossing of the phantom divide line is impossible for model 1 and model 2 . these results are also confirmed by the @xmath2 analysis given in the right panels . in order to further confirm our results , we redo our analysis with other values of @xmath66 , such as @xmath84 or @xmath85 , and find that the result remains unchanged . thus , we conclude that the phantom divide line is not crossed for both models . this is in conflict with what given in ref . @xcite where a crossing of the phantom line is found for model 2 . the constraints on model parameters of model 1 and model 2 will be discussed , respectively , in this section . three different kinds of observational data , i.e. , the type ia supernovae ( sne ia ) , the baryonic acoustic oscillation ( bao ) from the spectroscopic sloan digital sky survey ( sdss ) and the cosmic microwave background ( cmb ) radiation from wilkinson microwave anisotropy probe ( wmap ) , will be used in order to break the degeneracy between the model parameters . the fitting methods are summarized in the appendix . for the sne ia data , we use the union 2 compilation released by the supernova cosmology project collaboration recently @xcite . calculating the @xmath86 , we find that , for model 1 , the best fit values occur at @xmath87 , @xmath88 with @xmath89 , whereas , for model 2 , @xmath90 , @xmath91 with @xmath92 . then , we consider the constraints from the bao data . the parameter @xmath21 given by the bao peak in the distribution of sdss luminous red galaxies @xcite is used . the constraints from sne ia+bao are given by minimizing @xmath93 . the results are @xmath94 , @xmath95 ( at the @xmath5 confidence level ) with @xmath96 for model 1 and @xmath97 , @xmath98 ( at the @xmath5 confidence level ) with @xmath99 for model 2 . the contour diagrams are shown in fig . ( [ fig1 ] ) . the constraints on model 1 ( left ) and model 2 ( right ) from sne ia + bao . the red and blue+red regions correspond to @xmath100 and @xmath101 confidence regions , respectively.,title="fig:",width=226 ] the constraints on model 1 ( left ) and model 2 ( right ) from sne ia + bao . the red and blue+red regions correspond to @xmath100 and @xmath101 confidence regions , respectively.,title="fig:",width=226 ] furthermore , the cmb data is added in our analysis . the cmb shift parameter @xmath12 @xcite is used . the constraints from sne ia + bao + cmb are given by @xmath102 . ( [ fig2 ] ) shows the results . we find that , at the @xmath5 confidence level , @xmath6 , @xmath7 with @xmath103 for model 1 and @xmath8 , @xmath9 with @xmath104 for model 2 . the constraints on model 1 ( left ) and model 2 ( right ) from sne ia + bao + cmb . the red and blue+red regions correspond to @xmath100 and @xmath101 confidence regions , respectively.,title="fig:",width=226 ] the constraints on model 1 ( left ) and model 2 ( right ) from sne ia + bao + cmb . the red and blue+red regions correspond to @xmath100 and @xmath101 confidence regions , respectively.,title="fig:",width=226 ] with the observational data considered above , we also discuss the constraints on the @xmath10cdm and the results are @xmath105 with @xmath106 ( sne ia + bao ) and @xmath107 with @xmath108 ( sne ia + bao + cmb ) at the @xmath5 confidence level . a summary of constraint results on model 1 , model 2 and @xmath10cdm is given in table ( 1 ) . from figs . ( [ fig1 ] , [ fig2 ] ) and table ( 1 ) , one can see that the @xmath10cdm ( corresponding to @xmath49 for model 1 and @xmath58 for model 2 ) is consistence with the observations at the @xmath109 confidence level , while the dgp model ( corresponds to @xmath50 for model 1 ) is ruled out at the @xmath5 confidence level . meanwhile , using the @xmath11 ( dof : degree of freedom ) criterion , we find that the @xmath10cdm is favored by observations . 0.5pt .[tab1 ] summary of the constraint on model parameters and @xmath11 . in the table s+b+c represents sne ia + bao + cmb . [ cols="^,^,^,^,^,^,^,^,^,^ " , ] in addition , we study the evolution of the equation of state for the effective dark energy . the results are shown in fig . ( [ fig3 ] ) . the dashed , dotdashed and solid lines show the evolutionary curves with the model parameters at the best fit values from sne ia , sne ia+bao , and sne ia+bao+cmb , respectively . apparently , sne ia favors a phantom - like dark energy , while sne ia + bao + cmb favor a quintessence - like one . the evolutionary curves of the equation of state for the effective dark energy from model1 ( left ) and model2 ( right ) . the model parameters are set at the best fit values . the dashed , dotdashed and solid lines correspond to the constraints from sne ia , sne ia + bao , and sne ia + bao + cmb , respectively.,title="fig:",width=226 ] the evolutionary curves of the equation of state for the effective dark energy from model1 ( left ) and model2 ( right ) . the model parameters are set at the best fit values . the dashed , dotdashed and solid lines correspond to the constraints from sne ia , sne ia + bao , and sne ia + bao + cmb , respectively.,title="fig:",width=226 ] recently , the @xmath0 gravity theory is proposed to explain the present cosmic accelerating expansion without the need of dark energy . in this letter , we discuss firstly the statefinder geometrical analysis and @xmath2 diagnostic to the @xmath0 gravity . two concrete @xmath0 models proposed by linder @xcite are studied . from the @xmath2 diagnostic and the phase space analysis of the statefinder parameters @xmath71 and pair @xmath110 , we find that , for both model 1 and model 2 , a crossing of the phantom divide line is impossible , which conflicts with the result obtained in ref . @xcite where a crossing is found for model 2 . we then consider the constraints on model 1 and model 2 from the latest union 2 type ia supernova set released by the supernova cosmology project collaboration , the baryonic acoustic oscillation observation from the spectroscopic sloan digital sky survey data release galaxy sample , and the cosmic microwave background radiation observation from the seven - year wilkinson microwave anisotropy probe result . we find that at the @xmath5 confidence level , for model 1 , @xmath6 , @xmath7 with @xmath103 and for model 2 , @xmath8 , @xmath9 with @xmath104 . we also find that the @xmath10cdm ( corresponds to @xmath49 for model 1 and @xmath58 for model 2 ) is consistence with observations at @xmath100 confidence level and it is favored by observation through the @xmath11 ( dof : degree of freedom ) criterion . however , the dgp model , which corresponds to @xmath50 for model 1 , is ruled out by observations at the @xmath5 confidence level . finally , we study the evolution of the equation of state for the effective dark energy in the @xmath0 theory . our results show that sne ia favors a phantom - like dark energy , while sne ia + bao + cmb prefers a quintessence - like one . the analysis of the current paper also indicates that the @xmath0 theory can give the same background evolution as other models such as @xmath10cdm , although they have completely different theoretical basis . thus , it remains interesting to study other aspects of @xmath0 theory , such as the matter density growth , which may help us distinguish it from other gravity theories . this work was supported in part by the national natural science foundation of china under grants nos . 10775050 , 10705055 , 10935013 and 11075083 , zhejiang provincial natural science foundation of china under grant no . z6100077 , the srfdp under grant no . 20070542002 , the fanedd under grant no . 200922 , the national basic research program of china under grant no . 2010cb832803 , the ncet under grant no . 09 - 0144 , the pcsirt under grant no . irt0964 , and k.c . wong magna fund in ningbo university . recently , the supernova cosmology project collaboration @xcite released the union2 compilation , which consists of 557 sne ia data points and is the largest published and spectroscopically confirmed sna ia sample today . we use it to constrain the theoretical models in this paper . the results can be obtained by minimizing the @xmath111 value of the distance moduli @xmath112 ^ 2}{\sigma_{u , i}^2}\;,\end{aligned}\ ] ] where @xmath113 are the errors due to the flux uncertainties , intrinsic dispersion of sne ia absolute magnitude and peculiar velocity dispersion . @xmath114 is the observed distance moduli and @xmath115 is the theoretical one , which is defined as @xmath116 here @xmath117 , @xmath118 , and the luminosity distance @xmath119 can be calculated by @xmath120 with @xmath121 given in eqs . ( [ mod1ez ] , [ mod2ez ] ) . since @xmath122 ( or @xmath123 ) is a nuisance parameter , we marginalize over it by an effective approach given in ref . expanding @xmath124 to @xmath125 with @xmath126 , @xmath127/\sigma_{u , i}^2 $ ] and @xmath128 ^ 2/\sigma_{u , i}^2 $ ] , one can find that @xmath124 has a minimum value at @xmath129 , which is given by @xmath130 thus , we can minimize @xmath86 instead of @xmath124 to obtain constraints from sne ia . for bao data , the parameter @xmath21 given by the bao peak in the distribution of sdss luminous red galaxies @xcite is used . the results can be obtained by calculating : @xmath131 ^ 2}{\sigma_a^2}\end{aligned}\ ] ] where @xmath132 with the scalar spectral index @xmath133 from the wmap 7-year data @xcite and the theoretical value @xmath21 is defined as @xmath134^{2/3}\end{aligned}\ ] ] with @xmath135 . since the cmb shift parameter @xmath12 @xcite contains the main information of the observations of the cmb , it is used in our analysis . the wmap7 data gives the observed value of @xmath12 to be @xmath136 @xcite . the corresponding theoretical value is defined as @xmath137 where @xmath138 . therefore , the constraints on model parameters can be obtained by fitting the observed value with the corresponding theoretical one of parameter @xmath12 through the following expression @xmath139 ^ 2}{\sigma_r^2}.\end{aligned}\ ] ] 99 a. g. riess , et al . j. * 116 * , 1009 ( 1998 ) ; s. perlmutter , et al . , astrophys . j. * 517 * , 565 ( 1999 ) . d. n. spergel , et al . , apjs , * 148 * , 175 ( 2003 ) ; d. n. spergel , et al . , apjs , * 170 * , 377s ( 2007 ) . m. tegmark , et al . d * 69 * , 103501 ( 2004 ) . d. j. eisenstein , et al . , astrophys . j. * 633 * , 560 ( 2005 ) . s. nojiri , s.d . odintsov , arxiv:0807.0685 ; t.p . sotiriou , v. faraoni , rev . phys . * 82 * , 451 ( 2010 ) ; a. de felice and s. tsujikawa , living rev . rel . * 13 * , 3 ( 2010 ) . a. einstein , sitzungsber . kl . , 217 ( 1928 ) ; 401 ( 1930 ) ; a. einstein , math . ann . * 102 * , 685 ( 1930 ) ; k. hayashi and t. shirafuji , phys . d * 19 * , 3524 ( 1979 ) ; * 24 * , 3312 ( 1981 ) . e. v. linder , phys . d * 81 * , 127301 ( 2010 ) . r. myrzakulov , arxiv:1006.1120 ; k. k. yerzhanov , s. r. myrzakul , i. i. kulnazarov , r. myrzakulov , arxiv:1006.3879 ; p. wu and h. yu , phys . lett . b * 692 * , 176 ( 2010 ) ; r. yang , arxiv:1007.3571 , p. yu . tsyba , i. i. kulnazarov , k. k. yerzhanov , r. myrzakulov , arxiv:1008.0779 ; j. b. dent , s. dutta , e. n. saridakis , arxiv:1008.1250 ; g. r. bengochea , arxiv:1008.3188 ; p. wu and h. yu , arxiv:1008.3669 . k. bamba , c. q. geng , c. c. lee , arxiv:1008.4036 . r. ferraro and f. fiorini , phys . d * 75 * , 084031 ( 2007 ) . r. ferraro and f. fiorini , phys . d * 78 * , 124019 ( 2008 ) . g. dvali and m. s. turner , arxiv : astro - ph/0301510 . d. j. h. chung and k. freese , phys . d * 61 * , 023511 ( 1999 ) . g. dvali , g. gabadadze , m. porrati , phys . b * 485 * , 208 ( 2000 ) . e. v. linder , phys . d * 80 * , 123528 ( 2009 ) . k. bamba , c. q. geng and c. c. lee , arxiv:1005.4574 . v. sahni , t. d. saini , a. a. starobinsky and u. alam , etp lett . * 77 * , 201 ( 2003 ) [ pisma zh . * 77 * , 249 ( 2003 ) ] ; v. sahni , a. shafieloo and a. a. starobinsky , phys . d * 78 * , 103502 ( 2008 ) . p. wu and h. yu , int . d * 14 * , 1873 ( 2005 ) . v. sahni , a. shafieloo and a. a. starobinsky , phys . d * 78 * , 103502 ( 2008 ) . p. wu and h. yu , jcap * 02 * , 019 ( 2008 ) . a. ali , r. gannouji , m. sami , a. a. sen , phys . d * 81 * , 104029 ( 2010 ) . r. amanullah , et al . , arxiv:1004.1711 . s. nesseris and l. perivolaropoulos , phys . d * 72 * , 123519 ( 2005 ) . e. komatsu et al . , arxiv:1001.4538 . y. wang and p. mukherjee , astrophys . j. * 650 * , 1 ( 2006 ) . j. r. bond , g. efstathiou and m. tegmark , mon . not . soc . * 291 * , l33 ( 1997 )

the @xmath0 theory , which is an extension of teleparallel , or torsion scalar @xmath1 , gravity , is recently proposed to explain the present cosmic accelerating expansion with no need of dark energy . in this letter , we first perform the statefinder analysis and @xmath2 diagnostic to two concrete @xmath0 models , i.e. , @xmath3 and @xmath4 , and find that a crossing of phantom divide line is impossible for both models . this is contrary to an existing result where a crossing is claimed for the second model . we , then , study the constraints on them from the latest union 2 type ia supernova ( sne ia ) set , the baryonic acoustic oscillation ( bao ) , and the cosmic microwave background ( cmb ) radiation . our results show that at the @xmath5 confidence level @xmath6 , @xmath7 for model 1 and @xmath8 , @xmath9 for model 2 . a comparison of these two models with the @xmath10cdm by the @xmath11 ( dof : degree of freedom ) criterion indicates that @xmath10cdm is still favored by observations . we also study the evolution of the equation of state for the effective dark energy in the theory and find that sne ia favors a phantom - like dark energy , while sne ia + bao + cmb prefers a quintessence - like one .

the ( patlak ) keller - segel system is a tentative model to describe chemo - taxis phenomenon , an attractive chemical phenomenon between organisms . in two dimensions , the classical 2-d parabolic - elliptic keller - segel model reduces to a single non linear p.d.e . , @xmath2 with some initial @xmath3 . + here @xmath4 , @xmath5 and @xmath6 is the gradient of the harmonic kernel , i.e. @xmath7 . + it is not difficult to see that preserves positivity and mass , so that we may assume that @xmath3 is a density of probability , i.e. @xmath8 and @xmath9 . for an easy comparison with the p.d.e . literature , just remember that our parameter @xmath10 satisfies @xmath11 when @xmath12 denotes the total mass of @xmath13 ( the parameter @xmath14 being usually included in the definition of @xmath15 ) . + as usual , @xmath13 is modeling a density of cells , and @xmath16 is ( up to some constant ) the concentration of chemo - attractant . a very interesting property of such an equation is a blow - up phenomenon [ blow - up ] assume that @xmath17 and that @xmath18 . then if @xmath19 , the maximal time interval of existence of a classical solution of is @xmath20 with @xmath21 if @xmath22 then @xmath23 . for this result , a wonderful presentation of what keller - segel models are and an almost up to date state of the art , we refer to the unpublished hdr document of adrien blanchet ( available on adrien s webpage @xcite ) . we also apologize for not furnishing a more complete list of references on the topic , where beautiful results were obtained by brilliant mathematicians . but the present paper is intended to be a short note . actually is nothing else but a mc kean - vlasov type equation ( non linear fokker - planck equation if one prefers ) , involving a potential which is singular at @xmath24 . hence one can expect that the movement of a typical cell will be given by a non - linear diffusion process @xmath25 where @xmath26 denotes the distribution of probability of @xmath27 . the natural linearization of is through the limit of a linear system of stochastic differential equations in mean field interactions given for @xmath28 by @xmath29 for a well chosen initial distribution of the @xmath30 . here the @xmath31 are for each @xmath32 independent standard 2-d brownian motions . under some exchangeability assumptions , it is expected that the distribution of any particle ( say @xmath33 ) converges to a solution of as @xmath34 , yielding a solution to . this strategy ( including the celebrated propagation of chaos phenomenon ) has been well known for a long time . one can see @xcite for bounded and lipschitz potentials , @xcite for unbounded potentials connected with the granular media equation . the goal of the present note is the study of existence , uniqueness and non explosion for the system . that is , this is the very first step of the whole program we have described previously . moreover we will see how the @xmath32-particle system is feeling the blow - up property of the keller - segel equation . for such singular potentials very few fournier , hauray and mischler @xcite have tackled the case of the 2-d viscous vortex model , corresponding to @xmath35 for which no blow - up phenomenon occurs . in the same spirit the sub - critical keller - segel model corresponding to @xmath36 for some @xmath37 is studied in @xcite . the methods of both papers are close , and mainly based on some entropic controls . these methods seem to fail for the classical keller - segel model we are looking at . however , during the preparation of the manuscript , we received the paper by n. fournier and b. jourdain @xcite , who prove existence and some weak uniqueness by using approximations . though some intermediate results are the same , we shall here give a very different and much direct approach , at least for existence and some uniqueness . however , we shall use one result in @xcite to prove a more general uniqueness result . also notice that when we replace the attractive potential @xmath15 by a repulsive one ( say @xmath38 ) , we find models connected with random matrix theory ( like the dyson brownian motion ) . our main theorem in this paper is the following [ thmmain ] let @xmath39 . then , * for @xmath40 and @xmath41 , there exists a unique ( in distribution ) non explosive solution of , starting from any @xmath42 . moreover , the process is strong markov , lives in @xmath43 and admits a symmetric @xmath44-finite , invariant measure given by @xmath45 * for @xmath46 , if @xmath47 , the system does not admit any global solution ( i.e. defined on the whole time interval @xmath48 ) , * for @xmath46 , if @xmath49 , either the system explodes or the @xmath32 particles are glued in finite time . since later we will be interested in the limit @xmath50 , this theorem is in a sense optimal : for @xmath51 we have no asymptotic explosion while for @xmath47 the system explodes . also notice that in the limiting case @xmath52 , we have ( at least ) explosion for the density of the stochastic system and not for the equation . the proof of this theorem is partly `` pathwise '' , based on comparisons between one dimensional diffusion processes and the behavior of squared bessel processes , partly based on dirichlet forms theory and partly based on an uniqueness result for @xmath0 dimensional skew bessel processes obtained in @xcite . the latter is only used to get rid of a non allowed polar set of starting positions which appears when using dirichlet forms . the remaining part of the whole program will be the aim of future works . most of the proofs in this section will use comparison with squared bessel processes . let us recall some basic results on these processes . [ defbes ] let @xmath53 . the unique strong solution ( up to some explosion time @xmath54 ) of the following one dimensional stochastic integral equation @xmath55 is called the ( generalized ) squared bessel process of dimension @xmath56 starting from @xmath57 . in general squared bessel processes are only defined for @xmath58 , that is why we used the word generalized in the previous definition . for these processes the following properties are known [ propbes ] let @xmath59 be a generalized squared bessel process of dimension @xmath56 . let @xmath60 the first hitting time of the origin . * if @xmath61 , then @xmath60 is almost surely finite and equal to the explosion time , * if @xmath62 , then @xmath63 and @xmath64 for all @xmath65 almost surely , * if @xmath66 , then @xmath63 almost surely and the origin is instantaneously reflecting , * if @xmath67 , then the origin is polar ( hence @xmath68 almost surely ) . for all this see @xcite chapter xi , proposition 1.5 . now come back to . for simplicity we skip the index @xmath32 in the definition of the process . + since all coefficients are locally lipschitz outside the set @xmath69 and bounded when the distance to @xmath70 is bounded from below , the only problem is the one of collisions between particles . as usual we denote by @xmath71 the lifetime of the process . for simplicity we also assume , for the moment , that the starting point does not belong to @xmath70 , so that the lifetime is almost surely positive . for @xmath72 we define @xmath73 and @xmath74 . we shall say that a @xmath75-collision occurs at ( a random ) time @xmath76 if @xmath77 for all @xmath78 , @xmath79 for all @xmath80 . of course , there is no lack of generality when looking at the first @xmath75 indices , and we can also assume that at this peculiar time @xmath76 , any other collision involves at most @xmath75 other particles . + in what follows we denote @xmath81 , and @xmath82 of course a @xmath75-collision occurs at time @xmath76 if and only if @xmath83 and @xmath84 . let us study the process @xmath85 . applying ito s formula we get on @xmath86 @xmath87 we denote @xmath88 the martingale part and the non - constant drift part . let us compute the martingale bracket , using the immediate @xmath90 and @xmath91 . @xmath92 i.e. finally @xmath93 according to doob s representation theorem ( applied to @xmath94 ) , there exists ( on an extension of the initial probability space ) a one dimensional brownian motion @xmath95 such that almost surely for @xmath86 @xmath96 in order to study the drift term @xmath97 we will divide it into two sums : the first one , @xmath98 taking into consideration the @xmath99 and @xmath100 in @xmath15 , i.e. the pair of particles which will be directly involved in the eventual @xmath75-collision ; the other one @xmath101 , involving the remaining indices . more precisely @xmath102 with @xmath103 we will deal with @xmath101 later . first we ought to simplify the expression of @xmath98 . indeed @xmath104 so that using again @xmath105 the latter is still equal to @xmath106 and using again @xmath91 , we finally arrive at @xmath107 with the results obtained in and we may simplify , writing ( still on @xmath86 ) @xmath108 hence defining @xmath109 the process @xmath110 satisfies @xmath111 i.e. can be viewed as a perturbation of a squared bessel process of dimension @xmath112 we shall denote by @xmath113 in the sequel . if @xmath114 , @xmath115 so that @xmath116 is exactly the squared bessel process of dimension @xmath117 . hence , according to proposition [ propbes ] * if @xmath118 there is explosion in finite time for the process @xmath116 ( hence for @xmath119 also ) , * if @xmath120 , there is an almost sure @xmath32-collision in finite time , and then all the particles are glued , provided no explosion occurred before for the process @xmath119 , * if @xmath121 there is an almost sure @xmath32-collision in finite time , provided no explosion occurred before for the process @xmath119 , * if @xmath122 there is almost surely no @xmath32-collision ( before explosion ) . in particular we see that the particle system immediately feels the critical value @xmath52 , in particular explosion occurs in finite time as soon as @xmath118 . + for @xmath121 we know that @xmath116 is instantaneously reflected once it hits the origin , but it does not indicate whether all or only some particles will separate ( we only know that they are not all glued ) . notice that when @xmath123 this condition reduces to @xmath124 , and then both particles are separated . hence in this very specific case , there is no explosion ( for the distance between both particles ) in finite time almost surely , but there are always @xmath0-collisions . as we said before , the lifetime @xmath71 is greater than or equal to the first multiple collision time @xmath76 . + since we shall consider @xmath110 as a perturbation of @xmath113 , what happens for the latter ? * for @xmath125 , @xmath113 reaches @xmath24 in finite time a.s . and then explosion occurs , * for @xmath126 , @xmath113 reaches @xmath24 and is sticked , * for @xmath127 , @xmath113 reaches @xmath24 and is instantaneously reflected , * for @xmath128 , @xmath113 does not hit @xmath24 in finite time a.s . [ lemk ] for all @xmath129 , it holds @xmath130 introduce the function @xmath131 defined for @xmath132 . then @xmath133 is negative on @xmath134 , so that the lemma is proved for @xmath135 . for @xmath136 , it amounts to @xmath137 which is true for all @xmath138 ( with equality for @xmath139 and @xmath140 ) . in particular , since @xmath122 , @xmath113 never hits @xmath24 for @xmath129 , while it reaches @xmath24 but is instantaneously reflected for @xmath141 . what we expect is that the same occurs for @xmath110 . + in order to prove it , let @xmath76 be the first multiple collision time . with our convention ( changing indices if necessary ) there exists some @xmath72 such that @xmath76 is the first @xmath75-collision time @xmath142 . note that this does not prevent other @xmath143-collisions ( with @xmath144 ) possibly at the same time @xmath76 for the particles with indices larger than @xmath145 . but as we will see this will not change anything . the reasoning will be the same but the conclusion completely different for @xmath141 and for @xmath146 . introduce , for @xmath37 , the random set @xmath147 it holds @xmath148 in particular if @xmath149 there exists some @xmath37 so that @xmath150 . + we shall see that this is impossible when @xmath146 . indeed recall that @xmath151 so on @xmath152 , we have , for @xmath153 , latexmath:[\[\begin{aligned } the latter being a consequence of cauchy - schwarz inequality . thus on @xmath152 , for @xmath153 latexmath:[\[\label{eqr } hence , on @xmath152 for @xmath153 the drift @xmath156 ( which is not markovian ) of @xmath157 satisfies @xmath158 in particular for any @xmath159 , @xmath160 provided @xmath161 is small enough . thus the hitting time of the origin for the process with drift @xmath162 is larger than the one for the corresponding squared bessel process ( thanks to well known comparison results between one dimensional ito processes , see e.g. @xcite chap.6 , thm 1.1 ) , and since this holds for all @xmath163 , finally is larger than the one of @xmath113 . but as we already saw , @xmath113 never hits the origin for @xmath164 . using again the comparison theorem ( this time with @xmath156 and @xmath165 ) , @xmath110 does not hit the origin in finite time on @xmath166 which is in contradiction with @xmath167 . actually all we have done in the previous sub subsection is unchanged for @xmath141 , except the conclusion . indeed @xmath168 reaches the origin but is instantaneously reflected . so @xmath169 ( on @xmath170 ) can reach the origin too , but is also instantaneously reflected . actually using that @xmath171 together with the feller s explosion test , it is easily seen that @xmath169 will reach the origin with a ( strictly ) positive probability ( presumably equal to one , but this is not important for us ) . but this instantaneous reflection is not enough for the non explosion of the process @xmath119 , because @xmath172 is @xmath173 valued . before going further in the construction , let us notice another important fact : there are no multiple @xmath0-collisions at the same time , i.e. starting from @xmath174 the process lives in @xmath43 at least up to the explosion time @xmath71 . of course this is meaningful provided @xmath175 . + to prove the previous sentence , first look at @xmath176 assuming that @xmath177 and that no other @xmath0-collision happens at time @xmath76 . it is easily seen that ( just take care that we had an extra factor @xmath0 in our definition of @xmath178 ) @xmath179 so that , defining @xmath180 we get that @xmath181 where @xmath182 is a remaining term we can manage just as we did for @xmath178 . since for @xmath175 , @xmath183 , @xmath184 , hence @xmath185 does not hit the origin . notice that if we consider @xmath186 @xmath0-collisions , the same reasoning is still true , just replacing @xmath187 by @xmath188 , the final equation being unchanged except for @xmath182 . according to all what precedes what we need to prove is the existence of the solution of with an initial configuration satisfying @xmath189 with @xmath190 , all other coordinates being different and different from @xmath190 . indeed , on @xmath191 , @xmath192 the set of particles with exactly two glued particles , so that if we can prove that starting from any point of @xmath193 , we can build a strong solution on an interval @xmath194 $ ] for some strictly positive stopping time @xmath195 , it will show that @xmath196 almost surely . however we will not be able to prove the existence of such a strong solution . actually we think that it does not exist . we will thus build some weak solution and show uniqueness in some specific sense . this will be the goal of the next sections . recall that @xmath199 means that exactly two coordinates coincide ( say @xmath190 ) , all other coordinates being distinct and distinct from @xmath200 . we may thus define @xmath201 so that @xmath202 and points @xmath203 will satisfy @xmath204 . if @xmath205 , we may similarly define @xmath206 and then @xmath207 . in all cases the balls @xmath208 are the open balls . now if @xmath15 is some compact subset of @xmath43 we can cover @xmath15 by a finite number of sets @xmath207 , so that for any measure @xmath209 , a function @xmath210 belongs to @xmath211 if and only if @xmath212 for all @xmath213 in @xmath43 . it is clear that for @xmath205 , @xmath209 is a bounded measure on @xmath207 . when @xmath199 , say that @xmath190 and perform the change of variables @xmath215 in restriction to @xmath207 , @xmath209 can be written @xmath216 hence is a bounded measure on @xmath207 provided @xmath217 just looking at polar coordinates for @xmath218 . in this case it immediately follows that @xmath209 is a @xmath44 finite measure on @xmath43 . also remark that if @xmath219 is compactly supported by @xmath15 and belongs to @xmath220 then it belongs to @xmath221 and @xmath222 but we can say much more . to this end consider the symmetric form @xmath223 first we check that this form is closable in the sense of @xcite . to this end it is enough to show that it is a closable form when restricted to functions @xmath224 for all @xmath42 . if @xmath205 the form is equivalent to the usual scalar product on square lebesgue integrable functions , so that it is enough to look at @xmath199 . hence let @xmath225 be a sequence of functions in @xmath226 , converging to @xmath24 in @xmath227 and such that @xmath228 converges to some vector valued function @xmath210 in @xmath227 . what we need to prove is that @xmath210 is equal to @xmath24 . to this end consider a vector valued function @xmath229 which is smooth and compactly supported in @xmath230 for some @xmath37 . then a simple integration by parts shows that @xmath231 for some @xmath232 so is equal to @xmath24 . hence @xmath210 vanishes almost surely on @xmath230 , for all @xmath37 , so that @xmath210 is @xmath209-almost everywhere equal to @xmath24 . by construction , @xmath233 is regular and local . hence , its smallest closed extension @xmath234 is a dirichlet form , which is in addition regular and local . according to theorem 6.2.2 . in @xcite , there exists a @xmath209-symmetric diffusion process @xmath235 whose form is given by @xmath233 . notice that , integrating by parts , we see that the generator of this diffusion process coincides with the generator @xmath236 given by for the functions @xmath219 in @xmath238 such that @xmath239 . this is a core for the domain @xmath240 . the dirichlet form theory tells us that once @xmath241 , @xmath242 is a @xmath243 martingale for quasi every starting point @xmath213 , i.e. for all @xmath213 out of some subset @xmath244 of @xmath43 which is of zero @xmath209-capacity . but remark that for any @xmath245 and @xmath246 , the transition kernel @xmath247 of the markov semi - group is absolutely continuous with respect to @xmath209 . indeed using the local malliavin calculus as in @xcite ( or elliptic standard results ) , this transition kernel has a ( smooth ) density w.r.t . lebesgue measure ( hence w.r.t . @xmath209 ) on each open subset of @xmath248 for any @xmath37 . hence if @xmath249 , @xmath250 for all @xmath37 so that @xmath251 and finally using monotone convergence , @xmath252 . + since @xmath247 is absolutely continuous w.r.t . @xmath209 , we deduce from theorem 4.3.4 in @xcite that the sets of zero @xmath209 capacity are exactly the polar sets for the process . to prove the lemma , for all @xmath245 it is enough to show the martingale property starting from @xmath213 up to the exit time @xmath257 of @xmath207 ( since @xmath258 because @xmath244 is polar , see the discussion above ) . in the sequel , for notational convenience , we do not write the exit time @xmath257 ( all times @xmath259 have to be understood as @xmath260 ) and we simply write @xmath43 instead of @xmath207 . to show this result it is actually enough to look locally in the neighborhood of a point @xmath199 such that @xmath190 , and with our previous notation to look at both coordinates of @xmath261 . indeed @xmath262 belongs ( at least locally ) to @xmath240 as well as all other coordinates @xmath263 for @xmath264 . let @xmath265 for @xmath266 be the coordinate application of @xmath261 . clearly @xmath267 for @xmath268 , hence belongs to @xmath269 thanks to our assumption on @xmath270 . introduce the function defined on @xmath271 by , @xmath272 @xmath229 is of @xmath273 class except at @xmath274 . now define @xmath275 . we have @xmath276(x)\ ] ] the remaining term @xmath277 corresponding to the interactions with particles @xmath263 for @xmath264 . but it is easily seen that @xmath281 converges to @xmath282 in @xmath283 as @xmath284 . since @xmath278 converges to @xmath285 in @xmath283 too , we deduce that @xmath286 where @xmath287 denotes the natural filtration on the probability space , since the same property is true for @xmath278 . similarly the brackets converge to @xmath288 . since the same holds for @xmath289 , we get the desired result @xmath290 almost surely . actually this result holds true @xmath243 almost surely for @xmath209 almost all @xmath291 . but since @xmath247 is absolutely continuous w.r.t . @xmath209 for @xmath246 , it immediately follows using the markov property at time @xmath259 , that is true @xmath243 a.s . for all @xmath292 , but only for @xmath246 . hence for all @xmath292 and all @xmath246 , @xmath293 is a martingale defined on @xmath294 , whose bracket is given by @xmath295 , i.e. is ( @xmath0 times ) a brownian motion . in particular for a fixed @xmath259 , @xmath296 is bounded in @xmath297 . up to a sub - sequence it is thus weakly convergent in @xmath298 as @xmath299 so that @xmath300 is well defined @xmath243 a.s . , and satisfies for all @xmath301 this time . thus it is a martingale with a linear bracket , i.e. @xmath0 times a brownian motion . assume in addition that @xmath303 . then the previous diffusion process never hits @xmath302 since the latter is exactly the set where either some @xmath75-collision occurs for some @xmath146 or at least two @xmath0-collisions occur at the same time . so it is actually the unique @xmath209-symmetric markov diffusion defined on @xmath304 solving . indeed we could associate to any markovian extension of @xmath305 another diffusion process , which would coincide with the previous one up to the hitting time of @xmath302 which is almost surely infinite . we have thus obtained [ thmweak ] assume that @xmath303 and that @xmath40 . + then there exists a unique @xmath209-symmetric ( see ) diffusion process @xmath306 ( i.e. a hunt process with continuous paths ) , defined for @xmath301 and @xmath245 where @xmath307 is polar ( or equivalently of @xmath209 capacity equal to @xmath24 ) such that for all @xmath308 , @xmath309 is a @xmath243 martingale ( for the natural filtration ) with @xmath236 given by . furthermore @xmath235 lives in @xmath43 ( never hits @xmath302 ) . as for the previous lemma , it is enough to work locally in the neighborhood of the points in @xmath193 and to look at the new particles @xmath310 . let @xmath311 be written in these new coordinates . using a taylor expansion in @xmath312 ( @xmath313 and all the others @xmath263 being fixed ) and the fact that if the partial derivatives at @xmath314 of a smooth function of @xmath312 are vanishing , then this function belongs to the domain of the generator , we see that proving the martingale property for @xmath219 amounts to the corresponding martingale property for smooth functions @xmath210 written as @xmath315 i.e. amounts to the previous lemma ( and of course the remaining particles for which there is no problem ) . it remains to extend the martingale property we proved to hold for @xmath311 to @xmath308 . take @xmath308 and define @xmath316 as the first time the distance @xmath317 is less than @xmath318 . then replacing @xmath219 by some @xmath319 which coincides with @xmath219 on @xmath320 , we see that @xmath321 is a @xmath243 martingale . since @xmath316 growths to infinity the conclusion follows from lebesgue theorem . the main disadvantage of the previous construction is that it is not explicit and that it does not furnish a solution starting from all @xmath291 but only for all @xmath213 except those in some unknown polar set . in particular , proving the regularity of the markov transition kernels up to @xmath193 requires additional work . the advantage is that if we require @xmath209-symmetry , we get uniqueness of the diffusion process . this theorem is to be compared with theorem 7 in @xcite , where existence of a weak solution is shown by using approximation and tightness , in the same @xmath303 case ( take care of the normalization of @xmath10 which is not the same here and therein ) . note that the result in @xcite is concerned with existence starting from some initial absolutely continuous density and does not furnish a diffusion process . @xmath322 in this subsection we assume that @xmath303 and that @xmath40 . our aim is to build a solution starting from any point in @xmath43 , i.e. to get rid of the polar set @xmath244 in the previous sub - section . the construction will be very similar ( still using dirichlet forms ) but we shall here use one result in @xcite , namely the uniqueness result for a @xmath0 dimensional bessel process . we can cover @xmath193 by an enumerable union of @xmath324 ( @xmath325 ) . it is thus immediate that the lemma will be proved once we prove that @xmath326 but we have seen in the previous section that , when the process is in some @xmath324 ( where say @xmath204 ) , the process @xmath327 is larger than or equal to the square of a bessel process @xmath328 of index @xmath56 strictly between @xmath24 and @xmath0 . but ( see @xcite proof of proposition 1.5 p.442 ) , the time spent at the origin by the latter is equal to @xmath24 , i.e. @xmath329 almost surely . the same necessarily holds for @xmath330 , hence the result . we intend now to prove some uniqueness , when starting from a point in @xmath193 . actually , using some standard tools of concatenation of paths , it is enough to look at the behavior of our process starting at some @xmath325 with @xmath204 , up to the exit time of @xmath324 ( or some open non empty subset of @xmath324 ) . in this case the only difficulty is to control the pair @xmath331 since all other coordinates are defined through smooth coefficients . of course writing @xmath332 we have that @xmath333 and @xmath334 where @xmath335 and @xmath336 are smooth functions ( in @xmath324 ) , @xmath337 and @xmath338 being two independent @xmath0 dimensional brownian motions . define @xmath339 as we defined @xmath324 ( see , but replacing @xmath340 by @xmath341 and consider a smoothed version @xmath342 of the indicator of @xmath339 i.e. a smooth non negative function such that @xmath343 we may extend all coefficients ( except @xmath344 ) as smooth compactly supported functions outside @xmath324 , and replace @xmath344 by @xmath345 . if we can show uniqueness for this new system we will have shown uniqueness up to the exit time of @xmath339 for , and the remaining part of the initial system . thus , after a standard girsanov transform , we are reduced to prove uniqueness for @xmath349 hence for @xmath350 . @xmath351 is some type of @xmath0-dimensional skew bessel process with dimension @xmath352 ( see @xcite for the one dimensional version ) . its squared norm @xmath353 is a squared bessel process of dimension @xmath354 , so that the origin is not polar for the process @xmath350 . as we did in the previous sub - section , we can directly prove the existence and uniqueness of a symmetric hunt process ( here the reference measure is @xmath355 ) using the associated dirichlet form , and since the origin is not polar , we know the existence of a solution starting from @xmath356 . here we only need @xmath357 , but for the whole construction our initial assumption on @xmath10 is required . finally we can check that the occupation time formula of lemma [ lemlocal ] is still true . but as before , if now we have existence starting from every initial point , we only have uniqueness in the sense of symmetric markov processes . to get weak uniqueness we can use polar coordinates : the squared norm is a squared bessel process , so that strong uniqueness holds ( with the corresponding dimension we are looking at ) ; the polar angle is much tricky to handle . this is the main goal of lemma 19 in @xcite , and the final weak uniqueness then follows from the proof of theorem 17 in @xcite and the occupation time formula . [ remfin ] it can be noticed than this result is out of reach of the method developed by krylov and rckner in @xcite for a general brownian motion plus drift @xmath358 , since it requires that @xmath359 for some @xmath360 . also notice that one can not use standard girsanov transform for solving , since for a @xmath0-dimensional brownian motion starting from the origin , @xmath361 see @xcite . @xmath322

we introduce a stochastic system of interacting particles which is expected to furnish as the number of particles goes to infinity a stochastic approach of the @xmath0-d keller - segel model . in this note , we prove existence and some uniqueness for the stochastic model for the parabolic - elliptic keller - segel equation , for all regimes under the critical mass . prior results for existence and weak uniqueness have been very recently obtained by n. fournier and b. jourdain @xcite . @xmath1 universit de toulouse _ key words : keller - segel model , diffusion processes , bessel processes . _ _ msc 2010 : 35q92 , 60j60 , 60k35 .

diffractive higgs production may play an important role in identifying and studying a @xmath3- and @xmath4-even , light higgs boson at the lhc , see , for example , ref . there exist a wide range of predictions from a variety of models for the cross section for diffractive higgs production , which have yielded answers ranging over many orders of magnitude . one unfortunate consequence is to discredit diffractive higgs production as a possible way to identify a higgs boson . here we emphasize that the huge spread of predictions is _ either _ because different diffractive processes have been considered _ or _ because important effects have been neglected . one of the aims of this note is to guide the reader through the plethora of predictions , making critical comparisons between the different approaches wherever possible . let us consider a light higgs boson ( with mass less than 130 gev ) with the dominant @xmath5 decay . from an observational point of view , it is convenient to discuss three different diffractive production mechanisms , where we will use a + sign to indicate the presence of a rapidity gap . * * exclusive production * : @xmath6 + if the outgoing protons are tagged , this process has the advantage that the higgs mass may be measured in two independent ways ; first , by the missing mass to the outgoing protons and , second , by the @xmath5 decay . so the signal must satisfy @xmath7 , with allowance for experimental resolution . moreover , the @xmath8 background is suppressed by a spin ( @xmath9 ) selection rule , which leads to a favourable signal - to - background ratio . * * inclusive production * : @xmath10 + the advantage is a much larger cross section . however , there is no spin selection rule to suppress the @xmath8 background , and the signal - to - background ratio is unfavourable . moreover , the accuracy of the higgs mass determination is worse , as @xmath11 is not applicable . * * central inelastic production * : @xmath12 + there is additional radiation accompanying the higgs in the central region , which is separated from the outgoing protons by rapidity gaps . although this mechanism is often used for predictions , it has , in our view , no special advantages for higgs detection . we may regard each large rapidity gap as being generated by an effective pomeron exchange . it may be _ either _ a qcd pomeron , which at lowest order is a gluon gluon state , _ or _ a phenomenological pomeron with parameters fixed by data . the above information is summarised in fig . 1 , together with a leading order qcd diagram of each process . [ cols="^,^,^ " , ] we start the discussion of the results shown in table 1 with the calculation of the exclusive double - diffractive cross section by levin @xcite . he assumed the survival probability to soft rescattering @xmath13 . to account for qcd radiation he multiplied the final result by an effective @xmath14 factor which was estimated phenomenologically assuming a poisson probability @xmath15 , where @xmath16 is the mean multiplicity of mini - jets produced in hadron interactions with energy @xmath17 . this assumption overestimates the survival factor @xmath14 in comparison with the perturbative qcd calculation , since instead of getting the double - logarithmic sudakov - like suppression , his probability @xmath15 corresponds to a single logarithm . in the calculation by cudell and hernandez @xcite , both the soft @xmath18 and hard @xmath14 survival factors were neglected . in addition to the pure exclusive process , inclusive events where an incoming proton dissociates into @xmath19 resonances were allowed , so the predicted cross section becomes larger . a crucial point , both in this calculation and in that of levin , is the normalization of the two - gluon exchange amplitude . without the double - logarithmic @xmath20 factor inside the loop integration over the gluon transverse momentum @xmath21 , the integral is infrared divergent . to obtain a finite result the authors have to choose an infrared cut - off or to introduce a finite mass for the gluon . the value of the cut - off , or mass , is tuned to reproduce the total @xmath22 cross section , @xmath23 , in terms of the low nussinov two - gluon pomeron exchange . it has been noted @xcite that the use of such a prescription further overestimates the higgs production cross section . indeed , in terms of @xmath21 factorization , the higgs production forward amplitude is of the form @xmath24 where the factor @xmath25 represents the @xmath26 vertex and @xmath27 is the unintegrated skewed gluon density . the unintegrated gluon density embodies the @xmath20 factor @xcite which accounts for the fact that the gluon which participates in the hard @xmath26 subprocess remains untouched in the evolution from @xmath21 up to the hard scale , @xmath28 ; this hard scale is an implicit variable in the @xmath29 in ( [ eq : m_higgs ] ) . similarly , via the optical theorem , we may express the total cross section in terms of two - gluon exchange @xmath30 where @xmath31 , as follows from the internal kinematics of the process , and where the implicit scale in @xmath29 is now @xmath32 . at first sight it appears that ( [ eq : sigma_tot ] ) will give a precise normalisation of the higgs cross section , via ( [ eq : m_higgs ] ) . however , in addition to the different implicit scales , the typical values of @xmath33 sampled in ( [ eq : sigma_tot ] ) are about two orders of magnitude smaller than the values of @xmath34 sampled in ( [ eq : m_higgs ] ) . since @xmath29 grows as @xmath33 decreases and since @xmath35 , this normalisation considerably overestimates the cross section for higgs production . despite the fact that @xmath18 and @xmath14 factors were included in the prediction of the cross section given in @xcite , the result is close to that of @xcite . one reason is that these small survival factors are compensated by the use of a larger value , a much larger qcd coupling ( at low scale , @xmath36 gev ) is to be taken . however , the high - order evolution of the higgs vertex confirms the former choice @xcite . ] of @xmath37 in the @xmath26 vertex . the reliability of the prediction of the diffractive production of the higgs boson can be checked experimentally by measuring the much larger cross section for double - diffractive central production of a pair of high @xmath38 jets @xcite . the amplitude for this process has the same structure as ( [ eq : m_higgs ] ) , with the @xmath26 vertex replaced by the matrix element of the @xmath39 subprocess . the original calculation of dijet production was performed by berera and collins @xcite . the result @xcite , with @xmath18 and @xmath14 factors neglected and normalised to @xmath23 , is about 5600 nb for cdf dijets at the tevatron energy , in contrast with the experimental upper limit of less than 3.7 nb at 95% confidence level @xcite . the huge difference originates from the product of three factors the survival factors @xmath400.1 and @xmath410.2 should be included in the prediction , and the normalisation should be reduced by about a factor of 10 , since ( [ eq : sigma_tot ] ) should be compared to ( [ eq : m_higgs ] ) at much lower @xmath33 . indeed berera and collins @xcite had noted that the survival factors should be computed before their leading order calculation is compared with data . in fact , when account is taken of the survival factors , our perturbative approach @xcite leads to the prediction of about 1 nb @xcite for the exclusive production of dijets corresponding to the kinematics gev jets at the tevatron energy is far from as good as the factor of 2 uncertainty claimed for dijets of mass @xmath42 at the lhc . the contribution from the low @xmath21 domain is less under control for the cdf kinematics of ref . ] of the cdf dijet search @xcite , which leads to a dijet bound of less than 3.7 nb . let us return to the discussion of the predictions listed in table 1 . since cudell and hernandez @xcite do not include the @xmath18 and @xmath14 survival factors , and apply a @xmath23 normalisation , we may expect that the dijet cross section would be overestimated by a factor of about 1000 . levin @xcite includes estimates of the @xmath18 and @xmath14 factors and , following his prescription , we would expect a dijet cross section of about 1000 nb , which is still much larger than the experimental limit . there is no simple way of using these dijet overshoot factors to correct the predictions for higgs production given in refs . we can not simply scale down the predictions by dividing by the overshoot factors . the correction factor has , first , an energy dependence arising from the effective gluon density normalised to @xmath23 and , second , due to the energy dependence of the soft survival factor @xmath18 . moreover , the qcd radiative effects described by the @xmath20 factor depend strongly on the hard scale , and are quite different for dijet production , with jets of @xmath4310 gev , and higgs production with scale @xmath44 gev . instead of fixing the normalisation of the prediction for exclusive higgs production by using @xmath23 , a more reliable method is to use the gluon density given by global parton analyses and to include the sudakov - like survival factor @xmath45 inside the loop integral over @xmath21 in ( [ eq : m_higgs ] ) @xcite . this factor provides the infrared stability of the integral , while the known gluon distribution fixes the normalisation . more recently , the method has been further improved @xcite . first , the skewed effect is included ( using the prescription of refs . @xcite ) , that is the effect due to unequal longitudinal momentum fractions carried by the left and right @xmath46 channel gluons in fig . 1(a ) : explicitly , we have @xmath47 . second , the nlo corrections to the @xmath26 vertex , and the next - to - leading correction to the double - logarithmic @xmath20 factor ( that is the single log term in @xmath20 ) , are included is almost ` at rest ' and practically does not radiate . thus , the qcd radiation is associated with the hard @xmath26 subprocess . ] . this is the method used for obtaining the numbers quoted for the third entry @xcite in table 1 . the most delicate point , in the prediction of the cross section for diffractive higgs production , is the calculation of the probability , @xmath18 , that the rapidity gaps survive the soft rescattering . @xmath18 can not be determined using perturbative qcd and non - perturbative techniques have to be applied . to improve the accuracy of the prediction of @xmath18 , a detailed analysis - loop insertions in the pomeron trajectory ( to describe better the periphery of the proton ) . ] of all available soft high energy @xmath22 and @xmath48 data was performed @xcite . using the results of this analysis it is possible to compute the soft survival factor @xmath18 for a complete range of diffractive processes . the factors for higgs production are given in refs . @xcite . for exclusive higgs production at the lhc the soft survival factor @xmath18 is found to be 0.02 . after all the above effects are included , the uncertainty in the prediction of the cross section , @xmath49 fb , is estimated to be about a factor of two @xcite . if we allow the protons to dissociate , but still keep the rapidity gaps on either side of the produced higgs boson , then we enlarge the cross section by a factor of 310 , depending on the range of masses allowed for the dissociation @xcite . in addition to the larger available phase space for inclusive kinematics , also the gap survival factor is larger ; in fact using the formalism of ref . @xcite we find @xmath50 at the lhc , while for exclusive and central inelastic production we have @xmath51 . however , we lose all the advantages of exclusive double - diffractive higgs production . in particular , we lose the good missing mass resolution provided by the proton tagger , the equality @xmath52 from the @xmath5 decay , and the suppression of the @xmath8 qcd background and of the pile - up events . we therefore do not discuss this process any further here . so far we have considered processes where there are no secondaries accompanying higgs production in the central rapidity region . by ` central inelastic higgs production ' we mean that secondaries are allowed in some central rapidity interval . two contributions to the process are sketched in fig . 2 . as we shall see in a moment , we may call diagrams 2(a ) and 2(b ) lower and higher order @xmath37 contributions respectively . in fact , much attention is paid in the literature to higgs production in pomeron pomeron inelastic collisions , fig . 2(b ) , which , in our notation , corresponds to the higher order @xmath37 contribution to central inelastic production . so we discuss this first . the cross section for higgs production by pomeron pomeron collisions is larger than for exclusive diffractive production , but still much smaller than that for the normal inclusive production , @xmath53 . the expected signal - to - background ratio is practically the same as for normal inclusive production but at a lower energy , corresponding to the pomeron pomeron energy as measured by the missing mass method . we have effectively degraded the lhc energy down to energies comparable to the tevatron ! of course , the luminosity of the lhc is larger than that of the tevatron . however , the effective pomeron pomeron luminosity contains its own small factors . the only advantage - tagging may be easier due to the lower mean multiplicity of soft secondaries , see also @xcite . ] of higgs production by pomeron pomeron inelastic collisions , in comparison to normal inelastic production , at the lhc , is the possibility to use proton taggers to avoid pile - up problems ( associated with multiple interactions in each bunch crossing ) . usually the cross section for central inelastic production is estimated using the factorization hypothesis , la ingleman schlein @xcite . to account for the probability @xmath18 that the rapidity gaps survive the soft rescattering ( which violates factorization ) , the predictions are normalised to the observed rate of central inelastic double - diffractive dijet production at the tevatron @xcite . from a qcd viewpoint , the soft pomeron pomeron interaction , fig . 2(b ) , should be regarded as fig . 2(c ) where the soft pomerons are replaced by ( low nussinov ) two - gluon exchange . we note that fig . 2(c ) contains an extra factor of @xmath37 , as compared to fig . of course , this coupling occurs at low scale , but nevertheless we should not be surprised when we find that the contribution of pomeron pomeron collisions , fig . 2(c ) , is less than that of central inelastic production , fig . for example , in ref . @xcite the cross section corresponding to fig . 2(b , c ) was calculated using the h1 parameterisation of the pomeron flux and structure function @xcite . in the absence of dijet data at the lhc energy , the lhc prediction of @xcite is presented without accounting for the @xmath18 factor . if we multiply this result by the same @xmath54 as in refs . @xcite we obtain @xmath55 fb , which is an order of magnitude smaller than 50 fb the cross section corresponding to fig . central inelastic production , fig.1(c ) , may be regarded as higher - order qcd radiative corrections to exclusive higgs central production ( fig . allowing qcd radiation , in a central rapidity interval around the higgs boson , increases the probability of gap survival , but weakens the potential of the @xmath20 factor to provide infrared convergence of the loop integral over @xmath21 . the cross section is increased , with the extra contribution coming from the low @xmath21 region . in ref . @xcite , results were focused on central inelastic production allowing radiation only in a relatively small central rapidity interval , @xmath56 . in this case , the residual @xmath20 factor is still able to ensure infrared convergence of the loop integral . if proton taggers are installed , then the mass of the central system ( that is the higgs plus accompanying radiation ) can be measured by the missing mass method . for the large masses , up to @xmath57 tev that were considered in ref . @xcite , the @xmath20 factor approaches unity and almost any qcd radiation is allowed . in these circumstances there is no convergence of the @xmath21 integral , and the only possibility is to normalise the prediction to @xmath23 , recall ( [ eq : m_higgs ] ) and ( [ eq : sigma_tot ] ) . the typical values of @xmath33 sampled in ( [ eq : m_higgs ] ) are @xmath58 at the lhc . to evaluate ( [ eq : sigma_tot ] ) in a comparable @xmath33 domain , we use the value of @xmath23 at a much lower ( cern isr ) energy . based on this normalisation , and including a soft survival factor @xmath54 , we predict a central inelastic cross section of 50 fb , which is to be compared to the 320 fb predicted in ref . strictly speaking , the 320 fb in @xcite was calculated for pomeron pomeron inelastic collisions , fig . 2(b ) . for comparison , if for this latter process we were to use the donnachie landshoff @xcite parameterization for the pomeron flux , the pomeron structure function as measured by @xcite , and the known soft survival factor @xmath18 , we would obtain 1.7 fb at the lhc . however , in @xcite the cross section was normalized using the cdf dijet data @xcite at the tevatron , for which the dominant contribution comes from central inelastic diagrams of the type fig . 2(a ) ; so comparison with 50 fb is more relevant . note that the dijet mass distributions are driven ( modulo detector effects ) by the logarithmic structure of the available longitudinal phase space and by the value of the pomeron intercept , and so lead to a similar mass distribution for figs . 2(a ) and 2(b ) . the residual discrepancy between 320 fb and 50 fb may be traced , first , to the fact that the same gap survival factor , @xmath18 , is assumed in @xcite for lhc and tevatron energies , whereas it is expected @xcite that with collider energy reflects the rise of the total interaction cross section , and is in agreement with the d0 and cdf data for the production of jets separated by rapidity gaps , measured at 630 and 1800 gev @xcite . ] @xmath59 secondly , a smaller slope , @xmath60 , is used for higgs , as compared to dijet , production ; see eq . ( 1 ) of @xcite . neglecting the pomeron slope , @xmath61 , the cross section is proportional to @xmath62 . finally , at the tevatron energy , an extra contribution to fig . 2(b ) comes from reggeon reggeon and pomeron reggeon exchange interactions . allowing for all these effects would decrease the predicted higgs cross section of 320 fb by about a factor in the region of 510 , bringing the cross section of ref . @xcite into general agreement with our central inelastic prediction at the lhc . for the tevatron energy , instead of the number given in ref . @xcite we have entered the later prediction of ref . @xcite . for the central inelastic configuration , it was claimed in @xcite that , by tagging the outgoing protons , and by measuring the jets accompanying the higgs , it is possible to obtain a good missing mass resolution for the higgs . unfortunately this is only true for a centrally produced system of @xmath63 close to @xmath64 , which corresponds to a very small fraction of the events , comparable to the number for exclusive production . moreover , for the reasons listed above , the cross section was overestimated . the pomeron pomeron approach of cox et al . @xcite is close to that of boonekamp et al . the main difference is that , instead of using a soft pomeron intercept @xmath65 , a larger intercept @xmath66 was used , as given by the h1 diffractive deep inelastic data . again the prediction is normalised to the cdf dijet data @xcite . therefore the prediction at the tevatron energy is reasonable . cox et al . @xcite use the same parameters for the higgs and dijet production amplitudes . moreover , they use the h1 analysis of diffractive data to specify the flux and the gluon structure of the pomeron . they find that their normalisation is equivalent to a soft survival factor of @xmath67 at the tevatron . the theoretical expectation for @xmath18 is about 0.05 . this implies that a significant part of the cross sections must come from the larger fig . 2(a ) contributions , rather than fig . 2(b ) , to compensate for the smaller value of @xmath18 . unlike all the previous approaches , the predictions of the soft colour interaction ( sci ) model of enberg et al . @xcite are obtained from monte carlo simulations , rather than from an analytic approach . the model assumptions on soft interaction are implemented in pythia @xcite and embody the possibility of soft spectator rescattering and initial state qcd radiation . the sci model effectively incorporates the @xmath18 and @xmath14 survival factors generated within the framework of the pythia monte carlo @xcite . rapidity gaps are produced in the model @xcite by additional soft colour interactions in the final state , which are contrived to screen the colour flow across the gaps . the strength of these extra soft colour interactions was tuned to reproduce the diffractive deep inelastic data obtained at hera . it was demonstrated that the model , with the same parameters , describes reasonably well the single diffractive processes observed at the tevatron . however , the generator was created to simulate _ inelastic _ processes . it operates by starting from the hard subprocess and generates the parton showers by backward evolution . the generator never accounts for the important coherence between different parton showers , nor for the colourless nature of the initial particles . the incoming protons are just considered as a system of coloured partons and only the overall colour charge is conserved . as a consequence , the probability not to emit additional secondary jets ( and so to reproduce an exclusive process ) turns out to be negligibly small . in particular , such a generator is unable to reproduce the elastic cross section . originally these generators create many secondary minijets at the parton shower stage and the probability to screen all these minijets by colour interchange is extremely low . such generators were not constructed to reproduce exclusive processes , where the colour coherence effects and colourless nature of the incoming hadrons are important . for this reason we believe that the extremely low limit for the exclusive @xmath6 cross section , which would follow from such an approach , would not be trustworthy . it is informative to note that , in our perturbative qcd approach @xcite , the effective pomeron or two - gluon exchange has relatively compact transverse size . the sudakov - like @xmath20 factor occurs inside the loop integral over @xmath21 and , in this way , the large - size ( small @xmath21 ) component of the pomeron is strongly suppressed by qcd radiative effects . when the two - gluon system forms a large - size colour - dipole it emits numerous secondary gluon jets which completely fill the rapidity gap . there is a vanishing small probability @xmath14 for the gap to survive such emissions . the main contribution to the loop integral comes from relatively large @xmath21 in the region of the saddle point @xmath68 . the value of @xmath68 grows with both @xmath64 and the collider energy @xmath69 . for a higgs of mass @xmath70 gev produced at the lhc , the transverse size of the exchange is @xmath71 fm . on the contrary , in the approaches of references @xcite , a soft large - size pomeron is exchanged across the rapidity gaps with transverse size @xmath72 fm . this could cause a much stronger sudakov suppression if it were to be calculated by perturbative qcd . another consequence of the small size of the perturbative pomeron concerns the validity of the @xmath9 selection rule for the semi - forward hard diffractive production amplitudes . recall that this rule plays a crucial role in the suppression of the qcd background selection rule is still valid , and suppresses the @xmath8 background , even beyond leading order , arising from events where the @xmath8 pair is accompanied by one or more soft gluons @xcite . hence the qcd - induced @xmath8 background is expected to be suppressed both for the exclusive process and for low mass central - inelastic production where the missing mass @xmath73 is close to @xmath64 . ] @xcite . in the exact forward direction , @xmath9 by virtue of angular momentum conservation . however , violation of this rule can come from orbital angular momenta , @xmath74 , where @xmath75 is the transverse momentum of the leading proton and @xmath76 is the transverse size of the pomeron . therefore the admixture of the @xmath77 state is strongly suppressed for the small - size pomeron - exchange occurring in the exclusive amplitude @xcite . on the other hand , for c - inelastic production , where the @xmath20 factor becomes inactive and we deal with a large - size pomeron , we lose the @xmath9 selection rule and , as a result , have a much larger background . the same is true for monte - carlo - based models . the soft colour interaction , which screens the colour across the gap , takes place at large distances and therefore we have no @xmath9 selection rule . so the expected signal - to - background ratio is small . we compiled a representative range of different predictions of the cross sections for diffractive production of a higgs boson of mass about 120 gev at the tevatron and lhc . we critically compared the wide range of predictions and explained the origin of the differences . in summary , the wide spread of predictions occurs either because different processes have been considered or because important effects have been neglected . the cross sections for inclusive and central inelastic diffractive higgs production are larger than for exclusive production . however , for these non - exclusive processes it is hard to suppress the qcd @xmath8 background and the signal - to - background ratio is small . second , we can not improve significantly the accuracy of the measurement of the mass of the higgs boson by tagging the forward protons and measuring the missing mass . on the other hand , the cross section for exclusive diffractive production is known with sufficient accuracy to be sure that this channel can be used to play an important role in higgs detection via @xmath5 at the lhc , provided that forward proton taggers are installed . the mass of the higgs could then be accurately measured by the missing - mass method , @xmath78 gev @xcite . moreover , the leading order @xmath8 background is strongly suppressed by a @xmath9 selection rule . details of the calculation of the @xmath79 exclusive higgs production cross section are given in ref . the cross section is predicted to be 3 fb at the lhc , with a factor of two uncertainty @xcite . the main sources of the @xmath8 background are , at leading order , caused by gluon jets being misidentified as a @xmath8 pair , by a @xmath80 admixture due to non - forward protons and by a @xmath9 contribution arising from @xmath81 . also there is a background contribution from @xmath82 events in which the emitted gluon is approximately collinear with a @xmath83 jet . these backgrounds were considered in detail in ref . @xcite , leading to a signal - to - background ratio of about 3 . note that in @xcite only the @xmath84 hard subprocess was considered at nlo , and radiation for the spectator , screening gluon was not discussed . however , this latter process is numerically small because of the additional suppression of colour - octet @xmath8 production around @xmath85 ; rotational invariance around the @xmath83 quark direction causes the cross section to be proportional to @xmath86 in the @xmath8 c.m . frame @xcite . we may summarize the exclusive diffractive higgs signal ( @xmath87 with @xmath5 ) by the following example . consider the detection of a higgs of mass 120 gev with an integrated luminosity of 30 fb@xmath88 at the lhc . when account is taken of the efficiencies associated with proton tagging and with the identification of @xmath83 and @xmath89 jets , and allowance is made for the polar angle cuts and the @xmath5 branching ratio , then the original @xmath90 events is reduced to an observable signal of 11 @xmath5 events , with a background of 4 @xcite . we stress that the predicted value of the exclusive cross section can be checked experimentally . all the ingredients , except for the nlo correction to the @xmath26 vertex , are the same for our signal as for exclusive double - diffractive dijet production , @xmath91 , where the dijet system is chosen in the same kinematic domain as the higgs boson , that is @xmath92 @xcite . therefore by observing the larger dijet production rate , we can confirm , or correct , the estimate of the exclusive higgs signal . we thank brian cox , albert de roeck , rikard enberg , jeff forshaw , aliosha kaidalov , genya levin , leif lnnblad , risto orava , robi peschanski , christophe royon , torbjrn sjstrand and michael spira for useful discussions . one of us ( vak ) thanks the leverhulme trust for a fellowship . this work was partially supported by the uk particle physics and astronomy research council , by the russian fund for fundamental research ( grants 01 - 02 - 17095 and 00 - 15 - 96610 ) and by the eu framework tmr programme , contract fmrx - ct98 - 0194 ( dg 12-miht ) . xx a. de roeck , v.a . khoze , a.d . martin , r. orava and m.g . ryskin , hep - ph/0207042 . m.g . albrow and a. rostovtsev , hep - ph/0009336 . cdf collaboration : t. affolder et al . * 84 * ( 2000 ) 5043 . khoze , a.d . martin and m.g . ryskin , phys . lett . * b502 * ( 2001 ) 87 . cox , j.r . forshaw and b. heinemann , b540 * ( 2002 ) 263 . m. boonekamp , r. peschanski and c. royon , phys . * 87 * ( 2001 ) 251806 . m. boonekamp , a. de roeck , r. peschanski and c. royon , hep - 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a critical comparison is made between recent predictions of the cross sections for diffractive higgs production at the tevatron and the lhc . we show that the huge spread of the predictions arises either because different diffractive processes are studied or because important effects are overlooked . exclusive production offers a reliable , viable higgs signal at the lhc provided that proton taggers are installed . plus 2 mm minus 2 mm 23.0 cm 17.0 cm -1.0 in -42pt ippp/02/44 + dcpt/02/88 + 24 september 2002 + * diffractive higgs production : myths and reality * v.a . khoze@xmath0 , a.d . martin@xmath1 and m.g . ryskin@xmath0 + @xmath1 institute for particle physics phenomenology , university of durham , dh1 3le , uk + @xmath2 petersburg nuclear physics institute , gatchina , st . petersburg , 188300 , russia +

in recent years , a large number of investigations has been made into the control of chaotic dynamics , and many techniques have been applied in simulations and experiments ( see , for instance , @xcite for an overview ) . two techniques have been proposed for an _ open - loop _ control of chaos . the first approach is related to vibrational methods , as a scalar periodic perturbation is applied to the chaotic system . usually , the control signals are sinusoidal @xcite or two - mode forces @xcite . recently it has been shown that the method can be improved by use of optimized multimode signals which are more complex @xcite . the second method uses equations of motion and a specific goal dynamics to derive vector control forces . if the goal trajectory is suitably chosen , the chaotic system under control converges to the goal . therefore , this method is often referred to as entrainment control @xcite . in this paper it is shown how entrainment control can be improved with respect to small control forces . for this purpose , a specific property of chaotic systems is exploited : dense unstable periodic orbits ( upos ) . in fact , upos are common goal orbits of most feedback techniques employed for control of chaos ( see , e.g. , @xcite ) . because a upo is a natural , but unstable motion of the system , control forces have to be applied only for transfer to and stabilization of the orbit . in the ideal ( noiseless ) case , the stabilization forces tend to zero once the upo is actually reached . therefore , feedback control of upos can be maintained with very small forces in low noise systems . despite the power of such methods , however , there are situations where one wants to or has to dispense with a feedback from the system . then , open - loop techniques are needed . while there exists a theory for open - loop stabilization of unstable fixed points ( vibrational control , see @xcite ) , a counterpart for stabilization of unstable periodic orbits is still lacking to the author s knowledge . although it is supposed that the scalar periodic perturbation methods mentioned above usually stabilize periodic dynamics in the vicinity of upos , this has not been shown yet explicitly . the underlying mechanism , possibly some resonance phenomenon , as well as the exact final dynamics are still unknown . this is different for entrainment control , where the appropriate dynamics is given _ a priori _ , and which is described in the following . we start with a nonlinear dynamical system in the continuous time domain , which is influenced by a control signal vector . the equations of motion are supposed to be known , according to the ordinary differential equation @xmath0 with the system state @xmath1 in the state space @xmath2 , the control vector @xmath3 in the control signal space @xmath4 and the vector field @xmath5 . let @xmath6 be a goal dynamics generated by a vector field @xmath7 , @xmath8 , @xmath9 . to introduce the goal dynamics as solutions of eq . ( [ e1 ] ) , we have to apply forces @xmath10 which solve the equation @xmath11 thus , the vector field @xmath12 is simply changed to @xmath7 by the control forces . to do this , however , feedback from the system is necessary , as the actual system state @xmath13 appears in eq . ( [ e3 ] ) . the main point of the entrainment control method is to eliminate @xmath13 by the assumption that the system is already located in the initial goal state @xmath14 when the control is started at @xmath15 . if so , the correct control signal @xmath10 is given by the solution of the equation @xmath16 equation ( [ e4 ] ) can be solved without any system state measurement ; in fact , not even a generating vector field @xmath7 has to be given : it is sufficient to know the goal trajectory itself and its time derivative ( velocity ) for the control time interval , @xmath17}$ ] . there are several conditions that have to be fulfilled to make the entrainment control scheme work . first of all , eq . ( [ e4 ] ) has to be solvable for @xmath10 . this is trivially true for simple vector additive forces : @xmath18 we restrict our discussion to such forces , which are the most common in the literature on entrainment control . however , problems immediately arise if , e.g. , the control space dimension @xmath19 is less than the state space dimension @xmath20 ( a treatment of general control influence with @xmath19=@xmath21 can be found in @xcite ) . the next condition to be satisfied is asymptotic stability of the goal trajectory . while control forces according to eq . ( [ e4 ] ) ensure that the goal trajectory is a solution of the controlled system eq . ( [ e1 ] ) , there is no statement about whether nearby located system states are attracted by it in other words , whether entrainment occurs in a vicinity of @xmath22 according to @xmath23 . even if this is the case , one needs a large basin of attraction ( ideally the whole phase space ) to make the method work for a large set of possible initial states distant from @xmath22 . statements about basins are very hard to find ( compare @xcite ) , but a discussion of stability can more easily be made . for simple vector additive control , eq . ( [ e5 ] ) , we call all points in phase space where all eigenvalues of the jacobian of @xmath24 have negative real part , _ convergent regions _ ( see also @xcite ) . goal trajectories entirely located in convergent regions turn out to be asymptotically stable , if their time derivative @xmath25 is sufficiently bounded . because of the stated stability aspects , goal trajectories are usually chosen to be located in convergent regions . this results in a typical drawback of the method : due to very little overlap of convergent regions and unperturbed chaotic attractor , such a goal dynamics is quite different to the natural system dynamics . consequently , the system is strongly altered by control , and control forces are large . in fact , they have to be of about the magnitude of the velocities appearing in the uncontrolled system in order to pull the movement into convergent regions . to attack this problem , one has to realize that location in convergent regions is not a necessary condition for stability of a goal trajectory . the resulting dynamics of chaotic systems controlled by periodic perturbation methods indeed suggest that stable dynamics can also be achieved in the chaotic attractor region , especially near a upo . control forces are smaller then , as the natural dynamics is only slightly altered . an extreme case would be to consider a upo itself as a goal for entrainment control : we get zero control forces according to eq . ( [ e4 ] ) . however , such a goal is of course unstable . it has to be at least slightly changed to result in a stable one . to this end , a family of deformations of a upo is considered in the following . let @xmath26 denote a known upo of the chaotic system with period @xmath27=@xmath28 . we chose a finite fourier series as a deformation @xmath29 , and also include a linear time transformation by a factor @xmath30 . the family of goal trajectories now reads @xmath31 it is parametrized by @xmath32 and the fourier coefficient vectors @xmath33 , @xmath34 . since @xmath35 has no effect on @xmath22 , a deformation ( or a goal ) is characterized by a total of @xmath36 real numbers . now , we formulate the determination of advantageous deformation parameters that lead to a stable goal trajectory with small control forces as an optimization problem . a real number according to a cost function is assigned to each probed set of parameters . the cost assesses the stability of the chosen goal trajectory ( which is determined numerically ) as well as the magnitude of the resulting control forces : @xmath37 & \mu < 1 , \\ \|{\bf u}(t)\|_{max } + \gamma [ \exp(\mu)-1 ] + c & \mu \ge 1 \end{array } \right.\ ] ] here , @xmath38 is the maximum absolute value of the characteristic ( floquet ) multipliers of the goal orbit . these are well defined , as the goal is periodic , and they are calculated by integration of the variational equations @xcite . instability is indicated by @xmath39 and causes high cost via a large positive penalty term @xmath40 . the cost function further includes an @xmath41 and the maximum norm of the resulting forces . the weight of stability with respect to magnitude of forces can be adjusted by @xmath42 . the global minimum of the cost function in the deformation parameter space corresponds to the best goal trajectory in sense of the chosen balance between stability and small forces . for various reasons , a direct analytic treatment of the given optimization problem is usually not possible : the upo is not known in analytic form , a direct expression for stability of a deformed upo is missing ( the variational equations have to be integrated ) , and the cost function is not continuous . consequently , numerical methods are employed . the upo is represented by a periodic cubic spline interpolation , and @xmath43 is calculated via numerical integration of the variational equations . the optimization is done by a numerical technique that can handle high - dimensional problems with rough and rapidly varying cost functions . for this purpose , the stochastically guided algorithm amebsa from @xcite was chosen ; this is a combination of simulated annealing and the downhill simplex method . in this section , the control of a chaotic lorenz system is demonstrated . the equations with vector additive control read @xmath44 where time dependence of the forces is explicitly written . a given periodic goal trajectory @xmath45}$ ] yields control forces according to eq . ( [ e5 ] ) . parameters of the lorenz system are set to @xmath46 , @xmath47 , and @xmath48 . the resulting chaotic attractor is shown in fig . [ fig1](a ) . an embedded upo which corresponds to @xmath49 in eq . ( [ e7 ] ) is given in fig . [ fig1](b ) by the interrupted line . the deformation is defined by five fourier modes ( @xmath50=5 ) which leads to a 34-dimensional search space for optimization ; @xmath42 is set to 5 , @xmath40 to 100 in eq . ( [ e8 ] ) . the best result of several optimization runs is shown in fig . [ fig1](b ) by the solid line . it is a stable goal trajectory and therefore a stable periodic orbit ( spo ) of the controlled system . the actual values of deformation parameters can be found in tab . [ tab1 ] together with additional data of the upo and the spo . the deformation lies in the range of some percent , and it is plotted in fig . [ fig2](a ) . the resulting forces , shown in fig . [ fig2](b ) , change the vector field of the chaotic system less than about 10% . this is an improvement of more than a magnitude if compared to goals in convergent regions @xcite . numerical tests indicated that the spo is globally asymptotically stable ; the basin is the whole phase space . however , transient times until control is established depend strongly on the initial state , and range from just a few up to a few hundred control periods . a typical behavior is presented in fig . [ fig3 ] . after control is turned on , an intermittent transient appears . finally , the system settles down on the desired goal orbit , which is maintained . it has been shown how open - loop entrainment control in the vicinity of unstable periodic orbits can be realized . the search for suitable goal trajectories has been formulated in terms of an optimization problem with respect to upo deformations . feasibility has been demonstrated in an example , where a chaotic lorenz system has been successfully controlled to an optimized distortion of a upo . the locations of such goal orbits are independent of convergent regions , and thus the required forces are small compared to hitherto used goal dynamics far from the chaotic attractor . .[tab1 ] coordinates @xmath51 give a point of the original upo of the lorenz system , @xmath27 its period . maximum absolute values of the floquet multipliers are given by @xmath43 for the upo and for the optimized deformation ( spo ) . parameters of the spo are the time transformation coefficient @xmath30 and fourier coefficients @xmath52 , @xmath53 . subscripts of numbers indicate a decimal shift , i.e. , @xmath54 stands for @xmath55 . 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( 1990 ) _ phys . rev . lett . _ * 65 * ; jackson , e.a . ( 1990 ) _ physica d _ * 50 * ; shermer , r. _ et al . _ ( 1991 ) _ phys . a _ * 43 * ; breeden , j.l . ( 1994 ) _ phys . a _ * 190*. ott , e. , grebogi , c. , and yorke , y.a . ( 1990 ) controlling chaos , _ phys . * 64 * , 1196 - 1199 . bellman , r.e . , bentsman , j. , and meerkov , s.m . ( 1986 ) vibrational control of nonlinear systems : vibrational stabilizability , _ ieee trans . _ * ac-31 * , 710 - 716 . mettin , r. , hbler , a. , scheeline , a. , and lauterborn , w. ( 1995 ) parametric entrainment control of chaotic systems , _ phys . e _ * 51 * , 4065 - 4075 . jackson , e.a . ( 1991 ) controls of dynamic flows with attractors , _ phys . a. _ * 44 * , 48394853 . parker t.s . and chua , l.o . ( 1989 ) _ practical numerical algorithms for chaotic systems _ , springer - verlag , new york . press , w.h . , teukolsky , s.a . , vetterling , w.t . and flannery , b.r . ( 1992 ) _ numerical recipes in c _ , 2nd ed . , cambridge university press , cambridge

it is demonstrated that improved entrainment control of chaotic systems can maintain periodic goal dynamics near unstable periodic orbits without feedback . the method is based on the optimization of goal trajectories and leads to small open - loop control forces .

let @xmath0 be a domain in the euclidean space @xmath1 , @xmath2 be its boundary and @xmath3 be a differentiable function . a _ critical point _ of @xmath4 is a point in @xmath0 at which the gradient @xmath5 of @xmath4 is the zero vector . the importance of critical points is evident . at an elementary level , they help us to visualize the graph of @xmath4 , since they are some of its notable points ( they are local maximum , minimum , or inflection / saddle points of @xmath4 ) . at a more sophisticated level , if we interpret @xmath4 and @xmath5 as a gravitational , electrostatic or velocity potential and its underlying field of force or flow , the critical points are the positions of equilibrium for the field of force or stagnation points for the flow and give information on the topology of the equipotential lines or of the curves of steepest descent ( or stream lines ) related to @xmath4 . a merely differentiable function can be very complicated . for instance , whitney @xcite constructed a non - constant function of class @xmath6 on the plane with a connected set of critical values ( the images of critical points ) . if we allow enough smoothness , this is no longer possible as morse - sard s lemma informs us : indeed , if @xmath4 is at least of class @xmath7 , the set of its critical values must have zero lebesgue measure and hence the regular values of @xmath4 must be dense in the image of @xmath4 ( see @xcite for a proof ) . when the function @xmath4 is the solution of some partial differential equation , the situation improves . in this survey , we shall consider the four archetypical equations : @xmath8 that is the _ laplace s _ equation , the _ torsional creep _ equation , the _ eigenfunction _ equation and the _ heat _ equation . it should be noticed at this point some important differences between the first and the remaining three equations . one is that the critical points of harmonic functions the solutions of the laplace s equation are always `` saddle points '' as it is suggested by the maximum and minimum principles and the fact that @xmath9 is the sum of the eigenvalues of the hessian matrix @xmath10 . the other three equations instead admit solutions with maximum or minimum points . also , we know that the critical points of a non - constant harmonic function @xmath4 on an open set of @xmath11 are isolated and can be assigned a sort of finite multiplicity , for they are the zeroes of the holomorphic function @xmath12 . by means of the theory of quasi - conformal mappings and generalized analytic functions , this result can be extended to solutions of the elliptic equation @xmath13 ( with suitable smoothness assumptions on the coefficients ) or even to weak solutions of the an elliptic equation in divergence form , @xmath14 even allowing discontinuous coefficients . instead , solutions of the other three equations can show curves of critical points in @xmath11 , as one can be persuaded by looking at the solution of the torsional creep equation in a circular annulus with zero boundary values . these discrepancies extend to any dimension @xmath15 , in the sense that it has been shown that the set of the critical points of a non - constant harmonic function ( or of a solution of an elliptic equation with smooth coefficients modeled on the laplace equation ) has at most locally finite @xmath16-dimensional hausdorff measure , while solutions of equations fashioned on the other three equations have at most locally finite @xmath17-dimensional hausdorff measure . further assumptions on solutions of a partial differential equation , such as their behaviour on the boundary and the shape of the boundary itself , can give more detailed information on the number and location of critical points . in these notes , we shall consider the case of harmonic functions with various boundary behaviors and the solutions @xmath18 , @xmath19 and @xmath20 of the following three problems : @xmath21 @xmath22 @xmath23 where @xmath24 is a given function . we will refer to , , - , as the _ torsional creep problem _ , the _ dirichlet eigenvalue problem _ , and the _ initial - boundary value problem for the heat equation _ , respectively . a typical situation is that considered in theorem [ th : am1b ] : a harmonic function @xmath4 on a planar domain @xmath0 is given together with a vector field @xmath25 on @xmath2 of assigned topological degree @xmath26 ; the number of critical points in @xmath0 then is bounded in terms of @xmath26 , the euler characteristic of @xmath0 and the number of proper connected components of the set @xmath27 ( see theorem [ th : am1b ] for the exact statement ) . we shall also see how this type of theorem has recently been extended to obtain a bound for the number of critical points of the li - tam green s function of a non - compact riemanniann surface of finite type in terms of its genus and the number of its ends . owing to the theory of quasi - conformal mappings , theorem [ th : am1b ] can be extended to solutions of quite general elliptic equations and , thanks to the work of g. alessandrini and co - authors , has found effective applications to the study of inverse problems that have as a common denominator the reconstruction of the coefficients of an elliptic equation in a domain from measurements on the boundary of a set of its solutions . a paradigmatic example is that of electric impedence tomography ( eit ) in which a conductivity @xmath28 is reconstructed , as the coefficient of the elliptic equation @xmath29 from the so - called neumann - to - dirichlet ( or dirichlet - to - neumann ) operator on @xmath2 . in physical terms , an electrical current ( represented by the co - normal derivative @xmath30 ) is applied on @xmath2 generating a potential @xmath4 , that is measured on @xmath2 within a certain error . one wants to reconstruct the conductivity @xmath28 from some of these measurements . roughly speaking , one has to solve for the unknown @xmath28 the first order differential equation @xmath31 once the information about @xmath4 has been extended from @xmath2 to @xmath0 . it is clear that such an equation is singular at the critical points of @xmath4 . thus , it is helpful to know _ a priori _ that @xmath5 does not vanish and this can be done via ( appropriate generalizations of ) theorem [ th : am1b ] by choosing suitable currents on @xmath2 . the possible presence of maximum and/or minimum points for the solutions of , , or - makes the search for an estimate of the number of critical points a difficult task ( even in the planar case ) . in fact , the mere topological information only results in an estimate of the _ signed sum _ of the critical points , the sign depending on whether the relevant critical point is an extremal or saddle point . for example , for the solution of or , we only know that the difference between the number of its ( isolated ) maximum and saddle points ( minimum points are not allowed ) must equal @xmath32 , the _ euler characteristic _ of @xmath0 a morse - type theorem . thus , further assumptions , such as geometric information on @xmath0 , are needed . more information is also necessary even if we consider the case of harmonic functions in dimension @xmath33 . in the author s knowledge , results on the number of critical points of solutions of , , or - reduce to deduction that their solutions admit a critical point if @xmath0 is _ convex_. moreover , the proof of such results is somewhat indirect : the solution is shown to be _ quasi - concave _ indeed , log - concave for the cases of and - , and @xmath34-concave for the case and then its analyticity completes the argument . estimates of the number of critical points when the domain @xmath0 has more complex geometries would be a significant advance . in this survey , we will propose and justify some conjectures . the problem of locating critical points is also an interesting issue . the first work on this subject dates back to gauss @xcite , who proved that the critical points of a complex polynomial are its , if they are multiple , and the equilibrium points of the gravitational field of force generated by particles placed at the zeroes and with masses proportional to the zeroes multiplicities ( see section [ sec : location ] ) . later refinements are due to jensen @xcite and lucas @xcite , but the first treatises on this matter are marden s book @xcite and , primarily , walsh s monograph @xcite that collects most of the results on the number and location of critical points of complex polynomials and harmonic functions known at that date . in general dimension , even for harmonic functions , results are sporadic and rely on explicit formulae or symmetry arguments . two well known questions in this context concern the location of the _ hot spot _ in a heat conductor a hot spot is a point of ( absolute or relative ) maximum temperature in the conductor . the situation described by - corresponds with the case of a _ grounded _ conductor . by some asymptotic analysis , under appropriate assumptions on @xmath24 , one can show that the hot spots _ originate _ from the set of maximum points of the function @xmath35 the distance of @xmath36 from @xmath2 and tend to the maximum points of the unique positive solution of , as @xmath37 . in the case @xmath0 is convex , we have only one hot spot , as already observed . in section [ sec : location ] , we will describe three techniques to locate it ; some of them extend their validity to locate the maximum points of the solutions to and . we will also give an account of what it is known about convex conductors that admit a stationary hot spot ( that is the hot spot does not move with time ) . it has also been considered the case in which the homogeneous dirichlet boundary condition in is replaced by the homogeneous neumann condition : @xmath38 these settings describe the evolution of temperature in an _ insulated _ conductor of given constant initial temperature and has been made popular by a conjecture of j. rauch @xcite that would imply that the hot spot must tend to a boundary point . even if we now know that it is false for a general domain , the conjecture holds true for certain planar convex domains but it is still standing for unrestrained convex domains . the remainder of the paper is divided into three sections that reflect the aforementioned features . in section [ sec : harmonic ] , we shall describe the local properties of critical points of harmonic functions or , more generally , of solutions of elliptic equations , that lead to estimates of the size of critical sets . in section [ sec : number ] , we shall focus on bounds for the number of critical points that depend on the boundary behavior of the relevant solutions and/or the geometry of @xmath2 . finally , in section [ sec : location ] , we shall address the problem of locating the possible critical points . as customary for a survey , our presentation will stress ideas rather than proofs . this paper is dedicated with sincere gratitude to giovanni alessandrini an inspiring mentor , a supportive colleague and a genuine friend on the occasion of his @xmath39 birthday . much of the material presented here was either inspired by his ideas or actually carried out in his research with the author . a harmonic function in a domain @xmath0 is a solution of the laplace s equation @xmath40 it is well known that harmonic functions are analytic , so there is no difficulty to define their critical points or the _ critical set _ @xmath41 before getting into the heart of the matter , we present a relevant example . in dimension two , we have a powerful tool since we know that a harmonic function is ( locally ) the real or imaginary part of a holomorphic function . this remark provides our imagination with a reach set of examples on which we can speculate . for instance , the harmonic function @xmath42 , \ n\in{\mathbb{n}},\ ] ] already gives some insight on the properties of harmonic functions we are interested in . in fact , we have that @xmath43 thus , @xmath4 has only one distinct critical point , @xmath44 , but it is more convenient to say that @xmath4 has @xmath45 critical points at @xmath44 or that @xmath44 is a critical point with _ multiplicity _ @xmath46 with @xmath47 . by virtue of this choice , we can give a topological meaning to @xmath46 . to see that , it is advantageous to represent @xmath4 in polar coordinates : @xmath48 here , @xmath49 and @xmath50 is the principal branch of @xmath51 , that is we are assuming that @xmath52 . thus , the topological meaning of @xmath46 is manifest when we look at the level `` curve '' @xmath53 : it is made of @xmath54 straight lines passing through the critical point @xmath44 , divides the plane into @xmath55 _ cones _ ( angles ) , each of amplitude @xmath56 and the sign of @xmath4 changes across those lines ( see fig . one can also show that the signed angle @xmath57 formed by @xmath5 and the direction of the positive real semi - axis , since it equals @xmath58 , increases by @xmath59 while @xmath60 makes a complete loop clockwise around @xmath44 ; thus , @xmath46 is a sort of _ winding number _ for @xmath5 . [ fig : power ] at the critical point @xmath44 ; @xmath4 changes sign from positive to negative at dashed lines and from negative to positive at solid lines.,title="fig : " ] the critical set of a homogeneous polynomial @xmath61 is a cone in @xmath1 . moreover , if @xmath62 is also harmonic ( and non - constant ) one can show that @xmath63 if @xmath64 and @xmath4 is any harmonic function , the picture is similar to that outlined in the example . in fact , we can again consider the `` complex gradient '' of @xmath4 , @xmath65 and observe that @xmath66 is _ holomorphic _ in @xmath0 , since @xmath67 , and hence analytic . thus , the zeroes of @xmath66 ( and hence the critical points of @xmath4 ) in @xmath0 are _ isolated _ and have _ finite multiplicity_. if @xmath68 is a zero with multiplicity @xmath46 of @xmath66 , then we can write that @xmath69 where @xmath20 is holomorphic in @xmath0 and @xmath70 . on the other hand , we also know that @xmath4 is locally the real part of a holomorphic function @xmath71 and hence , since @xmath72 , by an obvious normalization , it is not difficult to infer that @xmath73 where @xmath74 and @xmath75 is holomorphic and @xmath76 . passing to polar coordinates by @xmath77 tells us that @xmath78 where @xmath79 . thus , we have that @xmath80 and hence , modulo a rotation by the angle @xmath81 , in a small neighborhood of @xmath68 , we can say that the critical level curve @xmath82 is very similar to that described in the example with @xmath83 replaced by @xmath68 . in particular , it is made of @xmath84 simple curves passing through @xmath68 and any two adjacent curves meet at @xmath68 with an angle that equals @xmath56 ( see fig . [ fig : harmonic ] . the curves meet with equal angles at the critical point.,title="fig : " ] if @xmath33 , similarly , a harmonic function can be approximated near a zero @xmath83 by a homogeneous harmonic polynomial of some degree @xmath84 : @xmath85 however , the structure of the set @xmath86 depends on whether @xmath83 is an isolated critical point of @xmath87 or not . in fact , if @xmath83 is not isolated , then @xmath86 and @xmath88 could not be diffeomorphic in general , as shown by the harmonic function @xmath89 indeed , if @xmath90 , @xmath91 is the @xmath60-axis , while @xmath86 is made of @xmath92 isolated points ( @xcite ) . these arguments can be repeated with some necessary modifications for solutions of uniformly elliptic equations of the type , where the variable coefficients @xmath93 are lipschitz continuous and @xmath94 are bounded measurable on @xmath0 and the uniform ellipticity is assumed to take the following form : @xmath95 now , the classical theory of _ quasi - conformal _ mappings comes in our aid ( see @xcite and also @xcite ) . by the _ uniformization theorem _ ( see @xcite ) , there exists a quasi - conformal mapping @xmath96 , satisfying the equation @xmath97 such that the function @xmath98 defined by @xmath99 satisfies the equation @xmath100 where @xmath62 and @xmath101 are real - valued functions depending on the coefficients in and are essentially bounded on @xmath102 . notice that , since the composition of @xmath103 with a conformal mapping is still quasi - conformal , if it is convenient , by the riemann mapping theorem , we can choose @xmath102 to be the unit disk @xmath104 . by setting @xmath105 , simple computations give that @xmath106 where @xmath107 is essentially bounded . this equation tells us that @xmath108 is a _ pseudo - analytic _ function for which the following _ similarity principle _ holds ( see @xcite ) : there exist two functions , @xmath109 holomorphic in @xmath104 and @xmath110 hlder continuous on the whole @xmath111 , such that @xmath112 owing to , it is clear that the critical points of @xmath4 , by means of the mapping @xmath113 , correspond to the zeroes of @xmath114 or , which is the same , of @xmath109 and hence we can claim that they are isolated and have a finite multiplicity . this analysis can be further extended if the coefficients @xmath115 and @xmath116 are zero , that is for the solutions of . in this case , we can even assume that the coefficients @xmath93 be merely essentially bounded on @xmath0 , provided that we agree that @xmath4 is a non - constant _ weak _ solution of . it is well known that , with these assumptions , solutions of are in general only hlder continuous and the usual definition of critical point is no longer possible . however , in @xcite we got around this difficulty by introducing a different notion of critical point , that is still consistent with the topological structure of the level curves of @xmath4 at its critical values . to see this , we look for a surrogate of the harmonic conjugate for @xmath4 . in fact , implies that the @xmath117-form @xmath118 is closed ( in the weak sense ) in @xmath0 and hence , thanks to the theory developed in @xcite , we can find a so - called _ stream function _ @xmath119 whose differential @xmath120 equals @xmath57 , in analogy with the theory of gas dynamics ( see @xcite ) . thus , in analogy with what we have done in subsection [ sub : harmonic ] , we find out that the function @xmath121 satisfies the equation @xmath122 where @xmath123 and @xmath124 is a lower bound for the smaller eigenvalue of the matrix of the coefficients : @xmath125 the fact that @xmath126 implies that @xmath71 is a _ quasi - regular _ mapping that can be factored as @xmath127 where @xmath128 is a _ quasi - conformal homeomorphism _ and @xmath129 is holomorphic in @xmath104 ( see @xcite ) . therefore , the following _ representation formula _ holds : @xmath130 where @xmath98 is the real part of @xmath129 . [ fig : elliptic ] . at that point , any two consecutive curves meet with positive angles , possibly not equal to one another.,title="fig : " ] this formula informs us that the level curves of @xmath4 can possibly be distorted by the homeomorphism @xmath131 , but preserve the topological structure of a harmonic function ( see fig . this remark gives grounds to the definition introduced in @xcite : @xmath132 is a _ geometric critical point _ of @xmath4 if the gradient of @xmath98 vanishes at @xmath133 . in particular , geometric critical points are isolated and can be classified by a sort of multiplicity . a similar local analysis can be replicated when @xmath64 for quasilinear equations of type @xmath134 where @xmath135 and @xmath136 for every @xmath137 and some constants @xmath138 and @xmath139 . [ fig : degenerate ] at a critical value.,title="fig : " ] these equations can be even degenerate , such as the @xmath140-laplace equation with @xmath141 ( see @xcite ) . it is worth mentioning that also the case in which @xmath142 , where @xmath143 is increasing , with @xmath144 , and superlinear and growing polynomially at infinity ( e.g. @xmath145 ) , has been studied in @xcite . in this case the function @xmath146 vanishes at @xmath147 and it turns out that the critical points of a solution @xmath4 ( if any ) are _ never _ isolated ( fig . 2.4 ) . as already observed , critical points of harmonic functions in dimension @xmath33 may not be isolated . besides the example given in section [ sub : harmonic ] , another concrete example is given by the function @xmath149 where @xmath150 is the first bessel function : the gradient of @xmath4 vanishes at the origin and on the circles on the plane @xmath44 having radii equal to the zeroes of the second bessel function @xmath151 . it is clear that a region @xmath0 can be found such that @xmath152 is a _ bounded continuum_. nevertheless , it can be proved that @xmath86 always has locally finite @xmath16-dimensional hausdorff measure @xmath153 . a nice argument to see this was suggested to me by d. peralta - salas @xcite . if @xmath4 is a non - constant harmonic function and we suppose that @xmath86 has dimension @xmath154 , then the general theory of analytic sets implies that there is an open and dense subset of @xmath86 which is an analytic sub - manifold ( see @xcite ) . since @xmath4 is constant on a connected component of the critical set , it is constant on @xmath86 , and its gradient vanishes . thus , by the cauchy - kowalewski theorem @xmath4 must be constant in a neighborhood of @xmath86 , and hence everywhere by unique continuation . of course , this argument would also work for solutions of an elliptic equation of type @xmath155 with analytic coefficients . when the coefficients @xmath156 in are of class @xmath157 , the result has been proved in @xcite ( see also @xcite ) : if @xmath4 is a non - constant solution of , then for any compact subset @xmath158 of @xmath0 it holds that @xmath159 the proof is based on an estimate similar to for the complex dimension of the singular set in @xmath160 of the complexification of the polynomial @xmath87 in the approximation . the same result does not hold for solutions of equation @xmath161 with @xmath162 . for instance the gradient of the first laplace - dirichlet eigenfunction for a spherical annulus vanishes exactly on a @xmath17-dimensional sphere . a more general counterexample is the following ( see ( * ? ? ? * remark p. 362 ) ) : let @xmath163 be of class @xmath164 and with non - vanishing gradient in the unit ball @xmath143 in @xmath1 ; the function @xmath165 satisfies the equation @xmath166 we have that @xmath167 and it has been proved that any closed subset of @xmath1 can be the zero set of a function of class @xmath164 ( see @xcite ) . however , once is settled , it is rather easy to show that the _ singular set _ @xmath168 of a non - constant solution of also has locally finite @xmath16-dimensional hausdorff measure ( * ? ? ? * corollary 1.1 ) . this can be done by a trick , since around any point in @xmath0 there always exists a _ solution @xmath169 of and it turns out that the function @xmath170 is a solution of an equation like and that @xmath171 . in particular the set of critical points on the nodal line of an eigenfunction of the laplace operator has locally finite @xmath16-dimensional hausdorff measure . nevertheless , for a solution of the set @xmath172 can be very complicated , as a simple example in @xcite ) shows : the function @xmath173 , where @xmath71 is a smooth function with @xmath174 that vanishes exactly on an arbitrary given closed subset @xmath158 of @xmath175 , is a solution of @xmath176 heuristically , as in the @xmath177-dimensional case , the proof of is essentially based on the observation that , by taylor s expansion , a harmonic function @xmath4 can be approximated near any of its zeroes by a homogeneous harmonic polynomial @xmath178 of degree @xmath179 . technically , the authors use the fact that the complex dimension of the critical set in @xmath160 of the complexified polynomial @xmath180 is bounded by @xmath181 . a @xmath164-perturbation argument and an inequality from geometric measure theory then yield that , near a zero of @xmath4 , the @xmath153-measure of @xmath86 can be bounded in terms of @xmath182 and @xmath46 . the extension of these arguments to the case of a solution of is then straightforward . recently in @xcite , has been extended to the case of solutions of elliptic equations of type @xmath183 where the coefficients @xmath184 and @xmath185 are assumed to be lipschitz continuous and essentially bounded , respectively . a more detailed description of the critical set @xmath86 of a harmonic function @xmath4 can be obtained if we assume to have some information on its behavior on the boundary @xmath2 of @xmath0 . while in section [ sec : harmonic ] the focus was on a qualitative description of the set @xmath86 , here we are concerned with establishing bounds on the number of critical points . an exact counting formula is given by the following result . [ th : am1 ] let @xmath0 be a bounded domain in the plane and let @xmath186 where @xmath187 are simple closed curves of class @xmath188 . consider a harmonic function @xmath189 that satisfies the dirichlet boundary condition @xmath190 where @xmath191 are given real numbers , not all equal . then @xmath4 has in @xmath192 a finite number of critical points @xmath193 ; if @xmath194 @xmath195 denote their multiplicities , then the following identity holds : @xmath196 [ fig : capacitor ] : the domain @xmath0 has @xmath197 holes ; @xmath4 has exactly @xmath177 critical points ; dashed and dotted are the level curves at critical values.,title="fig : " ] thanks to the analysis presented in subsection [ subsec : elliptic ] , this theorem still holds if we replace the laplace equation in by the general elliptic equation . in fact , modulo a suitable change of variables , we can use with @xmath198 on the boundary . the function considered in theorem [ th : am1 ] can be interpreted in physical terms as the potential in an electrical capacitor and hence its critical points are the points of equilibrium of the electrical field ( fig . the proof of theorem [ th : am1 ] relies on the fact that the critical points of @xmath4 are the zeroes of the holomorphic function @xmath199 and hence they can be counted with their multiplicities by applying the classical _ argument principle _ to @xmath71 with some necessary modifications . the important remark is that , since the boundary components are level curves for @xmath4 , the gradient of @xmath4 is parallel on them to the ( exterior ) unit normal @xmath200 to the boundary , and hence @xmath201 . thus , the situation is clear if @xmath4 does not have critical points on @xmath2 : the argument principle gives at once that @xmath202=j-2 , \end{gathered}\ ] ] where by @xmath203 we intend the _ increment _ of an angle on an oriented curve @xmath204 and by @xmath205 we mean that @xmath2 is trodden in such a way that @xmath0 is on the left - hand side . if @xmath2 contains critical points , we must first prove that they are also isolated . this is done , by observing that , if @xmath68 is a critical point belonging to some component @xmath206 , since @xmath4 is constant on @xmath206 , by the _ schwarz s reflection principle _ ( modulo a conformal transformation of @xmath0 ) , @xmath4 can be extended to a function @xmath207 which is harmonic in a whole neighborhood of @xmath68 . thus , @xmath68 is a zero of the holomorphic function @xmath208 and hence is isolated and with finite multiplicity . moreover , the increment of @xmath209 on an oriented closed simple curve @xmath204 around @xmath68 is exactly twice as much as that of @xmath210 on the part of @xmath204 inside @xmath0 . this explains the second addendum in . notice that condition can be re - written as @xmath211 where @xmath212 is the _ tangential _ unit vector field on @xmath2 . we can not hope to obtain an identity as if @xmath213 is not constant . however , a bound for the number of critical points of a harmonic function ( or a solution of ) can be derived in a quite general setting . in what follows , we assume that @xmath0 is as in theorem [ th : am1 ] and that @xmath214 denotes a ( unitary ) vector field of class @xmath215 of given topological degree @xmath26 , that can be defined as @xmath216 also , we will use the following definitions : a. if @xmath217 is a decomposition of @xmath2 into two disjoint subsets such that @xmath218 on @xmath219 and @xmath220 on @xmath221 , we denote by @xmath222 the number of connected components of @xmath219 which are _ proper subsets _ of some component @xmath206 of @xmath2 and set : @xmath223 b. if @xmath224 , by @xmath225 we denote the number of connected components of @xmath226 which are _ proper subsets _ of some component @xmath206 of @xmath2 . notice that in ( i ) the definition of @xmath227 does not change if we replace @xmath219 by @xmath221 . [ th : am1b ] let @xmath189 be harmonic in @xmath0 and denote by @xmath228 the multiplicity of a zero @xmath229 of @xmath199 . a. if @xmath227 is finite and @xmath4 has no critical point in @xmath2 , then @xmath230 b. if @xmath231 is finite , then @xmath232-d,\ ] ] where @xmath233 $ ] is the greatest integer @xmath234 . this theorem is clearly less sharp than theorem [ th : am1 ] since , in that setting , it does not give information about critical points on the boundary . however , it gives the same information on the number of interior critical points , since in the setting of theorem [ th : am1 ] the degree of the field @xmath18 on @xmath205 equals @xmath235 and @xmath236 . [ fig : oblique ] . here , @xmath237 ; @xmath238 ; @xmath239 if @xmath240 or @xmath18 ; @xmath241 if @xmath242 and the origin is in @xmath0 ; @xmath243 if @xmath244 or @xmath245 . , title="fig : " ] the possibility of choosing the vector field @xmath25 arbitrarily makes theorem [ th : am1b ] a very flexible tool : for instance , the number of critical points in @xmath0 can be estimated from information on the tangential , normal , co - normal , partial , or radial ( with respect to some origin ) derivatives ( see fig . 3.2 ) . as an illustration , it says that in a domain topologically equivalent to a disk , in order to have @xmath84 interior critical point the normal ( or tangential , or co - normal ) derivative of a harmonic function must change sign at least @xmath246 times and a partial derivative at least @xmath84 times . thus , theorem [ th : am1b ] helps to choose neumann data that insures the absence of critical points in @xmath0 . for this reason , in its general form for elliptic operators , it has been useful in the study of eit and other similar inverse problems . we give a sketch of the proof of ( a ) of theorem [ th : am1b ] , that hinges on the simple fact that , if we set @xmath247 and @xmath248 , then @xmath249 hence , if @xmath250 is a minimizing decomposition of @xmath2 as in ( i ) , then @xmath251 thus , two occurrences must be checked . if a component @xmath206 is contained in @xmath219 or @xmath252 , then @xmath253 that implies that @xmath57 and @xmath254 must have the same increment , being the right - hand side an integer . if @xmath206 contains points of both @xmath219 and @xmath221 , instead , if @xmath255 and @xmath256 are two consecutive components on @xmath206 , then @xmath257 therefore , if @xmath258 is the number of connected components of @xmath259 ( which equals that of @xmath259 ) , then @xmath260 and hence @xmath261 * the obstacle problem . * an estimate similar to that of theorem [ th : am1b ] has been obtained also for @xmath64 by sakaguchi @xcite for the _ obstacle problem_. let @xmath0 be bounded and simply connected and let @xmath262 be a given function in @xmath263 the obstacle . there exists a unique solution @xmath264 such that @xmath265 in @xmath0 of the obstacle problem @xmath266 it turns out that @xmath267 and @xmath4 is harmonic outside of the _ contact set _ @xmath268 . in @xcite it is proved that , if the number of connected components of local maximum points of @xmath262 equals @xmath269 , then @xmath270 with the usual meaning for @xmath271 and @xmath272 . in @xcite , this result is also shown to hold for a more general class of quasi - linear equations . the proof of this result is based on the analysis of the level sets of @xmath4 at critical values , in the wake of @xcite and @xcite . topological bounds as in theorems [ th : am1 ] or [ th : am1b ] are not possible in dimension greater than @xmath177 . we give two examples . [ fig : torus ] must have a critical point near the center of @xmath143 and one between the ends of @xmath273.,title="fig : " ] * the broken doughnut in a ball . * the first is an adaptation of one contained in @xcite and reproduces the situation of theorem [ th : am1 ] ( see fig . let @xmath143 be the unit ball centered at the origin in @xmath274 and @xmath273 an open torus with center of symmetry at the origin and such that @xmath275 . we can always choose coordinate axes in such a way that the @xmath276-axis is the axis of revolution for @xmath273 and hence define the set @xmath277 . @xmath278 is simply connected and tends to @xmath273 as @xmath279 . now , set @xmath280 and consider a capacity potential for @xmath0 , that is the harmonic function in @xmath281 with the following boundary values @xmath282 since @xmath281 has @xmath177 planes of symmetry ( the @xmath283 and @xmath284 planes ) , the partial derivatives @xmath285 and @xmath286 must be zero on the two segments that are the intersection of @xmath281 with the @xmath287-axis . if @xmath288 is the segment that contains the origin , the restriction of @xmath4 to @xmath289 equals @xmath117 at the point @xmath290 , is @xmath83 at the point @xmath291 , is bounded at the origin by a constant @xmath292 independent of @xmath293 , and can be made arbitrarily close to @xmath117 between the `` ends '' of @xmath294 , when @xmath279 , it follows that , if @xmath293 is sufficiently small , @xmath295 ( and hence @xmath5 ) must vanish twice on @xmath288 . it is clear that this argument does not depend on the size or on small deformations of @xmath273 . thus , we can construct in @xmath143 a ( simply connected ) `` chain '' @xmath296 of an arbitrary number @xmath84 of such tori , by gluing them together : the solution in the domain obtained by replacing @xmath294 by @xmath296 will then have at least @xmath55 critical points . * circles of critical points . * the second example shows that , in general dimension , a finite number of sign changes of some derivative of a harmonic function @xmath4 on the boundary does not even imply that @xmath4 has a finite number of critical points . to see this , consider the harmonic function is subsection [ sub : circles ] : @xmath297 it is easy to see that , for instance , on any sphere centered at the origin the normal derivative @xmath298 changes its sign a finite number of times . however , if the radius of the sphere is larger than the first positive zero of @xmath299 , the corresponding ball contains at least one circle of critical points . * star - shaped annuli . * nevertheless , if some additional geometric information is added , something can be done . suppose that @xmath300 , where @xmath301 and @xmath302 are two domains in @xmath1 , with boundaries of class @xmath6 and such that @xmath303 . suppose that @xmath301 and @xmath302 are _ star - shaped _ with respect to the same origin @xmath304 placed in @xmath302 , that is the segment @xmath305 is contained in the domain for every point @xmath62 chosen in it . then , the _ capacity potential _ @xmath4 defined as the solution of the dirichlet problem @xmath306 does not have critical points in @xmath192 . this is easily proved by considering the harmonic function @xmath307 since @xmath301 and @xmath302 are starshaped and of class @xmath6 , @xmath308 on @xmath309 . by the strong maximum principle , then @xmath310 in @xmath0 ; in particular , @xmath5 does not vanish in @xmath0 and all the sets @xmath311 turn out to be star shaped too ( see @xcite ) . this theorem can be extended to the capacity potential defined in @xmath312 as the solution of @xmath313 such results have been extended in @xcite to a very general class of nonlinear elliptic equations . with suitable restrictions on the coefficients , can be regarded as the _ laplace - beltrami equation _ on the riemannian surface @xmath11 equipped with the metric @xmath314 theorems [ th : am1 ] and [ th : am1b ] can then be interpreted accordingly . this point of view has been considered in a more general context in @xcite , where the focus is on green s functions of a @xmath177-dimensional complete riemannian surface @xmath315 of finite topological type ( that is , the first fundamental group of @xmath227 is finitely generated ) . a green s function is a symmetric function @xmath316 that satisfies in @xmath227 the equation @xmath317 where @xmath318 is the laplace - beltrami operator induced by the metric @xmath66 and @xmath319 is the _ dirac delta _ centered at a point @xmath320 . a symmetric green s function @xmath321 can always be constructed by an approximation argument introduced in @xcite : an increasing sequence of compact subsets @xmath322 containing @xmath323 and exhausting @xmath227 is introduced and @xmath321 is then defined as the limit on compact subsets of @xmath324 of the sequence @xmath325 , where @xmath326 is the solution of such that @xmath327 on @xmath328 and @xmath329 is a suitable constant . a green s function defined in this way is generally not unique , but has many properties in common with the fundamental solution for laplace s equation in the euclidean plane . with these premises , in @xcite it has been proved the following notable topological bound : @xmath330 where @xmath331 and @xmath332 are the _ genus _ and the _ number of ends _ of @xmath227 ; the number @xmath333 is known as the first _ betti number _ of @xmath227 . moreover , if the betti number is attained , then @xmath321 is _ morse _ , that is at its critical points the hessian matrix is non - degenerate . in @xcite , it is also shown that , in dimensions greater than two , an upper bound by topological invariants is impossible . two different proofs are constructed in @xcite and @xcite , respectively . both proofs are based on the following _ uniformization principle _ : since @xmath315 is a smooth manifold of finite topological type , it is well known ( see @xcite ) that there exists a compact surface @xmath334 endowed with a metric @xmath335 of constant curvature , a finite number @xmath336 of isolated points points @xmath337 and a finite number @xmath338 of ( analytic ) topological disks @xmath339 such that @xmath315 is conformally isometric to the manifold @xmath340 , where @xmath341 is interior of @xmath342 that means that there exist a diffeomorphism @xmath343 and a positive function @xmath71 on @xmath227 such that @xmath344 ; it turns out that the genus @xmath331 of @xmath334 and the number @xmath345 that equals the number @xmath332 ends of @xmath227 determine @xmath227 up to diffeomorphisms . the proof in @xcite then proceeds by analyzing the transformed green s function @xmath346 . it is proved that @xmath347 satisfies the problem @xmath348 where @xmath349 and the constants @xmath350 , possibly zero ( in which case @xmath347 would be @xmath335-harmonic near @xmath351 ) , sum up to @xmath117 . thus , a local blow up analysis of the _ hopf index _ @xmath352 , of the gradient of @xmath347 at the critical points @xmath353 ( isolated and with finite multiplicity ) , together with the _ hopf index theorem _ ( @xcite ) , yield the formula @xmath354 where @xmath355 is the euler characterstic of the manifold @xmath356 and @xmath357 is a sufficiently small disk around @xmath358 . since @xmath355 is readily computed as @xmath359 and @xmath360 , one then obtains that @xmath361 of course , the gradient of @xmath362 vanishes if and only if that of @xmath321 does . the proof contained in @xcite has a more geometrical flavor and focuses on the study of the integral curves of the gradient of @xmath321 . this point of view is motivated by the fact that in euclidean space the green s function ( the fundamental solution ) arises as the electric potential of a charged particle at @xmath323 , so that its critical points correspond to equilibria and the integral curves of its gradient field are the lines of force classically studied in the xix century . such a description relies on techniques of dynamical systems rather than on the toolkit of partial differential equations . we shall not get into the details of this proof , but we just mention that it gives a more satisfactory portrait of the integral curves connecting the various critical points of @xmath321 an issue that has rarely been studied . the bounds and identities on the critical points that we considered so far are based on a crucial topological tool : the index @xmath363 of a critical point @xmath68 . for a function @xmath364 , the integer @xmath363 is the _ winding number or degree _ of the vector field @xmath5 around @xmath68 and is related to the portrait of the set @xmath365 for a sufficiently small neighborhood @xmath366 of @xmath68 . as a matter of fact , if @xmath68 is an isolated critical point of @xmath4 , one can distinguish two situations ( see @xcite ) : a. if @xmath366 is sufficiently small , @xmath367 and @xmath368 ; b. if @xmath366 is sufficiently small , @xmath369 consists of @xmath84 simple curves and , if @xmath370 , each pair of such curves crosses at @xmath68 only ; it turns out that @xmath371 . critical points with index @xmath372 equal to @xmath117 , @xmath83 , or negative are called _ extremal , trivial _ , or _ saddle _ points , respectively ( see @xcite ) . a saddle point is _ simple _ or _ morse _ if the hessian matrix of @xmath4 at that point is not trivial . in the cases we examined so far , we always have that @xmath373 , that is @xmath68 is a saddle point , since ( i ) and ( ii ) with @xmath374 can not occur , by the maximum principle . the situation considerably changes when @xmath4 is a solution of , , or . here , we shall give an account of what can be said for solutions of . the same ideas can be used for solutions of the semilinear equation @xmath375 subject to a homogeneous dirichlet boundary condition , where the non - linearity @xmath376 satisfies the assumptions : @xmath377 ( see @xcite for details ) . we present here the following result that is in the spirit of theorem [ th : am1 ] . [ th : am1c ] let @xmath0 be as in theorem [ th : am1 ] and @xmath189 be a solution of . if @xmath378 is an isolated critical point of @xmath4 in @xmath192 , then a. either @xmath68 is a nodal critical point , that is @xmath379 , and the function @xmath380 is asymptotic to @xmath381 , as @xmath382 , for some @xmath383 and @xmath384 , b. or @xmath68 is an extremal , trivial , or simple saddle critical point . finally , if all the critical points of @xmath4 in @xmath192 are isolated is simply connected , by using the analyticity of @xmath4 ( see @xcite ) ] , the following identity holds : @xmath385 here , @xmath386 and @xmath387 denote the number of the simple saddle and extremal points of @xmath4 . thus , a bound on the number of critical points in topological terms is not possible additional information of different nature should be added . the proof of this theorem can be outlined as follows . first , one observes that , at a nodal critical point @xmath132 , @xmath9 vanishes , and hence the situation described in subsection [ sub : harmonic ] is in order , that is @xmath380 actually behaves as specified in ( a ) and the index @xmath363 equals @xmath388 . if @xmath389 , a reflection argument like the one used for theorem [ th : am1 ] can be used , so that @xmath68 can be treated as an interior nodal critical point of an extended function with vanishing laplacian at @xmath68 and ( a ) holds ; in this case , however , as done for theorem [ th : am1 ] , the contribution of @xmath68 must be counted as @xmath390 . secondly , one examines non - nodal critical points . at these points @xmath9 is either positive or negative . if , say , @xmath391 , then at least one eigenvalue of the hessian matrix of @xmath4 must be negative and the remaining eigenvalue is either positive ( and hence a simple saddle point arises ) , negative ( and hence a maximum point arises ) or zero ( and hence , with a little more effort , either a trivial or a simple saddle point arises ) . thus , the total index of these points sums up to @xmath392 . finally , identity is obtained by applying hopf s index theorem in a suitable manner . as emerged in the previous subsection , topology is not enough to control the number of critical points of an eigenfunction or a torsion function . here , we will explain how some geometrical information about @xmath0 can be helpful . convexity is a useful information . if the domain @xmath393 , @xmath15 , is convex , one can expect that the solution @xmath18 of and the only _ positive _ solution @xmath394 of it exists and , as is well known , corresponds to the first dirichlet eigenvalue @xmath395 have only one critical point ( the maximum point ) . this expectation is realistic , but a rigorous proof is not straightforward . in fact , one has to first show that @xmath18 and @xmath394 are _ quasi - concave _ , that is one shows the convexity of the level sets @xmath396 for @xmath397 or @xmath398 . it should be noted that @xmath394 is _ never concave _ and examples of convex domains @xmath0 can be constructed such that @xmath18 is not concave ( see @xcite ) . the quasi - concavity of @xmath18 and @xmath394 can be proved in several different ways ( see @xcite ) . here , we present the argument used in @xcite . there , the desired quasi - convexity is obtained by showing that the functions @xmath399 and @xmath400 are concave functions ( @xmath18 and @xmath394 are then said _ @xmath34-concave _ and _ log - concave _ , respectively ) . in fact , one shows that @xmath288 and @xmath262 satisfy the conditions @xmath401 and @xmath402 the concavity test established by korevaar in @xcite , based on a maximum principle for the so - called _ concavity function _ ( see also @xcite ) , applies to these two problems and guarantees that both @xmath288 and @xmath262 are concave . with similar arguments , one can also prove that the solution of - is @xmath403-concave in @xmath404 for any fixed time @xmath405 . the obtained quasi - concavity implies in particular that , for @xmath397 or @xmath394 , the set of critical points @xmath86 , that here coincides with the set @xmath406 is convex . this set can not contain more than one point , due to the analyticity of @xmath4 . in fact , if it contained a segment , being the restriction of @xmath4 analytic on the chord of @xmath192 containing that segment , @xmath4 would be a _ positive _ constant on this chord and this is impossible , since @xmath407 at the endpoints of this chord . this same argument makes sure that , if @xmath408 in a convex domain @xmath0 , then for any fixed @xmath409 there is a _ unique _ point @xmath410 the so - called _ hot spot _ at which the solution of - attains its maximum in @xmath192 , that is @xmath411 the location of @xmath412 in @xmath0 will be one of the issues in the next section . * a conjecture . * counting ( or estimating the number of ) the critical points of @xmath18 , @xmath394 , or @xmath20 when @xmath0 is not convex seems a difficult task . for instance , to the author s knowledge , it is not even known whether or not the uniqueness of the maximum point holds true if @xmath0 is assumed to be _ star - shaped _ with respect to some origin . we conclude this subsection by offering and justifying a conjecture on the number of hot spots in a bounded simply connected domain @xmath0 in @xmath11 . to this aim , we define for @xmath409 the set of hot spots as @xmath413 we shall suppose that the function @xmath24 in is continuous , non - negative and not identically equal to zero in @xmath0 , so that , by _ hopf s boundary point lemma _ , @xmath414 . also , by an argument based on the analyticity of @xmath20 similar to that used for the uniqueness of the maximum point in a convex domain , we can be sure that @xmath415 is made of isolated points ( see @xcite for details ) . ( a parabolic version of ) theorem [ th : am1c ] then yields that @xmath416 where @xmath417 and @xmath418 are the number extremal and simple saddle points of @xmath419 ; clearly @xmath417 is the cardinality of @xmath415 . an estimate on the total number of critical points of @xmath419 will then follow from one on @xmath417 . notice that , if @xmath420 and @xmath421 , @xmath422 , are dirichlet eigenvalues ( arranged in increasing order ) and eigenfunctions ( normalized in @xmath423 ) of the laplace s operator in @xmath0 , then the following _ spectral formula _ @xmath424 where @xmath425 is the fourier coefficient of @xmath24 corresponding to @xmath421 . then we can infer that @xmath426 as @xmath37 , with @xmath427 and the convergence is uniform on @xmath192 under sufficient assumptions on @xmath24 and @xmath0 . this information implies that , if @xmath428 , then @xmath429 where @xmath430 is the set of local maximum points of @xmath394 . now , our conjecture concerns the influence of the shape of @xmath0 on the number @xmath417 . to rule out the possible influence of the values of @xmath24 , we assume that @xmath408 : then we know that there holds the following asymptotic formula ( see @xcite ) : @xmath431=-d_{\gamma}(x)^2 \ \mbox { for } \ x\in{\overline}{{\omega}};\ ] ] here , @xmath432 is the _ distance _ of a point @xmath433 from the boundary @xmath2 . the convergence in is uniform on @xmath192 under suitable regularity assumptions on @xmath2 . [ fig : cocoon ] increases , @xmath415 goes from @xmath434 , the set of maximum points of @xmath435 , to @xmath430 , the set of maximum points of @xmath394.,title="fig : " ] now , suppose that @xmath435 has _ exactly _ @xmath46 distinct local ( strict ) maximum points in @xmath0 . formula suggests that , when @xmath405 is sufficiently small , @xmath419 has the same number @xmath46 of maximum points in @xmath0 . as time @xmath405 increases , one expects that the maximum points of @xmath419 do not increase in number . therefore , the following bounds should hold : @xmath436 from the asymptotic analysis performed on , we also derive that the _ total number of critical points of _ @xmath394 does not exceeds @xmath437 . we stress that can not always hold with the equality sign . in fact , if @xmath438 denotes the unit disk centered at @xmath439 and we consider the domain @xmath281 obtained from @xmath440 by `` smoothing out the corners '' ( see fig . 3.4 ) , we notice that @xmath441 for every @xmath442 , while @xmath281 tends to the unit ball centered at the origin and hence , if @xmath293 is small enough , @xmath394 has only one critical point , being @xmath281 `` almost convex '' . based on a similar argument , inequalities like should also hold for the number of critical points of the torsion function @xmath18 . in fact , if @xmath443 is the solution of the one - parameter family of problems @xmath444 where @xmath445 is a positive parameter , we have that @xmath446=-d_{\gamma},\ ] ] uniformly on @xmath192 ( see again @xcite ) . we finally point out that the asymptotic formulas presented here hold in any dimension ; thus , the bounds in may be generalized in some way . to conclude this section about the number of critical points of solutions of partial differential equations , we can not help mentioning a conjecture proposed in @xcite ( also see @xcite ) . this is motivated by the study of eigenfunctions of the laplace - beltrami operator @xmath318 in a compact riemannian manifold @xmath447 . let @xmath448 be a sequence of eigenfunctions , @xmath449 let @xmath450 be a point of maximum for @xmath451 in @xmath227 and @xmath452 a geodesic ball centered at @xmath453 and with radius @xmath454 . if we blow up @xmath452 to the unit disk in @xmath11 and let @xmath455 be the eigenfunction after that change of variables , then a subsequence of @xmath456 will converge to a solution @xmath4 of @xmath457 if we can prove that @xmath4 has _ infinitely many _ isolated critical points , then we can expect that their number be unbounded also for the sequence @xmath448 . a naive insight built up upon the available concrete examples of entire eigenfunctions ( the _ separated _ eigenfunctions in rectangular or polar coordinates ) may suggest that it would be enough to prove that any solution of has infinitely many nodal domains . it turns out that this is not always true , as a clever counterexample obtained in ( * ? ? ? * theorem 3.2 ) shows : _ there exists a solution of with exactly two nodal domains_. the counterexample is constructed by perturbing the solution of @xmath458 where @xmath459 are the usual polar coordinates and @xmath151 is the second bessel s function ; @xmath71 has infinitely many nodal domains . the desired example is thus obtained by the perturbation @xmath460 , where @xmath461 and @xmath462 is suitably chosen . as a result , if @xmath293 is sufficiently small , the set @xmath463 is made of two interlocked spiral - like domains ( see ( * ? ? ? * figure 3.1 ) ) . a related result was proved in @xcite , where it is shown that there is no topological upper bound for the number of critical points of the first eigenfunction on riemannian manifolds ( possibly with boundary ) of dimension larger than two . in fact , with no restriction on the topology of the manifold , it is possible to construct metrics whose first eigenfunction has as many isolated critical points as one wishes . recently , it has been proved in @xcite that , if @xmath315 is a non - positively curved surface with concave boundary , the number of nodal domains of @xmath451 diverges along a subsequence of eigenvalues of density @xmath117 ( see also @xcite for related results ) . the surface needs not have any symmetries . the number can also be shown to grow like @xmath464 ( @xcite ) . in light of such results , yau s conjecture was updated as follows : show that , for any ( generic ) @xmath447 there exists at least one sub - sequence of eigenfunctions for which the number of nodal domains ( and hence of the critical points ) tends to infinity ( @xcite ) . the first result that studies the critical points of a function is probably _ rolle s theorem _ : between two zeroes of a differentiable real - valued function there is _ at least one _ critical point . thus , a function that has @xmath84 distinct zeroes also has at least @xmath45 critical points an estimate from below and we roughly know where they are located . after rolle s theorem , the first general result concerning the zeroes of the derivative of a general polynomial is _ gauss s theorem _ : if @xmath465 is a polynomial of degree @xmath84 , then @xmath466 and hence the zeroes of @xmath467 are , in addition to the multiple zeroes of @xmath468 themselves , the roots of @xmath469 these roots can be interpreted as the _ equilibrium points _ of the gravitational field generated by the masses @xmath470 placed at the points @xmath471 , respectively . if the zeroes of @xmath468 are placed on the real line then , by rolle s theorem , it is not difficult to convince oneself that the zeroes of @xmath467 lie in the smallest interval of the real axis that contains the zeroes of @xmath468 . this simple result has a geometrically expressive generalization in _ lucas s theorem _ : the zeroes of @xmath467 lie in the _ convex hull _ @xmath472 of the set @xmath473 named _ lucas s polygon _ and no such zero lies on @xmath474 unless is a multiple zero @xmath271 of @xmath468 or all the zeroes of @xmath468 are collinear ( see fig . in fact , it is enough to observe that , if @xmath475 or @xmath476 , then all the @xmath271 lie in the closed half - plane @xmath477 containing them and the side of @xmath472 which is the closest to @xmath60 . thus , if @xmath478 is an outward direction to @xmath479 , we have that @xmath480= \sum_{k=1}^k m_k\,\frac{{\mathop{\mathrm{re}}}\bigl[{\overline}{(z - z_k)}\,\ell\bigr]}{|z - z_k|^2}>0,\ ] ] since all the addenda are non - negative and not all equal to zero , unless the @xmath271 s are collinear . if @xmath468 has real coefficients , we know that its non - real zeroes occur in conjugate pairs . using the circle whose diameter is the segment joining such a pair this is called a _ jensen s circle _ of @xmath468 one can obtain a sharper estimate of the location of the zeroes of @xmath467 : each non - real zero of @xmath467 lies on or within a jensen s circle of @xmath468 . this result goes under the name of jensen s theorem ( see @xcite for a proof ) . all these results can be found in walsh s treatise @xcite , that contains many other results about zeroes of complex polynomials or rational functions and their extensions to critical points of harmonic functions : among them restricted versions of theorem [ th : am1 ] give information ( i ) on the critical points of the green s function of an infinite region delimited by a finite collection of simple closed curves and ( ii ) of harmonic measures generated by collections of jordan arcs . besides the _ argument s principle _ already presented in these notes , a useful ingredient used in those extensions is a _ hurwitz s theorem _ ( based on the classical _ rouch s theorem _ ) : if @xmath481 and @xmath482 are holomorphic in a domain @xmath0 , continuous on @xmath192 , @xmath482 is non - zero on @xmath2 and @xmath481 converges uniformly to @xmath482 on @xmath192 , then there is a @xmath483 such that , for @xmath484 , @xmath481 and @xmath482 have the same number of zeroes in @xmath0 . this theorem admits at least two proofs and it is worth to present both of them . the former is somewhat reminiscent of lucas s proof and is based on an explicit formula for @xmath4 , @xmath492 that can be derived as a consequence of _ stokes s formula_. here , @xmath493 is the surface area of a unit sphere in @xmath1 , @xmath494 denotes the @xmath17-dimensional surface measure , and @xmath298 is the ( outward ) normal derivative of @xmath4 . by the hopf s boundary point lemma , @xmath495 on @xmath2 . also , if @xmath496 , we can choose a hyperplane @xmath497 passing through @xmath404 and supporting @xmath489 ( at some point ) . if @xmath25 is the unit vector orthogonal to @xmath497 at @xmath404 and pointing into the half - space containing @xmath489 , we have that @xmath498 is non - negative and is not identically zero for @xmath499 . therefore , @xmath500 which means that @xmath501 . the latter proof is based on a symmetry argument ( @xcite ) and , as it will be clear , can also be extended to more general non - linear equations . let @xmath497 be any hyperplane contained in @xmath192 and let @xmath477 be the open half - space containing @xmath489 and such that @xmath502 . let @xmath503 be the mirror reflection in @xmath497 of any point @xmath504 . then the function defined by @xmath505 is harmonic in @xmath506 , tends to @xmath83 as @xmath507 and @xmath508 therefore , by the hopf s boundary point lemma , @xmath509 at any @xmath510 for any direction @xmath25 not parallel to @xmath497 . of course , if @xmath511 , we obtain that @xmath512 by directly using the hopf s boundary point lemma . generalizations of lucas s theorem hold for other problems . here , we mention the well known result of chavel and karp @xcite for the minimal solution of the cauchy problem for the heat equation in a riemannian manifold @xmath447 : @xmath513 where @xmath24 is a bounded initial data with compact support in @xmath227 . in @xcite , it is shown that , if @xmath227 is complete , simply connected and of constant curvature , then the set of the _ hot spots _ of @xmath4 , @xmath514 is contained in the convex hull of the support of @xmath24 . the proof is based on an explicit formula for @xmath4 in terms of the initial values @xmath24 . for instance , when @xmath515 , we have the formula @xmath516 with this formula in hand , by looking at the second derivatives of @xmath4 , one can also prove that there is a time @xmath517 such that , for @xmath518 , @xmath415 reduces to the single point @xmath519 which is the center of mass of the measure space @xmath520 ( see @xcite ) . we also mention here the work of ishige and kabeya ( @xcite ) on the large time behavior of hot spots for solutions of the heat equation with a rapidly decaying potential and for the schrdinger equation . from a physical point of view , the solution describes the evolution of the temperature of @xmath227 when its initial value distribution is known on @xmath227 . the situation is more difficult if @xmath521 is not empty . we shall consider here the case of a _ grounded _ heat conductor , that is we will study the solution @xmath20 of the cauchy - dirichlet problem - . * bounded conductor . * as already seen , if @xmath522 , implies . for an arbitrary continuous function @xmath24 , from we can infer that , if @xmath46 is the first integer such that @xmath523 and @xmath524 are all the integers such that @xmath525 , then @xmath526 also , when @xmath408 , holds and hence @xmath527 where @xmath434 is the set of local ( strict ) maximum points of @xmath435 . these informations give a rough picture of the _ set of trajectories _ of the hot spots : @xmath528 notice in passing that , if @xmath0 is convex and has @xmath182 distinct hyperplanes of symmetry , it is clear that @xmath529 is made of the same single point the intersection of the hyperplanes that is the hot spot _ does not move _ or is _ stationary_. also , it is not difficult to show ( see @xcite ) that the hot spot does not move if @xmath0 is invariant under an _ essential _ group @xmath108 of orthogonal transformations ( that is for every @xmath530 there is @xmath531 such that @xmath532 ) . characterizing the class @xmath533 of convex domains that admit a stationary hot spot seems to be a difficult task : some partial results about convex polygons can be found in @xcite ( see also @xcite ) . there it is proved that : ( i ) the equilateral triangle and the parallelogram are the only polygons with @xmath197 or @xmath534 sides in @xmath533 ; ( ii ) the equilateral pentagon and the hexagons invariant under rotations of angles @xmath535 , or @xmath497 are the only polygons with @xmath92 or @xmath536 sides _ all _ touching the inscribed circle centered at the hot spot . to see this , it is enough to consider the _ half - disk _ ( see fig . 4.3 ) @xmath541 being @xmath538 convex , for each @xmath409 , there is a unique hot spot that , as @xmath537 , tends to the maximum point @xmath542 of @xmath435 . thus , it is enough to show that @xmath543 is not a spatial critical point of @xmath544 for some @xmath409 or , if you like , for @xmath394 . this is readily seen by _ alexandrov s reflection principle_. let @xmath545 and define @xmath546 @xmath539 is the reflection of @xmath20 in the line @xmath547 . we clearly have that @xmath548 thus , the strong maximum principle and the hopf s boundary point lemma imply that @xmath549 for @xmath550 , and hence @xmath543 can not be a critical point of @xmath20 . the alexandrov s principle just mentioned can also be employed to estimate the location of a hot spot . in fact , as shown in @xcite , by the same arguments one can prove that hot spots must belong to the subset @xmath551 of @xmath0 defined as follows . let @xmath552 be a hyperplane orthogonal to the direction @xmath553 and let @xmath554 and @xmath555 be the two half - spaces defined by @xmath552 ; let @xmath556 denote the mirror reflection of a point @xmath404 in @xmath552 . then , the _ heart _ has also been considered in @xcite under the name of _ minimal unfolded region_. ] of @xmath0 is defined by @xmath557 when @xmath0 is convex , then @xmath551 is also convex and , if @xmath2 is of class @xmath6 , we are sure that its distance from @xmath2 is positive ( see @xcite ) . also , we know that @xmath415 is made of only one point @xmath412 , so that @xmath558 the set @xmath551 contains many notable geometric points of the set @xmath0 , such as the _ center of mass _ , the _ incenter _ , the _ circumcenter _ , and others ; see @xcite , where further properties of the heart of a convex body are presented . see also @xcite for related research on this issue . as clear from @xcite , the estimate just presented is of purely geometric nature , that is it only depends on the lack of symmetry of @xmath0 and does not depend on the particular equation we are considering in @xmath0 , as long as the equation is invariant by reflections . a different way to estimate the location of the hot spot of a grounded convex heat conductor or the maximum point of the solution of certain elliptic equations is based on ideas related to alexandrov - bakelman - pucci s maximum principle and does take into account the information that comes from the relevant equation . for instance , in @xcite it is proved that the maximum point @xmath559 of @xmath394 in @xmath192 is such that @xmath560 where @xmath561 is a constant only depending on @xmath182 , @xmath562 is the _ inradius _ of @xmath0 ( the radius of a largest ball contained in @xmath0 ) and @xmath563 is the _ diameter _ of @xmath0 . the idea of the proof of is to compare the _ concave envelope _ @xmath71 of @xmath394 the smallest concave function above @xmath394 and the function @xmath66 whose graph is the surface of the ( truncated ) cone based on @xmath0 and having its tip at the point @xmath564 ( see fig . 4.4 ) . now , @xmath571 has a precise geometrical meaning : it is the set @xmath572 , that is a multiple of the _ polar set _ of @xmath0 with respect to @xmath559 defined by @xmath573 the volume @xmath574 can be estimated by the formula of change of variables to obtain : @xmath575 where @xmath576 is the _ contact set_. since the determinant and the trace of a matrix are the product and the sum of the eigenvalues of the matrix , by the _ arithmetic - geometric mean inequality _ , we have that @xmath577 , and hence we can infer that @xmath578^ndx= \int_c \left[\frac{{\lambda}_1({\omega})\,\phi_1}{n \phi_1(x_\infty)}\right]^ndx\le \left[\frac{{\lambda}_1({\omega})}{n}\right]^n |{\omega}|,\ ] ] being @xmath579 in @xmath192 . finally , in order to get explicitly , one has to bound @xmath580 from below by the volume of the polar set of a suitable half - ball containing @xmath0 , and @xmath581 from above by the _ isodiametric inequality _ ( see @xcite for details ) . the two methods we have seen so far , give estimates of how far the hot spot must be from the boundary . we now present a method , due to grieser and jerison @xcite , that gives an estimate of how far the hot spot can be from a specific point in the domain . the idea is to adapt the classical _ method of separation of variables _ to construct a suitable approximation @xmath4 of the first dirichlet eigenfunction @xmath394 in a planar convex domain . clearly , if @xmath0 were a rectangle , say @xmath582\times[0,1]$ ] , then that approximation would be exact : in fact @xmath583.\ ] ] if @xmath0 is not a rectangle , after some manipulations , we can suppose that @xmath584 where , in @xmath582 $ ] , @xmath585 is convex , @xmath586 is concave and @xmath587 } f_1=0 , \ \max_{[a , b ] } f_2=1\ ] ] ( see fig . 4.5 ) . the geometry of @xmath0 does not allow to find a solution by separation of variables as in the case of the rectangle . however , one can operate `` as if '' that separation were possible . to understand that , consider the length of the section of foot @xmath404 , parallel to the @xmath323-axis , by @xmath588 and notice that , if we set @xmath589 the function @xmath590 satisfies for fixed @xmath404 the problem @xmath591 thus , it is the first dirichlet eigenfunction in the interval @xmath592 , normalized in the space @xmath593)$ ] . the basic idea is then that @xmath594 should be ( and in fact it is ) well approximated by its lowest fourier mode in the @xmath323-direction , computed for each fixed @xmath404 , that is by the projection of @xmath394 along @xmath116 : @xmath595 to simplify matters , a further approximation is needed : it turns out that @xmath262 and its first derivative can be well approximated by @xmath596 and its derivative , where @xmath19 is the first eigenfunction of the problem @xmath597 \phi(x)=0 \ \mbox { for } \ a < x < b , \quad \phi(a)=\phi(b)=0.\ ] ] since near the maximum point @xmath598 of @xmath19 , @xmath599 can be bounded from below by a constant times @xmath600 , the constructed chain of approximations gives that , if @xmath601 is the maximum point of @xmath394 on @xmath192 , then there is an absolute constant @xmath566 such that @xmath602 @xmath566 is independent of @xmath0 , but the result has clearly no content unless @xmath603 . * unbounded conductor . * if @xmath0 is unbounded , by working with suitable barriers , one can still prove formula when @xmath408 ( see @xcite ) , the convergence holding uniformly on compact subsets of @xmath0 . thus , any hot spot @xmath412 will again satisfy . to the author s knowledge , @xcite is the only reference in which the behavior of hot spots for large times has been studied for some grounded unbounded conductors . there , the cases of a half - space @xmath604 and the exterior of a ball @xmath605 are considered . it is shown that there is a time @xmath517 such that for @xmath518 the set @xmath415 is made of only one hot spot @xmath606 and @xmath607 if @xmath608 , while for @xmath609 , if @xmath24 is radially symmetric , then there is a time @xmath517 such that @xmath610 , for @xmath518 , where @xmath611 is some smooth function of @xmath405 such that @xmath612 upper bounds for @xmath415 are also given in @xcite for the case of the exterior of a smooth bounded domain . we conclude this survey by giving an account on the so - called _ hot spot conjecture _ by j. rauch @xcite . this is related to the asymptotic behavior of hot spots in a _ perfectly insulated _ heat conductor modeled by the following initial - boundary value problem : @xmath613 observe that , similarly to , a spectral formula also holds for the solution of : @xmath614 here @xmath615 is the increasing sequence of neumann eigenvalues and @xmath616 is a complete orthonormal system in @xmath423 of eigenfunctions corresponding to the @xmath617 s , that is @xmath618 is a non - zero solution of @xmath619 with @xmath620 . the numbers @xmath425 are the fourier coefficients of @xmath24 corresponding to @xmath621 , that is @xmath622 since @xmath623 and @xmath624 , we can infer that @xmath625\to \sum_{n = m}^{m+k-1 } \widehat{{\varphi}}(n)\,\psi_n(x ) \ \mbox { as } \ t\to\infty,\ ] ] where @xmath46 is the first integer such that @xmath523 and @xmath524 are all the integers such that @xmath626 . thus , similarly to what happens for the case of a grounded conductor , as @xmath37 , a hot spot @xmath412 of @xmath20 tends to a maximum point of the function at the right - hand side of . now , roughly speaking , the conjecture states that , for `` most '' initial conditions @xmath24 , the distance from @xmath2 of any hot and cold spot of @xmath20 must tend to zero as @xmath37 , and hence it amounts to prove that the right - hand side of attains its maximum and minimum at points in @xmath2 . it should be noticed now that the quotes around the word _ most _ are justified by the fact that the conjecture does not hold for all initial conditions . in fact , as shown in @xcite , if @xmath627 , the function defined by @xmath628 is a solution of with @xmath629 that attains its maximum at @xmath630 for any @xmath409 . however , it turns out that in this case @xmath631 thus , it is wiser to rephrase the conjecture by asking whether or not the hot and cold spots tend to @xmath2 if the coefficient @xmath632 of the first non - constant eigenfunction @xmath633 is not zero or , which is the same , whether or not maximum and minimum points of @xmath633 in @xmath192 are attained only on @xmath2 . in @xcite , a weaker version of this last statement is proved to hold for domains of the form @xmath634 , where @xmath635 has a boundary of class @xmath636 . in @xcite , the conjecture has also been reformulated for convex domains . indeed , we now know that it is false for fairly general domains : in @xcite a planar domain with two holes is constructed , having a simple second eigenvalue and such that the corresponding eigenfunction attains its strict maximum at an interior point of the domain . it turns out that in that example the minimum point is on the boundary . nevertheless , in @xcite it is given an example of a domain whose second neumann eigenfunction attains both its maximum and minimum points at interior points . in both examples the conclusion is obtained by probabilistic methods . besides in @xcite , positive results on this conjecture can be found in @xcite . in @xcite , the conjecture is proved for planar convex domains @xmath0 with two orthogonal axis of symmetry and such that @xmath637 this restriction is removed in @xcite . in @xcite , @xmath0 is assumed to have only one axis of symmetry , but @xmath633 is assumed anti - symmetric in that axis . a more general result is contained in @xcite : the conjecture holds true for domains of the type @xmath638 where @xmath585 and @xmath586 have unitary lipschitz constant . in @xcite , a modified version is considered : it holds true for general domains , if _ vigorous maxima _ are considered ( see @xcite for the definition ) . if no symmetry is assumed for a convex domain @xmath0 , y. miyamoto @xcite has verified the conjecture when @xmath639 ( for a disk , this ratio is about 1.273 ) . for unbounded domains , the situation changes . for the half - space , jimbo and sakaguchi proved in @xcite that there is a time @xmath273 after which the hot spot equals a point on the boundary that depends on @xmath24 . in @xcite , the case of the exterior @xmath0 of a ball @xmath640 is also considered for a radially symmetric @xmath24 . for a suitably general @xmath24 , ishige @xcite has proved that the behavior of the hot spot is governed by the point @xmath641 if @xmath642 , then @xmath415 tends to the boundary point @xmath643 , while if @xmath644 , then @xmath415 tends to @xmath645 itself . , _ on a representation theorem for linear systems with discontinuous coefficients and its applications _ , in convegno internazionale sulle equazioni lineari alle derivate parziali , cremonese , roma 1955 . , _ on extensions of the brunn - minkowski and prkopa - leindler theorems , including inequalities for log concave functions , and with an application to the diffusion equation _ , j. funct . * 22 * ( 1976 ) , 366389 .

we give a survey at an introductory level of old and recent results in the study of critical points of solutions of elliptic and parabolic partial differential equations . to keep the presentation simple , we mainly consider four exemplary boundary value problems : the dirichlet problem for the laplace s equation ; the torsional creep problem ; the case of dirichlet eigenfunctions for the laplace s equation ; the initial - boundary value problem for the heat equation . we shall mostly address three issues : the estimation of the local size of the critical set ; the dependence of the number of critical points on the boundary values and the geometry of the domain ; the location of critical points in the domain . [ section ] [ thm]proposition [ thm]lemma [ thm]corollary [ thm]remark # 1 # 1#2#30= -.50 [ 1]*proof of # 1 . *

strongly correlated electron systems show a rich variety of unconventional phenomena such as high temperature superconductivity @xcite and quantum criticality @xcite and their theoretical description and understanding constitutes a particular challenge . the origin of these correlations is the strong coulomb interaction , as particularly found in materials with partially filled @xmath11- or @xmath12-bands , such as transition metals , their oxides , rare earth and lanthanide compounds . the coulomb interaction between two electrons , which scatter from orbitals @xmath13 , @xmath14 to @xmath15 , @xmath16 in the course of the interaction , is simply given by @xmath17 here , @xmath18 and vacuum permittivity @xmath19 ; @xmath20 is the electron wave function for orbital @xmath13 ; no screening by further electrons has been included in this bare interaction @xmath21 . we do not consider relativistic corrections such as the spin - orbit coupling here so that the one - electron eigenstates simply need to be multiplied with a spinor and the integrals @xmath21 are independent of spin ; the @xmath15 and @xmath13 one - electron eigenstates ( as well as @xmath16 and @xmath14 ) need to have the same spin though . for practical calculations , it is essential to reduce the number of interaction parameters . often , e.g. in dft+@xmath2 ( density - functional theory augmented by a hubbard-@xmath2 interaction in a static mean - field approximation ) @xcite and dft+dmft ( dynamical mean - field theory ) , @xcite one considers only the _ local _ interaction . that is , all orbitals @xmath22 in eq . ( [ eq : u ] ) are on the same site ; they might correspond to wannier orbitals @xcite localized around the same lattice site . this is justified not only because this on - site interaction is by far the largest interaction parameter , but also since non - local interactions between orbitals on different sites can be treated in simple ( hartree ) mean field theory in the limit of a large number of neighbors . @xcite certainly there are situations where such non - local interactions can be of importance , particularly in one- and two - dimensions , or also between transition metal @xmath11 and oxygen @xmath23 orbitals . @xcite a further reduction of parameters can be achieved using the so - called slater integrals @xcite @xmath24 here , the underlying assumption is spherical symmetry , which allows for an analytical angular integration so that eventually only the integrals eq . ( [ eq : slater ] ) over the radial part @xmath25 of the wave functions remain , see appendix . these slater integrals , the simpler kanamori @xcite interaction , and or even just a single @xmath2-parameter are commonly used in dft+@xmath2,@xcite dft+dmft,@xcite or full - multiplet configuration - interaction calculations . @xcite however , a crystal lattice is not spherically symmetric . it has a lower , e.g. cubic , symmetry . the aim of our paper is hence to analyze the nature and magnitude of the deviations from spherical interaction parameters . to this end , we study the specific and arguably most relevant case of transition metal oxides with a cubic perovskite ( ) structure . in section [ sec : analytic ] , we study analytically the structure of the coulomb matrix elements for a @xmath26o@xmath27 octahedron . for the low energy @xmath0orbitals , the cubic coulomb interaction requires three parameters instead of the two parameters for spherical symmetry . we explicitly derive the most relevant integrals that deviate from the slater integrals ( [ eq : slater ] ) . in section [ sec : numeric ] , we calculate the quantitative deviations from spherical symmetry by means of maximally localized wannier orbitals . while the bare interaction in @xmath28 is still described reasonably well by spherically symmetric interaction parameters , the stronger @xmath23-@xmath11 hybridization in @xmath29 results in larger deviations ( @xmath6 ) . in a wannier basis which includes both the transition metal @xmath0and the oxygen @xmath23 orbitals , working with spherically symmetric interactions is justified . even for deviations between cubic and spherical symmetric interactions are only @xmath30 in this case . the effect of screening within the thomas - fermi approximation is considered in section [ sec : screening ] . for short screening lengths , deviations from spherical symmetry are even larger than in the unscreened case ; for realistic screening lengths , deviations are reduced but still significant for in a 3-orbital wannier basis ( @xmath7 ) . we consider the typical situation for transition metal oxides with an octahedron of oxygens surrounding each transition metal atom as shown in fig . [ fig : octahedron ] . while an isolated transition metal atom would be spherically symmetric and the parameterization in terms of slater integrals exact , the oxygen octahedron reduces the symmetry to cubic point group symmetry @xcite around the transition metal atom . therefore , the fivefold degeneracy of the atomic @xmath11 level is partially lifted , leaving a threefold degenerate @xmath0and a twofold degenerate @xmath31level in the cubic environment . in the cases we consider , the octahedron vertices are occupied by negatively charged ions . in this case , the @xmath31states , which have a lot of weight along the lines , are higher in energy than the @xmath0states , whose weight resides predominantly in the space between the ions , see fig . [ fig : octahedron ] ( right ) . the effective @xmath0orbitals are a combination of predominantly transition metal @xmath11 orbitals admixed with oxygen @xmath23 orbitals . for many transition metal oxides , these @xmath0orbitals constitute the low - energy degrees of freedom for excitations around the fermi energy.@xcite for an analytical description we consider an atomic transition metal @xmath0orbital , denoted as @xmath32 with @xmath33 in the following . this @xmath32 orbital mixes with a linear combination of oxygen @xmath23 orbitals of the same symmetry , see , e.g. , ref . . it is convenient to define this linear combination as @xmath34 : e.g. , @xmath35 where @xmath36 is the @xmath37 orbital centered around the oxygen atom in the the positive @xmath38 direction , see fig . [ fig : octahedron ] . the orbitals @xmath39 and @xmath40 follow from eq . ( [ eq : oxy ] ) by cubic symmetry , i.e. , @xmath41 and @xmath42 , respectively . symmetry ensures that the orbital @xmath34 is orthogonal to @xmath43 , except when @xmath44 . thus to orthogonalize the set of orbitals @xmath45 , one has only to orthonormalize @xmath34 with respect to its associated @xmath32 . the orthonormalized orbitals are @xmath46 the mixing of transition metal @xmath11 orbitals and oxygen @xmath23 orbitals stems from hybridization ; by symmetry , there is a hybridization only between @xmath32 and @xmath47 with the same @xmath13 . hence we obtain the tight - binding hamiltonian @xmath48 where @xmath49 and @xmath50 are the @xmath11 and @xmath23 ( more precisely the orthogonalized @xmath51 ) energy level ; @xmath52 is the charge transfer energy . the predominantly @xmath11 eigenfunctions of this tight - binding hamiltonian , @xmath53 , are the effective low energy @xmath0orbitals @xmath54 with @xmath55 , @xmath56^{-1/2}$ ] , and @xmath57^{-1/2}$ ] . after defining the low energy @xmath0orbitals , we need to calculate the coulomb interaction between these one - particle eigenstates , i.e. , @xmath58 this is the relevant site - local coulomb interaction for the low energy degrees of freedom . note that in this context , @xmath59 is defined as matrix element between direct products of single - particle states denoted as @xmath60 , not between antisymmetrized fock - states . since often @xmath61 in transition metal oxides , we consider in the following only the leading terms in the limit of large distance between transition metal and oxygen site . in this limit , the direct overlap @xmath62 , @xmath63 ( which is the overlap with respect to the one - particle hamiltonian , @xmath64 ) , and coulomb integrals between orbitals on different sites are small . in the following we hence restrict ourselves to all terms of up to second order in ( any ) of the above off - site overlaps , and obtain the following three contributions : directly from the @xmath65 terms in eq . ( [ eq : dprime ] ) and from the orthogonalization of the @xmath66 we get a contribution @xmath67 this term is centered around the transition metal ion and can be expressed in terms of the slater integrals @xmath68 for the @xmath69 orbitals . hence , this term can still be parameterized with kanamori interaction parameters . from two @xmath70 s in eq . ( [ eq : dprime ] ) we get a contribution @xmath71 note , @xmath72 and @xmath73 have a contribution from the same oxygen site , so that the @xmath74 and @xmath75 integrals both include on - site overlaps . since for large oxygen - transition metal distances the inter - site overlap decays exponentially while the coulomb interaction decays like @xmath76 , we keep the term eq . ( [ eq : ppdd ] ) . finally , there is a contribution involving only one @xmath70 in eq . ( [ eq : dprime ] ) and a coulomb integral overlap between transition - metal and oxygen site : @xmath77 all other terms are of higher order in @xmath63 or the off - center overlap integrals . eqs . ( [ eq : ppdd ] ) and ( [ eq : pddd ] ) involve coulomb integrals with two distinct sites , oxygen and transition metal . hence , they can not be expressed in terms of slater integrals any longer . one can also envisage that from the orbital in fig . [ fig : octahedron ] ( right ) . while the spherical rotations around the @xmath78 or @xmath38 axis of the central @xmath79 part of the @xmath80 orbital in fig . [ fig : octahedron ] ( right ) map the @xmath79 orbital onto a linear combination of the three @xmath32 orbitals , this is not possible any longer with the oxygen admixture @xmath81 in @xmath80 except for 90 degrees rotations . non - cubic rotations will put the rotated orbitals into positions where there is actually no oxygen site . employing the cubic symmetry , we can further reduce the number of integrals needed in eqs . ( [ eq : dddd ] ) , ( [ eq : ppdd ] ) and ( [ eq : pddd ] ) ; or ( [ eq : u ] ) cf . . any integral involving an orbital index @xmath82 once or thrice is odd in one cubic direction and hence vanishes . this leaves us with integrals where all orbitals are the same , i.e. , the intra - orbital hubbard interaction @xmath83 and integrals where we have two distinct orbitals @xmath84 twice . for the latter we have the three possibilities : the inter - orbital interaction @xmath85 , the hund s exchange @xmath86 , the pair hopping term and @xmath87 which for real - valued wave functions has the same amplitude as @xmath4 . these symmetry considerations actually hold in general , but without spherical symmetry @xmath88 because of the terms eqs . ( [ eq : ppdd ] ) and ( [ eq : pddd ] ) . for spherical symmetry , the connection to the slater integrals is as follows , cf . : @xmath89 , @xmath90 , @xmath91 , so that @xmath1 holds . @xcite if we have instead only cubic symmetry , we can still parameterize the interaction in terms of @xmath2 , @xmath3 , and @xmath4 , but now with @xmath92 and no expression in terms of slater integrals . in second quantization , this kanamori hamiltonian,@xcite which is obtained from eq.([eq : u ] ) by including all valid spin combinations in eq.([eq : u ] ) , reads : @xmath93 \\ -\sum_{\alpha \neq \beta } j(c^\dagger_{\alpha,\downarrow}c^\dagger_{\beta,\uparrow}c^{\phantom{\dagger}}_{\beta,\downarrow}c^{\phantom{\dagger}}_{\alpha,\uparrow } + c^\dagger_{\beta,\uparrow}c^\dagger_{\beta,\downarrow}c^{\phantom{\dagger}}_{\alpha,\uparrow}c^{\phantom{\dagger}}_{\alpha,\downarrow } \ ! + \ ! \,.\end{gathered}\ ] ] here , @xmath94 ( @xmath95 ) creates ( annihilates ) an electron with spin @xmath96 in orbital @xmath13 ; @xmath97 . in contrast for the @xmath98 orbitals , which are proportional to @xmath99 and @xmath100 , the relation @xmath101 still holds for cubic symmetry : again because of cubic symmetry ( @xmath102 ) any term involving one or three @xmath103 orbitals vanishes ; only the @xmath83 , @xmath85 , @xmath104 terms remain . however now , instead of interchanging the orbitals , cubic symmetry operations such as ( @xmath105 , @xmath106 , @xmath107 ) , lead to mixed orbitals : @xmath108 . hence , the intra - orbital hubbard interaction @xmath2 for the @xmath98 orbitals is not a cubic invariant , and @xmath2 has to depend on the other parameters @xmath3 and @xmath4 through @xmath1 . we now aim to validate our analytical results and quantify the deviation from the spherical - symmetry relation eq . ( [ eq : u ] ) in real materials . to this end , we perform dft calculations@xcite using a generalized - gradient approximation to the exchange - correlation functional @xcite for two cubic perovskite materials , and construct low - energy effective models using maximally - localized wannier functions ( mlwf ) @xcite . in terms of the formalism of sec . [ sec : analytic ] , the role of the wannier functions is to provide the radial dependence of the orbitals which was irrelevant for our arguments from symmetry , but which must be provided to compute numerical values for the interaction parameters . the main difference is that we considered a local octahedron before , while wannier functions @xmath110 properly belong to a periodic crystal : they have finite hopping amplitudes @xmath111 also for @xmath112 ( or equivalently , they show a @xmath113-dispersion ) , and form an orthonormal set @xmath114 with respect to sites @xmath115and orbitals @xmath13 . our first example is , which is often used as a `` testbed '' strongly - correlated material ( for dft+dmft calculations , see e.g. ref . ; detailed discussions of wannier projections in this and related materials have been given in refs . ) . the cubic perovskite is a paramagnetic , correlated metal with electronic configuration @xmath28 , i.e. one of the @xmath0-derived states will be filled . secondly , we consider the recently synthesized compound . @xcite with a low - spin @xmath29 configuration , this is another paramagnetic metal . since the @xmath5 states are more extended than the @xmath8 states of , we expect to find greater @xmath23-@xmath11 hybridization and , in turn , greater deviation from @xmath1 in this case . for each material , we construct two sets of wannier functions : 1 . three `` @xmath11-only '' wannier functions corresponding to the @xmath53 of eq . ( [ eq : dprime ] ) , and 2 . twelve `` @xmath116 '' wannier functions corresponding to the atomic @xmath69 and @xmath117 states . it is instructive to compare these two approaches : the first set of wannier functions translates the three @xmath0-derived bands to three orbitals @xmath118 centered on the ion . direct and -mediated hopping processes are subsumed in an effective @xmath26-@xmath26 hopping @xmath119 . to account for this , the @xmath120 must have substantial weight not only at the but also at the atoms . ( in a band picture , the reason is the significant -@xmath23 contribution to the @xmath0-derived bands . ) which combinations of -@xmath23 and -@xmath11 orbitals mix is determined by symmetry as discussed in sec . [ sec : analytic ] , cf.fig . [ fig : dos ] . going beyond an effective single - particle description , the coulomb interaction is expected to be well represented by a site local `` multi - band hubbard '' term @xmath121 which can be parameterized by three independent quantities @xmath2 , @xmath3 , and @xmath4 , as we saw in the previous section . the second set of wannier functions spans nine additional bands , three @xmath23-derived bands per . with the @xmath23-states explicitly included , the @xmath11-like mlwfs are free to become more localized ; the weight at the sites will be carried by the @xmath23-like orbitals ( cf . fig [ fig : d+p ] ) . the downside is that the resulting model becomes significantly more complex , since a correct treatment of such a `` @xmath116 '' model must take into account not only the intra - atomic interactions on the ( @xmath122 ) and on the ( @xmath123 ) sites , but also inter - atomic ( @xmath124 ) interactions . @xcite this added complexity will increase the computational cost to solve the model in any numerical method , but it will also make the physical interpretation of the results more involved . before we turn to the results , note that the heavy ( @xmath125 ) element leads to an appreciable spin - orbit splitting in . because it would invalidate the symmetry analysis of sec . [ sec : analytic ] , we neglect this effect in the construction of the wannier functions . while our analysis could be extended to include spin - orbit coupling , a spin - orbit interaction term can also be added to the tight - binding model afterwards in any case @xcite . + fig . [ fig : dos ] shows the densities of states ( dos ) of and , and the wannier orbitals corresponding to the 3-band case . the dos around the fermi level is derived from the @xmath126-antibonding combinations of -@xmath23 and -@xmath0states ; correspondingly , the 3-band orbitals are composed of a @xmath11-like contribution at the site and @xmath23-like contributions at the sites sharing a plane with the @xmath11-like part , akin to the @xmath47 orbitals in section [ sec : analytic ] . these wannier orbitals are also referred to as `` @xmath11-only '' , where the quotation marks hint that these orbitals are actually not pure @xmath11-orbitals . the wannier functions are equivalent to each other under cubic symmetry . as expected , the @xmath23-@xmath11 hybridization is stronger in than . this is seen both in the dos ( more weight around the fermi energy @xmath127 ) and in the orbitals ( bigger lobes at the sites ) . we can quantify this observation by integrating over the shaded areas in the dos ; this yields an admixture of @xmath128 and @xmath129 , respectively . in this sense , the `` @xmath11-bands '' of consist in fact of almost equal parts and contributions . these values agree qualitatively with eq . [ eq : dprime ] , which yields @xmath130 and @xmath131 using the parameters from table [ tab : hopping12].@xcite quantitative differences have to be expected because ( i ) eq . ( [ eq : dprime ] ) holds for an isolated octahedron instead of the periodic crystal , ( ii ) there are further hopping integrals that would have to be considered , and ( iii ) the partial dos of fig . [ fig : dos ] are only projections within the muffin tin spheres @xcite . right ) . by symmetry , the twelve orbitals are grouped into three equivalent @xmath11-like orbitals ( top left ) ; and two types of @xmath23-like orbitals , three `` @xmath132 '' ( bottom ) whose symmetry axes point toward their neighbors , and six `` @xmath133 '' ( top right ) pointing away from the sites . with the -@xmath23 states explicitly included , no @xmath23-@xmath0hybridization is seen in these orbitals . correspondingly , the @xmath11 and @xmath133 orbitals are close to their atomic counterparts . conversely , the @xmath132 orbitals , which mediate the @xmath96-bonding between -@xmath23 and -@xmath31 states , are elongated along their symmetry axis and have large contributions at their neighbors . @xcite , title="fig : " ] right ) . by symmetry , the twelve orbitals are grouped into three equivalent @xmath11-like orbitals ( top left ) ; and two types of @xmath23-like orbitals , three `` @xmath132 '' ( bottom ) whose symmetry axes point toward their neighbors , and six `` @xmath133 '' ( top right ) pointing away from the sites . with the -@xmath23 states explicitly included , no @xmath23-@xmath0hybridization is seen in these orbitals . correspondingly , the @xmath11 and @xmath133 orbitals are close to their atomic counterparts . conversely , the @xmath132 orbitals , which mediate the @xmath96-bonding between -@xmath23 and -@xmath31 states , are elongated along their symmetry axis and have large contributions at their neighbors . @xcite , title="fig : " ] + right ) . by symmetry , the twelve orbitals are grouped into three equivalent @xmath11-like orbitals ( top left ) ; and two types of @xmath23-like orbitals , three `` @xmath132 '' ( bottom ) whose symmetry axes point toward their neighbors , and six `` @xmath133 '' ( top right ) pointing away from the sites . with the -@xmath23 states explicitly included , no @xmath23-@xmath0hybridization is seen in these orbitals . correspondingly , the @xmath11 and @xmath133 orbitals are close to their atomic counterparts . conversely , the @xmath132 orbitals , which mediate the @xmath96-bonding between -@xmath23 and -@xmath31 states , are elongated along their symmetry axis and have large contributions at their neighbors . @xcite , title="fig : " ] for these `` @xmath11-only '' wannier orbitals , we have calculated the coulomb interaction by spatial integration of eq . ( [ eq : u ] ) . @xcite table [ tableu ] summarizes the results obtained for the bare interaction . for the @xmath28 perovskite , deviations from the spherical symmetric relation @xmath134 are 6% . that is , calculations employing this relation can still be expected to yield quite reliable results . for the @xmath29 perovskite , on the other hand , deviations are 25% . the reason for this is the larger admixture of oxygen @xmath23 contributions , which according to section [ sec : analytic ] yields larger off - center coulomb integral overlaps and hence a larger deviation from spherical symmetry . lx r@.l@ evx r@.l@ ev interaction & & & & + @xmath2 & & 16&30 & & 10&54 + @xmath3 & & 15&14 & & 9&67 + @xmath4 & & 0&55 & & 0&33 + @xmath135 & & 0&58 & & 0&44 + recently , transition metal oxides with heavy @xmath136 or @xmath5 elements attract more and more attention . indeed in such systems electronic correlations are stronger than expected due to hund s rule coupling @xcite . all the more important is a correct hamiltonian and multiplet structure with hund s exchange . in this respect , our finding highlights the substantial difference between @xmath135 and @xmath4 . a kanamori hamiltonian with three independent coulomb interactions needs to be considered for obtaining the correct multiplet structure . next , we turn to the 12-orbital `` @xmath116 '' wannier functions . this is an alternative description of the low energy physics , where the oxygen @xmath23-orbitals are explicitly taken into account . the corresponding wannier functions for are shown in fig . [ fig : d+p ] . the @xmath11-like orbitals are again equivalent up to symmetry , but two inequivalent types of @xmath23-like orbitals appear . symmetry also greatly restricts the possible hopping processes between these states . the hopping amplitudes within the octahedron as well as selected longer - ranged ones for all four wannier projections are reported in app . [ app : hopping ] . we list the coulomb interaction parameters between the 12-band orbitals for and in tables [ tableupdsvo ] and [ tableupdboo ] , respectively . for the @xmath11-like orbitals , @xmath1 is fulfilled with a reasonable accuracy of 3% even in . having the additional degree of freedom regarding oxygen-@xmath23 wannier orbitals , the @xmath0 orbitals are now localized around the transition metal ion and have the spherically symmetric form , cf . [ fig : d+p ] . in this case , two parameters are sufficient for the @xmath11-@xmath11 kanamori interaction . as a side note , observe that in the 12-band case @xmath137 , while in the 3-band case @xmath138 . this is because @xmath3 and @xmath4 are more strongly reduced than @xmath2 by the shift of @xmath0 weight to the oxygens which occurs in the 3-band case , as @xmath3 and @xmath4 are _ inter-_orbital interactions that include more non - overlapping oxygens in the interaction integral . let us emphasize that the @xmath11-@xmath23 interaction also plays an important role @xcite . the @xmath11-@xmath23 interactions of density - density type are listed in tables [ tableupdsvo ] and [ tableupdboo ] ( right ) . there are two types of @xmath23 orbitals , denoted as @xmath133 and @xmath132 ( see fig . [ fig : d+p ] ) . interactions with @xmath23 orbitals centered on an oxygen atom outside the plane of the @xmath11 orbital lobes are denoted by @xmath139 . the @xmath11-@xmath132 interactions @xmath140 and @xmath141 with the @xmath132 orbitals oriented toward the transition metal site is considerably stronger than that with the more regular @xmath133 orbitals . there is only one @xmath142 , while two parameters arise from density - density interaction between @xmath11 and @xmath133 orbitals with the @xmath133 orbitals being centered around oxygen sites within the plane defined by the @xmath11 orbitals . we denote these as @xmath143 if the @xmath133 orbital lies within the same plane and @xmath144 if it is oriented perpendicular to it . the considerable differences in the @xmath11-@xmath23 coulomb interaction can be understood from the very different @xmath133 and @xmath132 orbitals in fig . [ fig : d+p ] . these differences are of relevance for @xmath116 dft+dmft calculations that include @xmath124 . @xcite lx r@.l@ ev xx lx r@.l@ ev interaction & & & & & interaction & & + @xmath2 & & 19&99 & & & @xmath145 & & 7&24 + @xmath3 & & 18&52 & & & @xmath146 & & 7&18 + @xmath4 & & 0&74 & & & @xmath147 & & 8&52 + & & & @xmath141 & & 6&87 + @xmath135 & & 0&74 & & & @xmath148 & & 8&06 + lx r@.l@ ev xx lx r@.l@ ev interaction & & & & & interaction & & + @xmath2 & & 14&90 & & & @xmath145 & & 6&94 + @xmath3 & & 13&65 & & & @xmath146 & & 6&80 + @xmath4 & & 0&64 & & & @xmath147 & & 7&85 + & & & @xmath141 & & 7&23 + @xmath135 & & 0&62 & & & @xmath148 & & 6&38 + the values reported above were calculated for a bare , unscreened coulomb interaction . in this section we include screening , within thomas - fermi theory . that is , we replace the bare interaction in eq . ( [ eq : u ] ) by @xmath149 where @xmath150 is the thomas - fermi screening length . let us emphasize that for a cubic crystal the screened interaction @xmath151 itself will be of cubic symmetry , and hence deviate from spherical symmetry . this effect is not taken into account in the following ; it will on its own generate further deviations of the kanamori interaction parameters from the spherical relation @xmath1 . in the following , we adjust the parameter @xmath150 to yield a screened coulomb interaction @xmath152ev for @xmath8 as calculated by constrained lda @xcite . this corresponds to a screening length @xmath153 ( the lattice parameters are @xmath154 @xcite and @xmath155 @xcite ) . we employ the same screening length also for as this yields an interaction parameter @xmath156ev , which is in the expected range for the @xmath5 . table [ tableuscreened ] shows the results obtained for the `` @xmath11-only '' models . in the case of @xmath8 orbitals as exemplified by , deviations from spherically symmetric interaction parameters are already small without screening and become negligible if screening is included . by contrast , for @xmath5 , @xmath157 is significantly violated even when screened . let us note that the degree of deviation is quite robust over a large range of screening lengths . for example with @xmath158 we obtain a similar deviation of 14% ( @xmath159ev , @xmath160ev , and @xmath161ev ) . lx r@.l@ evx r@.l@ ev interaction & & & & + @xmath2 & & 4&40 & & 2&44 + @xmath3 & & 3&47 & & 1&80 + @xmath4 & & 0&46 & & 0&28 + @xmath135 & & 0&47 & & 0&32 + interestingly , for weak screening ( large @xmath162 ) @xmath4 can even be enhanced whereas @xmath2 and @xmath3 are always reduced . the reason for this is that the exchange integral @xmath4 includes positive _ and _ negative contributions ; and for large @xmath150 , the negative contributions are more strongly reduced than the positive ones . for example at @xmath163 we obtain @xmath164ev for , which is larger than the unscreened @xmath165ev . as the increase is very small , the results are given to a higher precision than elsewhere in the paper . with @xmath166ev and @xmath167 ev , deviations are 6.2% for this screening strength . in the limit of infinite screening , i.e. , @xmath168 , one can show that @xmath169 . that is , one can describe this limit by one kanamori interaction parameter @xmath170 for spherical symmetry , and two ( @xmath2 and @xmath171 ) for cubic symmetry . numerically , we get however also for cubic `` @xmath11-only '' wannier functions @xmath172 for both and . the limits of strong and weak screening show that the idea that screening strongly reduces @xmath3 and hardly reduces @xmath4 is not true in general . for strong screening , @xmath4 is reduced as much as @xmath3 , since they are equal , while for weak screening @xmath4 is even enhanced . we have analyzed the physical origin and the magnitude of the difference between a spherically symmetric and a cubic interaction for @xmath0 orbitals . deviations are quite large for @xmath5 orbitals of heavy transition metals . since for these systems hund s exchange is paramount for electronic correlations,@xcite we conclude that a kanamori interaction with three instead of two independent parameters is necessary . unfortunately , this requires the calculation of one additional interaction parameter and hence a more thorough analysis of the interaction in dft+dmft calculations than customary hitherto . only if the oxygen degrees of freedom are included in the wannier projection , this is not necessary . in this case , however , the ( different ) @xmath11-@xmath23 coulomb interactions should be taken into account . for @xmath98 orbitals there is no such difference between spherical and cubic interaction . depending on the screening length , screening enhances or reduces the difference between spherically symmetric and cubic interaction parameters . screening can even enhance @xmath4 whereas @xmath2 and @xmath3 are always reduced . for thomas - fermi screening , @xmath173 is still significantly violated for @xmath5 . let us note that the simple thomas - fermi screening employed here is spherically symmetric , whereas the physical screening function obeys the cubic , not the spherical symmetry . this effect is an additional source of deviations from spherically symmetric interaction parameters . for both @xmath98-only and @xmath0-only low - energy effective models , we have a kanamori interaction for cubic symmetry . this makes continuous - time quantum monte - carlo simulations @xcite very efficient because of an additional local symmetry , see ref . . for the sake of completeness , let us briefly add the representation of the coulomb interaction eq . ( [ eq : u ] ) by slater integrals . expressing @xmath174 in terms of spherical harmonics @xmath175 and with @xmath176 where @xmath25 is independent of @xmath177 ( or @xmath13 ) , the coulomb interaction eq . ( [ eq : u ] ) becomes @xmath178 r(r ' ) y_{\beta } ( \theta ' , \varphi ' ) r(r ) y_{\alpha}(\theta , \varphi ) { r'}^2 r^2 \ ; . \end{gathered}\ ] ] this integral can be decomposed into a radial ( @xmath179)and an angular part ( @xmath180 ) , and the latter can be expressed in terms of clebsch - gordan coefficients . thus only the radial integrals aka slater parameters @xmath68 of eq . ( [ eq : slater ] ) need to be calculated . in this appendix , we report the numerical values of selected hopping amplitudes in our wannier projections.@xcite the values for may be compared to refs . . note that this is not an enumeration of the largest hopping amplitudes ; rather , the selection is meant to be illustrative . lx r@.lx r@.lx r@.lx r@.lx r@.lx r@.lx & & & & & & & & & & & & + & & @xmath1810&263 & & @xmath1810&027 & & @xmath1810&084 & & 0&009 & & 0&006 & & 0&580 + & & @xmath1810&394 & & @xmath1810&043 & & @xmath1810&112 & & 0&012 & & 0&013 & & @xmath1810&453 + for the 3-band wannier functions ( values in table [ tab : hopping3 ] ) , no hopping is possible within the unit cell . two nearest - neighbor hoppings are allowed , a @xmath126-type hopping @xmath182 when the displacement is in the same plane as the orbital lobes ( e.g. @xmath183 orbitals with displacement @xmath184 ) , and a smaller @xmath185 of @xmath186 type where the displacement is perpendicular ( e.g. @xmath183 and @xmath187 ) . inter - orbital nearest - neighbor hopping is forbidden by cubic symmetry . there are three second - nearest neighbor hopping parameters : @xmath188 when both orbitals and the displacement share a plane ( e.g. @xmath183 and @xmath189 ) ; @xmath190 when the orbitals planes are parallel ( e.g. @xmath191 and @xmath189 ) ; and @xmath192 when the planes are perpendicular ( e.g. @xmath193 and @xmath189 ) . l r@.lx r@.lx r@.lx r@.lx r@.lx r@.lx r@.lx r@.lx r@.lx r@.lx r@.lx r@.lx r@.lx & & & & & & & & & & & & & & & & & & & & & & & + & @xmath1810&128 & & @xmath1810&005 & & 1&099 & & 0&064 & & 0&369 & & 0&258 & & @xmath1810&044 & & @xmath1810&078 & & 0&671 & & @xmath1810&407 & & @xmath1813&780 & & @xmath1815&520 + & @xmath1810&187 & & @xmath1810&002 & & 1&240 & & 0&007 & & 0&204 & & 0&195 & & @xmath1810&024 & & @xmath1810&107 & & 0&903 & & @xmath1812&063 & & @xmath1813&896 & & @xmath1816&887 + for the 12-band case ( table [ tab : hopping12 ] ) , we report the nearest - neighbor @xmath194 hopping parameters @xmath195 analogous to those of the 3-band case , but not those to further neighbors . in any case , -mediated hopping , which was subsumed in the hoppings of the 3-band orbitals , now has to be taken into account explicitly . within the octahedron , the following hopping processes are possible : @xmath196 when the @xmath133 orbital resides in the plane defined by the @xmath11 ( e.g. @xmath197 ) ; @xmath198 between nearest neighbors , i.e. along an edge of the octahedron ( e.g. @xmath199 ) ; @xmath200 which is the same as the last , but between orbitals of different orientation ( e.g. @xmath201 ) ; @xmath202 along an edge ( e.g. @xmath203 ) ; @xmath204 along an edge ( e.g. @xmath205 ) ; @xmath206 across the octahedron ( e.g. @xmath207 ) ; and @xmath208 across the octahedron ( e.g. @xmath209 ) . comparing the sequences of values for the two materials , the same trends are observed ( with the exceptions of @xmath210 and @xmath211 ) . however , the values for the `` @xmath11-only '' orbitals , and for @xmath212 and @xmath213 processes in the 12-band orbitals , are in general larger for than , the larger lattice constant of notwithstanding ( @xmath214 versus @xmath215 for ) . this is reflective of the greater @xmath23-@xmath11 hybridization and spatial extent of the @xmath5 states . contrariwise , the 12-band @xmath216 hopping processes have larger amplitude in . our interpretation is that in this case , the larger spatial distance prevails ; indeed , the difference in wannier function spread @xmath217 between and is much more pronounced for the @xmath11 than for the @xmath23 orbitals . the exceptions to this rule , @xmath218 and @xmath219 ( hopping across the octahedron ) , may be explained by the stronger @xmath23-@xmath11 hybridization in . v. i. anisimov , a. i. poteryaev , m. a. korotin , a. o. anokhin and g. kotliar , j. phys . matter * 9 * 7359 ( 1997 ) ; a. i. lichtenstein and m. i. katsnelson , phys . b * 57 * 6884 ( 1998 ) ; k. held , i. a. nekrasov , g. keller , v. eyert , n. blmer , a. mcmahan , r. scalettar , t. pruschke , v. i. anisimov and d. vollhardt , phys . 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many - body calculations for multi - orbital systems at present typically employ slater or kanamori interactions which implicitly assume a full rotational invariance of the orbitals , whereas the real crystal has a lower symmetry . in cubic symmetry , the low - energy @xmath0 orbitals have an on - site kanamori interaction , albeit without the constraint @xmath1 implied by spherical symmetry ( @xmath2 : intra - orbital interaction , @xmath3 : inter - orbital interaction , @xmath4 : hund s exchange ) . using maximally localized wannier functions we show that deviations from the standard , spherically symmetric interactions are indeed significant for @xmath5 orbitals ( @xmath6 for ; @xmath7 if screening is included ) , but less important for @xmath8 orbitals ( @xmath9 for ; @xmath10 if screened ) .

let @xmath0 be a rational map of the riemann sphere to itself of degree @xmath1 . the map @xmath2 is hyperbolic if there are a smooth conformal metric @xmath3 defined on a neighborhood of the julia set @xmath4 and a constant @xmath5 such that @xmath6 for all @xmath7 . it is equivalent to every critical point of @xmath2 tends to an attracting periodic cycle under forward iteration . see @xcite and @xcite . let @xmath8 be the space of all the mbius equivalence classes of rational maps of degree @xmath9 . the space @xmath8 has dimension @xmath10 . a central problem in holomorphic dynamics is the following . the set of hyperbolic rational maps is open and dense in the space @xmath8 . openness of the set of hyperbolic rational maps is known , but density is only known in the family of real polynomials , see @xcite , @xcite , @xcite , @xcite , @xcite and @xcite . a rational map @xmath2 admits an _ invariant line field _ on the julia set @xmath4 if there is a measurable beltrami differential @xmath11 on @xmath12 such that @xmath13 a.e . , @xmath14 on a positive measure subset @xmath15 of @xmath4 and @xmath16 on @xmath17 . a rational map @xmath2 is called a _ latt@xmath18s example _ if it is doubly covered by an integral torus endmorphism . the julia set of such a rational map is @xmath12 and @xmath19 is an invariant line field on @xmath12 . a rational map @xmath2 carries no invariant line fields on its julia set , except when @xmath2 is a latt@xmath18s example . this conjecture is stronger than the density of hyperbolic dynamics . the no invariant line field conjecture implies the density of hyperbolic dynamics in the space of all rational maps . the absence of invariant line fields on the julia set is known in the following cases : 1 . non - infinitely renormalizable quadratic polynomials with no irrational indifferent periodic points , @xcite , @xcite and @xcite . robust infinitely renormalizable quadratic polynomials and real quadratic polynomials , @xcite . quadratic polynomials with a siegel cycle of bounded type rotation number , @xcite , @xcite and @xcite . real polynomials with only one non - escaping critical point which is real and has odd local degree , @xcite . 5 . real rational maps(non latt@xmath18s example ) whose critical points are all on the extended real axis and have even local degrees , @xcite . summable rational maps with completely invariant fatou domains , @xcite . summable rational maps with small exponents , @xcite . weakly hyperbolic rational maps , @xcite . in this paper we will prove let @xmath2 be a rational map with a cantor julia set . then @xmath2 carries no invariant line fields on its julia set . \(1 ) it is reasonable to conjecture that a cantor julia set always has measure zero . theorem 1 can be regarded as a step towards this conjecture . \(2 ) as a special case of theorem 1 , we disprove a question in the chapter 12 of @xcite . let @xmath20 be a complex manifold . a _ holomorphic family _ of rational maps @xmath21 is a holomorphic map @xmath22 , given by @xmath23 . let @xmath24 be the set of structurally stable parameters . that is , @xmath25 if and only if there is a neighborhood @xmath26 of @xmath27 such that @xmath28 and @xmath29 are topologically conjugate for all @xmath30 . the space @xmath31 of quasiconformally stable parameters is defined similarly , with conjugacy replaced by quasiconformality . using the harmonic @xmath32-lemma of bers and royden , mcmullen and sullivan proved the following result . in any holomorphic family of rational maps , @xmath33 is open and dense in @xmath20 . moreover , the structurally stable and quasiconformally stable parameters coincide , i.e. , @xmath34 . a structurally stable rational map is hyperbolic . combine theorem 1 and the theory of teichmller space of rational maps in @xcite , we can prove the following result . let @xmath2 be a rational map with a cantor julia set . if @xmath2 is structurally stable , then it is hyperbolic . the same result in theorem 2 was also proved by makienko under some additional assumptions , see @xcite . a nested sequence of some critical pieces constructed by kozlovski , shen , and van strien in @xcite , which we shall call kss nest " , will play a crucial rule in the proof of theorem 1 . principal nest and modified principal nest are used to study the dynamics of unicritical polynomials , see @xcite , @xcite , @xcite and @xcite . in @xcite , lyubich proved the linear growth of its principal moduli " for quadratic polynomials . this yields the density of hyperbolic maps in the real quadratic family . the same result is also obtained by graceyk and wiatek in @xcite . see also @xcite and @xcite . recently , the local connectivity of julia sets and combinatorial rigidity for unicritical polynomials are proved in @xcite and @xcite by means of principal nest and modified principal nest . this paper is organized as follows . in section 2 , we present some distortion lemmas which are used in section 4 . in section 3 , we introduce the branner - hubbard puzzle about rational maps with cantor julia set and the kss nest constructed in @xcite . by means of the kss nest and the distortion lemmas , we prove that the shapes of some critical puzzle pieces are bounded in section 4 . in section 5 , we give the proofs of theorem 1 and theorem 2 . any doubly connected domain @xmath35 on the complex plane is conformally equivalent to one of the following three types of typical domains : 1 . @xmath36 , 2 . @xmath37 , where @xmath38 , 3 . @xmath39 . in the case @xmath35 is conformally equivalent to @xmath40 , the modulus of @xmath35 is defined as @xmath41 . in the other two cases , @xmath42 . for @xmath43 , let @xmath44 $ ] . the modulus @xmath45 is decreasing in @xmath46 with @xmath47 and @xmath48 . let @xmath35 be a doubly connected domain in @xmath49 which separates the unit circle from the points @xmath50 . then @xmath51 . denote @xmath52 and @xmath53 the distance and diameter with respect to the euclidean metric respectively . let @xmath54 be two simply connected domains and @xmath55 . then there exists a constant @xmath56 such that @xmath57 for any @xmath58 . for any @xmath58 , let @xmath59 be a conformal map from @xmath49 onto @xmath26 with @xmath60 . then @xmath61 from grtzsch theorem , there exists a constant @xmath62 such that @xmath63 . by koebe distortion theorem , we have @xmath64 and @xmath65 so we can take @xmath66 , which satisfies the inequality @xmath67 let @xmath26 be a simply connected domain and @xmath68 . the _ shape _ of @xmath26 about @xmath69 , denoted by @xmath70 , is defined as @xmath71 let @xmath72 be a holomorphic proper map of degree @xmath9 with @xmath73 , @xmath74 . suppose that 1 . @xmath75 , 2 . @xmath76 . then there exists a constant @xmath77 such that @xmath78 let @xmath79 there are points @xmath80 and @xmath81 such that @xmath82 and @xmath83 . the holomorphic proper map @xmath84 can be written as @xmath85 where @xmath86 and @xmath87 by the first assumption , we have @xmath88 and @xmath89 for all @xmath90 . consider the annulus @xmath91 , we have @xmath92 by grtzsch theorem , there exists a constant @xmath93 such that @xmath94 for all @xmath90 . we have @xmath95 with @xmath96 . on the other hand , @xmath97 with @xmath98 in which @xmath99 and @xmath100 come from lemma 1 and its proof . it follows that @xmath101 with @xmath102 . from now on , we always assume that the julia set @xmath4 of a rational map @xmath2 is a cantor set . the fatou set @xmath103 has only one component . it is either an attracting basin or a parabolic basin . we first construct the branner - hubbard puzzle . * the attracting case*. we assume that @xmath104 is the fixed attracting point . take a simply connected neighborhood @xmath105 of @xmath104 such that @xmath106 . let @xmath107 be the component of @xmath108 containing @xmath104 . then @xmath109 and @xmath110 . for a large enough integer @xmath111 , @xmath112 has only one component for any @xmath113 . the set @xmath114 is the disjoint union of a finite number of topological disks . for each @xmath113 , let @xmath115 be the collection of all components of @xmath116 which are called puzzle pieces of depth @xmath117 . for any point @xmath118 and any @xmath113 , there is only one @xmath119 containing @xmath120 . thus each point @xmath118 determines a nested sequence @xmath121 and @xmath122 . * the parabolic case*. we suppose that @xmath123 is the parabolic fixed point and @xmath104 is in the fatou set . according to the leau - fatou flower theorem , there is a flower petal @xmath124 with @xmath125 such that @xmath126 . we can construct the puzzle as in the attracting case . each point @xmath127 determines a nested sequence @xmath128 and @xmath129 . take @xmath130 large enough such that @xmath131 contains all critical points in the fatou set and each puzzle piece contains at most one critical point . let @xmath132 in the attracting case and @xmath133 in the parabolic case . for each @xmath118 ( in parabolic case , @xmath134 respectively ) , the tableaux @xmath135 is defined in @xcite . it is a two dimension array @xmath136 with @xmath137 . the position @xmath138 is called critical if @xmath139 contains a critical point of @xmath2 . if @xmath139 contains a critical point @xmath99 , the position @xmath138 is called a @xmath99-position . the tableau @xmath140 of a critical point @xmath141 is called periodic if there is a positive integer @xmath142 such that @xmath143 for all @xmath113 . since the julia set is a cantor set , @xmath140 is not periodic for all @xmath141 . all the tableaus satisfy the following three rules 1 . if @xmath144 for some critical point @xmath99 , then @xmath145 for all @xmath146 . 2 . if @xmath144 for some critical point @xmath99 , then @xmath147 for @xmath148 . 3 . let @xmath140 be a tableau for some critical point @xmath99 and @xmath135 be any tableau . assume 1 . @xmath149 for some critical point @xmath150 and @xmath151 , and @xmath152 contains no critical points for @xmath153 . 2 . @xmath154 and @xmath155 for some @xmath156 . + then @xmath157 . \(1 ) the tableau @xmath135 for @xmath120 is _ non - critical _ if there exists an integer @xmath158 such that @xmath159 is not critical for all @xmath160 . \(2 ) we write @xmath161 if for any @xmath113 , there exists @xmath160 such that @xmath162 , i.e. , @xmath163 . it is clear that @xmath161 if and only if @xmath164 or @xmath165 , the limit set of the forward orbit of @xmath120 . if @xmath161 and @xmath166 , then @xmath167 . for each critical point @xmath168 , let @xmath169 and @xmath170=\{c^\prime\in\mathrm{crit}\,| \,\ , c\to c^\prime \textrm { and } c^\prime \to c \}.\ ] ] ( 3)we say @xmath171 is a _ child _ of @xmath172 if @xmath173 $ ] , @xmath174 , and @xmath175 is conformal . \(4 ) suppose @xmath176 , i.e. , @xmath177\not=\emptyset$ ] . we say @xmath140 is _ persistently recurrent _ if @xmath178 has only finitely many children for all @xmath113 and all @xmath179 $ ] . otherwise , @xmath140 is said to be _ reluctantly recurrent_. take @xmath111 large enough such that for any @xmath141 , there is no @xmath180-position in the first row of @xmath140 if @xmath181 . let @xmath182 then @xmath183 this is not a classification because these sets might intersect . the following lemma can be found in @xcite . if @xmath140 is persistently recurrent , then @xmath184 $ ] . now , we briefly introduce a critical nest which is constructed by kozlovski , shen , and van strien in @xcite . such nest will be called _ kss _ nest . let @xmath35 be an open set and @xmath185 . the connected component of @xmath35 containing @xmath120 will be denoted by comp@xmath186 . given a puzzle piece @xmath187 , let @xmath188 for any @xmath189 , let @xmath190 be the connected component of @xmath191 containing @xmath192 . we further define @xmath193 if @xmath194 and @xmath195 if @xmath196 . for any @xmath189 , let @xmath197 be the integer such that @xmath198 and let @xmath199 be the depth of @xmath187 . by tableau rules ( t1 ) and ( t2 ) , there is at most one @xmath99-position on the diagonal @xmath200 in the tableau @xmath201 for any @xmath202 . hence @xmath203 for some constant @xmath204 depending only on @xmath205 . suppose @xmath206 is persistently recurrent , then @xmath207 $ ] . let @xmath208,\,\,\,\,d_0=\deg_{c_0}f,\,\,\,\,d_{max}=\max\{\deg_cf @xmath209)=\bigcup_{n\geqslant 0}f^n([c_0]).\ ] ] for each puzzle piece @xmath210 , there are pullbacks @xmath211 of @xmath187 containing @xmath212 with the following properties 1 . @xmath213 and @xmath214 , 2 . @xmath215 , @xmath216 , @xmath217 is the smallest integer such that @xmath218 and @xmath219 , 3 . @xmath220 ) = \emptyset$ ] . for details , see @xcite or @xcite . given a puzzle piece @xmath221 containing @xmath212 , a _ successor _ of @xmath221 is a piece of the form @xmath222 , where @xmath223 is a child of @xmath224 for some @xmath225 $ ] . it is clear that @xmath226 is a successor of @xmath221 . since @xmath227 is persistently recurrent , @xmath221 has at least two successors and that @xmath221 has only finitely many successors . let @xmath228 be the last successor of @xmath221 . then there exists an integer @xmath229 , the largest among all of the successors of @xmath221 , such that @xmath230 . from the definition of successor , we have 1 . @xmath231 . now we can define the _ kss nest _ in the following way : @xmath232 is a given piece containing @xmath233 and for @xmath113 , @xmath234 with @xmath235 $ ] . see figure [ fig2 ] . suppose @xmath236 , @xmath237 , @xmath238 for @xmath239 , and @xmath240 . let @xmath241 and @xmath242 . then @xmath243)=\emptyset$ ] and @xmath244)=(\mathcal{b}(l_{n})\setminus \mathcal{a}(l_{n}))\cap \mathrm{orb}([c_{0}])=\emptyset.\ ] ] it follows that @xmath245 see figure [ fig2 ] . . then @xmath247 and @xmath248 for any puzzle piece @xmath249 containing @xmath212 and @xmath250)$ ] , let @xmath251 be the smallest integer such that @xmath252 and @xmath253)\}.\ ] ] it is obvious that 1 . @xmath254 if @xmath255 ; 2 . @xmath256 if @xmath257 , @xmath258 and @xmath259 for @xmath260 ; 3 . @xmath261 . the following lemma plays a crucial rule in the proof of our results and in @xcite . for any @xmath262 , 1 . @xmath263 ; 2 . @xmath264 ; 3 . @xmath265 for @xmath266 . the following is an immediately corollary . for any @xmath262 , 1 . @xmath267 2 . @xmath268 . we suppose @xmath206 is persistently recurrent and puzzle pieces @xmath269 are constructed as in the previous section . for the polynomial case , the following is the key lemma in @xcite . @xmath270 . there exist a constant @xmath156 depending only on @xmath271 and @xmath272 , and an integer @xmath199 such that @xmath273 and @xmath274 for all @xmath275 . in the attracting case , the annuli @xmath276 and @xmath277 are always non - degenerate . in the parabolic case , there exists an integer @xmath199 such that @xmath276 and @xmath278 are non - degenerate for @xmath279 . in fact , there is an integer @xmath280 such that @xmath281 is non - degenerate because @xmath282 and @xmath283 . take @xmath284 as @xmath232 in the construction of kss nest . by corollary 1 , @xmath285 and @xmath286 so there exists an integer @xmath287 such that @xmath288 and @xmath289 for @xmath275 . this implies that @xmath290 and @xmath277 are non - degenerate for @xmath275 because @xmath291 and @xmath292 are pullbacks of @xmath293 . by the same proof of lemma 5 , @xmath294 for some constant @xmath295 depending only on @xmath271 and @xmath272 when @xmath275 . see @xcite . suppose @xmath296 . let @xmath297 and @xmath298 . since @xmath299 and @xmath300 we conclude that @xmath301 from corollary 1 . from properties ( p1 ) , ( p2 ) and ( p4 ) , @xmath302 for some @xmath204 depending only on @xmath271 and @xmath272 . hence @xmath303 take @xmath304 . this @xmath305 depends only on @xmath271 and @xmath272 , and satisfied the conditions set out in this lemma . there exists a constant @xmath306 such that @xmath307 for @xmath275 , where @xmath199 is the integer in lemma 6 . since @xmath308 ) = ( k_{n}\setminus \widetilde{k}_{n})\cap \mathrm{orb}([c_{0 } ] ) = \emptyset$ ] and @xmath309 , it follows @xmath310 . let @xmath311 and @xmath312 . then @xmath313 for some constant @xmath314 depending only on @xmath271 and @xmath272 . let @xmath315 and @xmath316 be conformal maps with @xmath317 and @xmath318 . let @xmath319 . then @xmath72 is a properly holomorphic map with @xmath320 by koebe distortion theorem and lemma 2 , there exists a constant @xmath321 depending only on @xmath271 and @xmath272 such that @xmath322 hold for all @xmath275 . we conclude that @xmath323 for some constant @xmath306 when @xmath275 . recall the definition of @xmath324 in section 3 . let @xmath325 in the attracting case and @xmath326 in the parabolic case , where @xmath123 is the parabolic fixed point . \(1 ) @xmath135 is non - critical , i.e. @xmath336 . there exists an integer @xmath158 such that @xmath159 is not a critical position for all @xmath160 . for any @xmath262 , @xmath337 . the degrees of these maps @xmath338 has an upper bound @xmath204 . take a subsequence @xmath334 of @xmath339 such that @xmath340 for some fixed puzzle piece @xmath333 . then @xmath335 for all @xmath117 . \(2 ) @xmath341 for some @xmath342 . from ( 1 ) , there are a puzzle piece @xmath333 , a positive integer @xmath343 and infinitely many @xmath344 such that @xmath345 for all @xmath117 . for each @xmath117 , let @xmath346 be the first moment such that @xmath347 , @xmath348 is the first @xmath99-position on the @xmath344-th row in @xmath135 . by tableau rules ( t1 ) and ( t2 ) , there is at most one @xmath349-position on the diagonal @xmath350 for any @xmath351 there exists a positive integer @xmath352 depending on @xmath353 such that @xmath354 take @xmath355 and @xmath356 . then @xmath335 for all @xmath117 . \(3 ) @xmath341 for some @xmath357 . there exist an integer @xmath358 , @xmath359 $ ] , @xmath360 $ ] and infinitely many integers @xmath361 such that @xmath362 are children of @xmath363 . since @xmath359 $ ] , we have @xmath364 . for each @xmath117 , let @xmath365 be the first moment such that @xmath366 . there is at most one @xmath367-position on the diagonal @xmath368 in @xmath135 for any @xmath369 therefore , @xmath370 and there is an integer @xmath371 independent of @xmath117 such that @xmath372 for any @xmath262 . take @xmath373 , @xmath374 and @xmath375 . then @xmath335 for all @xmath117 . \(4 ) @xmath376 , i.e. @xmath377 and @xmath378 . take @xmath379 . let@xmath380 be all @xmath212-positions in @xmath135 . we claim that there is at most one @xmath99-position on the diagonal @xmath381 for all @xmath382 . if this is false , there are at least two @xmath99-positions on this diagonal for some @xmath383 . this means @xmath384 and @xmath385 . by lemma 1 , @xmath184 $ ] and @xmath386 . it contradicts with @xmath379 . so the above claim is true . there exists a positive integer @xmath387 such that @xmath388 take a subsequence @xmath389 of @xmath390 such that @xmath340 for a fixed @xmath333 . then @xmath335 for all @xmath117 . * the attracting case*. there is a subsequence of @xmath389 , say itself , such that @xmath394 converges to some point @xmath395 . we assume that @xmath396 for all @xmath117 . it is obvious that there is a disk @xmath397 in @xmath398 for some constant @xmath399 . by distortion results for holomorphic @xmath400-valent mappings(see @xcite , @xcite,@xcite,@xcite and @xcite ) , there are constants @xmath401 and @xmath402 depending on @xmath120 such that @xmath403 and @xmath404 for all @xmath117 . since @xmath405 , the point @xmath120 is not a density point of @xmath4 . * the parabolic case*. if there are a puzzle piece @xmath406 of depth @xmath199 compactly contained in @xmath333 and infinitely many @xmath334 such that @xmath407 , we can prove that @xmath120 is not a density point of @xmath4 by the same argument as in the attracting case . otherwise , there is a subsequence of @xmath394 , say itself , converges to some point @xmath408 . the point @xmath409 belongs to @xmath410 . we assume that @xmath123 is the parabolic fixed point as before . in the construction of the branner - hubbard puzzle , the flower petal @xmath411 can be chosen such that there exists a sector @xmath412 with the vertex at @xmath409 . for all @xmath117 , let @xmath413 be the distance from the point @xmath414 to the boundary of the sector @xmath415 . let @xmath416 and @xmath417 then @xmath418 by the leau - fatou flower theorem , there is a disk @xmath419 for large @xmath117 . by the same argument as above , @xmath120 is not a density point of @xmath4 . let @xmath2 be a rational map of degree @xmath425 and @xmath120 be a point in @xmath4 . if there are a constant @xmath5 , a positive integer @xmath426 and a sequence @xmath427 in @xmath420 with the following properties : 1 . @xmath428 , @xmath429 are topological disks and @xmath430 as @xmath431 . @xmath432 is a proper map of degree between @xmath433 and @xmath434 . 3 . for some @xmath435 such that @xmath436 and for @xmath437 we have @xmath438 and @xmath439 . 4 . @xmath440 for any @xmath275 , the annuli @xmath448 and @xmath449 are non - degenerate . let @xmath346 be the first moment such that @xmath450 . let @xmath451 , @xmath452 and @xmath453 . for large @xmath117 , the puzzle piece @xmath454 contains no critical points . let @xmath455 be the smallest integer such that @xmath456 contains a critical point @xmath225 $ ] . set @xmath457 , @xmath458 and @xmath459 . see figure [ fig3 ] . from the conditions @xmath460)=(k_{n}\setminus \widetilde{k}_{n})\cap \mathrm{orb}([c_{0}])=\emptyset\ ] ] and @xmath461 , we know that @xmath462 is conformal and @xmath463 for some constant @xmath464 depending only on @xmath353 . by proposition 1 and lemma 2 , @xmath465 and @xmath466 for some constant @xmath467 . let @xmath468 , @xmath469 and @xmath470 . further let @xmath471 , @xmath472 and @xmath473 . then @xmath474 for some constant @xmath475 depending only on @xmath205 . there exists a positive constant @xmath476 such that @xmath477 for each large @xmath117 , define @xmath478 . then @xmath479 is a properly holomorphic mapping of degree between @xmath433 and some constant @xmath434 . all conditions in proposition 3 are satisfied , hence the lemma holds . if @xmath2 has an invariant line field @xmath441 on the julia set @xmath4 , then there exists a positive measure subset @xmath15 of @xmath4 such that @xmath480 . since @xmath441 is measurable in @xmath393 , almost every point in @xmath393 is almost continuous . from proposition 4 and proposition 2 , @xmath481 @xmath482 . it is a contradiction . so @xmath2 carries no invariant line field on its julia set . by the implicit function theorem , any rational map with indifferent cycles is structurally unstable , therefore @xmath2 has no siegel disks or parabolic basins . moreover , any rational map with herman rings is also structurally unstable by a result due to ma , see @xcite . these imply that @xmath492 . from theorem 1 , @xmath493 . we conclude that @xmath494 and all critical points are in the attracting fatou components . it means that @xmath2 is hyperbolic . m. shishikura and tan lei , _ an alternative proof of ma s theorem on non - expanding julia sets_. in : the mandelbrot set , theme and variations , edited by tan lei , 265 - 279 , london math . lecture note ser . , no 274 , cambrige university press , cambrige , 2000 . y. yin , _ geometry and dimension of julia sets_. in : the mandelbrot set , theme and variations , edited by tan lei , 281 - 287 , london math . soc . lecture note ser . , no 274 , cambrige university press , cambrige , 2000 .

in this paper , we prove that a rational map with a cantor julia set carries no invariant line field on its julia set . it follows that a structurally stable rational map with a cantor julia set is hyperbolic .

consider communication over a discrete - time memoryless channel modeled by a conditional point mass function ( pmf ) or probability density function ( pdf ) @xmath0 , where @xmath1 and @xmath2 are the input and output symbols , @xmath3 and @xmath4 are the input and output alphabets , respectively . let @xmath5 be the shannon capacity . fano showed in @xcite that the minimum error probability @xmath6 for block channel codes of rate @xmath7 and length @xmath8 is bounded by @xmath9 where @xmath10 is a positive function of channel transition probabilities , known as the error exponent . for finite input and output alphabets , without coding complexity constraint , the maximum achievable @xmath10 is given by gallager in @xcite , @xmath11 where @xmath12 is the input distribution , and @xmath13 is given for different values of @xmath7 as follows , @xmath14 the definitions of other variables in ( [ gallagere1 ] ) can be found in @xcite . if we replace the pmf by pdf , the summations by integrals and the @xmath15 operators by @xmath16 in ( [ gallagere ] ) , ( [ gallagere1 ] ) , the maximum achievable error exponent for continuous channels , i.e. , channels whose input and/or output alphabets are the set of real numbers @xcite , is still given by ( [ gallagere ] ) . in @xcite , forney proposed a one - level concatenated coding scheme , which can achieve the following error exponent , known as forney s exponent , for any rate @xmath17 with a complexity of @xmath18 . @xmath19}(1-r_o)e\left ( \frac{r}{r_o}\right ) , \label{ecr}\ ] ] where @xmath20 and @xmath7 are the outer and the overall rates , respectively . forney s coding scheme concatenates a maximum distance separable ( mds ) outer error - correction code with well performed inner channel codes . to achieve @xmath21 , the decoder is required to exploit reliability information from the inner codes using a general minimum distance ( gmd ) decoding algorithm @xcite . forney s gmd algorithm essentially carries out outer code decoding , under various conditions , for @xmath22 times . the overall decoding complexity of @xmath18 is due to the fact that the outer code ( which is a reed - solomon code ) used in @xcite has a decoding complexity of @xmath23 . forney s concatenated codes were generalized to multi - level concatenated codes , also known as the generalized concatenated codes , by blokh and zyablov in @xcite . as the order of concatenation goes to infinity , the error exponent approaches the following blokh - zyablov bound ( or blokh - zyablov error exponent ) @xcite@xcite . @xmath24}}\left ( \frac{r}{r_o}-r\right)\left [ \int^{\frac{r}{r_o}}_0\frac{dx}{e_l(x , p_x)}\right ] ^{-1}.\ ] ] in @xcite , guruswami and indyk proposed a family of linear - time encodable / decodable nearly mds error - correction codes . by concatenating these codes ( as outer codes ) with _ fixed - lengthed _ binary inner codes , together with justesen s gmd algorithm @xcite , forney s error exponent was shown to be achievable over binary symmetric channels ( bscs ) with a complexity of @xmath22 @xcite , i.e. , linear in the codeword length . the number of outer code decodings required by justesen s gmd algorithm is only a constant , as opposed to @xmath22 in forney s case @xcite . since each outer code decoding has a complexity of @xmath22 , upper - bounding the number of outer code decodings by a constant is required for achieving the overall linear complexity . because justesen s gmd algorithm assumes binary channel outputs @xcite@xcite , achievability of forney s exponent was only proven for bscs in ( * ? ? ? * ; * ? ? ? * theorem 8) . in this paper , we show that forney s gmd algorithm can be revised to carry out outer code decoding for only a constant number of times . with the help of the revised gmd algorithm , by using guruswami - indyk s outer codes with fixed - lengthed inner codes , one - level and multi - level concatenated codes can arbitrarily approach forney s and blokh - zyablov exponents with linear complexity , over general discrete - time memoryless channels . consider one - level concatenated coding schemes . assume , for an arbitrarily small @xmath25 , we can construct a linear encodable / decodable outer error - correction code , with rate @xmath20 and length @xmath26 , which can correct @xmath27 symbol errors and @xmath28 symbol erasures so long as @xmath29 . note that this is possible for large @xmath26 as shown by guruswami and indyk in @xcite . to simplify the notations , we assume @xmath30 is an integer . the outer code is concatenated with suitable inner codes with rate @xmath31 and fixed length @xmath32 . the rate and length of the concatenated code are @xmath33 and @xmath34 , respectively . in forney s gmd decoding , inner codes forward not only the estimates @xmath35 $ ] but also a reliability vector @xmath36 $ ] to the outer code , where @xmath37 , @xmath38 and @xmath39 . let @xmath40 for any outer codeword @xmath41 $ ] , define a dot product @xmath42 as follows @xmath43 [ theorem1 ] there is at most one codeword @xmath44 that satisfies @xmath45 theorem [ theorem1 ] is implied by theorem 3.1 in @xcite . rearrange the weights in ascending order of their values and let @xmath46 be the indices such that @xmath47 define @xmath48 $ ] , for @xmath49 , where @xmath50 is a positive constant with @xmath51 being an integer , and @xmath52 is given by @xmath53 define dot product @xmath54 as @xmath55 then following theorem gives the key result that enables the revision of forney s gmd decoder . [ theorem2 ] if @xmath56 , then for some @xmath49 , @xmath57@xmath58 . define a set of values @xmath59 for @xmath60 and an integer @xmath61 , where @xmath62 . can not be @xmath63 . because if @xmath64 , i.e. , @xmath65 , then there are at least @xmath30 zeros in vector @xmath66 . consequently , @xmath67 , which contradicts the assumption that @xmath56 . ] let @xmath68 we have @xmath69 and @xmath70 define a new weight vector @xmath71 $ ] with @xmath72 define @xmath73 $ ] with @xmath74 such that for @xmath75 @xmath76 and for @xmath77 @xmath78 we have @xmath79 define a set of indices @xmath80 according to the definition of @xmath81 , for @xmath82 , @xmath83 . hence @xmath84 since @xmath85 , and @xmath86 , we have @xmath87 consequently , @xmath88 implies @xmath89 if @xmath90 for all @xmath91 s , then @xmath92 which contradicts ( [ alphax ] ) . therefore , there must be some @xmath91 that satisfies @xmath93 since for @xmath94 , @xmath91 has no more than @xmath95 number of @xmath96 s , which implies @xmath97 , the vectors that satisfy ( [ px ] ) must exist among @xmath91 with @xmath98 . in words , for some @xmath99 , @xmath100 . theorems [ theorem1 ] and [ theorem2 ] indicate that , if @xmath44 is transmitted and @xmath56 , for some @xmath49 , errors - and - erasures decoding specified by @xmath101 ( where symbols with @xmath102 are erased ) will output @xmath44 . since the total number of @xmath101 vectors is upper bounded by a constant @xmath51 , the outer code carries out errors - and - erasures decoding only for a constant number of times . consequently , a gmd decoding that carries out errors - and - erasures decoding for all @xmath101 s and compares their decoding outputs can recover @xmath44 with a complexity of @xmath103 . since the inner code length @xmath32 is fixed , the overall complexity is @xmath22 . the following theorem gives an error probability bound for one - level concatenated codes with the revised gmd decoder . [ theorem3 ] assume inner codes achieve gallager s error exponent given in ( [ gallagere ] ) . let the reliability vector @xmath66 be generated according to forney s algorithm presented in ( * ? ? ? * ; * ? ? ? * section 4.2 ) . let @xmath44 be the transmitted outer codeword . for large enough @xmath8 , error probability of the one - level concatenated codes is upper bounded by @xmath104,\end{aligned}\ ] ] where @xmath21 is forney s error exponent given by ( [ ecr ] ) and @xmath105 is a function of @xmath106 and @xmath107 with @xmath108 if @xmath109 . the proof of theorem [ theorem3 ] can be obtained by first replacing theorem 3.2 in @xcite with theorem [ theorem2 ] , and then following forney s analysis presented in ( * ? ? ? * ; * ? ? ? * section 4.2 ) . the difference between forney s and the revised gmd decoding schemes lies in the definition of errors - and - erasures decodable vectors @xmath101 , the number of which determines the decoding complexity . forney s gmd decoding needs to carry out errors - and - erasures decoding for a number of times linear in @xmath26 , whereas ours for a constant number of times . although the idea behind the revised gmd decoding is similar to justesen s gmd algorithm @xcite , justesen s work has focused on error - correction codes where inner codes forward hamming distance information ( in the form of an @xmath110 vector ) to the outer code . applying the revised gmd algorithm to multi - level concatenated codes @xcite@xcite is quite straightforward . achievable error exponent of an @xmath111-level concatenated codes is given in the following theorem . [ theorem4 ] for a discrete - time memoryless channel with capacity @xmath5 , for any @xmath112 and any integer @xmath113 , one can construct a sequence of @xmath111-level concatenated codes whose encoding / decoding complexity is linear in @xmath8 , and whose error probability is bounded by @xmath114}}\frac{\frac{r}{r_o}-r}{\frac{r}{r_om}\sum_{i=1}^m\left [ e_l\left((\frac { i}{m})\frac{r}{r_o},p_x\right ) \right ] ^{-1 } } \nonumber \\\end{aligned}\ ] ] the proof of theorem [ theorem4 ] can be obtained by combining theorem [ theorem3 ] and the derivation of @xmath115 in @xcite@xcite . note that @xmath116 , where @xmath117 is the blokh - zyablov error exponent given in ( [ bzbound ] ) . theorem [ theorem4 ] implies that , for discrete - time memoryless channels , blokh - zyablov error exponent can be arbitrarily approached with linear encoding / decoding complexity . we proposed a revised gmd decoding algorithm for concatenated codes over general discrete - time memoryless channels . by combining the gmd algorithm with guruswami and indyk s error correction codes , we showed that forney s and blokh - zyablov error exponents can be arbitrarily approached by one - level and multi - level concatenated coding schemes , respectively , with linear encoding / decoding complexity . the authors would like to thank professor alexander barg for his help on multi - level concatenated codes .

guruswami and indyk showed in @xcite that forney s error exponent can be achieved with linear coding complexity over binary symmetric channels . this paper extends this conclusion to general discrete - time memoryless channels and shows that forney s and blokh - zyablov error exponents can be arbitrarily approached by one - level and multi - level concatenated codes with linear encoding / decoding complexity . the key result is a revision to forney s general minimum distance decoding algorithm , which enables a low complexity integration of guruswami - indyk s outer codes into the concatenated coding schemes . coding complexity , concatenated code , error exponent

people have proposed various baryogensis mechanisms to understand the cosmic matter - antimatter asymmetry which is as same as a baryon asymmetry . the leptogenesis @xcite in seesaw @xcite context has become one of the most attractive baryogenesis mechanisms because it can simultaneously explain the generation of baryon asymmetry and the smallness of neutrino masses . however , the conventional leptogenesis scenario contains many free parameters so that it can not give the exact dependence of the baryon asymmetry on the neutrino mass matrix unless we do some assumptions on the texture of the relevant masses and couplings . for example , we can expect a successful leptogenesis in the canonical seesaw model even if the neutrino mass matrix does nt contain any cp phases @xcite . in this paper we shall propose a new leptogenesis scenario in an @xmath0 left - right symmetric @xcite model to give some predictions on the mass scale @xmath2 and the cp violation @xmath3 of the dirac neutrinos . motivated by solving the strong cp problem without introducing an unobserved axion @xcite , we will consider a softly broken parity symmetry under which the dimensionless couplings of the @xmath4 mirror fields are identified to the couplings of the @xmath5 ordinary fields . the mirror dirac neutrinos can have a heavy mass matrix proportional to the seesaw - suppressed @xcite mass matrix of the ordinary dirac neutrinos . through the @xmath6 gauge interactions , the mirror neutrinos can decay to produce a lepton asymmetry in the mirror muons and an opposite lepton asymmetry in the mirror electrons . both of the mirror muons and electrons can decay into the ordinary right - handed leptons with some dark matter scalars . but the lifetime of the mirror muons can be shorter than that of the mirror electrons . this means the mirror electron asymmetry can be expected not to participate in the @xmath7 sphaleron processes . in other words , only the mirror muon asymmetry can be partially converted to a baryon asymmetry . the @xmath8 charged fields include the following fermions , @xmath9 q'^{}_{r}(2,+\frac{1}{6}),~\ ! d'^{}_{l}(1,-\frac{1}{3}),~\ ! u'^{}_{l}(1,+\frac{2}{3}),~\!l'^{}_{r}(2,-\frac{1}{2}),~\ ! e'^{}_{l}(1,-1);\end{array}\nonumber\\ & & \end{aligned}\ ] ] and the following scalars , @xmath10 here the ordinary fields without prime are charged under the @xmath5 gauge groups while the mirror fields with prime are charged under the @xmath11 gauge symmetry . we also have three @xmath12$]-singlet scalars which are nontrivial under the @xmath13 symmetries , @xmath14 with the brackets being the @xmath13 charges . furthermore , we introduce the ordinary right - handed neutrinos and the mirror left - handed neutrinos , @xmath15 which are @xmath16 singlets but carry a quantum number @xmath17 under an additional @xmath18 gauge symmetry . the scalars @xmath19 and @xmath20 can be distinguished as they are assumed to have the @xmath18 charges @xmath21 and @xmath22 , respectively . the parity invariant yukawa interactions involving the above fields then can be written down , @xmath23 the ordinary fermions(scalars ) are odd(even ) under a @xmath24 discrete symmetry while the mirror fermions(scalars ) are odd(even ) under a @xmath25 discrete symmetry . this @xmath26 symmetry will not be broken at any scales . therefore , the scalars @xmath27 will not acquire any nontrivial vacuum expectation values . at least two of the scalars @xmath27 can keep stable because of their couplings , @xmath28 we can consider an @xmath12$]-singlet scalar @xmath29 with a proper @xmath13 charge such as @xmath30 to spontaneously break the present @xmath13 down to the usual @xmath31 . subsequently , the @xmath32$]-doublet scalar @xmath33 and the @xmath34$]-doublet scalar @xmath35 will drive the spontaneous symmetry breaking @xmath36 and @xmath37 , respectively . as for the @xmath18 , it will be spontaneously broken when an @xmath38$]-singlet scalar , @xmath39 develops its vacuum expectation value . in the presence of the cubic terms as below , @xmath40 the @xmath41$]-doublet scalars @xmath42 and @xmath43 can obtain the induced vacuum expectation values , @xmath44 as the discrete parity symmetry is softly broken in the scalar potential , the ordinary(mirror ) vacuum expectation values can have a large(mild ) hierarchy , @xmath45 after the above symmetry breaking , it is easy to read the following relation between the ordinary and mirror fermion mass matrices from the parity invariant yukawa interactions ( [ yukawa ] ) , @xmath46 \frac{\langle\eta'\rangle}{\langle\eta\rangle } & = & \frac{m_{\nu'^{}_{1}}^{}}{m_{\nu^{}_{1}}^{}}=\frac{m_{\nu'^{}_{2}}^{}}{m_{\nu^{}_{2}}^ { } } = \frac{m_{\nu'^{}_{3}}^{}}{m_{\nu^{}_{3}}^{}}\,,\nonumber\\ [ 2 mm ] v&=&v'^{}_{\textrm{ckm}}=v^{}_{\textrm{ckm}}\,,~~u = u'^{}_{\textrm{pmns}}=u^{}_{\textrm{pmns}}\,.\end{aligned}\ ] ] here @xmath47 and @xmath48 denote the ordinary and mirror fermion mass eigenvalues , while @xmath49 and @xmath50 are the ckm and pmns matrices . clearly , the mirror dirac neutrinos have a heavy mass matrix proportional to the seesaw - suppressed mass matrix of the ordinary dirac neutrinos . for the following demonstration , we also give the charged gauge boson masses @xmath51 as well as the pmns matrix which now does nt contain any majorana phases , @xmath52 -s_{12}^{}c_{23}^{}-c_{12}^{}s_{23}^{}s_{13}^{}e^{i\delta } _ { } & ~~c_{12}^{}c_{23}^{}-s_{12}^{}s_{23}^{}s_{13}^{}e^{i\delta } _ { } & s_{23}^{}c_{13}^{}\\ [ 2 mm ] ~~s_{12}^{}s_{23}^{}-c_{12}^{}c_{23}^{}s_{13}^{}e^{i\delta } _ { } & -c_{12}^{}s_{23}^{}-s_{12}^{}c_{23}^{}s_{13}^{}e^{i\delta } _ { } & c_{23}^{}c_{13}^ { } \end{array}}\right].\end{aligned}\ ] ] the mirror neutrinos can have the two - body and three - body decay modes as shown in fig . [ decay ] , when their masses are in the following range , @xmath53 we calculate the decay width at tree level , @xmath54 \!\!\!\!\!\!&&+\sum_{\alpha\beta\gamma}^{}\gamma(\nu'^{}_i\rightarrow e'^{-}_{\alpha}+q'^{{\tiny+1/3}}_{\beta}+q'^{{\tiny+2/3}}_{\gamma } ) \nonumber\\ [ 2 mm ] \!\!\!\!\!\!&\simeq&\frac{3g^4_{}}{256\pi^3_{}}m_{\nu'^{}_{i}}^{}\left[1+\frac{4\pi^2_{}}{g^2_{}}\left(|u_{ei}^{}|^2_{}+|u_{\mu i}^{}|^2_{}\right)\xi^2_{}\right]\,,\end{aligned}\ ] ] where the parameter @xmath55 is given by @xmath56 although the mirror neutrino decays exactly conserve the lepton number , they can generate a lepton asymmetry in the mirror muons and an opposite lepton asymmetry in the mirror electrons at one - loop level , @xmath57 the relevant diagrams are shown in fig . [ mndecay ] . if the mirror neutrinos have a quasi - degenerate mass spectrum , the self - energy corrections will dominate the cp asymmetry with a resonant enhancement @xcite , @xmath58 it is easy to read the cp asymmetries in the decays of the quasi - degenerate mirror neutrinos , @xmath59 here we have quoted the cp - violating parameter , @xmath60 in the case the mirror neutrinos have a hierarchical mass spectrum , both the self - energy corrections and the vertex corrections will significantly contribute to the cp asymmetry , @xmath61 the cp asymmetry in the decays of the lightest mirror neutrino ( @xmath62 for normal hierarchy or @xmath63 for inverted hierarchy ) then should be given by @xmath64 in the presence of the scalars @xmath27 , the induced mirror electron and muon asymmetries will be immediately cancelled each other if the decaying and scattering processes shown in fig . [ conversation ] are very fast . we expect such processes to go into equilibrium at some low temperatures such as @xmath65 where the @xmath66 sphaleron processes have not been active no longer . for this purpose we can require the interaction rates smaller than the hubble constant at the crucial temperature @xmath67 , @xmath68 & = & \frac{\left(f^\dagger_{e}f^{}_e\right)_{\mu'\mu'}^ { } \left(f^\dagger_{e}f^{}_e\right)_{e'e'}^{}}{6144\pi^3_{}}\frac{m_{\mu'}^{5}}{m_{\chi_e^{}}^4 } < h(t)\left|^{}_{t_{\textrm{sph}}}\right.\,,\nonumber\\ & & \\ \gamma_{s}^{\mu'\rightarrow e'}&= & \sum_{\alpha\beta}^{}\gamma(\mu'^{-}_{}+e^{+}_{\alpha}\rightarrow e'^{-}_{}+e^{+}_{\beta})\nonumber\\ [ 2 mm ] & \simeq&\frac{3\left(f^\dagger_{e}f^{}_e\right)_{\mu'\mu'}^ { } \left(f^\dagger_{e}f^{}_e\right)_{e'e'}^{}}{4\pi^3_{}}\frac{t_{}^{5}}{m_{\chi_e^{}}^4 } < h(t)\left|^{}_{t_{\textrm{sph}}}\right.\,,\nonumber\\ & & \\ \gamma_{t}^{\mu'\rightarrow e'}&=&\sum_{\alpha\beta}^{}\gamma(\mu'^{-}_{}+e^{-}_{\alpha}\rightarrow e'^{-}_{}+e^{-}_{\beta})\nonumber\\ [ 2 mm ] & \simeq&\frac{\left(f^\dagger_{e}f^{}_e\right)_{\mu'\mu'}^ { } \left(f^\dagger_{e}f^{}_e\right)_{e'e'}^{}}{4\pi^3_{}}\frac{t_{}^{5}}{m_{\chi_e^{}}^4 } < h(t)\left|^{}_{t_{\textrm{sph}}}\right.\,,\nonumber\\ & & \\ \gamma_{t}^{\mu'+e'}&=&\sum_{\alpha\beta}^{}\gamma(\mu'^{-}_{}+e^{+}_{\alpha}\rightarrow e'^{-}_{}+e^{+}_{\beta})\nonumber\\ [ 2 mm ] & \simeq&\frac{\left(f^\dagger_{e}f^{}_e\right)_{\mu'\mu'}^ { } \left(f^\dagger_{e}f^{}_e\right)_{e'e'}^{}}{4\pi^3_{}}\frac{t_{}^{5}}{m_{\chi_e^{}}^4 } < h(t)\left|^{}_{t_{\textrm{sph}}}\right.\,.\nonumber\\ & & \end{aligned}\ ] ] here the hubble constant is given by @xmath69 with @xmath70 being the planck mass and @xmath71 being the relativistic degrees of freedom . it is well known we can obtain a baryon asymmetry from a lepton asymmetry produced before the @xmath66 sphaleron processes stop working at a temperature around @xmath65 @xcite . on the other hand , we have already got a lepton asymmetry stored in the mirror muons and an opposite lepton asymmetry stored in the mirror electrons . these mirror muons and electrons will decay into the ordinary right - handed leptons with some stable scalars . the mirror muon and electron decays have been shown in fig . [ emudecay ] . if the mirror muons and electrons both decay efficiently above the scale @xmath67 , the mirror muon and electron asymmetries will both participate in the sphalerons . in consequence , we will fail in getting a nonzero baryon asymmetry from the induced mirror lepton asymmetries . however , it is allowed that the mirror muons can have a shorter life time while the mirror electrons can have a longer life time , i.e. @xmath72 & & \left.+\gamma(\mu'^{-}_{}\rightarrow e^{-}_{\alpha}+\chi^{}_d+\chi^\ast_u+\chi^\ast_u)\right]\nonumber\\ [ 2 mm ] & = & \frac{\left(f^\dagger_{e}f^{}_e\right)_{\mu'\mu'}^{}}{32768\pi^5 _ { } } \left(|\kappa_3^{}|^2+\frac{1}{3}|\kappa_2^{}|^2\right)\frac{m_{\mu'}^{5}}{m_{\chi_e^{}}^4 } > h(t_{\textrm{sph}}^{})\,,\nonumber\\ & & \\ \gamma_{e'}^{}&=&\sum_{\alpha}^{}\left[\gamma(e'^{-}_{}\rightarrow e^{-}_{\alpha}+\chi^\ast_d+\chi^\ast_d+\chi^\ast_d)\right.\nonumber\\ [ 2 mm ] & & \left.+\gamma(e'^{-}_{}\rightarrow e^{-}_{\alpha}+\chi^{}_d+\chi^\ast_u+\chi^\ast_u)\right]\nonumber\\ [ 2 mm ] & = & \frac{\left(f^\dagger_{e}f^{}_e\right)_{e'e'}^{}}{32768\pi^5 _ { } } \left(|\kappa_3^{}|^2+\frac{1}{3}|\kappa_2^{}|^2\right)\frac{m_{e'}^{5}}{m_{\chi_e^{}}^4}<h(t_{\textrm{sph}}^{})\,.\nonumber\\ & & \end{aligned}\ ] ] in this case , similar to the left - handed lepton asymmetry and the opposite right - handed neutrino asymmetry in the neutrinogenesis scenario @xcite , the mirror muon asymmetry rather than the mirror electron asymmetry will be partially converted to an ordinary baryon asymmetry through the sphalerons @xcite , @xmath73 \frac{\varepsilon_{\nu'^{}_1}^\mu}{g_\ast^ { } } & \textrm{for~normal~hierarchy } \,,\\ [ 2 mm ] \frac{\varepsilon_{\nu'^{}_3}^\mu}{g_\ast^{}}&\textrm{for~inverted~hierarchy}\,.\end{array}\right.\end{aligned}\ ] ] here we have taken the weak washout condition @xmath74 into account . this can be achieved by choosing the mirror neutrinos heavy enough . by inputting the cosmic baryon asymmetry @xcite , @xmath75 and the neutrino oscillation data @xcite , @xmath76 \!\!\!\!&&\!\!\!\!s_{12}^2=0.308\,,~s_{23}^2=0.437\,,~s_{13}^2=0.0234\,;\\ [ 2 mm ] \!\!\!\!&&\textrm{or}\nonumber\\ [ 2 mm ] \!\!\!\!&&\!\!\!\!\delta m_{21}^2=7.54\times 10^{-5}_{}\,\textrm{ev}^2_{}\,,~\delta m_{31}^2=-2.38\times 10^{-3}_{}\,\textrm{ev}^2_{}\,,\nonumber\\ [ 2 mm ] \!\!\!\!&&\!\!\!\!s_{12}^2=0.308\,,~s_{23}^2=0.455\,,~s_{13}^2=0.0240\,,\end{aligned}\ ] ] we can drive a low limit on the cp violation for the quasi degenerate neutrinos , @xmath77 & & m_{1}^{}\simeq m_{2}^{}\simeq m_{3}^{}=0.2\,\textrm{ev } \longrightarrow \sum m_\nu^{}=0.6\,\textrm{ev}\,,\quad\end{aligned}\ ] ] alternatively , a low limit on the mass scale for the maximal cp violation , @xmath78 \!\!\!\!\!\!\!\!\!\!\!\!&&\longrightarrow \sum m_\nu^{}=0.0629\,\textrm{ev}\quad\textrm{for}\quad \sin\delta = -1\,.\end{aligned}\ ] ] the present model give a non - perturbative qcd lagrangian as follows , @xmath79 where @xmath80 is the original qcd phase while @xmath81 and @xmath82 are the mass matrices of the ordinary and mirror down- and up - type quarks , respectively , @xmath83m_d^{}\!\left[\begin{array}{c}d^{}_r\\ d'^{}_r\end{array}\right]\!-[\bar{u}^{}_l , \bar{u}'^{}_l]m_u^{}\!\left[\begin{array}{c}u^{}_r\\ u'^{}_r\end{array}\right]+\textrm{h.c.}~\textrm{with}\nonumber\\ & & \!\!m_d^{}=\ ! \left[\!\begin{array}{cc}y_d^{}\langle\phi\rangle&0\\ 0&y_d^{\dagger}\langle\phi'\rangle\end{array}\!\right],~m_u^{}=\ ! \left[\!\begin{array}{cc}y_u^{}\langle\phi\rangle&0\\ 0&y_u^{\dagger}\langle\phi'\rangle\end{array}\!\right].\end{aligned}\ ] ] when the @xmath80-term is removed as a result of the parity invariance , the real determinants @xmath84 and @xmath85 will lead to a zero @xmath86 . we hence can obtain a vanishing strong cp phase @xmath87 . the model also contains two stable scalars @xmath88 and @xmath89 because of the unbroken @xmath26 symmetry . these scalars can annihilate into the ordinary species through their yukawa interactions and higgs portal . for a proper choice of their masses and couplings , they can obtain a right relic density to explain the dark matter puzzle . this type of scalar dark matter has been studied in a lot of literatures @xcite . in this paper we have proposed a new leptogenesis scenario in the left - right symmetric framework to predict the low limits on the neutrino mass scale and cp violation from the cosmic baryon asymmetry . these predictions may be verified by future neutrino oscillation experiments and cosmological observations . our model contains a softly broken parity to solve the strong cp problem without introducing an unobserved axion . the dirac neutrinos obtain a seesaw - suppressed mass matrix . some dark matter scalars also participate in the leptogenesis processes . * acknowledgement * : this work was supported by the shanghai jiao tong university under grant no . wf220407201 and by the shanghai laboratory for particle physics and cosmology under grant no . 11dz2260700 . p. minkowski , phys . b * 67 * , 421 ( 1977 ) ; t. yanagida , in _ proceedings of the workshop on unified theory and the baryon number of the universe _ , edited by o. sawada and a. sugamoto ( kek , tsukuba , 1979 ) , p. 95 ; m. gell - mann , p. ramond , and r. slansky , in _ supergravity _ , edited by f. van nieuwenhuizen and d. freedman ( north holland , amsterdam , 1979 ) , p. 315 ; s.l . glashow , in _ quarks and leptons _ , edited by m. lvy _ ( plenum , new york , 1980 ) , p. 707 mohapatra and g. senjanovi , phys . lett . * 44 * , 912 ( 1980 ) . m. magg and c. wetterich , phys . b * 94 * , 61 ( 1980 ) ; j. schechter and j.w.f . valle , phys . d * 22 * , 2227 ( 1980 ) ; t.p . cheng and l.f . li , phys . d * 22 * , 2860 ( 1980 ) ; g. lazarides , q. shafi , and c. wetterich , nucl . b * 181 * , 287 ( 1981 ) ; r.n . mohapatra and g. senjanovi , phys . d * 23 * , 165 ( 1981 ) . j.c . pati and a. salam , phys . d * 10 * , 275 ( 1974 ) ; r.n . mohapatra and j.c . pati , phys . d * 11 * , 566 ( 1975 ) ; r.n . mohapatra and j.c . pati , phys . d * 11 * , 2558 ( 1975 ) ; r.n . mohapatra and g. senjanovi , phys . d * 12 * , 1502 ( 1975 ) . m. flanz , e.a . paschos , and u. sarkar , phys . b * 345 * , 248 ( 1995 ) ; m. flanz , e.a . paschos , u. sarkar , and j. weiss , phys . b * 389 * , 693 ( 1996 ) ; l. covi , e. roulet , and f. vissani , phys . lett . b * 384 * , 169 ( 1996 ) ; a. pilaftsis , phys . rev . d * 56 * , 5431 ( 1997 ) . v. silveira and a. zee , phys . b * 161 * , 136 ( 1985 ) ; j. mcdonald , phys . d * 50 * , 3637 ( 1994 ) ; c.p . burgess , m. pospelov , and t. ter veldhuis , nucl . b * 619 * , 709 ( 2001 ) ; v. barger , p. langacker , m. mccaskey , m.j . ramsey - musolf , and g. shaughnessy , phys . d * 79 * , 015018 ( 2009 ) ; m. gonderinger , y. li , h. patel , and m.j . ramsey - musolf , jhep * 1001 * , 053 ( 2010 ) ; w.l . guo and y.l . wu , jhep * 1010 * , 083 ( 2010 ) .

in an @xmath0 left - right symmetric framework with spontaneous breaking @xmath1 , we present a new leptogenesis scenario to predict low limits on neutrinos mass scale @xmath2 and cp violation @xmath3 . benefited from a softly broken parity symmetry , which is motivated by solving the strong cp problem without introducing an unobserved axion , the dimensionless couplings of the mirror fields charged under @xmath4 are mapped to the couplings of the ordinary fields charged under @xmath5 . the mirror dirac neutrinos can have a heavy mass matrix proportional to the seesaw - suppressed mass matrix of the ordinary dirac neutrinos . through the @xmath6 gauge interactions , the mirror neutrinos can decay to generate a lepton asymmetry in the mirror muons and an opposite lepton asymmetry in the mirror electrons . before the @xmath7 sphaleron processes go out of equilibrium , the mirror muons rather than the mirror electrons can efficiently decay into the ordinary right - handed leptons with some dark matter scalars and hence the mirror muon asymmetry can be partially converted to an expected baryon asymmetry .

biological invasions of alien species are commonly divided into three stages : arrival , establishment , and expansion ( liebhold and tobin 2008 ) . the precise circumstances of an alien species arrival , which refers to the transport of an alien species to new areas outside of its native range , are generally not known and are not the purpose of this article . the establishment stage refers to a growth phase of the population density up to some threshold above which it is usually assumed that natural extinction is highly unlikely . however , if during the expansion stage , which refers to the spreading of the alien species to nearby new areas , the population expands in space through dispersal without significantly increasing its size , thus leading to a drop of its density , there might be a risk of extinction for species subject to an allee effect . the allee effect refers to a certain process that leads to decreasing net population growth with decreasing density , thus inducing the existence of a so - called allee threshold below which populations are driven toward extinction . the causes of allee effect identified by ecologists are numerous . they include failure to locate mates ( hopper and roush 1993 ; berec _ et al _ 2001 ) , inbreeding depression ( lande 1998 ) , failure to satiate predators ( gascoigne and lipcius 2004 ) , lack of cooperative feeding ( clark and faeth 1997 ) , etc . stochasticity , e.g. , demographic and/or environmental stochasticity , may also play an important role during the critical time period when an alien species already established in one area starts to spread its population into a new area through dispersal . in this article , we think of the establishment stage as a local expansion of the population in a given geographical location , which involves an increase of population density in this location , while we think of the expansion stage as a global expansion of the population in space into nearby geographical locations regardless of its density . we call a global expansion successful if it leads to the population being established in nearby geographical locations , and unsuccessful if on the contrary the population fails to get established in new locations which may also lead to a global extinction ( the population goes extinct in all patches ) . the main purpose of this article is to study the critical time period when a species already established in a specific geographical location starts to expand in space , and determine whether the expansion stage is successful or not . both allee effect and stochasticity are central to better understand why some alien species successfully expand into new geographical areas , and there has been recently a growing recognition of the importance of these two components in biological invasions ( drake 2004 ; leung _ et al _ 2004 ; taylor and hastings 2005 ; ackleh _ et al _ 2007 ) . understanding their role and strength is of critical importance to gain some insight into why some species are more invasive than others , and may suggest some proper biological control strategies to regulate some populations ( liebhold and tobin 2008 ) . if an alien species subject to an allee effect establishes its population in one area , i.e. , its population is above the allee threshold in this area , then the first step of population expansion is to spread to a nearby new area where the population is either absent or at least below the allee threshold . a natural way to model this situation is to consider a two - patch model with heterogeneous initial conditions such that 1 . both patches are coupled by interacting through dispersal , and 2 . in the absence of interactions , i.e. , when the patches are uncoupled , the initial conditions lead to establishment in one patch and extinction in the other patch . this approach has been used previously by alder ( 1993 ) and kang _ et al _ ( 2009 ) . in this article , we follow this modeling strategy to study the global expansion ( population above the threshold in both patches ) and global extinction ( population below the threshold in both patches ) of an alien species subject to an allee effect during the critical time period between the establishment stage and the expansion stage by employing both a deterministic two - patch model and its stochastic analog . the objectives of our study are twofold : the first is to study the consequences of the inclusion of dispersal and allee effect on the extinction and expansion for both deterministic and stochastic models with heterogeneous initial conditions ; the second is to understand the effects of stochasticity by comparing the results based on both models . there is a copious amount of literature on the invasion and extinction of populations subject to allee effects ( e.g. , dennis 1989 , 2002 ; veit and lewis 1996 ; mccarthy 1997 ; shigesada and kawasaki 1997 ; greene and stamps 2001 ; keitt _ et al _ 2001 ; fagan _ et al _ 2002 ; wang _ et al _ 2002 ; liebhold and bascompte 2003 ; schreiber 2003 ; zhou _ et al _ 2004 ; petrovskii _ et al _ 2005 ; taylor and hastings 2005 ) which also includes various models in patchy environment ( e.g. , amarasekare 1998a , 1998b ; gyllenberg _ et al _ 1999 ; ackleh _ et al _ 2007 ; kang _ et al _ 2009 ) . in the deterministic side , amarasekare ( 1998a , 1998b ) investigated how an interaction between local density dependence , dispersal , and spatial heterogeneity influence population persistence in patchy environments . in particular , she studied how allee ( or allee - like ) effects arise from these patchy models . gyllenberg _ et al _ ( 1999 ) studied a deterministic model of a symmetric two - patch metapopulation to determine conditions that allow the allee effect to conserve and create spatial heterogeneities in population densities . rather than exploring the global dynamics of their models , both amarasekare ( 1998a , 1998b ) and gyllenberg _ et al _ ( 1999 ) studied the influence of an allee effect on local dynamics , e.g. , number of equilibriums and local stability . there are few studies regarding the influence of an allee effect on the extinction versus expansion of populations in patchy environments ( e.g. , ackleh _ et al _ 2007 ; kang _ et al _ 2009 ) . et al _ ( 2009 ) studied the influence of an _ allee - like effect _ for a discrete - time two - patch model on plant - herbivore interactions where patches are coupled through a dispersal . their study suggests that for a certain range of dispersal parameters the population of herbivores in both patches drops under the allee threshold , thus leading to an extinction of the herbivores in both patches , for the majority of positive initial conditions . in the stochastic side , the recent work by ackleh _ et al _ ( 2007 ) focuses on a multi - patch population model combining stochasticity and allee effect . their numerical simulations show that populations with initial sizes below but near their allee threshold in each patch can still become established and invasive if stochastic processes affect life history parameters . the closer the population to its allee threshold , the greater the probability of invasion . a more theoretical approach based on interacting particle systems can be found in krone ( 1999 ) . in his model , each site of the infinite integer lattice has to be thought of as a patch which is either empty , occupied by a small colony with a high risk of going extinct , or occupied by a full colony with a longer life span . if successful , a small colony gets established to become a full colony , while empty patches get colonized by a small colony due to invasions from adjacent full colonies , making space explicit . in this paper , although we model the population dynamics deterministically following the approach of amarasekare ( 1998a , 1998b ) , gyllenberg _ et al _ ( 1999 ) and ackleh _ et al _ ( 2007 ) , our stochastic process as well as analytical results for both models are new . for the deterministic model , our focus is on the global dynamics of the system combining dispersal and allee effects . in particular , we give analytical results on how allee threshold and dispersal affect the geometry of the basins of attraction of the stable equilibriums . the stochastic model is derived from the deterministic one using a process that has two absorbing states corresponding to global extinction and global expansion , which allows to have a rigorous definition of successful invasion . in particular , our model is designed to study analytically the probability that a fully occupied patch successfully invade a nearby empty patch . to gain insight into the effects of stochasticity on the population dynamics , we will compare in details the results obtained for both models . the rest of the article is organized as follows . in section [ sec : deterministic ] , we introduce the deterministic two - patch model with allee effect coupled by dispersal . based on the analysis of the invariant sets , we give a complete picture of the global dynamics of the system including the existence of the nontrivial locally stable equilibriums and the geometry of their basin of attraction . numerical simulations have been performed to gain some insight into how dispersal and allee threshold affect the exact basin of attraction of the equilibriums . in section [ sec : stochastic ] , we introduce and analyze mathematically the stochastic model focusing on the time to fixation of the process , the existence of metastable states and the probability of a successful invasion when starting from heterogeneous initial conditions . simulations of the stochastic model have also been performed to better understand these aspects . in section [ sec : comparison ] , we compare the predictions based on both models , and describe the biological implications of our analytical and numerical results . finally , section [ sec : proofs ] is devoted proofs . the first step in constructing the deterministic two - patch model is to consider single - species dynamics including an allee affect as potential candidates to describe the evolution in a single patch . the two - patch model is then naturally derived by looking at a two - dimensional system in which both components are coupled through dispersal . the ecological dynamics of a single species population subject to an allee effect that can mimic the dynamics in the absence of dispersal is usually described by the model @xmath0 where @xmath1 denotes the population density at time @xmath2 . the function @xmath3 measures the logistic component of population growth , which is given by @xmath4 where @xmath5 is the per capita intrinsic growth rate and @xmath6 measures the extra mortality caused by intraspecific competition . in general , the bistability of the differential equation is triggered by combining the negative density - dependence of the logistic growth @xmath3 with the positive density - dependence of an additional demographic factor represented here by the function @xmath7 . the decreasing reproduction due to a shortage of mating encountered in low population density and the decreasing mortality due to the weakening predation risk in higher population density are two important examples of such factors ( stephens and sutherland 1999 ) which , following dercole _ et al _ ( 2003 ) , can be modeled by a holling type ii functional response : @xmath8 . the resulting population model creates , under suitable parameter values , a threshold below which the population goes extinct eventually and above which the population density approaches a positive equilibrium . the simplest and generic model that captures the population dynamics of a single species with allee effects can be described by @xmath9 where @xmath5 is the per capita intrinsic growth rate after rescaling and @xmath10 is a threshold that lies between 0 and 1 after rescaling . the latter , called allee threshold , determines whether the population goes extinct or establishes itself . more precisely , the population dynamics of can be summarized as follows . [ l_sae ] if the population of a single species is described by , then it goes extinct when @xmath11 while its density goes to 1 when @xmath12 . thinking of model as describing the population dynamics in one patch , the dynamics of two interacting identical patches with dispersal @xmath13 can be modeled by @xmath14 where @xmath15 $ ] is a dispersal parameter , representing the fraction of population migrating from one patch to another per unit of time . although the system - is symmetric in @xmath16 and @xmath17 , asymmetry will be introduced by considering different initial conditions in each patch : @xmath18 . we will pay a particular attention to situations where one patch is initially below and the other patch above the allee threshold , in which case , in the absence of dispersal , the population goes extinct in the first patch but establishes itself in the second one . the main objective is to understand , based on analytical and numerical results , how the dispersal parameter @xmath13 and the allee threshold @xmath10 affect the global dynamics , i.e. , the limit sets of the system - and the geometry of their basin of attraction . our analytical results suggest the following picture of the global dynamics . recall first that , in the absence of dispersal , the system has four locally stable equilibrium points , which correspond to cases when the population in each patch either goes extinct or gets established . in the presence of dispersal , the existence of ( stable ) limit cycles is also excluded : starting from almost every initial condition in @xmath19 , the system converges to an equilibrium point . this is partly proved analytically in theorem [ th : simple ] and supported by the numerical simulations of figure [ fig : ode ] . therefore , we focus our attention on the existence , stability and basins of attraction of the equilibrium points . the effects of the dispersal parameter and the value of the allee threshold are as follows . first , the dynamics of the deterministic two - patch model in the presence of weak dispersal are similar to that of the uncoupled system , having four locally stable equilibriums ( theorem [ th : attractors ] ) , i.e. , the extinction state @xmath20 , the expansion state @xmath21 and two asymmetric interior equilibriums @xmath22 , @xmath23 , which is not retained after the inclusion of demographic stochasticity , which induces two absorbing states and two metastable states . while increasing the dispersal parameter from 0 , the basin of attraction of the extinction state @xmath20 and expansion state @xmath21 increase until a certain critical value at which both patches interact enough to synchronize , which drives the system to either global extinction @xmath20 or global expansion @xmath21 : there are only two attractors ( theorems [ th : basin ] and [ th : attractors ] ) . above this critical value , dispersal promotes extinction when the allee threshold is below one half but promotes survival when the allee threshold is above one half ( theorems [ th : basin ] and [ th : dispersal ] ) . finally , in the presence of a strong dispersal , both patches synchronize fast enough so that the global dynamics reduce to that of a single - patch model : if the initial global density , i.e. , the average of the densities in both patches , is below the allee threshold then the population goes extinct whereas if it exceeds the allee threshold then the population expands globally ( theorem [ th : dispersal ] ) . in other respects , for any value of the dispersal parameter , increasing the allee threshold promotes extinction , and populations initially below the allee threshold in both patches are doomed to extinction , whereas populations initially above the allee threshold in both patches expand globally . these results are stated rigorously in the following two subsections . simulation results and detailed summary are given in the last subsection . in order to understand the global dynamics of the deterministic two - patch model , the first step is to identify its omega limit sets . since the model is simply a two - dimensional ode , its omega limit sets are either equilibrium points or limit cycles according to the poincar - bendixson theorem . as stated in the next theorem , when the dispersal parameter is sufficiently large , an application of the dulac s criterion reveals simple dynamics by excluding the existence of limit cycles : for any initial condition , the system converges to an equilibrium point . [ th : simple ] for any @xmath24 , @xmath25 and @xmath26 , if @xmath27 then every trajectory of - converges to an equilibrium point . theorem [ th : simple ] indicates for instance that every trajectory of the system - converges to an equilibrium point under the condition @xmath28 if one takes @xmath29 . in addition , theorem [ th : basin ] below implies that if limit cycles emerge for smaller values of the dispersal parameter then each of them is included in one of the two regions of the phase space in which the population lies above the allee threshold in one patch but below the allee threshold in the other patch , i.e. , @xmath30 where @xmath31 . numerical simulations ( see figure [ fig : ode ] ) further suggest that , for any value of the dispersal parameter , there is no stable limit cycle , which implies that locally stable equilibriums are the only possible attractors of the system , so we focus our attention on the existence , stability and basins of attraction of the equilibrium points . we also would like to point out that if the system has no allee effect , e.g. , a metapopulation model coupled by both competition and migration with uniparental reproduction , then this system admits no periodic solutions ( proposition 1 in gyllenberg _ et al _ 1999 ) . it can be easily seen that the system - has three symmetric equilibriums for all positive values of the parameters : one boundary equilibrium given by @xmath32 and two interior equilibriums given respectively by @xmath33 and @xmath34 . for obvious reasons , we call @xmath35 the extinction state of the system and @xmath36 the expansion state . theorem [ th : basin ] below indicate that , for all parameter values , these two trivial equilibriums are locally stable whereas the interior equilibrium point @xmath37 is unstable . hence , to understand the global dynamics of the system , the next step is to study the geometry of the basins of attraction of the two trivial equilibriums , i.e. , @xmath38 letting @xmath39 and @xmath40 denote the subsets @xmath41 lemma [ l_sae ] indicates that , in the absence of dispersal , the basins of attraction of @xmath35 and @xmath36 for the ( uncoupled ) system are given by @xmath42 and @xmath43 . the following theorem shows how the inclusion of a dispersal affects the basins of attraction . [ th : basin ] 1 . the extinction state @xmath35 and expansion state @xmath36 are always locally stable whereas the interior fixed point @xmath37 is always unstable . 2 . if @xmath44 then @xmath37 is a saddle while if @xmath45 then @xmath37 is a source . 3 . @xmath46 . if in addition @xmath47 then @xmath48 4 . @xmath49 . if in addition @xmath50 then @xmath51 theorem [ th : basin ] indicates that the inclusion of dispersal promotes both global extinction and global expansion of the system , as the basins of attraction of both equilibrium points @xmath35 and @xmath36 are larger in the presence than in the absence of dispersal . numerical simulations further suggest that , up to a certain critical value , increasing the dispersal parameter translates into an increase of @xmath52 and @xmath53 . the value of the allee threshold @xmath10 also plays an important role in the global dynamics . when the allee threshold lies below one half , which is common in nature , the largest possible basin of attraction of @xmath35 is @xmath54 moreover , according to numerical simulations ( see figure [ fig : ode ] ) , increasing the allee threshold promotes extinction of the system in the sense that , the dispersal parameter being fixed , the smaller the allee threshold , the smaller the basin attraction of the extinction state @xmath35 and the larger the basin attraction of the expansion state @xmath36 . finally , we would like to point out that parts 1 and 2 of the theorem hold for the system - but not always for two - patch models with allee effect . a counter example is provided by the metapopulation model coupled by both competition and migration with biparental reproduction studied by gyllenberg _ et al _ ( 1999 ) . the first step to prove parts 3 and 4 of theorem [ th : basin ] will be to identify the positive invariant sets of the system - included into the upper right quadrant @xmath31 . recall that a set is called positive invariant if any trajectory starting from this set stays in this set at all future times . since we are interested in the global dynamics of the system , our objective will be to find all the possible invariant sets in @xmath55 . notice that the union and the intersect of positive invariant sets are also positive invariant . all these positive invariant sets have an important role in understanding the dynamics in regions of the phase space where the population is below the allee threshold in one patch but above the allee threshold in the other patch . in particular , they will give us means of decomposing the phase space by restricting our attention to the dynamics on each invariant set and then sewing together a global solution from the invariant pieces . in this subsection , we study the effects of the dispersal parameter on the dynamics of the two - patch model when the allee threshold is fixed . theorem [ th : simple ] suggests that the number of attractors is also equal to the number of locally stable equilibriums . our study shows that the value of the dispersal parameter determines the number of equilibriums , thus the possible number of attractors . let @xmath56 denote the stable manifold of the unstable interior equilibrium @xmath37 , i.e. , @xmath57 the following theorem indicates that , when the dispersal is sufficiently large , both patches interact enough to synchronize , which drives the system to either global extinction or global expansion : there are only two stable equilibriums , the extinction state @xmath35 and the expansion state @xmath36 . [ th : dispersal ] assume that @xmath58 then , the system - has only two attractors : @xmath35 and @xmath36 . moreover , 1 . if @xmath47 and holds then @xmath59 2 . if @xmath50 and holds then @xmath60 if the inequality holds , we can consider that the system has a very strong dispersal . then theorem [ th : dispersal ] indicates that , when @xmath47 , both patches synchronize fast enough so that the global dynamics reduce to the one of a single - patch model : if the initial global density , i.e. , the average of the densities in both patches , is below the allee threshold then the population goes extinct whereas if it exceeds the allee threshold then the population expands globally . in addition , the theoretical results in theorems [ th : basin ] and [ th : dispersal ] , suggest that the smaller the allee threshold , the smaller the basin of attraction of the extinction state and the larger the basin of attraction of the expansion state . this agrees with the simulation results of figure [ fig : ode ] . finally , in order to explore the number of locally stable equilibriums when the dispersal is small , we now look at the nullclines of the system . define @xmath61 then , the nullclines of the system - are given by @xmath62 and @xmath63 . the interior equilibriums are determined by the positive roots of @xmath64 , which is a polynomial with degree 9 . this implies that the system has at most 8 interior equilibriums since 0 is always a solution . according to the expression of the nullclines @xmath65 and @xmath66 ( see figure [ fig : nullclines ] page ) , we can see that the number of interior equilibriums strongly depends upon the value of the dispersal parameter : in the presence of strong dispersal , both patches synchronize and the system has only two positive interior equilibriums @xmath37 and @xmath36 , which is confirmed by theorem [ th : dispersal ] , while in the presence of weak dispersal , there is enough independence between both patches so that the system has 8 positive interior equilibriums . we are interested in the locally stable equilibriums since the possible number of attractors is intimately connected to the number of stable equilibriums . the following theorem summarizes the properties of the equilibriums and their stability for different parameters values . [ th : attractors ] 1 . if @xmath25 , @xmath67 $ ] then every trajectory converges in @xmath68 ^ 2 $ ] so all the equilibriums @xmath69 ^ 2 $ ] . 2 . if @xmath70 then there are only three equilibriums : @xmath35 , @xmath37 and @xmath36 , with @xmath35 and @xmath36 locally stable and @xmath37 saddle . 3 . if @xmath71 then the nullcline @xmath72 has exactly two positive roots that we denote by @xmath73 . let @xmath74 . 1 . if @xmath75 then the system has five fixed points with only two locally stable : @xmath35 and @xmath36 . 2 . if @xmath76 then the system achieves its maximum number of equilibriums which is equal to 9 ; only four of them are locally stable : two symmetric equilibriums @xmath35 and @xmath36 and two asymmetric interior equilibriums @xmath77 and @xmath23 . part 1 of theorem [ th : attractors ] suggests that we can restrict our analysis of the basin of attraction of locally stable equilibriums to the compact space @xmath78 ^ 2 $ ] . moreover , from theorem [ th : attractors ] , we can see that when the dispersal parameter is small enough , the system has 9 equilibriums , including four locally stable equilibriums . in the presence of an allee effect , small dispersal may promote survival : patches that are below the allee threshold are rescued by immigrants from adjacent patches above the allee threshold . this implies that when dispersal is introduced to a system with an allee effect , populations can exist at intermediate densities , corresponding to the equilibriums @xmath77 and @xmath23 , as a source - sink system , or expand to high density @xmath36 . moreover , according to perturbation theory ( simon 1974 ; amarasekare 2000 ) , both asymmetric interior equilibriums appear from the equilibriums @xmath79 and @xmath80 of the uncoupled system , i.e. , in the absence of dispersal , caused by the small perturbation @xmath13 . therefore , we have @xmath81 and @xmath82 . finally , note that the absence of limit cycles given by theorem [ th : simple ] when holds combined with theorem [ th : attractors ] implies that [ c:4a ] if the system - has four locally stable equilibriums and inequality holds for some @xmath24 , then the system has exactly four attractors . when the system has four attractors as stated in corollary [ c:4a ] , the simulations in figure [ fig : ode ] suggest that the smaller the dispersal , the smaller the basin of attraction of the extinction state and the expansion state , but the larger the basin attraction of the asymmetric interior equilibriums . in particular , if @xmath83 , then @xmath84 where @xmath85 denotes the basin of attraction of the asymmetric interior equilibriums . theorem [ th : dispersal ] suggests that the larger the allee threshold , the larger the basin of attraction of the extinction state and the smaller the basin of attraction of the expansion state when inequality holds . the simulations shown in figure [ fig : ode ] confirm this and give us a more complete picture of how the dispersal @xmath13 and allee threshold @xmath10 affect the exact basin of attraction of the locally stable equilibriums including asymmetric interior equilibriums : * effects of dispersal @xmath13 * fix allee threshold @xmath10 and growth rate @xmath5 , let dispersal @xmath13 vary . 1 . when @xmath13 is small so that the system has four locally stable equilibriums ( @xmath13 smaller than some critical value @xmath86 ) , the smaller the dispersal , the smaller the basin of attraction of the extinction state and the expansion state , but the larger the basin of attraction of the asymmetric interior equilibriums ( see ( d ) and ( f ) of figure [ fig : ode ] ) . this indicates that smaller dispersals promote persistence of the populations in both patches by creating sink - source dynamics . 2 . when @xmath13 is large so that the system has only two attractors @xmath35 and @xmath36 ( @xmath13 larger than the critical value @xmath86 ) , the larger the dispersal , the larger the basin of attraction of the extinction state but the smaller the basin of attraction of the expansion state when @xmath47 ( see ( a ) and ( c ) of figure [ fig : ode ] ) . when @xmath50 , the monotonicity is flipped due to the symmetry of the system ( see figure [ fig : ode2 ] ) . extreme cases : when @xmath13 is very small , the two - patch model behaves nearly like the uncoupled system , having four attractors and almost the same basins of attraction , while when @xmath87 , the global population behaves according to a one - patch system with allee threshold @xmath88 , in particular @xmath89 * effects of allee threshold @xmath10 * fix dispersal @xmath13 and growth rate @xmath5 , let allee threshold @xmath10 vary . 1 . regardless of the number of locally stable equilibriums , the larger the allee threshold , the larger the basin of attraction of the extinction state but the smaller the basin of attraction of the expansion state ( see ( a ) , ( b ) , ( e ) and ( f ) of figure [ fig : ode ] ) . * effects of growth rate @xmath5 * fix dispersal @xmath13 and allee threshold @xmath10 , let growth rate @xmath5 vary . 1 . by introducing the new time @xmath90 , we can scale off the parameter @xmath5 of the system - so the dispersal @xmath13 becomes @xmath91 . this implies that the growth rate @xmath5 and the dispersal parameter @xmath13 have opposite effects on the basin of attraction of the locally stable equilibriums , i.e. , increasing the value of @xmath5 is equivalent to decreasing the value of @xmath13 . tables [ tab : comparisondeter1]-[tab : comparisondeter2 ] give a complete picture , based on our analytical and numerical results , of how dispersal and allee threshold affect the basin of attraction of the locally stable equilibriums . we only focus on the case @xmath47 but similar results can be deduced when @xmath50 using the symmetry of the system - . parameters & basin of attraction of @xmath35 & basin of attraction of @xmath36 + dispersal @xmath92 & @xmath52 @xmath93 & @xmath53 @xmath94 + allee threshold @xmath95 & @xmath52 @xmath93 & @xmath53 @xmath94 + + parameters & basin of attr . of @xmath35 & basin of attr . of asymmetric equilibriums & basin of attr . of @xmath36 + dispersal @xmath96 & @xmath97 & @xmath98 & @xmath99 + allee threshold @xmath100 & @xmath97 & no monotonicity & @xmath101 + @xmath83 & @xmath102 & @xmath103 & @xmath104 + while the deterministic model is similar to the one in ackleh _ et al _ ( 2007 ) , our stochastic model differs from theirs , which is derived naturally from the deterministic model by including independent poisson increments , i.e. , variability in birth , death and migration events . this gives rise to a multi - patch individual - based model for which they study numerically the probability of a successful invasion , defined as the event that the population size in one patch exceeds some denominated threshold . however , well - known results about irreducible markov chains imply that the population is driven almost surely to extinction which corresponds to the unique absorbing state of their stochastic process . in contrast , we model stochastically the two - patch system via a process that has two absorbing states corresponding to a global extinction and a global expansion , respectively . this allows to have a definition of successful invasion more rigorous and more tractable mathematically . in particular , while their stochastic model is designed to study numerically the probability that a population starting near the allee threshold in each patch gets successfully established , our model is designed to study analytically the probability that a fully occupied patch successfully invade a nearby empty patch . more precisely , to understand the effect of stochasticity on the interactions between both patches , we introduce a markov jump process that , similarly to the deterministic model , keeps track of the evolution of the population size in each patch . to obtain a markov process , the state is updated at random times represented by the points of a poisson process with a certain intensity making the times between consecutive updates independent exponentially distributed random variables . motivated by the fact that the unit square @xmath105 ^ 2 $ ] is positive invariant for the deterministic model , we will choose this set as the state space , i.e. , the state at time @xmath2 is a random vector @xmath106 , where the first and second coordinates represent the population size in the first and second patch , respectively . following the deterministic model , the stochastic dynamics involve three mechanisms : expansion , extinction , and migration . to model the presence of an allee affect , we again introduce a threshold parameter @xmath26 that can be seen as a critical size under which the population undergoes extinction and above which the population undergoes expansion , i.e. , allee threshold . this aspect is modeled by assuming that each component of the stochastic process jumps independently at rate @xmath25 to either 0 ( extinction ) or 1 ( expansion ) depending on whether it lies below or above the allee threshold . recall that an event `` happens at rate @xmath5 '' if the probability that it happens during a short time interval of length @xmath107 approaches @xmath108 as @xmath109 . in particular , expansion and extinction are formally described by the conditional probabilities @xmath110 this is also equivalent to saying that the waiting time for an expansion or an extinction is exponentially distributed with mean @xmath111 . given that the population size in a given patch is at the allee threshold , we flip a fair coin to decide whether an expansion or an extension event occurs at that patch which , in view of well - known properties of poisson processes , implies that @xmath112 to understand the effects of inter - patch interactions on the evolution of the system , we also include migration events consisting of the displacement of a fraction @xmath13 of the population of each patch to the other patch . we assume that these events occur at the normalized rate 1 , therefore migrations are described by @xmath113 we refer to figure [ fig : dynamics ] for a schematic illustration of the dynamics , where dark rectangles represent parts of the populations which are interchanged in the event of a migration . to analyze mathematically the stochastic process , it will be useful to look at the model as a simple example of interacting particle system . interacting particle systems are continuous - time markov processes whose state space maps the vertex set of a connected graph into a set representing the possible states at each vertex . the evolution is described by local interactions as the rate of change at a given vertex only depends on the configuration in its neighborhood . in particular , the markov process @xmath114 can be seen as an interacting particle system evolving on a very simple graph that consists of only two vertices , representing both patches , connected by one edge , indicating that patches interact . the reason for looking at the stochastic model as an example of interacting particle system is that this will allow us to construct the process graphically from a collection of independent poisson processes based on an idea of harris ( 1972 ) , which is a powerful tool to analyze the process mathematically . we now describe in details the behavior of the process along with our main results . note that , considering a stochastic model rather than a deterministic one , the long - term behavior is described by a set of invariant measures on the state space rather than single point equilibriums . to the two trivial equilibriums of the deterministic model , @xmath35 and @xmath36 , correspond two invariant measures which are dirac measures that concentrate on those two points , respectively . these two measures are two absorbing states : the configuration in which both patches are empty and the configuration in which both patches are fully occupied . we call global extinction and global expansion the events that the process eventually fixates to the first and the second absorbing state , respectively . interestingly , to the two asymmetric equilibriums of the deterministic model in the presence of weak dispersal correspond two quasi - stationary distributions representing two metastable states of the stochastic process ( see theorem [ metastable ] ) : depending on the initial configuration , the transient behavior might be described by one of these two quasi - stationary distributions , but after a long random time in the presence of weak dispersal ( see theorem [ metastability ] ) , the system fixates to one of the two absorbing states , suggesting that situations predicted by the deterministic model in which a small population can live next to a large population are artificially stable . another important question is how stochasticity affects the geometry of the basins of attraction of the two absorbing states , although strictly speaking there is no basin of attraction for the stochastic model since the limiting behavior might be unpredictable , and how fast the system fixates . we will see that there is a set of initial configurations for which the limiting behavior of the stochastic process is predictable , and fixation to one of the two absorbing states occurs quickly ( see theorem [ fixation ] ) . starting from any other configuration , the limiting behavior becomes unpredictable in the sense that the process may reach any of the two absorbing states with positive probability . in the presence of weak dispersal , however , the limit is almost predictable in the sense that the probability that the system undergoes a global expansion after exiting one of its metastable states approaches zero or one ( see theorem [ probabilities ] ) . whether the system fixates to one or the other absorbing state strongly depends on the value of the allee threshold . the limit is less and less predictable and the time to fixation shorter and shorter as the dispersal parameter increases . in order to describe rigorously the behavior of the stochastic model introduced above , our main objective is to estimate the times to fixation @xmath115 and the corresponding probabilities of fixation , @xmath116 as a function of the initial configuration and the three parameters of the system . as previously explained , in contrast with the deterministic model which can have up to four distinct attractors , with probability one , either global expansion or global extinction occurs for the stochastic process , i.e. , @xmath117 the state space can be divided into four subsets . starting from only two of these subsets the limit is predictable in the sense that @xmath118 we call an upper configuration any configuration of the system in which the population size in each patch exceeds the allee threshold , and a lower configuration any configuration in which the population size in each patch lies below the allee threshold . these sets are denoted respectively by @xmath119 note that the set of upper configurations is closed under the dynamics , i.e. , once the system hits an upper configuration , the configuration at any later time is also an upper configuration . this implies that , starting from an upper configuration , global expansion occurs with probability one . similarly , starting from a lower configuration , global extinction occurs with probability one . by representing the process graphically , the time to fixation can be computed explicitly , as stated in the following theorem . [ fixation ] we have @xmath120 \ = \ { \mathbb{e}}\,[\,\tau^- \,| \ ( x_0 , y_0 ) \in \omega^- ] \ = \ \frac{6r + 1}{2 r^2}.\ ] ] the previous theorem indicates that , starting from an upper configuration , the system converges with probability one to the absorbing state @xmath21 , whereas starting from a lower configuration , it converges with probability one to the other absorbing state @xmath20 . this result can be seen as the analog of theorem [ th : basin ] which states that the sets of upper and lower configurations are included in the basin of attraction of the equilibrium points @xmath36 and @xmath35 , respectively . theorem [ fixation ] also indicates that , when the rates at which expansions , extinctions , and migrations occur are of the same order , the expected time to fixation is quite short . the long - term behavior of the process starting from a configuration which is neither an upper configuration nor a lower configuration is more difficult to study as the probabilities of global expansion and global extinction are both strictly positive , which we shall refer to as unpredictable behavior . we will prove that , in any case , the system hits either an upper or a lower configuration at a random time which is almost surely finite , after which it evolves as indicated by theorem [ fixation ] . hence , the time to fixation and probabilities of global expansion and extinction can be determined by estimating the hitting times @xmath121 and the corresponding hitting probabilities @xmath122 since theorem [ fixation ] implies that @xmath123 \ = \ { \mathbb{e}}\,[t ] \ + \ \frac{6r + 1}{2r^2 } \quad \hbox{and } \quad p \,(\tau = \tau^+ ) \ = \ p \,(t = t^+).\ ] ] even though our next results hold for any values of the parameters , they indicate that interesting behaviors emerge when the dispersal parameter @xmath13 is small . in contrast with the deterministic model which , in this case , has four attractors , as indicated by theorem [ th : attractors ] , the stochastic model first exhibits a metastable behavior by oscillating for an arbitrarily long time around one of the two nontrivial equilibriums of the deterministic model , and then fixates to one of its two absorbing states . the limit is almost predictable as the probability of global expansion approaches either 0 or 1 depending on the value of the threshold parameter . for simplicity and since the system is symmetric , we shall assume that @xmath124 and @xmath125 but the proofs of our results easily extend to the more general case when @xmath126 recall that , starting from an upper configuration or a lower configuration , the time to fixation is rather small . in contrast , when @xmath124 , @xmath125 and @xmath13 is small , the stochastic process converges to a quasi - stationary distribution in which the population size at patch @xmath127 is relatively close to 0 and the population size at patch @xmath128 relatively close to 1 , and stays at its quasi - stationary distribution for a very long time , i.e. , the expected value of @xmath129 is large . however , due to stochasticity , the system reaches eventually an upper or a lower configuration , and then fixates rapidly . the next theorem gives an explicit lower bound of the expected value of the hitting time , which is the time the system stays at its quasi - stationary distribution . [ metastability ] for any initial configuration , we have @xmath130 moreover , @xmath131 \ \geq \ \frac{n_0}{2 + 4r } \ \bigg(\frac{1 + 2r}{1 + r } \bigg)^{n_0}\ ] ] where @xmath132 note that , when the allee threshold is bounded away from 0 and 1 , and the dispersal parameter is small , @xmath133 is large , and so is the expected value of the hitting time @xmath129 . note also that , before the hitting time , no expansion event can occur at patch @xmath127 while no extinction event can occur at patch @xmath128 . this indicates that the metastable state of the stochastic two - patch model is described by the stationary distribution of the markov process @xmath134 with state space @xmath105 ^ 2 $ ] , and whose evolution is given by @xmath135 where @xmath107 is a small time interval . that is , the process @xmath136 is obtained from @xmath114 by assuming that only extinction events at patch @xmath127 and only expansion events at patch @xmath128 can occur , which indeed describes the evolution of the original process @xmath114 before it reaches an upper or a lower configuration . letting @xmath137 denote the stationary distribution of this new process , the behavior of the stochastic two - patch model before the hitting time @xmath129 is described by the following theorem . [ metastable ] under the measure @xmath137 we have @xmath138 this indicates that , when @xmath13 is small , the population size at patch @xmath127 is close to 0 ( i.e. , @xmath139 ) and the population size at patch @xmath128 close to 1 ( i.e. , @xmath140 ) . the expected values above have to be thought of as the analog of the two asymmetric equilibriums of the deterministic model : @xmath77 and @xmath23 . after evolving a long time according to the quasi - stationary distribution @xmath137 , the process hits either an upper or a lower configuration , so the last question we would like to answer is whether global expansion or global extinction occurs after the system exits its metastable state . starting from an upper or a lower configuration , the answer is given by theorem [ fixation ] . starting from @xmath124 and @xmath125 , the symmetry of the model implies that @xmath141 our last result shows that , when @xmath142 and @xmath143 is small , the limiting behavior of the system is almost predictable in the sense that the probability of global expansion approaches either 0 or 1 . [ probabilities ] assume that @xmath47 . then @xmath144 where @xmath145 the previous theorem indicates that , when @xmath143 is small , @xmath146 in particular , in contrast with the deterministic model for which the limit depends on the initial condition and the geometry of the basins of attraction , starting from any initial configuration but an upper or a lower configuration , the limiting behavior of the stochastic model is only sensitive to the value of the parameters , with the allee threshold @xmath10 playing a central role . while theorem [ fixation ] gives an exact estimate of the time to fixation starting from particular initial conditions , the other results provide theoretical lower and upper bounds that allows us to gain a valuable insight into the long - term behavior of the stochastic two - patch model in the presence of weak dispersal . to better understand the combined effect of the allee threshold and dispersal parameter when starting from heterogeneous initial conditions , we refer the reader to the numerical simulations of figure [ fig : proba - time ] and tables [ tab : proba]-[tab : time ] . the left panel of the figure represents the probability of a global extinction , with the probability increasing with the darkness , and the right panel the expected time to fixation , with time increasing with the darkness , as a function of the dispersal parameter and the allee threshold . the tables provide some numerical values of the probability of extinction and expected time to fixation averaged over 10,000 independent realizations of the stochastic process for specific values of the parameters . the predictions based on theorems [ metastability ] and [ probabilities ] that the time to extinction blows up and the probability of extinction approaches either zero or one in the presence of weak dispersal appears clearly looking at the left side of both panels and the left column of the tables for which @xmath147 . the left panel and table [ tab : proba ] further indicate that the probability of a global extinction depends non - monotonically upon the dispersal parameter : when the allee threshold is below one half , the probability of extinction first increases with the dispersal parameter and then decreases after the dispersal reaches a critical value that depends on @xmath10 , which can be easily seen in the row @xmath148 of the table . when the allee threshold exceeds one half , the monotonicity is flipped . simulations also indicate that , the dispersal parameter being fixed , the probability of extinction increases as the allee threshold increases . although we omit the details of the proof , this can be easily shown analytically invoking a standard coupling argument to compare two processes , the first one with allee threshold @xmath149 and the second one with @xmath150 , the other parameters being the same for both processes . the black triangle labeled 1 in the upper right corner of the left picture reveals that global extinction occurs almost surely when @xmath151 . indeed , starting from the heterogeneous condition @xmath124 and @xmath125 , after the first migration event , we have @xmath152 in particular , both patches are below the allee threshold from which it follows that the population goes extinct eventually . almost sure global expansion in the parameter region corresponding to the lower right white triangle labeled 2 can be proved similarly . finally , as suggested by theorem [ metastability ] , the right picture and table [ tab : time ] indicate that the expected value of the time to fixation increases as the dispersal parameter decreases but also as the allee threshold gets closer to one half , which can again be proved analytically based on standard coupling arguments even through we omit the details of the proof . . [ cols="^,^,^,^,^,^,^,^",options="header " , ] [ tab : comparison ] recall that , in the absence of interactions between patches , both the deterministic model and the stochastic model predict a local expansion in patches where the initial population size is above the allee threshold and a local extinction in patches where the initial population size is below the allee threshold . this induces the existence of four locally stable equilibriums for the deterministic model , and four absorbing states for the stochastic model , which correspond to cases when the population in each patch either goes extinct or gets established . including interactions between patches , our results for the deterministic model indicate that , in the presence of weak dispersal , the dynamics retain four attractors , just as in the absence of interactions , up to a critical value @xmath86 when the patches synchronize : the two asymmetric equilibrium points are lost so that only global expansion and global extinction can happen . in contrast , including both stochasticity and even weak interactions , only the two absorbing states corresponding to global expansion and global extinction are retained . the most interesting behaviors emerge when the dispersal is weak , in which case , to the two asymmetric locally stable equilibriums of the deterministic model , correspond two metastable states for the stochastic model . looking at the global dynamics , the predictions based on the analysis of the deterministic two - patch model indicate that below the critical value @xmath86 dispersal promotes global expansion and global extinction in the sense that the basins of attraction of the two trivial fixed points expands while increasing the dispersal parameter . above the critical value @xmath86 dispersal promotes a global expansion when the allee threshold exceeds one half but promotes global extinction in the more realistic case when the allee threshold lies below one half . as mentioned above , in the presence of weak dispersal , both asymmetric equilibrium points become two metastable states , i.e. , quasi - stationary distributions , after the inclusion of stochasticity , suggesting that situations in which a small population lives next to a large population are artificially stable : in such a context , the two - patch system evolves first as dictated by one of the two quasi - stationary distributions then , after a long random time , experiences either a global expansion or a global extinction . in addition , the long - term behavior of the stochastic model becomes almost predictable in the sense that , with very high probability , the system will undergo a global expansion when the allee threshold lies below one half and a global extinction when the allee threshold exceeds one half , which is of primary importance to predict the destiny of heterogeneous two - patch systems in the presence of weak dispersal . while increasing the dispersal parameter , the stochastic model no longer exhibits a metastable behavior , the time to fixation decreases , and the long - term behavior becomes more and more unpredictable . in the presence of a very strong dispersal , however , the analysis of the deterministic model and the stochastic model starting from a heterogeneous configuration give the same predictions . in this case , both patches synchronize enough so that the global dynamics reduce to that of a single - patch model : if the initial global density , i.e. , the average of the densities in both patches , is below the allee threshold then the population goes extinct whereas if it exceeds the allee threshold then the population expands globally . our analysis of idealized two - patch models is an important first step to understand more realistic multi - patch systems . empirical data indicate that allee thresholds in nature vary accross species and habitat types but are typically much smaller than one half . the predictions , based on the deterministic model in the presence of enough dispersal so that patches synchronize and on the stochastic model in the general case , that populations usually expand successfully when the allee threshold is small is due to the fact that only two patches interact . literally , the critical threshold @xmath153 has to be thought of as one divided by the number of patches . looking at a multi - patch model in which @xmath154 patches interact all together , our analytical results suggest that a critical behavior should emerge for allee thresholds near @xmath155 when starting with a population established in only one patch , and more generally the number of patches where the population is initially established divided by the number of interacting patches . therefore , even for realistic values of the allee threshold , the long - term behavior is no longer straightforward in the presence of a large number of patches . numerical simulations can also provide a valuable insight into the long - term behavior of multi - patch models including additional refinements such as density - dependent dispersals , heterogeneous environments with possibly different allee thresholds in different patches , and more importantly the inclusion of a spatial structure through a network of interactions represented by a two - dimensional regular lattice or more general planar graphs rather than a complete graph where patches interact all together . as previously explained , the key to proving our main results is to first identify a number of sets which are positive invariant for the system - . this will they give us means of decomposing the phase space by restricting our attention to the dynamics on each invariant set and then sewing together a global solution from the invariant pieces . our first preliminary result indicates that , starting from any biologically meaningful initial condition , that is any condition belonging to @xmath156 , the trajectory of the system stays in the upper right quadrant and is bounded . assuming by contradiction that the system - is not positive invariant in upper right quadrant , we can find @xmath157 and a time @xmath158 such that @xmath159 let @xmath160 denote the boundary of @xmath55 , i.e. , @xmath161 by continuity of the trajectories , the intermediate value theorem implies the existence of a time @xmath162 such that @xmath163 therefore @xmath164 is well defined and @xmath165 for all @xmath166 $ ] . then , we have the following alternative . 1 . if @xmath167 then @xmath168 for all @xmath169 , which contradicts the existence of @xmath129 . 2 . if @xmath170 and @xmath171 then @xmath172 so @xmath173 this contradicts the existence of @xmath174 . 3 . if @xmath175 and @xmath176 , the same argument exchanging the roles of the functions @xmath16 and @xmath17 leads again to a contradiction . in conclusion , if @xmath177 and @xmath178 then @xmath179 at all positive times @xmath2 , which establishes the first part of the lemma . this also implies that , starting from any initial condition in @xmath55 , @xmath180 \ < \ 0 \end{array}\ ] ] whenever @xmath181 is larger than some @xmath182 . therefore , @xmath183 this completes the proof of lemma [ p_b ] . @xmath184 it follows from the previous lemma that , excluding the initial condition in which both patches are initially empty , the population densities in both patches are simultaneously positive at any positive time . this implies in particular that the trivial equilibrium @xmath35 is the only boundary equilibrium . by symmetry , we may assume that @xmath189 and @xmath178 . we first apply lemma [ p_b ] to get @xmath190 where @xmath191 is as in the proof of lemma [ p_b ] . in particular , @xmath192 \ x \vspace{4pt } \\ \ & \geq & \ - \ [ r \max \,(\theta , ( m - \theta ) ( m - 1 ) ) + \mu ] \ x \ \geq \ - k \,x \end{array}\ ] ] for some constant @xmath193 . therefore , @xmath194 finally , if @xmath195 then the same holds for @xmath196 , while if @xmath197 then @xmath198 which implies that @xmath187 for all @xmath199 for some small @xmath200 . the fact that this holds at all times follows from the same reasoning as before based on the fact that both functions are bounded . @xmath184 first , we assume that the initial condition @xmath208 and introduce @xmath209 then , the system - can be rewritten as @xmath210 with initial condition @xmath211 . now , the arguments of the proof of lemma [ p_b ] imply that @xmath55 is positive invariant for - . moreover , @xmath212 so the set @xmath40 is positive invariant for the system - . the fact that @xmath213 is positive invariant follows from the same argument but applied to @xmath214 to prove the positive invariance of @xmath39 we first observe that lemma [ p_b ] implies that any trajectory starting from a point in the square @xmath39 can not exit the square crossing its left of bottom side . moreover , the same arguments as in the proof of lemma [ p_b ] imply that it can not exit the square crossing its right or top sides either because of the following three properties . this proves that @xmath39 is positive invariant . the fact that the square @xmath221 is also positive invariant follows from the same argument , looking at the derivatives along each side and using that the four corners are equilibriums . to prove the positive invariance of the last three sets , we introduce the new functions @xmath222 a straightforward calculation shows that @xmath223 from , we see that @xmath224 is an invariant manifold of @xmath225 , i.e , @xmath226 from which it follows that the set @xmath202 is positive invariant for the original system - . in particular , if @xmath227 then reduces to @xmath228 therefore , by applying lemma [ l_sae ] , we can conclude that which , in view of the definition of @xmath233 and @xmath225 , and the fact that @xmath202 is positive invariant , is equivalent to the last two statements of lemma [ th : invariant ] . finally , for any initial condition @xmath234 , lemma [ p_b ] implies that @xmath235 and @xmath236 are both bounded uniformly in time so , using equation and the same argument as in the proof of lemma [ c1 ] , we can deduce that @xmath237 for all @xmath238 and some constant @xmath193 . this proves that @xmath239 is positive invariant . by symmetry , the same holds for the set @xmath240 . @xmath184 by poincar - bendixson theorem , the omega limit set of the system - is either a fixed point or a limit cycle . if the inequality holds , we can use dulac s criterion to exclude the existence of a limit cycle . let @xmath24 and define the scalar function @xmath241 on @xmath242 . then , @xmath243 \\ \hspace{120pt } + \ \displaystyle \frac{\partial}{\partial y } \ [ ( r y \,(y - \theta ) ( 1 - y ) + \mu \,(x - y ) ) \,p(x , y ) ] \vspace{8pt } \\ = \ \displaystyle ( xy)^{-c } \ [ r ( c - 3 ) ( x^2 + y^2 ) + r ( 2 - c ) ( 1 + \theta ) ( x + y ) \\ \vspace{-5pt } \\ \hspace{120pt } + \ 2 r \theta ( c - 1 ) - 2 \mu + c \mu ( 2 - x y^{-1 } - y x^{-1 } ) ] \vspace{8pt } \\ \leq \ \displaystyle ( xy)^{-c } \ \bigg[r ( c - 3 ) \bigg(x + \frac{(2 - c)(1 + \theta)}{2 ( c - 3 ) } \bigg)^2 - \frac{r ( 2 - c)^2 ( 1 + \theta)^2}{4 ( c - 3 ) } \vspace{8pt } \\ \hspace{5pt } + \ \displaystyle r ( c - 3 ) \bigg(y + \frac{(2 - c)(1 + \theta)}{2 ( c - 3 ) } \bigg)^2 - \frac{r ( 2 - c)^2 ( 1 + \theta)^2}{4 ( c - 3 ) } \displaystyle + 2 r \theta ( c - 1 ) - 2 \mu \bigg ] . \end{array}\ ] ] in particular , if holds then the equation above is strictly negative for any @xmath244 . therefore , by dulac s criterion , the system has no limit cycle , i.e. , any trajectory of - starting with a nonnegative initial condition converges to a fixed point . @xmath184 @xmath35 the jacobian matrix associated with this equilibrium is @xmath245 with eigenvalues @xmath246 and @xmath247 associated with @xmath21 and @xmath248 as their eigenvectors , respectively . we can easily conclude that the trivial boundary equilibrium @xmath35 is locally stable since both eigenvalues of are negative . @xmath37 the jacobian matrix associated with this equilibrium is @xmath249 with eigenvalues @xmath250 and @xmath251 associated with @xmath21 and @xmath248 as their eigenvectors , respectively . we can easily conclude that the equilibrium @xmath37 is always unstable on the invariant set @xmath202 . moreover , if @xmath44 then @xmath37 is a saddle , while if @xmath45 then @xmath37 is a source . @xmath36 the jacobian matrix associated with this equilibrium is @xmath252 with two negative eigenvalues @xmath253 and @xmath254 since @xmath255 . therefore , the equilibrium @xmath36 is also locally stable . to prove the third part of the theorem , we first define the function @xmath256 . then @xmath257 to prove that @xmath258 we first assume that @xmath259 since the set @xmath39 is positive invariant , we have @xmath260 for all @xmath238 . using in addition that @xmath261 we can conclude that @xmath235 converges to zero . recalling the definition of @xmath233 and invoking again the positive invariance of @xmath39 , we can deduce that @xmath262 and @xmath196 converge to zero so holds . to prove that @xmath263 we now assume that @xmath264 then , we have the following alternative . 1 . @xmath265 . since @xmath221 is positive invariant , we may use the same argument as before to see that the derivative of @xmath233 is nonnegative and the system converges to the equilibrium point @xmath36 . 2 . @xmath266 . repeating again the same argument but with the positive invariant set @xmath213 implies that the system converges to @xmath36 . 3 . @xmath267 . we may assume that @xmath268 without loss of generality since the system is symmetric . then , using the positive invariance of the set @xmath240 we have @xmath269 for all @xmath238 so @xmath270 this indicates that the trajectory starting at @xmath271 can only exit the infinite rectangle @xmath272 \times [ 1 , \infty)$ ] by crossing its bottom or right side . therefore , we have the following three possibilities . 1 . no exit : @xmath273 for all @xmath238 . in this case , the sign of the derivatives implies convergence to @xmath36 . bottom side : @xmath274 for some time @xmath238 . in this case , point 1 above implies convergence to the equilibrium point @xmath36 . 3 . right side : @xmath275 for some time @xmath238 . in this case , point 2 above implies convergence to the equilibrium point @xmath36 . combining 1 - 3 above implies . now assume that @xmath47 . defining @xmath276 recall that the system - can be rewritten as @xmath277 to prove that @xmath278 it suffices to prove that @xmath279 assume by contradiction that is not satisfied . then , there exists an initial condition with @xmath280 and a time @xmath158 such that @xmath281 by continuity of the trajectories , the intermediate value theorem implies the existence of a time @xmath162 such that @xmath282 therefore @xmath283 is well defined and @xmath284 for all @xmath166 $ ] . to prove that this leads to a contradiction , we consider the following two cases . 1 . if @xmath285 , the invariance of @xmath240 and @xmath239 implies that @xmath286 at any time , from which it follows that @xmath287 in particular , there exists @xmath200 such that @xmath288 for all @xmath289 , which contradicts the existence of time @xmath174 . 2 . if @xmath290 the result directly follows from the fact that @xmath291 is positive invariant , as it is the intersection of two invariant sets . define as previously the functions @xmath222 using above , we obtain @xmath292 assume first that @xmath234 . using the fact that the set @xmath239 is positive invariant by lemma [ th : invariant ] , we deduce that @xmath293 at all positive times @xmath2 . in particular , recalling the definition of @xmath225 , and using the expression of the derivative @xmath294 above and the fact that holds , we obtain that @xmath295 since @xmath296 if and only if @xmath224 , we deduce that @xmath236 converges to 0 . by symmetry , the same can be proved of the system starting with any initial conditions such that @xmath268 . since @xmath202 is positive invariant , we have the same conclusion when the initial condition satisfies @xmath290 , which can also be seen from the expression of the derivative @xmath294 . therefore , if holds then @xmath297 so for any @xmath298 and any @xmath299 , there exists @xmath300 such that @xmath301 it follows that any trajectory of the system converges to one of the symmetric equilibriums @xmath35 , @xmath37 or @xmath36 . now , observing that @xmath302 we obtain that @xmath303 in view of the expression of the jacobian matrix , this implies that the equilibrium @xmath215 is a saddle with unstable manifold @xmath304 in particular , it follows from hartman - grobman theorem and the second part of lemma [ th : invariant ] that there are only two attractors : @xmath35 and @xmath36 . hence , the system starting from any initial condition not belonging to the manifold @xmath56 converges to either @xmath35 or @xmath36 . to conclude the proof , it suffices to observe that , by in the proof of theorem [ th : basin ] , if @xmath47 then the set @xmath305 is positive invariant so the system starting from any initial condition in this set converges to @xmath36 , the only attractor in this invariant set . similarly , the last statement follows from the fact that , if @xmath50 then the set @xmath306 is positive invariant and contains only one attractor : @xmath35 . @xmath184 we first prove that all the equilibriums of the system - belong to the unit square @xmath68 \times [ 0 , 1]$ ] . since the system is symmetric and , by lemma [ th : invariant ] , the omega limit set of any initial condition @xmath307 belongs to the unit square ( either @xmath35 , @xmath37 or @xmath36 ) , it suffices to focus on the case @xmath308 since @xmath290 only happens when starting from @xmath215 , to avoid trivialities , we shall assume in addition that @xmath268 . then , applying lemma [ th : invariant ] , we obtain that @xmath269 for all @xmath238 which , together with , implies that @xmath309 excluding the trivial case when the initial condition belongs to the stable manifold of @xmath215 , in which case its omega limit set reduces to @xmath215 , we have the following alternative . 1 . if @xmath310 for some @xmath238 then , by the second part of theorem [ th : basin ] , the omega limit set of the initial condition is @xmath36 . 2 . if @xmath311 for all @xmath238 then and the fact that @xmath240 is positive invariant imply that @xmath312 for some @xmath158 . using as previously the continuity of the trajectories and the fact that @xmath313 allows to invoke the intermediate value theorem and prove by contradiction that @xmath314 at any time @xmath315 . this establishes the first part of theorem [ th : attractors ] . the second part follows directly from the proof of theorem [ th : dispersal ] . to prove the third part , we first observe that the equation of the nullcline @xmath65 can be rewritten as @xmath316.\ ] ] in particular , if @xmath317 then the nullcline intersects the x - axis at the three points with coordinates @xmath20 , @xmath318 and @xmath319 where @xmath320 finally , a phase - plane analysis based on figure [ fig : nullclines ] shows that 1 . if @xmath75 then the system has five fixed points with only two locally stable equilibriums : @xmath35 and @xmath36 . 2 . if @xmath76 then the system achieves maximum number of equilibriums which is nine , with only four locally stable equilibriums . if the system has less than nine equilibriums , then it has only two local stable equilibriums : @xmath35 and @xmath36 . the first step is to prove that the set of upper configurations is closed under the dynamics . we observe that , condition on the event that the configuration is an upper configuration , only expansions and migrations can occur . furthermore , migration events can only result in an increase of the lowest density and a decrease of the highest density , i.e. , if a migration event occurs at time @xmath2 and the configuration at time @xmath325 is an upper configuration then @xmath326 it follows that the set of upper configurations ( and similarly the set of lower configurations ) is closed under the dynamics , i.e. , @xmath327 for all times @xmath2 . since , starting from an upper configuration , the system jumps to @xmath21 whenever two expansion events at @xmath127 and @xmath128 occur consecutively ( they are not separated by a migration event ) , we deduce that the stopping time @xmath328 is almost surely finite . the same holds for the stopping time @xmath329 when starting from a lower configuration . hence , @xmath330 to compute the expected value of the time to fixation , we now construct the stochastic process graphically from a collection of poisson processes , relying on an idea of harris ( 1972 ) . two poisson processes , each with parameter @xmath5 , are attached to each of the patches @xmath127 and @xmath128 , and an additional poisson process with parameter one is attached to the edge connecting the patches . all three processes are independent . let @xmath331 denote these poisson processes . at any time of the process @xmath332 the population size at patch @xmath127 jumps to either 0 or 1 depending on whether it is smaller or larger than @xmath10 by this time , respectively . the evolution at patch @xmath128 is defined similarly but using the poisson process @xmath333 . at each time in @xmath334 , a fraction @xmath13 of the population at each patch is displaced to the other patch . to compute the expected value , we let @xmath238 and introduce the stopping times @xmath335 then , @xmath336 is the probability that two consecutive migration events are separated by at least one extinction - expansion event at patch @xmath127 and one extinction - expansion event at patch @xmath128 . to compute this probability , we first observe that @xmath337 and @xmath338 are independent exponentially distributed random variables with parameter @xmath5 , from which it follows that @xmath339 since @xmath340 is exponentially distributed with parameter 1 , @xmath341 hence , the last time a migration event occurs before fixation is equal in distribution to @xmath342 where the random variable @xmath343 is a geometrically distributed with parameter @xmath344 from which we deduce that @xmath345 \ = \ { \mathbb{e}}\,[t_e ] \times { \mathbb{e}}\,[j - 1 ] \ + \ { \mathbb{e}}\,[\max ( t_x , t_y ) ] \vspace{8pt } \\ \hspace{50pt } = \ \displaystyle \frac{(r + 1)(2r + 1)}{2r^2 } \ - \ 1 \ + \ \int_0^{\infty } p \,(\max ( t_x , t_y ) > u ) \,du \vspace{8pt } \\ \hspace{50pt } = \ \displaystyle \frac{3r + 1}{2r^2 } \ + \ \int_0^{\infty } 1 - ( 1 - \exp ( -ru))^2 \,du \ = \ \frac{6r + 1}{2r^2}. \end{array}\ ] ] the same holds for the stopping time @xmath329 when starting the process from a lower configuration . this completes the proof of theorem [ fixation ] . @xmath184 we first prove that @xmath346 . let @xmath299 small . then , for almost all realizations of the process , there exists an increasing sequence of random times @xmath347 such that @xmath348 moreover , there exists @xmath193 that does not depend on @xmath349 such that , if after @xmath350 a sequence of @xmath351 migration events occur before any expansion or extinction events then the system hits either an upper configuration or a lower configuration . since @xmath351 is finite , such an event has a strictly positive probability , so the borel - cantelli lemma implies that the process hits either an upper configuration or a lower configuration after a random time which is almost surely finite : @xmath352 . theorem [ fixation ] then implies that @xmath353 to estimate the expected value of @xmath129 , we observe that the transition rates of the process indicate that if at time @xmath2 exactly @xmath154 migration events but neither expansion nor extinction events have occurred then @xmath354 so that @xmath355 and @xmath356 where @xmath357 and @xmath358 are defined recursively by @xmath359 a straightforward calculation shows that @xmath360 and @xmath361 , therefore @xmath362 where @xmath363 is for the integer part . now , let @xmath364 be the markov process with state space @xmath365 and transition rates @xmath366 and starting at @xmath367 . we call @xmath368- , @xmath369- , and @xmath370-jumps , the jumps described by the three transition rates above , respectively , and refer the reader to the left - hand side of figure [ fig : grid ] for an illustration of the process . by construction of the sequences @xmath371 and @xmath372 , we have @xmath373 for all @xmath374 $ ] , i.e. , before the process hits an upper or a lower configuration , @xmath375 is stochastically smaller than @xmath376 while @xmath377 is stochastically larger than @xmath378 . this implies that @xmath379 \geq { \mathbb{e}}\,[t^*]$ ] where @xmath380 @xmath381 then , @xmath382 is the first time @xmath383 exits the set @xmath384 , i.e. , @xmath385 so , to bound @xmath386 $ ] from below , it suffices to prove that @xmath387 for an arbitrarily long time . the idea is to prove that , when starting from the smaller rectangle @xmath36 , the process stays in @xmath384 and comes back to @xmath36 after @xmath133 jumps with probability close to 1 . using in addition the markov property , we obtain that the number of jumps required to exit @xmath384 is stochastically larger than @xmath133 times a geometric random variable with small success probability . to make this argument precise , we let @xmath388 denote the embedded discrete - time markov chain associated with the process @xmath383 . to count the number of steps needed to exit the rectangle @xmath384 , we define a sequence of bernoulli random variables @xmath389 associated to @xmath388 by setting @xmath390 since @xmath388 is a discrete - time markov chain , the random variables @xmath391 are independent bernoulli random variables , and a straightforward calculation shows that the success probability is given by @xmath392 moreover , since @xmath393 , we have that @xmath394 see the right - hand side of figure [ fig : grid ] . this indicates that @xmath395 finally , using that @xmath383 jumps at rate @xmath396 and that @xmath397 is stochastically larger than a geometric random variable @xmath398 with success probability @xmath399 we can conclude that @xmath400 \ \geq \ { \mathbb{e}}\,[t^ * ] \ \geq \ \frac{n_0}{1 + 2r } \ { \mathbb{e}}\,[z ] \ = \ \frac{n_0}{2 + 4r } \ \bigg(\frac{1 + 2r}{1 + r } \bigg)^{n_0}.\ ] ] this completes the proof . @xmath184 first , we observe that the process @xmath376 introduced in the proof of theorem [ metastability ] is stochastically larger than @xmath401 so to prove the first inequality it suffices to establish its analog for the expected value @xmath402 where @xmath403 is the stationary distribution of the stochastic process @xmath376 . note that the infinitesimal matrix of the markov process @xmath376 expressed in the basis @xmath404 is given by @xmath405 by solving @xmath406 , we find that @xmath407 this implies that @xmath408 the proof of the second inequality is similar . @xmath184 we first observe that the processes @xmath409 and @xmath410 can be constructed on the same probability space starting from the same initial configuration in such a way that @xmath411 and @xmath412 until the hitting time @xmath129 , which we assume from now on . let @xmath413 and , for all @xmath414 , let @xmath350 denote the time of the @xmath349th jump of the process @xmath415 . since migration events do not change the value of @xmath416 , time @xmath350 corresponds to the time of an extinction event at @xmath127 or an expansion event at @xmath128 , therefore we have @xmath417 let @xmath299 small such that @xmath418 , and consider the events @xmath419 first , since @xmath420 , migration events between @xmath350 and @xmath421 displace less individuals on the event @xmath422 than on @xmath423 so @xmath424 second , note that @xmath425 if and only if we have @xmath426 in particular , if @xmath350 is the time of an extinction event at @xmath127 then @xmath427 only if at least @xmath428 migration events have occurred since the last expansion event at patch @xmath128 . this implies that @xmath429 since by symmetry the random variables @xmath401 and @xmath430 are identically distributed , and @xmath431 is stochastically smaller than @xmath432 , we deduce that @xmath433 finally , observing that @xmath434 we can conclude that @xmath435 this completes the proof . @xmath184 a. s. ackleh , l. j. s. allen and j. carter , 2007 . establishing a beachhead : a stochastic population model with an allee effect applied to species invasion . _ theoretical population biology _ , * 71 * 290300 .

we investigate the impact of allee effect and dispersal on the long - term evolution of a population in a patchy environment , focusing on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global in - all - patch extinction . our study is based on the combination of analytical and numerical results for both a deterministic two - patch model and its stochastic analog . the deterministic model has either two or four attractors . in the presence of weak dispersal , the analysis of the deterministic model shows that a high - density and a low - density populations can coexist at equilibrium in nearby patches , whereas the analysis of the stochastic model indicates that this equilibrium is metastable , thus leading after a large random time to either an in - all - patch expansion or an in - all - patch extinction . up to some critical dispersal , increasing the intensity of the interactions leads to an increase of both the basin of attraction of the in - all - patch extinction and the basin of attraction of the in - all - patch expansion . above this threshold , while increasing the intensity of the dispersal , both deterministic and stochastic models predict a synchronization of the patches resulting in either a global expansion or a global extinction : for the deterministic model , two of the four attractors present when the dispersal is weak are lost , while the stochastic model no longer exhibits a metastable behavior . in the presence of strong dispersal , the limiting behavior is entirely determined by the value of the allee threshold as the global population size in the deterministic and the stochastic two - patch models evolves as dictated by the single - patch counterparts . for all values of the dispersal parameter , allee effects promote in - all - patch extinction in terms of an expansion of the basin of attraction of the extinction equilibrium for the deterministic model and an increase of the probability of extinction for the stochastic model . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

the observed anomalies of atmospheric @xcite and solar @xcite neutrinos strongly suggest that neutrinos be massive and lepton flavors be mixed . in the framework of three charged leptons and three active neutrinos , the phenomena of flavor mixing and @xmath5 violation are described by a unitary matrix @xmath1 , which relates the neutrino mass eigenstates @xmath6 to the neutrino flavor eigenstates @xmath7 : @xmath8 the unitarity of @xmath1 represents two sets of normalization and orthogonality conditions : @xmath9 where greek and latin subscripts run over @xmath10 and @xmath11 , respectively . if neutrinos are dirac particles , a full parametrization of @xmath1 requires four independent parameters three mixing angles and one @xmath5-violating phase , for example . if neutrinos are majorana particles , however , two additional @xmath5-violating phases need be introduced for a complete parametrization of @xmath1 . in both cases , @xmath5 and @xmath12 violation in normal neutrino oscillations depends only upon a single rephasing - invariant parameter @xmath13 @xcite , defined through @xmath14 where @xmath15 and @xmath16 run respectively over @xmath10 and @xmath11 . a major goal of the future long - baseline neutrino oscillation experiments is to measure @xmath17 and @xmath13 as precisely as possible @xcite . once the matrix elements of @xmath1 are determined to a good degree of accuracy , a stringent test of its unitarity will become available . as a straightforward consequence of the unitarity of @xmath1 , two interesting relations can be derived from the normalization conditions in eq . ( 2 ) : @xmath18 and @xmath19 the off - diagonal asymmetries @xmath20 and @xmath21 characterize the geometrical structure of @xmath1 about its @xmath22-@xmath3-@xmath23 and @xmath24-@xmath3-@xmath4 axes , respectively . if @xmath25 held , @xmath1 would be symmetric about the @xmath22-@xmath3-@xmath23 axis . indeed the counterpart of @xmath20 in the quark sector is very small ( of order @xmath26 @xcite ) ; i.e. , the @xmath0 quark mixing matrix is almost symmetric about its @xmath27-@xmath28-@xmath29 axis . an exactly symmetric flavor mixing matrix may hint at an underlying flavor symmetry , from which some deeper understanding of the fermion mass texture can be achieved @xcite . in this sense , the tiny off - diagonal asymmetry of the quark flavor mixing matrix is likely to arise from a slight breakdown of certain flavor symmetries of quark mass matrices . the purpose of this paper is to examine whether the lepton flavor mixing matrix @xmath1 is really symmetric or not . in section ii , we find that current neutrino oscillation data strongly favor @xmath30 ; i.e. , @xmath1 is possible to be symmetric about its @xmath24-@xmath3-@xmath4 axis . it remains too early to get any phenomenological constraints on @xmath20 , unless very special assumptions are made . in section iii , we point out that the off - diagonal symmetry @xmath31 corresponds to three pairs of _ congruent _ unitarity triangles in the complex plane . taken realistic long - baseline experiments of neutrino oscillations into account , the terrestrial matter effects on @xmath13 , @xmath20 and @xmath21 are briefly discussed in section iv . section v is devoted to some further discussions about possible implications of @xmath31 on specific textures of lepton mass matrices . finally we summarize our main results in section vi . current experimental data @xcite strongly favor the hypothesis that atmospheric and solar neutrino oscillations are dominated by @xmath32 and @xmath33 transitions , respectively . thus their mixing factors @xmath34 and @xmath35 have rather simple relations with the elements of the lepton flavor mixing matrix @xmath1 . the mixing factor associated with the chooz ( or palo verde ) reactor neutrino oscillation experiment @xcite , denoted as @xmath36 , is also a simple function of @xmath17 in the same hypothesis . the explicit expressions of @xmath35 , @xmath34 and @xmath36 read as follows : @xmath37 an analysis of the super - kamiokande data on atmospheric neutrino oscillations @xcite yields @xmath38 and @xmath39 at the @xmath40 confidence level . corresponding to @xmath41 , @xmath42 can be drawn from the chooz experiment @xcite . we restrict ourselves to the large - angle mikheyev - smirnov - wolfenstein ( msw ) solution to the solar neutrino problem @xcite , as it gives the best global fit of present data . at the @xmath43 confidence level , @xmath44 and @xmath45 have been obtained @xcite . with the help of eqs . ( 2 ) and ( 6 ) , one may express @xmath46 , @xmath47 , @xmath48 and @xmath49 in terms of @xmath50 , @xmath51 and @xmath52 : @xmath53 without loss of generality , three mixing angles ( @xmath50 , @xmath51 and @xmath52 ) can all be arranged to lie in the first quadrant . then we need only adopt the solution @xmath54 @xcite , in accord with @xmath42 . we may also express @xmath55 in terms of @xmath51 and @xmath52 , once the normalization relation @xmath56 is taken into account . it turns out that useful experimental constraints are achievable for those matrix elements in the first row and in the third column of @xmath1 . however , it is impossible to get any constraints on the other four matrix elements of @xmath1 , unless some special assumptions are made .- violating phase of @xmath1 to vary between 0 and @xmath57 , fukugita and tanimoto @xcite have presented the numerical ranges of all nine @xmath17 by use of current neutrino oscillation data . this rough construction of the lepton flavor mixing matrix is actually unable to shed light on its off - diagonal asymmetries and @xmath5-violating features . ] this observation means that it remains too early to get any instructive information on the off - diagonal asymmetry @xmath20 from current neutrino oscillation experiments , but it is already possible to examine whether @xmath30 coincides with current data and what its implications can be on leptonic @xmath5 violation and unitarity triangles . to see whether @xmath30 is compatible with the present data of solar , atmospheric and reactor neutrino oscillations , we simply set @xmath58 in eq . ( 7 ) and then obtain @xmath59 as @xmath42 @xcite , the second term on the right - hand side of eq . ( 8) serves as a small correction to the leading term @xmath34 . the difference between @xmath60 and @xmath61 is therefore insignificant . indeed @xmath62 leads definitely to @xmath63 , as a straightfoward result of @xmath30 . allowing @xmath34 to vary in the experimental range @xmath38 , we plot the numerical dependence of @xmath64 on @xmath36 in fig . 1 . one can observe that the values of @xmath64 predicted from eq . ( 8) are consistent very well with current experimental data . thus we conclude that a vanishing or tiny off - diagonal asymmetry of @xmath1 about its @xmath2-@xmath3-@xmath4 axis is strongly favored . let us proceed to discuss possible implications of @xmath30 on the leptonic unitarity triangles . it is known that six orthogonality relations of @xmath1 in eq . ( 2 ) correspond to six triangles in the complex plane @xcite , as illustrated in fig . these six triangles totally have eighteen different sides and nine different inner angles . unitarity requires that all six triangles have the same area amounting to @xmath65 , where @xmath13 is just the rephasing - invariant measure of leptonic @xmath5 violation defined in eq . now that the off - diagonal asymmetries @xmath20 and @xmath21 describe the geometrical structure of @xmath1 , they must have direct relations with the unitarity triangles in the complex plane . indeed it is easy to show that @xmath66 or @xmath30 corresponds to the congruence between two unitarity triangles ; i.e. , @xmath67 and @xmath68 as @xmath30 is expected to be rather close to reality , we draw the conclusion that the unitarity triangles @xmath69 and @xmath70 must be approximately congruent with each other . a similar conclusion can be drawn for the unitarity triangles @xmath71 and @xmath72 as well as @xmath73 and @xmath74 . the long - baseline experiments of neutrino oscillations in the near future will tell whether an approximate congruence exists between @xmath69 and @xmath74 or between @xmath73 and @xmath70 . a particularly interesting possibility would be @xmath75 ; i.e. , only two of the six unitarity triangles are essentially distinct . next we examine how large the area of each unitarity triangle ( i.e. , @xmath65 ) can maximally be in the limit @xmath30 , in which @xmath1 is parametrized as follows : @xmath76 where @xmath77 , @xmath78 , and so on . the merit of this phase choice is that the dirac - type @xmath5-violating phase @xmath79 does not appear in the effective mass term of the neutrinoless double beta decay @xcite , which depends only upon the majorana phases @xmath80 and @xmath81 . without loss of generality , one may arrange the mixing angles @xmath82 and @xmath83 to lie in the first quadrant . three @xmath5-violating phases ( @xmath84 ) can take arbitrary values from 0 to @xmath85 . clearly @xmath86 holds . with the help of eq . ( 6 ) or ( 7 ) , we are able to figure out the relations between @xmath87 and @xmath88 . the result is @xmath89 then we obtain @xmath90 where @xmath64 has been given in eq . again the difference between @xmath91 and @xmath92 is insignificant . if @xmath93 or @xmath57 held , we would arrive at @xmath94 . in general , however , @xmath5 symmetry is expected to break down in the lepton sector . for illustration , we plot the numerical dependence of @xmath95 on @xmath36 in fig . 3 , where the experimentally allowed values of @xmath34 are used . it is obvious that the upper bound of @xmath96 ( when @xmath97 ) can be as large as a few percent , only if @xmath98 . this result implies that leptonic @xmath5 and @xmath12 violation might be observable in the future long - baseline neutrino oscillation experiments . in realistic long - baseline experiments of neutrino oscillations , the terrestrial matter effects must be taken into account @xcite . the pattern of neutrino oscillations in matter can be expressed in the same form as that in vacuum , however , if we define the _ effective _ neutrino masses @xmath99 and the _ effective _ lepton flavor mixing matrix @xmath100 in which the terrestrial matter effects are already included @xcite . note that @xmath99 are functions of @xmath101 , @xmath102 and @xmath103 ; and @xmath104 are functions of @xmath101 , @xmath105 and @xmath103 , where @xmath103 is the matter parameter and its magnitude depends upon the neutrino beam energy @xmath106 and the background density of electrons @xmath107 . the analytically exact relations between @xmath108 and @xmath109 can be found in ref . @xcite , if @xmath107 is assumed to be a constant . in analogy to the definitions of @xmath13 , @xmath20 and @xmath21 , the _ effective _ @xmath5-violating parameter @xmath110 and off - diagonal asymmetries @xmath111 and @xmath112 can be defined as follows : @xmath113 where @xmath15 and @xmath16 run respectively over @xmath10 and @xmath11 ; @xmath114 and @xmath115 it is then interesting to examine possible departures of @xmath116 from @xmath117 in a concrete experimental scenario . to illustrate , we compute @xmath110 , @xmath111 and @xmath112 using the typical inputs @xmath118 , @xmath119 and @xmath120 , which yield @xmath121 , @xmath122 and @xmath30 in vacuum . the neutrino mass - squared differences are taken to be @xmath123 and @xmath124 . the explicit formulas relevant to our calculation have been given in ref . we plot the numerical dependence of @xmath125 , @xmath111 and @xmath112 on the matter parameter @xmath103 in fig . 4 , where both the cases of neutrinos ( @xmath126 ) and antineutrinos ( @xmath127 ) are taken into account . one can see that the off - diagonal symmetry @xmath31 in vacuum can substantially be spoiled by terrestrial matter effects . the deviation of @xmath111 from @xmath20 and that of @xmath110 from @xmath13 are remarkably large , if @xmath128 . as emphasized in ref . @xcite , there exists the simple reversibility between the fundamental neutrino mixing parameters in vacuum and their effective counterparts in matter . the former can therefore be expressed in terms of the latter , allowing more straightforward extraction of the genuine lepton mixing quantities ( including @xmath13 , @xmath20 and @xmath21 ) from a variety of long - baseline neutrino oscillation experiments . if @xmath17 are determined to a very high degree of accuracy , it will be possible to test the unitarity of @xmath1 @xcite and establish leptonic @xmath5 violation through the non - zero area of six unitarity triangles @xcite even in the absence of a direct measurement of @xmath13 . if @xmath31 really holds , one may wonder whether this off - diagonal symmetry of @xmath1 hints at very special textures of the neutrino mass matrix @xmath129 and ( or ) the charged lepton mass matrix @xmath130 . in the following , we take two simple but instructive examples to illustrate possible implications of @xmath31 on @xmath130 and @xmath129 . * example a * in no conflict with current data on atmospheric @xcite , solar @xcite and reactor @xcite neutrino oscillations , a remarkably simplified form of @xmath1 with @xmath30 and @xmath131 is @xmath132 where @xmath133 and @xmath134 @xcite . in the flavor basis where @xmath130 is diagonal , @xmath129 can be given as @xmath135 where symmetric matrices @xmath136 , @xmath137 and @xmath138 read @xmath139 the texture of @xmath129 is rather complicated , hence it is difficult to observe any hidden flavor symmetry associated with lepton mass matrices . * example b * the result of @xmath129 in example a will become simpler , if @xmath140 is further taken ( i.e. , @xmath141 is of the bi - maximal mixing form @xcite ) . in this special case , however , simple textures of @xmath130 and @xmath129 can be written out in a more general flavor basis . it is easy to show that @xmath130 and @xmath129 of the following textures lead to @xmath141 with @xmath142 : @xmath143 \ ; , \nonumber \\ m_\nu & = & \frac{c_\nu}{2 } \left [ \left ( \matrix { 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } \right ) + \left ( \matrix { 0 & \varepsilon_\nu & 0 \cr \varepsilon_\nu & 0 & 0 \cr 0 & 0 & \delta_\nu \cr } \right ) \right ] \ ; , % ( 20)\end{aligned}\ ] ] where @xmath144 and @xmath145 are small perturbative parameters @xcite . in the limit @xmath146 , @xmath130 has the @xmath147 symmetry and @xmath129 displays the @xmath148 symmetry @xcite . the perturbative corrections in @xmath130 allow electron and muon to acquire their masses : @xmath149 we then arrive at @xmath150 gev , @xmath151 and @xmath152 . the perturbative corrections in @xmath129 make three neutrino masses non - degenerate : @xmath153 as a result , @xmath154 for the large - angle msw solution to the solar neutrino problem . examples a and b illustrate how the off - diagonal symmetry of @xmath1 ( @xmath30 ) can be reproduced from specific textures of lepton mass matrices . in view of current experimental data on solar and atmospheric neutrino oscillations , we have discussed the geometrical structure of the @xmath0 lepton flavor mixing matrix @xmath1 . we find that the present data strongly favor the off - diagonal symmetry of @xmath1 about its @xmath2-@xmath3-@xmath4 axis . this symmetry , if really exists , will correspond to three pairs of congruent unitarity triangles in the complex plane . it remains too early to tell whether @xmath1 is symmetric or not about its @xmath22-@xmath3-@xmath23 axis . a brief analysis of terrestrial matter effects on the universal @xmath5-violating parameter and off - diagonal asymmetries of @xmath1 has also been made . we expect that future long - baseline experiments of neutrino oscillations can help establish the texture of the lepton flavor mixing matrix , from which one could get some insights into the underlying flavor symmetries responsible for the charged lepton and neutrino mass matrices . fogli , e. lisi , d. montanino , and a. palazzo , phys . d * 64 * , 093007 ( 2001 ) ; j.n . bahcall , m.c . gonzalez - garcia , and c. pena - garay , jhep * 0108 * , 014 ( 2001 ) ; p.i . krastev and a.yu . smirnov , hep - ph/0108177 ; p. aliani , v. antonelli , m. picariello , and e. torrente - lujan , hep - ph/0111418 . v. barger , k. whisnant , s. pakvasa , and r.j . phillips , phys . d * 22 * , 2718 ( 1980 ) ; h.w . zaglauer and k.h . schwarzer , z. phys . c * 40 * , 273 ( 1988 ) ; t.k . kuo and j. pantaleone , rev . 61 * , 937 ( 1989 ) .

current neutrino oscillation data indicate that the @xmath0 lepton flavor mixing matrix @xmath1 is likely to be symmetric about its @xmath2-@xmath3-@xmath4 axis . this off - diagonal symmetry corresponds to three pairs of _ congruent _ unitarity triangles in the complex plane . terrestrial matter effects can substantially modify the genuine @xmath5-violating parameter and off - diagonal asymmetries of @xmath1 in realistic long - baseline experiments of neutrino oscillations .

the proposal of doping c@xmath0 crystals in a field - effect device ( fet ) and the possibility of metallic conduction and even superconductivity in such devices had raised wide - spread interest . while the revelation of dishonest data handling in some cases@xcite led to a severe damping of the initial enthusiasm , fundamental aspects of field effect doping remains a timely and interesting problem . for field - effect transistors made with self - assembled monolayers this question was addressed in ref . and for the reported enhancement of the superconducting transition temperature in c@xmath0 crystals intercalated with haloform molecules in refs . . attempts to observe the field - effect in graphite were reported in ref . . here we address , from a theoretical point of view , the question how strongly c@xmath0 can be doped in an electric field before its electronic structure is substantially changed and how this structure changes in even stronger fields . this is relevant for understanding the fundamental features of field - effect devices based on c@xmath0 and involves a number of interesting physical problems . it appears that doping c@xmath0 crystals in a field - effect device ( fet ) is very hard to achieve in practice , one of the reasons being the exceptionally strong fields that are required . very strong fields , however , not only induce charge carriers , but also polarize the molecules , and , due to the stark effect , in general , lift degeneracies . these effects are of particular importance , as c@xmath0 is quite polarizable , and as its molecular levels are highly degenerate . the term `` field - doping '' naively implies that these effects are small , such that the main effect of the external field is inducing charge carriers into electronic levels which are essentially unaffected by the field . it is clear that if the external field is strong enough , the electronic structure of the crystal will be strongly modified by the field , and one thus can no longer speak of doping . a fundamental question about field - effect devices is therefore connected with the doping levels achievable , before the electronic structure of the active material is substantially altered . moreover we discuss in detail how the electronic structure is changed , when the external field is beyond this `` doping limit '' . in this regime the commonly used analogy of the field - doped c@xmath0 with the alkali - doped fullerenes no longer holds . nevertheless a field - effect device with high carrier concentration would be an interesting device in its own right . one might , e.g. , speculate that for c@xmath0 in a strong electric field the electron - phonon coupling is enhanced compared to unperturbed c@xmath0 , as for molecular orbitals of lower symmetry less couplings are forbidden by symmetry . for a first estimate of the fields involved in field - doping , consider a simple capacitor . given a charge per area @xmath1 on the plates , the electric field _ between _ the plates is @xmath2 . assuming that in field - doped c@xmath0 the induced charge resides in the top - most layer,@xcite inducing @xmath3 elementary charges per molecules requires an _ external _ field ( originating from the gate electrode ) of @xmath4 , where @xmath5 is the area per molecule in the top - most layer of the crystal . for a c@xmath0 crystal with lattice constant @xmath6 14 typical areas per molecule are @xmath7 for the ( 111)-plane and @xmath8 for the ( 001)-plane . even though these areas are quite large , the external fields necessary for field doping are substantial , being of the order of 1 v / per induced elementary charge per molecule . this is , however , not the field experienced by a molecule . as c@xmath0 is highly polarizable ( @xmath9 @xmath10 ) , in the solid , the external field at the site of a molecule is screened by the polarization of the neighboring molecules : a monolayer of dipoles @xmath11 centered on the lattice sites @xmath12 , generates a field @xmath13 at @xmath14 , where the sum is over all sites in the monolayer , except @xmath14 . for the ( 111 ) layer , this sum is about @xmath15 , for the less dense ( 001 ) layer it is about @xmath16 . as the dipole moments @xmath11 of the molecules are induced by the screened field @xmath17 at the site of the molecule ( @xmath18 ) , we find , by self - consistent solution , that the external field is reduced by about a factor of two . inducing charge and polarizing the molecules is , however , not the only effect of the external field . in addition it also leads to a splitting of the molecular levels the stark effect . as the molecular orbitals of c@xmath0 have a definite parity , a homogeneous field splits the levels only in second order . thus for low fields the splitting is quite small , but increases quickly for larger fields . we can expect that the splitting of the molecular levels disrupts the electronic structure of the crystal only when it becomes of the order of the band width , which is about @xmath19 ev in c@xmath0 . calculations indicate that the splittings of the @xmath20- and @xmath21-levels in a _ homogeneous _ field are surprisingly small , being less than @xmath19 ev up to @xmath22 v / ( cf . figure [ splitting ] ) . given the crudeness of the argument , it thus seems entirely feasible that doping of a few elementary charges per molecule could be achieved before the electronic structure of the c@xmath0 is substantially changed . in the argument above we have described the c@xmath0 molecules as polarizable points . it is then natural to refine the model by considering also higher multipoles . such a multipole expansion is particularly suitable for c@xmath0 as the molecules are nearly spherical . the approach is then the following : first we calculate the response of a c@xmath0 molecule to external multipole fields . then we self - consistently solve the electrostatic problem for a lattice of molecules in a homogeneous external field . this provides us with the multipole expansion of the screened field acting on each molecule in the solid . given that screened field , we can then determine the splitting of the molecular levels as a function of the induced charge . the organization of the paper reflects this approach : in section [ sec : dft ] we describe the density functional calculations for determining the multipole response and the stark splitting for a c@xmath0 molecule in a multipole field for several symmetrical configuration . using group theory , in section [ sec : irresp ] , the irreducible parameters for the multipole response are determined . this allows the calculation of the polarization for arbitrary configurations . in section [ sec : split ] we give an analogous treatment for the stark splittings and explicitly show how the splitting changes as the molecule is rotated relative to the external field . in section [ sec : mol_in_layer ] we use these ingredients to self - consistently solve for the screened field seen by a molecule in a charged monolayer . the splitting of the molecular levels in this self - consistent multipole field and the effect of this splitting on the density of states is presented in section [ sec : scsplit ] . our conclusions are presented in section [ sec : concl ] . the methods for calculating the response and splitting for an arbitrarily oriented external field from the results of the density functional calculations that were performed only for special orientations are described in the appendices . appendix [ sec : rsh ] gives an example of how to calculate the response of a molecule using the irreducible parameters derived in section [ sec : irresp ] . in appendix [ sec : coupling ] we derive the coupling matrices needed for the calculating the level splitting when the molecule is rotated in the external field . finally , appendix [ sec : shtranslation ] gives the derivation of the matrix describing the field generated by a lattice of identical multipoles at the origin , which is needed for finding the self - consistent electrostatic field . to determine the response of a c@xmath0 molecule to external multipole fields we have performed all - electron density functional calculations using gaussian - orbitals.@xcite the basis set comprises 5s4p3d for carbon@xcite and we use the perdew - burke - ernzerhof functional.@xcite in our calculations we apply an external multipole field and study the change in the multipole moments of the charge density and the splitting of the molecular levels as a function of the strength of the external field . to take advantage of the molecule s symmetry we consider multipole fields with the @xmath23-axis along the 2- , 3- , and 5-fold axis of the molecule ( cf . figure [ orient ] ) . as these axes are each contained in a mirror plane , which we chose to be the @xmath24-@xmath23-plane , we can treat the thus oriented molecule as having symmetry group @xmath25 , @xmath26 , and @xmath27 respectively . applying external multipole fields with @xmath28 , this symmetry is maintained if the fields are proportional to the real part of the spherical harmonic @xmath29 , where @xmath30 is an integer multiple of the order ( @xmath312 , 3 , or 5 ) of the symmetry axis . likewise , the response of the charge density will only have multipole components proportional to @xmath32 , with @xmath30 an integer multiple of @xmath3 . calculations were done for such symmetry conserving multipole fields up to @xmath33 . for the 3-fold axis oriented along @xmath23 , we have , in addition , calculated the response to external fields proportional to @xmath34 , @xmath35 , and @xmath36 , i.e. , with a symmetry lowered to @xmath37 . as we are interested in the linear response , we have considered small multipole fields and made sure that the calculated response of the charge density is indeed proportional to the strength of the external field . we find that the linear response of a c@xmath0 molecule is very similar to that of a metallic sphere of radius 4.4 . this effective sphere radius shows a slight increase with @xmath38 . in addition there are weak off - diagonal terms . to judge the accuracy of our calculation , we have checked how well the selection rules , that are not already imposed by the @xmath39 symmetry , are fulfilled for these off - diagonal terms . from the calculated response , we have determined the irreducible linear response coefficients , which will be given in the next section . in addition to polarizing the molecule , the external field also splits the degenerate molecular levels of the c@xmath0 molecule . as the unperturbed molecular orbitals have a definite parity , for a multipole field with odd @xmath38 there will be no first - order splitting the quadratic stark effect . on the other hand , a multipole field with even @xmath38 can couple states of like parity , so in that case the splitting is linear . this is shown in figure [ splitting ] . in section [ sec : split ] it will be demonstrated how the splitting of the homo- and lumo - orbitals that were calculated for high - symmetry geometries can be extended by group theory to arbitrary orientations of the molecule relative to the external multipole field . all calculations have been performed for the equilibrium geometry of the unperturbed , neutral c@xmath0 molecule . to estimate the effect of an external field on the shape of the molecule , we have relaxed the structure in the presence of homogeneous external fields of up to 1 v / . we find only small changes ( up to about 0.005 ) in the lengths of the bonds ( 1.40 for the short and 1.45 for the long bonds ) . likewise , the polarizability changes by less than 1.5% . finally , we have calculated the total energy of the isolated c@xmath0 ion as a function of its charge @xmath40 ( spin unpolarized calculation with relaxed geometries ) and extracted the second order term @xmath41 . to compare with previous calculations,@xcite we find for the polarizability ( multipole field with @xmath42 ) of the neutral molecule @xmath439.3 @xmath10/atom , and @xmath443.2 ev . the polarizability @xmath45 of a molecule describes the linear dependence of the induced dipole moment @xmath46 on the applied electric field @xmath47 . for a multipole expansion , @xmath45 becomes a matrix @xmath48 describing the response to all multipole fields.@xcite to fix the notation ( which follows ref . ) we briefly review the definition of the multipole response matrix . the solutions of the laplace equation @xmath49 are given by @xmath50 where the two terms denote the external potential ( @xmath51 ) , and the induced potential ( @xmath52 ) due to a charge distribution @xmath53 located around @xmath54 . note that both , the laplace equation as well as the expansion of @xmath55 into multipoles only holds for @xmath56 which lie outside the charge distribution . we have introduced the regular and irregular spherical harmonics,@xcite @xmath57 special cases for the regular spherical harmonics are @xmath58 and @xmath59 , hence , the external field , @xmath60 corresponds to a constant shift , and @xmath61 is the @xmath23-component of the electric field . for the irregular spherical harmonics we have @xmath62 and @xmath63 , thus for the induced potential , @xmath64 gives the monopole charge while @xmath65 is the dipole moment . generally , the coefficients @xmath66 are the multipole moments of the charge distribution @xmath53 @xmath67 decomposing the charge distribution @xmath68 into the unperturbed charge density and the change in the charge density due to the external potential , we obtain a decomposition of the multipole moments @xmath69 . within linear response , the coefficients @xmath70 of the induced multipole moments depend linearly on the coefficients @xmath71 of the external potential , which defines the linear - response matrix @xmath48 : @xmath72 where the sign takes into account that the induced fields oppose the external fields . then @xmath73 gives the dipolar response tensor , while @xmath74 is the self - capacitance @xmath75 . the interaction energy of the molecule with the external potential is @xmath76 , which , using the previous definitions , reduces to @xmath77 therefore , @xmath78 and @xmath66 are pairs of conjugate variables and the _ total _ energy of the molecule as a function of the external field is given by @xmath79 since this is a quadratic form , we see that the matrix @xmath45 is hermitian . we can make @xmath45 real and symmetric by unitarily transforming to a real basis ( using @xmath80 and @xmath81 instead of @xmath29 and @xmath82 for @xmath83 ) . the structure of the response matrix @xmath45 depends on the symmetry of the molecule . for a metallic sphere of radius @xmath84 the response is isotropic , i.e. , @xmath45 is diagonal in the basis of the spherical harmonics : @xmath85 . lowering the symmetry to icosahedral , @xmath86 , introduces some anisotropy . to understand the response matrix for c@xmath0 we have to consider how the irreducible representations ( ir ) of the rotation group @xmath87 split into irs of the @xmath86 ( see table [ tab : ldecompi ] ) . an external multipole field of angular momentum @xmath38 can only give rise to a response with angular momentum @xmath88 , if both irs of the @xmath87 contain a common ir of the @xmath86 . in particular , because of parity , fields with even ( odd ) @xmath38 can only give rise to responses with even ( odd ) @xmath88 . furthermore , as the irreducible representations with @xmath89 are also irreducible with respect to the @xmath86 , for @xmath89 we have @xmath90 . thus , restricting the multipole expansion to @xmath89 , the response of c@xmath0 is isotropic , with @xmath91 , @xmath92 , and @xmath93 . for @xmath94 the space spanned by the spherical harmonics @xmath29 is no longer irreducible with respect to the @xmath86 . thus we need to find linear combinations of the spherical harmonics that span the irreducible representations of the icosahedral group . we call them @xmath95 where @xmath38 and @xmath24 denote the ir of the @xmath87 and @xmath86 , respectively , while the index @xmath96 labels the functions within an irreducible representation of the @xmath86 . if in the decomposition an ir @xmath24 should occur several times , we would have to introduce an additional multiplicity label . however , as can be seen from table [ tab : ldecompi ] , up to @xmath33 each ir appears at most once . we therefore suppress the multiplicity label here . explicit expressions for the basis functions @xmath95 , can , e.g. , be found in ref . , chapter 16 , or ref . , table 4.2 . in the new basis , the matrix @xmath45 is built of blocks of diagonal matrices @xmath97 where @xmath98 constitute the minimal set of parameters , and , @xmath45 being real symmetric , @xmath99 . the matrix elements of @xmath45 were calculated up to @xmath33 using the results of the density functional calculations described in section [ sec : dft ] . the @xmath98 are listed in table [ tab : lr ] . from this minimal set of independent parameters we can determine the response for arbitrary orientations of the c@xmath0 molecule relative to the external multipole field . an example of how to do this is given in appendix [ sec : rsh ] . the practical advantage of this procedure is clear : we only need to perform density functional calculations for a number of highly symmetric configurations , for which the numerical effort is much reduced . using group theory the response for arbitrary configurations can then be determined from these special cases . note that the group theoretical approach presented in this section is particularly elegant in the case of a neutral molecule which has icosahedral symmetry . upon charging , orbitals become partially filled and the symmetry is reduced which leads to a higher number of irreducible response coefficients . furthermore , the symmetry of a charged molecule depends on how the additional charge arranges in the degenerate orbitals . this is a subtle question in the case of an isolated c@xmath0 molecule and involves jahn - teller effects and coulomb interaction in competition @xcite . in the present work we restrict the analysis to neutral molecules . .[tab : ldecompi ] decomposition of the irreducible representations ( ir ) of the rotation group @xmath87 into the ir of the icosahedral group @xmath100 . [ cols="^,^",options="header " , ] @xmath101 \\ \sqrt 8\,[\alpha_{33}(t_{2u})-\alpha_{33}(h_u ) ] & \alpha_{33}(h_u)+ 8\,\alpha_{33}(t_{2u } ) \end{array } \right ) .\ ] ] we thus find @xmath102\,v_{30},\\ \delta q_{3c3}&= & \frac{\sqrt 8}{9}\ , [ \alpha_{33}(t_{2u})-\alpha_{33}(h_u)]\,v_{30}.\end{aligned}\ ] ] in this section we discuss the calculation of the level splitting for arbitrary directions within perturbation theory using the coupling constants of table [ tab : cc ] . as discussed above , we restrict the analysis to @xmath42 and @xmath103 external potentials , which corresponds to @xmath104 and @xmath105 potentials in the icosahedral symmetry @xmath86 . in first order perturbation theory , the splitting of the levels in the degenerate subspace @xmath106 is given by the eigenvalues of the matrix @xmath107 this matrix vanishes in the case of an odd potential and the splitting is given by the second order expression @xmath108 the matrix - elements in ( [ eq : firstorder ] ) and ( [ eq : secondorder ] ) are given in ( [ eq : cpl ] ) and involve the icosahedral clebsch - gordan coefficients @xmath109 . in order for the @xmath96-indices to be defined we consider the molecule oriented with the 5-fold axis parallel to the @xmath23-axis ( see figure [ orient ] ) . this allows to label the states within a multiplet unambiguously with its @xmath110 index @xmath96 . the ordered basis of a @xmath20 subspace has @xmath96-indices @xmath111 whereas the ordered basis of an @xmath21 subspace is given by @xmath112 . note that in the case of applied @xmath42 or @xmath103 potential , the @xmath96 index corresponds to the @xmath30 index of the spherical harmonics . for a detailed discussion we refer to ref . . we will present the coefficients @xmath109 as matrices with respect to the indices @xmath113 and @xmath114 . in order to reduce the number of matrices , we will give the coupling matrices for @xmath115 potentials , which are rotated around the @xmath116-axis by an angle @xmath117 . the resulting matrices are then given by @xmath118 where @xmath119 is the rotation matrix of the spherical harmonics in a given @xmath38-subspace . using the previous relations and equation ( [ eq : cpl ] ) , the coupling matrix for an even @xmath103 potential is given by @xmath120 note that the multiplicity label @xmath121 is only relevant for the homo @xmath21 because @xmath122 occurs twice in the product @xmath123 . the coupling matrix for the odd @xmath42 potential is @xmath124 in the following we restrict the sum over subspaces @xmath125 closest in energy to @xmath106 which is the @xmath126 subspace in the case of the lumo and the @xmath127 and @xmath128 subspaces in the case of the the homo ( see fig [ fig : c60spectrum ] ) . below , the coupling matrices @xmath129 which are needed to calculate the splitting of the homo and lumo are given . they are traceless @xmath130 and normalized such that @xmath131 . the coupling matrices which describe the splitting of the lumo are @xmath132 the eigenvalues of these two matrices are independent of @xmath117 and given by @xmath133 and @xmath134 which implies that the splitting is independent of the orientation of the molecule with respect to the direction of the applied @xmath42 and @xmath103 potentials . the coupling of the homo ( @xmath21 ) to the lower lying @xmath122 and @xmath135 levels is given by @xmath136 the coupling of the homo among themselves is given by @xmath137 in equation ( [ eq : secondorderapp ] ) the product @xmath138 enters . for the matrices given in ( [ eq : l1t1ucpl ] ) , ( [ eq : l1hgcpl ] ) and ( [ eq : l1ggcpl ] ) these products can be expressed in terms of the @xmath103 coupling matrices ( [ eq : l2t1ucpl ] ) , ( [ eq : l2hu1cpl ] ) and ( [ eq : l2hu2cpl ] ) : @xmath139 using these relations , the total coupling matrix ( neglecting the constant terms in the previous relations ) due to the applied @xmath42 and @xmath103 potential , is given by @xmath140 + \\ & & v_{20}\,d_2\ ; \big[\cos\delta_2\,{\mathcal{c}}^{\theta}(1;h_u h_u;h_g ) + \sin\delta_2\,{\mathcal{c}}^{\theta}(2;h_u h_u;h_g ) \big ] , \end{aligned}\ ] ] where @xmath141 and @xmath142 . similarly we have @xmath143 $ ] , @xmath144 $ ] and @xmath145 , @xmath146 . equation ( [ eq : lumocpl ] ) implies that the contributions of the @xmath42 and @xmath103 to the splitting of the lumo add up trivially . using the values in table [ tab : cc ] and the energies of figure [ fig : c60spectrum ] yields the values @xmath147 and @xmath148 , which are almost equal when compared to @xmath149 . this can be understood by the following remarks : @xmath150 in the case of @xmath151 and @xmath152 . furthermore , it can be shown that @xmath153 assuming that the angular dependence of the homo is given by @xmath154 spherical harmonics . taking an average value of @xmath155 yields the approximate relation @xmath156 , \end{aligned}\ ] ] which shows that , to a good approximation , the contributions of the @xmath42 and @xmath103 potential to the splitting of the homo add up trivially . in this section it is shown how to calculate the matrix @xmath157 appearing in equation ( [ eq : vscr1 ] ) . the second term on the right side of this equation describes the coefficients of the term @xmath158 which enters the screened potential @xmath159 and which describes the potential induced by all neighboring sites . using the definition ( [ eq : v ] ) , we can rewrite this expression as @xmath160 the function @xmath161 can be decomposed using the translation formula ( for @xmath162 ) @xmath163 where @xmath164 denote the clebsch - gordan coefficients . this formula can be found in different forms in the literature , see for example ref . . substituting ( [ eq : shrel ] ) in the sum ( [ eq : neighborpot ] ) with @xmath165 and @xmath166 yields @xmath167 where the matrix @xmath157 is given by @xmath168!}{(2l_1)!(2l_2)!}}\ , c^{l_1\!+\!l_2\,m_1\!-\!m_2}_{l_1m_1\,l_2 -\!m_2}\ , \sum_{{{\bf r}}_i\ne 0}i_{l_1\!+\!l_2\ , m_1\!-\!m_2}({{\bf r}}_i)\\ & = & ( -1)^{m_2+l_2}\sqrt{\frac{(l_1\!+\!l_2\!+\!m_1\!-\!m_2 ) ! 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we investigate the possibility of doping c@xmath0 crystals by applying a strong electric field . for an accurate description of a c@xmath0 field - effect device we introduce a multipole expansion of the field , the response of the c@xmath0 molecules , and the stark splitting of the molecular levels . the relevant response coefficients and splittings are calculated _ ab initio _ for several high symmetry orientations . using a group theoretic analysis we extend these results to arbitrary orientations of the c@xmath0 molecules with respect to the external field . we find that , surprisingly , for the highest occupied ( homo ) and the lowest unoccupied molecular orbital ( lumo ) , respectively , the two leading multipole components lift the degeneracy of the molecular level in the same way . moreover the relative signs of the splittings turn out to be such that the splittings add up when the external field induces charge into the respective level . that means that when charge carriers are put into a level , its electronic structure is strongly modified . therefore , in general , in c@xmath0 field - effect devices charge is not simply put into otherwise unchanged bands , so already because of this their physics should be quite different from that of the alkali - doped fullerenes .

_ biracks _ were first introduced in @xcite as a generalization of _ racks _ , algebraic structures whose axioms encode blackboard - framed isotopy of oriented knots and links in @xmath1 . initially called `` wracks '' alluding to the `` wrack and ruin '' resulting from keeping only conjugation in a group , racks were first considered by conway and wraith in the 1950s @xcite . the current term `` rack '' ( without the `` w '' ) is due to fenn and rourke in @xcite , who dropped the `` w '' to denote dropping invariance under writhe - changing type i reidemeister moves . in @xcite , an integer - valued invariant of knots and links associated to a finite birack @xmath2 known as the _ integral birack counting invariant _ , @xmath3 was introduced by the second author , generalizing the rack counting invariant from @xcite . in @xcite an associative algebra known as the _ rack algebra _ @xmath0 $ ] was defined for every finite rack , and in @xcite the rack counting invariant from @xcite was enhanced with representations ( known as _ rack modules _ ) of a modified form of @xmath0 $ ] . in this paper , for every finite birack @xmath2 we define an associative algebra @xmath4 $ ] we call the _ birack algebra_. we use representations of @xmath4 $ ] , known as or @xmath2-modules , to enhance the birack counting invariant from @xcite . the new invariant is defined for classical and virtual knots and links . the paper is organized as follows : in section [ b ] we review the basics of biracks and the birack counting invariant . in section [ ba ] we define the birack algebra and give examples of birack modules . in section [ i ] we use birack modules to enhance the birack counting invariant and give examples to demonstrate that the birack module enhanced invariant is not determined by the jones or alexander polynomials and is strictly stronger than the unenhanced birack counting invariant . we conclude with a few questions for future research in section [ q ] . we begin with a definition ( see @xcite ) . it is frequently useful to specify a birack structure on a finite set @xmath5 by listing the operation tables for the components of @xmath6 viewed as binary operations @xmath7 and @xmath8 . note the reversed order of the operands in @xmath9 ; this is for compatibility with previous work representing these operations as right actions . then a birack operation @xmath6 on @xmath2 is specified by the @xmath10 matrix @xmath11 $ ] with @xmath12 and @xmath13 where @xmath14 and @xmath15 . conversely , such a matrix specifies a birack structure iff the operation it defines via @xmath16},x_{l[i , j]})\ ] ] satisfies the birack axioms . as with other algebraic structures , we have the following standard notions : the birack axioms encode the blackboard - framed reidemeister moves with semiarcs labeled according to the rules below . @xmath17 at a positive kink , the invertibility properties of @xmath6 define bijections @xmath18 and @xmath19 defined by @xmath20 and @xmath21 which give us the labels on the other semiarcs in a blackboard framed type i move . @xmath22 the exponent @xmath23 of the bijection @xmath24 , that is , the smallest positive integer @xmath23 satisfying @xmath25 for all @xmath26 , is known as the _ birack rank _ or _ birack characteristic _ of @xmath2 . if @xmath2 is a finite set , then @xmath23 is guaranteed to exist , but infinite biracks may have infinite rank . @xmath27 a labeling of semiarcs in a framed oriented link diagram with elements of a birack @xmath2 determines a homomorphism @xmath28 if and only if the birack labeling condition is satisfied by the labels in @xmath2 at every crossing . conversely , every homomorphism @xmath28 determines a unique labeling of the semiarcs of @xmath29 . by construction , we have the following standard result : if @xmath29 and @xmath30 are oriented link diagrams which are related by blackboard framed reidemeister moves and @xmath2 is a finite birack , then the sets of labelings of the semiarcs of @xmath29 and @xmath30 by elements of @xmath2 satisfying the birack labeling condition , @xmath31 and @xmath32 , are in bijective correspondence . thus , the cardinality of the set of birack labelings of an oriented link diagram by a finite birack @xmath2 is a positive integer valued invariant of framed isotopy . to obtain an invariant of unframed ambient isotopy , we observe as in @xcite that the cardinalities of the sets of birack labelings are periodic in the birack rank @xmath33 since birack labelings are preserved by the _ @xmath23-phone cord move _ : @xmath34 thus , summing these cardinalities over a complete period of framings mod @xmath23 yields an invariant of ambient isotopy known as the _ integral birack counting invariant _ @xmath35 more formally , we have @xmath36 [ ex1 ] @xmath37 @xmath38 the integral birack counting invariant is also defined for virtual links via the usual technique of ignoring virtual crossings when dividing the link into semiarcs or , equivalently , regarding our virtual link diagrams as being drawn on a surface with sufficient genus to avoid virtual crossings . we would like to enhance the integral birack counting invariant by finding a way to distinguish between labelings rather than just counting how many total labeling we have . to this end , we will use a scheme analogous to the @xmath40-birack structure to define an associative algebra we call the _ birack algebra _ associated to a finite birack @xmath2 . we will discuss the motivation for this definition later in this section . * @xmath41 * @xmath42 * @xmath43 * @xmath44 * @xmath45 * @xmath46 * @xmath47 the motivation behind the birack algebra definition is to define secondary labelings of @xmath2-labeled oriented blackboard framed link diagrams with a `` bead '' at every semiarc and to use the @xmath40-birack operations on the beads at a crossing with each @xmath48 and @xmath49 coefficient indexed by the @xmath2-labels on the input strands at a positive crossing and the output strands at a negative crossing . @xmath50 the birack algebra relations are chosen to preserve bead labelings under the blackboard - framed reidemeister moves and the @xmath23-phone cord move . the choice of bead labeling rules guarantees that for every blackboard framed @xmath2-labeled diagram , the number of bead labelings is the same before and after type ii moves and framed type i moves : @xmath51 @xmath52 the other type ii and framed type i cases are similar . six of the seven birack algebra relations come from the type iii move : @xmath53 comparing coefficients on the output beads @xmath54 and @xmath55 yields the first six relations . the final birack algebra relation comes from the @xmath23-phone cord move . pushing a bead on a strand labeled with birack element @xmath56 though a positive kink multiplies the bead by @xmath57 ; thus , we need the product of these over a complete period of framings mod @xmath23 to be @xmath58 . @xmath59.\ ] ] @xmath60= \left[\begin{array}{cccccccc } t_{2,1 } & -1 & 0 & 0 & s_{2,1 } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & r_{2,1 } & -1 & 0 & 0 \\ 0 & s_{2,1 } & 0 & 0 & 0 & t_{2,1 } & 0 & -1 \\ 0 & r_{2,1 } & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & t_{2,1 } & -1 & 0 & 0 & s_{2,1 } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & r_{2,1 } & -1 \\ 0 & 0 & 0 & s_{2,1 } & -1 & 0 & 0 & t_{2,1 } \\ -1 & 0 & 0 & r_{2,1 } & 0 & 0 & 0 & 0 \\ \end{array}\right]$}\ ] ] we will now use an @xmath2-module @xmath61 to enhance the integral birack counting invariant by taking the set of @xmath4$]-module homomorphisms @xmath62}(\mathbb{z}_f[x],r)$ ] as a signature for each birack labeling @xmath63 . since each homomorphism in @xmath62}(\mathbb{z}_f[x],r)$ ] can be understood as a labeling of the beads by elements of @xmath61 , we are effectively counting bead labelings of @xmath2-labelings of @xmath29 , enhancing the birack counting invariant with a bead counting invariant of @xmath2-labeling of @xmath29 . @xmath64}(\mathbb{z}_f[x],r)\ |\ f\in \mathrm{hom}(fb((l,\mathbf{w})),x ) , \mathbf{w}\in w\}\ ] ] @xmath65}(\mathbb{z}_f[x],r)|}\right).\ ] ] by construction , we have if @xmath29 and @xmath30 are ambient isotopic oriented classical or virtual links , then @xmath66 and @xmath67 . note that the integral birack counting invariant @xmath68 is recovered from the birack module enhanced invariant @xmath69 by evaluating at @xmath70 and from @xmath71 by taking cardinality . in our next example we show that @xmath72 is stronger in general than @xmath73 . @xmath74\ ] ] @xmath75\ ] ] @xmath76}=\left[\begin{array}{cccccc } 0 & -1 & 0 & s_{u , v } & t_{u , v } & 0 \\ 0 & 0 & -1 & r_{u , v } & 0 & 0 \\ -1 & 0 & s_{z , w } & 0 & 0 & t_{z , w } \\ 0 & 0 & r_{z , w } & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & s_{v , w } & t_{v , w } \\ -1 & 0 & 0 & 0 & r_{v , w } & 0 \\ \end{array}\right ] \\ \end{array}$}\ ] ] @xmath77 \rightarrow \left[\begin{array}{cccccc } 0 & 4 & 0 & 3 & 4 & 0 \\ 0 & 0 & 4 & 2 & 0 & 0 \\ 4 & 0 & 3 & 0 & 0 & 4 \\ 0 & 0 & 2 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 & 3 & 4 \\ 4 & 0 & 0 & 0 & 2 & 0 \\ \end{array}\right ] \rightarrow \left[\begin{array}{cccccc } 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array}\right]\ ] ] @xmath78\ ] ] @xmath79\ ] ] @xmath80 @xmath81\ ] ] we end with a few open questions for future research . are there examples of biracks @xmath2 and birack modules @xmath61 whose @xmath82 invariants detect mutation or homflypt - equivalent knots ? can @xmath82 distinguish knots and links with the same khovanov or knot floer homology ? what , if any , is the relationship between birack module invariants and vassiliev invariants ? in @xcite , biquandle labelings are extended to define counting invariants of knotted surfaces in @xmath83 , and in particular it is shown that surface biquandles are just biquandles . does the same hold for the _ surface biquandle algebra _ obtained by requiring bead - labeling invariance under roseman moves ?

we extend the rack algebra @xmath0 $ ] defined by andruskiewitsch and graa to the case of biracks , enabling a notion of birack modules . we use these birack modules to define an enhancement of the birack counting invariant generalizing the birack module counting invariant in @xcite . we provide examples demonstrating that the enhanced invariant is not determined by the jones or alexander polynomials and is strictly stronger than the unenhanced birack counting invariant .

the past few years have seen considerable growth in the theory of what might be called `` knot homologies . '' roughly speaking , these invariants are homological versions of the now classical knot polynomials the alexander polynomial , the jones polynomial , and their mutual generalization , the homfly polynomial . as a first approximation , we might say that a knot homology is a bigraded homology group @xmath0 associated to a knot @xmath1 . the two gradings are a `` homological grading '' ( the usual sort of grading one expects on a chain complex ) and a `` filtration grading . '' if we take the filtered euler characteristic of @xmath2 , we recover the corresponding knot polynomial . as our understanding of these objects evolves , it seems likely that the definition of a knot homology will evolve with it . as a first approximation , however , we offer the following : a knot homology is a theory which assigns to an oriented link @xmath3 together with some auxiliary data @xmath4 a filtered chain complex @xmath5 satisfying the following properties : 1 . the filtered euler characteristic of @xmath6 is a `` classical '' polynomial invariant of @xmath7 . 2 . the filtration on @xmath5 gives rise to a spectral sequence @xmath8 . for all @xmath9 , @xmath10 does not depend on the choice of auxiliary data @xmath11 , and is thus an invariant of the link @xmath7 . the homology of the total complex @xmath5 depends only on coarse information about @xmath7 , such as the number of components and their linking numbers . in this formulation , the group @xmath2 mentioned above is the @xmath12 term of the spectral sequence . one thing that makes this definition attractive is the fact that the known examples arise from rather different areas of mathematics . the idea that such an object might exist at all is due to mikhail khovanov . in @xcite he constructed a bigraded homology theory which i ll call @xmath13 , whose filtered euler characteristic is the unnormalized jones polynomial of @xmath14 . more recently , khovanov and rozansky @xcite have constructed an infinite family of of such knot homologies , one for each @xmath15 . their filtered euler characteristics give certain specializations of the homfly polynomial . ( when @xmath16 one recovers @xmath13 . ) our other example of a knot homology comes from gauge theory ; more precisely , from the heegaard floer homology introduced by peter ozsvth and zoltn szab @xcite . this theory naturally gives rise to a bigraded homology theory known as the knot floer homology @xcite , @xcite , which i ll denote by @xmath17 . its filtered euler characteristic is the alexander polynomial of @xmath14 , which corresponds to the case @xmath18 missing from khovanov and rozansky s construction . at a first glance , the two types of knot homologies appear to be quite different . although they share the formal properties listed above , they are defined and computed in very different ways , and things which are easy to see in one theory may be quite unexpected in the other . for example , it was obvious from the start that @xmath19 is the @xmath12 term of a spectral sequence , but the corresponding fact for @xmath20 was discovered by lee in @xcite , several years after the appearance of @xcite . ( for the khovanov - rozansky theories , this result is due to gornik @xcite . ) on closer inspection , however , a more subtle correspondence between the two theories begins to appear . this correspondence has guided much of my own research in this area . in particular , it led to the discovery of a relation between the khovanov homology and the slice genus , which was the subject of my talk at mcmaster . rather than simply rehash this material , which is already covered in @xcite , i thought i would try to explain where it came from . the main goal of this paper , then , is to describe the above - mentioned correspondence between the khovanov homology and the knot floer homology , and to give some examples to convince the reader that it is an interesting one . this correspondence does not hold for all knots , but it is common enough that the author feels that there must be some sort of explanation . perhaps someone who reads this paper will be able to provide one . to properly explain the correspondence between the two theories , one must summarize a number of basic facts about them . as a secondary goal , we have tried to make this summary self - contained and accessible to anyone interested in learning about knot homologies . the rest of the paper is organized as follows . we begin in section [ sec : filt ] with a brief review of some facts about filtered chain complexes . sections [ sec : hfk ] and [ sec : kh ] describe the basic properties of the knot floer homology and the khovanov homology , respectively . we do not attempt to give definitions , but instead focus on the formal properties of these theories , their relation with classical models for the alexander and jones polynomials , and methods of computation . in section [ sec : correspondence ] , we describe the correspondence we have in mind , and give some reasons for believing that it is interesting . finally , in sections [ sec : examples ] and [ sec : examples2 ] , we describe some examples of knots for which the correspondence is known to hold , as well as a few cases for which it fails . the author would like to thank matthew hedden , peter kronheimer , ciprian manolescu , peter ozsvth , and zoltn szab for many helpful conversations , and hans boden , ian hambleton , andrew nicas , and doug park for putting together a great conference . we begin by establishing some notation and conventions related to filtered chain complexes . let @xmath21 be a chain complex freely generated over @xmath22 by a finite set of generators @xmath23 . we say that @xmath24 is a bigraded complex with homogenous generators @xmath23 if we are given two gradings @xmath25 and @xmath26 with the property that if @xmath27 then @xmath28 and @xmath29 whenever @xmath30 . we refer to @xmath31 as the _ homological grading _ on @xmath32 , and @xmath33 as the _ filtration grading . _ the filtration grading defines a filtration @xmath34 on @xmath32 , simply by setting @xmath35 we refer to @xmath36 as a _ downward _ filtration . ( if @xmath37 was always greater than @xmath38 , the result would be an _ upward _ filtration . ) the filtered euler characteristic of @xmath39 is defined to be the sum @xmath40 it is an element of the laurent series ring @xmath41 $ ] . the filtration @xmath36 on @xmath42 gives rise to a spectral sequence , which can be explicitly described as follows . we decompose the differential @xmath43 in terms of the preferred basis @xmath44 , setting @xmath45 where @xmath46 with @xmath47 . since the number of @xmath48 s is finite , all but finitely many of the @xmath49 are @xmath50 . from the identity @xmath51 , we conclude that @xmath52 as well , so @xmath53 is a chain complex . then the identity @xmath54 implies that @xmath55 is a chain complex . repeating , we obtain a sequence of chain complexes @xmath56 which eventually converges to @xmath57 . the complex @xmath58 is generally referred to as the @xmath59 term of the spectral sequence . if we are working with coefficients in a field , it is not difficult to show ( _ c.f . _ section 5.1 of @xcite ) that @xmath60 can be endowed with a differential @xmath61 in such a way that @xmath62 is chain homotopy equivalent to @xmath63 . @xmath62 is again a bigraded complex , and the resulting spectral sequence is isomorphic to our original spectral sequence @xmath64 . in the definition of the known knot homologies , the following situation arises . starting from a link @xmath7 plus a choice of some additional data @xmath4 , one obtains a bigraded complex @xmath32 . a different choice of data @xmath65 gives rise to a different chain complex @xmath66 together with a filtered chain map @xmath67 . it is easy to see that @xmath68 induces maps @xmath69 . one checks directly that the map @xmath70 is an isomorphism ; it then follows from general principles @xcite that @xmath71 is an isomorphism for all @xmath72 . the knot homology is the bigraded complex @xmath73 ; it is well - defined up to filtered isomorphism . since @xmath74 will be the primary object of our attention , we make a few comments specific to it . to begin with , observe that @xmath75 decomposes as a direct sum of complexes @xmath76 where @xmath77 is generated by those @xmath48 with @xmath78 . as a group , then , we can decompose @xmath79 . when we want to distinguish the summands in @xmath74 , we denote @xmath80 by @xmath81 . it is often convenient to represent @xmath82 by its _ filtered poincar polynomial _ : the filtered poincar polynomial of @xmath83 is given by _ @xmath84 _ _ _ it is a laurent polynomial in @xmath31 and @xmath33 . if we substitute @xmath85 , the filtered poincar polynomial reduces to the filtered euler characteristic . when @xmath86 , we will often use the shorthand @xmath87 to refer to a generator of this group . let @xmath14 be a knot in @xmath88 . the knot floer homology @xmath89 is a bigraded chain complex equipped with a homological grading @xmath90 and a filtration grading @xmath91 , which is also known as the _ alexander _ grading . conventionally , the alexander grading is chosen so as to define a downward filtration on @xmath17 . the filtered euler characteristic of @xmath92 is the alexander polynomial @xmath93 , and the homology of the complex @xmath89 is a single copy of @xmath22 in homological grading @xmath50 . the positive trefoil and hopf link . ] let @xmath94 be the unknot . the complex @xmath95 is generated by a single element whose alexander and homological gradings are both equal to zero . the differential on this complex is necessarily trivial . let @xmath96 be the positive trefoil knot shown in figure [ fig : trefoil ] . then @xmath97 is generated by three elements @xmath98 , and @xmath99 , where @xmath100 and @xmath101 . the filtered poincar polynomial is given by @xmath102 substituting @xmath85 , we recover the filtered euler characteristic , which is the alexander polynomial of @xmath96 : @xmath103 the action of @xmath43 is given by @xmath104 , @xmath105 . it is well known that the alexander polynomial is symmetric under the involution which sends @xmath107 . @xmath19 is endowed with an analogous symmetry . to descibe the behavior of the homological grading under this symmetry , it is convenient to introduce a third grading on the knot floer homology . suppose @xmath108 is a homogenous element of @xmath92 . we define the @xmath106-grading on @xmath92 by @xmath109 , and denote by @xmath110 the filtered subquotient of @xmath111 with alexander grading @xmath112 and @xmath113-grading @xmath114 . since @xmath115 and @xmath116 , it follows that the @xmath113-grading induces a filtration on @xmath89 . we do not really get any new information from this filtration . in fact , it is easy to see that the induced spectral sequence is the same as the one induced by the alexander grading . nonetheless , the @xmath106-grading turns out to be a convenient and natural thing to consider . our first piece of evidence for this fact is provided by _ ( @xcite , @xcite ) _ @xmath117 this symmetry is easily seen to hold in the example of the trefoil , where all the generators have @xmath106-grading @xmath118 . knots for which all generators of @xmath119 have the same @xmath106-grading ( like the trefoil ) have particularly simple knot floer homologies . we say that a knot @xmath14 is _ @xmath113-thin _ with @xmath120 if all generators of @xmath92 have delta - grading @xmath121 . if @xmath14 is @xmath106-thin , all generators of @xmath89 in a given alexander grading have the same homological grading as well . it follows that the isomorphism class of @xmath89 is completely determined by the alexander polynomial of @xmath14 and the value of the @xmath106-grading in which it is supported . a large class of @xmath106-thin knots is provided by _ ( @xcite ) _ [ thm : alexalt ] alternating knots are @xmath106-thin with @xmath122 . ( in @xcite and @xcite , such knots were called _ perfect_. ) we use the sign convention of @xcite , namely that positive knots have positive signature . ( a _ positive link _ is one which admits a planar diagram in which all crossings are positive . ) this is the opposite of the convention used in @xcite . many small nonalternating knots are @xmath113-thin as well . among the 53 nonalternating knots with 10 or fewer crossings , at least 39 are known to be @xmath106-thin . the simplest example of a knot which is not @xmath106-thin is the @xmath123 torus knot , whose poincar polynomial is given by @xmath124 @xmath111 carries a lot of geometric information about the knot @xmath14 and the manifolds obtained by surgery on it . in particular , it provides lower bounds on the genus of embedded surfaces bounding @xmath14 , both in @xmath125 and @xmath126 dimensions . in three dimensions , this bound is actually sharp : @xcite let @xmath127 denote the seifert genus of @xmath14 . then for all @xmath128 , @xmath129 , while @xmath130 . this generalizes the well - known fact that @xmath127 is greater than or equal to the degree of @xmath131 . the alexander filtration on @xmath17 gives rise to a spectral sequence , all of whose terms are invariants of @xmath14 . this spectral sequence converges to the homology of the total complex , which is @xmath22 . _ ( @xcite , @xcite ) _ let @xmath133 be the alexander grading of the surviving copy of @xmath22 in the spectral sequence for @xmath17 . since the spectral sequence is an invariant of @xmath14 , @xmath132 is clearly an invariant as well . below , we summarize some interesting properties of @xmath134 : [ prop : tauprop ] the invariant @xmath135 satisfies the following : 1 . ( additivity ) @xmath136 . if @xmath137 is the mirror image of @xmath14 , then @xmath138 . 2 . ( adjunction ) @xmath139 , where @xmath140 denotes the slice genus of @xmath14 . 3 . if @xmath14 is an alternating knot , then @xmath141 . 4 . if @xmath14 is a positive knot , then @xmath142 . properties ( 1)(3 ) are due to ozsvth and szab , and may be found in @xcite . ( property ( 3 ) is a corollary of proposition [ thm : alexalt ] . ) they also proved property ( 4 ) for the special case of torus knots @xcite . the general case follows from this special one , together with work of livingston @xcite and rudolph @xcite . property ( 2 ) is an application of the adjunction inequality in ozsvth - szab theory . this inequality is familiar from classical gauge theory , and was first applied in this context by kronheimer and mrowka in their proof of the milnor conjecture @xcite . let @xmath143 be an oriented @xmath121-component link . in @xcite , ozsvth and szab show how @xmath7 can naturally be thought of as a knot in @xmath144 . this construction gives rise to a knot floer homology group @xmath145 , which is again a filtered complex . its filtered euler characteristic is given by @xmath146 and its total homology has rank @xmath147 . the poincar polynomial of the total homology is given by @xmath148 ( when @xmath121 is odd , the homological grading on @xmath149 is naturally an element of @xmath150 rather than of @xmath22 . ) @xmath151 has the following elementary properties : 1 . @xmath152 , where @xmath153 denotes @xmath7 with the orientations of all components reversed . @xmath154 , where @xmath155 is the mirror image of @xmath7 , and @xmath156 denotes the operation of taking the dual complex . @xmath157 , where @xmath158 is the link obtained by taking the oriented connected sum of any component of @xmath159 with any component of @xmath160 . @xmath161 , where @xmath162 is the rank two complex with poincar polynomial @xmath163 and trivial differential . in addition , @xmath119 satisfies a _ skein exact sequence _ , which is a generalization of the skein relation for the alexander polynomial : [ prop : skeinalex ] _ ( @xcite ) _ there are long exact sequences @xmath164 ( when the middle term has more components than the other two terms ) and @xmath165 ( when the middle term has fewer components . ) here @xmath166 is the complex with filtered poincar polynomial @xmath167 ^ 2 $ ] and trivial differential . all the maps in these sequences respect the alexander filtration . a generalization of theorem [ thm : alexalt ] holds as well : _ nonsplit _ alternating links are @xmath106-thin @xcite . let @xmath168 denote the positive hopf link of figure [ fig : trefoil ] . @xmath169 is free of rank 4 , and its filtered poincar polynomial is given by @xmath170 in the skein exact sequence @xmath171 the map @xmath33 is the @xmath50 map . recall that @xmath92 arises as the second term in the spectral sequence of a certain bigraded chain complex , which we will call @xmath172 . the extra data needed to define @xmath173 is a heegaard splitting of the complement of @xmath14 together with a preferred meridian for this splitting . given this data , the generators of @xmath172 can be computed by a process which is more or less the same as computing the alexander polynomial via fox calculus @xcite . in contrast , the differentials in the complex @xmath172 are determined by counting the number of elements in certain zero - dimensional moduli spaces of pseudoholomorphic disks in a symplectic manifold . although it is sometimes possible to determine the number of points in these moduli spaces , it is in general a difficult problem . as a result , there is currently no known algorithm for computing the knot floer homology of a given knot . in most cases , successful computation of @xmath19 depends on finding a nice heegaard splitting for the knot complement . two particularly nice classes of splittings were described by ozsvth and szab in @xcite and @xcite . the first type of splitting is applicable to a specific class of knots those which can be represented by a doubly pointed heegaard diagram of genus @xmath118 . in @xcite , goda , matsuda , and morifuji observed that these knots are precisely those which admit @xmath174 bridge decompositions . ( see section [ subsec:(1,1 ) ] for more details . ) for such knots , the methods of @xcite provide a completely algorithmic way of computing the knot floer homology . the method of @xcite is based on the kauffman state model for the alexander polynomial @xcite . it is potentially applicable to any knot , but is most effective for alternating knots it is used to prove theorem [ thm : alexalt ] and for knots with relatively small crossing number . it has been used by ozsvth , szab @xcite , and eftekhary @xcite to compute the knot floer homology of three - strand pretzel knots . as a rule of thumb , it tends to be effective at computing @xmath175 when @xmath112 is close to @xmath127 , but rather less so when @xmath112 is close to @xmath50 . other special heegaard splittings have been used by eftekhary @xcite and hedden @xcite to make some computations for whitehead doubles and cabled knots . we close this section by mentioning two indirect computational techniques , which do not rely on a choice of heegaard splitting . the first is to use the skein exact sequence of proposition [ prop : skeinalex ] . this can be an effective method for proving a knot is @xmath106-thin , especially when the knot is only mildly nonalternating or has a small number of crossings . the second method applies if @xmath14 has a lens space , or , more generally an @xmath7-space surgery . in this case , @xmath17 is completely determined by the alexander polynomial of @xmath14 @xcite . let @xmath143 be an oriented @xmath121component link . the khovanov homology @xmath176 is a bigraded chain complex equipped with a homological grading @xmath177 and a filtration grading @xmath178 , which is also known as _ jones _ grading . as a group , @xmath176 was defined by khovanov in @xcite ; the chain complex structure was described by lee in @xcite . conventionally , @xmath20 is defined to be a _ cohomology _ theory with an upward filtration . the filtered euler characteristic of @xmath176 is given by @xmath179 where @xmath180 is the jones polynomial of @xmath7 . the homology of the complex @xmath176 has rational rank @xmath181 and has no @xmath182torsion for @xmath183 , but its @xmath184torsion can be rather complicated . if @xmath185 ( so @xmath7 is actually a knot ) then both rational generators have homological grading @xmath50 . more generally , for @xmath186 , the homological gradings of the generators are determined by the pairwise linking numbers of the components of @xmath7 . ( see @xcite for details . ) some elementary properties of the khovanov homology are stated below : @xmath187 satisfies 1 . 2 . @xmath189 . 3 . @xmath190 . let @xmath94 be the unknot . @xmath191 is generated by two elements @xmath192 . both generators have homological grading 0 , and their @xmath178-grading is given by @xmath193 . the graded poincar polynomial is @xmath194 the differential on @xmath195 is necessarily trivial . let @xmath168 be the hopf link of figure [ fig : trefoil ] . then @xmath196 has rank @xmath126 , and its graded poincar polynomial is @xmath197 as a complex @xmath198 is trivial , so its homology has rank @xmath126 . let @xmath96 be the trefoil of figure [ fig : trefoil ] . then @xmath199 has rational rank @xmath126 , and its graded poincar polynomial is @xmath200 if we use @xmath201 coefficients , however , the rank is @xmath202 : @xmath203 we have @xmath204 there is a single nonzero differential in the complex @xmath199 , which takes @xmath205 to @xmath206 . the total homology thus has rank @xmath184 , with both generators having homological grading zero . the khovanov homology is constructed using the kauffman state model for the jones polynomial . as such , it is naturally endowed with a skein exact sequence based on kauffman s _ unoriented _ skein relation for the jones polynomial . [ prop : khskein ] there are long exact sequences @xmath207 and @xmath208 where @xmath209 is the difference between the number of negative crossings in the unoriented resolution @xmath210 and the number of such crossings in the original diagram . here , notation such as @xmath211 should be understood to indicate the complex @xmath212 shifted in such a way as to multiply its poincar polynomial by @xmath178 . the arrow marked with @xmath213 is the boundary map in the long exact sequence ; it raises the homological grading by @xmath118 . let @xmath168 be the positive hopf link . then both resolutions of a crossing yield the unknot , and the first exact sequence becomes @xmath214 which splits to give a short exact sequence @xmath215 just as the unoriented skein relation for the jones polynomial can be used to show that it satisfies the oriented skein relation @xmath216 the skein exact sequence above can be used to show that @xmath20 satisfies an oriented skein exact sequence analogous to that of proposition [ prop : skeinalex ] . in section 3 of @xcite , khovanov describes a slight variant of his construction which results in a related bigraded homology theory known as the _ reduced khovanov homology . _ this group is an invariant of a link @xmath7 together with a particular marked component @xmath217 of @xmath7 . we denote it by @xmath218 , or @xmath219 if @xmath14 is a knot . like @xmath20 , @xmath220 is endowed with a homological grading @xmath90 and a jones grading @xmath178 . its graded euler characteristic is given by the jones polynomial : @xmath221 recent work of bar - natan @xcite and turner @xcite implies that @xmath222 may be endowed with a differential analogous to lee s , but without the problems with @xmath184-torsion . the total homology of the complex @xmath222 is @xmath223 , where @xmath121 is the number of components of @xmath7 . when @xmath7 has more than one component , @xmath222 suffers from the disadvantage that it is not really a link invariant : it depends on the choice of marked component . for knots , however , @xmath224 seems to be an interesting and natural invariant in its own right . the reduced khovanov homology satisfies skein exact sequences analogous to the ones described above for @xmath225 . the two theories are related by there is a long exact sequence @xmath226 let @xmath96 be the positive trefoil knot . then @xmath227 ; its graded poincar polynomial is @xmath228 the boundary map @xmath229 takes @xmath230 , which has poincar polynomial @xmath231 to @xmath232 , which has poincar polynomial @xmath233 the only possible nontrival component of @xmath234 is the one which takes @xmath235 to @xmath236 . it is not difficult to see that this component is multiplication by @xmath184 . in fact , it follows from work of ozsvth and szab @xcite or shumakovitch @xcite that the map @xmath229 is always congruent to @xmath50 @xmath237 . just as in the case of the knot floer homology , it turns out that there is a third interesting grading on @xmath20 and @xmath224 . suppose @xmath238 is a homogenous element of @xmath239 . we define the @xmath240-grading on @xmath241 by @xmath242 , and denote by @xmath243 the filtered subquotient of @xmath244 with jones grading @xmath112 and @xmath113-grading @xmath114 . if @xmath14 is a knot , the @xmath245-grading on @xmath246 is defined similarly . @xmath106-gradings of elements of @xmath247 are always odd , while @xmath106-gradings of elements of @xmath246 are always even . a knot @xmath14 is @xmath20-thin with @xmath248 if @xmath246 is free over @xmath22 and all its generators have @xmath106-grading @xmath121 . it follows that @xmath224 of a @xmath20thin knot is determined by its jones polynomial and the value of @xmath106 that @xmath224 is supported in . in fact , using the chain complex structure on @xmath249 , lee has shown that the rational @xmath225 of a @xmath20-thin knot is determined by this information as well . as evidence that the preceding two definitions are interesting , we have the following theorem , which is essentially due to lee , although she phrased it in terms of @xmath20 rather than @xmath250 . _ ( @xcite ) _ [ thm : jonesalt ] if @xmath7 is a nonsplit alternating link , then @xmath7 is @xmath20-thin with @xmath251 . let @xmath14 be a knot . if we use rational coefficients , the spectral sequence induced on @xmath253 by the jones filtration converges to @xmath254 . in analogy with the @xmath132 invariant , we can define invariants of @xmath14 by looking at the @xmath178-gradings of the surviving terms in the spectral sequence . at first glance , it appears that we get two such invariants , since there are two surviving generators in the spectral sequence . in reality , it can be shown that the @xmath178-gradings of these generators always differ by @xmath184 , so there is really only one invariant : if @xmath14 is a knot in @xmath88 , we let @xmath255 be the average of the @xmath178 gradings of the two surviving rational generators in the spectral sequence for @xmath256 . since the two @xmath178 gradings are odd integers , @xmath255 is even . the invariant @xmath252 can be shown to have the following properties , which are exact analogs of the properties of @xmath132 described in proposition [ prop : tauprop ] . _ ( @xcite ) _ [ prop : sprop ] the invariant @xmath257 satisfies the following : 1 . ( additivity ) @xmath258 . if @xmath137 is the mirror image of @xmath14 , then @xmath259 . @xmath260 , where @xmath140 denotes the slice genus of @xmath14 . 3 . if @xmath14 is an alternating knot , then @xmath261 . 4 . if @xmath14 is a positive knot then @xmath262 . in @xcite , @xmath187 is defined as the homology of a finite dimensional chain complex @xmath263 , which is defined using a planar diagram of @xmath7 . the generators of @xmath264 correspond to states in the kauffman state model for the jones polynomial . the differentials are completely explicit as well , so @xmath176 is by definition algorithmically computable . the size of @xmath265 grows exponentially with the number of crossings in the planar diagram of @xmath7 , so it is only in the simplest cases that the homology can be computed directly by hand . on the other hand , the problem is well - suited to computer computation . the first program for this purpose was written by bar - natan @xcite ; it could be used to compute @xmath241 for links of up to @xmath266 or @xmath267 crossings . based on his calculations , bar - natan formulated some influential conjectures , which formed the basis for much of the early work on khovanov homology . more recently , shumakovitch has written a substantially faster program known as _ khoho _ @xcite which can effectively compute @xmath20 and @xmath224 for links with as many as @xmath268 crossings . despite this fact , many basic computational questions remain unanswered . for example , the khovanov homology of the @xmath269 torus knot is still unknown . we are now in a position to describe the correspondence alluded to in the introduction . first , we need one more bit of notation . we denote by @xmath270 the group generated by all generators of @xmath271 which have @xmath106-grading equal to @xmath114 . in other words , we have @xmath272 the notation @xmath273 should be understood similarly . we say that a knot @xmath14 has _ property fk _ ( for floer - khovanov ) if it satisfies the following two conditions : 1 . for each value of @xmath114 , we have @xmath274 2 . @xmath275 . the definition is motivated by the following alternating knots have property fk . this follows trivially from theorems [ thm : alexalt ] and [ thm : jonesalt ] , since for an alternating knot @xmath276 what is perhaps more surprising is that a great many non - alternating knots have property fk as well . in fact , i spent about six months under the impression that property fk might hold for all knots before discovering that part ( 1 ) of the property fails for the @xmath277 torus knot . part ( 2 ) still holds in this case , and to the best of my knowledge , there still no examples known for which it fails . in section [ sec : examples ] , we describe several examples of knots known to have property fk . many have floer homologies and khovanov homologies which seem quite nontrivial . although it now seems likely that property fk fails for all knots which are sufficiently complicated in some sense , the level of complication needed seems rather high . when one considers how different the definitions of the two theories seem , the correspondence seems to demand an explanation . in this section , we describe a number of arguments which might be advanced to explain the correspondence described above . although they are all at least vaguely plausible , none of them seem truly satisfactory . * * skein theory : * the two theories share some similar basic properties . they agree for alternating knots and links , both satisfy skein exact sequences ( if one wants , the unoriented skein sequence of proposition [ prop : khskein ] can be used to prove an oriented skein sequence similar to that of proposition [ prop : skeinalex ] . ) and are both constrained by the requirement that they be a complex with simple homology . perhaps these requirements are enough to force property fk to hold for a large number of knots . the arguments against this idea are twofold . first , the requirements described above are fairly weak in practice . skein theory can be used to prove some things , but there are many knots for which it seems ineffective . second , the floer homology of the branched double cover satisfies at least the first two of the properties described above , but quickly begins to differ from them once you leave the realm of alternating knots . * * a master theory : * it is tempting to imagine that the two theories can be subsumed as special cases of a single construction , like that of khovanov and rozansky @xcite . the similarity might become evident in this more complicated theory . the major objection to this idea is that the khovanov - rozansky theories do not share the simple behavior that @xmath224 and @xmath119 exhibit for alternating knots . ( this follows from the fact that the homfly polynomial of an alternating knot need not be alternating . ) * * a spectral sequence : * a third possibility is that the two theories are related by a spectral sequence . this is particularly attractive in light of the work in @xcite , which showed that there is a spectral sequence starting at the reduced khovanov homology and converging to the floer homology of the branched double cover . it is also supported by the fact that in all the known examples of knots which do not have property fk , the rank of the @xmath224 is greater than that of @xmath19 . this is perhaps the most attractive possibility of the three , but it is not clear where such a spectral sequence might come from . whatever its origins , the correspondence between the floer homology and the khovanov homology has proved to be a useful guide to the study of both . the two have complementary strengths and weaknesses . on the one hand , the khovanov homology is very simple and easy to compute with , but we have relatively little geometric intuition into its behavior . on the other , the floer homology can be difficult to compute , but comes with twenty years worth of geometric intuition developed by gauge theory . even though we have no direct evidence of a relation between the two theories , the fk correspondence can be a useful guide , suggesting that we try to prove analogs of statements which are known in one theory directly in the other . for example , the possibility that alternating knots might be @xmath106-thin was first suggested by lee s proof of the analogous result for the khovanov homology . conversely , the fact that @xmath132 was known to be a lower bound for the slice genus suggested that one should try to prove the same thing for its counterpart @xmath252 , in the khovanov theory @xcite . as a corollary , one obtains topological proofs of some results which were previously only known via gauge theory . the classical milnor conjecture , which states that the slice genus of a torus knot is equal to its seifert genus , is one such case . ( since torus knots are positive , the result is a consequence of the third property in proposition [ prop : sprop ] . ) while we were at mcmaster , bob gompf kindly pointed out another such application . namely , @xmath252 can also be used to give a gauge - theory free proof of the existance of an exotic @xmath278 . indeed , gompf has shown that to construct such a manifold , it suffices to exhibit a knot @xmath14 which is smoothly but not topologically slice . ( see @xcite p. 522 for a proof . ) by a theorem of freedman , any knot with alexander polynomial @xmath118 is topologically slice @xcite , so we need only find a knot @xmath14 with @xmath279 and @xmath280 . it is not difficult to produce such a knot for example , the @xmath281 pretzel knot will do . the khovanov homology of this knot can be calculated , either by _ khoho _ or using the skein exact sequence , and from there it can easily be determined that @xmath282 . in this section , we describe some examples of knots which can be seen to have property fk . our main goal is to convince the reader that this property is both interesting and common . to this end , we have tried to describe a variety of knots for which it holds , including some for which @xmath224 and @xmath19 seem quite complicated . in this category we include all knots with 10 or fewer crossings , numbered as in rolfsen @xcite . it seems very likely that all of these knots have property fk . more precisely , @xmath119 of the non - alternating @xmath283 and @xmath284 crossing knots was computed by ozsvth and szab in @xcite . all of these knots are @xmath106-thin except for @xmath285 ( the @xmath123 torus knot ) and @xmath286 . for the 10 crossing knots , goda , matsuda , and morifuji computed @xmath119 for @xmath287@xmath288 , @xmath289 and @xmath290 , using the fact that these are @xmath174 knots @xcite . of the remainder , skein theory can be used to show that @xmath291,@xmath292 , @xmath293 , @xmath294@xmath295,@xmath296,@xmath297 , @xmath298 , and @xmath299@xmath300 are @xmath106-thin with @xmath301 . finally , the methods of @xcite can be used to show that @xmath302 , @xmath303,@xmath304,@xmath305 , and @xmath306 are @xmath106-thin with @xmath307 , and to compute @xmath119 of @xmath308 , @xmath309 and @xmath310 , which are not @xmath113-thin . this leaves two knots @xmath302 and @xmath311 for which the author was unable to determine @xmath312 . the khovanov homology of all these knots can be computed using either bar - natan s program or _ and in all cases it is easy to see that property fk holds . perhaps the most interesting example is provided by the knot @xmath289 , for which the ranks of both @xmath250 and @xmath119 are equal to @xmath267 . in both cases , this knot exhibits a lot of `` hidden '' homology the sum of the absolute values of coefficients of the alexander polynomial is only 7 , and the corresponding sum for the jones polynomial is only 5 . of course , it is easy to compute the khovanov homology for @xmath302 and @xmath311 as well they are both @xmath313-thin with @xmath314 . it would be very surprising if they were not @xmath240-thin too . a knot @xmath315 is said to be a @xmath174 knot if there is a genus @xmath118 heegaard splitting @xmath316 of @xmath88 with the property that @xmath317 is a single trivially embedded arc . the knot floer homology of these knots is algorithmically computable @xcite , @xcite , so they offer us a good opportunity to compare @xmath119 and @xmath224 on a large set of `` complicated '' knots . by looking at examples of this type , we were able to turn up a small sample of knots which do not have property fk , as well as a rather larger number of knots that do . we briefly sketch the method of computation here . a doubly pointed genus one heegaard diagram for the knot @xmath318 . the `` rainbow '' on either side contains @xmath178 strands of @xmath319 , the middle band contains @xmath320 , and the lower band contains @xmath321 . ] as described in @xcite , a @xmath174 decomposition of a knot determines and is determined by a doubly pointed diagram of @xmath88 . such a diagram is composed of a pair of curves @xmath322 and @xmath323 on the doubly punctured torus @xmath324 , with the property that the algebraic intersection number @xmath325 . after an isotopy , we may assume that @xmath326 and @xmath327 are in the form shown in figure [ fig:(1,1)knot ] . such a diagram is specified by four non - negative integers @xmath328 and @xmath252 . here @xmath182 is the total number of intersection points of @xmath326 with @xmath327 , @xmath178 is the number of strands in each `` rainbow , '' @xmath320 is the number of strands running from below the left - hand rainbow to above the right - hand one , and @xmath252 is the `` twist parameter '' : if we label the intersection points on either side of the diagram starting from the top , then the @xmath112-th point on the right - hand side is identified with the @xmath329-th point on the left - hand side . conversely , suppose we are given @xmath330 satisfying @xmath331 and @xmath332 , and with the property that the resulting curve @xmath326 has intersection number @xmath333 with @xmath327 . then we can recover the corresponding knot in @xmath88 as follows . first , let @xmath334 be the standard genus one heegaard splitting of @xmath88 , and let @xmath335 be a curve on the boundary torus that bounds a compressing disk in @xmath336 . we identify @xmath337 with the boundary of the standard solid torus in @xmath88 in such a way that @xmath327 is identified with @xmath338 and @xmath326 has intersection number @xmath50 with @xmath339 . connect @xmath238 to @xmath340 by an embedded curve in @xmath337 disjoint from @xmath319 and @xmath340 to @xmath238 by an embedded curve in @xmath337 disjoint from @xmath327 . to obtain the knot , push the interior of the second curve into the solid torus , so that it becomes disjoint from the second curve . we denote the resulting knot by @xmath318 . note that this identification is not unique different values of @xmath341 , and @xmath252 may well produce the same knot . the knot floer homology of @xmath318 can be computed by the method of @xcite . its total rank is always @xmath182 . to find the khovanov homology , we used the above method to produce a planar diagram of @xmath14 . the diagram was then simplified ( and , if possible , identified ) using _ knotscape_. finally , the khovanov homology was computed using _ khoho_. the table below contains a list of those @xmath174 knots which were examined and found to have property fk . the knots in the table are of the form @xmath318 where @xmath342 . this is an admittedly unscientific sample ; it was chosen by sorting all the knots with a given value of @xmath182 by their alexander polynomial , and then selecting one representative for each alexander polynomial . the idea was to avoid duplicates , since the same knot ( or its mirror image ) is usually represented by several different @xmath318 s . the selection was further narrowed by discarding knots with fewer than @xmath343 crossings , those which appeared likely to be @xmath113-thin ( _ i.e. _ @xmath344 ) and those whose alexander polynomials suggested that they had @xmath7-space surgeries . ( these tended to be too big for _ khoho _ to handle . ) the entries in the table may be explained as follows . the first and second columns identify the knot as @xmath345 , and by its _ the next column shows the alexander polynomial . to save space , we have abbreviated by writing only the coefficients of non - negative powers of @xmath91 . for example , the first entry in the table indicates an alexander polynomial of @xmath346 . the next column contains the @xmath245-polynomial , which is the poincar polynomial of @xmath17 with respect to the @xmath106-grading . for example @xmath347 has @xmath125 generators with @xmath106-grading @xmath348 , and @xmath283 with @xmath106-grading @xmath349 . the last column shows @xmath132 , which in all cases is equal to @xmath350 . in this section , we give some examples of knots which are known not to satisfy property fk . initially , such examples were rather hard to come by , but the appearance of _ khoho _ has made them substantially easier find . although the size of the sample here is too small to support even an optimistic conjecture , it is worth noting that all the examples described here share the following properties . * _ large bridge number : _ although this invariant is hard to compute , it appears that all examples have bridge number @xmath352 . * _ large @xmath224 : _ in all the examples , the rank of @xmath250 is greater than the rank of @xmath119 . * _ torsion in @xmath224 : _ all examples have @xmath353 torsion in their reduced khovanov homology . for small knots , at least , this phenomenon is quite rare . @xmath250 is free for all knots with fewer than 13 crossings , and there are only four 13 crossing knots which have @xmath354 torsion in @xmath250 . ( they are @xmath355 , @xmath356 , @xmath357 , and @xmath358 . ) two of these knots appear in the list below . it would certainly be interesting to determine if the other two have property fk . the @xmath277 and @xmath359 torus knots do not have property fk . in fact , it seems likely that this is the case for all @xmath269 torus knots with @xmath360 , but these are the only ones for which we were able to determine @xmath224 . for these two knots , the knot floer homology is free with poincar polynomial @xmath361 in particular , the ranks of @xmath19 are @xmath362 and @xmath343 respectively . in contrast , the ranks of @xmath250 are @xmath284 and @xmath363 . their poincar polynomials are @xmath364 in addition , @xmath365 has a @xmath201 summand in degrees @xmath366 and @xmath367 , and @xmath368 has a @xmath354 summand in degrees @xmath369 , @xmath370 , @xmath371 , and @xmath372 . ( the author would like to thank alexander shumakovitch for providing the information for the @xmath373 torus knot . ) the knot floer homology of certain cabled knots was computed by hedden in @xcite . although it seems difficult to compute the khovanov homology of most cabled knots , the @xmath375 cables of the trefoil are small enough to be attacked directly using _ khoho_. the @xmath376 and @xmath377 cables of the positive trefoil each have planar diagrams with @xmath267 crossings . ( their _ knotscape _ numbers are @xmath357 and @xmath356 . ) using the methods of @xcite , their knot floer homologies can be seen to have ranks @xmath378 and @xmath362 , respectively . ( the author would like to thank matt hedden for sharing this fact . ) more precisely , their poincar polynomials are given by @xmath379 on the other hand , their reduced khovanov homologies both have rank @xmath343 . their poincar polynomials are @xmath380 and @xmath381 both knots have @xmath201 torsion in @xmath224 ; for the @xmath376 cable it is in degrees @xmath382 , @xmath383 , @xmath384 , and @xmath385 , while for the @xmath377 cable it is in degrees @xmath386 , @xmath387 , and @xmath388 . in both cases , it is easy to check that @xmath389 . more generally , it can be seen that for any odd @xmath390 , the rank of the knot floer homology of the @xmath375 cable of the trefoil is less than the rank of its @xmath250 . ( the rank of @xmath119 is computed by @xcite , while the the rank of @xmath224 can be bounded below using the skein exact sequence . ) these knots have positive diagrams , so they automatically satisfy @xmath391 . he we include three other @xmath174 knots which do not have property fk . in the notation of section [ subsec:(1,1 ) ] these are @xmath392 , @xmath393 , and @xmath394 , which _ knotscape _ was unable to reduce to fewer than @xmath363 crossings . below , we give individual details for each knot . @xmath395 : @xmath119 is rank @xmath267 with poincar polynomial @xmath396 @xmath250 is rank @xmath397 , with poincar polynomial @xmath398 and @xmath201 summands in degrees @xmath399 , and @xmath400 . this knot has @xmath401 and @xmath402 . @xmath403 : @xmath119 is rank @xmath397 with poincar polynomial @xmath404 @xmath250 is rank @xmath363 , with poincar polynomial @xmath405 and @xmath201 summands in degrees @xmath406 and @xmath407 . this knot has @xmath408 . @xmath409 : @xmath119 is rank @xmath397 with poincar polynomial @xmath410 @xmath250 is rank @xmath411 , with poincar polynomial @xmath412 and @xmath201 summands in degrees @xmath413 and @xmath414 . this knot has @xmath415 .

this is an expository paper discussing some parallels between the khovanov and knot floer homologies . we describe the formal similarities between the theories , and give some examples which illustrate a somewhat mysterious correspondence between them .

ground - state kohn - sham ( ks ) density functional theory ( dft ) is a widely - used tool for electronic structure calculations of atoms , molecules , and solids @xcite , in which only the density functional for the exchange - correlation energy , @xmath1 $ ] , must be approximated . but a direct , orbital - free density functional theory could be constructed if only the non - interacting kinetic energy , @xmath2 , were known sufficiently accurately as an explicit functional of the density @xcite . using it would lead automatically to an electronic structure method that scales linearly with the number of electrons @xmath3 ( with the possible exception of the evaluation of the hartree energy ) . thus the ks kinetic energy functional is something of a holy grail of density functional purists , and interest in it has recently revived @xcite . in this work , we exploit the unreasonable accuracy " of asymptotic expansions @xcite , in this case for large neutral atoms , to show that there is a very simple exact condition that approximations to @xmath2 must satisfy , if they are to attain high accuracy for total energies of matter . by matter , we mean all atoms , molecules , and solids that consist of electrons in the field of nuclei , attracted by a coulomb potential . the exact condition is the ( known ) asymptotic expansion of @xmath4 for neutral atoms , in powers of @xmath5 . by careful extrapolation from accurate numerical calculations up to @xmath6 , we calculate the coefficients of this expansion . we find that the usual gradient expansion , derived from the slowly - varying gas , but applied to essentially exact densities , yields only a good approximation to these coefficients . thus , all new approximations should either build in these coefficients , or be tested to see how well they approximate them . we perform several tests , using atoms , molecules , jellium surfaces , and jellium spheres , and analyze two existing approximations . in ref . @xcite , a related method was used to derive the gradient coefficient in modern generalized gradient approximations ( gga s ) for exchange . given this importance of @xmath7 as a condition on functionals , we revisited and improved upon the existing parametrizations of the neutral - atom thomas - fermi ( tf ) density . the second - half of the paper is devoted to testing its accuracy . for an @xmath3-electron system , the hamiltonian is = + + , [ eq : hamil ] where @xmath8 is the kinetic energy operator , @xmath9 the external potential , and @xmath10 the electron - electron interaction , respectively . the electron density @xmath11 yields @xmath12 , where @xmath3 is the particle number . to explain asymptotic exactness , we ( re)-introduce the @xmath13-scaled potential @xcite ( which is further discussed in ref . @xcite ) , given by v^ ( ) = ^4/3 v(^1/3 ) , nn , [ vzeta ] where @xmath14 is the external potential , and the thomas - fermi expectation value is @xmath15=\zeta^{7/3}v\ext[n]$ ] . in this @xmath13-scaling scheme , nuclear positions @xmath16 and charges @xmath17 of molecules are scaled into @xmath18 and @xmath19 respectively . in a uniform electric field , @xmath20 . for neutral atoms , scaling @xmath13 is the same as scaling @xmath21 , and this gives schwinger s asymptotic expansion for the total energy of neutral atoms @xcite , e = - c_0 z^7/3 - c_1 z^2 - c_2 z^5/3 + , [ ezasymp ] where @xmath22 , @xmath23 , @xmath24 , and @xmath21 is the atomic number . this large @xmath21-expansion gives a remarkably good approximation to the hartree - fock energy of the neutral atoms , with less than a 10% error for h , and less than 0.5% error for ne . by the virial theorem for neutral atoms , @xmath25 , and @xmath26 to this order in the expansion ( since the correlation energy is roughly @xmath27 ) . hence , the non - interacting kinetic energy has the following asymptotic expansion . t= c_0 z^7/3 + c_1 z^2 + c_2 z^5/3 + [ tszasymp ] we say that an approximation to the kinetic energy functional is _ asymptotically exact _ to the @xmath28-th degree if it can reproduce the exact @xmath29 . the three displayed terms in eq . ( [ ezasymp ] ) constitute the second - order asymptotic expansion for the total energy of neutral atoms , and we expect that this asymptotic expansion is a better starting point for constructing a more accurate approximation to the kinetic energy functional than the traditional gradient expansion approximation ( gea ) . the leading term in eq . ( [ tszasymp ] ) is given _ exactly _ by a local approximation to @xmath2 ( tf theory ) , but the leading _ correction _ is due to higher - order quantum effects , and only approximately given by the gradient expansion evaluated on the density . however , these coefficients are _ vital _ to finding accurate kinetic energies . since we know that @xmath30 becomes exact as @xmath7 , we define @xmath31 and investigate @xmath32 as a function of @xmath21 . how accurate is the asymptotic expansion for @xmath32 ? in figure [ f : deltats ] , we evaluate @xmath2 for atoms within the optimized effective potential ( oep ) @xcite using the exact exchange functional and plot the percentage error in @xmath32 , for all atoms and the asymptotic series with just two terms . the series is incredibly accurate , with only a 13% error for @xmath3=2 ( he ) , and 14% for @xmath3=1 . thus , any approximation that reproduces the correct asymptotic series ( up to and including the @xmath33 term ) is likely to produce a highly accurate @xmath2 . and @xmath34.,height=207 ] to demonstrate the power and the significance of this approach , we apply it directly to the first term ( where the answer is already known , but perhaps not yet fully appreciated in the dft community ) . using any ( all - electron ) electronic structure code , one calculates the total energies of atoms for a sequence running down a column . by sticking with a specific column , one reduces the oscillatory contributions across rows , and the alkali - earth column yields the most accurate results . by then fitting the resulting curve of @xmath4 as a function of @xmath5 to a parabola , one finds @xmath35 . now assume one wishes to make a local density approximation ( lda ) to @xmath2 , but knows nothing about the uniform electron gas . dimensional analysis yields @xcite t= a i , i= d^3r n^5/3 ( ) , [ eq : tloc ] but does not determine the constant , @xmath36 . a similar fitting of @xmath37 , based on the self - consistent densities evaluated using the oep exact exchange functional , gives a leading term of 0.2677 @xmath38 , yielding @xmath39 . thus we have deduced the local approximation to the non - interacting kinetic energy . a careful inspection of the above argument reveals that the uniform electron gas is never mentioned . as @xmath3 grows , the wavelength of the majority of the particles becomes short relative to the scale on which the potential is changing , loosely speaking , and semiclassical behavior dominates . the local approximation is a universal semiclassical result , which is exact for a uniform gas simply because that system has a constant potential . on the basis of that argument , we know the exact value is @xmath36=@xmath40 , demonstrating that ( for this case ) our result is accurate to about 0.1% . this argument tells us that the reliability of the local approximation is no indicator of how rapidly the density varies . that this argument is correct for neutral atoms was carefully proven by lieb and simon in 1973 @xcite and later generalized by lieb to all matter @xcite . the focus of the first part of this paper is on the remaining two known coefficients ( @xmath41 and @xmath33 ) and how well the gea performs for them . we evaluate those gradient terms by fitting asymptotic series exactly and we find that the gradient expansion does well , but is not exact . from this information , we develop a modified gradient expansion approximation that reproduces the exact asymptotic coefficients @xmath41 and @xmath33 , merely as an illustration of the power of asymptotic exactness . we test it on a variety of systems , finding the expected behavior . in section [ sec : mtf ] , we present a parametrization of the tf density which is more accurate than previous parametrizations . the tf density has a simple scaling with @xmath21 and becomes relatively exact and slowly - varying for a neutral atom as @xmath42 , breaking down only near the nucleus and in the tail . we compare various quantities of our parametrization with exact values and earlier parametrizations , and analyze the properties of the tf density . we begin with a careful methodology for extracting the asymptotic behavior from highly accurate numerical calculations . fully numerical dft calculations were performed using the opmks code @xcite to calculate the total energies of neutral atoms using the oep exact exchange functional . the spin - density functional version of @xmath2 has been used for all systems @xcite . and @xmath43 as a function of @xmath5 with exact asymptotic coefficients.,height=207 ] to attain maximum accuracy for @xmath41 and @xmath33 , we need to suppress the oscillations which come from the next term , @xmath44 . consider first the oep results . we investigate the differences between @xmath45 and @xmath43 with exact asymptotic coefficients in figure [ f : deltaosc ] . we extract 6 data points ( @xmath21=24 ( cr ) , 25 ( mn ) , 30 ( zn ) , 31 ( ga ) , 61 ( pm ) , and 74 ( w ) ) which have the smallest differences , i.e. , nearest to where the curve crosses the horizontal axis . we then make a least - squares fit with a parabolic form in @xmath5 , ignoring the oscillation term , = 0.768745+c_1z^-1/3+c_2z^-2/3 . [ eq : fitting ] effectively , we solve two linear equations for @xmath41 and @xmath33 . we explicitly include the exact @xmath46 , since we do nt have enough data points to extract @xmath47 accurately , especially in the region @xmath48 . it is important to control the behavior of the fitting line at @xmath0 . this fitting yields a good estimate of @xmath49 and @xmath50 , with error less than 1% . this demonstrates the accuracy of our method for @xmath51 and @xmath33 ( by construction ) . we repeat the same procedure to extract @xmath51 and @xmath52 coefficients of tf and second- and fourth - order gea s which are given by t^gea2= t+ t^(2 ) , and @xcite : t^gea4= t+ t^(2)+t^(4 ) . these gradient corrections to the local approximation are given by t^(2 ) = d^3r ( ) s^2 ( ) , [ t2 ] and t^(4 ) = d^3r ( ) , [ t4 ] where @xmath53 , @xmath54 , and @xmath55 are defined as ( ) = k^2()n ( ) , [ t0 ] s()= , [ s ] q()= , [ q2 ] and @xmath56 . we have also applied this procedure to both @xmath57 and @xmath58 . since the asymptotic expansions of these energies begin at @xmath59 , we extract only a @xmath51 and a @xmath52 for each using the following equations : & = & c_1z^-1/3 + c_2z^-2/3 , + & = & c_1z^-1/3 + c_2z^-2/3 . these results are also included in table [ t : coeffalea ] , and are of course consistent with our results from eq . ( [ eq : fitting ] ) . .[t : coeffalea ] the coefficients in the asymptotic expansion of the exact kinetic energy and various local and semilocal functionals . the fit was made to @xmath21=24 ( cr ) , 25 ( mn ) , 30 ( zn ) , 31 ( ga ) , 61 ( pm ) , and 74 ( w ) . the functionals of the last two rows are defined in section [ s : interp ] . [ cols="^,^,^",options="header " , ] [ table2 ] table [ table2 ] shows that our modern parametrization is far more accurate than existing models by all measures , and that our simple pedagogical model is roughly correct for many features . finally , we make some comparisons with densities of real atoms to illustrate those features of real atoms that are captured by tf . the radial density , @xmath54 ( eq . ( [ s ] ) ) , and @xmath55 ( eq . ( [ q2 ] ) ) are given by 4r^2n(r ) = z^4/3f(x)/a , [ eq : tfdens ] where @xmath60 , s(r ) = , a_1= ( 9/2 ) ^1/3 /2 , [ eq : tfs ] and q(r ) = , [ eq : tfq ] where @xmath61 is defined as @xmath62 . the gradient relative to the screening length is t ( ) = , k ( ) = , and here t(r)= , a_2 = = 0.6124 . [ eq : tft ] we also show large- and small-@xmath63 limit behaviors of various quantities using @xmath64 as @xmath65 and @xmath66 as @xmath67 . & n(r ) & , + & 4r^2n(r ) & , + & s(r ) & , + & q(r ) & , + & t(r ) & . ) and scf densities with oep exact exchange . tf scaled densities of ba and ra are on top of each other.,height=207 ] ( relative to the local fermi wavelength ) vs. @xmath68.,height=207 ] we plot the @xmath21-scaled exact ( self - consistent densities with oep exact exchange functional ) and tf radial densities of ba ( @xmath69 ) and ra ( @xmath70 ) in figure [ f : rd ] . although the shell structure is missing , and the decay at a large distance is wrong , the overall shape of the tf density is relatively correct . in figures [ f : s2 ] , [ f : q2 ] , and [ f : t2 ] , we plot the scaled @xmath71 , @xmath72 , and @xmath73 using the exact and the tf densities of ba and ra . in particular , @xmath73 measures how fast the density changes on the scale of the tf screening length , and its magnitude does not vary with @xmath21 in tf theory . from these figures , we see that @xmath71 , @xmath72 and @xmath73 of the tf density diverge near the nucleus , since the tf density does not satisfy kato s cusp condition . ( relative to the local fermi wavelength ) vs. @xmath68.,height=207 ] when @xmath7 for a realistic density , @xmath71 is small except in the density tail ( @xmath74 over most of the density ) , and @xmath72 is small except in the tail and @xmath75 core regions ( @xmath76 over most of the density ) . this is why gradient expansions for the kinetic and exchange energies , applied to realistic densities , work as well as they do in this limit . the kinetic and exchange energies have only one characteristic length scale , the local fermi wavelength , but the correlation energy also has a different one , the local screening length . since @xmath73 is not and does not become small in this limit , gradient expansions do not work well at all for the correlation energies of atoms @xcite . the standard of smallness " for @xmath77 and @xmath78 , and the more severe standard of smallness for @xmath79 , are explained in refs . @xcite and @xcite . finally we evaluate @xmath80 on the tf density . we find the correct @xmath81 in the @xmath42 expansion from @xmath82 , but @xmath51 vanishes , due to the absence of a proper nuclear cusp , and @xmath52 diverges because @xmath57 diverges at its lower limit of integration . we have shown the importance of the large-@xmath3 limit for density functional construction of the kinetic energy ( with the functional evaluated on a kohn - sham density ) , and also provided a modern , highly accurate parameterization of the neutral - atom tf density . our results should prove useful in the never - ending search for improved density functionals . for atoms and molecules , the large-@xmath3 limit seems more important than the slowly - varying limit . on the ladder @xcite of density - functional approximations , there are three rungs of semilocal approximations ( followed by higher rungs of fully nonlocal ones ) . the lda uses only the local density , the gga uses also the density gradient , and the meta - gga uses in addition the orbital kinetic energy density or the laplacian of the density . for the exchange - correlation energy , the gga rung can not @xcite simultaneously describe the slowly - varying limit and the @xmath7 limit for an atom , and we have found here that the same is true ( but less severely by percent error of a given energy component ) for the kinetic energy . this follows because , as @xmath7 , the reduced gradient @xmath71 of eq . ( [ s ] ) becomes small over the energetically important regions of the atom , as can be inferred from fig . [ f : s2 ] , so that a gga reduces to its own second - order gradient expansion even in regions where a meta - gga does not @xcite ( e.g. , near a nucleus , where @xmath72 diverges but @xmath71 does not , as shown in figs . [ f : s2 ] and [ f : q2 ] ) . for the kinetic as for the exchange - correlation energy , meta - gga s @xcite can recover both the slowly - varying and large-@xmath21 limits ; it remains to be seen how well fully nonlocal approximations @xcite can do this .

we study the asymptotic expansion of the neutral - atom energy as the atomic number @xmath0 , presenting a new method to extract the coefficients from oscillating numerical data . we find that recovery of the correct expansion is an exact condition on the kohn - sham kinetic energy that is important for the accuracy of approximate kinetic energy functionals for atoms , molecules and solids , when evaluated on a kohn - sham density . for example , this determines the small gradient limit of any generalized gradient approximation , and conflicts somewhat with the standard gradient expansion . tests are performed on atoms , molecules , and jellium clusters . we also give a modern , highly accurate parametrization of the thomas - fermi density of neutral atoms . ^ # 1 var(_1 ... ) _ c _ h # 1#1 _ int d^3 r

both the belle @xcite and babar @xcite experiments have measured the branching fractions for and decays @xcite . these decays proceed via the annihilation diagram of fig . [ fig : diagram ] . within the standard model ( sm ) , the predicted rates are @xmath3 for the @xmath1 , all parameters on the right - hand - side except for are well - known . the cabibbo - kobayashi - maskawa ( ckm ) matrix element @xmath4 is well - constrained by a global fit to several experimental observables and unitarity of the ckm matrix . thus a measurement of @xmath5 allows one to determine the decay constant @xmath6 . this can be compared to qcd lattice calculations , which have become relatively precise . for the @xmath0 , the ckm matrix element @xmath7 is known to only 9% ; this is nonetheless more precise than measurements of @xmath8 and allows one to extract @xmath9 . belle has done two analyses @xcite ; the most recent one used 605 fb@xmath10 of data and obtained evidence for a signal . this analysis employed a semileptonic tag : one @xmath11 in an event is fully reconstructed as @xmath12 , where @xmath13 and @xmath14 . the signal decays @xmath15 and @xmath16 are then searched for by reconstructing a single track not associated with the tag side . the signal yield is obtained by fitting the distribution of @xmath17 , which is the energy sum of calorimeter clusters not associated with a charged track . a peak near @xmath18 indicates a @xmath19 or @xmath16 decay . the main backgrounds are @xmath20 processes and @xmath21 continuum events . the fit is unbinned and uses a likelihood function @xmath22 where @xmath23 runs over all events ( @xmath24 ) , @xmath25 runs over all signal and background categories , @xmath26 is the yield of category @xmath25 , and @xmath27 is the probability density function ( pdf ) for category @xmath25 . the branching fraction is calculated as @xmath28 , where @xmath29 is the reconstruction efficiency as calculated from monte carlo ( mc ) simulation . the fit results are listed in table [ tab : btaunu_belle ] , and the fit projections are shown in fig . [ fig : btaunu_belle ] . the systematic errors are dominated by uncertainty in the background pdf and the tag reconstruction efficiency . the overall result is @xmath30 where the first error is statistical and the second is systematic . .fit results for , from belle @xcite . the data sample corresponds to 605 fb@xmath10 . [ cols="<,^,^,^",options="header " , ] distribution of data events ( points ) and fit projections for , from belle @xcite . _ ( a ) _ all @xmath31 decay modes combined ; _ ( b ) _ @xmath32 ; _ ( c ) _ @xmath33 ; _ ( d ) _ @xmath34 . the open and hatched histograms correspond to signal and background , respectively . ] babar has published two analyses of decays : one using a semileptonic tag @xcite and the other using a hadronic tag @xcite . both analyses use data samples consisting of @xmath35 @xmath36 pairs . the former is similar to that used in the belle analysis : the tag side is reconstructed as @xmath37 , where @xmath38 and @xmath39 . babar searches for @xmath19 , @xmath16 , and also @xmath40 , where for the last mode the @xmath41 mass is required to be near that of the @xmath42 . the signal is identified by plotting @xmath43 , the energy sum of calorimeter clusters not associated with a charged track ; a peak near zero indicates @xmath44 decay . the signal yield is obtained by counting events in a signal region , e.g. , @xmath45 , and subtracting off background as estimated from @xmath43 sidebands . the number of events in the final sample is 245 , the background estimate is @xmath46 , and the resulting branching fraction is @xmath47 . the babar hadronic - tag analysis is more complicated . the tagging side is reconstructed as @xmath48 , where @xmath49 denotes a hadronic system of total charge @xmath50 composed of @xmath51 , @xmath52 , @xmath53 , and @xmath54 , where @xmath55 , @xmath56 , and @xmath57 . the @xmath58 is reconstructed in the same channels as those used for the semileptonic analysis , as is the @xmath44 on the signal side . a background subtraction is done on the tag side . the signal yield is obtained by counting events in an @xmath43 signal region and subtracting off background as estimated from an @xmath43 sideband . there are 24 signal candidates and @xmath59 estimated background events ; the resulting branching fraction is @xmath60 , where the first error is statistical , the second is due to the background uncertainty , and the third is due to other systematic sources . the data is shown in fig . [ fig : btaunu_babar ] along with projections of the fit . this result is consistent with the semileptonic - tagged result ; combining the two gives @xmath61 this is consistent with the belle result , eq . ( [ eqn : btaunu_belle ] ) . distribution of data events ( points ) and fit projections for , from babar using a hadronic tag @xcite . _ ( a ) _ @xmath32 ; _ ( b ) _ @xmath33 ; _ ( c ) _ @xmath34 ; _ ( d ) _ @xmath62 . the hatched histogram shows the combinatorial background component , and , for comparison , the open histogram shows signal for a branching fraction of 0.3% . ] the belle analysis of decays @xcite uses 548 fb@xmath10 of data and searches for @xmath63 , where the primary @xmath64 and @xmath65 can be charged or neutral ; the @xmath66 is `` reconstructed '' ( see below ) via @xmath67 ; @xmath68 signifies any number of additional pions and up to one photon ( from fragmentation ) ; the @xmath64 is reconstructed via @xmath69 , where @xmath70 ; and neutral kaons are reconstructed via @xmath71 . if the primary @xmath65 is charged , both it and the @xmath64 must have flavors opposite to that of the @xmath1 ; these constitute a `` right - sign '' ( rs ) sample . if the flavors are not both opposite , the event is categorized as `` wrong - sign '' ( ws ) and used to parameterize the background . the same classification applies to primary neutral @xmath65 events , except for these only the @xmath64 flavor must be opposite to that of the @xmath1 for the event to be classified as rs . the decay sequence is identified via a recoil mass technique . first , the recoil mass of the @xmath64 , @xmath65 , and @xmath72 particles is calculated and required to be within 150 of @xmath73 ; then the @xmath74 is included and the recoil mass is required to be within 150 of @xmath75 ; and finally , the @xmath76 is included and the recoil mass required to be within 0.55 of zero . the final @xmath77 recoil mass distribution is shown in fig . [ fig : widhalm1 ] ; a sharp peak is observed near zero , indicating decay . , from belle @xcite . ] the analysis is complicated by the fact that the recoil mass technique is very sensitive to the number of tracks in an event and the track reconstruction efficiency , as all tracks must be reconstructed for the recoil mass to be accurate . as it is difficult to simulate track multiplicity accurately due to uncertainties in quark fragmentation , the data is divided into bins of @xmath78 , the number of `` primary particles '' reconstructed in an event . here , a primary particle is one that is not a daughter of any particle reconstructed in the event . the minimum value for @xmath78 is three , corresponding to @xmath79 without any additional particles from fragmentation . the data is then fit in two dimensions , @xmath80 recoil mass vs. @xmath78 ( see fig . [ fig : widhalm2 ] ) . the signal pdf is obtained from mc and modeled separately for different values of @xmath81 , the _ true _ number of primary particles in an event ( @xmath81 can differ from @xmath78 due to particles being lost or incorrectly assigned ) . the branching fraction is obtained from two fits : the first fit uses the @xmath80 recoil mass spectrum and yields the number of @xmath1 candidates ; the result is @xmath82 . for this fit the background shape is taken from the ws sample and the background levels floated in the fit . the second fit uses the @xmath77 recoil mass spectrum and yields the number of candidates ; the result is @xmath83 . for this fit the background shape is taken from a rs `` @xmath84 '' sample , i.e. , all selection criteria are the same except that an electron candidate is required instead of a muon candidate . as true @xmath84 decays are suppressed by @xmath85 , this sample provides a good model of the background . the systematic errors listed are dominated by uncertainties in the signal and background pdfs and are obtained by varying the shapes of these pdfs . the branching fraction is the ratio @xmath86 , corrected for the ratio of reconstruction efficiencies . the result is @xmath87 the babar experiment searches for @xcite using 230 fb@xmath10 of data by fully reconstructing a flavor - specific @xmath88 , or @xmath89 decay on the tagging side . tag candidates are reconstructed in the following modes : @xmath90 ; @xmath91 ; @xmath92 ; and @xmath93 with @xmath94 . an isolated @xmath76 track is required . the neutrino momentum is taken to be the missing momentum in the event : @xmath95 . a photon is required and paired with the @xmath1 candidate to make a @xmath96 , and the mass difference @xmath97 is calculated . the data is subsequently divided into four subsamples : a tag - side mass sideband and a tag - side signal region for @xmath76 and @xmath98 candidates . for both lepton samples , the tag - side - sideband @xmath99 spectrum is subtracted from the tag - side - signal @xmath99 spectrum ( fig . [ fig : dsmunu_babar1 ] ) , and then the sideband - subtracted @xmath98 spectrum is subtracted from the sideband - subtracted @xmath76 spectrum . the final @xmath99 distribution ( fig . [ fig : dsmunu_babar2]a ) is fit with signal and background pdfs ; the signal yield obtained is @xmath100 events . to determine the branching fraction , the signal yield is normalized to decays . like the signal mode , the @xmath1 candidate is required to originate from @xmath101 . the tag - side - sideband @xmath99 spectrum is subtracted from the tag - side - signal @xmath99 spectrum , and the resulting spectrum is fit with signal and background pdfs ( fig . [ fig : dsmunu_babar2]b ) . the signal yield obtained is @xmath102 events . dividing @xmath103 by @xmath104 and correcting for the ratio of reconstruction efficiencies gives @xmath105 for this analysis , the @xmath106 is reconstructed via @xmath107 with @xmath108 @xcite . conveniently , cleo has measured the branching fraction @xmath109 for @xmath110 ; the result is @xmath111 @xcite . to multiply the two results together to obtain @xmath5 requires dividing eq . ( [ eqn : dsratio_babar ] ) by @xmath112 and subtracting ( in quadrature ) the 1.2% uncertainty in @xmath113 from the systematic error . in addition , babar has subtracted off a small amount of @xmath114 background ( 48 events ) ; as this process is included in the cleo measurement , these events must be added back in to babar s @xmath115 yield . thus the babar result becomes @xmath116 multiplying this by cleo s measurement gives @xmath117 spectra for the tag - side sideband region ( dashed ) and tag - side signal region ( solid ) for @xmath76 ( top ) and @xmath98 ( bottom ) samples , from babar @xcite . ] the belle and babar collaborations have used their measurements of @xmath118 and eq . ( [ eqn : fb ] ) to calculate the product of the @xmath11 decay constant @xmath9 and the ckm matrix element @xmath7 . the results are @xmath119 taking a weighted average gives @xmath120 and dividing by the particle data group value @xmath121 @xcite gives @xmath122 this value is @xmath123 higher than the most recent lattice qcd results , that of the hpqcd collaboration ( @xmath124 @xcite ) and that of the fermilab / milc collaboration ( @xmath125 @xcite ) . 0.20 in the heavy flavor averaging group ( hfag ) has calculated a world average ( wa ) value for @xmath5 and used this to determine a wa value for the @xmath1 decay constant @xcite . this value can be compared to recent lattice qcd calculations ; a significant difference could indicate new physics . the wa value for is obtained by inverting eq . ( [ eqn : fds ] ) : @xmath126 where , for @xmath127 , the wa value is inserted . the error on is calculated as follows : values for variables on the right - hand - side of eq . ( [ eqn : fds_inverted ] ) are sampled from gaussian distributions having means equal to the central values and standard deviations equal to their respective errors . the resulting values of are plotted , and the distribution is fit to a bifurcated gaussian to obtain the @xmath128 errors . the results of this procedure are shown in fig . [ fig : hfag_fds ] . also included are measurements of @xmath5 @xcite and @xmath129 @xcite from cleo . thus there are three types of measurements : from the absolute branching fraction , from the absolute branching fraction , and from the @xmath130 ratio . the overall wa value is obtained by averaging the three results , carefully accounting for correlations such as the input values for @xmath4 and @xmath131 . the result is @xmath132 . this value is higher than the two most precise lattice qcd results , that of the hpqcd ( @xmath133 @xcite ) and fermilab / milc ( @xmath134 @xcite ) collaborations . the weighted average of the theory results is @xmath135 , which differs from the hfag result by @xmath2 . ) . ] in summary , belle has observed with @xmath136 significance . from the measured branching fraction they determine the product @xmath137 . babar has observed with @xmath138 significance and has also measured the branching fraction to determine @xmath137 . the results from the two experiments are consistent ; the weighted average has 13% precision and is consistent with lattice qcd calculations . for decays , belle has observed this mode using a recoil mass technique and has measured the branching fraction with 15% precision . babar has also observed this mode and has measured the branching fraction relative to that for with 13% precision . dividing this by the branching fraction for @xmath107 and including @xmath114 decays allows one to multiply by cleo s measurement of @xmath109 to obtain @xmath5 . the heavy flavor averaging group has used the belle and babar measurements and also measurements from cleo to calculate a world average value for ; the result is @xmath132 . this value is @xmath2 higher than the average of two recent lattice qcd calculations ; the difference could indicate new physics . we thank the organizers of cipanp 2009 for a stimulating scientific program and excellent hospitality . we thank laurenz widhalm , yoshihide sakai , and andreas kronfeld for reviewing this manuscript and suggesting many improvements .

the belle and babar experiments have measured the branching fractions for and decays . from these measurements one can extract the @xmath0 and @xmath1 decay constants , which can be compared to lattice qcd calculations . for the @xmath1 decay constant , there is currently a @xmath2 difference between the calculated value and the measured value . uchep-09 - 02 address = department of physics , university of cincinnati , p.o . box 210011 , cincinnati , ohio 45221

the contradiction of mean - field theories ( mfts ) with experiment near a critical point was apparent , even from the days of andrews ( 1869)@xcite . problems of mfts were well established by 1960s and were resolved by the advent of renormalization group theory due to wilson and others in early 1970s@xcite . mft fails to predict correct values of critical indices , for example , the exponent @xmath0 of magnetization @xmath1 differs in mft and in experiment : @xmath2 and @xmath3 . one can generally say that mfts are qualitatively correct but quantitatively wrong . in the traditional viewpoint , this failure of mft is assigned to diverging fluctuations near a critical point@xcite . one generally argues as follows . consider landau s formulation of mfts@xcite and write free energy per unit volume in the presence of magnetic field @xmath4 as @xmath5 as magnetization per unit volume @xmath6 is a thermodynamical variable , probability of its fluctuation , from @xmath6 to @xmath7 with respect to unfluctuated state is : @xmath8 by taylor expanding free energy about unfluctuated value , one will have @xmath9 as @xmath10 the magnetic susceptibility is defined as @xmath11 , differentiating the free energy ( above ) twice wrt @xmath6 one can write @xmath10 as @xmath12 this gives the root - mean - square ( rms ) value ( @xmath13 ) of @xmath6 at @xmath14 : @xmath15 as the fluctuations are gaussian fluctuations . thus , one notices that as @xmath10 diverges at @xmath16 , the rms fluctuation in @xmath6 also diverges ! or within mean - field theory near the critical point , fluctuations dominate on the average behavior . this diverging fluctuations near a critical point is viewed as a problem of mfts , and thus mfts predict wrong values of the critical indices . or in more appropriate language mft predicts its own demise@xcite . contrary to this we argue that diverging fluctuations in real physical systems near a critical point are genuine ( they are actually present in real physical systems ) and the mft theory faithfully reproduce that . the prediction of diverging fluctuations is an other success of mfts ( including its qualitative description of various phases ) . the reason why mfts fail to predict correct values of critical indices is different one and is explained below using wilsonian renormalization group ideas . to support our above argument of faithful reproduction of diverging fluctuations by mft near a critical point we give two justifications . first one is based on the breakdown of a very important property of statistical independence ( si ) when the correlation length diverges near a critical point . from the breakdown of si one can show the divergence of fluctuations . the other one is based on the fluctuation - dissipation theorem and power law correlation functions . divergence of correlation length can be seen very easily from ginzburg - landau formulation@xcite in which order parameter @xmath17 vary weakly on atomic dimensions : @xmath18 one can calculate the correlation length @xmath19 ( length scale over which the effect of a fixed test spin extends out to other spins ) by considering a delta function magnetic field : @xmath20 . by standard procedure of minimizing @xmath21 one can easily show@xcite : @xmath22 above differential equation can be solved easily with fourier transforms : @xmath23 and the correlation length can be defined : @xmath24 . the correlation length diverges at the critical point , a standard result . the diverging correlation length is intimately connected to diverging fluctuations . this can be very clearly seen with the breakdown of statistical independence ( si ) . the fundamental reason is that due to large correlation length various sub - parts of the system respond in a correlated way . this renders the system non - self - averaging ! the fundamental hypothesis of si ( actually an important property of ordinary statistical mechanical systems , i.e. , systems with short range inter - particle interactions ) as advocated by landau@xcite breaks down and fluctuations in the sum - function observables ( @xmath25 ) do not obey @xmath26 law@xcite near criticality . or fluctuations do not converge : let a given system is divided into two parts . let @xmath27 be the distribution function for part 1 and @xmath28 for part 2 . if the two parts of the system are correlated ( due to large correlation length ) , then @xmath29 where @xmath30 is the distribution function of whole system . this is the breakdown of si . one immediate consequence is that statistical average of two physical quantities will obey : @xmath31 if @xmath32 ( system containing @xmath33 sub - parts ) , then @xmath34 as @xmath35 . from this it follows that the relative fluctuation ( @xmath36 ) does not obey @xmath26 ( whereas in standard statistical mechanical systems due to large @xmath33 fluctuations are negligible ) . thus systems near criticality , due to the presence of long correlation length , lose the important property of self - averaging ! _ from above we observe a very important connection : diverging correlation length and breakdown of si are connected with each other . _ diverging fluctuations due to diverging correlation length can also be seen through fluctuation - dissipation theorem : by fluctuation - dissipation theorem ( fdt ) the magnetic susceptibility @xmath10 and the correlation function @xmath37 are related to each other@xcite : @xmath38 at the critical point the correlation function assumes a power law as the correlation length diverges ( @xmath39 ) . due to the power law correlation function the above integral do not converge and thus susceptibility diverges which from above analysis of gaussian fluctuations leads to diverging rms value of @xmath6 ( @xmath13 ) . above two justifications one based upon si and the other on fdt do not involve mean - field ideas and thus independently verify what is predicted by the mfts and seen in actual experiments . to understand why mft fails to reproduce critical data one has to consider the construction of mft . consider the standard ising model ( figure [ ising ] ) : [ cols="^,^ " , ] let the system be immersed in an external magnetic field @xmath4 . the hamiltonian in dimensionless form ( @xmath40 and @xmath41 ) is : @xmath42 there are two competing tendencies ( 1 ) lining - up of spins due to @xmath43 and @xmath44 , and ( 2 ) disruption due to thermal agitation . at low temperatures first prevails and at high the second . ising model mean - field theory is done in standard way@xcite . first , consider only one spin immersed in magnetic field . ensemble average is : @xmath45 next , consider the immersed test spin interacting with many neighboring spins : @xmath46 with @xmath47 here the exact field @xmath48 due to nearest neighbor spins is replaced by an effective field @xmath49 ( @xmath50 is called the coordination number ( number of nearest spins ) . this constitutes the mf approximation . right here one makes an error when one is near the critical point : near the critical point fluctuations become long ranged ( diverging correlation length ) . these long ranged correlations enhance the effective field seen by our test spin . thus @xmath51 or the mfts fail to account for the effective field generated due to weak non - zero values of magnetization on a much larger length scales ( as compared to lattice constant ) when temperature is only slightly less than the critical temperature . in other words in mfts averages are done on a length scale ( of the order of lattice constant ) much smaller than the correlation length which is quite large as compared to lattice constant near criticality . and averages on a smaller length scale gives @xmath52 when temperature is slightly less than the critical temperature , but on a larger length scale of the order of correlation length it is not zero . to properly take into account this important long distance effects one must average out fluctuations on all length scales step by step ( see for example@xcite ) . one very visual procedure is the kadanoff blocking method@xcite and results are in very good agreement with experiment and with exact solutions in special cases@xcite . see also the last section in@xcite . a common misconception that mean - field theory predicts its own demise is clarified . near a critical point diverging fluctuations are _ actually _ present in real physical systems , and the mfts very faithfully reproduce those . thus , this should be viewed as another success of mfts , not their failure . the reason why mfts predict wrong values of critical indices is that the mfts fail to account for weak non - zero values of order parameter on a much larger length scales ( as compared to lattice constant ) when temperature is only slightly less than the critical temperature . in other words in mfts averages are done on a length scale much smaller than the correlation length which is quite large as compared to lattice constant near criticality .

it is well known that mean - field theories fail to reproduce the experimentally known critical exponents . the traditional argument which explain this failure of mean - field theories near a critical point is the ginsburg criterion in which diverging fluctuations of the order parameter is the root cause . we argue , contrary to the above mentioned traditional view , that diverging fluctuations in real physical systems near a critical point are genuine consequence of the breakdown of the property of statistical independence , and are faithfully reproduced by the mean - field theory . by looking at the problem from the point of view of `` statistical independence '' the divergence of fluctuations in real physical systems near criticality becomes immediately apparent as a connection can be established between diverging correlation length and diverging fluctuations . to address the question of why mean - field theories , much successful qualitatively , fail to reproduce the known values of critical indices we argue , using the essential ideas of the wilsonian renormalization group , that mean - field theories fail to capture the long length scale averages of an order parameter near a critical point

the kondo effect was first observed , in the 1930s , for iron impurities in gold and silver @xcite , as an anomalous rise in the resistivity with decreasing temperature . kondo@xcite showed that this effect is caused by an antiferromagnetic exchange coupling between the localized magnetic impurity spins and the spins of the delocalized conduction electrons @xcite , and based his arguments on a spin-@xmath4 , one - band model . while this model undoubtedly captures the essential physics correctly in a qualitative way , it has recently been shown@xcite that a quantitatively correct description of the kondo physics of dilute fe impurities in au or ag requires a fully screened kondo model involving three channels and a spin-@xmath5 impurity . this conclusion was based on a comparison of temperature and magnetic field dependent transport measurements@xcite to theoretical predictions for fully screened kondo models with channel number @xmath0 and local spin @xmath6 related by @xmath7 , with @xmath8 yielding much better agreement than @xmath9 or 2 . the theoretical results in ref . were obtained using the numerical renormalization group ( nrg),@xcite and for @xmath10 various non - abelian symmetries@xcite , such as su(2)@xmath11u(1)@xmath11su@xmath12 , had to be exploited to achieve reliable results at finite magnetic field . the technology for implementing non - abelian symmetries with @xmath13 in nrg calculations has been developed only recently.@xcite given the complexity of such calculations , it is desirable to benchmark their quality by comparing their predictions to exact results . the motivation for the present paper was to perform such a comparison for the low - energy fermi - liquid behavior of fully screened kondo models , as elaborated upon below . all fully screened kondo models feature a ground state in which the impurity spin is screened by the conduction electrons into a spin singlet . the low - energy behavior of these models can be described by a phenomenological fermi - liquid theory ( flt ) formulated in terms of the phase shift experienced by conduction electrons that scatter elastically off the screened singlet . such a description was first devised for the simplest case of @xmath9 by nozires@xcite in 1974 , and generalized to the case of arbitrary @xmath0 by nozires and blandin ( nb)@xcite in 1980 . their results were confirmed and elaborated by various authors and methods , including nrg,@xcite field - theoretic calculations,@xcite the bethe ansatz,@xcite conformal field theory ( cft),@xcite renormalized perturbation theory,@xcite and reformulations@xcite and generalizations@xcite of nozires approach in the context of kondo quantum dots . in the present paper , we focus on three particular fermi - liquid coefficients , @xmath1 , @xmath2 and @xmath3 , characterizing the leading dependence of the resistivity on magnetic field ( @xmath14 ) and temperature ( @xmath15 ) , and the curvature of the equilibrium kondo resonance as function of excitation energy @xmath16 ) , respectively . explicit formulas for all three of these coefficients are available in the literature for @xmath9 , but for general @xmath0 only for the case of @xmath2 . given the wealth of previous studies of fully - screened kondo models , the lack of corresponding formulas for @xmath1 and @xmath3 was somewhat unexpected . thus , we offer here a unified derivation of all three fermi - liquid coefficients , @xmath2 , @xmath1 and @xmath3 . we follow the strategy which affleck and ludwig ( al)@xcite have used to reproduce nozires results@xcite for @xmath9 , namely doing perturbation theory in the leading irrelevant operator , and generalize it to the case of arbitrary @xmath0 . our formulation of this strategy follows that used by pustilnik and glazman ( pg)@xcite for their discussion of kondo quantum dots . while all pertinent ideas used here can be found in the literature , we hope that our rather compact way of combining them will be found useful . for our numerical work , we faced two challenges : first , the complexity of the numerical calculations increases rapidly with increasing @xmath0 ; this was dealt with by exploiting non - abelian symmetries . second , numerical calculations do not achieve the scaling limit that is implicitely presumed in analytical calculations ; its absence was compensated by using suitable definitions of the kondo temperature , following ref . . the paper is organized as follows . in sec . [ sec : model ] we define the model and summarize our key results for the fermi - liquid coefficients @xmath1 , @xmath2 and @xmath3 . section [ sec : fermi - liquid ] compactly summarizes relevant elements of flt and uses them to calculate these coefficients . section [ sec : nrg ] describes our numerical work and results . section [ sec : conclusions ] summarizes our conclusions . the fully - screened kondo model for @xmath0 conduction bands coupled to a single magnetic impurity at the origin is defined by the hamiltonian @xmath17 , with [ eq : kondomodel ] @xmath18 here @xmath19 describes @xmath0 channels of free conduction electrons , with spin index @xmath20 and channel index @xmath21 . we take the dispersion @xmath22 to be linear and symmetric around the fermi energy , @xmath23 . each channel has exchange coupling @xmath24 to a local su(2 ) spin of size @xmath25 with spin operators @xmath26 , and @xmath14 describes a local zeeman field in the @xmath27-direction ( we use units @xmath28 ) . the overall symmetry of the model@xcite is su(2)@xmath11 sp@xmath29 for @xmath30 , and u(1)@xmath11sp@xmath29 for @xmath31 ( see sec . [ sec : nrg - details ] for details ) . the model is characterized by a low - energy scale , the kondo temperature , @xmath32 $ ] , where @xmath33 is the density of states per channel and spin species and @xmath34 is of the order of the conduction electron bandwidth . for a disordered metal containing a dilute concentration of magnetic impurities , the magnetic - impurity contribution to the resisitivity has the form@xcite @xmath35 here @xmath36 is the fermi function , and the impurity spectral function @xmath37 is the imaginary part of the @xmath15 matrix @xmath38 describing scattering off a magnetic impurity . the latter is defined through@xcite @xmath39 with @xmath40 and @xmath41 the full and bare conduction electron green s functions , respectively . [ for a kondo quantum dot tuned such that the low - energy physics is described by eq.([eq : kondomodel ] ) , the conductance @xmath42 through the dot has a form similar to eq.([eq : g(t , b ) ] ) , with @xmath43 replaced by @xmath42.@xcite ] as mentioned in the introduction , the ground state of the fully screened kondo model is a spin singlet , and the regime of low - energy excitations below @xmath44 shows fermi - liquid behavior.@xcite one characteristic fermi - liquid property is that the leading dependence of the @xmath15 matrix on its arguments , when they are small relative to @xmath44 , is quadratic , @xmath45 ( particle - hole and spin symmetries forbid terms linear in @xmath46 or @xmath14 . ) this implies the same for the resistivity , @xmath47 with @xmath48 . the so - called fermi - liquid coefficients @xmath3 , @xmath2 and @xmath1 are universal , @xmath0-dependent numbers , characteristic of the fully screened fermi - liquid fixed point . for @xmath9 , the coefficients @xmath2 and @xmath1 have recently been measured experimentally in transport studies through quantum dots and compared to theoretical predictions.@xcite the coefficient @xmath3 is , in principle , also measurable via the non - linear conductance of a kondo dot coupled strongly to one lead and very weakly to another.@xcite ( the latter condition corresponds to the limit of a weak tunneling probe ; it ensures that the non - linear conductance probes the _ equilibrium _ shape of the kondo resonance , and hence the equilibrium fermi - liquid coefficient @xmath3 . ) the goal of this paper is twofold : first , to analytically establish the @xmath0 dependence of @xmath3 , @xmath2 and @xmath1 using fermi - liquid theory similar to nb ; and second , to numerically calculate them using an nrg code that exploits non - abelian symmetries , in order to establish a benchmark for the quality of the latter . our main results are as follows : first , if the kondo temperature is defined by @xmath49 where @xmath50 is the static impurity susceptibility at zero temperature , the fermi - liquid coefficients are given by @xmath51 for general @xmath0 , the formula for @xmath2 has first been found by yoshimori,@xcite while those for @xmath1 and @xmath3 are new ( though not difficult to obtain ) . second , our numerical results for @xmath52 and @xmath53 are found to agree with the predictions of eq.([eq : cbteresults ] ) to within 5% . in this section , we analytically calculate the fermi - liquid coefficients @xmath1 , @xmath2 and @xmath3 for general @xmath0 . with the benefit of hindsight , we selectively combine various elements of the work on flt of nozires,@xcite nb,@xcite al@xcite and pg@xcite . detailed justifications for the underlying assumptions are given by these authors in their original publications and hence will not be repeated here . instead , our goal is to assemble their ideas in such a way that the route to the desired results is short and sweet . we begin by summarizing nozires ideas for expressing the @xmath15 matrix in terms of scattering phase shifts and expanding the latter in terms of phenomenological fermi - liquid parameters . next , we recount al s insight that this expansion can be reproduced systematically by doing perturbation theory in the leading irrelevant operator of the model s zero - temperature fixed point . then we adopt pg s strategy of performing the expansion in a quasiparticle basis in which the contant part of the phase shift has already been taken into account , which considerably simplifies the calculation . our own calculation is presented using notation analogous to that of pg , while taking care to highlight the extra terms that arise for @xmath54 . it turns out that their extra contributions can be found with very little extra effort . since the ground state of the fully screened kondo model is a spin singlet , a low - energy quasiparticle scattering off the impurity experiences strong elastic scattering as if the impurity were nonmagnetic . moreover , it also experiences a weak local interaction if some energy @xmath55 is available to weakly excite the singlet , causing some inelastic scattering . since the singlet binding energy is @xmath44 , the strength of this local interaction is proportional to @xmath56 . nozires@xcite realized that this combination of strong elastic scattering and a weak local interaction can naturally be treated in terms of scattering phase shifts . the phase shift of a quasiparticle with quantum numbers @xmath57 and excitation energy @xmath46 relative to the fermi energy can be written as @xmath58 here @xmath59 is the phase shift for @xmath60 ; it has the maximum possible value for scattering off a non - magnetic impurity , namely @xmath61 . finite - energy corrections arising from weak excitations of the singlet are described by @xmath62 , which is proportional to @xmath56 . if inelastic scattering is weak , unitarity of the @xmath6 matrix can be exploited@xcite to write the @xmath15 matrix in the following form ( we use the notation pg@xcite ; for a detailed analysis , see al s discussion@xcite of the terms arising from their figs . 6 and 7 ) : @xmath63 \ ; . \qquad \end{aligned}\ ] ] here @xmath64 accounts for weak inelastic two - body scattering processes , and is proportional to @xmath65 . it is to be calculated in a basis of quasiparticle states in which the phase shift @xmath66 has already been accounted for . ( here and below , tildes will be used on quantities defined with respect to the new basis if they differ from corresponding ones in the original basis . ) expanding eq.([eq : define - phase - shift - tmatrix - inelastic ] ) in the small ( real ) number @xmath67 and recalling that @xmath68 , one finds that the imaginary part of the @xmath15 matrix , which determines the spectral function , can be expressed as @xmath69 \ ; , { \qquad \phantom{.}}\end{aligned}\ ] ] to order @xmath65 . comparing this to eq.([eq : aexpand ] ) , we conclude that knowing @xmath70 to order @xmath56 and @xmath71 to order @xmath65 suffices to fully determine the fermi - liquid coefficients @xmath1 , @xmath2 and @xmath3 . now , a systematic calculation of @xmath70 and @xmath64 requires a detailed theory for the strong - coupling fixed point , which became available only with the work of al in the early 1990s . nevertheless , nozires succeeded in treating the case @xmath9 already in 1974,@xcite using a phenomenological expansion of @xmath72 in powers of @xmath73 [ @xmath74 represents the zeeman energy of quasiparticles in a magnetic field , see eq.([eq : hpg ] ) below ] and @xmath75 , the deviation of the total quasiparticle number @xmath76 from its ground - state value . the prefactors in this expansion have the status of phenomenological fermi - liquid parameters . using various ingenious heuristic arguments , he was able to show that all these parameters , and also @xmath64 , are related to each other and can be expressed in terms of a single energy scale , namely the kondo temperature . moreover , by choosing the prefactor of @xmath46 in this expansion to be @xmath56 , he suggested a definition of the kondo temperature that also fixes its numerical prefactor . ( our paper adopts this definition , too . ) in 1980 , nb generalized this strategy @xcite to general @xmath0 , finding an expansion of the form @xmath77 where @xmath78 and @xmath79 are phenomenological fermi - liquid parameters related by @xmath80 . [ nb s initial version of eq.([eq : nb - phase - expansion ] ) [ their eq . ( 34 ) ] does not contain the zeeman contribution @xmath74 , but the latter is implicit in their subsequent treatment of the zeeman field before their eq . ( 37 ) . ] in the following subsections , we show how nb s expansion for @xmath70 can be derived systematically . al@xcite and pg@xcite have shown how to do this for @xmath9 ; we will generalize their discussion to arbirtrary @xmath0 . al showed@xcite that nb s heuristic results can be derived in a systematic fashion by doing perturbation theory in the leading irrelevant operator of the model s zero - temperature fixed point . as perturbation , they took the operator with the lowest scaling dimension satisfying the requirements of being ( i ) local , ( ii ) independent of the impurity spin operator @xmath26 , since the latter is fully screened , ( iii ) su(2)-spin - invariant , ( iv ) and independent of the local charge density , just as the kondo interaction . the operator sastifying these criteria has the form@xcite @xmath81 where @xmath82 is the quasiparticle spin density at the impurity site , and @xmath83 denotes the point - splitting regularization procedure ( see appendix ) . in appendix d of ref . , al showed in great detail how nb s phase shifts can be computed using eq.([eq : al - start - jj ] ) , for the single - channel case of @xmath9 . they did not devote as much attention to the case of general @xmath0 , though the needed generalizations are clearly implied in their work . we here present the corresponding calculation in some detail , following the notational conventions of pg , which differ from those of al in some regards ( see appendix ) . the main difference is that pg formulate the perturbation expansion in a new basis of quasiparticle states , in which the phase shift @xmath66 has already been accounted for , which somewhat simplifies the discussion . ( we remark that pg chose @xmath84 rather than @xmath61 as used by nb and us , but the extra @xmath85 has no consequences for the ensuing arguments . ) the quasiparticle hamiltonian describing the vicinity of the strong - coupling fixed point ( fp ) has the form @xmath86 where @xmath87 describes free quasiparticles in a magnetic field @xmath14 , with zeeman energy @xmath74 . note that although the zeeman term in the bare hamiltonian ( [ eq : kondomodel ] ) is local , it is global in eq.([eq : hpg ] ) , because the effective quasiparticle hamiltonian @xmath88 contains no local spin . using standard point - splitting techniques , which we review in pedagogical detail in the appendix , the leading irrelevant operator ( [ eq : al - start - jj ] ) can be written as @xmath89 , with [ eq : h - momentum - main ] @xmath90 where @xmath91 here we have expressed the coupling constant @xmath92 in terms of the inverse kondo temperature using [ cf . eq.([eq : identify - alpha1-phi1 ] ) ] @xmath93 with the numerical proportionality factor chosen such that @xmath44 agrees with definition of the kondo temperature used by nb and pg , as discussed below . importantly , the point - splitting procedure fixes the relative prefactors arising in @xmath94 , @xmath95 and @xmath96 ( whereas nb s approach requires heuristic arguments to fix them ) . our notation for @xmath94 and @xmath95 coincides with that used by pg . @xmath96 contains all new contributions that enter additionally for @xmath97 . figure [ fig : one ] gives a diagrammtic depiction of all three terms . , @xmath95 and @xmath96 , respectively . ( d)-(f ) nonzero second - order contributions to the quasiparticle self - energy , @xmath98 , involving @xmath99 , @xmath100 and @xmath101 , respectively . the contributions involving @xmath102 , @xmath103 and @xmath104 all vanish , the former two due to the odd power of energy in the two - leg vertex . ] our first goal is to recover nb s expansion of the phase shift @xmath70 to leading order in @xmath105 and @xmath106 . following pg , this can be done by calculating @xmath70 perturbatively to first order order in @xmath56 , in the new basis of quasiparticle states that already incorporate the phase shift @xmath107 . to order @xmath56 , no inelastic scattering occurs , and @xmath70 is related to the elastic @xmath15 matrix by @xmath108 the elastic @xmath15 matrix , in turn , equals the real part of the quasiparticle self - energy , @xmath109 . ( actually , to order @xmath56 , the self - energy is purely real . ) by expanding eq.([eq : define - phase - shift - tmatrix ] ) for small @xmath70 , the phase shift is thus seen to be given by the real part of the self - energy : @xmath110 now , as pointed out already by nozires in 1974,@xcite a first - order perturbation calculation of the self - energy is equivalent to treating interaction terms in the mean - field ( mf ) approximation . they then take the form @xmath111 where @xmath112 , the quasiparticle number relative to the @xmath30 ground state , is given by @xmath113 the mean - field version of the leading irrelevant operator thus has the form @xmath114 \ ; . \end{aligned}\ ] ] for such a single - particle perturbation , the self - energy can be directly read off from @xmath115 using @xmath116 because @xmath117 sums of the type @xmath118 yield residues involving @xmath119 . using eq.([eq : read - off - sigma - from - h ] ) in eq.([eq : expandphaseshift - t - matrix ] ) for the phase shift , we find @xmath120 \ ; .\end{aligned}\ ] ] this fully agrees with the expansion ( [ eq : nb - phase - expansion ] ) of nb if we make the identification @xmath121 , thus confirming the validity of nb s heuristic arguments . note that the coefficient of @xmath105 in eq.([eq : gp - phase - shift - a - la - nb ] ) comes out as @xmath56 , in agreement with the conventions of nb and pg , as intended by our choice of numerical prefactor in eq.([eq : identify - alpha - tk ] ) . as consistency check , let us review how nb used eq.([eq : gp - phase - shift - a - la - nb ] ) to calculate the wilson ratio . first , eq.([eq : gp - phase - shift - a - la - nb ] ) implies an impurity - induced change in the density of states per spin and channel of @xmath122 . this yields a corresponding impurity - induced change in the specific heat , @xmath123 . at zero field ( where @xmath74 and @xmath124 vanish ) , the change relative to the bulk is given by @xmath125 second , the friedel sum rule for the impurity - induced change in local charge in channel @xmath126 for spin @xmath85 at @xmath127 gives @xmath128 and eq.([eq : gp - phase - shift - a - la - nb ] ) , together with eq.([eq : quasiparticle - density - in - field ] ) for @xmath129 , leads to @xmath130 = \frac{\sigma b ( n+2)}{3 { t_{\rm k } } } \ ; . { \qquad \phantom{.}}\end{aligned}\ ] ] the linear response of the impurity - induced magnetization , @xmath131 , then gives the impurity contribution to the spin susceptibility as @xmath132 ( for all expressions involving @xmath50 here and below , the limit @xmath133 is implied . ) the corresponding bulk contribution is @xmath134 . thus , the wilson ratio is found to be @xmath135 in agreement with more elaborate calculations by yoshimori@xcite and by mihly and zawadowski.@xcite note that eq.([eq : friedel - magnetization ] ) relates nozires definition of the kondo temperature to an observable quantity , @xmath50 , that can be calculated numerically . we used this as a precise way of defining @xmath44 in our numerical work . ( subtleties involved in calculating @xmath50 are discussed in sec . [ sec : define - tk ] . ) note that up to a prefactor , eq.([eq : friedel - magnetization ] ) for @xmath50 has the form @xmath136 , where @xmath137 is the static susceptibility of a free spin @xmath6 at temperature @xmath15 . we are now in a position to extract our first fermi - liquid coefficient , @xmath1 . for this , it suffices to know the spectral function @xmath138 in eq.([eq : aexpand ] ) to quadratic order in @xmath14 , at @xmath139 , where @xmath140 . inserting the corresponding expression ( [ eq : tdeltab ] ) for @xmath141 into eq.([eq : imtoriginal ] ) for @xmath142 , we find @xmath143 \ ; . \end{aligned}\ ] ] comparing this to eq.([eq : aexpand ] ) , we read off @xmath144 . note that if the definition ( [ eq : friedel - magnetization ] ) of @xmath44 in terms of @xmath50 is taken as given , @xmath1 can actually be derived on the back of an envelope : for a fully screened kondo model , the impurity - induced spin susceptibility gets equal contributions from all @xmath0 channels , @xmath145 , and the friedel sum rule relates the contribution from each channel to phase shifts , @xmath146/(2 \pi b)$ ] , implying @xmath147 . using this in eq.([eq : imtoriginal ] ) yields @xmath148 \ ; , \end{aligned}\ ] ] which is equivalent to eq.([eq : a - b - dependence ] ) if eq.([eq : friedel - magnetization ] ) holds . we next discuss inelastic scattering for @xmath30 , but at finite temperature . to order @xmath65 , inelastic scattering is described by the imaginary part of the quasiparticle self - energy arising from the second - order contributions of @xmath94 , @xmath95 and @xmath96 , shown in diagrams ( d)-(f ) of fig . [ fig : one ] , respectively . these diagrams give [ eq : tinelastic123 ] @xmath149 the first two can also be found in the discussion of pg , whose strategy we follow here . ( they also appear , in slightly different guise , in the discussion of al@xcite . ) the third is proportional to the second , and the factor @xmath150 originates from @xmath151 with @xmath152 , since the relative prefactor between @xmath96 and @xmath95 brings in two powers of @xmath150 , and the algebra of pauli matrices yields a factor @xmath153 . now , the term called @xmath64 in eq.([eq : define - phase - shift - tmatrix - inelastic ] ) by definition describes the contribution of the _ two - body _ terms @xmath95 and @xmath96 to inelastic scattering : @xmath154 \ ; .\end{aligned}\ ] ] the contribution @xmath155 from @xmath94 is _ not _ included in @xmath156 here , since it actually equals @xmath157 , and hence is already contained in the factor @xmath158 in eq.([eq : define - phase - shift - tmatrix - inelastic ] ) . indeed , in eq.([eq : imtoriginal ] ) for the imaginary part of the @xmath15 matrix in the original basis , the @xmath159 term equals @xmath160 . collecting all ingredients , eq.([eq : imtoriginal ] ) gives @xmath161 \nonumber \\ \label{eq : aet - original } & = & \frac{1}{\nu \pi^2 } \left [ 1 - \frac{(2n+7)\varepsilon^2 + ( 2n+1)\pi^2 t^2}{6 { t_{\rm k}}^2 } \right ] \ ; . { \qquad \phantom{.}}\end{aligned}\ ] ] for @xmath9 , the second term reduces to the familiar form @xmath162 found by al@xcite and gp@xcite . comparing eqs.([eq : aet - original ] ) and ( [ eq : aexpand ] ) and ( [ eq : resistivity - taylor ] ) we read off @xmath163 and @xmath164 , implying @xmath165 . in this section , we describe our nrg work . we had set ourselves the goal of achieving an accuracy of better than 5% for the fermi - liquid coefficients . to achieve this , two ingredients were essential . first , exploiting non - abelian symmetries ; and second , defining the kondo temperature with due care . the latter is a matter of some subtlety @xcite because the wide - band limit assumed in analytical work does not apply in numerical calculations . we begin below by giving the lehmann representation for the desired spectral function . we then discuss the non - abelian symmetries used in our nrg calculations and explain how the kondo temperature was extracted numerically . finally , we present our numerical results . to numerically calculate the @xmath15 matrix of eq.([eq : define - t - matrix ] ) , we use equations of motion@xcite to express it as @xmath166 = j_{\rm k}\sum_{\sigma'}{\vec s } \cdot \frac{{\vec \tau}_{\sigma \sigma'}}{2 } \psi_{m\sigma'}(0 ) . { \qquad \phantom{.}}\end{aligned}\ ] ] here @xmath167 denotes a retarded correlation function , and @xmath168 , where @xmath169 is the number of discrete levels in the band ( and hence proportional to the system size ) . the spectral function is then calculated in its lehmann - representation , @xmath170 with @xmath171 , using the full density matrix ( fdm ) approach of nrg.@xcite for our numerical work , we take the conduction band energies to lie within the interval @xmath172 $ ] , with fermi energy at 0 and half - bandwidth @xmath173 , and take the density of states per spin , channel and unit length to be constant , as @xmath174 . ( it is related to the extensive density of states used in sec . [ sec : fermi - liquid ] by @xmath175 . ) for the calculations used to determine the fermi - liquid parameters , we use exchange coupling @xmath176 , so that the kondo temperature @xmath177 $ ] has the same order of magnitude for @xmath52 and 3 , namely @xmath178 . following standard nrg protocol,@xcite the conduction band is discretized logarithmically with discretization parameter @xmath179 , mapped onto a wilson chain , and diagonalized iteratively . nrg truncation at each iteration step is controlled by either specifying the number of kept states per shell , @xmath180 , or the truncation energy , @xmath181 ( in rescaled units , as defined in ref . ) , corresponding to the highest kept energy per shell . spectral data are averaged over @xmath182 different , interleaving logarithmic discretization meshes.@xcite the values for nrg - specific parameters used here are given in legends in the figures below . for the fully screened @xmath0-channel kondo model , the dimension of the local hilbert space of each supersite of the wilson chain is @xmath183 . since this increases exponentially with the number of channels , it is essential , specifically so for @xmath10 , to reduce computational costs by exploiting non - abelian symmetries @xcite to combine degenerate states into multiplets . several large symmetries are available@xcite : for @xmath30 , the model has su(2)@xmath11u(1)@xmath11su@xmath12 spin - charge - channel symmetry . if the bands desribed by @xmath19 are particle - hole symmetric , as assumed here , the model also has a su(2)@xmath11[su(2)]@xmath184 spin-(charge)@xmath184 symmetry , involving su(2 ) mixing of particles and holes in each of the @xmath0 channels . the u(1)@xmath11su@xmath12 and [ su(2)]@xmath184 symmetries are not mutually compatible ( their generators do not all commute ) , however , implying that both are subgroups of a larger symmetry group , the symplectic sp@xmath29 . thus the full symmetry of the model for @xmath30 is su(2)@xmath11sp@xmath29 . for @xmath31 it is u(1)@xmath11sp@xmath29 , since a finite magnetic field breaks the su(2 ) spin symmetry to the abelian u(1 ) @xmath185 symmetry . when the model s _ full _ symmetry is exploited , the multiplet spaces encountered in nrg calculations exhibit _ no _ more degeneracies in energy at all . using only abelian symmetries turned out to be clearly insufficient to obtain well converged numerical data for @xmath10 , despite having a relatively large @xmath179 . this , however , is required for accurate fermi - liquid coefficients with errors below a few percent . for numerically converged data , therefore , it was essential to use non - abelian symmetries . for our @xmath30 calculations , it turned out to be sufficient to use su(2)@xmath11u(1)@xmath11su@xmath12 symmetry for calculating @xmath2 , but the full su(2)@xmath11sp@xmath29 symmetry was needed for calculating @xmath3 . likewise , for our @xmath31 calculations of @xmath1 , we needed to use the full u(1)@xmath11sp@xmath29 symmetry . doing so led to an enormous reduction in memory requirements , the more so the larger the rank of the symmetry group [ sp@xmath29 has rank @xmath0 , and su@xmath12 has rank @xmath186 . for @xmath10 , for example , we kept @xmath187 multiplets for su(2)@xmath11u(1)@xmath11su@xmath188 or @xmath189 multiplets for su(2)@xmath11sp@xmath190 during nrg truncation , which , in effect , amounts to keeping @xmath191 individual states.@xcite the fermi - liquid theory of sec . [ sec : fermi - liquid ] implicitly assumes that the model is considered in the so - called scaling limit , in which the ratio of kondo temperature to bandwidth vanishes , @xmath192 . in this limit , physical quantities such as @xmath193 are universal scaling functions , which depend on their arguments only in the combinations @xmath194 and @xmath195 . since the shape of such a scaling function , say @xmath196 plotted versus @xmath194 , is universal , i.e. independent of the bare parameters ( coupling @xmath24 and bandwidth @xmath197 ) used to calculate it , curves generated by different combinations of bare parameters can all be made to collapse onto each other by suitably adjusting the parameter @xmath44 for each . in the same sense the fermi - liquid parameters @xmath1 , @xmath2 and @xmath3 , being taylor - coefficients of universal curves , are universal , too . one common way to achieve a scaling collapse , popular particularly in experimental studies , is to identify the kondo temperature with the field @xmath198 or temperature @xmath199 at which the impurity contribution to the resisitivity has decreased to half its unitary value , @xmath200 however , this is approach is not suitable for the purpose of extracting fermi - liquid coefficients , for which @xmath44 has to be defined in terms of ( analytically accessible ) low - energy properties characteristic of the strong - coupling fixed point . in sec . [ sec : fermi - liquid ] we have therefore adopted nozires definition of @xmath44 in terms of the leading energy dependence of the phase shift @xmath201 [ eq.([eq : gp - phase - shift - a - la - nb ] ) ] , implying that it can be expressed in terms of @xmath50 , of the local static spin susceptibility at zero temperature [ eq.([eq : friedel - magnetization ] ) ] . in the scaling limit , this definition of @xmath44 matches @xmath198 or @xmath199 up to prefactors , i.e. @xmath202 and @xmath203 are universal , @xmath0-dependent numerical constants , independent of the model s bare parameters . in numerical work , however , the scaling limit is never fully realized , since the bandwidth is always finite . it may thus happen that a scaling collapse expected analytically is not found when the corresponding curves are calculated numerically . for example , if the kondo temperature is defined , as seems natural , in terms of a purely local susceptibility , @xmath204 , involving only the response of the local spin to a local field , @xmath205 then curves expected to show a scaling collapse actually do not collapse onto each other , as pointed out recently in ref . ( see figs . 2(d)-(f ) there ) . that paper also showed how to remedy this problem : the static spin susceptibility used to calculate @xmath44 has to be defined more carefully , and two slightly different definitions have to be used , depending on the context . the first option is needed when studying zero - temperature ( i.e. ground state ) properties as a function of some external parameter , such as the field dependence of the resisitivity ( needed for @xmath1 ) . in this case , a corresponding susceptibility defined in terms of the response of the system s _ total _ spin to a local field should be used : @xmath206 the superscript fs stands for `` friedel sum rule '' , to highlight the fact that using this rule to calculate the linear response of @xmath207 to a local field directly leads to relation ( [ eq : friedel - magnetization ] ) between @xmath50 and @xmath44 . the second option is needed when studying dynamical or thermal quantities that depend on the system s many - body excitations for given fixed external parameters ( e.g. fixed @xmath30 ) , such as the temperature - dependence of the resistivity ( needed for @xmath2 ) , or the curvature of the kondo resonance ( needed for @xmath3 ) . in this case , one should use @xmath208 the superscript sc stands for `` scaling '' , to indicate that this definition of the kondo temperature ensures@xcite a scaling collapse of dynamical or thermal properties . figure [ fig : scaling ] demonstrates that a scaling collapse is indeed found when the field- or temperature - dependent resistivity , plotted versus @xmath209 or @xmath210 , respectively , is calculated for two different values of @xmath24 ( solid and dashed lines , respectively ) . note that this works equally well for @xmath52 and @xmath53 . ( for @xmath9 , such scaling collapses had already been shown in ref . . ) we remark that the three kondo temperatures defined in eqs.([eq : chi_d])-([eq : chi_sc ] ) differ quite significantly from each other for the kondo hamiltonian of eq.([eq : kondomodel ] ) , with differences as large as 12% , 31% and 55% for @xmath9 , 2 and 3 , respectively , for the parameters used in fig . [ fig : scaling ] . this indicates that although we have chosen bare paramters for which @xmath211 is smaller than @xmath212 , we have still not reached the scaling limit [ in which the definitions eq . ( [ eq : chi_d])-([eq : chi_sc ] ) of the kondo temperature should all coincide numerically@xcite ] . we have checked that the differences between @xmath213 , @xmath214 and @xmath215 decrease when @xmath216 is reduced in an attempt to get closer to the scaling limit , but estimate that truly reaching that limit would require @xmath217 for the kondo model , implying @xmath218 . thus , reaching the scaling limit by brute force is numerically unfeasible . therefore , using @xmath214 and @xmath215 rather than @xmath213 is absolutely essential for obtaining scaling collapses . it is similarly essential for an accurate determination of the fermi - liquid parameters . correspondingly , for the results discussed below , we have used @xmath214 as definition of the kondo temperature when extracting @xmath1 , and @xmath215 when extracting @xmath2 and @xmath3 . ( dashed or solid ) , and for @xmath52 and @xmath53 . for each @xmath0 , the dashed and solid curves overlap so well that they are almost indistinguishable . the insets compare the energy scales @xmath198 and @xmath199 at which the resistivity has decreased to half its unitary value [ cf . eq.([eq : thalfbhalf ] ) ] , to the scales @xmath214 and @xmath215 [ cf . eqs.([eq : chi_m ] ) and ( [ eq : chi_sc ] ) ] , respectively . the shown ratios are universal numbers of order unity , but not necessarily very close to 1 , with a significant dependence on @xmath0 : @xmath219 and @xmath220 for @xmath9 , 2 and 3 , respectively . the legend in the lower left of panel ( b ) specifies the nrg parameters used for both panels . ] when one is interested in spectral properties , one typically has to broaden the discrete data . for the determination of the fermi - liquid coefficients , however , where high numerical accuracy is required , it is desirable to avoid standard broadening . for the calculation of @xmath2 and @xmath1 this can be achieved@xcite by directly inserting the lehmann sum over @xmath221 functions for the spectral function @xmath222 [ eq . ( [ eq : discdata ] ) ] into the energy integral for @xmath223 [ eq . ( [ eq : g(t , b ) ] ) ] , resulting in a sum over discrete data points that produces a smooth curve . the curve is smooth because eq.([eq : g(t , b ) ] ) in effect thermally broadens the @xmath221 peaks in the lehmann representation . this is true even in the limit @xmath224 , because in nrg calculations it is realized by taking @xmath15 nonzero , but much smaller than all other energy scales . for the determination of @xmath225 , in contrast , one faces the problem that @xmath226 is represented not as an integral of a sum over discrete @xmath221 functions , but directly in terms of the latter . to avoid having to broaden these by hand , it is desirable to find a way to extract @xmath225 from an expression involving an integral over the discrete spectral data , as for @xmath1 and @xmath2 . this can be achieved as follows . first , note that @xmath225 is , by definition , a coefficient in the general taylor expansion of the normalized spectral function @xmath227 for small frequencies , @xmath228 due to particle - hole symmetry , @xmath229 for all @xmath230 odd , and by definition @xmath231 . to determine @xmath232 from an integral over discrete data , we consider a weighted average of @xmath233 over @xmath46 , @xmath234 where @xmath235 is a symmetric weighting function of width @xmath236 and weight 1 , and moments defined by @xmath237 for integer @xmath238 ( with @xmath239 ) . here we use @xmath240 but other choices are possible , too ( e.g. , a gaussian peak ) . clearly , the leading @xmath241 dependence of @xmath242 for small @xmath241 reflects the leading @xmath243 dependence of @xmath233 and allows for an accurate determination of @xmath232 . indeed , using eqs.([eq : aomegataylor])-([eq : deffprime ] ) , we obtain a power - series expansion for @xmath242 of the form @xmath244 . thus , by fitting @xmath245 to the nrg data for @xmath242 , one can determine the desired coefficients in ( [ eq : aomegataylor ] ) using @xmath246 . in particular , the fermi - liquid coefficient of present interest is given by @xmath247 . [ fig : flcoeff](a)-(c ) show our nrg data ( heavy solid lines ) for the resistivity plotted versus @xmath209 at zero temperature or plotted versus @xmath210 at zero field , and for the weighted spectral function plotted versus @xmath248 , respectively . we determined the fermi - liquid coefficients @xmath1 , @xmath2 and @xmath225 from the quadratic terms of fourth - order polynomial fits to these curves . including the fourth - order term allows the fitting range to be extended towards somewhat larger values of the argument , thus increasing the accuracy of the fit . for each solid curve , the quadratic term from the fit is shown by heavy dashed lines ; these are found to agree well with the corresponding predictions from flt , shown by light lines of matching colors . the level of agreement is quite remarkable , given the rather limited range in which the behavior is purely quadratic : with increasing argument , quartic contributions become increasingly important , as reflected by the growing deviations between dashed and solid lines ; and at very small values of the argument ( @xmath249 ) , the nrg data become unreliable due to known nrg artefacts . numerical values for the extracted fermi - liquid coefficients are given in table [ tab : nrg_results ] ; they agree with those predicted analytically to within @xmath250 . this can be considered excellent agreement , especially for the numerically very challenging case of @xmath10 . .numerically extracted values of @xmath1 , @xmath2 and @xmath225 , given here relative to the corresponding predictions from flt of eq.([eq : cbteresults ] ) . the deviations between nrg and flt values are @xmath250 in all cases . to numerically determine these coefficients , we used the quadratic coefficient of a fourth - order polynomial fit to the corresponding nrg data . error bars were estimated by comparing the quartic fits to polynomial fits of different higher orders . [ cols="^,^,^,^",options="header " , ] our two main results can be summarized as follows . first , we have presented a compact derivation of three fermi - liquid coefficients for the fully - screened @xmath0-channel kondo model , by generalizing well - established calculations for @xmath9 to general @xmath0 . the corresponding calculations , building on ideas of nozires , affleck and ludwig , and pustilnik and glazman , are elementary . we hope that our way of presenting them emphasizes this fact , and perhaps paves the way for similar calculations in less trivial quantum impurity problems that also show fermi - liquid behavior , such as the asymmetric single - impurity anderson hamiltonian , or the 0.7-anomaly in quantum point contacts.@xcite second , we have established a benchmark for the quality of nrg results for the fully screened @xmath0-channel kondo model , by showing that it is possible to numerically calculate equilibrium fermi - liquid coefficients with an accuracy of better than 5% for @xmath9 , 2 and 3 . to achieve numerical results of this quality , two technical ingredients were essential , both of which became available only recently : first , exploiting larger - rank non - abelian symmetries in the numerics;@xcite and second , carefully defining the kondo temperature@xcite in such a way that numerically - calculated universal scaling curves are indeed universal , in the sense of showing a proper scaling collapse , despite the fact that the scaling limit @xmath251 is typically not achieved in numerical work . we acknowledge helpful discussions with k. kikoin , c. mora , a. ludwig and g. zarnd . we are grateful to d. schuricht for drawing our attention to ref . , and for sending us a preprint of ref . prior to its submission . the latter work , which we received in the final stages of this work , also uses @xmath252 of eq.([eq : al - start - jj ] ) as starting point for calculating fermi - liquid coefficients for the @xmath0-channel kondo model , and its result for @xmath253 is consistent with our own . we gratefully acknowledge financial support from the dfg ( we4819/1 - 1 for a.w . , and sfb - tr12 , sfb-631 and the cluster of excellence nanosystems initiative munich vor j.v.d . , m.h . , and a.w . ) this appendix offers a pedagogical derivation of the hamiltonian @xmath252 given in eq.([eq : h - momentum - main ] ) of the main text using the point - splitting regularization strategy , following al ( appendix d of ) . its main purpose is to show how the relation @xmath254 between fermi - liquid parameters that nb had found by intuitive arguments@xcite follows simply and naturally from point splitting . for a detailed discussion of the point - splitting strategy , see refs . . according to al , the leading irrelevant operator for the fully screened @xmath0-channel kondo model has the form @xmath255 here @xmath256 is the total ( point - split ) spin density from all channels at position @xmath257 ( the impurity or dot sits at @xmath258 ) , and @xmath259 is the corresponding ( non - point - split ) spin density for channel @xmath126 . here @xmath260 denotes point splitting , @xmath261{}{a}{(x+\eta)}{b}a(x+\eta)b(x ) \bigr ] , { \qquad \phantom{.}}\end{aligned}\ ] ] a field - theoretic scheme for regularizing products of operators at the same point by subtracting their ground state expecation value , @xmath262{}{a}{}{b}ab = \langle a b\rangle$ ] . ( in most cases , point splitting is equivalent to normal ordering . ) for present purposes , we follow al@xcite and take @xmath263 to be free fermion fields with linear dispersion ( @xmath264 ) in a box of length @xmath265 ( with @xmath266 , @xmath267 ) , with normalization @xmath268 and free ground state correlators @xmath269 note that we follow pg in our choice of field normalization , which differs from that used by al@xcite by @xmath270 . consequently , our coupling constant is related to theirs by @xmath271 . in the definition of @xmath252 , point splitting is needed because the product of two spin densities , @xmath272 , diverges with decreasing seperation @xmath273 between their arguments . to make this explicit , we use wick s theorem , @xmath274{a}{b}{}{c } ab cd \ ! : \ ! + \ ! : \ ! \ ! \bcontraction[0.6ex]{}{a}{bc}{d } ab cd \ ! : \ ! + \ ! : \ ! \ ! \bcontraction[0.6ex]{a}{b}{}{c } \bcontraction[1ex]{}{a}{bc}{d } ab cd \ ! : , \nonumber \end{aligned}\ ] ] to rewrite the product of spin densities as follows : the point - splitting prescription in eq.([eq : al - startingpoint ] ) subtracts off the @xmath276 divergence of the last term of eq.([eq : wick ] ) . the contributions of the second and first terms to @xmath252 can be organized as @xmath277 , describing single - particle elastic scattering and two - particle interactions , respectively . taking @xmath278 and @xmath279 , we find : @xmath280 \ ! : \label{eq : hel-1 } \\ & = & - \frac{3 \lambda}{8 \pi i } \lim_{\eta \to 0 } \sum_{m \sigma } : \ ! \left [ \frac{1}{\eta } \bigl(\psi^\dagger_{m\sigma}(\eta ) - \psi^\dagger_{m\sigma}(0 ) \bigr ) \psi_{m\sigma}(0 ) - \psi^\dagger_{m \sigma}(0)\frac{1}{\eta } \bigl ( \psi_{m\sigma}(\eta ) - \psi_{m\sigma}(0 ) \bigr ) \right ] \ ! : \ ; \label{eq : hel-2 } \\ & = & - \frac { 3\lambda}{8 \pi i } \sum_{m\sigma } : \ ! \bigl [ \bigl ( \partial_x \psi^{\dagger}_{m\sigma } \bigr ) ( 0 ) \psi_{m\sigma } ( 0 ) - \psi^{\dagger}_{m\sigma } ( 0 ) \bigl(\partial_x \psi_{m\sigma})(0 ) \bigr ] \ ! : , \label{eq : hel-3}\end{aligned}\ ] ] to obtain eq.([eq : hel-2 ] ) , we used @xmath282 and subtracted and added : @xmath283 : inside the square brackets . now pass to the momentum representation , using eq.([eq : definefermions ] ) and the shorthand notations ( following pg@xcite ) here @xmath286 is the extensive 1d density of states per spin and channel , and the prefactors were expressed in terms of the constants @xmath287 ( this notation is consistent with that of ref . , where @xmath252 served starting point for calculating fermi - liquid corrections , too . ) checking dimensions , with @xmath288$]=@xmath289 and @xmath290$]=@xmath291 ( @xmath289 stands for energy , @xmath292 for length ) , we see that @xmath293$]=@xmath294 . since @xmath295$]=@xmath296 , @xmath297$]=@xmath298 , we have @xmath299= [ \phi_1 ] = 1/\mathcal{e}$ ] , thus , @xmath300 and @xmath301 have dimensions of inverse energy . in the main text , they are identified with @xmath56 ; in fact , the numerical prefactor in eq.([eq : identify - alpha1-phi1 ] ) is purposefully chosen such that the leading term in the expansion ( [ eq : gp - phase - shift - a - la - nb ] ) of the phase shift @xmath67 turns out to take the form @xmath302 . to elucidate how the case @xmath54 differs from @xmath303 , we write @xmath304 in the main text , with @xmath95 and @xmath96 given in eqs.([eq : htwo - momentum ] ) and ( [ eq : hthree - momentum ] ) , respectively , where @xmath96 occurs only for @xmath54 . the drude conductivity of a disordered metal has the form @xmath305 . in the presence of magnetic impurities ( but neglecting phonons ) , the total scattering rate is the sum of the scattering rates due static disorder and magnetic impurities , @xmath306 , with @xmath307 . eq.([eq : g(t , b ) ] ) is obtained@xcite by expanding @xmath308 in powers of the ratio @xmath309 , and correspondingly expanding the resisitity as @xmath310 , where the second contribution , due to magnetic impurities , is much smaller than the first .

we analytically and numerically compute three equilibrium fermi - liquid coefficients of the fully screened @xmath0-channel kondo model , namely @xmath1 , @xmath2 and @xmath3 , characterizing the magnetic field and temperature - dependence of the resisitivity , and the curvature of the equilibrium kondo resonance , respectively . we present a compact , unified derivation of the @xmath0-dependence of these coefficients , combining elements from various previous treatments of this model . we numerically compute these coefficients using the numerical renormalization group , with non - abelian symmetries implemented explicitly , finding agreement with fermi - liquid predictions on the order of 5% or better .

pre - swift observations of gamma - ray burst ( grb ) afterglows have led to great strides in their theoretical interpretation , while leaving some major unanswered questions . the radio , optical , and @xmath0-ray emission of grb afterglows exhibit a power - law decrease with time ( @xmath8 ) from hours to tens of days after the burst , with the temporal index @xmath9 consistent with the slope @xmath10 of the power - law continuum ( @xmath11 ) within the framework of relativistic spherical blast - waves ( mszros & rees 1997 ) or of spreading relativistic jets ( rhoads 1999 ) . the collimation of the grb ejecta yields a steepening of the power - law decay when the relativistic beaming has decreased sufficiently that the jet boundary becomes visible . such a steepening has been observed for the first time in the optical light - curve of the afterglow 990123 ( kulkarni 1999 ) . since then about 10 other afterglows have displayed an optical light - curve break at about 1 day after the burst . the achromaticity of a jet light - curve break has not been clearly proven by pre - swift observations because the @xmath0-ray light - curves were not monitored over a time long enough to capture the jet - break . furthermore , the radio light - curves were usually poorly sampled during the first day and strongly affected by interstellar scintillation . the observations of many @xmath0-ray afterglows by swift , together with ground - based optical observations , will enable us to test achromaticity of the afterglow light - curve break , as appears to be the case for the afterglow 050525a ( blustin 2005 ) . the quenching of the interstellar scintillation of the radio afterglow 970508 ( frail , waxman & kulkarni 2000 ) has confirmed that the source size increases as expected for a relativistic blast - wave , providing another test for this model . the decrease of the scintillation has also been observed in the radio afterglows 991208 , 021004 , and 030329 . further testing has been prompted by the detection of the optical afterglows of grbs 990123 and 021211 at very early times ( akerlof 1999 , fox 2003 , li 2003 ) , starting at about 100 s after the burst . the steep decays ( @xmath12 and 1.6 , respectively ) exhibited by these afterglows in the first 20 minutes can be attributed to the grb ejecta energized by internal shocks ( mszros & rees 1997 , 1999 ) or by the reverse shockcaused by the interaction with the circumburst medium ( sari & piran 1999 ) . a significant discrepancy between afterglow observations and theoretical expectations exists for the radio afterglows of grbs 991208 , 991216 , 000301c , and 010222 , whose decay over 12 decades in time is substantially slower than that of the optical emission ( frail 2004 , panaitescu & kumar 2004 ) . a change in the blast - waves dynamics , such as the transition to semi - relativistic dynamics , is not a possible explanation , because the different radio and optical decays are observed over time ranges which overlap substantially . our analysis ( panaitescu & kumar 2004 ) of these afterglows shows that evolving microphysical parameters can not decouple the optical and radio decays . this decoupling may be achieved if there is an extra radio emission arising from some late ejecta , energized by a reverse shock . for the optical afterglow to remain unaffected , the incoming ejecta should not alter the dynamics of the blast - wave , they should carry less kinetic than that already existing in the swept - up circumburst medium . the swift measurements of the @xmath0-ray afterglow emission , starting from 100 s after the burst , opens new possibilities for testing the blast - wave model and for refining its details . the xrt 0.210 kev light - curves of the 9 @xmath0-ray afterglows ( 050126 , 050128 , 050219a , 050315 , 050318 , 050319 , 050401 , 050408 , 050505 ) presented by campana ( 2005 ) , chincarini ( 2005 ) , and tagliaferri ( 2005 ) , have shown that some @xmath0-ray afterglows decay very fast ( @xmath13 ) within the first few minutes after the burst , as reported previously for the afterglows 990510 ( pian 2001 ) and 010222 ( int zand 2001 ) , followed by a slower decay phase ( @xmath14 ) , and a break to a steeper decay ( @xmath15 ) at a later time , ranging from 1 hour to 1 day . the purpose of this paper is to investigate what features of the blast - wave model are required to accommodate the various decays of these swift @xmath0-ray afterglows . barthelmy ( 2005 ) have shown that the very early fast decay of the @xmath0-ray emission of the afterglows 050315 and 050319 can be understood as the grb emission from the fluid moving at angles larger than the inverse of the forward shock s lorentz factor . due to the curvature of the emitting surface , this large angle emission arrives at the observer at an ever increasing time and ever decreasing frequency ( kumar & panaitescu 2000 ) . however , for two other swift afterglows with an early , fast decaying @xmath0-ray emission ( 050126 and 050219a ) , tagliaferri ( 2005 ) have found that the 15350 kev grb emission extrapolated to the xrt 0.210 kev band ( under the assumption that the burst power - law spectrum extends unbroken to lower energies ) falls short of the flux measured at the beginning of the @xmath0-ray observations . if the burst spectrum has a break below 15 kev , below which it is harder , then the grb extrapolated flux would be even less . this suggests that the fast falling - off @xmath0-ray emission does not arise from the same mechanism as the burst itself , and that it may arise in the forward shock . therefore , for at least these two last bursts , we shall test whether the very early @xmath0-ray emission can have the same origin as the rest of the afterglow . the steepening observed at later times is most naturally attributed to a collimated outflow ( jet ) , hence we shall test if the pre- and post - break @xmath0-ray light - curve indices and the spectral slopes are consistent with this interpretation . for the dynamics and collimation of the relativistic blast - wave , we consider three cases : @xmath16 a spherical grb remnant , in the sense that observations were done at a time when the afterglow lorentz factor @xmath17 was larger than the inverse of the jet opening @xmath18 and , hence , the effects associated with collimation were not yet detectable , @xmath19 a jet whose edge is visible ( @xmath20 ) and which does not expand laterally ( because it is embedded in an outer outflow , but whose emission is dimmer ) , and @xmath21 a jet with sharp edges , which spreads laterally and is observed when @xmath20 . these models will be named , , and , respectively . at a frequency above that of the synchrotron peak , @xmath22 , the index @xmath9 of the light - curve power - law decay depends on + @xmath16 the index @xmath23 of the power - law electron distribution with energy @xmath24 @xmath19 the density stratification of the circumburst medium ( cbm ) , for which we assume a power - law profile @xmath25 which comprises a homogeneous cbm ( @xmath26 ) and a pre - ejected wind at constant speed and mass - loss rate ( @xmath27 ) , the condition @xmath28 being required for a decelerating blast - wave , + @xmath21 the location of the cooling frequency @xmath29 relative to the observing band . the @xmath29 is the synchrotron characteristic frequency corresponding to an electron energy for which the radiative ( synchrotron + inverse compton ) timescale is equal to the electron age . + the expressions for @xmath30 for the model are given in mszros & rees ( 1997 ) and sari , narayan & piran ( 1998 ) for @xmath26 and in chevalier & li ( 2000 ) for @xmath27 . rhoads ( 1999 ) and sari , piran & halpern ( 1999 ) have shown that , for the model , @xmath31 , irrespective of the location of @xmath29 and cbm stratification . these and other results for the and models are summarized below . because we will determine from observations the required structure of the cbm , the parameter @xmath32 , we start from the most general expressions for the evolution with observer time @xmath33 of the afterglow spectral properties : peak flux @xmath34 , and frequencies @xmath22 and @xmath29 . as derived in panaitescu & kumar ( 2004 ) , they are : @xmath35 for both the and models ( note that the evolution of @xmath22 is in independent of @xmath32 and that @xmath29 increases for @xmath36 , but decreases for @xmath37 ) , @xmath38 for the model and @xmath39 for the model . for the latter , the faster decay is due to that the jet area is a factor @xmath40 smaller than that visible to the observer in the case of a spherical outflow . the synchrotron afterglow continuum is @xmath11 ( sari , narayan & piran 1998 ) , where @xmath41 we restrict our attention to the @xmath42 cases , for which @xmath43 , as observed by xrt for the swift @xmath0-ray afterglows . from equations ( [ nuic])([fpj ] ) , it is easy to obtain the synchrotron light - curve decay @xmath8 : @xmath44 [ sa ] @xmath45 @xmath46 @xmath47 from the above equations , it can be seen that the passage of the cooling frequency through the observing band steepens the afterglow decay by @xmath48 , which is at most 1/4,in addition to softening the spectrum by @xmath49 for @xmath36 or hardening it by @xmath50 for @xmath37 . the representative values chosen for @xmath32 these equations show that the observable quantity @xmath51 has a stronger dependence on the cbm structure for @xmath52 ( winds ) than for @xmath53 . the case @xmath54 should be taken only as the @xmath55 limit ; for @xmath54 the outflow deceleration is not a power - law in the observer time , instead @xmath56 . for the model , the ( @xmath57 ) closure relation is : @xmath58 @xmath59 equations ( [ sc ] ) , ( [ jc ] ) , and ( [ jc ] ) are valid whatever is the location of the injection frequency . however , there are further constraints for the applicability of the @xmath60 case : @xmath61 for all models and @xmath62 for the and models respectively . the models , , and with @xmath63 will be designated as , , and ( the letter `` a '' indicating that the electrons radiating synchrotron emission at the observing frequency are losing energy adiabatically ) , while the models with @xmath64 will be called , , and ( where the letter `` c '' shows that the electrons radiating at @xmath65 are cooling radiatively ) . note from equations ( [ sc ] ) , ( [ ja ] ) , and ( [ jc ] ) that for the and models , the index @xmath9 is independent of the medium structure , hence the type of cbm can not be determined for these models . in the derivation of equation ( [ nuic ] ) we have ignored a multiplicative factor @xmath66 ( where @xmath67 is the compton parameter ) in the expression of @xmath29 . therefore equation ( [ nuic ] ) is valid if @xmath68 ( the radiative cooling of the electrons emitting at @xmath29 is synchrotron - dominated ) or if @xmath67 constant ( which corresponds to the @xmath69 case , where the @xmath67 parameter depends only on the ratio of the electron and magnetic field energies ) . if @xmath70 and @xmath71 , the decrease of the compton parameter with time leads to a faster increase or a slower decrease of @xmath29 than given in equation ( [ nuic ] ) and to a slower decay of the afterglow emission at @xmath72 . this case is most likely relevant for the @xmath73 @xmath0-ray afterglow phase , between the flattening and steepening times @xmath74 and @xmath75 , when the @xmath0-ray light - curve may exhibit a slower decay than that resulting from equations ( [ sc ] ) , ( [ jc ] ) , and ( [ jc ] ) . the equations for the afterglow light - curve at @xmath72 for the ( @xmath70 , @xmath71 ) case , derived by panaitescu & kumar ( 2001 ) , lead to : + + @xmath76 @xmath77 @xmath78 @xmath79 for simplicity , the results in equations ( [ sc0])([jc2 ] ) are given for the two most likely types of cbm structure @xmath26 and @xmath27 and not for any @xmath32 . for the model , @xmath9 is quasi - independent on the stratification of the cbm : @xmath80 the results given for @xmath81 in equations ( [ sc0])([jcy ] ) are valid also for @xmath82 as long as the total electron energy is a constant fraction of the post - shock energy , which is equivalent to saying that the high - energy cut - off of the electron distribution , which must exist for a finite total electron energy , has the same evolution as the minimum electron energy , @xmath83 . the above equations show that the passage of the cooling frequency through the observing domain slows the afterglow decay by @xmath84 for @xmath26 and @xmath85 . the temporal evolutions given in equations ( [ nuic ] ) and ( [ fpj ] ) were derived under the assumption of an adiabatic blast - wave . if the electron fractional energy is around 50 percent and if the electrons cool radiatively ( @xmath69 ) , then radiative losses become important . in this case the afterglow emission decays faster than for an adiabatic grb remnant , given the stronger deceleration , therefore radiative blast - waves should be of importance for the fast decaying , very early swift @xmath0-ray afterglows . from @xmath16 the dynamics of a fully radiative blast - wave ( @xmath86 , where @xmath87 is the mass of the swept - up cbm ) and using @xmath19 the scalings for the spectral characteristics ( @xmath88 , where @xmath89 is the post - shock magnetic field strength , @xmath83 is the electron energy , and @xmath90 is the energy of the electrons whose radiative cooling timescale is equal to the dynamical timescale ; @xmath91 for the model and @xmath92 for the model ) , and @xmath21 the relation between the observer time and blast - wave radius @xmath93 , the following evolutions of the spectral characteristics can be derived : @xmath94 @xmath95 @xmath96 hereafter , radiative afterglows will be indicated with the letter `` r '' preceding the specific model . note from equation ( [ rnuic ] ) that , just as for an adiabatic afterglow , the cooling frequency increases for @xmath36 and decreases for @xmath37 . the light - curve decay indices resulting from equations ( [ beta ] ) and ( [ rnuic])([rfpj ] ) are : @xmath97 @xmath98 @xmath99 @xmath100 the condition @xmath69 required by radiative dynamics guarantees that the compton parameter @xmath67 is constant , hence there are no further complications with the inverse compton - dominated electron cooling , as it was the case for an adiabatic blast - wave . given that , in the model , the jet lorentz factor decreases exponentially with radius ( rhoads 1999 ) , the dynamics and light - curves of a radiative jet should be close to those for an adiabatic jet ( and [ [ jc ] ] ) . there are two other factors which can alter the afterglow decay index @xmath9 . one is that the relativistic outflow can be endowed with an angular structure , where the ejecta kinetic energy per solid angle , @xmath101 , is not constant ( mszros , rees & wijers 1998 ) . the light - curve decay indices for an axially - symmetric outflow with a power - law structure @xmath102 where the angle @xmath103 is measured from the symmetry axis ( which , for simplicity , is assumed to be also the direction toward the observer ) are given in mszros , rees & wijers ( 1998 ) and panaitescu & kumar ( 2003 ) . in this work , recourse to a structured outflow will be made only to explain afterglow decays which are slower than that expected for the model . evidently , such structured outflows require @xmath104 . if the slow @xmath0-ray decay is preceded by a faster fall - off , then the index @xmath105 changes to @xmath106 close to the outflow axis , corresponding to the or models . if the slow @xmath0-ray decay is followed by a steepening , then , going away from the outflow axis , the index @xmath105 changes to either @xmath107 ( if the steeper decay is accommodated by the model ) or to @xmath106 ( if that steeper decay can be explained with the and models ) . therefore , in the most general case , where the @xmath0-ray light - curve exhibits a sharp decay followed by a slow fall - off and then a steeper dimming , the outflow should have a bright spot moving toward the observer , surrounded by a dim envelope ( so that a steep decay is obtained when the spot edge becomes visible to the observer ) , which is embedded in a more energetic outer outflow ( yielding the slower decay ) , whose collimation leads to the late steepening when the outflow boundary becomes visible . the decay index for the synchrotron emission from a structured outflow can be derived as described in panaitescu & kumar ( 2003 ) . for a power - law radial structure of the cbm and angular structure of the outflow , we obtain : @xmath108 } { \ds 8 - 2s + q } \label{soa}\ ] ] @xmath109 } { \ds 8 - 2s + q } \;. \label{soc}\ ] ] the above results are valid for @xmath110^{-1 } & \nu_i < \nu < \nu_c \\ \left [ 3 + 2\beta - 0.5 s ( \beta + 1 ) \right]^{-1 } & \nu_i,\nu_c < \nu \end{array } \right . \;,\ ] ] because for @xmath111 the emission from the outflow axis ( @xmath112 ) , where the energy per solid angle would formally diverge , becomes dominant and sets another light - curve decay index . from equations ( [ soa ] ) and ( [ soc ] ) it follows that , for a given cbm structure , the slowest decay that a structured outflow can produce is that obtained in the @xmath113 limit : @xmath114 hence , for a homogeneous medium ( @xmath26 ) , the light - curve from a structured outflow could rise ( @xmath115 ) . evidently , the structured outflow model can be at work only if the above decay is slower than that observed , the condition @xmath116 leading to a constraint on the cbm structure : @xmath117 equations ( [ soa ] ) and ( [ soc ] ) give the outflow structural parameter which accommodates the observed light - curve index @xmath9 and spectral slope @xmath10 : @xmath118 another process which can reduce the afterglow dimming rate is the injection of energy in the blast - wave ( paczyski 1998 , rees & mszros 1998 ) by means of some ejecta which were ejected later than the grb ejecta ( a long - lived engine ) or at the same time but with a smaller lorentz factor , thus reaching the decelerating grb ejecta during the afterglow phase ( a short - lived engine ) . a delayed injection of energy into the afterglow can be due to the absorption of the dipole electromagnetic radiation emitted by a millisecond pulsar ( dai & lu 1998 , zhang & mszros 2001 ) if such a pulsar was formed . the addition of energy in the blast - wave mitigates its deceleration and , implicitly , the afterglow decay rate . rees & mszros ( 1998 ) have derived the decay index @xmath9 for an energy injection that is a power - law in the ejecta lorentz factor . the expressions for the index @xmath9 for an energy injection which is a power - law in the observer time , @xmath119 are given in ( 23 ) , ( 24 ) , and ( 30 ) of panaitescu & kumar ( 2004 ) . from those equations , it follows that energy injection reduces the light - curve decay indices given in equations ( [ sa])([jc ] ) by @xmath120 @xmath121 @xmath122 for the adiabatic , , and models . lastly , all the decay indices given in the above equations were derived assuming that the microphysical parameters which determine the spectral characteristics ( @xmath22 , @xmath29 , @xmath34 ) and the continuum slope ( @xmath10 ) , the parameters for the typical post - shock electron energy & magnetic field strengthand the power - law index @xmath23 of the electron distribution with energy , are constant . this possibility is not investigated in this work . as described in the introduction , the swift @xmath0-ray afterglows exhibit three phases : the @xmath123 phase , lasting until @xmath124 s , is characterized by a sharp decay , the @xmath73 phase , lasting until @xmath125 s , is marked by a much slower fall - off , while in the @xmath126 phase , the @xmath0-ray light - curve displays a faster decay . the light - curve decay indices @xmath9 and the spectral slopes @xmath10 are listed for each phase in table 1 . the closure relations between @xmath9 and @xmath10 presented in section [ theory ] provide either a criterion for distinguishing among the various models that can accommodate the observed afterglow properties or allow the determination of the cbm structure . since @xmath28 is required for a decelerating blast - wave , this also serves as a test of the various models . table 1 lists the models for which the closure relations given in section [ theory ] between the light - curve decay index @xmath9 and spectral slope @xmath10 are satisfied within @xmath127 , for each afterglow decay phase . to find a model for the entire afterglow , these piece - wise models must now be put together in a sequence that makes sense and is not contrived . the criteria by which we construct a model for the entire @xmath0-ray afterglow are : + * i ) * models relying on coincidences to accommodate two adjacent @xmath0-ray phases are excluded , only one factor ( cooling frequency passage , change of cbm structure , region of non - monotonic variation in the energy per solid angle becoming visible , beginning / cessation of energy injection ) at a time is employed to explain a variation of the @xmath0-ray decay index , + * ii ) * radiative outflows can evolve into adiabatic ones , but not the other way around , + * iii ) * any of the three dynamical models ( , , ) can be followed by the same model , but only the model can be followed by the and models , allowing for a collimated outflow , spreading or non - spreading , whose edge becomes visible to the observer , + * iv ) * the evolution of the cooling frequency @xmath29 required to join two models at @xmath74 or @xmath75 must be compatible with the cbm structural index @xmath32 , @xmath29 can increase only if @xmath36 and can decrease only if @xmath37 ( modulo the effect of a decreasing compton parameter when electron cooling is dominated by inverse compton scatterings ) . [ cols="<,^,^,<,^,^,<,^,^ , < " , ] model coding + ( 1 ) : exponent of radial density profile ( ) ; for the so model , the upper limit on @xmath32 is that resulting from equation ( [ smax ] ) + ( 2 ) : exponent of the energy injection law ( ) obtained from ( [ se])([je ] ) for the index @xmath32 required at @xmath128 or at @xmath129 + ( 3 ) : exponent of the angular distribution of the energy per solid angle ( ) obtained from equation ( [ q ] ) for the index @xmath32 required at @xmath128 or at @xmath129 + ( 4 ) : exponent of the power - law distribution of electrons with energy ( ) ; this value is for all @xmath0-ray phases except la grb * 050126 . * the xrt light - curve of this afterglow exhibits a steep fall - off until @xmath130 s , followed by a slower decay . tagliaferri ( 2005 ) have shown that extrapolation of the 15350 kev bat emission to the 0.210 kev xrt band is dimmer at 100 s than the observed xrt flux . furthermore , the xrt spectrum ( @xmath131 ) during the @xmath123 phase is softer than the bat spectrum ( @xmath132 ) , hence the early @xmath0-ray afterglow is not the large - angle grb emission and must be attributed to the forward shock . if there is no spectral evolution ( @xmath133 ) across @xmath74 , as indicated by tagliaferri ( 2005 ) , then the slow @xmath0-ray decay of the @xmath73 phase can not be explained by a change in the structure of the cbm medium for any of the models ( , , , ) which accommodate the @xmath123 phase . conversely , if the cbm structure does not change across @xmath74 , then the slower decay at the @xmath73 phase requires a substantial hardening of the spectrum , corresponding to a rising one ( @xmath134 ) for the models , , and , or one with @xmath135 for the model , both of which are inconsistent with the xrt observations . furthermore , for the possible models for the @xmath123 afterglow phase , the passage of the cooling frequency through the @xmath0-ray band can only steepen the afterglow decay . hence , the most plausible models that can explain the flattening @xmath0-ray light - curve of 050126 require energy injection or a structured outflow . the model with either energy injection or a structured outflow does not satisfy conditions @xmath136 and @xmath137 above . * although xrt observations started at 100 s after the burst , a steep early decay has not been observed ( campana 2005 ) . its decay steepens at @xmath138 s , without a spectral evolution . of the many possible combinations of models for the @xmath73 and @xmath126 phases , the most plausible is that of a collimated outflow ( and models ) , leading to a steepening of the @xmath0-ray decay when the boundary of the jet becomes visible . another possibility is that of non - spreading jet ( model ) which transits from a @xmath7 wind into a region of increasing density at @xmath75 . we note that all these models require a rather hard electron distribution , with @xmath139 . * 050219a . * the features of this afterglow are similar to those of 050126 . it exhibits a fast fall - off until @xmath124 s , followed by a slower decay . the extrapolation of the 15350 kev bat emission to the 0.210 kev xrt band underpredicts the observed flux at 100 s ( tagliaferri 2005 ) and the @xmath0-ray spectral slope ( @xmath140 ) is much softer than that of the burst ( @xmath141 ) , hence the rapid , early fall - off of the 050219a @xmath0-ray afterglow is not the grb large - angle emission . just as for the afterglow 050126 , a change in the cbm structure can not explain the @xmath0-ray light - curve flattening . if the cbm structure is considered unchanged across @xmath74 , then the slowing of the @xmath0-ray decay would require a rising spectrum ( @xmath134 ) for the @xmath73 phase , which is inconsistent with the xrt observations . because all models for the @xmath123 afterglow phase require that the cooling frequency is above the @xmath0-ray domain , its passage is either impossible or it would steepen the light - curve decay . consequently , the slower decay observed for the @xmath0-ray afterglow 050219a after @xmath74 requires either energy injection or a structured outflow . condition @xmath136 is not satisfied by either the and models and a structured outflow , while the model with energy injection requires too much energy . * 050315 . * the @xmath0-ray emission exhibits a flattening at @xmath142 s , accompanied by a hardening of the spectrum ( @xmath143 ) , and followed by a steeper decay after @xmath144 s , across which there is no spectral evolution . barthelmy ( 2005 ) have shown that the early , steep fall - off is consistent with the large - angle grb emission : the extrapolation of the 15350 kev bat emission to the 0.2 - 10 kev xrt band matches the xrt flux measured at 100 s , the @xmath0-ray spectral slope ( @xmath145 ) is comparable to that of the burst ( @xmath146 ) , and the @xmath0-ray decay index ( @xmath147 ) is close to the expected value ( @xmath148 ) . the steepening at @xmath75 can be easily understood as due to a collimated outflow ( the or models ) . a radiative non - spreading jet interacting with @xmath52 cbm could also accommodate the steepening , if the cbm is a wind , but it is less likely that the radiative phase could last until later than 1 day after the burst . * 050318 . * because xrt observations started at @xmath149 h after the burst , the fast decay phase may have been missed . a steepening of the @xmath0-ray light - curve decay occurs at @xmath150 s without a spectral evolution . this steepening can be due to seeing the boundary of a jet ( spreading or not ) . there are other possible models that can accommodate the steepening , all involving a variation in the cbm structural index @xmath32 . they are the outflow exiting a shell of a sharply increasing density and entering a @xmath151 wind and jc outflow transiting from a @xmath7 wind to a shell with sharply increasing density at @xmath75 . * 050319 . * this afterglow is similar to 050315 , the hardening of the @xmath0-ray spectrum across the light - curve flattening , which occurs at @xmath152 , being stronger . barthelmy ( 2005 ) have shown that the bat grb emission extrapolated to the xrt band matches the @xmath0-ray flux measured at 200 s. if the origin of time for the @xmath0-ray emission is set at the beginning of the second ( and last ) grb pulse , then the decay index ( @xmath153 ) of the early @xmath0-ray emission is consistent with the expectations for the large - angle grb emission ( @xmath154 ) . however , the early @xmath0-ray spectrum ( @xmath155 ) is rather soft compared to that of the burst . on the other hand , the substantial hardening of the @xmath0-ray spectrum across @xmath74 , with @xmath156 , exceeds that which the passage of the cooling frequency through the observing band can produce ( @xmath157 ) , suggesting that the @xmath0-ray emissions during the @xmath123 and @xmath73 afterglow phases arise from different mechanisms . we note that the @xmath0-ray light - curve for both phases may be explained in the structured outflow framework if we make the ad - hoc assumption that the spectrum of the spot emission ( dominating the afterglow flux before @xmath74 ) is softer than that from the surrounding outflow ( which overtakes the spot emission after @xmath74 ) . the steepening of the @xmath0-ray light - curve at @xmath150 s can be explained by seeing the edge of a jet ( spreading or not ) , or with the model and a cbm structure changing from a @xmath7 wind to a homogeneous medium at @xmath75 . all these models require a hard electron distribution , with @xmath158 . * although the xrt observations started 100 s after the burst , a steeply falling - off phase was not seen . until @xmath159 s , it exhibits a decay so slow that it can not be explained without energy injection or a structured outflow . the model with energy injection requires too much energy , while the structured outflow does not satisfy condition @xmath136 for the model . then the steepening of the @xmath0-ray light - curve at @xmath75 can be understood either as resulting from the cessation of the energy injection or from seeing the outflow boundary . in the latter case , the light - curve decay should be steeper than for the model and slower than for the model . that the steeper decay after @xmath75 can be accommodated by either the and models ( table 1 ) supports a structured outflow as the source of the @xmath0-ray light - curve steepening . * 050408 . * this afterglow is very similar to 050401 , except that the steepening occurs later , at @xmath160 s. its light - curve decay before @xmath75 is also too slow and requires an energy injection episode or a structured outflow . the @xmath0-ray spectral slope after @xmath75 is not known , but if we assume that there is no spectral evolution across @xmath75 ( as is the case for all other afterglows ) , then the light - curve decay index and spectral slope measured after @xmath75 can be accommodated by the and models . the and models are also allowed , though it is unlikely that the radiative phase lasts until days after the burst . * 050505 . * this afterglow is similar to 050318 , its @xmath0-ray light - curve steepening at @xmath161 s without a spectral evolution . however , in contrast to 050318 , the spectral slopes and decay indices before and after @xmath75 can not be reconciled within any model other than , even if we allow for a varying cbm structure . besides that the radiative phase is unlikely to last until 1 day after the burst , the model requires a @xmath7 cbm profile , for which the closure equations given in section [ radiative ] are not accurate . hence , it seems more plausible that the slow decay of this afterglow before @xmath75 is due to energy injection or a structured outflow . the model fails to satisfy conditions @xmath136 and @xmath137 for these two case . as for the afterglows 050401 and 050408 , the steepening of the @xmath0-ray light - curve could then be attributed to the end of the energy injection or to the outflow axis becoming visible to the observer . as shown in table 1 , the three decay phases of the swift @xmath0-ray afterglows can be understood in the following way : @xmath16 the hardening of the 0.210 kev spectrum of the @xmath0-ray afterglows 050315 and 050319 from @xmath6 s ( when a fast decaying @xmath0-ray emission is observed ) to @xmath162 s ( when the @xmath0-ray light - curve exhibits a slow decay ) indicates that the early , fast - falling off @xmath0-ray emission arises from a different mechanism than the rest of the afterglow . barthelmy ( 2005 ) have argued that this mechanism is the same as for the grb emission . the results of tagliaferri ( 2005 ) suggest that this explanation does not work well for the afterglows 050126 and 05018 . in these cases , the steep @xmath0-ray decay require a very narrow outflow whose edge is seen as early as 100 s after the burst , the following , slower decay phase being explained by energy injection in the blast - wave or by a rather contrived ( see below ) angular structure of the blast - wave , + @xmath19 the @xmath0-ray decay measured until 1h , 0.3d , and 0.5d for the afterglows 050401 , 050408 , and 050505 , respectively , is too slow to be explained by the simplest blast - wave model . such a slow decay can be produced by an outflow endowed with angular structure or by a continuous injection of energy in the forward shock , + @xmath21 the steepening of the @xmath0-ray decay observed at 1h2d for the afterglows 050128 , 050315 , 050318 , and 050319 , at a time comparable to that of the steepening of the optical decay of many pre - swift afterglows , can be explained by seeing the edge of a jet . for the remaining 5 afterglows , whose pre - break @xmath0-ray decay requires energy injection , the steepening can be attributed to the cessation of injection . in the large - angle grb emission model for the early , fast falling - off phase , the @xmath0-ray emission arises from the same mechanism as the grb itself , but arrives at observer later because it comes from the shocked gas moving slightly off the direction toward the observer . for this model to be at work , three conditions must be satisfied . first , the 15350 kev grb emission extrapolated to the 0.210 kev @xmath0-ray band , under the assumption that the power - law burst spectrum @xmath163 extends unbroken down to 0.2 kev , should match or exceed the @xmath0-ray flux at the first epoch of observations . second , the spectral slope of the early afterglow should be the same as that of the grb . third , the @xmath0-ray light - curve decay index should be equal to @xmath164 ( kumar & panaitescu 2000 ) . for completeness , we present here a short derivation of this result . if the gr emission stops suddenly at some radius @xmath165 and blast - wave lorentz factor @xmath17 , then the received flux is @xmath166 where @xmath167 is the outflow comoving frame surface - brightness at frequency @xmath168 , @xmath169 is the elementary area whose radiation is received over an observer time @xmath170 , @xmath103 is the angle ( measured from the direction toward the observer ) of the fluid element from which radiation is received at time @xmath171 ( hence @xmath172 ) , @xmath173 is the relativistic doppler factor , and the expressions for @xmath174 and @xmath175 have been derived for @xmath176 , for the large - angle emission . the last factor @xmath177 in the expression of @xmath178 accounts for the beaming of radiation from a relativistic source . after substitutions , one obtains that @xmath179 . barthelmy ( 2005 ) found that the above three conditions for the large - angle grb emission as the source of the very early , fast @xmath0-ray decay are met for the afterglows 050315 and 050319 . for two other afterglows , 050421 ( sakamoto 2005 , godet 2005 ) and 050713b ( parsons 2005 , page 2005 ) , we find that their fast @xmath0-ray decays can not be reconciled with their hard @xmath0-ray continua by any of the blast - wave models considered here , but they do satisfy the last two conditions above for the large - angle grb emission interpretation . tagliaferri ( 2005 ) showed that the early @xmath0-ray emissions of the afterglows 050126 and 050219a are brighter than the grb extrapolated fluxes and softer than the burst emission , therefore their early @xmath0-ray afterglows can not be immediately identified with the large - angle grb emission . kumar ( 2005 ) discuss the conditions under which the fast @xmath0-ray decay of these last two afterglows can be reconciled with the large - angle grb emission . the outflow structure required to explain a light - curve flattening followed by a steepening must contain a bright spot ( moving toward the observer ) surrounded by a dimmer envelope where the ejecta have a lower energy per solid angle @xmath101 , so that the emission from the spot exhibits a fast decay after its boundary becomes visible . further , the envelope should be embedded in a wider outflow whose emission overtakes that of the spot when the blast - wave lorentz factor has decreased sufficiently . to explain the light - curve flattening , the @xmath101 in this wider outflow should rise away from the spot as @xmath180 to @xmath181 , where @xmath103 is the angle measured from the outflow s symmetry axis . finally , to explain the light - curve steepening , the @xmath101 should stop increasing with angle ( for the model ) or peak and then decrease ( for the and models ) . the decrease could be gradual , with the @xmath0-ray light - curve steepening occurring when the fluid at the peak of @xmath101 becomes visible and the outer outflow contributing to the post - break emission ( this is the light - curve break mechanism proposed by rossi , lazzati & rees 2002 ) . if the decrease of @xmath101 is sharp , then the post - break @xmath0-ray light - curve decay will be faster , particularly if the outflow undergoes lateral spreading ( this is the light - curve break mechanism proposed by rhoads 1999 ) . in the energy injection model , the forward shock energy increases due to some relativistic ejecta which catch up with the decelerating blast - wave ( rees & mszros 1998 ) . the energy injection reduces the blast - wave deceleration rate and mitigates the decay of the afterglow emission . in principle , during the slow @xmath0-ray decay phase , there could be an energy injection for all afterglows considered here ; table 2 lists only the cases when it is required . as shown in table 2 , we find that , to explain the slow phase of the @xmath0-ray afterglow decay , the blast - wave energy should increase with observer time faster than @xmath182 and slower than @xmath183 . the arrival of new ejecta at the forward shock could be due either to the spread in the lorentz factor of ejecta released simultaneously ( short - lived engine ) or to a long - lived grb engine , releasing a relativistic outflow for a source - frame duration comparable to the observer - frame duration of the slow @xmath0-ray decay phase . in the former case , the ejecta forward - shock contrast lorentz factor is @xmath184^{1/2 } \siml 2 $ ] ( panaitescu & kumar 2004 ) , where @xmath185 is defined by equation ( [ ei ] ) , while in the latter case the lorentz factor ratio can be much larger . consequently , we expect that , for a short - lived engine , the reverse shock propagating in the incoming ejecta is only mildly relativistic and radiates mostly at radio frequencies while for a long - lived engine the reverse shock could be very relativistic and radiate in the infrared - optical . as mentioned above , the early , fast decay of the @xmath0-ray afterglows 050126 and 050219a can not be readily identified with the large - angle grb emission and could be attributed to the forward shock . the fast @xmath0-ray decay displayed by these afterglows at @xmath2 s can then be explained only within the structured outflow and energy injection models . in addition , two other afterglows , not considered in this work , 050712 ( grupe 2005 ) and 050713a ( morris 2005 ) , exhibit an @xmath0-ray decay which is too slow and incompatible with the reported @xmath0-ray spectral slope , both requiring either an energy injection or a structured outflow . for the afterglows 050128 , 050318 , and 050319 , we find that a change in the circumburst density profile provides an alternate model to structured outflows and jets for the steepening of the @xmath0-ray decay observed by xrt after 0.1 d. for all three afterglows , the changing external density corresponds to a transition from a @xmath7 wind to a region of uniform or increasing density . the @xmath7 density structure requires a time - varying mass - loss rate and/or speed of the wind of the massive star grb progenitor , while the uniform or increasing density shell could result from the internal interactions in a variable wind ( ramirez - ruiz 2005 ) . the self - similar solutions of chevalier & imamura ( 1983 ) for wind - wind interactions indicate that a uniform shell results from a substantial decrease of the star s mass - loss rate accompanied by a large increase in the wind speed . these major changes in the wind properties would have to occur @xmath186 yrs before the grb explosion , if the radius where the @xmath7 circumburst density profile terminates is that of the forward shock at 0.11 d. to answer the question of how can we distinguish between the above three models ( energy injection , structured outflow , non - monotonic circumburst density ) for flattenings and steepenings of the afterglow light - curve , we note that , if the cooling frequency located between the optical and @xmath0-ray domains , each of those models yields a specific difference @xmath187 between the changes @xmath188 and @xmath189 of the @xmath0-ray and optical light - curve decay indices . therefore , to discriminate among the few possible models discussed here , it is very important to monitor the optical afterglow emission over a wide range of times , from minutes to days after the burst . the most often encountered feature resulting from the analysis of the nine @xmath0-ray afterglows analyzed in this work is the existence of a substantial energy injection in the blast - wave at hours to 1 d after the burst . this energy injection is necessary to reconcile the slowness of the @xmath0-ray decay with the hardness of the @xmath0-ray continuum for five out of nine afterglows and is possible for the other four as well . the injection should increase the blast - wave energy as @xmath5 , @xmath33 being the observer time , leading to a shock energy which is eventually larger by a factor 101,000 than that at the beginning of the slow @xmath0-ray decay phase . the exponent @xmath185 of power - law energy injection identified in table [ t2 ] is generally inconsistent with that expected for the absorption of the dipole radiation from a millisecond pulsar ( see also zhang 2005 ) . in this case , an important energy input in the blast - wave is obtained only for the first few thousand seconds , when the pulsar electromagnetic luminosity is constant ( zhang & mszros 2001 ) , which leads to @xmath190 . hence the energy injection must be mediated by the arrival of new ejecta at the forward - shock . if the grb engine were so long - lived that only a small fraction of the total outflow energy yielded the burst emission , then such a large energy injection would imply an even higher grb efficiency than previously inferred ( above 30 percent lloyd - ronning & zhang 2004 ) from afterglow energetics and would pose a serious issue for the grb mechanism ( nousek 2005 ) . however , we can not yet tell if the energy injection lasts for hours because the central engine is long - lived or because there is a sufficiently wide distribution of the initial lorentz factor of the ejecta expelled by a short - lived engine . in the latter case , the entire outflow could be emitting both the grb and afterglow emission and the grb efficiency remains unchanged . if the fast decaying @xmath0-ray emission preceding the slow @xmath0-ray fall - off arises from the forward shock as well , then energy injection must start at the end of the steep decay phase , it must be a well defined episode . however , in this scenario the outflow must be very tightly collimated to yield a fast decay at only 100 seconds after the burst . the blast - wave lorentz factor is @xmath191 ( @xmath192 being the shock energy in @xmath193 ergs and @xmath194 the circumburst medium particle density in @xmath195 ) , thus the jet opening must be less than @xmath196 . alternatively , the early fast falling - off @xmath0-ray emission could be the large - angle emission from the burst phase , overshining the forward shock emission . this interpretation is favoured by barthelmy ( 2005 ) and hill ( 2005 ) for the afterglows 050117 , 050315 , 0509319 , although for other afterglows ( 050126 and 050219a tagliaferri 2005 ) the spectral properties of the burst and @xmath0-ray afterglow emissions are not readily consistent with each other . for a structured outflow to explain all the three decay phases of the swift @xmath0-ray afterglows , the distribution of the ejecta kinetic energy with angle must be non - monotonic . this is somewhat contrived and inconsistent with the results obtained by macfadyen , woosley & heger ( 2001 ) from simulations of jet propagation in the collapsar model . thus structured outflows do not appear to provide a natural explanation for the features of the @xmath0-ray afterglow light - curves . we note that the inferred indices of the power - law electron distribution with energy , which are given in table 2 , range from 1.3 to 2.8 . one would have to ignore 4 of these 9 swift afterglows to obtain a unique electron index , @xmath197 . this is a puzzling feature of relativistic shocks in grb afterglows : the shock - accelerated electrons do not have a universal distribution with energy , a fact which is also proven by the wide spread of the high - energy spectral slopes of batse bursts ( @xmath198 in fig . 9 of preece 2000 ) , which is equal to @xmath199 or @xmath200 , and the by wide range of the optical post - break decay indices of the bepposax afterglows ( @xmath201 in fig . 3 of stanek 2001 and in fig.2 of zeh , klose & kann 2005 ) , which is equal to @xmath23 . however , from the @xmath0-ray spectral slope of 15 bepposax afterglows , de pasquale ( 2005 ) conclude that the electron index @xmath23 has an universal value of @xmath202 ( see their fig . 3 ) . akerlof c. , 1999 , nature , 398 , 400 barthelmy s. , 2005 , apjl , in press ( astro - ph/0511576 ) blustin a. , 2005 , apj , accepted ( astro - ph/0507515 ) campana s. etal , 2005 , apj , 625 , l23 chevalier l. , imamura j. , 1983 , apj , 270 , 554 chevalier r. , li z. , 2000 , apj , 536 , 195 chincarini g. , 2005 , apj , submitted ( astro - ph/0506453 ) dai z. , lu t. , 1998 , a&a , 333 , l87 de pasquale m. , 2005 , a&a , submitted ( astro - ph/0507708 ) fan y. , dai z. , huang y. , lu t. , 2002 , chinese j. a&a , 2 , 449 fox d. , 2003 , apj , 586 , l5 frail d. , waxman e. , kulkarni s. , 2000 , apj , 537 , 191 frail d. , 2004 , apj , 600 , 828 godet o. , 2005 , gcn circ . 3301 grupe d. , 2005 , gcn circ . 3579 hill j. , 2005 , apj , accepted ( astro - ph/0510008 ) int zand j. , 2001 , apj , 559 , 710 kulkarni s. , 1999 , nature , 398 , 389 kumar p. , panaitescu a. , 2000 , apj , 541 , l51 kumar p. , panaitescu a. , 2003 , mnras , 346 , 905 kumar p. , 2005 , science , submitted li w. , filippenko a. , chornock r. , jha s. , 2003 , apj , 586 , l9 lloyd - ronning n. , zhang b. , 2004 , apj , 613 , 477 mszros p. , rees m.j . , 1997 , apj , 476 , 232 mszros p. , rees m.j . , 1999 , mnras , 306 , l39 mszros p. , rees m.j . , wijers r. , 1998 , apj , 499 , 301 morris d. , 2005 , gcn circ . 3606 nousek j. , 2005 , apj , submitted ( astro - ph/0508332 ) paczyski b. , 1998 , apj , 494 , l45 page k. , 2005 , gcn circ . 3602 parsons a. , 2005 , gcn circ . 3600 panaitescu a. , kumar p. , 2001 , apj , 554 , 667 panaitescu a. , kumar p. , 2003 , apj , 592 , 390 panaitescu a. , kumar p. , 2004 , mnras , 350 , 213 pian e. , 2001 , a&a , 372 , 456 preece r. , 2000 , apj , 126 , s19 ramirez - ruiz e. , garcia - segura g. , salmonson j. , perez - rendon b. , 2005 , apj , 631 , 435 rees m.j . , mszros p. , 1998 , apj , 496 , l1 rhoads j. , 1999 , apj , 525 , 737 rossi e. , lazzati d. , rees m.j . , 2002 , mnras , 332 , 945 sakamoto t. , 2005 , gcn circ . 3305 sari r. , piran t. , 1999 , apj , 520 , 641 sari r. , narayan r. , piran t. , 1998 , apj , 497 , l1 sari r. , piran t. , halpern j. , 1999 , apj , 519 , l17 stanek k. , 2001 , apj , 563 , 592 tagliaferri g. , 2005 , nature , 436 , 985 macfadyen a. , woosley s. , heger a. , 2001 , apj , 550 , 410 yost s. , harrison f. , sari r. , frail d. , 2003 , apj , 597 , 459 zeh a. , klose s. , kann d. , 2005 , apj , accepted ( astro - ph/0509299 ) zhang b. , mszros p. , 2001 , apj , 552 , l35 zhang b. , kobayashi s. , mszros p. , 2003 , apj , 595 , 950 zhang b. , 2005 , apj , submitted ( astro - ph/0508321 )

the @xmath0-ray light - curves of 9 swift xrt afterglows ( 050126 , 050128 , 050219a , 050315 , 050318 , 050319 , 050401 , 050408 , 050505 ) display a complex behaviour : a steep @xmath1 decay until @xmath2 s , followed by a significantly slower @xmath3 fall - off , which at 0.22 d after the burst evolves into a @xmath4 decay . we consider three possible models for the geometry of relativistic blast - waves ( spherical outflows , non - spreading jets , and spreading jets ) , two possible dynamical regimes for the forward shock ( adiabatic and fully radiative ) , and we take into account a possible angular structure of the outflow and delayed energy injection in the blast - wave , to identify the models which reconcile the @xmath0-ray light - curve decay with the slope of the @xmath0-ray continuum for each of the above three afterglow phases . by piecing together the various models for each phase in a way that makes physical sense , we identify possible models for the entire @xmath0-ray afterglow . the major conclusion of this work is that a long - lived episode of energy injection in the blast - wave , during which the shock energy increases at @xmath5 , is required for 5 afterglows and could be at work in the other 4 as well . for some afterglows , there may be other mechanisms that can explain the @xmath6 s fast falling - off @xmath0-ray light - curve ( the large - angle grb emission ) , the 400 s5 h slow decay ( a structured outflow ) , or the steepening at 0.22d ( a jet - break , a collimated outflow transiting from a wind with a @xmath7 radial density profile to a homogeneous or outward - increasing density region ) . optical observations in conjunction with the @xmath0-ray can distinguish among these various models . our simple tests allow the determination of the location of the cooling frequency relative to the @xmath0-ray domain and , thus , of the index of the electron power - law distribution with energy in the blast - wave . the resulting indices are clearly inconsistent with an universal value . _ s _ gamma - rays : bursts - ism : jets and outflows - radiation mechanisms : non - thermal - shock waves

we consider a largely ignored metric which belongs to a class of vacuum solutions referred to as degenerate solutions of class a @xcite given by @xmath0 which are axisymmetric solutions , but where the usual 2-dimensional spheres are replaced by pseudo - spheres , @xmath1 , i.e. , by surfaces of negative , constant curvature . these are still surfaces of revolution around an axis , and @xmath2 represents the corresponding rotation angle . for the vacuum case we get @xmath3 where @xmath4 is a constant . we immediately see that the static solution holds for @xmath5 and that there is a coordinate singularity at @xmath6 ( note that @xmath7 neither vanishes nor becomes @xmath8 at @xmath6)@xcite . this is the complementary domain of the exterior schwarzschild solution . in the region @xmath9 , as in the latter solution , the @xmath10 and @xmath11 metric coefficients swap signs . defining @xmath12 and @xmath13 , we obtain @xmath14 with the following parametric definitions @xmath15 , @xmath16 and @xmath17 , which is a particular case of a bianchi iii axisymetric universe . using pseudo - spherical coordinates @xmath18 , the spatial part of the metric ( [ metric_constnc ] ) can be related to the hyperboloid @xmath19 embedded in a 4-dimensional flat space . we then have @xmath20\,{\rm d}r^2 + r^2 \left ( { \rm d}u^2+\sinh^2u\ , { \rm d}v^2 \right ) \ ; .\ ] ] where the prime stands for differentiation with respect to @xmath21 , and @xmath22 . we can recast metric ( [ metric_constnc ] ) into the following @xmath23 \,{\rm d}\tau^2 + \left(\frac{2\mu}{\bar r}\right)^2\ , \cos ^4\left[\ln\left ( \bar r\right)^{\mp 1}\right]\ , \left[{\rm d}\bar{r}^2\,+ \bar r^2\,({\rm d}u^2 + \sinh^2{u}\,{\rm d}v^2 ) \right]\ ; , \label{isotrop3}\end{aligned}\ ] ] which is the analogue of the isotropic form of the schwarzschild solution . the spatial surfaces are conformally flat , but the flat metric is not euclidean . indeed , the 3dim spatial metric @xmath24 is foliated by 2-dimensional surfaces of negative curvature , since @xmath25 , and it corresponds to @xmath26 . we thus can not recover the usual newtonian weak - field limit . analysing the `` radial '' motion of test particles , we have the following equation @xmath27 where @xmath28 and @xmath29 are constants of motion defined by @xmath30 and @xmath31 , for fixed @xmath32 . the former and latter constants represent the energy and angular momentum per unit mass , respectively . we thus define the potential @xmath33 this potential is manifestly repulsive , crosses the @xmath21-axis at @xmath6 , and for sufficiently high values of @xmath29 it has a minimum at @xmath34 . however this minimum falls outside the @xmath6 divide . so a test particle is subject to a repulsive potential forcing it to inevitably cross the event horizon at @xmath6 attracted either by some mass at the minimum or by masses at infinity . in it is hinted that the non - existence of a clear newtonian analogue is related to the existence of mass sources at @xmath8 , but no definite conclusions were drawn . a natural extension of the solution ( [ metric_constnc - antiscwarz ] ) would be to add exotic matter to obtain static and pseudo - spherically symmetric traversable wormhole solutions @xcite . consider the metric ( [ metric_constnc ] ) given by @xmath35 and @xmath36 $ ] . the coordinate @xmath21 decreases from a constant value @xmath4 to a minimum value @xmath37 , representing the location of the throat of the wormhole , where @xmath38 , and then it increases from @xmath37 back to the value @xmath4 . the condition @xmath39 imposes that @xmath40 , contrary to the morris - thorne counterpart @xcite . the solution provides the following stress - energy scenario @xmath41 \label{radialpwh}\,,\\ p_t(r)&=&\frac{1}{8\pi } \left(\frac{b}{r}-1\right)\bigg[\phi ' ' + ( \phi')^2 + \frac{b'r+b - r}{2r(b - r)}\phi ' + \frac{b'r - b}{2r^2(b - r ) } \bigg ] \label{lateralpwh}\,,\end{aligned}\ ] ] in which @xmath42 is the energy density , @xmath43 is the radial pressure , @xmath44 is the pressure measured in the tangential directions . note that the radial pressure is always positive at the throat , i.e , @xmath45 , contrary to the morris - thorne wormhole , where a radial tension at the throat is needed to sustain the wormhole . in addition to this , the mathematics of embedding imposes that @xmath46 at the throat , which implies a negative energy density at the throat ( see @xcite for more details ) . this condition is another significant difference to the morris - thorne wormhole , where the existence of negative energy densities at the throat is not a necessary condition . several interesting equations of state were considered in @xcite , and we refer the reader to this work for more details . a theorem by buchdahl @xcite establishes the reciprocity between any static solution of einstein s vacuum field equations and a one - parameter family of solutions of einstein s equations with a ( massless ) scalar field . in the conformally transformed einstein frame , note that scalar - tensor gravity theories are described by @xmath47 + \left ( 16\pi g_n/\phi^{-2}(\varphi)\right)\ , l_{matter}\right ] \label{e_st_action } $ ] . in this representation we have gr plus a massles scalar field which is now coupled to the matter fields . different scalar - tensor theories correspond to different couplings . in the absence of matter we can use buchdahl s theorem . so , given the metric ( [ metric_constnc ] ) , we derive the corresponding scalar - tensor solution @xmath48 where @xmath49 and @xmath50 . this clearly reduces to our anti - schwarzschild metric ( [ metric_constnc ] ) in the gr limit when @xmath51 , and hence @xmath52 implying that @xmath53 is constant . reverting @xmath54 , and the conformal transformation , @xmath55 , we can recast this solution in the original frame in which the scalar - field is coupled to the geometry and the content is vacuum , the so - called jordan frame . the @xmath6 limit is no longer just a coordinate singularity , but rather a true singularity as it can be verified from the analysis of the curvature invariants . in the einstein frame this occurs because the energy density of the scalar field diverges likewise in the schwarzschild case@xcite . of paramount importance is that , once again , the st - solution has no newtonian limit ( as its gr limit does not have one ) . this implies that the usual parametrized post - newtonian formalism that assesses the departures of modified gravity theories from gr does not hold for this class of metrics ( see @xcite ) . we have outlined the exotic features of the vacuum static solution with a pseudo - spherical foliation of space . we have revealed the existence of generalised wormholes , and derived its extension to scalar - tensor gravity theories . a fundamental feature of these solutions is the absence of a newtonian weak field limit , which reminds us of a quotation from john barrow @xcite _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the miracle of general relativity is that a purely mathematical assembly of second - rank tensors should have anything to do with newtonian gravity in any limit_. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the authors are grateful to raul vera and guillermo a. gonzlez for helpful discussions . jpm also acknowledges the loc members , ruth , raul , jesus and jos for a very enjoyable conference . 99 w. b. bonnor and m- a. p. martins , classical and quantum gravity , 8 , 727 ( 1991 ) ; m. a. p. martins , gen . ( 1996 ) 1309 . l. a. anchordoqui , j. d. edelstein , c. nunez and g. s. birman , arxiv : gr - qc/9509018 . f. s. n. lobo and j. p. mimoso , arxiv:0907.3811 [ gr - qc ] . m. s. morris and k. s. thorne , am . j. phys . * 56 * , 395 ( 1988 ) . h. a. buchdahl , phys . 115 , 1325 ( 1959 ) .

we investigate a static solution with an hyperbolic nature , characterised by a pseudo - spherical foliation of space . this space - time metric can be perceived as an anti - schwarzschild solution , and exhibits repulsive features . it belongs to the class of static vacuum solutions termed `` a degenerate static solution of class a '' ( see [ 1 ] ) . in the present work we review its fundamental features , discuss the existence of generalised wormholes , and derive its extension to scalar - tensor gravity theories in general .

the powerful effective - field theory ( eft ) method allows for a perturbative treatment of the strong interaction at low energies @xcite . the perturbative expansion is made in powers of energy and momentum of the lightest relevant degrees of freedom ( pions , nucleons , etc . ) , rather than in powers of the coupling constant as is usually done in ( renormalizable ) field theory . to illustrate the idea , imagine we would like to describe the properties of cosmic microwave background ( cmb ) , i.e. , a gas of cold photons . the proper theory for this system is of course qed , and the interaction of photons arises there through the electron loops . the typical energy of a cmb photon is 1 mev , which is much smaller ( by six orders of magnitude ! ) than the electron mass @xmath0 mev . a spontaneous pair production is therefore excluded ; the fermion degrees of freedom are too heavy and will never appear explicitly in this system . it is possible to eliminate them from the description altogether by integrating out the fermion fields and expanding the result in powers of inverse electron mass . another way of doing that is to write an `` effective theory '' containing only the relevant degrees of freedom , _ viz . _ , the photon field @xmath1 . the most general lagrangian of such theory would begin with = -f_f^ + a_1 _ f _ ^f^ + c_1 ( f_f^)^2 + c_2 ( f_f^)^2 + , where @xmath2 is the field strength and @xmath3 is its dual . the first term in this lagrangian is just the free electromagnetic ( e.m . ) radiation and the rest represents the effect of electron loops at low energies . even if this effect is not known to us precisely , we know that it will satisfy all the symmetries of qed , such as the lorentz- and discrete symmetries , as well as the electromagnetic gauge invariance . these symmetries allow the effective theory to be written in terms of @xmath4 only , and as the result , the lagrangian can be ordered in powers of derivatives of the photon field . a derivative translates into the momentum , or energy , and hence the derivative expansion translates into the expansion in energy . the parameters of the effective theory , in this case @xmath5 s and @xmath6 s , must be expressed in terms of the qed parameters electron mass and charge . this can be achieved by matching " : calculating the same quantity in the effective theory and in qed , and equating the results . for example , @xmath7 can be determined from the vacuum polarization , while @xmath8 and @xmath9 can be matched at the level of light - by - light scattering amplitudes . the effective framework is especially useful when the underlying theory is non - perturbative , as it is in the case of qcd at low energies . in our example , the hadronic effects in the photon gas can still be presented in the form of . it is still hard to calculate from first principles what the hadronic contribution to the low - energy constants is , but we can measure it ! for example , the constants @xmath8 and @xmath9 , describing the low - energy photon self - interactions , can be related to linearly - polarized cross - sections of photon - photon fusion , @xmath10 and @xmath11 @xcite : c_1 = _ 0^ s , c_2 = _ 0^ s , where @xmath12 is the total invariant energy of the @xmath13 collision . from this we know already that these constants are positive definite and so the low - energy photons attract . taking the cross - sections of @xmath14hadrons , we can obtain the hadronic contribution to these quantities , and with that we can calculate the most important hadronic effects in the photon gas . i have to note though that the polarized @xmath13 fusion cross - sections have not yet been measured and so this example for now is [ line - through]*academic * served for illustration purpose only . if we replace photons with pions in the above example , and electrons with quarks and gluons , we must end up with the chiral eft , commonly referred to as chiral perturbation theory ( @xmath15pt ) . the name @xmath15pt was originally assigned to the expansion of static quantities , such as masses , electromagnetic moments , in powers of the pion mass @xcite . in modern language , @xmath15pt is a simultaneous expansion in powers of pion mass and momentum the chiral expansion . the break - down scale of this expansion is believed to be set at around 1 gev . introducing the nucleon into the picture , in the words of first paper attempting it @xcite , complicates life a lot . " first of all , the nucleon is heavy , and its mass seems to pop out in the places it should nt , violating some power - counting arguments ; we will see one example in a moment . secondly , the nucleon is easily excited into the @xmath16 , the excitation energy being @xmath17 mev . in attempts to find a systematic treatment of these issues , several different versions of baryon @xmath15pt were born : * heavy - baryon @xmath15pt ( hb@xmath15pt ) @xcite , where in addition to the chiral expansion , a semi - relativistic expansion in the inverse nucleon mass is made . * infrared regularization ( ir - b@xmath15pt ) @xcite , where the negative - energy pole is removed from the baryon propagators . * extended on - mass - shell scheme ( eoms - b@xmath15pt ) @xcite , which recognizes that certain renormalizations must be done to implement consistent power counting . the chiral expansion of the nucleon mass to order @xmath18 is given by m_n = - 4 m_^2 + ^(3)_n , where @xmath19 and @xmath20 are low - energy constants ( lecs ) from the chiral effective lagrangian , and @xmath21 is the nucleon self - energy . the nucleon self - energy may have ( infinitely ) many chiral - loop contributions of the type shown in , however the weinberg s power counting tells us that a graph with @xmath22 loops , @xmath23 pion and @xmath24 nucleon lines , @xmath25 vertices from the lagrangian of order @xmath26 , contributes to order @xmath27 , with n = 4l - 2n_-n_n + _ k k v_k . the leading @xmath28 couplings ( pseudovector , weinberg tomozawa , etc . ) are of the first order ( @xmath29 ) and therefore , to order @xmath18 only the graph ( a ) contributes . evaluating this graph yields @xcite : ^(3)_n & = & \ { - l_m_n^3 + ( 1-l _ ) m_n m_^2 + & & - m_^3 - } . where @xmath30 exhibits the uv divergence as @xmath31 , with @xmath32 being the number of space - time dimensions , @xmath33 the scale of dimensional regularization , and @xmath34 the euler s constant . for simplicity we have assumed the physical values for the parameters entering the loop : @xmath35 mev , @xmath36 , @xmath37 mev ; the difference with the chiral - limit values leads to higher - order effects . the expression is an exact result of a textbook calculation and so it is disappointing to see that it seems to invalidate the power counting formula . the power counting estimates this loop is of @xmath18 size , in this case @xmath38 , but the first two terms are obviously larger . on the other hand , this expression begs for a renormalization . we do have the two low - energy constants in , which can be renormalized to absorb the uv divergencies and remove the dependence of the nucleon mass on the renormalization scale . however , only having @xmath39 in , which corresponds to the @xmath40 scheme applied in the original paper @xcite , does not work : there is the @xmath41 term remaining , which violates the power counting . the essence of the eoms @xcite is to absorb this term too in the course of the renormalization . as the result we have , & & : m_n = - 4 m_^2 - \ { + } , + & & : m_n = - 4 m_^2 - \ { ( - ) + + & & + ( l_-1 ) } , + & & : m_n = - 4 m_^2 - , where @xmath19 and @xmath20 are now the renormalized ( physical ) values of these parameters , and where , for comparison , i displayed also the @xmath18 result of the ir and hb versions of b@xmath15pt . the eoms result is consistent with power counting and the @xmath40 is not , but can one renormalization scheme be better than the other ? in eft it can . observe that the whole difference between the two schemes at this order is expressed as the difference in the value of @xmath8 : ^ ( ) - = ^ ( ) suppose now we match @xmath20 to qcd by fitting , for instance , to the lattice qcd data . in the two schemes the fit will be identical but the values for the lec will differ according to . it is still possible that both @xmath8 are of _ natural _ size , i.e. , of oder of one in gev units . now , after calculating to one order higher , we would refit the data and find a new value of the lec . in eoms the value of @xmath8 would not change much from one order to another , since @xmath42 as a function of @xmath43 would indeed change only in the higher - order terms . in the @xmath40 scheme the value of @xmath8 could change a lot , since any loop can produce a large shift in that @xmath44 term . the latter situation is not satisfactory , especially if we want to have our lecs to represent some physical quantities . for example , we would like @xmath19 to be the nucleon mass in the chiral limit , while @xmath20 could represent the value of the @xmath45-term in the chiral limit . in this case the eoms is indeed favored over the @xmath40 . looking back at the other results displayed in , we ought to dismiss the ir scheme for not having the correct analytic properties . the non - analyticity of the square root at @xmath46 is canceled in the eoms due to the factor of @xmath47 , and is not canceled in the ir , because @xmath48 . of course , at small pion masses this pathology is barely seen , as the ir result is different from the eoms by the following term alone : m_n = - \ { + ( l_-1 ) } . one can argue that this term is of order @xmath49 , which is beyond the accuracy of this calculations . but then the question is why do we need this term at all , especially when it occurs as a result of violation of the analytic properties . we finally come to the point that the eoms expression in contains an infinite amount of terms which are nominally of higher order in @xmath50 , while in the hb expression these terms are happily dropped . i deliberately used the word nominally " , because a term going as @xmath51 is not necessarily smaller then a @xmath38 term depends on the coefficients . one assumes the coefficients to be all of natural size , but is this always true ? we can see this to be pretty much true for our case . expanding the factor in curly brackets in the eoms expression of , we find \ { } = + ( -1 ) - + o(m_^3 ) , and hence the hb result , @xmath52 , is a very good _ but then there are other examples . the magnetic polarizability of the proton @xmath53 at order @xmath18 in b@xmath15pt expands as @xcite : _ p = \ { + + - + o(m_^2 ) } . one can see that the nominally - higher - order terms have unnaturally large coefficients and can not be neglected . the same situation is observed in the other polarizabilities of the nucleon . in such cases the hb expansion fails . a comprehensive comparison of the various schemes has recently been done by geng et al . @xcite in the context of the su(3 ) b@xmath15pt study of the baryon magnetic moments . the eoms comes out to be favored by this study as well . as for the other recent applications of b@xmath15pt in the on - mass - shell scheme , i would like to mention a next - to - next - to - leading order calculation of the proton compton scattering @xcite , and of the nucleon and @xmath16-isobar electromagnetic form factors @xcite . 9 s. weinberg , physica a * 96 * , 327 ( 1979 ) . j. gasser and h. leutwyler , annals phys . * 158 * ( 1984 ) 142 . v. pascalutsa and m. vanderhaeghen , phys . lett . * 105 * , 201603 ( 2010 ) [ arxiv:1008.1088 ] . h. pagels , phys . * 16 * , 219 ( 1975 ) . j. gasser , m. e. sainio and a. svarc , nucl . b * 307 * , 779 ( 1988 ) . e. jenkins and a. v. manohar , phys . b * 255 * , 558 ( 1991 ) . t. becher and h. leutwyler , eur . j. c * 9 * , 643 ( 1999 ) [ arxiv : hep - ph/9901384 ] . t. fuchs , j. gegelia , g. japaridze and s. scherer , phys . d * 68 * , 056005 ( 2003 ) . t. ledwig , v. pascalutsa and m. vanderhaeghen , phys . b * 690 * , 129 ( 2010 ) . v. bernard , n. kaiser and u .- g . meiner , phys . lett . * 67 * , 1515 ( 1991 ) . v. lensky , v. pascalutsa , eur . j. c * 65 * , 195 ( 2010 ) . l. s. geng , j. martin camalich , m. j. vicente vacas , phys . * b676 * , 63 - 68 ( 2009 ) . l. s. geng , j. martin camalich , m. j. vicente vacas , phys . rev . * d80 * , 034027 ( 2009 ) . t. ledwig , j. martin - camalich , v. pascalutsa , m. vanderhaeghen , arxiv:1105.0468 ; in preparation .

the issue of consistent power counting in baryon chiral perturbation theory is revisited . address = institut fr kernphysik , johannes gutenberg universitt , mainz d-55099 , germany

over the last few years , there has been a tremendous increase in the study of galaxy clusters as cosmological probes , initially through the use of x - ray emission observations , and in recent years , through the use of sunyaev - zeldovich ( sz ) effect . briefly , the sz effect is a distortion of the cosmic microwave background ( cmb ) radiation by inverse - compton scattering of thermal electrons within the hot intracluster medium ( sunyaev & zeldovich 1980 ; see birkinshaw 1998 for a recent review ) . by combining the sz intensity change and the x - ray emission observations , the angular diameter distance , @xmath2 , to a cluster can be derived ( e.g. , cavaliere _ et al . _ 1977 ) . combining the distance measurement with redshift allows a determination of the hubble constant , h@xmath3 . on the other hand , angular diameter distances with redshift can be used to constrain cosmological world models . the accuracy of the hubble constant determined from a sz and x - ray analysis depends on the assumptions . using numerical simulations , inagaki et al . ( 1995 ) and roettiger _ ( 1997 ) showed that the hubble constant measured through the sz effect can seriously be affected by systematic effects , which include the assumption of isothermality , cluster gas clumping , and asphericity . the hubble constant can also be affected by statistical effects , including cluster peculiar velocities and astrophysical confusions , such as radio sources & cmb primary anisotropies . the latter statistical effects are expected to produce a broad distribution in the hubble constant measured for a sample of galaxy clusters , while the former systematic effects are expected to offset the hubble constant from the true value . in recent years , several other effects have also been suggested to explain the difference between the sz and x - ray hubble constant and the ones derived from other techniques . these include the preferential removal of the lensed background radio sources in sz surveys ( loeb & refregier 1997 ) , which would systematically lower the hubble constant by as much as 13% for sz observations at 15 ghz , and gravitational lensing of the arcminute scale cmb anisotropy ( cen 1998 ) , which would broaden the hubble constant distribution for a sample of galaxy clusters . the first effect is in opposite direction to the radio source contamination in sz observations due to galaxy cluster member radio sources , which dominate the radio source number counts towards galaxy clusters . as discussed in cooray _ ( 1998a ) , the two radio source effects are likely to cancel out . the loeb & refregier ( 1997 ) effect is also not expected to occur for sz observations at high frequencies . the second effect , due to gravitational lensing of cmb anisotropy through galaxy cluster potential , is not expected to be a dominant source of error in the hubble constant , given that cen ( 1998 ) considered the largest upper limits to arcminute scale anisotropies , which have not yet been detected . apart from the sz and x - ray hubble constant , the gas mass fraction , @xmath4 , measurements from x - ray ( also using sz , gravitational lensing and optical velocity dispersion measurements ) , can also be used to constrain the cosmological parameters . the primary assumption in such an analysis is that the gas mass fraction , when measured out to a standard ( hydrostatic ) radius is constant . evrard ( 1997 ) applied these arguments to a sample of galaxy clusters using x - ray data , and put constraints on the cosmological mass density of the universe , @xmath5 , with some dependence on the hubble constant . under the assumption that the cluster gas mass fraction is constant in a sample of galaxy clusters , the apparent redshift evolution of the baryonic fraction can also be used to constrain the cosmological parameters ( e.g. , pen 1997 ) . cooray ( 1998 ) and danos & pen ( 1998 ) used the present x - ray gas mass fraction data to derive @xmath6 in a flat universe ( @xmath7 ) and @xmath8 in an open universe ( @xmath9 ; 90% c.i . ) . in shimasaku ( 1997 ) , the assumption of constant gas mass fraction in galaxy clusters was used to put constrains on @xmath10 , the rms linear fluctuations on scales of 8 h@xmath0 mpc , and on @xmath11 , the slope of the fluctuation spectrum . given the importance of sz and x - ray emission observations in cosmological studies , we initiated a program to study the systematic effects in the present sz and x - ray hubble constant measurements and gas mass fraction measurements . as part of this study , we found a negative correlation between the broad distribution of the hubble constant and the gas mass fraction measurements . we explain this observation as due to a projection effect of aspherical clusters modeled with a spherical geometry . in section 2 , we present the effects of projection on the hubble constant and the gas mass fraction by projecting triaxial ellipsoidal clusters and extending the work of fabricant et al . the observational evidence for projection effects in the present hubble constant values based on sz and x - ray route are presented in section 3 . in section 4 , we outline an alternative method to calculate the hubble constant , by combining sz , x - ray , gravitational lensing , and velocity dispersion measurements of clusters , and which is subjected to less projection effects than current method involving only the sz and x - ray observations . we apply this technique to a2163 based on the published observational data , and derive a new hubble constant . a summary and conclusions are presented in section 5 . in order to study the effect of aspherical clusters in present sz and x - ray hubble constant , we extend the work of fabricant et al . ( 1984 ) to calculate the x - ray surface brightness and the sz temperature change produced by clusters with ellipsoidal geometries . independent of the cluster shape , the x - ray surface brightness towards a clusters is given by : @xmath12 where @xmath13 . in order to model the electron number density profile within clusters , we consider the @xmath14-model , which can be written as : @xmath15^{-\frac{3 \beta}{2}},\ ] ] where @xmath16 and @xmath17 are coordinates of the ellipsoid axes , while @xmath18 and @xmath19 are the observed semi - major and semi - minor axes . to simplify the calculations , we assume that the symmetry axis @xmath17 is at an inclination angle @xmath20 to the line of sight along the observer , which we take to be the @xmath21-axis . following fabricant et al . ( 1984 , appendix a ) , we integrate along the z - axis to derive : @xmath22^{\frac{1}{2}-3 \beta}.\end{aligned}\ ] ] the other important observable towards clusters is the sz effect , which is given by : @xmath23 where @xmath24\ ] ] is the frequency dependence with @xmath25 , @xmath26 ( fixsen _ et al . _ 1994 ) and @xmath27 is the cross section for thomson scattering . the integral is performed along the line of sight through the cluster . as with the x - ray surface brightness , we consider the same ellipsoidal shape to evaluate the observed sz temperature change . again by integrating along the line of sight , @xmath21-axis , we derive : @xmath28^{\frac{1}{2}-\frac{3 \beta}{2}}. \end{aligned}\ ] ] the hubble constant is usually derived by combining the x - ray brightness and the sz temperature change to eliminate the central number density @xmath29 . by this combination , one can derive the observed length of one of the axis , e.g. : @xmath30 \nonumber \\ & \times z,\end{aligned}\ ] ] where @xmath31 is the scale factor first introduced in birkinshaw et al . ( 1991 ) , which can now be written as : @xmath32 when the symmetry axis of the cluster is along the line of sight ( @xmath33 ) , then @xmath34 which is directly related to the observed cluster ellipticity , while when the cluster is spherical ( @xmath35 ) , @xmath36 , and no effects due to projection is present in the data . in eq . 7 , we know from sz and x - ray observations all the quantities except the scale factor z. therefore , the length of the cluster along the line of sight can be known up to a multiplicative factor . the hubble constant is derived based on the angular diameter distance to the cluster , @xmath37 , using an assumed cosmological model , and the observed size of the axis , @xmath38 , used to calculate the distance in eq . the derived hubble constant can be written as : @xmath39 based on observed ellipticities of galaxy clusters , we can estimate the expected error in the hubble constant . using x - ray emission from a sample of clusters , mohr et al . ( 1995 ) showed that the median ellipticity is @xmath40 0.25 . this suggest that the ratio @xmath41 is @xmath40 0.7 if clusters are intrinsically prolate or @xmath40 1.5 if clusters oblate . therefore , ignoring the effects due to inclination , the hubble constant as measured from sz and x - ray observations of an individual cluster can be offseted as much as 30% to 50% , based on a spherical model of clusters where asphericity is ignored . here , we have assumed that clusters are ellipsoids . the derived scale factor in eq . 8 , as well as the numerical values , are likely to be different if clusters are biaxial or triaxial . recently , zaroubi et al . ( 1998 ) studied the projection effects of biaxial clusters and determined @xmath42 , where @xmath20 is the inclination angle . the observational evidence which suggest clusters are biaxial is limited . for ellipsoidal clusters , we have determined that @xmath43 varies with both the inclination angle and the sizes of semi - major and semi - minor axes . for triaxial clusters , it is likely that @xmath43 will vary with all three rotation angles and the length scales of the three axes that define the cluster . in a future paper , we plan to study the projection effects of triaxial clusters ; for the purpose of this paper , we will only consider ellipsoids . apart from the hubble constant , the projection effects are also present in the total gas mass derived from the x - ray emission observations with @xmath44 , and the total mass based on the virial theorem using x - ray temperature as @xmath45 . then , the gas mass fraction can be written as @xmath46 . since @xmath47 and the @xmath48 , we expect the @xmath49 and @xmath4 to exhibit a negative correlation , if both measurements are affected by the projection effect . = ? llclc cluster & redshift & @xmath49 ( km s@xmath0 mpc@xmath0 ) & @xmath49 reference & @xmath4 ( @xmath50 ) ( h@xmath51 ) + a2256 & 0.0581 & 68@xmath52 & myers _ et al . _ 1997 & @xmath53 + a478 & 0.0881 & 30@xmath54 & myers _ et al . _ 1997 & @xmath55 + a2142 & 0.0899 & 46@xmath56 & myers _ et al . _ 1997 & 0.21@xmath57 + a1413 & 0.143 & 44@xmath58 & saunders 1996 & 0.12@xmath59 + a2218 & 0.171 & 59 @xmath1 23 & birkinshaw & hughes 1994 & 0.16@xmath60 + a2218 & 0.171 & 34@xmath61 & jones 1995 & 0.16@xmath60 + a665 & 0.182 & 46 @xmath1 16 & hughes & birkinshaw 1998b & 0.14@xmath60 + a665 & 0.182 & 48@xmath62 & cooray _ _ 1998c & 0.14@xmath60 + a2163 & 0.201 & 58@xmath63 & holzapfel _ et al . _ 1997 & 0.15@xmath59 + cl0016 + 16 & 0.5455 & 47@xmath64 & hughes & birkinshaw 1998a & 0.17@xmath65 + table 1 lists the hubble constant values that have so far been obtained from sz observations ( cooray _ et al . _ 1998b , see also hughes 1997 ) . these values have been calculated under the assumption of a spherical gas distribution with a @xmath14 profile for the electron number density and an isothermal atmosphere . for the same clusters , we compiled a list of gas mass fraction measurements using x - ray , sz , and gravitational lensing observations . most of the clusters in table 1 have been analyzed by allen & fabian ( 1998 ) , where they included cooling flow corrections to the x - ray luminosity and the gas temperature . for the two clusters ( a2256 & cl0016 + 16 ) for which @xmath49 measurements are available , but not analyzed in allen & fabian ( 1998 ) , we used the results from buote & canizares ( 1996 ) and neumann & bhringer ( 1996 ) , respectively . the gas mass fractions in allen & fabian ( 1998 ) have been calculated to a radius of 500 kpc , while for the a2256 and cl0016 + 16 , they have been calculated to different radii , and also under different cosmological models . using the angular diameter distance dependence on the gas mass fraction measurements with redshift ( cooray 1998 ) , we converted all the gas mass fraction measurements to a cosmology of @xmath66 , @xmath9 , and h@xmath67 @xmath68 km s@xmath0 mpc@xmath0 . in order to facilitate comparison between the gas mass fractions measured at various radii , we scaled them to the @xmath50 radius based on relations presented by evrard ( 1997 ) . the @xmath50 radius has been shown to be a good approximation to the outer hydrostatic boundary of galaxy clusters ( evrard , metzler , navarro 1996 ) . we list the derived cluster gas mass fraction at the @xmath50 radius in table 1 . in fig . 1 , we show the calculated @xmath4 against @xmath49 values for each of the clusters . as shown , the gas fraction measurements have a broad distribution with a scatter of @xmath40 40% from the mean value . a similar broadening of the hubble constant , from 30 to 70 km s@xmath0 mpc@xmath0 with a mean of @xmath40 50 km s@xmath0 mpc@xmath0 is observed . the correlation is negative , and suggest that clusters with high gas mass fraction measurements produces hubble constant values at the low end of the distribution , while the opposite is seen for clusters with high gas mass fraction . the solid line in fig . 1 is the best - fit relation between @xmath43 and @xmath4 assuming @xmath69 . for values in table 1 , the best - fit line , when the slope between @xmath43 and @xmath4 is allowed to vary , scales as @xmath70 , which is fully consistent with the expected relation . since the current sz cluster sample is small , a careful study of a complete sample of galaxy clusters are need to fully justify the projection effects between sz and x - ray derived hubble constant and gas mass fractions values . we derived a similar negative correlation between @xmath43 and @xmath4 when the cluster gas mass fraction is measured from sz . for example , myers _ ( 1997 ) derived a gas mass fraction of ( @xmath71 ) @xmath68 for a2256 , which is at the low end of the gas mass fraction values , while a gas mass fraction of ( @xmath72 ) @xmath68 was derived for a478 , which is the cluster at the high end . we note here that , as we discuss later , the sz derived gas mass fractions scale with @xmath43 as only @xmath73 , while x - ray derived gas mass fractions , which are presented in table 1 , scale with @xmath43 as @xmath74 . in comparison , the gas mass fractions derived from sz and x - ray observations may be affected similar to the measurements based on only the x - ray data . additional probes of the total mass are the gravitational lensing measurements and the optical virial analysis of internal galaxy velocity dispersion measurements . in the present sz / x - ray sample , a2218 ( kneib _ et al . _ 1995 ) and a2163 ( squires _ et al . _ 1997 ) have lensing mass measurements . in both these clusters total virial masses when measured using x - ray gas temperature , agrees with the weak lensing mass measurements at large radii , and since these two clusters are not the ones which are primarily responsible for the observed negative correlation , we can not state the effect of lensing mass measurements on the above data . also , in the present sz cluster sample , a2256 and a2142 ( girardi _ et al . _ 1998 ) , and cl0016 + 16 ( carlberg _ et al . _ 1997 ) have measured total masses from optical virial analysis . these virial masses are in good agreement with x - ray masses , allowing an independent robust measurement of the total mass ( girardi _ et al . _ 1998 ) . finally , there is a slight possibility that the observed broad distribution and negative correlation in @xmath49 and @xmath4 is not really present . the negative correlation is only present at a level of @xmath40 2 @xmath75 , assuming that the errors in @xmath43 and @xmath4 are independent . the removal of either one of the clusters at high or low end reduces the negative correlation , decreasing the significance of the observed correlation . however , both the hubble constant and , possibly , the gas mass fraction is expected to be constant , suggesting that a point , or a region when considering errors in @xmath49 and @xmath4 , is preferred . we rule out the possibility that both @xmath49 and @xmath4 are constants in the present data with a confidence greater than 95% . usually , the broad distribution of the sz and x - ray hubble constants has been explained in literature based on the expected systematic effects . the systematic effects in the gas mass fraction measurements are reviewed in evrard ( 1997 ) and cooray ( 1998 ) . we briefly discuss these systematic uncertainties in the context of their combined effects on @xmath49 and @xmath4 . it has been suggested that cluster gas clumping may overestimate @xmath49 from the true value . as reviewed in evrard ( 1997 ) , cluster gas clumping also overestimates @xmath4 , suggesting that if gas clumping is responsible for the observed trend , a positive correlation should be present . the nonisothermality underestimates @xmath49 by as much as 25% ( e.g. roettiger _ et al . to explain the distribution of @xmath49 values , the cluster temperature profile from one cluster to another is expected to be different . however , markevitch _ ( 1997 ) showed the similarity between temperature profiles of 30 clusters based on asca data ( including a478 , a2142 & a2256 in present sample ) . since sz and x - ray structural fits weigh the gas distribution differently , even a similar temperature profile between clusters can be expected to cause the change in the hubble constant from one cluster to another . another result from the markevitch _ ( 1997 ) study is that the @xmath4 measurements as measured using @xmath14-models and standard isothermal assumption is underestimated . the similarity of cluster temperature profiles also suggests that the gas mass fractions are affected by changes in temperature from one cluster to another . it is likely that the present isothermal assumption has underestimated both @xmath49 and @xmath4 , and that temperature profiles are responsible for the observed behavior . a large sample of clusters , perhaps the same cluster sample studied by markevitch _ et al . ( 1997 ) , should be studied in sz to determine the exact effect of radial temperature profiles on @xmath49 , and its distribution . the third possibility is the cluster asphericity . the effect of cluster projection on @xmath49 was first suggested by birkinshaw _ ( 1991 ) , who showed that the derived values for @xmath49 can be offset by as much as a factor of 2 if the line of sight along the cluster is different by the same amount . the present cluster isophotal ellipticities suggest that @xmath49 may be offset as much as @xmath1 27% ( e.g. , holzapfel _ et al . the present @xmath4 distribution is suggestive of this behavior . cen ( 1997 ) , using numerical simulations , studied the effects of cluster projection on gas mass fraction measurements , and suggested differences of the order @xmath40 40% . the @xmath4 distribution is similar to what has been seen in cen ( 1997 ) . it is more likely that the projection effects are causing the distribution of @xmath49 and @xmath4 values , unless a systematic effect still not seen in numerical simulations is physically present in galaxy clusters . such effects could come from effects due to variations in the temperature profiles from one cluster to another . for the rest of the discussion , we assume that the present values are affected by projection effects , rather than temperature profiles . here , we consider the possibility of deriving the hubble constant in a meaningful manner without any biases due to cluster projections . it has been suggested in literature that observations of a large sample of galaxy clusters can be used to average out the dependence on the scale factor @xmath31 and to produce the true value of the hubble constant , which we define as @xmath76 from individual hubble constant measurements , @xmath77 , in a large sample of clusters . we investigate the possibility of such an averaging by considering the different projections of clusters at different inclination angles . assuming the previously described ellipsoidal shape and the effect of the scale factor @xmath31 in the hubble constant , we can over the all possible inclination angles @xmath20 and the ratio @xmath41 to derive the expected average value of the hubble constant @xmath78 : @xmath79 \times h_0^{\rm true},\ ] ] if all clusters are prolate , and @xmath80 \times h_0^{\rm true},\ ] ] if all clusters are oblate . here when all clusters are prolate and that the semi - major axis used to calculate the hubble constant , then the distribution has a mean of @xmath76 . however , if the semi - minor axis is used , then the average hubble constant is underestimated from the true value by about @xmath40 10% , assuming that the mean @xmath41 is 0.7 for prolate clusters . if all clusters are oblate , and the semi - major axis is used to derive the hubble constant , then the mean of the distribution overestimates the true value of the hubble constant by as much as @xmath40 20% , if the mean @xmath41 is 1.5 for oblate clusters . for oblate clusters , the true value of the hubble constant can be obtained when the semi - minor axis is used . however , in both oblate and prolate cases , the distribution has a large scatter requiring a large sample of galaxy clusters to derive a reliable value of the hubble constant . a similar calculation can also be performed for the gas mass fraction to estimate the nature of the value derived by averaging out a gas mass fraction measurements for a large sample of clusters . here again , a similar offset as in the hubble constant is present , and measurements of gas mass fraction in a large sample of clusters are needed to put reliable limits on the cosmological parameters , especially the mass density of the universe based on cosmological baryon density ( e.g. , evrard 1997 ) . so far , we have only considered the sz and x - ray observations of galaxy clusters . by combining weak lensing observations towards galaxy clusters , we show that it may be possible to derive a reliable value of the hubble constant based on observations of a single cluster . the gravitational lensing observations of galaxy clusters measure the total mass along the line of sight through the cluster . the sz effect measures the gas mass along the line of sight , and thus , the ratio of sz gas mass to gravitational lensing total mass should yield a measurement of the gas mass fraction independent of cluster shape assumptions and asphericity . here , we assume that the cluster gas distribution exactly traces the cluster gravitational potential due to dark matter , and that these two measurements are affected equally by cluster shape . this is a reasonable assumption , but however , it is likely that gas distribution does not follow the dark matter potential , and that there may be some dependence on the cluster shape between the two quantities . for now , assuming that the gas mass fraction from sz and gravitational lensing is not affected by cluster projection , we outline a method to estimate the hubble constant independent of the scale factor @xmath31 . the gas mass fraction based on sz and lensing is @xmath82 , while the gas mass fraction based on x - ray emission gas mass and the total mass based on x - ray temperature is @xmath83 . since the two gas mass fraction measurements are expected to be the same , then one can solve for a combination of @xmath43 and @xmath31 . however to break the degeneracy between @xmath43 and @xmath31 an additional observation or an assumption is needed . in general , there are large number of clusters with x - ray measurements and x - ray based gas mass fraction measurements . by averaging out the gas mass fraction for such a large sample of clusters , we can estimate the universal gas mass fraction value for clusters , e.g. @xmath84 ( evrard 1997 ; cooray 1998 ) . if assumed that this gas fraction is valid for the cluster for which sz and weak lensing observations are available , we can then calculate the hubble constant . we applied this to sz , x - ray and weak lensing observations of galaxy cluster a2163 . the sz observations of a2163 are presented in holzapfel et al . ( 1997 ) , while weak lensing and x - ray observations are presented in squires et al . the sz effect towards a2163 can be described with a @xmath85 ( @xmath86 ) parameter of @xmath87 , which includes various uncertainties described in holzapfel et al . the weak lensing observations of a2163 has been used to derive the total cluster mass in squires et al . ( 1996 ) , and the lensing observations are most sensitive out to a radius of @xmath40 200@xmath88 ( 0.423 h@xmath0 mpc ) from the cluster center , where the total mass is @xmath89 @xmath73 @xmath90 . using the cluster model ( @xmath14 and @xmath91 ) in holzapfel et al . ( 1997 ) , we integrated the sz temperature change to this radius from cluster center along the line of sight to derive a gas mass of @xmath92 @xmath93 @xmath90 . this represents the gas mass within the cylindrical cut across the cluster , and effectively probes the same region as the weak lensing observations . the gas mass fraction based on the sz effect and the weak lensing total mass is @xmath94 . when this gas mass is compared to the effective gas mass fraction of clusters , @xmath95 ( evrard 1997 ; cooray 1998 ) , we obtain @xmath96 . we have slightly overestimated the error in the average gas mass fraction to take into account the fact that this fraction is measured at the outer hydrostatic radius ( @xmath40 1 mpc ) , and may not correspond to the value at the observed radius of a2163 . in squires ( 1997 ) , the gas mass fraction was measured to be @xmath97 for a2163 , which is in agreement with our universal value , but the value in squires et al . ( 1997 ) may be subjected to a scaling factor . the combined sz / lensing gas mass fraction and the average gas mass fraction for clusters result in a hubble constant of @xmath98 km s@xmath0 mpc@xmath0 . given that we used data from 2 different papers in deriving this hubble constant , it is likely that this value may be subjected to unknown systematic effects between the two studies . we strongly recommend that a careful analysis of cluster data be carried out to derive the hubble constant based on sz , x - ray and weak lensing observations . in addition , total virial masses from velocity dispersion analysis should also be considered in such an analysis to constrain the cluster shape . it is likely that much stronger and reliable result may be obtained through this method , instead of just sz and x - ray observations . in holzapfel et al . ( 1997 ) , the hubble constant was derived to be @xmath40 60 km s@xmath0 mpc@xmath0 for an isothermal temperature model and @xmath40 78 km s@xmath0 mpc@xmath0 for a hybrid temperature model . our value is lower than these two values , but is in good agreement with the average value of @xmath49 based on sz and x - ray as tabulated in table 1 , which is in agreement with the average gas mass fraction value . using the hubble constant measurements based on sz and x - ray , and the gas mass fraction measurements , we have suggested a possible systematic effect due to cluster projection . even though , cluster projection had been suggested as a possible systematic bias in @xmath49 measurements , more attention has recently been given to various _ exotic _ effects as a way to explain the broad distribution of hubble constant values . we have shown here the presence of projection effects in the present @xmath49 and @xmath4 measurements and have analytically calculated the effect of cluster projection in deriving the hubble constant . it is also assumed in literature that for a large sample of clusters , the average of the individual hubble constants , after making various corrections , can be used to determine the true hubble constant . we have shown here that this may not be easily possible , and that when a random and large sample is available with a mix of prolate and oblate clusters , the best that one could expect to obtain is a hubble constant value within 10% of the true value , unless the distribution of ellipticities for cluster sample is carefully taken into account . thus , we strongly recommend that more attention be given to the cluster asphericity in deriving cosmologically important measurements such as hubble constant and the cluster gas mass fraction . for individual clusters , for which sz observations are available , we have shown that a combined study of sz , x - ray , velocity dispersion measurements and weak lensing observations can be used in a more physical manner to derive the hubble constant . thus , we have demonstrated the usefulness of gravitational lensing observations of galaxy clusters for cosmologically important studies , and when combined , more meaningful results are expected to be produced instead of just combining sz and x - ray observations . we strongly recommend that weak lensing observations and velocity dispersion measurements be carried out to test the reliability of hubble constant values in table 1 , and to complement sz observations of clusters . i would like to acknowledge useful discussions with john carlstrom and bill holzapfel . i would also like to thank the two referees , mark birkinshaw and an anonymous referee , for detailed comments on the manuscript . this study was partially supported by the mccormick fellowship at the university of chicago , and a grant - in - aid of research from the national academy of sciences , awarded through sigma xi , the scientific research society .

it is well known that a combined analysis of the sunyaev - zeldovich ( sz ) effect and the x - ray emission observations can be used to determine the angular diameter distance to galaxy clusters , from which the hubble constant is derived . the present values of the hubble constant derived through the sz / x - ray route have a broad distribution ranging from 30 to 70 km s@xmath0 mpc@xmath0 . we show that this broad distribution is primarily due to the projection effect of aspherical clusters which have been modeled using spherical geometries . the projection effect is also expected to broaden the measured gas mass fraction in galaxy clusters . however , the projection effect either under- or overestimate the hubble constant and the gas mass fraction in an opposite manner , producing an anticorrelation . using the published data for sz / x - ray clusters , we show that the current hubble constant distribution is negatively correlated with the measured gas mass fraction for same clusters , suggesting that the projection effects are present in current results . if the gas mass fraction of galaxy clusters , when measured out to an outer hydrostatic radius is constant , it may be possible to account for the line of sight geometry of galaxy clusters . however , to perform such an analysis , an independent measurement of the total mass of galaxy clusters , such as from weak lensing , is needed . using the weak lensing , optical velocity dispersion , sz and x - ray data , we outline an alternative method to calculate the hubble constant , which is subjected less to projection effect than the present method based on only the sz and x - ray data . for a2163 , the hubble constant based on published sz , x - ray and weak lensing observations is 49 @xmath1 29 km s@xmath0 mpc@xmath0 .

consider a locally compact group @xmath0 acting continuously on a locally compact second countable space @xmath8 and @xmath4 a probability measure on @xmath0 . the _ associated random walk on @xmath8 _ is the markov chain over @xmath8 defined by the transition probabilities @xmath9 for all @xmath10 . our aim is to study the recurrence properties of such a random walk . we will not focus here on the _ almost sure recurrence _ as in @xcite and @xcite but on the _ recurrence in law _ as in @xcite , @xcite and@xcite . [ defrectrans ] the random walk on @xmath8 is _ recurrent in law _ at a point @xmath10 if for all @xmath11 , there exists a compact set @xmath12 and @xmath13 such that for all @xmath14 : @xmath15 the random walk on @xmath8 is _ uniformly recurrent in law _ if the same compact set @xmath16 can be chosen for all the starting points @xmath17 . a probability measure @xmath5 on @xmath8 is said to be _ @xmath4-stationary _ or _ @xmath4-invariant _ if one has @xmath18 those definitions are tightly linked . indeed , there exists a @xmath4-stationary probability measure on @xmath8 if and only if the random walk on @xmath8 is recurrent in law at some point @xmath10 ( see lemma [ urloimesinv ] for one implication ) . in this paper , @xmath0 will always be a real algebraic group acting algebraically on a real algebraic variety @xmath8 ; the measure @xmath4 will be compactly supported and _ zariski dense _ , which means that its support spans a zariski dense subgroup in @xmath0 . when @xmath0 is a reductive group and @xmath19 is an algebraic homogeneous space , it is proven in @xcite that there exists a @xmath4-stationary probability measure on @xmath8 if and only if @xmath8 is compact . the aim of our article is to focus on situations where the algebraic group @xmath0 is not reductive . in particular , in corollary [ cassl ] , we will exhibit examples of non - compact homogeneous spaces on which there always exists a @xmath4-stationary probability measure . the key tool in our analysis will be to link the recurrence properties of these random walks to the lyapunov exponents of @xmath4 . the definition of these lyapunov exponents depends on the choice of a linear action of @xmath0 on @xmath1 . [ deflyap ] given a linear action of @xmath0 on @xmath1 , the _ lyapunov exponents _ of @xmath4 are the real numbers @xmath20 such that , for all @xmath21 , we have @xmath22 { \ensuremath{\operatorname{d}\!{\mu}}}^{*n } ( g).\ ] ] the sequence of lyapunov exponents is always decreasing : @xmath23 ( see ( * ? ? ? * prop 1.2 ) ) . more properties of these exponents are given in @xcite , @xcite ; their use in the context of reductive groups is detailed in @xcite , @xcite , @xcite and @xcite . we assume now that @xmath0 is either the affine group @xmath24 or the special affine group @xmath25 . for @xmath26 , we denote by @xmath27 the @xmath28-lyapunov exponent corresponding to the linear action of @xmath0 on @xmath1 . for instance , in dimension @xmath29 , one has @xmath30 { \ensuremath{\operatorname{d}\!{\mu}}}(a , u)\ ] ] where @xmath31 . for any @xmath32 , bougerol and picard have shown in @xcite that there exists a @xmath4-stationary probability measure on @xmath1 if and only if the first lyapunov exponent of @xmath4 is strictly negative : @xmath33 . the main result of this paper is the following theorem [ recgrassaff ] , which extends this equivalence to the affine grassmannians @xmath34 where @xmath35 . by definition the affine grassmannian @xmath34 is the space of @xmath3-dimensional affine subspaces of @xmath1 . the group @xmath0 acts transitively on @xmath34 . [ recgrassaff ] let @xmath0 be the affine group or the special affine group of @xmath1 , let @xmath4 be a zariski dense probability measure with compact support on @xmath0 and let @xmath36 . + a ) if @xmath37 , then the random walk on @xmath34 is nowhere recurrent in law , there exists no @xmath4-stationary probability measure on @xmath34 , and for all @xmath17 in @xmath34 the sequence of means of transition probabilities weakly converges to @xmath38 : @xmath39 { } 0.\ ] ] b ) if @xmath40 , then the random walk on @xmath34 is uniformly recurrent in law , there exists a unique @xmath4-stationary probability measure @xmath5 on @xmath34 , and for all @xmath17 in @xmath34 the sequence of means of transition probabilities weakly converges to @xmath5 : @xmath39 { } \nu.\ ] ] the result of bougerol and picard in @xcite covers the @xmath41 case . in fact , their proof uses only the weaker assumption that @xmath4 has a finite first moment and that its support does not preserve any proper affine subspace of @xmath1 . the following corollary , which is particularly noteworthy insofar as it does not mention lyapunov exponents , is deduced from theorem [ recgrassaff ] . [ cassymetrique ] assume @xmath4 is symmetric . then there exists a @xmath4-stationary probability measure @xmath5 on @xmath34 if and only if @xmath7 . in this case , @xmath5 is unique . since @xmath4 is symmetric , for all @xmath26 , the lyapunov exponents satisfy the equalities @xmath42 . moreover , since @xmath4 is zariski dense in @xmath0 , it follows from the guivarch - raugi simplicity theorem that the sequence of lyapunov exponents is strictly decreasing : @xmath43 ( see ( * ? ? ? * corol . 10.15 ) ) . therefore one has the equivalence @xmath44 . [ cassl ] let @xmath45 . when @xmath0 is the special affine group and @xmath46 , there exists a unique @xmath4-stationary probability measure on @xmath34 . in this case , the sum of the lyapunov exponents is zero . hence , the simplicity of the lyapunov exponents implies @xmath47 . for instance , when @xmath0 is the special affine group of @xmath48 , the random walk on the space of affine lines of @xmath48 is always uniformly recurrent in law while the random walk on the space of points of @xmath48 is nowhere recurrent in law . by embedding the affine grassmannian @xmath34 of @xmath1 in the projective space of a suitable exterior power @xmath50 of @xmath51 , we will deduce theorem [ recgrassaff ] from the following theorem [ thmeq ] : we first need two definitions . an algebraic group @xmath0 is _ zariski connected _ if it is connected for the zariski topology . a linear action of @xmath0 on a vector space @xmath52 is _ proximal _ if there exists a rank @xmath53 linear endomorphism @xmath54 of @xmath52 which is a limit of a sequence @xmath55 with @xmath56 and @xmath57 in @xmath58 . [ thmeq ] let @xmath50 be a finite - dimensional real vector space , @xmath0 a zariski connected algebraic subgroup of @xmath59 , @xmath52 a @xmath0-invariant subspace of @xmath50 such that + @xmath60 @xmath0 acts irreducibly and proximally on @xmath52 and on @xmath61 . + @xmath62 the representations of @xmath0 in @xmath52 and @xmath63 are not equivalent . + @xmath64 @xmath52 has no @xmath0-invariant complementary subspace in @xmath50 . + let @xmath65\smallsetminus\pj[w]$ ] , let @xmath4 be a zariski dense probability measure with compact support on @xmath0 and let @xmath66 and @xmath67 be the first lyapunov exponents of @xmath4 in @xmath52 and @xmath63 respectively . + a ) if @xmath68 , then the random walkon @xmath49 is nowhere recurrent in law , there exists no @xmath4-stationary probability measure on @xmath49 , and for all @xmath17 in @xmath49 one has the weak convergence @xmath69 { } 0.$ ] + b ) if @xmath70 , then the random walkon @xmath49 is uniformly recurrent in law , there exists a unique @xmath4-stationary probability measure @xmath5 on @xmath49 , and for all @xmath17 in @xmath49 , one has the weak convergence @xmath69 { } \nu.$ ] in chapter [ passage ] , we explain how to embed the affine grassmannian @xmath71 in the variety @xmath72\smallsetminus\pj[\lambda^{k+1}{\mathbb{r}}^{d}]$ ] and we deduce theorem [ recgrassaff ] from theorem [ thmeq ] . the last three chapters will deal with the proof of theorem [ thmeq ] . in chapter [ thmeqrecloi ] , we prove the uniform recurrence in law when @xmath73 ( corollary [ l1lp1urloi ] ) . the crux of the proof is the construction of a proper function on @xmath49 which is contracted by the averaging operator ( proposition [ pmuuch ] ) . in chapter [ l1lp1nrecloi ] , we prove the non - recurrence in law when @xmath68 ( proposition [ l1lp1nonrecloiprop ] ) . the key point is the study of the ratio of the norms in @xmath52 and in @xmath63 of a random product @xmath74 . on the one hand , the existence of a @xmath4-stationary probability measure on @xmath49 would imply that these ratios are bounded ( lemma [ liminfdnanfinie ] ) . on the other hand , when @xmath68 , the law of large numbers and the law of iterated logarithms for these products prevent these ratios from being bounded ( lemma [ liminfdnaninfinie ] ) . in chapter [ thmequnicite ] , we prove the uniqueness of the @xmath4-stationary measure on @xmath49 ( proposition [ unicitedirac ] ) . indeed , using the joining measure ( corollary [ corjoista ] ) of two distinct @xmath4-stationary probability measures on @xmath49 , we construct ( lemma [ lemyvw ] ) a @xmath4-stationary measure @xmath75 on the space @xmath76\smallsetminus ( \pj[w]\cup\pj[w'])$ ] . this contradicts the classification of stationary measures in @xcite since this space does not contain compact @xmath0-orbits ( lemma [ pasdorbitescompactesdansx ] ) . the weak convergence of the sequence of means of transition probabilities follows easily ( corollary [ corthmeqlunlpun ] ) . in appendix [ seclimlaw ] , we collect known facts on random walks on reductive groups . in this paper , all the vector spaces will be finite dimensional real vector spaces , all the measures will be borel measures and we will not distinguish between a real algebraic group and its group of real points . we explain first how to deduce theorem [ thmeq ] from theorem [ recgrassaff ] we use the notation of theorem [ thmeq ] . the group @xmath0 is the affine group or the special affine group of @xmath1 , the space @xmath34 is the affine grassmannian of @xmath1 , the probability measure @xmath4 on @xmath0 is zariski dense and compactly supported . let us construct @xmath0-vector spaces @xmath77 to which we will apply theorem [ thmeq ] . we identify the affine space @xmath1 with the affine hyperplane of @xmath78 : @xmath79 the group @xmath0 is then a subgroup of @xmath80 , which stabilizes @xmath81 , and we have @xmath82 where @xmath83 and @xmath84 are the grassmannians of @xmath85-dimensional vector subspaces of @xmath51 and of @xmath1 respectively . now , let @xmath86 the group @xmath0 acts linearly on the vector space @xmath50 and leaves invariant its vector subspace @xmath52 . the plcker map @xmath87 \;\ ; ; \;\ ; u\longmapsto \lambda^{k+1}u.\ ] ] is an embedding of the grassmannian variety in the projective space of @xmath50 . it induces a @xmath0-equivariant injection @xmath88\smallsetminus\pj[w].\ ] ] [ constructionvwwp ] with the above notations , + a ) hypotheses @xmath60 , @xmath62 , @xmath64 hold for these @xmath50 , @xmath52 and @xmath89 . + b ) the @xmath0-equivariant inclusion @xmath90 has closed image . + c ) we have the equality @xmath91 . \a ) @xmath60 : the representation of @xmath92 in @xmath93 is irreducible by ( * ? ? ? 8.13.1.4 ) . this representation is proximal since the image in @xmath94 of a diagonal element of @xmath0 with positive distinct eigenvalues is a proximal element of @xmath94 . the same is true for the representation in @xmath95 . + @xmath62 : the fact that the representations of @xmath92 in @xmath52 and @xmath63 are not equivalent is also proven in ( * ? ? ? 8.13.1.4 ) . + @xmath64 : since the representation of @xmath92 in @xmath52 and @xmath63 are irreducible and are not equivalent , by schur s lemma , the only @xmath92-invariant complementary subspace of @xmath52 in @xmath96 is @xmath63 . but @xmath63 is not invariant by the translations of @xmath0 . \b ) the image @xmath97 is closed in @xmath49 since @xmath98)={\mathrm{gr}}_{k+1}(d)$ ] . \c ) this equality is the difference of the equalities @xmath99 which follow from the very definition of the lyapunov exponents . we use proposition [ constructionvwwp ] . if @xmath37 , then we can apply theorem [ thmeq ] in the case where @xmath100 , and there can be no @xmath4-stationary probability measure on @xmath34 . conversely , if @xmath40 , we are in the case where @xmath101 . since @xmath34 is a @xmath0-invariant closed subset of @xmath49 , we obtain uniform recurrence in law on @xmath34 . lemma [ urloimesinv ] then ensures the existence of a @xmath4-stationary probability measure on @xmath34 , which is thus the unique @xmath4-stationary probability measure on @xmath49 . the goal of this chapter is to show that the random walk on @xmath49 is uniformly recurrent in law when @xmath102 ( corollary [ l1lp1urloi ] ) . we recall in this section the uniform contraction hypothesis and why this condition implies the uniform recurrence in law . the setting is very general ( see @xcite , @xcite or @xcite for more details ) . let @xmath8 be a locally compact second - countable space and @xmath103 a markov - feller operator on @xmath8 . [ defuch ] the operator @xmath103 satisfies the _ uniform contraction hypothesis _ ( uch ) if there exists a proper map @xmath104 and two constants @xmath105 and @xmath106 such that , over @xmath8 , @xmath107 we recall that a map is _ proper _ if the inverse image of every compact set is relatively compact . the definition of recurrence in law extends to markov chains on @xmath8 . uniform recurrence in law is fundamentally linked with ( uch ) : [ uchurloi ] if @xmath103 satisfies ( uch ) , then the associated markov chain on @xmath8 is uniformly recurrent in law . see ( * ? ? ? * thm 15.0.1 ) , ( * ? ? ? * lem 3.1 ) or ( * ? ? ? * lem 2.1 ) . [ urloimesinv ] if @xmath103 is recurrent in law at point @xmath10 , there exists a @xmath103-invariant probability measure on @xmath8 . by the banach - alaoglu theorem , the sequence of means of transition probabilities @xmath108 has at least one accumulation point @xmath109 for the weak-@xmath110 topology . this finite measure @xmath111 is @xmath103-invariant . since @xmath103 is recurrent in law at @xmath17 , there is no escape of mass and @xmath111 is a probability measure . the following lemma is a useful tool to check ( uch ) . [ lempnouch ] let @xmath112 . if @xmath113 satisfies @xmath114 then @xmath103 satisfies @xmath114 too . let @xmath115 be the proper map and @xmath116,@xmath117 the constants such that @xmath118 over @xmath8 . let @xmath119 for @xmath120 , @xmath121 , @xmath122 . then the proper map @xmath123 defined by @xmath124 satisfies the inequality @xmath125 on @xmath8 , and thus @xmath103 satisfies(uch ) . in this section , we use again the notations and assumptions of theorem [ thmeq ] . we will prove that the averaging operator satisfies the uniform contraction hypothesis . we recall that @xmath77 are real vector spaces , @xmath0 is a zariski connected algebraic subgroup of @xmath59 preserving @xmath52 and satisfying @xmath60 , @xmath62 , @xmath64 . we identify the quotient @xmath89 with a complementary subspace @xmath126 of @xmath52 in @xmath50 . note that this subspace @xmath126 is not @xmath0-invariant . we recall also that @xmath4 is a zariski dense probability measure on @xmath0 with compact support and that @xmath127 and @xmath128 are the first lyapunov exponents of @xmath4 in @xmath52 and @xmath63 , and that we are studying the associated random walk on the @xmath0-space @xmath65\smallsetminus\pj[w]$ ] . the corresponding markov operator @xmath129\longrightarrow\cont[x_{v , w}]$ ] is given by @xmath130 [ pmuuch ] same notations and assumptions as in theorem [ thmeq ] . + if @xmath70 , then the markov operator @xmath131 satisfies ( uch ) . the space @xmath49 can be seen as the set @xmath132\,|\ , w\in w,\ , w'\in w_s\smallsetminus\{0\}\}.\ ] ] choose a norm on @xmath50 , and , for @xmath133 , consider the functions @xmath134\longmapsto \tfrac{\n[w]^{\delta}}{\n[w']^{\delta}}.\ ] ] these functions are proper and well - defined . we want to find @xmath133 , @xmath1350,\,1[$ ] , @xmath106 , @xmath13 such that , over @xmath49 , one has the inequality @xmath136 since @xmath52 is @xmath0-invariant , we can write @xmath137 as @xmath138 let @xmath139 . then , by a lemma due to furstenberg ( cf . * thm 4.28 ) , @xcite ) since @xmath0 acts irreducibly on @xmath52 and @xmath63 there exists @xmath13 such that for all @xmath140 , for all non - zero @xmath141 , the following inequalities hold : @xmath142}{\n[w]}{\ensuremath{\operatorname{d}\!{\mu}}}^{*n}(g ) \leq { \lambda_1}+\varepsilon , \\ \label{inegfurstlpun } { \lambda^{\prime}_1}-\varepsilon & \leq \frac{1}{n}\int_g \log \frac{\n[d_g w']}{\n[w']}{\ensuremath{\operatorname{d}\!{\mu}}}^{*n}(g ) \leq { \lambda^{\prime}_1}+\varepsilon.\end{aligned}\ ] ] for @xmath143 and @xmath144\in x_{v , w}$ ] , one computes @xmath145 we will give an upper bound for the right - hand integral for all @xmath17 in the complementary set of some compact @xmath146 in @xmath8 ; since map @xmath147 is bounded on the compact set @xmath146 , this will give inequality ( [ inequch ] ) . let @xmath148 be the constant defined by @xmath149\,\n[a_g^{-1 } ] { \ensuremath{\operatorname{d}\!{\mu}}}^{*n_0}(g).\ ] ] let @xmath146 be the compact subset of @xmath49 given by @xmath150\,|\ , w\in w , \ ; w'\in w_s,\ ; \n[w']\geq c\ , \n[w ] \}.\ ] ] for @xmath151-almost every @xmath137 , for all @xmath152 , the following ratio is bounded : @xmath153}{\n[w]}\ , \frac{\n[w']}{\n[d_gw ' ] } \;\leq\ ; \sup_{g\in \operatorname{supp}\mu^{*n_0 } } \n[d_g^{-1}](\n[a_g]+c\n[c_g]).\ ] ] therefore , we can find some constant @xmath154 such that for all @xmath133 , for all @xmath152 , for @xmath151-almost every @xmath137 , we can write @xmath155 for all @xmath152 , for @xmath151-almost every @xmath137 , the following upper bound holds : @xmath156}{\n[w ] } - \log\frac{\n[d_g w']}{\n[w']}+ \log \frac{\n[a_gw+c_gw']}{\n[a_gw]}\\ & \leq \log\frac{\n[a_g w]}{\n[w ] } - \log\frac{\n[d_g w']}{\n[w']}+\frac{\n[c_gw']}{\n[a_gw ] } \\ & \leq \log\frac{\n[a_g w]}{\n[w ] } - \log\frac{\n[d_g w']}{\n[w']}+ \n[c_g]\n[a_g^{-1}]c . \end{aligned}\ ] ] using inequalities ( [ inegfurstlun ] ) , ( [ inegfurstlpun ] ) and the definition of @xmath157 , we get the inequality @xmath158 let @xmath159 . we then get the upper bound , for all @xmath152 , @xmath160 choose @xmath143 such that @xmath161 is strictly between @xmath38 and @xmath53 . therefore , since @xmath146 is compact , there exists a constant @xmath162 such that for all @xmath10 : @xmath163 and , by lemma [ lempnouch ] , the operator @xmath131 satisfies ( uch ) . [ l1lp1urloi ] same notations and assumptions as in theorem [ thmeq ] . + if @xmath70 , then the random walkon @xmath8 is uniformly recurrent in law . this is a direct consequence of proposition [ pmuuch ] : since @xmath131 satisfies ( uch ) , we only need to apply proposition [ uchurloi ] . the goal of this chapter is to show that the random walk on @xmath49 is nowhere recurrent in law when @xmath165 ( proposition [ l1lp1nonrecloiprop ] ) . we recall in this section the definition and the properties of the limit probability measures associated to a stationary measure . the setting is very general . let @xmath0 be a locally compact group acting on a second countable locally compact space @xmath8 and @xmath4 be a probability measure on @xmath0 . let @xmath166 be the product space @xmath167 and @xmath168 be the product measure @xmath169 . the following lemma is due to furstenberg . see ( * ? ? ? * lem 3.2 ) or ( * ? ? ? * lemma 2.17 ) . [ nub ] let @xmath5 be a @xmath4-stationary probability measure on @xmath8 . for @xmath168-almost every @xmath170 , the sequence @xmath171 of probability measures on @xmath8 has a limit @xmath172 , which we will call _ limit probability_. moreover , we have @xmath173 the following construction will be useful in chapter [ secunista ] . see ( * ? ? ? * cor 3.5 ) for a proof . [ corjoista ] let @xmath174 and @xmath175 be two @xmath4-stationary probability measures on @xmath8 . then the probability measure on @xmath176 @xmath177 is @xmath4-stationary . it is called the _ joining measure _ of @xmath174 and @xmath175 . this corollary will be used in combination with the following basic lemma . [ diagonalesansatome ] let @xmath178 , @xmath179 be probability measures on a topological space @xmath8 and let @xmath180 be the diagonal of @xmath8 . if @xmath181 , then @xmath178 and @xmath179 are identical dirac measures . by assumption , we have @xmath182 hence , for @xmath179-almost every @xmath10 , we have @xmath183 , which implies that measures @xmath178 and @xmath179 are identical dirac measures . in this section , we again use the same notations and assumptions as in theorem [ thmeq ] . we will prove that the space @xmath49 supports no @xmath4-stationary measures . recall that @xmath77 are real vector spaces , @xmath0 is a zariski connected algebraic subgroup of @xmath59 preserving @xmath52 and satisfying @xmath60 , @xmath62 , @xmath64 , also recall that @xmath4 is a zariski dense probability measure on @xmath0 with compact support , that @xmath127 and @xmath128 are the first lyapunov exponents of @xmath4 in @xmath52 and in @xmath61 , and that we are studying the associated random walk on the @xmath0-space @xmath65\smallsetminus\pj[w]$ ] . [ l1lp1nonrecloiprop ] same notations and assumptions as in theorem [ thmeq ] . + if @xmath68 , then the random walkon @xmath49 is nowhere recurrent in law , and there exists no @xmath4-stationary probability measure on @xmath49 . by lemma [ urloimesinv ] the first assertion follows from the second one . this second assertion is a consequence of the following lemmas [ liminfdnanfinie ] and [ liminfdnaninfinie ] . let @xmath167 and @xmath169 . for @xmath184 in @xmath166 we write as in : @xmath185 [ liminfdnanfinie ] same notations and assumptions as in theorem [ thmeq ] . if there exists a @xmath4-stationary probability measure on @xmath49 , then for @xmath168-almost every @xmath170 , we have @xmath186/\n[d_n]\ ; < \ ; \infty.\ ] ] the proof of lemma [ liminfdnanfinie ] will be given in section [ secaccpoi ] . it relies on the properties of the limit probability measures @xmath172 . [ liminfdnaninfinie ] same notations and assumptions as in theorem [ thmeq ] . if @xmath68 , then for @xmath168-almost every @xmath170 , one has @xmath187/\n[d_n]\ ; = \ ; \infty.\ ] ] the proof of lemma [ liminfdnaninfinie ] will be given in section [ secprocar ] . it relies on the law of large numbers and on the law of the iterated logarithm for the random variables @xmath188 . the aim of this section is to prove lemma [ liminfdnanfinie ] . we will need the following analog of ( * ? ? ? 3.7 ) for a non - irreducible action . [ pasdemasseausousespace ] same notations and assumptions as in theorem [ thmeq ] . let @xmath5 be a @xmath4-stationary probability measure on @xmath189 $ ] such that @xmath190)=0 $ ] . then for every proper subspace @xmath191 of @xmath50 , we have @xmath192)=0 . $ ] assume there exists a proper subspace @xmath191 of @xmath50 such that @xmath193)>0 $ ] . let @xmath194 be the minimal dimension of such a subspace @xmath191 . if @xmath195 and @xmath196 are two distinct vector subspaces of dimension @xmath194 , one has the equality @xmath197\cup\pj[u_2])=\nu(\pj[u_1])+\nu(\pj[u_2]).\ ] ] let @xmath198 ) \,|\ , u\subset v,\ , \dim u = r_0\}>0\;\;$ ] and consider the set @xmath199)=\alpha,\ , \dim u = r_0\}.\ ] ] this set is finite and non - empty . by @xmath4-stationarity of @xmath5 , for @xmath4-almost every @xmath137 , we have @xmath200 . therefore , since @xmath4 is zariski dense in @xmath0 , this set @xmath201 is @xmath0-invariant . since @xmath0 is zariski connected , all the subspaces @xmath191 belonging to @xmath201 are @xmath0-invariant . but by @xmath60 , @xmath62 and @xmath64 , the only proper @xmath0-invariant subspace of @xmath50 is @xmath52 . this is contradictory since , by assumption , we have @xmath190)=0 $ ] . we assume also that there exists a @xmath4-stationary probability measure @xmath5 on @xmath49 . in order to prove , it is enough to check that for @xmath168-almost every @xmath170 , for all accumulation points @xmath54 in @xmath202 of the sequence @xmath203}$ ] , the image of @xmath54 is not included in @xmath52 : @xmath204 lemma [ pasdemasseausousespace ] shows that @xmath205)=0 $ ] , hence the image probability measure @xmath206 is well - defined and the sequence @xmath207 weakly converges to @xmath206 . by lemma [ nub ] this sequence @xmath207 also weakly converges to @xmath172 , and therefore we have @xmath208 therefore , for @xmath168-almost all @xmath117 in @xmath166 , one has , for all accumulation point @xmath54 , @xmath209 ) = 1.\ ] ] since @xmath190)=0 $ ] , one also has , for @xmath168-almost all @xmath117 in @xmath166 , @xmath210)=0,\ ] ] and hence the images @xmath211 are not contained in @xmath52 . this proves . the aim of this section is to prove lemma [ liminfdnaninfinie ] . let @xmath212 be the natural projection @xmath213 the image group @xmath214 is a reductive subgroup of @xmath215 . the image measure @xmath216 is a zariski dense probability measure on @xmath217 . the proof of lemma [ liminfdnaninfinie ] will use the notations of appendix [ seclimlaw ] with the reductive group @xmath217 and its probability measure @xmath218 . in particular , @xmath219 $ ] is the lie algebra of @xmath217 , @xmath220 $ ] is the lie algebra of a maximal split torus of @xmath217 , @xmath221 is the cartan projection , @xmath222 is the lyapunov vector , @xmath223 is the covariance @xmath224-tensor , @xmath220_{\overline{\mu}}$ ] is its linear span , and @xmath225 is the unit ball of @xmath220_{\overline{\mu}}$ ] . we will also use the following two lemmas . we set @xmath226 and @xmath227 . [ lemhigdis ] the highest weights @xmath228 and @xmath229 of the representations of @xmath217 in @xmath52 and @xmath63 are distinct . since @xmath217 is zariski connected , condition @xmath60 tells us that @xmath52 and @xmath63 are irreducible representations of @xmath219 $ ] and that their highest weight spaces are one dimensional . condition @xmath62 tells us that these representations of @xmath219 $ ] are not equivalent . therefore as in ( * ? ? ? * chap 8.6.3 ) , the highest weights @xmath228 and @xmath229 must be distinct . [ centreoverg ] the center @xmath230 of @xmath217 is equal to @xmath231 by schur s lemma , the commutant of @xmath217 in @xmath232 is a division algebra . since the representation of @xmath217 in @xmath52 is proximal , this commutant is the field @xmath233 of scalar matrices . therefore @xmath230 acts on @xmath52 ( and also on @xmath63 ) by scalar matrices . fix norms on @xmath52 and @xmath63 as in lemma [ representationgeometriquecartaniwas ] , so that , for any element @xmath234 in @xmath217 with @xmath235 , @xmath236 , one has @xmath237 in particular , the first lyapunov exponents in @xmath52 and @xmath63 are given by @xmath238 let @xmath239 and @xmath240 . for @xmath241 , we write @xmath242 we distinguish three cases : * first case :* @xmath243 . in this case one has @xmath244 . according to and the law of large numbers [ corinv ] , for @xmath245-almost every @xmath246 , we have @xmath247/\n[d_n ] ) \ ; = \ ; \lim_{n\rightarrow\infty}(\chi-\chi')(\kappa(b_1\cdots b_n ) ) \ ; = \ ; \infty .\ ] ] * second case :* @xmath248 and @xmath249_{\overline{\mu}})\neq 0 $ ] . in this case , one has @xmath250 and there exists @xmath17 in the unit ball @xmath225 of @xmath220_{\overline{\mu}}$ ] such that @xmath251 according to the law of the iterated logarithm [ corinv ] , for @xmath245-almost every @xmath252 , there exists an increasing sequence of integers @xmath253 such that @xmath254 and therefore such that @xmath255/\n[d_{n_i } ] ) \ ; = \ ; \lim_{i\rightarrow\infty}(\chi-\chi')(\kappa(b_1\cdots b_{n_i } ) ) \ ; = \ ; \infty .\ ] ] * third case :* @xmath248 and @xmath256_{\overline{\mu}})= 0 $ ] . let @xmath257 since the group @xmath217 is reductive , by lemma [ centreoverg ] , the subgroup @xmath258 is semisimple . let @xmath259 $ ] be the lie algebra of @xmath258 . by ( * ? ? ? * thm 13.19 ) , we have @xmath220\cap \lie[s]\subset \lie[a]_{\overline{\mu}}$ ] , and thus also @xmath260\cap \lie[s])=0.\ ] ] we introduce the group morphism @xmath261 defined by : @xmath262 for every @xmath234 in @xmath217 , we can write @xmath263 with @xmath264 and @xmath265 . using equations , and the equality @xmath266 , we compute , @xmath267/\n[d]\right ) \ ; = \ ; ( \chi-\chi')(\kappa(g ) ) \ ; = \ ; ( \chi-\chi')(\kappa(z ) ) \ ; = \ ; \delta(z ) \ ; = \ ; \delta(g).\ ] ] we want to describe the behavior of the random variable @xmath268/\n[d_n]\right ) $ ] on @xmath269 where as above @xmath270 . using equation , we see that @xmath271 is the sum of @xmath272 real - valued independent and identically distributed random variables @xmath273 . note that the law of the variable @xmath274 has compact support . since @xmath248 , we have @xmath275 { } 0 $ ] . thus the variable @xmath274 is centered . if this random variable @xmath274 were almost surely @xmath38 , it would mean that for @xmath218-almost every @xmath276 , we have @xmath277 . since @xmath218 is zariski dense in @xmath217 , this would imply @xmath278 , or , equivalently , @xmath279)=0,\ ] ] where @xmath280\subset\lie[a]$ ] is the lie algebra of @xmath230 . equalities and would tell us that the highest weights @xmath228 and @xmath229 were equal . this would contradict lemma [ lemhigdis ] . therefore this centered variable @xmath274 is not almost surely @xmath38 . thus the classical recurrence properties of real random walks ( cf e.g. ( * ? ? ? * thm 3.38 ) ) tell us that @xmath281 almost surely . in each of these three cases , we have checked . [ secunista ] the main aim of this chapter is to prove the uniqueness of the stationary measure on @xmath49 ( proposition [ unicitedirac ] ) . the proof of uniqueness will rely on the following lemma [ lemyvw ] . we keep the notations and assumptions of theorem [ thmeq ] . let @xmath283 be the projection @xmath284 \;\ ; ; \;\ ; [ v]\longmapsto [ v+w]\ ] ] and let @xmath282 be the @xmath0-invariant subvariety of @xmath285 @xmath286 [ lemyvw ] same notations and assumptions as in theorem [ thmeq ] . + there is no @xmath4-stationary probability measure @xmath287 on @xmath282 . suppose that such a measure @xmath287 does exist . consider again the natural projection @xmath288 introduced in . let @xmath214 be the image of @xmath0 by @xmath212 , a reductive subgroup of @xmath215 , and let @xmath216 be the image of @xmath4 by @xmath212 , a zariski dense probability measure on @xmath217 . now consider the map @xmath289 where @xmath290\smallsetminus ( \pj[w]\cup\pj[w'])$ ] . let @xmath291 be the probability measure on @xmath292 that is the image of @xmath287 by @xmath212 . since the map @xmath293 is equivariant , the probability measure @xmath75 is @xmath218-stationary . according to proposition [ bqorbitescompactesmeasures ] such a measure @xmath75 is supported by a compact @xmath217-orbit in @xmath292 . this contradicts the following lemma [ pasdorbitescompactesdansx ] . [ pasdorbitescompactesdansx ] there are no compact @xmath217-orbits in @xmath292 . such a compact orbit would be of the form @xmath294 , where @xmath295 is an algebraic subgroup of @xmath217 containing a conjugate of the group @xmath296 with @xmath297 a maximal split subtorus of @xmath217 and @xmath298 a maximal unipotent subgroup normalized by @xmath297 . since @xmath52 and @xmath63 are proximal irreducible representations of @xmath217 , there is only one @xmath298-invariant line @xmath299 in @xmath52 and one @xmath300 in @xmath63 . hence the @xmath298-invariant lines in @xmath301 are included in the plane @xmath302 . since , by lemma [ lemhigdis ] , the highest weights @xmath228 and @xmath229 of @xmath52 and @xmath63 are distinct , the lines @xmath299 and @xmath300 are the only @xmath297-invariant lines in @xmath303 . therefore , a compact @xmath217-orbit in @xmath76 $ ] is contained in @xmath304\cup\pj[w']$ ] . we can now show the uniqueness of the @xmath4-stationary probability measure @xmath5 on @xmath49 . the same proof will tell us that its limit probability measures @xmath172 are dirac measures . [ unicitedirac ] same notations and assumptions as in theorem [ thmeq ] . + if @xmath70 , the @xmath4-stationary probability measure @xmath5 on @xmath49 is unique . + moreover , the limit measures @xmath172 are @xmath168-almost surely dirac measures . let @xmath174 and @xmath175 be two @xmath4-stationary probability measures on @xmath49 . by corollary [ corjoista ] the joining measure @xmath305 on @xmath285 is @xmath4-stationary . let us show that its support is contained in the subvariety @xmath306 where @xmath307 $ ] is again the canonical projection . since the action of @xmath0 on @xmath63 is irreducible and proximal , there exists a unique @xmath4-stationary measure @xmath308 on @xmath309 $ ] called the _ furstenberg measure_. its limit probability measures @xmath310 are @xmath168-almost surely dirac measures @xmath311 for some @xmath312 $ ] . see ( * ? ? ? 3.7 ) for more detail on the furstenberg measure . since @xmath308 is unique , we have the equalities @xmath313 therefore , for @xmath168-almost every @xmath170 , we have @xmath314 and hence @xmath315 by the very definition of the joining measure , integrating this equality gives @xmath316 by definition , this set @xmath317 is the union @xmath318 . by lemma [ lemyvw ] , the @xmath0-variety @xmath282 does not support @xmath4-stationary measures . therefore the joining measure @xmath305 is supported on the diagonal @xmath319 . hence , for @xmath168 almost every @xmath117 in @xmath166 , the measure @xmath320 is also supported on the diagonal : @xmath321 therefore , by lemma [ diagonalesansatome ] , the limit probability measures @xmath322 and @xmath323 are both equal to the same dirac measures . hence , by lemma [ nub ] , one has @xmath324 . in this section we prove that the sequence of means of the transition probabilities @xmath325 on @xmath49 always has a limit . [ corthmeqlunlpun ] same notations and assumptions as in theorem . [ thmeq ] . let @xmath326 . + a ) when @xmath327 , one has the weak convergence @xmath328 { } 0.$ ] + b ) when @xmath70 , one has the weak convergence @xmath328 { } \nu.$ ] every accumulation point of the sequence of probability measures @xmath329 is a @xmath4-stationary finite measure . when @xmath327 , by proposition [ l1lp1nonrecloiprop ] , such a measure is necessarily @xmath38 . when @xmath70 , by corollary [ l1lp1urloi ] , the corresponding markov chain is recurrent in law ; hence , no mass is lost and the accumulation points are thus @xmath4-stationary probability measures . by proposition [ unicitedirac ] , there is only one such measure . in this appendix , we recall some facts about random walks on reductive groups , which are mainly detailed in @xcite . let @xmath0 be a zariski connected real algebraic reductive group . let @xmath330 be a maximal split subtorus of @xmath0 , @xmath220 $ ] be the lie algebra of @xmath330 , @xmath220_+\subset\lie[a]$ ] be a weyl chamber and @xmath331_+$ ] . there exists a maximal compact subgroup @xmath146 of @xmath0 such that @xmath0 has a _ cartan decomposition _ @xmath332 the _ cartan projection _ of @xmath0 is the unique map @xmath333_+$ ] such that , for all @xmath137 , @xmath334 [ representationgeometriquecartaniwas ] ( ( * ? ? ? * lem . 6.33 ) ) let @xmath0 be a zariski connected real algebraic reductive group , @xmath212 be an irreducible algebraic representation of @xmath0 in a real vector space @xmath52 and @xmath335^*$ ] be the _ highest weight of @xmath52_. there exists a norm on @xmath52 such that , for all @xmath137 , @xmath336).\ ] ] [ thmlgn](law of large numbers , ( * ? ? ? * thm 10.9 ) ) let @xmath0 be a zariski connected real reductive group and @xmath4 a zariski dense probability measure with compact support on @xmath0 . for @xmath168-almost every @xmath170 , we have the convergence @xmath339{}{\sigma_{\mu}}.\ ] ] define the _ covariance 2-tensor _ ( see ( * ? ? ? * prop.14.18 ) ) @xmath340 $ ] as the limit @xmath341 @xmath342 is a symmetric @xmath224-tensor on @xmath220 $ ] . we denote by @xmath220_{\mu}\subset\lie[a]$ ] the _ linear span of @xmath342 _ which is the smallest subspace @xmath220_{\mu}$ ] of @xmath220 $ ] such that we have @xmath340_{\mu}$ ] . we can see @xmath342 as an inner product over @xmath220_{\mu}$ ] . we then denote by @xmath343 the closed unit ball of @xmath220_{\mu}$ ] for the metric corresponding to @xmath342 . when @xmath0 is semisimple , the covariance @xmath224-tensor @xmath342 is _ non degenerate _ i.e. one has @xmath220_{\mu}=\lie[a]$ ] . [ thmlil ] ( law of the iterated logarithm , ( * ? ? ? * thm 13.17 ) ) let @xmath0 be a zariski connected real reductive group and @xmath4 a zariski dense probability measure with compact support on @xmath0 . then , for @xmath168-almost every @xmath170 , the set of accumulation points of the sequence @xmath344 [ corinv ] let @xmath0 be a zariski connected real reductive group and @xmath4 a zariski dense probability measure with compact support on @xmath0 . + a ) for @xmath168-almost every @xmath170 , we have the convergence @xmath346{}{\sigma_{\mu}}.\ ] ] b ) for @xmath168-almost every @xmath170 , the set of accumulation points of the sequence @xmath347 the proof will use the probability measure @xmath348 on @xmath0 which is the image of @xmath4 by the map @xmath349 . recall that there exists a linear map @xmath350\rightarrow\lie[a]$ ] called the _ opposition involution _ ( see @xcite ) such that , for all @xmath137 , we have @xmath351 [ iotasigmamu ] the lyapunov vector @xmath352 of @xmath348 , its covariance @xmath224-tensor @xmath353 and the closed unit ball @xmath354 of the linear span @xmath220_{\check{\mu}}$ ] of @xmath353 are equal to : @xmath355 we apply the law of large numbers [ thmlgn ] to the probability measure @xmath348 . for @xmath168-almost every @xmath170 , the sequence @xmath356 converges to @xmath352 . applying the opposition involution @xmath357 , we find that , for @xmath168-almost every @xmath170 , the sequence @xmath358 converges to @xmath359 which is equal to @xmath360 by lemma [ iotasigmamu ] .

we study the action of the affine group @xmath0 of @xmath1 on the space @xmath2 of @xmath3-dimensional affine subspaces . given a compactly - supported zariski dense probability measure @xmath4 on @xmath0 , we show that @xmath2 supports a @xmath4-stationary measure @xmath5 if and only if the @xmath6-lyapunov exponent of @xmath4 is strictly negative . in particular , when @xmath4 is symmetric , @xmath5 exists if and only if @xmath7 .

virtually all estimates of the current star formation rates ( sfrs ) for galaxies in the universe rely on stellar population synthesis models for calibration ( as reviewed by @xcite ) . these models predict global observables , such as the total h@xmath1 or infrared flux , that can be compared to observations ( e.g. * ? ? ? * ; * ? ? ? this is necessary because individual forming stars can not be resolved in any galaxy other than our own . until recently , even estimates of the sfr of the milky - way have relied on global observables . such studies generally rely on indirect tracers of massive ( o- and early - b - type ) stars to determine a massive sfr . this value is then extrapolated to lower masses to derive a global sfr for the galaxy . for example , @xcite found a value of 5@xmath0yr@xmath2 , by making use of the fact that the integrated flux density from an region is a direct measure of the number of ionizing photons required to maintain that region , and is therefore an indirect measure of the number of o and early - b - type stars . a more recent example includes the estimate of 4@xmath0yr@xmath2from @xcite who use a measurement of the total mass of 2.8@xmath0of @xmath3al in the galaxy from the @xmath4-ray flux measured by integral as a proxy for the massive star population of the galaxy . @xcite find a value of 2.7@xmath0yr@xmath2 , by using the total 100@xmath5mflux of the galaxy and a conversion factor that depends on the same population synthesis model as used for other galaxies @xcite . finally , @xcite use the total free - free emission in the wmap foreground map as a probe of the massive star population , and derive a global sfr of 1.3@xmath0yr@xmath2 . the difference between the values may in part be due to the fact that these studies do not all assume the same initial mass function ( imf ) . for a recent review of the tracers and determinations of the galactic sfr , normalized to the same imf , we refer the reader to @xcite . in general , any total flux measure that is related to the sfr of a galaxy ( including the milky - way ) is completely dominated by the high mass stars , since these are responsible for virtually all of the luminosity of a galaxy . therefore , all of these methods are fundamentally limited by an extrapolation to lower masses , and higher sensitivity observations will not improve the estimates . for example , the ionizing flux for the regions or the @xmath4-ray flux from @xmath3al mentioned above are entirely dominated by the massive stars . in this letter we propose to measure the present - day galactic sfr by directly observing and counting pre - main sequence stars . estimates of the sfr of individual regions within a few hundred parsecs of the sun have recently been derived in this way @xcite , yet no such measurement exists on the scale of the galaxy . while galactic plane surveys with finite sensitivity preferentially favor higher mass stars , as surveys reach higher and higher sensitivities , the imf extrapolation to lower masses will gradually account for less and less of the resulting sfr , and the accuracy of this method will be much higher than for values derived from massive star formation indicators . in fact , this method may ultimately provide a more accurate measurement than for any other galaxy , because it is a direct measure of the very young , just formed stellar population , which is unresolved in most other galaxies . in this letter we make use of the _ spitzer_/irac glimpse survey of the galactic mid - plane @xcite , specifically glimpse i ( covering @xmath6@xmath7@xmath8@xmath7and @xmath9@xmath7 ) and glimpse ii ( which fills in the region for @xmath10@xmath7 , with @xmath9@xmath7for @xmath11@xmath7 , @xmath12@xmath7for 2@xmath7@xmath13@xmath7 , and @xmath14@xmath7for @xmath15@xmath7 ) . the total area covered by these two surveys is 274deg@xmath16 . we use the census of intrinsically red mid - infrared sources from ( * ? ? ? * hereafter r08 ) which includes 18,949 sources from these two surveys . the sources from r08 were selected using simple brightness and color selection criteria , namely 13.89@xmath17[4.5]@xmath176.50 , 9.52@xmath17[8.0]@xmath17 4.01 , and [ 4.5]-[8.0]@xmath171 . extended sources are excluded . in total , 18,949 sources were selected , and the completeness was estimated to be at least 65.7 . as shown in r08 , these sources consist mostly of young stellar objects ( ysos ; @xmath18 ) and asymptotic giant branch ( agb ) stars ( @xmath19 ) , and may include a small number of planetary nebulae ( at most @xmath20 ) . thus , there are likely to be approximately 9,500 to 13,300 ysos in the r08 census . taking into account the fact that the 65.7completeness is a lower limit , the actual number of ysos that would have been detected if the survey had been complete is in the range 9,500 to 20,200 . to derive a galactic sfr using the r08 census , we construct a population synthesis model for ysos in the galaxy , and apply the same observational constraints as for the r08 census , varying the model sfr such that the number of ` detected ' synthetic ysos matches the observed number . the model is constructed as follows . first a 3d distribution for star formation is assumed , from which the yso positions are randomly sampled . each yso is then assigned a random age and mass by assuming an imf and a lower and upper stellar age . the number of synthetic ysos sampled is thus directly related to the sfr for that model , which is given by the total mass of synthetic ysos divided by the difference between the upper and lower age . each synthetic yso is then assigned intrinsic magnitudes at irac wavelengths ( based on the stellar mass and age ) , which are scaled to its distance from the earth . a 3d dust distribution and an extinction law are assumed in order to compute the extinction along the line of sight . finally , the number of synthetic ysos that fall inside the glimpse i and ii survey areas , satisfy the brightness and color selection criteria from r08 , and are point sources , is compared to the observed number , and the sfr of the model is adjusted so that the two values match . the following paragraphs describe the assumptions made for each of these steps . the exact 3d distribution of ysos in the galaxy is uncertain , but there exist statistical estimates of where we expect stars to be forming . in particular , @xcite used the distribution of h and h@xmath21 as a function of galactocentric radius to estimate the star formation rate relative to that at the solar radius , by assuming a schmidt - type law relating the gas density and the star formation rate ( sfr@xmath22 ) , and taking into account the rate of passage of the gas through the spiral arms . the resulting radial profile was in good agreement with the distribution of tracers of massive stars ( specifically regions , pulsars , and supernova remnants ) found by previous studies . this radial function peaks around 4 to 5kpc . the values used were obtained from digitizing the solid line in figure 2 of @xcite . we assume that the azimuthal distribution of ysos is uniform , even though most star formation occurs in spiral arms . we make this assumption because the biggest effect of including spiral arms in the model would be to cause the longitude distribution of the sources to change on small scales , but would not change the overall number of detected synthetic sources significantly . therefore , the azimuthally symmetric distribution we assume is still statistically representative of the overall distribution of ysos . we assume a scale height of 80pc , approximately the height of the galactic thin disk . we take the sun to be located at a galactocentric radius of 8.5kpc@xcite , and 27pcabove the galactic plane @xcite . we adopt the imf from ( * ? ? ? * equations ( 1 ) and ( 2 ) ) . since the imf is being used to implicitly extrapolate the observed mass to lower masses , it is also necessary to choose a minimum and maximum mass . the exact limits used are not important , as long as virtually all the _ observed _ ysos fall inside that range . the sfr derived will then be the total sfr inside that range . we choose a lower mass of 0.1@xmath0and an upper mass of 50@xmath0 , since we do not expect to see any objects outside these limits 0.1@xmath0sources would be too faint , and 50@xmath0sources are very rare ( and saturated in glimpse ) . recent observations suggest that primordial circumstellar disks are present for a few myr then disappear on relatively short timescales ( e.g. * ? ? ? therefore , since we are considering ysos with infrared excesses typical of primordial circumstellar disks ( and of younger stages of evolution ) we can assume that most of the r08 sources are younger than a few myr . we choose a upper limit on the age of the synthetic ysos of 2myr , and a lower limit on the age of 1,000yr , as we do not expect to be able to see any sources younger than this . it is unlikely that the overall star formation has changed significantly over the last few myr , which is a small timescale compared to the galactic rotation period of over 200myr . therefore , we do not need to worry about time variations in the sfr , in contrast to methods based on main - sequence stars , where choosing a constant sfr is a strong assumption . the 3d distribution of dust in the galaxy is the subject of active research ( e.g. @xcite ) . we use the distribution from ( * ? ? ? * equations ( 1 ) , ( 3 ) , and table ( 2 ) ) , which is a simple double exponential model based on modeling the far - ir dust emission , and is accurate on large scales . there exist more complex models for the distribution of dust , such as that from @xcite which includes spiral arms . the effect of a more clumpy extinction distribution will be to introduce scatter in the extinction values , but the overall effect is not likely to be significant . in recent years there have been a number of determinations of the extinction law at mid - infrared wavelengths for both the interstellar medium @xcite and for star forming regions @xcite . for this work , we are modeling the large - scale extinction from the interstellar medium and therefore we use the values from @xcite for the irac bands . the final piece of the puzzle that is needed to construct our model is a prescription to convert the masses and ages of the sources into observable magnitudes . unfortunately , no simple prescription exists for this , as the brightness and color depend on the amount of circumstellar dust and the properties of the central source , which in turn are far from certain for all stellar masses and ages . of all the assumptions made in this letter , this is likely to be most crucial one . in ( * hereafter r06 ) we computed a large set of model seds for ysos with a large range of evolutionary stages ( from heavily embedded sources to transitional disks ) and stellar masses ( from 0.1@xmath0to 50@xmath0 ) . to sample the parameters , each source was assigned a mass and age , and the stellar temperature and radius were found from evolutionary tracks . subsequently , the disk and envelope parameters were sampled as a function of these stellar parameters . the sampling attempted to take into account observational and theoretical constraints on pre - main - sequence evolution . it is important to note that the parameter space in r06 does not constitute an evolutionary scenario , but rather allows for different scenarios , some of which may be unrealistic . however , the general trends are nevertheless close to how we believe stars form . for each synthetic yso , we choose the model with the closest stellar mass and age , and sample a random viewing angle . since we are assuming that the sources in the r08 census are younger than 2myr , the r06 models with ages between 2 and 10myr are not used . the irac magnitudes of the models we do use are dominated by the stellar luminosity and temperature , and the envelope infall rate ( which determines the envelope mass and therefore the degree of obscuration of the central star ) . therefore , the most important assumptions from r06 that will impact this work are the choice of stellar evolutionary tracks ( which fix the stellar luminosity and temperature for a given age and mass ) and the choice of the time dependence of the envelope infall rate . the latter is sampled from a range that is constant before 0.1myr and decreases to zero by 1myr . the r06 models were computed in 50 circular apertures from 100 to 100,000au , effectively providing radial brightness profiles at all wavelengths . therefore , in the population synthesis model , we can find the flux for each synthetic source in an aperture corresponding to the resolution of the irac observations ( 2 ) . in addition , we use this information to reject sources that would have been extended in the glimpse survey , and would therefore have been excluded from the r08 census . we do this by selecting only sources where 99of the flux falls inside an aperture 2 in radius . this reduced the number of final ` detected ' ysos by approximately 10 . ignoring uncertainties in the input assumption , we find that a sfr in the range 0.68 to 1.45@xmath0/yr reproduces the number of observed ysos , where the range of values accounts for the uncertainty in the actual number of ysos , which is in the range 9,500 to 20,200 ( [ sec : observations ] ) . to compare the properties of the ` detected ' synthetic ysos with those of the 11,649 selected candidate ysos from r08 , we computed a model with @xmath23 synthetic ysos corresponding to a sfr of 0.83@xmath0/yr of which 11,919 were ` detected ' . this model is the one shown in figures [ fig : imf ] to [ fig : lonlat ] . figure [ fig : imf ] shows the mass distribution of all the synthetic ysos , the subset that fall in the survey area , and the subset that would have been detected by _ spitzer _ and included in the r08 census . the distribution of ` detected ' synthetic sources is biased towards high masses , and peaks just under 10@xmath0 . around 50of all 10@xmath0synthetic ysos are detected . the drop - off at lower and higher masses is due to the lower and upper brightness cutoffs . since approximately 2.7 million synthetic ysos between 0.1 and 50@xmath0are required galaxy - wide to explain the observed numbers , the r08 census represents less than 0.5% of all ysos in the galaxy . figure [ fig : spatial ] shows the 3d spatial distribution of all the synthetic ysos that would have been included in the r08 census , color - coded by the interstellar extinction to earth . most sources are seen within 10kpcbut many are seen out to the far side of the galaxy , even though the interstellar extinction is significant . the ring with a 5kpcradius is due to the peak in the radial sfr distribution assumed ( [ sec : sfrdist ] ) , but we note that the real distribution of ysos does not necessarily trace a ring . finally , figure [ fig : lonlat ] shows the longitude and latitude distribution , as well as the @xmath24 $ ] and @xmath25-[8.0]$ ] distributions for all the synthetic ysos that would have been included in the r08 census , and for all the yso candidates in the r08 census . even though no effort was made to fit the model distributions to the observed ones , the two are in reasonable agreement . the [ 8.0 ] magnitude distributions do diverge at the faint end , but the difference never exceeds 50 , and this difference may be due to remaining contaminating agb stars . both the observed and model longitude distribution drop off around longitudes of 30 to 40@xmath7 on either side of the galactic center . these longitudes correspond to those where the line of sight is tangent to the peak in the radial distribution of star formation ( [ sec : sfrdist ] ) . also of interest is that the observed latitude distribution of sources is asymmetric , with @xmath26 of sources at negative latitudes . the model also predicts a similar asymmetry ( albeit slightly larger ) , with @xmath27@xmath28 of sources at negative latitudes . the reason that the model presents an asymmetry is entirely due to the fact that the sun is displaced 27pcabove the galactic mid - plane . therefore , it is safe to infer from this that the asymmetry in the latitude distribution of observed sources is also due to the displacement of the sun above the galactic mid - plane . by developing a population synthesis model for the milky - way , and adjusting the star formation rate such that the number of synthetic ysos that would have been present in the r08 census matched the actual number of observed ysos , we obtain a star formation rate of 0.68 to 1.45@xmath0/yr . the uncertainties on this value take into account only the uncertainty in the actual number of observed ysos ( since these are not perfectly separable from agb stars ) and the uncertainty in the completeness . our result is of the same order as but lower than previous estimates ( [ sec : introduction ] ) , but this can be explained in part by differences in the assumed imf . the previous values quoted in [ sec : introduction ] assume either a @xcite or @xcite imf , which over - predict the number of low - mass stars by approximately 50compared to a more realistic @xcite imf . if we assume a salpeter imf from 0.1 to 50@xmath0 , we obtain a sfr in the range 0.98 to 2.09@xmath0/yr , in better agreement with previous values . we find that the ysos in the r08 census are dominated by sources in the range 3 to 20@xmath0 , and that these are seen out to very large distances ( 10 to 15kpc ) in some cases . the longitude and latitude distribution of the synthetic ysos are in reasonable agreement with the observed distribution , and we find that the asymmetry in the latitude distribution can be entirely explained by the fact that the sun is not located exactly in the mid - plane . the input assumptions for the model are simple , consisting of an axisymmetric distribution of synthetic ysos peaking around 4 to 5kpc , a double exponential distribution for the dust , a kroupa imf , a standard mid - infrared extinction law , and a prescription for the magnitudes of ysos based on the stellar mass and age , using the models from r06 . the assumption that is likely to matter the most is the latter , namely knowing what brightness and color to assign to ysos as a function of age and mass . this is because while there are many examples of well - studied low - mass ysos with approximate ages , the pre - main - sequence evolution of 10@xmath0stars , which dominate the ` detected ' synthetic ysos , is poorly determined . the model will be developed into a much more powerful diagnostic tool in the future . in particular , we plan to fit the angular , brightness , and color distributions of the sources to improve the estimate of the sfr and place constraints on other input parameters and assumptions . for example , the angular distribution of the sources will help place constraints on the model for the sfr as a function of galactocentric radius , and the color and brightness distributions will help understand whether the assumptions for the intrinsic magnitudes of ysos and the extinction properties are sensible , or whether they need to be modified . in addition , we will thoroughly explore the parameter space in order to understand how well constrained the input assumptions and parameters are , and what additional data could resolve degeneracies . finally , we plan to model multiple datasets simultaneously , to reproduce not only the observable properties of sources seen in any one survey , but all available infrared or sub - mm surveys of the galactic plane , such as the ukidss and herschel surveys . we would like to thank the anonymous referee for useful suggestions which helped improve this letter . support for this work was provided by nasa through the spitzer space telescope fellowship ( tr ) and theoretical research ( tr , bw ) programs , through a contract issued by the jet propulsion laboratory , california institute of technology under a contract with nasa .

we present initial results from a population synthesis model aimed at determining the star formation rate of the milky - way . we find that a total star formation rate of 0.68 to 1.45@xmath0/yr is able to reproduce the observed number of young stellar objects in the _ spitzer_/irac glimpse survey of the galactic plane , assuming simple prescriptions for the 3d galactic distributions of ysos and interstellar dust , and using model seds to predict the brightness and color of the synthetic ysos at different wavelengths . this is the first galaxy - wide measurement derived from pre - main - sequence objects themselves , rather than global observables such as the total radio continuum , h@xmath1 , or fir flux . the value obtained is slightly lower than , but generally consistent with previously determined values . we will extend this method in the future to fit the brightness , color , and angular distribution of ysos , and simultaneously make use of multiple surveys , to place constraints on the input assumptions , and reduce uncertainties in the star formation rate estimate . ultimately , this will be one of the most accurate methods for determining the galactic star formation rate , as it makes use of stars of all masses ( limited only by sensitivity ) rather than solely massive stars or indirect tracers of massive stars .

the effect of electron electron ( @xmath2 ) interactions on transport in bulk - like systems of various dimensionalities is still an area of active research to this day , both experimentally@xcite and theoretically@xcite . on the face of it , it would seem that in the case of doped parabolic band semiconductors where umklapp processes are negligible , @xmath2 scattering should not affect the linear transport properties of bulk - like systems ( purely quantum effects such as weak - localization corrections excepted ) since an @xmath2 scattering event conserves the total current in the system . nevertheless , it has been appreciated for a long time that @xmath2 scattering _ can _ affect the mobility of a system semiclassically by scattering carriers into or out of parts of the brillouin zone which are strongly affected by the other available scattering mechanisms.@xcite in ref . , it was shown that the expression for mobility in the presence of quasi - elastic scatters takes on different forms in the limits of zero and infinitely strong @xmath2 scattering . given a quasi - elastic energy - dependent transport ( i.e. , calculated with the @xmath3 term ) scattering time @xmath4 due to other scattering processes in the system such as acoustic phonons or impurities , the mobilities of the system for the cases of zero and infinite @xmath2 scattering rates , respectively , are given by @xmath5 here , @xmath6 where @xmath1 is the carrier density of the system , @xmath7 , @xmath8 is the fermi - dirac distribution function , @xmath9 is the dimensionality of the system , and we are assuming an isotropic parabolic band system . clearly , for the case when that temperature @xmath0 is small on the scale of the energy scale over which @xmath4 varies significantly , @xmath10^{-1}$ ] . conversely , in the case where @xmath0 is large on the scale over which @xmath4 varies , there can be significant differences in the calculated mobility using the two different methods . for example , we show below that in particular cases in gaas quantum wells , @xmath11 . thus , for accurate theoretical determination of the mobility at the semiclassical level , it is important that @xmath2 scattering effects are included . furthermore , it has been shown@xcite that experiments measuring the drag rate between electron gases between two coupled quantum wells is sensitive of the exact details of the linear response distribution function @xmath12 in each layer . since @xmath12 is strongly affected by @xmath2 scattering in the intermediate temperature regime @xmath13 ( where @xmath14 is the fermi temperature ) , it is important to include the effects of @xmath2 scattering in calculations of the drag rate . in this paper , we demonstrate an efficient way of including @xmath2 scattering in the calculation of for linear transport for two - dimensional cylindrically symmetric systems , within the semiclassical boltzmann equation formalism . a similar calculation has been presented in three - dimensions.@xcite within this formalism , the boltzmann equation for linear response can be solved exactly ( within numerical accuracy ) . we have included full effects of finite - temperature dynamical screening , which automatically includes phenomena such as landau damping and collective mode enhancements to scattering . the description of the @xmath2 scattering formalism for the boltzmann equation is given in sections [ sec : formal ] and [ sec : scatop ] , and section [ sec : results ] contains the results and discussion . throughout this paper , we assume that bands are isotropic and parabolic . the @xmath2 scattering occurs in the presence of other conduction electrons , and hence the bare interparticle coulomb interaction @xmath15 is screened . furthermore , at the intermediate temperatures in which we are interested , the energy transfer between the electrons in a scattering event is often a substantial fraction of the kinetic energy of the electrons , and hence the scattering matrix elements for @xmath2 interactions should be calculated using the _ dynamically _ screened coulomb interaction @xmath16 , where @xmath17 is the dielectric function . in this paper , we use @xmath17 given by the random phase approximation ( rpa ) , which we evaluate using a method described previously by us,@xcite and we use the born approximation for the scattering probability . the scattering probability @xmath18 for a pair of electrons initially in states @xmath19 to be scattered to @xmath20 depends on whether or not the electrons have the same or opposite spins . for electrons with the same spin , say @xmath21 , @xmath22 the fraction @xmath23 in eq . ( [ likespin ] ) is due to double counting , since @xmath24 and @xmath25 describe exactly the same process . for opposite spins , @xmath26 there is an equal probability that an electron scatters off another electron with equal or opposite spin , so one can sum over the eqs . ( [ likespin ] ) and ( [ unlikespin ] ) to obtain an average " scattering probability@xcite @xmath27\bigr\}. \label{averagew}\end{aligned}\ ] ] the first and second terms in eq . ( [ averagew ] ) referred to as the direct and exchange terms , respecitively . in practice , the exchange term often makes calculations considerably more complicated and is usually ignored . the physical grounds for doing so are as follows . first , @xmath2 collisions are usually dominated by small @xmath28 scattering ( because @xmath29 falls quickly with @xmath28 for finite @xmath30 ) . the direct [ exchange ] term has the form @xmath31 [ @xmath32 with @xmath33 , which implies that the direct term usually dominates over the exchange . also , the sign of exchange term can sometimes be negative , which leads to cancellation of this term within collision integral . the effect of the exchange term was studied in a 3d system with scatically screened interaction.@xcite there it was found that the exchange term was significant for @xmath34 but not for @xmath35 . in the calculations that follows we have used @xmath36 , and since we furthermore include dynamical screening , which leads to a peaked interaction at small @xmath28 , we can assume that the direct interaction dominates in our case . we write down the formal expressions for the electron electron scattering operator both including and excluding the exchange term . however , in the actual numerical evaluation , we ignore the exchange interaction . the boltzmann equation for electrons in uniform electric field @xmath37 producing a force @xmath38 is @xmath39 where the subscripts @xmath40 and p , i are for scattering due to electron electron interactions and the phonon+impurity interactions , respectively . we define the function @xmath41 , related to the deviation of the distribution function from equilibrium , as @xmath42 \psi(\mbox{\boldmath{$k$}}).\ ] ] this function can be written in terms of a sum of angular components @xmath43 where @xmath44 is the angle from an axis of symmetry ( here , the direction of the electric field ) . by the assumption of cylindrical symmetry of the system , the scattering terms in the boltzmann equation do not mix different @xmath45 components , and one can isolate and concentrate on the @xmath46 component , @xmath47 , since this is the one that affects the current and hence the mobility . the @xmath46 component of the linearized electron electron collision operator ( i.e. , neglecting higher powers in @xmath41 ) , which we denote @xmath48 $ ] , is@xcite @xmath49(k ) & = & -2\int \frac{d\mbox{\boldmath{$k$}}'}{(2\pi)^2 } \int \frac{d\mbox{\boldmath{$q$}}}{(2\pi)^2}\ \overline{w}(\mbox{\boldmath{$k$}}+\mbox{\boldmath{$q$ } } , % % following line can not be broken before 80 char \mbox{\boldmath{$k$}}'-\mbox{\boldmath{$q$}};\mbox{\boldmath{$k$}},\mbox{\boldmath{$k$}}')\times \nonumber\\ & & \phantom{xxx } f^0(\mbox{\boldmath{$k$ } } ) f^0(\mbox{\boldmath{$k$ } } ' ) \bigl[1 - f^0(\mbox{\boldmath{$k$}}+\mbox{\boldmath{$q$ } } ) \bigr ] \bigl[1 - f^0(\mbox{\boldmath{$k$}}'-\mbox{\boldmath{$q$}})\bigl]\nonumber\\ & & \phantom{xxx } \delta\left(\epsilon_{\mbox{\boldmath{$k$ } } } + \epsilon_{\mbox{\boldmath{$k$ } } ' } - \epsilon_{\mbox{\boldmath{$k$}}+\mbox{\boldmath{$q$ } } } - \epsilon_{\mbox{\boldmath{$k$ } } ' - \mbox{\boldmath{$q$}}}\right ) \nonumber\\ & & \phantom{xxx } \bigl\{\psi_1(k ) x_{\mbox{\boldmath{$k$}},\mbox{\boldmath{$f$ } } } + \psi_1(k')x_{\mbox{\boldmath{$k$}}',\mbox{\boldmath{$f$ } } } \nonumber\\ & & \phantom{xxx } - \psi_1(|\mbox{\boldmath{$k$}}+\mbox{\boldmath{$q$}}| ) x_{\mbox{\boldmath{$k$}}+\mbox{\boldmath{$k$}},\mbox{\boldmath{$f$ } } } - \psi_1(|\mbox{\boldmath{$k$}}'-\mbox{\boldmath{$q$}}| ) x_{\mbox{\boldmath{$k$}}'-\mbox{\boldmath{$q$}},\mbox{\boldmath{$f$ } } } \bigr\}. \label{eecoll}\end{aligned}\ ] ] here , @xmath50 is the cosine of the angle between @xmath51 and @xmath52 . the goal is to write the operator @xmath53 in the form @xmath54(k ) = x_{\mbox{\boldmath{$k$}},\mbox{\boldmath{$f$}}}\int_0^\infty dp\ \ p\ ; k(k , p)\ \psi_1(p).\ ] ] the kernal @xmath55 is symmetric , from detailed balance,@xcite and the extra factor of @xmath56 in the integral comes from phase - space . thus , in order to incorporate electron electron scattering for a particular density and temperature into a calculation , one need only generate @xmath55 once and store it ; it can then be used for all calculations involving electron electron scattering at that density and temperature . the four @xmath57 s in eq . ( [ eecoll ] ) give four terms , each of which give a contribution to the kernel , @xmath58 . in the following subsections , we explicitly write down the form of each of these kernels . the @xmath59 can be factored out , and we obtain @xmath60 \nonumber\\ & & \phantom{hell}2\int \frac{d\mbox{\boldmath{$k$}}'}{(2\pi)^2 } f_0(\mbox{\boldmath{$k$}}')\,[1-f_0(\mbox{\boldmath{$k$}}'- \mbox{\boldmath{$q$ } } ) ] \overline{w}(\mbox{\boldmath{$k$}}+\mbox{\boldmath{$q$ } } , \mbox{\boldmath{$k$}}';\mbox{\boldmath{$k$ } } , \mbox{\boldmath{$k$}}'+\mbox{\boldmath{$q$}})\nonumber\\ & & \phantom{hell}\delta\left(\varepsilon_{\mbox{\boldmath{$k$}}'+ \mbox{\boldmath{$q$ } } } - \varepsilon_{\mbox{\boldmath{$k$ } } ' } - \{\varepsilon_{\mbox{\boldmath{$k$}}+\mbox{\boldmath{$q$}}}- \varepsilon_{\mbox{\boldmath{$k$}}}\}\right).\end{aligned}\ ] ] in the event where the exchange interaction can be neglected , one obtains , as in ref . ( we denote the scattering integral which neglects the exchange interaction with an asterisk ) @xmath61\ ; \frac{2\pi}{\hbar } \mbox{\boldmath{$q$}}}-\varepsilon_{\mbox{\boldmath{$k$}}})|^2\nonumber\\ & & \ \ \ \ \mathrm{im}[\chi(\mbox{\boldmath{$q$}},\varepsilon_{\mbox{\boldmath{$k$ } } } - \varepsilon_{\mbox{\boldmath{$k$}}+\mbox{\boldmath{$q$ } } } ) ] \ ; n_b(\varepsilon_{\mbox{\boldmath{$k$}}+\mbox{\boldmath{$q$ } } } -\varepsilon_{\mbox{\boldmath{$k$}}})\bigr].\end{aligned}\ ] ] where @xmath62^{-1}$ ] is the bose function , and @xmath63 is the rpa polarizability . since @xmath65 and the @xmath66 terms vanish from symmetry considerations , we can write @xmath67 . then , the second kernel is @xmath68 [ 1-f^0(\mbox{\boldmath{$p$}}-\mbox{\boldmath{$q$}})]\nonumber\\ & & \phantom{\ \ \ \ \ } \delta(\varepsilon_{\mbox{\boldmath{$k$ } } } + \varepsilon_{\mbox{\boldmath{$p$ } } } - \varepsilon_{\mbox{\boldmath{$k$}}+\mbox{\boldmath{$q$ } } } - \varepsilon_{\mbox{\boldmath{$k$}}-\mbox{\boldmath{$q$ } } } ) . \label{i2}\end{aligned}\ ] ] the @xmath69 integration can be evaluated by the change of variables @xmath70 then , the @xmath71-function in eq . ( [ i2 ] ) becomes @xmath72\right),\ ] ] which gives @xmath73\ ; [ 1-f^0(\varepsilon_{\overline{\mbox{\boldmath{$k$}}}-\mbox{\boldmath{$q$}}})],\end{aligned}\ ] ] where @xmath74 is the angle between @xmath75 and @xmath76 . using @xmath78 in the integrand and letting @xmath79 , gives @xmath80}{(2\pi)^2 } \int_0^{2\pi } d\theta_{\mbox{\boldmath{$p$}},\mbox{\boldmath{$k$}}}\ \cos\theta_{\mbox{\boldmath{$p$}},\mbox{\boldmath{$k$ } } } \phantom{\ \ \ } \int \frac{d\mbox{\boldmath{$k$}}'}{(2\pi)^2 } f^0(\mbox{\boldmath{$k$}}'+\mbox{\boldmath{$p$}}-\mbox{\boldmath{$k$ } } ) [ 1-f^0(\mbox{\boldmath{$k$ } } ' ) ] \nonumber\\ & & \overline{w}(\mbox{\boldmath{$p$}},\mbox{\boldmath{$k$ } } ' ; \mbox{\boldmath{$k$}},\mbox{\boldmath{$k$}}'+\mbox{\boldmath{$p$ } } -\mbox{\boldmath{$k$ } } ) \delta(\varepsilon_{\mbox{\boldmath{$p$ } } } + \varepsilon_{\mbox{\boldmath{$k$ } } ' } - \varepsilon_{\mbox{\boldmath{$k$ } } } -\varepsilon_{\mbox{\boldmath{$k$}}'+\mbox{\boldmath{$p$}}- \mbox{\boldmath{$k$}}}).\end{aligned}\ ] ] the @xmath71-function @xmath81 reduces the @xmath82 dimensional integral to one dimension . if one neglects exchange , then as with the first term the @xmath82 integral can be done , giving @xmath83 } { 4 \sinh^2[(\varepsilon_{\mbox{\boldmath{$p$}}}- \varepsilon_{\mbox{\boldmath{$k$}}})/(2k_b t ) ] } \nonumber\\ & & \int_0^\pi d\theta_{\mbox{\boldmath{$k$}},\mbox{\boldmath{$p$ } } } cos\theta_{\mbox{\boldmath{$k$}},\mbox{\boldmath{$p$ } } } \ \frac{2\pi}{\hbar}|v(\mbox{\boldmath{$p$}}- \mbox{\boldmath{$k$}},\varepsilon_{\mbox{\boldmath{$k$}}}- \varepsilon_{\mbox{\boldmath{$p$}}})|^2\ ; \mbox{im}[\chi(\mbox{\boldmath{$p$}}-\mbox{\boldmath{$k$ } } , \varepsilon_{\mbox{\boldmath{$k$}}}-\varepsilon_{\mbox{\boldmath{$p$}}})]\end{aligned}\ ] ] the kernel is @xmath85\ f^0(\mbox{\boldmath{$p$}}+\mbox{\boldmath{$q$}})\right.\nonumber\\ & & \phantom{\ \ \ } \left.\delta(\varepsilon_{\mbox{\boldmath{$k$ } } } + \varepsilon_{\mbox{\boldmath{$p$}}+\mbox{\boldmath{$q$ } } } - \varepsilon_{\mbox{\boldmath{$k$ } } + \mbox{\boldmath{$q$ } } } - \varepsilon_{\mbox{\boldmath{$p$}}})\right\}\end{aligned}\ ] ] the term in the @xmath71-function goes as @xmath86 which reduces the @xmath69 integration down to one dimension . in fact , the kernels for higher order components are very similar to the ones given above . for an angular variation proportional to @xmath87 , the @xmath88 term is identical for all @xmath1 , whereas with @xmath89 , @xmath90 and @xmath91 one simply replaces @xmath46 with @xmath92 in the @xmath44 integration . we have shown that one can calculate the matrix @xmath55 which gives the electron electron scattering term for small deviations from the equilibrium . once the matrix @xmath55 has been calculated , one simply needs to iterate the equation for @xmath47 until convergence is obtained . in order to calculate @xmath47 for the case when the @xmath2 scattering rate dominates , it is often useful to use the fact that electron electron scattering leaves a drifted fermi - dirac distribution invariant.@xcite thus , for the case of elastic or quasi - elastic collisions , one can define @xmath93 where @xmath94 is for a drifted fermi - dirac distribution . any @xmath95 can be used . in our case , because we cut off the matrix @xmath55 at a @xmath96 , which implicitly sets @xmath97 , we chose @xmath95 so that @xmath98 so that the distribution function is continuous at @xmath96 . we write the linearized @xmath2 scattering term as @xmath99 = -\frac{\psi(k)}{\tau_{ee } } + j[\psi_1(k)]\ ] ] where the first term on the right hand side corresponds to the diagonal @xmath88 term and @xmath100 corresponds to @xmath101 . the boltzmann equation for @xmath47 in the case when the other scattering mechanisms are quasielastic ( which might include acoustic phonon scattering , which generally involves very small energy electron loss ) is @xmath102\left(\tilde{\psi}_1(k)+ \psi^{\mathrm{df}}_1(k)\right ) } { \tau_{\mathrm{el}}(k ) } -\frac{\tilde{\psi}_1(k)}{\tau_{e - e}(k ) } + j[\tilde{\psi}_1](k),\ ] ] where @xmath103 is the quasi - elastic scattering rate . this implies that one must iterate the equation @xmath104 + i^*[\tilde{\psi}_1 ] - f_0(k)[1-f_0(k ) ] \psi^{\mathrm{df}}_1(k)\tau^{-1}_{\mathrm{el}}(k ) } { f_0(k)[1-f_0(k)]\tau^{-1}_{\mathrm{el}}(k ) + \tau_{e - e}^{-1}(k ) } \ ] ] to find @xmath47 . we study the case of electrons confined in a 100 wide square gaas quantum well with infinite barriers . we assume that the there is a @xmath71-doping layer of ( uncorrelated ) charged impurities , equal in density to that of the electrons in the well , situated a distance @xmath9 away from the center of the well . we included three scattering mechanisms : @xmath2 , charged - impurity and acousitic - phonon scattering , and we approximated the acousitic - phonon scattering as being elastic . we calculated the matrix in the form on 200 by 200 grid - points from @xmath105 to @xmath106 , and we used spline routines to interpolate between the grid points . the @xmath107 and @xmath108 diverge logarithmically as @xmath109 , which complicates the splining procedure , but we got around this problem by splining @xmath110 , which is a smooth function . in fig . 1 , we show the deviation function @xmath111 for a fixed density @xmath112 and temperature @xmath113 , but with several different distances @xmath9 of the ionized impurities from the center of the quantum well . note that when the distribution is a drifted fermi - dirac function , @xmath114 . thus , as the impurities are moved further away , the impurity scattering becomes weaker and the @xmath2 scattering starts to dominate@xcite and drive the distribution function closer to a drifted fermi - dirac function . the inset shows @xmath47 calculated both including and excluding electron electron scattering for @xmath115 , which shows more clearly the effect of @xmath2 on @xmath47 . while transport experiments in a single layer are not particularly dependent on the details of the shape of @xmath47 , it has been shown@xcite that drag experiments in coupled quantum wells are quite sensitive to the details of @xmath47 . in particular , when @xmath47 rises faster than @xmath116 ( which implies that there are more carriers in the high energy region than for a drifted fermi - dirac distribution ) , the drag rate increases because high energy particles give a larger contribution to the overall drag rate , and there is greater opportunity for coupling to the plasmons of the system , which also enhances the drag rate . therefore , for the purpose of calculating the drag rate in coupled quantum wells in intermediate temperatures , it is crucial to calculate the actual form of @xmath47 accurately , including all salient scattering mechanisms . 2 shows the mobility @xmath117 as a function of ionized impurity distance @xmath9 from the center of the quantum well . also shown are the mobilities @xmath118 and @xmath119 , for the limits of zero and infinite electron electron scattering , respectively . the @xmath120 is generally larger than @xmath121 because @xmath2 scattering tends to scatter runaway " electrons with large velocities ( where the impurity scattering rate is small ) back into lower velocity states . the inset shows that @xmath120 for this case can be almost twice @xmath121 . as @xmath9 becomes larger , the @xmath2 scattering dominates over all other scattering mechanisms and @xmath122 . conversely , for small @xmath9 , the impurity scattering is relatively large compared to the electron scattering , and @xmath117 is closer to the @xmath120 than @xmath121 . the crossover from @xmath120 to @xmath121 is shown with the open squares in the inset of fig . 2 . the crossover occurs when the impurity scattering and electron electron mean free paths become equivalent . the transport scattering rate , for impurity scattering is given by @xmath123 . the electron electron scattering in two - dimensional systems is approximately given by@xcite @xmath124 ^ 2 \left[\ln\left[\frac{e_f}{k_b t}\right ] + \ln \left[2\frac{q_{tf}}{p_f}\right ] + 1 \right].\ ] ] for this system , this is on the order of @xmath125 . thus , the crossover point , which should occur when these two are equal , is given by @xmath126 . an inspection of fig . 2 also shows this to hold . chabasseur - molyneux _ et al._@xcite have experimentally found this crossover in gaas / algaas heterojunctions . finally , at @xmath127 , @xmath128 , implying the @xmath2 scattering has caused a substantial reduction in the mobility . to summarize , in this paper we have described a method of including electron electron scattering , including full finite - temperature dynamical screening , exactly in the boltzmann equation , for small deviations of the distribution function from equilibrium . using this method to calculate the distribution function and mobilities for electrons in a gaas quantum well , we find a well - defined crossover from @xmath120 to @xmath121 ( which can be significantly different from each other ) when the @xmath2 and impurity scattering mean free paths are equivalent . for certain parameters studied , @xmath2 is responsible for reduction in the mobility of up to 40% . we thank karim el - sayed for useful discussions . kf was supported by the carlsberg foundation . the @xmath129 in eq . ( [ likespin ] ) term has been transformed to the @xmath130 term by a change of variables , @xmath131 . this transformation leaves the product of other terms in the electron electron scattering integral invariant . strictly , a `` transport '' @xmath2 scattering rate ( as opposed to lifetime rate ) , equivalent to the impurity _ transport _ scattering rate , should be computed in order to make the comparison between the two . while the transport time merely involves an extra factor of @xmath134 for impurities , it is not so simple for @xmath2 because of the complex structure of the collision integral . hence , for heuristic purposes , we use the lifetime , which seems to give the right order of magnitude in this case .

we describe a method for numerically incorporating electron electron scattering in quantum wells for small deviations of the distribution function from equilibrium , within the framework of the boltzmann equation . for a given temperature @xmath0 and density @xmath1 , a symmetric matrix needs to be evaluated only once , and henceforth it can be used to describe electron electron scattering in any boltzmann equation linear - response calculation for that particular @xmath0 and @xmath1 . using this method , we calculate the distribution function and mobility for electrons in a quantum - well , including full finite - temperature dynamic screening effects . we find that at some parameters which we investigated , electron electron scattering reduces mobility by approximately 40% .

previous work@xcite on dilute emulsions has shown that , when coarsening is solely caused by a diffusive flux of dissolved dispersed - phase molecules between droplets ( ostwald ripening@xcite ) , coarsening may be prevented by the addition of a sufficient number of molecules that are insoluble in the continuous phase and hence _ trapped _ within droplets . the trapped molecules provide an osmotic pressure which counteracts the laplace pressure due to surface tension@xcite ( which drives coarsening ) ; resulting in ` osmotic stabilisation ' . a quantitative criterion for stability , valid even for emulsions with polydispersity in both the droplets sizes and number of trapped molecules they contain , was given in @xcite . it was also found in @xcite that for an ` insufficiently stabilised ' emulsion ( without enough trapped species to obey the required criterion ) , the subsequent coarsening was qualitatively unaltered from that without any of the trapped species @xcite . the only effect was is to reduce the effective volume fraction of the coarsening droplets by an amount corresponding to the final volume of a population of small droplets that attain coexistence with the coarsened bulk phase . the latter are prevented from entirely dissolving by the trapped molecules they contain . when the effective volume fraction is reduced to zero , full stability is achieved . in the present paper we discuss whether analogous conditions exist for the osmotic stabilisation of concentrated emulsions and foams containing trapped insoluble molecules in the dispersed phase ; we again consider both the monodisperse and the polydisperse case . previously the use of nitrogen as a trapped species to stabilise foams has been investigated with theory@xcite , experiment@xcite , and computer simulations@xcite , but theoretical conditions for stability of foams with nonspherical bubbles were not found . the use of other trapped gases to inhibit dissolution ( as opposed to coarsening ) of spherical bubbles is addressed in@xcite where the effect of condensation of the included gases is also addressed ; we do not consider this here . in our work , we treat the idealised case of a fully insoluble trapped species ( perfect trapping ) whereas in many cases the results will be modified by residual solubility effects ( considered for the dilute case in@xcite ) . some gases are , however , practically insoluble in water , for example c@xmath6f@xmath7 @xcite ; and in the case of emulsions , the insoluble limit is easily achieved ( by using oligomeric species ) . for simplicity we do not consider the effects of residual solubility in the present work . from now on we use mainly the language of foams , in which the trapped and soluble gas are treated as ideal , but the work is also applicable to dense emulsions whose droplets contain an ideal mixture of soluble and trapped molecules . the only distinction between these cases is that foams are compressible , but in fact much of the paper concerns _ effectively incompressible _ foams , in which pressure differences between bubbles are negligible compared with their mean pressures , and the total bubble volume is conserved . the osmotic stabilisation of foams is a far more complex proposition than for dilute emulsions . in a dilute emulsion , each droplet is spherical with an internal pressure that is increased by the droplets laplace pressure , which is directly related to the bubble s volume through its spherical geometry , and equals @xmath8 for a droplet of radius @xmath9 and surface tension @xmath10 . since nonspherical foam bubbles in contact have no simple relationship between their volume and surface area , there is no direct relationship , in a concentrated foam , between a bubble s pressure and its volume . a bubble @xmath1 with a radius of curvature @xmath11 at its plateau borders has a laplace pressure of @xmath12 , and an internal pressure of @xmath13 . given a connected domain of the liquid phase , the atmospheric pressure @xmath3 equals the pressure in the plateau borders ( we ignore the effects of gravity throughout ) . the drier the foam , the smaller the borders and the larger the laplace pressure , so that if osmotic stabilisation of a foam or emulsion required the pressure of trapped molecules to balance the laplace pressure ( as it does in the dilute case ) osmotic stabilisation would be hard to achieve . however , another relevant length scale is @xmath14 , where @xmath15 is the volume of bubble @xmath1 . since the curvature of adjacent bubble - bubble faces are of order @xmath16 , the pressure differences between adjacent bubbles ( which are responsible for coarsening ) , are of order @xmath17 . so if the osmotic stabilisation of a foam merely requires the partial pressure of trapped molecules to balance @xmath18 , then osmotic stabilisation is a reasonable proposition . we shall confirm below that this is so . the reduction in the driving force for coarsening from the level suggested by the laplace pressure has long been recognised ( see e.g. , @xcite ) but the implications of this for stabilisation by trapped species has not previously been addressed in detail . in what follows we first give in section [ dilutefoams ] some results for the dilute case ( based on @xcite ) , and then investigate what factors determine the pressure within concentrated foam bubbles . we follow the approach of princen@xcite , and study the osmotic compression ( at atmospheric pressure @xmath3 ) of previously spherical foam bubbles by an osmotic pressure @xmath4 . section [ compressed / drained ] discusses the disjoining pressures between bubbles , the uniformity or otherwise of the osmotic pressure @xmath4 , and the condition for mechanical equilibrium with an excess bulk gas phase created by coarsening . the increase in a bubble s pressure above that of a bulk gas is defined by @xmath19 , and we argue that typically @xmath20 . this is explicitly confirmed in section [ 2dmodel ] for princen s monodisperse 2d model . we call @xmath5 the _ geometric pressure _ and identify it ( rather than laplace pressure ) as the driving force for coarsening , in general . a formal condition for osmotic stabilisation is derived in section [ stabfoamcond ] , and examined for various limiting geometries . we discuss @xmath5 for polydisperse foams in section [ 2dpolysystems ] , and consider exceptions to our estimate that @xmath20 which might arise under certain conditions ( which we argue to be uncommon ) . we then use the geometric pressure to further investigate the stability requirements of polydisperse foams . the coarsening of insufficiently stabilised foams is studied in sections [ coarsenfchapt]-[fys ] , starting with a simple mean - field model in section [ rapidr ] . by considering dissipation rates for the diffusive flux of gas between bubbles and for the viscous rearrangement of bubbles , we are able to predict ( as a function of various parameters ) the rate - limiting mechanism and the associated growth rate ( section [ dissiprates ] ) . section [ elasticd ] shows that when bubble rearrangements are sufficiently rare , elastic stresses may arrest coarsening . section [ fys ] extends this to the case of a finite yield strain beyond which bubble rearrangements will cause the foam to flow , finding that a foam s initial state then determines whether coarsening will still occur . we conclude in section [ conclusions ] with a brief summary and discussion of our results . a `` dilute foam '' comprises of spherical gas bubbles floating freely in a solvent . following @xcite , we treat the bubbles as macroscopic objects and neglect their entropy of translation . firstly we consider the size ( and hence composition ) at which spherical bubbles , containing both soluble and trapped gas molecules , may coexist with a bulk gas phase ( for example , formed by one bubble in the foam becoming macroscopic ) . the gas pressure @xmath21 within a spherical bubble labelled @xmath1 is increased above the atmospheric pressure by its laplace pressure @xmath8 : @xmath22 we consider bubbles with @xmath23 soluble gas molecules and @xmath24 trapped gas molecules , and treat the gases as ideal . then @xmath25 , where @xmath26 , @xmath27 are the partial pressures of the soluble and trapped gas molecules respectively , and for the soluble gas molecules @xmath28 where @xmath29 is the chemical potential of a bulk gas phase of soluble molecules at atmospheric pressure @xmath3 . using eqs.[dilp_i],[dilmu1 ] and @xmath30 , we may write @xmath31 as @xmath32 so that when @xmath33 , @xmath34 and bubbles may coexist with a bulk gas phase ( at pressure @xmath3 ) . using the ideal gas law for the trapped species @xmath35 , such coexistence requires @xmath36 where @xmath37 . this expression determines a ` coexistence volume ' , @xmath38 , at which the laplace pressure and the partial pressure of trapped gas balance . solving for @xmath38 we have @xmath39 this is identical to the coexistence volume for a dilute incompressible emulsion droplet , of interfacial tension @xmath10 , containing @xmath24 molecules of a trapped species @xcite . at coexistence with a bulk gas phase , @xmath40 and @xmath41 , so from the ideal gas law the number of soluble gas molecules in bubbles is @xmath42 similarly in two dimensions , dilute ( circular ) bubbles will coexist with a bulk gas phase only if their areas @xmath43 , with @xmath44 they then contain @xmath45 soluble gas molecules with @xmath46 now we consider `` nondilute foams '' in which bubbles press on one another and are distorted into nonspherical shapes@xcite . following princen@xcite , we consider the compression of a previously dilute foam under an osmotic pressure @xmath4 . in a nondilute foam a typical bubble s interface consists of gently curved faces which contact adjacent bubbles , and highly curved regions at the plateau borders . although bubble faces press on one another with a disjoining pressure , this typically results in a negligible direct contribution to the free energy@xcite ; we may assume that the disjoining forces only _ indirectly _ affects a bubble s free energy , by distorting its shape and increasing surface area . put differently , throughout the foam the surface tension @xmath10 is taken constant , independent of the volume fraction of gas present . following equation [ dilmu1 ] ( the ideal gas law ) we obtain @xmath47 for nondilute bubbles as @xmath48 , where @xmath29 is the chemical potential of a bulk gas of soluble molecules subject to pressure @xmath49 . were such a bulk gas to arise by coarsening , @xmath49 would balance both the atmospheric pressure @xmath3 and the osmotic pressure @xmath4 ; defining @xmath50 we require @xmath51 ( see figure [ bulkgasp ] ) @xcite . so for nondilute foam bubbles we have @xmath52 as we compress the system with a semipermeable membrane , the previously spherical bubbles will distort in shape and the continuous phase will flow@xcite so that the additional pressure is evenly distributed amongst bubbles . ( this contrasts with granular materials for example@xcite . ) in the simplest scenario of 2d , monodisperse foams , the bubbles are compressed into monodisperse hexagons with ` rounded ' corners , and equal internal pressures . however , compression of a general polydisperse foam will result in bubbles deforming into various shapes ; a given bubble will have a pressure which depends not only on its volume ( as for spherical bubbles ) , but on the arrangement and pressures of _ all _ of its neighbours . the above discussion may be clarified by noting that bubble interfaces which press on one another , do so with a radius of curvature at most of order @xmath16 ( where @xmath9 is the radius of a bubble with the same volume in an uncompressed state ) . hence pressure differences between bubbles are of order @xmath18 . so if @xmath4 exceeds @xmath18 then the increase in bubbles pressures will ( to within terms of order @xmath18 ) , be homogeneous throughout the foam @xcite . so we define the increase in a bubble s pressure above that of bulk gas ( @xmath53 ) , by @xmath54 and expect @xmath5 to be of order @xmath18 . here @xmath5 reduces to the laplace pressure of the @xmath1th dilute , uncompressed ( @xmath55 ) , spherical 3d bubble , for which the increase in pressure above a coexisting bulk gas is @xmath56 . but in a compressed state @xmath5 is no longer the laplace pressure , for the latter is @xmath57 , with @xmath11 the radius of curvature at a plateau border of droplet @xmath1 . since we consider an ideal mixture of soluble and trapped gases , then eq . [ pgdef ] requires @xmath58 , which substituting into eq . [ munonideal ] gives @xmath59 so since a bulk gas formed by coarsening has @xmath60 , a bubble can coexist with a coarsened bulk gas phase when @xmath61 . we note @xmath5 as defined by eq . [ pgdef ] is generally not easy to calculate from geometric considerations . nonetheless the origin of @xmath5 for a spherical bubble is geometrical , and @xmath5 of a nondilute foam bubble is determined by packing geometry , so we refer to @xmath5 as a bubble s `` geometric pressure '' . to confirm the reasonableness of our arguments for the magnitude of @xmath5 , the following section considers a 2d model for which @xmath21 , @xmath4 , and @xmath5 are exactly calculable . we first clarify what it means for a foam to be `` incompressible '' . if @xmath62 then variations in a bubble s pressure are negligible compared with its actual internal pressure . so a bubble s gas density is approximately unaffected by its geometric pressure , and hence we may treat such bubbles as effectively incompressible . we emphasise that bubbles with @xmath63 may only be treated as incompressible with respect to _ coarsening _ ( which changes their geometric pressures ) ; such systems are _ not _ incompressible under changes in @xmath3 or @xmath4 ( which changes their laplace pressures as well ) . we now study a monodisperse , incompressible 2-dimensional foam , at osmotic pressure @xmath4 . ( in appendix [ fbalance ] an alternative argument confirms the results for an equivalent , but compressible model . ) for @xmath64 monodisperse bubbles will form an hexagonal array , with bubbles distorted into approximately hexagonal shapes but with rounded corners ( see figure [ 2dmodelfig ] ) . for simplicity the film thickness between adjacent bubbles is taken to be negligible . we note that an equivalent system of osmotically compressed , monodisperse _ cylindrical _ emulsion droplets was considered by princen@xcite . princen s calculation for the osmotic pressure is used later . since the bubbles are taken to be incompressible , the osmotic pressure @xmath4 does work by removal of the continuous liquid phase from bubbles plateau borders . writing the area of liquid associated with each bubble as @xmath65 ( with @xmath65 given as the sum of one third of the volume of liquid at each of its plateau borders ) , then since we consider a monodisperse system with bubbles of area @xmath66 , the osmotic pressure is given by@xcite @xmath67 where @xmath68 is the interfacial length of bubble @xmath1 . ( we keep a separate label @xmath1 for each bubble , although they are identical , for clarity later on . ) in eq . [ pgdef ] we defined @xmath69 . since the monodisperse system considered here has hexagonal symmetry , by analogy with the laplace pressure of a circular bubble we propose that @xmath5 may be calculated by considering the increase in a bubble s interfacial length with an isotropic expansion at fixed liquid area @xmath65 . so for this system we postulate that @xmath70 with @xmath71 calculated for an isotropic expansion , and with @xmath72 the same for all bubbles ( since the system is monodisperse ) . @xmath72 will be different for different lattice arrangements , and also varies ( along with bubble shape ) , with liquid content . only in the absence of bubble - bubble contacts will @xmath72 equal the laplace pressure , with @xmath73 . since @xmath72 is calculated at a fixed volume of liquid per gas bubble , we have fixed radius of curvature at the plateau borders . [ pghyp ] gives @xmath74 where @xmath75 is the length of the flat bubble - bubble faces ( all equal ) . in appendix [ bubblegeom ] we obtain the exact expressions for the interfacial length , and area of a nearly hexagonal bubble as @xmath76 @xmath77 after differentiation and some algebra , eqs [ 2dgp2 ] , [ intlength ] , and [ areacons ] give @xmath78 we note that @xmath72 may also be written as @xmath79 where @xmath80 is the area of liquid in plateau borders , per bubble . princen@xcite calculated the osmotic pressure of monodisperse , cylindrical emulsion droplets by equating the work done by the osmotic pressure @xmath4 with the increase in interfacial energy as droplets distort ( at fixed droplet volume ) . for this quasi-2d geometry he obtained @xmath81\ ] ] with @xmath82 the volume fraction of emulsion droplets , @xmath83 their volume fraction at first contact , and @xmath9 the corresponding radius . princens@xcite calculation applies equally to our _ incompressible _ , monodisperse 2d foam : the area @xmath84 and volume @xmath15 of monodisperse cylindrical drops of length @xmath85 obey @xmath86 , @xmath87 . therefore @xmath88 and @xmath89 . in the variables used in this paper we have @xmath90 and @xmath91 where comparison of eq . [ princenvar1 ] with eq . [ pg2 ] shows that @xmath92 . so using eqs . [ princenvar1 ] and [ princenvar2 ] , eq . [ princenpi ] becomes @xmath93 so since @xmath94 , we have the _ exact _ result that @xmath95 , which for the system studied confirms both our physical argument for the pressure in an osmotically compressed bubble , and our hypothesis for this 2d monodisperse system that the geometric pressure should be calculated for an isotropic expansion at fixed plateau border radius @xmath96 ( eq . [ pghyp ] ) . from eq . [ 2dgp4 ] , in the dilute limit of circular bubbles @xmath97 @xmath98 , and in the dry limit of hexagonal bubbles @xmath99 . so since @xmath100 , we find the surprising result that @xmath72 ( and hence @xmath21 ) , is only weakly affected by @xmath4 . we return to this in section [ sdf ] . the area @xmath101 at which bubbles in a hexagonal packing may coexist with a bulk gas ( so that @xmath102 ) , obeys @xmath103 . so using eq . [ pg2 ] and @xmath104 we obtain @xmath105 where @xmath106 , is the area with which an entirely dry foam with hexagonal bubbles ( @xmath107 ) may coexist with a bulk gas . we then obtain the number of soluble species at coexistence from @xmath108 . having established a general definition of geometric pressure ( eq . [ pgdef ] ) , and shown that it corresponds ( in at least one special case ) to an isotropic expansion at fixed liquid content , we now use conservation of the total number of gas molecules and total number of bubbles to derive a criterion to ensure the formation of a stable distribution of foam bubbles . we take the bubble size distribution to be composed of two parts , a ` coarsening ' part of the distribution , and a ` stable ' part consisting of bubbles which have shrunk to a stable size at which they coexist with the coarsening bubbles . for a sufficient quantity of trapped species the assumption of the bubble distribution having a coarsening part is found to be inconsistent , enabling us to derive a stability criterion for foams . these arguments have strong similarities to those in @xcite , but are considerably generalised . we take @xmath109 , @xmath110 , and @xmath111 as the number densities of bubbles overall , in the stable part of the distribution , and in the coarsening part of the distribution respectively . conservation of bubble number ( no coalescence ) gives @xmath112 where we note that @xmath113 and @xmath114 may be time - dependent . @xmath45 , the number of soluble species present in bubble @xmath1 when coexisting with a bulk gas phase , is determined by @xmath115 . however since relations between @xmath5 and @xmath15 may vary during coarsening ( due to changes in the bubble s environment ) , we take @xmath45 as time - dependent . we define the following * @xmath116 : the average number of soluble molecules per bubble . this is _ time independent_. * @xmath117 : the average number of soluble molecules per bubble in the stable distribution . as coarsening proceeds , @xmath118 and the bubbles in the stable distribution have @xmath119 . then @xmath120 tends toward the average number of soluble molecules per bubble at coexistence with a bulk gas phase . * @xmath121 : the average number of soluble molecules per bubble , within the coarsening distribution , at time @xmath122 . as a coarsening foam tends towards an equilibrium state , conservation of the number of soluble molecules requires @xmath123 combining eqs . [ consbub1 ] and [ consmol1 ] , gives @xmath124 so if @xmath125 then eq . [ coarseningvsstable ] requires @xmath126 . but since the larger , coarsening bubbles have @xmath127 , then @xmath128 , and the foam must be stable against coarsening . since @xmath117 depends on @xmath129 , whose relation to @xmath15 is not known , the derivation of a stability condition in closed form is not always possible , unlike the dilute case @xcite . however , the above condition enables us to investigate the requirements for stability . this is done in sections [ exact ] and [ sdf ] below . for both dilute foams and the model of a monodisperse 2d foam , the geometric pressures are calculable exactly , giving exact expressions for @xmath130 . these are : eq . [ nsb3ddil ] ; eq . [ nsb2ddil ] ; and @xmath131 with @xmath101 given by eq . [ 2dab ] ; for dilute 3d foams , dilute 2d foams , and the monodisperse 2d model respectively . stability conditions are calculated by averaging over the relevant equation for @xmath45 ( where appropriate ) , and ensuring that eq . [ tdepfstab ] is satisfied . for dilute 3d foams the requirement is @xmath132 , with eq . [ nsb3ddil ] giving @xmath133 which resembles , but generalises a result in @xcite . similarly for dilute 2d foams we require @xmath132 , but with eq . [ nsb2ddil ] giving @xmath134 finally , since the model of 2d foams considers monodisperse bubbles we merely require that @xmath135 , with eq . [ 2dab ] giving @xmath136 recall that @xmath106 so that @xmath45 depends on @xmath10 as expected . the area of liquid associated with each bubble becomes negligible compared to @xmath137 as the foam becomes increasingly dry . note also that the dry monodisperse hexagonal foam is , even without trapped species , already dynamically stable with respect to infinitesimal volume changes of a single cell but not with respect to geometric reorganisation so as to create five - sided and seven - sided cells . such a foam is , however , unstable with respect to homogenous cell shrinkage throughout the system ( with the excess gas forming a bulk coexisiting phase ) , whereas sufficient trapped species , as calculated above , will restore full thermodynamic stability in this sense . we accept that the distinction between stability and metastability becomes blurred , especially as the geometry of the foam becomes more complex . the edges of bubbles in a reasonably dry 2d foam ( @xmath138 ) , meet with an angle approximately equal to @xmath139 . if the edges met with an angle of _ exactly _ @xmath139 , then bubbles would need to have _ exactly equal _ radii of curvature at their plateau borders , and hence _ equal _ bubble pressures @xmath94 . since the curvatures between adjacent bubbles remain of order @xmath140 , then as a foam becomes increasingly dry , bubbles pressure differences become increasingly small compared with the mean bubble pressure ( @xmath141 ) . so provided a polydisperse foam is sufficiently dry , then @xmath20 and @xmath142 . in a sufficiently _ wet _ and polydisperse foam , very small bubbles might reside within the plateau borders of larger bubbles without mechanically experiencing an osmotic pressure @xmath4 . such a bubble in a 3d foam will be spherical with @xmath143 and hence @xmath144 . for such bubbles @xmath5 need not necessarily be of order @xmath18 . in fact even for the same bubble size distribution and the same osmotic pressure @xmath4 , a foam s _ volume _ can depend on the arrangement of bubbles it contains ( see figure [ voldependence ] ) . however , given a reasonable osmotic compression @xmath4 , then a reasonably narrow distribution of @xmath24 will be sufficient to ensure that such bubbles occupy a negligible volume fraction , and hence may be neglected when calculating the stability condition eq . [ tdepfstab ] . this is shown in appendix [ polydappendix ] . to address the stability condition [ tdepfstab ] , we need to know the size at which shrunken bubbles in the stable distribution will coexist with a bulk phase of soluble gas . we will assume that for a given @xmath4 the distribution of @xmath24 is sufficiently narrow that we may neglect any tiny bubbles residing wholly within plateau borders of larger ones ( see above , and appendix [ polydappendix ] ) . then for a shrunken bubble in @xmath145 dimensions of volume @xmath15 , we define a quantity @xmath146 such that @xmath147 and expect @xmath148 . ( any coarsening or rearrangements will make @xmath146 time dependent . ) for example , our model 2d foam has ( from eq . [ pg2 ] ) @xmath149 which is graphed in figure [ gammapict ] . in this example @xmath146 varies monotonically between @xmath150 for an entirely dilute foam ( circular bubbles ) , and @xmath151 for an entirely dry foam ( hexagonal bubbles ) , but always remains of order @xmath152 . we may calculate an approximate value for the volumes of shrunken bubbles by taking @xmath146 as independent of @xmath15 ( although changes in a bubble s environment mean that @xmath146 may change with time ) . using @xmath153 , and obtaining @xmath38 by equating @xmath154 with @xmath155 , gives @xmath156 in appendix [ reqapp ] we use eq . [ ansatzeq ] to investigate the stability requirement eq . [ tdepfstab ] in more detail . by assuming that @xmath146 and @xmath24 are uncorrelated , it is shown that unless the average value @xmath157 may increase _ without bound _ , then a stability threshold does exist . ( that is , there will always be some number of trapped molecules per bubble beyond which coarsening will be prevented . ) equation [ gammacurve ] shows that , for a monodisperse 2d foam , @xmath158 is bounded above even in the dry limit ( in contrast to the laplace pressure ) . also since @xmath159 ( see appendix [ fbalance ] ) , then for @xmath157 to increase without bound , @xmath146 and @xmath9 will need to be correlated in a very specific way . on balance , all these arguments suggests that @xmath157 remains bounded as a polydisperse foam evolves . hence our results suggest it should always be possible to osmotically stabilise a polydisperse foam by adding enough trapped species . as a rough estimate , stability requires a sufficient pressure of trapped gas that @xmath160 , which for @xmath161nm@xmath162 , and @xmath163 m requires @xmath164nm@xmath165 , ie of order @xmath166 of atmospheric pressure . now we consider the coarsening of incompletely stabilised , non - dilute foams . previous work on coarsening of dilute emulsions in @xcite also applies to dilute foams _ mutatis mutandis_. here we concentrate on non - dilute foams in which bubbles impinge on one another and are distorted from their otherwise spherical shape . as with dilute emulsions@xcite , the trapped molecules prevent bubbles from entirely disappearing . the resulting foam morphology and coarsening kinetics will be determined by two main factors . the first is the ` excess volume fraction ' of dispersed phase , defined as the total volume fraction of gas which ultimately will coexist with a stable ensemble of shrunken bubbles , in osmotic and mechanical equilibrium with it : this is the amount of gas actually available for coarsening . the second factor is the ease with which shrunken bubbles may rearrange to allow larger bubbles to coarsen . the excess volume fraction of disperse phase , determines a foam s expected late - stage morphology ( see figure [ excessfig ] ) : 1 . _ no excess volume fraction : _ the foam is stable . _ very low excess volume fractions : _ larger bubbles are surrounded by a ` sea ' of shrunken bubbles . competitive coarsening between larger bubbles requires a gas flux _ through _ the sea of smaller bubbles . _ very high excess volume fraction : _ larger bubbles are _ decorated _ by collections of smaller bubbles at their _ vertices _ , with their _ faces _ impinging on other large bubbles . _ intermediate volume fractions : _ structures between the previous two extremes . [ lowexcess ] [ highexcess ] in what follows , we focus on the case of low but nonzero excess volume fraction . here coarsening of larger bubbles will require rearrangements of the shrunken bubbles to prevent the build up of excess elastic strains ( which will otherwise halt coarsening , as shown below ) . we envisage four scenarios : 1 . _ inviscid rearrangements : _ bubble rearrangements occur easily , and with negligible dissipation of energy . viscous rearrangements : _ bubbles can rearrange , but resist doing so and hence slow the rate of coarsening . _ negligible rearrangements ( elastic medium ) : _ bubbles grow within an effectively elastic medium , which may eventually arrest coarsening . _ elasto plastic rearrangements : _ there is a maximum yield strain beyond which rearrangements allow flow of the shrunken bubble sea , but below which it behaves as an elastic medium . we expect rearrangements to occur easily in a sufficiently wet foam , but rearrangements in a very dry foam to occur rarely or not at all . so we expect the scenarios from @xmath167 , to become more applicable as foams become increasingly dry . since _ both _ rearrangements and diffusion of disperse phase are required for coarsening to occur , coarsening will proceed with a rate determined by the _ slowest _ process . this contrasts , for example , with phase separation in a binary fluid , where the coarsening rate is governed by the _ fastest _ process ( with diffusive coarsening at early times , viscous hydrodynamic coarsening at intermediate times , and inertial hydrodynamic coarsening at late times @xcite ) . in the present case rapid coarsening may initially be limited by viscous forces , then later as coarsening slows , viscous forces will become negligible and coarsening diffusion - limited . note that a similar classification of kinetic regimes may in part be applicable to the coarsening dynamics of foams containing no trapped species . it would be appealing , perhaps , to study this case in detail first , before addressing the situation where trapped species are present . however , the latter case is actually a lot simpler , at least in the case of low excess volume fractions ( figure [ excessfig ] , left ) considered here . this is because the actively coarsening bubbles are effectively decoupled by a sea of passive shrunken bubbles ; this simplification is absent at excess volume fractions approaching unity ( which recovers the unstabilised case ) . we consider a mean - field model for the coarsening of a small excess volume fraction , in which grown bubbles are already sufficiently large that they contain a negligible quantity of trapped molecules , and have an approximately spherical shape ( figure [ lowexcess ] ) . hence large bubbles have @xmath168 and @xmath169 . we restrict ourselves to incompressible foams ( in the sense described in section [ compressed / drained ] ) , and take shrunken bubbles to have an approximately constant size @xmath170 , with @xmath171 . an average pressure @xmath172 of soluble gas in shrunken bubbles at distance @xmath96 from a grown bubble s centre is obtained by coarse - graining over bubbles at @xmath96 , and we assume @xmath173 is the same for all grown bubbles . we at first take rearrangements to be inviscid , so that bubble growth is determined by the rate at which soluble gas diffuses through bubble - bubble interfaces in the shrunken bubble ` sea ' ; this assumption is relaxed in section [ dissiprates ] . consider the flux of gas from shrunken bubbles at radius @xmath96 to adjacent bubbles at radius @xmath174 , see figure [ continuum_sea ] . following the approach of von neumann ( see @xcite ) , we take the flux of gas between two bubbles as proportional to both the pressure difference of their soluble gas , and the surface area through which the gas may pass . defining a flux velocity per unit pressure @xmath175 , we obtain an average volume flux of gas from bubbles at @xmath96 to bubbles at @xmath176 of @xmath177 since @xmath178 then @xmath179 , so solving for a steady state gas flux with @xmath180 , we obtain a radial flux @xmath181 and a droplet growth rate @xmath182 where we used @xmath183 , and wrote @xmath184 . for an incompressible system all bubbles have a volume per gas molecule equal to that of a bulk gas , @xmath185 . so with a little algebra we get @xmath186 where @xmath187 . previously the thickness of liquid films between bubble faces was taken as zero . now we take such films to have a small but finite thickness @xmath188 . then assuming the rate of flux through liquid films to be diffusion limited , we may calculate @xmath175 in terms of the diffusion constant for dissolved gas molecules @xmath189 , the volume per gas molecule @xmath190 , @xmath191 , and @xmath188 . this gives@xcite @xmath192 so the growth rate may then be written as @xmath193 note that for a _ dilute _ foam@xcite the factor @xmath194 is absent from eq . [ rrgrowrate ] . the increase in growth rate is due to the reduced volume fraction of liquid through which gas molecules actually need diffuse , a reduction of order @xmath195 . we continue to study the mean - field model of section [ rapidr ] , but no longer require inviscid rearrangements . when bubble rearrangements _ are _ inviscid there is dissipation from the _ diffusion resistance _ of dissolved - molecules diffusing through liquid films between bubbles , and if bubble rearrangements are viscous there is also dissipation as bubbles rearrange . by equating the rate of dissipation with the rate of decrease in free energy , in sections [ lsdissip ] , [ invisciddissip ] , and [ viscousdissip ] we obtain the order of magnitude for a droplet s growth rate . greatest _ source of dissipation limits the coarsening rate , so by comparing the dissipation rates we can estimate ( section [ viscousdissip ] ) when each type of coarsening will occur . a similar approach applied to emulsion rheology ( not coarsening ) , is found in @xcite . as an example , we firstly consider coarsening in the traditional lsw@xcite scenario of a vanishingly small volume fraction of bubbles in a liquid ( without trapped molecules ) . here dissipation is from the diffusion resistance of diffusing , dissolved gas molecules . in a steady state the radial flux of dissolved gas molecules , at a distance @xmath96 from the centre of a bubble of radius @xmath196 is @xmath197 we write @xmath198 , with @xmath199 and @xmath200 , the concentration and average radial velocity of dissolved gas molecules at @xmath96 respectively . since @xmath201 , where @xmath202 is the concentration of dissolved gas molecules adjacent to a bulk gas phase , then @xmath203 the number of dissolved gas molecules within spheres of radii @xmath96 and @xmath204 is of order @xmath205 . the average dissipation rate per molecule at @xmath96 is @xmath206 , where @xmath207 is the viscous drag coefficient on a molecule of disperse phase moving through the liquid . hence the total dissipation arising from diffusion to a drop is of order @xmath208 . using the einstein relation@xcite @xmath209 , eq . [ diffvel ] , and integrating , we obtain the rate of dissipation due to dissolved molecules diffusing between bubbles @xmath210 , as @xmath211 coarsening occurs to reduce interfacial energy , with a rate of reduction in energy of order @xmath212 . so equating @xmath213 with eq . [ dissipation1 ] and rearranging , we get @xmath214 in agreement with the traditional analysis of lsw@xcite for coarsening of dilute emulsion droplets ( as opposed to foam bubbles as studied here ) . we now apply the method to the mean - field model of section [ rapidr ] , where inviscid rearrangements were assumed . the argument in section [ lsdissip ] gives the average velocity with which dissolved gas molecules will diffuse through liquid films , in eq . [ diffvel ] , and the dissipation per molecule in liquid films remains @xmath215 . but now dissipation only occurs in the liquid films between bubble faces . to integrate only over the volume of such films , the spatial volume element @xmath216 is reduced by a factor of @xmath195 . so the dissipation due to diffusion resistance @xmath210 is @xmath217 and we obtain @xmath218 in agreement with the mean - field calculation ( eq . [ rrgrowrate ] ) . this gives a growth law for the mean droplet size , @xmath219 . we now consider the rate of dissipation due to viscous stresses in the thin liquid films , as shrunken bubbles rearrange . adjacent bubble faces are again taken to be separated by a distance @xmath188 , determined by the disjoining pressure between bubble membranes . since the excess volume fraction is small , we assume that bubble growth results in fluid flow that is approximately radial . taking @xmath220 as the velocity of the shrunken - bubble fluid , then incompressibility of that fluid requires @xmath221 , so that in the spherically symmetric case there is a radial velocity @xmath222 at distance @xmath96 from the centre of a bubble of radius @xmath196 , growing with velocity @xmath223 . consider now two adjacent shrunken bubbles at distances @xmath96 and @xmath224 respectively from the centre of the growing bubble . the differing velocities at @xmath96 and @xmath224 will mean that bubbles must rearrange , and slide past one another . for @xmath225 the relative velocity of the bubbles is of order @xmath226 . the shear rate of the liquid between bubbles is of the order of this relative velocity divided by the film thickness @xmath188 . within the film the viscous stress is therefore @xmath227 where @xmath228 is the viscosity of the continuous liquid phase , and @xmath229 is the shear rate in a liquid film at @xmath96 . the volume - averaged viscous dissipation is dominated by the contribution from within the films @xcite and , per unit volume , is of order @xmath195 times the dissipation rate ( of order @xmath230 ) within each film . so we obtain the total dissipation rate due to viscous bubble rearrangements , @xmath231 , as @xmath232 using eq . [ rfluidvel ] and integrating , this gives @xmath233 as in the previous calculations ( sections [ lsdissip ] and [ invisciddissip ] ) , the rate of decrease in the surface free energy of the growing bubble is of order @xmath234 . equating @xmath234 with eq . [ viscousdissip ] , and rearranging gives @xmath235 and hence a linear growth law , @xmath236 . it is interesting to ask why the above argument and the resulting linear growth law does _ not _ apply to the conventional lsw coarsening of dilute emulsion droplets in a structureless fluid continuum . in emulsions the volume of a molecule in a droplet is similar to that in the continuous phase , so when a dissolved molecule is incorporated into a drop , the increase in the droplet s volume equals the volume of liquid previously displaced by the molecule . hence the only displacement of liquid is that already accounted for by the stokes - einstein drag on the molecule as it diffuses through the continuous liquid phase . on the other hand , since the volume per gas molecule in a bubble is much larger than its volume when dissolved in solution , the viscous dissipation _ may _ be relevant for coarsening of dilute foam bubbles in a structureless fluid . ( in this case , arrival of gas molecules at the surface of a growing bubble causes a net fluid flow radially outward . ) a simple order of magnitude estimate gives @xmath237 and @xmath238 . this differs from the case where the surrounding medium is a ` shrunken bubble ' fluid by the absence of the enhancement factor @xmath194 . in a coarsening dense foam , both rearrangements and diffusion are necessary for bubble growth , so the growth rate will be limited by the _ greatest _ source of dissipation . so when @xmath239 coarsening is diffusion limited , and when @xmath240 coarsening is limited by viscous dissipation . here we estimate the ratio for plausible foam parameters . comparing eqs . [ invisciddissip ] and [ viscousdissip ] , we see that @xmath241 in terms of a molar concentration @xmath242 , the molar volume @xmath243 of gas in bubbles at pressure @xmath244 , and the gas constant @xmath245 , we have @xmath246 at room temperature and atmospheric pressure we take typical values of @xmath247j , @xmath248m@xmath249 , @xmath250m@xmath251 , @xmath252m@xmath253 , and @xmath254nm@xmath165s , ( eg . for co@xmath6 gas bubbles in water @xcite ) . taking @xmath255 m , gives @xmath256 so that for micron - sized foam bubbles , viscous dissipation will be observed for @xmath195 smaller than @xmath257 . thus for sufficiently small bubbles , thin liquid films , and high liquid viscosity , viscous limited growth may be observed ( giving @xmath258 ) . however at room temperature and pressure we expect diffusion limited coarsening to be more common , and moreover this will always dominate once the radius @xmath196 of the coarsening droplets becomes sufficiently large . note that for foams @xmath259 , so @xmath260 is proportional to @xmath261 : at low pressures and high temperatures the prospect of observing viscous limited coarsening is increased . for example if @xmath262nm@xmath165 ( @xmath263 atmosphere ) then @xmath264 , which for @xmath265 predicts viscous limited growth . for dilute foams @xmath210 and @xmath231 are given by eqs . [ dissipation1 ] and [ diltsdotv ] respectively , so that @xmath266 hence at room temperature and pressure , taking @xmath267 , @xmath189 , @xmath268 , @xmath228 , @xmath243 , and @xmath196 as above , @xmath269 and coarsening will be diffusion limited@xcite . writing @xmath259 we find that viscous limited growth requires @xmath270 nm which at room temperature requires @xmath271 atmosphere . suppose we no longer allow bubbles to rearrange , so that the sea of shrunken bubbles acts as an elastic medium . by estimating the increase in elastic energy as a growing bubble changes volume , we show that coarsening will halt , with the foam now ` elastically ' ( as opposed to ` osmotically ' ) stabilised . the assumption of negligible rearrangement requires detailed explanation , since in any foam ( containing trapped species or otherwise ) local rearrangement follows inevitable when certain conditions are reached at the junctions between adjacent cells . however , it is well known that , _ in the absence of coarsening _ , a foam can exhibit a finite macroscopic yield strain , below which there may be occasional local rearrangements but these are insufficient to allow plastic deformation of the medium as a whole @xcite . our physical picture is that the shrunken bubble sea , which is fully stabilised against coarsening by virtue of the trapped species that it contains , can then offer elastic resistance to the growth of an isolated large bubble , even though the latter is at lower chemical potential of gas than other , less large bubbles that have coarsened elsewhere in the system . the elastic coarsening inhibition mechanism that we advance is thus only possible because of the stability against coarsening of the shrunken bubble sea . if the bubbles in this sea were themselves members of the coarsening population , then the rearrangement conditions at the junctions between them would be constantly met within it , and there would be an incessant rearrangement of bubbles caused by their local volume changes . such a medium could not exhibit a yield stress in any conventional sense , and would be unable to elastically stabilise the growth of isolated large bubbles . this remark is consistent with von neumann s theorem , which ( in two dimensions ) establishes that in the absence of trapped species , coarsening can never cease for any structure other than a perfect hexagonal lattice . the theorem does not apply in the presence of trapped species @xcite , and so the elastic arrest mechanism considered below leads to no contradiction with it . for definiteness we consider an initial state comprised of spherical _ ready - grown _ bubbles in an elastically unstrained sea of shrunken bubbles ( see figure [ excessfig ] ) . such a system could be formed by osmotically compressing a previously dilute , partially coarsened foam . consider a layer of shrunken bubbles initially at distance @xmath272 from the centre of a grown bubble of initial radius @xmath273 ( see figure [ bubblelayer ] ) . then growth of the larger bubble from radius @xmath273 to radius @xmath196 will require the given layer to move so as to leave a volume equal to the change in the larger bubble s volume , which is @xmath274 . so a layer with initial inner radius @xmath272 must move so that @xmath272 is increased to @xmath96 , with @xmath275 hence we obtain a bubble layer s new position @xmath276 as @xmath277 in the absence of rearrangements , the linear extension @xmath278 of bubbles in a layer with initial inner radius @xmath272 is @xmath279 where @xmath280 is the distance by which shrunken bubbles at @xmath96 will move ( see figure [ bubblelayer ] ) , due to the larger bubbles growth . so using @xmath281 , @xmath282 , and eq . [ rexact ] , we expand in terms of @xmath283 , which to lowest order gives @xmath284 if we again use eq . [ rexact ] , then for small strains or large distances @xmath272 from a bubble we may expand in terms of @xmath285 , to obtain @xmath286 now consider the energy of extending or contracting along one axis of a single shrunken bubble , with its length changed from @xmath287 to @xmath288 . restricting ourselves to small strains ( or equivalently @xmath289 ) , we may expand the associated increase in energy @xmath290 in terms of a power series in @xmath278 , @xmath291 since we consider an initial state with _ unstrained _ bubbles , equilibrium requires that both positive and negative values of @xmath278 will give positive @xmath292 , requiring @xmath293 . so to lowest order @xmath294 . initially we have of order @xmath295 shrunken bubbles between spheres at @xmath272 and @xmath296 . so the change in elastic energy due to a large bubble s growth or shrinking is @xmath297 which upon substitution of eq . [ 8.52 ] , @xmath298 , and integrating gives @xmath299 the asymptote of @xmath300 for @xmath301 means that in the absence of bubble rearrangements the elastic energy in the surrounding foam will ultimately prevent coarsening , and elastically stabilise the system . the approximation of @xmath302 is strictly only valid for large @xmath272 or sufficiently small strains . for large strains we might expect plastic rearrangements to occur . consider the following simple model : the shrunken bubbles may elastically support a maximum ( yield ) strain @xmath303 , beyond which macroscopic rearrangements will occur . then @xmath304 with @xmath305 the plastic threshold for bubble rearrangement . in the absence of rearrangements [ 8.52 ] gives @xmath306 so that @xmath305 implicitly defines a radius @xmath307 within which rearrangements occur , but beyond which the medium behaves elastically . so from eq . [ 8.52.2 ] , we may define @xmath308 , which since @xmath309 gives @xmath310 for a plastic region to exist around a bubble , we require @xmath311 ; eq . [ r0 * 1 ] then requires @xmath312 . whether sufficient growth is possible for this threshold to be reached depends on the radius @xmath313 at which bubble growth would halt , in a purely elastic system as considered in the previous section . if the inequality @xmath314 is not satisfied ( as occurs for large enough @xmath303 ) one requires a large fluctuation , either thermally or in the initial condition , to establish a plastic zone around a droplet , allowing it to coarsen further . in that case the foam is at least metastable and in practice this may be sufficient . to get a stability condition , we consider next the elastic energy stored around a droplet that has grown from size @xmath273 to size @xmath196 . taking deformations within any plastic region to be of order @xmath305 , we may calculate the elastic energy as @xmath315 giving @xmath316 where , since @xmath317 , both terms on the right are positive . we now assume that a grown bubble adopts a state of mechanical equilibrium with the surrounding shrunken - bubble sea , on a timescale fast compared to any coarsening process . then for a reversible fluctuation in bubble volume by @xmath318 , the work done by the bubble s gas equals the work done on the bubbles surrounding environment . hence @xmath319 so differentiating eq . [ elp ] , with @xmath320 given by eq . [ el2 ] for @xmath321 and by eq . [ elastope ] for @xmath312 , we get @xmath322 this means that any bubble that has successfully coarsened to become a bulk gas phase ( whose pressure clearly can not be infinite as @xmath323 ) has overcome the plastic threshold and has a pressure @xmath324 obeying @xmath325 now we are able to estimate a stability threshold for a foam that combines a finite amount of trapped species with a finite yield strain ( under the simplified initial condition we chose above , that the initial state has a dilute array of partially grown bubbles in an unstrained , shrunken - bubble sea ) . in this case the lowest possible free energy obtainable by coarsening is that of a bulk gas at pressure @xmath324 . so unless the initial pressure @xmath21 of grown bubbles exceeds @xmath324 , then the foam _ must _ be stable@xcite . an initially unstrained bubble has @xmath326 so instability requires @xmath327 , and hence @xmath328 so since @xmath329 , such a foam may only be unstable if @xmath330 . for example , considering foams formed by the osmotic compression of a dilute partially coarsened foam , then if @xmath273 is small enough that @xmath331 then coarsening may occur , but if it has already coarsened sufficiently that @xmath332 , then after osmotic compression it will be elastically stabilised . we have considered dilute foams with spherical bubbles , and nondilute foams in which bubbles are compressed by an osmotic pressure @xmath4 ( which distorts bubbles and increases their gas pressure ) . prior to osmotic compression , variations in bubble pressures are of order @xmath18 , so bubbles may only support large osmotic pressures ( @xmath333 ) by deforming and pressing on one another . pressure differences between such osmotically compressed bubbles generally remain limited to @xmath18 ( with a possible exception in the case of extreme polydispersity in initial droplet size ) . if coarsening occurs , a coexisting bulk gas is created that has pressure @xmath334 , with @xmath3 the ambient atmospheric pressure . we defined the excess in a bubble s pressure above that of such a bulk gas by @xmath335 , and argued that @xmath336 . this was confirmed quantitatively and explicitly for a monodisperse 2d model of foam in section [ 2dmodel ] . @xmath5 , which we call the geometric pressure , provides the driving force behind coarsening in the case of a compressed foam or emulsion ; in general it is very different from the laplace pressure but reduces to it in the dilute limit . for dilute 2d and 3d systems and also for a concentrated but monodisperse 2d foam , @xmath5 can be found explicitly . this allows one to find rigorous conditions ( eqs . [ dilcond ] , [ 2ddilcond ] , and [ nsb2d ] ) for their stability . more generally we have argued that for a reasonably large @xmath4 and a reasonably narrow distribution of @xmath24 , osmotic stabilisation should generally be possible , and typically requires @xmath337 . the morphology of an insufficiently stabilised foam ( containing inadequate levels of a trapped species ) , is determined by the ` excess volume fraction ' of dispersed phase ; this will form a bulk gas phase if the system is allowed to reach full equilibrium ( _ i.e. , _ subject only to the constraint that the trapped species stay in their designated droplets and coalescence is absent , which with trapped species present will ensure conservation of droplet number ) . the kinetics of coarsening will be determined either by the rate of diffusion of gas between bubbles or the rate at which shrunken bubbles rearrange , with the slowest process determining the bubble growth rate . we studied coarsening of a small excess volume fraction , finding diffusion limited growth to result in @xmath338 ( as opposed to @xmath339 as arises in foams without trapped molecules@xcite ) , and viscous limited growth to give @xmath340 . for typical parameters we expect diffusion limited growth to be more common , with viscous limited growth more likely to be observed if the liquid viscosity is high , the temperature is high , and the pressure is low@xcite . under conditions of negligible bubble rearrangements , the buildup of elastic stresses among the sea of shrunken , stabilised bubbles will arrest coarsening of well - separated large bubbles and ` elastically stabilise ' the entire foam . this can sometimes occur when there is a finite yield strain ; whether a foam will be elastically stabilised is then determined by the foam s initial state ( and the yield strain @xmath303 ) . thus osmotic compression of a partially coarsened _ dilute _ foam , may _ halt _ coarsening and elastically stabilise the foam ; the further the foam has coarsened initially , the more likely this is to occur . our work applies , substantially unchanged , to concentrated emulsions . as with nondilute foams the disjoining energy between repelling droplets in emulsions may usually be neglected @xcite , and the osmotic pressure of trapped molecules in droplets equals the partial pressure of trapped gas molecules in bubbles ( so long as both are treated as ideal @xcite ) . hence if we replace the volume per gas molecule @xmath341 with the molecular volume of disperse phase in droplets @xmath342 , and also replace @xmath343 with @xmath344 , then most of the above work applies equally well to concentrated emulsions . we adopt a similar approach to one previously used by princen@xcite , which will enable us to determine the order of magnitude of @xmath72 when bubble sizes are polydisperse . unlike the one in the main text , this approach is equally valid for compressible and incompressible bubbles . consider a semipermeable membrane with a shape which follows the top surface of a line of hexagonal bubbles ( see figure [ monobrane ] ) , so that bubbles remain hexagonal in shape . then since the membrane does nt move , the total force on the membrane due to @xmath345 must balance the total force on the membrane acting from below . then we consider a typical bubble adjacent to the membrane ( see figure [ monobrane ] ) . such a bubble will have a geometry as in figure [ nhexpict ] but with the bubble rotated by @xmath346 to leave two flat edges vertical . the force acting from below the membrane is @xmath21 multiplied by the projection onto the horizontal axis of the flat bubble edges(@xmath347 ) , plus @xmath3 multiplied by the projection of the plateau borders onto the horizontal axis ( @xmath348 ) . the force acting from above the membrane is simply @xmath349 multiplied by the bubble s width @xmath350 . then for the forces above and below to balance we require@xcite @xmath351 writing @xmath352 and rearranging , we obtain @xmath353 which writing in terms of @xmath354 and @xmath355 and comparing with eq . [ pg2 ] gives @xmath356 as before . note that since the method may be applied to compressible systems , both @xmath21 and @xmath72 for the monodisperse 2d model will remain the same for both compressible and incompressible bubbles . we now generalise the argument to a polydisperse system . the projections of the bubble faces and plateau borders onto the horizontal axis are written as @xmath357 and @xmath358 respectively . then considering a suitably shaped membrane , force balance requires @xmath359 so writing @xmath360 and restricting ourselves to fairly dry foams with @xmath361 , then @xmath362 where @xmath363 , and we can obtain @xmath364 now we note that since @xmath365 and @xmath366 , where @xmath367 , then @xmath368 so that if @xmath369 then @xmath370 . in 3d the same argument but with projected areas @xmath371 , and @xmath372 leads to @xmath373 which for @xmath374 gives @xmath375 finally we note that in higher dimensions the equivalent argument will continue to give @xmath376 . the area of a hexagon of side @xmath75 is six times the area of the equilateral triangles of which it is composed . the area of each triangle is @xmath377 , with @xmath378 . hence we obtain the area of a hexagon as @xmath379 . its interfacial length is simply given by @xmath380 . we now describe a nearly hexagonal bubble in terms of the length of the flattened faces @xmath75 , and the radius of curvature of the plateau border @xmath96 ( see figure [ nhexpict ] ) . then the area @xmath354 of a nearly hexagonal bubble is given by the sum of six sections of a circle , each of angle @xmath381 and radius @xmath96 , plus six rectangles of length @xmath75 and height @xmath96 , plus the area of a hexagon of side length @xmath75 . so @xmath382 and the surface length obeys @xmath383 what is required for states such as that in figure [ voldependence ] to exist ? using @xmath384 and the ideal gas law ( for the trapped gas ) , the pressure of soluble gas in a spherical bubble residing in a plateau border is @xmath385 , which has a maximum value of @xmath386 . so in a system which coarsens with @xmath387 , the existence of spherical bubbles in plateau borders requires @xmath388 hence such bubbles will need a sufficiently small quantity of trapped species that @xmath389 which writing @xmath4 in terms of the laplace pressure of a bulk gas , @xmath390 , requires @xmath391 we may compare @xmath392 with that for a typical stable ( and compressed ) bubble of radius @xmath393 , containing @xmath394 trapped molecules . then since @xmath395 one requires @xmath396 a spherical bubble in a plateau border has @xmath397 considering @xmath4 for which @xmath398 or smaller , then we require @xmath399 . so for a reasonable @xmath4 of order @xmath400 , and a reasonably narrow distribution for @xmath24 , then there will be a negligible volume fraction of bubbles residing in plateau borders . we examine the form of @xmath401 , the number of soluble species per bubble below which a stable foam is ensured . noting that @xmath402 , we write it out in full to obtain , @xmath403 we note that @xmath404 so that @xmath405 we assume that @xmath146 and @xmath24 are uncorrelated , that is for a given bubble volume we assume that variations in @xmath5 are independent of @xmath24 . then using eqs . [ 90 ] and [ 91 ] , we have @xmath406 in section [ fbalance ] we found that @xmath407 , which suggests @xmath157 is of order a constant @xmath408 . then since @xmath409 a foam formed with an initial number of soluble species @xmath410 such that @xmath411 will be stable . similarly if @xmath157 is bounded by @xmath412 , then a stability condition is obtained by replacing @xmath408 with @xmath412 in eq . [ nsbbound1 ] . alternately , if @xmath157 is a decreasing function of time , then @xmath413 and hence @xmath414 , so provided @xmath415 then we could again guarantee a stable foam . in contrast to the above cases , if @xmath157 _ increases _ without bound , then @xmath117 will _ decrease _ without bound , and a stability condition will never strictly exist . the only way a stability condition can fail to exist is if @xmath157 can increase in time _ without bound_. @xmath416 : area of two - dimensional bubble @xmath1 ; area of surrounding liquid ( within hexagonal unit cell ) ; area at which such a bubble ( containing trapped species ) coexists osmotically with a bulk gas phase ; the same , but for a completely dry hexagonal foam . @xmath422 : number of trapped molecules in @xmath1th bubble ; number of soluble molecules within @xmath1th bubble ; the number of soluble molecules within @xmath1th bubble when coexisting with a bulk gas phase . @xmath423 : average number of soluble gas molecules per bubble ; average number of soluble molecules per bubble within the stable ( noncoarsening ) population ; average number of soluble molecules per bubble within the ( unstable ) coarsening population . @xmath429 : radius of @xmath1th bubble ; mean bubble radius ; initial radius of a coarsening bubble ; its radius at time @xmath122 ; radius at which its growth is arrested in absence of rearrangement ; radius of a shrunken bubble at or near coexistence with a bulk gas phase . @xmath430 : radial distance of a given layer of shrunken bubbles from the centre of a growing bubble ; its initial value ; the latter quantity for bubbles whose local strain coincides with the yield strain of the shrunken bubble sea . nondilute foams resemble concentrated emulsions , which consist of incompressible emulsion droplets which are pressed together by the action of a semipermeable membrane ( through which only continuous phase may pass ) . von neumann s law gives in two dimensions the rate of change of area of the @xmath1th bubble as @xmath448 where @xmath449 is its number of edges . with trapped species present , one has @xmath450 where @xmath451 is a neighbouring cell and @xmath452 the length of the common edge . the sign of the extra term depends on the volumes of the neighbouring cells ( and the number of trapped species they contain ) so that static equilibrium may be reached in a system where not all cells have @xmath453 neighbours . see @xcite . we note that molecules in dilute _ emulsion droplets _ have a much smaller value of @xmath243 than do molecules in gaseous foam bubbles . hence even if there is a significant discrepancy between the molecular volumes of dissolved disperse phase and disperse phase in droplets , coarsening of dilute emulsions will be diffusion limited . it is interesting that the viscous limited and diffusion limited growth laws eqs . [ mce10.22 ] , [ mce10.30 ] may be predicted ( up to constant prefactors ) , by the most naive order of magnitude arguments for the phase ordering kinetics of binary fluids ( see @xcite , page @xmath454 ) . however such arguments _ do not _ correctly predict when each regime will apply , this depends on the structure of the shrunken bubble fluid .

in the absence of coalescence , coarsening of emulsions ( and foams ) is controlled by molecular diffusion of the dispersed phase species from one emulsion droplet ( or foam bubble ) to another . previous studies of dilute emulsions have shown how the osmotic pressure of a trapped species within droplets can overcome the laplace pressure differences that drive coarsening , and `` osmotically stabilise '' the emulsion . webster and cates ( _ langmuir _ , * 1998 * , _ 14 _ , @xmath0 ) gave rigorous criteria for osmotic stabilisation of mono- and polydisperse emulsions , in the dilute regime . we consider here whether analogous criteria exist for the osmotic stabilisation of mono- and polydisperse _ concentrated _ emulsions and foams . we argue that in such systems the pressure differences driving coarsening are small compared to the mean laplace pressure . this is confirmed for a monodisperse 2d model , for which an exact calculation gives the pressure in bubble @xmath1 as @xmath2 , with @xmath3 the atmospheric and @xmath4 the osmotic pressure , and @xmath5 a ` geometric pressure ' that reduces to the laplace pressure only for a spherical bubble , and depends much less strongly on bubble deformation than the laplace pressure itself . in fact , for princen s 2d emulsion model , @xmath5 is only 5% larger in the dry limit than the dilute limit . we conclude that osmotic stabilisation of dense systems typically requires a pressure of trapped molecules in each droplet that is comparable to the laplace pressures the same droplets would have if they were spherical , as opposed to the much larger laplace pressures actually present in the system . we study the coarsening of foams and dense emulsions when there is insufficient of the trapped species present . various rate - limiting mechanisms are considered , and their domain of applicability and associated droplet growth rates discussed . in a concentrated foam or emulsion , a finite yield threshold for droplet rearrangement among stable droplets may be enough to prevent coarsening of the remainder .

in the last decade , the generalizations of several operators to quantum variant have been introduced and their approximation behavior have been discussed [ see for instance @xcite , @xcite , @xcite , @xcite , @xcite etc ] . the further generalization of quantum calculus is the post - quantum calculus , denoted by @xmath0-calculus . recently , some researchers started working in this direction ( cf . @xcite , @xcite , @xcite , @xcite ) . some basic definitions and theorems , which are mentioned below may be found in these papers and references therein . @xmath3 _ { p , q}:=p^{n-1}+p^{n-2}q+p^{n-3}q^2+\cdots + pq^{n-2}+q^{n-1}=\frac{p^{n}-q^{n}}{p - q } \cdot\ ] ] the @xmath4-factorial is given by @xmath5 _ { p , q}!=\prod\limits_{k=1}^{n}\left [ k\right ] _ { p , q } , \ \ n\ge 1 , \ \ \left[0\right ] _ { p , q}!=1 $ ] . the @xmath6-binomial coefficient satisfies@xmath7_{p , q}!}{\left [ n - k\right ] _ { p , q}!\left [ k\right ] _ { p , q } ! } , \ \ 0\le k\le n.\ ] ] the @xmath0-power basis is defined as @xmath8 the @xmath0-derivative of the function @xmath9 is defined as @xmath10 as a special case when @xmath11 , the @xmath0-derivative reduces to the @xmath2-derivative . the @xmath0-derivative fulfils the following product rules@xmath12 d_{p , q}(f(x)g(x ) ) & = & g(px)d_{p , q}f(x)+f(qx)d_{p , q}g(x).\end{aligned}\ ] ] obviously @xmath13 and for @xmath14 we have @xmath15 _ { p , q}(px\ominus a)_{p , q}^{n-1 } , \\[2 mm ] d_{p , q}(a\ominus x)_{p , q}^{n } & = & -\left [ n\right ] _ { p , q}(a\ominus qx)_{p , q}^{n-1}.\end{aligned}\ ] ] let @xmath9 be an arbitrary function and @xmath16 . the @xmath0-integral of @xmath17 on @xmath18 $ ] ( see @xcite ) is defined as @xmath19 and @xmath20 the formula of @xmath0-integration by part is given by @xmath21 very recently gupta and aral @xcite proposed the @xmath0 analogue of usual durrmeyer operators by considering some other form of @xmath0 beta functions , which is not commutative . in the present article , we define different @xmath0-variant of beta function of first kind and find an identity relation with @xmath0-gamma functions . it is observed that @xmath0-beta functions may satisfy the commutative property , by multiplying the appropriate factor while choosing @xmath0 beta function . as far as the approximation is concerned , order is important in post - quantum calculus . we propose a generalization of durrmeyer type operators and establish some direct results . let @xmath22 is a nonnegative integer , we define the @xmath0-gamma function as@xmath23 _ { p , q } ! , \quad \ 0<q < p.\ ] ] let @xmath24 we define @xmath6-beta integral as@xmath25 the @xmath0-gamma and @xmath0-beta functions fulfil the following fundamental relation@xmath26where @xmath27 . for any @xmath28 @xmath29 since@xmath30using ( [ a1 ] ) for @xmath31 and @xmath32 _ { p , q}}$ ] with the equality @xmath33 _ { p , q}\left ( 1\ominus qx\right ) ^{n-1}$ ] we have @xmath34 _ { p , q}}{p^{m-1}\left [ n\right ] _ { p , q}}\int\limits_{0}^{1}x^{m-2}\left ( 1\ominus qx\right ) _ { p , q}^{n}{{\mathrm{d}}}_{p , q}x \notag \\ & = & \frac{\left [ m-1\right ] _ { p , q}}{p^{m-1}\left [ n\right ] _ { p , q}}b_{p , q}\left ( m-1,n+1\right ) . \label{a2}\end{aligned}\]]also we can write for positive integer @xmath22@xmath35using ( [ a2 ] ) , we have @xmath36 _ { p , q}}{p^{m}\left [ n\right ] _ { p , q}}b_{p , q}\left ( m , n+1\right),\]]which implies that @xmath37further , by definition of @xmath0 integration @xmath38 _ { p , q } } \cdot\end{aligned}\]]we immediately have@xmath39 _ { p , q } } \notag \\ \!\!\!&=&\!\!\!p^{m+(m+1)+\cdots+(m+n-2)}\frac{\left ( p\ominus q\right ) _ { p , q}^{n-1}}{\left ( p^{m}\ominus q^{m}\right ) _ { p , q}^{n}}\left ( p - q\right ) \notag \\ \!\!\!&=&\!\!\!p^{(n-1)(2m+n-2)/2}\frac{\left ( p\ominus q\right ) _ { p , q}^{n-1}}{\left ( p^{m}\ominus q^{m}\right ) _ { p , q}^{n}}\left ( p - q\right ) . \label{a3}\end{aligned}\ ] ] following @xcite , we have @xmath40 thus ( [ a3 ] ) leads to @xmath41 this completes the proof of the theorem . the following observations have been made for @xmath0 beta functions : * for @xmath27 , we have @xmath42 * the @xmath6-beta integral defined by ( [ a5 ] ) is not commutative . in order to make commutative , we may consider the following form @xmath43 for this form , @xmath0-gamma and @xmath0-beta functions fulfill the following fundamental relation @xmath44where @xmath45 obviously for form ( [ a - com ] ) , we get @xmath46 for @xmath47 and @xmath48 we have the following identity , which can be easily verified using the principle of mathematical induction : @xmath49 using the above identity , we consider the @xmath4-analogue of bernstein operators for @xmath50 $ ] and @xmath51 as @xmath52_{p , q}}{[n]_{p , q}}\right ) , \ ] ] where the @xmath0-bernstein basis is defined as @xmath53/2 } x^{k}(1\ominus x)_{p , q}^{n - k } .\ ] ] [ r0 ] other form of the @xmath0-analogue of bernstein polynomials has been recently considered by mursaleen et al . @xcite . [ r1 ] using the identity ( [ eb0 ] ) and the following recurrence relation ( for @xmath54 : @xmath55_{p , q } u_{n , m+1}^{p , q}(px)=p^{n } x ( 1-px ) d_{p , q}[u_{n , m}^{p , q}(x ) ] + [ n]_{p , q } px u_{n , m}^{p , q}(px),\ ] ] the @xmath0-bernstein polynomial satisfy @xmath56_{p , q}},\ ] ] where @xmath57 @xmath58 recently , gupta and wang ( see @xcite ) discussed the @xmath2-variant of certain bernstein - durrmeyer type operators . we now extend these studies and propose the following @xmath0-bernstein - durrmeyer operators based on @xmath0-beta function . for @xmath59 $ ] and @xmath60 the @xmath4-analogue of bernstein - durrmeyer operators is defined as @xmath61 _ { p , q } \sum\limits_{k=1}^{n}p^{-(n - k+1)(n+k)/2}b_{n , k}^{p , q}(1,x)\nonumber\\[2 mm ] & & \qquad\quad\times\,\int\limits_{0}^{1}b_{n , k-1}^{p , q}(t)f ( t){{\mathrm{d}}}_{p , q}t + b_{n,0}^{p , q}(1,x)f(0),\end{aligned}\ ] ] where @xmath62 is defined by ( [ eb ] ) and @xmath63 it may be remarked here that for @xmath64 these operators will not reduce to the @xmath2-durrmeyer operators ; but for @xmath65 these will reduce to the durrmeyer operators . [ l1 ] let @xmath66 @xmath67 then for @xmath68 $ ] and @xmath60 we have @xmath69_{p , q}x}{[n+2]_{p , q}},\ ] ] @xmath70_{p , q}x}{[n+2]_{p , q}[n+3]_{p , q}}+\frac{([n]_{p , q}-p^{n-1})p^{2}q[n]_{p , q}x^2}{[n+2]_{p , q}[n+3]_{p , q } } \cdot\ ] ] using ( [ a4 ] ) and ( [ a5 ] ) and remark [ r1 ] , we have @xmath71 _ { p , q}\sum\limits_{k=1}^{n}p^{-[(n+1-k ) ( n+k)/2 ] } b_{n , k}^{p , q}(1,x)\\ & & \quad\times\,\int\limits_{0}^{1}\qfrac{n}{k-1}_{p , q } t^{k-1 } \ , ( 1\ominus qt)_{p , q}^{n+1-k}{{\mathrm{d}}}_{p , q}t+b_{n,0}^{p , q}(1,x ) \\ & = & \left [ n+1\right ] _ { p , q}\sum\limits_{k=1}^{n}p^{-[(n+1-k ) ( n+k)/2 ] } b_{n , k}^{p , q}(1,x)\qfrac{n}{k-1}\\ & & \quad\times\ , b_{p , q}(k , n - k+2)+b_{n,0}^{p , q}(1,x ) \\ & = & \left [ n+1\right ] _ { p , q}\sum\limits_{k=1}^{n}p^{-[(n+1-k ) ( n+k)/2 ] } b_{n , k}^{p , q}(1,x ) \frac{[n]_{p , q}![k-1]_{p , q}!}{[n+1-k]_{p , q}!}\\ & & \quad\times \ p^{[(n+1-k ) ( n+k)/2 ] } \ \frac{[k-1]_{p , q}![n - k+1]_{p , q}!}{[n+1]_{p , q}!}+b_{n,0}^{p , q}(1,x ) \\ & = & b_{n , p , q}\left ( 1,x\right ) = 1.\end{aligned}\ ] ] next , applying remark [ r1 ] , we have @xmath72 _ { p , q}\sum\limits_{k=1}^{n}p^{-[(n+1-k)(n+k)/2 ] } b_{n , k}^{p , q}(1,x)\\ & & \quad\times\,\int\limits_{0}^{1}\qfrac{n}{k-1}_{p , q } t^{k } \ , ( 1\ominus qt)_{p , q}^{n+1-k } { { \mathrm{d}}}_{p , q}t \\ & = & \left [ n+1\right ] _ { p , q}\sum\limits_{k=1}^{n}p^{-[(n+1-k ) ( n+k)/2 ] } b_{n , k}^{p , q}(1,x)\\ & & \quad\times\,\qfrac{n}{k-1}_{p , q } b_{p , q}(k+1,n - k+2 ) \\ & = & \left [ n+1\right ] _ { p , q}\sum\limits_{k=1}^{n}p^{-[(n+1-k ) ( n+k)/2 ] } \ b_{n , k}^{p , q}(1,x)\qfrac{n}{k-1}_{p , q}\\ & & \quad\times \ p^{(n+1-k)(n+k+2)/2 } \ \frac{[k]_{p , q}![n - k+1]_{p , q}!}{[n+2]_{p , q } ! } \\ & = & \sum\limits_{k=1}^{n}p^{n - k+1}b_{n , k}^{p , q}(1,x ) \ \frac{[k]_{p , q}}{[n+2]_{p , q } } \\ & = & \frac{p[n]_{p , q}}{[n+2]_{p , q}}\sum\limits_{k=1}^{n}b_{n , k}^{p , q}(1,x)\frac{p^{n - k}[k]_{p , q}}{[n]_{p , q}}=\frac{p[n]_{p , q}x}{[n+2]_{p , q } } \cdot\end{aligned}\ ] ] further , using the identity @xmath73_{p , q}=p^k+q[k]_{p , q}$ ] and by remark [ r1 ] , we get @xmath74 _ { p , q}\sum\limits_{k=1}^{n}p^{-[(n+1-k ) ( n+k)/2 ] } \ b_{n , k}^{p , q}(1,x)\\ & & \quad\times\,\int\limits_{0}^{1}\qfrac{n}{k-1}_{p , q}t^{k+1 } \ , ( 1\ominus qt)_{p , q}^{n+1-k } { { \mathrm{d}}}_{p , q}t \\ & = & \left [ n+1\right ] _ { p , q}\sum\limits_{k=1}^{n}p^{-[(n+1-k ) ( n+k)/2 ] } \ b_{n , k}^{p , q}(1,x)\\ & & \quad\times\,\qfrac{n}{k-1}_{p , q } b_{p , q}(k+2,n - k+2 ) \\ & = & \left [ n+1\right ] _ { p , q}\sum\limits_{k=1}^{n}p^{-[(n+1-k ) ( n+k)/2 ] } \ b_{n , k}^{p , q}(1,x)\qfrac{n}{k-1}_{p , q}\\ & & \quad\times \ , p^{(n+1-k)(n+k+4)/2}\frac{[k+1]_{p , q}![n - k+1]_{p , q}!}{[n+3]_{p , q}!}\\ & = & \sum\limits_{k=1}^{n}p^{2(n - k+1 ) } \ b_{n , k}^{p , q}(1,x ) \ , \frac{[k]_{p , q}[k+1]_{p , q}}{[n+2]_{p , q}[n+3]_{p , q}}\end{aligned}\ ] ] i.e. , @xmath75_{p , q}(p^k+q[k]_{p , q})}{[n+2]_{p , q}[n+3]_{p , q } } \\ & = & \frac{p^{n+2}[n]_{p , q}}{[n+2]_{p , q}[n+3]_{p , q}}\sum\limits_{k=1}^{n}b_{n , k}^{p , q}(1,x ) \ \frac{p^{n - k}[k]_{p , q}}{[n]_{p , q}}\\ & & \qquad+\frac{p^{2}q[n]_{p , q}^2}{[n+2]_{p , q}[n+3]_{p , q}}\sum\limits_{k=1}^{n}b_{n , k}^{p , q}(1,x)\left(\frac{p^{n - k}[k]_{p , q}}{[n]_{p , q}}\right)^2\\ & = & \frac{p^{n+2}[n]_{p , q}x}{[n+2]_{p , q}[n+3]_{p , q}}+\frac{p^{2}q[n]_{p , q}^2}{[n+2]_{p , q}[n+3]_{p , q}}\left(x^2+\frac{p^{n-1}x(1-x)}{[n]_{p , q}}\right)\\ & = & \frac{p^{n+2}[n]_{p , q}x}{[n+2]_{p , q}[n+3]_{p , q}}+\frac{p^{2}q[n]_{p , q}^2x^2}{[n+2]_{p , q}[n+3]_{p , q}}+\frac{p^{n+1}q[n]_{p , q}x(1-x)}{[n+2]_{p , q } [ n+3]_{p , q}}\\ & = & \frac{(p+q)p^{n+1}[n]_{p , q}x}{[n+2]_{p , q}[n+3]_{p , q}}+\frac{([n]_{p , q}-p^{n-1})p^{2}q[n]_{p , q}x^2}{[n+2]_{p , q}[n+3]_{p , q } } \cdot\end{aligned}\ ] ] [ cm ] using above lemma , we can obtain the following central moments : @xmath76_{p , q}-[n+2]_{p , q})x}{[n+2]_{p , q}}\\[2 mm ] 2^\circ\!\!\!&&\!\!\ ! q}\left((t - x)^2,x\right)=\frac{(p+q)p^{n+1}[n]_{p , q}x}{[n+2]_{p , q}[n+3]_{p , q}}\\[2 mm ] & & \quad + \frac{[([n]_{p , q}-p^{n-1})p^{2}q[n]_{p , q}-2p[n]_{p , q}[n+3]_{p , q}+[n+2]_{p , q } [ n+3]_{p , q}]x^2}{[n+2]_{p , q}[n+3]_{p , q}}.\end{aligned}\ ] ] [ l2 ] let @xmath22 be a given natural number , then @xmath77_{p , q } } \ \left ( \varphi^{2}(x ) + \frac{1}{[n+2]_{p , q } } \right),\ ] ] where @xmath78 @xmath79.$ ] in view of lemma [ l1 ] , we obtain @xmath80_{p , q}x}{[n+2]_{p , q}[n+3]_{p , q}}\\[2 mm ] & & \quad + \frac{[([n]_{p , q}-p^{n-1})p^{2}q[n]_{p , q}-2p[n]_{p , q}[n+3]_{p , q}+[n+2]_{p , q } [ n+3]_{p , q}]x^2}{[n+2]_{p , q}[n+3]_{p , q}}.\qquad\end{aligned}\ ] ] by direct computations , using the definition of the @xmath0-numbers , we get @xmath81_{p , q}= p^{n+1 } ( p + q)(p^{n-1}+p^{n-2}q+p^{n-3}q^2+\cdots + pq^{n-2}+q^{n-1})>0\ ] ] for every @xmath82 . furthermore , the expression @xmath81_{p , q}+([n]_{p , q}-p^{n-1})p^{2}q[n]_{p , q}-2p[n]_{p , q}[n+3]_{p , q}+[n+2]_{p , q } [ n+3]_{p , q}\ ] ] is equal to @xmath83_{p , q}+ p^{2}q[n]_{p , q}^2-p^{n+1}q[n]_{p , q}\\[2 mm ] & & -2p[n]_{p , q}(p^{n+2}+qp^{n+1}+q^2p^n+q^3[n]_{p , q})\\[2 mm ] & & + ( p^{n+1}+qp^n+q^2[n]_{p , q})(p^{n+2}+qp^{n+1}+q^2p^n+q^3[n]_{p , q})\le 6.\end{aligned}\ ] ] in conclusion , for @xmath79,$ ] we have @xmath84_{p , q } } \ \delta_n^2(x),\end{aligned}\ ] ] which was to be proved . in this section , we estimate some direct results , viz . , local and global approximation in terms of modulus of continuity . our first main result is a local theorem . + for this , we denote @xmath85 : g''\in c[0,1]\bigr\},$ ] for @xmath86 , @xmath87functional is defined as @xmath88where norm-@xmath89 denotes the uniform norm on @xmath90 $ ] . following the well known inequality due to devore and lorentz @xcite , there exists an absolute constant @xmath91 such that @xmath92where the second order modulus of smoothness for @xmath93 $ ] is defined as @xmath94}|f(x+h)-f(x)|.\]]the usual modulus of continuity for @xmath93 $ ] is defined as @xmath95}|f(x+h)-f(x)|.\ ] ] [ t - d1 ] let @xmath96 be a natural number and let @xmath60 @xmath97 be defined as in lemma [ l2 ] . then , there exists an absolute constant @xmath98 such that @xmath99_{p , q}^{-1/2 } \delta_{n}(x)\right ) + \omega \left(f,\frac{2x}{[n+2]_{p , q}}\right),\ ] ] where @xmath100,$ ] @xmath101_{p , q}}$ ] , @xmath102 , @xmath79 $ ] and @xmath82 . for @xmath100 $ ] we define @xmath103_{p , q}x}{[n+2]_{p , q } } \right).\ ] ] then , by lemma [ l1 ] , we immediately observe that @xmath104 and @xmath105_{p , q}x}{[n+2]_{p , q } } = x.\ ] ] by applying taylor s formula @xmath106 we get @xmath107 & & \qquad - \int\limits_{x}^{\frac{p[n]_{p , q}x}{[n+2]_{p , q } } } \left(\frac{p[n]_{p , q}x}{[n+2]_{p , q } } - u \right ) g''(u)\,{{\mathrm{d}}}u.\end{aligned}\ ] ] thus , @xmath108 @xmath109_{p , q}x}{[n+2]_{p , q } } } \biggm \vert \frac{p[n]_{p , q}x}{[n+2]_{p , q } } - u \biggm \vert \vert g''(u ) \vert { { \mathrm{d}}}u \biggm \vert \notag \\[2 mm ] \!\!\!&\le & \!\!\ ! d_{n}^{p , q } ( ( t - x)^{2 } , x ) \vert g''\vert + \left ( \frac{p[n]_{p , q}x}{[n+2]_{p , q } } - x \right)^{2}\vert g''\vert.\end{aligned}\ ] ] also , we have @xmath110_{p , q}x}{[n+2]_{p , q } } - x \right)^{2 } \notag \\ & & \qquad \le \frac{6}{[n+2]_{p , q } } \left ( \varphi^{2}(x ) + \frac{1}{[n+2]_{p , q } } \right ) + \left(\frac{(p[n]_{p , q } - [ n+2]_{p , q } ) x}{[n+2]_{p , q } } \right)^{2}\qquad \notag \\ & & \qquad \le \frac{10}{[n+2]_{p , q } } \left ( \varphi^{2}(x ) + \frac{1}{[n+2]_{p , q } } \right).\end{aligned}\ ] ] hence , by ( [ 3.3.12 ] ) and with the condition @xmath96 and @xmath111,$ ] we have @xmath112_{p , q } } \ \delta_{n}^{2}(x ) \ \vert g''\vert.\ ] ] furthermore , for @xmath93 $ ] we have @xmath113_{p , q } x}{[n+2]_{p , q } } \right ) \biggm \vert \le 3 \vert f \vert.\ ] ] for all @xmath100.$ ] now , for @xmath100 $ ] and @xmath114 we obtain @xmath115 @xmath116_{p , q } x}{[n+2]_{p , q } } \right ) - f(x ) \biggm \vert \notag \\ & \le & \!\!\ ! \vert \widetilde{d}_{n}^{p , q}(f - g , x ) \vert + \vert \widetilde{d}_{n}^{p , q}(g , x ) - g(x ) \vert + \vert g(x ) - f(x ) \vert \\[2 mm ] & & \qquad\qquad\qquad\qquad\qquad\quad + \biggm \vert f\left ( \frac{p[n]_{p , q } x}{[n+2]_{p , q } } \right ) - f(x ) \biggm \vert \notag \\ & \le & \!\!\ ! 4 \ \vert f - g \vert + \frac{10}{[n+2]_{p , q } } \delta_{n}^{2}(x ) \vert g''\vert + \omega \left ( f , \biggm \vert \frac { ( p[n]_{p , q } - [ n+2]_{p , q } ) x}{[n+2]_{p , q } } \biggm \vert \right ) , \notag \\ \notag\end{aligned}\ ] ] where we have used ( [ 3.3.17 ] ) and ( [ 3.3.18 ] ) . taking the infimum on the right hand side over all @xmath114 we obtain at once @xmath117_{p , q } } \delta_{n}^{2}(x ) \right ) + \omega \left ( f , \frac{2x}{[n+2]_{p , q } } \right).\ ] ] finally , in view of ( [ 3.3.9 ] ) , we find @xmath118_{p , q}^{-1/2}\delta _ { n}(x)\right ) + \omega \left ( f,\frac{2x}{[n+2]_{p , q}}\right).\ ] ] this completes the proof of the theorem . the weighted modulus of continuity of second order is defined as : @xmath119}|f(x+h\varphi ( x))-2f(x)+f(x - h\varphi ( x))|\]]where @xmath120 . the corresponding @xmath121-functional is defined by @xmath122where @xmath123:g'\in ac_{{\mathop{\mathrm{loc}}}}[0,1],\varphi ^{2}g''\in c[0,1]\bigr\}\]]and @xmath124 $ ] means that @xmath125 is differentiable and @xmath126 is absolutely continuous on every closed interval @xmath127\subset \lbrack 0,1]$ ] . it is well - known due to ditzian - totik ( see ( * ? ? ? * , theorem 1.3.1 ) ) that @xmath128for some absolute constant @xmath129 moreover , the ditzian - totik moduli of first order is given by @xmath130}|f(x+h\psi ( x))-f(x)|,\ ] ] where @xmath131 is an admissible step - weight function on @xmath132.$ ] + now , we state our next main result , i.e. , the global estimate . [ td2]let @xmath133 be a natural number and let @xmath134 @xmath135 be defined as in lemma [ l2 ] . then , there exists an absolute constant @xmath91 such that @xmath136_{q}^{-1/2 } ) + \vec{\omega}_{\psi } ( f,[n+2]_{q}^{-1}),\]]where @xmath93,$ ] @xmath137 , and @xmath138 @xmath139 . $ ] we again consider @xmath140_{p , q}x}{[n+2]_{p , q}}\right ) , \ ] ] where @xmath93.$ ] now , using taylor s formula , we have @xmath141 using ( [ l1 ] ) , we obtain @xmath142 & & \qquad\qquad -\int\limits_{x}^{\frac{p[n]_{p , q}x}{[n+2]_{p , q}}}\ \left ( \frac{p[n]_{p , q}x}{[n+2]_{p , q}}-u\right ) g''(u)\ , { { \mathrm{d}}}u.\end{aligned}\ ] ] thus , @xmath143_{p , q}x}{[n+2]_{p , q}}}\biggm \vert \frac{p[n]_{p , q}x}{[n+2]_{p , q}}-u\biggm \vert |g''(u)|\,{{\mathrm{d}}}u\biggm \vert .\qquad\end{aligned}\ ] ] since @xmath144 is concave on @xmath132,$ ] therefore , for @xmath145 @xmath146,$ ] the following estimate holds : @xmath147 thus , using ( [ 3.3.20 ] ) , we obtain @xmath148 @xmath149_{p , q}x}{[n+2]_{p , q } } } \frac{\bigm \vert \frac{p[n]_{p , q}x}{[n+2]_{p , q}}-u\bigm \vert}{\delta _ { n}^{2}(u)}\,{{\mathrm{d}}}u\!\!\biggm \vert \ ! \vert \delta _ { n}^{2}g''\vert \notag \\ \!\!\!&\le & \!\!\!\frac{1}{\delta _ { n}^{2}(x ) } d_{n}^{p , q}((t - x)^{2},x ) \vert \delta _ { n}^{2}g''\vert + \frac{1}{\delta _ { n}^{2}(x ) } \left ( \frac{p[n]_{p , q}x}{[n+2]_{p , q}}-x\right ) ^{2}\vert \delta _ { n}^{2}g''\vert . \notag\end{aligned}\ ] ] for @xmath68,$ ] in view of ( [ ee ] ) and @xmath150_{p , q}}\ |g''(x)| \le \vert \varphi^{2 } \ , g''\vert + \frac{1}{[n+2]_{p , q}}\ \vert g''\vert , \ ] ] we have @xmath151_{p , q}}\left ( \vert \varphi ^{2}g''\vert + \frac{1}{[n+2]_{p , q}}\ \vert g''\vert \right ) . \label{3.3.21}\ ] ] using the fact that @xmath152_{p , q } \le [ n+2]_{p , q}$ ] , ( [ 3.3.18 ] ) and ( [ 3.3.21 ] ) , for @xmath100,$ ] we get @xmath153_{p , q}x}{[n+2]_{p , q}}\right ) -f(x)\biggm \vert \\ \!\!\!&\le&\!\!\!4\vert f - g\vert + \frac{10}{[n+2]_{p , q}}\vert \varphi ^{2}g''\vert + \frac{10}{[n+2]_{p , q}}\vert g''\vert\\[2 mm ] & & \qquad\qquad\qquad\ \,+\biggm \vert f\left ( \frac{p[n]_{p , q}x}{[n+2]_{p , q}}\right ) -f(x)\biggm \vert .\end{aligned}\ ] ] on taking the infimum on the right hand side over all @xmath154 we obtain @xmath155_{p , q}x } { [ n+2]_{p , q}}\right ) -f(x)\biggm \vert = \biggm \vert f\left ( x+\psi ( x)\ \frac{x \ , \left(p[n]_{p , q}-[n+2]_{p , q}\right ) } { [ n+2]_{p , q } \ \psi ( x)}\right ) -f(x)\biggm \vert\qquad\qquad\ ] ] @xmath156_{p , q}-[n+2]_{p , q}\right ) } { [ n+2]_{p , q } \ \psi ( x)}\in \lbrack 0,1]}\ \biggm \vert f\left ( t+\psi ( t ) \frac{x \ , \left(p[n]_{p , q}-[n+2]_{p , q}\right)}{[n+2]_{p , q } \psi ( x)}\right ) -f(t)\biggm \vert \\ \vec{\omega}_{\psi } \left ( f,\frac{|x \ , \left(p[n]_{p , q}-[n+2]_{p , q}\right)|}{[n+2]_{p , q } \psi ( x)}\right)\\[2 mm ] \!\!\!&\le&\!\!\ ! \vec{\omega}_{\psi } \left ( f,\frac{x}{[n+2]_{p , q } \psi ( x)}\right ) = \vec{\omega}_{\psi } \left ( f,\frac{1}{[n+2]_{p , q}}\right ) .\end{aligned}\ ] ] hence , by ( [ 3.3.22 ] ) and ( [ 3.3.19 ] ) , we finally get @xmath157_{p , q}^{-1/2})\ + \ \omega _ { \psi } ( f,[n+2]_{p , q}^{-1}).\ ] ] this completes the proof of the theorem . for @xmath158 and @xmath159 $ ] it is obvious that @xmath160 _ { p , q}=\frac{1}{p - q}.$ ] an example of such choice for @xmath161 depending on @xmath22 is recently given in @xcite . now , we show comparisons and some illustrative graphs for the convergence of @xmath0-analogue of bernstein - durrmeyer operators @xmath162 for different values of the parameters @xmath1 and @xmath163 such that @xmath164 . for @xmath165 $ ] , @xmath166 and @xmath167 , the convergence of the difference of the operators @xmath162 to the function @xmath9 , where @xmath168 , for different values of @xmath22 is illustrated in fig . [ fig1 ] . for @xmath59 $ ] , when @xmath168 and @xmath169 and @xmath170,scaledwidth=80.0% ] for @xmath59 $ ] , when @xmath171 and @xmath169 and @xmath170,scaledwidth=80.0% ] the convergence of the difference of the operators @xmath162 to the function @xmath9 , where @xmath172 for different values of @xmath22 and @xmath165 $ ] is illustrated in fig . [ fig2 ] . for the function @xmath173 ( and @xmath174 , @xmath175 ) , the limit of @xmath176 , when @xmath177 , is @xmath178 . graphics of @xmath179 , for @xmath180 , are presented in fig . [ fig3 ] . , when @xmath173 and @xmath178 , for @xmath180,scaledwidth=80.0% ] about a decade ago , king @xcite proposed a technique to obtain better approximation for the well known bernstein polynomials . in this technique , these operators approximate each continuous function @xmath181,$ ] while preserving the function @xmath182 these were basically compared with estimates of approximation by bernstein polynomials . various standard linear positive operators preserve @xmath183 and @xmath184 i.e. , preserve constant and linear functions , but this approach helps in reproducing the quadratic functions as well . so , using king s technique , we modify the operators ( [ operator ] ) as follows : @xmath185 _ { p , q}\sum\limits_{k=1}^{n}p^{-(n - k+1)(n+k)/2 } b_{n , k}^{p , q}(1,r_{n}(x))\\ & & \qquad\times\,\int\limits_{0}^{1}b_{n , k-1}^{p , q}(t ) f\left ( t\right ) { { \mathrm{d}}}_{p , q}t + b_{n,0}^{p , q}(1,r_{n}(x ) ) f(0),\end{aligned}\ ] ] where @xmath186_{p , q } \ , x}{p [ n]_{p , q}}$ ] and @xmath187_{p , q}}{p [ n]_{p , q}}\right]$ ] . then , we have @xmath188 let @xmath96 be a natural number and let @xmath60 @xmath97 be defined as in lemma [ l2 ] . then , there exists an absolute constant @xmath98 such that @xmath190 where @xmath187_{p , q}}{p [ n]_{p , q}}\right],$ ] @xmath82 , and @xmath191_{p , q } } + \frac{\{([n]_{p , q}-p^{n-1 } ) \ , q \ , [ n+2]_{p , q } - [ n]_{p , q } [ n+3]_{p , q}\}x^{2}}{[n]_{p , q } [ n+3]_{p , q } } \cdot\end{aligned}\ ] ] for @xmath193_{p , q}}{p [ n]_{p , q}}\right],$ ] @xmath166 and @xmath167 , the convergence of the difference of the operators @xmath194 to the function @xmath9 , where @xmath168 , for different values of @xmath22 is illustrated in fig . [ fig4 ] .

in the present paper , we consider @xmath0-analogue of the beta operators and using it , we propose the integral modification of the generalized bernstein polynomials . we estimate some direct results on local and global approximation . also , we illustrate some graphs for the convergence of @xmath0-bernstein - durrmeyer operators for different values of the parameters @xmath1 and @xmath2 using mathematica package .

we study the @xmath149 edwards - anderson model , whose hamiltonian is given by @xmath150 the spins @xmath151 are placed on the nodes , @xmath152 , of a cubic lattice of linear size @xmath9 and we set periodic boundary conditions . the couplings @xmath153 , which join nearest neighbors only , are chosen randomly with @xmath154 probability and are quenched variables . for each choice of the couplings ( one `` sample '' ) , we simulate two independent copies of the system , @xmath155 and @xmath156 . we denote by @xmath157 the average over the thermal noise and by @xmath158 the _ subsequent _ average over the samples . the model described by eq . undergoes a sg transition at @xmath29 and @xmath159 @xcite . for our dynamical data we have run new non - equilibrium simulations on memento , janus and janus ii . we use heat - bath dynamics , in which one monte carlo step roughly corresponds to one picosecond of the experimental system @xcite . see appendix [ sec : simulations ] for technical details of these simulations . the two main dynamical observables are the magnetization density @xmath160 and the spin temporal correlation function @xmath161 . equilibrium results at @xmath44 are available for @xmath162 @xcite . in this case the main quantity is the probability density function @xmath34 of the spin overlap @xmath163 : @xmath164 in particular , we are interested in the integral @xmath165 the @xmath34 curves are easily described for finite @xmath9 . they are symmetric under @xmath166 , with two maxima at @xmath167 and a flat central region . in the thermodynamic limit , the two peaks turn into delta functions at @xmath168 , which mark the maximum possible value of @xmath169 . the size evolutions , as checked for @xmath170 @xcite , are as follows : @xmath171 ( at @xmath44 , @xmath172 @xcite ) , the width of the peaks at @xmath167 scales as @xmath173 while @xmath174 turns out to be greater than zero and @xmath9-independent . using heat - bath dynamics on the janus , janus ii and memento supercomputers , we consider the following numerical experiment . starting from a completely random configuration of the spins at @xmath44 , we first let the system evolve in absence of a magnetic field , i.e. @xmath29 , for a waiting time @xmath51 . as this @xmath51 grows , the spins rearrange in amorphous magnetic domains of increasing average size @xmath118 , as we show in fig . [ fig : fssxi ] ( @xmath118 is computed with the @xmath175 integral estimator described in refs . @xcite ) . after this time @xmath51 , we turn on a tiny field @xmath176 and follow the response at a later time @xmath177 . versus waiting time @xmath51 at @xmath44 for different lattice sizes : @xmath178 ( data taken from @xcite ) , @xmath54 ( new simulations ) and @xmath179 ( metropolis dynamics from @xcite , rescaling the @xmath180 axis by a factor of 4 to compare with our heat - bath dynamics ) . the dashed lines aim to point out the different @xmath51 ( and their corresponding @xmath118 ) considered in this work . [ fig : fssxi ] ] we have considered five different values of @xmath51 : @xmath49 and @xmath50 were simulated on janus ii ; @xmath181 and @xmath182 on janus ( smaller systems were simulated on memento , see below our study of size effects ) . times are measured in units of monte carlo sweeps . the measuring times @xmath7 were chosen as the integer part of @xmath183 for integer @xmath184 ( discarding repetitions ) . for each @xmath51 we repeat the procedure described above for four values of the magnetic field : @xmath185 in the case of memento and janus i supercomputers and @xmath186 on janus ii . we considered exactly the same set of samples with each @xmath20 and reused the same sequences of random numbers in an effort to eliminate sources of fluctuations . depending on the computer used , we simulated different system sizes , either @xmath178 ( on memento and janus i ) or @xmath54 ( on janus ii ) . we simulated 647 samples for @xmath178 ( all @xmath3 and @xmath20 values ) . for @xmath54 , we used 55 samples for @xmath49 and 335 samples for @xmath50 [ we also simulated 336 samples at @xmath29 in order to compute @xmath14 ] . notice that self - averaging means that one needs fewer samples for larger sizes . previous works at @xmath29 suggested that finite - size effects should be negligible , compared to our typical statistical accuracy , as long as we ensure that @xmath188 @xcite . as a new test of the validity of this statement , we compare our new results of @xmath14 obtained with janus ii and @xmath54 with previous works corresponding to @xmath178 @xcite and @xmath179 @xcite ( see fig . [ fig : fssxi ] ) finding no significant dependence on @xmath9 in the studied range of @xmath51 . the discussion on the violations of fdt requires the computation of the linear susceptibility , that is , of @xmath189 with this aim , we measure @xmath190 at several values of the external field , and use them to extract the @xmath191 limit . indeed , since the edwards - anderson hamiltonian is odd in the field around @xmath29 , one can write the magnetization in terms of odd powers of @xmath20 , which allows us to separate the linear response @xmath27 from the non - linear responses @xmath192 in order to make some progress , we taylor - expand @xmath193 , thus finding : @xmath194 therefore , if we measure @xmath195 for three small fields and neglect higher - order contributions in @xmath20 , we can extract @xmath146 from a set of three equations and three unknowns [ by the same token , we obtain @xmath196 and @xmath197 as well , but these magnitudes will not be discussed herein ] . we show in fig . [ fig : mh2e26 ] @xmath198 and @xmath146 for one of our values of @xmath51 . from the @xmath198 data obtained at @xmath199 and @xmath200 . data shown here corresponds to @xmath201 . for the sake of visibility , only one every two measured times have been plotted in points . [ fig : mh2e26 ] ] alternatively , instead of performing simulations at different @xmath20 , one could have obtained @xmath146 directly from simulations at @xmath29 using methods such as those described in refs . the drawback of this approach is that it would have required a much larger amount of samples in order to get equivalent statistical errors . the original data consisted of pairs @xmath202 , where @xmath7 takes some discrete values . however , if we reproduce fig . 1 in the main text but using the raw measurements ( see fig . [ fig : fdtoriginal ] ) we find much noisier curves . indeed , data for successive times , although very correlated , displays random fluctuations . besides , the statistical errors for @xmath203 and @xmath204 are completely negligible compared to the errors in @xmath205 ( they are indistinguishable in the figure ) . we used these two facts to our benefit in order to smooth and reduce the statistical errors of these curves . let us describe our smoothing procedure step by step . versus @xmath43 at @xmath44 and five values of @xmath51 using raw processed data ( to be compared with fig . 1 in the main text , which was obtained only after the smoothing of the simulation data at fixed @xmath20 and an extrapolation to @xmath191 ) . [ fig : fdtoriginal ] ] we fit our data for @xmath205 to a smooth function of @xmath206 this choice [ instead of just @xmath43 ] , although irrelevant in the @xmath191 limit , turns out to reduce the non - linear corrections in @xmath20 as we show in fig . [ fig : fdtdiffc ] , and yields easier and more accurate fits . to @xmath40 when plotted versus @xmath43 ( left ) or @xmath208 ( right ) . data corresponds to @xmath201 and @xmath44 . [ fig : fdtdiffc ] ] our chosen functional form is as follows . let the quantity @xmath205 be approximated by @xmath209 ( @xmath210 depends on @xmath20 and @xmath3 , but we will write @xmath210 nevertheless , to keep the notation as light as possible ) : @xmath211 } { 2}\\ & + , f_\mathrm{s}(\hat x ) \frac{1-\tanh[q(\hat x)]}{2}\ , , \end{split}\ ] ] with @xmath212 . in other words , there are two functional forms : @xmath213 , adequate for small @xmath214 and @xmath215 , good for large @xmath214 . the crossover between the two functional forms takes place at @xmath216 in an interval of half - width @xmath217 ( although we keep @xmath218 and @xmath219 as fitting parameters ) . the functional form for small @xmath214 are diagonal [ @xmath220 pad approximants , @xmath221 as for the region where violations of the fluctuation - dissipation theorem are tiny , we chose a polynomial in @xmath222 @xmath223 we keep @xmath224 as fitting variables . following refs . @xcite , we perform a fit considering only the diagonal part of the covariance matrix ( we obtain @xmath225 significantly smaller than one , probably due to data correlation ) . errors are computed following a jackknife procedure [ we perform an independent fit for each jackknife block , and compute errors from the jackknife fluctuations of the fitted @xmath209 ] . our fits are reported in table [ table : fits ] . .information about the fits to eqs . ( [ eq : fit1],[eq : fit2],[eq : fit3]).[table : fits ] [ cols="^,^,^,^,>",options="header " , ] once each curve @xmath227 is smoothed at each @xmath20 , we extract the linear susceptibility following the procedure described in the previous section . we show a comparison between the original and smoothed data in fig . [ fig : fdtoriginal - smoothed ] . we found that in most the cases the extrapolated linear response @xmath40 was compatible within the error with the smaller field considered . however , the extrapolation @xmath191 becomes particularly delicate and even changes the shape of the curve at large values of the @xmath228 ratio , as we show in fig . [ fig:2e11 ] . versus @xmath43 curves . data corresponds to @xmath44 and five values of @xmath51 . [ fig : fdtoriginal - smoothed ] ] versus @xmath229 for several values of @xmath20 ( in color empty dots ) and @xmath230 , together with the extrapolation @xmath191 ( in black crosses ) . the inset is a blow up of the region for large @xmath228 in the square box . [ fig:2e11 ] ] part of our discussion in the main text seeks to find a relation between the linear response at finite @xmath51 with the overlap distribution @xmath34 in equilibrium at a finite size @xmath35 . that is , @xmath231 we computed @xmath33 by means of a numerical integration of the @xmath34 discussed in ref . @xcite for @xmath232 and @xmath233 . we show @xmath33 in the main panel of fig . [ fig : scl ] . in order to identify @xmath35 we needed a function @xmath234 that is continuous both in @xmath28 and in @xmath9 , which we construct by computing a cubic spline of the data along both variables ( first in @xmath28 and only then in @xmath9 ) . errors are computed using the jackknife method . we show some interpolation curves along the @xmath180 variable in the inset of fig . [ fig : scl ] . once @xmath234 is at hand , @xmath38 can be extracted by looking for the @xmath180 value that satisfies the eq . at each time @xmath7 , fixing the off - equilibrium data @xmath40 and @xmath235 . versus @xmath28 for different system sizes obtained using eq . and data from ref . ( inset ) orthogonal cuts to the figure in the main panel plotted as function of @xmath9 in color points together with the interpolating cubic spline curve along this variable.[fig : scl ] ] up to now , finite - size effects have been investigated only for single - time correlation functions [ and the related extraction of @xmath14 ] . as far as we know , size effects were not studied previously in the response to a magnetic field @xmath146 . in this context , it is somewhat worrying that we have identified a large length scale @xmath236 ( discussed below ) in the regime where violations of the fdt are incipient . for this reason , we have explicitly checked that our data does not suffer from finite - size effects in that region ( as we show in fig . [ fig : fss ] ) by comparing results from three system sizes , @xmath237 and @xmath238 , in the case of @xmath239 , finding no finite - size dependence . for the smaller system sizes we considered 28000 samples for @xmath240 and 12000 samples for @xmath241 . versus @xmath43 at @xmath44 . data from @xmath237 and @xmath238 are compared in the case of @xmath239 . all the points are compatible within the error bars.[fig : fss ] ] in the main text , we have used several times the inequality @xmath242 our purpose here is to remind the reader of its derivation , for the sake of completeness . let us first recall the notations used in the main text : @xmath243 we start by noticing @xmath244 due to the inequality @xmath245 for the cumulative distribution . next , we integrate by parts to find [ recall that @xmath246 @xmath247 finally , to obtain the upper bound in ( [ eq : desigualdad ] ) , we remark that @xmath248 is monotonically decreasing in @xmath9 for a system with periodic boundary conditions . . in the top panel , we compare the dynamic response @xmath249 with the equilibrium curve @xmath250 . the range of @xmath28 covers the peak width of @xmath251 @xcite . since the curvature is clearly larger for @xmath252 than for @xmath250 , eq . tells us that that the maximum of @xmath253 is higher than the maximum of @xmath251 . the lines correspond to diagonal fits to fourth order polynomials in @xmath28 ( we increased the order of the polynomial until the figure of merit diagonal-@xmath254 for the fit of the dynamic response no longer decreased ) . the bottom panel shows the second derivative of the interpolating polynomials of the top panel , multiplied by @xmath255 . according to eq . , these derivatives should give us @xmath256 and @xmath34 . indeed , the peak position and height in @xmath251 is very reasonably reproduced by this approach , see ref . we have shown in the main text that , for every @xmath3 and small enough @xmath7 , @xmath257 can be very large . this _ short - time but large - size _ effect arises when @xmath258 . in fact , for @xmath50 ( our largest ) we can compute @xmath52 without extrapolations only for the largest @xmath7 . the above observation begs the question : how large can @xmath52 be in this small-@xmath7 regime ? we provide here a crude extrapolation for our @xmath50 data , mostly based on the scaling laws found in @xcite . we start by noticing that one could be tempted to extract the spin - overlap probability directly from the aging response . one can define the _ dynamic overlap _ probability density function : @xmath259 then , one could compare @xmath260 with the equilibrium @xmath34 at @xmath261 . the weak point in this approach is that taking two derivatives of the curve @xmath262 , which is subject to random errors , is very difficult . our way out will be to recall that the area under the peak of the @xmath34 is approximately @xmath9-independent @xcite . therefore , we shall estimate the peak height ( rather than the peak width ) . our efforts to locate the maximum ( let alone the full curve ) for @xmath253 are documented in fig . [ fig : p - dyn ] ( but the reader is warned to take the results _ cum grano salis _ ) . we note from fig . [ fig : p - dyn ] that the ratio of the height of the maxima for @xmath50 and @xmath263 is @xmath264 . therefore , from the scaling of the peak width , @xmath265 , we extrapolate @xmath266 which is certainly larger than our maximum equilibrium size , @xmath263 . in the main text , we wondered about the consequences of having at our disposal only a simplified approximation for @xmath33 : @xmath267\,.\ ] ] in the above equation , @xmath121 and @xmath122 are @xmath9-independent constants . all the depedence on the system size is in @xmath268 . in fact , @xmath268 was obtained by fitting the actual data @xmath269 to a quadratic polynomial in @xmath270 . we took @xmath271 from ref . @xcite [ recall that the maximum of the spin - overlap probability , @xmath34 scales with @xmath9 as @xmath272 . once @xmath268 was known , we determined the constants @xmath121 and @xmath122 from a least - squares minimization of the difference between @xmath128 and the actual data . 63ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) , in @noop _ _ , , ( , ) link:\doibase 10.1103/physrevlett.91.037203 [ * * , ( ) ] link:\doibase 10.1007/bf02704172 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.87.087204 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.157201 [ * * , ( ) ] , link:\doibase 10.1088/1742 - 5468/2010/06/p06026 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.105.177202 [ * * , ( ) ] , link:\doibase 10.1038/nphys1482 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.106.150603 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.71.173 [ * * , ( ) ] link:\doibase 10.1007/bf02184881 [ * * , ( ) ] link:\doibase 10.1088/0305 - 4470/31/11/011 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.81.1758 [ * * , ( ) ] \doibase 10.1023/a:1004602906332 [ * * , ( ) ] link:\doibase 10.1088/0305 - 4470/33/12/305 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.88.257202 [ * * , ( ) ] , link:\doibase 10.1103/physrevb.67.214425 [ * * , ( ) ] link:\doibase 10.1140/epjb / e2004 - 00278 - 6 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.43.1754 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.90.237201 [ * * , ( ) ] link:\doibase 10.1103/physrevb.72.104407 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.79.3660 [ * * , ( ) ] link:\doibase 10.1209/epl / i1999 - 00313 - 4 [ * * , ( ) ] link:\doibase 10.1103/physreve.63.012503 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.220601 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.035701 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.83.5038 [ * * , ( ) ] link:\doibase 10.1209/epl / i2001 - 00182 - 9 [ * * , ( ) ] link:\doibase 10.1103/physrevb.81.104201 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.109.097401 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.103.040601 [ * * , ( ) ] link:\doibase 10.1088/1742 - 5468/2009/04/p04012 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.97.265702 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1532349 [ * * , ( ) ] link:\doibase 10.1038/nphys3435 [ * * , ( ) ] link:\doibase 10.1016/j.cpc.2007.09.006 [ * * , ( ) ] , link:\doibase 10.1016/j.cpc.2013.10.019 [ * * , ( ) ] , link:\doibase 10.1103/physrevb.43.8199 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.82.5128 [ * * , ( ) ] , link:\doibase 10.1103/physrevb.62.14237 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.82.438 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.92.167203 [ * * , ( ) ] link:\doibase 10.1126/science.1120714 [ * * , ( ) ] @noop _ _ ( , , ) @noop * * , ( ) , link:\doibase 10.1103/physrevb.32.7384 [ * * , ( ) ] link:\doibase 10.1109/mcse.2009.11 [ * * , ( ) ] link:\doibase 10.1007/s10955 - 009 - 9727-z [ * * , ( ) ] , link:\doibase 10.1007/s100510050942 [ * * , ( ) ] , link:\doibase 10.1103/physreve.68.065104 [ * * , ( ) ] link:\doibase 10.1103/physrevb.91.174202 [ * * , ( ) ] link:\doibase 10.1103/physrevb.91.104430 [ * * , ( ) ] , http://stacks.iop.org/1742-5468/2016/i=1/a=013301 [ * * , ( ) ] @noop _ _ ( , , ) \doibase 10.1023/a:1018607809852 [ * * , ( ) ] , link:\doibase 10.1103/physreve.93.032126 [ * * , ( ) ] link:\doibase 10.1103/physrevb.88.224416 [ * * , ( ) ] , @noop _ _ ( , , ) @noop * * , ( )

the unifying feature of glass formers ( such as polymers , supercooled liquids , colloids , granulars , spin glasses , superconductors , ... ) is a sluggish dynamics at low temperatures . indeed , their dynamics is so slow that thermal equilibrium is never reached in macroscopic samples : in analogy with living beings , glasses are said to age . here , we show how to relate experimentally relevant quantities with the experimentally unreachable low - temperature equilibrium phase . we have performed a very accurate computation of the non - equilibrium fluctuation - dissipation ratio for the three - dimensional edwards - anderson ising spin glass , by means of large - scale simulations on the special - purpose computers janus and janus ii . this ratio ( computed for finite times on very large , effectively infinite , systems ) is compared with the equilibrium probability distribution of the spin overlap for finite sizes . the resulting quantitative statics - dynamics dictionary , based on observables that can be measured with current experimental methods , could allow the experimental exploration of important features of the spin - glass phase without uncontrollable extrapolations to infinite times or system sizes . theory and experiment follow apparently diverging paths when studying the glass transition . on the one hand , experimental glass formers ( spin glasses , fragile molecular glasses , polymers , colloids , ) undergo a dramatic increase of characteristic times when cooled down to their glass temperature , @xmath0 @xcite . below @xmath0 , the glass is always out of equilibrium and _ aging _ appears @xcite . consider a rapid quench from a high temperature to the working temperature @xmath1 ( @xmath2 ) , where the system is left to equilibrate for time @xmath3 and probed at a later time @xmath4 . response functions such as the magnetic susceptibility turn out to depend on @xmath5 , with @xmath6 @xcite . the age of the glass , @xmath3 , remains the relevant time scale even for @xmath3 as large as several days . on the other hand , theoreticians find it easier to compute equilibrium properties ( e.g. , non - linear magnetic susceptibilities ) , either for finite or infinite systems . relating this equilibrium information to the aging experimental responses is an open problem . a promising way to fill the gap is to establish a statics - dynamics dictionary ( sdd ) @xcite : non - equilibrium properties at _ finite times _ @xmath7 , @xmath3 , as obtained on samples of macroscopic size @xmath8 , are quantitatively matched to equilibrium quantities computed on systems of _ finite size _ @xmath9 [ the sdd is an @xmath10 correspondence ] . clearly , in order for it to be of any value , an sdd can not strongly depend on the particular pair of aging and equilibrium quantities that are matched . some time ago , we proposed one such a sdd @xcite . however , this sdd was unsatisfactory in two respects . first , @xmath9 was matched only to @xmath3 ( irrespectively of the probing time @xmath4 ) . second , our sdd matched spatial correlation functions whose experimental study is only incipient @xcite . a different approach was advocated by barrat and berthier @xcite , who suggested building an sdd through the pattern of violations of the fluctuation - dissipation theorem ( fdt ) @xcite . however , the feasibility ( for glasses ) of this approach is a non - trivial statement that needs to be established . indeed , as we argue below the fdt - based sdd does _ not _ exist for ferromagnets . let us remark that fdt violations carry crucial information @xcite : they provide a promising experimental path towards measuring parisi s functional order parameter @xcite . as a consequence the pattern of fdt violations has attracted much attention . one encounters numerical studies for both ising @xcite and heisenberg @xcite spin glasses , as well as for structural glasses @xcite . on the experimental side , we have studies on atomic spin glasses @xcite , superspin glasses @xcite , polymers @xcite , colloids @xcite or dna @xcite . here , we perform a detailed simulation of fdt violations in the three - dimensional ising spin glass employing the custom - made supercomputers janus @xcite and janus ii @xcite . in fact , this has been the launching simulation campaign of the janus ii machine , which was designed with this sort of dynamical studies in mind . our simulations stand out by the spanned time range ( 11 orders of magnitude ) , by our high statistical accuracy and by the range of system sizes , enabling us to control size effects ( @xmath11 and @xmath12 ) . thus armed , we assess whether or not an sdd can be built from these violations , and compare the sdd proposed in this paper with other proposals . we focus on spin glasses , rather than on other model glasses , for a number of reasons : ( i ) their sluggish dynamics is known to be due to a thermodynamic phase transition at @xmath13 @xcite ; ( ii ) the size of the _ glassy _ magnetic domains , @xmath14 , is experimentally accessible @xcite ( @xmath15 lattice spacings @xcite , much larger than comparable measurements for structural glasses @xcite ) ; ( iii ) a restricted fdt - based sdd , see eq . eq . below , has been well established @xcite ; ( iv ) fdt violations have been studied experimentally @xcite ; ( v ) well developed , yet mutually contrasting , theoretical scenarios are available for spin glasses in equilibrium @xcite ; ( vi ) magnetic systems are notably easier to model and to simulate numerically ( in fact , special - purpose computers have been built for the simulation of spin glasses @xcite ) . [ [ fdt - violations - and - the - sdd ] ] fdt violations and the sdd + + + + + + + + + + + + + + + + + + + + + + + + + + + + we suddenly cool a three - dimensional spin - glass sample of size @xmath16 from high temperature to the working ( sub - critical ) temperature @xmath17 at the initial time @xmath18 ( see appendix [ sec : methods ] , below , for more details and definitions ) . during the non - equilibrium relaxation a coherence length @xmath19 ) grows @xcite , which is representative of the size of the spin - glass domains . then , from the waiting time @xmath3 on , we place the system under a magnetic field of strength @xmath20 , and consider the response function at a later measuring time @xmath4 @xmath21 where @xmath22 is the magnetization density in a sample of linear size @xmath9 . this susceptibility is then compared with the spin temporal correlation function @xmath23 . from now on , we shall take the limits @xmath24 which are easy to control numerically : if @xmath25 size effects are negligible @xcite ( see also appendix [ sec : finite - size ] ) . the fdt states that @xmath26 , with both @xmath27 and @xmath28 computed at @xmath29 . however , for @xmath30 the fdt does not hold . in fact , fdt violations take the form @xcite ( the order of limits is crucial ) : @xmath31\,,\label{eq : fdtv-1 } \end{split}\ ] ] where @xmath7 is scaled as @xmath3 grows , to ensure that the full range @xmath32 gets covered , and @xmath33 is given by a double integral of @xmath34 , the equilibrium distribution function of the spin overlap [ see eq . ] . here , we mimic an experimental protocol @xcite in that we consider the non - equilibrium response on a very large system but at _ finite times_. we try to relate this response with the equilibrium overlap for a system of finite effective size @xmath35 @xmath36 where we have assumed that both @xmath27 and @xmath28 have reached their thermodynamic limit . the same approach was followed for a two - dimensional spin glass by barrat and berthier @xcite ( note , however , that there is no stable spin - glass phase at @xmath37 in two spatial dimensions ) . . provides a statics - dynamics dictionary ( sdd ) relating both times @xmath7 and @xmath3 with a single effective equilibrium size @xmath38 . note that it is not obvious a priori that our program can be carried out . for instance , our sdd does not exist for ferromagnets : one can readily show that @xmath39 , see appendix [ sec : inequality ] , but @xmath40 grows in ferromagnets well above this bound @xcite . sdds based on the comparison of aging and equilibrium correlation functions ( rather than on fdt violations ) have been studied in some detail @xcite . it was found that the effective length depends solely on @xmath3 . indeed , @xmath41 with @xmath42 , was accurate enough to match the correlation functions @xcite . ref . @xcite also agreed with eq . . in fact , eq . also underlies the analysis of refs . @xcite . yet , we shall show below that eq . is oversimplified . versus @xmath43 at @xmath44 [ for fixed @xmath3 , @xmath43 monotonically decreases from @xmath45 at @xmath46 to @xmath47 at @xmath48 . data for @xmath49 and @xmath50 was obtained on janus ii ( the other @xmath51 are from janus ) . the five values of @xmath3 correspond to effective equilibrium sizes @xmath52 that , according to eq . , span the size range investigated in ref . @xcite ( namely , @xmath53 ) . * inset : * growth of the spin - glass coherence length @xmath14 as a function of time , computed at zero magnetic field and following refs . @xcite , from simulations of @xmath54 lattices at @xmath44 on janus ii . in dashed lines we plot the scaling @xmath55 with @xmath56 from ref . @xcite . ] [ [ numerical - data ] ] numerical data + + + + + + + + + + + + + + + + the three basic quantities computed in this work , namely @xmath57 and @xmath14 are displayed in fig . [ fig : fdt - xi ] . full details about this computation are provided in appendix [ sec : susclin ] . let us remark that the janus ii supercomputer allows us to probe unexplored dynamical regimes , either @xmath58 as large as @xmath59 ( i.e. , we follow the magnetic response for a very long time , after the field was switched on at @xmath49 ) or @xmath60 as large as @xmath61 ( i.e. , we study the response of a very old spin glass , but we are limited to @xmath62 in this case ) . it is also remarkable that we are able to compute both the susceptibility @xmath27 and the correlation function @xmath28 without worrying about finite - size effects . indeed , size effects become visible when the coherence length reaches the threshold @xmath63 @xcite which in our @xmath54 lattice translates to @xmath64 lattice spacings . as fig . [ fig : fdt - xi]inset shows , we are quite far from this safety threshold . with respect to previous measurements of the fdt ratio , it is worth stressing that now we are able to take the @xmath65 limit in a more controlled way . this is far from trivial , given that the linear response regime shrinks to very small field when @xmath51 increases ( see appendix [ sec : susclin ] ) . the data in fig . [ fig : fdt - xi ] also stands out by its statistical accuracy ( due to the large number of samples and large system sizes we simulated , but also thanks to the analysis method described in appendix [ sec : smooth ] . as a consequence , the violations to the fdt @xmath66 can be studied in great detail . in particular , the reader might be stricken by the linear behavior at @xmath67 . in fact , following refs . @xcite , this linear behavior could be interpreted as evidence for one step of replica - symmetry breaking ( see , for instance , ref . @xcite ) . however , we shall argue below that the effective length in eq . evolves as time @xmath7 grows , thus producing an upturn in the response which is probably responsible for the linear behavior in fig . [ fig : fdt - xi ] . let us make a final remark . we know that @xmath33 is upper bounded by @xmath68 . now , at @xmath44 we know that @xmath69 @xcite ( or 0.46(3 ) @xcite ) . therefore , the dynamic responses @xmath70 in fig . [ fig : fdt - xi ] are well below @xmath71 , at variance with ferromagnets @xcite . it follows that the effective length in eq . can be perfectly defined for three - dimensional spin glasses . ( we only show data for three @xmath3 , for the sake of clarity ) . lines are @xmath72 , recall eq . , with the effective equilibrium size as in eq . : @xmath73 . dotted lines correspond to @xmath74 , which is the proportionality constant that was found by matching equilibrium and non - equilibrium correlation functions @xcite . the continuous lines were found by choosing the best possible @xmath75 for each @xmath3 . this representation shows that the single - time statics - dynamics dictionary @xmath76 breaks down for large @xmath7 , when @xmath77 is much larger than @xmath78 . [ fig : fdt - no - one - time ] ] , we show the effective equilibrium side @xmath38 in units of the coherence length at the measuring time @xmath79 versus the ratio of coherence lengths @xmath80 ( recall that @xmath7 is the time elapsed since switching - on the magnetic field ) . the ratio of coherence lengths is 1 for @xmath46 and goes as @xmath81 for large time , with @xmath82 @xcite . let us stress that there is no extrapolation in this figure , only interpolation ( i.e. , @xmath83 falls within the simulated equilibrium sizes , @xmath84 ) . the solid line is a fit to the scaling function @xmath85 in eq . and eq . . * inset : * @xmath38 data from the main panel in units of the coherence length at the initial time time @xmath14 , as a function of the time ratio @xmath58 . ] [ [ the - effective - equilibrium - size ] ] the effective equilibrium size + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + as we show in fig . [ fig : fdt - no - one - time ] , our data are too accurate to be quantitatively described by combining eq . with eq . this simple description fails both at short times @xmath7 ( i.e. , when @xmath86}$ ] ) and also at very long @xmath7 , although one can find a constant @xmath75 that works well for intermediate @xmath7 . the discrepancy for long @xmath7 seems easy to rationalize : since the growth of @xmath14 is very slow , recall fig . [ fig : fdt - xi]inset , @xmath79 and @xmath14 are very similar to each other for small @xmath7 and , therefore , @xmath87 makes sense . however , since @xmath14 grows without bounds in the spin - glass phase , one should eventually have @xmath88 . under these circumstances , it is only natural that @xmath89 . we can test this proposal by computing an exact @xmath83 for each @xmath90 pair ( see appendix [ sec : leff ] for details ) , which we plot in fig . [ fig : fdt - leff ] : in the main panel in units of @xmath77 and in the inset in units of @xmath78 . the first important observation from the main panel in fig . [ fig : fdt - leff ] is that , for long enough times , we find @xmath91 , in agreement with the intuition exposed above . this is definitely different from eq . , used until now . the data in the inset of fig . [ fig : fdt - leff ] explain why the previous relation in eq . passed many numerical tests until now : the non - monotonic behavior of @xmath92 for short times @xmath7 makes this ratio roughly compatible with a constant @xmath93 as long as @xmath94 . surprisingly , the ratio @xmath95 , or equivalently @xmath92 , becomes large as well when @xmath96 , thus explaining the inability of eq . in describing dynamical data at short times @xmath7 ( see fig . [ fig : fdt - no - one - time ] ) . nonetheless in the limit @xmath96 , i.e. @xmath97 , the effective equilibrium size @xmath52 seems to reach a finite value ; a divergence of @xmath98 in this limit seems unlikely ( see appendix [ sec : pdyn ] ) . [ [ l_mathrmeff - and - the - spin - glass - coherence - length ] ] @xmath52 and the spin - glass coherence length + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + now that it is clear that both @xmath14 and @xmath79 are relevant for @xmath52 one may ask about the crossover between the @xmath14-dominated regime and the @xmath79-dominated regime . fig . [ fig : fdt - leff ] tells us that @xmath99 is , to a good approximation , a function of the ratio @xmath80 are slightly off , in fig . [ fig : fdt - leff ] . we attribute the effect to a strong statistical fluctuation , enhanced by the fact that all data points with the same @xmath3 are extremely correlated . ] . thus , we attempted to fit the crossover with the functional form @xmath100 where the scaling function is @xmath101 interpolation of data shown in fig . [ fig : fdt - leff ] returns : @xmath102 , @xmath103 and @xmath104 . noticing that @xmath105 and @xmath106 , where @xmath107 is the exponent for the time growth of the coherence length , @xmath82 ( see fig . [ fig : fdt - xi]inset , and refs . @xcite ) , the scaling function @xmath85 can be also rewritten in a much simpler form as @xmath108 fitting data in fig . [ fig : fdt - leff ] with this simpler scaling function returns @xmath109 ( see full curve in fig . [ fig : fdt - leff ] ) . given that the fit with 3 adjustable parameters in eq . and the one in eq . with just 1 adjustable parameter have practically the same quality - of - fit , we tend to prefer the simpler ansatz , as long as it interpolates the numerical data well enough . the ultimate check for the success of eq . and eq . in reproducing the aging response is provided by fig . [ fig : dominance - large - power ] , where the dynamical measurements ( data points with errors ) are plotted together with the equilibrium function @xmath110 . the very good agreement in the whole range gives a strong support in favor of an sdd based on eq . and eq . . note as well that eq . explains the previous success of the simpler sdd in eq . . in fact , at short times @xmath7 , the two coherence lengths @xmath79 and @xmath14 are very similar to each other , and the amplitude @xmath75 in eq . is essentially @xmath111 . the ansatz of eq . provides as well a simple explanation for the upturn of the aging response at small values of @xmath28 , recall fig . [ fig : fdt - xi ] . indeed , as time @xmath7 increases , the correlation function decays as @xmath112 @xcite . but , from @xmath113 we conclude that , even at fixed @xmath3 , @xmath52 diverges for large @xmath7 as @xmath114 . now , to a first approximation , one may expect that @xmath115 ( see the description of the overlap distribution function in appendix [ sec : methods ] ) . we thus expect the susceptibility to approach its @xmath47 limit in a singular way , as @xmath116 . but @xmath52 is taken from the ansatz in eq . and eq . , which improves on the single - time statics - dynamics dictionary based on @xmath78 by considering a crossover to a @xmath77-dominated regime . [ fig : dominance - large - power ] ] [ [ which - features - of - the - pq - can - be - obtained - from - dynamic - measurements ] ] which features of the @xmath117 can be obtained from dynamic measurements ? + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + one of the major gains of the present analysis would be to obtain parisi s functional order parameter @xmath117 from experimental dynamic data . in an ideal situation , one would have data for @xmath27 , @xmath28 and @xmath118 , complemented by the ansatz in eq . . then , one would like to know which features of the underlying @xmath33 can be retrieved from these dynamic measurements . in order to answer this question , we have considered a very simplified @xmath119 , that possesses the main features of the @xmath34 measured in numerical simulations ( see appendix [ sec : methods ] ) : @xmath120 where @xmath121 and @xmath122 are constants , @xmath123 is the heaviside step function and @xmath124 is a weight enforcing normalization . ] . integrating @xmath119 twice we get @xmath125.\ ] ] @xmath126 is taken from the true @xmath34 . recall that @xmath127 , eq . . instead , the @xmath9-independent @xmath121 and @xmath122 are fitted in order to obtain a @xmath128 as similar as possible to the true @xmath33 . in other words , @xmath129 shares with the true distribution only four numeric features : normalization , first moment @xmath130 , @xmath131 which is essentially @xmath9-independent , and the second derivative @xmath132 . in particular , note that having @xmath133 is a crucial feature of the mean - field solution @xcite . the outcome of this analysis is given in fig . [ fig : dominance - large - power - syn ] . it turns out that the simplified @xmath134 in eq . , is almost as effective as the true @xmath33 in representing the non - equilibrium data through the effective size @xmath52 in eq . . the only obvious disagreement is that eq . predicts a non - analytic behavior for the susceptibility @xmath27 at @xmath135 , which is not found in our non - equilibrium data . in other words , the effective size for times such that @xmath136}$ ] is large , but certainly @xmath52 is not infinite as demanded by eq . . fortunately , even the crude description in eq . could lead to some interesting analysis . for instance , one could select pair of times @xmath90 such that @xmath137 . then , @xmath138 will be the same for all those points . now , we note from eq . that @xmath79 can vary by as much as a factor of two , for such points . it follows that @xmath139 should vary significantly over this set of times with fixed @xmath140 . hence , the crucial parameters @xmath121 and @xmath122 could be extracted . for instance , if the susceptibility @xmath141 would turn out not to depend on @xmath139 ( while keeping @xmath52 fixed ) , this would mean @xmath142 , in contrast with the mean field prediction @xmath133 . but this time we use the simplified @xmath128 from eq . note that dynamic data are well reproduced by eq . and eq . , even in this simple approximation . [ fig : dominance - large - power - syn ] ] [ [ discussion ] ] discussion + + + + + + + + + + + + it was discovered some twenty years ago that experimental aging response functions carry information on parisi s functional order parameter @xcite . we now know that this connection between non - equilibrium and equilibrium physics relies on a very general mathematical property , stochastic stability @xcite , shared by many glass models . however , experimental attempts to explore this connection encountered a major problem @xcite : an essentially uncontrolled extrapolation to infinite waiting time @xmath3 is required . here , we have proposed employing a statics - dynamics dictionary @xcite to avoid uncontrolled extrapolations . indeed , we have shown that the aging responses at finite @xmath3 can be connected to the parisi s order parameter as computed at equilibrium in a system of finite size . we have shown that this fdt - based sdd is essentially consistent with previous proposals @xcite that focused on spatial correlation functions . this is an important consistency test . there is a caveat , though : when the probing time @xmath4 is such that one has @xmath143 for the coherence lengths , the fdt - based sdd disagrees from previous dictionaries in that the size of the equivalent equilibrium system is @xmath144 ( rather than @xmath145 ) . in fact , we have found that the @xmath52 dependence on both length scales can be simply parameterized , recall eq . and eq . . on the other hand , the only previous sdd known to us that was based on eq . misses the @xmath144 behavior @xcite . there are a couple of possible reasons for this failure . for one , the time scales in ref . @xcite do not allow for length - scale separation @xmath143 . besides , the sdd from ref . @xcite was obtained for two - dimensional spin glasses ( which only have a paramagnetic phase ) . therefore , the results of ref . @xcite are probably a manifestation of finite - time / finite - size scaling @xcite . let us conclude by stressing that the three basic quantities analyzed in this work , namely the susceptibility @xmath146 , the correlation function @xmath147 and the coherence length @xmath79 , have been obtained experimentally in a dynamic setting very similar to simulations ( for @xmath27 and @xmath28 , see refs . @xcite , for @xmath118 see refs . @xcite ) . we thus think that it should be possible to extract the spin - glass functional order parameter from already existing experimental data . furthermore , fdt violations have been studied as well in superspin glasses @xcite and in a variety of soft condensed - matter systems @xcite . we therefore expect that our analysis will be of interest beyond the realm of spin glasses . [ [ acknowledgments ] ] acknowledgments + + + + + + + + + + + + + + + + + + some of the simulations in this work ( the @xmath148 systems , to check for size effects ) where carried out on the _ memento _ cluster : we thank staff from bifi s supercomputing center for their assistance . we thank giancarlo ruocco for guidance on the experimental literature . we warmly thank m. pivanti for his contribution to the early stages of the development of the janus ii computer . we also thank link engineering ( bologna , italy ) for their precious role in the technical aspects related to the construction of janus ii . we thank eu , government of spain and government of aragon for the financial support ( feder ) of janus ii development . this work was partially supported by mineco ( spain ) through grant nos . fis2012 - 35719-c02 , fis2013 - 42840-p , fis2015 - 65078-c2 , and by the junta de extremadura ( spain ) through grant no . gru10158 ( partially funded by feder ) . this project has received funding from the european union s horizon 2020 research and innovation program under the marie skodowska - curie grant agreement no . 654971 . this project has received funding from the european research council ( erc ) under the european union s horizon 2020 research and innovation program ( grant agreement no 694925 ) . dy acknowledges support by nsf - dmr-305184 and by the soft matter program at syracuse university . mbj acknowledges the financial support from erc grant nprgglass .

many new or improved measurements of the spin asymmetry @xmath0},\ ] ] @xmath1 , in longitudinally polarized dis off various different targets @xmath2 have become available in the past two years : e142 , e143 , and smc have presented their final data sets @xcite , and new results from hermes , e154 , and e155 have been reported recently @xcite . since none of these measurements were included in our original nlo qcd analysis @xcite , it seems to be appropriate to reanalyse these data within the framework of @xcite . in nlo the polarized structure function @xmath3 in ( [ eq : a1 ] ) reads ( suppressing all @xmath4 and @xmath5 dependence ) @xmath6\,\,,\end{aligned}\ ] ] where @xmath7 are the spin - dependent wilson coefficients and the symbol @xmath8 denotes the usual convolution in @xmath4 space . from ( [ eq : g1 ] ) it is obvious that inclusive dis data @xcite can reveal only information on @xmath9 but neither on @xmath10 _ and _ @xmath11 nor on @xmath12 , which enters ( [ eq : g1 ] ) only as an @xmath13 correction . thus one can either stick to a comprehensive analysis of polarized dis @xcite or one has to impose certain _ assumptions _ about the flavor decomposition in order to be able to estimate processes other than dis for upcoming experiments like rhic . as in @xcite , and other more recent qcd analyses @xcite , we follow the latter option . due to the lack of space we shall be rather brief and concentrate only on the most important changes since @xcite . the limited amount of data demands a reasonably simple , but flexible enough ansatz for the polarized densities such as @xmath14 with @xmath15 . note that in @xcite we actually used @xmath16 in ( [ eq : ansatz ] ) instead of @xmath17 and @xmath18 , however the positivity constraint @xmath19 which we want to exploit in our analysis , does not necessarily hold for the valence densities . of course , the bound ( [ eq : pos ] ) is strictly valid only in lo and is subject to nlo corrections @xcite because the @xmath20 become unphysical , scheme - dependent objects in nlo . however the corrections are not very pronounced , in particular at large @xmath4 @xcite , the only region where ( [ eq : pos ] ) imposes some restrictions in practice . we therefore use ( [ eq : pos ] ) also in nlo . to further simplify ( [ eq : ansatz ] ) we _ assume _ that @xmath21 , fix @xmath22 by the relations between the first moments of the non - singlet combinations @xmath23 and the @xmath24 and @xmath25 values ( using the updated value for @xmath26 @xcite ) , and take @xmath27 . in the latter relation we _ choose _ @xmath28 ( @xmath29 symmetric sea ) , but similarly agreeable fits are obtained , e.g. , for @xmath30 ( see also @xcite ) , as well as by using an independent @xmath4 shape for @xmath31 , reflecting the above mentioned uncertainty in the flavor separation . for the unpolarized distributions @xmath32 in ( [ eq : ansatz ] ) and ( [ eq : pos ] ) we use the updated grv densities @xcite and also adopt their values for the input scale @xmath33 ( @xmath34 in nlo ) and @xmath35 . note that in @xcite the rg equation for @xmath36 is now solved exactly instead of using the approximative nlo formula ( see , e.g. , @xcite ) , which is more appropriate at low @xmath5 where many of the polarized data lie . to fix the remaining parameters in ( [ eq : ansatz ] ) we perform fits to the directly measured spin asymmetry ( [ eq : a1 ] ) in lo _ and _ nlo , which is mandatory if one wants to adopt these distributions in a consistent analysis of the perturbative stability of polarized processes . also most mc programs only contain lo matrix elements , and hence lo densities are more appropriate . apart from the above outlined ` standard scenario ' fit , equally good fits can be performed in a ` valence scenario ' ( see @xcite ) which is based on the assumption @xcite that the @xmath37 values fix only the valence parts of @xmath23 . the main feature of such a model is the possibility to describe the data with a vanishing strange sea . due to the lack of space we do not pursue this scenario here and restrict ourselves in what follows to the results obtained in the nlo @xmath38 ` standard scenario ' framework . a comparison of our new nlo fit with the available @xmath39 data @xcite and with our previous analysis @xcite is presented in fig . 1 . the total @xmath40 values of the new and old fits are 147.4 and 183 , respectively , for 185 data points and adding statistical and systematical errors in quadrature . as can be seen , sizeable differences appear only in case of the neutron asymmetry , which leaves its footprint also in the individual parton densities @xmath20 shown in fig . 2 . since the neutron data mainly probe @xmath41 , the most prominent changes are observed here . the differences in the sea and in @xmath12 only reflect the fact that they are constrained to a much lesser extent by the data than @xmath17 and @xmath18 . in particular @xmath12 remains to be hardly constrained at all , which is not surprising due to the lack of any direct information on @xmath12 so far . therefore we show in fig . 2 also the results of two other fits , which are based on additional constraints on @xmath12 . for the ` @xmath42 ' fit we start from a vanishing gluon input in ( [ eq : ansatz ] ) , and the ` static @xmath12 ' is chosen in such a way that its first moment becomes independent of @xmath5 . this can be achieved by setting @xmath43 and yields in lo @xmath44 ( see @xcite ) , where @xmath45 is the total helicity carried by quarks ( the relation is only subject to a small nlo correction ) . both gluons give also excellent fits to the available data and do not affect the results for @xmath17 and @xmath18 ( see fig . 2 ) . in fact we can obtain fits without any significant change in @xmath40 for @xmath46 in the range -0.3 0.6 ( our best fit has 0.28 , corresponding to @xmath47 ) . the uncertainty in @xmath12 is compatible to the one found in @xcite , although our gluons extend also to slightly negative first moments . it is interesting to observe that for our best fit gluon the spin of the nucleon @xmath48 is dominantly carried by quarks and gluons at our input scale @xmath33 , and only during the @xmath5 evolution a large negative @xmath49 is being built up in order to compensate for the strong rise of @xmath50 , see fig . 5 in @xcite . for the ` static @xmath12 ' the situation is completely different : by construction the quark and gluon contributions to ( [ eq : spinsum ] ) cancel each other implying that for _ all _ values of @xmath5 the spin is entirely of angular momentum origin , contrary to what is intuitively expected . inevitably the large uncertainty in @xmath12 implies that the small @xmath4 behaviour of @xmath3 is completely uncertain and not predictable ( see fig . 3 in @xcite ) , which translates also into a large theoretical error from the @xmath51 extrapolation when calculating first moments of @xmath3 . taking our best fit we obtain @xmath52 and @xmath53 at @xmath54 in agreement with a recent smc qcd analysis @xcite . the two challenging questions concerning polarized parton densities are still @xmath12 and the flavor decomposition . recent semi - inclusive dis results @xcite may help to unravel the latter , but major progress , in particular on @xmath12 , can be only expected from rhic . a realization of the optional upgrade of hera to a polarized collider would be also very helpful to gain more insight into the spin structure of the nucleon _ and _ the photon , with the latter being completely unmeasured so far . it is a pleasure to thank m. glck , e. reya , and w. vogelsang for a fruitful collaboration . anthony et al . , e142 collab . , phys . d 54 ( 1996 ) 6620 ; k. abe et al . , e143 collab . d 58 ( 1998 ) 112003 ; b. adeva et al . , sm collab . , d 58 ( 1998 ) 112001 . k. abe et al . , e154 collab . , phys . 79 ( 1997 ) 26 ; k. ackerstaff et al . , hermes collab . b 404 ( 1997 ) 383 ; a. airapetian et al . , hermes collab . , phys . lett . b 442 ( 1998 ) 484 ; p.l . anthony et al . , e155 collab . , slac - pub-8041 , 1999 ( phys . rev . lett . ) . m. glck , e. reya , m. stratmann , and w. vogelsang , phys . d 53 ( 1996 ) 4775 . g. altarelli , r. ball , s. forte , and g. ridolfi , nucl . phys . b 496 ( 1997 ) 337 ; acta phys . polon . b 29 ( 1998 ) 1145 . b. adeva et al . , sm collab . , phys . d 58 ( 1998 ) 112002 . d. de florian , o.a . sampayo , and r. sassot , phys . d 57 ( 1998 ) 5803 . e. leader , a.v . siderov , and d.b . stamenov , phys . d 58 ( 1998 ) 114028 . g. altarelli , s. forte , and g. ridolfi , nucl . b 534 ( 1998 ) 277 . particle data group , eur . j. c 3 ( 1998 ) 1 . m. glck , e. reya , and a. vogt , eur . j. c 5 ( 1998 ) 461 . j. lichtenstadt and h.j . lipkin , phys . b 353 ( 1995 ) 119 . m. stratmann , in proceedings of the ` 2nd topical workshop on deep inelastic scattering off polarized targets ' , zeuthen , 1997 , j. blmlein and w .- d . nowak ( eds . ) , p. 94 . m. ruh ( hermes ) , these proceedings .

an updated next - to - leading order ( nlo ) qcd analysis of all presently available longitudinally polarized deep - inelastic scattering ( dis ) data is presented in the framework of the radiative parton model .

figures 5(a ) and ( b ) show the positional dependence of the dimer - peak energy at 80 and 70 k , respectively , obtained from the same sample shown in the main manuscript . figure 5(c ) depicts the positional dependence of the dimer - peak energy in the dotted outlined area shown in fig . the dimer - peak energy is spatially homogeneous above 70 k. this clearly shows that the sample temperature is also spatially homogeneous above 70 k , because the dimer - peak energy exhibits strong temperature dependence at the high - temperature phase as seen in fig . 2(a ) in the main manuscript . in fact , the 80-k and 70-k results in fig . 5 are explicitly distinguished with each other . if there is a temperature gradient inside this sample , a position - dependent dimer peak energy will be obtained as shown by the dashed lines in fig . on the other hand , our experimental results show no spatial variations of the dimer peak energy . this evidences that our sample has negligible temperature gradient . in fig . 6 , we show the imaging data measured on other single crystals . in the sample # 2 , the blue - color charge - order state emerges from the left side , indicating that the gold deposition for the reference measurement does not affect the observed inhomogeneity . in the sample # 3 , the charge - order state appears from the left and right sides . these samples show spatial inhomogeneity below 70 k , indicating that the inhomogeneity is an intrinsic property in @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 . in @xmath12-(et)@xmath1cu[n(cn)@xmath1]cl , grease coating induces a trace of superconductivity due to the extrinsic stress originating from the difference of expansion coefficients between samples and grease @xcite . here we fixed the sample on the cold head by using the carbon paste , which may affect the transition temperature to charge ordering because the external pressure reduces @xmath4 in @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 @xcite . to clarify this extrinsic effect , we fixed a part of the crystal on the cryostat by using paste , and measured the positional variation of local reflectivity spectrum . figure 7 shows the local reflectivity spectra measured with 50-@xmath17 m interval on sample # 4 . the charge order state with split peak was developed from the fixed area , clearly showing that the survived dimer - mott state far below @xmath4 does not originate from the stress effect due to the carbon paste . moreover , the charge order appears just below @xmath4 in all samples we measured , excluding such extrinsic stress effect . the present spatial pattern does not depend on the past environment and shows no hysteresis . the experiment shown in the main text was done as follows : the sample was firstly cooled - down to 30 k , and then the mapping data was measured from 30 k to 80 k with heating . then , the sample was cooled - down from 80 k to 50 k , and 50-k data was measured again . after that , 10-k data was measured . figures 8(a ) and ( b ) show the reflectivity - ratio mapping measured at 50 k in the first heating and second cooling processes , respectively . similar pattern is found in both results . figure 9 displays the reflectivity - ratio mapping measured on the same sample in the main paper with fast scanning rate ( 40 minutes per one imaging ) . the results are almost same as those measured with slow scanning rate ( 3 hours per one imaging ) shown in fig . 4 ( main paper ) , indicating that the sample is essentially in equilibrium at each measurement temperature .

we report a novel insulator - insulator transition arising from the internal charge degrees of freedom in the two - dimensional quarter - filled organic salt @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 . the optical conductivity spectra above @xmath3 k display a prominent feature of the dimer - mott insulator , characterized by a substantial growth of a dimer peak near 0.6 ev with decreasing temperature . the dimer - peak growth is rapidly quenched as soon as a peak of the charge order shows up below @xmath4 , indicating a competition between the two insulating phases . our infrared imaging spectroscopy has further revealed a spatially competitive electronic phases far below @xmath4 , suggesting a nature of quantum phase transition driven by material - parameter variations . = 10000 organic molecular conductors exhibit complex electronic phase diagram owing to their unique internal degrees of freedom coupled with correlation effects . among them , the quasi two - dimensional ( 2d ) quarter - filled salts @xmath5 , where @xmath6 is a dimer organic molecule and @xmath7 is a monovalent anion , are attracting much interest @xcite . in the weakly dimerized materials , the correlated carrier is localized on the molecular site due to long - range nature of coulomb interaction , leading to charge order as seen in @xmath8-(et)@xmath1rbzn(scn)@xmath9 @xcite and @xmath10-(et)@xmath1i@xmath11 @xcite [ et = bis(ethylenedithio)-tetrathiafulvalene ] . on the other hand , under strong dimerization , the hole is localized on the dimer to act as a mott insulator ( the dimer - mott insulator ) , as realized in @xmath12-(et)@xmath13 , where @xmath7=cu@xmath1(cn)@xmath11 , cu[n(cn)@xmath1]cl . recently , intriguing behaviors have been found in materials located near the border of such insulating phases . in the dimer - mott insulator @xmath12-(et)@xmath1cu@xmath1(cn)@xmath11 , the localized hole behaves as a magnetic dipole but shows no magnetic order down to very low temperatures , leading to a novel spin - liquid state @xcite . interestingly , in sharply contrast to conventional mott insulators , this dimer mott insulator exhibits unusual temperature and frequency variations of dielectric constant @xcite . this experimental fact strongly suggests that the et dimer is _ electrically _ polarized owing to the intra - dimer charge degree of freedom , which hotta called `` dipolar liquid '' @xcite . the dielectric anomaly is also observed in the dimer - mott insulators @xmath12-(et)@xmath1cu[n(cn)@xmath1]cl @xcite and @xmath15-(et)@xmath1icl@xmath1 @xcite . these results have revealed that the instability to electric dipole order ( charge order ) is hidden in the dimer - mott insulators @xcite . the collective mode of the electric dipole order is theoretically calculated to be gapless or extremely narrow - gapped @xcite , and can affect the low - temperature specific heat and thermal conductivity @xcite . on the other hand , in these dimer - mott insulators , there is no direct spectroscopic evidence of charge disproportionation @xcite , which can be sensitively probed through optical conductivity @xcite or nuclear magnetic resonance measurements @xcite . these results indicate that the observed dielectric anomaly is not simply attributed to charge order , and then raise a crucial question whether the instability toward the charge order exists in such dimer - mott phase . the aim of this paper is to explore the opposite case where the instability to the dimer - mott phase is hidden in a well established charge - ordered dimer - type organic conductor . here we show such a phenomenon in the dimer - type quarter - filled organic salt @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 through the infrared optical study including a spatial imaging measurement . the polarized reflectivity spectra in a large area ( @xmath16 @xmath17m@xmath18 ) on the sample surface were measured for energies between 90 mev and 1.4 ev using a fourier transform infrared spectrometer ( ftir ) equipped with an infrared microscope . the details of measurement area will be shown later . the sample was fixed with a conductive carbon paste on the cold head of helium flow - type refrigerator . the sample cooling rate was about 1 k / min . we used a standard gold overcoating technique for measuring the reference spectrum at each temperature . the complex optical conductivity is obtained from the kramers - kronig ( kk ) analysis . standard extrapolation of @xmath19 dependence was employed above 1.4 ev . in low energies , we extrapolated the reflectivity using several methods , but the peak shapes and positions are negligibly affected . infrared - imaging measurements using a synchrotron radiation ( sr ) light were performed at bl43ir , spring-8 , japan @xcite . we measured the positional dependence of the @xmath20-axis - polarized local reflectivity spectra on a crystal surface of @xmath21 @xmath17m@xmath18 by using an ftir for energies between 0.1 ev and 0.8 ev . high spatial resolution of @xmath22 10 @xmath17 m was achieved with high - brilliance sr light ( spot diameter of 10 @xmath17 m ) and an infrared microscope equipped with a precision _ xy_-scanning stage . -(_meso_-dmbedt - ttf)@xmath1pf@xmath2 , where dmbedt - ttf = 2-(5,6-dihydro-1,3-dithiolo[4,5-_b_][1,4]dithiin-2-ylidene)-5,6-dihydro-5,6-dimethyl-1,3-dithiolo[4,5-_b_][1,4]dithiin . hydrogen atoms are omitted for clarity . two donor molecules circled by the dotted ellipsoid are dimerized . ( b ) a checkerboard - type charge - ordered state below @xmath3 k. ( c ) a dimer - mott insulating state above @xmath4 . ] the single crystals of @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 were grown by the electrochemical method @xcite , where dmbedt - ttf stands for 2-(5,6-dihydro-1,3-dithiolo[4,5-_b_][1,4]dithiin-2-ylidene)-5,6-dihydro-5,6-dimethyl-1,3-dithiolo[4,5-_b_][1,4]dithiin . the crystal structure is composed of the stacking of conducting donor layers separated by insulating anionic ones @xcite . in the conducting layer , two donor molecules are weakly dimerized as shown in fig . 1(a ) . the x - ray diffraction @xcite as well as the infrared and raman studies @xcite clearly resolve a charge disproportionation below @xmath3 k. the charge ordering pattern is of checkerboard type [ fig . 1(b ) ] @xcite , which can be regarded as an antiferro - type electric dipole order , contrast to stripe - type charge ordered states in other 2d quarter - filled salts @xcite . of @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 single crystal measured in the large area on the sample surface . ( a ) temperature - dependent @xmath23 spectra measured with polarization parallel to the _ c _ * axis . the dc conductivity @xmath24 is plotted with the circles . ( b ) expanded view of a temperature - dependent vibrational molecular modes around 180 mev in @xmath23 spectra measured with polarization parallel to the _ b _ * axis . ] figure 2(a ) displays the real part of the optical conductivity spectra @xmath23 obtained from the kk transformation of the reflectivity spectra measured in the large area on the sample surface with polarization parallel to the conducting _ c _ * axis . a low - energy spectrum below 0.2 ev is gradually enhanced with decreasing temperature above @xmath4 but is suddenly suppressed and replaced by a sharp peak structure at @xmath25 0.2 ev at @xmath4 . this indicates that the formation of charge order drastically modifies the electronic structure in a wide energy range and such spectroscopic feature has also been observed in other charge - ordered material @xcite . the vibrational molecular modes in @xmath23 measured with polarization parallel to the insulating _ b _ * axis provide evidence for charge order below @xmath4 [ fig . 2(b ) ] , consistent with previous results @xcite . as shown in fig . 2(b ) , the @xmath26 mode peak ( out - of - phase stretching mode of ring c = c ) of dmbedt - ttf@xmath27 at @xmath28 mev splits into two bands ( charge - rich site at @xmath29 mev and charge - poor site @xmath30 mev ) owing to the charge disproportionation below @xmath4 . firstly we discuss the high - temperature phase above @xmath4 . the distinct feature newly found in @xmath23 is a pronounced peak structure at @xmath31 ev , which exhibits a strong enhancement with lowering temperature from 300 k down to @xmath4 . this mid - infrared peak can be assigned to a dimer peak , a transition from bonding to anti - bonding orbitals of dimerized molecules , as observed in @xmath12-type dimerized et salts @xcite . most importantly , the dimer - peak intensity is enhanced with decreasing temperature in the dimer - mott insulating phase , while it is reduced in a correlated metallic phase @xcite . thus the observed enhancement of dimer - peak intensity down to @xmath4 strongly indicates that the high - temperature phase in this material should be regarded as a dimer - mott insulating phase [ fig . 1(c ) ] , rather than a conventional metal . below 200 k , @xmath23 seems to exhibit a drude - like response below 0.3 ev , but it should show a peak structure centered at a finite energy to connect low @xmath32 values plotted in fig . 2(a ) . such a low - energy response strikingly resembles those of the quarter - filled salt @xmath8-(et)@xmath1i@xmath11 , in which an incoherent transport is realized at high temperatures @xcite . per formula unit calculated up to 0.3 ev . several results obtained using different low - energy extrapolations are displayed . inset : temperature variation of dc resistivity @xmath33 near @xmath4 . ] to shed further light on the unusual electronic state in high - temperature dimer - mott phase of @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 , we discuss the effective carrier number @xmath34 expressed by , @xmath35 where @xmath36 is the free electron mass , @xmath37 is the charge of an electron , and @xmath38 is the volume occupied by one formula unit of @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 . to evaluate the low - energy spectral weight , the cut - off energy @xmath39 was adopted to be 0.3 ev at which the spectra exhibit a minimum above @xmath4 [ fig . 2(a ) ] . figure 3 shows the temperature variations of @xmath34 obtained using several extrapolation methods for reflectivity spectra . note that difference in the extrapolation methods is negligible in following discussion . now two important features are noticed : firstly , @xmath34 is increased with lowering temperature above @xmath4 . this indicates that a temperature - dependent transfer energy probably due to a shrinkage of the sample volume contributes to the conductive nature in this phase . this differs from the situation in conventional metals , in which temperature variation of resistivity is mostly governed by the reduced scattering rate . secondly , @xmath34 is significantly smaller than unity , showing that the low - energy metallic weight is about 10% of what is expected in conventional metals . we stress that this @xmath34 involves a sizable contribution from high - energy transitions including the dimer peak as seen in fig . 2(a ) , and therefore the contribution from conduction electrons should be smaller than the @xmath34 values shown in fig . 3 . furthermore , the magnitudes of @xmath23 and the resulting @xmath34 in @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 are roughly one order smaller than those in @xmath8-(et)@xmath1i@xmath11 @xcite and such considerable suppression of low - energy spectral weight has been observed in the dimer - mott insulator @xmath12-(et)@xmath1cu[n(cn)@xmath1]cl @xcite and the mott insulating phase of nis@xmath40se@xmath41 @xcite . these results capture an insulating nature at the high - temperature phase in @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 , and thus indicate that the charge order transition at @xmath4 is _ a transition from dimer - mott to charge - order phase_. -(_meso_-dmbedt - ttf)@xmath1pf@xmath2 . ( a - d ) the vibrational molecular modes near 180 mev in the local reflectivity spectra @xmath42 measured with polarization parallel to the _ c _ * axis . on each panel , 60 local reflectivity spectra , which were measured in the region surrounded by solid - line rectangular box ( from @xmath43 to @xmath44 point ) shown in ( e ) , are displayed . these spectra are separated by 20 @xmath17 m and span a total distance of 1.2 mm as represented with the vertical offset . ( e - k ) the reflectivity - ratio mapping on the crystal surface of @xmath21 @xmath17m@xmath18 . the color scale shows the reflectivity ratio @xmath45(184.5 mev)@xmath46(181 mev ) . the right - side white region is a gold - coated area for measuring the reference spectra . the dashed - line rectangular box in ( e ) shows a region in which the large - area reflectivity measurements were performed . ] next let us discuss the low - temperature charge - ordered phase . as seen in fig . 2(a ) , the growth of dimer peak is completely quenched by the formation of charge order below @xmath4 , indicating a competitive nature between dimer - mott and charge - order phases . an intriguing question is how these insulating phases compete in real space . here we show the positional dependence of local reflectivity spectrum . figures 4(a - d ) show the vibrational molecular modes near 180 mev in the local reflectivity @xmath42 measured at several temperatures . each panel displays 60 local reflectivity spectra measured at from @xmath43 to @xmath44 points shown in fig . 4(e ) . as seen in fig . 4(a ) , reflectivity spectra are spatially homogeneous at @xmath4 : almost all spectra possess the single peak near 181 mev ( red - color spectra ) as expected in the dimer - mott phase [ fig . 2(b ) ] . charge ordering characterized by the split peak at 178.2 and 184.5 mev emerges at a tiny portion inside the crystal as shown by the blue - color spectra . meanwhile , the spectra measured below @xmath4 [ figs . 4(b - d ) ] are obviously inhomogeneous and can be sorted into two groups , red - color spectra having one single peak originating from the dimer - mott insulating state and blue - color spectra with split peaks from the charge - ordered state . here we evaluate the reflectivity ratio @xmath45(184.5 mev)@xmath46(181 mev ) at each point , which gives a relative strength between the charge - ordered state with 184.5-mev peak and the dimer - mott state with 181-mev peak , and plot its spatial distribution in figs . 4(e - k ) . the red- and blue - color areas indicate the dimer - mott and the charge - ordered states , respectively . note that the large - area reflectivity spectra shown in figs . 2(a ) and ( b ) were measured in the rectangular box surrounded by dashed line in figs . 4(e ) , from which the charge order appears just below @xmath4 . in contrast to spatially - homogeneous spectra above @xmath4 , the low - temperature spectra are highly inhomogeneous . note that there is negligible temperature gradient inside the crystal @xcite . the observed inhomogeneity does not originate from the phase separation in the hysteresis region near first - order phase transitions since the resistive hysteresis loop is closed within 10 k near @xmath4 as shown in the inset of fig . 3 , while the observed inhomogeneity survives far below @xmath4 . indeed , the present spatial pattern does not depend on the past environment and shows no hysteresis @xcite . an extrinsic stress effect is also excluded @xcite . we note that our results are obtained at ambient pressure , in sharply contrast to spatial inhomogeneity observed only in pressure @xcite . below @xmath4 , the charge - ordered state gradually invades the dimer - mott insulating state with lowering temperature , significantly different from conventional phase transitions occurring at finite temperatures , at which a high - entropy phase at high temperatures is immediately replaced by a low - temperature low - entropy phase . in this material , most surprisingly , the dimer - mott state survives even at 10 k far below @xmath4 . this experimental fact strongly indicates that the phase transition in this material is not driven by entropy term in the free energy . but rather , this transition seems to occur , when the materials parameters reach a critical value through their temperature variation . we suggest that this type of transition is of quantum nature in the sense that the transition is driven by the materials parameters . in the present system , it has been found that the interdimer transfer integral is doubly increased from room temperature to 11.5 k @xcite . such a considerable change of interdimer integral drives a phase transition from dimer - mott to antiferro - type electric dipole order @xcite . now the phase competition expands an inhomogeneous region near the border between those two insulating ground states in the parameter space @xcite . the present compound may locate near the border at low temperatures . here we stress that the observed inhomogeneity well explains previous results . while the peak splitting in the local optical spectroscopy is abrupt @xcite , the superlattice intensity of bulk x - ray measurement exhibits a gradual increase with temperature @xcite . our results show that the gradual increase originates from the volume - fraction change of charge - ordered state . below @xmath4 , the magnetic susceptibility sets to a finite value @xcite , indicating the survived dimer - mott state inside the crystal , well consistent with the present results . let us finally discuss the nucleation mechanism of charge order . in the previous samples , the superlattice peak intensity and the resistivity are gradually increased below 90 k @xcite , while a recent sample of higher quality shows a sharp increase of resistivity at @xmath4 @xcite . this difference may originate from the sample quality . also , the seeding position of charge order depends on sample @xcite . thus we speculate that the seeding is unavoidable crystalline imperfections , as also indicated from other competing correlated system @xcite . similar nucleation has also been proposed in a ferroelectric relaxor : a polar domain is created near the nano - sized chemically ordering regions @xcite . after the nucleation , the charge - ordered state is gradually expanded into the whole region with temperature . in the present compound , @xmath47 ( @xmath48 : pressure ) is negative @xcite , indicating that the volume of charge - ordered state is larger than that of dimer - mott state . thus , the induced charge - ordered state pressurizes the dimer - mott state near the boundary locally , leading to @xmath4 reduction in such a boundary region . this indicates a large energy to move the boundary , that may cause the observed macroscopic inhomogeneity . in summary , the infrared measurements on @xmath0-(_meso_-dmbedt - ttf)@xmath1pf@xmath2 reveal anomalous phase transition phenomena from dimer - mott to charge - order state . we suggest a quantum nature of this transition driven by variations of temperature - dependent materials parameters , which possibly induce a spatial inhomogeneity owing to competitive nature between two insulating states . this quantum nature might be ubiquitous among organic systems which exhibit phase transitions at high temperatures , where the materials parameters considerably change with temperature . we thank y. nogami , v. robert , h. seo , y. suzumura , h. taniguchi and m. tsuchiizu for fruitful discussion . the imaging experiments using synchrotron radiation were performed at bl43ir in spring-8 with the approvals of jasri ( no . 2011b1221 , 2011b1232 , 2012a1082 , 2012a1141 , 2012b1352 , 2012b1223 ) . this work was supported by a grant - 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since the experimental realization of bose - einstein condensates ( bec s ) in 1995 @xcite , a wide variety of properties and effects in bec s has been studied . among the major topics of interest are topological defects such as vortices @xcite and solitons @xcite . more recently , the realization of bec s in optical traps @xcite has opened up interesting new avenues of research in this area . this development makes it possible to create multicomponent condensates where all magnetic substates @xmath0 are trapped . the resulting spin degree of freedom permits the existence of topological defects that can not appear in scalar condensates , such as skyrmions @xcite and monopoles @xcite . important properties of monopoles , such as stability and dynamics , have already been examined @xcite . here we propose an explicit method for creating a monopole in bec s , focusing on the experimentally relevant case of @xmath1 . our proposal extends the method of `` phase imprinting '' used to generate topological defects in scalar condensates @xcite . in particular , we demonstrate that the energy shift experienced by each component of the condensate depends on both @xmath0 and the polarization of the laser beam , and that by applying a sequence of different polarizations the phases of the components can be engineered in a broad manner . a spin-1 bec can be described by the condensate wavefunction @xmath2 , where @xmath3 is the total density and @xmath4 is a normalized spinor describing the local spin of the condensate . for an anti - ferromagnetic bec such as @xmath5 , the ground state spinor is given by @xcite [ eq : ground ] _ g()= ( c 0 + 1 + 0 ) . the spinor corresponding to a monopole has the form @xcite [ eq : monopolespinor ] _ m ( ) = ( c -m_x()+im_y ( ) + m_z ( ) + m_x()+im_y ( ) ) . @xmath6 is a unit vector field that describes the local rotation of the ground state spinor . by considering the energy and winding number of such a condensate , it can be shown @xcite that @xmath7 uniquely characterizes the monopole structure . we see that @xmath6 assumes the familiar hedgehog structure associated with a monopole . for the purpose of phase engineering , it is convenient to also write this spinor in spherical coordinates as [ eq : spherical ] _ m ( ) = ( c - e^-i + + e^i + ) , where @xmath8 is the azimuthal angle and @xmath9 the polar angle . in addition , we define the phase of each component @xmath10 . for example , the phases of the components in the monopole are given by [ eq : monopolephase ] _ m= ( c -+ + ( -z ) + ) , where @xmath11 is the step function . the hamiltonian of the condensate in second quantized form is @xcite @xmath12where @xmath13 is the field annihilation operator for spin state @xmath14 , @xmath15 are the spin matrices , @xmath16 is the trapping potential , and @xmath17 and @xmath18 are constants that depend on the scattering lengths @xmath19 and @xmath20 . @xmath21 and @xmath22 describe respectively the applied magnetic field and the laser beams that will be used to create the monopole . during the times that the applied magnetic field and lasers are turned on , we assume that their respective hamiltonians dominate @xmath23 and are the only terms that need to be considered in the evolution of the system . the hamiltonian for the applied magnetic field is given by @xmath24 , where @xmath25 is the gyromagnetic ratio . when we ignore all other terms in @xmath23 , the magnetic field leads to the following equations of motion : [ eq : b - motion ] = ( _ ij)_j . when a laser beam is applied , the ground state energy of magnetic substate @xmath14 is shifted by an amount @xcite [ eq : lightshift ] e_i = i_j , where the sum is carried out over all excited states @xmath26 . here , @xmath27 is the detuning of the laser from the @xmath28 transition , @xmath29 the laser intensity , @xmath30 the rate of spontaneous decay , @xmath31 the atomic resonance frequency , and @xmath32 is a transition strength characterizing the coupling between atom and laser . the coefficients @xmath32 depend on the ground substate @xmath0 and on the polarization of the laser beam and are given in figure [ fig : transitionstrengths ] . the concept of phase engineering via imprinting was suggested @xcite and experimentally demonstrated @xcite as a means of creating topological defects in scalar condensates . we extend the method to spinor condensates , focusing on the case of @xmath1 . we show that the energy shifts induced by different beam polarizations allow for wide capabilities in phase engineering . suppose that a laser is detuned by an amount @xmath33 from the @xmath34 transition , and that the states @xmath35 and @xmath36 are separated in frequency by an amount @xmath37 , as shown in figure [ fig : frequencydiagram ] . from eq . ( [ eq : lightshift ] ) , the energy of the ground state is shifted by an amount e_i = i_^jp_1/2(f=1)_kp_1/2(f=2)(+ ) . we assume that the laser pulse has a square envelope with a duration @xmath38 that is short compared to the condensate s correlation time @xmath39 , where @xmath40 is the chemical potential . in this case the primary effect of the light shift is to imprint a phase onto each component @xcite : [ eq : phaseshift ] _ i()_i()[-ie_it/ ] . using the coefficients given in figure [ fig : transitionstrengths ] and the expansion @xmath41 , one can obtain explicit results for the phase shifts generated in the condensate . for @xmath42 , the shifts corresponding to @xmath43- , @xmath44- , and @xmath45-polarized beams are @xmath46\label{eq : pi - phaseshift } , \\ \delta\varphi^{+}(t)^{t } & \approx & -{\alpha}t(3\quad 2\quad 1)\label{eq : plus - phaseshift } , \\ \delta\varphi^{-}(t)^{t } & \approx & -{\alpha}t(1\quad 2\quad 3)\label{eq : minus - phaseshift}.\end{aligned}\ ] ] here @xmath47 . equations ( [ eq : pi - phaseshift])-([eq : minus - phaseshift ] ) suggest a basic technique for phase engineering of the spinor condensate . to shift the phase of @xmath48 relative to @xmath49 , one can use beams with polarizations @xmath50 and @xmath51 . once the phase between these two states is fixed as desired , one can use @xmath43-polarized light to adjust the phase of @xmath52 relative to the other states . although it is not used in our proposal for monopole creation , we note that this phase shift is due to a second order term in eq . ( [ eq : pi - phaseshift ] ) , and we thus expect that it will be more difficult to implement in an experiment . it may be useful , in addition , to alter the global phase of the condensate , which can be realized approximately by using the first order term in eq . ( [ eq : pi - phaseshift ] ) . finally , one can incorporate spatial dependence into the phases by using the above pulses in conjunction with absorption plates to create non - uniform intensity profiles @xmath53 @xcite . writing @xmath54 , where @xmath55 is the maximum intensity in the profile and @xmath56 , eqs . ( [ eq : pi - phaseshift])-([eq : minus - phaseshift ] ) remain valid with the replacements @xmath57 and @xmath58 . martikainen et al . point out @xcite that the monopole spinor given in eq . ( [ eq : monopolespinor ] ) resembles a combination of other topological defects . indeed , the form of @xmath59 presented in eq . ( [ eq : spherical ] ) makes it clear that the @xmath60 state contains a vortex , the @xmath49 state an anti - vortex , and the @xmath52 state a soliton . they thus suggest that one can experimentally create monopoles by creating the appropriate defects in the individual components . here we provide a method to accomplish this using the phase imprinting concepts discussed above . starting from the ground state of eq . ( [ eq : ground ] ) , we first apply a constant magnetic field of magnitude @xmath61 along the negative @xmath62-axis for a time @xmath63 . defining @xmath64 , eq . ( [ eq : b - motion ] ) readily yields _ b(t_b)= ( c -(_bt_b ) + ( _ bt_b ) + ( _ bt_b ) ) . the phases of @xmath65 ( for @xmath66 ) are _ b^t=(00 ) . the effect of the magnetic field is to populate the @xmath67 states prior to phase engineering . experimentally , @xmath68 should be chosen to best realize the monopole structure . to create the monopole , we assume that we have blue - detuned lasers , all of which have equal intensities @xmath55 and values of @xmath33 , and which can be applied in square - shaped pulses . the results can be accordingly modified to cover more general situations . in addition , we assume that we have absorption plates that can create the following intensity profiles : @xmath69 the pulse sequence to create the monopole is given in table [ table : pulsesequence ] . pulses a and b use intensity profiles with azimuthal dependence to create a vortex in the @xmath48 component and an anti - vortex in the @xmath49 component . pulses c and d together imprint the necessary phases to generate the soliton in the @xmath52 component . it can be readily verified that the resulting phases of the components are exactly those of the monopole given in eq . ( [ eq : monopolephase ] ) . the total time of the laser pulses , assuming that pulses with profiles @xmath70 and @xmath71 can be executed simultaneously , is @xmath72 . strictly speaking , pulse c is not physically possible the profile @xmath70 requires an axis of propagation @xmath73 in the @xmath74 plane , which is inconsistent with a polarization vector of @xmath75 . however , it is possible to create an effective interaction term resembling @xmath76 using a beam that does propagate in the @xmath74 plane . indeed , consider a pulse with polarization vector @xmath77 sandwiched between magnetic field pulses that generate the rotations @xmath78 and @xmath79 . one can readily check that ( -if_y ) ^ ( if_y)=. physically , the @xmath0 eigenstates along the @xmath80-axis are transformed by rotation to become @xmath0 eigenstates along the @xmath81-axis . the phase shifts are subsequently performed on these states , where @xmath70 and @xmath82 are consistent , before the system is rotated back . [ cols="^,^,^,^,^",options="header " , ] experimentally , one should seek to make @xmath33 as large as possible while increasing @xmath55 to maintain a given potential depth . this is because the photon scattering rate @xmath83 of an individual atom scales as @xmath84 compared to @xmath85 for the dipole potential @xcite , and also because the approximation errors in eqs . ( [ eq : pi - phaseshift])-([eq : minus - phaseshift ] ) scale as @xmath86 . to maintain the validity of eqs . ( [ eq : pi - phaseshift])-([eq : minus - phaseshift ] ) , however , one must impose the limit @xmath87 , where @xmath88 is the fine structure splitting between @xmath89 and @xmath90 . physically , this means that the light shifts due to the excited @xmath90 states remain negligible compared to those originating from the @xmath89 states . as a concrete example , we consider the case @xmath91 and @xmath92 , where @xmath93 is the saturation intensity . this results in a pulse sequence length of @xmath94 and a scattering time of an individual atom of @xmath95 . we also mention that the ratio @xmath96 can be enhanced by pairing every beam in the original sequence with an appropriately chosen second beam . if both beams have the same polarization and are detuned equally above and below the @xmath97 transition , the light shifts from the @xmath90 states are approximately cancelled out , while those from the @xmath89 states are reinforced . this makes it possible to abandon the requirement @xmath87 and choose much larger detunings . finally , we note that the solution ( [ eq : phaseshift ] ) for the dynamics of the system under the laser beams is only approximate and overlooks subtler but important points . clearly such a transformation is unphysical in our case at the origin as it would introduce an infinite phase gradient energy @xcite , and a more careful analysis of the dynamics near the center would have to include other terms in the hamiltonian . also , in a real system , an absorption plate can not perfectly generate the intensity profiles given in ( [ eq:+azimuthalintensity ] ) and ( [ eq :- azimuthalintensity ] ) , which leads to various degrees of imperfection in vortex creation @xcite . in addition , experimental efforts to create a soliton in a scalar condensate using an intensity profile like that of ( [ eq : stepintensity ] ) show that the dipole potential also creates a density wave in addition to the soliton @xcite . we refer the reader to the cited references for further discussion . ultimately , however , it does not appear that these imperfections preclude the creation of vortices and solitons , which suggests that the creation of monopoles via phase imprinting should also be viable in spite of such imperfections . we have proposed a method to create a monopole in an antiferromagnetic spin-1 bec by extending the phase imprinting concept used in scalar bec s . we demonstrate that laser beams with different polarizations generate energy shifts that can be used to engineer the phases of the magnetic substates in a broad manner . we also present an explicit pulse sequence that , starting from the ground state of the condensate , produces phases in the substates corresponding to those of a monopole .

we present a method for creating a monopole in an antiferromagnetic spin-1 bose - einstein condensate . the required phase engineering of the multicomponent condensate is achieved using light shifts , which depend on both the magnetic substate @xmath0 and polarization of the incident laser beam .

the path integral approach to quantum mechanics proposed by feynman @xcite today is the efficient tool of theoretical physics . the path integrals play an important role in quantum field theory @xcite , @xcite , @xcite , @xcite statistical physics and theory of critical phenomena @xcite , @xcite , @xcite quantum optics @xcite , theory of stochastic processes @xcite , @xcite . however , up to now path integrals have not been applied to atomic physics problem - electron charge transfer processes . we generalize the path integral approach to the problem of the rearragemenet collisions and , in particularly , to the capture of an electron by an ion passing another atom . it is so - called the electron charge transfer process . the charge transfer process is an important atomic physics process . besides the intrinsic interest in the charge transfer as a fundamental physical process , a knowledge of its mechanisms is a prerequisite for an understanding of radiation detectors , radiation damage in matter , injection into thermonuclear fusion systems , astrophysical processes , gas discharges , mass spectrometry and numerous other practical devices . the theory of the charge transfer reactions based on the traditional quantum - mechanical approaches can be found in the set of monographs ( see , for example , @xcite , @xcite ) . the recent experimental data and references can be found in @xcite , @xcite . in the framework of the path integral approach we have reformulated the problem of calculation of charge transfer cross section . we have developed two new alternative quantum mechanical influence functionals in the problem of the electron charge transfer . we treat the electron charge transfer process either as an electron transition problem or as an reciprocal elastic scattering of ion and atom in the some effective potential field . the paper is organized as follows . in sec.2 we develop the path integral approach to elastic scattering problem . as an application of the developed approach we have reproduced the well known equation for differential cross section in the first - order born approximation . in sec.3 we have generalized the path integral approach to atomic physics rearragemenet collisions problem . the cross section for the electron charge transfer process is expressed in the terms of the path integral . as was shown by r. feynman @xcite , it is possible to describe the quantum mechanical system in terms of the transition amplitude @xmath0 , @xmath1 where @xmath2 is the start point and @xmath3 is the end point of the quantum - mechanical evolution , @xmath4 means integration over all possible trajectories @xmath5 , @xmath6 is the classical action of the mechanical system as a functional of its trajectory @xmath5 , and @xmath7 is the planck s constant . in order to clarify the meaning of the above expressions we observe the particle of the mass @xmath8 moving in the potential field @xmath9 . the classical action for this mechanical system is given by @xmath10 we divide the time interval @xmath11 into @xmath12 steps of width @xmath13 . then the functional @xmath14 can be represented by the function @xmath15 , where @xmath16 are the points which coincide with the trajectory @xmath17 at the moments @xmath18 ( @xmath19 @xmath20 @xmath21 . hence , the path integral given by eq.([eq1 ] ) can be written as follows @xmath22 @xmath23 where @xmath24 , @xmath25 . the eq.([eq3 ] ) presents the transition quantum mechanical amplitude @xmath26 from the point @xmath27 to the point @xmath28 . let us show how to apply the path integral approach to some known atomic physics problems . consider the scattering of a non relativistic electron on the central potential @xmath29 . being outside of the interaction region ( we assume that the potential is different from zero only in some local area ) the electron moves as a free particle . hence the wave function of the incident electron is just a plane wave @xmath30 where @xmath31 is the electron momentum and @xmath32 is the electron energy . the normalization of the wave function @xmath33 is chosen in such way that the incident flux @xmath34 will be equal to the electron velocity @xmath35 then the knowledge of the transition amplitude and fixing of initial state of physical system give the general solution for the evolution of a quantum mechanical system in the potential @xmath36 @xmath37 since we are only interested in the elastic scattering cross section we should exclude un - scattered component of the total flux . this component is due to unperturbed ( or free ) evolution of the incident particles . hence , we have the following expression for the wave function @xmath38 of the final states of our interest @xmath39 where the free ( unperturbed ) particle transition amplitude @xmath40 is given by @xmath41 @xmath42 @xmath43 taking into account the definitions given by eqs.([eq6 ] ) - ( [ eq8 ] ) the wave function @xmath44 can be written as @xmath45 @xmath46 physical characteristic of the scattering process is the differential scattering cross section @xmath47 . it is defined as the ratio of the number of particles scattered into the solid angle @xmath48 per unit time to the flux density of the incident particles . there are @xmath49 particles ( electrons ) that pass through the elementary area @xmath50 per unit time . here @xmath51 is the radial component of the particle flux density @xmath52 hence the differential cross section is defined by the following equation @xcite @xmath53 where @xmath34 and @xmath51 are given by eq.([eq5 ] ) and eq.([eq10 ] ) reciprocally . thus , eqs.([eq9])-([eq11 ] ) are the path integral approach to the non - relativistic quantum mechanical problem of elastic scattering . let us show that in first - order perturbation theory eqs.([eq9])-([eq11 ] ) give the well known result of the quantum mechanical scattering theory ( the first - order born approximation , @xcite ) . in the first order on the @xmath54 eq.([eq9 ] ) can be written as @xmath55 @xmath56 if the wave function @xmath57 is defined according to eq.([eq4 ] ) then eq.([eq12 ] ) is read @xmath58 @xmath59 here @xmath60 is the free particle quantum mechanical transition amplitude given by eq.([eq8 ] ) . it is convenient to present eq.([eq8 ] ) as the fourier integral @xmath61 substituting the above equation for @xmath62 in eq.([eq13 ] ) yields @xmath63 @xmath64 @xmath65 further , by integrating over the @xmath66 we find @xmath67 when the distance from the origin is much bigger then the effective range for the potential @xmath54 it is possible to approximate @xmath68 where @xmath69 . then the eq.([eq14 ] ) will transformed into the following one @xmath70 @xmath71 thus , with the help of eqs.([eq10 ] ) and ( [ eq15 ] ) the differential cross section defined by eq.([eq11 ] ) can be read as @xmath72 this is the well known the first - order born approximation @xcite for the differential scattering cross section . introducing the notation @xmath73 for the transferred momentum @xmath74 allows for eq.([eq16 ] ) to be written as @xmath75 where @xmath76 is a fourier transform of the scattering potential that corresponds to the transferred momentum @xmath73 . thus , we have shown how the path integral approach to elastic scattering problem allows to obtain the cross section eq.([eq17 ] ) in the first - order born approximation . in this section we apply the approach developed in the sec.2 to the electron charge transfer problem . the classical mechanical action for a system of two heavy particles ( resting in the lab system neutral atom a and a moving positive ion b@xmath77 ) that exchange an electron between each other can be written as follows @xmath78 @xmath79 where @xmath80 is the coordinate of the moving ion b@xmath77 , m is the mass of the ion b@xmath77 , @xmath81 is the coordinate of the electron in the lab frame placed on the resting neutral atom a. in eq.([eq18 ] ) @xmath82 is the interaction potential for the system ( e@xmath83+a@xmath77 ) . @xmath84 is the interaction potential for the system ( e@xmath83+b@xmath77 ) and , at last , @xmath85 is the interaction potential for the system ( b@xmath86+a@xmath77 ) . the transition amplitude @xmath87 from the initial point @xmath88 to the final point @xmath89 has the following form @xmath90 the path integral over two paths @xmath91 and @xmath5 can be treated by means of the influence functionals . the term influence functional was introduced by feynman @xcite . indeed suppose we carry out the path integration over the ion trajectories @xmath91 . then the result can be written as @xmath92 @xmath93 where the influence functional @xmath94 has the form @xmath95 @xmath96 the influence functional @xmath97 defined by eq.([eq21 ] ) is in fact transition amplitude for ion under the influence of a potential @xmath98 which is computed assuming @xmath5 is held to be a fixed path as @xmath99 changes . it is obviously that the influence functional @xmath100 is a functional of @xmath101 . let us represent the influence functional @xmath102 in the following way @xmath103 or @xmath104 @xmath96 where the effective potential @xmath105 has been introduced . then eq.([eq20 ] ) can be rewritten as @xmath106 @xmath107 as a result we have just the path integral over the electron path @xmath108 . thus , eq.([eq24 ] ) allows treat the electron charge transfer problem as the electron transition under the influence of the perturbation @xmath109 . it is possible to introduce another influence functional in the problem of electron charge transfer . indeed , by integrating over all electron trajectories @xmath17 we find for the transition amplitude @xmath110 the following equation @xmath111 @xmath112 where the influence functional @xmath113 is given by @xmath114 @xmath115 the influence functional @xmath116 defined by eq.([eq26 ] ) is in fact the electron transition amplitude under the influence of a potential @xmath117 which is computed assuming @xmath91 is held to be a fixed path as @xmath5 changes . the influence functional @xmath118 is a functional of ion trajectory @xmath99 . it is conveniently to represent the influence functional @xmath119 in the following way @xmath120 or @xmath121 @xmath115 where the effective potential @xmath122 has been introduced . the transition amplitude @xmath87 can now be written to have only path integration over the ion trajectories @xmath123 @xmath124 @xmath125 this equation allows to consider the charge transfer problem as the elastic scattering of the ion on the effective scattering potential @xmath126 , where @xmath122 is introduced by eq.([eq28 ] ) . the eqs.([eq22])-([eq24 ] ) and ( [ eq27])-([eq29 ] ) represent the electron charge transfer problem in the terms of the influence functionals @xmath102 or @xmath119 . let us describe the rearrangement collision by means of the path integral approach . consider the following charge exchange reaction @xmath127 where b@xmath77 is a projectile and a is a neutral atom target . assuming the center of mass for the system above is at rest , then a classical mechanical action can be written in two alternative ways @xmath128 @xmath129 where @xmath130 are the lagrangians of the systems _ a , b_@xmath77 _ , a_@xmath77 _ , b. _ here _ t_@xmath131 and _ t_@xmath132 are the kinetic energies of relative motion of systems a and b@xmath77 , and systems a@xmath77 and b respectively and , at last , @xmath133 and @xmath134 are the perturbations described below . the geometry of the electron charge transfer problem is shown in fig . 1 . we consider the electron ( charge @xmath135 and mass @xmath8 ) initially bound to the heavy particle a of mass @xmath136 , where @xmath137 is the mass of a proton . the vector @xmath80 is the position vector of particle b@xmath77 relative to the centre of mass of the electron and the particle a , while @xmath138 is the position vector of the particle a relative to the centre of mass of the electron and the particle b ( see fig . 1 ) . as a result of the interaction with the passing ion b@xmath77 of mass @xmath139 the electron is captured into a bound state around the particle b. the action @xmath140 can be written initially in the form of eq . ( [ eq30 ] ) with @xmath141 @xmath142 where @xmath82 is the interaction potential between a@xmath77 and e@xmath83 . @xmath143 describes the interaction between the system ( a@xmath144+e@xmath83 ) and b@xmath77 . @xmath145 is irrelevant since the ion b@xmath77 is assumed structureless . in the eq.([eq30 ] ) the coordinates @xmath81 and @xmath80 are in the frame attached to the center of mass of the system ( a+b@xmath77 ) . the action @xmath146 can be rearranged for the final configuration into the form of eq.([eq31 ] ) with @xmath147 @xmath148 where @xmath149 is the interaction potential between b@xmath77 and e@xmath83 . @xmath150 describes the interaction between a@xmath77 and the system ( b@xmath77+e@xmath83 ) . @xmath151 is irrelevant since the ion a@xmath77 is assumed structureless . again , in the eq.([eq31 ] ) the coordinates @xmath152 and @xmath138 are in the frame attached to the center of mass of the system ( a@xmath77+b ) . once the action is known it is possible to construct the transition amplitude @xmath153 that describes the relative motion of the neutral atom , ion and the transition of the electron from the neutral atom to the positive ion , @xmath154 @xmath155 or @xmath156 @xmath157 further it is possible to represent the wave function @xmath158 of the interacting ( rearrangement collision ) particles as follows @xmath159 @xmath160 here @xmath161 is the wave function of the electron bound to the nucleus a , @xmath162 is the wave function of the relative motion ( a@xmath163e@xmath164 and b@xmath77 , @xmath31 and @xmath165 are the momentum and the energy respectively of the relative motion ( a@xmath163e@xmath164 and b@xmath86 . the total energy of the system that enters the reaction is @xmath166 here @xmath167 is the energy of the electron bound to the nucleus a@xmath86 . as a result of the rearrangement collision the system in the outgoing canal consists of the a@xmath77 and the electron bound to the ion b@xmath163e@xmath168 . in this final state the total energy of the system is @xmath169 where @xmath170 is the energy of the relative motion of the a@xmath77 and @xmath171b@xmath172e@xmath164 after the charge transfer reaction and @xmath173 is the binding energy of the electron to the ion b@xmath77 . due to the energy conservation @xmath174 it follows that the energy of the relative motion after the charge transfer reaction is @xmath175 different final states of the system ( states with different quantum numbers ) are called the reaction channels . the channel is called open channel if the following condition is satisfied . @xmath176 in this case the energy of the relative motion of the particles after the reaction is positive and hence they can fly apart to infinity . if there is the opposite sign in the above equation @xmath177 then the channel is called closed channel . the evolution of the quantum mechanical system is defined by the wave function @xmath178 . the transition amplitude ( [ eq36 ] ) or ( [ eq37 ] ) and the initial state of the system ( wave function ( [ eq38 ] ) ) will fully define the @xmath178 for the system @xmath171a@xmath163e@xmath164 and b@xmath77 @xmath179 where the transition amplitude @xmath180 is defined according to ( [ eq37 ] ) . in order to specify the charge transfer process it is necessary to transform the wave function ( [ eq40 ] ) in such a way that it behaves as the scattered spherical wave in the limit @xmath181 . taking into consideration eq.([eq40 ] ) we can define the new wave function @xmath182 as follows @xmath183 where the transition amplitude @xmath184 is given by @xmath185 the charge transfer cross section can be defined as the ratio of the flux of ions ( that went through the charge change process and scattered into the solid angle @xmath186 per unit time ) to the initial flux of the ions . there are @xmath187 ions ( went through the charge exchange process region ) go through the area @xmath188 per unit time . here @xmath51 is the radial component of the flux defined as follows @xmath189 where @xmath190 is given by the eq.([eq41 ] ) and @xmath191 is defined in according to eq.([eq34 ] ) . the initial ion flux is equal to the velocity of the ions due to the normalization of the wave function eq.([eq39 ] ) @xmath192 hence we have come to the following definition for the charge transfer differential cross section @xmath193 where @xmath194 is given by eq.([eq32 ] ) . the eqs.([eq38 ] ) , ( [ eq41 ] ) , ( [ eq43])-([eq45 ] ) represent the formulation of the charge transfer process in terms of the feynman path integral . let us show that the developed approach used in the first order of the perturbation theory will lead to the well known charge transfer cross section formula @xcite , @xcite . if the eq.([eq41 ] ) is expanded in series of the potential only to the first order then the transition amplitude @xmath195 can be written as below @xmath196 @xmath197 or as @xmath198 @xmath199 then @xmath200 @xmath201 where @xmath62 is defined by eq.([eq42 ] ) . further , following almost exactly the same procedure that led from eq.([eq13 ] ) to eq.([eq16 ] ) we write @xmath190 in the form @xmath202 @xmath203 where @xmath204 is the wave function of the electron bound to nucleus a and @xmath205 is the wave function of the electron bound to the nucleus b , and @xmath191 is given in according to eq.([eq34 ] ) . substituting the eq.([eq49 ] ) for @xmath206 in the definition ( [ eq43 ] ) and taking into account eqs.([eq44 ] ) and ( [ eq45 ] ) we have the following equation @xmath207 where @xmath208 and @xmath209 are the absolute values of the initial and the final momentums of moving ion . thus we obtain the differential cross section for the electron charge transfer process in the first - order born approximation @xcite , @xcite , @xcite , @xcite . hence we have shown that the path integral approach allows to obtain the well - known results for the electron charge transfer problem . the path integrals approach is applied to atomic physics problem . the electron charge transfer has been studied by means of the path integral approach . we have developed the influence functional treatment of the electron charge transfer process either as an electron transition problem or as an elastic scattering of ion and atom in the some effective potential . it has been shown how the first born approximation for the elastic scattering cross section can be derived by path integral approach . for the charge transfer reaction the differential cross section in the first born approximation has been obtained to prove the adequacy of the path integrals approach to this problem . we would like to thank the isotrace lab . and especially to prof . ted litherland for his interest in the development of new ideas , numerous stimulating discussions and the financial support of this work . for references to the various experiments , see h. s. w. massey and h. b. gilbody . _ electronic and ionic impact phenomena vol . 4 : recombination and fast collisions of heavy particles . ( _ oxford university press , new york , 1974 ) .

a path integral approach has been generalized for the non - relativistic electron charge transfer processes . the charge transfer - the capture of an electron by an ion passing another atom , or more generally the problem of rearrangement collisions is formulated in terms of influence functionals . it has been shown that the electron charge transfer process can be treated either as electron transition problem or as elastic scattering of ion and atom in the some effective potential field . the first - order born approximation for the electron charge transfer cross section has been reproduced to prove the adequacy of the path integral approach for this problem . _ pacs _ number(s ) : 03.65.db , 03.65.nk , 34.70.+e tcilatex

stochastic approximation algorithms and their variants are commonly found in control , communication and related fields . popularity has grown due to increased computing power , and the interest in various ` machine learning ' algorithms @xcite . when the algorithm is linear , then the error equations take the following linear recursive form : @xmath0 x_t + w_{t+1 } , \elabel{rlsalpha}\ ] ] where @xmath1 is an error sequence , @xmath2 is a sequence of @xmath3 random matrices , @xmath4 is a `` disturbance '' , and @xmath5 is the @xmath3 identity matrix . an important example is the lms ( least mean square ) algorithm . consider the discrete linear time - varying model : @xmath6 where @xmath7 and @xmath8 are the sequences of ( scalar ) observations and noise , respectively , and @xmath9^t$ ] and @xmath10^t$ ] denote the @xmath11-dimensional regression vector and time varying parameters , respectively . the lms algorithm is given by the recursion @xmath12 where @xmath13 , and the parameter @xmath14 $ ] is the _ step size_. hence , @xmath15 , \ ] ] where @xmath16 . this is of the form rlsalpha/ with @xmath17 , @xmath18 , and @xmath19 . on iterating rlsalpha/ we obtain the representation , x_t+1 & = & ( i - m_t ) x_t + w_t+1 + & = & ( i - m_t ) + w_t+1 [ e : esterr ] + & = & _ i = t^0 ( i - m_i ) x_0 + _ i = t^1 ( i - m_i ) w_1 + + ( i - m_t)w_t + w_t+1 . from the last expression it is clear that the matrix products @xmath20 play an important role in the behavior of rlsalpha/. properties of products of random matrices are of interest in a wide range of fields . application areas include numerical analysis @xcite , statistical physics @xcite , recursive algorithms @xcite , perturbation theory for dynamical systems @xcite , queueing theory @xcite , and even botany @xcite . seminal results are contained in @xcite . a complementary and popular research area concerns the eigenstructure of _ large _ random matrices ( see e.g. @xcite for recent application to capacity of communication channels ) . although the results of the present paper do not address these issues , they provide justification for simplified models in communication theory , leading to bounds on the capacity for time - varying communication channels @xcite . the relationship with dynamical systems theory is particularly relevant to the issues addressed here . consider a _ nonlinear _ dynamical system described by the equations , @xmath21 where @xmath22 is an ergodic markov process , evolving on a state space @xmath23 , and @xmath24 is smooth and lipschitz continuous . although it is , of course , impossible to iterate a nonlinear model of this general form , we can construct a random linear model to address many interesting issues . viewing the initial condition @xmath25 as a continuous variable , we write @xmath26 as the resulting state trajectory and consider the sensitivity matrix , @xmath27 from nonlin/ we have the linear recursion , @xmath28 s_t , \elabel{sensitivity}\ ] ] where @xmath29 , @xmath30 . if @xmath31 is suitably stable then the same is true for the nonlinear model , and we find that trajectories couple to a steady state process @xmath32 : @xmath33 these ideas are related to issues developed in . the traditional analytic technique for addressing the stability of nonlin/ or of rlsalpha/ is the _ ode method _ of @xcite . for linear models , the basic idea is that , for small values of @xmath34 , the behavior of rlsalpha/ should mimic that of the linear ode , @xmath35 where @xmath36 and @xmath37 are means of @xmath38 and @xmath39 , respectively . to obtain a finer performance analysis one can instead compare rlsalpha/ to the linear diffusion model , @xmath40 where @xmath41 is a brownian motion . under certain assumptions one may show that , if the ode ode/ is stable , then the stochastic model rlsalpha/ is stable in a statistical sense , and comparisons with sde/ are possible under still stronger assumptions ( see e.g. @xcite for results concerning both linear and nonlinear recursions ) . in @xcite an alternative point of view was proposed where the stability verification problem for rlsalpha/ is cast in terms of the spectral radius of an associated discrete - time semigroup of linear operators . this approach is based on the functional analytic setting of @xcite , and analogous techniques are used in the treatment of multiplicative ergodic theory and spectral theory in @xcite . the main results of @xcite may be interpreted as a significant extension of the ode method for linear recursions . our present results give a unified treatment of both the linear and nonlinear models treated in @xcite and @xcite , respectively . utilizing the operator - theoretic framework developed in @xcite also makes it possible to offer a transparent treatment , and also significantly weaken the assumptions used in earlier results . we provide answers to the following questions : ( i ) : : for what range of @xmath42 is the random linear system rlsalpha/ @xmath43-stable , in the sense that @xmath44 $ ] is bounded in @xmath45 ? ( ii ) : : what does the averaged model ode/ tell us about the behavior of the original stochastic model ? ( iii ) : : what is the impact of variability on _ performance _ of recursive algorithms ? in this section we develop stability theory and structural results for the linear model rlsalpha/ where @xmath46 is a fixed constant . it is assumed that an underlying markov chain @xmath47 , with general state - space @xmath23 , governs the statistics of rlsalpha/ in the sense that @xmath48 and @xmath49 are functions of the markov chain : @xmath50 we assume that the entries of the @xmath51-matrix valued function @xmath52 are bounded functions of @xmath53 . conditions on the vector - valued function @xmath54 are given below . we begin with some basic assumptions on @xmath47 , required to construct a linear operator with useful properties . we assume throughout that the markov chain @xmath47 is _ geometrically ergodic _ or , equivalently , _ @xmath55-uniformly ergodic_. this is equivalent to assuming the validity of the following two conditions : _ irreducibility & aperiodicity : _ there exists a @xmath56-finite measure @xmath57 on the state space @xmath23 such that , for any @xmath53 and any measurable @xmath58 with @xmath59 , @xmath60 _ geometric drift : _ there exists a _ lyapunov function _ @xmath61 , @xmath62 , @xmath63 , @xmath64 , a ` small set ' @xmath65 , and a ` small measure ' @xmath66 , satisfying @xmath67 under these assumptions it is known that @xmath47 is ergodic and has a unique invariant probability measure @xmath68 , to which it converges geometrically fast , and without loss of generality we can assume that @xmath69 for a detailed development of geometrically ergodic markov processes see @xcite . we let @xmath70 denote the set of measurable _ vector - valued _ functions @xmath71 satisfying @xmath72 where @xmath73 is the euclidean norm on @xmath74 , and @xmath75 is the lyapunov function as above . for a linear operator @xmath76 we define the induced operator norm via @xmath77 where the supremum is over all non - zero @xmath78 . we say that @xmath79 is a bounded linear operator if @xmath80 , and its spectral radius is then given by @xmath81 the _ spectrum _ @xmath82 of the linear operator @xmath79 is @xmath83 if @xmath84 is a finite matrix , its spectrum is just the collection of all its eigenvalues . generally , for the linear operators considered in this paper , the dimension of @xmath84 and its spectrum will be infinite . the family of linear operators @xmath85 , @xmath86 , that will be used to analyze the recursion rlsalpha/ are defined by , @xmath87 \\[.25 cm ] & = & \expect_x \left[(i-\alpha m_1)^\transpose f(\phi_1 ) \right]\ , , \end{array}\ ] ] and we let @xmath88 denote the spectral radius of @xmath89 . we assume throughout the paper that @xmath90 is a bounded function . under these conditions we obtain the following result as in @xcite . there exists @xmath91 such that for @xmath92 , @xmath93 , and @xmath94 . to ensure that the recursion rlsalpha/ is stable it is necessary that the spectral radius satisfy @xmath95 . under this condition it is obvious that the mean @xmath96 $ ] is uniformly bounded in @xmath45 . the following result summarizes additional conclusions obtained below . suppose that the eigenvalues of @xmath97 are all positive , and that @xmath98 , where the square is interpreted component - wise . then , there exists a bounded open set @xmath99 containing @xmath100 , where @xmath101 is given in , such that : ( i ) : : for all @xmath102 we have @xmath103 , and for any initial condition @xmath104 , @xmath105 , @xmath106 \to \sigma^2_\alpha<\infty , \;\;\mbox{geometrically fast , as $ t\to\infty$.}\ ] ] ( ii ) : : if @xmath47 is stationary , then for @xmath102 there exists a stationary process @xmath107 such that for any initial condition @xmath104 , @xmath105 , @xmath108\to 0 , \;\;\mbox{geometrically fast , as $ t\to\infty$.}\ ] ] if @xmath109 and @xmath49 is i.i.d . with @xmath110 then @xmath111 $ ] is unbounded . proof outline for iterating the system equation esterr/ we may express the expectation @xmath112 $ ] as a sum of terms of the form , @xmath113,\quad j , k=0,\dots , t\ , . \elabel{qerr}\ ] ] for simplicity consider the case @xmath114 . taking conditional expectations at time @xmath115 , one can then express the expectation qerr/ as @xmath116 \bigr)\ ] ] where @xmath117 is defined in clq/ , and @xmath118 . we define @xmath119 as the set of @xmath34 such that the spectral radius of this linear operator is strictly less than unity . thus , for @xmath102 we have , for some @xmath120 , @xmath121 similar reasoning may be applied for arbitrary @xmath122 , and this shows that @xmath123 $ ] is bounded in @xmath30 for any deterministic initial conditions @xmath104 , @xmath105 . to construct the stationary process @xmath124 we apply backward coupling as developed in @xcite . consider the system starting at time @xmath125 , initialized at @xmath126 , and let @xmath127 , @xmath128 , denote the resulting state trajectory . we then have for all @xmath129 , @xmath130 , \qquad t\ge 0\ , , \ ] ] which implies convergence in @xmath43 to a stationary process : @xmath131 , @xmath30 . we can then compare to the process initialized at @xmath132 , @xmath133,\qquad t\ge 0\ , , \ ] ] and the same reasoning as before gives ( ii ) . next we show that @xmath134 is in fact an eigenvalue of @xmath89 for a range of @xmath135 , and we use this fact to obtain a multiplicative ergodic theorem . the maximal eigenvalue @xmath136 in is a generalization of the perron - frobenius eigenvalue ; c.f . @xcite . suppose that the eigenvalues @xmath137 of @xmath138 are distinct . then , ( i ) : : there exists @xmath139 such that the linear operator @xmath140 has @xmath11 distinct eigenvalues @xmath141 for all @xmath142 , and @xmath143 is an analytic function of @xmath144 in this domain for each @xmath145 . ( ii ) : : for @xmath146 there are associated eigenfunctions @xmath147 and eigenmeasures @xmath148 satisfying @xmath149 moreover , for each @xmath145 , @xmath53 , @xmath150 , @xmath151 are analytic functions on @xmath152 . ( iii ) : : suppose moreover that the eigenvalues @xmath153 are real . then we may take @xmath139 sufficiently small so that @xmath154 are real for @xmath155 . the maximal eigenvalue @xmath156 is equal to @xmath88 , and the corresponding eigenfunction and eigenmeasure may be scaled so that the following limit holds : @xmath157 where the convergence is in the @xmath55-norm . + in fact , there exists @xmath158 and @xmath159 such that for any @xmath160 the following limit holds : @xmath161 = h_{\alpha } ( x ) \mu_\alpha(f ) + b_0 e^{-\delta_0 t } v(x)\ , .\ ] ] the linear operator @xmath162 possesses a @xmath11-dimensional eigenspace corresponding to the eigenvalue @xmath163 . this eigenspace is precisely the set of constant functions , with a corresponding basis of eigenfunctions given by @xmath164 , where @xmath165 is the @xmath145th basis element in @xmath166 . the @xmath11-dimensional set of vector - valued eigenmeasures @xmath167 given by @xmath168 spans the set of all eigenmeasures with eigenvalue @xmath169 . consider the linear operator defined by @xmath170 f , \qquad f\in\lv.\ ] ] it is obvious that @xmath171 is a rank-@xmath11 linear operator , and for @xmath172 we have from the @xmath55-uniform ergodic theorem of @xcite , @xmath173^t \to 0,\qquad t\to\infty,\ ] ] where the convergence is in norm , and hence takes place exponentially fast . it follows that the spectral radius of @xmath174 is strictly less than unity . by standard arguments it follows that , for some @xmath139 , the spectral radius of @xmath175 is also strictly less than unity . the results then follow as in theorem 3 of @xcite . conditions under which the bound @xmath95 is satisfied are given in , where we also provide formulae for the derivatives of @xmath136 : suppose that the eigenvalues @xmath153 are real and distinct . then , the maximal eigenvalue @xmath176 satisfies , ( i ) : : @xmath177 . ( ii ) : : the second derivative is given by , @xmath178 r_0 \ , , \ ] ] where @xmath179 is a right eigenvector of @xmath36 corresponding to @xmath180 , and @xmath181 is the left eigenvector , normalized so that @xmath182 . ( iii ) : : suppose that @xmath183 , @xmath53 . then we may take @xmath184 in ( ii ) , and the second derivative may be expressed , @xmath185 where an @xmath186 is the central limit theorem covariance for the stationary vector - valued stochastic process @xmath187 v_0 $ ] , and @xmath188 $ ] is its variance . to prove ( i ) , we differentiate the eigenfunction equation @xmath189 to obtain @xmath190 setting @xmath191 then gives a version of _ poisson s equation _ , @xmath192 where @xmath193 $ ] . since @xmath194 we may integrate both sides with respect to the invariant probability @xmath68 to obtain @xmath195h_0= -\barm^\transpose h_0=\lo'\ho.\ ] ] this shows that @xmath196 is an eigenvalue of @xmath197 , and @xmath198 is an associated eigenvector for @xmath199 . it follows that @xmath200 by maximality of @xmath136 . we note that poisson s equation matrixfish/ combined with equation ( 17.39 ) of @xcite implies the formula , @xmath201 - \sum_{l=0}^{\infty}\expect_{x}[(m_{l+1 } - \barm)^\transpose ] h_0 \ , .\ ] ] to prove ( ii ) we consider the second - derivative formula , @xmath202 evaluating these expressions at @xmath191 and integrating with respect to @xmath68 then gives the steady state expression , @xmath203 . \elabel{pplambda}\ ] ] in deriving this identity we have used the expressions , @xmath204,\quad \cll_0''f\ , ( x ) = 0,\qquad f\in\lv,\ x\in\state.\ ] ] this combined with pplambda/ gives the desired formula since we may take @xmath205 in ( ii ) . to prove ( iii ) we simply note that in the symmetric case the formula in ( ii ) becomes , @xmath206 = \trace(\gamma - \sigma)\ , .\ ] ] in order to understand the second - order statistics of @xmath207 it is convenient to introduce another linear operator @xmath117 as follows , @xmath208 \\[.25 cm ] & = & \expect_x \left[(i-\alpha m_1)^\transpose f(\phi_1 ) ( i-\alpha m_1 ) \right ] , \end{array } \elabel{clq}\ ] ] where the domain of @xmath209 is the collection of matrix - valued functions @xmath210 . when considering @xmath117 we redefine @xmath211 accordingly . it is clear that @xmath212 is a bounded linear operator under the geometric drift condition and the boundedness assumption on @xmath52 . let @xmath213 denote the spectral radius of @xmath117 . we can again argue that @xmath213 is smooth in a neighborhood of the origin , and the following follows as in : assume that the eigenvalues of @xmath36 are real and distinct . then there exists @xmath139 such that for each @xmath214 there exists an eigenvalue @xmath215 for @xmath216 satisfying @xmath217 , and @xmath218 is real for real @xmath155 . the eigenvalue @xmath219 is smooth on @xmath152 and satisfies , @xmath220 this is again based on differentiation of the eigenfunction equation given by @xmath221 , where @xmath218 and @xmath222 are the eigenvalue and matrix - valued eigenfunction , respectively . taking derivatives on both sides gives @xmath223 where @xmath224 $ ] . as before , we then obtain the steady - state expression , @xmath225=-\barm^\transpose h_0-h_0 \barm=\eta'_0 \ho.\ ] ] and , as before , we may conclude that @xmath226 . consider the discrete - time , linear time - varying model @xmath227 where @xmath228 is a sequence of scalar observations , @xmath229 is a noise process , @xmath230 is the sequence of @xmath11-dimensional regression vectors , and @xmath231 are @xmath11-dimensional time - varying parameters . in this section we illustrate the results above using the lms ( least mean square ) parameter estimation algorithm , @xmath232 where @xmath233 is the error sequence , @xmath234 , @xmath30 . as in the introduction , writing @xmath235 we obtain @xmath236 \ , .\ ] ] this is of the form rlsalpha/ with @xmath237 , @xmath238 and @xmath239 . for the sake of simplicity and to facilitate explicit numerical calculations , we consider the following special case : we assume that @xmath240 is of the form @xmath241 , where the sequence @xmath242 is bernoulli ( @xmath243 with equal probability ) and take @xmath244 to be an i.i.d . noise sequence . in analyzing the random linear system we may ignore the noise @xmath244 and take @xmath245 . this is clearly geometrically ergodic since it is an ergodic , finite state space markov chain , with four possible states . in fact , @xmath47 is geometrically ergodic with lyapunov function @xmath246 . viewing @xmath247 as a vector in @xmath248 , the eigenfunction equation for @xmath89 becomes @xmath249 \ha = \la\ha\ ] ] where @xmath250 $ ] , @xmath251 $ ] , @xmath252 $ ] . in this case , we have the following local behavior : in a neighbor of @xmath253 , the spectral radii of @xmath89 , @xmath117 satisfy @xmath254 \frac{d^n}{d\alpha^n } \xi_\alpha\big|_{\alpha=0 } & = & 0 , n \geq 2 ; \qquad & \frac{d^n}{d\alpha^n } \xi^q_\alpha\big|_{\alpha=0 } & = & 0 , n \geq 3 . \end{array}\ ] ] so @xmath136 and @xmath218 are linear and quadratic around @xmath253 , respectively . this follows from differentiating the respective eigenfunction equations . here we only show the proof for operator @xmath255 , the proof for operator @xmath79 is similar . taking derivatives on both sides of the eigenfunction equation for @xmath117 gives , @xmath256 setting @xmath191 gives a version of _ poisson s equation _ , @xmath257 using the identities of @xmath258 and @xmath259 $ ] , we obtain the steady state expression @xmath260 since @xmath261 , we have @xmath262 . now , taking the 2nd derivatives on both sides of qfirst/ gives , @xmath263 letting @xmath191 and considering the steady state , we obtain @xmath264=\eta''_0\ho+2\eta'_0 \expect_{\pi}[\ho ' ] . \elabel{qsecond_0}\ ] ] poisson s equation qfirst_0/ combined with equation qfirst_steady/ and equation ( 17.39 ) of @xcite implies the formula , @xmath265 \\ & = & \expect_{\pi}(\ho')+\sum_{l=0}^{\infty}\expect_{x}[(\barm - m_{l+1})^\transpose\ho+\ho ( \barm - m_{l+1 } ) ] . \end{array}\ ] ] so , from @xmath261 , @xmath266 and qsecond_0/ we have @xmath267 . in order to show @xmath218 is quadratic near zero , we take the 3rd derivative on both sides of qpprime/ and consider the steady state at @xmath191 , @xmath268 with equation ( 17.39 ) of @xcite and @xmath266 and @xmath267 , we can show @xmath269 and @xmath270 for @xmath271 , hence @xmath218 is quadratic around @xmath253 . we now turn to the nonlinear model shown in nonlin/. we take the special form , @xmath272\ , , \elabel{nonlin2}\ ] ] we continue to assume that @xmath47 is geometrically ergodic , and that @xmath273 , @xmath30 , with @xmath98 . the associated ode is given by @xmath274 where @xmath275 , @xmath276 . ( n1 ) : : the function @xmath278 is lipschitz , and there exists a function @xmath279 such that @xmath280 furthermore , the origin in @xmath281 is an asymptotically stable equilibrium point for the ode , @xmath282 ( n2 ) : : there exists @xmath283 such that @xmath284 . ( n3 ) : : there exists a unique stationary point @xmath285 for the ode nonlinode/ that is a globally asymptotically stable equilibrium . ( i ) : : for any @xmath288 , there exists @xmath289 such that @xmath290 ( ii ) : : if the origin is a globally exponentially asymptotically stable equilibrium for the ode nonlinode/ , then there exists @xmath291 such that for every initial condition @xmath104 , @xmath105 , @xmath292 \le b_2 \alpha.\ ] ] proof outline for the continuous - time process @xmath293 is defined to be the interpolated version of @xmath207 given as follows : let @xmath294 , @xmath295 , and define @xmath296 , with @xmath297 defined by linear interpolation on the remainder of @xmath298 $ ] to form a piecewise linear function . using geometric ergodicity we can bound the error between @xmath297 and solutions to the ode nonlinode/ as in @xcite , and we may conclude that the joint process @xmath299 is geometrically ergodic with lyapunov function @xmath300 . assume that ( n1)(n3 ) hold , and that the eigenvalues of the matrix @xmath36 have strictly positive real part , where @xmath302 then there exists @xmath303 such that for any @xmath304 , the conclusions of ( ii ) hold , and , in addition : ( i ) : : the spectral radius @xmath88 of the random linear system sensitivity/ describing the evolution of the sensitivity process is strictly less than one . ( ii ) : : there exists a stationary process @xmath107 such that for any initial condition @xmath104 , @xmath105 , @xmath305\to 0,\qquad t\to\infty\ , .\ ] ] p. bougerol . limit theorem for products of random matrices with markovian dependence . in _ proceedings of the 1st world congress of the bernoulli society , vol . 1 ( tashkent , 1986 ) _ , pages 767770 , utrecht , 1987 . vnu sci . press . o. dabeer and e. masry . the lms adaptive algorithm : asymptotic error analysis . in _ proceedings of the 34th annual conference on information sciences and systems , ciss 2000 _ , pages wp16 wp17 , princeton , nj , march 2000 .

we give a development of the ode method for the analysis of recursive algorithms described by a stochastic recursion . with variability modelled via an underlying markov process , and under general assumptions , the following results are obtained : ( i ) : : stability of an associated ode implies that the stochastic recursion is stable in a strong sense when a gain parameter is small . ( ii ) : : the range of gain - values is quantified through a spectral analysis of an associated linear operator , providing a non - local theory . ( iii ) : : a second - order analysis shows precisely how variability leads to sensitivity of the algorithm with respect to the gain parameter . all results are obtained within the natural operator - theoretic framework of geometrically ergodic markov processes .

the top quark is the heaviest particle of the standard model ( sm ) of elementary particle physics and its short lifetime implies that it decays before hadronization takes place , therefore it retains its full polarization content when it decays . this allows one to study the top spin state using the angular distributions of its decay products . highly polarized top quarks become available at hadron colliders through single top production processes , which occurs at the @xmath4 level of the @xmath5 pair production rate @xcite . near maximal and minimal values of top quark polarization at a linear @xmath6 collider can be achieved in @xmath7 production by tuning the longitudinal polarization of the beam polarization @xcite so that a polarized linear @xmath6 collider may be viewed as a copious source of close to zero and close to @xmath8 polarized top quarks . a first study of polarization in @xmath7 events was performed by the @xmath9 collaboration @xcite which showed good agreement between the sm prediction and data . + the top quark total width , @xmath10 , is proportional to the third power of its mass and is much larger than the typical qcd scale @xmath11 . therefore it enables us to treat the top quark almost as a free particle and to apply perturbative methods to evaluate the quantum corrections to its decay process . + the large hadron collider ( lhc ) is a formidable top factory which is designed to produce about 90 million top quark pairs per year of running at design c.m . energy @xmath12 tev and design luminosity @xmath13 in each of the four experiments @xcite . this will allow one to specify the properties of the top quark , such as its total decay width @xmath10 , mass @xmath14 and branching fractions to high accuracy . the theoretical aspects of top quark physics at the lhc are summarized in @xcite . + due to the element @xmath15 of the ckm @xcite quark mixing matrix , top quarks almost exclusively decay to bottom quarks , via @xmath16 followed by bottom quarks hadronization , @xmath17 , before b - quarks decay . here , @xmath18 stands for the observed final state hadron fragmented from the b - quark so that its production process regarded as the nonperturbative aspect of @xmath18-hadron formation from top decays . of particular interest are the distribution in the scaled @xmath18-hadron energy @xmath19 in the top quark rest frame as reliably as possible . in ref . @xcite we have calculated the doubly differential distribution @xmath20 of the partial width of the decay @xmath21 where @xmath22 is the decay angle of the positron in the w - boson rest frame and @xmath23 is the scaled - energy of bottom - flavored hadrons b. in ref . @xcite , we also evaluated the first order qcd corrections to the energy distribution of b - hadrons from the decay of an unpolarized top quark into a charged - higgs boson , via @xmath24 , in the theories beyond - the - sm with an extended higgs sector . in ref . @xcite , is mentioned that a clear separation between the decay modes @xmath25 and @xmath26 can be achieved in both the @xmath27 pair production and the @xmath28 single top production at the lhc . + since bottom quarks hadronize before they decay , the particular purpose of this paper is the evaluation of the nlo qcd corrections to the energy distribution of charmed - flavored ( d ) and bottom - flavored ( b ) hadrons from the decay of a polarized top quark into a bottom quark , via @xmath29 where d stands for one of the mesons @xmath30 , @xmath31 and @xmath32 . these measurements will be important to deepen our understanding of the nonperturbative aspects of d- and b - mesons formation which are described by realistic and nonperturbative fragmentation functions ( ffs ) . the @xmath33 ff is obtained through a global fit to @xmath6 data from cern lep1 and slac slc in ref . @xcite and the @xmath34 ffs are determined in @xcite by fitting the @xmath6 experimental data from the belle , cleo , opal and aleph collaborations . as was demonstrated in @xcite , the finite-@xmath35 corrections are rather small and thus to study the distributions in the heavy meson scaled - energy ( @xmath23 for b - meson and @xmath36 for d - mesons ) , we employ the massless scheme or zero - mass variable - flavor - number ( zm - vfn ) scheme @xcite in the top quark rest frame , where the zero mass parton approximation is also applied to the bottom quark . the non - zero value of the b - quark mass only enter through the initial condition of the nonperturbative ff . this paper is structured as follows . in sec . [ sec : one ] , we introduce the angular structure of differential decay width by defining the technical details of our calculations . in sec . [ sec : two ] , our analytic results for the @xmath37 qcd corrections to the angular distributions of partial decay rates are presented . in sec . [ sec : three ] , we present our numerical analysis . in sec . [ sec : four ] , our conclusions are summarized . in this section we explain the @xmath37 radiative corrections to the partial decay rate @xmath38 . the dynamics of the current - induced @xmath39 transition is depicted in the hadron tensor @xmath40 which is introduced by @xmath41 where @xmath42 refers to the lorentz - invariant phase space factor . in the sm the weak current is given by @xmath43 . since we are not summing over the top quark spin the hadron tensor @xmath40 also depends on the top spin @xmath44 . in this work we shall be concerned only with two types of intermediate state in eq . ( [ tensor ] ) , i.e. @xmath45 for born term and @xmath37 one - loop contributions and @xmath46 for @xmath37 tree graph contribution . the decay @xmath38 is analyzed in the rest frame of the top quark ( fig . [ lo ] ) where the three - momentum of the @xmath47-quark points into the direction of the positive z - axis . the polar angle @xmath48 is defined as the angle between the top quark polarization vector @xmath49 and the z - axis . we shall closely follow the notation of @xcite where we discussed the @xmath37 radiative corrections to the partial decay rate of unpolarized top quarks . the angular distribution of the top quark differential decay width @xmath50 is given by the following simple expression to clarify the correlations between the top quark decay products and the top spin @xmath51 where we have defined the b - quark scaled - energy as @xmath52 that @xmath53 being @xmath54 . neglecting the b - quark mass one has @xmath55 . in eq . ( [ widthdefine ] ) , p is the magnitude of the top quark polarization with @xmath56 such that @xmath57 corresponds to an unpolarized top quark and @xmath58 corresponds to @xmath8 top quark polarization . @xmath59 and @xmath60 stand for the unpolarized and polarized differential rates , respectively . in the following we discuss the technical details of our calculation for the @xmath37 radiative corrections to the tree - level decay rate of @xmath61 using dimensional regularization . definition of the polar angle @xmath48 in the top quark rest frame . @xmath49 is the polarization vector of the top quark . ] the born term tensor is calculated by squaring the born term amplitude which is given by @xmath62 where @xmath63 stands for the w - boson polarization vector , the angle @xmath64 is the weak mixing angle and we take @xmath65 @xcite , @xmath66 and @xmath67 stand for the spin and four - momenta of particles , respectively . the squared born amplitude is expressed as @xmath68,\nonumber\\\end{aligned}\ ] ] where we replaced @xmath69 in the unpolarized dirac string by @xmath70 in the polarized state . + considering fig . [ lo ] , we parameterize the four - momenta and the polarization vector in the top rest frame as @xmath71 where the parameter p is the degree of polarization . therefore , the lo amplitude squared reads @xmath72 is related to the fermi s constant @xmath73 as @xmath74 . + the decay rate of a particle with a mass @xmath75 and momentum @xmath67 into a given final state of particles @xmath76 is @xmath77 that in the two - body phase space ( e.g. @xmath78 ) , using the following relation @xmath79 one has @xmath80 substituting ( [ amplitude ] ) and ( [ phase ] ) into ( [ gamma ] ) , one has @xmath81 where the polarized and unpolarized born term rates read @xmath82 and @xmath83 the polarization asymmetry @xmath84 is defined by @xmath85 which is simplified to @xmath86 if we set @xmath87 gev and @xmath88 gev @xcite . the virtual one - loop contributions to the fermionic ( v - a ) transitions have a long history . since at the one - loop level , qed and qcd have the same structure then the history even dates back to qed times . in this section we will investigate the one - loop corrections and describe the method applied to extract the singularities at zero - mass scheme . in the zm - vfn scheme , where @xmath89 is set right from the start , both the soft and collinear singularities are regularized by dimensional regularization in @xmath90 space - time dimensions to become single poles in @xmath91 . these singularities are subtracted at factorization scale @xmath92 and absorbed into the bare ffs according to the modified minimal - subtraction @xmath93 scheme . this renormalizes the ffs and creates in @xmath94 finite terms including the term @xmath95 which are rendered perturbatively small by choosing @xmath96 . + virtual ( a , b ) and real gluon ( c , d ) contributions to @xmath61 at nlo . ] the virtual contributions exhibit both the infrared ( ir ) and ultraviolet ( uv ) singularities which are regularized in d - dimensions . to evaluate the one - loop contributions to @xmath61 we consider the feynman diagrams drawn in fig . [ feynmandiagrams ] . the renormalized amplitude of the virtual corrections can be written as @xmath97 considering fig . [ feynmandiagrams]a , the counter term of the vertex is given by @xmath98 where the wave - function renormalization constants of the top ( @xmath99 ) and bottom ( @xmath100 ) can be found in @xcite . from fig . [ feynmandiagrams]b , for the one - loop vertex correction one has @xmath101 where @xmath102 stands for the color factor and @xmath103 refers to the gluon four - momenta . the integral ( [ oneloop ] ) is both ir- and uv - divergent that we use dimensional regularization to extract singularities taking the replacement @xmath104 at @xmath37 the full amplitude is the sum of the amplitudes of the born term , virtual one - loop and the real contributions @xmath105 squaring the full amplitude we have @xmath106 and @xmath107 . considering eqs . [ gamma ] and [ phase ] , the virtual corrections to the doubly differential decay rate is then given by @xmath108 where @xmath109 is defined in ( [ scale ] ) . the one - loop vertex correction and the wave - function renormalization contain uv- and ir - singularities that all uv - singularities are canceled after summing all virtual corrections up whereas the ir - divergences are remaining which are now labeled by @xmath110 . therefore , the virtual doubly differential distribution reads @xmath111 where the unpolarized differential decay rate normalized to the unpolarized born term is @xmath112 and the polarized differential width normalized to the polarized born width reads @xmath113 with @xmath114 here @xmath115 is the euler constant and the dilogarithmic function @xmath116 is defined as @xmath117 as is seen , the one - loop contribution is purely real . this can be got from an inspection of the one - loop feynman diagram fig . [ feynmandiagrams]b , which does not accept any nonvanishing physical two - particle cut . definition of the azimuthal angle @xmath118 and the polar angles @xmath22 and @xmath48 . @xmath49 is the polarization vector of the top quark . ] the @xmath37 real graph contribution results from the square of the real gluon emission graphs shown in figs . [ feynmandiagrams](c ) and [ feynmandiagrams](d ) . the real amplitude for the decay process @xmath119 reads @xmath120 where @xmath121 is the strong coupling constant , n is the color index @xmath122 so @xmath123 and the first and second terms in the curly brackets refer to real gluon emission from the top quark and the bottom quark , respectively . the polarization vector of the gluon is denoted by @xmath124 . by working in the massless scheme , the mass of b - quark is set to zero thus the ir - divergences arise from the soft- and collinear gluon emission . as before to regulate the ir - singularities we work in d - dimensions . in the top quark rest frame ( fig . [ feynnlo ] ) the momenta and the polarization vector are defined as @xmath125 considering ( [ gamma ] ) the real differential rate in d - dimensions is given by @xmath126 to calculate the differential decay rate @xmath127 normalized to the born width , we fix the momentum of b - quark and integrate over the energy of gluon which ranges from @xmath128 and @xmath129 and to obtain the angular distribution of differential width @xmath130 , the angular integral in d - dimensions is written as @xmath131 in the massless scheme , the real and virtual differential widths contain the poles @xmath132 and @xmath133 which disappear only in the total nlo width . this requires that to get the correct finite terms in the normalized doubly differential distribution @xmath134 , the polarized and unpolarized born widths will have to be evaluated in dimensional regularization at @xmath135 . we find @xmath136\bigg\},\nonumber\\ \gamma_0^{pol}&=&\frac{m_t^3 g_f}{8\sqrt{2}\pi}(1 - 2\omega)(1-\omega)^2\bigg\{1+\epsilon \big[h+\frac{2\omega}{1 - 2\omega}\big]+\nonumber\\ & & \epsilon^2 \big[\frac{1}{2}(h+\frac{5}{2})^2-h\frac{5 - 14\omega}{2(1 - 2\omega)}-\frac{\pi^2}{12}-\frac{25}{8}\big]\bigg\},\nonumber\\\end{aligned}\ ] ] with @xmath137 now the real gluon contribution is given by @xmath138 where , by defining @xmath139 , one has @xmath140 + 2(1+x_b^2)\big(\frac{\ln(1-x_b)}{1-x_b}\big)_++\nonumber\\ & & \frac{1}{(1-x_b)_+}\bigg[1 - 4x_b+x_b^2+\nonumber\\ & & \frac{4x_b\omega(1-\omega)(1-x_b)^2}{(1 + 2\omega)(1-x_b(1-\omega))}+\nonumber\\ & & ( 1+x_b^2)\bigg(\ln[\frac{x_b^2(1-\omega)^2m_t^2}{4\pi\mu_f^2}]+\nonumber\\ & & \gamma_e-\frac{1}{\epsilon}\bigg)\bigg]\bigg\},\nonumber\\\end{aligned}\ ] ] and @xmath141\nonumber\\ & & + 2(1+x_b^2)\big(\frac{\ln(1-x_b)}{1-x_b}\big)_++\nonumber\\ & & \frac{1}{(1-x_b)_+}\bigg[-1-x_b^2+\frac{8\omega(1-x_b)^2}{1 - 2\omega}+\nonumber\\ & & 4\frac{x_b\omega(1-\omega)(1-x_b)^2}{(1 - 2\omega)(1-x_b ( 1-\omega))}+\nonumber\\ & & ( 1+x_b^2)\bigg(\ln[\frac{x_b^2(1-\omega)^2m_t^2}{4\pi\mu_f^2}]+\gamma_e-\frac{1}{\epsilon}\bigg)\nonumber\\ & & + \frac{8\omega(1-x_b)^2}{x_b(1-\omega)(1 - 2\omega)}\ln(1-x_b(1-\omega))\bigg]\bigg\}.\nonumber\\\end{aligned}\ ] ] now we are in a situation to present our analytic results for the angular distribution of the differential decay rate , by summing the born - level ( [ bornfinal ] ) , the virtual ( [ virtualfinal ] ) and real gluon ( [ realfinal ] ) contributions . according to the lee - nauenberg theorem , all singularities cancel each other after summing all contributions up and the final result is free of ir - singularities . therefore , the complete @xmath37 results are @xmath142 that we presented @xmath143 in ref . @xcite and @xmath144 in the @xmath145 scheme is expressed , for the first time , as @xmath146\nonumber\\ & & + 2(1+x_b^2)\big(\frac{\ln(1-x_b)}{1-x_b}\big)_++\nonumber\\ & & \frac{1}{(1-x_b)_+}\bigg[-1-x_b^2+\frac{8\omega(1-x_b)^2}{1 - 2\omega}+\nonumber\\ & & 4\frac{x_b\omega(1-\omega)(1-x_b)^2}{(1 - 2\omega)(1-xb ( 1-\omega))}+\nonumber\\ & & ( 1+x_b^2)\ln[x_b^2(1-\omega)^2\frac{m_t^2}{\mu_f^2}]+\nonumber\\ & & \frac{8\omega(1-x_b)^2}{x_b(1-\omega)(1 - 2\omega)}\ln(1-x_b(1-\omega))\bigg]\bigg\}.\nonumber\\\end{aligned}\ ] ] since the observed mesons can be also produced through a fragmenting gluon , therefore , to obtain the most accurate result for the energy spectrum of meson we have to add the contribution of gluon fragmentation to the b - quark to produce the outgoing meson . from fig . [ fig1 ] , it is seen that the contribution of gluon decreases the size of decay rate up to @xmath147 at the threshold , thus this contribution can be important at low energy of the observed meson . therefore , the differential decay rate @xmath148 is also required where @xmath149 is defined as @xmath150 as in ( [ scale ] ) . to obtain the doubly differential distribution @xmath151 , we integrate over the b - quark energy by fixing the gluon momentum in the phase space . the result is @xmath152 where @xmath153 can be found in our previous work @xcite and @xmath154 is listed here @xmath155+\nonumber\\ & & \frac{1}{2(1 - 2\omega)}\big[2(1 - 2\omega)+\frac{4(1+\omega)}{1-\omega}-\nonumber\\ & & x_g(1 + 6\omega)-\frac{\omega(6\omega^2+\omega+2)}{(1-\omega)(1-x_g(1-\omega))}\nonumber\\ & & -\frac{\omega^2(1 - 2\omega)}{(1-\omega)(1-x_g(1-\omega))^2}\big]+\nonumber\\ & & \frac{1+(1-x_g)^2}{x_g}\big[2\ln(x_g(1-x_g))-\ln\frac{\mu_f^2}{m_t^2}\nonumber\\ & & -\ln(1-x_g(1-\omega))+2\ln(1-\omega)]\bigg\}.\nonumber\\\end{aligned}\ ] ] in this section , performing a numerical analysis we present our phenomenological results for the energy spectrum of the heavy mesons b and d from polarized top decays and compare them with the unpolarized one in @xcite . we define the normalized - energy fractions of the outgoing mesons similarly to the parton - level one in ( [ scale ] ) as @xmath156 for the b - meson and @xmath157 for the d - meson . considering the factorization theorem of the qcd - improved parton model @xcite , the energy distribution of a hadron can be expressed as the convolution of the parton - level spectrum with the nonperturbative fragmentation function @xmath158 @xmath159 where @xmath160 stands for @xmath161 and @xmath162-mesons and the allowed @xmath19 ranges are @xmath163 . the integral convolution appearing in ( [ convolute ] ) is defined as @xmath164 in ( [ convolute ] ) , @xmath92 and @xmath165 are the factorization and the renormalization scales , respectively , and one can use two different values for these scales ; however , a choice often made consists of setting @xmath166 and we adopt the convention @xmath167 for our results . in ( [ convolute ] ) , @xmath168 are the parton - level differential rates presented in ( [ first ] ) and ( [ second ] ) and @xmath169 are the nonperturbative ffs describing the hadronizations @xmath170 and @xmath171 which are process independent . several models have been yet proposed to describe the ffs . in ref . @xcite , authors calculated the ffs for @xmath172 and @xmath32 mesons by fitting the experimental data from the belle , cleo , opal , and aleph collaborations in the modified minimal - subtraction ( @xmath145 ) factorization scheme . they have parameterized the @xmath173 distributions of the @xmath174 ffs at their starting scale @xmath175 , as suggested by bowler @xcite , as @xmath176 while the ff of the gluon is set to zero and this ff is evolved to higher scales using the dglap equations @xcite . as in @xcite is claimed , this parametrization yields the best fit to the belle data @xcite in a comparative analysis using the monte - carlo event generator jetset / pythia . the values of fit parameters together with the achieved values of @xmath177 are presented in table [ fit ] . .[fit ] values of fit parameters for @xmath178 , @xmath179 and @xmath180 ffs at the starting scale @xmath175 resulting from the global fit in the zm approach together with the values of @xmath177 achieved . [ cols="^,^,^,^,^",options="header " , ] from ref . @xcite we employ the @xmath33 ff determined at nlo in the zm - vfn approach through a global fit to @xmath6-annihilation data taken by aleph @xcite and opal @xcite at cern lep1 and by sld @xcite at slac slc . the power ansatz @xmath181 was used as the initial condition for the @xmath182 ff at @xmath175 , while the gluon ff was generated via the dglap evolution . the fit parameters are @xmath183 , @xmath184 , and @xmath185 with @xmath186 . following ref . @xcite we adopt the input values @xmath187 gev@xmath188 , @xmath189 gev , @xmath190 gev , @xmath191 gev , @xmath192 gev , @xmath193 gev , and @xmath194 mev with @xmath195 active quark flavors and adjusted such that @xmath196 . + to study the scaled - energy ( @xmath23 and @xmath36 ) distributions of the bottom- and charmed - flavored hadrons produced in the polarized top decay , we consider the quantities @xmath197 and @xmath198 in the zm - vfn scheme . in fig . [ fig1 ] , our prediction for the b - meson is shown by studying the size of the nlo corrections , by comparing the lo ( dotted line ) and nlo ( solid line ) results , and the relative importance of the @xmath33 ( dashed line ) and @xmath199 ( dot - dashed line ) fragmentation channels at nlo . we evaluated the lo result using the same nlo ffs . the @xmath199 contribution is negative and appreciable only in the low-@xmath23 region . for higher values of @xmath23 , as is expected @xcite , the nlo result is practically exhausted by the @xmath33 contribution . note that the contribution of the gluon can not be discriminated . it is calculated to see where it contributes to @xmath200 . so this part of the paper is of more theoretical relevance . in the scaled - energy of mesons as a experimental quantity , all contributions including the b quark , gluon and light quarks contribute . @xmath201 as a function of @xmath23 in the zm - vfn ( @xmath89 ) scheme . the nlo result ( solid line ) is compared to the lo one ( dotted line ) and broken up into the contributions due to @xmath182 ( dashed line ) and @xmath202 ( dot - dashed line ) fragmentation . we set @xmath203 and @xmath204 . ] in fig [ fig2 ] , the scaled - energy ( @xmath23 ) distribution of b - hadrons produced in unpolarized ( dashed line ) and polarized ( solid line ) top quark decays at nlo are studied . as is seen , in the unpolarized top decay the partial decay width at hadron - level is around @xmath205 higher than the one in the polarized top decay in the peak region . in fig . [ fig3 ] the same comparison is also down for the transition @xmath178 applying the bowler model ( [ bw ] ) for the ffs . [ fig3 ] shows that the probability to produce the charmed - flavored mesons through top quark decays in the high-@xmath36 range ( @xmath206 ) is zero . in fig . [ fig4 ] we study the scaled - energy ( @xmath36 ) distribution of charmed - flavored hadrons produced in polarized top decays as @xmath207 ( solid line ) , @xmath208 ( dotted line ) and @xmath209 ( dashed line ) . note that our results are valid just for @xmath210 and @xmath211 . @xmath200 as a function of @xmath23 in the zm - vfn ( @xmath89 ) scheme considering the unpolarized ( dashed line ) and polarized ( solid line ) partial decay rates at nlo . ] @xmath36 spectrum in top decay , with the hadronization modeled according to the bowler model considering the unpolarized ( dashed line ) and polarized ( solid line ) decay rates at nlo . we set @xmath203 and @xmath204 . ] @xmath212 as a function of @xmath36 at nlo for @xmath213 ( solid line ) , @xmath214 ( dotted line ) and @xmath215 ( dashed line ) . ] the top quark decays rapidly so that it has no time to hadronize and passes on its full spin information to its decay products . the cern lhc , as a superlative top factory , allows us to study top quark decays that within the sm are completely dominated by the mode @xmath216 , followed by @xmath217 . therefore , the distribution in the scaled h - hadron energy @xmath19 in the top rest frame are of particular interest . in fact , the @xmath19 distribution provides direct access to the h - hadron ffs , and its @xmath218 distribution allows one to analyze the top quark polarization where the polar angle @xmath48 refers to the angle between the polarization vector of the top and the z - axis . + in @xcite we have studied the scaled - energy ( @xmath23 ) distribution of the b - meson in unpolarized top quark decays and in the present work we made our predictions for the scaled - energy ( @xmath219 ) distributions of the b- and d - mesons in the polarized top quark rest frame by studying the quantity @xmath220 . as was mentioned , the scaled - energy distribution of hadron enables us to deepen our knowledge of the hadronization process and to pin down the @xmath221 ffs while the angular analysis of the polarized top decay constrain these ffs even further . furthermore , the polarization state of top quarks can be specified from the angular distribution of the outgoing hadron energy . the universality and scaling violations of the b- and d - hadron ffs will be able to test at lhc by comparing our nlo predictions with future measurements of @xmath222 and @xmath223 . one can also test the sm and/or non - sm couplings through polarization measurements involving top quark decays ( mostly @xmath61 ) . the formalism made here is also applicable to the production of hadron species other than b and d hadrons , such as pions and kaons , through the polarized top quark decay using the @xmath224 ffs presented in our recently paper @xcite . g. mahlon and s. j. parke , phys . d * 55 * , 7249 ( 1997 ) . s. groote , j. g. korner , b. melic and s. prelovsek , phys . d * 83 * , 054018 ( 2011 ) . v. m. abazov _ et al . _ [ d0 collaboration ] , phys . d * 87 * , 011103 ( 2013 ) . n. cabibbo , phys . rev . lett . * 10 * , 531 ( 1963 ) ; m. kobayashi and t. maskawa , prog . phys . * 49 * , 652 ( 1973 ) . b. a. kniehl , g. kramer and s. m. moosavi nejad , nucl . b * 862 * , 720 ( 2012 ) . s. m. moosavi nejad , phys . d * 85 * , 054010 ( 2012 ) ; s. m. moosavi nejad , eur . phys . j. c * 72 * , 2224 ( 2012 ) . t. kneesch , b. a. kniehl , g. kramer and i. schienbein , nucl . b * 799 * , 34 ( 2008 ) . j. binnewies , b.a . kniehl , and g. kramer , phys . rev . d * 58 * , 034016 ( 1998 ) ; + m. cacciari and m. greco , nucl . b*421 * , 530(1994 ) . g. corcella and a. d. mitov , nucl . b * 623 * , 247 ( 2002 ) . c. caso _ et al . _ [ particle data group collaboration ] , eur . j. c * 3 * , 1 ( 1998 ) . k. nakamura _ et al . _ ( particle data group ) , j. phys . g * 37 * , 075021 ( 2010 ) . j. c. collins , phys . d * 66 * ( 1998 ) 094002 . m. g. bowler , z. phys . c * 11 * ( 1981 ) 169 . v. n. gribov and l. n. lipatov , sov . j. nucl . * 15 * , 438 ( 1972 ) [ yad . fiz . * 15 * , 781 ( 1972 ) ] ; g. altarelli and g. parisi , nucl . * b126 * , 298 ( 1977 ) ; yu . l. dokshitzer , sov . jetp * 46 * , 641 ( 1977 ) [ zh . eksp . * 73 * , 1216 ( 1977 ) ] . belle collaboration , r. seuster , et al . d * 73 * , 032002 ( 2006 ) . a. heister _ et al . _ ( aleph collaboration ) , phys . lett . b * 512 * , 30 ( 2001 ) . g. abbiendi _ et al . _ ( opal collaboration ) , eur . j. c * 29 * , 463 ( 2003 ) . k. abe _ et al . _ ( sld collaboration ) , phys . rev . * 84 * , 4300 ( 2000 ) ; phys . d * 65 * , 092006 ( 2002 ) .

we consider the decay of a polarized top quark into a stable @xmath0 boson and charmed - flavored ( d ) or bottom - flavored ( b ) hadrons , via @xmath1 . we study the angular distribution of the scaled - energy of b / d - hadrons at next - to - leading order ( nlo ) considering the contribution of bottom and gluon fragmentations into the heavy mesons b and d. to obtain the energy spectrum of b / d - hadrons we present our analytical expressions for the parton - level differential decay widths of @xmath2 at nlo . comparison of our predictions with data at the lhc enable us to test the universality and scaling violations of the b- and d - hadron fragmentation functions ( ffs ) . these can also be used to determine the polarization states of top quarks and since the energy distributions depend on the ratio @xmath3 we advocate the use of such angular decay measurements for the determination of the top quark s mass .

an ability to produce more then one electron - hole pairs ( excitons ) per single absorbed photon in semiconductor materials can potentially enhance the solar energy conversion efficiency in photovoltaic devices beyond the fundamental shockley - queisser limit.@xcite in the literature this process is referred to as carrier multiplication ( cm ) or multiple exciton generation , and has been extensively studied in the _ bulk _ semiconductors for decades.@xcite recent ultrafast optical studies of cm in semiconductor _ nanocrystals _ ( nc ) reported significantly higher qe and much lower activation energy threshold ( aet ) as compared to the bulk materials on the photon energy scale normalized by corresponding band gap values.@xcite these reports stimulated extensive experimental and theoretical studies of the cm in the nanostructured materials in which the exciton dynamics is affected by the quantum confinement . it was initially expected that the quantum confinement enhancement of the coulomb interactions between the carriers could result in more efficient multi - exciton production . furthermore , a relaxation of the quasi - momentum conservation constrains by breaking the translational symmetry should open additional pathways for cm and reduce the aet . finally , the presence of the phonon bottleneck in ncs should slow down the intraband phonon - assisted cooling and further increase the qe . however , the subsequent reports claimed significantly lower qe and even the absence of the cm in ncs.@xcite this controversy possibly rises from experimental inaccuracy,@xcite sample - to - sample variation in surface preparation,@xcite and contribution from extraneous effects such as photocharging.@xcite the optical measurements of qe in bulk pbs and pbse semiconductors@xcite show that , in fact , the _ bulk _ qe exceeds validated values in ncs@xcite if compared on the absolute photon energy scale . @xcite however , due to confinement induced increase in the band gap energy , @xmath0 , the benefits to photovoltaics is higher in ncs than in bulk . the controversy calls for development of new sensitive spectroscopic tools,@xcite alternative photo - current measurements,@xcite and the reassessment of the quantum confinement role in the cm dynamics based on a unified theoretical model.@xcite accepted theoretical models of cm are based on the many - body coulomb interactions which correspond to the valence - conduction band transitions conserving the total charge but not the number of electrons and holes.@xcite in the exciton picture , this represents the transitions between the exciton bands of different multiplicity . hence , we will refer to these transitions below as the _ interband exciton transitions _ , and to the corresponding coulomb interactions as the _ interband coulomb interactions_. within adopted nomenclature , the _ intraband coulomb interactions _ ( i.e. , restricted to an exciton band of a certain multiplicity ) conserve the total number of electrons and holes and determine the multi - exciton interaction ( e.g. , binding ) energy . the simplest theoretical model initially developed to explain cm in bulk materials is the impact ionization ( ii ) model.@xcite it treats cm dynamics following high energy photoexcitation as the lowest order interband transitions whose rate is given by the fermi s golden rule . this rate depends on the interband coulomb matrix elements and the final multiple - carriers ( e.g. , biexciton ) density of states ( dos).@xcite the model has been further applied to interpret the cm dynamics in ncs along with various methods for the electronic structure calculations . franceschetti , an and zunger performed atomistic pseudopotential calculations of the ii and auger recombination rates in a pbse nc . assuming the absence of the quasi - momentum constraint and rapid growth of the biexciton dos , they estimated low value ( @xmath1 ) of aet defining this quantity as the energy above which the ii rate exceeds the auger recombination one.@xcite rabani and baer performed screened semiempirical pseudopotential calculations of the ii rates in cdse and inas and found significant reduction of the qe with the ncs size increase . observed trend makes cm already inefficient in ncs of diameter @xmath2 nm . the observation has been rationalized by strong size dependence of the interband coulomb interactions and the trion dos behavior.@xcite using the semiempirical tight binding model allan and delerue performed extensive calculations for pbse and pbse , inas and si ncs.@xcite they found that although the coulomb interactions are enhanced by the quantum confinement , the quantum - confinement - induced reduction in the biexciton dos facilitates the _ reduction _ of the ii rate and subsequently of qe . the comparison of the ii rates calculated in pbse bulk and pbse ncs shows that the ii rates in the ncs does not exceed the bulk one.@xcite based on these calculations , the authors of ref . [ ] argue that the experimentally observed drop of qe in ncs can be attributed to the dominant effect of the reduced carrier s dos . lack of the phonon bottleneck , leads to the rapid intraband phonon - assisted relaxation which further reduces the qe.@xcite schaller , agranovich , and klimov , first pointed out that in addition to ii , the direct photogeneration of biexcitons can take place in ncs through virtual - exciton channel".@xcite subsequently , rupasov and klimov suggested that additional contribution to photogeneration qe can rise from the vitual biexciton channel".@xcite using the effective mass model for the electronic structure of pbs and pbse ncs due to kang and wise,@xcite the authors estimated the photogeneration qe and argued that this process dominates the cm dynamics in ncs . recently , silvestri and agranovich using the same model , performed detailed calculations for relatively small radii ( specifically for 1.95 nm and 3 nm ) pbse ncs.@xcite they concluded that the contribution of the photogeneration processes in ncs is much weaker than it was claimed before , and clarified that the overestimated qe results from disregarding the effect of selection rules for the interband coulomb matrix elements and the oscillator strength factors weighting optically allowed transitions . here , we demonstrate that the absence of the interference between the photogeneration pathways and the presence of small size dispersion further reduces the photogeneration qe . another model describing the _ coherent _ photogeneration of biexcitons from the resonant exciton states initially proposed by shabaev , efros , and nozik considered idealized five level system,@xcite and was subsequently refined to account for the biexciton dos effect.@xcite using the effective mass @xmath3 hamiltonian , the authors of ref . [ ] performed calculations for small radii ( i.e. , 2 nm and 3 nm ) pbse ncs , and found that dense biexciton dos leads to the vanishing coherent oscillations . they also pointed out that efficient cm can be expected in considered small ncs . the drawback of the coherent superposition model is the lack of the pure - dephasing effects and the inhomogeneous broadening which as we demonstrate here play important role in the cm dynamics reducing the qe . finally , the _ ab initio _ calculations on small _ clusters _ ( @xmath4 nm ) demonstrated the role of strong coulomb correlations and fast exciton - biexciton dephasing in multiexciton photogeneration , @xcite and suggested contributions of the phonon - assisted auger processes to cm.@xcite the atomistic calculations mostly focused on ii dynamics occurring during the population relaxation are limited to the small diameter ( @xmath5 nm ) ncs . however , reported experimental studies consider larger ncs with the diameter varying up to 10 nm . to close this gap and to perform comparison with bulk , an extrapolation procedure combined with atomistic calculations has been used.@xcite the photogeneration dynamics has been investigated using effective mass models for larger but still small diameter ( @xmath6 nm ) ncs . recently , we have published a letter summarizing the results of our numerical investigation on the direct photogeneration and population relaxation processes contributing to cm in a broad diameter range pbse ncs and in pbse bulk.@xcite the calculations employ our earlier proposed interband exciton scattering model ( iesm)@xcite which recovers the models discussed above as limiting cases and which we further parameterized using the effective mass , @xmath3 , electronic structure model proposed by kang and wise.@xcite@xmath7 in the letter , we argue that in both cases of the photogeneration and population relaxation the ii is the main mechanisms of cm . this explains weak contribution of the direct photogeneration to the total quantum qe . analyzing the scaling of the total qe with the nc size , we found that qe in ncs plotted on absolute energy scale does not exceed that in the bulk . this is in agreement with reported experimental data and some theoretical studies , and confirms that the quantum - confinement - induced reduction in the biexciton dos makes a dominant contribution to qe . present paper discusses in great details the theoretical method and provides extended analysis of the numerical results summarized in ref . in addition , we consider the effect of the optical pump pulse duration on qe , and provide a comparative analysis of cm in pbs and pbse materials . the paper is organized as follows . in sec . [ model ] , we review the weak coulomb interaction limit of the iesm used for numerical calculations , introduce convenient quasicontinuous representation , and further discuss the details of the model parameterization . in sec . [ numerics ] an extensive analysis of our numerical results on pbse ncs is given . comparison of the key cm features in pbse and pbs is performed in sec . [ disc ] concludes the paper by a discussion on the limitations of the adopted model , connection with experiment , and possible improvements of qe in ncs . the central quantity describing cm efficiency is qe which can be calculated as @xmath8 here @xmath9 and @xmath10 are the total exciton and biexciton populations , respectively , produced by single absorbed photon in the limit of vanishing pump fluencies . they depend on the delay time measured from the center of the pump pulse and allow one to determine both the qe due to the photogeneration event and the total qe after the population relaxation . a complimentary quantity , often used to describe cm yield , is the biexciton quantum yield , @xmath11 , defined as a number of biexcitons produced per single absorbed photon . in this section , we describe the weak coulomb limit of the iesm@xcite which we use to calculate the time evolution of the exciton and biexciton populations entering eq . ( [ qe ] ) . we also introduce the quasicontinuous frequency representation which is convenient to analyze the quantum confinement signatures and the size scaling of the quantities determining the qe in transition from nc to the bulk limit . finally , we discuss the model parameterization for the numerical calculations of qe in pbse and pbs materials . in the weak coulomb interaction limit , the leading contribution to the population of the @xmath12-th exciton state generated by the pump pulse is given by @xmath13 where @xmath14 is the transition dipole moment between the ground state , @xmath15 , ( i.e. , filled valence band ) and an exciton state @xmath16 , and @xmath17 { \cal e}_{pm}(t^ { ' } ) { \cal e}_{pm}(t^{'}-t_1 ) , \end{aligned}\ ] ] is the pulse self - convolution function depending on the ground state to exciton state transition frequency , @xmath18 , and dephasing rate , @xmath19 . the pump pulse is characterized by the envelope amplitude , @xmath20 , giving the temporal profile , and the central frequency , @xmath21 . in our calculations , we use the gaussian form of the pulse envelope function , @xmath22 , with the mean amplitude , @xmath23 , and pulse duration , @xmath24 . for the pulses longer than typical dephasing time , @xmath25 , the continuous wave ( cw ) limit is recovered and the pulse self - convolution function becomes proportional to the lorentzian line shape function . to be consistent with the calculations of the photogenerated _ biexciton _ population appearing in the second order interband coulomb coupling , we account for the first and second order corrections to the _ exciton _ population and disregard the effect of the exciton coherences both introduced in ref . [ ] . the negligible contribution of the latter quantities has been validated by our direct numerical calculations . the photogenerated population of the @xmath26-th biexciton state is@xcite @xmath27 where @xmath28 is the intraband transition dipole between the biexciton states @xmath29 and @xmath30 , and @xmath31},\end{aligned}\ ] ] describes the transition amplitude associated with the single ( born ) scattering event between exciton sate , @xmath16 , and biexciton state , @xmath29 . the quantity @xmath32 , also entering eq . ( [ nxx ] ) , is the transition amplitude between the ground state , @xmath15 ( @xmath33 ) , and biexciton state , @xmath30 . @xmath34 is the interband coulomb matrix element in the exciton representation giving rise to the interband scattering , and @xmath35 is the interband dephasing rate . the pulse self - convolution function , @xmath36 , entering eq . ( [ nxx ] ) contains @xmath37 and @xmath38 instead of @xmath39 and @xmath40 , respectively . according to our calculations the coherence contributions to the biexciton populations@xcite are also negligible . our numerical calculations show that the rapid sign - variation of the interfering terms in eq . ( [ nxx ] ) leads to their cancelation allowing us to write eq . ( [ nxx ] ) as @xmath41 the resulting three _ non - interfering _ photogeneration pathways are illustrated in fig . [ fig - spath ] . the first and the second pathways , given by the first and the second terms in eq . ( [ nxxi ] ) , are shown in panels ( a ) and ( b ) , respectively . these terms describe redistribution of the exciton oscillator strengths ( @xmath42 ) between exciton and biexciton bands mediated by the interband born scattering ( @xmath43 ) . first pathway ( panel ( a ) ) , involves the ground - state - to - exciton resonant optical transition and further scattering to final biexciton states distributed around @xmath44 according to the non - zero components of @xmath43 . we will refer to this process as the _ indirect biexciton photogeneration _ throughout this paper.@xcite panel ( b ) illustrates the second pathway , where the exciton is virtual and final biexciton state is in resonance with the optical pulse . thus , this pathway will be refereed to as the _ direct biexciton photogeneration via virtual exciton states_.@xcite the third pathway ( panel ( c ) ) consists of born scattering between the ground state and biexciton states ( @xmath45 ) stabilized by the intraband dipole transition , @xmath46 . accordingly , the final biexciton state is in resonance with the optical pulse . this process , refereed to as the _ direct biexciton photogeneration via biexciton states_,@xcite becomes prohibited in the bulk limit by the momentum conservation constraint for optical valence - conduction band transitions . equation ( [ nx ] ) with the higher order corrections and eqs . ( [ nxx])([nxxi ] ) fully describe the exciton and biexciton populations prepared by the pump pulse . we use them for the numerical evaluation of the photogenerated exciton , @xmath47 , and biexciton , @xmath48 , populations and qe ( eq . ( [ qe ] ) ) . they also provide the initial conditions to model the population relaxation dynamics using a set of rate equations.@xcite using this computational approach , our goal is to clarify the effect of quantum confinement on qe in transition from ncs to the bulk limit . since the cm dynamics in ncs takes place in the energy region characterized by high electron and hole dos , we expect that the cm dynamics in ncs and in the bulk should have common features . thus , we intend to see how strongly these features are affected by the presence of the confinement potential . quantitatively , we are going to look at the interplay between the size scalings of the interband coulomb interaction and exciton / biexciton dos determining the qe variation in transition from nc to the bulk limit . first , we define the bulk limit as the thermodynamic limit : @xmath49 , @xmath50 , and @xmath51 , where @xmath52 is a crystal volume , @xmath53 is the unit cell volume , and the ratio @xmath54 gives the number of unit cells in the crystal . then eqs . ( [ nx ] ) and ( [ nxxi ] ) , and the quantities determining the population relaxation should be represented in such a form that the effect of the interband coulomb interactions and exciton / biexciton dos are clearly distinguished . this can be achieved by using the quasicontinuous energy representation . since some quantities of interest in the bulk limit have volume scaling , it is convenient to introduce associated intensive ( i.e. , volume - independent in the bulk limit ) variables . their deviation from the well defined bulk values will provide us with the convenient measure of the quantum confinement effects . we start with the exciton and biexciton populations which in the quasicontinuous energy representation read@xmath55 respectively . both of them have linear scaling with the volume in the bulk limit . therefore , we eliminate the latter scaling by multiplying the total populations with the dimensionless prefactors ( @xmath56 ) and end up with the following intensive quantities @xmath57 although the prefactor @xmath56 cancels out in the expression for qe ( eq . ( [ qe ] ) ) , indicating that the latter quantity is indeed intensive , it is convenient to keep it in eqs . ( [ nx ] ) and ( [ nxx ] ) for consistency . according to appendix [ appx_vscl ] , we define the intensive exciton and biexciton dos as @xmath58 respectively . the associated _ optically allowed _ exciton and joint biexciton dos read @xmath59 respectively . they carry information on the optical selection rules reducing the number of states participating in the photogeneration process . our central quantity is the effective coulomb term defined as the r.m.s . of the interband coulomb matrix elements which couple states within the frequency intervals @xmath60 $ ] and @xmath61$],@xmath62^{1/2}.\end{aligned}\ ] ] related effective coulomb term , coupling the ground state and biexciton states , can also be defined as @xmath63^{1/2}.\end{aligned}\ ] ] as we discuss in appendix [ appx_vscl ] , the volume prefactor @xmath64 [ @xmath65 in eq . ( [ veff ] ) [ eq . ( [ veff0 ] ) ] corresponds to a finite effective coulomb value in the bulk limit . therefore , the size scaling of such defined interaction with the nc diameter , @xmath66 , provides quantitative measure of the quantum confinement effects . in general , a deviation from the bulk value for the effective coulomb reflects the net result of the scaling of the coulomb matrix elements with @xmath66 , relaxation of the momentum conservation constraints , and the appearance of new selection rules associated with symmetry of the confinement potential . assuming the cw excitation , the exciton population produced by the pump pulse is simply proportional to the corresponding optically allowed dos ( eq . ( [ xodos ] ) ) , @xmath67 where @xmath68 . the biexciton population as a function of the pump frequency in the adopted representation reads @xmath69 ^ 2 \frac{\tilde\rho_{x}(\omega_{pm})\rho_{xx}(\omega^{'})}{(\omega^{'}-\omega_{pm})^2+\gamma^2 } \\\nonumber&+ & \frac{{\cal a}}{\hbar^2}\int d\omega^{'}[v^{x , xx}_{eff}(\omega^{'},\omega_{pm})]^2 \frac{\tilde\rho_{x}(\omega^{'})\rho_{xx}(\omega_{pm})}{(\omega^{'}-\omega_{pm})^2+\gamma^2 } \\\nonumber&+ & \frac{{\cal a}}{\hbar^2}\int d\omega^{'}[v^{xx}_{eff}(\omega^{'})]^2 \frac{\tilde\rho_{xx}(\omega^{'}\omega_{pm})}{{\omega^{'}}^2}.\end{aligned}\ ] ] as desired , eq . ( [ nxxe ] ) is volume independent and clearly distinguish the contributions from the effective coulomb interactions and variously defined dos . accordingly , eqs . ( [ nxe ] ) and ( [ nxxe ] ) provide central expressions for the interpretation of the numerical simulation results on the photogeneration qe as discussed in the subsequent section . the population relaxation dynamics is described by the following set of kinetic equations @xmath70 these equations contain both the ii and auger recombination rates @xmath71 ^ 2\rho_{xx}(\omega ) , \\\label{kar } k_{ar}(\omega)&=&\frac{2\pi}{\hbar^2}\left(\frac{v}{v}\right)^2[v^{x , xx}_{eff}(\omega)]^2\rho_{x}(\omega),\end{aligned}\ ] ] respectively . here the shorthand notation @xmath72 stands for the diagonal component of the effective coulomb term , i.e. , @xmath73 . according to eq . ( [ kii ] ) , the ii rate is an intensive variable . in contrast , the auger recombination rate ( eq . ( [ kar ] ) ) vanishes in the the bulk limit as @xmath74 . in the case where the ii processes dominate the cm dynamics , the ratio of the ii to auger recombination rates@xmath75 should determine qe . since this ratio is proportional to the ratio of the corresponding dos , it has been proposed as a selection criterion for the materials showing efficient cm.@xcite however , eq . ( [ kr ] ) clearly shows that besides the material - specific signatures given by the ratio of the _ intensive _ dos , it also contains the volume scaling factor . therefore , we argue that eq . ( [ kr ] ) can only be used for a material selection criterion , after the volume prefactor is eliminated . the intraband relaxation rates , @xmath76 , @xmath77 , in eqs . ( [ nxr ] ) and ( [ nxxr ] ) describe the phonon - assisted cooling . in the absence of the phonon bottleneck and in the region of high exciton dos , it is expected that single - phonon processes dominate the exciton and biexciton intraband relaxation . accordingly , we calculate these quantities by using the following expression@xcite @xmath78 - 1},\ ] ] where @xmath79 . @xmath80 is thermal energy , and @xmath81 is the phonon spectral density approximated by the ohmic form with exponential cutoff @xmath82 here , the adjustable parameters are electron - phonon coupling , @xmath83 , with the constraint @xmath84 and the phonon frequency cutoff , @xmath85 . ( [ nxr])-([jph ] ) represent the quasicontinuous representation of the discrete kinetic equations introduced in ref . [ ] . we found that the former representation is more suitable for numerical integration . the solution of these equations with the initial conditions determined by the photogenerated populations ( eqs . ( [ nx])([nxxi ] ) ) provide closed computational scheme that we use to determine the total qe . next , we discuss the parameterization of the introduced model and some details on the adopted numerical techniques . an accurate knowledge of single electron and hole wave functions is required to construct the exciton and biexciton states and to evaluate transition dipoles and the interband coulomb matrix elements . for this purpose , we adopt the effective mass @xmath3 formalism , originally developed by mitchell and wallis@xcite and dimmock@xcite for the bulk lead chalocogenide semiconductors , and further advanced by kang and wise@xcite for the spherically symmetric ncs . the formalism is based on a four - band envelope function model explicitly taking into account spin - orbit interaction between valence and conduction bands . in considered pbse and pbs materials , the band structure anisotropy is small , and therefore , neglected in our calculations . an electron and hole wave function obtained within this formalism reads @xmath86 here @xmath87 is @xmath88-th component of the bulk bloch wave function associated with the band - edge states in @xmath89-valley whose index , @xmath90 , denotes the bands.@xcite the envelope eigenfunctions @xmath91 and the corresponding eigenenergies , @xmath92 , are found by solving the @xmath3 hamiltonian eigenvalue problem with the infinite wall boundary condition , @xmath93 , at the surface of a spherical nc of radius @xmath94.@xcite if @xmath95 ( @xmath96 ) we identify the state as a conduction band electron ( valance band hole ) state . the eigenstate index represents a set of quantum numbers , @xmath97 , such as primary quantum number describing number of the wave function nods , parity , total angular momentum , and its projection , respectively . in what follows , we will refer to the single particle states , described by eq . ( [ kw_spinor ] ) , as the kang - wise ( kw ) states . in the bulk limit , we set the envelope functions , @xmath91 , to the plane waves . a natural way to introduce the exciton and biexciton states is to use the second quantization representation within the basis of the kw - states ( eq . ( [ kw - sq ] ) ) . taking into account that the electron - hole coulomb interactions are weak compared to their kinetic energies ( i.e. , strong confinement regime ) , we introduce exciton and bi - exciton states as the following configurations of uncorrelated electron - hole pairs latexmath:[\ ] ] eqs . ( [ vk1 ] ) and ( [ vk2 ] ) clearly demonstrate a general property that the coulomb matrix elements used in the numerical calculations of the bulk limit scale inversely proportional to the volume.@xcite the nonparabolicity in the employed @xmath236-hamiltonian is crucial for the accurate evaluation of the above coulomb matrix elements . specifically , summations over the spinor components in eq . ( [ cint ] ) implies that the interband coulomb scattering amplitudes vanish exactly if the non - diagonal terms of the @xmath236-hamiltonian are set to zero , i.e. , in strictly parabolic case.@xcite in the bulk , where quasimomentum @xmath26 is a `` good '' quantum number , diagonal and off - diagonal matrix elements of the @xmath236-hamiltonian scale as @xmath237 and @xmath26 , respectively . at high energies , where the diagonal elements dominate over the off - diagonal ones , the latter can be treated perturbatively giving @xmath238 as the contribution of the hole ( electron ) states to a high energy electron ( hole ) wave function .

we report on systematic numerical study of carrier multiplication ( cm ) processes in spherically symmetric nanocrystal ( nc ) and bulk forms of pbse and pbs representing the test bed for understanding basic aspects of cm dynamics . the adopted numerical method integrates our previously developed interband exciton scattering model ( iesm ) and the effective mass based electronic structure model for the lead chalocogenide semiconductors . the analysis of the iesm predicted cm pathways shows complete lack of the pathway interference during the biexciton photogeneration . this allows us to interpret the biexciton photogeneration as a single impact ionization ( ii ) event and to explain major contribution of the multiple ii events during the phonon - induced population relaxation into the total quantum efficiency ( qe ) . we investigate the role of quantum confinement on qe , and find that the reduction in the biexciton density of states ( dos ) overruns weak enhancement of the coulomb interactions leading to lower qe values in ncs as compared to the bulk on the absolute photon energy scale . however , represented on the photon energy scale normalized by corresponding band gap energies the trend in qe is opposite demonstrating the advantage of ncs for photovoltaic applications . comparison to experiment allows us to interpret the observed features and to validate the applicability range of our model . modeling of qe as a function of pulse duration shows weak dependence for the gaussian pulses . finally , comparison of the key quantities determining qe in pbse and pbs demonstrates the enhancement of ii rate in the latter materials . however , the fast phonon - induced carrier relaxation processes in pbs can lead to the experimentally observed reduction in qe in ncs as compared to pbse ncs .

extraordinary progress has been made in the last half - dozen years @xcite towards the goal of cooling a small mechanical resonator down to its quantum ground state and hence to realise quantum behavior in a macroscopic system . implementations include cavity cooling of micromirrors on cantilevers @xcite ; dielectric membranes in fabry - perot cavities @xcite ; radial and whispering gallery modes of optical microcavities @xcite and nano - electromechanical systems @xcite . indeed the realizations span 12 orders of magnitude in size @xcite , up to and including the ligo gravity wave experiments . in 2011 two separate experiments @xcite achieved sideband cooling of micromechanical and nanomechanical oscillators to the quantum ground state . in ref . @xcite , spectral signatures ( in the form of asymmetric displacement noise spectra ) of quantum ground state cooling were further investigated . corresponding advances in the theory of optomechanical cooling have also been made @xcite . over the last year or so , a promising new paradigm has been attracting much interest : several groups @xcite have now investigated schemes for optomechanical cooling of levitated dielectric particles , including nanospheres , microspheres and even viruses . the important advantage is the elimination of the mechanical support , a dominant source of environmental noise which can heat and decohere the system . in general , these proposals involve two fields , one for trapping and one for cooling . this may involve an optical cavity mode plus a separate trap , or two optical cavity modes : the so - called `` self - trapping '' scenario . mechanical oscillators in the self - trapping regime differ from other optomechanically - cooled devices in a second fundamental respect ( in addition to the absence of mechanical support ) : the mechanical frequency , @xmath0 , associated with centre - of - mass oscillations is not an intrinsic feature of the resonator but is determined by the optical field . in particular , it is a function of one or both of the detuning frequencies , @xmath4 and @xmath5 , of the optical modes . in previous work @xcite , we analysed cooling in the self - trapped regime and found that the optimal condition for cooling occurs where both fields competitively cool and trap the nanosphere . this happens when @xmath0 is resonantly red detuned from both the detuning frequencies i.e. @xmath6 so the relevant resonant frequencies are mutually interdependent . most significantly , the effective light - matter coupling strength @xmath1 also depends on the detunings . the effective coupling strength , @xmath7 ( the optomechanical coupling rescaled by the square root of photon number ) determines whether one can attain strong coupling regimes in levitated systems such as recently observed in a non - levitated set - up @xcite . it determines too whether one may access other interesting dynamics , both in the semiclassical and quantum regimes . we consider in particular the possibility of simultaneous hybridization of the two optical modes with the mechanical mode ; here , we consider also the implications of a static bistability , which , unusually , occurs also in the limit of quite weak driving in the levitated self - trapped system . in the present work , we investigate theoretically and experimentally the strength of the optomechanical coupling . in particular , we present experimental measurements of the mechanical frequency of a nanosphere trapped in an optical standing wave in order to investigate the optical coupling as a function of the size of the nanosphere . in section 1 we review the theory of the cavity cooling and dynamics of a self - trapped system , and in section 2 we employ the experimentally measured size dependence of the coupling to determine the range of optomechanical coupling strengths accessible in a cavity . the data suggests that the most effective means to attain stronger coupling will be to employ larger nanospheres of typical radii @xmath8 nm . our work suggests that increasing photon number by stronger driving ( and by implication increasing rescaled coupling strengths ) will not prove an effective alternative , since in the present system we show @xmath9 rather than @xmath10 , so the rescaled coupling increases very slowly with laser input power . in section 3 , we in investigate the cooling and dynamics . in sec . 3.1 , we review the corresponding cooling rate expressions obtained from quantum perturbation theory ( or linear response theory ) . in sec . 3.2 we report a study of the corresponding semiclassical langevin equations and compare them with fully quantum noise spectra ; we compare also quantum , semiclassical and perturbation theory results for levitated nanospheres . in section 4 we investigate novel regimes of triple mode hybridization , coincident with static bistabilities , which the present study shows are experimentally accessible given the large optomechanical coupling strengths associated with @xmath11 nm nanospheres . in section 5 we describe the experimental study which provides data from which the size - dependence of the coupling may be inferred . in section 6 , we conclude . we approximate the equivalent cavity model by a one - dimensional system , with centre - of - mass motion confined to the axial dimension . in this simplified study , we consider only the axial dynamics : for the cavity system , we will have a much smaller tranverse frequency relative to the axial frequency , i.e. @xmath12 and there is little mixing between these transverse and axial degrees of freedom . we consider the dynamics of the following hamiltonian : @xmath13 two optical field modes of a high finesse cavity @xmath14 are coupled to a nanosphere with centre - of - mass position @xmath15 . the parameter @xmath16 ( dependent on the nanosphere polarizability ) , determines the depth of the optical standing - wave potentials . we investigate the case where both modes competitively cool and trap the nanosphere , in contrast to previous schemes @xcite where one optical field is exclusively responsible for trapping , while the other is exclusively responsible for cooling . @xmath17 is given in the rotating frame of the laser which drives the modes with amplitudes @xmath18 and @xmath19 respectively , where @xmath20 represents the ratio of driving amplitudes for the two modes . we restrict ourselves to the regime @xmath21 , since we consider the most general case where both optical modes contribute to the trapping as well as the cooling . thus we can define mode 1 simply as the mode which is more strongly driven and mode 2 as the mode which is ( except where @xmath22 ) more weakly driven . the detunings @xmath23 for @xmath24 are between the input lasers and the corresponding cavity mode of interest , and @xmath25 represents the phases of the optical potentials . the two fields could represent two modes generated by the same laser field , or they could be generated by two independent lasers . nonetheless , since the particle motion is confined to within one wavelegth , one can make the approximation @xmath26 . previous studies generally consider @xmath27 to be convenient , since then the anti - node of one field coincides with a purely linear potential of the other optical field , but we may also consider general values of @xmath28 . one can write corresponding equations of motion : @xmath29 where @xmath24 for the two optical - mode realisation . additional damping terms have also been added : @xmath30 accounts for photon losses due to mirror imperfections and the @xmath31 term for mechanical damping . the above should also include quantum noise terms arising from ( say ) shot noise or gas collisions : for brevity , the quantum noise terms are left out until sec.3 . we consider here the linearised dynamics ; we replace operators by their expectation values and perform the shifts about equilibrium values such as @xmath32 , and @xmath33 . the values for the equilibrium photon fields ( e.g. for the two - mode case ) are @xmath34^{-1 } \ $ ] and @xmath35^{-1}$ ] . the equilibrium position is then given by the relation @xmath36 , by numerical solution of the equation , @xmath37 where @xmath38 . as usual we consider the dynamics of the fluctuations via the linearised equations . to first order , the linearised equations of motion , in the shifted frame , are : @xmath39 the resulting effective mechanical harmonic oscillator frequency is : @xmath40 we can restrict ourselves to real equilibrium fields . we take @xmath41 then transform @xmath42 . thus @xmath43 . we also rescale the mechanical oscillator coordinates @xmath44 and @xmath45 , where @xmath46 is the zero - point fluctuation length scale . hence , @xmath47 . + below we drop the tilde so the equilibrium field values @xmath48 are real . using field operators @xmath49 , the linearised dynamics for a two - optical mode system would correspond to an effective hamiltonian : @xmath50 . it is assumed that cavity parameters would correspond to @xmath51hz at @xmath52 nm , for photon numbers @xmath53 . for comparison , the value of @xmath16 is also shown , as are the experimental and simulated frequencies @xmath54 . the @xmath0 are scaled by a factor of 10 for clarity and in fact , values of @xmath55mhz are quite realistic in optical cavities.,height=316 ] the hamiltonian in eq . [ linham ] appears analogous in form to standard , well - studied optomechanical hamiltonians , albeit with two optical modes rather than one . however , it differs in one important respect : in this case , both the mechanical frequency @xmath56 and the optomechanical coupling strengths @xmath57 are not fixed and depend on the detunings . the fact that the frequencies of the three modes ( two optical , one mechanical ) are interdependent makes the dynamics different from other optomechanical set - ups , where the equilibrium mechanical frequency ( i.e. excluding shifts arising from the fluctuations ) is intrinsic to the mechanical oscillator . there is considerable interest in achieving strong - coupling , which leads to regimes of light - matter hybridization . the corresponding mode splitting has been observed experimentally @xcite . in typical set - ups , these regimes are reached if the rescaled effective optomechanical coupling exceeds the damping rates i.e. @xmath58 , where @xmath59 is the cavity photon number . even if the unnormalized coupling is weak , strong - coupling regimes may be achieved by increasing the driving power and thus increasing intracavity photon numbers . in the present levitated system , a particularly interesting regime would involve triple - mode hybridization enabling , for example , the coupling of the two modes of light via the mechanical mode . however , here , mode hybridization ( for which @xmath60 ) depends non - trivially on the detunings . we argue that large coupling can not be easily achieved by increasing the driving power , since @xmath61 and thus increases slowly with the driving strength . we can show , by a simple argument , that increasing the nanosphere size provides the most effective means to attain strong coupling . for the self - trapped system , optomechanical coupling strengths are @xmath62 and depend on the detunings via @xmath63 . note also that @xmath64 here too depends on the detunings via @xmath0 . for triple mode hybridization , @xmath65 . for convenience , we also take @xmath66 . then , since @xmath67 , we can re - write eq . [ freqs ] : @xmath68 we consider near symmetric driving of the two optical modes for which @xmath69 and thus @xmath70 so @xmath71 . hence , @xmath72 since the optomechanical coupling increases only very slowly with cavity photon number the most effective means to reach strong coupling regimes is to increase the nanosphere size to the maximum practical value ( @xmath73 nm ) . for the ideal case where the nanosphere radius @xmath74 is small @xcite ie . @xmath75 , then @xmath76 where the small nanosphere coupling takes the form : @xmath77 where @xmath78 is the sphere volume ( and hence @xmath79 where the density @xmath80kgm@xmath81 for silica ) . in turn , @xmath82 is the cavity volume , where @xmath83 m is the cavity waist and @xmath84 cm is the cavity length . for larger nanospheres , the measured size - dependent corrections must be applied . in the experiments described below , we find that the mechanical oscillation frequency is modulated by a finite size correction @xmath85 ( see fig . [ exp_trap_freq ] and description of the measurement of @xmath86 in section 5 below ) . thus , since : @xmath87 then @xmath88 and the coupling is in turn modulated by the finite size correction . the experimental results suggest that for @xmath89 nm , then @xmath90 and @xmath91 . for example , for @xmath52 nm , @xmath92 cm and @xmath93 m , then @xmath94hz . for reasonable values of cavity decay constants @xmath95hz , then for @xmath96 , @xmath97 for @xmath98 nm , @xmath99 . thus a 200 nm sphere provides an optomechanical coupling about an order of magnitude larger than a 40 - 50 nm sphere . to achieve a comparable increase in coupling by photon number enhancement would require increasing the driving power by a factor of order @xmath100 . a more careful analysis , including the effects of the finite - size correction function @xmath101 is shown in fig . [ beadg ] . we see that @xmath102 ( and for @xmath103 , also @xmath104 ) reaches a maximum value for @xmath105 nm before falling to zero . other maxima for larger @xmath74 do not provide a larger value of @xmath106 . furthermore , they have the disadvantage that they may enhance photon recoil heating effects . for comparison , the value of @xmath16 is also shown , overlaid on the experimental and simulated frequencies . a previous study @xcite , using rescaled coordinates , investigated the full parameter space of two optical mode cooling . here we investigate more carefully the effect of non - zero mechanical damping . using linear response theory , we can extract cooling rates from the equations eq . [ a2s ] : @xmath107 , \label{gamma}\ ] ] where @xmath108 ^ 2 + \frac{\kappa^2}{4 } } , \label{gamma12}\ ] ] for @xmath24 . net cooling occurs for @xmath109 . although the above is quite similar in form to standard optomechanical expressions , as explained previously , rather different behaviour is observed since here @xmath0 and @xmath110 are both dependent on the @xmath111 . from quantum perturbation theory we can show that @xmath112 , the rate of transition from state @xmath113 to @xmath114 is : @xmath115 while @xmath116 for @xmath117 , then @xmath118 gives the cooling rate of eq.[gamma ] . however , with the exact expressions we can show that the equilibrium mean phonon number is @xmath119 for parameters @xmath120 and @xmath121 . blue corresponds to cooling , yellow / white to heating . the white lines indicate the locus of the single field resonances ( where @xmath122 or where @xmath123 ) . the detunings are given in units of @xmath16 and are dimensionless . for @xmath22 it is clear that there is a deep , maximum cooling region at a double resonance where the two white lines intersect and both optical fields cool simultaneously . it is also evident that there is a strong cooling resonance for @xmath124 . for @xmath121 , three cooling resonances @xmath125 , @xmath126 and @xmath127 merge to give a very broad strong - cooling _ , quite insensitive to detuning @xmath5 over a range of over 1mhz . here @xmath128mhz and the input power into mode 1 corresponds to 2mw.,height=316 ] in fig . [ cool ] we show colour maps comparing the cooling and minimum phonon numbers for both @xmath121 and @xmath22 respectively . the cooling behaviour was investigated previously in @xcite . in this case , for each fixed detuning @xmath4 there are up to three cooling resonances ( at three different values of @xmath5 ) , where strong damping is observed ( and similarly for each fixed @xmath5 ) . this is in contrast to single optical mode schemes where there is a single cooling resonance for which @xmath129 or @xmath130 . for the @xmath121 map the three cooling resonances merge , giving a single extended cooling region of about 1mhz width . for @xmath22 the map has a high degree of symmetry , since the role of the two optical modes is interchangeable . the figures show that the largest cooling rates are found in the double resonance region , making it the most favourable region to work in . the equilibrium phonon number in eq . [ neq ] concerns only the idealised situation where there is a very good vacuum , negligible photon recoil heating and thus no mechanical damping or heating effects . for small @xmath98 nm spheres , we assume recoil heating is negligible @xcite and the dominant source of mechanical damping is background gas collisions , which provide an effective mechanical damping @xmath131 @xcite where @xmath132 is ratio of the gas particle s mass to that of the sphere , @xmath133 is the gas number density and @xmath134 is the mean gas velocity for a room temperature thermal distribution . it can be shown that the perturbation theory argument above can be adapted to obtain equilibrium phonon numbers for a given cooling rate @xmath135 : @xmath136 } { \gamma_m + |\gamma_{opt}| } , \label{pt}\end{aligned}\ ] ] where @xmath137k . alternatively , the final equilibrium temperatures @xmath138 , where @xmath139 is the equilibrium oscillator temperature which would have been obtained in a perfect vacuum . although we investigate only a two optical mode system , generalization to more optical modes is straightforward . we consider a set of equations of motion , for @xmath140 : @xmath141 where the optomechanical strengths @xmath142 depend on the detunings ( as does the mechanical frequency @xmath0 ) . in the two mode case we consider , we take @xmath143 and @xmath144 . the optical modes are subject to photon shot noise , while the mechanical modes are subject to brownian noise from collisions with gas molecules in the cavity . for the photon shot noise , we assume independent lasers and uncorrelated zero temperature noise for which @xmath145 , while @xmath146 . for the gas collisions , we take @xmath147 and @xmath148 where the number of surrounding bath phonons @xmath149 . the above equations can be integrated in frequency space to obtain analytical expressions for the displacement noise spectra for the arbitrary mode case . we can evaluate the displacement spectrum @xmath150 . we obtain : @xmath151 where the @xmath152 represent optical and mechanical susceptibilities : @xmath153^{-1 } ; \ \chi_m(\omega ) = \left[-i(\omega-\omega_m)+\frac{\gamma_m}{2}\right]^{-1}.\end{aligned}\ ] ] with @xmath154 and @xmath155 ; then also @xmath156 . mw , @xmath157hz , @xmath158mhz , @xmath159mhz , @xmath121 . some hybridization between the mechanical mode and optical mode 1 is seen in the characteristic double - peak sideband structure.,height=288 ] ( b ) , we show equilibrium phonon numbers obtained from eqs . ( [ pt ] ) , ( [ qnoise ] ) and ( [ ssc ] ) i.e. perturbation theory , the analytical quantum noise formula and semiclassical langevin equations respectively . agreement between quantum and semiclassical results is excellent , less so for perturbation theory at low pressures.,height=288 ] mhz , even at quite high pressures ( here 1 mbar ) , mode splitting is seen in the noise spectra of the optical modes . in all the plots , @xmath160mhz is held fixed while @xmath5 is swept from 0 to -1.6mhz ( for an input power of 2mw into mode 1 , while @xmath121 ) . ( a ) shows noise spectra for both optical mode 1 and mode 2 . three way hybridization between the mechanical and both optical modes appears clearly ( highlighted in the bold blue line ) . for clarity , some of the strongest peaks have been truncated in height . in ( b ) three avoided crossings are apparent . the dominant character of each normal mode is indicated by the colour ( black is mechanical , blue is optical mode 1 , red is optical mode 2 ) . when @xmath161 is large , there is no mixing . however as @xmath162 , there is strong mixing and the dominant character of each normal mode changes from light to matter ( or vice versa ) as an avoided crossing is encountered . panel ( c ) shows the cooling and indicates strong cooling at each of the avoided crossings . , height=336 ] we compare the quantum displacement with corresponding semiclassical solutions in the steady state . the linearised two mode system eq . [ linham ] , in matrix form corresponds to a standard problem @xcite . inclusion of the noise arising from gas collisions or laser shot noise yields a set of corresponding langevin equations : @xmath163 , where @xmath164 is termed the drift matrix . its eigenvalues give the stabilities and eigenfrequencies of the system s normal modes , while the noise is determined by @xmath165 , a constant diagonal matrix . the elements of the random noise matrix are assumed to be @xmath166-correlated @xmath167 . methods for obtaining the solution for the steady state correlation functions of this system , under conditions of stability , i.e. where all the eigenvalues of @xmath164 have negative real parts , are well - known @xcite . the required noise spectra , or autocorrelation functions , in frequency space are : @xmath168 where , @xmath169 where the diagonal matrix @xmath170 has elements @xmath171 . from the above , the noise spectra of all modes may be calculated . eq.ssc yields semiclassical sideband spectra , symmetrical in @xmath172 . in fig . [ noise ] , we compare semiclassical displacement spectra calculated from eq . [ ssc ] with corresponding quantum results obtained from eq . [ qnoise ] . at high pressures ( and hence high phonon occupancy ) there is excellent agreement between semiclassical and quantum results . at low pressures ( near ground state cooling ) however , the quantum spectrum shows a characteristic asymmetry , such as was observed recently in experiments on photonic cavities @xcite . , showing mode splitting for similar parameters to fig . [ noise ] , near the quantum limit , except that here @xmath173mhz is held fixed while @xmath5 ( vertical axes ) is swept . ( a ) shows the semiclassical spectrum which is symmetric in frequency @xmath172 . ( b ) shows the quantum spectrum which is asymmetric . in both cases , triple hybridization appears clearly and is seen near @xmath174mhz . ( c ) shows the corresponding behaviour for @xmath175 , showing that the triple peak structure has disappeared . note that @xmath176 is plotted.,height=288 ] the equilibrium variance ( and hence the final phonon number ) of the mechanical oscillator is , @xmath177 . noting the rescaling @xmath178 and setting @xmath179 , we can write @xmath180 . using eqs . ( [ qm ] ) , ( [ ssc ] ) and ( [ pt ] ) , we can investigate final equilibrium phonon numbers ( and the minimum achievable for levitated self - trapped spheres ) comparing quantum , semiclassical and perturbation theory respectively . in fig . [ phonon ] , we compare the corresponding equilibrium phonon numbers , @xmath181 , @xmath182 and @xmath183 respectively for the unusual triple cooling resonance region shown in fig . [ cool ] . cooling to near the ground state @xmath184 is possible for a pressure of order @xmath185 mbar , even for modest driving powers of 2mw and values of @xmath186hz corresponding to spheres of order @xmath187 nm . we consider a relatively low input power of 0.37mw . ( a ) plots the optomechanical cooling rate ( blue indicates cooling , brown indicates heating ) . the red dashed line indicates the locus of bistability as a function of the detunings @xmath4 and @xmath5 . the discontinuity in the cooling can be discerned near the strong doubly resonant cooling region . ( b ) shows displacement noise spectra @xmath188 as a function of @xmath172 plotted along @xmath189mhz ( along black horizontal line in upper panel ) . we sweep in increasing @xmath5 . at the point where @xmath189mhz intersects the bistability , a discontinuity in the noise spectra is apparent . on one side of the discontinuity there is strong hybridization between the mechanical mode and optical mode 2 ; this changes abruptly across the discontinuity to hybridization between the mechanical mode and optical mode 1 for larger @xmath5 . this may allow for control of entanglement between the modes . the map corresponds to near - vacuum conditions so the system is near the quantum ground state in this regime ( as evidenced by the asymmetric sidebands ) . ] the multimode , or at least two mode , self - trapping regime may permit new possibilities for position sensing and for controlling entanglement between two optical modes and the mechanical resonator . here we investigate regimes where such effects are clearly apparent . the implications of the measurement for the accessible range of optomechanical coupling strengths suggests that multiple hybridization and bistability are quite accessible with reasonable cavity parameters . in fig . [ eigensplit ] we investigate the complex behaviour of the eigenmode frequencies of the self - trapped , levitated system . on the left panels ( fig . [ eigensplit ] ( a ) ) we plot the noise spectra of the two optical modes , which exibit sidebands near @xmath190 since the corresponding optical fields are modulated by the motion of the mechanical oscillator . here we fix one detuning ( @xmath191mhz ) and look at the behaviour as the other detuning is varied . the sidebands are displaced in frequency and split : one effect is simply due to the dependence of @xmath0 on @xmath192 ( unique to the levitated system ) ; it arises from the calculation of the equilibrium fields and frequencies . the other effect is due to normal mode mixing ( hybridization of light and matter modes ) arising from the linearised equations . if @xmath193 simultaneous hybridization is observed , provided @xmath194 . figure [ eigensplit ] ( b ) shows that there are several distinct avoided anti - crossings , where the dominant character of each eigenmode changes ; if two crossings coincide , the spectra show a characteristic triple - peak structure ( symmetric about @xmath195 in the semiclassical regime shown here ) . panel ( c ) shows that the corresponding cooling rate is enhanced at each avoided level crossing . in fig . [ split ] we plot displacement spectra corresponding to fig . [ noise ] , but over a range of values of @xmath5 . a log - scale is used for @xmath188 . the triple mixing which can appear when two avoided crossings nearly coincide is clearly apparent at @xmath196mhz . static bistability in a cavity of varying length has been seen experimentally @xcite . the potential for generating entanglement has recently been investigated in an optomechanical system @xcite ; however , a relatively high laser power @xmath197mw is required . for the self - trapped systems , the incoherent sum of the optical standing - wave potentials @xmath198 and @xmath199 does not by itself produce a double - well structure ; nevertheless , as we see below , in combination with optomechanical shifts , bistabilities are observed , even for weak driving . whether a double - well structure emerges , or not , is completely independent of the driving power ( where @xmath200 ) and can emerge at very low input powers , as we demonstrate below . it is easy to see that the levitated particle moves in an effective static potential @xmath201 where : @xmath202 , \nonumber\\ \label{pot}\end{aligned}\ ] ] and @xmath203 . \nonumber \\ \label{pot1}\end{aligned}\ ] ] here , note that the shifted detuning @xmath204 is dependent on @xmath15 , not the equilibrium displacement @xmath63 . this potential admits two stable equilibrium points over parameter regimes where @xmath205 ( in practice , such bistability is observed already for @xmath206 ) . it is evident that the driving power factors out , so does not affect the shape of the potential , providing only a scaling factor . we show in fig . [ bi ] that for a high @xmath207 ratio simultaneous hybridisation and bistability co - exist : we show that that for @xmath208mw , @xmath209 , @xmath210 and modest photon numbers @xmath211 we can switch discontinuously from hybridisation between the mechanical mode and optical mode 1 , to hybrization between the mechanical mode and mode 2 . we take @xmath66 . in the noise spectra , the switch is heralded by a large zero - frequency peak in the displacement spectra , which is clearly apparent in fig . we have built a simple standing wave dipole trap to develop protocols for loading a single nanosphere into the trap to confirm that nanospheres with a range of radii around 100 nm can be trapped . importantly , we have measured the variation in trap frequency with sphere radius so that realistic values of optomechanical coupling strengths , for a given nanosphere radius , can be included in our models . in this section we explain how the size - dependent modulation function @xmath86 , used in the theory section above to obtain @xmath212 and the optomechanical coupling strengths , was measured . a schematic of the standing wave trap used in our experiments is shown in fig . the standing wave trap consists of two focused beams that counter - propagate and overlap near their foci . the two laser beams , derived from the same laser at a wavelength of 1064 nm , enter the trapping region via optical fibres . the light exiting the fibres is focused using aspheric lenses ( thorlabs c140tme ) with a focal length of 1.45 mm and numerical aperture 0.55 . the power in each trapping beam after it has passed through each lens is measured to be @xmath213mw , and the best focused beam waist ( radius ) is theoretically 1.7@xmath214 m . to optimize the alignment of the trap , we maximize the light through - coupled from one fibre into the other . this is accomplished by mounting one optical fibre and its aspheric lens on an xyz flexure stage . the alignment is done inside the vacuum chamber at atmospheric pressure . a long working distance microscope ( navitar zoom 6000 system , with up to 45x zoom ) is used to image the trapped sphere . the image is split into two using a beamsplitting plate , with one image directed to a ccd camera for diagnostics and the other aligned onto a quadrant cell photodiode ( qpd ) which measures position fluctuations as a function of time in two orthogonal axes . we define the axial direction as that along which the trapping light propagates , and the transverse direction as the orthogonal axis in the focal plane of our imaging system . light at 532 nm is used to illuminate the trapped sphere , as the qpd is more sensitive at this wavelength . the green beam enters the system via one of the optical fibres , as shown in fig . [ trap ] , and a filter is used to stop 1064 nm light reaching the detectors . the power of the 532 nm beam is 10 mw . an image of a string of 100 nm diameter beads trapped in the standing wave trap . a single bead is trapped by continually blocking and unblocking one of the trapping beams until only one sphere is trapped . ] silica ( sio@xmath215 ) nanospheres , manufactured by microspheres - nanospheres and bangs laboratories , are introduced into the trapping region at atmospheric pressure via an ultrasonic nebulizer ( omron ne - u22 ) . these spheres range in radius from 26 nm to 510 nm and are suspended in methanol . the nanosphere solution is sonicated using an ultrasonic bath for at least an hour before trapping to prevent clumping . once introduced into the trapping region the methanol surrounding the spheres rapidly evaporates and the spheres are trapped over many fringes of the standing wave , as shown in fig . [ beads ] . as our imaging system does not have single fringe resolution we can not determine if more than one sphere is trapped in a single fringe by this method . however , this information can be inferred from the relative intensity of the light scattered from the trapped spheres and also by the reduced stability of the particles in the trap when more than one particle is trapped . to reduce the number of trapped particles the trapping light is briefly blocked and unblocked . this is repeated until a single sphere is visible in the trap . at this pressure , where there is a strong damping force from air the sphere can be held in the trap indefinitely . to measure the trap frequency the air is pumped from the system , and at this point no more spheres enter the trap , as without air - damping their velocity is too high . the air pressure in the trap is reduced to 5 mbar , so that clear trap frequencies can be obtained from the power spectrum of the position fluctuations of the trapped sphere , as recorded on the qpd . example power spectra are shown in fig . [ fft ] . above 5 mbar the damping of the motion in the trap due to air broadens the peak in the power spectrum so that finding an accurate trap frequency is difficult . below pressures of 5 mbar the spheres become unstable in the trap and escape . this is most likely due to radiometric forces which have been compensated for in other experiments using feedback techniques @xcite . at 5 mbar the damping rate due to gas collisions is significantly less than our lowest measured trap frequencies , and thus the measured frequency at this pressure is a good approximation to the bare trap frequency which would be measured in vacuum without damping . the angular axial trap frequency for a small polarizable particle in a standing wave is @xmath216 , where the polarizability of a sphere of refractive index @xmath113 is @xmath217 . the maximum intensity in the radial center of each equal intensity beam is given by @xmath218 , and @xmath219 is the magnitude of the wavevector of each beam . the sphere has mass @xmath220 , radius @xmath74 and density @xmath221kgm@xmath81 . the transverse trap frequency is given by @xmath222 , where @xmath223 is the focused spot size ( radius ) of the two counter - propagating beams . from these expressions the ratio of the trap frequencies is given by @xmath224 . the trap frequencies in each axis are determined by fitting measured position fluctuation power spectra using @xmath225 , where @xmath226 is boltzmann s constant and @xmath227 is the damping rate . the fit to the data is shown in fig . [ fft ] with @xmath228khz for the axial trap frequency . several sets of data were taken for different spheres of the same nominal radius and the measured axial and transverse trap frequencies for each size sphere are shown in fig . [ exp_trap_freq ] . the derived trap frequencies for each sphere radius are the average over different experiments at each radius , and the errors are the standard errors in the mean . the uncertainty in the sphere radius is taken from the information supplied by the manufacturer . two axial ( red and green data in fig . [ exp_trap_freq ] ) frequencies and one transverse frequency ( blue data points ) were measured . when a particle is tightly trapped by the optical field only one axial frequency is expected from a single sphere in a standing wave . the lower axial frequency ( in green in figure [ exp_trap_freq ] ) is always observed in the data and this is taken as the true axial frequency . the higher frequency , which is often present in the data , may be due to the trapping of two spheres in a single anti - node , with the higher frequency occurring due to optical binding , which requires further study @xcite . the higher axial frequency also changes rapidly with sphere size , indicating that it is not the true axial trap frequency , which should be almost constant for the small spheres . the presence of a single axial frequency is , we believe indicitave of having trapped a single sphere . although we do nt know the radial dimensions of the beam within the trap we can estimate this value from the ratio of the axial to transverse trap frequencies for small spheres the spot size from this ratio is @xmath229 and for @xmath230=9.8 this gives a spot size of @xmath231 m . since the trap frequency with two overlapping beams of size 2.3@xmath214 m would be equal to 207khz with a power in each beam of 150mw , and we only measure a maximum axial trap frequency of approximately 40 khz , we conclude that the particles are trapped in a standing wave formed where the waist of one beam is much larger . if one spot size is 2.3@xmath214 m the other would have to be 15@xmath214 m . a plot of the calculated axial trapping frequency , found by calculating maxwell s stress tensor ( @xcite ) , is also shown in fig . [ exp_trap_freq ] . like the experimental data the trapping frequency is constant for small spheres and decreases to approximately zero when the particle size is comparable to the size of the interference pattern produced by the standing wave . at larger radii the force on the particle changes sign and a stable trap is formed in a node of the standing wave , as shown for the particle of radius 510 nm . our measurements confirm that for particle radii less that 200 nm the simple dipole model for the nanospheres is adequate for modelling the cooling and dynamics of the nanospheres in an optical cavity utilising 1064 nm radiation . our experiments have shown that optical traps without feedback are currently limited to operation at pressures down to a few millibar for all particles that we have measured . in addition this limiting pressure did not change by reducing the intensity by 50% . this radiometric force is due to localized heating of nanosphere and the subsequent heating of the surrounding air . at low pressures , when the mean free path is comparable to the size of the nanosphere , the radiometric force competes with and eventually dominates the dipole force which traps the particle . while feedback techniques have been successful @xcite , decreasing the absorption of the nanospheres is another route to minimising radiometric effects . this is feasible since all the spheres we have used in this study are not made of optical quality glass but from colloidally grown nanospheres . finally , we have also successfully trapped silica spheres in an ion trap at pressures of @xmath185 mbar which could be used to load an optical trap formed by a cavity at lower pressures where radiometric forces are not significant . we have described a study of the dynamics and noise spectra of self - trapped levitated optomechanical systems . we have been able to show , by combining experimental measurements and theoretical calculations that strong light - matter coupling is attainable over a wide range of particle sizes , and that these can be trapped . the interdependence of the mechanical and optical mode frequencies , unique to self - trapped levitated systems provides a complex and interesting side - band structure , including multi - mode mixing and bistabilities which we aim to explore experimentally . these conclusions are supported by measurements of trap frequency made in an optical standing trap where we have demonstrated a protocol for loading a single nanosphere in a single antinode . _ acknowledgements _ : we acknowledge support for the uk s engineering and physical sciences research council . 99 f. marquardt and s girvin , physics * 2 * 40 ( 2009 ) . t. kippenberg and k. vahala , science , * 321 * 1172 ( 2008 ) . metzger and k. karrai , nature , * 432 * 1002 ( 2004 ) . o. arcizet et al , nature , * 444 * 71 ( 2006 ) . s. gigan et al , nature , * 444 * 67 ( 2006 ) . c. a. regal et al , nature phys,*4 * 555 ( 2008 ) . j. d. thompson et al , nature , * 452 * 72 ( 2008 ) ; 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the use of levitated nanospheres represents a new paradigm for the optomechanical cooling of a small mechanical oscillator , with the prospect of realising quantum oscillators with unprecedentedly high quality factors . we investigate the dynamics of this system , especially in the so - called self - trapping regimes , where one or more optical fields simultaneously trap and cool the mechanical oscillator . the determining characteristic of this regime is that both the mechanical frequency @xmath0 and single - photon optomechanical coupling strength parameters @xmath1 are a function of the optical field intensities , in contrast to usual set - ups where @xmath0 and @xmath1 are constant for the given system . we also measure the characteristic transverse and axial trapping frequencies of different sized silica nanospheres in a simple optical standing wave potential , for spheres of radii @xmath2 nm , illustrating a protocol for loading single nanospheres into a standing wave optical trap that would be formed by an optical cavity . we use this data to confirm the dependence of the effective optomechanical coupling strength on sphere radius for levitated nanospheres in an optical cavity and discuss the prospects for reaching regimes of strong light - matter coupling . theoretical semiclassical and quantum displacement noise spectra show that for larger nanospheres with @xmath3 nm a range of interesting and novel dynamical regimes can be accessed . these include simultaneous hybridization of the two optical modes with the mechanical modes and parameter regimes where the system is bistable . we show that here , in contrast to typical single - optical mode optomechanical systems , bistabilities are independent of intracavity intensity and can occur for very weak laser driving amplitudes .

evidence for evolution in galaxies at intermediate redshifts has been found in a number of pioneering studies , both in clusters ( e.g. , butcher & oemler 1978 , 1984 ; dressler & gunn 1983 , couch & sharples 1987 ) and in the field ( e.g. , kron 1980 , broadhurst , ellis & shanks 1988 ) . large spectroscopic surveys with 4 m class telescopes , coupled with hst images , have begun to clarify the nature of this evolution . the cfrs redshift survey of @xmath15 field galaxies ( le fvre 1995 ; lilly 1995 ) is one such example . here we pursue a complementary approach involving the detailed structural and kinematic study of smaller samples of individual galaxies using hst images and higher resolution spectroscopy with the keck telescope . with these data , we can exploit the power of the tully - fisher relation for spirals ( _ e.g. _ , vogt 1996 ) and the fundamental plane relation for early - type ( e / s0 ) galaxies ( see franx 1993 ) . the fundamental plane ( djorgovski & davis 1987 ; dressler 1987 ) is particularly valuable due to its low intrinsic scatter ( jrgensen , franx and kjrgaard 1993 ) . for the coma cluster , the fundamental plane relation , @xmath16 where @xmath17 is the effective radius , @xmath18 is the surface brightness at that effective radius in the visible , and @xmath19 is the central velocity dispersion , has a scatter of only 17% _ rms _ ( jrgensen , franx and kjrgaard 1996 [ jfk96 ] ) . this low scatter implies that the following well - defined relation exists for early - type galaxies ( faber 1987 ) : @xmath20 thus , the fundamental plane is valuable because it explicitly incorporates galaxy masses . franx ( 1995 ) and van dokkum and franx ( 1996 [ vdf ] ) have demonstrated the value of this relation for studying evolution in early - type galaxies at intermediate redshift . the latter authors showed that the fundamental plane in the rich cluster cl0024 + 16 is well defined , and consistent with simple evolutionary models , but the observed sample was very small . here we present new results for two additional clusters at intermediate redshifts . these new data triple the galaxy sample , and extend the observed redshift range to @xmath1 . the spectroscopic sample was selected on the basis of @xmath21-band magnitude . blue galaxies were rejected to avoid field contamination , though the color restriction was chosen such that star - forming and post - starburst cluster members were not excluded . slit masks were designed to include as many bright galaxies as possible , though we only present data here for galaxies with hst imaging ( see figure [ mosaic ] [ plate 1 ] ) . in addition , two known `` e+a '' ( dressler & gunn 1983 ) galaxies were added to the cl1358 mask . these galaxies are not included in the general sample , but are discussed separately . thus , ten galaxies in cl1358 , and five in ms2053 , are analyzed in this paper . the spectroscopic observations were made using multi - slit masks with the low resolution imaging spectrograph ( lris ) at the keck telescope . we observed at a typical resolution of @xmath22 60 - 85 . the data reduction was very similar to the data reduction of vdf for cl0024 . the resulting spectra were very high s / n ( typically 20 - 60 per resolution element ) . we show spectra of two galaxies in figure [ mosaic](c , d ) . we modeled the spectral resolution of the spectrograph in great detail , for the template stars as well as the galaxies . this is necessary to ensure that the template stars used for the determination of the velocity dispersions have the correct spectral resolution . this procedure is the most essential technical aspect of measuring velocity dispersions of galaxies at intermediate redshift . some galaxies showed peculiar features in their spectra . these features , either strong balmer absorption lines , emission lines , residual sky lines , or atmospheric absorption bands , were given zero weight in the template fitting . we corrected the central velocity dispersions to an aperture of 34 at the distance of coma , using the procedure of jrgensen , franx , & kjrgaard ( 1995b ) . the corrections are small , 1.065 for cl1358 + 62 and 1.066 for ms205304 ( @xmath23 ) . the resulting velocity dispersions and random errors are listed in table 1 . we used wfpc2 hst images to measure the structural parameters . observations were taken in the filters f606w and f814w for cl1358 + 62 , and in f702w and f814w for ms205304 . these data were processed in the usual way for cosmic rays and removal of the sky background . the field for cl1358 + 62 was very large , @xmath24 ( franx 1997 ) . for ms2053 - 04 , only 1 central pointing was available . as a result we have more fundamental plane measurements for cl1358 + 62 . we determined the photometric parameters in two different ways , following the procedures used by vdf . we first used point - spread function images to fit convolved @xmath25-law profiles to the galaxy images . in addition , we deconvolved the images with the clean procedure ( hgbom 1974 ) , and derived growth curves for the galaxies . the results from both methods were compared to estimate the errors . it is worth noting that the median differences in @xmath26 and @xmath27 were @xmath28 and @xmath29 , but the combined parameter @xmath30 only differed by @xmath31 for cl1358 + 62 and @xmath32 for ms205304 . this is the combination of parameters that enters the fundamental plane , and as a result , our subsequent analysis is insensitive to the individual errors in @xmath17 and @xmath18 . because the coma data were derived from growth curves , we proceeded to use the growth curve results in the following analysis . after calibration using holtzman ( 1995 ) , the photometry was transformed to the redshifted @xmath33-band , for direct comparison to the coma photometry . this is possible , because we have observations in multiple passbands close to the redshifted @xmath33-band . colors were measured within an aperture of @xmath34 , and galactic extinctions were derived from burstein and heiles ( 1982 ) and cardelli , clayton & mathis ( 1989 ) . errors have been determined directly from the spectroscopic , structural and photometric fits , as well as the photometric transformations . the random errors are listed in table [ tab : params ] , and the typical random error bars are shown in figure [ fpz ] ( as thin error bars ) . we have considered possible sources of systematic errors and we have estimated their contribution ( shown as thick error bars in figure [ fpz ] ) . we have several sources of systematic errors : ( i ) photometric transformations at @xmath35 mag , which is dominated by the uncertainties in the absolute zeropoint of the f814w passband ; ( ii ) velocity dispersions , where our procedures may have relative errors of @xmath36 with respect to similar measurements at low redshift ( the absolute velocity dispersions may be in error systematically by up to @xmath37 - 7% ) ; ( iii ) structural parameters ( @xmath26 and @xmath27 ) , where deviations from an model could cause systematic errors on the level of @xmath38 in the combined parameter @xmath39 . another source of uncertainty is due to departures from homology ( see , _ e.g. _ , capelato 1995 , ciotti 1996 ) . non - homology can affect our measurement of evolution through the aperture correction for the velocity dispersions . jorgensen ( 1995b ) determined the aperture corrections empirically , by using long slit data on nearby galaxies . they found no strong effect out to an effective radius . therefore , these aperture corrections are likely to be appropriate for most of our galaxies . however , for the smallest galaxies , this correction is more uncertain and may require future observations of velocity dispersion profiles to large radii in a broad sample of nearby galaxies ( _ e.g. _ , corollo 1995 ) . the fundamental plane for the clusters are shown in figure [ fpz ] along with the fps for cl0024 + 16 ( vdf ) and coma ( jfk96 ) . we use the coefficients for the fp determined by jfk96 from a large sample of 225 early - type galaxies in ten nearby clusters . the figure shows clearly that a well defined fundamental plane exists , _ despite the fact that the galaxies in the intermediate redshift clusters were chosen without morphological information_. furthermore , the sample is large enough to derive the scatter about the coma fundamental plane . we find surprisingly low _ rms _ scatters in @xmath40 of @xmath41 , @xmath42 , @xmath43 , and @xmath44 for coma , cl1358 + 62 , cl0024 + 16 , and ms205304 , respectively . the galaxies also show a large offset from the coma relation , due mainly to cosmological surface brightness dimming . one interesting question is whether the coefficients of the fp are the same in higher redshift clusters , _ i.e. _ , are the luminous and less luminous galaxies evolving at the same rate ? however , the current sample is too small to provide a definitive answer . the weak indication that the slope is flatter when the distant galaxies are taken together needs to be verified with larger samples before any conclusions should be made ( see also vdf ) . we determined the mean @xmath4 ratio for each cluster directly from the fundamental plane zeropoint , adopting the slopes of the fundamental plane of jfk96 and @xmath23 . the resulting evolution of @xmath4 ratio is shown in figure [ mlz ] . the errors are taken from 2.3 and have been added in quadrature . weighting the individual galaxies by their random errors does not change the results significantly . clearly , the @xmath4 ratio is lower at higher redshift , consistent with evolution of the stellar populations . we have drawn simple , single - burst model predictions in the same plot , adopting formation redshifts @xmath45 of infinity , and @xmath46 . the current data are not consistent with the predictions for co - eval populations which have formed recently . more data are needed to test whether more complex models with recent star formation can be accommodated ( see , e.g. , franx and van dokkum 1996 and poggianti & barbaro 1996 ) . we have measured structural parameters and central velocity dispersions for galaxies in two clusters at intermediate redshift , cl1358 + 62 at @xmath0 and ms205304 at @xmath1 . the fundamental plane relations in the intermediate redshift clusters are very similar to that found in coma . this observation demonstrates that mature early - type galaxies existed in these clusters at @xmath47 ; their primary epochs of star formation must have occurred at much higher redshifts . the sample is also large enough to measure the scatter in the fundamental plane relation reliably . we find it to be low : @xmath48 in @xmath40 , or @xmath49 in @xmath4 . this suggests that the populations are very homogeneous , and that the age differences between the galaxies are not very large ( ciotti , lanzoni , & renzini 1996 ) . the mean @xmath4 ratio of the galaxies was clearly lower several gyr ago , consistent with passively evolving stellar populations . this evolution depends on the formation redshift(s ) of the population , the imf(s ) , and cosmological model ( _ e.g. _ , franx 1995 ) . we show model predictions for formation redshifts of @xmath50 and @xmath51 in figure [ mlz ] ( tinsley & gunn 1976 ) using @xmath23 . the new data are fully consistent with a high formation redshift , @xmath52 , strengthening the conclusion of vdf . this result is a lower limit ; the constraints are even stronger if @xmath53 . this interpretation , however , is complicated by the fact that mergers , interactions , starbursts , and other processes may continue to transform late - type galaxies into early - type galaxies : the early - type galaxies we observe at high redshift may only be a subset of the early - type galaxies we observe at low redshift . in this case , the early - type galaxies observed at high redshift ( if they remained undisturbed until the present ) should be compared to the oldest early - type galaxies locally . in some sense , these high redshift early - type galaxies have been compared to a _ mean _ low redshift counterpart , probably not as old as the comparison requires . thus , the formation redshift estimated from the @xmath54 evolution can be biased upwards ( see , _ e.g. _ , franx and van dokkum 1996 ) . we included two cl1358 + 62 `` e+a '' galaxies in our sample to test whether these galaxies are progenitors of early - type galaxies in low redshift clusters . they are shown in figure [ fpz ] by the `` x '' symbols . we can use the coma fundamental plane to measure the @xmath4 ratios of these `` e+a '' galaxies , assuming that their structural properties are similar to nearby early - type systems . this assumption may not necessarily be correct , as the franx ( 1993 ) analysis of an e+a in abell 665 ( @xmath55 ) shows that it is essentially bulgeless ; it is not clear whether it will become an s0 or remain a spiral system . with this caveat in mind , we show the mean @xmath4 offset for the two e+a galaxies in figure [ mlz ] as an `` x. '' this @xmath4 is consistent with a `` formation '' redshift of @xmath56 , a @xmath33-band luminosity weighted mean of the formation redshifts of the subcomponents . this is consistent with the hypothesis that the e+as have undergone a burst of star formation 1 - 2 gyr before they have been observed ( dressler and gunn 1983 ) . we conclude that a fraction of nearby cluster early - type galaxies has undergone secondary bursts of star formation at @xmath8 . at this point , more work is needed to assess the relevance of the e+a galaxies to the evolution of early - type galaxies and we will study the e+as in these clusters in greater detail elsewhere . observations of higher redshift clusters will be needed to test whether star formation and starbursts were even more prevalent at earlier times , as suggested by lubin ( 1996 ) and rakos & schombert ( 1995 ) . our sample can also be compared to surveys of nearby field galaxies , in which the data of gonzlez ( 1993 ) and faber ( 1995 ) suggest that a large age spread ( @xmath57 - 18 gyr ) exists . thus , a large fraction of these experienced bursts of star formation at @xmath8 . our data suggest that it might be hard , but not impossible , to model the evolutionary history of the cluster galaxies in the same way as the field galaxies . we appreciate the effort of all those in the hst program that made this unique observatory work as well as it does . the assistance of those at stsci who helped with the acquisition of the hst data is gratefully acknowledged . we also appreciate the effort of those at the keck , mmt and kpno telescopes who developed and supported the facility and the instruments that made this program possible . support from stsci grants go05989.01 - 94a , go05991.01 - 94a , and ar05798.01 - 94a is gratefully acknowledged . broadhurst t. j. , ellis r. s. , & shanks t. 1988 , , 235 , 827 burstein d. , & heiles c. 1982 , , 87 , 1165 butcher h. , & oemler a. 1978 , , 219 , 18 butcher h. , & oemler a. 1984 , , 285 , 426 capelato , h. v. , de carvalho , r. r. , & carlberg , r. g. 1995 , , 451 , 525 cardelli , j. a. , clayton , g. c. , & mathis , j. s. 1989 , , 345 , 245 ciotti , l. , lanzoni , b. , & renzini , a. 1996 , , 282 , 1 corollo , c. m. , de zeeuw , p. t. , van der marel , r. p. , danziger , i. j. , & qian , e. e. 1995 , , 441 , l25 couch w. j. , & sharples r. m. 1987 , , 229 , 423 djorgovski s. , & davis m. 1987 , , 313 , 59 dressler a. , & gunn j. e. 1983 , , 270 , 7 dressler a. , lynden - bell d. , burstein d. , davies r. l. , faber s. m. , terlevich r. j. , & wegner g. 1987 , , 313 , 42 ellis r. s. , smail , i. , dressler , a. , couch , w.j . , oemler , a. , butcher , h. , & sharples , r.m . 1996 , preprint faber s. m. , dressler a. , davies r. l. , burstein d. , lynden - bell d. , terlevich r. , & wegner g. 1987 , faber s. m. , ed . , nearly normal galaxies . springer , new york , p. 175 faber , s. m. , trager , s. c. , gonzlez j. j. , & worthey , g. 1995 , iau symp . 164 , stellar populations ( dordrecht : kluwer ) , 255 franx m. 1993 , , 105 , 1058 franx m. 1995 iau symp . 164 , stellar populations ( dordrecht : kluwer ) , 269 franx m. , & van dokkum , p. g. 1996 , _ new light on galaxy evolution _ , eds . bender , r. , & davies , r. l. , in press . franx m. , 1997 , in preparation gonzlez , j. j. 1993 , ph.d . thesis , univ . california , santa cruz h " ogbom j. a. 1974 , a&as , 15,417 holtzman , j. a. , burrows , c. j. , casertano , s. , hester , j. j. , trauger , j. t. , watson , a. m. , & worthey , g. 1995 , , 107 , 1065 jrgensen i. , franx m. , & kjrgaard p. 1993 , , 411 , 34 jrgensen i. , franx m. , & kjrgaard p. 1995a , , 273 , 1097 jrgensen i. , franx m. , & kjrgaard p. 1995b , , 276 , 1341 jrgensen i. , franx m. , & kjrgaard p. 1996 , , 280 , 167 [ jfk96 ] kron , r. 1980 , , 43 , 305 le fvre , o. , 1995 , , 461 , 534 lilly s. j. , tresse l. , hammer f. , crampton d. , & le fvre o. 1995 , , 455 , 50 lubin , l. 1996 , , accepted for publication poggianti , b. m. , & barbaro , g. 1996 , , in press rakos , k. d. , & schombert , j. m. 1995 , apj , 439 , 47 tinsley , b. m. , & gunn , j. e. 1976 , , 203 , 52 van dokkum , p. g. , & franx m. 1996 , , 281 , 985 [ vdf ] vogt , n. p. , forbes , d. a. , phillips , a. c. , gronwall , c. , faber , s. m. , illingworth , g. d. , & koo , d. c. 1996 , , 465 , l15 l c c l c c 200&18.70&0.68&@xmath58&0.913&22.50236&17.98&0.80&@xmath59&0.541&21.93256&17.56&0.78&@xmath60&1.024&21.81269&17.85&0.82&@xmath61&0.826&21.55298&18.25&0.82&@xmath62&0.448&20.77328&19.08&0.60&@xmath63&0.712&22.24375&17.44&0.81&@xmath64&3.910&23.83408&19.04&0.77&@xmath65&0.287&20.74454&18.71&0.73&@xmath66&0.780&22.23470&18.41&0.79&@xmath67&0.738&22.23311&20.52&1.12&@xmath68&0.402&22.53197&18.59&1.20&@xmath69&2.182&23.99422&20.59&1.20&@xmath70&0.413&22.57551&20.89&1.14&@xmath71&0.157&20.93432&20.41&1.08&@xmath72&0.483&22.82

we present new results on the fundamental plane of galaxies in two rich clusters , cl1358 + 62 at @xmath0 and ms205304 at @xmath1 , based on keck and hst observations . our new data triple the sample of galaxies with measured fundamental plane parameters at intermediate redshift . the early - type galaxies in these clusters define very clear fundamental plane relations , confirming an earlier result for cl0024 + 16 at @xmath2 . this large sample allows us to estimate the scatter reliably . we find it to be low , at 0.067 dex in @xmath3 , or 17% in re , similar to that observed in comparable low redshift clusters . this suggests that the structure of the older galaxies has changed little since @xmath1 . the @xmath4 ratios of early - type galaxies clearly evolve with redshift ; the evolution is consistent with @xmath5 . the @xmath4 ratios of two e+a galaxies in cl1358 + 62 are also lower by a factor of @xmath6 , consistent with the hypothesis that they underwent a starburst 1 gyr previously . we conclude that the fundamental plane can therefore be used as a sensitive diagnostic of the evolutionary history of galaxies . our data , when compared to the predictions of simple stellar population models , imply that the oldest cluster galaxies formed at high redshift ( @xmath7 ) . we infer a different evolutionary history for the e+a galaxies , in which a large fraction of stars formed at @xmath8 . larger samples spanning a larger redshift range are needed to determine the influence of starbursts on the general cluster population . -1kpc@xmath9 -1mpc@xmath9 -1sec@xmath9 2deg@xmath10 0h@xmath11 0@xmath12 # 1@xmath13 # 1 km s@xmath9 mpc@xmath9 # 1 @xmath14#1

the presence of magnetic fields in objects of astrophysical interest is evident . starting from the earth s magnetic field , to the solar magnetically driven phenomena and at the most extreme from magnetars to the recently observed filaments in the perseus cluster of galaxies @xcite , magnetic fields play an important rle in the evolution of the object they occur in . the study of the magnetic fields is complicated due to the form of partial differential equations that have to be solved and their non - linear behaviour . for this reason most of the research has been done on equilibrium configurations . even in the non - relativistic limit a few of the published studies are analytical , for instance @xcite whereas most of them are computational i.e. @xcite and references therein . in the relativistic case the studies are mainly motivated by pulsar magnetospheres , @xmath1-ray bursts , and relativistic jets . most of the solutions are numerical , for instance @xcite and only a few of them are analytic @xcite . in this paper we study the problem of a force - free relativistic electromagnetic field inspired by the studies previously made by @xcite for the geometry and the symmetry used ; and from @xcite for the relativistic generalisation we make . @xcite showed that it is possible to study magnetic fields that expand uniformly within a sphere . these fields emerge from a point , obey the ideal mhd relation @xmath2 and the net electromagnetic force is zero , then he solved the partial differential equation he found assuming a linear form . in this paper we show that an analogue case that permits analytical solutions and allows us to include time evolution exists for cylindrical magnetic fields . the field emerges from a linear singularity as in @xcite who studied this type of fields in the static limit , but in our problem the field expands radially . the expansion velocity is proportional to the distance from the axis and inversely proportional to time , thus every element of the plasma moves at constant speed . the expansion is slow near the axis and reaches the speed of light on a cylindrical surface . there is no field in the space outside this surface . the field viewed in the co - moving frame of reference obeys the force - free relation in the non - relativistic sense @xmath3 as there is only magnetic field present , whereas in the frame of reference at rest with respect to the axis the net electromagnetic force by taking into account both the interaction of the magnetic fields with the currents and the interaction of the electric fields with the charges is zero . in the limit of low velocities the equation obtained is similar to the grad - shafranov partial differential equation studied for the case of magnetic fields in equilibrium in the absence of pressure , but for relativistic velocities they differ considerably . we proceed on the derivation of the equation and then we solve it for various cases . assume a non - resistive plasma of negligible inertia containing a magnetic field @xmath4 and an electric field @xmath5 in cylindrical geometry @xmath6 , these fields do not depend on @xmath7 . the electric field is related to the magnetic by the definition of ideal mhd @xmath2 , where @xmath8 is the velocity of the plasma normalized to the speed of light . in addition assume that the plasma expands with @xmath9 the magnetic field can be described by a flux function @xmath10 for the @xmath11 and @xmath12 components , and leave without any constraint the @xmath13 component yet , apart from @xmath14 @xmath15 or @xmath16 the term arising from the gradient of @xmath17 will be called coplanar field @xmath18 . from the ideal mhd relation the electric field is @xmath19 the magnetic and the electric field must satisfy maxwell s equations . indeed , a magnetic field of the above form is by construction divergence - free . the induction equation is @xmath20 then by substituting equations ( [ magnetic ] ) and ( [ electric ] ) into equation ( [ induction0 ] ) we find that the equations for the @xmath21 and @xmath22 components are satisfied . in order the @xmath7 component of equation ( [ induction0 ] ) to be equal to zero , @xmath13 has to satisfy the following partial differential equation @xmath23 we re - express @xmath24 , substituting this into equation ( [ diffbz ] ) gives @xmath25 therefore @xmath26 , so @xmath13 can be written as @xmath27 , where @xmath28 is an arbitrary function of @xmath29 and @xmath22 . the two remaining maxwell s equations are used to evaluate the charge ( @xmath30 ) and the current density ( @xmath31 ) , @xmath32 @xmath33 as we are looking for fields which are force - free in this generalized sense we demand that the total force the electric and the magnetic fields exert to charge and current densities is zero . this force is @xmath34 thus , by substituting the current and charge density found above into equation ( [ force ] ) we evaluate all three components of the force . from the @xmath35 component we take @xmath36 then we multiply equation ( [ zforce ] ) by @xmath37 and re - arragne the terms @xmath38^{3/2}w)}{\partial \phi}\frac{\partial p}{\partial v } \nonumber \\ -\big[(1-v^{2})^{3/2}\frac{\partial w}{\partial v}-3v(1-v^{2})^{1/2}w\big]\frac{\partial p}{\partial \phi}=0\ , . \label{jacobian1}\end{aligned}\ ] ] setting @xmath39 equation ( [ jacobian1 ] ) reduces to @xmath40 which is the jacobian of @xmath41 and @xmath17 with respect to @xmath29 and @xmath22 . as it equals zero , @xmath42 , thus @xmath43 , where @xmath44 is an arbitrary function of @xmath17 . given that , the @xmath7 component of the magnetic field becomes @xmath45 using equation ( [ bz ] ) in the force equation ( [ force ] ) the @xmath21 and @xmath22 components of the force are both zero when @xmath46 this is the cylindrical version of prendergast s equation , which is the basic equation deduced in this paper and our major aim hereafter is to find analytical and semi - analytical solutions . it simplifies if we apply the transformation @xmath47 @xmath48 this is a convenient form of the force - free equation , which we shall use later in this paper to find analytical solutions . in what follows we study equation ( [ eqn ] ) and forms that allow separable solutions of the form @xmath49 in order to do so , we investigate forms of @xmath41 which permit separation of variables . @xmath41 is any function of @xmath17 and this gives a great freedom on the solutions of the basic equation ( [ eqn ] ) , which may be highly non - linear if such a form is chosen . at first we study the form of the equation in the non - relativistic limit , then solutions in the absence of a @xmath7 component of the magnetic field and then solutions of the linearized form of equation ( [ eqn ] ) . for @xmath50 equation ( [ eqn ] ) reduces to @xmath51 separable solutions are allowed in two cases . in the linear case for @xmath52 , equation ( [ nonrel ] ) becomes @xmath53 where @xmath54 is the constant of separation and it is a real number as @xmath55 has to be periodic . the solution to the angular part is @xmath56 the differential equation for @xmath57 admits bessel functions of order @xmath54 for solutions @xmath58 the constants @xmath59 , @xmath60 , @xmath61 and @xmath62 can be evaluated subject to the boundary conditions . the horizontal shear causing the @xmath7 component of the field is parametrized by the constant @xmath63 . the other class of separable solutions is for @xmath64 and @xmath65derived by @xcite , who studied a problem of the same geometry in the static case . indeed if we substitute the forms above in equation ( [ nonrel ] ) , the terms involving @xmath29 cancel and we take an ordinary differential equation for @xmath55 @xmath66 this equation can be solved numerically , subject to the boundary conditions and the parameters @xmath63 , @xmath67 and @xmath68 . it is possible to by - pass the numerical solution of equation ( [ fnonrel ] ) and construct an approximate solution for @xmath67 close to zero as the one found by @xcite , by normalizing to unity @xmath69 we find @xmath70 where @xmath71 and @xmath72 . if @xmath67 is no more small then we can use the more accurate and compicated form described in appendix a of @xcite . although the mathematical treatment described above is the same to aly s case , the two equations describe distinct physical configurations . the fields in our problem are not static but expand while being force - free . this growth can also be conceived as some flux generated from the base and expanding with velocity equal to @xmath73 . in aly s case the fields are static , however it is possible to achieve expansion due to the shearing of the field lines on the surface of the cylinder . that problem is studied as a sequence of static solutions where the transition from one solution to the next one takes place slowly enough so that the displacement current is negligible . in our case the shearing does not change explicitly with time as it is imposed from the beginning . however it is possible to assume the field lines to be sheared slowly , indeed as discussed by @xcite mhd systems where the boundary conditions change slowly have negligible induction currents and pass through a sequence of equilibrium stages . in our case , if the shearing of the field lines takes place slowly , then a series of force - free solutions of expanding fields describes this process . thus the shearing of the field lines may take place simultaneously with the expansion . here we investigate solutions of the fully relativistic equation ( [ eqn ] ) without neglecting any term . first we study solutions in the absence of a @xmath7 component for the magnetic field which gives a coplanar magnetic field without any current and charge density and then solutions containing all three components by assuming a linear form for the right hand side of the differential equation . the simplest non - trivial solution of equation ( [ eqn ] ) exists in the absence of a @xmath7 component of the magnetic field , @xmath74 . it is a magnetic field that emerges from the surface of a cylinder where no shear is imposed . in this configuration there is no current or electric charge . we shall use equation ( [ eqn1 ] ) as this form is more convenient , let @xmath75 and derivatives to be denoted by dash , equation ( [ eqn1 ] ) becomes @xmath76 the solution of the angular part is of sinusoidal form @xmath77 the solution for @xmath78 is @xmath79 this corresponds to a coplanar magnetic field , where every magnetic field line lies on a plane normal to the axis of the cylinder . unlike the non - relativistic equation , which admits two classes of separable solutions when @xmath80 , in the relativistic regime , separable solutions exist only in the linear case , and there is no analogue to the non - relativistic powerlaws . to show this assume separable solutions of the standard form ; in division of equation ( [ eqn1 ] ) by @xmath17 it is @xmath81 the first term of the left hand side is only a function of @xmath82 and can be expressed as @xmath83 , the second term similarly is @xmath84 , as for the right hand side let @xmath85 . the next step is to act with the operators @xmath86 and in the resulting expression with the operator @xmath87 ; the final equation is @xmath88 then name @xmath89 and substitute above , @xmath90 from equation ( [ separation ] ) it is @xmath91 which leads to @xmath92 and the form of @xmath93 that permits separation of variables is @xmath94 . when substituting these forms in the right hand side of equation ( [ eqn1 ] ) it becomes a sum of a term that only depends on @xmath82 and one that only depends on @xmath22 , however when the left hand side of the equation is considered the terms involving @xmath82 do not sum up to a constant thus it is impossible to separate the equation by using power laws . it is only the linear form for @xmath95 , which permits separation of variables , indeed @xmath96 the angular part admits again sinusoidal solutions , as for the @xmath78 the equation to solve is @xmath97 which can not be integrated analytically and we solve it numerically . the numerical solution of equation ( [ linearv ] ) depends on the choice of the parameters and the boundary conditions . assume that some flux emerges from a cylindrical surface @xmath98 at time @xmath99 which in velocity space lies @xmath100 , the boundary conditions for @xmath57 on this surface are determined by the fields . the fields are expressed in terms of @xmath57 and @xmath55 , they are @xmath101 @xmath102 @xmath103 the value of @xmath57 on the surface the flux emerges from is related to the intensity of the @xmath21 component of the magnetic field and the derivative @xmath104 to the intensity of @xmath22 component , in what follows we use the normalised value @xmath105 . then the parameters @xmath63 and @xmath54 are chosen and we integrate the differential equation . the physical meaning of the parameter @xmath63 is related to the horizontal shear of the field lines and therefore since @xmath13 is proportional to @xmath63 , the more shear is induced on the base of the field the larger this parameter is . an inspection of the differential equation even without proceeding to the numerical solution demonstrates the decelerating effect of this term , greater shear leads to a stronger @xmath13 component of the magnetic field . this component is proportional to @xmath106 and exerts a strong force to the magnetic field . thus , the flux decreases rapidly for @xmath29 close to unity . after this point the solution undergoes oscillations ( figures 1 , 2 ) and the field forms disconnected appendages . these structures are the result of the linear assumption made previously . it is now possible to draw the field lines for any time @xmath107 ( figures 3 , 4 ) as we have the expressions for @xmath57 and @xmath55 and the field lines are given by substituting expressions ( [ br ] ) , ( [ bf ] ) and ( [ b_z ] ) in @xmath108 ) . @xmath57 is normalized to unity at @xmath109 , @xmath110 and @xmath111 , the solid line is for @xmath112 , the dashed for @xmath113 , the dotted for @xmath114 and the dash - dotted for @xmath115 . this decrease in the absolute value of the derivative leads to a weaker @xmath12 at @xmath116 , thus the field emerges closer to the perpendicular and the flux function reaches a greater @xmath117 before starting the oscillations.,scaledwidth=50.0% ] ) . @xmath57 is normalized to unity at @xmath109 , @xmath118 and @xmath111 , the solid line is for @xmath112 , the dashed for @xmath113 , the dotted for @xmath114 and the dash - dotted for @xmath115 . this decrease in the absolute value of the derivative leads to a weaker @xmath12 at @xmath116 , thus the field emerges closer to the perpendicular and the flux function reaches a greater @xmath117 before starting the oscillations . compared to the lines plotted in figure 1 the @xmath7 field is stronger here and the oscillations start earlier.,scaledwidth=50.0% ] , the line of sight is parallel to the axis of the cylinder . the inner circle corresponds to a cylinder of radius @xmath119 and the outer circle to one of radius @xmath120.,scaledwidth=50.0% ] the other parameter @xmath54 defines the order of the multipole chosen , for instance @xmath111 is a cylindrical dipole and @xmath121 gives a quadrupole etc . if a non - integer value is chosen , then the problem can be studied within a confined arcade , of any opening angle , however the solution will not have a physical significance outside this domain . finally the value of the derivative @xmath104 is chosen and the differential equation is integrated numerically . in the solutions plotted we choose four values for it , we find that for a smaller absolute value of @xmath104 the flux reaches greater distances in @xmath29 space ; thus fields where the ratio @xmath122 is bigger have fluxes that reach greater distances , so the more radial the flux is at the base the further it gets before it starts oscillating . in this section we study the shear and the energy stored in this configuration subject to @xmath63 . we assume that the same radial flux emerges from the surface by keeping @xmath123 the same and it reaches a maximum velocity @xmath117 also constant , where @xmath124 , given these boundary conditions we solve equation ( [ linearv ] ) for various values of the parameter @xmath63 . then we use the solutions to study the shear and the energy . the amount of shear for a given field line can be evaluated by integrating equation ( [ fieldlines ] ) after substituting the fields from equations ( [ br]-[b_z ] ) . an element of a field line labeled by @xmath125 that lies between @xmath29 and @xmath126 in velocity space is sheared by @xmath127 an important feature of this quantity is that it does not change as the field expands . we evaluate the total shear of a field line by integrating equation ( [ shear ] ) from @xmath116 where the field line emerges , up to the maximum @xmath128 it reaches , this @xmath128 is given by solving equation @xmath129 , then by symmetry the shear from the top to the other foot point is the same . inspection of equation ( [ shear ] ) shows that the shear depends on @xmath63 as it appears directly and indirectly in the integral through the form of @xmath57 which is a numerical solution depending on the parameter @xmath63 , there is also a dependence on the choice of the field line which also appears directly in the equation but also indirectly through the end point of integration @xmath128 which depends on @xmath125 . in table 1 we use the results of the linear solution to evaluate the shear @xmath130 of the field line which has been stretched the most for a given @xmath63 . the numerical integration shows actually that as @xmath63 increases the shear increases , a proportionality relation between @xmath63 and @xmath131 is a first approximation , and the deviation of this approximation is due to the indirect dependence on @xmath63 through the other quantities appearing in ( [ shear ] ) . the total energy carried by an electromagnetic field in a volume @xmath132 is @xmath133 we can express the forms of the electric and the magnetic field in terms only of @xmath134 and @xmath29 and study the energy contained in a volume in velocity space and how it evolves with time . by substituting the forms of the magnetic fields from equations ( [ br]-[b_z ] ) and by the fact that @xmath2 the energy contained in a volume in velocity space with boundaries @xmath116 and @xmath117 , @xmath7 from @xmath135 to @xmath136 and @xmath22 from @xmath135 to @xmath137 , is @xmath138\big\}vdvd\phi\ , . \label{energy}\end{aligned}\ ] ] the form of the integral in equation ( [ energy ] ) suggests that the energy within a constant volume in velocity space changes with time . it is constant only when there is no @xmath7 component of the magnetic field . the poynting vector of an electromagnetic field is @xmath139 and by substituting in terms of @xmath57 and @xmath55 we find @xmath140\bm{\hat{r}}\nonumber \\ + \frac{vv'\phi\phi'}{v(ct)^{2}}\bm{\hat{\phi}}-\frac{c_{0}v^{2}\phi\phi'}{v(ct)^{3}(1-v^{2})^{3/2}}\bm{\hat{z}}\big\}\ , . \label{poynting}\end{aligned}\ ] ] as the @xmath21 component of the poynting vector is positive , the energy flows outwards as the fields expand . we have integrated equation ( [ energy ] ) from @xmath119 to @xmath141 for a range of values of @xmath63 at @xmath142 . we used the form of @xmath57 found by solving the differential equation ( [ linearv ] ) subject to the boundary conditions stated in this section ; the results appear in table 1 . the overall conclusion is that the energy increases with @xmath63 . this is due to two reasons , the @xmath7 component of the magnetic field increases , thus it carries more energy . this increase on @xmath13 has also a side effect , for the field to remain force - free the coplanar component of the magnetic field @xmath18 and @xmath143 have to increase to balance the extra force . thus the energy carried by them becomes larger as well . it is evident from table 1 that both @xmath144 and @xmath145 increase with @xmath63 and so does their sum @xmath146 . .the physical quantities of the system for @xmath63 ranging from @xmath135 to @xmath147 , evaluated by using the solution of equation ( [ linearv ] ) subject to the boundary conditions @xmath148 , @xmath149 and @xmath111 . as @xmath63 increases the shearing of the field lines @xmath130 increases . the energy carried by the field also increases , where @xmath144 is the energy carried by the coplanar component of the magnetic field and the @xmath150 of the electric ; @xmath145 is the energy carried by @xmath13 and @xmath151 . [ cols=">,>,>,>,>",options="header " , ] observations of @xmath1-ray flares @xcite suggest that they are associated to strong magnetic fields expanding with relativistic velocities . these flares are thought as potential origins of short duration @xmath1-ray bursts @xcite . the relativistic solution of the equations studied in this paper can be applied to the initial stages of magnetar giant flares emerging . it has been proposed @xcite that giant flares from @xmath1-ray repeaters are formed through processes similar to coronal mass ejections on the sun . as opposed to solar arcades , they reach high lorentz factors . our model describes the electromagnetic field of arcades which expand and reach relativistic velocities . when the flare expands with a great velocity it is essential to take into account the relativistic effects as extra terms appear on equation ( [ eqn ] ) compared to the non - relativistic form , equation ( [ nonrel ] ) . in our study the fields expand only in one dimension , this is a reasonable assumption when the magnetic arcades are not large compared to the radius of the object where they emerge from or when there is a primary direction of expansion . an other issue is the neglect of inertia forces , this is fine when the plasma is underdense and most energy is carried by the magnetic field . however very close to the speed of light , no matter how small the density is , it will be no more negligible as it will be multiplied by a large lorentz factor . when the expansion in a second dimension becomes important then the properties of the spherical geometry have to be taken into account as they are presented by @xcite . in this paper we derived the force - free analogue of prendergast s equation @xcite in the case of cylidrically expanding magnetic fields . these solutions offer a theoretical insight to both relativistic and non - relativistic expanding systems . the non - relativistic solutions have structural similarities to static ones where time is not explicitly included , but it is implied via a series of static solutions where the expansion is merely due to shearing of the magnetic field lines whereas the flux emerging from the surface is constant . in our case the field expands due to two reasons : shearing of the field lines as in the static problem and due to an increase of the flux emerging from the surface . the increase of flux appears directly in the equations . the plane parallel geometry also sets constraints to the applications of these solutions . astrophysical systems are more likely to occur in spherical geometry , a case studied previously @xcite , however this particular geometry leads to results of satisfactory accuracy compared to those of the spherical geometry when the radius the magnetic flux extends is comparable to the radius of curvature of the surface the magnetic field emerges . systems where the expansion is mainly in one dimension are also described by this geometry . an astrophysical system of interest is that of solar arcades , this model describes the combined effect of the extra flux emerging from the base and of the shearing of the field lines provided that the size of the arcade is small compared to the radius of the sun . a detailed comparison of plane parallel models and spherical ones can be found in @xcite and the loss of equilibrium in these models leading to the opening of the field lines has been studied by @xcite . it is also shown that the energy contained in the magnetic arcade increases when more shear is imposed . in relativistic systems the demand of simultaneous expansion and shearing has some problems as the field at the top expands very fast whereas the shearing of the field lines at the base has to take place slowly . thus the message for the @xmath13 component to increase will arrive much later at the top . for this reason we consider that the magnetic field is sheared before the expansion takes place . the author is grateful to professor donald lynden - bell for the inspiring discussions and guidance .

we study relativistically expanding electromagnetic fields of cylindrical geometry . the fields emerge from the side surface of a cylinder and are invariant under translations parallel to the axis of the cylinder . the expansion velocity is in the radial direction and is parametrized by @xmath0 . we consider force - free magnetic fields by setting the total force the electromagnetic field exerts on the charges and the currents equal to zero . analytical and semi - analytical separable solutions are found for the relativistic problem . in the non - relativistic limit the mathematical form of the equations is similar to equations that have already been studied in static systems of the same geometry . [ firstpage ] gamma - rays : bursts ; methods : analytical ; mhd ; stars : magnetic fields ; stars : neutron ;

detailed spectroscopic studies of the nucleon excitation spectrum and the structure of these excited states have played a central role in the development of our understanding of the dynamics of the strong interaction . the concept of quarks that emerged through such studies led to the development of the constituent quark model @xcite ( cqm ) in the 1980s . as a result of intense experimental and theoretical effort over the past 30 years , it is now apparent that the structure of the nucleon and its spectrum of excited states ( @xmath0 ) are much more complex than what can be described in terms of models based on constituent quarks alone . at the typical energy and distance scales found within the @xmath0 states , the quark - gluon coupling is large . therefore , we are confronted with the fact that quark - gluon confinement , and hence the dynamics of the @xmath0 spectrum , can not be understood through application of perturbative quantum chromodynamics ( qcd ) techniques . the need to understand qcd in this non - perturbative domain is a fundamental issue in nuclear physics that the study of @xmath0 structure can help to address . such studies , in fact , represent the necessary first steps toward understanding how qcd generates mass , i.e. how mesons , baryons , and atomic nuclei are formed . studies of low - lying baryon states , as revealed by electromagnetic probes at low four - momentum transfer ( @xmath9 gev@xmath8 ) , have revealed @xmath0 structure as a complex interplay between the internal core of three dressed quarks and an external meson - baryon cloud . @xmath0 states of different quantum numbers have notably different relative contributions from these two components , demonstrating distinctly different manifestations of the non - perturbative strong interaction in their generation . the relative contribution of the quark core increases with @xmath7 in a gradual transition to a dominance of quark degrees of freedom for @xmath10 gev@xmath8 . this kinematics area still remains almost unexplored in exclusive reactions . studies of the @xmath7 evolution of @xmath0 structure from low to high @xmath7 offer access to the strong interaction between dressed quarks in the non - perturbative regime that is responsible for @xmath0 formation . electroproduction reactions provide for a probe of the inner structure of the contributing @xmath0 resonances through the extraction of the amplitudes for the transition between the incident virtual photon - nucleon state and the final @xmath0 state , i.e. the @xmath5 electrocoupling amplitudes , which describe the physics . among these amplitudes are @xmath11 and @xmath12 , which describe the @xmath0 resonance electroexcitation for the two different helicity configurations of a transverse photon and the nucleon , as well as @xmath13 , which describes the @xmath0 resonance electroexcitation by longitudinal photons of zero helicity . detailed comparisons of the theoretical predictions for these amplitudes with their experimental measurements is the basis of progress toward understanding non - perturbative qcd . the extraction of the @xmath5 electrocouplings is needed in order to gain access to the dynamical momentum - dependent mass and structure of the dressed quark in the non - perturbative domain where the quark - gluon coupling is large @xcite . this is critical in exploring the nature of quark - gluon confinement and dynamical chiral symmetry breaking ( dcsb ) in baryons . current theoretical approaches to understand @xmath0 structure fall into two broad categories . in the first category are those that enable direct connection to the qcd lagrangian , such as lattice qcd ( lqcd ) and qcd applications of the dyson - schwinger equations ( dse ) . in the second category are those that use models inspired by or derived from our knowledge of qcd , such as quark - hadron duality , light - front holographic qcd ( ads / qcd ) , light - cone sum rules ( lcsr ) , and cqms . see ref . @xcite for an overview of these different approaches . it is important to realize that even those approaches that attempt to solve qcd directly can only do so approximately , and these approximations ultimately represent limitations that need careful consideration . as such , it is imperative that whenever possible the results of these intensive and challenging calculations be compared directly to the data from electroproduction experiments over a broad range of @xmath7 for @xmath0 states with different quantum numbers . comparisons of the experimental results on the @xmath5 electrocouplings to the theoretical predictions provide for crucial insights into many aspects of the dynamics , including confinement and dcsb , through mapping of the dressed quark mass function @xcite and extractions of the quark distribution amplitudes for the @xmath0 states of different quantum numbers @xcite . the @xmath0 program is one of the key cornerstones of the physics program in hall b at jefferson laboratory ( jlab ) . the large acceptance spectrometer clas @xcite was designed to measure photo- and electroproduction cross sections and spin observables over a broad kinematic range for a host of different exclusive reaction channels . consistent determination of @xmath0 properties from different exclusive channels with different couplings and non - resonant backgrounds offers model - independent support for the findings . to date photoproduction data sets from clas and elsewhere have been used extensively to constrain coupled - channel fits and advanced single - channel models . however , data at @xmath7=0 allows us to identify new states and determine their quantum numbers , but they tell us very little about the structure of these states . it is the @xmath7 dependence of the @xmath5 electrocouplings that reveals these details . in addition , electrocoupling data is promising for both spectrum and structure studies as the ratio of resonant to non - resonant amplitudes increases with increasing @xmath7 . finally , the electroproduction data are an effective tool to confirm the existence of new @xmath0 states as the data must be described by @xmath7-independent resonance masses and hadronic decay widths . the program goal is the study of the spectrum of @xmath0 states and their associated structure over a broad range of distance scales through studies of the @xmath7 dependence of the @xmath5 electrocouplings . for each final state this goal is realized employing two distinct phases . the first phase consists of the measurements of the experimental observables , cross sections and spin observables , in as fine a binning in the relevant kinematic variables ( @xmath7 , @xmath6 , @xmath14 ) as possible with the data . the second phase consists of developing advanced reaction models that fully describe the data in order to then extract the electrocoupling amplitudes for the dominant contributing @xmath0 states . electrocoupling amplitudes for most @xmath0 states below 1.8 gev have been extracted for the first time from analysis of clas data in the exclusive @xmath15 and @xmath16 channels for @xmath7 up to 5 gev@xmath8 , in @xmath17 for @xmath7 up to 4 gev@xmath8 , and for @xmath18 for @xmath7 up to 1.5 gev@xmath8 . electrocoupling amplitudes of the @xmath19 ( left ) and @xmath20 ( right ) @xmath0 states from the analyses of the clas @xmath1 ( circles ) and @xmath21 ( triangles , squares ) data . ( left ) calculation from a non - relativistic light - front quark model with a running quark mass ( red line ) and calculation of the quark core from the dse approach ( blue line ) . ( right ) calculation from the hypercentral constituent quark model ( blue line ) . the magnitude of the meson - baryon cloud contributions are shown by the magenta line in both plots . see refs . @xcite for details on the data and the models . the electrocouplings have units of 10@xmath22 gev@xmath23 . ] figure [ low - lying ] shows representative clas data for the @xmath24 electrocouplings for the @xmath19 and @xmath20 @xcite . studies of the electrocouplings for @xmath0 states of different quantum numbers at lower @xmath7 have revealed a very different interplay between the inner quark core and the meson - baryon cloud as a function of @xmath7 . structure studies of the low - lying @xmath0 states , e.g. @xmath25 , @xmath19 , @xmath20 , and @xmath26 , have made significant progress in recent years due to the agreement of results from independent analyses of the clas @xmath1 and @xmath21 final states @xcite . the good agreement of the extracted electrocouplings from both the @xmath1 and @xmath3 exclusive channels is non - trivial in that these channels have very different mechanisms for the non - resonant backgrounds . the agreement thus provides compelling evidence for the reliability of the results . the size of the meson - baryon dressing amplitudes are maximal for @xmath27 gev@xmath8 ( see fig . [ low - lying ] ) . for higher @xmath7 , there is a gradual transition to the domain where the quark degrees of freedom just begin to dominate , as seen by the improved description of the @xmath0 electrocouplings obtained within the dse approach , which accounts only for the quark core contributions . for @xmath10 gev@xmath8 , the quark degrees of freedom are expected to fully dominate the @xmath0 states @xcite . therefore , in the @xmath5 electrocoupling studies for @xmath10 gev@xmath8 expected with the clas12 program , the quark degrees of freedom will be probed more directly with only small contributions from the meson - baryon cloud . analysis of clas data for the @xmath3 channel has provided the only detailed structural information regarding higher - lying @xmath0 states , e.g. @xmath28 , @xmath29 , @xmath30 , @xmath31 , and @xmath32 . [ high - lying ] shows a representative set of illustrative examples for @xmath33 for the @xmath28 @xcite , @xmath24 for the @xmath31 @xcite , and @xmath34 for the @xmath32 @xcite . here the analysis for each @xmath0 state was carried out independently in different bins in @xmath6 across the width of the state for @xmath7 up to 1.5 gev@xmath8 with very good correspondence within each @xmath7 bin . note that most of the @xmath0 states with masses above 1.6 gev decay preferentially through the @xmath3 channel instead of the @xmath1 channel . final state as a function of @xmath7 . ( left ) @xmath33 of the @xmath28 @xcite , ( middle ) preliminary extraction of @xmath24 for the @xmath31 @xcite , and ( right ) preliminary extraction of @xmath34 for the @xmath32 @xcite . each electrocoupling amplitude was extracted in independent fits in different bins of @xmath6 across the resonance peak as shown for each @xmath7 bin ( points in each @xmath7 bin offset for clarity ) . the electrocouplings have units of 10@xmath22 gev@xmath23 . ] with a goal to have independent confirmation of the extracted electrocouplings for each @xmath0 state from multiple exclusive final states , a natural avenue to investigate for the higher - lying @xmath0 states is the strangeness channels @xmath35 and @xmath36 . in fact , data from the @xmath4 channels is critical to provide an independent extraction of the electrocoupling amplitudes for the higher - lying @xmath0 states . the clas program has yielded by far the most extensive and precise measurements of @xmath4 electroproduction data ever measured across the nucleon resonance region . these measurements have included the separated structure functions @xmath37 , @xmath38 , @xmath39 , @xmath40 , @xmath41 , and @xmath42 for @xmath35 and @xmath36 @xcite , recoil polarization for @xmath35 @xcite , and beam - recoil transferred polarization for @xmath35 and @xmath36 @xcite . these measurements span @xmath7 from 0.5 to 4.5 gev@xmath8 , @xmath6 from 1.6 to 3.0 gev , and the full center - of - mass angular range of the @xmath43 . these final states , due to the creation of an @xmath44 quark pair in the intermediate state , are naturally sensitive to coupling to higher - lying @xmath45-channel resonance states at @xmath46 gev . note also that although the two ground - state hyperons have the same valence quark structure ( @xmath47 ) , they differ in isospin , such that intermediate @xmath0 resonances can decay strongly to @xmath35 final states , but intermediate @xmath48 states can not . because @xmath36 final states can have contributions from both @xmath0 and @xmath48 states , the hyperon final state selection constitutes an isospin filter . shown in fig . [ ky - data ] is a small sample of the available data in the form of the @xmath35 and @xmath36 structure functions , illustrating its typical statistical precision . ( top ) and @xmath36 ( bottom ) final states at @xmath7=1.80 gev@xmath8 and @xmath49=0.5 from ref . the curves show the isobar model from maxwell @xcite ( red line ) and the regge plus resonance model from ghent @xcite ( for rpr-2007 ) and @xcite ( for rpr-2011 ) ( blue lines ) that were constrained by fits to the clas photoproduction data . ] [ ky - data ] includes two of the more advanced single channel models for the electromagnetic production of @xmath4 final states . the mx model is the isobar model from maxwell @xcite , and the rpr-2007 @xcite and rpr-2011 @xcite models are the regge plus resonance framework developed at ghent . both the mx and rpr models were developed based on fits to the extensive and precise photoproduction data from clas and elsewhere and describe those data well . however , they utterly fail to describe the electroproduction data in any of the kinematic phase space . reliable information on @xmath4 hadronic decays from @xmath0s is not yet available due to the lack of an adequate reaction model . however , after such a model is developed , the @xmath0 electrocoupling amplitudes for states that couple to @xmath4 can be obtained from fits to the extensive existing clas @xmath4 electroproduction data over the range @xmath50 gev@xmath8 , which should be carried out independently in different bins of @xmath7 . the development of reaction models for the extraction of the @xmath5 electrocouplings from the @xmath4 electroproduction channels is urgently needed . the electrocoupling parameters determined from the data involving the pionic channels for several low - lying @xmath0 states for photon virtualities up to @xmath51 gev@xmath8 have already provided valuable information . at these distance scales , the resonance structure is determined by both meson - baryon dressing and dressed quark contributions . the @xmath0 program with the new clas12 spectrometer in hall b @xcite is designed to study excited nucleon structure up to @xmath7=12 gev@xmath8 , the highest photon virtualities ever probed in exclusive electroproduction reactions . in the kinematic domain of @xmath7 from 5 to 12 gev@xmath8 , the data can probe more directly the inner quark core and map out the transition from the confinement to the perturbative qcd domains . the @xmath0 program with clas12 consists of two approved experiments . e12 - 09 - 003 @xcite will focus on the non - strange final states ( primarily @xmath1 , @xmath2 , @xmath3 ) and e12 - 06 - 108a @xcite will focus on the strange final states ( primarily @xmath35 and @xmath36 ) . these experiments will allow for the determination of the @xmath7 evolution of the electrocoupling parameters for @xmath0 states with masses in the range up to 3 gev in the regime up to @xmath7=12 gev@xmath8 . these experiments will be part of the first production physics running period with clas12 in 2017 . the experiments will collect data simultaneously using a longitudinally polarized 11 gev electron beam on an unpolarized liquid - hydrogen target at a nominal luminosity of @xmath52@xmath53s@xmath54 . the program of @xmath0 studies with the clas12 detector has a number of important objectives . these include : 0.3 cm i ) . to map out the quark structure of the dominant @xmath0 and @xmath48 states from the acquired electroproduction data through the exclusive final states including @xmath16 , @xmath15 , @xmath17 , @xmath55 , @xmath35 , and @xmath36 . this objective is motivated by results from existing analyses such as those shown in fig . [ low - lying ] , where it is seen that the meson - baryon dressing contribution to the @xmath0 structure decreases rapidly with increasing @xmath7 . the data can be described approximately in terms of dressed quarks already for @xmath7 up to @xmath563 gev@xmath8 . it is therefore expected that the data at @xmath10 gev@xmath8 can be used more directly to probe the quark substructure of the @xmath0 and @xmath48 states @xcite . the comparison of the extracted resonance electrocoupling parameters from this new higher @xmath7 regime to the predictions from lqcd and dse calculations will allow for a much improved understanding of how the internal dressed quark core emerges from qcd and how the dynamics of the strong interaction are responsible for the formation of the @xmath0 and @xmath48 states of different quantum numbers . 0.15 cm ii ) . to investigate the dynamics of dressed quark interactions and how they emerge from qcd . this work is motivated by recent advances in the dse approach , which have provided links between the dressed quark propagator , the dressed quark scattering amplitudes , and the qcd lagrangian . dse analyses of the extracted @xmath0 electrocoupling parameters have the potential to allow for investigation of the origin of dressed quark confinement in baryons and the nature of dcsb , since both of these phenomena are rigorously incorporated into dse approaches @xcite . 0.15 cm iii ) . to study the @xmath7-dependence of the non - perturbative dynamics of qcd . this is motivated by studies of the momentum dependence of the dressed quark mass function of the quark propagator within lqcd @xcite and dse @xcite . the calculated mass function approaches the current quark mass of a few mev only in the high @xmath7 regime of perturbative qcd . however , for decreasing momenta , the current quark acquires a constituent mass of 300 mev as it is dressed by quarks and gluons . verification of this momentum dependence would further advance understanding of non - perturbative dynamics . efforts are currently underway to study the sensitivity of the proposed transition form factor measurements to different parameterizations of the momentum dependence of the quark mass @xcite . 0.15 cm iv ) . to access the quark distribution amplitudes in @xmath0 states of different quantum numbers based on the lcsr approach and relating these amplitudes to the qcd lagrangian within lqcd @xcite . 0.15 cm v ) . to offer constraints from resonance transition form factors for the @xmath57 gpds . we note that a key aspect of the clas12 measurement program is the characterization of exclusive reactions at high @xmath7 in terms of gpds . the elastic and @xmath5 transition form factors represent the first moments of the gpds @xcite , and they provide for unique constraints on the structure of nucleons and their excited states . thus the @xmath0 program at high @xmath7 represents the initial step in a reliable parameterization of the transition @xmath57 gpds and is an important part of the larger overall clas12 program studying exclusive reactions . 0.3 cm it is also important to note that the @xmath1 and @xmath21 electroproduction channels represent the two dominating exclusive channels in the resonance region . the knowledge of the electroproduction mechanisms for these channels is critically important for @xmath0 studies in channels with smaller cross sections such as @xmath35 and @xmath36 production , as they can be significantly affected in leading order by coupled - channel effects produced by their hadronic interactions in the pionic channels . the study of the spectrum and structure of the excited nucleon states represents one of the key physics foundations for the measurement program in hall b with the clas spectrometer . to date measurements with clas have provided a dominant amount of precision data ( cross sections and spin observables ) for a number of different exclusive final states for @xmath7 from 0 to 4.5 gev@xmath8 . from these data the electrocouplings of most @xmath0 states up to @xmath561.8 gev have been extracted for the first time . the @xmath0 program with the new clas12 spectrometer will extend these studies up to @xmath7 of 12 gev@xmath8 , the highest photon virtualities ever probed in exclusive reactions . this program will ultimately focus on the extraction of the @xmath5 electrocouplings for the @xmath45-channel resonances that couple strongly to the @xmath1 , @xmath2 , @xmath3 , and @xmath4 final states . these studies in concert with theoretical developments will allow for insight into the strong interaction dynamics of dressed quarks and their confinement in baryons over a broad @xmath7 range . the data will address the most challenging and open problems of the standard model on the nature of hadron mass , quark - gluon confinement , and the emergence of the @xmath0 states of different quantum numbers from qcd . this work was supported by the u.s . department of energy . the author is grateful for many lengthy and fruitful discussions on this topic with victor mokeev . the author also thanks the organizers of the hadron 2015 conference for the opportunity to present this work .

studying excited nucleon structure through exclusive electroproduction reactions is an important avenue for exploring the nature of the non - perturbative strong interaction . electrocouplings for @xmath0 states in the mass range below 1.8 gev have been determined from analyses of clas @xmath1 , @xmath2 , and @xmath3 data . this work made it clear that consistency of independent analyses of exclusive channels with different couplings and non - resonant backgrounds but the same @xmath0 electro - excitation amplitudes , is essential to have confidence in the extracted results . in terms of hadronic coupling , many high - lying @xmath0 states preferentially decay through the @xmath3 channel instead of @xmath1 . data from the @xmath4 channels will therefore be critical to provide an independent analysis to compare the extracted electrocouplings for the high - lying @xmath0 states against those determined from the @xmath1 and @xmath3 channels . a program to study excited @xmath0 decays to non - strange and strange exclusive final states using clas12 will measure differential cross sections to be used as input to extract the @xmath5 transition form factors for the most prominent @xmath0 states in the range of invariant energy @xmath6 up 3 gev in the virtually unexplored domain of momentum transfers @xmath7 up to 12 gev@xmath8 .

dense starless cores in nearby low - mass star - forming regions such as taurus represent the simplest areas in which to study the initial conditions of star formation . the dominant component of starless cores , h@xmath5 , is largely invisible in the quiescent interstellar medium , so astronomers typically rely on spectral line maps of trace molecules and continuum observations of the thermal emission from dust to derive their kinematics and physical state . however , it is now well established that different species and transitions trace different regions of dense cores , so that a comprehensive multi line observations , together with detailed millimeter and sub millimeter continuum mapping are required to understand the structure and the evolutionary status of an object which will eventually form a protostar and a protoplanetary system . previous studies of starless cores in taurus , as well as other nearby star - forming regions , have shown that the relative abundance of many molecules varies significantly between the warmer , less dense envelopes and the colder , denser interiors ( see ceccarelli et al . 2006 and di francesco et al . 2006 for detailed reviews on this topic ) . for instance , @xcite , @xcite and @xcite have shown that carbon - bearing species such as c@xmath11o , c@xmath12o , c@xmath13s , and cs are largely absent from the cores l1544 , l1498 and l1517b at densities larger than a few 10@xmath3 @xmath14 , while nitrogen - bearing species such as n@xmath5h@xmath6 and ammonia are preferentially seen at high densities . the chemical variations within a starless core are likely the result of molecular freeze out onto the surfaces of dust grains at high densities and low temperatures , followed by gas phase chemical processes , which are profoundly affected by the abundance drop of important species , in particular co ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in the past few years it has also been found that not all starless cores show a similar pattern of molecular abundances and physical structure . indeed , there is a subsample of starless cores ( often called pre - stellar cores ) , which are particularly centrally concentrated , that shows kinematic and chemical features typical of evolved objects on the verge of star formation . these features include large values of co depletion and deuterium fractionation , and evidence of `` central '' infall , i.e. presence of infall asymmetry in high density tracers in a restricted region surrounding the mm continuum dust peak @xcite . it is interesting that not all _ physically _ evolved cores show _ chemically _ evolved compositions , as shown by @xcite and @xcite . it is thus important to study in detail a larger number of cores to understand what is causing the chemical differentiation in objects with apparently similar physical ages . this is why we decided to focus our attention on tmc1c , a dense core in taurus , with physical properties quite similar to the prototypical pre stellar core l1544 ( also in taurus ) , to study possible differences and try to understand their nature . tmc1c is a starless core in the taurus molecular cloud , with a distance estimated at 140 pc @xcite . in a previous study , we have shown that tmc1c has a mass of 6 m@xmath15 within a radius of 0.06 pc from the column density peak , which is a factor of two larger than the virial mass derived from the n@xmath5h@xmath6(10 ) line width , and we have shown that there is evidence for sub - sonic inward motions @xcite as well as a velocity gradient consistent with solid body rotation at a rate of 0.3 km s@xmath16 pc@xmath16 @xcite . tmc1c is a coherent core with a roughly constant velocity dispersion , slightly higher than the sound speed , over a radius of 0.1 pc @xcite . using scuba and mambo bolometer maps of tmc-1c at 450 , 850 and 1200 , we have mapped the dust temperature and column density and shown that the dust temperature at the center of the core is very low ( @xmath176 k ) @xcite . in order to disentangle the physical and chemical information that can be gleaned from a combination of gas and dust observations of a dense core , we have now mapped tmc-1c at three continuum wavelengths @xcite and seven molecular lines . in sec . [ observations ] , continuum and line observations are described . spectra and maps are presented in sec . [ results ] . the analysis of the data , along with the discussion , has been divided in three parts : kinematics , including line width variations across the cloud , velocity gradients and inward velocities , is in sec . [ analysis_i ] ; gas and dust column density and temperature are in sect . [ analysis_ii ] ; molecular depletion and chemical processes are discussed in sect . [ analysis_iii ] . a summary can be found in sect . [ summary ] . to map the density and temperature structure of tmc1c , we have observed thermal dust emission at 450 and 850 with scuba and at 1200 with mambo2 . we observed a 10@xmath210 region around tmc-1c using scuba @xcite on the jcmt at 450 and 850 . we used the standard scan - mapping mode , recording 450 and 850 data simultaneously @xcite . chop throws of 30 , 44 and 68 were used in both the right ascension and declination directions . the resolution at 450 and 850 is 7.5 and 14 respectively . the absolute flux calibration is @xmath1712% at 450 and @xmath174% at 850 . the noise in the 450 and 850 maps are 13 and 9 mjy / beam , respectively . the data reduction is described in detail in @xcite . kauffmann et al . ( in prep . ) used the mambo-2 array @xcite on the iram 30meter telescope on pico veleta ( spain ) to map tmc1c at 1200 . the mambo beam size is 107 . the source was mapped on - the - fly , chopping in azimuth by 60 to 70 at a rate of 2 hz . the absolute flux calibration is uncertain to @xmath1710% , and the noise in the 1200 map is 3 mjy / beam . the data reduction is described in detail in kauffmann et al . ( in prep . ) . we have used the iram 30-m telescope to map out emission from several spectral lines in order to understand the kinematic and spectral structure of tmc1c . in november 1998 , we mapped the spectral line maps of the c@xmath11o(10 ) , c@xmath11o(21 ) , c@xmath12o(21 ) , c@xmath13s(21 ) , dco@xmath6(21 ) , dco@xmath6(32 ) , n@xmath5h@xmath6(10 ) transitions . the inner 2of tmc-1c were observed with 20 spacing in frequency - switching mode , and outside of this radius the data were collected with 40 sampling . the data were reduced using the class package , with second - order polynomial baselines subtracted from the spectra . the system temperatures , velocity resolution , beam size and beam efficiencies are listed in table [ iramobs ] . the spectra taken at the peak of the dust column density map are shown in figure [ iramspec ] . the integrated intensity , velocity , line width and rms noise for each transition is given in table [ fittable ] . from the figure it is evident that self absorption is present everywhere , except in and lines . clear signatures of inward motions ( brighter blue peak ; e.g. myers et al . 1997 , see also sect . [ infall ] ) are only present in the high density tracers and , which typically probe the inner portion of dense cores ( e.g. caselli et al . 2002b ; lee et al . the c@xmath13s(21 ) line appears to be self absorbed at the cloud velocity , but the spectrum is too noisy to confirm this . to highlight the extent of the `` infall '' asymmetry in ( 10 ) , fig . [ spectra_map ] shows the profile of the main hyperfine component of the ( 10 ) transition ( f@xmath18,f = 2,3 @xmath19 1,2 ) across the whole mapped area ( see sect . [ maps ] ) . we will discuss these spectra in more detail in sect . [ infall ] , but for now it is interesting to see how the profile shows complex structure , consistent with inward motion ( red boxes ) as well as outflow ( blue boxes ) and absorption from a static envelope . from fig . [ iramspec ] and [ spectra_map ] , it is clear that the main ( 10 ) hyperfine components are self - absorbed around the dust peak position . therefore , a map of the ( 10 ) intensity integrated under the seven components will not reflect the column density distribution . however , the weakest component ( f@xmath18 f = 10 @xmath19 11 ) is not affected by self - absorption , as shown in fig . [ infall_dust_peak ] , where the weakest and the main ( f@xmath18 f = 23 @xmath19 12 ) components toward the dust peak position ( the most affected by self - absorption ) are plotted together for comparison . the two hyperfines in fig . [ infall_dust_peak ] have very different profiles : the main component is blue - shifted , suggestive of inward motions ( see sect . [ infall ] ) , whereas the weak component is symmetric and its velocity centroid is red shifted compared to the main component , indicating optically thin emission . given that the self - absorption is more pronounced at the position of the dust peak , where the ( 10 ) optical depth is largest , we conclude that the weak component is likely to be optically thin across the whole tmc1c core . thus , in the case of ( 10 ) self absorption , we used the weak hyperfine component , divided by 1/27 ( its relative intensity compared to the sum , normalized to unity , of the seven hyperfines ) , to determine the ( 10 ) integrated intensity , line width , and , as shown in sec . [ analysis_ii ] , the column density . in this analysis , only spectra with signal to noise ( s / n ) for the weak component @xmath20 3 have been considered . in all other cases , ( 10 ) did not show signs of self absorption and a normal integration below the 7 hyperfines has been performed . based on hyperfine fits to the c@xmath11o(10 ) transition and a comparison of the relative strengths of the three components , we see that the c@xmath11o(10 ) emission is optically thin throughout tmc1c . although the noise is generally too high in the c@xmath11o(21 ) data to make indisputable hyperfine fits , the results of such an attempt suggest that the c@xmath11o(21 ) lines are also optically thin , which is expected for thin c@xmath11o(10 ) emission and temperatures of @xmath1710 k. thus , the integrated intensity maps of lines will reflect the column density distribution . in order to estimate the optical depth of the c@xmath12o(21 ) lines , we compare the integrated intensity of c@xmath12o(21 ) to that of c@xmath11o(21 ) . if both lines are thin , then the observed ratio should be equal to the cosmic abundance ratio , @xmath21 [ @xmath12o]/[@xmath11o ] = 3.65 @xcite . we observe that the ratio of the integrated intensities @xmath22 , which corresponds to an optical depth of the c@xmath12o(21 ) line of @xmath23 . integrated intensity maps of c@xmath11o(10 ) , c@xmath11o(21 ) , c@xmath12o(21 ) , dco@xmath6(21 ) and n@xmath5h@xmath6(10 ) are shown in fig . [ intmaps ] . note that the n@xmath5h@xmath6 integrated intensity map peaks right around the position of the dust column peaks , which is not true for c@xmath11o and c@xmath12o . we do not present integrated intensity maps of c@xmath13s(21 ) or dco@xmath6(32 ) , which have lower signal to noise . ammonia observations have shown that tmc1c is a coherent core , having a constant line width across the core at a value slightly higher than the thermal width , and increasing outside the `` coherent '' radius , @xmath170.1 pc @xcite . our n@xmath5h@xmath6 observations of tmc1c show that the line width remains constant , at a value @xmath172 times higher than the thermal line width , over the entire core ( see fig . [ fwhmbins ] for a map of the line width and a plot of line width vs. radius ) , though the dispersion in the n@xmath5h@xmath6 line width is very large . this result is in agreement with the nh@xmath24 observations of tmc1c , and is not consistent with the decreasing n@xmath5h@xmath6 and n@xmath5d@xmath6 linewidths at larger radii seen in l1544 and l694 - 2 , which in other important ways ( density and temperature structure , velocity asymmetry seen in ( 10 ) ) closely resemble tmc1c . to make sure that the lack of correlation of the ( 10 ) line width is not due to geometric effects , considering the elongated structure of tmc1c , we also plotted the ( 10 ) line width as a function of antenna temperature , and found similar results . this behavior may be due to the coherence of the central portion of the core , which has nearly constant length along the line of sight , and thus the velocity dispersion comes from regions of the core that have similar scales ( see fig . 4 in goodman et al . cores formed by compressions in a supersonic turbulent flow naturally develop these regions of constant length at their centers @xcite . another reason that the n@xmath5h@xmath6 line widths appear constant across the cloud could be the different `` infall '' velocity profile , with the velocity peaking farther away from the dust peak than in l1544 and l694 - 2 ( though the projected velocity would still be at its maximum at the dust peak ) . in the case of l1544 , @xcite showed that the ( 10 ) line profile is consistent with the @xcite model at a certain time in the cloud evolution , where the `` infall '' velocity profile peaks at a radius of about 3,000 au ( see also myers 2005 for alternative models with similar radial velocities ) . indeed , in sec . [ infall ] we show that the extent of the asymmetry seen in ( 10 ) suggests that the peak of the inward motions is at about 7,000 au , so that one does not expect to see broader line widths within this radius . in fact , the binned data in fig . [ fwhmbins ] show a hint of a peak at about 50 ( 7000 au at the distance of taurus ) . in order to compare the thermal and non - thermal line widths in tmc-1c , we assume that the gas temperature is equal to 10 k and use the formulae : @xmath25 @xmath26 where @xmath27 is the molecular weight of the species and @xmath28 is the mass of hydrogen . figure [ fwhmratio ] shows the non - thermal line width plotted against the thermal line width at the position of the dust column density peak . to allow a fair comparison , all the data in the figure have been first spatially smoothed at the same resolution of the map ( 1 ) . although c@xmath11o , c@xmath12o , dco@xmath6 and n@xmath5h@xmath6 all have similar molecular weights , they have significantly different values for their non - thermal line widths . the thermal line width is much smaller than the non - thermal line width for the molecules c@xmath11o and c@xmath12o , while the ratio is closer to unity for dco@xmath6 and n@xmath5h@xmath6 . this suggests that the isotopologues of co are tracing material at larger distances from the center , with a larger turbulent line width , than are dco@xmath6 and n@xmath5h@xmath6 , which presumably are tracing the higher density material closer to the center of tmc-1c . the c@xmath12o(21 ) line is slightly thick , and this is probably the reason of its slightly larger line width when compared to the thin c@xmath11o(21 ) line , as shown in figure [ fwhmratio ] . for each transition observed , we see no clear correlation between the observed line width and the thermal line width ( and therefore with temperature , column density and distance from the peak column density , see section [ dustcolumn ] ) . as in the study of depletion , the lower signal to noise in c@xmath13s(21 ) , dco@xmath6(21 ) and dco@xmath6(32 ) make any possible trends between @xmath29 and @xmath30 more difficult to determine . from fig . [ fwhmratio ] we note that nh@xmath24 and n@xmath5h@xmath6 have similar non - thermal line widths , which makes sense given that and are expected to trace similar material ( e.g. benson et al . 1998 ) . however , this result is in contrast with the findings of @xcite who found narrower line widths towards l1498 and l1517b . we finally note that the line widths that we measure in c@xmath11o and c@xmath12o are larger than the n@xmath5h@xmath6 linewidths throughout tmc1c , which contrasts with the results seen in c@xmath12o and n@xmath5h@xmath6 in b68 @xcite . this is consistent with the fact that tmc1c , unlike b68 , is embedded in a molecular cloud complex and it is not an isolated core . thus , co lines in tmc1c also trace the ( lower density and more extended ) molecular material , part of the taurus complex , where larger ranges of velocities are present along the line of sight . in order to study the velocity field of tmc-1c , we determine the centroid velocities for c@xmath11o(21 ) , c@xmath12o(21 ) , c@xmath13s(21 ) , dco@xmath6(21 ) and dco@xmath6(32 ) with gaussian fits . the centroid velocities of the c@xmath11o(10 ) and n@xmath5h@xmath6(10 ) lines are determined by hyperfine spectral fits . for those n@xmath5h@xmath6 spectra that show evidence of self - absorption , the velocity is derived from a gaussian fit to the thinnest component . the velocity gradient at each position is calculated by fitting the velocity field with the function : @xmath31 where @xmath32 is the bulk motion along the line of sight , @xmath33 and @xmath34 are ra and dec offsets from the position of the central pixel , @xmath35 is the magnitude of the velocity gradient in the plane of the sky , and @xmath36 is direction of the velocity gradient . the fit to the velocity gradient is based on fitting a plane through the position position velocity cube as in @xcite ( for the `` total '' gradient across the cloud ) and in @xcite ( for the `` local '' gradient at each position ) . the fit for the `` total '' velocity gradient gives a single direction and magnitude for the entire velocity field analyzed . the `` local '' velocity gradient is calculated at each position in the spectral line maps based on the centroid velocities of the center position and its nearest neighbors , with the weight given to the neighbors decreasing exponentially with their distance from the central position . analysis of ammonia observations with @xmath37 resolution of tmc1c indicate an overall velocity gradient of 0.3 km s@xmath16 pc@xmath16 directed 129 degrees east of north @xcite . the velocity field that we measure in tmc1c has spatial resolution three times greater ( @xmath1720 ) than the ammonia study , and reveals a pattern more complicated than that of solid body or differential rotation . the velocity fields measured by c@xmath11o(10 ) , c@xmath11o(21 ) and c@xmath12o(21 ) are shown in fig . [ coarrows ] . although there is a region that closely resembles the velocity field expected from rotation ( gradient arrows of approximately equal length pointing in the same direction ) , the measured velocities vary from blue to red to blue along a nw - se axis . the n@xmath5h@xmath6(10 ) velocity fields ( shown in fig . [ coarrows ] ) also follow the same blue to red to blue pattern along the nw - se axis , but the observations cover a somewhat different area than the co observations , which complicates making a direct comparison . taken as a whole , it is clear that there is an ordered velocity field in portions of the tmc-1c core , and that the lower density co tracers `` see '' a velocity field similar to that probed by n@xmath5h@xmath6 lines , which trace higher density material . in any case , the velocity field that looks like rotation reported in @xcite turns out to be more complicated when seen over a larger area with finer resolution . the direction and magnitude of the velocity gradient in the region that resembles solid body rotation is shown in fig . [ centergrad ] for each transition . to quantify the velocity of the inward motion from the ( 10 ) line across the tmc1c cloud , we use a simple two layer model , similar to that described by @xcite . this model assumes that the cloud can be divided in two parts with uniform excitation temperature ( @xmath38 , gradients in @xmath38 between the two layers as in de vries & myers 2005 , are not considered here ) , line width ( @xmath29 ) , optical depth ( @xmath39 ) and lsr velocity ( @xmath40 ) and that the foreground layer has a lower excitation temperature . for simplicity , we also assume that the seven hyperfines have the same @xmath38 , which is a very rough assumption in regions of large optical depth , as recently found by @xcite . despite of the simplicity of the model , we find good fits to the seven hyperfine lines and determine the value of the velocity difference between the two layers , which can be related to the `` infall '' velocity . in fig . [ two_layer ] we present five spectra which represent a cut across the major axis of the core , passing through the dust peak . for display purposes , the spectra have been centered to 0 velocity , subtracting the lsr velocity obtained from a gaussian fit to the weak hyperfine component ( for offsets [ -40,60 ] , [ -20,40 ] , [ 0,20 ] , where the self - absorption in present ) or hfs fits in class ( for offsets [ 20,0 ] and [ 40,-20 ] ) . the @xmath41 velocity is shown in the top right of each panel . the cut is from south - east ( offset [ 40,-20 ] , see fig . [ spectra_map ] ) to north - west ( offset [ -40,60 ] ) . the first thing to note in the figure is that clear signs of self - absorption and asymmetry are present toward the dust peak and in the north - west , but not in the two southern positions . this trend can also be seen as a general feature in fig . [ spectra_map ] , where it is evident that asymmetric lines are more numerous north west of the dust peak . the excitation temperature , total optical depth , line width and the velocity ( @xmath42 , see fig . [ two_layer ] ) of the foreground ( f ) and background ( b ) layers are reported in table [ table_layer ] . to find the best fit parameters , we first performed an hfs fit to the [ 20,0 ] spectrum , which is the closest spectrum to the dust peak not showing self - absorption . the values of @xmath43 , @xmath38 and line width obtained from this fit have been adopted for the background emission at the dust peak position and the best fit has been found by adding the foreground layer and minimising the residuals . for the two spectra north west of the [ 0,20 ] position , adjustment to the parameters of the background layer were necessary to obtain a good fit . we point out that the five spectra we chose for this analysis are representative of the whole area surrounding the tmc1c dust peak , where a mixture of symmetric , blue shifted and red shifted spectra are present . as already stated , the majority of the asymmetric spectra show inward motions and extend over a region with radius @xmath177000 au ( see fig . [ spectra_map ] ) . the properties derived for the foreground layer ( @xmath38 @xmath13.33.5 k , @xmath43 @xmath11015 , and @xmath440.2 ) have been used as input parameters in a large velocity gradient ( lvg ) codemoldata / radex.php ] for a uniform medium and found to be consistent with the ( 10 ) tracing gas at a density @xmath8 @xmath1 5@xmath210@xmath4 , kinetic temperature @xmath45 @xmath1 10 k and with column density @xmath46 @xmath1 5@xmath210@xmath47 , values comparable to those found for the background layer ( see table [ fdtable ] and sec . [ gascolumn ] ) . it is interesting that the maximum of the line - of - sight component of the inward velocity ( @xmath170.15 ) is found toward the dust peak , whereas one pixel away from it , the inward velocity drops to 0.05 @xmath48 . this is suggestive of a geometric effect , in which the inward velocity vector is directed toward the dust peak , so that only a fraction @xmath49 ( with @xmath36 the angle between the l.o.s . and the infall velocity direction ) of the total velocity is directed along the line of sight in those positions away from the dust peak . of course , our simplistic model prevents us to go further than this , i.e. the uncertainties are too large to build a 3d model of the velocity profile within the cloud . as shown in fig . [ spectra_map ] , in the north west end of the tmc1c core ( around offset [ -250,150 ] ) , there are other signatures of inward motions , which may indicate the presence of another gravitational potential well . this suggestion is indeed reinforced by fig . 8 of @xcite , which shows high extinction and low temperatures in the same direction . unfortunately , the continuum coverage is not good enough to attempt a detailed analysis , but it appears evident that the extension toward the north west is another dense core connected to the main tmc1c condensation with lower density and warmer gas and dust . to derive the column density of gas from each molecule , we assume that all rotation levels are characterized by the same excitation temperature @xmath50 ( the ctex method , described in caselli et al . 2002b ) . in case of optically thin emission , @xmath51 where @xmath52 and @xmath53 are the wavelength and frequency of the transition , @xmath54 is the boltzmann constant , @xmath55 is the planck constant , @xmath56 is the einstein coefficient , @xmath57 and @xmath58 are the statistical weights of the lower and upper levels , @xmath59 and @xmath60 are the equivalent rayleigh - jeans excitation and background temperatures , w is the integrated intensity of the line . the partition function ( @xmath61 ) and the energy of the lower level ( @xmath62 ) for linear molecules are given by : @xmath63 and @xmath64 and @xmath65 is the rotational constant ( see table [ constants5 ] for the values of the constants ) . the ( 10 ) and ( 21 ) lines have hyperfine structure , enabling the measurement of the optical depth . we find that the lines are optically thin throughout the core . to determine the column density , we assume an excitation temperature of 11 k , which is the average value of @xmath38 found from our data around the dust peak position , as explained in sect . [ gastemp ] . in the case of ( 21 ) lines , we correct for optical depth before determining the column density , using the correction factor : @xmath66 as explained in sect . [ results_spectra ] , ( 10 ) lines show clear signs of self absorption in an extended area around the dust peak . to determine the column density across the core , first we select spectra without self absorption , and those with high s / n ( i.e. with @xmath67 @xmath20 20 , with @xmath68 @xmath69 integrated intensity ; see caselli et al . 2002b ) have been fitted in class to find @xmath38 and @xmath39 . the mean value of @xmath38 found with this analysis ( 4.4@xmath700.1 k ) has been used for all other positions where an independent estimate of @xmath38 was not possible ( i.e. for self - absorbed or thin lines ) . for optically thin ( 10 ) transitions , the intensity was integrated below the seven hyperfine and the expression ( [ ntot ] ) used to determine the total column density . in cases of self - absorbed spectra , the column density has been estimated from the integrated intensity of the weakest ( and lowest frequency ) hyperfine component ( @xmath71 = 1 0@xmath191 1 ) , using eq . [ ntot ] ( assuming @xmath38=4.4 k ) and multiplying by 27 ( the inverse of the hyperfine relative intensity ) . the weakest component is not affected by self absorption , as shown in fig . [ infall_dust_peak ] , suggesting that its optical depth is low . we have checked that these two different methods approximately give the same results by measuring the column density with both procedures in those cases where self - absorption is not present and where the weakest hyperfine component has a s / n ratio of at least 4 . we found that the two column density values agree to within 10% . ( 21 ) and ( 32 ) lines are clearly self absorbed and the column density determination is very uncertain ( given that there are no clues about their optical depth and excitation temperature ) . the estimates listed in table [ fdtable ] should be considered lower limits . @xmath38 = 4.4 k has beed assumed , based on the fact that the lines are expected to trace similar conditions than . c@xmath13s(21 ) spectra have low sensitivity , and the lines are affected by self absorption ( see fig . [ iramspec ] ) , so the derived c@xmath13s column density is highly uncertain . the abundance , @xmath72 ( @xmath69 @xmath73 ) , toward the dust peak is 1.6@xmath210@xmath10 , identical ( within the errors ) to that derived toward the l1544 dust peak ( * ? ? ? this is interesting considering that @xmath74 in tmc1c is 1.6 times lower than in l1544 , in which closely follows the dust column ( as already found in previous work ) . however , unlike l1544 , where the abundance appears constant with impact parameters ( e.g. tafalla et al . 2002 and vastel et al . 2006 ) , in tmc1c the abundance increases away from the dust peak by a factor of about two within 50 , as shown in fig . [ cut ] ( see also fig . [ iramfdnh ] in sect . [ smol ] ) . [ cut ] displays @xmath75 ( see sect . [ dustcolumn ] ) , @xmath76 and @xmath72 , normalized to the corresponding maximum values ( 63.2 mag , 1.1@xmath210@xmath77 , and 4.8@xmath210@xmath10 , respectively ) in two cuts ( one in right ascension and one in declination ) passing through the dust peak . one point to note is that the abundance derived at the dust peak ( marked by the black dotted line ) is the _ minimum _ value observed , indicating moderate ( factor of @xmath172 ) depletion . in @xcite we used scuba and mambo maps at 450 , 850 and 1200 to create column density and dust temperature maps of tmc1c . in this paper we smooth the dust continuum emission maps to the 20 spacing of the iram maps and then derive @xmath75 and @xmath78 to facilitate a direct comparison of the gas and dust properties . at each position , we make a non - linear least squares fit for the dust temperature and column density such that the difference between the predicted and observed 450 , 850 and 1200 observations is minimized . the errors associated with such a fitting procedure are described in @xcite . dust column density and temperature maps of tmc1c are shown in fig . 8 of @xcite . from fig . 8 in @xcite , it is clear that there is an anti - correlation between extinction and dust temperature ( as also predicted by theory , e.g. evans et al . 2001 , zucconi et al . 2001 , galli et al . 2002 ) . to better show this , the two quantities are plotted in fig . [ tdvsav ] . the data in fig . [ tdvsav ] are not smoothed to the iram 30 m beam at 3 mm , since this is only a dust property intercomparison and does not refer to the gas properties . higher @xmath75 ( 60 @xmath79 90 mag ) and lower dust temperatures ( 5 @xmath80 6 k ) are detectable at the higher resolution . we compare the @xmath81 relationship seen in tmc1c with that predicted by @xcite for an externally heated pre - protostellar core ( the solid red line in fig . [ tdvsav ] ) . we find that at high column density ( @xmath82 ) the observed dust temperature in tmc1c is lower than that of the model core , while at low column density ( @xmath83 ) the observed dust temperature is higher than the model predicts . however , given that the model predicts the dust temperature at the center of a _ spherical _ cloud , and that the geometry of tmc1c is certainly not spherical , only a rough agreement between the model and observations should be expected . because of its low dipole moment , co is a good gas thermometer , given that it is easily thermalized at typical core densities . however , it is now well established that co is significantly frozen onto dust grains at densities @xmath84 10@xmath85 @xmath14 ( one exception being l1521e ; tafalla & santiago , 2004b ) and this is also the case in tmc1c . therefore , at the dust peak we do not expect to measure a gas temperature from co of @xmath177 k , but instead a higher value reflecting the temperature in the outer layers of the cloud . the lines available for this analysis are : c@xmath11o(10 ) , c@xmath11o(21 ) , and c@xmath12o(21 ) . the c@xmath11o(21)/c@xmath12o(21 ) brightness temperature ratio has been used to derive the excitation temperature of the c@xmath12o line , which is coincident with the kinetic temperature if the line is thermalized . to test the hyposthesis of thermalization we use an lvg ( large - velocity - gradient ) program to determine at which volume density and kinetic temperature the observed c@xmath11o(10 ) and c@xmath11o(21 ) brightness temperatures can be reproduced . we use a one - dimensional non - lte radiative transfer code @xcite available at http://www.strw.leidenuniv.nl/ moldata / radex.html . these two lines have similar frequencies , so the corresponding angular resolution is almost identical and no convolution is needed . following a similar analysis done with the j=10 transition of the two co isotopologues ( myers et al . 1983 ) , the optical depth of the c@xmath12o(21 ) line ( @xmath86 ) can be found from : @xmath87}{t_{\rm mb}[\cseo ( 2 - 1 ) ] } & = & 3.65 \times \frac{1 - e^{-\tau_{18}}}{\tau_{18 } } , \end{aligned}\ ] ] where @xmath88 $ ] is the main beam brightness temperature of transition @xmath89 ( assuming a unity filling factor ) . the last term in the right hand side is the optical depth correction which is used to determine the total column density of in a plane parallel geometry , which most likely applies to co emitting regions , i.e. the external core layers . once @xmath86 is measured , the excitation temperature ( @xmath90 ) of the corresponding transition ( thus the gas kinetic temperature , if the line is in local thermodynamic equilibrium ) can be estimated from the radiative transfer equation : @xmath91 ( 1 - e^{-\tau } ) , \label{erad}\end{aligned}\ ] ] where @xmath92 and @xmath93 are the equivalent reyleigh jeans temperatures , with @xmath94 @xmath95 = @xmath96 , and @xmath53 the frequency of the ( 21 ) line ( see table [ iramobs ] ) . figure [ ftemp ] ( left panel ) shows the results of this analysis . the set of data points in fig . [ ftemp ] is limited to only those spectra with @xmath97/t_{\rm rms}$ ] @xmath84 10 , to avoid scatter due to noise . the error associated with the gas temperature has been calculated by propagating the errors on @xmath86 and @xmath98[(21 ) ] into eq . [ erad ] and its expression is given in the appendix . it is interesting to note that the ( 21 ) excitation temperature appears to decrease away from the center , and , if this line is thermalized , it suggests that the kinetic temperature also drops away from the center , in contrast with the dust temperature . indeed , the two quantities are completely uncorrelated in the 12 common positions with large s / n ( 21 ) spectra ( not shown ) . @xmath38 is close to 12 k at the core center , whereas it drops to 910 k one arcmin away from the dust peak . is this drop due to a decreasing gas temperature , as recently found by @xcite in the bok globule b68 ? unlike b68 , we believe that our result is due to the volume density decrease . indeed , the critical densities of the j = 21 lines of and are a few @xmath2 10@xmath99 , so that only if the volume density traced by one of the two isotopologues is larger than , say , 5@xmath210@xmath3 can the j = 21 lines be considered good gas thermometers . in the next subsection , we investigate this point more quantitatively . in fig . [ ftemp ] ( right panel ) , the brightness temperature ratio of the ( 10 ) and ( 21 ) lines is plotted as a function of distance from the dust peak . because of the different angular resolutions at the 21 and 10 frequencies , the 1 mm data have been smoothed to the 3 mm resolution and both data cubes have then been regridded , to allow a proper comparison . the ratio is indeed increasing towards the edge of the cloud , consistent with our previous finding of a @xmath38[(21 ) ] drop in the same direction ( see left panel ) . both ( 10 ) and ( 21 ) lines possess hyperfine structure , which provides a direct estimate of the line optical depth . using the hfs fit procedure available in class , we found that all over the tmc1c cloud , both lines are optically thin . this means that it is not possible to derive the excitation temperature in an analytic way , so we use the lvg code introduced in sect . [ gastemp ] . this code assumes homogeneous conditions , which is likely to be a good approximation for the region traced by co isotopologues . in fact , because of freeze - out , co does not trace the regions with densities larger than about 10@xmath85 @xmath14 ( see below and sect . [ smol ] ) , so that the physical conditions traced by co around the dust peak are likely to be close to uniform ( n(h@xmath5 ) @xmath17 a few times 10@xmath3 and about constant temperature ) . this is also supported by the integrated intensity co maps , which appear extended and uniform around the dust peak ( see fig . [ intmaps ] ) . to better understand this result , the lvg code has been run to see how changes in volume density and gas temperature affect the line ratio . this is shown in fig . [ flvg ] , where the top panel shows the @xmath98 ratio as a function of @xmath45 for a fixed value of the volume density ( @xmath8 = 2@xmath210@xmath3 ) , a column density of 10@xmath100 ( as found in section [ gascolumn ] ) , and a line width of 0.4 , as observed . the horizontal dashed lines enclose the range of @xmath98 ratios observed in tmc1c and reported in fig . [ ftemp ] . thus , the observed @xmath98 range ( at this volume density ) corresponds to a range of gas temperature between 11 k toward the dust peak and @xmath177 k away from it , thus confirming our previous findings of a decreasing ( 21 ) excitation temperature away from the dust peak . in the bottom panel of fig . [ flvg ] , the same brightness temperature ratio is plotted as a function of @xmath8 , for a fixed kinetic temperature ( @xmath45=11 k ) , @xmath101 = 10@xmath100 and @xmath29=0.4 , as before . the black curve shows this variation and , not surprisingly , the observed range of @xmath98 ratios can also be explained if the volume density ( traced by the lines ) decreases from @xmath102 toward the dust peak to @xmath103 away from it ( the point farthest away being at a projected distance of 80 , or 11,000 au , see fig . [ ftemp ] ) . note that the volume density traced by the line towards the dust peak is significantly lower than the central density of tmc-1c ( @xmath15@xmath210@xmath85 ; see schnee et al . 2007 ) , once again demonstrating that co is not a good tracer of dense cores . the bottom panel of fig . [ ftemp ] is consistent with a volume density _ decrease _ away from the dust peak , or , more precisely , a lower fraction of ( relatively ) dense gas intercepted by the lines along the line of sights . in the same plot , the red curves show the excitation temperatures of the ( 10 ) and ( 21 ) lines vs. @xmath8 . note that @xmath38[(21 ) ] = @xmath45 only when the density becomes larger than @xmath1710@xmath85 . thus , the ( 21 ) line is sub - thermally excited in tmc1c . in summary , the rise in the ( 10)/(21 ) brigthness temperature ratio away from the dust peak ratio can be caused by _ either _ a gas temperature decrease _ or _ a volume density decrease ( or both ) . considering that the dust ( and likely the gas ; see the recent paper by crapsi et al . 2007 ) temperature is clearly increasing away from the dust peak , we believe that the drop in @xmath38 observed both using the and lines is more likely due to a drop in the volume density traced by these species . this is reasonable in the case of a core embedded in a molecular cloud complex , such as tmc1c , where the fraction of low density material intercepted along the line of sight by and observations is significantly larger than in isolated bok globules such as b68 ( see bergin et al . 2006 ) . in any case , a detailed study of the volume density structure of the outer layers of dense cores will definitely help in assessing this point . by comparing the integrated intensity maps of co isotopologues and n@xmath5h@xmath6(10 ) ( fig . [ intmaps ] ) in tmc-1c with the column density implied by dust emission ( fig . 8 in @xcite ) , we see that at the location of the dust column density peak the co emission is not peaked at all . the n@xmath5h@xmath6(10 ) emission peaks in a ridge around the dust column density maximum , not at the peak , but in general n@xmath5h@xmath6traces the dust better than the c@xmath12o emission does . below , we measure the depletion of each observed molecule and compare our results to similar cores . previous molecular line observations of starless cores such as l1512 , l1544 , l1498 and l1517b , consistently show that co and its isotopologues are significantly depleted , e.g. @xcite . however , other molecules such as dco@xmath6 and n@xmath5h@xmath6 are typically found to trace the dust emission well , ( e.g. caselli et al . 2002b ; tafalla et al . 2002 , 2004 ) , although there is some evidence of their depletion in the center of chemically evolved cores , such as b68 @xcite , l1544 @xcite and l1512 @xcite . in order to measure the depletion in tmc-1c , we define the depletion factor of species @xmath89 : @xmath104 where @xmath105 is the `` canonical '' ( or undepleted ) fraction abundance of species @xmath89 with respect to h@xmath5 ( see table [ constants5 ] ) , @xmath106 is the column density of molecular hydrogen as derived from dust emission , and @xmath107 is the column density of the molecular species as derived in section [ gascolumn ] . the derived depletion factors for each molecule ( except for and c@xmath13s , where the column density determination is quite uncertain , as explained in sec . [ gascolumn ] ) in each position with signal to noise @xmath20 3 are plotted against the dust - derived column density in figure [ iramfdnh ] . the typical random error in the derived depletion is shown in each panel , and is derived from the noise in the spectra . uncertainty in the derived column density from dust emission is dominated by calibration uncertainties in the bolometer maps , and is not included in this calculation , nor is the uncertainty in the calibration of the spectra ( @xmath1720% ) , which would adjust the derived depletion factors systematically , but would not alter the observed trends . depletions factors are found to increase with higher dust colunm density ( fig . [ iramfdnh ] ) . in tmc-1c we see a linear relationship between c@xmath11o and c@xmath12o depletion and dust - derived column density , which has also been seen in c@xmath12o by @xcite in the core l1521f , which contains a very low luminosity object @xcite . to check the impact of resolution on the derived depletion , we compare the c@xmath12o(21 ) depletion when smoothed to 20 ( the resolution of the c@xmath11o(10 ) data ) with that derived from smoothing c@xmath12o(21 ) to 14 ( the resolution of the bolometer data ) . we find no systematic difference between the two calculations of the depletion , and a 13% standard deviation in the ratio of the derived depletions . the depletion factor and column density at the position of the dust peak is listed in table [ fdtable ] for each tracer . the depletion factor measured in clearly follows a different trend compared to the co isotopologues . first of all , in fig . [ iramfdnh ] the depletion factor is allowed to have values below 1 , because of our ( arbitrary ) choice for the `` canonical '' abundance of assumed here to be equal to 1.4@xmath108 , the average value across tmc1c . we point out that a `` canonical '' abundance for is much harder to derive than for co , because lines are much harder to excite ( and thus detect ) in low density regions where depletion is negligible . nevertheless , fig . 20 shows that the depletion factor monotonically increases ( as in the case of co ) for n(h@xmath5 ) @xmath109 @xmath110 ( or a@xmath111 mag ) . _ this is clear evidence of depletion in the core nuclei , in a central region with radius @xmath176000 au , where a@xmath111 mag _ ( see also fig . [ cut ] ) . the dispersion in the depletion factor vs. n(h@xmath5 ) relation is very large with no obvious trend at lower a@xmath112 values ( n(h@xmath5 ) @xmath113 @xmath110 ) , which we believe is due to our choice of excitation temperature where the ( 10 ) line is optically thin . as explained in sect . [ gascolumn ] , in the case of optically thin lines , t@xmath114 has been assumed equal to 4.4 k , the mean value derived from the optically thick spectra which do not show self - absorption ( and which trace regions with a@xmath115 mag ) . therefore , in all positions with n(h@xmath5 ) below 2@xmath116 @xmath110 , where the volume density is also likely to be low , the assumed ( 10 ) excitation temperature is likely to be an overestimate of the real t@xmath114 . to see if this can indeed be the cause of the observed scatter , consider a cloud with kinetic temperature of 10 k , volume density of 3@xmath117 @xmath14 , line width of 0.3 km s@xmath16 and an excitation temperature of 4.4 k for the ( 10 ) line . using the radex lvg program , this corresponds to a column density of 10@xmath47 @xmath110 . if the density drops by a factor of two ( whereas all the other parameters are fixed ) , the ( 10 ) excitation temperature drops to 3.6 k. in these conditions , using t@xmath114 = 4.4 k instead of 3.6 k , in our analytic column density determination ( see sect . [ gascolumn ] ) , implies underestimating n ( ) by 50% . therefore , our assumption of constant t@xmath114 can be the main cause of the observed f@xmath118 scatter at low extinctions . because of the anti - correlation between dust temperature and column density ( see fig . [ tdvsav ] ) , we expect that there will also be an anti - correlation between the depletion factor , @xmath119 , and dust temperature . figure [ iramfdnh ] ( right panels ) shows the depletion factor for each molecule plotted against the line - of - sight averaged dust temperature . as expected , the depletion is highest in the low temperature regions though the lower signal to noise in c@xmath13s and dco@xmath6 make this somewhat harder to see . the anti - correlation between the depletion factor and dust temperature has also been seen by @xcite in ic5146 in c@xmath12o , though in tmc-1c the temperatures are somewhat lower . our data clearly suggest that there is an increasing depletion of with increasing column ( and volume ) density . in previous work ( e.g. tafalla et al . 2002 , 2004 ; vastel et al . 2006 ) , the observed abundance appears constant across the core , although the data are also consistent with chemical models in which the abundance decreases by factors of a few @xcite . @xcite also deduce small depletion factors for when comparing data to models and @xcite found clear signs of depletions at densities above @xmath1710@xmath85 . there is also evidence of depletion towards the class 0 protostar iram 04191 + 1522 @xcite in taurus . the average abundance that we find in tmc-1c , relative to h@xmath5 , is 1.4@xmath120 . what appears to be different from previous work is that the co depletion factor towards the dust peak is relatively low ( compared to , e.g. , l1544 ) , and , _ at the same time _ , ( 10 ) lines are bright over an extended region . if tmc1c were chemically young ( such as l1521e ; tafalla & santiago 2004 , hirota et al . 2002 ) , then there would be negligible co freeze out and low abundances of , given that is a `` late - type '' molecule ( i.e. its formation requires significantly longer times ( factors @xmath2010 ) than co and other c bearing species ) . in tmc1c we observe moderate co _ and _ depletions , as well as extended emission with derived fractional abundances around 10@xmath10 . to derive an approximate value of the average gas number density of the region where emission is present , we first sum all the observed ( 10 ) spectra ( over the whole mapped area , with size @xmath1450@xmath2170 , corresponding to a linear geometric mean of about 40,000 au ; see fig . [ coarrows ] ) and then perform an hfs fit in class to derive the excitation temperature . we find @xmath38 = 3.6@xmath700.02 k and @xmath46 = 4.84@xmath700.03 @xmath121 , which can be reproduced with the lvg code if @xmath8 = 5@xmath210@xmath4 and @xmath45 = 11 k , as found in previous sections ( see sect . [ infall ] and [ gastemp ] ) . the average value of the extinction across the whole tmc1c core is 23 mag , so that the corresponding abundance is 2@xmath108 , close to the average value found before . how long does it take to form with fractional abundances of @xmath122 in regions with volume densities @xmath123 @xmath14 ? @xcite derive times @xmath124 yr at n(h@xmath5 ) = 10@xmath3 @xmath14 , so that this can be considered _ a lower limit to the age of the tmc1c core_. in summary , all the above observational evidence suggests that the majority of the gas observed towards tmc1c has been at densities @xmath1710@xmath3 @xmath14 for at least a few times 10@xmath85 yr and that material is accreting toward the region marked by the mm dust emission peak . we finally note that the density profile of the region centered at the dust peak position is _ steeper _ ( consistent with a power law ; see fig . 13 of schnee & goodman 2005 ) than found in other cores , so that bonnor - ebert spheres may not be the unique structure of dense cores in their early stages of evolution . although tmc1c is more massive than l1544 by a factor of about two , the physical structures of the two cores are similar : the central density of tmc1c is @xmath125 ( factor of @xmath17 2 lower than l1544 , according to tafalla et al . 2002 ) and the central temperature is @xmath177 k , similar to the dust temperature deduced by evans et al . ( 2001 ) and zucconi et al . ( 2001 ) in the center of l1544 , and close to the gas temperature recently measured by crapsi et al . ( 2007 ) , again toward the l1544 center . however , the chemical characteristics of the two cores appear quite different . in tmc1c : 1 . the observed co depletion factor is about 4.5 times smaller than in l1544 ( see section [ smol ] and @xcite ) ; 2 . the deuterium fractionation is three times lower @xcite than in l1544 ; 3 . the column density at the dust peak is two times lower and the column density is 1.7 times larger than in l1544 @xcite . all this is consistent with a younger chemical ( and dynamical ) age @xcite . to understand this chemical differentiation in objects in apparently similar dynamical phases , we used the simple chemical model originally described in @xcite and more recently updated by @xcite . the model consists of a spherical cloud with density and temperature gradients as determined by @xcite . the model starts with , , co and o in the gas phase , a gas to dust mass ratio of 100 , and a mathis , rumpl & nordsiek ( 1977 ; mrn ) grain size distribution . molecules and atoms are allowed to freeze out onto dust grains and desorb via cosmic ray impulsive heating @xcite . the adopted binding energies of co and are 1100 k and 982.3 k , respectively . the co binding energy is intermediate between the one measured for co onto ( i ) icy mantles ( 1180 k ; collings et al . 2003 and fraser et al . 2004 ) and ( ii ) co mantles ( 885 k ; berg et al . the adopted value ( 1100 k ) is the weighted mean of the two measured values , assuming that water is about four times more abundant than co in the taurus molecular cloud ( see table 2 of ehrenfreund & charnley 2000 and references therein ) . see @xcite for adsorption onto icy mantles . for the atomic oxygen binding energy we used 750 k , as in @xcite . the following parameters have also been assumed from @xcite : ( i ) the cosmic ray ionization rate ( 1.3@xmath210@xmath126 s@xmath16 ) ; ( ii ) the minimum size of dust grains ( @xmath127 = 5@xmath128 cm ) ; ( iii ) the `` canonical '' abundance of co ( 9.5@xmath210@xmath129 , from @xcite ; ( iv ) the sticking coefficient ( @xmath130=1 , as recently found by @xcite for co and ) ; ( v ) the initial abundance of equal to 4@xmath131 , i.e. about 50% the total abundance of nitrogen observed in the interstellar medium @xcite ; ( vi ) the initial abundance of `` metals '' ( m@xmath6 , in fig . [ fchem ] ) of 10@xmath132 ( from @xcite ) ; and ( vii ) the initial abundance of oxygen , fixed at a half the canonical abundance of co ( i.e. 13 times lower than the cosmic abundance ; meyer et al . 1998 ) . the model is run until the column density toward the center of the cloud reaches the observed value ( @xmath133 = 8@xmath210@xmath4 yr ) . during this time , the abundance of molecular ions is calculated within the cloud using steady state chemical equations with the instantaneous abundances of the neutral species . to determine @xmath134 , the reaction scheme of @xcite is used , where the abundance of the generic molecular ion `` mh@xmath6 '' ( essentially the sum of , , and their deuterated forms ) is calculated ( see @xcite for more details ) . the calculated abundance profiles of the various species have then been convolved with the hpbws of the 30 m antenna at the corresponding frequencies and the derived column densities are in very good agreement with the observed quantities ( within factors of 2 for , and , of course , co isotopologues ) , which is very encouraging , considering the simplicity of the model . the best - fit chemical structure of tmc1c , reached after 10,000 yr , is shown in the left panel of fig . [ fchem ] . note that despite the similar binding energies of co and , the and drops are steeper than those of and , which is due to the fact that the co freeze out ( although lower than in l1544 ) enhances the production rate , as pointed out by previous chemical models @xcite . finally , we note that the co depletion factor _ within _ the cloud , @xmath135(co)(co ) , used here to indicate the co depletion _ within _ the cloud , should not be confused with @xmath136(co ) , the _ observed _ ( or integrated - along - the - line - of - site ) co depletion factor ( @xmath136 = @xmath137 ; see also crapsi et al . 2004 ) . ] , is significantly lower than in l1544 at radii @xmath138 5,000 au , which reflects the different density profile ( see right bottom panel in fig . [ fchem ] ) . the present data , together with previous work , show that there are significant chemical variations among apparently similar cloud cores and that co is not always heavily depleted when the volume density becomes larger than a few @xmath210@xmath3 ( as found in l1544 , l1498 and l1517b ; caselli et al . 2002 , tafalla et al . 2002 , 2004 ) . indeed , the case of l1521e , a taurus starless core , where the central density is 10@xmath85 but no co freeze out is observed @xcite , suggests that cloud cores in similar dynamical stages can have different chemical compositions . this point has been further discussed by @xcite , who underline the importance of the environment in setting the chemical / dynamical stage of a core , so that a core like l1521e ( and l1689b ) may have experienced a recent contraction phase , where the chemistry has not yet had the time to adjust to the new physical structure . on the other hand , the bok globule b68 , being close to equilibrium , may have achieved the present structure a long time ago , so that both co and had time to freeze out ( as found by @xcite ) . the present detailed study of tmc1c adds a new piece to the puzzle : cores which are currently accreting material from the surrounding cloud appear chemically younger , with lower co depletion factors . if the accreting cloud material , at densities @xmath13810@xmath3 , is old enough ( @xmath139 yr ) to have formed observable abundances of ( as in tmc1c ) , then lines toward the dust peak will be bright . on the other hand , the chemical structure of cores such as l1521e ( rich in co , but poor in n bearing species such as and ) may be understood as young condensations which are accreting either lower density material ( where the chemical times scales for formation are significantly longer ) or material which spent only a small fraction of 10@xmath85 yr at relatively high densities ( @xmath1710@xmath3 ) . the former hypothesis may be valid in environments less massive than those associated with tmc1c ( and it does not require large contraction speeds ) , whereas the latter hypothesis needs dynamical time scales shorter than or at most comparable to chemical time scales ( which are about 10@xmath3 yr at densities of @xmath1710@xmath85 , as can be found from the freeze out time scale of species such as co ) . in summary , tmc-1c , being more massive than l1544 and other typical low - mass cores , has a _ larger reservoir _ of undepleted material at densities close to 10@xmath3 , where both co and are abundant . detailed chemical models suggest that tmc1c must be at least @xmath140 yr old , to reproduce the observed abundances across the cloud . our simple chemical code tell us that the observed co depletion factors can be reached in only 10,000 yr . therefore , the core nucleus is either significantly younger than the surrounding material or the surrounding ( undepleted ) material has accreted toward the core nucleus in the past 10,000 yr . in either case , this is evidence that _ the densest part of tmc1c has recently accreted material_. from the velocity gradients presented in section [ svel ] it is hard to see a clear pattern of flowing material towards the dust peak , but the `` chaotic '' pattern is reminiscent of a turbulent flow that may be funnelling towards the densest region , aided by gravity . in any case , the extended inward motions deduced from observations ( see sect . [ infall ] ) is consistent with material at about ( 510)@xmath210@xmath4 _ currently accreting _ toward the dust peak position at velocities around 0.1 . it will be extremely important to compare this velocity field with those predicted by turbulent simulations of molecular cloud evolution , especially considering the possibility of competitive accretion ( bonnell & bate 2006 ) . observation of cs will also be important to check our prediction of a larger `` extended infall '' velocity , when compared to l1544 . a detailed observational study of the starless core tmc1c , embedded in the taurus molecular cloud , has been carried out with the iram 30 m antenna . we have determined that tmc-1c is a relatively young core ( t @xmath139 yr ) , with evidence of material accreting toward the core nucleus ( located at the dust emission peak ) . the core material at densities @xmath14110@xmath85 is embedded in a cloud condensation with total mass of about 14 m@xmath142 and average density of @xmath110@xmath3 , where co is mostly in the gas phase and had the time to reach the observed abundances of @xmath110@xmath10 . the overall structure is suggestive of ongoing inflow of material toward the central condensation . in addition , we have found that : \1 . ( 10 ) lines show signs of inward asymmetry over a region of about 7,000 au in radius . this is the most extended inward asymmetry observed in so far . the data are consistent with simple two layer models , where the line - of - sight component of the relative ( infall ) velocities range from @xmath170.15 ( toward the dust peak ) to @xmath170.05 ( at a distance from the dust peak of about 7,000 au ) . \2 . co isotopologues and show increasing depletion as @xmath143 increases and @xmath78 decreases . the amount of co depletion that we observe is a factor of @xmath175 lower than that of l1544 , whereas column densities are only a factor of two lower . also , show clear signs of moderate depletion toward the dust peak position . the gas temperature determined from ( 21 ) is 12 k at the dust peak , indicating that co is not tracing the dense ( @xmath144 ) and cold ( @xmath145 10 k ) regions of dense cores . the ( 21 ) excitation temperature drops outside the dust peak , and this is consistent with a roughly constant kinetic temperature and a dropping volume density ( _ traced by co isotopologues _ ) from @xmath146 toward the dust peak to @xmath147 at a projected distance from the dust peak of about 11,000 au ( no high s / n data are available to probe larger size scales ) . n@xmath5h@xmath6(1 - 0 ) line widths are constant across the core , which is consistent with previous nh@xmath24 measurements @xcite , but different from what has been found with n@xmath5h@xmath6 and n@xmath5d@xmath6 observations of l1544 and l1521f @xcite , where line widths are increasing toward the core center . the increase in line width with radius seen in c@xmath12o and n@xmath5h@xmath6 in b68 @xcite is not seen in tmc1c , and unlike b68 , the c@xmath12o line width is significantly larger than the n@xmath5h@xmath6 line width throughout tmc-1c . this is consistent with the fact that tmc1c , unlike b68 , is embedded in a molecular cloud complex , so that co lines trace more material along the line of sight of tmc1c . the velocity field that we see in tmc1c does not show the global signs of rotation that were seen in nh@xmath24 observations over a somewhat different area at arcminute resolution in @xcite . nevertheless , one portion of tmc1c encompassing the dust peak position does have a more coherent velocity field , suggestive of solid body rotation with magnitude @xmath14 pc@xmath16 , in the tracers c@xmath11o(1 - 0 ) , c@xmath11o(2 - 1 ) , c@xmath12o(2 - 1 ) and ( 1 - 0 ) . the observed chemical structure of the tmc1c core can be reproduced with a simple chemical model , adopting the co and n@xmath5 binding energies recently measured in the laboratory . we argue here that `` chemically young and physically evolved '' cores like l1521e and l1698b ( those with low co depletion , faint lines , central densities above @xmath148 @xmath14 and centrally concentrated structure ) have lower density envelopes than tmc1c in which the abundance did not have the time to reach equilibrium values . on the other hand , `` chemically and physically evolved cores '' like l1544 , l694 - 2 and l183 ( those with high co depletion and bright lines ) are likely to have lower rates of accretion of material from the envelope to the nucleus than in tmc1c ( or it has ended ) , and with a core nucleus undergoing contraction . finally , `` chemically evolved '' but less centrally concentrated cores ( e.g. l1498 , l1512 , b68 ) , can just be older objects ( age @xmath149 yr ) , close to equilibrium , as suggested by lada et al . ( 2003 ) . in the case of tmc1c , there is evidence that the core is at least @xmath150 yr old and has recently accreted less chemically evolved material . more comprehensive chemical models , taking into account the accretion of chemically young material , as well as a comparison between the observed velocity patterns and turbulent models of cloud core formation are sorely needed to test our conclusions . our anonymous referee has provided valuable comments and suggestions which have improved the content and clarity of this paper . we would like to thank phil myers , ramesh narayan , david wilner and doug johnstone for their suggestions , assistance , and insights . the james clerk maxwell telescope is operated by the joint astronomy centre on behalf of the particle physics and astronomy research council of the united kingdom , the netherlands organisation for scientific research , and the national research council of canada . iram is supported by insu / cnrs ( france ) , mpg ( germany ) , and ign ( spain ) . this material is based upon work supported under a national science foundation graduate research fellowship . from the equation of radiative transfer ( see eq . [ erad ] ) , once @xmath98 , @xmath39 and the corresponding errors are known , the error on @xmath38 ( @xmath151 ) can be determined following the rules of error propagation : @xmath152 where @xmath153 and @xmath154 are the errors associated with @xmath98 and @xmath39 , respectively . the expression of @xmath38 is found by inverting eq . [ erad ] : @xmath155 } \ , , \ , { \rm where } \\ \nonumber a & = & \frac{t_{\rm mb}}{1 - e^{-\tau } } + j_{\nu}(t_{\rm bg } ) .\end{aligned}\ ] ] thus , the partial derivatives in eq . 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0 ) & 353 & 0.052 & 18.4 & 0.66 & 112.3592837 & 5.2 + c@xmath11o(2 - 1 ) & 1893 & 0.052 & 9.2 & 0.40 & 224.7143850 & 5.2 + c@xmath12o(2 - 1 ) & 1050 & 0.053 & 9.4 & 0.41 & 219.5603541 & 5.2 + c@xmath13s(2 - 1 ) & 202 & 0.061 & 21.4 & 0.72 & 96.4129495 & 5.2 + dco@xmath6(2 - 1 ) & 479 & 0.041 & 14.3 & 0.54 & 144.0773190 & 5.2 + dco@xmath6(3 - 2 ) & 622 & 0.054 & 9.5 & 0.42 & 216.1126045 & 5.2 + n@xmath5h@xmath6(1 - 0 ) & 185 & 0.031 & 22.1 & 0.73 & 93.1737725 & 5.2 lrrrr c@xmath11o(1 - 0 ) & 5.33 @xmath70 0.03 & 0.37 @xmath70 0.07 & 0.45 @xmath70 0.11 & 0.089 + c@xmath11o(2 - 1 ) & 5.49 @xmath70 0.01 & 0.71 @xmath70 0.05 & 0.43 @xmath70 0.04 & 0.144 + c@xmath12o(2 - 1 ) & 5.20 @xmath70 0.01 & 2.07 @xmath70 0.01 & 0.41 @xmath70 0.01 & 0.250 + c@xmath13s(2 - 1 ) & 5.26 @xmath70 0.04 & 0.17 @xmath70 0.03 & 0.50 @xmath70 0.10 & 0.092 + dco@xmath6(2 - 1 ) & 5.14 @xmath70 0.01 & 0.28 @xmath70 0.03 & 0.16 @xmath70 0.02 & 0.204 + dco@xmath6(3 - 2 ) & 5.25 @xmath70 0.02 & 0.34 @xmath70 0.04 & 0.26 @xmath70 0.04 & 0.179 + n@xmath5h@xmath6(1 - 0 ) & 5.20 @xmath70 0.01 & 0.20 @xmath70 0.03 & 0.19 @xmath70 0.03 & 0.104 ccccc -40,60 b & 4.5 & 17 & 0.25 & 0.0 + -40,60 f & 3.3 & 10 & 0.15 & -0.05 + -20,40 b & 4.5 & 15 & 0.25 & 0.0 + -20,40 f & 3.5 & 15 & 0.20 & 0.05 + 0,20 b & 4.4 & 23 & 0.17 & 0.0 + 0,20 f & 3.3 & 10 & 0.23 & 0.15 + 20,0 & 4.4@xmath700.3 & 23@xmath702 & 0.17@xmath700.01 & ... + 40,-20 & 4.9@xmath700.3 & 8.8@xmath700.8 & 0.22@xmath700.01 & ... lrrrrr c@xmath11o(1 - 0 ) & 6.697e-8 & 56.179990 & 1 & 3 & @xmath158 + c@xmath11o(2 - 1 ) & 6.425e-7 & 56.179990 & 3 & 5 & @xmath158 + c@xmath12o(2 - 1 ) & 6.011e-7 & 54.891420 & 3 & 5 & @xmath159 + c@xmath13s(2 - 1 ) & 1.600e-5 & 24.103548 & 3 & 5 & @xmath160 + dco@xmath6(2 - 1 ) & 2.136e-4 & 36.01976 & 3 & 5 & @xmath161 + dco@xmath6(3 - 2 ) & 7.722e-4 & 36.01976 & 5 & 7 & @xmath161 + n@xmath5h@xmath6(1 - 0 ) & 3.628e-5 & 46.586867 & 1 & 3 & @xmath162 lrrr c@xmath11o & 1.0e15 & 2.8 & 5 + c@xmath11o & 7.8e14 & 3.6 & 7 + c@xmath12o & 2.6e15 & 3.8 & 3 + c@xmath13s & 5.4e11 & 12 & 20 + dco@xmath6 & 1.3e12 & 10 & 14 + dco@xmath6 & 2.8e12 & 5.9 & 13 + n@xmath5h@xmath6 & 9.4e12 & 0.9 & 6 , see the scale in the right y - axis ) and , for comparison , within l1544 ( dashed red curve , from vastel et al . the discontinuity in the fractional abundance profiles is due to the particular density profile used . ( _ right panel _ ) temperature and density profiles in tmc1c , derived by @xcite and used in the chemical model . dashed curves refer to l1544 ( adopted by vastel et al . [ fchem],width=576 ]

we have mapped the starless core tmc-1c in a variety of molecular lines with the iram 30 m telescope . high density tracers show clear signs of self - absorption and sub - sonic infall asymmetries are present in ( 10 ) and ( 21 ) lines . the inward velocity profile in ( 10 ) is extended over a region of about 7,000 au in radius around the dust continuum peak , which is the most extended `` infalling '' region observed in a starless core with this tracer . the kinetic temperature ( @xmath0 k ) measured from and suggests that their emission comes from a shell outside the colder interior traced by the mm continuum dust . the ( 21 ) excitation temperature drops from 12 k to @xmath110 k away from the center . this is consistent with a volume density drop of the gas traced by the lines , from @xmath14@xmath210@xmath3 towards the dust peak to @xmath16@xmath210@xmath4 at a projected distance from the dust peak of 80 ( or 11,000 au ) . the column density implied by the gas and dust show similar n@xmath5h@xmath6 and co depletion factors ( @xmath7 ) . this can be explained with a simple scenario in which : ( i ) the tmc1c core is embedded in a relatively dense environment ( @xmath8 @xmath1 10@xmath3 ) , where co is mostly in the gas phase and the abundance had time to reach equilibrium values ; ( ii ) the surrounding material ( rich in co and ) is _ accreting _ onto the dense core nucleus ; ( iii ) tmc-1c is older than 3@xmath9 yr , to account for the observed abundance of across the core ( @xmath110@xmath10 w.r.t . ) ; and ( iv ) the core nucleus is either much younger ( @xmath1 10@xmath3 yr ) or `` undepleted '' material from the surrounding envelope has fallen towards it in the past 10,000 yr .

conceptual model of the processes involved in membrane excitability and intracellular ip@xmath0-mediated calcium oscillations by calcium release through ip@xmath0-receptors in the er membrane in nrk fibroblasts . cell - membrane excitability is supported by inwardly rectifying potassium channels ( g@xmath108 ) , ca - dependent cl - channels ( g@xmath109 ) , l - type ca - channels ( g@xmath70 ) , store - dependent calcium ( sdc ) channels ( g@xmath110 ) , a pmca pump and leak channels ( g@xmath111 ) . the total flux of calcium through the er - membrane is the result of the contribution by the serca pump , by the ip@xmath0-receptor ( @xmath112 ) and by leak channels in the @xmath94 membrane ( @xmath29 ) . the membrane excitability and ip@xmath0-mediated calcium oscillations are coupled by the cytosolic calcium concentration , which is also affected by a calcium buffer b. stable and unstable states for the excitable membrane using @xmath49 as a control parameter . the intracellular ca - oscillator was silenced by setting the ip@xmath0 concentration to zero . thick ( thin ) lines correspond to the stable steady - state solutions for the [ @xmath46 ( panel @xmath90 ) and for the membrane potential ( panel @xmath91 ) for increasing ( decreasing ) values of for @xmath49 . the set of parameter values in this model was as reported in @xcite , with @xmath113 . the bifurcation diagram for the intracellular calcium oscillator in the single - cell model as a function of [ ip@xmath0 ] after elimination of action potentials ( @xmath73 ) and with @xmath114 set to 2 x @xmath75 @xmath85 x @xmath77 in panel a and to 8 x @xmath75 @xmath85 x @xmath77 in panel b. analogous to fig . [ fig2 ] , thick and thin solid lines correspond to the stable states for increasing and decreasing values for [ ip@xmath115 $ ] , respectively . the three insets in panel a show the ca - concentration as a function of time for [ ip@xmath0 ] values at 0.01 , 2 and 4 @xmath88 m . the inset in b shows the stable ( solid lines ) and unstable ( dashed - dotted lines ) states for a large range of [ ip@xmath0 ] values . the set of all other parameter values in this model was as reported in @xcite , with @xmath116 . the bifurcation diagram for the single - cell model . the figure shows the stable ( solid lines ) and unstable ( dashed - dotted lines ) states for [ @xmath46 ( panel @xmath90 ) and the membrane potential ( panel @xmath91 ) as a function of ip@xmath0 concentration . panel c and d show the membrane potential for [ ip@xmath0 ] at 0.7 @xmath88 m in case of increasing and decreasing [ ip@xmath0 ] , respectively . the small arrows on the curves show the direction of change of the stable modes for increasing and decreasing values of @xmath117 $ ] . the set of parameter values in this model was as reported in @xcite . bifurcation diagrams for the single - cell model as shown in fig . [ fig4]a for different values of the @xmath118 channel conductance from @xmath119 ns ( bottom ) to @xmath120 ns ( top ) . at 0.04 ns ( second graph from bottom ) , the hysteresis loop has a maximum in the [ ip@xmath0 ] range from 0.5 to 1.95 @xmath88 m . ip@xmath0 range of the hysteresis loop in @xmath88 m ( cf . [ fig5 ] ) as a function on sdc conductance ( @xmath104 ) for a nrk cell with a capacitance of 20 pf reveals a value in which the hysteresis area is maximum ( dots are simulated data ) . below the peak , we illustrate the @xmath118 conductance measured in experiments and separately reported in @xcite ( solid line ) and @xcite ( dashed line ) .

many cell types exhibit oscillatory activity , such as repetitive action potential firing due to the hodgkin - huxley dynamics of ion channels in the cell membrane or reveal intracellular inositol triphosphate ( ip@xmath0 ) mediated calcium oscillations ( caos ) by calcium - induced calcium release channels ( ip@xmath0-receptor ) in the membrane of the endoplasmic reticulum ( er ) . the dynamics of the excitable membrane and that of the ip@xmath0-mediated caos have been the subject of many studies . however , the interaction between the excitable cell membrane and ip@xmath0-mediated caos , which are coupled by cytosolic calcium which affects the dynamics of both , has not been studied . this study for the first time applied stability analysis to investigate the dynamic behavior of a model , which includes both an excitable membrane and an intracellular ip@xmath0-mediated calcium oscillator . taking the ip@xmath0 concentration as a control parameter , the model exhibits a novel rich spectrum of stable and unstable states with hysteresis . the four stable states of the model correspond in detail to previously reported growth - state dependent states of the membrane potential of normal rat kidney fibroblasts in cell culture . the hysteresis is most pronounced for experimentally observed parameter values of the model , suggesting a functional importance of hysteresis . this study shows that the four growth - dependent cell states may not reflect the behavior of cells that have differentiated into different cell types with different properties , but simply reflect four different states of a single cell type , that is characterized by a single model . + _ key words : hysteresis ; bistability ; calcium oscillations ; cell signaling _ l + + + + + + + + + [ cols= " < " , ] complexity and multiple transitions among behaviorial states are ubiquitous in biological systems @xcite . in physics instabilities and hysteresis are well known to play an important role in collective properties and have been studied since many years @xcite . recently , multi - stability with hysteresis has also awakened a large interest in biological systems @xcite . instabilities , for instance , are crucial for efficient information processing in the brain , such as in odor encoding @xcite . moreover , unstable dynamic attractors have been demonstrated in cortical networks , with critical relevance to working memory and attention @xcite . in a wide sense , multistable systems allow changes among different stable solutions where the system takes advantage of instabilities as gateways to switch between different stable branches @xcite . bistability driven by instabilities prevents the system from reaching intermediate states , e.g. partial mitosis . hysteresis prevents the system from changing its state when parameter values , that characterize the system , vary . this is of relevance , for instance , in cell mitosis . once initiated , mitosis should not be terminated before completion @xcite . thus , hysteresis may lock the cell into a fixed state , preventing it from sliding back to another state @xcite . at the level of cell networks , multistability , and in particular bistability , plays an important role in cell signaling as well @xcite . for example , communication between neurons takes place at synaptic contacts , where arrival of an action potential stimulates release of a neurotransmitter , thus affecting the post - synaptic potential of the target cell . typically , each cell receives input from thousands of other cells mediated by different neurotransmitters , which modify the post - synaptic potential by excitation or inhibition at different time scales @xcite . this information at the cell membrane may be transferred to the cell nucleus by so - called second messengers to affect the nucleus in controlling dna - expression , protein synthesis , mitosis , etc . calcium is one such second messenger and calcium oscillations have been reported over a wide range of frequencies with a chaotic or regular pattern @xcite . in many biological systems , cells display spontaneous calcium oscillations ( caos ) and repetitive action - potential firing . these phenomena have been described separately by models for intracellular inositol trisphosphate ( ip@xmath0)-mediated caos @xcite and for plasma membrane excitability @xcite . we have recently presented a single - cell model that combines an excitable membrane with an ip@xmath0-mediated intracellular calcium oscillator @xcite . the ip@xmath0-receptor is described as an endoplasmic reticulum ( er ) calcium channel with open and close probabilities that depend on the cytoplasmic concentrations of calcium ( @xmath1 $ ] ) and ip@xmath0 ( [ ip@xmath0 ] ) . an essential component of this model relates to store - operated calcium channels in the plasma membrane . since it is not known whether multiple types of store - operated calcium channels are involved in normal rat kidney ( nrk ) fibroblasts , we will use the general terminology of store - dependent calcium ( sdc ) channels . nrk fibroblasts in cell culture exhibit growth - state dependent changes in their electrophysiological behavior @xcite . subconfluent - grown serum - deprived quiescent cells exhibit a stable resting membrane potential near -70 mv ( `` resting state '' ) . upon subsequent treatment with epidermal growth factor the cells re - enter the cell cycle , undergo density - dependent growth - arrest ( contact inhibition ) at confluency and spontaneously fire action potentials associated with intracellular calcium oscillations ( `` ap - firing state '' ) . subsequent addition of retinoic acid or transforming growth factor ( tgf)@xmath3 to the contact inhibited cells causes the cells to become phenotypically transformed and to depolarize the cell to approximately -20 mv ( `` depolarized state '' ) . this depolarization has been shown to be caused by an elevation of the concentration of prostaglandin ( pg)f@xmath4 secreted by the unrestricted proliferating transformed cells . washout of the medium conditioned by the transformed cells by perfusion with fresh serum - free medium causes the cells to slowly repolarize , and , preceded by a short period of fast small - amplitude spiking of their membrane potential ( `` fast oscillating state '' ) , to regain spontaneous repetitive action potential firing activity ( `` ap - firing state '' ) similar to that of the contact inhibited cells . these phenomena have been described in detail @xcite and are very similar to the behavior of other cell types with calcium oscillations and action potential firing , such as interstitial cells of cajal @xcite and hepatocytes @xcite . in this study we have analyzed the model reported in @xcite . this model , which is shown schematically in fig . [ fig1 ] , illustrates the basic characteristics of nrk fibroblasts . it reproduces , on the basis of single - cell data @xcite , the dynamics of both the plasma membrane excitability and that of the intracellular calcium oscillator . we have recently shown that ( pg)f@xmath4 dose - dependently induces ip@xmath0-dependent intracellular calcium oscillations in nrk fibroblasts @xcite . since the growth - state dependent modulation of the membrane potential of nrk fibroblasts is related to the concentration of ( pg)f@xmath4 in their culture medium @xcite and since this prostaglandin dose - dependently increases [ ip@xmath0 ] , we took [ ip@xmath0 ] as a control parameter to analyze the stability of the single - cell model . the stability analysis shows how coupling of an excitable membrane with an intracellular calcium oscillator leads to a rich behavior of a cell with multiple stable and unstable states with hysteresis . we show that the growth - state dependent modulations of the membrane potential of nrk fibroblasts in cell culture described above can be understood as the stable states of the single - cell model with membrane excitability and calcium oscillations of these cells . the stable states of the model reproduce the four growth - dependent states of nrk cells , corresponding to the resting state at @xmath5 mv , the ap - firing state for spontaneous action potential firing , the depolarized state at @xmath6 mv and the fast oscillating state with small - amplitude spiking around @xmath6 mv . therefore , the four growth - dependent states of nrk fibroblasts may not reflect the behavior of cells that have differentiated into different cell types with different properties , but reflect four different states of a single cell type , that is characterized by a single model . [ [ section ] ] the dynamics of nrk cell membrane excitability is given by a set of equations which describe the active and passive ion transport systems in the plasma membrane and the endoplasmic reticulum , as illustrated in fig . [ fig1 ] ( see @xcite for a detailed description ) . the change in the membrane potential as a function of time due to the currents through inwardly rectifying potassium channels ( @xmath7 ) , l - type ca - channels ( @xmath8 ) , ca - dependent cl - channels ( @xmath9 ) , leak channels @xmath10 , and sdc - channels ( @xmath11 ) is given by @xmath12 @xmath7 and @xmath13 determine the membrane potential of the cell at rest near -70 mv and are specified in @xcite . the equation describing the l - type ca - current @xmath14 in terms of the hodgkin - huxley kinetics of the l - type ca - channel , is given by @xmath15 where m is the voltage - dependent activation variable , h is the voltage - dependent inactivation variable and @xmath16 is the inactivation parameter . the dynamics of the variables m and h are described by first order differential equations of the hodgkin - huxley type @xcite . the calcium - dependent inactivation is given by @xmath17+k_{vca})$ ] . the ca - dependent cl - current @xmath9 is given by @xmath18}{[ca_{cyt}^{2 + } ] + k_{cl(ca)}}\ : g_{cl(ca)}\:(v_m -e_{cl(ca)})\label{dicldt}\end{aligned}\ ] ] the chloride current increases with the cytosolic calcium concentration @xmath1 $ ] , causing a depolarization to the nernst potential of chloride ions ( @xmath19 ) near -20 mv in nrk fibroblasts for sufficiently high values of @xmath1 $ ] . the store - dependent calcium current @xmath11 is described by @xmath20 + k_{sdc } } \ : g_{sdc}\:(v_m - e_{sdc})\label{isdc}.\end{aligned}\ ] ] this store - dependent calcium channel allows calcium ions to flow from the extracellular space into the cytosol at a rate inversely proportional to the calcium concentration in the er @xcite . sdc channels are thought to play a major role in the control of ca - homeostasis in the cell @xcite . + the rate of change of ca - content of the cytosol of the cell due to inflow through the cell membrane and from the er store , and by buffering is described by @xmath21}{dt } & = & a_{pm}\:j_{pm}+a_{er } ( j_{ip_3r}+j_{lker } - j_{serca } ) - vol_{cyt}\ : \frac{d[bca]}{dt},\label{dcacytdt}\end{aligned}\ ] ] where @xmath22 represents the cytoplasmic volume and @xmath23 and @xmath24 the area of the cell membrane and of the er membrane , respectively . the term [ bca ] denotes the buffer - calcium complex in the cytosol and will be explained later . the flux of calcium through the membrane ( @xmath25 ) is the sum of the influxes of @xmath26 ions through the l - type ca - channel , through the sdc - channel , and of the extrusion by the pmca - pump @xcite , and is given by @xmath27 . the dynamics for the intracellular calcium oscillator is described by the flux of calcium through the er membrane . the rate of change of calcium content in the er depends on the sum of flux through the ip@xmath0-receptor ( @xmath28 ) , flux by leak through the er - membrane ( @xmath29 ) and flux by removal by the serca pump ( @xmath30 ) , which results in @xmath31}{dt } = a_{er } ( - j_{ip_3r } - j_{lker } + j_{serca}),\end{aligned}\ ] ] where @xmath32 represents the volume of the er . the flux through the ip@xmath0-receptor is described by @xmath33 -[ca_{cyt}^{2+}])\label{jip3r}\end{aligned}\ ] ] where @xmath34 -[ca_{cyt}^{2+}]$ ] is the concentration difference between calcium in the er and in the cytosol . @xmath35 is the rate constant per unit area of ip@xmath0-receptor mediated release . the terms @xmath36 and @xmath37 represent the fraction of open activation and inactivation gates , respectively . @xmath36 and @xmath38 depend both on the cytosolic calcium concentration and are described by @xmath39}{k_{fip_3 } + [ ca_{cyt}^{2+}]}\end{aligned}\ ] ] and @xmath40}{k_{wip_3 } + [ ip_3 ] } } { \frac{[ip_3]}{k_{wip_3 } + [ ip_3]}+k_{w(ca)}[ca_{cyt}^{2+}]}\label{winfty}.\end{aligned}\ ] ] the inactivation time constant of the ip@xmath0-receptor is defined by @xmath41}{k_{wip_3 } + [ ip_3]}+k_{w(ca)}[ca_{cyt}^{2+}]}.\label{tauw}\end{aligned}\ ] ] @xmath42 , @xmath43 , @xmath44 and @xmath45 are constants . the fraction of open activation gates ( f ) is independent of the ip@xmath0 concentration , but increases when the calcium concentration in the cytosol increases . the fraction of open inactivation gates ( w ) depends on the ip@xmath0 concentration and on [ @xmath46 . @xmath47 determines the duration of the de - inactivation of w. @xmath29 is a passive leak of @xmath26 from the er into the cytosol which is not mediated by the ip@xmath0-receptor , but by an additional ca - channel in the er membrane , presumably the translocon . experimental evidence for a role of the translocon complex as a passive @xmath26 leak channel has been presented recently @xcite . @xmath29 is given by @xmath48 - [ ca_{cyt}^{2+}])$ ] . we used the leakage parameter @xmath49 as a control parameter to study the dynamics of the plasma membrane , because changes in the leak of ca - ions through the er membrane produce proportional changes in [ @xmath46 . @xmath30 represents the flux of calcium into the er by the serca pump and is given by @xmath50 ^ 2 / ( k_{serca}^2+[ca_{cyt}^{2+}]^2)\}\label{jserca}$ ] . finally , calcium in the cytosol is buffered by proteins in the cytosol . the dynamics of buffering is given by @xmath51/dt \:=\ : k_{on}([t_b ] - [ bca])[ca_{cyt}^{2 + } ] -k_{off}\:[bca]\label{buffer}$ ] , where @xmath52 $ ] is the total concentration of buffer in the cytosol and @xmath53 and @xmath54 are the buffer rates @xcite . the excitable membrane and the ip@xmath0-mediated intracellular calcium oscillator are coupled by the ca - concentration @xmath1 $ ] in the cytosol as explained in @xcite . during an action potential , opening of the l - type ca - channel causes a large inward current of ca - ions through the plasma membrane . the increased @xmath1 $ ] activates the ip@xmath0-receptor ( calcium release channel ) , causing calcium release from the er , which further contributes to the intracellular cytosolic calcium transient . in the reverse process , ip@xmath0-mediated calcium oscillations cause periodic calcium transients , which lead to periodic opening of the ca - dependent cl - channels . the depolarization of the membrane potential towards the nernst potential of the ca - dependent cl - channels near -20 mv causes activation of the l - type ca - channels in the plasma membrane and excitation @xcite . after an action potential or ca - transient the reduction of cytosolic calcium by the activity of the serca and pmca pumps reduces @xmath9 ( see eq . [ dicldt ] ) , enough to allow the membrane to return to the membrane potential at rest near @xmath5 mv . the dynamics of the single - cell model depends on seven variables ( @xmath55 , @xmath56 , @xmath37 , @xmath57 $ ] , @xmath58 , @xmath1 $ ] and @xmath34 $ ] ) , which were defined above . to study the stability of the complete system we have determined the singular states for the system and calculated the floquet multipliers of these singular states @xcite . [ [ section-1 ] ] we will first analyze the bifurcations and local stability of both the excitable membrane and intracellular calcium oscillator separately , and then compare the results with the properties of the single - cell model including both the membrane dynamics and intracellular calcium oscillator . two different analyses , namely , our own implementation in @xmath59 , and the software package @xmath60 @xcite , which includes an @xmath61 @xcite interface , gave the same results . in the single - cell model the intracellular calcium oscillations can be eliminated by setting the ip@xmath0 concentration ( [ ip@xmath0 ] ) to zero . this allows the study of the excitable cell membrane separately from the calcium oscillator . the dynamics of the plasma membrane depends on the cytosolic calcium concentration [ @xmath46 , which opens the ca - dependent cl - channel . since the leak of ca - ions from the er affects the mean value of [ @xmath46 , the dynamics of the membrane is studied as a function of the leakage parameter @xmath49 . fig . [ fig2 ] shows a hysteresis diagram for the excitable cell membrane with the steady states of the calcium concentration in the cytosol ( [ @xmath46 , panel a ) and of the membrane potential ( @xmath58 , panel b ) . the thick and thin solid lines refer to the stable states for increasing and decreasing values of @xmath49 , respectively . the dashed - dotted lines reflect the transitions between the two stable branches for increasing and decreasing values of @xmath49 . starting at the value zero for @xmath49 , the inwardly rectifying k - channels keep the membrane potential at the resting membrane potential of the nrk fibroblasts near @xmath5 mv , where the membrane is able to produce an action potential upon electrical stimulation @xcite . for increasing values of @xmath49 , [ @xmath46 and @xmath58 increase gradually , causing a decreasing threshold for activation . the gradual increase of @xmath58 is due to gradual opening of the ca - dependent cl - channels for increasing [ @xmath46 ( see eq . [ dicldt ] ) . at @xmath62 @xmath63 , [ @xmath46 is large enough to open the ca - dependent cl - channels driving the membrane potential towards the nernst potential for cl@xmath64-ions which is near @xmath6 @xmath65 in nrk fibroblasts ( see fig . [ fig2]b ) . the resulting depolarization causes closure of the inwardly rectifying k - channels and opening of the l - type ca - channels which leads to an increase of calcium inflow from the extracellular medium into the cytosol . the positive feedback via the membrane potential between ca - dependent cl - channels and l - type ca - channels explains the abrupt increase of [ @xmath46 ( dashed - dotted line ) to @xmath66 @xmath67 . when we decrease @xmath49 starting from @xmath68 @xmath63 ( thin solid line ) , the cell remains depolarized near -20 mv far below the value of @xmath49 at @xmath69 @xmath63 . this is caused by the feedback between the ca - dependent cl - channels and l - type ca - channels . when @xmath49 decreases , [ @xmath46 also decreases , which reduces the fraction of open ca - dependent cl - channels . as a consequence , the membrane potential slightly decreases just below -20 mv , which leads to an increased fraction of open l - type ca - channels , since the product of mh of steady - state activation and inactivation ( see eq . [ ical ] ) reaches a maximum just below -20 mv . the increment of the fraction of open l - type ca - channels leads to an extra inflow of calcium in the cytosol , which increases the fraction of the open ca - dependent cl - channels and prevents the system from falling back to a membrane potential near @xmath5 mv . thus , in spite of the slow decrease of calcium concentration and membrane potential caused by the leak channels , the feedback by the l - type ca - channels keeps the system at an elevated [ @xmath46 and membrane potential near @xmath6 mv until low values @xmath49 . the calcium in the cytosol returns to a low concentration , only when the @xmath49 is decreased to very low values . then the ca - dependent cl - channels close and the membrane potential repolarizes to @xmath5 @xmath65 . separate simulations showed that the inward rectifier contributes to the transitions , but not to the hysteresis . [ [ section-2 ] ] following a similar plan as for the excitable cell membrane , we obtained a bifurcation diagram for the intracellular calcium oscillator as a function of the ip@xmath0 concentration under conditions that the l - type ca - channels are blocked such as with nifedipine . this was achieved by setting g@xmath70 to zero and k@xmath71 to its physiological value of @xmath72 @xmath63 . in this way , we eliminate the contribution of calcium inflow by the l - type ca - channels and remove a principal influence of the membrane model on the intracellular calcium oscillator . therefore , we only take into account the ca - flux through the sdc - channels and pmca pump in the plasma membrane . as explained in @xcite , the relative strength of the pmca and serca pump is crucial to reproduce the steady state calcium concentrations in the cytosol and in the er . by eliminating the calcium inflow by the l - type ca - channels , less calcium flows into the cell . therefore , we have to change the relative strength of the pmca and/or serca pump to maintain the proper balance between calcium concentration in the cytosol and er . in this model study , we choose to decrease the strength of the serca pump . by doing so , the system reveals a bifurcation diagram ( fig . [ fig3]a ) similar to that observed in other models @xcite . fig . [ fig3]a shows the dynamical behavior of [ @xmath46 as a function of [ ip@xmath0 ] for @xmath73 and with @xmath74 set to 2 x @xmath75 @xmath76 x @xmath77 . fig . [ fig3]a shows a single stable steady state for small values of [ ip@xmath0 ] ( range 0.0 - 0.2 @xmath67 ) . at [ ip@xmath0 ] near 0.2 @xmath67 the dynamics reveals a supercritical hopf bifurcation ( thick solid line ) , and the system becomes a calcium oscillator in the range of ip@xmath0 concentrations between 0.2 and 3.6 @xmath67 . for [ ip@xmath0 ] values near @xmath78 @xmath67 the system meets a supercritical hopf bifurcation and remains stable for higher ip@xmath0 concentration at a ca - concentration near @xmath79 @xmath67 . in the range for [ ip@xmath0 ] above 3.5 @xmath67 , the elevated mean level of [ @xmath46 gives rise to a short time constant @xmath47 for the inactivation parameter @xmath37 ( eq . [ tauw ] ) . due to this small time constant the inactivation @xmath37 recovers relatively fast compared to the removal of [ @xmath46 , i.e. before the activation parameter @xmath80 de - activates to small values . as a result the product @xmath81 does not reach small values and the ip@xmath0-receptor remains open ( see eq . [ jip3r ] ) , causing a constant leak of calcium . for decreasing [ ip@xmath0 ] values ( thin solid line ) , the system starts at a stable fixed point which remains stable until @xmath82 @xmath67 . in the range [ ip@xmath0 ] between 3.5 and 3.6 @xmath67 , the system exhibits bistability and a hysteresis over a small range of ip@xmath0-values . this hysteresis is caused by the positive feedback between [ @xmath46 and the activation gate ( @xmath80 ) . for decreasing [ ip@xmath0 ] , the [ @xmath46 is already elevated and so a large fraction of activation gates @xmath80 is already open and the time constant @xmath47 is short . due to the short time constant @xmath47 , the time for de - inactivation ( w ) is faster than for de - activation ( f ) . as a consequence , the product of @xmath80 and @xmath37 does not reach small values and calcium passes continuously through the ip@xmath0-receptor from the store into the cytosol . this hysteresis did not show up in the figures presented by li @xmath83 rinzel @xcite , but appears in their model if we insert the parameter values which apply to the nrk fibroblasts ( see @xcite ) . for [ ip@xmath0 ] values below 3.5 @xmath67 , the stable fixed point disappears , and the system starts to operate as an oscillator , until [ ip@xmath0 ] values smaller than @xmath84 @xmath67 , where the system returns to a single stable steady state . as a next step , we have set the strength of the serca pump back to its default value 8.@xmath75 @xmath85 x @xmath77 which corresponds to the value in the single - cell model with an excitable membrane and ip@xmath0-mediated calcium oscillations . this results in the bifurcation diagram shown in fig . [ fig3]b . fig . [ fig3]b shows a major hysteresis in the [ ip@xmath0 ] range between 8 and 53 @xmath86 ( see inset ) . to compare the results with those in fig . [ fig3]a we scaled fig . [ fig3]b in the same [ ip@xmath0 ] range as in fig . [ fig3]a . when the strength of the serca pump is increased to 8.@xmath75 @xmath87 x @xmath77 , [ @xmath46 decreases more rapidly after a calcium transient . this affects the time constant @xmath47 of the inactivation parameter @xmath37 ( see eq . [ tauw ] ) . for small [ @xmath46 levels , this time constant is relatively large , ensuring a slow de - inactivation . this explains why a more powerful serca pump gives rise to calcium oscillations over a much larger range of ip@xmath0 concentrations . only at sufficiently large [ ip@xmath0 ] values does @xmath47 become sufficiently small such that de - inactivation ( @xmath37 ) takes place more rapidly than de - activation ( @xmath80 ) . for these high ip@xmath0-values , the product fw of the activation parameter ( f ) and the inactivation parameter ( w ) is large enough to allow a continuous leak of calcium through the ip@xmath0-receptor . the inset in fig . [ fig3]b shows a single stable steady state for small values of [ ip@xmath0 ] . at [ ip@xmath0 ] near 0.2 @xmath67 , the dynamics reveals a subcritical hopf bifurcation ( thick solid line ) , and the system becomes a calcium oscillator in the range of ip@xmath0 concentrations between 0.2 and 53 @xmath67 . for [ ip@xmath0 ] above 53 @xmath67 , [ @xmath46 is elevated at a steady state concentration near 4 @xmath88 m . for decreasing [ ip@xmath0 ] values ( thin solid line ) , the system starts at a stable elevated [ @xmath46 which remains stable until [ ip@xmath0 ] is near 8 @xmath86 . for [ ip@xmath0 ] values below 8 @xmath67 , the stable fixed point disappears , and the system starts to operate as an oscillator , until [ ip@xmath0 ] values smaller than 0.2 @xmath67 . we conclude that increasing the activity of the serca pump makes it more easy for the cell to oscillate at higher [ ip@xmath0 ] values and causes a hysteresis over a larger range of [ ip@xmath0 ] values . [ [ section-3 ] ] unblocking the l - type ca - channels ( @xmath890.7 @xmath67 ) transforms the bifurcation diagram of [ fig3]b into that of fig . [ fig4]a . fig . [ fig4 ] shows [ @xmath46 ( panel @xmath90 ) and the membrane potential ( panel @xmath91 ) as a function of ip@xmath0 concentration in the cell . the solid and dashed - dotted lines represent stable and unstable states , respectively . for small [ ip@xmath0 ] values in the range from 0.00 to 0.15 @xmath67 , the cell has a single stable steady state ( `` resting state '' ) with a membrane potential near -70 mv . for [ @xmath92 > 0.15 $ ] @xmath67 , the stable fixed point becomes unstable in a subcritical hopf bifurcation . calcium oscillations together with action potentials occur for ip@xmath0 concentrations in the range between 0.15 and 1.75 @xmath67 ( `` ap - firing state '' ) ( see panel @xmath59 which shows the membrane potential as a function of time for [ ip@xmath0 ] @xmath93 0.7 @xmath88 m ) . in this regime , a rapid calcium inflow from the @xmath94 into the cytosol opens the ca - dependent cl - channel , causing an inward current towards the cl - nernst potential close to @xmath6 @xmath65 . this depolarization activates the l - type ca - channels leading to an ap . after closure of the ip@xmath0-receptor , calcium is removed from the cytosol by the ca - pumps in the cell membrane and er , leading to repolarization to @xmath5 mv . for [ ip@xmath0 ] @xmath95 1.75 @xmath67 , the fixed point @xmath96,v_m)$ ] near ( 3.00 @xmath86,-20 @xmath97 becomes stable in a subcritical hopf bifurcation ( `` depolarized state '' ) . this can be understood from the fact that the time - constant @xmath47 ( see eq . [ tauw ] ) for calcium - dependent ( de-)inactivation of the ip@xmath0-receptor decreases for increasing values of [ ip@xmath0 ] and for increasing values of the mean @xmath1 $ ] . near [ ip@xmath0 ] @xmath93 1.75 , the time - constant @xmath47 is relatively short . during a cytosolic calcium transient the fast inactivation of the inactivation gates @xmath37 of the ip@xmath0-receptor is followed by a fast de - inactivation of the inactivation gates of the ip@xmath0-receptor . during the fast de - inactivation , the fraction of open activation gates @xmath80 of the ip@xmath0-receptor is still high due to high @xmath1 $ ] ( because removal of calcium through the serca and pmca pump is not fast enough ) . as a consequence the ip@xmath0-receptor remains open . now the @xmath98-receptor acts as a constant leak channel , like @xmath29 . this leak of calcium into the cytosol opens the ca - dependent cl - channels , causing a maintained depolarization to the cl - nernst potential near @xmath6 @xmath65 ( `` depolarized state '' ) ( panel b ) . if [ ip@xmath0 ] is decreased starting from [ ip@xmath0 ] @xmath93 2.5 @xmath67 , the cell with both the excitable membrane and intracellular calcium oscillator active exhibits a complex hysteresis pattern . for decreasing ip@xmath0 concentrations , the system stays in a single stable state ( solid line ) at an elevated @xmath1 $ ] near 3 @xmath67 and a membrane potential near @xmath6 @xmath65 until [ ip@xmath0 ] @xmath99 0.85 @xmath67 ( `` depolarized state '' ) . then , the cell goes through a hopf bifurcation ( dashed line ) forcing the system to behave as a stable oscillator with small calcium oscillations with an amplitude of about @xmath100 @xmath88 m and with small membrane potential oscillations around @xmath101 mv ( `` fast oscillating state '' ) . these small oscillations of the membrane potential just below @xmath6 mv are illustrated in more detail in panel d. note that the oscillations of the membrane potential in panels c and d are both obtained for [ ip@xmath0 ] @xmath93 0.7 @xmath88 m , illustrating the hysteresis . the oscillations shown in panel d are due to small ip@xmath0-mediated calcium oscillations with active involvement of the l - type ca - channel dynamics . due to decreasing [ ip@xmath0 ] , the de - inactivation time constant @xmath47 of the ip@xmath0-receptor increases gradually . this makes it possible for the cell to generate calcium oscillations . setting @xmath102 in the equation for the l - type ca - current to 1 does not change the bifurcation diagram of fig . [ fig4 ] . the shape of the bifurcation scheme in fig . [ fig4]a remains the same , but the calcium oscillations extend over a larger range of @xmath1$]-values ( approximately twice as large ) . at [ ip@xmath0 ] @xmath99 0.45 @xmath67 the stable small - amplitude oscillator becomes unstable ( dashed line ) , returning the system to the stable oscillations with large amplitude ca - oscillations with a peak value near @xmath103 @xmath67 and with action potentials in the range between -70 and -10 @xmath65 ( `` ap - firing state '' ) . finally , for [ ip@xmath0 ] values smaller than @xmath84 @xmath67 the system returns to a single stable state ( `` resting state '' ) . in comparison with the simple dynamics of the cell membrane and intracellular ca - oscillator , shown in figs . [ fig2 ] and [ fig3 ] , it is remarkable to see the complex behavior of the single - cell model shown in fig . [ fig4 ] . since the sdc channels in the plasma membrane play a crucial role in stabilization of the calcium dynamics @xcite , we studied the dynamics of the cell as a function of the sdc conductance in a range between 0.00 and 0.20 ns . fig . [ fig5 ] shows the hysteresis diagrams for five different values of @xmath104 . as explained in @xcite , the calcium homeostasis of the cell is unstable for @xmath105 ns . for small values of @xmath104 bistability and hysteresis appears . the ip@xmath0 range with hysteresis is largest for a @xmath104 value near 0.04 ns ( see fig . [ fig5 ] ) . for higher values of @xmath104 , the range of hysteresis decreases until the typical hopf - bifurcation for the intracellular ip@xmath0-mediated calcium oscillations remains for @xmath106 ns . the ip@xmath0 range of the hysteresis as a function on the sdc conductance channel is shown in fig . [ fig6 ] . we define the ip@xmath0 range of hysteresis as the range of [ ip@xmath0 ] in which multiple states are found for increasing and decreasing [ ip@xmath0 ] . for example , in fig . [ fig4 ] hysteresis takes place for [ ip@xmath0 ] values between 0.45 and 1.75 @xmath67 , giving an ip@xmath0 range of hysteresis of 1.3 @xmath67 . recent data in the literature show that the sdc conductance , which was found to give the largest range for hysteresis in our study ( near 0.04 ns ) , corresponds to the observed sdc conductance in other studies @xcite . the sdc conductance reported in table 1 of @xcite and in @xcite was in the range between 0.04 and 0.05 ns ( solid line below the peak in fig . [ fig6 ] ) . for hepatocytes @xcite a sdc conductance was reported in the range between 0.08 and 0.14 ns . however , since the density of all ion channels in hepatocytes is twice as high as in fibroblast @xcite , the ratio of conductances for the ion channels is the same in hepatocytes and nrk fibroblasts . if we correct for this higher density , rescaling all conductances for those of nrk fibroblasts , we obtain the dotted line in [ fig6 ] . therefore , the sdc conductance , for which hysteresis is found over the largest range of ip@xmath0- values in our study ( see figs . [ fig5 ] and [ fig6 ] ) , is in agreement with experimental observations for sdc conductance . [ [ section-4 ] ] in this study we have analyzed a relatively simple model with an excitable membrane and with ip@xmath0-mediated calcium oscillations . the interaction between these mechanisms in a single - cell model revealed a surprisingly rich behavior with stable / instable states with hysteresis . the hysteresis and bistability of the membrane potential and the intracellular calcium concentration in fig . [ fig4]a and b obtained by stability analysis of the single - cell model provide an explanation for the various growth - state dependent changes in the electrophysiological behavior of normal rat kidney ( nrk ) fibroblasts in cell culture @xcite . the stability analysis of the single - cell model ( fig . [ fig4 ] ) reveals for low [ ip@xmath0 ] values ( range 0.0 - 0.2 @xmath67 ) a cell in the `` resting state '' . increasing the [ ip@xmath0 ] leads to spontaneous ap firing ( `` ap - firing state '' ) and at high [ ip@xmath0 ] values above 1.75 @xmath67 , the cell depolarizes ( `` depolarized state '' ) . when we start at an [ ip@xmath0 ] value of 2.0 @xmath67 and decrease [ ip@xmath0 ] , the system is in the `` depolarized state '' and changes from the `` fast oscillating state '' ( range 0.45 - 0.8 @xmath67 ) to the `` ap - firing state''(range 0.2 - 0.45 @xmath67 ) back to its `` resting state '' ( range 0.0 - 0.2 @xmath67 ) , which is in agreement with experimental data shown by harks et al . @xcite . in the study of harks et al . @xcite washout of the medium conditioned by the transformed cells by perfusion with fresh serum - free medium , causes the cells to slowly repolarize , and , preceded by a short period of fast small - amplitude spiking of their membrane potential ( `` fast oscillating state '' ) , to regain spontaneous repetitive action potential firing activity similar to that of the contact inhibited cells . this compares well with the results in fig . [ fig4]b , which shows for decreasing [ ip@xmath0 ] a very similar behavior . therefore , we conclude that the stable states of the model as revealed by stability analysis of the single - cell model correspond in great detail to the observed growth - state dependent modulations of the membrane potential of nrk fibroblasts in cell culture . this strongly suggests that these growth - state dependent modulations of the membrane potential of nrk cells reflect just different states of the same cell , rather than the behavior of cells , that have differentiated to different cell types with different properties during the various stages of growth - factor stimulated development in vitro . most of the parameter values in our model were taken from the literature ( see @xcite ) for a detailed overview ) . interestingly , the parameter values for the excitable membrane and for the ip@xmath0-mediated calcium oscillator , which are very different mechanisms , are not independent . this can be understood from the fact that the dynamics of the excitable membrane and of the ip@xmath0 receptor are coupled by the cytosolic calcium concentration . changing one parameter of the excitable membrane or calcium oscillator affects the other mechanism by changes in the cytosolic calcium concentration . this is illustrated , for example , by fig . [ fig3 ] . changing the strength of the serca pump causes large differences in the range of hysteresis in the dynamics of cytosolic calcium ( fig . [ fig3 ] ) , and therefore also in the dynamics of the membrane potential ( see fig . [ fig4 ] , which illustrates the relation between the dynamics of cytosolic calcium and the membrane potential ) . although the parameter values for the excitable membrane and for the ip@xmath0-mediated calcium oscillator were taken from different studies , they fit nicely together to explain the behavior of nrk cells both qualitatively and quantitatively . this provides strong evidence for the reliability of these parameter values . moreover , this suggests that cells should have complicated regulatory mechanisms to control all parameter values within a proper range of parameter values to ensure the proper cell dynamics . summarizing , we explored the dynamical properties of a single - cell model reproducing experimental observations on calcium oscillations and action potential generation in nrk fibroblasts . a bifurcation analysis revealed hysteresis and a complex spectrum of stable and unstable states , which allows the system to switch among different stable branches . stability of the cell behavior is dominated by the homeostatic function of the sdc channel . the conductance , which provides the largest ip@xmath0 range for hysteresis , compares well with experimental values for this conductance @xcite . experimental observations in nrk fibroblasts revealed the same kind of hysteresis as shown by this study . + we acknowledge financial support from the nederlandse organisatie voor wetenschappelijk onderzoek ( nwo ) , ministerio de educacion y ciencia ( mec ) , junta de andalucia ( ja ) and engineering and physical sciences research council ( epsrc ) , projects nwo 805.47.066 , mec fis2005 - 00791 , ja fqm-165 and epsrc ep / c0 10841/1 . 10 murray , j. d. 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baryon number violation is a very sensitive probe of physics beyond the standard model ( sm ) . interactions which violate baryon number ( @xmath8 ) are not present in the renormalizable part of the sm lagrangian , but they can arise as effective higher dimensional operators . the lowest @xmath8violating operators @xcite have @xmath9 and are suppressed by two powers of an inverse mass scale . these operators are realized naturally when sm is embedded in a grand unified theory ( gut ) such as @xmath10 and @xmath0 upon integrating out the heavy vector gauge bosons and colored scalar bosons . they lead to the decay of the nucleon into modes such as @xmath11 and @xmath12 . present experimental limits on nucleon lifetime constrain the masses of the mediators ( vector gauge boson or scalar bosons ) to be larger than about @xmath13 gev , which is close to the unification scale determined from the approximate meeting of the three gauge couplings when extrapolated to higher energies . an interesting feature of the @xmath9 baryon number violating operators is that they conserve baryon number minus lepton number ( @xmath14 ) symmetry , leading to the selection rule @xmath15 for nucleon decay@xcite . thus , observation of any decays which violate @xmath15 rule would hint at new dynamics different from those responsible for the @xmath9 operators . decay modes in this category include @xmath3 , @xmath16 etc , which all obey the selection rule @xmath17 . observation of these decay modes would thus furnish evidence against the simple gut picture with one step breaking to the sm . in this paper we study the next to leading @xmath18 operators , which obey the selection rule @xmath19 for nucleon decay @xcite , and show that they arise naturally within @xmath0 grand unified theories . in non supersymmetric @xmath0 models with an intermediate scale we find the nucleon lifetime for decay modes such as @xmath20 to be within reach of ongoing and proposed experiments . we also identify susy @xmath0 models where these decays may be within reach , consistent with gauge coupling unification . while we focus mainly on renormalizable @xmath0 models with @xmath4 of higgs bosons employed for rank reduction , we show that our results also hold for models with @xmath5 used for this purpose . the second main result of this paper is a mechanism for generating the baryon asymmetry of the universe at the gut epoch . the way it comes about is as follows . there are heavy scalar bosons and gauge bosons in @xmath0 theories which generate the @xmath18 operators . these particles have @xmath1violating two body decays , which can generate the observed baryon asymmetry of the universe naturally , as we show here . this would not be possible in the case of @xmath1preserving decays of gut scale particles such as the ones in @xmath10 . although grand unified theories were thought to be the natural stage for implementing the sakharov s conditions for baryogenesis @xcite up until the mid-1980 s @xcite , this idea was practically abandoned after the realization that the sphalerons @xcite , which violate @xmath21 symmetry , would erase any baryon asymmetry that obeyed the @xmath22 selection rule . this is because the effective interactions generated by sphalerons , the non - perturbative configuration of the weak interactions , are in thermal equilibrium for temperatures in the range @xmath23 gev , and violate @xmath24 symmetry . however , if baryon asymmetry was generated by @xmath1violating decays of gut scale particles , they would be immune to sphaleron destruction . we show that this mechanism of baryogenesis , which also induces the @xmath18 @xmath8violating operators , is very efficient and occurs quite generically in @xmath0 models . in minimal models there is a tight connection between the induced baryon asymmetry and the masses of quarks , leptons and the neutrinos . we also note that the minimal renormalizable versions of these models @xcite have been extremely successful in describing neutrino masses and mixings @xcite , and in particular predicted relatively large value for the neutrino mixing angle @xmath25 , which is consistent with recent results from daya bay , t2k , double - chooz and minos experiments @xcite . the results of the present paper show that these models can also explain the observed baryon asymmetry in a manner closely connected to the neutrino oscillation parameters . this paper is organized as follows . in sec . 2 we discuss the @xmath18 @xmath8 and @xmath1violating operators . in sec . 3 we show how these operators arise in unified @xmath0 theories , both in the non - supersymmetric version and in the susy version . in sec . 4 we address nucleon decay lifetime for the @xmath1violating modes in @xmath0 models with an intermediate scale . 5 is devoted to gut scale baryogenesis mechanism tied to the @xmath18 operators . here we show the close connection between baryon asymmetry and fermion masses and mixings . finally , we conclude in sec . we begin by recalling the leading baryon number violating operators in the standard model which have @xmath9 . there are five such operators with baryon number @xmath26 @xcite : @xmath27 here we have not shown the color contractions ( which is unique in each term via @xmath28 ) , and we have suppressed the flavor indices . we have followed the standard notation for fermion fields with all fields being left handed . thus @xmath29 stands for the left handed antiparticle of @xmath30 . the spinor indices are contracted via the charge conjugation matrix between fields in parentheses . @xmath31 are the @xmath32 indices . the complex conjugate operators of eq . ( [ dim6 ] ) would of course carry @xmath33 . an interesting feature of the @xmath9 baryon number violating operators on eq . ( [ dim6 ] ) is that they all carry lepton number @xmath34 along with @xmath35 . consequently these operators preserve @xmath1 . thus nucleon decay mediated by these operators would obey the selection rule @xmath36 . the decays @xmath37 and @xmath38 would be allowed by this selection rule , while decays such as @xmath39 and @xmath20 , which require @xmath17 would be forbidden . grand unified theories based on @xmath10 and @xmath0 gauge symmetries generate the operators of eq . ( [ dim6 ] ) suppressed by two inverse powers of gut scale masses . with the gut scale near @xmath13 gev , as suggested by the approximate unification of the three gauge couplings , these operators lead to nucleon lifetimes of order @xmath40 years for @xmath1 conserving modes , which are in the range that is currently being probed by experiments . discovery of nucleon decay into @xmath17 channels such as @xmath41 would however suggest that the underlying dynamics is quite different from that of the @xmath9 effective operators of eq . ( [ dim6 ] ) . as already noted in the introduction , while gut scale particles can generate a baryon asymmetry in their @xmath8violating decays , it was realized that interactions of the electroweak sphalerons would wash out any such asymmetry that conserves @xmath1 . gut scale baryogenesis thus went out of fashion after the discovery of sphalerons . this was also in part due to the leptogenesis mechanism @xcite discovered soon thereafter , which can elegantly explain the observed baryon asymmetry with a connection to the small neutrino masses induced via the seesaw mechanism@xcite . now we turn to the next to leading @xmath8violating operators beyond those of eq . ( [ dim6 ] ) , which are of dimension seven . these operators are interesting in that they carry @xmath42 @xcite . while they are suppressed by one additional power of a heavy mass scale , they can naturally lead to sphaleron proof baryogenesis , as we show here . in several instances we also find that these operators may lead to observable @xmath1 violating nucleon decay . there are nine @xmath18 baryon number violation operators with @xmath43 listed below @xcite : @xmath44 we have used the same notation as in eq . ( [ dim6 ] ) . here @xmath45 is the standard model higgs doublet transforming under @xmath46 as @xmath47 . @xmath48 stands for the covariant derivative with respect to the @xmath46 gauge symmetry . note that @xmath49 , @xmath50 and @xmath51 must be antisymmetric in the down flavor indices . operators of the type @xmath52 are not written , since they are related to those listed in eq . ( [ dim7 ] ) by the equations of motion . all vector and tensor operators can be fierz transformed into the set of operators in eq . ( [ dim7 ] ) . note that all operators of eq . ( [ dim7 ] ) carry @xmath35 and @xmath53 , and thus @xmath54 , with the complex conjugates operators carrying @xmath55 . it is these operators which can mediate nucleon decay of the type @xmath56 and @xmath3 . the higher dimensionality of these operators would suggest naively that the nucleon decay lifetime would be much longer than the ones obtained from eq . ( [ dim6 ] ) . however , as we show below , in unified theories based on @xmath0 with an intermediate scale , these decays may be accessible to experiments . most interestingly , these operators can naturally generate baryon asymmetry of the universe at the gut scale , which is facilitated by the fact that the electroweak sphaleron interactions do not wash out a @xmath1 asymmetry generated at such a scale . in the supersymmetric version of the standard model , baryon number violation can arise through operators in the superpotential analogous to eq . ( [ dim7 ] ) . these superpotential operators would have dimension six . holomorphicity of the superpotential would however constrain the allowed operators . there is a single operator of dimension six given by the superpotential coupling @xmath57 which carries @xmath33 and @xmath58 . this operator must be antisymmetric in the down - flavor indices owing to bose symmetry . @xmath59 here is the up type higgs doublet of mssm . in the superpotential , this operator will appear with two inverse powers of a heavy mass scale . terms in the lagrangian resulting from eq . ( [ w ] ) would have two fermion fields , one higgs field and two superpartner scalar fields , for example . when the superpartner scalar fields are converted to standard model fermions by a gaugino loop , effective @xmath18 operators of eq . ( [ dim7 ] ) would be generated , suppressed by a factor @xmath60 , rather than @xmath61 that occurs for eq . ( [ dim7 ] ) without susy , where @xmath62 is the heavy mass scale . therefore , potentially these susy contributions can be more significant for nucleon decay . we can now present the complete list of @xmath55 effective operators through @xmath18 in the sm by adding to eq . ( [ dim7 ] ) operators with @xmath63 . these operators have been classified in ref . @xcite . while not directly related to nucleon decay , these operators arise along with the @xmath18 operators of eq . ( [ dim7 ] ) in @xmath0 unified theories , and they are also relevant for gut scale baryogenesis . here we collect the linearly independent set of these operators through @xmath18 . the leading operator of course is the well - known @xmath64 seesaw operator @xcite @xmath65 the next to leading operators are of @xmath18 , and there are ten of them , as listed below . @xmath66 we shall see the appearance of some of these operators in the embedding of eq . ( [ dim7 ] ) in @xmath0 models . in the supersymmetric standard model , the four holomorphic operators @xmath67 would be allowed in the superpotential ( with no significance attributed to the spinor contractions of eq . ( [ lep ] ) when applied to the superfields ) . in this section , we show that the @xmath18 baryon number violating operators of eq . ( [ dim7 ] ) arise naturally in the context of @xmath0 unified theories after the spontaneous breaking of @xmath1 , which is a part of the gauge symmetry . the @xmath1 symmetry may break at the gut scale so that @xmath0 breaks directly to the standard model gauge symmetry , or it may break at an intermediate scale @xmath68 below the gut scale . in the latter case the intermediate symmetry could be one among several possibilities : @xmath69 ; @xmath70 ; @xmath71 ; @xmath72 ; or @xmath73 , with or without left right parity symmetry . in the non supersymmetric version an intermediate scale is necessary to be compatible with gauge coupling unification@xcite , while with supersymmetry the direct breaking of @xmath0 down to the mssm is preferable . even in the latter case , there is room for intermediate scale particles , provided that they form complete multiplets of the @xmath10 subgroup , since such particles do not spoil the unification of gauge couplings observed with the mssm spectrum . to see how the @xmath18 operators of eq . ( [ dim7 ] ) arise within @xmath0 , we first focus on the scalar mediated operators and write down the yukawa couplings in the most general setup . the higgs fields which can couple to the fermion bi - linears @xmath74 are @xmath75 , @xmath76 and @xmath77 , with the couplings of the @xmath75 and @xmath76 being symmetric in flavor indices @xmath78 and those of the @xmath77 being antisymmetric . the terms in these yukawa couplings that are relevant to the generation of the @xmath18 operators are given below @xcite . @xmath79 , \label{yuk10}\end{aligned}\ ] ] @xmath80 , \label{yuk126}\end{aligned}\ ] ] @xmath81 . \label{yuk120}\end{aligned}\ ] ] these terms are written in terms of the standard model decomposition of the sub - multiplets . we have followed the phase convention of ref . @xmath82 stands for the @xmath83 tensor @xmath28 . here we have not displayed terms that are irrelevant for inducing the @xmath18 baryon number violating operators . ( specifically , we have omitted color singlet , color octet , and color sextet couplings . ) the yukawa couplings obey @xmath84 , @xmath85 and @xmath86 . the @xmath46 quantum numbers of the various sub - multiplets are given as follows . @xmath87 different fields with the same sm quantum numbers appear in some couplings , they are distinguished by subscripts @xmath88 etc . we have used the same notation for fields with the same sm quantum numbers in @xmath75 , @xmath76 and @xmath77 , but it should be understood that these are distinct fields . after gut symmetry breaking various subfields with the same sm quantum number would mix . some of these mixings would involve the vacuum expectation value of the sm singlet field from the @xmath76 , denoted by @xmath89 carrying @xmath55 . it is this field that supplies large majorana mass for the right handed neutrino through the coupling @xmath90 . with @xmath91 , trilinear scalar couplings of the type @xmath92 , @xmath93 , @xmath94 and @xmath95 , will develop . this issue will be addressed in more detail below , but we note that such couplings are invariant under the unbroken sm gauge symmetry . when combined with the yukawa couplings of eqs . ( [ yuk10])-([yuk120 ] ) , they would induce the @xmath18 baryon number violating operators of eq . ( [ dim7 ] ) . operators induced by the symmetric yukawa couplings of @xmath75 and @xmath76 of @xmath0 . here the sm quantum numbers of the various fields are @xmath96 , and @xmath97 . ] to see how the @xmath18 operators arise in more detail , let us focus of the flavor symmetric yukawa couplings of eqs . ( [ yuk10])-([yuk126 ] ) . these couplings generate two of the @xmath18 operators as shown in fig . [ sym ] . here @xmath97 is the sm higgs doublet , which is a linear combination of the @xmath98 and @xmath99 fields from @xmath75 , @xmath76 as well as any other higgs sub - multiplet with the quantum number of @xmath100 in the theory with which these fields mix . similarly , @xmath101 generically stands for any linear combination of @xmath102 and @xmath103 from the @xmath75 , @xmath104 and @xmath105 from the @xmath76 , etc . before estimating the strength of these operators , let us examine the origin of the trilinear scalar couplings that appear in these diagrams in @xmath0 . to see the origin of @xmath92 and similar vertices , let us recall first the decomposition of various @xmath0 fields under the subgroups @xmath69 and @xmath73 . under @xmath106 , we have the following decomposition : @xmath107 under the @xmath108 subgroup various fields decompose as follows . @xmath109 now , the quartic coupling @xmath110 , which is invariant ( there is a single such coupling ) , contains the term @xmath111 under @xmath112 . the @xmath113 field is a subset of @xmath114 fragment . the @xmath97 field is part of @xmath114 , while @xmath115 and one such field is also part of @xmath116 . thus one sees that the coupling @xmath111 would contain the term @xmath117 , where @xmath118 denotes the sm singlet field from @xmath4 that acquires a gut scale vev . in the @xmath119 decomposition , @xmath120 , @xmath121 , and @xmath122 . thus @xmath110 contains the term @xmath123 , which has the piece @xmath117 . there are three non - trivial invariants of he type @xmath124 . these couplings also contain the term @xmath125 , as can be seen by examining the decomposition under @xmath112 and separately under @xmath119 . coupling would also contain a term @xmath126 under @xmath112 , which has a @xmath117 term in it . however , the @xmath102 appearing here is from the @xmath127 , which is @xmath128 of eq . ( [ yuk126 ] ) . @xmath128 coupling by itself does not violate baryon number @xcite , as is evident from eq . ( [ yuk126 ] ) . ] in an analogous fashion one sees that the coupling @xmath129 contains @xmath125 , with the @xmath130 arising from the @xmath131 . to complete this discussion we also note that there are three invariants of the type @xmath132 , which contain @xmath133 , with the @xmath134 and one @xmath114 taken from the @xmath135 . the @xmath134 fragment contains the @xmath136 of eq . ( [ higgs ] ) , which would enter the @xmath18 operators arising by integrating out the flavor antisymmetric @xmath135 fragments . in particular , this term would induce the needed @xmath137 vertex . the @xmath133 term also contains @xmath95 vertex , with @xmath138 , @xmath139 from the @xmath135 . the @xmath132 invariant also contains the term @xmath140 , with the @xmath141 and one @xmath114 taken from the @xmath135 . this piece generates the terms @xmath142 vertex , with @xmath143 . finally , there are two invariants of the type @xmath144 , which also contain the term @xmath133 . in order to complete @xmath0 symmetry breaking , additional higgs fields such as a @xmath145 , @xmath146 or a @xmath147 is needed . the interactions of these fields with @xmath148 and @xmath135 can generate further trilinear and quadrilinear couplings . take for example the case of a @xmath146 employed for completing the symmetry breaking . the trilinear couplings @xmath149 and @xmath150 are then invariant . noting that under @xmath73 subgroup , @xmath151 , we see that the latter coupling would contain a term @xmath152 . this has a piece @xmath153 , which would mix the two @xmath154 fields once @xmath155 develops , breaking @xmath1 by two units . ( note that the @xmath156 and @xmath157 carry different @xmath1 charges . ) the cubic coupling @xmath150 also contains the terms @xmath158 and @xmath159 under @xmath73 . these terms contain the couplings @xmath160 and @xmath161 respectively . these are the desired trilinear couplings for the generation of the @xmath18 operators . in this case , the mixing of @xmath162 with the @xmath163 is utilized in order to connect the @xmath154 field with the fermion fields in fig . similar results follow from the quartic couplings @xmath164 and @xmath165 . while trilinear scalar couplings of the type @xmath93 and @xmath166 do arise for the @xmath76 sub - multiplets , these couplings do not directly lead to baryon number violation . the @xmath167 field from @xmath76 has the coupling @xmath168 , while @xmath169 has the coupling @xmath170 ( see eq . ( [ yuk126 ] ) ) . the exchange of @xmath171 from @xmath76 would lead to an effective operator @xmath172 , while that of @xmath173 would generate the operator @xmath174 , which is operator @xmath175 of eq . ( [ lep ] ) . the @xmath93 and @xmath166 couplings of the @xmath76 sub - multiplets would however be relevant for gut scale baryogenesis , since they also violate @xmath1 symmetry . baryon number violating operators obtained by integrating our fields from the @xmath135 . here the sm quantum numbers of the scalar fields are : @xmath176 and @xmath177 in fig . [ antisym ] we display the effective @xmath18 operators obtained by integrating out the flavor antisymmetric @xmath135 coupling to fermions . we see that four operators are induced this way . as already noted , the required trilinear vertices to complete these diagrams arise from quartic couplings . it is also possible to replace the @xmath154 fields in fig . [ antisym ] by a @xmath154 field from the @xmath178 , in which case the sum of such diagrams would have no definite symmetry property in two of the flavor indices . note that all of the @xmath18 operators arising from fig . [ sym ] and [ antisym ] respect @xmath14 symmetry , as can be seen by assigning @xmath179 . it should be noted that operators @xmath180 of eq . ( [ dim7 ] ) do not arise at tree level by integrating out superheavy particles . they can arise via loops , with suppressed strength . in the estimate of nucleon lifetime these operators play a subleading role , since the amplitude for the decay would be further suppressed by the nucleon momentum ( rather than the electroweak vev for operators @xmath181 ) . what if the higgs field employed for reducing the rank of @xmath0 is a @xmath5 rather than a @xmath4 ? in this case the @xmath18 diagrams of fig . [ sym ] and [ antisym ] would still arise , albeit in a slightly different way . the @xmath1 quantum numbers of the higgs doublets @xmath98 from the @xmath75 , @xmath4 and @xmath77 fields are all zero . the @xmath5 contains a sm singlet filed with @xmath182 which acquires a gut scale vev . it also contains a @xmath183 field with @xmath184 . similarly , @xmath185 contains a sm singlet field with @xmath186 and a @xmath98 field with @xmath187 . the trilinear scalar couplings of the type @xmath188 and @xmath189 would mix the @xmath190 higgs doublet @xmath98 from the @xmath75 and the @xmath98 higgs from the @xmath185 which has @xmath182 . the light sm higgs doublet then would have no definite @xmath14 quantum number . the @xmath191 component of @xmath185 under @xmath112 contains the field @xmath113 , and the @xmath192 of @xmath185 under @xmath112 contains @xmath193 , and thus the coupling @xmath92 is generated via the @xmath194 coupling . one could also take @xmath101 from the @xmath75 , @xmath113 from the @xmath185 and @xmath97 from the @xmath185 to generate the @xmath92 coupling from @xmath189 . it should be noted that the @xmath195 field from the @xmath185 does have yukawa couplings to fermions , since in this type of models the heavy majorana masses of the @xmath196 fields arise from the couplings @xmath197 , and upon insertion of one vev for the sm singlet field here , the coupling of @xmath198 to fermions is realized . we point out that the @xmath199 field is partly in the goldstone mode , associated with the breaking of @xmath0 to @xmath10 . however , since other fields such as @xmath145 , @xmath146 or @xmath147 should be employed to complete the symmetry breaking down to the sm , one such physical @xmath199 will remain in the spectrum . this is because the @xmath145 , @xmath146 and @xmath147 all contain a @xmath199 field , and only one @xmath154 is absorbed by the gauge multiplet . baryon number violating operators via the exchange of vector gauge bosons @xmath200 and @xmath201 of @xmath0 . ] the @xmath18 operators of eq . ( [ dim7 ] ) can also arise by integrating out the gauge bosons of @xmath0 . these vector operators are related by fierz identities to the scalar operators displayed in eq . ( [ dim7 ] ) . the relevant diagrams are shown in fig . [ gauge ] . [ gauge ] ( d ) conserves @xmath8 , but violates @xmath202 and @xmath14 @xcite . the effective operator from this diagram , after a fierz rearrangement can be identified with @xmath203 of eq . ( [ lep ] ) . ) in these diagrams @xmath204 denotes the gauge boson with the sm quantum numbers @xmath205 ( same as the quark doublet @xmath206 ) , and @xmath207 is the gauge boson that transforms as @xmath208 , the same as that of @xmath29 fermion . in fig . [ gauge ] , each vertex conserves @xmath14 as it should , which can be seen by assigning @xmath209 to @xmath210 . the covariant derivative for the rank reducing field @xmath4 would contain the term @xmath211 which enters the @xmath18 operators . when @xmath5 is used instead of the @xmath4 , the covariant derivative would contain a similar term , but now the @xmath45 and @xmath89 fields would carry @xmath187 and @xmath212 respectively . baryon number violating operator of eq . ( [ w ] ) in @xmath0 . diagram ( a ) arises from integrating out colored fields from @xmath75 and @xmath76 , while diagram ( b ) arises from @xmath77 . ] we have identified in eq . ( [ w ] ) a single holomorphic operator of dimension six in the superpotential which violates @xmath8 and @xmath1 . in fig . [ susy ] we show how this operator can arise in susy @xmath0 . fig . [ susy ] ( a ) is obtained by integrating out colored fields in the @xmath75 and @xmath76 , while fig . [ susy ] ( b ) is obtained by integrating out such fields from the @xmath77 . these are superfield diagrams , effectively generating @xmath213terms in the lagrangian . note that the effective superpotential is antisymmetric in down - quark flavor in both diagrams ( a ) and ( b ) . this can be explicitly verified , by making use of the @xmath214 vertex in fig . [ susy ] ( a ) which is color antisymmetric . in fig . [ susy ] ( a ) , the vertex @xmath215 is contained in the yukawa coupling of eq . ( [ yuk126 ] ) and the vertex @xmath216 can be either from eq . ( [ yuk10 ] ) or from eq . ( [ yuk126 ] ) . the @xmath217 and @xmath218 transitions occur through @xmath219 ( or @xmath220 ) mass term , for example . as for the vertex @xmath221 here is a concrete example . the coupling @xmath222 is invariant , and contains such a term , where @xmath223 , @xmath224 and @xmath225 . note that @xmath225 carries @xmath226 , since it is part of the fragment @xmath227 under @xmath112 . of course , the mssm field @xmath59 in this case is an admixture of @xmath98 from @xmath75 , @xmath4 , @xmath76 and @xmath228 and carries no definite @xmath1 charge . as for fig . [ susy ] ( b ) , the vertices @xmath229 and @xmath230 are contained in the @xmath77 yukawa coupling of eq . ( [ yuk120 ] ) . the vertex @xmath231 can arise from the superpotential coupling @xmath232 , which is invariant . suporpotential operators of eq . ( [ w ] ) can also arise in susy @xmath0 models which utilize low dimensional higgs representations , for e.g. , @xmath233 . such models have been widely discussed in the literature @xcite . while @xmath234parity is no longer automatic in these models ( unlike in the case of @xmath4 models ) , it can be ensured by a discrete @xmath235 symmetry which distinguishes @xmath5 from the chiral fermions @xmath236 and which remains unbroken . the doublet triplet splitting problem can be addressed without fine - tuning in these models via the dimopoulos wilczek mechanism @xcite . generation of fermion masses in these models relies on higher dimensional operators such as @xmath197 which induce heavy majorana masses for the @xmath196 , and @xmath237 which induce lighter family masses and ckm mixings . ( @xmath5 and @xmath185 acquire gut scale vevs along their sm singlet components , while @xmath238 and @xmath239 do not . the @xmath32 doublet components of @xmath5 and @xmath238 acquire weak scale vevs , see for e.g. , discussions in ref . @xcite . ) to see the origin of eq . ( [ w ] ) in such models , consider generation of fig . [ susy ] ( a ) . the vertex @xmath240 arises from the yukawa coupling @xmath237 after a gut scale vev is inserted for the @xmath5 ( note that @xmath238 contains a @xmath154 field ) , and the vertex @xmath241 is contained in the coupling @xmath242 . now , @xmath243 can convert itself into @xmath244 via the coupling @xmath245 when the @xmath1preserving vev of @xmath246 is inserted , while the @xmath154 can transition into @xmath247 via the coupling @xmath248 with the insertion of a @xmath185 vev . the @xmath249 term is contained in the coupling @xmath250 , which completes the diagram . explicit models @xcite contain all these terms necessary for generating the @xmath9 superpotential operator . before discussing the rates for @xmath1violating nucleon decay , let us note that certain scalar bosons and certain gauge bosons would induce the more dominant @xmath9 baryon number violating operators of eq . ( [ dim6 ] ) . scalar bosons with sm quantum numbers @xmath101 , @xmath251 and @xmath252 and vector gauge bosons with sm quantum numbers @xmath253 and @xmath200 can can induce these @xmath1 preserving @xmath9 operators . current nucleon lifetime limits restrict the masses of these gauge bosons to be larger than about @xmath13 gev , and those for the scalar bosons to be heavier than about @xmath254 gev ( for yukawa couplings of order @xmath255 ) . since some of the @xmath18 operators arise in @xmath0 via the exchange of these particles , the above limits have to be met in our estimate of @xmath18 decay rates . we proceed to estimate the lifetime of the nucleon for its @xmath1violating decays in @xmath0 a straightforward way.preserving nucleon decay has been studied in the context of @xmath234parity breaking susy in ref . the diagrams of fig . [ sym ] lead to the estimate @xmath256 here we have defined the yukawa couplings of @xmath102 and @xmath154 fields appearing in fig . [ sym ] to be @xmath257 and @xmath258 . these couplings are linear combinations of the yukawa coupling matrices @xmath259 , @xmath260 and @xmath261 of eqs . ( [ yuk10])-([yuk120 ] ) with flavor indices corresponding to the first family fermions . the factors @xmath262 and @xmath213 are chiral lagrangian factors , @xmath263 and @xmath264 . @xmath265 is the nucleon decay matrix element @xcite , @xmath266 , and @xmath267 gev . we have defined the trilinear coupling of fig . [ sym ] to have a coefficient @xmath268 . as expected , the rate is suppressed by six powers of inverse mass , owing to the higher dimensionality of the effective operator . the mass of @xmath101 is constrained to be relatively large , as it mediates @xmath9 nucleon decay . for @xmath269 , @xmath270 gev must be met from the @xmath9 decays . as an illustration , choose @xmath271 , @xmath272 gev , @xmath273 gev , and @xmath274 gev in eq . ( [ tau ] ) . this choice would result in @xmath275 yrs . such a spectrum is motivated by the intermediate symmetry @xmath276 ( without discrete parity ) , which is found to be realized at @xmath277 gev from gauge coupling unification @xcite . as a second example , take @xmath278 gev , @xmath279 gev , @xmath280 gev , @xmath281 . this choice of spectrum leads to @xmath282 yrs . and two @xmath283 scalar multiplets at @xmath284 gev ( right panel ) . the left panel corresponds to having one @xmath199 and one @xmath283 scalar fields at the weak scale.,title="fig : " ] and two @xmath283 scalar multiplets at @xmath284 gev ( right panel ) . the left panel corresponds to having one @xmath199 and one @xmath283 scalar fields at the weak scale.,title="fig : " ] this second choice for the spectrum can be motivated as follows . the unification of gauge couplings may occur without any particular intermediate symmetry , but with certain particles surviving to an intermediate scale . we have found two examples of this type where the @xmath18 nucleon decay is within observable range . suppose the @xmath199 particle , along with a pair of @xmath283 scalar particles ( contained in the @xmath246 , @xmath285 or @xmath228 needed for symmetry breaking ) survive down to @xmath286 gev . the three sm gauge couplings are found to unify at a scale @xmath287 gev in this case . this is shown in fig . [ unif ] right panel . this figure is obtained by the one - loop renormalization group evolution of the sm gauge couplings above @xmath68 with beta function coefficients @xmath288 , where @xmath289 . since the @xmath9 nucleon decay lifetime would also be near @xmath290 yrs with such a unification scale , this scenario would predict observable rates for both the @xmath1conserving and @xmath1violating nucleon decay modes . another possible scenario for consistent gauge coupling unification is to assume that @xmath199 and one @xmath283 scalar multiplet survive down to the weak scale . again , the gauge couplings unify around @xmath287 gev , as shown in the left panel of fig . [ unif ] . here the @xmath291function coefficients used are @xmath292 , appropriate for this spectrum . the estimate @xmath293 yrs would follow , as in the second example above , but now with @xmath294 . nucleon decay rates arising from diagrams of fig . [ antisym ] are similar , but with a significant difference . take for example fig . [ antisym ] ( b ) with the @xmath295 trilinear vertex . neither the @xmath136 nor the @xmath199 field would mediate @xmath9 nucleon decay , and may be considerably lighter than the gut scale . in non supersymmetric @xmath0 models it is natural that some scalars survive down to an intermediate scale . typically the intermediate scale is of order @xmath296 gev if the associated symmetry contains @xmath297 . thus it is possible that both @xmath167 and @xmath154 have masses of order @xmath298 gev . nucleon decay rate is now estimated to be @xmath299 here @xmath268 is defined as the coefficient of the trilinear scalar vertex . for @xmath300 gev , @xmath301 gev , @xmath302 , @xmath303 yrs , which is in the observable range . the gauge boson exchange diagrams of fig . [ gauge ] would also induce @xmath304)violating nucleon decay . however , we find the rates for these decays to be suppressed . [ gauge ] ( a ) for example . the vector gauge boson @xmath204 must have mass of order @xmath13 gev , since it mediates @xmath9 nucleon decay , while @xmath207 could be much lighter as it does not have @xmath8violating interaction by itself . the @xmath305 vertex necessary for connecting the @xmath18 diagram has a coefficient of order @xmath306 , while the mass of @xmath207 is of order @xmath307 . the amplitude for @xmath18 nucleon decay arising from fig . [ gauge ] is then given by @xmath308 . this amplitude is a factor @xmath309 smaller compared to the standard @xmath9 nucleon decay amplitude originating from gut scale gauge bosons . unless the @xmath1breaking scale @xmath310 is close to the weak scale @xmath311 , not a likely scenario based on gauge coupling unification , nucleon lifetime for @xmath1violating modes from these diagrams would be beyond the reach of experiments . however , as we shall see in the next section , the @xmath1violating decays of the gauge boson can naturally explain the observed baryon asymmetry of the universe . as for the supersymmetric diagram of fig . [ susy ] , the @xmath9 operator in the superpotential eq . ( [ w ] ) has a strength @xmath312 . the lagrangian would contain a term @xmath313 , upon insertion of the vev of @xmath59 . the scalars can be converted to @xmath314 and @xmath315 quarks via a gluino loop , and would result in the following estimate for processes such as @xmath316 : @xmath317 here @xmath318 is a typical susy breaking mass scale . for @xmath319 , @xmath320 , @xmath321 gev , @xmath322 gev , @xmath323 tev , one obtains @xmath324 yrs . as for the consistency of such an intermediate scale mass for @xmath154 with gauge coupling unification in susy models , we note that @xmath199 forms a complete @xmath10 @xmath131plet along with a @xmath208 and a @xmath325 fields . if these fields also have an intermediate scale mass , unification of gauge coupling would work as in the mssm . it should be noted that in susy models , the decay @xmath326 would be suppressed , since the vev of @xmath59 picks a neutrino field in eq . ( [ w ] ) . discovery of @xmath326 decay would thus hint at a deeper non supersymmetric dynamics . we now proceed to the computation of the baryon asymmetry of the universe induced at the gut epoch . the @xmath1violating decays of the scalars @xmath101 and @xmath136 and of the vector gauge boson @xmath200 will be used to illustrate the mechanism . we shall see that in each case , the out of equilibrium condition can be satisfied , and that there is enough cp violation . these decays generate an asymmetry in @xmath1 , which is not destroyed by the effective interactions induced by the electroweak sphalerons , and would survive to low temperatures . this is in contrast with the induced baryon asymmetry in the @xmath1preserving decays of gut scale scalars and gauge bosons in unified models such as @xmath10 , which is washed out by the spharleron interactions . we begin with the @xmath1violating decay of the scalar @xmath101 which is assumed to have a mass of order the gut scale . ( @xmath14 asymmetry in decays of specific heavy particles has recently been discussed in ref . @xcite . ) to be concrete , we shall work in the framework of non supersymmetric @xmath0 , although our results would hold for susy @xmath0 as well , with some minor modifications . we identify @xmath102 to be the lightest of the various @xmath327 scalar fields in the @xmath0 theory . the yukawa couplings of eq . ( [ yuk10])-([yuk120 ] ) imply that @xmath102 ( which is in general a linear combination of @xmath328 from the @xmath75 and @xmath76 and @xmath77 fields ) has two body decays into fermions of the type @xmath329 . these decays preserve @xmath1 , as can be seen by assigning @xmath330 . now , @xmath102 also has a two body scalar decay , @xmath331 as shown in fig . [ baryo1 ] ( a ) , which uses the @xmath1 breaking vev of @xmath89 . the scalar field @xmath154 has two body fermionic decays of the type @xmath332 ( the latter if kinematically allowed ) , which define @xmath1 charge of @xmath154 to be @xmath333 . thus the decay @xmath331 would violate @xmath1 by @xmath334 ( recall that @xmath45 has zero @xmath1 charge ) . asymmetry in @xmath102 decay . ] focussing on the @xmath1violating decay @xmath331 , we define a @xmath1 asymmetry parameter @xmath335 as follows . let the branching ratio for @xmath331 be @xmath336 which produces a net @xmath1 number of @xmath337 , and that for @xmath338 be @xmath339 , with net @xmath340 . the branching ratio for the two fermion decays @xmath341 is then @xmath342 which has @xmath343 , and that for @xmath344 is @xmath345 which has @xmath346 . thus in the decay of a @xmath347 pair , a net @xmath1 number , defined as @xmath335 , is induced , with @xmath348 note that an interplay between the @xmath1conserving decays and the @xmath1preserving decays of @xmath102 is necessary for inducing this asymmetry . in addition , cp violation is required , otherwise @xmath349 and thus @xmath350 . nonzero @xmath335 also requires a loop diagram which has an absorptive part . all these conditions are realized in @xmath0 models . the loop diagrams for @xmath351 are shown in fig . [ baryo1 ] ( b)-(d ) , which involve the exchange of fermions . since @xmath102 can also decay to two on shell fermions , these loop diagrams have absorptive parts . note that fig . [ baryo1 ] ( b)-(c ) are proportional to the majorana masses for the @xmath196 fields , while fig [ baryo1 ] ( d ) is not . these diagrams have cp violating phases , ensuring a nonzero @xmath335 . we evaluate fig . [ baryo1 ] in a basis where the majorana mass matrix of the @xmath196 fields is diagonal and real . the contribution of fig . [ baryo1 ] ( b ) to @xmath335 is found to be @xmath352 { \rm br}. \label{asymb}\ ] ] here we have defined the trilinear scalar vertex of fig . [ baryo1 ] ( a ) to have a coefficient @xmath268 in the lagrangian . @xmath353 is the yukawa coupling matrix corresponding to the coupling @xmath354 , etc . @xmath355 is the diagonal and real mass matrix of @xmath196 fields . @xmath356 stands for the branching ratio @xmath357 . a factor of @xmath358 has been included here for the two @xmath32 final states in the decay . the function @xmath359 is defined as @xmath360 with @xmath361 denoting the mass of @xmath362 . here @xmath363 stands for the step function , signalling additional ways of cutting the diagram when @xmath364 in fig . [ baryo1 ] ( b ) . [ baryo1 ] ( c ) yields the following contribution to @xmath335 : @xmath365{\rm br } , \label{asymc}\ ] ] where @xmath366 is defined as @xmath367 fig . [ baryo1 ] ( d ) arises because in any realistic @xmath0 model there are at least two @xmath102 fields . the heavier @xmath102 field is denoted as @xmath368 . in principle one can sum over all such @xmath368 contributions , but here we have kept only one such @xmath368 field . its contribution to @xmath335 is found to be @xmath369 { \rm br } , \label{asymd}\ ] ] where @xmath370 is defined as @xmath371 this contribution , which is non - vanishing even in the limit of vanishing @xmath361 , requires @xmath372 . here we have defined the trilinear coupling @xmath373 to have a coefficient @xmath374 in the lagrangian . we have also assumed that @xmath375 , so that there is no resonant enhancement for the decay . ( when @xmath102 and @xmath368 are nearly degenerate in mass , such a resonant enhancement is possible . in this case , the expression for the decay rate will be smoothened by the width of these particles . appropriate expressions in this case can be found in ref . @xcite . ) the branching ratio factor @xmath376 appearing in eqs . ( [ asymb ] ) , ( [ asymc ] ) , ( [ asymd ] ) can be estimated as follows . for this purpose let us assume that @xmath102 is the field @xmath102 from @xmath75 with yukawa couplings as given in eq . ( [ yuk10 ] ) . the partial widths for the decays @xmath377 and @xmath378 are then given by @xmath379 in the expression for @xmath380 we have assumed that @xmath196 is much lighter than @xmath102 . in terms of these partial widths , the branching ratio that appears in eqs . ( [ asymb ] ) , ( [ asymb ] ) , ( [ asymd ] ) is given as @xmath381 . to get a feeling for numbers , let us choose a realistic set of parameters : @xmath279 gev , @xmath382 ( corresponding to the top quark yukawa coupling at gut scale ) with other @xmath383 negligible , and @xmath384 gev . this would correspond to @xmath385 , which shows a strong dependence on @xmath268 . the total @xmath1 asymmetry in @xmath331 and its conjugate decay is given by @xmath386 this will result in the baryon to entropy ratio @xmath387 given by @xmath388 where @xmath389 is the total number of relativistic degrees of freedom at the epoch when these decays occur . in our present example @xmath390 which includes the sm particles and the @xmath154 and @xmath102 scalar fields . the factor @xmath314 in eq . ( [ eta ] ) is the dilution factor which takes into account back reactions that would partially wash out the induced baryon asymmetry . @xmath314 is determined by solving the boltzmann equations numerically , but simple analytic approximations are available suitable to the present setup . defining a ratio @xmath391 where @xmath392 is the hubble expansion rate , @xmath393 the dilution factor can be written as @xcite @xmath394 these approximations work well for @xmath395 or so , beyond which @xmath314 would be exponentially suppressed . for @xmath279 gev , @xmath384 gev , we find @xmath396 , with the corresponding dilution factors being @xmath397 . thus we see that there is not much dilution with this choice of parameters , although @xmath385 can become small for smaller values of @xmath268 . if we choose @xmath398 gev instead , and vary @xmath384 gev , then we find @xmath399 and the corresponding dilution factors to be @xmath400 , with @xmath401 . although the electroweak sphaleron interactions would not wash away the gut scale induced @xmath1 asymmetry , partial wash out can occur via the @xmath1violating interactions of the right handed neutrinos . this is possible because the @xmath196 fields acquire @xmath1violating majorana masses , and their interactions with the higgs field and the lepton fields can erase part of the gut induced asymmetry . in some cases it may be desirable to have partial wash out . one can also prevent any wash out by decoupling the @xmath196 fields at the same temperature as the @xmath102 field ( which would happen if @xmath402 ) . for the surviving light @xmath196 fields , the condition @xmath403 would guarantee no further wash out , where @xmath404 is the dirac yukawa coupling of the light @xmath196 field . while we do not present the calculation of @xmath387 in the supersymmetric version of the @xmath0 model , results in that case would be similar to the non susy case . more diagrams contribute to the generation of @xmath1 asymmetry with superparticle decays included . the number of relativistic degrees of freedom @xmath389 would also double in this case . the @xmath1 asymmetry induced in the decay @xmath405 is similar to the one induced in the decay @xmath331 . the tree level @xmath1violating decay and the one loop corrections are shown in fig . [ baryo2 ] . @xmath167 has a fermionic decay mode @xmath406 shown in fig . [ baryo2 ] ( a ) , which can be used to define its @xmath1 quantum number as @xmath407 . the decay @xmath408 ( fig . [ baryo2 ] ( b ) ) would then violate @xmath1 by @xmath409 units , since @xmath410 obtained from the decays @xmath411 . the one loop correction to this decay is shown in fig . [ baryo2 ] ( c ) , which is evaluated in analogy to fig . [ baryo1 ] ( c ) to be @xmath412{\rm br}. \label{asymeta}\ ] ] here the function @xmath413 is defined as @xmath414 and @xmath268 identified as the coefficient of the trilinear vertex @xmath137 . asymmetry in @xmath415 decay . ] a noteworthy feature of baryon asymmetry generated in the decay @xmath415 is that both the scalars @xmath167 and @xmath154 can be relatively light , since they do not induce @xmath9 baryon number violating operators of eq . ( [ dim6 ] ) . in supersymmetric extensions of @xmath0 , this allows for the possibility that the gravitino abundance problem of supergravity models can be evaded . typically in supergravity models , the reheat temperature after inflation is required to be @xmath416 gev , in order to sufficiently dilute the gravitino abundance in the universe . if @xmath417 gev , this requirement would be compatible with the @xmath1 asymmetry generation . the @xmath18 baryon number violating operators of eq . ( [ dim7 ] ) can arise by integrating out the @xmath200 and @xmath418 gauge bosons of @xmath0 ( see fig . [ gauge ] ) , which lie outside of the @xmath10 subgroup . the decay @xmath419 and the conjugate decay @xmath420 can produce a primordial @xmath1 asymmetry at the gut scale , which would survive down to low temperatures without being washed out by the sphaleron interactions . level decay diagram and the one loop correction are shown in fig . [ baryo3 ] . the vector gauge boson @xmath204 has two fermion decays into the following channels : @xmath421 for the charge @xmath422 component , and @xmath423 for the charge @xmath424 component . these decays conserve @xmath1 , as can be seen by assigning @xmath425 . the gauge boson @xmath426 has the fermionic decays @xmath427 , suggesting that @xmath428 . the decay @xmath419 would then change @xmath1 by @xmath409 , as in the case of the scalar decay @xmath331 . the @xmath1 asymmetry arising from fig . [ baryo3 ] is found to be @xmath429{\rm br } , \label{asymv}\ ] ] with the function @xmath430 defined as @xmath431 here we have defined the lagrangian coefficient of the @xmath305 vertex to be @xmath432 . this is a consistent definition , since the lighter gauge boson @xmath207 would acquire a mass of order @xmath307 . here @xmath433 is a clebsch factor of order unity , with its value depending on the higgs representation used for rank reduction ( @xmath4 or @xmath5 ) . asymmetry in the decay of vector gauge boson @xmath204 . ] the factor @xmath356 that appears in eq . ( [ asymv ] ) is the branching ratio @xmath434 . it is determined in terms of the two partial widths as @xmath435 , where @xmath436 if we choose @xmath437 gev , @xmath438 ( with @xmath439 and @xmath440 ) , so that the dilution factor is @xmath441 . there is a modest suppression in @xmath442 arising from the branching ratio , since @xmath443 with this choice . we now show how the gut scale induced asymmetry in @xmath331 decay can consistently explain the observed value of @xmath444 , in a class of minimal @xmath0 models . in this class of models , a single @xmath75 and a single @xmath76 couple to fermions , as in eqs . ( [ yuk10])-([yuk126 ] ) . in susy models this is automatic with a single @xmath75 and @xmath76 employed . in non susy models the @xmath445 can also couple ( @xmath75 must be complexified to generate realistic fermion masses ) , however if a peccei quinn symmetry is assumed , the @xmath445 coupling would be absent . this class of models is highly constrained due to the small number of parameters that describe the fermion masses and mixings , and leads to predictions for the neutrino oscillation parameters @xcite . in addition to generating large mixing angles for solar neutrino oscillations and for atmospheric neutrino oscillations , these models predict a relatively large value of @xmath25 , viz . , @xmath446 , both in the non supersymmetric and the supersymmetric versions , consistent with recent results from daya bay and other experiments @xcite . to illustrate how realistic choice of parameters generate acceptable @xmath387 , we choose the @xmath102 field to be almost entirely in the @xmath75 . we also choose @xmath447 that appears in fig . [ baryo1 ] ( d ) to be small , so that the leading contribution to @xmath335 is from fig . [ baryo1 ] ( c ) , as given in eq . ( [ asymc ] ) . in this limit , we find @xmath448 here we have kept only the third family yukawa couplings , which is the leading contribution , and we have defined @xmath449 . choosing @xmath450 ( the top quark yukawa coupling at the gut scale ) , and @xmath451 , @xmath452 gev , @xmath453 ( so that @xmath454 gev , consistent with the light @xmath455 mass arising via the seesaw mechanism ) , @xmath456 , we find @xmath457 . if @xmath398 gev , then @xmath458 , @xmath459 so that the dilution factor is @xmath460 . this results in a net @xmath461 , consistent with observations . while natural choices of parameters can generate acceptable @xmath387 , due to the high sensitivity of dilution factor on the masses of the heavy particles , precise predictions are difficult to make . for @xmath419 decay we find the process to be typically out of equilibrium so that @xmath462 for @xmath463 gev . natural values of the asymmetry parameter in this case is @xmath464 . some dilution effects from the @xmath196 interactions would be welcome in this case . it should be mentioned that the @xmath18 operators of eq . ( [ lep ] ) also arise naturally in @xmath0 models , as already noted . the @xmath93 and the @xmath166 vertices arising from the @xmath4 couplings can be used for gut scale @xmath1genesis without generating @xmath18 nucleon decay operators . the decays @xmath465 has already been analyzed , but if these particles arise from @xmath4 they do not lead to @xmath1violating nucleon decay . in conclusion , we have pointed out that the complete set of @xmath18 baryon number violating operators that lead to the selection rule @xmath466 in nucleon decay can emerge as effective low energy operators in @xmath0 unified theories with either a @xmath4 or a @xmath5 higgs field used for breaking the @xmath14 gauge symmetry . the strength of these operators is unobservable in single step models where @xmath0 breaks directly down to the standard model . in non supersymmetric @xmath0 models , an intermediate symmetry is required in order for the gauge couplings to unify correctly . we have shown that in several instances with such an intermediate scale , the @xmath18 baryon number violating operators can lead to observable nucleon decay rates . the decay modes are distinct from the conventional gut motivated modes , and include @xmath467 , etc . we have also identified supersymmetric scenarios where such modes may be within reach of experiments , consistent with gauge coupling unification . a second major result of this paper is a new way of generating @xmath1 asymmetry in the early universe by the decay of gut mass particles . it is these particles which also induce the @xmath18 nucleon decay operators . such an asymmetry is sphaleron proof , in that it does not get erased by the effective interactions of the electroweak sphalerons . we present several examples where consistent asymmetry can be generated with the gut scale decays of particles obeying the @xmath468 selection rule . further , we show that in minimal @xmath0 models which explain the large neutrino mixing angles and predict relative large value for @xmath25 , consistent with recent experimental results , that the induced baryon asymmetry via the proposed gut scale mechanism is compatible with observations . there is thus a strong connection between neutrino oscillation parameters and baryon asymmetry in this class of models . the work of ksb is supported in part the us department of energy , grant numbers de - fg02 - 04er41306 and that of rnm is supported in part by the national science foundation grant number phy-0968854 . ksb acknowledges helpful discussions with j. julio . a. d. sakharov , pisma zh . fiz . * 5 * , 32 ( 1967 ) [ jetp lett . * 5 * , 24 ( 1967 ) ] . for reviews on gut scale baryogenesis before the discovery of the sphaleron and for original references see : p. langacker , phys . rept . * 72 * , 185 ( 1981 ) ; e. w. kolb and m. s. turner , ann . nucl . part . sci . * 33 * , 645 ( 1983 ) ; _ the early universe _ , by e. kolb and m. turner , ( frontiers in physics ) ( 1986 ) . k. s. babu , r. n. mohapatra , phys . lett . * 70 * , 2845 ( 1993 ) ; b. bajc , g. senjanovic and f. vissani , hep - ph/0110310 ; phys . rev . lett . * 90 * , 051802 ( 2003 ) ; t. fukuyama and n. okada , jhep * 0211 * , 011 ( 2002 ) ; h. s. goh , r. n. mohapatra , s. p. ng , phys b570 * , 215 ( 2003 ) ; k.s . babu , c. macesanu , phys . rev . * d72 * , 115003 ( 2005 ) ; s. bertolini , t. schwetz , m. malinsky , phys . rev . * d73 * , 115012 ( 2006 ) ; a. s. joshipura , k. m. patel , arxiv:1105.5943 [ hep - ph ] . c. s. aulakh , b. bajc , a. melfo , g. senjanovic , f. vissani , phys . * b588 * , 196 - 202 ( 2004 ) ; t. fukuyama , a. ilakovac , t. kikuchi , s. meljanac , n. okada , phys . * d72 * , 051701 ( 2005 ) ; c. s. aulakh and s. k. garg , nucl . b * 857 * , 101 ( 2012 ) . f. p. an _ et al . _ [ daya - bay collaboration ] , arxiv:1203.1669 [ hep - ex ] . k. abe _ et al . _ [ t2k collaboration ] , phys . 107 * , 041801 ( 2011 ) ; p. adamson _ [ minos collaboration ] , phys . lett . * 107 * , 181802 ( 2011 ) ; h. de . kerrect [ double chooz collaboration ] , talk at the lownu conference in seoul , korea ( 2011 ) , http://workshop.kias.re.kr/lownu11/?program . p. minkowski , phys . b * 67 * , 421 ( 1977 ) ; m. gell - mann , p. ramond , and r. slansky , in _ supergravity _ , eds . d. freedman _ et al . _ , ( north - holland , amsterdam , 1980 ) ; t. yanagida , in _ proceedings of the workshop on baryon number in the universe _ , eds . o. sawada and a. sugamoto , ( kek , 1979 ) ; r. mohapatra and g. senjanovi , phys . lett . * 44 * , 912 ( 1980 ) . d. chang , r. n. mohapatra , j. gipson , r. e. marshak and m. k. parida , phys . d * 31 * , 1718 ( 1985 ) ; r. n. mohapatra and m. k. parida , phys . d * 47 * , 264 ( 1993 ) ; n. g. deshpande , e. keith and p. b. pal , phys . rev . d * 46 * , 2261 ( 1993 ) ; s. bertolini , l. di luzio and m. malinsky , phys . d * 80 * , 015013 ( 2009 ) ; s. bertolini , l. di luzio and m. malinsky , arxiv:1202.0807 [ hep - ph ] . c. h. albright and s. m. barr , phys . d * 58 * , 013002 ( 1998 ) ; 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we show that grand unified theories based on @xmath0 generate naturally the next to leading baryon number violating operators of dimension seven . these operators , which violate @xmath1 , lead to unconventional decays of the nucleon such as @xmath2 and @xmath3 . in two step breaking schemes of non - supersymmetric @xmath0 , nucleon lifetime for decays into these modes is found to be within reach of experiments . we also identify supersymmetric scenarios where these decays may be accessible , consistent with gauge coupling unification . further , we show that the @xmath1asymmetry generated in the decays of gut scale scalar bosons and/or gauge bosons can explain consistently the observed baryon asymmetry of the universe . the induced @xmath1 asymmetry is sphaleron proof , and survives down to the weak scale without being erased by the electroweak interactions . this mechanism works efficiently in a large class of non susy and susy @xmath0 models , with either a @xmath4 or a @xmath5 higgs field employed for rank reduction . in minimal models the induced baryon asymmetry is tightly connected to the masses of quarks , leptons and neutrinos and is found to be compatible with observations . osu - hep-12 - 04 + umd - pp-012 - 004 * violating nucleon decay and + gut scale baryogenesis in * + * k.s . babu*@xmath6 and * r.n . mohapatra*@xmath7 _ @xmath6department of physics , oklahoma state university , stillwater , ok 74078 , usa _ _ @xmath7maryland center for fundamental physics , department of physics , + university of maryland , college park , md 20742 , usa _

spontaneous pair production of oppositely charged particles was first discussed by heisenberg and euler @xcite by making use of the dirac picture of the vacuum . it also provided the solution of the klein paradox of relativistic quantum mechanics @xcite for the scattering on a potential step . within second quantized field theory it was first discussed by hund @xcite , as a precursor of the later famous calculation of schwinger @xcite , whose generalization to gravitational fields was made by hawking @xcite . schwinger calculated the one - loop effective action of qed in a constant electromagnetic field by the proper time method . the imaginary part of this effective action yields directly the probability of pair production from the vacuum . the result confirms the fact anticipated by sauter @xcite twenty years earlier that particles can pass through strong repulsive potentials without the exponential damping expected in quantum tunneling processes . a detailed review and the relevant references can be found in refs . @xcite and @xcite . another method to describe the problem of pair production is due to @xcite . it is based on the use of causal green functions @xcite to relate the particle production directly to the scattering process . it can be shown that the rate of pair production can be expressed as an ordinary energy - momentum integral over the logarithm of the reflection coefficient . this formula allows one to connect in a most transparent and efficient way the one - particle dirac approach in which the klein paradox first appeared with the second quantized field theory in which the problem was first solved satisfactorily . it has been used for many other developments including semiclassical approximations @xcite . in the same papers @xcite , also the scattering and pair production processes by the sauter potential @xmath0 were analyzed in detail . the exact solutions of the both dirac and klein - gordon equations were found in this potential , and the causal green functions were constructed to define the pair production rates of fermions and bosons . however , the calculations were never carried on to derive an exact expression for the pair production rate . it is the purpose of this paper to complete this gap . consider a relativistic scalar particle of charge @xmath1 and mass @xmath2 moving in an external electromagnetic potential @xmath3 corresponding to nonuniform electric field along @xmath4-direction with the strength @xmath5 . its kinetic energy and momentum are @xmath6 in the transverse direction the particle propagates freely as a plane wave @xmath7 $ ] . thus we represent the solution of klein - gordon equation in the potential @xmath8 as a product of this plane wave with the @xmath4-dependent wave function @xmath9 satisfying the schrdinger - like equation ( @xmath10 ) : @xmath11\psi ( z ) = 0\,.\label{1.2}\end{aligned}\ ] ] we shall consider the smooth step potential of the sauter type @xmath12 with @xmath13 , where the sharp step potential is recovered in the limit @xmath14 , while the limit @xmath15 with fixed @xmath16 reproduces the linear potential due to a constant electric field . in the potential ( [ 1.3 ] ) , eq . ( [ 1.2 ] ) can be solvable exactly @xcite . the solution describes the barrier scattering of a particle impinging the left with asymptotic boundary conditions @xmath17 where @xmath18 , @xmath19 and @xmath20 are normalization constants . let us introduce the initial and final values of the particle energy far to the left and to the right of the potential ( [ 1.3 ] ) : @xmath21 and the corresponding momenta @xmath22 in the scattering process , the momenta @xmath23 in eq . ( [ 1.5 ] ) are real thus restricting the asymptotic energies to @xmath24 . to solve eq . ( [ 1.2 ] ) with the asymptotic conditions ( [ 1.4 ] ) we set @xmath25 where @xmath26 we further replace @xmath4 by the dimensionless variable @xmath27 running from @xmath28 to @xmath29 . as usual , we extract the asymptotic behavior at @xmath30 with the help of the substitution @xmath31 this brings eq . ( [ 1.2 ] ) to the following differential equation for @xmath32 : @xmath33 f ( \y ) = 0 \ , . \label{1.10}\end{aligned}\ ] ] the singularity at @xmath34 suggests replacing @xmath35 with @xmath36 leading to a hypergeometric equation for the function @xmath37 : @xmath38\y\right\ } w ' ( \y ) - \left[\delta^2 -2i\nu\delta + \frac{v p_0}{k^2 \hbar^2}\right ] w ( \y ) = 0 . \label{1.13}\end{aligned}\ ] ] the solution is the hypergeometric function @xmath39\ , \label{1.14}\end{aligned}\ ] ] up to some normalization factor . for @xmath30 , this function tends to @xmath40 , and the solution of eq . ( [ 1.2 ] ) contains only the transmitted wave @xmath41 $ ] satisfying the asymptotic condition in eq . ( [ 1.4 ] ) with @xmath42 . for @xmath43 , we find the asymptotic form of the function @xmath44 in accordance with the asymptotic condition of eq . ( [ 1.4 ] ) via the kummer transformation of the hypergeometric function @xmath37 , with the coefficients @xmath45\,\gamma [ 1-\delta - i(\mu+\nu])}\,,\\ a_2 & = & \frac{\gamma ( 1 - 2i\nu)\,\gamma ( 2i\mu)}{\gamma [ \delta+i(\mu-\nu)]\,\gamma [ 1-\delta+i(\mu-\nu])}\ , . \label{1.15}\end{aligned}\ ] ] from these we determine the reflection and transmission coefficients for the sauter potential : @xmath46 in our discussion of pair production , we shall focus mostly on the reflection coefficient @xmath47 . to find a simple expression for it , we relate the total kinetic energy @xmath48 to the parameters @xmath49 of the sauter potential ( [ 1.3 ] ) via eq . ( [ 1.12 ] ) . for @xmath50 , the square root @xmath51 in eq . ( [ 1.12 ] ) is real . after substituting @xmath18 and @xmath19 from eq . ( [ 1.15 ] ) into eq . ( [ 1.16 ] ) , the reflection coefficient takes the form @xmath52 where we have used the relation @xmath53 . in the limit @xmath14 where @xmath54 , eq . ( [ 1.17 ] ) reproduces the well - known coefficient @xcite : @xmath55 ^ 2\ , \label{1.18}\end{aligned}\ ] ] for the reflection of a relativistic particle off a potential step @xmath56 = \left\ { \begin{array}{ccc } a&\mbox{,}&z > 0\\ \!\!- a&\mbox{,}&z < 0 \end{array}\right.\,,\label{1.19}\end{aligned}\ ] ] which arises from eq . ( [ 1.3 ] ) in this limit . for @xmath57 , the square root @xmath58 in eq . ( [ 1.12 ] ) becomes purely imaginary , @xmath59 , where @xmath60 is real . then the reflection coefficient in eq . ( [ 1.16 ] ) with @xmath18 and @xmath19 of eq . ( [ 1.15 ] ) takes the form @xmath61 where we have used the relation @xmath62 . a special case of uniform electric field along the @xmath4-direction is included here in the limit @xmath63 with @xmath64 const . it corresponds to a linear potential @xmath65 in eq . ( [ 1.3 ] ) . in this limit , @xmath66 becomes very large , and @xmath67 , so that the reflection coefficient ( [ 1.20a ] ) reduces to @xmath68\right\}^{-1}\ , . \label{1.22}\end{aligned}\ ] ] we have assumed in this section that the particles can pass through repulsive potential barrier with the same ( positive ) sign of the initial and final energies @xmath69 and @xmath70 , respectively . in the opposite situation , when the initial energy @xmath71 is positive but the final @xmath72 becomes negative , the famous _ klein paradox _ arises . for a particle moving from left to right this means that the region of large positive @xmath4 can only be accessible to antiparticle . then a non - zero transmission coefficient must be present even for a strong potential @xmath73 , where @xmath74 is the potential energy difference at infinity . as has been explained in refs . @xcite , this happens for @xmath75 and @xmath76 , implying the spontaneous production of particle - antiparticle pairs with the total energy difference @xmath77 . the pair production from the vacuum is now derived as follows . the average number of created pairs in the scattering process yields the same result as the imaginary part of the effective action in schwinger calculation @xcite . this means that the probability for vacuum to remain a vacuum under the influence of the external potential , i.e. , the vacuum persistence probability , is related to the reflection coefficient by @xmath78 where the product and the sums are taken over all relevant quantum numbers @xmath79 and @xmath80 of the created particles . correspondingly , the pair production probability is @xmath81 in a box - like volume @xmath82 with @xmath83 being the area of the potential step transverse to the @xmath4-direction , and @xmath84 the total time , the sum over @xmath79 and @xmath80 becomes an integral @xmath85 where the reflection coefficient @xmath86 is defined by eq . ( [ 1.17 ] ) or ( [ 1.20 ] ) , and @xmath87 . the rotation invariance around the third axis reduces the integral @xmath88 to @xmath89 . the remaining integral over @xmath79 and @xmath90 is done over the klein region @xmath91 the change of the sign of @xmath92 is necessary for a vacuum pair production by the sauter potential ( [ 1.3 ] ) with @xmath93 . the pair production probability is now completely defined by eqs . ( [ 2.2])([2.4 ] ) . the probability per unit area and unit time is @xmath94 where the integration region in the @xmath95-plane is shown in fig . 1 . @xmath96 in the @xmath95-plane , the integration covers the positive region restricted by two intersecting parabolas @xmath97 and @xmath98 with horizontal axes of symmetry above and below the @xmath90-axis for @xmath93.,title="fig:",width=207 ] ( 0,10 ) ( -173,107)@xmath99 ( -170,49)@xmath29 ( -12,37)@xmath100 ( -135,115)@xmath101 ( -140,-6)@xmath102 ( -21,75)@xmath103 for the actual calculation , we interchange the order of integration in eq . ( [ 2.5 ] ) to @xmath104\!\ln r ( p_0,p_{\perp}^2 ) . \label{2.6}\end{aligned}\ ] ] we now replace @xmath105 in the first integral of eq . ( [ 2.6 ] ) and make use of the symmetry of the reflection coefficient @xmath106 under the interchanging @xmath107 in eqs . ( [ 1.17 ] ) and ( [ 1.20 ] ) . the new integration region in the @xmath108-plane is shown in fig . 2 . as a result , we obtain for the pair production rate per unit area in the sauter potential ( [ 1.3 ] ) the integral representation @xmath109 in the @xmath110-plane , the integration covers the region under the left branch of the parabola in the first quadrant for @xmath111.,width=226 ] ( 0,12 ) ( -185,105)@xmath100 ( -165,4)@xmath29 ( -10,4)@xmath99 ( -15,48)@xmath101 ( -126,115)@xmath103 for a potential step of the sauter type ( [ 1.3 ] ) , the reflection coefficient as a function of @xmath79 and @xmath90 is given by eq . ( [ 1.18 ] ) . its logarithm reads @xmath112\ , , \label{3.1}\end{aligned}\ ] ] where @xmath113 with @xmath114 for @xmath115 . as in eq . ( [ 1.7 ] ) , the functions @xmath116 satisfy the constraint @xmath117 ^ 2 - \left[q_{- } ( p_0,p_{\perp}^2)\right]^2 = 4\ev p_0\ , . \label{3.2}\end{aligned}\ ] ] the constraint suggests introducing a parameter @xmath118 so that @xmath119 this allows us to express eq . ( [ 3.1 ] ) in terms of @xmath118 as @xmath120 where @xmath121\ , . \label{3.5}\end{aligned}\ ] ] with @xmath122 for @xmath123 . it is useful to eliminate the variable @xmath124 in favor of @xmath118 in the first integral of eq . ( [ 2.7 ] ) . alternatively we could have defined @xmath125\ , , \label{3.6}\end{aligned}\ ] ] with @xmath126 for @xmath123 where @xmath127 due to eq . ( [ 3.2 ] ) . this definition corresponds to eqs . ( [ 3.3 ] ) and ( [ 3.4 ] ) with @xmath128 . the change of the measure in the first integral of eq . ( [ 2.7 ] ) due to substituting @xmath129 is @xmath130 with eqs . ( [ 3.4 ] ) and ( [ 3.7 ] ) , the pair production rate ( [ 2.7 ] ) takes the form @xmath131 where the integration region in the @xmath132-plane is shown in fig . the integration covers the upper region restricted by the logarithmic curve @xmath133 and the positive @xmath79-axis in the @xmath132-plane . below the @xmath79-axis lies the alternative region restricted by the mirrored curve @xmath134.,width=264 ] ( 0,10 ) ( -203,133)@xmath118 ( -204,64)@xmath29 ( -10,62)@xmath99 ( -26,98)@xmath101 the integrals in eq . ( [ 3.8 ] ) are now straightforward to do . the right - hand integral yields @xmath135_{p_{\perp}^2 = 0}. \label{3.9}\end{aligned}\ ] ] after this , the remaining integral in eq . ( [ 3.8 ] ) becomes a combination of elliptic integrals via the substitution @xmath136 with @xmath137 . the last term in ( [ 3.9 ] ) leads to @xmath138 where @xmath139 is the dimensionless @xmath140-integral @xmath141\ , , \label{3.10a}\end{aligned}\ ] ] and @xmath142 , @xmath143 are complete elliptic integrals of the first and second kind , respectively , with the argument @xmath144 , where @xmath145 . with a little more effort we transform @xmath79-integral over the first term in ( [ 3.9 ] ) into a @xmath140-integral : @xmath146\ln\left[\frac{q_{+}(p_0,0 ) + q_{-}(p_0,0)}{\sqrt{4\ev p_0}}\right ] = \alpha_{+}\alpha_{-}^2\,i_1 ( \alpha)\,,\label{3.11}\end{aligned}\ ] ] where @xmath147 is the dimensionless integral @xmath148 - \ln t - \ln 2(1 + \alpha)\right\}\,.\label{3.11a}\end{aligned}\ ] ] this can be expressed in terms of the elliptic integrals of the first and second kind @xmath149 and @xmath150 as follows : @xmath151\,,\label{3.12}\end{aligned}\ ] ] with @xmath152 . finally , collecting all contributions in eq . ( [ 3.8 ] ) yields the pair production rate per area @xmath153 \right.\nonumber\\ & + & \left.\frac{\alpha}{3}\left[(1 - \alpha)(4 + \alpha)\,f \left(\varphi , \frac{1}{\alpha}\right ) + \left(1 + 3 \alpha + \alpha^2\right ) e \left(\varphi,\frac{1}{\alpha}\right)\right]\right\ } \,.\label{3.13}\end{aligned}\ ] ] we employ now eq . ( [ 2.7 ] ) to compute the pair production probability for the sauter potential ( [ 1.3 ] ) where the reflection coefficient is defined by eqs . ( [ 1.17 ] ) and ( [ 1.20 ] ) for all values of the parameters @xmath154 and @xmath155 . in order to illustrate the calculation , we specify these , for example , as @xmath156 . in this case , the parameter @xmath155 defines the inverse width of the electric field , whereas the parameter @xmath154 governs its size @xmath157 , whose maximum is @xmath158v / m . the limit @xmath15 with fixed @xmath16 reproduces the linear potential due to a constant electric field . the reflection coefficient of the sauter potential with @xmath57 is given by eq . ( [ 1.20 ] ) . an equivalent form of this is @xmath159 where @xmath160 and @xmath161 are the functions of @xmath79 and @xmath90 defined by eqs . ( [ 1.5a ] ) , ( [ 1.5 ] ) and ( [ 1.6 ] ) with the constraint ( [ 1.7 ] ) , while @xmath162 is a constant . taking logarithms of eq . ( [ 4.1 ] ) leads to the expansion @xmath163 the right hand side is found by replacing each logarithm of the hyperbolic functions as @xmath164 , and combining all sums . with eq . ( [ 4.2 ] ) , the pair production rate per area ( [ 2.7 ] ) takes the form @xmath165 where @xmath166 are the integrals @xmath167 the region of integration is shown in fig . 2 . a physically more instructive quantity than the production rate ( [ 2.7 ] ) can be obtained by dividing @xmath168 by the width of the potential step . for the sauter potential the width is defined by the ratio @xmath169 thus we obtain the pair creation rate per volume of nonzero field @xmath170 we perform the @xmath171-integration in eq . ( [ 4.3 ] ) by changing , for each @xmath172 separately , from the variable @xmath90 to the dimensionless one @xmath118 defined by @xmath173 ^ 2 \label{4.4a}\ ] ] with @xmath174 , where @xmath175 ^ 2 = ( n\pi / k\hbar)^{2}\left[\left(p_0 - v \right)^2 -m^2\right]\ , . \label{4.4b}\end{aligned}\ ] ] for each @xmath172 , eq . ( [ 4.4a ] ) yields @xmath176 and @xmath177 . let us also introduce the dimensionless functions @xmath178 so that @xmath179 due to eq . ( [ 1.7 ] ) . then @xmath171-integrals take the form @xmath180 the two terms in eq . ( [ 4.4 ] ) can now be combined into a single integral as follows . we substitute with @xmath181 in the first term , and with @xmath182 in the second , where @xmath183 . \label{4.4d}\end{aligned}\ ] ] noting that @xmath184 $ ] in the first and the second substitution , respectively , while @xmath185 ^ 2/t^4\}\,d t$ ] in both cases , we obtain @xmath186 ^ 2}{t^4}\right\}\,\cosh t\ , . \label{4.5}\end{aligned}\ ] ] evaluating this integral yields @xmath187 with @xmath188\ ! \mp\frac{\left[\theta^{(n)}_{+}\theta^{(n)}_{-}\right]^2}{2}\!\!\left[{\rm chi } \,\theta^{(n)}_{\pm}\!- \frac{\sinh\!\theta^{(n)}_{\pm}}{\theta^{(n)}_{\pm } } - \frac{\cosh\!\theta^{(n)}_{\pm}}{\left[\theta^{(n)}_{\pm}\right]^2}\right]\!\right\}\ ! , \label{4.7}\end{aligned}\ ] ] where @xmath189 is short for @xmath190 , @xmath191 are the hyperbolic cosine integrals , and the last two terms represent the leading terms in their asymptotic expansions for large arguments @xmath189 . having obtained @xmath192 , we are left in eq . ( [ 4.3 ] ) with the sum @xmath193 of the integrals over the rather lengthy functions ( [ 4.7 ] ) : @xmath194 however , this sum can be combined into a single integral by subjecting @xmath195 in eq . ( [ 4.9 ] ) to a change of variables @xmath196 provided that we define the new dimensionless integration variable @xmath197 as follows . in order to transform the integral @xmath198 , we introduce the dimensionless variable @xmath199 with @xmath200 for and @xmath111 , where @xmath201 and @xmath202 . noting that @xmath203 $ ] within these limits , we resolve eq . ( [ 4.10 ] ) in terms of @xmath79 as @xmath204}\right\}^{1/2}\ , , \label{4.11}\end{aligned}\ ] ] with positive @xmath197 due to @xmath115 . to determine @xmath190 of eq . ( [ 4.4d ] ) in terms of @xmath197 by means of eq . ( [ 4.11 ] ) , we find first for a given @xmath197 the positive square roots @xmath205}\right\}^{1/2}\geq 0\ , . \label{4.12}\end{aligned}\ ] ] combining these yields @xmath206}\right\}^{1/2}\ , . \label{4.13}\end{aligned}\ ] ] from eq . ( [ 4.13 ] ) we obtain , finally , the functions @xmath207 to be substituted instead of @xmath190 into the first integral @xmath198 as @xmath208}\right\}^{1/2}\ , . \label{4.14}\end{aligned}\ ] ] note that the inequality @xmath203 $ ] ensures the positivity of expressions under the square roots in eqs . ( [ 4.11])([4.14 ] ) . in order to treat the second integral @xmath209 in eq . ( [ 4.9 ] ) , we define a new integration variable @xmath197 similar to eq . ( [ 4.10 ] ) : @xmath210 with @xmath211 for and @xmath93 , where @xmath201 and @xmath202 . since @xmath212 $ ] in these limits , we solve eq . ( [ 4.15 ] ) in terms of @xmath79 in the same way as in eq . ( [ 4.11 ] ) , leading again to @xmath213}\right\}^{1/2}\ , , \label{4.16}\end{aligned}\ ] ] with positive @xmath197 due to @xmath115 . to determine @xmath190 in terms of @xmath197 by means of eq . ( [ 4.16 ] ) , we find now for a given @xmath197 the positive square roots @xmath214}\right\}^{1/2}\geq 0\ , . \label{4.17}\end{aligned}\ ] ] it follows from eq . ( [ 4.17 ] ) that @xmath215}\right\}^{1/2}\ , . \label{4.18}\end{aligned}\ ] ] this yields @xmath216 which replaces @xmath190 in the second integral @xmath209 as follows : @xmath217}\right\}^{1/2}\,\,\ , , \quad \theta^{(n)}_{-}(\xi ) = { 2\pi n}\xi \ , . \label{4.19}\end{aligned}\ ] ] again , the inequality @xmath218 $ ] ensures the positiveness of the expressions under the square roots in eqs . ( [ 4.16])([4.19 ] ) . we now go from the integration variable @xmath79 to @xmath197 in eq . ( [ 4.9 ] ) for @xmath198 and @xmath209 separately . substituting these in eq . ( [ 4.8 ] ) yields @xmath219\nonumber\\ \!\!\!\!&+&\frac{\hbar^2 k^2}{(2\pi n)^2 } \int^{{\bar\xi } } _ { 0}\!d \xi\,\frac{d p_{0 } ( \xi)}{d \xi}\ , \left[\cosh\left({2\pi n}\,\xi\right ) -{2\pi n } \xi\,\sinh\left({2\pi n}\,\xi\right)\right]\ , , \label{4.20}\end{aligned}\ ] ] where @xmath220 and the function @xmath221 is given by eq . ( [ 4.11 ] ) [ or ( [ 4.16 ] ) ] . equation ( [ 4.20 ] ) can be simplified by partial integration thanks to the vanishing of the function @xmath221 at the endpoints of integration . this yields @xmath222 where the function @xmath223 has the form @xmath224 after inserting @xmath225 from eq . ( [ 4.11 ] ) [ or ( [ 4.16 ] ) ] , this reads explicitly @xmath226^{1/2}\ ! - \frac{{\bar\xi } ^2 + \left(m/\hbar k\right)^2}{3}\ ! \left[\frac{{\bar\xi } ^2 -\xi^2}{{\bar\xi } ^2 - \xi^2 + \left(m/\hbar k\right)^2}\right]^{3/2}\ ! = f(-\xi ) . \label{4.23}\end{aligned}\ ] ] note that the integral over this function vanishes : @xmath227 , so that @xmath228 . alternatively , we may introduce the variable @xmath229 to arrive at the integral @xmath230 with @xmath231^{1/2 } \!\!- \frac{1 + \gamma^2}{3}\left[\frac{1-\eta^2}{1 - \eta^2 + \gamma^2}\right]^{3/2}\!\ ! = g(-\eta)\ , , \label{4.25}\end{aligned}\ ] ] where @xmath232 . the integrals @xmath166 are all functions of @xmath233 , and @xmath155 . let us first check our final expression ( [ 4.3a ] ) with ( [ 4.24 ] ) by going to the limit of a constant electric field @xmath63 with @xmath234 , where the exact result is known . in this limit , the parameter @xmath235 becomes small and can be neglected , and the integrals ( [ 4.24 ] ) become approximately @xmath236 \ , . \label{5.1}\end{aligned}\ ] ] inserting these into eq . ( [ 4.3a ] ) leads to the following hyperbolic sums @xmath237 with @xmath238 , where @xmath239 and @xmath240 are the polylogarithm functions @xmath241 note that the constant field limit corresponds to large arguments in eq . ( [ 5.2 ] ) , since @xmath242 , @xmath243 , where @xmath244 we must exploit therefore the analytic continuation of the polylogarithm functions defined by the series ( [ 5.2a ] ) into the region @xmath245 . by taking advantage of the formula @xcite : @xmath246 where @xmath247 is the hurwitz zeta function @xmath248 we bring the right - hand side of ( [ 5.2 ] ) to the form @xmath249 for @xmath238 this reads explicitly , @xmath250 substituting eq . ( [ 4.3a ] ) with eqs . ( [ 5.1 ] ) and ( [ 5.3 ] ) into eq . ( [ 2.7a ] ) , we obtain the approximate pair production rate per nonzero field volume @xmath251\right.\nonumber\\ & \!\!\!\!+\!\!\!\!&\left.\frac{1}{(2\pi{\bar\xi } ) ^2}\!\left[\frac{\pi^4}{3}(\kappa{\bar\xi } ) + \frac{4\pi^4}{3}(\kappa{\bar\xi } ) \!\!\left ( \kappa^2 + { \bar\xi } ^{2 } \right)+\frac{1}{2}{\rm li}_{4}(-e^{-\lambda_{+}})- \frac{1}{2}{\rm li}_{4}(-e^{-\lambda_{-}})\right]\!\right\}\!. \label{5.4a}\end{aligned}\ ] ] we now take the constant - field limit @xmath63 at @xmath234 fixed , where @xmath252 remains finite , while @xmath253 tends to infinity , so that @xmath254 with @xmath238 vanishes . moreover , the polylogarithm functions @xmath255 with @xmath256 do not contribute because of vanishing prefactors . all divergent terms cancel each other . thus we obtain the pair production rate per nonzero field volume , which for constant field is the total volume : @xmath257 here we have inserted @xmath258 from eq . ( [ 4.3b ] ) . the division by @xmath259 is essential for getting a finite result in the constant - field limit . for completeness , we have reinserted in the final expression the light velocity @xmath260 to verify the complete agreement with the result of heisenberg and euler @xcite , schwinger @xcite , nikishov @xcite , and many others ( see e.g. @xcite and references therein ) . for arbitrary @xmath155 , the integral ( [ 4.21 ] ) can not be evaluated in closed analytic form . in order to obtain an approximate rate formula we insert @xmath166 from eq . ( [ 4.21 ] ) into eq . ( [ 4.3a ] ) and interchange the order of summation and integration to find the expansion @xmath261 + \cosh \left [ 2\pi n ( \xi + \kappa)\right ] \right\}\ , . \label{5.5}\end{aligned}\ ] ] the integral is simplified with the help of the summation formula @xmath262\ , , \label{5.6}\end{aligned}\ ] ] which permits us to bring the general rate to the form @xmath263 + \ln\left[1 + e^{2\pi ( \xi + \kappa)}\right]\right\}\ , , \label{5.7a}\end{aligned}\ ] ] or , by the symmetry of the function @xmath264 , to the more symmetric form @xmath265\ , . \label{5.7}\end{aligned}\ ] ] finally , integrating this by parts , we find @xmath266 where the function @xmath267 vanishes on both ends . explicitly , it reads @xmath268 remarkably , the second function under the integral ( [ 5.8 ] ) resembles a fermi distribution . indeed , we are going to show that the calculation of the integral ( [ 5.8 ] ) can be done by a method familiar to low - temperature expansions in statistical physics @xcite . the condition necessary for pair production @xmath269 implies that the parameter @xmath155 lies in the interval @xmath270 , where @xmath271 and @xmath272 ( in natural units with @xmath273 ) is the so - called critical field for which the work over two compton wavelengths @xmath274 can produce the energy @xmath275 of a pair . at the upper end @xmath276 of the above interval , the rate ( [ 5.8 ] ) vanishes , since @xmath277 becomes zero . for the calculation of the exact pair production rate ( [ 2.7a ] ) from eq . ( [ 5.7 ] ) , we introduce the dimensionless parameter @xmath278 , where @xmath279 , to rewrite eq . ( [ 5.8 ] ) in terms of the dimensionless variable @xmath280 . this brings the production rate ( [ 2.7a ] ) to the form @xmath281 where @xmath282 and the dimensionless function @xmath283 reads @xmath284 we expand this function into a power series @xmath285 with the coefficients @xmath286 substituting the expansion ( [ 5.14 ] ) back into eq . ( [ 5.11 ] ) , we encounter the odd - moment integrals of the fermi distribution @xmath287 these can all be found exactly . performing the integrals yields a binomial expansion @xmath288 here @xmath289 are the linear combinations of the polylogarithm functions @xmath290 with arguments @xmath291 where @xmath292 the exact production rate of eq . ( [ 5.11 ] ) becomes now the sum @xmath293 by making use of eqs . ( [ 5.17 ] ) , we rewrite this as an expansion over the polylogarithm functions @xmath294 where the coefficients @xmath295 are polynomials of @xmath296 @xmath297 together with eq . ( [ 5.14 ] ) these read explicitly , @xmath298 the series expansion given by eq . ( [ 5.22 ] ) converges well for small @xmath299 . here the parameter @xmath300 in eq . ( [ 5.20 ] ) becomes @xmath301 , where the first term is equal to @xmath302 of eq . ( [ 5.2b ] ) , and the polylogarithm functions @xmath303 with @xmath304 will be suppressed by powers of @xmath299 . the parameter @xmath305 tends to minus infinity , so that the polylogarithm functions @xmath306 for all @xmath2 yield the exponentially small contributions . by means of eq . ( [ 5.12 ] ) , the coefficients ( [ 5.24 ] ) are the polynomials of small @xmath299 @xmath307 finally , this yields the probability rate ( [ 5.22 ] ) as a series expansion in powers of small @xmath299 @xmath308\right.\nonumber\\ & + & \left . li}_{1}(-e^{-\tilde\rho } ) - \frac{3}{8\pi^2}\,{\rm li}_{2}(-e^{-\tilde\rho } ) - \frac{3\epsilon}{4\pi^3}\,{\rm li}_{3}(-e^{-\tilde\rho } ) \right ] + \cdots\right\}\ , , \label{5.26}\end{aligned}\ ] ] where the leading term is already an excellent approximation . note the coincidence of the second term in the first brackets with the probability rate ( [ 5.4 ] ) for a constant - field limit @xmath15 . we have calculated an exact expression for the production rate of charged scalar particle - antiparticle pairs from the vacuum by the sauter potential . for an arbitrary potential barrier , the rate was related to the scattering amplitude on the barrier , and expressed as an energy - momentum integral over the logarithm of the reflection coefficient . for the sauter potential , we have evaluated this integral and checked the result by recovering the known limits of a sharp step potential and of a uniform electric field .

we derive the exact rate of pair production of oppositely charged scalar particles by a smooth potential step @xmath0 in three dimensions . as a check we recover from this the known results for an infinitely sharp step as well as for a uniform electric field .

in recent years , twisted commutative algebras ( tca s ) have played an important role in the nascent field of representation stability . the best known example is the twisted commutative algebra @xmath4 freely generated by a single indeterminate of degree one . modules over this tca are equivalent to the @xmath3-modules of church ellenberg farb @xcite , and have received a great deal of attention . in @xcite , we studied the module theory of this tca , and established a number of fundamental structural results . the purpose of this paper is to extend these results to tca s freely generated by any number of degree one generators . this is , we believe , an important step in the development of tca theory , and connects to a number of concrete applications . let @xmath5 be the tca @xmath6 , where @xmath7 ; this is the tca freely generated by @xmath1 elements of degree 1 . we identify @xmath5 with the polynomial ring @xmath8 in variables @xmath9 , equipped with its natural @xmath10-action ; @xmath5-modules are required to admit a compatible polynomial @xmath10-action . ( see [ s : tca ] for complete definitions . ) the goal of this paper is to understand the structure of the module category @xmath11 as best we can . as a first step , we introduce the prime spectrum of a tca . this is defined similarly to the spectrum of a commutative ring , but ( as far as we are concerned in this paper ) is just a topological space . the spectrum of a tca gives a coarse view of its module category , so determining the spectrum is a good first step in analyzing the structure of modules . we explicitly determine the spectrum of @xmath5 . to state our result , we must make a definition . the * total grassmannian * of @xmath12 , denoted @xmath13 , is the following topological space . as a set , it is the disjoint union of the topological spaces @xmath14 for @xmath15 . ( here we are using the topological space underlying the scheme @xmath14 parametrizing rank @xmath16 quotients of @xmath12 . ) a set @xmath17 is closed if each @xmath18 is closed and moreover @xmath19 is downwards closed in the sense that if a quotient @xmath20 belongs to @xmath19 ( meaning the closed point of @xmath14 it corresponds to belongs to @xmath19 ) then any quotient of @xmath21 also belongs to @xmath19 . we prove : [ thm : spec ] the spectrum of @xmath5 is canonically homeomorphic to @xmath13 . in the course of proving this theorem , we classify the irreducible closed subsets of @xmath13 : each is the closure of a unique irreducible closed subset of @xmath14 , for some @xmath16 . this provides a wealth of interesting prime ideals in @xmath5 : for example , when @xmath22 the space @xmath23 is @xmath24 , and so each irreducible planar curve gives a prime ideal of @xmath5 . this shows that for @xmath25 there is interesting geometry contained in @xmath5 , contrary to the more rigid structure when @xmath26 . in joint work with rohit nagpal ( which he kindly allowed us to include in this paper ) , we show : the space @xmath13 has krull dimension @xmath27 . from this , we deduce : the category @xmath11 has krull gabriel dimension @xmath27 . let @xmath28 be the @xmath16th determinantal ideal . if we think of the variables @xmath29 as the entries of a @xmath30 matrix , then @xmath31 is generated by the @xmath32 minors of this matrix . alternatively , in terms of representation theory , @xmath31 is generated by the representation @xmath33 occurring in the cauchy decomposition of @xmath5 . let @xmath34 be the full subcategory of @xmath11 spanned by modules supported on ( i.e. , locally annihilated by ) @xmath31 . equivalently , @xmath35 is the category of modules whose support in @xmath13 is contained in @xmath36 . these categories give a filtration of @xmath11 : @xmath37 we call this the * rank stratification*. let @xmath38 be the serre quotient category . intuitively , @xmath39 is the piece of @xmath11 corresponding to @xmath40 . our approach to studying @xmath11 is to first understand the structure of the pieces @xmath39 , and then understand how these pieces fit together to build @xmath11 . every object of @xmath39 is locally annihilated by a power of @xmath31 . we concentrate on the subcategory @xmath41 $ ] consisting of objects annihilated by @xmath31 . the following theorem completely describes this category : let @xmath42 be the tautological bundle on @xmath14 and let @xmath43 be the tca @xmath44 on @xmath14 . then @xmath41 $ ] is equivalent to @xmath45 , the category of @xmath43-modules locally annihilated by a power of @xmath46 . every finitely generated object of @xmath39 admits a finite length filtration with graded pieces in @xmath41 $ ] . thus , for many purposes , the above theorem is sufficient for understanding @xmath39 . for example , it immediately implies : the grothendieck group of @xmath39 is canonically isomorphic to @xmath47 , where @xmath48 is the ring of symmetric functions , and thus is free of rank @xmath49 over @xmath48 . we now describe how @xmath11 is built from its graded pieces . for this we introduce two functors . let @xmath50 be an @xmath5-module . we define @xmath51 to be the maximal submodule of @xmath50 supported on @xmath31 , and we define @xmath52 to be the universal module to which @xmath50 maps that has no non - zero submodule supported on @xmath31 . we call @xmath52 the * saturation * of @xmath50 with respect to @xmath31 . the functor @xmath53 can be identified with the composition @xmath54 where the first functor is the localization functor and the second is the section functor ( i.e. , the right adjoint to localization ) . the functors @xmath55 and @xmath53 are left - exact , and we consider their right derived functors . we refer to @xmath56 as * local cohomology * with respect to the ideal @xmath31 . the most important result in this paper is the following finiteness theorem : [ thm : fin ] let @xmath50 be a finitely generated @xmath5-module . then @xmath57 and @xmath58 are finitely generated for all @xmath59 and vanish for @xmath60 . this result has a number of important corollaries . write @xmath61 for the bounded derived category with finitely generated cohomology groups . write @xmath62 to indicate that the triangulated category @xmath63 admits a semi - orthogonal decomposition into subcategories @xmath64 . we have a semi - orthogonal decomposition : @xmath65 here @xmath66 is identified with a subcategory of @xmath67 via the functor @xmath68 . we note that without finiteness conditions , such a decomposition follows almost formally ; to get the decomposition with finiteness conditions imposed requires the theorem . the functor @xmath69 is essentially the projection onto the subcategory @xmath70 , while the functor @xmath71 is the projection onto @xmath72 . ( this point of view explains the subscripts on these functors . ) we introduce the functor @xmath73 , which projects onto @xmath74 . we have a canonical isomorphism @xmath75 the projection onto the @xmath16th factor is given by @xmath76 . in particular , @xmath77 is free of rank @xmath78 as a @xmath48-module . finally , we prove a structure theorem for @xmath67 that refines the above corollary . for an integer @xmath15 , let @xmath79 denote the set of partitions @xmath80 contained in the @xmath81 rectangle ( i.e. , @xmath82 and @xmath83 ) . for @xmath84 , put @xmath85 where @xmath42 is the tautological bundle on @xmath14 . alternatively , @xmath86 is the quotient of @xmath87 by the ideal spanned by those copies of @xmath88 where @xmath89 has more than @xmath16 parts . the classes @xmath90 $ ] with @xmath84 form a @xmath91-basis for @xmath92 , while the classes @xmath93 $ ] form a @xmath48-basis for @xmath77 . our structure theorem is : [ thm : struc ] the objects @xmath94 , with @xmath89 arbitrary and @xmath84 , generate @xmath67 , in the sense of triangulated categories . thus every object of @xmath67 admits a finite filtration where the graded pieces are shifts of modules of this form . in fact , our results are more precise than this : for instance , we show that the @xmath95 s all appear in a certain order , with the @xmath96 s first , then the @xmath97 s , and so on . see remark [ rmk : proof - thm : struc ] for a proof . when @xmath26 , we showed in @xcite that every object @xmath50 of @xmath98 fits into an exact triangle @xmath99 where @xmath100 is a finite length complex of finitely generated torsion modules and @xmath101 is a finite length complex of finitely generated projective modules . a finitely generated torsion module admits a finite filtration where the graded pieces have the form @xmath102 , while a finitely generated projective module admits a finite filtration where the graded pieces have the form @xmath103 . ( we note that @xmath104 and @xmath105 . ) thus theorem [ thm : struc ] is essentially a generalization of the structure theorem from @xcite . we prove several other results about the structure of @xmath11 . we mention a few here : * the extremal pieces of the rank stratification @xmath106 and @xmath107 are equivalent . * projective @xmath5-modules are injective . * finitely generated @xmath5-modules have finite injective dimension . lest the reader extrapolate too far , we offer two warnings : ( a ) for @xmath26 , every finitely generated @xmath5-modules injects into a finitely generated injective @xmath5-module . this is no longer true for @xmath25 . ( b ) we believe that @xmath39 and @xmath108 are inequivalent for @xmath109 , though we do not have a rigorous proof of this . koszul duality gives an equivalence between the derived category of @xmath110 modules and the derived category of @xmath111 modules ( assuming some finiteness ) . the category of polynomial representations of @xmath10 has a transpose functor , which induces an equivalence between @xmath111 modules and @xmath112 modules . we call the resulting equivalence @xmath113 the * fourier transform*. ( here the `` dfg '' subscript means the @xmath10-multiplicity space of each cohomology sheaf is coherent . ) our main result on it is : [ thm : ft ] the fourier transform induces an equivalence between @xmath67 and @xmath114 . this theorem can be unpackaged into a much more concrete statement . let @xmath50 be an @xmath5-module , and let @xmath115 be its minimal projective resolution . write @xmath116 , where @xmath117 is a representation of @xmath10 , and let @xmath118 be the degree @xmath119 piece of @xmath117 . then @xmath120 is called the @xmath119th * linear strand * of the resolution . up to a duality and transpose , @xmath121 is @xmath122 . thus the above theorem implies that if @xmath50 is a finitely generated @xmath5-module then its resolution has only finitely many non - zero linear strands , and each linear strand ( after applying duality and transpose ) admits the structure of a finitely generated @xmath123-module . thus theorem [ thm : ft ] is a strong statement about the structure of projective resolutions of @xmath5-modules . in particular , it implies : a finitely generated @xmath5-module has finite regularity . we also prove a duality theorem for local cohomology and saturation with respect to the fourier transform . we just mention the following version of this result here : we have @xmath124 . in other words , the fourier transform reverses the rank stratification of @xmath11 . the results of this paper are of a foundational nature . we have additional , more concrete results , that build on this foundation ; for reasons of length , we have deferred them to companion papers . we summarize the main results here . the first group of results concerns hilbert series . in @xcite , the second author introduced a notion of hilbert series for twisted commutative algebras and their modules , and proved a rationality result for the tca s considered in this paper . in @xcite , we introduced an `` enhanced '' hilbert series that records much more information , and proved a rationality result in the @xmath26 case . using the tools of this paper , we have greatly extended this theory . we can now prove a rationality result for the enhanced hilbert series for arbitrary @xmath1 . moreover , we understand how the pieces of the hilbert series match up with the structure of the category @xmath11 . we have similar results on the far more subtle poincar series as well . the second group of results concerns depth and local cohomology . suppose that @xmath50 is an @xmath5-module . we can then consider the local cohomology group @xmath57 defined in this paper and , treating it as a polynomial functor , evaluate on @xmath125 . alternatively , we can take the local cohomology of the @xmath126-module @xmath127 with respect to the ideal @xmath128 . we show that these two constructions are canonically isomorphic for @xmath129 when @xmath50 is finitely generated . in particular , this shows that the local cohomology of @xmath127 with respect to determinantal ideals is finitely generated for @xmath129 , and exhibits representation stability in the sense of church farb @xcite . we also study the depth of @xmath127 with respect to @xmath128 and show that , for @xmath129 , it has the form @xmath130 for integers @xmath131 and @xmath132 . moreover , we show that if @xmath133 then @xmath134 , and if @xmath135 then the first non - zero local cohomology @xmath57 occurs for @xmath136 . the third group of results concerns regularity . in @xcite , church and ellenberg show that the regularity of an @xmath3-module can be controlled in terms of its presentation . we generalize this result to arbitrary @xmath1 . our theorem states that the regularity of a finitely generated @xmath5-module @xmath137 can be controlled in terms of @xmath138 for @xmath59 up to approximately @xmath139 . as a corollary , we find that the regularity of the @xmath126-module @xmath127 stabilizes as @xmath140 at a predictable value of @xmath119 ( approximately @xmath139 plus the number of rows in @xmath50 ) . the second author used the tca s appearing in this paper to study @xmath141-modules , which served as the primary tool in his study of syzygies of the segre embeddings and related varieties @xcite . in @xcite , we showed that the category of @xmath5-modules is equivalent to the category of @xmath142-modules , where @xmath142 is the category whose objects are finite sets and whose morphisms are injections together with a @xmath1-coloring on the complement of the image . ramos further studied @xmath142-modules in @xcite , and recently used them to study configuration spaces of graphs in @xcite . @xmath142-modules are also used in the first author s study of equations and syzygies of secant varieties of veronese embeddings @xcite , where they play a crucial role . we hope that the results of this paper will lead to additional insight related to the applications mentioned here . the equivariant structure of the ring @xmath5 has been intensively studied in the literature from combinatorial and algebraic perspectives , and we refer the reader to @xcite for some background and additional references . the homological aspects of this ring were shown to be closely related to the representation theory of the general linear lie superalgebra in @xcite , and this motivates the study of resolutions of its equivariant ideals . we refer the reader to @xcite for further information and calculations . our results imply that one can expect certain patterns and universal bounds to appear as the size of the matrix increases . in [ s : tca ] , we recall the requisite background on the representation theory of @xmath10 and tca s , and prove some general results about tca s . in [ sec : spec ] , we introduce the spectrum of a tca and study the spectrum of @xmath5 . in [ s : formalism ] , we develop a formalism of local cohomology and saturation functors with respect to a filtration of an abelian category . these results are mostly well - known ; we include this material simply to recall salient facts and set notation . in [ s : at0 ] , we study the two extremal pieces of the filtration of @xmath11 , namely the category @xmath106 of modules supported at 0 , and what we call the `` generic category '' @xmath143 , which is just another name for @xmath107 . these are important special cases since the other pieces of the category will be described using these pieces . in [ s : rank ] , we study the full rank stratification of @xmath11 , and prove the primary theorems of the paper . in [ s : koszul ] , we treat koszul duality and develop the theory of the fourier transform . we also include two appendices : appendix [ s : grass ] proves some well - known results about grassmannians for which we could not find a suitable reference , and appendix [ ss : oldkoszul ] gives a different , more direct , proof of the finiteness properties of koszul duality . * all schemes in this paper are noetherian , of finite krull dimension , and separated over @xmath144 . for a scheme @xmath145 , we use the term `` @xmath146-module '' in place of `` quasi - coherent @xmath146-module , '' and we use the term `` finitely generated @xmath146-module '' in place of `` coherent @xmath146-module . '' we write @xmath147 for the category of @xmath146-modules . @xmath148 denotes the underlying topological space of @xmath145 . * for a vector bundle @xmath149 over a scheme @xmath145 , we write @xmath150 for the relative grassmannian parametrizing rank @xmath16 quotients of @xmath149 . we often write @xmath151 for @xmath150 . we write @xmath42 for the tautological quotient bundle on @xmath150 and @xmath152 for the subbundle . * we let @xmath153 be the standard representation of @xmath10 . we write @xmath154 for the schur functor associated to the partition @xmath80 . * for a vector bundle @xmath149 on a scheme @xmath145 , we let @xmath155 be the tca @xmath156 . we let @xmath157 be the @xmath16th determinantal ideal . * for an abelian category @xmath158 ( typically grothendieck ) , we write @xmath159 for the category of finitely generated objects in @xmath158 . we write @xmath160 for the derived category , @xmath161 for the bounded derived category , @xmath162 for the bounded below derived category , and @xmath163 for the subcategory of the derived category on objects with finitely generated cohomologies . we always use cochain complexes and cohomological indexing . * if @xmath164 is an object for which @xmath165 is defined ( and locally noetherian ) , we write @xmath166 for the grothendieck group of the category @xmath167 . in particular , if @xmath145 is a noetherian scheme then @xmath168 is the grothendieck group of coherent sheaves , and if @xmath169 then @xmath170 is the grothendieck group of the category of finitely generated @xmath5-modules . we thank rohit nagpal for helpful discussions , and for allowing us to include the joint material appearing in [ ss : rohit ] . a representation of @xmath171 is * polynomial * if it occurs as a subquotient of a direct sum of tensor powers of the standard representation @xmath172 . let @xmath173 be the category of such representations . equivalently , @xmath173 can be described as the category of polynomial functors , and this will be a perspective we often employ ( see @xcite for details ) . the category @xmath173 is semi - simple abelian , and the simple objects are the representations @xmath174 indexed by partitions @xmath80 . from the perspective of polynomial functors , the simple objects are just the schur functors @xmath154 . the category @xmath173 is closed under tensor product . the tensor product of simple objects is computed using the littlewood richardson rule . every object @xmath175 of @xmath173 admits a decomposition @xmath176 where @xmath177 is a vector space . we refer to @xmath178 as the * @xmath80-isotypic piece * of @xmath175 , and to @xmath177 as the * @xmath80-multiplicity space * of @xmath175 . we let @xmath179 , and call this the degree @xmath119 piece of @xmath175 ; in this way , every object of @xmath173 is canonically graded . we say that @xmath80 * occurs * in @xmath175 if @xmath180 . for a partition @xmath80 , we let @xmath181 be the number of non - zero parts in @xmath80 . we let @xmath182 be the supremum of the @xmath181 over those @xmath80 that occur in @xmath175 , and we say that @xmath175 is * bounded * if @xmath183 . we have @xmath184 by the littlewood richardson rule ; in particular , a tensor product of bounded representations is bounded . let @xmath185 be the full subcategory of @xmath173 on objects @xmath175 with @xmath186 . the functor @xmath187 given by @xmath188 is fully faithful , and its image consists of all polynomial representations of @xmath189 . this is an extremely important fact , since it implies that in @xmath185 one can evaluate on @xmath125and thus reduce to a familiar finite dimensional setting without losing information . let @xmath190 be the category whose objects are sequences @xmath191 , where @xmath192 is a representation of the symmetric group @xmath193 . weyl duality provides an equivalence between @xmath190 and @xmath173 ; see @xcite for details . this perspective will appear in a few places in this paper . suppose that @xmath145 is a scheme over @xmath144 . we then let @xmath194 be the category of polynomial representations of @xmath10 on @xmath146-modules . every object of this category @xmath175 admits a decomposition @xmath195 where @xmath177 is an @xmath146-module . if @xmath196 is a map of schemes then there are induced functors @xmath197 and @xmath198 computed by applying @xmath199 and @xmath200 to the multiplicity spaces . we also have the derived functors @xmath201 , computed by applying @xmath202 to the multiplicity spaces . for the purposes of this paper , a * twisted commutative algebra * ( tca ) is a commutative algebra object in the category @xmath173 , or more generally , in @xmath194 for some scheme @xmath145 . explicitly , a tca is a commutative associative unital @xmath144-algebra equipped with an action of @xmath10 by algebra automorphisms , under which it forms a polynomial representation . a * module * over a tca @xmath5 is a module object in the category @xmath173 ( or @xmath194 ) , that is , an @xmath5-module equipped with a compatible @xmath10 action under which it forms a polynomial representation . we write @xmath11 for the category of @xmath5-modules . this is a grothendieck abelian category . an * ideal * of @xmath5 is an @xmath5-submodule of @xmath5 . if @xmath50 is an @xmath5-module then , treating @xmath50 and @xmath5 as schur functors , @xmath127 is an @xmath126-module with a compatible action of @xmath189 . let @xmath149 be a vector bundle of rank @xmath1 on @xmath145 . we define @xmath169 to be the tca @xmath203 on @xmath145 . as a schur functor , we have @xmath204 . in particular , if @xmath145 is a point then @xmath126 is just a polynomial ring in @xmath205 variables over @xmath144 . the cauchy formula gives a decomposition @xmath206 since @xmath207 if @xmath80 has more than @xmath1 rows , we see that @xmath208 . thus @xmath5 is bounded . it follows that any finitely generated @xmath5-module is bounded , as such a module is a quotient of @xmath209 for some finite length ( and thus bounded ) object @xmath175 of @xmath194 . we recall the following well - known result ( first proved in @xcite ) : the tca @xmath5 is noetherian , that is , any submodule of a finitely generated module is finitely generated . suppose @xmath50 is a finitely generated @xmath5-module , and consider an ascending chain @xmath210 of @xmath5-submodules of @xmath50 . let @xmath211 , which is finite by the above remarks ; of course , @xmath212 for all @xmath59 as well . since @xmath127 is a finitely generated @xmath126-module , it is noetherian , as @xmath126 is a finitely generated over @xmath146 . thus the chain @xmath213 stabilizes , which implies that @xmath210 stabilizes . we let @xmath28 be the @xmath16th determinantal ideal of @xmath5 ; it is generated by @xmath214 . the tca @xmath5 and its ideals @xmath31 are the main focus of this paper . we let @xmath215 be the internal hom in the category of @xmath146-modules . for @xmath216 , we let @xmath217 be the sheaf of @xmath218-equivariant homomorphisms . explicitly , @xmath219 we define the internal @xmath220 on @xmath194 by @xmath221 this is again an object of @xmath194 . we have the adjunction @xmath222 the trivial multiplicity space in @xmath223 is @xmath217 . when @xmath145 is affine , we write @xmath224 in place of @xmath225 . suppose @xmath145 is a point . let @xmath226 and let @xmath227 be its schur weyl dual . then @xmath228 the coefficient of @xmath174 in @xmath229 is @xmath230 . we can compute this @xmath220 space after applying schur weyl duality . weyl converts @xmath231 to @xmath232 $ ] , the regular representation in degree @xmath119 , and converts @xmath233 to @xmath234 , where @xmath235 . we thus find @xmath236 ) \otimes \bs_{\lambda}(\bv).\ ] ] since @xmath232 $ ] is concentrated in degree @xmath119 , only the terms with @xmath237 contribute . for an @xmath193-representation @xmath238 , we have @xmath239)=w^*$ ] . thus , via the canonical auto - duality of @xmath240 , the above becomes @xmath241 using the formula @xmath242 , the result follows . let @xmath169 . suppose that @xmath50 and @xmath243 are @xmath5-modules . then @xmath244 is naturally an @xmath245-module , and thus an @xmath5-module via the comultiplication map @xmath246 . we denote this @xmath5-module by @xmath247 . the operation @xmath248 endows @xmath11 with a new symmetric tensor product . in general , this operation does not preserve finiteness properties of @xmath50 and @xmath243 . however , if @xmath50 and @xmath243 are finitely generated and annihilated by a power of the maximal ideal of @xmath5 then then @xmath247 is again finitely generated and annihilated by a power of the maximal ideal . [ prop : uhom - ten ] the functor @xmath249 given by @xmath250 is a tensor functor , using the @xmath248 tensor product on @xmath11 . let @xmath216 . we have canonical maps @xmath251 we show that this map is an isomorphism . it suffices to work zariski locally on @xmath145 , so we may assume @xmath149 is trivial . since both sides are bi - additive in @xmath175 and @xmath238 , it suffices to treat the case where each has the form @xmath252 , where @xmath253 is an @xmath146-module . but @xmath146-modules pull out of these @xmath220 s , and so we may as well assume @xmath254 . the map in question is then pulled back from a point . it thus suffices to treat the case where @xmath145 is a point and @xmath175 and @xmath238 are irreducible ; we write @xmath12 in place of @xmath149 . we make one more reduction : instead of taking @xmath175 and @xmath238 irreducible , we can assume each is a tensor power of the standard representation , since every irreducible is a summand of such a tensor power . let @xmath255 be the schur weyl dual of @xmath5 . we have @xmath256 . thus @xmath257 thus with @xmath258 and @xmath259 , the map takes the form @xmath260 we can thus regard it as an endomorphism of the target . by adjunction , to give a map of @xmath5-modules @xmath261 is the same as to give a map @xmath262 ; in particular , an endomorphism of @xmath263 is an isomorphism if and only if it is so in degree @xmath264 . thus , to prove that the above map is an isomorphism , it suffices to prove that it is an isomorphism in degree @xmath264 . now , the degree @xmath264 piece of each side is obtained by replacing @xmath5 with @xmath144 in . this map is clearly an isomorphism , and so the proof is complete . we let @xmath265 be the sheaf version of @xmath266 for @xmath146-modules . [ prop : ext - specialize ] let @xmath5 be a tca over @xmath145 and let @xmath50 and @xmath243 be @xmath5-modules with @xmath267 . then there is a natural isomorphism @xmath268 let us clarify one point here : @xmath269 is the underlying algebra of @xmath126 without any equivariance issues . hence , @xmath270 deals with @xmath218-equivariant extensions of the algebra @xmath5 , while @xmath271 deals with extensions of the underlying algebra @xmath126 . the latter space carries an action of @xmath189 . evaluation gives a map @xmath272 and it suffices to prove that it is an isomorphism over some affine cover , so we now assume that @xmath145 is affine . let @xmath273 be a locally free resolution . then @xmath274 computes @xmath275 . since @xmath276 , the natural map @xmath277 is an isomorphism . note that @xmath278 is an algebraic representation of @xmath189 , and thus is semi - simple as a @xmath189-representation , and so formation of @xmath189 invariants commutes with formation of cohomology . thus the target complex computes @xmath279 and so the result follows . we write @xmath280 for the injective dimension of an object @xmath50 in an abelian category . for a scheme @xmath145 , we write @xmath281 for the cohomological dimension of @xmath145 : this is the maximum @xmath59 for which @xmath282 is non - zero on quasi - coherent sheaves . we note that @xmath283 ( grothendieck vanishing ) and @xmath284 if @xmath145 is affine ( serre vanishing ) . suppose @xmath145 is smooth . let @xmath285 , where @xmath286 has all multiplicity spaces locally free of finite rank . if @xmath50 is an @xmath5-module with @xmath211 then @xmath287 in particular , every bounded @xmath5-module has finite injective dimension . ( recall our standing assumption that @xmath288 . ) since @xmath289 is smooth , being an affine bundle over the smooth scheme @xmath145 , we have @xmath290 identically for @xmath291 . from the previous proposition , we thus have @xmath292 for @xmath293 . next , we have a local - to - global spectral sequence @xmath294 which implies that @xmath295 whenever @xmath296 . as @xmath297 , the result follows . [ cor : fin - inj - dim ] suppose @xmath145 is smooth . let @xmath169 for a vector bundle @xmath149 on @xmath145 of rank @xmath1 , and let @xmath50 be a bounded @xmath5-module . then @xmath298 in particular , all finitely generated @xmath5-modules have finite injective dimension . let @xmath5 be a tca over @xmath145 . let @xmath299 be the category of @xmath5-modules and let @xmath300 be the full subcategory on @xmath5-modules @xmath50 with @xmath301 . we have an inclusion functor @xmath302 and a truncation functor @xmath303 mapping @xmath50 to @xmath304 . these functors are both exact , and the inclusion functor is the right adjoint to the truncation functor . it follows that the inclusion functor preserves injectives , that is , if @xmath305 is an injective object of @xmath300 then it is also an injective object of @xmath299 . from this , we deduce the following useful result : [ prop : bdinj ] let @xmath5 be a tca and let @xmath50 be an @xmath5-module with @xmath306 . then there exists an injection @xmath307 where @xmath305 is an injective @xmath5-module with @xmath308 . the category @xmath300 is grothendieck and therefore has enough injectives . we can thus find an injection @xmath307 where @xmath305 is an injective object of @xmath300 . by the above observation , @xmath305 is injective in the category of all @xmath5-modules . in particular , we get an injective resolution @xmath309 of @xmath50 with @xmath310 for all @xmath59 , so we also get bounds on @xmath311 of some derived functors , like local cohomology ( see [ s : rank ] ) . [ prop : injox ] let @xmath169 and let @xmath305 be an injective @xmath5-module . then the @xmath312-isotypic component @xmath313 of @xmath305 is an injective @xmath146-module for all @xmath80 . the forgetful functor @xmath314 is right adjoint to the exact functor @xmath315 , and therefore takes injectives to injectives . thus @xmath305 is injective as an object of @xmath194 . as an abelian category , @xmath194 is simply the product of the categories @xmath147 indexed by partitions , and so the injectivity of @xmath305 in @xmath194 implies the injectivity of @xmath313 in @xmath147 . let @xmath196 be a proper map of schemes , let @xmath316 be a vector bundle on @xmath145 , and let @xmath317 be its pullback to @xmath151 . let @xmath318 and @xmath319 . if @xmath50 is a @xmath320-module then @xmath321 is naturally an @xmath322-module . [ prop : finpushfwd ] suppose @xmath50 is a finitely generated @xmath320-module . then @xmath321 is a finitely generated @xmath322-module , for all @xmath323 . we prove the result by descending induction on @xmath59 . to begin , note that @xmath324 for @xmath325 . ( recall our assumption that @xmath326 is finite . ) now suppose the result has been proved for @xmath327 and let us prove it for @xmath59 . since @xmath50 is finitely generated , we can pick a short exact sequence @xmath328 where @xmath175 is a finitely generated object of @xmath329 . note that @xmath243 is a finitely generated @xmath320-module by noetherianity . from the above , we obtain an exact sequence @xmath330 now , @xmath331 is finitely generated by induction , and so the image of @xmath321 in it is finitely generated by noetherianity . since @xmath332 , the projection formula gives @xmath333 . since @xmath334 is proper , @xmath335 is a finitely generated object of @xmath194 , and so the result follows . [ cor : finpushfwd ] the functor @xmath336 carries @xmath337 into @xmath338 . let @xmath5 be a tca . an ideal @xmath339 is * prime * if for any ideals @xmath340 , we have that @xmath341 implies @xmath342 or @xmath343 . we define @xmath344 to be the set of prime ideals of @xmath5 , and equip it with the zariski topology . if @xmath345 , then @xmath305 is prime if and only if @xmath346 is a prime ideal in @xmath347 ( see @xcite , and note that `` domain '' and `` weak domain '' coincide in the bounded case ) . in particular , @xmath344 coincides with the set of @xmath189 fixed points in @xmath348 , given the subspace topology . the spectrum of @xmath5 is a useful tool for obtaining a coarse picture of the module theory of @xmath5 . in this section , we will determine the spectrum of @xmath155 , and deduce some consequences for modules . let @xmath145 be a scheme and let @xmath149 be a vector bundle of rank @xmath1 over @xmath145 . for each @xmath15 we have the grassmannian @xmath150 of @xmath16-dimensional quotients of @xmath149 , which is a scheme over @xmath145 . to be precise , given an @xmath145-scheme @xmath349 , a morphism @xmath350 is given by the datum of a short exact sequence @xmath351 of locally free sheaves on @xmath100 such that the rank of @xmath352 is @xmath16 . given @xmath353 , let @xmath354 be the partial flag variety parametrizing surjections @xmath355 where @xmath356 has rank @xmath16 and @xmath357 has rank @xmath358 ( we mean this in the functor of points language as above ) . there are projection maps @xmath359 and @xmath360 . given a subset @xmath19 of @xmath361 , we let @xmath362 be @xmath363 . if @xmath19 is closed then so is @xmath364 ( since @xmath365 is proper ) , and if @xmath19 is irreducible then so is @xmath364 ( since @xmath366 is irreducible , as @xmath367 is a fiber bundle with irreducible fibers ) . explicitly , @xmath364 consists of all @xmath16-dimensional quotients of a space in @xmath19 . we now define a topological space @xmath368 called the * total grassmannian * of @xmath149 . as a set , @xmath368 is the disjoint union of the @xmath150 for @xmath15 . a subset @xmath19 of @xmath368 is closed if each set @xmath369 is zariski closed in @xmath150 and @xmath19 is downwards closed in the sense that @xmath370 for all @xmath371 . there is a natural continuous map @xmath372 . the discussion above gives : [ lem : closure ] let @xmath373 be closed . then the closure of @xmath19 in @xmath368 is the set of all quotients of a space in @xmath19 . [ prop : grirred ] suppose @xmath373 is zariski closed and irreducible . then the closure @xmath374 of @xmath19 in @xmath368 is irreducible , and all irreducible closed subsets of @xmath368 are obtained in this way . the closure of an irreducible set is irreducible , so @xmath374 is irreducible . now suppose that @xmath375 is a given irreducible set . let @xmath16 be minimal so that @xmath151 meets @xmath150 . then @xmath376 is a non - empty open subset of @xmath151 . thus @xmath19 is irreducible , and @xmath151 is the closure of @xmath19 . of course , @xmath19 is also a closed subset of @xmath150 , which completes the proof . fix a scheme @xmath145 and a vector bundle @xmath149 of rank @xmath1 on @xmath145 . our goal is to prove the following theorem : we have a canonical identification @xmath377 . in what follows , let @xmath169 and @xmath378 . recall that we have determinantal ideals @xmath28 . we let @xmath379 be the closed subset @xmath380 of @xmath151 , and we let @xmath381 . suppose that a connected algebraic group @xmath382 acts freely on a scheme @xmath145 and that the quotient scheme @xmath383 exists . then the natural map @xmath384 is a homeomorphism . let @xmath385 be the quotient map . given a point @xmath386 , let @xmath19 be its closure , an irreducible closed subscheme of @xmath383 . then @xmath387 is an irreducible closed subscheme of @xmath145 that is @xmath382-stable ( it is irreducible because all fibers are irreducible ) . we define a map @xmath388 by sending @xmath389 to the generic point of @xmath387 . suppose @xmath390 . let @xmath391 be the closure of @xmath392 , an irreducible closed @xmath382-stable subset . then @xmath393 is an irreducible closed subset of @xmath383 and @xmath394 . thus if @xmath395 is the generic point of @xmath393 then @xmath396 . thus @xmath397 . similarly , if @xmath386 with closure @xmath19 and @xmath398 is the generic point of @xmath399 then @xmath400 , and so @xmath395 . thus @xmath401 . we therefore see that @xmath402 and @xmath403 are mutually inverse bijections . suppose now that @xmath404 is a closed subset . then @xmath405 , where @xmath406 is the closure of @xmath151 in @xmath407 . the set @xmath151 is @xmath382-stable , and so @xmath408 is closed in @xmath409 . we thus see that @xmath403 is a closed mapping . it is immediate from the definition of @xmath402 that specializations lift along @xmath403 : if @xmath410 belongs to the closure of @xmath389 in @xmath409 then @xmath411 belongs to the closure of @xmath412 in @xmath413 . it is clear that @xmath413 is a noetherian spectral space , and so @xmath403 is a homeomorphism by ( * ? ? ? * tag 09xu ) . we have a canonical homeomorphism @xmath414 . the space @xmath379 is the spectrum of the tca @xmath415 , which has @xmath416 rows . thus @xmath379 is identified with the @xmath417 fixed space of @xmath418 , which is identified with the space of maps @xmath419 . the complement of @xmath420 is the locus where the map is surjective . the group @xmath417 acts freely on this locus , and the quotient is the scheme @xmath150 . the lemma thus follows from the previous lemma . [ lem : irred - y ] suppose @xmath421 is zariski closed and irreducible . then the closure @xmath374 of @xmath19 in @xmath151 is irreducible , and all irreducible closed subsets @xmath422 of @xmath151 are obtained in this way , and @xmath16 can be recovered as the smallest index such that @xmath423 . same as proposition [ prop : grirred ] . the homeomorphisms @xmath424 yield a bijective function @xmath425 . the map @xmath334 is a homeomorphism . let @xmath373 be a closed set , and let @xmath426 be its closure . let @xmath427 , a closed subset of @xmath428 , and let @xmath429 be its closure . we claim that @xmath430 . it suffices to check this on each fiber of @xmath407 , so we may as well assume @xmath145 is a single point and @xmath431 is a vector space . it then suffices to check on closed points after intersecting with each @xmath432 . a closed point of @xmath433 is a rank @xmath358 quotient @xmath175 of a rank @xmath16 quotient @xmath21 belonging to @xmath19 . by definition , there is a point in @xmath434 , thought of as a map @xmath435 , with coimage @xmath21 . ( recall that the coimage of @xmath334 is @xmath436 . ) it is easy to construct a map in the orbit closure of @xmath334 with coimage @xmath175 . it follows that @xmath429 contains @xmath437 . for the reverse inclusion , the locus @xmath434 where the image is contained in @xmath19 is a closed set , and so the closure of @xmath422 is contained in @xmath437 . it follows from the previous paragraph that @xmath334 and @xmath438 take closed sets to closed sets . indeed , every closed set is a finite union of irreducible closed sets , and each irreducible closed set is the closure of an irreducible closed set in @xmath428 or @xmath150 ( lemma [ lem : irred - y ] ) . this completes the proof . the next result compares the krull dimension of @xmath344 , which is typically easy to calculate , to the krull gabriel dimension of the category @xmath11 , which is harder to compute directly . ( see @xcite for the definition of krull gabriel dimension , though the following proof effectively contains a definition as well . ) [ prop : krulldim ] let @xmath5 be a noetherian tca . suppose that the following condition holds : @xmath439 then the krull gabriel dimension of the category @xmath11 agrees with the krull dimension of the topological space @xmath344 . let @xmath158 be the category of finitely generated @xmath5-modules . let @xmath440 be the category of finite length objects in @xmath158 , and having defined @xmath441 , let @xmath442 be the category consisting of objects in @xmath158 that become finite length in @xmath443 . let @xmath444 be the subcategory of @xmath158 on objects whose support locus in @xmath344 has krull dimension at most @xmath59 . we claim @xmath445 . this is clear for @xmath446 : a finitely generated @xmath5-module has finite length if and only if it is supported at the maximal ideal . suppose now we have shown @xmath447 , and let us prove @xmath445 . if @xmath50 is in @xmath444 then ( p ) shows that @xmath50 has finite length in @xmath448 , and so @xmath50 belongs to @xmath441 . conversely , suppose that @xmath50 belongs to @xmath441 , and let us show that @xmath50 belongs to @xmath444 . we may as well suppose @xmath50 is simple in @xmath449 and contains no non - zero subobject in @xmath447 . suppose that @xmath450 is a prime ideal such that @xmath451 has dimension @xmath59 and is contained in the support of @xmath50 . ( if no such @xmath450 exists then @xmath452 . ) we claim @xmath450 annihilates @xmath50 . suppose not . then @xmath453 is a non - zero subobject of @xmath50 and so does not belong to @xmath454 . since @xmath50 is simple modulo @xmath454 , it follows that @xmath455 belongs to @xmath447 . but this is a contradiction , since the support of @xmath455 is @xmath451 , but @xmath455 belongs to @xmath456 and therefore has support of dimension @xmath457 . we conclude that @xmath458 , and so @xmath50 has support of dimension @xmath459 , and thus belongs to @xmath444 . to finish , let @xmath1 be the krull dimension of @xmath344 . then @xmath460 but @xmath461 . it follows that @xmath462 but @xmath463 , and so @xmath1 is also the krull gabriel dimension of @xmath11 . let @xmath5 be a finitely generated bounded tca . then the condition holds . let @xmath464 for a prime ideal @xmath450 , let @xmath50 be a finitely generated @xmath43-module , and let @xmath210 be a descending chain . let @xmath465 . so @xmath466 implies that @xmath467 . then @xmath468 is a descending chain of finite dimensional vector spaces , and therefore stabilizes ; suppose it is stable for @xmath469 . then @xmath470 has non - zero annihilator in @xmath471 for all @xmath469 . it follows that @xmath472 has non - zero annihilator in @xmath43 for @xmath469 . the condition can be rephrased as : for every prime ideal @xmath450 , the category @xmath473 has krull gabriel dimension 0 , where @xmath473 is the quotient of @xmath474 by the serre subcategory of modules with non - zero annihilator . ( one thinks of @xmath473 as modules over a hypothetical residue field @xmath475 . ) proposition [ prop : krulldim ] is not specific to tca s , and holds for any tensor category satisfying similar conditions . fix a vector bundle @xmath149 on @xmath145 of rank @xmath1 . the goal of this section is to prove the following theorem : [ thm : gru - dim ] the space @xmath368 has krull dimension @xmath477 . if @xmath145 is universally catenary , then @xmath368 is catenary . combined with the other results of this section , we obtain : let @xmath169 . the category @xmath11 has krull gabriel dimension @xmath477 . [ lem : incidence ] let @xmath478 and @xmath373 be irreducible closed sets such that @xmath479 in @xmath13 . then @xmath480 . recall the definition of @xmath481 and @xmath365 and @xmath482 from [ ss : total - grass ] . by lemma [ lem : closure ] , @xmath483 is @xmath484 . the space @xmath485 is a @xmath486-bundle over @xmath151 , and therefore has dimension @xmath487 . since this space surjects onto a closed set containing @xmath19 , we obtain the stated inequality . [ lem : krull - dim ] the krull dimension of @xmath368 is at most @xmath477 . let @xmath488 be a maximal strict chain of irreducible closed subsets in @xmath368 . let @xmath489 be the set of indices @xmath59 for which @xmath490 meets @xmath150 but not @xmath491 , so that for @xmath492 we have @xmath493 for some irreducible closed set @xmath494 by lemma [ lem : irred - y ] . we note that each @xmath495 is non - empty by maximality of the chain . let @xmath496 ( resp . @xmath497 ) be the maximum ( resp . minimum ) dimension of @xmath498 with @xmath492 . since @xmath499 for all @xmath500 and @xmath501 , we have @xmath502 by the previous lemma ( for @xmath503 ) . we thus find @xmath504 ( in fact , the first inequality is an equality by the maximality of the chain . ) therefore , @xmath505 now , we can regard @xmath506 as a descending chain of irreducible closed sets in @xmath145 , as @xmath507 . the smallest member of this chain has dimension @xmath508 . thus we have @xmath509 , and the theorem follows . [ eg : krull - dim ] here is a chain of irreducible closed sets of length @xmath27 in @xmath13 for a vector space @xmath12 . let @xmath510 be a complete flag in @xmath12 . for @xmath511 and @xmath512 , let @xmath513 be the set of all @xmath16-dimensional subspaces of @xmath514 containing @xmath515 . by replacing a subspace by its quotient , these give irreducible closed subsets of @xmath516 , and form a chain @xmath517 let @xmath518 be the closure of @xmath513 in @xmath13 . this is irreducible by general principles . furthermore , by lemma [ lem : closure ] , we have a strict chain @xmath519 there are @xmath520 sets in this chain . for a general base @xmath145 , let @xmath151 be the reduced subscheme of an irreducible component of largest possible dimension . over the generic point of @xmath151 , @xmath149 becomes a vector space and we can build the chain as above . now take closures of these subvarieties to get a chain starting at @xmath151 of length @xmath27 . now concatenate this with a maximal chain of irreducible subspaces ending at @xmath151 to get a chain of length @xmath477 . combining lemma [ lem : krull - dim ] and example [ eg : krull - dim ] shows that @xmath521 . so it remains to show @xmath150 is catenary when @xmath145 is universally catenary . without loss of generality , we may replace @xmath145 with one of its irreducible components . we need to show that for any irreducible subspaces @xmath522 , every maximal chain of irreducible closed subsets between @xmath151 and @xmath523 has the same length . by extending these to maximal chains in the whole space ( which is irreducible ) , it suffices to consider the case @xmath524 and @xmath525 . use the notation from the previous proof . consider a chain @xmath488 which is maximal . suppose @xmath490 meets @xmath526 but not @xmath491 and that @xmath527 meets @xmath491 . write @xmath528 and @xmath529 for irreducible closed subsets @xmath494 and @xmath530 . by lemma [ lem : incidence ] , @xmath531 . it suffices to check that this must be an equality , since @xmath150 is catenary . now use the notation from lemma [ lem : incidence ] . so we have a map @xmath532 whose image is irreducible and contains @xmath498 . if the image strictly contains @xmath498 , we can insert its closure in between @xmath490 and @xmath527 and get a longer chain , which is a contradiction . so we conclude that the image is equal to @xmath498 . if @xmath533 , then the fibers of @xmath365 all have positive dimension . in that case , let @xmath534 be the fiber over @xmath535 , i.e. , the set of quotients in @xmath536 which further quotient to @xmath238 . it is easy to check that @xmath534 is closed and from what we have assumed , @xmath537 . now pick an open affine subset of @xmath145 that trivializes @xmath149 and that has nonempty intersection with the image of @xmath498 and @xmath536 in @xmath145 ; let @xmath12 denote the restriction of @xmath149 to this open affine . let @xmath538 be a schubert divisor in @xmath539 , i.e. , the intersection of @xmath539 with a hyperplane in the plcker embedding . if we choose @xmath538 generically , then it does not contain @xmath536 and @xmath540 has codimension @xmath541 in @xmath536 . also , the intersection @xmath542 is always nonempty ( this is easy to see by considering the plcker embedding ) . so @xmath543 . since @xmath498 is irreducible and @xmath544 is a projective bundle , we can choose an irreducible component @xmath545 such that @xmath546 . so @xmath422 is an irreducible subvariety of one smaller dimension whose closure can be inserted in between @xmath490 and @xmath527 , which gives a contradiction . let @xmath158 be a grothendieck abelian category and let @xmath547 be a localizing subcategory . we assume the following hypothesis holds : @xmath548 this is not automatic ( see examples [ ex : nonh ] and [ ex : nonh2 ] ) . let @xmath549 be the localization functor and let @xmath550 be its right adjoint ( the section functor ) . we define the * saturation * of @xmath551 ( with respect to @xmath547 ) to be @xmath552 . we also define the * torsion * of @xmath50 ( with respect to @xmath547 ) , denoted @xmath553 , to be the maximal subobject of @xmath50 that belongs to @xmath547 . ( this exists since @xmath547 is localizing . ) we say that @xmath551 is * saturated * if the natural map @xmath554 is an isomorphism . we say that @xmath555 is * derived saturated * if the natural map @xmath556 is an isomorphism . we note that @xmath557 is always saturated and @xmath558 is always derived saturated . we refer to @xmath559 as * local cohomology*. [ prop : satcrit ] let @xmath551 . the following are equivalent : 1 . @xmath50 is saturated . 2 . @xmath560 for @xmath561 and all @xmath562 . @xmath563 for all @xmath564 . the equivalence of ( a ) and ( b ) is ( * ? ? ? * , corollaire ) ( objects satisfying ( b ) are called @xmath547-ferm ) . the equivalence of ( b ) and ( c ) is ( * ? ? ? * , lemma ) . if @xmath565 then @xmath566 for @xmath567 . let @xmath568 be an injective resolution in @xmath547 . by , this remains an injective resolution in @xmath158 . thus @xmath569 is an isomorphism . but @xmath570 since each @xmath571 belongs to @xmath547 , and so the result follows . [ prop : injses ] let @xmath572 be injective . then we have a short exact sequence @xmath573 with @xmath574 and @xmath575 both injective . moreover , @xmath576 is injective . since @xmath577 is right adjoint to the inclusion @xmath578 , it takes injectives to injectives . thus @xmath574 is injective in @xmath547 , and so injective in @xmath158 by . let @xmath579 . then @xmath580 is injective and @xmath581 , and so @xmath580 is saturated by proposition [ prop : satcrit](b ) . since the map @xmath582 has kernel and cokernel in @xmath547 , it follows that @xmath580 is the saturation of @xmath305 , that is , the natural map @xmath583 is an isomorphism . finally , note that @xmath584 is exact , which implies that @xmath585 is exact , and so @xmath586 is injective . the functors @xmath100 and @xmath587 give mutually quasi - inverse equivalences between the category of torsion - free injectives in @xmath158 and the category of injectives in @xmath588 . proposition [ prop : injses ] shows that @xmath100 carries injectives to injectives , while this is true for @xmath587 for general reasons . if @xmath305 is a torsion - free injective in @xmath158 then proposition [ prop : injses ] shows that the map @xmath589 is an isomorphism . on the other hand , for any object @xmath305 of @xmath588 the map @xmath590 is an isomorphism . thus @xmath100 and @xmath587 are quasi - inverse . let @xmath591 be the set of isomorphism classes of indecomposable injectives in @xmath158 . from the previous proposition , we find : we have @xmath592 . [ prop : triangle ] let @xmath555 . then we have an exact triangle @xmath593 where the first two maps are the canonical ones . work in the homotopy category of injective complexes . let @xmath50 be an object in this category . then from proposition [ prop : injses ] , we have a short exact sequence of complexes @xmath594 and this gives the requisite triangle . [ prop : derived - sat ] an object @xmath555 is derived saturated if and only if @xmath595 for all @xmath596 . suppose @xmath50 is derived saturated . choose an injective resolution @xmath597 in @xmath588 . applying @xmath587 , we get a quasi - isomorphism @xmath598 . given @xmath596 , we have @xmath599 , and the latter is @xmath600 by proposition [ prop : satcrit ] since @xmath601 consists of saturated objects . conversely , if @xmath50 is not derived saturated , proposition [ prop : triangle ] implies that @xmath602 , and so @xmath603 . since @xmath604 , we are done . given a triangulated category @xmath605 , a collection of full triangulated subcategories @xmath606 is a * semi - orthogonal decomposition * of @xmath605 if 1 . @xmath607 whenever @xmath608 and @xmath609 and @xmath610 , and 2 . the smallest triangulated subcategory of @xmath605 containing @xmath606 is @xmath605 . in that case , we write @xmath611 . let @xmath612 ( resp . @xmath613 ) be the full subcategory of @xmath162 on objects @xmath50 such that @xmath614 ( resp . @xmath615 ) . we have the following : 1 . the inclusion @xmath616 is an equivalence , with quasi - inverse @xmath559 . the functor @xmath617 is an equivalence , with quasi - inverse @xmath618 . we have a semi - orthogonal decomposition @xmath619 . \(a ) pick @xmath620 . by proposition [ prop : triangle ] , we have a naturally given quasi - isomorphism @xmath621 . so the composition and @xmath622 are isomorphic to the identity . for @xmath623 , @xmath624 can be computed by applying @xmath577 termwise , so it is immediate that the composition @xmath625 is also isomorphic to the identity . \(b ) for @xmath626 , proposition [ prop : triangle ] gives a natural quasi - isomorphism @xmath556 , so @xmath627 . for @xmath628 , the homology of @xmath629 lives in @xmath547 , so @xmath630 can be computed by applying @xmath631 termwise , and we get @xmath632 . \(c ) since @xmath613 is the subcategory of derived saturated objects , we have @xmath633 whenever @xmath634 and @xmath635 by proposition [ prop : derived - sat ] . by proposition [ prop : triangle ] , @xmath162 is generated by @xmath612 and @xmath613 . in many familiar situations , the above functors and decompositions do not preserve finiteness . for example , suppose @xmath158 is the category of @xmath636$]-modules and @xmath547 is the category of torsion modules . let @xmath637 $ ] . then @xmath638)[1]$ ] and @xmath639 . thus the projection of @xmath50 to the two pieces of the semi - orthogonal decomposition are not finitely generated objects of @xmath158 . nonetheless , in our eventual application of this formalism to tca s , finiteness _ will _ be preserved ! let @xmath158 be a grothendieck abelian category and let @xmath640 be a chain of localizing subcategories ( we do not assume @xmath641 ) . we assume that each @xmath642 satisfies . we put @xmath643 we let @xmath644 be the localization functor , @xmath645 its right adjoint , and @xmath646 , and we let @xmath55 be the right adjoint to the inclusion @xmath647 . the subscripts in these functors are meant to indicate that they are truncating in certain ways : intuitively @xmath51 keeps the part of @xmath50 in @xmath648 and discards the rest , while @xmath649 discards the part of @xmath50 in @xmath648 and keeps the rest . by convention , we put @xmath650 for @xmath651 and @xmath652 for @xmath653 , and put @xmath654 for @xmath651 and @xmath655 for @xmath653 . we also put @xmath656 for @xmath651 and @xmath657 for @xmath653 . we have the following connections between these functors : we have the following : 1 . any pair of functors in the set @xmath658 commutes . we have @xmath659 . we have @xmath660 . we have @xmath661 if @xmath662 . these results hold for the derived versions of the functors as well . let @xmath610 , let @xmath663 be the localization functor , and let @xmath664 be its right adjoint . 1 . @xmath664 coincides with the restriction of @xmath665 to @xmath666 . 2 . injectives in @xmath667 remain injective in @xmath668 . @xmath669 coincides with the restriction of @xmath670 to @xmath671 . \(a ) suppose @xmath672 and write @xmath673 with @xmath674 . the natural map @xmath675 has kernel and cokernel in @xmath676 . since @xmath677 is a serre subcategory , it follows that @xmath678 belongs to @xmath677 . thus @xmath665 maps @xmath666 into @xmath677 , from which it easily follows that it is the adjoint to @xmath679 . \(b ) let @xmath305 be injective in @xmath666 . then @xmath680 is injective in @xmath677 since section functors always preserve injectives . by ( a ) , thus , by , we see that @xmath682 is injective in @xmath158 . but @xmath683 takes injectives to injectives , and so @xmath684 is injective in @xmath668 . \(c ) this follows immediately from ( a ) and ( b ) ( compute with injective resolutions ) . [ prop : ind - inj ] we have a natural bijection @xmath685 . let @xmath686 be the full subcategory of @xmath162 on objects @xmath50 such that @xmath687 and @xmath688 vanish . [ prop : derived - decomp ] we have the following : 1 . we have an equivalence @xmath689 . we have a semi - orthogonal decomposition @xmath690 . we now define a functor @xmath691 by @xmath692 . this is just the derived functor of @xmath693 . the functor @xmath694 is idempotent , and projects onto the @xmath59th piece of the semi - orthogonal decomposition . maintain the set - up from the previous section . we now assume the following hypothesis : * if @xmath551 is finitely generated then @xmath688 and @xmath695 are finitely generated for all @xmath59 and vanish for @xmath60 . we deduce a few consequences of this . 1 . we have an equivalence @xmath696 . the inverse is induced by @xmath697 . we have a semi - orthogonal decomposition @xmath698 . [ cor : kisom ] the functors @xmath699 and @xmath700 induce inverse isomorphisms @xmath701 . [ prop : kdecomp ] we have an isomorphism @xmath702 the projection onto the @xmath16th factor is given by @xmath76 . the map above is surjective : the composition @xmath703 , where the first map comes from the inclusion , is an isomorphism . suppose it is not injective . pick an object @xmath704 whose image is @xmath600 but such that @xmath705 \in \rk(\ca)$ ] is nonzero . let @xmath59 be minimal so that @xmath705 $ ] is in the image of @xmath706 but not in the image of @xmath707 . also , suppose we have chosen @xmath50 so that this index @xmath59 is as small as possible . note that @xmath708 = [ \rr \sigma_{\ge i } m]$ ] , so by our choice of @xmath50 , @xmath709 = 0 $ ] . proposition [ prop : triangle ] then implies that @xmath705 = [ \rr \gamma_{<i } m]$ ] , which contradicts our choice of @xmath59 , so no such object @xmath50 exists . we first give some positive results on property . recall that a grothendieck abelian category is * locally noetherian * if it has a set of generators which are noetherian objects . let @xmath158 be a locally noetherian grothendieck category and let @xmath547 be a localizing subcategory satisfying the following condition : @xmath710 then @xmath711 satisfies . let @xmath305 be an injective object of @xmath547 . we first note that @xmath712 if @xmath565 . indeed , a class in @xmath713 is represented by an extension @xmath714 and @xmath12 necessarily belongs to @xmath547 since this is a serre subcategory . since @xmath305 is injective in @xmath547 , the above sequence splits in @xmath547 , and this gives a splitting in @xmath158 . the result follows . now suppose that @xmath50 is a finitely generated object of @xmath158 and we have an extension @xmath715 let @xmath243 be a finitely generated submodule of @xmath12 that surjects onto @xmath50 ; this exists because @xmath12 is the sum of its finitely generated subobjects . using , pick @xmath716 such that @xmath717 and @xmath718 has no subobject in @xmath547 . let @xmath719 be the image of @xmath718 in @xmath50 , and let @xmath720 , so that we have an extension @xmath721 now , @xmath722 is a subobject of @xmath718 belonging to @xmath547 , and thus is zero . thus the map @xmath723 is an isomorphism , and so the above extension splits . from the exact sequence @xmath724 we obtain a sequence @xmath725 as @xmath726 is a quotient of @xmath727 , it belongs to @xmath547 , and so the leftmost group vanishes by the first paragraph . thus @xmath59 is injective . we have shown that the image of the class of @xmath12 under @xmath59 vanishes , and so the class of @xmath12 is in fact 0 . however , @xmath12 was an arbitrary extension , so we conclude that @xmath728 . we have thus shown that @xmath712 whenever @xmath551 is a finitely generated . a variant of baer s criterion now shows that @xmath305 is injective . let @xmath158 be a locally noetherian grothendieck category with a right - exact symmetric tensor product . let @xmath729 be an ideal , that is , a subobject of the unit object , and let @xmath711 be the full subcategory spanned by objects that are locally annihilated by a power of @xmath729 . suppose that the artin rees lemma holds , that is : @xmath730 then @xmath547 satisfies , and thus as well . let @xmath551 be finitely generated and let @xmath100 be the maximal subobject of @xmath50 in @xmath547 . the subobjects @xmath731 $ ] form an ascending chain in @xmath100 that union to all of @xmath100 , and so @xmath732 $ ] for some @xmath119 by noetherianity . let @xmath16 be the artin rees constant for @xmath733 . put @xmath734 . then @xmath726 belongs to @xmath547 ( this uses right - exactness of the tensor product ) . by artin rees , we have @xmath735 , and so @xmath719 has no subobject belonging to @xmath547 . [ cor : b - proph ] let @xmath5 be the tca @xmath155 and let @xmath729 be an ideal in @xmath5 . then the artin rees lemma holds . in particular , @xmath736 \subset \mod_a$ ] satisfies . let @xmath50 be a finitely generated @xmath5-module with submodule @xmath243 . pick @xmath737 . given @xmath16 , the equality @xmath738 holds for all @xmath739 if and only if it holds when we evaluate on @xmath125 . the existence of @xmath16 after we evaluate on @xmath125 can be deduced from the standard artin rees lemma which guarantees that such an @xmath16 exists locally ; since our space is noetherian , one can find a value of @xmath16 that works globally . we now give two examples where does not hold . [ ex : nonh ] let @xmath158 be the category of @xmath740-modules for a commutative ring @xmath740 . fix a multiplicative subset @xmath741 , and let @xmath547 be the category of modules @xmath50 such that @xmath742 . then . if @xmath740 is noetherian then it satisfies artin rees , and so holds by the above results . thus proposition [ prop : injses ] implies that the localization of an injective @xmath740-module with respect to @xmath587 remains injective . however , there are examples of non - noetherian rings @xmath740 where injectives localize to non - injectives @xcite . in such a case , must fail . [ ex : nonh2 ] let @xmath158 be the category of graded @xmath636$]-modules supported in degrees 0 and 1 ( where @xmath744 has degree 1 ) , and let @xmath547 be the subcategory of modules supported in degree 1 . then @xmath547 is equivalent to the category of vector spaces , and thus is semi - simple . however , except for 0 , no object of @xmath547 is injective in @xmath158 , and so does not hold . in this example , @xmath158 is locally noetherian , but does not hold . furthermore , @xmath745 and @xmath100 carry injectives to injectives , so these properties are weaker than . we fix ( for all of [ s : at0 ] ) a scheme @xmath145 over @xmath144 ( noetherian , separated , and of finite krull dimension , as always ) and a vector bundle @xmath149 on @xmath145 of rank @xmath1 . we let @xmath5 be the tca @xmath155 . we let @xmath746 denote the category of @xmath5-modules supported at 0 : precisely , this consists of those @xmath5-modules @xmath50 that are locally annihilated by a power of @xmath747 . we say that an @xmath5-module is * torsion * if every element is annihilated by a non - zero element of @xmath5 of positive degree . we let @xmath143 ( the `` generic category '' ) be the serre quotient of @xmath11 by the subcategory of torsion modules . the goal of [ s : at0 ] is to analyze the two categories @xmath746 and @xmath143 . recall from [ ss : internalhom ] the tensor product @xmath248 on @xmath11 . the object @xmath748 is naturally a finitely generated @xmath5-module supported at 0 . it follows that @xmath749 is again a finitely generated module supported at 0 . the @xmath193-action permuting the @xmath248 factors is @xmath5-linear , and so @xmath750 is a finitely generated torsion @xmath5-module . [ prop : tors ] we have the following : 1 . we have an isomorphism of @xmath5-modules @xmath751 . 2 . if @xmath50 is an @xmath5-module and @xmath253 is an @xmath146-module then @xmath752 3 . if @xmath253 is an injective @xmath146-module then @xmath753 is an injective @xmath5-module . every finitely generated object @xmath50 in @xmath746 admits a resolution @xmath754 where each @xmath755 is a finite sum of modules of the form @xmath753 with @xmath756 and @xmath757 for @xmath60 . every finitely generated object of @xmath746 admits a finite length filtration where the graded pieces have the form @xmath252 , where @xmath756 and @xmath758 acts by zero on @xmath174 . moreover , if @xmath145 is connected then the module @xmath174 has no non - trivial @xmath146-flat quotients . \(a ) this is clear for @xmath759 . the general case then follows from proposition [ prop : uhom - ten ] . \(b ) this follows immediately from ( a ) , and the fact that @xmath760 since the multiplicity spaces of @xmath5 are locally free of finite rank as @xmath146-modules . \(c ) this follows immediately from ( b ) . \(d ) suppose @xmath50 is supported in degrees @xmath761 . let @xmath762 . by part ( b ) , there is a canonical map @xmath763 , the cokernel of which is supported in degrees @xmath764 . the result follows from induction on @xmath119 . \(e ) given @xmath765 , first consider the filtration @xmath766 . by nakayama s lemma , @xmath767 for some @xmath16 , in which case the common value is @xmath600 since @xmath50 is torsion . each quotient @xmath768 has a trivial action of @xmath5 , so is a direct sum of modules of the form @xmath769 . these can be used to refine our filtration to the desired form . the last statement follows from irreducibility of @xmath770 . if @xmath771 then part ( e ) says that every finitely generated object of @xmath746 has finite length and the @xmath174 are the simple objects . in general , the statement in part ( e ) is a relative version of this , taking into account the non - trivial structure of @xmath146-modules . the category @xmath772 is naturally a module for the tensor category @xmath773 . thus @xmath774 is a module for @xmath775 . the following result describes @xmath774 as a @xmath48-module . [ cor : ktors ] the map @xmath776 taking @xmath777 $ ] to the class of the trivial @xmath5-module @xmath175 induces an isomorphism @xmath778 . this follows from proposition [ prop : tors](e ) . define @xmath779 , which is an ( infinite dimensional ) algebraic group over @xmath145 . a * representation * of @xmath780 is a quasi - coherent sheaf on @xmath145 on which @xmath780 acts @xmath146-linearly . for example , @xmath781 is naturally a representation of @xmath780 , which we call the * standard representation*. a representation is * polynomial * if it is a subquotient of a direct sum of representations of the form @xmath782 with @xmath253 is a quasi - coherent sheaf on @xmath145 and @xmath783 . we note that if @xmath175 and @xmath238 are polynomial representations then @xmath784 ( tensor product as @xmath146-modules ) is again a polynomial representation . we write @xmath785 for the category of polynomial representations . [ prop : gators ] we have a natural equivalence of categories @xmath786 , under which the tensor product @xmath248 corresponds to the tensor product @xmath787 . let @xmath50 be an @xmath155-module . since @xmath155 is the universal enveloping algebra of the abelian lie algebra @xmath788 , the @xmath155-module structure on @xmath50 gives a representation of the algebraic group @xmath788 on @xmath50 . moreover , the @xmath789-action on @xmath50 interacts with the @xmath788 action in the appropriate way to define an action of @xmath780 on @xmath50 . we show that this construction induces the equivalence of categories . we first observe that this construction is compatible with tensor products , that is , if @xmath50 and @xmath243 are @xmath155-modules then the @xmath780-representation on @xmath247 is just the tensor product of the @xmath780-representations on @xmath50 and @xmath243 . this follows immediately from the definitions . next we show that if @xmath50 is a torsion @xmath155-module then the associated @xmath780-representation is polynomial . it suffices to treat the case where @xmath50 is finitely generated , since a direct limit of polynomial representations is still polynomial . in this case , we can embed @xmath50 into a module of the form @xmath753 by proposition [ prop : tors ] . by the previous paragraph , we see that @xmath790 as a representation of @xmath780 , and thus is polynomial . thus @xmath50 embeds into a polynomial representation , and is therefore polynomial . it is clear from the construction that one can recover the @xmath155-module structure on @xmath50 from the @xmath780-representation , and so the functor in question is fully faithful . moreover , it follows that if @xmath50 is an @xmath155-module and @xmath243 is a @xmath780-subrepresentation then @xmath243 is in fact an @xmath155-submodule . from this , it follows that the essential image of the functor in question is closed under formation of subquotients . since the essential image includes all representations of the form @xmath791 , it follows that our functor is essentially surjective . [ cor : gators ] we have the following : 1 . if @xmath253 is an injective @xmath146-module then @xmath792 is an injective object of @xmath785 . every finitely generated object @xmath50 of @xmath785 admits a resolution @xmath754 where each @xmath755 is a finite direct sum of representations of the form @xmath792 with @xmath756 and @xmath757 for @xmath60 . 3 . every finitely generated object of @xmath785 admits a finite length filtration where the graded pieces have the form @xmath252 , where @xmath756 . moreover , if @xmath145 is connected then @xmath174 admits no non - trivial @xmath146-flat quotients . we now study the category @xmath143 . the key result is : there exists an @xmath146-linear equivalence of categories @xmath793 that is compatible with tensor products and carries @xmath794 to the standard representation @xmath748 . here is the idea of the proof : an @xmath5-module is a quasi - coherent equivariant sheaf on the space @xmath795 . the category @xmath143 can be identified with quasi - coherent equivariant sheaves on the open subscheme @xmath796 where the map is injective . the group @xmath797 acts transitively on this space with stabilizer @xmath798 ( almost ) , and so @xmath143 is equivalent to @xmath799 . we now carry out the details rigorously . this is unfortunately lengthy . first suppose that @xmath149 is trivial . let @xmath1 be the rank of @xmath149 , and choose a decomposition @xmath800 where @xmath801 has dimension @xmath1 . also choose an isomorphism @xmath802 . this isomorphism induces a pairing @xmath803 , which in turn induces a homomorphism @xmath804 . for an @xmath5-module @xmath50 , let @xmath805 . suppose now that @xmath806 is a second choice of isomorphism , and write @xmath807 where @xmath808 is a section of @xmath809 . regard @xmath810 as a subgroup of @xmath789 in the obvious manner . since @xmath789 acts on @xmath50 , there is an induced map @xmath811 . one readily verifies that this induces an isomorphism @xmath812 . we thus see that @xmath813 is canonically independent of @xmath59 , and so denote it by @xmath814 . ( to be more canonical , one could define @xmath814 as the limit of the @xmath813 over the category of isomorphisms @xmath59 . ) let @xmath815 be defined like @xmath798 , but using @xmath816 , that is , @xmath817 . let @xmath818 be the subgroup of @xmath797 consisting of elements @xmath808 such that @xmath819 and @xmath820 . note that @xmath821 , and so @xmath59 induces an isomorphism @xmath822 . the group @xmath818 stabilizes the pairing @xmath803 , and thus acts on @xmath813 . the group @xmath810 acts on @xmath818 , via its action on @xmath801 . if @xmath807 then the induced isomorphism @xmath823 is compatible with the @xmath818 actions in the sense that @xmath824 for @xmath825 . it follows that if we let @xmath815 act on @xmath813 via the isomorphism @xmath822 induced by @xmath59 , then @xmath826 for @xmath827 . we thus see that @xmath815 canonically acts on @xmath814 . now suppose that @xmath149 is arbitrary . then we can define @xmath814 with its @xmath815 action over a cover trivializing @xmath149 . since everything is canonical , the pieces patch to define @xmath814 over all of @xmath145 . we have thus defined a functor @xmath828 . we will eventually deduce the desired equivalence @xmath829 from this functor . it is clear that @xmath830 is a tensor functor : indeed , working locally , @xmath831 working locally , we have @xmath832 . this globalizes to @xmath833 , the standard representation of @xmath815 . we thus see that @xmath834 is a polynomial representation of @xmath815 . since every @xmath5-module is a quotient of a sum of modules of the form @xmath835 , it follows that @xmath814 is a polynomial representation of @xmath815 for any @xmath5-module @xmath50 . suppose now that @xmath149 is trivial and @xmath50 is finitely generated . let @xmath175 be a sufficiently large finite dimensional vector space , and choose a decomposition @xmath836 . then @xmath837 is identified with @xmath838 . now , the spectrum of @xmath839 is identified with the space @xmath840 over @xmath145 , and so @xmath837 is identified with the pullback of the coherent sheaf @xmath841 along the section @xmath842 induced by @xmath59 . this section lands in the open subscheme @xmath843 consisting of injective maps . we now claim that @xmath830 is exact . this can be checked locally . furthermore , since @xmath830 commutes with direct limits , it suffices to check on finitely generated modules . we can therefore place ourselves in the situation of the previous paragraph . we can compute @xmath844 in two steps : first restrict from @xmath840 to @xmath843 , and then restrict again to @xmath145 . the first step is exact since restriction to an open subscheme is always exact . now the key point : @xmath843 is a @xmath845-torsor over @xmath145 , and so any equivariant sheaf or equivariant map of such sheaves is pulled back from @xmath145 . thus pullback of such modules to @xmath145 is again exact . this completes the proof of the claim . we next claim that @xmath830 kills torsion modules . again , we can work locally and assume @xmath50 is finitely generated . if @xmath50 is torsion then the support of @xmath841 in @xmath840 does not meet @xmath843 , and so the pullback to @xmath145 vanishes . this proves the claim . we thus see that @xmath830 induces an exact tensor functor @xmath846 . we claim that @xmath847 is fully faithful . this can again be checked locally for finitely generated modules after evaluating on @xmath175 of dimension @xmath129 . since @xmath845 acts transitively on @xmath843 with stabilizer @xmath815 , giving a map of @xmath845-equivariant quasi - coherent sheaves on @xmath843 is the same as giving maps at the fibers at @xmath59 , as @xmath815-representations . suppose that @xmath848 is a map of @xmath5-modules such that the induced map @xmath849 vanishes . then the map @xmath850 vanishes over @xmath843 . this implies that the image of @xmath850 is torsion , and so the image of @xmath848 is torsion , and so the map @xmath848 is 0 in @xmath143 . this proves faithfulness of @xmath847 . now suppose that a map @xmath851 of @xmath815-representations is given . this is induced from a map @xmath850 over @xmath843 . this induces a map of quasi - coherent sheaves @xmath852 , where @xmath853 is the open immersion . now , @xmath854 is a @xmath855-equivariant @xmath839-module , but may not be polynomial . let @xmath856 be the maximal polynomial subrepresentation , which is an @xmath839-submodule containing @xmath857 . let @xmath858 be the canonical @xmath5-module with @xmath859 satisfying @xmath860 . the map @xmath861 is induced from a map of @xmath5-modules @xmath862 . now , @xmath863 pulls back to 0 under @xmath864 , and is thus torsion . it follows that @xmath865 is torsion , and so @xmath866 in @xmath143 . thus the constructed map @xmath862 of @xmath5-modules gives the required map @xmath848 in @xmath143 . this proves fullness of @xmath847 . we now claim that @xmath847 is essentially surjective . since all categories are cocomplete and @xmath830 is cocontinuous and fully faithful , it suffices to show that all finitely generated objects are in the essential image . by proposition [ cor : gators](b ) , a finitely generated object @xmath50 of @xmath867 can be realized as the kernel of a map @xmath868 , where @xmath101 and @xmath869 are each sums of representations of the form @xmath791 with @xmath253 an @xmath146-module . we have already shown that such modules are in the essential image of @xmath830 . thus @xmath870 and @xmath871 for @xmath872 and @xmath873 in @xmath143 . since @xmath847 is full , @xmath874 for some @xmath875 in @xmath143 . finally , since @xmath847 is exact , @xmath876 , which shows that @xmath50 is in the essential image of @xmath847 . we have thus shown that @xmath847 is an equivalence of categories @xmath877 . combining this with the obvious equivalence @xmath878 coming from a choice of isomorphism @xmath879 , we obtain the desired equivalence @xmath829 . there is a canonical map @xmath880 . we let @xmath881 be the kernel of the corresponding map in @xmath143 . under the equivalence @xmath829 in the proposition , we have @xmath882 . combining the proposition with corollary [ cor : gators ] , we obtain : [ cor : generic - inj ] we have the following : 1 . if @xmath253 is an injective object in @xmath194 then @xmath883 is injective in @xmath143 . every finitely generated object @xmath50 of @xmath143 admits a resolution @xmath754 where each @xmath755 has the form @xmath883 with @xmath884 and @xmath757 for @xmath60 . every finitely generated object of @xmath143 admits a finite length filtration where the graded pieces have the form @xmath885 where @xmath756 . moreover , if @xmath145 is connected then @xmath886 has no non - trivial @xmath146-flat quotients . the category @xmath143 is naturally a module for the tensor category @xmath173 . thus @xmath887 is a module for @xmath888 . the following result describes @xmath887 as a @xmath48-module . [ cor : ktheory - generic ] the map @xmath889 taking @xmath777 $ ] to @xmath890 $ ] induces an isomorphism @xmath891 . this follows from corollary [ cor : generic - inj](c ) . combining the results of the previous several sections , we obtain an equivalence @xmath892 . note that this equivalence is _ not _ @xmath173-linear ! indeed , @xmath893 , and this is not isomorphic to @xmath894 . this computation also shows that the isomorphism @xmath895 induced by @xmath830 is not @xmath48-linear . we now study the section functor @xmath896 using a geometric approach . let @xmath897 be an integer . let @xmath898 be the space of linear maps @xmath899 , thought of as a scheme over @xmath145 ; in fact , @xmath898 is just @xmath289 . let @xmath900 be the open subscheme of @xmath898 where the map is surjective , and write @xmath901 for the inclusion . by a * polynomially ( resp . algebraically ) equivariant sheaf * on @xmath898 we mean a @xmath189-equivariant quasi - coherent sheaf that is a subquotient of a direct sum of sheaves of the form @xmath902 , where @xmath253 is a quasi - coherent sheaf on @xmath145 and @xmath175 is a polynomial ( resp . algebraic ) representation of @xmath189 . we write @xmath903 ( resp . @xmath904 ) for the category of polynomially ( resp . algebraically ) equivariant sheaves . we make similar definitions for @xmath900 , though we will only use polynomially equivariant sheaves on @xmath900 . we can identify @xmath903 ( resp . @xmath905 ) with the category of @xmath189-equivariant @xmath126-modules that decompose as a polynomial ( resp . algebraic ) representation of @xmath189 . if @xmath175 is an algebraic representation of @xmath189 over @xmath146 then it has a maximal polynomial subrepresentation @xmath906 , and the construction @xmath907 is exact . this construction induces an exact functor @xmath908 , denoted @xmath909 . let @xmath910 , let @xmath911 be the structure map , and let @xmath42 be the tautological bundle on @xmath151 . a point in @xmath900 is a surjection @xmath912 . we thus obtain a map @xmath913 by associating to @xmath334 the quotient @xmath914 of @xmath125 . in fact , specifying @xmath334 is the same as specifying an isomorphism @xmath915 , and so we see that @xmath900 is identified with the space @xmath916 over @xmath917 . in particular , the map @xmath402 is affine : @xmath916 is the relative spectrum of the algebra @xmath918 the sum taken over all dominant weights @xmath80 . we thus see that @xmath919 can be identified with @xmath920 , where we identify @xmath898-modules with @xmath126-modules on @xmath145 . [ lem : algcoh ] let @xmath50 be a @xmath189-equivariant quasi - coherent sheaf on @xmath151 that is a subquotient of a direct sum of sheaves of the form @xmath921 where @xmath253 is an @xmath146-module . then @xmath922 is an algebraic representation of @xmath189 over @xmath146 , for any @xmath59 . this can be checked locally on @xmath145 , so we can assume @xmath145 is affine . write @xmath923 where @xmath924 and @xmath538 is an appropriate parabolic subgroup of @xmath382 . then @xmath50 corresponds to an algebraic representation @xmath243 of @xmath538 over @xmath146 . the push - forward @xmath922 is then identified with the derived induction from @xmath538 to @xmath382 of @xmath243 by ( * ? ? ? * proposition i.5.12(a ) ) , which is an algebraic representation of @xmath382 by definition . . then @xmath926 . by definition , @xmath50 is a subquotient of a sheaf of the form @xmath927 , where @xmath175 is a polynomial representation of @xmath189 over @xmath146 . we thus see that @xmath928 is a subquotient of @xmath929 thus @xmath928 is a @xmath189-equivariant quasi - coherent sheaf on @xmath151 that is a subquotient of a direct sum of sheaves of the form @xmath930 where @xmath253 is an @xmath146-module . the result now follows from lemma [ lem : algcoh ] . [ lem : jsupp ] 1 . if @xmath931 then the natural map @xmath932 has kernel and cokernel supported on the complement of @xmath900 , as does the cokernel of the inclusion @xmath933 . 2 . if @xmath925 then the inclusion @xmath934 has cokernel supported on the complement of @xmath900 . \(a ) the map @xmath935 has kernel and cokernel supported on the complement of @xmath900 , and factors through the inclusion @xmath933 . the result follows . \(b ) if @xmath50 has the form @xmath936 for @xmath937 then the result follows from ( a ) . since every object of @xmath938 is , by definition , a subquotient of one this form , it suffices to show that if ( b ) holds for @xmath50 then it holds for subs and quotients of @xmath50 . thus let @xmath50 be given and let @xmath243 be a submodule of @xmath50 . then @xmath939 is a submodule of @xmath940 , and @xmath941 . thus the map @xmath942 is injective . since the target is supported on the complement of @xmath900 , it follows that the source is as well . now let @xmath243 be a quotient of @xmath50 . the cokernel of the map @xmath943 is then supported on the complement of @xmath900 , by general theory . thus the same is true for the cokernel of the map @xmath944 . since the source is supported on the complement of @xmath900 , the same is thus true for the target . the restriction functor @xmath945 identifies @xmath938 with the serre quotient of @xmath903 by the subcategory of sheaves supported on the complement of @xmath900 . let @xmath299 be the subcategory of sheaves supported on @xmath946 . restriction to the open subscheme @xmath900 is an exact functor and annihilates @xmath299 , so we get a functor @xmath947 . to see that it is faithful , consider a morphism @xmath948 of sheaves on @xmath898 whose restriction to @xmath900 is @xmath600 . this means that the image of @xmath334 is supported on @xmath946 , so @xmath949 in the serre quotient @xmath950 . to get fullness , let @xmath951 be a morphism of sheaves . then we get @xmath952 , which induces a map @xmath953 . by lemma [ lem : jsupp](a ) , we have @xmath954 . also by lemma [ lem : jsupp](a ) , the map @xmath932 is an isomorphism in @xmath955 , and similarly for @xmath243 , and so @xmath956 actually defines a map @xmath957 in the quotient category . finally , for essential surjectivity , let @xmath925 be given . by lemma [ lem : jsupp](b ) , the natural map @xmath958 is an isomorphism . by general theory , there is a natural isomorphism @xmath959 . we thus see that @xmath960 where @xmath961 is an object of @xmath903 . suppose @xmath50 is an @xmath5-module . then @xmath127 is a polynomially @xmath189-equivariant @xmath126-module , and thus defines an @xmath898-module . this gives an exact functor @xmath962 . [ lem : geogen ] we have a commutative @xmath963up to isomorphism@xmath964 diagram @xmath965 ^ -t \ar[d ] & \mod_a^{{{\rm gen } } } \ar@{ .. >}[d ] \\ \mod_{\fh_n } \ar[r]^{j^ * } & \mod_{\fu_n } } \ ] ] where the left map is @xmath966 . by the universal property of serre quotients , it suffices to show that if @xmath50 is an @xmath5-module with @xmath967 then the @xmath898-module @xmath127 is supported on the complement of @xmath900 . if @xmath967 then the annihilator @xmath729 of @xmath50 is non - zero , and so @xmath968 is also non - zero ( since @xmath969 ) . we thus see that the support of @xmath127 is a proper closed subset of @xmath898 . it is therefore contained in the complement of @xmath900 , as this is the maximal proper closed @xmath189-stable subset . the following theorem is the key to our understanding of the saturation functor and its derived functors . [ thm : geos ] we have a diagram @xmath970^{\rr^n s } \ar[d ] & & \mod_a \ar[d ] \\ \mod_{\fu_n } \ar[rr]^{(\rr^n j_*)^{\pol } } & & \mod_{\fh_n } } \ ] ] that commutes up to a canonical isomorphism . here the vertical maps are as in lemma [ lem : geogen ] . in this proof , a torsion @xmath5-module is one localizing to 0 in @xmath143 , and a torsion @xmath898-module is one restricting to 0 on @xmath900 . we first construct a canonical injection @xmath971 for @xmath972 . first suppose that @xmath243 is an @xmath5-module . then @xmath973 is torsion - free and so @xmath974 is as well . thus @xmath975 is a map from @xmath976 to a torsion - free object with torsion kernel and cokernel . however , @xmath977 is the universal such map , and so we obtain a canonical map @xmath978 , which is necessarily injective . now , let @xmath979 . then @xmath980 and @xmath981 , by definition . we thus obtain the desired map . we now claim that the map just constructed is an isomorphism if @xmath982 , with @xmath286 . we have maps @xmath983 since the second map is injective , it suffices to show that the composite map is an isomorphism . for this , we will compute the rightmost object . the @xmath898-module @xmath984 is @xmath985 , and so we see that the @xmath900-module @xmath127 is @xmath986 . we thus have @xmath987 where the sum is over all dominant weights @xmath80 . we now apply @xmath988 , and use the fact that @xmath989 if @xmath80 is a partition and @xmath990 otherwise . we obtain @xmath991 where now the sum is over all partitions @xmath80 . we leave to the reader the verification that the natural map @xmath992 is the identity with the above identification . we now claim that the map @xmath971 is an isomorphism for all @xmath972 . to see this , let @xmath50 be given and choose an exact sequence @xmath993 where @xmath994 and @xmath995 have the form @xmath996 for @xmath286 . we then obtain a commutative diagram @xmath997 \ar[d ] & s(i^0)(\bc^n ) \ar[r ] \ar[d ] & s(i^1)(\bc^n ) \ar[d ] \\ 0 \to j_*(m(\bc^n ) ) \ar[r ] & j_*(i^0(\bc^n ) ) \ar[r ] & j_*(i^1(\bc^n ) ) } \ ] ] with exact rows . since the right two vertical maps are isomorphisms , so is the left vertical map . we have thus proved the result for @xmath998 . ( in fact , we showed that one does not even need to take the polynomial piece in this case . ) we now prove the result for arbitrary @xmath119 . the functors @xmath999 form a cohomological @xmath1000-functor . since formation of the polynomial subrepresentation is exact on @xmath904 , it follows that the functors @xmath1001 also form a cohomological @xmath1000-functor . since evaluation on @xmath125 is exact , the functors @xmath1002 and @xmath1003 are both cohomological @xmath1000-functors @xmath1004 . the first is clearly universal , since the higher derived functors kill injective objects of @xmath143 . thus to prove the result , it suffices to show that the second one is universal , and for this it suffices to show that it is coeffaceable . since every object of @xmath143 injects into an object of the form @xmath982 with @xmath286 , it suffices to show that @xmath1005 for @xmath1006 . applying @xmath1007 to , and using the projection formula , we find @xmath1008 we now come to the point : @xmath1009 for all @xmath1006 . indeed , if @xmath80 is a partition then @xmath1010 for @xmath1006 . now suppose @xmath80 is not a partition . by borel bott ( theorem [ thm : bwb ] ) , either @xmath1011 vanishes for all @xmath119 , or vanishes for all @xmath1012 and for @xmath1013 has the form @xmath1014 . in the latter case , @xmath1015 where @xmath1016 and @xmath1017 . if @xmath80 is not a partition then @xmath1018 , and it is clear from the formulation of theorem [ thm : bwb ] that @xmath1019 has a negative entry . thus @xmath1019 is not a partition , and so @xmath1020 . the functor @xmath1021 is @xmath194-linear . precisely , if @xmath286 is @xmath146-flat then there is a canonical isomorphism @xmath1022 for all @xmath972 . more generally , we have a canonical isomorphism @xmath1023 . let @xmath1024 be an @xmath146-flat complex quasi - isomorphic to @xmath175 and let @xmath1025 be an injective resolution of @xmath50 in @xmath1026 . for each @xmath59 , choose an injective resolution @xmath1027 , the iterated mapping cone of these complexes is denoted @xmath1028 . then both @xmath1029 and @xmath1028 are quasi - isomorphic to @xmath1030 and we can lift the identity map on @xmath1030 to get a morphism @xmath1031 . apply @xmath587 to both sides to get @xmath1032 now evaluate this map on @xmath125 . by theorem [ thm : geos ] , this replaces @xmath1026 by @xmath938 and @xmath11 by @xmath903 , in which case @xmath618 is identified with @xmath1033 . then the map above is a quasi - isomorphism by the usual projection formula . [ cor : dersat ] if @xmath286 then @xmath209 is derived saturated , that is , the natural map @xmath1034 is an isomorphism . it suffices to show this after evaluating on @xmath125 for all @xmath119 . this was shown in the course of the proof of theorem [ thm : geos ] . if @xmath286 is injective then @xmath1035 is an injective @xmath5-module . in particular , if @xmath771 then all projective @xmath5-modules are also injective . the functor @xmath587 takes injectives to injectives . by corollary [ cor : generic - inj ] , @xmath996 is injective in @xmath143 , and we have just shown that @xmath1036 is @xmath1035 . [ cor : sfin ] if @xmath972 is finitely generated then @xmath1037 is represented by a finite length complex of modules of the form @xmath1035 with @xmath1038 . in particular , @xmath1039 is finitely generated for all @xmath1040 and vanishes for @xmath129 . using corollary [ cor : generic - inj ] , pick a resolution @xmath1041 where @xmath1042 and @xmath1043 for @xmath60 . since @xmath1044 is @xmath745-acyclic , it follows that @xmath1045 is @xmath587-acyclic . we can thus use this resolution to compute @xmath618 . since @xmath1046 , we obtain a quasi - isomorphism @xmath1047 . the result follows . finally , we compute the derived saturation of the objects @xmath1048 . for a weight @xmath80 and partition @xmath1019 , write @xmath1049 if bott s algorithm applied to @xmath80 terminates after @xmath119 steps on @xmath1019 , see remark [ rmk : bott - algorithm ] . for a partition @xmath89 , we have @xmath1050 \xrightarrow{i } \nu } \bs_{\nu}(\bv ) \otimes \bs_{\lambda}(\ce),\ ] ] where the sum is over all partitions @xmath80 and @xmath1019 with @xmath1051 and thus related , and @xmath1052 $ ] is the weight @xmath1053 . in particular , @xmath1054}(\bv ) \otimes \bs_{\lambda}(\ce).\ ] ] take @xmath1055 . using theorem [ thm : geos ] , we have @xmath1056 where @xmath901 is the inclusion . as discussed above , we have a factorization @xmath1057 where @xmath1058 is the structure map and @xmath1059 sends a map in @xmath900 to its cokernel . note that @xmath1060 , and since pullback commutes with tensor operations , we get @xmath1061 where the sum is over all dominant weights @xmath80 . hence , the desired result follows from borel weil bott ( theorem [ thm : bwb ] ) , noting that any @xmath80 with negative entries are deleted from the final computation since we need to take the polynomial piece . when @xmath26 and @xmath1062 , this essentially recovers ( * ? ? ? * proposition 7.4.3 ) . to be precise , corollary [ cor : generic - inj](d ) says that the @xmath1063 are the simple objects of @xmath1026 , so they are the simple objects @xmath1064 defined in @xcite . the local cohomology calculation there for @xmath1065 agrees with @xmath1066 by ( * ? ? ? * corollary 4.4.3 ) , and the discussion in @xcite connects the border strip combinatorics mentioned there with borel weil bott . we fix , for all of [ s : rank ] , a scheme @xmath145 over @xmath144 ( noetherian , separated , and of finite krull dimension , as always ) and a vector bundle @xmath149 of rank @xmath1 on @xmath145 . we let @xmath169 . we introduce some notation mirroring that from [ ss : formalism2 ] . we write @xmath34 for the category of @xmath5-modules supported on @xmath380 , i.e. , that are locally annihilated by powers of @xmath31 . this gives an ascending chain of serre subcategories @xmath1067 that we refer to as the * rank stratification*. we define quotient categories @xmath1068 we let @xmath1069 be the localization functor and @xmath645 its right adjoint . we put @xmath646 , as usual , and let @xmath1070 be the functor that assigns to a module the maximal submodule supported on @xmath380 . we let @xmath1071 be the full subcategory of @xmath1072 on objects @xmath50 such that @xmath1073 , and we let @xmath1074 be the full subcategory on objects @xmath50 such that @xmath134 . we also put @xmath1075 . these are all triangulated subcategories of @xmath1072 . by [ s : formalism ] , we have a semi - orthogonal decomposition @xmath1076 . we let @xmath1077 $ ] be the category of @xmath5-modules annihilated by @xmath31 . this is a subcategory of @xmath34 . we let @xmath41 $ ] be the subcategory of @xmath39 on objects of the form @xmath1078 , where @xmath50 is an @xmath5-module such that @xmath1079 is supported on @xmath420 . obviously , @xmath699 carries @xmath1077 $ ] into @xmath41 $ ] . in fact : the functor @xmath1080 \to \mod_{a , r}[\fa_r]$ ] identifies @xmath41 $ ] with the serre quotient of @xmath1077 $ ] by @xmath1081 $ ] . the functor @xmath699 is exact and kills @xmath1081 $ ] , and thus induces a functor @xmath1082}{\mod_{a,<r}[\fa_r ] } \to \mod_{a , r}[\fa_r].\ ] ] we must show that the functor @xmath829 is an equivalence . we write @xmath299 for the domain of @xmath829 . we first claim that the functor @xmath829 is essentially surjective . it suffices to show that the functor @xmath1080 \to \mod_{a , r}[\fa_r]$ ] is essentially surjective . thus let @xmath1083 $ ] be a typical object , so that @xmath50 is an @xmath5-module such that @xmath1079 is supported on @xmath420 . then @xmath1084 belongs to @xmath1085 $ ] . since the map @xmath1086 is surjective and has kernel supported on @xmath420 , it follows that @xmath1087 is an isomorphism . since @xmath1088 $ ] , this establishes the claim . we now show that @xmath829 is fully faithful . let @xmath1089 $ ] . let @xmath1090 \to \mod_{a , r}[\fa_r]$ ] be the localization functor . then @xmath1091}(t(m ) , t(n ) ) = \varinjlim \hom_{\mod_a[\fa_r]}(m ' , n'),\ ] ] where the colimit is over @xmath1092 such that @xmath1093 $ ] and quotients @xmath1094 with kernel in @xmath1081 $ ] . on the other hand , @xmath1091}(t_{\ge r}(m ) , t_{\ge r}(n ) ) = \varinjlim \hom_a(m '' , n''),\ ] ] where the colimit is over @xmath1095 such that @xmath1096 and quotients @xmath1097 with kernel in @xmath1098 . since @xmath50 and @xmath243 are killed by @xmath31 , it follows that @xmath1099 and @xmath1100 are as well , and so this colimit is exactly the same as the previous one . we write @xmath1101 \to \mod_a[\fa_r]$ ] for the right adjoint of the localization functor @xmath699 appearing in the proposition . we write @xmath1102 for the derived functor of @xmath1103 . the notation @xmath1104 always means the derived functor of @xmath1105 . thus for @xmath1106 $ ] one computes @xmath1107 by using an injective resolution of @xmath50 in the category @xmath41 $ ] , while one computes @xmath1108 by using an injective resolution in @xmath1109 ( or simply @xmath39 ) . injective objects in these two categories are quite different ; nonetheless , we have : [ prop : equivs ] the functor @xmath1102 is isomorphic to the restriction of @xmath1104 to the derived category of @xmath41 $ ] . let @xmath1110 be an injective of @xmath41 $ ] , and put @xmath1111 , an injective of @xmath1085 $ ] . it suffices to show that the map @xmath1112 is an isomorphism . indeed , suppose this is the case . then @xmath1113 is @xmath700-acyclic and satisfies @xmath1114 . thus if @xmath568 is an injective resolution in @xmath41 $ ] then @xmath1115 computes @xmath1108 , since the objects @xmath1116 are @xmath700-acyclic , and equals @xmath1117 , which computes @xmath1102 . to prove that @xmath1112 is an isomorphism , it suffices ( by proposition [ prop : satcrit ] ) to show @xmath1118 for all @xmath1119 and @xmath1120 . we first treat the @xmath1121 case , i.e. , we show that any map @xmath1122 with @xmath1119 is zero . it suffices to treat the case where @xmath243 is finitely generated and thus annihilated by a power of @xmath1123 . by dvissage , we can assume @xmath1124 . but then @xmath1125 as well , and so @xmath1126 $ ] . since @xmath305 is saturated with respect to this category , the result follows . we now consider the case @xmath1127 . since @xmath305 is an @xmath415-module , derived adjunction gives @xmath1128 as @xmath305 is injective as an @xmath415-module , this @xmath1129 can be changed to @xmath220 . we find @xmath1130 since @xmath243 is supported on @xmath420 , so are the @xmath1131 s . thus , by the @xmath1121 case , the above @xmath220 vanishes . this completes the proof . let @xmath50 be an @xmath5-module annihilated by @xmath31 . then @xmath1132 is annihilated by @xmath31 for all @xmath1040 . indeed , @xmath1132 is by definition @xmath1133 , which by proposition [ prop : equivs ] is identified with @xmath1134 , and @xmath1103 ( and its derived functors ) take values in @xmath1077 $ ] . we note that , a priori , @xmath1132 is supported on @xmath420 for @xmath1006 , and thus locally annihilated by a power of @xmath1123 . however , this does not directly imply that @xmath1132 is annihilated by @xmath31 . we now give a complete description of the category @xmath41 $ ] . let @xmath1135 be the grassmannian of rank @xmath16 quotients of @xmath149 . let @xmath911 be the natural map , and let @xmath42 be the tautological rank @xmath16 quotient of @xmath1136 . let @xmath1137 . we let @xmath664 and @xmath679 be the usual functors between @xmath1138 and @xmath1139 . we have a natural map @xmath1140 , which induces a functor @xmath1141 via @xmath1142 . [ thm : abequiv ] the functor @xmath1143 \to \mod_b^{{{\rm gen}}}$ ] is exact and kills @xmath1081 $ ] . the induced functor @xmath1144 \to \mod_b^{\rm gen}\ ] ] is an equivalence and compatible with tensor products . let @xmath898 be defined as in [ ss : section ] , let @xmath1145 be the closed subscheme defined by @xmath128 , and let @xmath1146 be the complement of @xmath1147 in @xmath1145 . specialization defines a functor @xmath1077 \to \mod_{\fh_n^{\le r}}$ ] , which induces a functor @xmath41 \to \mod_{\fh_n^{=r}}$ ] , just as in lemma [ lem : geogen ] . let @xmath1148 be defined like @xmath900 as in [ ss : section ] but with respect to @xmath43 ; thus @xmath1148 is the scheme of surjections @xmath1149 . there is an isomorphism of schemes @xmath1150 , since a map @xmath899 of rank @xmath16 determines a rank @xmath16 quotient of @xmath1151 . consider the diagram @xmath1152 \ar[r ] \ar[d ] \ar@/^2em/[rr]^{t ' \circ \phi } & \mod_{a , r}[\fa_r ] \ar@{ .. >}[r]^-{\psi } \ar[d ] & \mod_b^{{{\rm gen } } } \ar[d ] \\ \mod_{\fh_n^{\le r } } \ar[r ] & \mod_{\fh_n^{=r } } \ar[r ] & \mod_{\fu'_n } } \ ] ] both functors in the bottom row are exact . it follows that @xmath1153 is exact . indeed , it is right exact , so it suffices to verify that it preserves injections . if @xmath848 were an injection such that @xmath1154 were not injective , then for @xmath129 the specialization of the kernel to @xmath125 would be a non - zero object of @xmath1155 , contradicting exactness of the bottom row . thus @xmath1153 is exact . it follows from the above diagram that @xmath1153 kills @xmath1081 $ ] : indeed , if @xmath50 were in this category then its specialization to @xmath125 would restrict to 0 on @xmath1146 for all @xmath119 , and so @xmath1156 . we thus get the induced functor @xmath830 as in the diagram . we first show that @xmath830 is fully faithful . let @xmath1157 $ ] be finitely generated , and thus bounded . to verify that @xmath1158}(m , n ) \to \hom_{\mod_b^{{{\rm gen}}}}(\psi ( m ) , \psi(n))$ ] is an isomorphism , we can do so after specializing to @xmath125 for @xmath119 sufficiently large . but this is clear , since the bottom right map in the above diagram is an equivalence . we now claim that every object of @xmath1139 is a quotient of one of the form @xmath1159 with @xmath286 . by definition , a @xmath43-module is a quotient of @xmath1160 for some @xmath1161 . it thus suffices to show that if @xmath253 is an @xmath1162-module then @xmath1163 is a quotient of @xmath1159 for some @xmath286 . we note that the natural map @xmath1164 is surjective . indeed , under the equivalence @xmath1165 , this corresponds to the natural surjection @xmath1166 . we thus have a surjection @xmath1167 , where @xmath1168 . since @xmath1169 is an ample line bundle relative to @xmath145 , any @xmath1162-module @xmath253 can be written as a quotient of a sum of @xmath1162-modules of the form @xmath1170 where @xmath1171 is an @xmath146-module and @xmath1006 is an integer . in this way , we obtain a surjection @xmath1172 where @xmath286 . we now verify that @xmath830 is essentially surjective . let @xmath1173 be given . choose a presentation @xmath1174 with @xmath216 , which is possible by the previous paragraph . since @xmath830 is fully faithful , we can write @xmath1175 for some morphism @xmath1176 in @xmath41 $ ] . since @xmath830 is exact , we have @xmath1177 . it is clear from the construction that @xmath830 is a compatible with tensor products . [ prop : pi - adjoint ] the functor @xmath1178 $ ] is the right adjoint to the functor @xmath1179 \to \mod_b$ ] . moreover , @xmath1180 is the derived functor of @xmath988 on @xmath1138 . we can identify @xmath11 and @xmath1138 with categories of quasi - coherent sheaves on the schemes @xmath1181 and @xmath1182 . the map @xmath1140 induces a map @xmath1183 . under the previous identifications , @xmath988 corresponds to @xmath199 and @xmath829 to @xmath200 . the adjointness statement follows from the usual adjointness of @xmath199 and @xmath200 . we now show that the @xmath1184 is the derived functor of @xmath988 on @xmath1138 . it suffices to show that injective @xmath43-modules are @xmath988-acyclic . thus let @xmath305 be an injective @xmath43-module . then each multiplicity space @xmath313 is injective as an @xmath146-module by proposition [ prop : injox ] , and therefore acyclic for @xmath988 ( see ( * ? ? ? * tag 0bdy ) ) . since @xmath988 is computed on @xmath194 simply by applying @xmath988 to each multiplicity space , it follows that @xmath305 is @xmath988-acyclic . the following diagram summarizes the picture : @xmath1185 \ar@<3pt>[d]^{t_{\ge r } } \ar@<3pt>[r]^{\phi } & \mod_b \ar@<3pt>[l]^{\pi _ * } \ar@<3pt>[d]^{t ' } \\ \mod_{a , r}[\fa_r ] \ar[r]^{\psi } \ar@<3pt>[u]^{s_{\ge r } } & \mod_b^{{{\rm gen } } } \ar@<3pt>[u]^{s ' } } \ ] ] [ lem : sr - formula ] @xmath1186 . the two paths from @xmath1077 $ ] to @xmath1139 commute by definition of @xmath830 . since @xmath830 is an equivalence , it follows that @xmath1187 is the right adjoint to @xmath1188 . on the other hand , since @xmath664 is right adjoint to @xmath679 and @xmath988 is right adjoint to @xmath829 , it follows that @xmath1189 is right adjoint to @xmath1153 . thus the two paths from @xmath1139 to @xmath1077 $ ] ( one of which uses the undrawn @xmath1190 ) also agree . [ prop : srformula ] let @xmath1106 $ ] , and let @xmath1191 be the corresponding object of @xmath1139 . then @xmath1108 is canonically isomorphic to @xmath1192 . by lemma [ lem : sr - formula ] , @xmath1186 . we thus see that @xmath1193 . here we have used the fact that @xmath1180 is the derived functor of @xmath988 on @xmath1138 ( proposition [ prop : pi - adjoint ] ) and the fact that @xmath1104 is the derived functor of @xmath700 on @xmath41 $ ] ( proposition [ prop : equivs ] ) . [ cor : srfin ] let @xmath1194 . then @xmath1195 is a finitely generated @xmath5-module for all @xmath1040 , and vanishes for @xmath129 . by dvissage , we can reduce to the case @xmath1196 $ ] . by proposition [ prop : srformula ] , we have @xmath1197 , where @xmath1191 is a finitely generated object of @xmath1139 . since @xmath669 carries @xmath1198 into @xmath1199 ( corollary [ cor : sfin ] ) and @xmath1180 carries @xmath1199 to @xmath67 ( corollary [ cor : finpushfwd ] ) , the result follows . the following theorem is one of the fundamental results of this paper . [ thm : satfin ] let @xmath1200 . then @xmath1201 and @xmath1202 also belong to @xmath98 . we proceed by descending induction on @xmath16 . when @xmath1203 , we have that @xmath55 is the identity functor and @xmath655 , so the statement is clear . now let us prove the statement for @xmath16 , assuming it has been proved for @xmath1204 . consider the triangle @xmath1205 applying @xmath71 , we obtain a triangle @xmath1206 but @xmath1207 , so the rightmost term is @xmath1208 , which belongs to @xmath98 by the inductive hypothesis . since @xmath1209 belongs to @xmath98 and is supported on @xmath1210 , it follows from corollary [ cor : srfin ] that @xmath1211 belongs to @xmath98 . it now follows from the above triangle that @xmath1201 belongs to @xmath98 . from the canonical triangle relating @xmath71 and @xmath69 , we see that @xmath1202 also belongs to @xmath98 . the theorem exactly states that the hypothesis ( fin ) from [ ss : fin ] holds , and so all the consequences of ( fin ) given there hold as well . we summarize the proof of theorem [ thm : satfin ] . there are two parts . the first is that we can compute @xmath1212 if @xmath50 is an @xmath415-module since we can relate it to cohomology of sheaves on grassmannians by proposition [ prop : srformula ] . ( note that in the formula in that proposition , @xmath1180 is sheaf cohomology on @xmath150 , while @xmath669 is essentially sheaf cohomology on @xmath1213 by theorem [ thm : geos ] . ) the second is that we can formally deduce the full result from this particular case via the inductive procedure in the above proof . [ rmk : loccohfin ] one can define local cohomology functor with respect to any ideal of @xmath5 . however , the finiteness observed in the theorem for determinantal ideals does not hold in general . in fact , it seems plausible that finiteness essentially holds only for determinantal ideals ( essentially because the property only depends on the radical ) . let @xmath605 be a triangulated category and let @xmath587 be a collection of objects in @xmath605 . the triangulated subcategory of @xmath605 * generated * by @xmath587 is the smallest triangulated subcategory of @xmath605 containing @xmath587 . the following result gives a useful set of generators for @xmath1214 . we use notation as in [ ss : modar ] : @xmath1135 , @xmath42 is the tautological bundle , @xmath1137 , and @xmath911 is the structure map . [ prop : dr ] the category @xmath1214 is the triangulated subcategory of @xmath98 generated by the objects @xmath1215 with @xmath1216 . by proposition [ prop : derived - decomp ] , the functor @xmath1217 is an equivalence . now , @xmath66 is generated by @xmath1218 ( thought of as complexes in degree 0 ) . every object of @xmath1218 has a finite length filtration where the graded pieces belong to @xmath1219 $ ] , and so it follows that @xmath1219 $ ] generates @xmath66 . by theorem [ thm : abequiv ] , @xmath1219 $ ] is equivalent to @xmath1220 , and under this equivalence , @xmath1104 corresponds to @xmath1221 ( proposition [ prop : srformula ] ) . we thus see that the image of @xmath1220 in @xmath98 under @xmath1221 generates @xmath1214 . now , by corollary [ cor : generic - inj ] , every object of @xmath1220 admits a finite length forward resolution by objects of the form @xmath1222 with @xmath1216 . it follows that the objects @xmath1223 generate @xmath1214 . by corollary [ cor : generic - inj ] , @xmath1224 , so the proposition follows . let us spell out a little more precisely what proposition [ prop : dr ] means . given @xmath1225 , proposition [ prop : dr ] implies that there are objects @xmath1226 , objects @xmath1227 , integers @xmath1228 , and exact triangles @xmath1229 \to.\ ] ] this gives a way of inductively building arbitrary objects of @xmath1214 from objects of the form @xmath1230 . one often has tools to study these more simple objects , which is why proposition [ prop : dr ] is useful . [ rmk : proof - thm : struc ] if one takes @xmath1231 in proposition [ prop : dr ] , then there are no higher pushforwards , and @xmath1232 is the module @xmath86 appearing in theorem [ thm : struc ] . since the objects @xmath1233 , with @xmath1234 and @xmath253 a finitely generated @xmath146-module , generate @xmath1235 ( see corollary [ cor : grass - gen ] ) , we find that the objects @xmath1236 generate @xmath1214 . this proves ( a generalization of ) theorem [ thm : struc ] . [ cor : dba - gens ] the category @xmath98 is generated by the objects from proposition [ prop : dr ] , allowing @xmath16 to vary . this is immediate since , by theorem [ thm : satfin ] , @xmath98 is generated by the @xmath1214 . using proposition [ prop : dr ] , we now formulate an axiomatic approach to proving results about @xmath1237-modules . by a * property of @xmath1237-modules * we mean a rule that assigns to every triple @xmath1238 consisting of a scheme @xmath145 , a locally free coherent sheaf @xmath149 on @xmath145 , and an object @xmath50 of @xmath1239 a boolean value @xmath1240 . [ prop : axiomatic ] let @xmath1241 be a property of @xmath1237-modules . suppose the following : 1 . if @xmath1242 is true for two terms in an exact triangle then it is true for the third . 2 . if @xmath1240 is true then so is @xmath1243)$ ] for all @xmath1244 . if @xmath1245 is a surjection then @xmath1246 for @xmath1247 . 4 . suppose @xmath196 is a proper map of schemes and @xmath149 is a locally free coherent sheaf on @xmath145 . then @xmath1248 for @xmath1249 . 5 . @xmath1242 is true for modules of the form @xmath1250 with @xmath1038 . then @xmath1240 is true for all @xmath145 , @xmath149 , and @xmath50 . let @xmath145 and @xmath149 be given , and let us prove @xmath1242 holds on all of @xmath1239 . we note that by ( a ) and ( b ) , the full subcategory on objects for which @xmath1242 holds is a triangulated subcategory of @xmath1239 . let @xmath1135 , let @xmath42 be the tautological bundle on @xmath151 , and let @xmath196 be the structure map . by ( e ) , @xmath1251 holds for all modules of the form @xmath1252 with @xmath1216 . thus by ( c ) , @xmath1253 holds for all such modules as well . by ( d ) , we see that @xmath1242 holds for all modules of the form @xmath1254 with @xmath1216 . by proposition [ prop : dr ] , it follows that @xmath1242 holds for all objects in @xmath1255 , for all @xmath16 . finally , @xmath1239 is generated by the categories @xmath1255 as @xmath16 varies ( corollary [ cor : dba - gens ] ) , so @xmath1242 holds on all of @xmath1239 , completing the proof . it is clear from the proof that the conditions in proposition [ prop : axiomatic ] are stronger than what is actually needed : for instance , in ( d ) it is enough to consider @xmath151 that are relative grassmannians . for our applications , the above proposition is enough though . the category @xmath11 is naturally a @xmath173-module , and so @xmath170 is a @xmath48-module . we now describe its structure as a @xmath48-module . let @xmath1256 be the structure map , and let @xmath356 be the tautological quotient bundle on @xmath150 . define @xmath1257 ) = [ \rr \pi_r{}_*(v \otimes \ba(\cq_r))].\ ] ] the main result is then : [ thm : groth ] the maps @xmath1258 induce an isomorphism of @xmath48-modules @xmath1259 we have @xmath1260 by proposition [ prop : kdecomp ] . we now have identifications @xmath1261)=\rk(\mod_{\ba(\cq_r)}^{{{\rm gen}}})=\lambda \otimes \rk(\gr_r(\ce)).\ ] ] the first follows from proposition [ cor : kisom ] ; the second from the fact that everything in @xmath39 has a filtration with graded pieces in @xmath41 $ ] ; the third from the equivalence of @xmath41 $ ] with @xmath1262 ( theorem [ thm : abequiv ] ) ; and the fourth from proposition [ cor : ktheory - generic ] . we thus have an isomorphism @xmath1263 . it only remains to verify that this isomorphism is given by the claimed formula . both isomorphisms are @xmath48-linear , so it suffices to check that they agree on @xmath777 \in \rk(\gr_r(\ce))$ ] . we now trace @xmath777 $ ] backwards through the identifications in , using notation as in [ ss : modar ] . it gives @xmath705 $ ] in @xmath1264 , with @xmath1265 ; which gives @xmath1266 $ ] in @xmath1267 ; which gives @xmath1268 $ ] in @xmath1269 . from proposition [ prop : srformula ] we have an isomorphism @xmath1270 by corollary [ cor : dersat ] we have @xmath1271 . we thus see that @xmath777 $ ] gives @xmath1272 $ ] in @xmath170 , which is exactly @xmath1273)$ ] . @xmath170 is isomorphic to a direct sum of @xmath78 copies of @xmath1274 . in particular , if @xmath771 , then @xmath170 is free of rank @xmath78 as a @xmath48-module . by corollary [ cor : grass - ktheory ] , @xmath1275 , and @xmath1276 . suppose @xmath771 . since @xmath1277 , there is a natural map @xmath1278 given by taking the external tensor product of modules . one can take the tensor product on the left as @xmath48-modules , and so both sides are free @xmath48-modules of rank @xmath78 . however , this map is not an isomorphism . we explain for @xmath1279 . write @xmath1280 and @xmath1281 for two copies of @xmath144 . the @xmath48-module @xmath1282 is free of rank 2 , and the classes of @xmath144 and @xmath1283 form a basis . thus the image of the above map is the @xmath48-module spanned by the external tensor product of these modules . these products are @xmath144 , @xmath1284 , @xmath1285 , and @xmath1286 . however , the classes of @xmath1284 and @xmath1285 coincide : indeed , under the description of @xmath1287 in terms of grassmannians the class of @xmath1284 corresponds to the class of the point @xmath1288 ( or rather , its structure sheaf ) , while the class of @xmath1285 corresponds to the class of the point @xmath1289 . since all points in @xmath1290 represent the same class in @xmath1291-theory , we see that @xmath1292=[\ba(l_2)]$ ] in the @xmath1291-groups of @xmath1293 . thus the image of the external tensor product map on @xmath1291-theory has rank at most 3 over @xmath48 . we let @xmath145 and @xmath149 and @xmath169 be as in previous sections . let @xmath1294 . we note that @xmath43 is naturally a coalgebra ; this structure will be relevant . let @xmath1295 be the koszul complex resolving @xmath1296 : this has @xmath1297 for @xmath1298 and @xmath1299 for @xmath567 , and has the usual koszul differential . given a complex @xmath1300 of @xmath5-modules , the tensor product complex @xmath1301 is naturally a dg - comodule over @xmath1302)$ ] . we now modify this construction to get a complex of @xmath43-comodules . for a complex @xmath50 of objects in @xmath194 , define the * right shear * by @xmath1303 here @xmath1304 denotes the degree @xmath59 piece of @xmath1305 . the right shear shifts the degree @xmath59 piece of the complex @xmath59 units to the right . we also define the * left shear * by @xmath1306 this is inverse to the right shear . for a complex @xmath50 of @xmath5-modules we now define @xmath1307 since the right shear of @xmath1302)$ ] is @xmath43 , this is a complex of @xmath43-comodules . to be completely explicit , we have @xmath1308 the @xmath43-comodule structure on @xmath1309 is the obvious one ( it is cofree ) . the differentials are given by @xmath1310 on an element in @xmath1311 where @xmath1312 is the differential on @xmath43 and @xmath1313 is the differential on @xmath50 . we note that if @xmath50 is bounded below then so is @xmath1314 . furthermore , @xmath1315 induces a functor @xmath1316 . we now define a functor in the reverse direction . the degree 0 copy of @xmath146 in @xmath43 is a subcomodule . let @xmath1317 be the koszul complex resolving it : this has @xmath1318 for @xmath323 and @xmath1319 for @xmath1320 . suppose that @xmath243 is a complex of @xmath43-comodules . we put @xmath1321 explicitly , @xmath1322 the @xmath5-module structure on @xmath1323 is the obvious one ( it is free ) . the differentials are given by @xmath1324 on an element in @xmath1325 where @xmath1326 is the differential on @xmath5 and @xmath1327 is the differential on @xmath243 . if @xmath243 is bounded above then so is @xmath1328 . furthermore , @xmath1329 induces a functor @xmath1330 . given how the differentials were defined , we can identify @xmath1331 with the complex @xmath1332 with the cohomological grading @xmath1333 and differential @xmath1334 since @xmath1335 is the koszul complex , there is a canonical quasi - isomorphism @xmath1336 , and hence we have a canonical quasi - isomorphism @xmath1337 for any complex of @xmath5-modules . similarly , there is a canonical map @xmath1338 of complexes of @xmath43-comodules that is always a quasi - isomorphism . thus @xmath1315 and @xmath1329 induce mutually quasi - inverse equivalences of @xmath1339 and @xmath1340 . for the benefit of later use , we record the following simple result here . [ prop : kozhom ] let @xmath50 be a complex of @xmath5-modules . then @xmath1341 we now want to modify the constructions of the previous section to replace the comodules that appear with modules . we do this by applying a duality to @xmath194 that interchanges @xmath43-modules and @xmath43-comodules . in this section , we assume that @xmath145 has a dualizing complex @xmath1342 ( see ( * ? ? ? * chapter v ) for definitions and basic properties ) . then @xmath1342 induces a duality @xmath1343 of @xmath1344 via @xmath1345 . we say that @xmath1346 is * degree - wise finitely generated * ( dfg ) if @xmath1347 is a coherent sheaf on @xmath145 for all @xmath80 . similarly , we say that a complex @xmath50 in @xmath194 is dfg if each @xmath1348 is . we let @xmath1349 be the full subcategory of @xmath1350 on the dfg objects . we extend @xmath1343 to @xmath1349 by simply applying @xmath1343 to the multiplicity spaces . that is , for @xmath1351 we write @xmath1352 , where @xmath1353 , and then define @xmath1354 . [ lem : duality - schur ] @xmath1355 is canonically isomorphic to @xmath1356 . the @xmath1357 multiplicity space of @xmath1358 is @xmath1359 , so it suffices to construct a canonical isomorphism @xmath1360 . the former can be identified with @xmath1361 . note that there is a duality on @xmath173 given ( in the polynomial functor perspective ) by @xmath1362 for finite - dimensional @xmath175 . when @xmath1363 is the identity , we canonically have @xmath1364 , and hence , we get a canonical identification for any schur functor and their tensor products . suppose now that @xmath50 is a complex of @xmath43-comodules . we thus have a comultiplication map @xmath1365 . applying @xmath1343 to this map yields a map @xmath1366 . recall that @xmath1367 is just @xmath1368 . then @xmath1369 pulls out of @xmath1343 by lemma [ lem : duality - schur ] . since @xmath149 is a locally free coherent sheaf , we have @xmath1370 . we thus have a map @xmath1371 . in fact , one can show that @xmath1372 naturally has the structure of a @xmath1373-module , where @xmath1374 . this construction gives an equivalence between @xmath1375 and @xmath1376 . it interchanges the bounded below and bounded above subcategories , and preserves the bounded subcategory . we now define @xmath1377 and @xmath1378 ( we note that the functors @xmath1315 and @xmath1329 preserve the dfg condition . ) it is clear that @xmath1379 and @xmath1380 are mutually quasi - inverse equivalences . we note that both @xmath1379 and @xmath1380 take the bounded below subcategory to the bounded above subcategory . we also note that @xmath1379 and @xmath1380 depend on the choice of dualizing complex @xmath1342 , though this dependence is absent from the notation . we now want to modify the constructions of the previous section to replace modules over the exterior algebra with modules over the symmetric algebra . we do this by applying the transpose functor to @xmath194 . recall that this is a covariant functor @xmath1381 that is @xmath147-linear and satisfies @xmath1382 , where @xmath1383 is the transpose of the partition @xmath80 . furthermore , while it is a tensor functor , it is not a _ symmetric _ tensor functor : it interchanges the usual symmetry and the graded symmetry of the tensor product on @xmath194 ( see @xcite ) . let @xmath1384 . then we have an equivalence of categories @xmath1385 via @xmath1386 . we now define @xmath1387 and @xmath1388 once again , it is clear that @xmath1389 and @xmath1390 are mutually quasi - inverse equivalences . [ prop : kosz - sym ] we have @xmath1391 and @xmath1392 . we have @xmath1393 the second equality uses that @xmath1394 commutes with @xmath1343 and is a tensor functor . the third equality uses that @xmath1395 is a complex of locally free sheaves . finally , @xmath1396 , so we see that @xmath1397 . the other identity is similar . we now define the * fourier transform * @xmath1398 to be the functor @xmath1399 . it is an equivalence of categories . we gather some of its basic properties here . [ prop : ftprop ] we have the following : 1 . @xmath1400 and @xmath1401 are canonically quasi - inverse to each other . 2 . @xmath1400 carries @xmath1402 into @xmath1403 . @xmath1404 \otimes \sf_{\ce}(-)$ ] , where @xmath1405 . 4 . if @xmath1406 is a locally free coherent sheaf on @xmath145 then @xmath1407 . if @xmath1245 is a surjection of vector bundles and @xmath1408 then @xmath1409 is canonically isomorphic to @xmath1410 . 6 . if @xmath1406 is a coherent sheaf on @xmath145 then @xmath1411 , regarded as a trivial @xmath1412-module . if @xmath1413 is defined with respect to a different dualizing complex then there is an integer @xmath1 and a line bundle @xmath1169 on @xmath145 such that @xmath1414 \otimes \cl$ ] . \(a ) this follows from proposition [ prop : kosz - sym ] . \(b ) we have already noted this for @xmath1399 . \(c ) we have @xmath1415 \otimes \sk(-)$ ] . thus @xmath1416 \otimes \sk^*(-)$ ] . finally , taking transposes yields the stated formula . \(d ) this is clear . \(e ) we have @xmath1417 \(f ) @xmath1418 is quasi - isomorphic to @xmath1406 concentrated in degree @xmath600 , so @xmath1419 , which gives @xmath1420 . \(g ) follows from @xcite . we now examine how the fourier transform interacts with pushforwards . we first set some notation . let @xmath196 be a proper map of schemes , let @xmath316 be a vector bundle @xmath145 , and let @xmath317 be its pullback to @xmath151 . put @xmath319 and @xmath318 . we let @xmath1421 , @xmath1422 , and @xmath1423 be defined as in previous sections . let @xmath1424 be a dualizing sheaf on @xmath145 and let @xmath1425 be the corresponding one on @xmath151 . write @xmath1426 and @xmath1427 for the duality functors they give . [ koszul - pushfwd ] we have canonical functorial isomorphisms of functors on @xmath1428 : 1 . 2 . @xmath1430 . @xmath1431 . \(a ) let @xmath1432 and pick a quasi - isomorphism @xmath307 with @xmath305 a bounded - below complex of injective @xmath320-modules . note that each multiplicity space of an injective @xmath320-module is injective as an @xmath1162-module ( proposition [ prop : injox ] ) . thus @xmath1433 . recall that @xmath1434 this is a bounded - below complex . as @xmath1435 , the projection formula implies that the sheaf @xmath1436 is @xmath199-acyclic . we can thus compute @xmath336 of the above complex by simply applying @xmath199 . doing this , and using the projection formula again , gives @xmath1437 however , this exactly coincides with @xmath1438 . \(b ) precompose the identity in part ( a ) with @xmath1427 and use the duality theorem @xmath1439 . \(c ) simply apply transpose to the identity in part ( b ) . we now carry out a fundamental computation . let @xmath1135 , let @xmath42 and @xmath152 be the usual bundles on @xmath151 , and let @xmath911 be the structure map . let @xmath1440 , and let @xmath1441 , @xmath1442 , and @xmath1443 be defined analogously . we note that @xmath151 and @xmath523 are canonically isomorphic . [ prop : ftgrass ] let @xmath1406 be a finitely generated @xmath1162-module , and let @xmath1444 be the corresponding @xmath1445-module . let @xmath80 be a partition of size @xmath119 . then there is a canonical isomorphism @xmath1446 \otimes \bd(\cm ' ) \otimes \ba(\cq')).\ ] ] we compute the left side . we first note that we can switch @xmath1400 and @xmath1180 by proposition [ koszul - pushfwd ] . next , @xmath174 pulls out of @xmath1400 and becomes @xmath1447 $ ] by proposition [ prop : ftprop](c ) . we have @xmath1448 , a trivial @xmath1449-module , by proposition [ prop : ftprop](f ) , and so @xmath1450 by proposition [ prop : ftprop](e ) . we have thus shown @xmath1451 \otimes \bd(\cm ) \otimes \ba(\cr^*)).\ ] ] we now move everything to @xmath523 via the isomorphism between @xmath151 and @xmath523 . this changes @xmath1452 to @xmath1443 and @xmath1406 to @xmath1444 and @xmath1453 to @xmath1441 . this yields the stated result . the following is the fundamental finiteness result about the fourier transform : [ thm : fourierfinite ] the fourier transform @xmath1400 carries @xmath1239 into @xmath1454 . let @xmath1240 be the truth - value of the statement `` @xmath1409 is bounded with finitely generated cohomology . '' ( note that while @xmath1400 depends on the choice of a dualizing sheaf on @xmath145 , the value of @xmath1242 does not by proposition [ prop : ftprop](g ) . ) then @xmath1241 is a property of @xmath1237-modules . we show that @xmath1241 holds for all modules by verifying the five conditions in proposition [ prop : axiomatic ] . the first two conditions are clear . we now consider the other three . \(c ) let @xmath1245 be a surjection of locally free coherent sheaves on @xmath145 and let @xmath1247 . then @xmath1409 is isomorphic to @xmath1410 by proposition [ prop : ftprop](e ) . thus if @xmath1455 holds then so does @xmath1240 . \(d ) suppose @xmath196 is a proper morphism of schemes , @xmath149 is a locally free coherent sheaf on @xmath145 , and @xmath1249 . proposition [ koszul - pushfwd](c ) gives an isomorphism @xmath1456 . ( we assume here that @xmath1457 is chosen to be @xmath1458 . ) so @xmath1459 by corollary [ cor : finpushfwd ] . \(e ) this follows from proposition [ prop : ftprop](c , f ) . a finitely generated @xmath5-module has finite regularity . the following is a sort of duality theorem involving the fourier transform and the rank stratification . [ thm : ftduality ] set @xmath1460 . we have natural identifications of functors @xmath1461 : 1 . @xmath1462 . 2 . @xmath1463 . 3 . @xmath1464 . it follows immediately from propositions [ prop : dr ] and [ prop : ftgrass ] that @xmath1400 carries @xmath1214 into @xmath1465 . now , let @xmath1200 . we then have an exact triangle @xmath1466 applying @xmath1467 yields an exact triangle @xmath1468 since @xmath1202 belongs to @xmath1469 , it follows that @xmath1470 belongs to @xmath1471 . similarly , @xmath1472 belongs to @xmath1473 . we also have an exact triangle @xmath1474 since @xmath1475 admits a semi - orthogonal decomposition @xmath1476 , it follows that there are canonical isomorphisms @xmath1477 and @xmath1478 . this proves ( a ) and ( b ) . as for ( c ) , we have @xmath1479 in the first and fourth lines we used the definition of @xmath76 , in the second line we used part ( a ) , and in the third line we used part ( b ) . let @xmath1480 be the map taking @xmath1481 to @xmath1482 . this is a ring homomorphism . since @xmath1400 is an equivalence @xmath1461 , it induces an isomorphism @xmath1483 . this map is @xmath164-linear , meaning @xmath1484 for @xmath1485 and @xmath1486 , by proposition [ prop : ftprop](c ) . the following result gives a complete description of @xmath1487 . we have a commutative diagram @xmath1488 \ar[d ] & \rk(\ba(\ce ) ) \ar[d]^{\varphi } \\ \bigoplus_{r=0}^d \lambda \otimes \rk(\gr_r(\ce^ * ) ) \ar[r ] & \rk(\ba(\ce^ * ) ) } \ ] ] where the horizontal maps are the ones from theorem [ thm : groth ] , and the left vertical map is @xmath1489 on the @xmath48 factors , and takes @xmath1490 \in \rk(\gr_r(\ce))$ ] to @xmath1491 \in \rk(\gr_{d - r}(\ce^*))$ ] , where @xmath1444 corresponds to @xmath1406 under the isomorphism @xmath1492 . this follows immediately from the description of the maps in theorem [ thm : groth ] and the calculation in proposition [ prop : ftgrass ] . let @xmath145 be a noetherian separated scheme of finite krull dimension over a field of characteristic @xmath600 and let @xmath149 be a vector bundle of rank @xmath1 . let @xmath1135 , let @xmath911 be the structure map , and let @xmath42 and @xmath152 be the tautological bundles . let @xmath1493 denote the symmetric group on @xmath1 letters , more precisely the group of bijections of @xmath1494 = \{1,\dots , d\}$ ] . given @xmath1495 , define its length to be @xmath1496 also define @xmath1497 given @xmath1498 , define @xmath1499 and @xmath1500 . note that given any @xmath1498 , either there exists @xmath1501 such that @xmath1502 , or there exists a unique @xmath1503 such that @xmath1504 is weakly decreasing . let @xmath1505 and @xmath1506 be weakly decreasing and set @xmath1507 . for the following , see ( * ? ? ? * corollary 4.1.9 ) . [ thm : bwb ] exactly one of the following two cases happens : 1 . if there exists @xmath1501 such that @xmath1502 , then @xmath1508 for all @xmath853 . otherwise , there exists unique @xmath1503 such that @xmath1509 is weakly decreasing , and @xmath1510 note that @xmath1511 where @xmath1512 , and similarly for any vector bundle . [ rmk : bott - algorithm ] the length @xmath1513 of a permutation is also equal to the minimal number of adjacent transpositions @xmath1514 needed to write @xmath1503 , i.e. , the minimal @xmath311 such that we can write @xmath1515 . the operation @xmath1516 has the effect of replacing @xmath1517 with @xmath1518 . so in ( b ) above , the process of getting @xmath1519 from @xmath1520 can be thought of in terms of a bubble sorting procedure : if @xmath1521 , apply @xmath1522 to get a new sequence with @xmath1523 ; the number of times needed to do this is @xmath1513 . we will refer to this procedure as `` bott s algorithm '' , and keeping the notation of ( b ) , we write @xmath1524 where @xmath1525 . [ cor : dual - basis ] suppose @xmath1526 and @xmath253 is a coherent sheaf on @xmath145 . then @xmath1527 using the projection formula , we may assume that @xmath1528 . in that case , this is ( * ? ? ? * lemma 3.2 ) when @xmath145 is a point , but the combinatorics is exactly the same in the general setting . here is a sketch of how this can be proven . pick @xmath1495 and consider @xmath1529 . then @xmath1530 and @xmath1531 if and only if @xmath1532 and @xmath1533 ; furthermore , @xmath1534 where @xmath1234 , so we write @xmath1535 ; also @xmath1536 . then what remains to show is : if @xmath1537 , then @xmath1538 has a repeated element , and this follows since we have @xmath1539 for some @xmath1540 and @xmath1541 . this is adapted from @xcite . let @xmath1542 denote the projection maps for @xmath1543 . given sheaves @xmath1544 on @xmath151 , define @xmath1545 . we have the following maps : @xmath1546 the composition corresponds to a section of @xmath1547 , whose zero locus is the diagonal @xmath1548 of @xmath1549 , and has codimension equal to the rank of @xmath1547 . hence the following koszul complex is exact : @xmath1550 using the cauchy identity , we can write @xmath1551 given @xmath1552 , we have a quasi - isomorphism @xmath1553 this is a formal verification : let @xmath1554 be the inclusion . then @xmath1555 in the second equality , we used the projection formula ; in the final equality , we used that @xmath1556 . the right side of can be computed using the koszul complex , which gives a spectral sequence @xmath1557 which converges to @xmath50 concentrated in degree @xmath1558 . so we conclude the following : [ prop : grass - gen ] @xmath1559 is generated by objects of the form @xmath1560 where @xmath1561 and @xmath1234 . [ cor : grass - gen ] @xmath1235 is generated by objects of the form @xmath1233 where @xmath1562 and @xmath1234 . let @xmath1563 be the complement of @xmath80 in the @xmath81 rectangle , thought of as a partition . then @xmath1564 is isomorphic to @xmath1565 , and tensoring with @xmath1566 is an automorphism of the derived category . for each @xmath1234 , define @xmath1567 by @xmath1568 . define @xmath1569 as the sum @xmath1570 . [ cor : grass - ktheory ] @xmath1571 is an isomorphism , so @xmath1572 . in particular , if @xmath145 is a point , then @xmath1573 . for @xmath1234 , define @xmath1574 by @xmath1575 and define @xmath1576 using @xmath1577 as the components . it follows from corollary [ cor : dual - basis ] that @xmath1578 is a diagonal matrix whose diagonals are @xmath1579 , so @xmath1571 is injective . it follows from proposition [ prop : grass - gen ] that @xmath1571 is also surjective , so we are done . in this appendix , we outline an alternative , direct approach to proving finiteness properties of resolutions of finitely generated @xmath1580-modules in the case that @xmath12 is a @xmath144-vector space . let @xmath1581 be a graded coalgebra with finite - dimensional components and let @xmath243 be a graded @xmath1581-comodule with finite - dimensional components . we say that @xmath243 is * finitely cogenerated * if there is a finite length quotient @xmath1094 such that the composition @xmath1582 is injective . this is equivalent to saying that the graded dual of @xmath243 is a finitely generated module over the graded dual of @xmath1581 . 1 . the module @xmath1585 has finite regularity . if @xmath1586 , then the regularity is @xmath600 , and otherwise , the regularity is at most @xmath1587 . @xmath1588 is finitely generated over @xmath1589 . let @xmath145 be the grassmannian of rank @xmath119 quotients of the space @xmath1590 . then we have the tautological exact sequence @xmath1591 where @xmath42 has rank @xmath119 . by theorem [ thm : bwb ] , for any partition @xmath89 , we have @xmath1592 , and all higher cohomology vanishes . in particular , @xmath1593 as an @xmath5-module . let @xmath1594 . using ( * ? ? ? * theorem 5.1.2 ) , the minimal free resolution @xmath1595 of @xmath1585 is given by @xmath1596 in particular , the regularity is the supremum over @xmath853 such that @xmath1597 is nonzero . by ( * corollary 2.3.3 ) , we have to calculate the cohomology of @xmath1599 , consider the sequence @xmath1600 and define @xmath1601 . we have an action of @xmath1602 coming from @xmath1603 . by borel weil bott ( theorem [ thm : bwb ] ) , if there is a non - identity @xmath1604 so that @xmath1605 , then all cohomology vanishes , and otherwise , there is a unique such @xmath1606 so that @xmath1607 is a partition , and the cohomology is @xmath1608 concentrated in degree @xmath1609 . if @xmath1586 , then by , any @xmath1610 that comes from a summand of @xmath1611 is a partition , so the resolution @xmath1595 is linear and we are done . otherwise , let @xmath1612 . we will show that the cohomology of @xmath1599 vanishes above degree @xmath1613 . assume that @xmath1614 has no repeated entries , otherwise the cohomology vanishes . then @xmath1615 and the permutation @xmath1606 that sorts @xmath1614 is in @xmath1616 . since @xmath1606 satisfies @xmath1617 and @xmath1618 , its length is at most @xmath1613 . this proves ( a ) . for ( b ) , we will instead prove that @xmath1619 is a finitely cogenerated comodule over @xmath1620 . from ( a ) , we know that there are finitely many linear strands . we will focus on the @xmath853th linear strand . first , consider the comultiplication map @xmath1621 we can rewrite this as @xmath1622 recall that over a local ring @xmath740 with residue field @xmath1625 , and an @xmath740-module @xmath50 , we construct the comodule structure on @xmath1626 as follows ( this is a modification of assmus description @xcite of the coalgebra structure on @xmath1627 ) . let @xmath1628 be an @xmath740-free resolution of @xmath50 and let @xmath1629 be an @xmath740-free resolution of @xmath1625 . tensoring both @xmath1595 and @xmath1630 with the residue field , we get a map @xmath1631 and taking homology , and using knneth s formula , this becomes @xmath1632 let @xmath149 be the total space of the trivial bundle @xmath1633 over @xmath145 . we have a twisted koszul complex @xmath1634 on @xmath149 . let @xmath1635 be the koszul resolution of @xmath146 over @xmath1636 ( here @xmath145 is the zero section in @xmath149 ) . then we have the relative version of @xmath1637 now we take the hypercohomology of both sides . since @xmath1638 is a complex of free @xmath1639-modules , this is a map of the form with @xmath1640 . we can calculate hypercohomology of a complex of sheaves in two different ways : either first calculate cohomology ( in the complex sense ) and then calculate sheaf cohomology , or else calculate sheaf cohomology first and then cohomology ( in the complex sense ) . the two different approaches form the @xmath1641 page of a spectral sequence which converges to the hypercohomology . if we first calculate cohomology in the sense of complexes , then we get a relative tor comultiplication map @xmath1642 the maps are graded pieces of this map . our goal is to understand the map we get by taking sheaf cohomology of both sides . note that taking sheaf cohomology commutes with the tensor product on the right hand side since @xmath1643 consists of free @xmath1639-modules . taking sheaf cohomology gives us a map of the form , so the spectral sequence degenerates on the @xmath1641 page . if we instead calculate sheaf cohomology first , then we get . a few remarks are in order : @xmath1638 is free over @xmath1639 , so tensoring with it commutes with taking cohomology ; the sheaves @xmath1644 are pullbacks of sheaves @xmath1645 from @xmath146 , so the sheaf cohomology of @xmath1646 is the same as the sheaf cohomology of @xmath1645 ( similarly for @xmath1647 ) . taking homology gives us a map of the form , so again this spectral sequence degenerates on the @xmath1641 page . recall that above we have seen that the shifted weyl group action that we must perform to calculate the cohomology of @xmath1599 only depends on the first @xmath1648 parts of @xmath89 . since we have @xmath1649 , there are only finitely many possibilities for this subpartition . write @xmath1650 where @xmath1651 is the first @xmath59 parts of @xmath89 , and @xmath1019 is the rest ( the symbol @xmath1652 denotes concatenation ) . so from now on , we will focus only on @xmath1599 where @xmath1651 is a fixed partition . let @xmath1653 . then in the composition @xmath1654 we see the subsheaves @xmath1655 and this restriction is an inclusion . in fact , since we are in characteristic 0 , the first map is a direct summand , so applying @xmath1597 , we still get an inclusion . the cokernel of the second map is @xmath1656 and we can see from theorem [ thm : bwb ] that it will not contain the sections of the first module ( namely because any partition @xmath1657 that appears in its cohomology will have @xmath1658 ) . so applying @xmath1597 to the composition also gives an inclusion . so for a cogenerating set of @xmath1659 , we take @xmath1660 where @xmath1661 ( this is the largest possible size of @xmath1651 as above ) . in particular , @xmath1662 is cogenerated in ( homological ) degrees @xmath1663 , which finishes the proof of ( b ) . let @xmath50 be an @xmath5-module . let @xmath119 be bigger than the number of rows in the partitions that appear in the presentation of @xmath50 . let @xmath1595 be a finite free resolution of @xmath127 over @xmath126 . considered as modules over @xmath5 , the @xmath1667 are direct sums of modules of the form @xmath1585 . we can construct an @xmath5-free resolution of @xmath50 using a mapping cone on @xmath1595 and @xmath5-free resolutions on these modules . since @xmath1595 is finite and each @xmath1667 has finite regularity over @xmath5 by proposition [ prop : truncationregularity ] , we conclude that @xmath50 has finite regularity .

twisted commutative algebras ( tca s ) have played an important role in the nascent field of representation stability . let @xmath0 be the complex tca freely generated by @xmath1 indeterminates of degree 1 . in a previous paper , we determined the structure of the category of @xmath2-modules ( which is equivalent to the category of @xmath3-modules ) . in this paper , we establish analogous results for the category of @xmath0-modules , for any @xmath1 . modules over @xmath0 are closely related to the structures used by the authors in previous works studying syzygies of segre and veronese embeddings , and we hope the results of this paper will eventually lead to improvements on those works . our results also have implications in asymptotic commutative algebra .

the domain - wall induced fermion mass in the free case was first calculated by shamir @xcite using green s function approach . vranas stated in his paper @xcite that he obtained the same result by diagonalizing the domain - wall dirac operator without showing the actual calculation . an explicit derivation of @xmath1 in the @xmath79 case was later provided by neuberger @xcite . here we show a complete derivation with the inclusion of @xmath44 . where @xmath95 and @xmath96 . mass matrices @xmath97 are defined as @xmath98 where @xmath99 and @xmath44 is the explicit fermion mass . our goal is to calculate the smallest eigenvalue of the bilinear hermitian domain - wall dirac operator @xmath100 since @xmath101 and @xmath102 have the same eigenvalue spectrum , it is sufficient to consider @xmath103 the second to @xmath104-th row of the secular equation @xmath105 , or , @xmath106 is solved by @xmath107\ ( s= 1 , \cdots n_s)\ , $ ] provided @xmath108 and @xmath109 satisfy @xmath110 the first and the last rows of the secular equation can be satisfied by a linear combination of exponential solutions , @xmath111 + a \exp[-\alpha(n_s - s)]\ $ ] , where @xmath112 is a constant to be determined : @xmath113 using eq . ( [ eq : lambda ] ) to eliminate @xmath108 from the above , we get @xmath114 eliminating a from eq . ( [ eq : a ] ) and rearranging terms , we have @xmath115 in the @xmath116 limit , @xmath117 . assuming @xmath118 and keeping terms linear in @xmath119 , we find @xmath120 finally , substituting @xmath118 into eq . ( [ eq : lambda ] ) , we get the eigenvalue , @xmath121 or @xmath122 , as quoted in ref . @xcite . r. narayanan and h. neuberger , phys . b * 302 * , 62 ( 1993 ) ; + r. narayanan and h. neuberger , nucl . b412 * , 574 ( 1994 ) ; + r. narayanan and h. neuberger , nucl . phys . * b443 * , 305 ( 1995 ) ; + h. neuberger , phys . b * 417 * , 141 ( 1998 ) . t. blum and a. soni , phys rev . d * 56 * , 174 ( 1997 ) ; + t. blum and a. soni , phys . 79 * , 3595 ( 1997 ) ; + t. blum , a. soni , and m. wingate , phys . d * 60 * , 114507 ( 1999 ) . p. chen et al . , presented at 29th intl . conf . on high - energy physics ( ichep 98 ) , vancouver , canada , 23 - 29 jul 1998 . in * vancouver 1998 , high energy physics , vol . 2 * 1802 - 1808 . hep - lat/9812011 ; + p. chen et al . , . suppl . * 73 * 204 , 207 , 405 ( 1999 )

in the domain - wall formulation of chiral fermion , the finite separation between domain - walls ( @xmath0 ) induces an effective quark mass ( @xmath1 ) which complicates the chiral limit . in this work , we study the size of the effective mass as the function of @xmath0 and the domain - wall height @xmath2 by calculating the smallest eigenvalue of the hermitian domain - wall dirac operator in the topologically - nontrivial background fields . we find that , just like in the free case , @xmath1 decreases exponentially in @xmath0 with a rate depending on @xmath2 . however , quantum fluctuations amplify the wall effects significantly . our numerical result is consistent with a previous study of the effective mass from the gell - mann - oakes - renner relation . chiral symmetry and its explicit and/or spontaneous breakings are important aspects of strong interaction phenomenology . chiral dynamics dominates the low - energy hadron structure and interactions . the chiral phase transition at finite temperature has been sought after experimentally for a long time . in addition , the weak interaction probes couple directly to the chiral currents , and the matrix elements of which sensitively depend on the chiral properties of hadron systems . on the theoretical frontier , however , massless fermions defy the naive nonperturbative treatments . indeed , for more than two decades , finding an appropriate fermion formulation has been one of the most difficult challenges in lattice quantum chromodynamics ( qcd ) . in the last few years , kaplan and shamir s domain - wall construction @xcite and narayanan and neuberger s overlap fermion formalism @xcite have emerged as promising approaches to simulating massless quarks . in this paper , we aim to study the effectiveness of the domain - wall approach which has already been used in a number of realistic numerical investigations @xcite . following previous studies , we adopt shamir s version of the domain wall fermion formulation @xcite , in which the five dimensional wilson fermion is first introduced . the finite fifth dimension with @xmath0 lattice sites extends from @xmath3 to @xmath4 . dirichlet boundary condition on the quark fields is applied to the four - dimensional slices at @xmath5 and @xmath6 . the fifth component of the gauge potential is identically zero and the other four components are the same at every @xmath7 slice . the above construction in the gauge - field - free case yields nearly - massless modes when the negative wilson mass @xmath2 ( the wall height ) is taken between 0 and 10 . these chiral fermions appear as the surface modes at @xmath3 and @xmath4 with opposite chirality . for @xmath8 , one flavor dirac fermion can be constructed as @xmath9 where @xmath10 are the chiral projection operators and @xmath11 labels four - dimensional lattice sites . for the finite wall separation ( @xmath12 ) , the chiral mode on the @xmath3 wall couples with the one with the opposite chirality on the @xmath4 wall in an exponentially small way . because of this coupling , a finite residual fermion mass is produced . the goal of this paper is to understand the size and dependence of this induced fermion mass on @xmath2 , @xmath0 when realistic background gauge fields are introduced . in the absence of the gauge potentials , the effective mass can be defined in terms of the pole of the free green s function . for a large @xmath0 , it has a simple analytical form @xcite , @xmath13 one can also obtain @xmath1 by either diagonalizing the hermitian domain - wall dirac operator @xmath14 or @xmath15 @xcite , where @xmath16 is the domain - wall dirac operator and @xmath17 is the reflection along the fifth dimension . on a lattice with the periodic boundary condition , the lowest four - momentum of a fermion is zero , and the lowest eigenvalue of @xmath18 is just @xmath1 . to study the effective mass in an external gauge field , we have constructed a code to diagonalize @xmath19 numerically . to test the code , we have computed @xmath1 on an @xmath20 lattice with @xmath21 and 16 , and the result is shown in fig . [ fig:1 ] together with the exact answer from eq . ( 1 ) . an ideal massless fermion is obtained at @xmath22 where @xmath1 vanishes identically . for @xmath23 , the induced quark mass decays exponentially in @xmath0 with a rate depending strongly on @xmath2 . the exponential decay slows down significantly as @xmath2 approaches 0 and 2 . for any given accuracy @xmath24 and any @xmath0 , there is a window surrounding @xmath22 in which @xmath25 . in the presence of a realistic gauge potential , the effective quark mass result from the finite wall separation may depend on how it is defined . different definitions shall yield results consistent up to a factor of order unity . one approach is to exploit the explicit quark mass dependence in chiral ward identities such as the gell - mann - oakes - renner ( gmor ) relation as done in ref . @xcite . here we explore the effective mass in an alternative way . in continuum field theory , the atiyah - singer theorem@xcite states that the dirac operator has a zero eigenvalue in the presence of an external background with topological charge @xmath26 . the explicit form of the solution was found by t hooft in 1976 @xcite . on the lattice , however , the notion of topological charge is ill defined : any gauge configuration can be continuously deformed into a null gauge field . moreover , the discretization of an instanton field can introduce finite lattice - spacing effects lifting any exact zero eigenvalue . therefore , a test of the atiyah - singer theorem on lattice is usually complicated with various lattice artifacts . there exists , however , a definition of lattice topology and fermion zero mode which largely avoids this complication . in the overlap formalism , the dirac operator is constructed from the overlap of two many - fermion ground states @xcite . according to their recipe , one starts from a four - dimensional wilson - dirac operator with a negative wilson mass @xmath2 and calculates its eigenvalues . for @xmath2 small and positive , the number of positive eigenvalues is equal to that of negative ones . when @xmath2 increases , a level might cross from positive to negative or vice versa . when this happens , the gauge field is regarded to have a net topological charge @xmath26 . then the overlap determinant is exactly zero by construction . this definition of lattice topology and zero mode do depend on , for instance , the wilson parameters @xmath27 and @xmath2 . however , the zero eigenvalue is exact , independent of the lattice spacing @xmath28 and volume @xmath29 . the domain - wall formulation can be regarded as an approximation to the overlap formalism @xcite . indeed , in the limit of @xmath30 , one recovers the overlap formalism apart from some unimportant discretization effect in the fifth dimension . for a fixed gauge configuration and wilson mass @xmath2 , if the overlap determinant is zero , @xmath19 has an exact zero mode in the limit of @xmath30 . in short , in a background gauge field , if the hermitian wilson - dirac operator has a level crossing , the gauge field is considered to have a nontrivial topology . edwards , heller , and narayanan have done extensive studies of the topological properties of lattice gauge configurations in this way @xcite . in a topologically - nontrivial background thus defined , the domain - wall dirac operator with a finite @xmath0 has small eigenvalues , nonvanishing only because of the finite wall separation . in the remainder of this paper , we are mainly interested in such domain - wall effects on the fermion zero - mode . we _ define _ the smallest eigenvalues of the hermitian domain - wall operator @xmath19 as the wall - induced effective fermion mass . as a first nontrivial example , we have shown in figs . [ fig:2a ] and [ fig:2b ] the results in a smooth instanton field configuration on an @xmath20 lattice . a similar study has been reported in refs . @xcite . in our case , the instanton configuration was generated according to the prescription in ref . @xcite : the size of the instanton @xmath31 is 10 and the cutoff parameter @xmath32 is 3 . the flow of the small eigenvalues of the hermitian wilson - dirac operator is shown in fig . [ fig:2a ] . a level crossing from positive to negative is seen at @xmath33 . four separate crossings in the opposite direction happen near @xmath34 . only three of the crossings are plotted . in the overlap formalism , the dirac operator has an exact zero eigenvalue in the region between the crossings . the lowest eigenvalue of the hermitian domain - wall dirac operator @xmath35 is shown in logarithmic scale in fig . [ fig:2b ] . the overall profile of the eigenvalue as a function of @xmath2 is similar to the free case in fig . [ fig:1 ] . for a fixed @xmath0 , the smallest eigenvalue occurs at around @xmath36 , shifted upward from @xmath37 . this shift reflects the renormalization of the wilson mass in the presence of the external gluon field . as @xmath2 deviates from @xmath38 , the domain - wall effects grow stronger . for a fixed @xmath2 , the effective fermion mass decreases exponentially as @xmath0 increases from 8 to 12 and 12 to 16 . [ hbt ] [ hbt ] [ hbt ] our result is quantitatively consistent with the chiral condensate @xmath39 calculation in ref . @xcite . for instance , at @xmath40 and @xmath41 , we have @xmath42 . we expect then @xmath39 grows like @xmath43 as the explicit quark mass parameter @xmath44 reduces to @xmath45 . for a fixed @xmath46 and @xmath40 , there is a window in @xmath47 $ ] in fig . [ fig:2b ] where the induced quark mass @xmath1 is smaller than @xmath44 . hence we expect @xmath39 is approximately independent of @xmath2 there . at @xmath48 , the effective quark mass is about @xmath49 with @xmath50 . we expect the chiral condensate to have little sensitivity on @xmath44 when @xmath51 . all of the above are in accordance with those reported in ref . @xcite . a smooth instanton on the lattice is far from a typical equilibrium gauge configuration entering in the feynman path integral . a more realistic study of the induced quark masses requires gauge configurations with quantum fluctuations fully included . in the following , we work on a set of monte carlo configurations generated on a four - dimensional @xmath20 lattice and with @xmath52 . the physical volume is somewhat small , but we suspect that the domain - wall effects have a weak dependence on it . we pick a lattice configuration in which the lowest eigenvalue of the hermitian wilson - dirac operator crosses from the positive to the negative at @xmath2 near 0.9 . in fig.[fig:3a ] , we have shown the flow of the lowest few eigenvalues as a function of @xmath2 . between @xmath53 and @xmath54 , four levels cross from the negative to positive region . although not shown explicitly , six level crossings occur at around @xmath55 . as pointed out in ref . @xcite , the hermitian wilson - dirac operator is symmetric with respect to @xmath55 and hence the flow diagram has the same symmetry . according to the overlap fermion formalism , the neuberger - dirac operator in the above gauge background has one zero eigenvalue when @xmath2 is in the interval @xmath56 $ ] , three zero eigenvalues in @xmath57 $ ] , three again in @xmath58 $ ] , and finally one in @xmath59 $ ] . in fig.[fig:3b ] , we have shown the smallest eigenvalue of the hermitian domain - wall dirac operator @xmath19 . different symbols correspond to three different @xmath60 ( pluses ) , @xmath61 ( circles ) , and @xmath62 ( squares ) . over a large region of @xmath2 , the effective quark mass decreases exponentially in @xmath0 , as is clear from the approximate equal spacings between pluses , circles , and squares . the fastest decay occurs at @xmath2 around @xmath63 , compared with 1.0 in the free case and 1.3 in the smooth instanton field . the wall effects become strong again near @xmath64 beyond which we find four small eigenvalues ( not shown ) . to our surprise , this transition point to the doubler region is at higher @xmath2 compared with the prediction from the spectral flow of the hermitian wilson - dirac operator . this may indicate some subtle differences between the eigenvalues of the transfer matrix in the domain - wall formalism and those of the hermitian wilson - dirac operator in the overlap formalism . the important point about fig.[fig:3b ] is that the magnitude of the effective mass is much enhanced relative to the case of the smooth instanton configuration . with a best choice of @xmath2 near 2.0 , the domain - wall dirac operator has the smallest eigenvalue @xmath65 at @xmath66 . for the same @xmath0 and with @xmath2 of 1.3 , the smooth instanton configuration yields an eigenvalue @xmath67 . this dramatic increase of the effective mass comes from the ultraviolet fluctuations . as we will discuss further below , the ultraviolet fluctuations at strong coupling can cause great trouble for the domain - wall formalism . in fig . [ fig:3c ] , we have shown the smallest eigenvalue of @xmath19 in another topological gauge configuration . the result is qualitatively similar to that in fig . [ fig:3b ] . in particular , for @xmath66 and @xmath68 , the induced quark mass is around @xmath69 . since the effective quark mass depends on particular gauge fields , it is useful to get an average over an ensemble of gauge configurations . for this purpose , we have generated a set of 150 configurations on a @xmath20 lattice at @xmath70 . by studying the spectral flow of the hermitian wilson - dirac operators , we found 12 topologically - nontrivial configurations . the second column in table i shows @xmath2 at which the first level crossing occurs . all the crossing points are entirely concentrated in the interval between 0.8 and 1 , a fact consistent with a similar study in a slightly larger lattice @xmath71 @xcite . we proceed to calculate the lowest eigenvalue of the hermitian domain - wall dirac operator with @xmath21 and @xmath62 at @xmath72 . the result is shown as the remaining columns in table 1 . we find the average effective quark mass at @xmath66 is @xmath73 . the dependence of the average effective mass in @xmath0 is consistent with the exponential within the error . our result can be compared with a previous study of the wall - induced quark mass using the gell - mann - oakes - renner relation on a @xmath74 lattice at the same value of @xmath75 @xcite . in the domain - wall formulation , the quenched chiral condensate is related to the pion susceptibility @xmath76 @xcite by @xmath77 where @xmath78 is a constant vanishing in the @xmath30 limit . the effective quark mass @xmath1 can be extracted from @xmath44-dependence of the condensate and @xmath76 . for @xmath66 and @xmath72 , an extrapolation to @xmath79 limit yields @xmath80 . note that the smallest @xmath44 at which the data is taken is @xmath81 . to understand the physical significance of @xmath82 , we recall the spectral version of the gmor relation in the overlap formulation @xcite : @xmath83 where @xmath84 are a conjugating pair of eigenvalues of @xmath85(@xmath86 = overlap dirac operator ) . the first term signifies the contribution from the topological zero modes ; for sufficiently large volume v and/or large @xmath44 , this topological charge term is negligible . the second term measures the chiral symmetry breaking effects on the conjugating pairs of eigenvalues . in the study quoted above @xcite , the topological charge term is insignificant even at the smallest @xmath87 , and the extracted @xmath88 undoubtly measures the explicit chiral - symmetry breaking effects in conjugating pairs of eigenvalues as induced by the domain walls . one can push the quenched calculation in ref . @xcite to the limit @xmath79 . in this case , both the chiral condensate @xmath39 and pion susceptibility are dominated by the zero modes . the quark mass obtained from the ratio of the two observables is just the smallest eigenvalue of the dirac operator in the topologically - nontrivial configurations . therefore in the quenched chiral limit , the effective quark mass determined from the gmor relation coincides with what we have considered in this paper . to be sure , the effective mass defined from the effects on the fermion zero modes is not the same as the one defined from the effects on the conjugating pairs of eigenvalues . nonetheless , both definitions shall be consistent within a factor of order unity . in this spirit , our average effective quark mass @xmath89 is indeed in accordance with @xmath90 from ref . @xcite . to be completely sure about the consistency of the two approaches , further numerical studies are needed . for instance , along the line of study in ref . @xcite one can attempt to subtract the zero - mode contribution to the chiral condensate and pion susceptibility , and then both quantities can be measured in the @xmath91 limit . on the other hand , the present study can be repeated at a larger physical volume ; a @xmath74 lattice will be more suitable for comparison . . average effective mass for the topologically nontrivial configurations [ cols="^,^,^,^,^",options="header " , ] we finally return to the central question of the domain - wall fermion formalism : how large an @xmath0 is needed for a practical simulation ? clearly , the results for the free field or artificially smooth gauge fields are of no help here . the answer depends on the size of quantum fluctuations . for large values of @xmath75 ( and perhaps small physical volumes as well ) , such as the case we have presented , one can work with @xmath92 or 16 and keep the induced quark mass under control . however , in going to smaller @xmath75 and larger physical volume , level crossing happens continuously in a region of @xmath2 above some critical value @xcite . from figs . [ fig:3b ] and [ fig:3c ] , we expect that if a level crossing occurs slightly before the @xmath2 where the domain - wall dirac operator is defined , the wall - induced quark mass will be huge . [ when the level crossing happens slightly above @xmath2 , the chiral symmetry breaking effects in the conjugation pairs of eigenvalues is expected to be strong . again this leads to a large effective quark mass . ] then the average effective quark mass can be strongly influenced by this type of accidental configurations depending on the frequency they occur . the level crossings at large @xmath2 reflect strong quantum fluctuations at the scale of the lattice spacing @xcite . therefore , it is not surprising that at @xmath93 , one needs to have very large @xmath0 ( 30 to 40 ) to keep @xmath1 small ; of course , this is the price that one has to pay to keep the physical volume large . to summarize , we have studied the induced quark mass resulted from the finite domain wall separation by diagonalizing the hermitian domain - wall dirac operator in topologically nontrivial configurations . we find the quantum fluctuation strongly enhances the domain - wall effects . however , the effective mass does show an exponential decay as a function of @xmath0 . our result on an @xmath20 lattice with @xmath52 is consistent with the effective fermion masses from the gmor relation , although a detailed analysis shows that the two definitions of the effective mass are not the same . finally , we comment on the size of @xmath0 needed in a practical monte carlo simulation . we thank n. christ , r. edwards , and j. negele for useful discussions related to the subject of this paper . the numerical calculation reported here was performed on the calico alpha linux cluster at the jefferson laboratory , virginia . this work is supported in part by funds provided by the u.s . department of energy ( d.o.e . ) under cooperative agreement doe - fg02 - 93er-40762 .

we thank s.f . pessoa for kindly providing us with the tight - binding parameters of bcc cu , d.m . edwards for helpful discussions and t.j.p . penna for helping with the figures . this work has been financially supported by cnpq and finep of brazil , and serc of u.k .

the exchange coupling between fe layers separated by bcc cu is calculated for fe / cu / fe ( 001 ) trilayers . it is shown that the coupling is basically regulated by three extrema of the bulk bcc cu fermi surface . the contributions from those extrema are all of the same order of magnitude , but that associated with the `` belly '' at the @xmath0-point dominates . the calculated temperature dependence of the coupling varies considerably with spacer layer thickness . individually , the amplitudes of these extrema contribution decrease with temperature , each according to a different rate . such an effect may cause an actual increase of coupling with temperature for some cu thicknesses . although the common crystal phase of bulk cu is fcc , it is possible to grow thin films of bcc cu on fe ( 001 ) . the bcc stacking proceeds for up to 12 or 20 atomic planes approximately , but for larger thicknesses significant lattice modifications occur , leading to a structural transformation.@xcite the exchange coupling between fe layers separated by bcc cu has been measured by groups at simon fraser university ( sfu ) and philips.@xcite both have found that the coupling oscillates with decreasing amplitude as a function of the cu thickness , but their results disagree in several important aspects.@xcite the philips group data show well - defined short - period oscillations@xcite whereas the sfu group originally observed a long - period oscillatory coupling.@xcite later , the sfu group found some indication of a short - wavelength oscillation in samples with smoother interfaces.@xcite the exchange coupling in multilayers can be strongly affected by sample interface quality.@xcite it is widely accepted that interface roughness tends to suppress short - wavelength oscillations and reduce the coupling amplitude . therefore , as pointed out in ref . , it is rather puzzling that the values obtained at philips are substantially smaller than those of the sfu group . motivated by these apparently conflicting experimental results , we have undertaken a theoretical analysis of the exchange coupling between fe layers across bcc cu in fe / cu / fe ( 001 ) trilayers . the coupling @xmath1 is calculated for several temperatures @xmath2 and spacer layer thicknesses @xmath3 , using an extension of the formulation developed in ref .. for suficiently large @xmath3 , we divide @xmath4 into oscillatory components coming from extrema which are related to the spacer fermi surface ( fs ) . these oscillatory contributions to the coupling are calculated separately . our results show that for perfectly smooth interfaces @xmath4 is dominated by short - period oscillations . we find that the temperature dependence of the coupling changes significantly with spacer layer thickness . the amplitude of each oscillatory component decreases with temperature , but they do so at different rates . we show that this may cause a surprising effect which is the increase of the coupling with temperature for some cu thicknesses . the interlayer exchange coupling , defined as the total energy difference per surface atom between the antiferromagnetic and ferromagnetic configurations of the trilayer , is given by @xcite : @xmath5 where @xmath6,\ ] ] @xmath7 and @xmath8 . in the equations above @xmath9 are the wave vectors parallel to the layers , @xmath10 is the usual fermi - dirac distribution function , and @xmath11 is a plane index . as in ref . we consider an imaginary cleavage plane across the spacer between planes @xmath11 and @xmath12 , separating the trilayer into two semi - infinite systems . @xmath13 and @xmath14 are matrices in orbital indices representing the surface one - electron green functions of the left and right cleaved systems , respectively . the trace is taken over orbital indices and @xmath15 denotes the spacer hopping matrix . this formula for the coupling is an extension of the result previously obtained in ref . and , for the one band model , reduces to the torque formula of edwards et al . @xcite in deriving it we have assumed that the electrons are noninteracting in the spacer and experience exchange - split one - electron potentials in the ferromagnetic layers . most of the experimental results are for the bilinear exchange coupling term @xmath16 which , for perfectly smooth fe / cu ( 001 ) interfaces , is virtually equal to @xmath17.@xcite we have calculated the required green functions within the tight - binding model with @xmath18 orbitals and hopping to second nearest neighbours . the tight - binding parameters for all bcc cu planes were determined from a first - principles lmto - tight - binding electronic structure calculation of paramagnetic bulk bcc cu . the parameters for ferromagnetic fe were obtained from paramagnetic bulk fe@xcite , by self - consistently adjusting the on - site energies , assuming charge neutrality . the effective intra - atomic electron - electron interactions were taken to be @xmath19 and @xmath20 . @xcite we neglect atomic potential differences due to the magnetic configuration change , thus making the approximation known as the `` force theorem '' . the @xmath9 sum in eq.(1 ) is performed numerically and the energy integral is evaluated in the complex plane by summing over matsubara frequencies at finite @xmath2 . the calculated results at @xmath2=300k for the bilinear exchange coupling @xmath21 as a function of cu thickness are presented in fig.[fig1 ] ( full circles ) . our results clearly show a short - period oscillatory exchange coupling , in excellent agreement with the philips group data as far as the period of oscillation is concerned . for sufficiently large spacer thickness it is possible to express the coupling as a sum of oscillatory components whose periods are determined by extrema that are related to the spacer fs . @xcite it is essential to use a non - perturbative treatment , as in the quantum well approach , to analyze the relative importance of these contributions . @xcite they depend upon the degree of confinement experienced by the carriers of both spin orientations in the corresponding extremum states , in the ferromagnetic and anti - ferromagnetic configurations of the magnetic layers.@xcite the widespread practice of considering only the periods predicted by rkky theory , and treating the amplitudes and phases of these contributions as adjustable parameters may be inadequate and rather misleading . the fitting usually involves several parameters and , in some cases , is not unique . besides , when the spacer fs has to be regarded as consisting of more than one sheet , periods not predicted by rkky theory may exist.@xcite moreover , at finite temperatures , the decrease of the oscillation amplitudes as @xmath3 increases is different for each extremum and may deviate strongly from the @xmath22 asymptotic regime@xcite usually assumed in that sort of fitting . to identify the periods of oscillations of @xmath4 it is useful to look at the spacer fs . in bcc cu only one energy band @xmath23 crosses the fermi energy @xmath24 . its calculated fs , shown in fig.[fig2](a ) , is basically a sphere with twelve `` necks '' developing at each face centre of the bulk bcc first brillouin zone . in the ( 001 ) direction of growth , three sets of @xmath25 associated with the fs extrema contribute to the coupling . the first set consists of a single wave vector @xmath26 ( @xmath0-point ) related to the fs `` belly '' . the other two are associated with the `` necks '' . set 2 consists of four vectors @xmath25 : @xmath27 and @xmath28 , and set 3 of the @xmath29-points located at @xmath30 . here all wave vectors are given in units of @xmath31 where @xmath32 is the lattice constant . due to the layered stucture of the system , it is useful to work with the layer adapted bulk brillouin zone instead of the usual bz . the former is defined as a prism whose base is the two - dimensional first bz and whose height is @xmath33 , where @xmath34 is the interplane distance perpendicular to the layers . the relevant cross sections of the spacer fs , together with the corresponding extremal wave vectors @xmath35 , are shown in fig.[fig2](b ) . we must distinguish sets 1 and 2 from set 3 because , for the latter , the fs can be regarded as consisting of more than one sheet . this is because more than one extrema occurs in the first prismatic bulk bz for each wave vector @xmath25 of set 3 . considering that the integrand @xmath36 in eq.[j ] is an oscillatory function of @xmath3 we can expand it in a fourier series . however , it is necessary to generalize the expansion to a multiple fourier series @xcite , when the equation @xmath37 has more than one pair of solutions @xmath38 . in this case , the general expansion of @xmath36 is @xmath39 for @xmath40 the exponential in eq.[j2 ] oscillates rapidly as a function of @xmath41 and @xmath42 . thus , the stationary phase method can be applied , and the dominant contribution to the coupling comes from @xmath43 and @xmath41 in the neighborhood of points at which the argument of the exponential is stationary . in this limit both the sum in @xmath41 and the energy integral in eq.[j ] can be evaluated analytically . the stationary points @xmath25 are the solutions of @xmath44 where @xmath45 is the two - dimensional gradient in @xmath46 space . for @xmath25 belonging to sets 1 and 2 only one fs sheet occurs in the first prismatic bulk bz ; the analysis then proceeds exactly as in ref . . the corresponding periods are : @xmath47=2.69 atomic planes and @xmath48=2.36 atomic planes respectively . however , for the @xmath29-points the fs can be regarded as consisting of two sheets . the two values of @xmath49 ( @xmath50 and @xmath51 associated with @xmath52 , shown by arrows in fig . [ fig2](b ) , correspond to equivalent periods @xmath53 atomic planes and @xmath54 atomic planes which can not be distinguished just by looking at discrete values of @xmath3 . however , @xmath55 and @xmath56 vanish simultaneously when calculated at @xmath57 . thus , eq . [ j3 ] is satisfied for any values of @xmath58 and @xmath59 , yielding other periods besides @xmath60 and @xmath61 . the relative contributions of these extrema depend on the comparative values of the corresponding coefficients @xmath62 . the situation is very similar to that discussed in ref . . nevertheless , our calculations have shown that the fundamental period @xmath60 ( which is equivalent to @xmath61 ) and its harmonics dominate . this is because the coefficients associated with them are far larger than those corresponding to alternative combinations of @xmath58 and @xmath63 . the calculated contributions to the coupling at t=300k coming separately from each set of @xmath25 are shown in fig . [ fig1 ] . we note that all three contributions are comparable , but the belly ( full continuous line ) clearly dominates . this contrasts with fcc co / cu ( 001 ) trilayers where the belly contribution is negligible in comparison with those given by the necks.@xcite the main reason for such difference is that minority carriers from the vicinity of the fs belly are fully confined in bcc cu by the fe layers in the ferromagnetic configuration . the physical origin of such confinement is that the sp - like bcc cu band which intersects the fs has no counterpart at the minority - spin fe fs . a similar situation happens for the cu fs states in the vicinity of @xmath64 belonging to set 2 , as shown in fig.[fig3 ] . on the other hand , the cu fs @xmath29-states of either spin can evolve into the corresponding fe fs states because they have sp - character , due to the existence of a small but finite sp - d hybridization in this @xmath65-space region . the degree of confinement experienced by the carriers in this case depends on the relative sp - d hybridization strengths . the agreement between the stationary phase approximation and numerical results verifies that the exchange coupling at room temperature in fe / cu ( 001 ) trilayers with perfect interfaces oscillates with a short period strongly influenced by the belly contribution . this is in accordance with the philips group observations , as far as the period is concerned . however , the calculated strength is an order of magnitude larger than what they have observed . on the other hand the amplitude of our long period component is about three times smaller than the short period contribution . we believe that the discrepancy between experimental and theoretical results is due to interface roughness which affects the amplitude and overall phase of the coupling . @xcite the reason why the philips results are much smaller than those of the sfu group remains unexplained . motivated by recent measurements of the sfu group,@xcite we have calculated the temperature variation of the coupling for different cu thicknesses . our results , shown in fig.[fig4 ] , are for perfect interfaces where the bilinear exchange coupling @xmath21 is much larger than the intrinsic biquadratic term . the rate of variation of @xmath66 with temperature clearly changes with spacer thickness . the calculated slope for fe/12cu / fe agrees very well with the measured value for fe/10cu / fe . the most striking result however , reproduced in the inset of fig.[fig4 ] , is the increase of the coupling with increasing temperature for some cu thicknesses . the temperature dependence of @xmath21 is governed not only by the spacer fs but also by the confining strength of the ferromagnetic layers @xcite , which differs for the three sets of extrema . as pointed out in ref . , the energy dependence of the phases @xmath67 of the `` fourier '' coefficients in eq . [ j2 ] varies according to the confinement strength and is very important in determining the temperature dependence of the coupling . it turns out that the values of @xmath68 calculated at the second set of extrema are about four and a half times larger than at the belly and the @xmath29-points . the temperature dependence of the former contribution is then stronger than the others . hence , the coupling is approximately given by the sum of three oscillatory functions of @xmath3 with comparable amplitudes which decay differently with temperature . therefore , at some values of @xmath3 , as in fe/13cu / fe , the balance is such that an overall increase in the coupling is obtained even though the amplitude of each contribution separately decreases with temperature . such increase was not detected by the sfu group . one possible explanation is that they have observed basically just a long period component . another reason could be the influence of spin fluctuations in the ferromagnetic layers , which is neglected in our calculations . nevertheless , this is an interesting temperature effect which may be observed under suitable conditions .

several exotic qcd states , such as glueballs , hybrids or multiquarks , are expected to have masses between 1 and 2.5 gev/@xmath14 @xcite . radiative @xmath15 and @xmath1 decays are traditionally regarded as gluon - rich environments ; such decays , although well investigated , are still promising for the discovery and identification of qcd exotica . for example , in radiative decays into pseudoscalars , two of the most promising glueball candidates , the @xmath6 and the @xmath7 , are seen @xcite . several high precision partial wave analyses of radiative @xmath15 decays have been performed @xcite , and a recent bes - paper presented an analysis of a high statistics sample of @xmath1 decays , without a partial wave analysis @xcite . this paper updates the @xmath1 radiative decays @xcite with @xmath2 and @xmath3 final states , giving access to scalar and tensor intermediate states . within the quark model @xcite , the @xmath1 is believed to have 2 @xmath16 as its main component , and the @xmath15 is the 1 @xmath16 @xmath17-state . this gives a @xmath1 with a behavior that differs from @xmath15 only due to the difference in energy scale and in the radial wave function . the so - called 12%-rule from perturbative qcd ( pqcd ) relates the branching fractions of the two states into a particular hadronic final state ( @xmath18 ) : @xmath19 where the branching fractions are from ref . a radiative decay mode would be similar , but would proceed via one photon and two gluons , exchanging one power of @xmath20 with @xmath21 in the coupling @xcite . a large violation of the 12%-rule was first observed in 1983 in @xmath1 decays to @xmath22 and @xmath23 by markii ; it became known as the @xmath22-puzzle @xcite . since then , many two - body decay modes have been compared , of which some obey and some violate the rule @xcite . the analysis uses the besii data sample of @xmath24 @xmath1-events @xcite , corresponding to an integrated luminosity of @xmath25 pb@xmath26 @xcite , and a set of continuum data at @xmath27 gev , ( @xmath28 ) pb@xmath26 @xcite , also measured with the besii detector . besii is a cylindrical multi - component detector , described in detail in refs . @xcite and @xcite . around the beam - pipe is a 12-layer straw vertex chamber , which provides trigger and track information . located radially outside , there is a 40-layer open - cell geometry main drift chamber ( mdc ) , covering 85% of the solid angle . the mdc is used for tracking and particle identification using @xmath29-techniques . the momentum resolution is @xmath30}$ ] , and the mdc @xmath31 resolution for hadron tracks is @xmath328% . particle identification by energy loss techniques is complemented by a measurement of the time - of - flight ( tof ) from the interaction point to 48 scintillation counters surrounding the mdc ( time resolution @xmath32200 ps for hadrons ) . outside the tof - scintillators is a 12 radiation length lead - gas barrel shower counter ( bsc ) , which measures the energy of electrons and photons over @xmath3280% of the total solid angle with a resolution of @xmath33}$ ] . in addition , the iron flux return , outside the solenoid coil ( 0.4 t axial field ) , is instrumented with three double layers of counters for identification of muons with momenta larger than 0.5 gev/@xmath34 . the cylindrical structure is closed by end - caps . events with at least one photon and two oppositely charged tracks with good helix fits and complete covariance matrices are selected . end - cap information is used to reject background with extra charged particles in the end - cap region , but only events within the well - understood barrel detectors are included in the analysis . the polar angles of the charged particles are required to fulfill @xmath35 for @xmath12 and @xmath36 for @xmath8 , where @xmath37 is the angle of the charged particle with respect to the beam axis . the vertex of the two charged tracks is required to satisfy @xmath38 cm and @xmath39 cm , where @xmath40 , @xmath41 , and @xmath42 are the @xmath43 , @xmath44 , and @xmath45 coordinates of the point of closest approach of each charged track to the beam axis . the photon with the highest energy is tested for quality : it must be within the barrel detector , it must have its first hit in the bsc inside the first 12 layers out of 24 , it must be separated from charged tracks by at least 15@xmath46 in the xy - plane at the entrance of the bsc , and the photon direction from the origin to the bsc must agree within 35@xmath46 with the shower direction in the bsc . the most abundant @xmath47 reactions at gev - energies are qed processes , mainly @xmath48 and @xmath49 . in cases where there is initial or final state radiation , the event could be misidentified as @xmath3 or @xmath2 . muons are rejected using muon chamber information ; this is especially important in the @xmath3-channel , therefore the stricter angular cut . bhabha events , @xmath50 , are rejected using the @xmath29 @xmath51-value under the @xmath52 hypothesis ( @xmath53 ) combined with the energy deposited in the calorimeter @xmath54 and the momentum @xmath55 . to reject background from @xmath56 , the selected sample is required to satisfy @xmath57 the tof and @xmath29 measurements of the two charged tracks and a kinematic fit with four degrees of freedom are used to calculate @xmath58-values for the overall event - hypothesis : @xmath3 or @xmath2 . a confidence level of better than 1% is required for each event sample , and for ambiguous cases , the most likely assignment is chosen . after all requirements , the overall detection and event selection efficiency is @xmath3215% ( see table [ tab : pinumevents ] and [ tab : knumevents ] for exact numbers ) . the probability that a @xmath2 event is selected as a @xmath59 event is 0.17% , and the probability that a @xmath59 event is selected as a @xmath2 event is 0.14% , from monte carlo simulation . a carefully checked geant3-based monte carlo program @xcite is used to determine invariant mass resolution ( @xmath6010 mev/@xmath14 for @xmath61 and for @xmath62 ) , angular efficiency , absolute efficiency , and background suppression . using this program , the contributions from qed - processes after event selection are found to be small ( dominated by 15@xmath632 @xmath64 events in @xmath3 ) . however , both qed backgrounds and non - resonant hadronic signals are estimated using the continuum data sample , where the main contributor to the non - resonant background is found to be initial - state - radiation and subsequent @xmath65 or @xmath66-formation . this was checked for the @xmath65 , and measurement agreed with calculations using a structure function approach @xcite . the full continuum contribution is assumed to be incoherent with the signal . resonant background sources are checked individually using known cross sections and simulation to determine the acceptance of the selection criteria @xcite . the main resonant background to both channels is found to be @xmath67 , which has been measured with great precision by besii @xcite using partial wave analysis . a monte carlo generated sample with the same angular and energy distribution as found in ref . @xcite is used , and 72@xmath635 misidentified background events in @xmath3 and 4.9@xmath630.5 misidentified events in @xmath68 are obtained . the signal and the background estimates are shown in figs . [ fig : pi ] and [ fig : kk08 ] . distribution for @xmath69 candidate events ( dots with error bars ) , along with the estimated background . the thick line corresponds to the background from @xmath70 using the branching fraction from @xcite , the filled histogram corresponds to the continuum data , and the hatched histogram to a background estimate including the continuum and misidentified @xmath70-events . all contributions are scaled to correspond to the integrated luminosity of the @xmath1 data set . the histogram is not acceptance corrected . ] distribution in @xmath71 candidate events ( dots with error bars ) along with the extimated background . the thick line corresponds to the maximum background from @xmath72 , which could be important if it is close to its upper limit @xcite . the filled histogram corresponds to the scaled continuum , and the hatched histogram to a background estimate including the continuum and misidentified @xmath70-events . all contributions are scaled to correspond to the integrated luminosity of the @xmath1 data set . the histogram is not acceptance corrected . ] from spin - parity conservation , any intermediate resonance @xmath73 in @xmath74 , @xmath3 must have @xmath75 , @xmath76 , or higher even spins . to investigate the resonance structure , the invariant mass spectra of the two pseudoscalar systems are fitted . a partial wave analysis of besii @xmath77-data @xcite required an @xmath9 at @xmath78 mev/@xmath14 , a @xmath79-state at @xmath80 mev/@xmath14 , and a @xmath79-state at @xmath81 gev/@xmath14 to fit the data properly . due to the large @xmath22-background in @xmath15 decays , the focus was on structures below 2 gev/@xmath14 , but it is worth mentioning that a peak around 2.1 gev/@xmath14 was observed ; this was the case also in @xmath2 . in @xmath2 , the @xmath10 and @xmath7 were prominent @xcite . an earlier analysis , using a smaller sample of @xmath82 good quality @xmath1-events from besi and using a @xmath83 background sample at @xmath84 gev , showed a clear @xmath9 [ @xmath85 , an @xmath7 signal in @xmath86 [ @xmath87 , and a clear @xmath7 [ @xmath88 , and a less clear @xmath10 in the @xmath79 resonances are modeled as constant width breit wigner functions , with @xmath89 angular distributions . the @xmath76 resonances are modeled as blatt weisskopf dampened d - wave shapes ( with parameters as in ref . @xcite and angular distributions according to eq . ( [ eq : wdist ] ) ) . masses and widths are taken from the particle data group compilation @xcite . for simplicity any possible nonresonant production and background component is assumed to follow three - body phase space . all resonances are treated as incoherent . the unbinned meson - meson invariant mass spectrum is fitted with a log likelihood method , and the scaled background component from the continuum data sample and the simulated @xmath67 background are included in the fit but with the opposite sign . the quality of different log likelihood fits to the same data set can be compared since the ratio @xmath90 follows a @xmath58-distribution @xcite . the number of degrees of freedom for this @xmath58-distribution is given by the number of free parameters in @xmath91 minus the number of free parameters in @xmath92 . in this way , the difference in log likelihood is used for an hypothesis test , for instance to determine the necessity of weakly needed resonances such as the @xmath6 in @xmath93 and the @xmath10 in @xmath2 . another measure of the quality of the overall fit is obtained using a pearson s @xmath58-test . the fit to the two - pion invariant mass spectrum between 1 and 3 gev/@xmath14 allows the determination of the number of events and the product decay branching fractions of @xmath94 where @xmath5 , @xmath6 , @xmath7 , or a high mass component modeled here with @xmath95 and @xmath96 ( table [ tab : pinumevents ] ) . the full fit , binned as in fig . [ fig : pipifit ] , has a pearson s @xmath97 of @xmath98 . .number of selected events and product branching fractions in the @xmath99 channel . the efficiency at the central resonance mass is determined by monte carlo and losses due to the limited mass range are taken into account . the first uncertainties in the branching fractions are statistical and the second systematic . [ cols="<,<,<,<",options="header " , ] the branching fraction of @xmath100 is measured in both the @xmath8 and the @xmath12 decays . both results are close to the one previous measurement from bes @xcite . the ratio between @xmath101 and @xmath102 is found to be ( 77@xmath6372)% , with a large uncertainty mainly from the @xmath12 measurement . the potential glueball , the @xmath6 , is accepted at the 5%-level in @xmath3 . it has been seen in @xmath103 @xcite , but this is a first measurement in @xmath1 radiative decays . the @xmath10 is at the verge of being accepted at the 5%-level in @xmath2 . the @xmath10 has a strong signal in @xmath104 @xcite and was part of the fit in the besi @xmath105 analysis @xcite , but a branching fraction was never given . the region above 2 gev/@xmath14 has an enhancement both in @xmath8 and @xmath12 . in @xmath8 , the region is fitted with two resonances : the @xmath95 and the @xmath96 . both of these have been detected before in pseudoscalar final states in radiative @xmath15-decays @xcite . in @xmath12 , the same region has a rather uniform distribution with an enhancement around the @xmath96 . the invariant mass distribution from @xmath1 decays is similar to the previously measured @xmath15 case but due to the limited number of events here , a partial wave analysis is not feasible . one problem when comparing the @xmath15 and the @xmath1 fit in @xmath61 is that in @xmath15 there is destructive interference between the @xmath106 and the @xmath7 causing a dip at @xmath321.8 gev/@xmath14 . there is also destructive interference around 1.5 gev/@xmath14 , which shifts the peak of the @xmath6 to lower values than the pole position @xcite . this pattern is not reproduced if we , as in this analysis , assume incoherence . the ratio @xmath107 is presented in table [ tab:12procent ] for a few final states @xmath73 for comparison with the 12%-rule . can we use the branching fractions to understand what goes on during the decay process ? in an old prediction by lipkin @xcite , different pqcd and pqed @xmath108 decay diagrams are compared under the assumption of isospin symmetry . has an affinity for @xmath109 and @xmath110 , which depends on the preferred diagram @xcite . lipkin predicted a ratio between @xmath109 and @xmath110 of 50% for the upper diagram and 8% for the lower , without taking phase space into account . ] the upper diagram in fig . [ fig : tensorprocess ] gives the ratio @xmath111 and the lower gives @xmath112 . the analysis presented here gives a ratio of @xmath113 whereas corresponding measurement in @xmath15 gave @xmath114% . from another pqcd diagram with photon radiation and subsequent hadron formation , close predicted @xcite @xmath115 under the assumption of quark - antiquark pair formation with @xmath116 and identical coupling between glue and scalar and glue and tensor . this can unfortunately not be compared to the present data since @xmath117 is unknown . the bes collaboration thanks the staff of bepc and computing center for their hard efforts . this work is supported in part by the national natural science foundation of china under contracts nos . 10491300 , 10225524 , 10225525 , 10425523 , the chinese academy of sciences under contract no . kj 95t-03 , the 100 talents program of cas under contract nos . u-11 , u-24 , u-25 , and the knowledge innovation project of cas under contract nos . u-602 , u-34 ( ihep ) , the national science foundation of china under contract no . 10225522 ( tsinghua university ) , the department of energy under contract no . de - fg02 - 04er41291 ( u. hawaii ) , the royal physiographic society in lund sweden , the uppsala university graduate school @xmath118 , the sederholm foundation , and the gertrud thelin foundation at uppsala university . * * w. m. yao _ et al . _ , journal of physics g * 33 * , 1 ( 2006 ) . n. brambilla _ et al . _ ( qwg and topic conveners ) , _ heavy quarkonium physics _ ( cern yellow reports , 2004 ) . m. ablikim _ et al . _ ( bes collaboration ) , phys . rev . lett . * 99 * , 011802 ( 2007 ) . j. z. bai _ et al . _ ( bes collaboration ) , phys . rev . d * 67 * , 032004 ( 2003 ) . e. eichten , k. gottfried , t. kinoshita , k. d. lane , and t. m. yan , phys . d * 17 * , 3090 ( 1978 ) . t. appelquist , a. de rjula , h. d. politzer , and s. l. glashow , phys . lett . * 34 * , 365 ( 1975 ) . m. chanowitz , phys . d * 12 * , 918 ( 1975 ) . l. okun and m. voloshin , itep-95 - 1976 ( 1976 ) , ( unpublished ) . s. j. brodsky , t. a. degrand , r. r. horgun , and d. g. coyne , phys . b * 73 * , 203 ( 1978 ) . k. koller and t. walsh , nucl . b * 140 * , 449 ( 1978 ) . m. e. b. franklin _ et al . _ ( markii collaboration ) , phys . 51 * , 963 ( 1983 ) . m. ablikim _ et al . _ ( bes collaboration ) , phys . b * 619 * , 247 ( 2005 ) . x. h. mo _ et al . _ , high energy phys . and nucl * 27 * , 455 ( 2004 ) , hep - ex/0407055 . s. p. chi , high energy phys . and nucl * 28 * , 1135 ( 2004 ) , in chinese . m. ablikim _ et al . _ ( bes collaboration ) , phys . d * 72 * , 012002 ( 2005 ) , hep - ex/0503040 . j. z. bai _ et al . _ ( bes collaboration ) , nucl . instr . and meth . a * 344 * , 319 ( 1994 ) . j. z. bai _ et al . _ ( bes collaboration ) , nucl . instr . and meth . a * 458 * , 627 ( 2001 ) m. ablikim _ et al . _ ( bes collaboration ) , nucl . instr . and meth . a * 552 * , 344 ( 2005 ) , physics/0503001 a. lundborg , ph . d. thesis , uppsala university ( 2007 ) , http://publications.uu.theses/abstract.xsql?dbid=7460 . m. ablikim _ et al . _ ( bes collaboration ) , phys . b * 614 * , 37 ( 2005 ) , hep - ex/0407037 . m. ablikim _ et al . _ ( bes collaboration ) , phys . rev . d * 70 * , 112007 ( 2004 ) , hep - ex/0410031 . m. ablikim _ et al . _ ( bes collaboration ) , phys . b * 642 * , 441 ( 2006 ) , hep - ex/0603048 . j. z. bai _ et al . _ ( bes collaboration ) , phys . rev . d * 68 * , 052003 ( 2003 ) , hep - ex/0307058 . b. s. zou , eur . j. a. * 16 * , 537 ( 2003 ) . r. m. baltrusaitis _ et al . _ ( markiii collaboration ) , phys . d * 35 * , 2077 ( 1987 ) . j. z. bai _ et al _ ( bes collaboration ) , phys . d * 69 * , 012003 ( 2004 ) . j. z. bai _ et al . _ ( bes collaboration ) , phys . rev . d * 69 * , 072001 ( 2004 ) . p. k. kabir and a. j. g. hey , phys . d * 13 * , 3161 ( 1976 ) . c. edwards _ et al . _ ( crystal ball collaboration ) , phys . d * 25 * , 3065 ( 1982 ) . m. krammer , phys . b * 74 * , 361 ( 1978 ) . f. e. close , phys . d * 27 * , 311 ( 1983 ) . b. f. l. ward , phys . d * 33 * , 1900 ( 1986 ) . g. alexander _ et al . _ ( pluto collaboration ) , phys . b * 76 * , 652 ( 1978 ) . m. ablikim _ et al . _ ( bes collaboration ) , phys . d * 70 * , 092004 ( 2004 ) . h. j. lipkin and h. r. rubinstein , phys . lett , b * 76 * , 324 ( 1978 ) .

radiative charmonium decays from the besii sample of 14@xmath0 @xmath1-events into two different final states , @xmath2 and @xmath3 , are analyzed . product branching fractions for @xmath4 , @xmath2 are given , where @xmath5 , @xmath6 , and @xmath7 in @xmath8 and @xmath9 , @xmath10 , and @xmath11 in @xmath12 . an angular analysis gives the ratios of the helicity projections for the @xmath9 in @xmath13 .

considerable interest has been devoted in the past decades to the study of intermetallic compounds based on anomalous lanthanides ( ce , yb , tm , eu and sm ) because of the rich variety of properties they have ( intermediate valence , heavy fermion behaviour , @xmath3 ) . more recently , particular attention was focused on 4__f _ _ intermetallics which are close to a magnetic instability , because of the observation of unconventional superconductivity in the vicinity of a quantum critical point ( qcp ) , where the magnetic order is destroyed @xcite . in the specific case of sm , the competition between the two possible valence states , nonmagnetic sm@xmath4 with a 4__f__@xmath5:@xmath6 configuration and magnetic sm@xmath2 with a 4__f__@xmath7:@xmath8 configuration , can sometimes lead to the formation of an intermediate valent ground state which shows peculiar electronic and magnetic properties @xcite . the charge state can be eventually tuned towards integer trivalency by the application of external pressure , with the consequent onset of long - range magnetic order owing to the fact that sm@xmath2 is a kramer s ion . well - known examples of intermediate valent sm compounds are the high - pressure `` golden '' phases of the sm monochalcogenides ( sms , smse and smte ) @xcite and smb@xmath0 @xcite . they belong to the class of kondo insulators or narrow - gap semiconductors , which behave at high temperature like an array of independent localized moments interacting with itinerant conduction electrons , whereas at low temperature they develop clear narrow - gap properties . at ambient pressure sms is a nonmagnetic semiconductor ( black phase ) , with nearly divalent sm ions , crystallizing in the nacl - type structure . at a very low pressure @xmath9 @xmath1 0.65 gpa at room temperature it undergoes an isostructural first order phase transition to an intermediate valent metallic state ( golden phase ) , with a large volume collapse ( @xmath18% ) and a valence @xmath10 @xmath11 2.6 just above @xmath9 @xcite . the temperature dependence of the electrical resistivity shows the opening of a small gap at low temperatures for pressures between @xmath9 and @xmath12 = 2 gpa . the gap closes continuously as pressure is increased and disappears for @xmath13 @xmath14 @xmath12 , leaving the system in a metallic ground state @xcite . in the golden phase the valence increases continuously with pressure , with an inflection point at @xmath12 ( @xmath10 @xmath1 2.7 ) @xcite , and the trivalent state is reached only at considerably higher pressure ( @xmath15 @xmath14 10 gpa ) @xcite . recently a first order transition at @xmath12 with the appearance of long - range magnetic order associated with intermediate valence has been evidenced by nuclear forward scattering ( nfs ) of synchrotron radiation @xcite and specific heat @xcite . the magnetically ordered state behaves like that of a stable trivalent compound , with values of the hyperfine parameters as expected for a @xmath16 ground state ( produced by the action of a cubic crystal field on the sm@xmath2 ions ) and almost independent from pressure in the range 2 - 19 gpa , while the ordering temperature @xmath17 increases smoothly from 15 k at @xmath12 to 24 k at 8 gpa . considerable short - range magnetic correlations develop above @xmath17 and persist up to about 2@xmath17 . in analogy to golden sms , smb@xmath0 is an intermediate valent ( @xmath10 = 2.6 at ambient pressure and room temperature ) insulator @xcite . however at temperatures above @xmath1 70 k the electrical resistivity and the hall effect are rather typical of a bad metal , and only for @xmath18 @xmath19 70 k the conductivity starts to decrease with temperature by several orders of magnitude ( two to five , depending on the quality of the sample ) because of the opening of an insulating gap of the order of 10 - 20 mev @xcite . despite the intermediate valent ground state of smb@xmath0 , the form factor as determined by neutron scattering is that expected for sm@xmath4 @xcite , as already observed for similar systems like golden sms and tmse @xcite . however the magnetic excitation spectrum shows that the `` true '' ground state of smb@xmath0 is an unusual quantum mechanical superposition of a divalent component of 4__f__@xmath5 configuration with a 4__f__@xmath75_d _ loosely bound state , where a 4__f _ _ electron is only partially delocalised and free to move within a limited volume @xcite . although this ground state is nonmagnetic , @xmath20b nmr measurements have evidenced the onset of dynamical magnetic correlations below @xmath1 15 k @xcite . when external pressure is applied , the semiconducting gap decreases and finally closes at a critical pressure @xmath12 @xmath11 4 - 7 gpa , above which the system behaves as a metal at all temperatures @xcite . at @xmath12 the resistivity has a less than quadratic dependence on temperature down to 1.5 k , suggesting that a qcp is crossed when the gap closes @xcite . the valence has been shown to increase smoothly from 2.6 at ambient pressure to 2.8 at 10 gpa , with an inflection point at @xmath1 5.5 gpa at the gap closure @xcite , in analogy with sms at 2 gpa . contrary to sms and smb@xmath0 , metallic sm is at ambient pressure in a trivalent state with long - range antiferromagnetic order below @xmath1 106 k. however , recent theoretical calculations predict that the 4__f _ _ electrons should become completely delocalized with a volume contraction corresponding to a pressure of the order of @xmath1 100 gpa @xcite , which can be achieved with the diamond anvil cell ( dac ) technique . sm crystallizes at ambient pressure in the sm - type structure , with a unit cell consisting of the periodic stacking of nine hexagonal planes ( a@xmath21b@xmath22a@xmath22b@xmath21c@xmath22b@xmath22c@xmath21a@xmath22c@xmath22 @xmath3 ) @xcite . the sublattice formed by the six planes with hexagonal nearest neighbour environment ( indicated by the subscript @xmath23 ) orders antiferromagnetically at @xmath24 = 106 k , while the planes with approximately cubic local symmetry ( indicated by the subscript @xmath25 ) order independently only below @xmath26 = 14 k @xcite . the 4__f _ _ ordered magnetic moment is @xmath1 0.6 @xmath27 @xcite , lower than the free ion value of 0.71 @xmath27 for the sm@xmath2 ion . high pressure resistivity studies performed up to 43 gpa show that , in the range between ambient pressure and 6 gpa , @xmath24 decreases while @xmath26 increases with pressure @xcite . above 6 gpa the two nel transitions occur at the same temperature , increasing with pressure and reaching @xmath1 135 k at 43 gpa . at higher pressure , only structural investigations have been performed , showing the presence of five successive structural phase transitions between ambient pressure and 189 gpa @xcite . in particular above 91 gpa sm adopts a body - centred tetragonal structure , in which calculations show that the 4__f _ _ electrons are delocalized and that sm is an itinerant magnet with spin moment of @xmath28 4 @xmath27 @xcite . this moment is very sensitive to pressure and disappears when the volume is sufficiently reduced . in this article we present further results on golden sms ( a detailed discussion of previous recent results is given in references @xcite ) and new results on smb@xmath0 and metallic sm at high pressures . these results were obtained by combining @xmath29sm nuclear forward scattering ( nfs ) of synchrotron radiation with high pressure specific heat measurements . from the nfs data we obtained information about the magnetic hyperfine field at the @xmath29sm nuclei as a function of pressure and temperature , and from the combination of nfs and specific heat we could determine the ordering temperature @xmath17 of the sm atoms and its pressure dependence . in particular we have obtained clear evidence that smb@xmath0 , in analogy with golden sms , develops long - range magnetic order at pressures above @xmath12 . these magnetically ordered states are stable up to at least 19 and 26 gpa for sms and smb@xmath0 , respectively . moreover , in the low pressure intermediate valent state of both compounds ( and down to ambient pressure for smb@xmath0 ) short - range magnetic correlations are present at low temperatures . in the case of metallic sm , the magnetic hyperfine field decreases monotonically with pressure and is reduced by almost 40 % at 46 gpa with respect to ambient pressure . the sms sample was prepared as described in reference @xcite . samarium hexaboride single crystals were grown by a standard aluminum flux technique @xcite . smb@xmath0 powder was pre - synthesized by reacting boron on samarium oxyde ( borothermal reduction ) under high vacuum using rf heating . the quality of the sample was checked by x - ray powder diffraction and showed the presence of pure smb@xmath0 with no trace of parasitic phases . for both sms and smb@xmath0 the @xmath29sm nfs measurements have been performed on powders made with isotopically enriched ( to 97 % ) sm , while the specific heat measurements have been performed on single crystals made with natural sm . the metallic sm sample was a polycrystalline foil , isotopically enriched to 97 % in @xmath29sm , commercially available from oak ridge national laboratories . high pressure was applied to the samples using the diamond anvil cell technique , with argon or nitrogen used as pressure transmitting media . the pressure was measured and changed always at room temperature for the nfs measurements , whereas it was measured and changed in situ using the setup described in reference @xcite for the specific heat measurements performed at the cea grenoble . the @xmath29sm nfs measurements ( resonant energy @xmath30 = 22.494 kev ; 5/2 - 7/2 transition ) were performed at the undulator beamline id22n @xcite of the european synchrotron radiation facility , grenoble , france , and a more detailed description of the experimental setup is given in reference @xcite . nfs is a technique related to the mssbauer effect , thus similar microscopic information to that inferred from conventional mssbauer spectroscopy can be obtained . this allows one to determine the pressure dependence of the magnetic hyperfine field @xmath31 , of the ordering temperature @xmath17 and of the electric field gradient ( efg ) @xmath32 at the @xmath29sm nuclei . the change of the efg with @xmath13 is obtained from the pressure dependence of the electric quadrupole splitting @xmath33@xmath34 = @xmath35@xmath32@xmath36 where @xmath36 is the nuclear quadrupole moment of the @xmath37 = 7/2 ground state . figures [ fig : one ] and [ fig : two ] show typical nfs spectra recorded at various temperatures and pressures for golden sms and smb@xmath0 , respectively . at high temperatures ( left panel in the two figures ) and for all pressures the spectra are characteristic of unsplit nuclear levels , as expected for sm ions in the absence of magnetic order and in an environment of cubic symmetry ( nacl - type stucture for sms and cscl - type structure for smb@xmath0 ) . a clear quantum beat pattern , due to the combined action of magnetic dipole and electric quadrupole interactions on the nuclear levels of @xmath29sm , appears at low temperatures ( right panel of the two figures , @xmath18 = 3 k ) for pressures higher than @xmath38 @xmath11 2 gpa for sms and @xmath39 @xmath11 9 gpa for smb@xmath0 . this is a clear indication that magnetic order sets in in both compounds at high pressure and low temperature . the analysis of the spectra , performed with the software motif @xcite using the full dynamical theory of nuclear resonance scattering , including the diagonalization of the complete hyperfine hamiltonian , reveals that for sms at 2.35 gpa and 3 k only a fraction ( corresponding to @xmath1 72 % ) of the sm atoms shows magnetic order , the rest being paramagnetic . however this fraction increases rapidly with pressure and reaches 100 % above @xmath1 3 gpa . at 2.35 gpa we deduce a value of 261(10 ) t and -1.50(6 ) mm / s for the magnetic hyperfine field @xmath31 and the induced quadrupole interaction @xmath33@xmath34 , respectively @xcite . for smb@xmath0 at 9.7 gpa and 3 k the analysis reveals that all sm ions are magnetically ordered . the saturation values of the hyperfine parameters at this pressure and 3 k are @xmath31 = 246(20 ) t and @xmath33@xmath34 = -1.27(12 ) mm / s . in the vicinity of the critical pressures @xmath38 and @xmath39 the low temperature nfs spectra can not be accounted for by a single set of hyperfine parameters , but broad distributions of both @xmath31 and @xmath33@xmath34 are present at the @xmath29sm nuclei , larger in smb@xmath0 than in sms . the average values of the hyperfine parameters are considerably reduced with respect to the free ion values for sm@xmath2 ( @xmath31 = 338 t and @xmath33@xmath34 = -2.1 mm / s @xcite ) . this can be ascribed to the effect of the cubic crystal field acting on the sm@xmath2 ions : the lowest multiplet @xmath8 is split into a @xmath40 doublet and a @xmath16 quartet ( here and in the following we will neglect the possible mixing of the ground multiplet with excited multiplets through exchange interactions ) , associated to completely different values of the hyperfine parameters ( @xmath31 = 113 t and @xmath33@xmath34 = 0 for the @xmath40 , while @xmath31 = 250 t and @xmath33@xmath34 = -1.7 mm / s for the @xmath16 ) . our measured values for both compounds , especially those for the magnetic hyperfine field , compare well with those calculated for a @xmath16 state . however , one must keep in mind that , at least for the lowest pressures above @xmath38 and @xmath39 , the magnetic order coexists with intermediate valence and therefore the ground state wavefunction can be considerably more complicated than the simple one describing the @xmath16 quartet . for both compounds the hyperfine parameters in the magnetically ordered state have only a very weak dependence upon pressure in the range studied , showing however a slight continuous increase of magnitude as pressure increases . at the same time , the widths of the distributions of @xmath31 and @xmath33@xmath34 decrease with increasing pressure but remain finite even at the highest pressures ( the distribution of @xmath31 has a relative width of @xmath1 3 % at @xmath13 @xmath14 5 gpa for sms and of @xmath1 7 % at @xmath13 @xmath14 11 gpa for smb@xmath0 ) . these facts prove that the pressure induced magnetic order in these two compounds is very stable and that the transition into the trivalent state ( which occurs for @xmath13 @xmath14 10 - 12 gpa in the case of sms @xcite ) is not associated to any anomaly in the pressure dependence of the hyperfine parameters . this fact and the good agreement between the measured values of @xmath31 and @xmath33@xmath34 and those expected for sm@xmath2 ions in a cubic crystal field might be an indication of the fact that , in the high pressure metallic state , these two compounds behave like stable trivalent compounds , independently of @xmath10 and similarly to tmse above 3 gpa @xcite . in order to shed more light on the properties of the high - pressure magnetically ordered state , we have performed temperature dependent nfs and specific heat measurements at various pressures . the temperature and pressure dependences of the specific heat for golden sms and smb@xmath0 are shown in figure [ fig : three](a ) and ( b ) , respectively . the presence of sharp anomalies at temperatures @xmath41 for pressures higher than 2 gpa for sms and @xmath1 9 gpa for smb@xmath0 is a clear indication that the order observed by nfs in both systems at high pressure is long - range magnetic order . @xmath17 is found to increase from 15 k at 2 gpa to 24 k at 8 gpa in sms , whereas it has a value of @xmath1 12 k , independent from pressure up to 16 gpa , in smb@xmath0 . the temperature dependence of the nfs spectra of sms for @xmath13 @xmath14 2 gpa does not reveal any sharp transition at @xmath17 as for what concerns the values of @xmath31 ( see figure [ fig : four ] ) and @xmath33@xmath34 , but a continuous decrease with increasing temperature up to @xmath1 2@xmath17 , where they become zero . the presence of hyperfine interactions at temperatures higher than @xmath17 can be seen as due to the persistence of short - range magnetic correlations ( probably of dynamical nature ) in the paramagnetic phase . however , an anomaly in the temperature dependence of the nfs countrate ( which is a complicated function of the physical properties of the compound under study ) is clearly visible ( see figure [ fig : five ] ) , with a minimum in the vicinity of @xmath17 and the following steep increase upon entrance into the paramagnetic state . a similar behaviour is found in smb@xmath0 , with a minimum in the countrate at the value of @xmath17 determined by the specific heat measurements and the subsequent increase as temperature increases , as shown in figure [ fig : five ] . however , in this case a clear transition from the long - range magnetic order below @xmath17 to a phase characterized by short - range magnetic correlations is directly observed in the nfs spectra , as shown in figure [ fig : four ] : above @xmath17 the average value of @xmath31 and @xmath33@xmath34 is considerably reduced , their distribution is much broader and finally only a fraction of the sm ions feels these interactions , which slowly decrease and disappear at temperatures higher than @xmath1 50 k for all pressures up to 26 gpa . in the top panels of figures [ fig : one ] and [ fig : two ] spectra are shown which are characteristic of the low pressure ( @xmath13 = 1.7 gpa ) semiconducting intermediate valent states of golden sms and smb@xmath0 , respectively . at 3 k , the nfs spectra can not be analyzed properly if the assumption is made that the nuclear levels are not split by any hyperfine interaction , as would be expected for a nonmagnetic ground state : this is clear from the fact that the best fit ( shown in the figures as a line ) to the two nfs spectra at 3 k and 1.7 gpa simply does not reproduce the measured data . the only reasonable agreement between calculated and experimental curves can be obtained in the hypothesis that at least a fraction of the sm ions feels hyperfine interactions due to the presence of ( short - range ) magnetic correlations , which can be seen as a precursor of the incipient onset of long - range magnetic order above @xmath38 or @xmath39 . because of the extremely broad distributions of these hyperfine interactions , they do not show up as a clear beat pattern in the nfs spectra but rather only as a slowing down of the nuclear decay and prevent a precise analysis of their strength . the broad maxima present in the temperature dependence of the specific heat for pressures 1.3 @xmath19 @xmath13 @xmath19 2 gpa in the case of sms and 7 @xmath19 @xmath13 @xmath19 9 gpa for smb@xmath0 can as well be interpreted in the framework of the onset of low temperature short - range slow magnetic correlations for pressures just below @xmath38 and @xmath39 . although the nfs spectra of smb@xmath0 are characterized by the presence of short - range correlations even at ambient pressure , appearing always below 50 - 100 k independently of pressure , two regimes can be distinguished in the temperature dependence of the nfs countrate : for 0 @xmath42 @xmath13 @xmath42 5 gpa there is no appreciable variation between 3 and 300 k , whereas for 5 @xmath42 @xmath13 @xmath42 9 gpa a steep increase is observed between @xmath1 10 k and @xmath1 50 k , similar to the one observed for higher pressures , but no minimum is present ( see figure [ fig : five ] ) . the interval between 5 and 9 gpa , approximately where the semiconducting gap closes , can therefore be seen as a region of transition between the low - pressure gapped state and the high - pressure state with long - range magnetic order . the ground state of intermediate valent golden sms and of ambient - pressure smb@xmath0 is generally considered as a nonmagnetic state : the magnetic susceptibility does not show any paramagnetic curie - weiss divergence at low temperatures , as would be expected for sm@xmath2 ions , and neutron diffraction experiments do not show any trace of magnetic correlations . however , the presence of strong sm@xmath4 - sm@xmath4 ( ferromagnetic ) exchange interactions already in the nearly divalent black phase of sms has been demonstrated by esr measurements @xcite and an anomalous temperature dependence of the @xmath20b relaxation rate in nmr experiments below 15 k points towards the presence of magnetic correlations in smb@xmath0 at ambient pressure too @xcite . our studies therefore seem to confirm the presence of slowly fluctuating moments in the semiconducting phase of smb@xmath0 . owing to the divalent - like character of the ground state wavefunction of smb@xmath0 , the exchange interactions mentioned above can induce a strong mixing of the nonmagnetic ground multiplet @xmath6 of divalent sm with its magnetic first excited multiplet @xmath43 , which lies only @xmath1 36 mev ( @xmath1 420 k ) higher in energy . moreover , the magnetic excitation spectrum of smb@xmath0 , as measured by inelastic neutron scattering @xcite , shows the presence below @xmath1 50 - 100 k of a sharp transition at only 14 mev ( @xmath1 160 k ) , which is interpreted as an excitation from the `` true '' intermediate valent ground state ( @xmath44 = 0 ) into its first excited state ( @xmath44 = 1 ) . in this temperature range the @xmath44 mixing could therefore be even much stronger than expected for the free sm@xmath4 ion , thus making the appearance of short - range magnetic correlations possible . recently a simple expression has been proposed @xcite for the extension to a lattice of the kondo temperature ( @xmath45 ) formula for a single ce impurity [ @xmath45 = ( 1-@xmath46)@xmath33 , with @xmath46 and @xmath33 the occupation number of the lower valent configuration ( trivalent for ce and divalent for sm and yb ) and the width of the virtual bound state , respectively ] . for a lattice , @xmath33 can be replaced by the fermi temperature [ @xmath47 = @xmath48@xmath49 . in the case of sm ( where the equilibrium of the valence is given by the expression sm@xmath4 @xmath50 sm@xmath2 + 5_d _ and the valence can be expressed as @xmath10 = 3 - @xmath46 ) , this gives @xmath51 = @xmath46@xmath48 : this implies that @xmath51 will reach a maximum as @xmath46 goes from 0 to 1 ( whereas it increases continuously in the case of ce ) . when @xmath52@xmath51 becomes lower than the crystal field splitting , slow relaxation processes are expected and thus magnetic ordering occurs . nfs measurements of metallic samarium have been performed at temperatures between 3 and 300 k in the pressure range 0 - 46 gpa , spanning over five different crystallographic structures . typical measured spectra are shown in figure [ fig : six ] , while the pressure dependence of the saturation ( @xmath18 = 3 k ) hyperfine parameters is shown in figure [ fig : seven ] . at ambient pressure , in the sm - type structure , two inequivalent sm sites with hexagonal or cubic coordination are present , in a ratio of 2:1 , respectively . at high temperatures , above @xmath24 = 106 k , both sites are expected to show an electric quadrupole interaction due to the non cubic environment of all sm atoms , owing to the fact that the ratio @xmath25/@xmath53 of the lattice parameters is smaller than its ideal value for the sm - type structure . by comparison with the measured values of the electric field gradients in dhcp la metal @xcite , one can roughly estimate the lattice contribution to the electric quadrupole interaction of both sites in sm metal to be very small , of the order of only @xmath1 + 0.15 mm / s . although the nfs spectra are not very well resolved at high temperatures , the best fit to the data gives @xmath33@xmath54 = 0.0(1 ) mm / s for the hexagonal sites and @xmath33@xmath55 = + 0.2(1 ) mm / s for the cubic sites at @xmath18 = 200 k. at low temperatures , below @xmath26 = 14 k , it is expected that both sites show a combination of magnetic hyperfine and electric quadrupole interactions . here @xmath56@xmath33@xmath34@xmath56 should be considerably enhanced with respect to its high temperature value , because a dominant 4__f _ _ term adds to the lattice contribution ( which has ususally only a slight dependence on temperature ) . the analysis of the nfs spectra reveals that the magnetic hyperfine field for the hexagonal sites at @xmath13 = 0 and @xmath18 = 3 k , @xmath57 = 344(5 ) t , is very close to the value expected for a free sm@xmath2 ion ( 338(6 ) t , reference @xcite ) , and it compares well with the average value found at 4.2 k by mssbauer spectroscopy ( 345(18 ) t ) @xcite . however , the electric quadrupole interaction @xmath33@xmath54 = -1.80(3 ) mm / s is considerably reduced with respect to the free ion value of -2.1 mm / s @xcite and the lattice contribution alone can not account for the difference . moon and koehler ( reference @xcite ) have calculated approximate ground state wavefunctions for the cubic and hexagonal sites of sm taking into account the crystal field and exchange interactions , in order to explain their measured neutron scattering amplitudes . for the hexagonal sites , in particular , their calculations reveal that the relatively strong exchange interactions , responsible for the high nel temperature of 106 k , induce a mixing of the @xmath44 = 5/2 ground state with the @xmath44 = 7/2 excited state , with a reduction of the 4__f _ _ magnetic moment of these sites ( 0.56 @xmath27 ) with respect to the sm@xmath2 free ion value ( 0.71 @xmath27 ) . using the same wavefunction , we have calculated the expected values of the magnetic hyperfine field and of the electric quadrupole interaction for this ground state , which amount to @xmath1 350 t and @xmath1 - 1.8 mm / s and are therefore in good agreement with our measured values . this is therefore one example of the complicated relationship between magnetic moment and hyperfine magnetic field which is quite typical for sm compounds , where the mixing of multiplets with different @xmath44 can at the same time induce a reduction of the magnetic moment as compared to the free ion value and an increase of the magnetic hyperfine field . at the cubic sites , the analysis of the nfs spectra indicates that the hyperfine parameters are considerably reduced with respect to those of the hexagonal sites : @xmath58 = 317(5 ) t and @xmath33@xmath55 = -1.51(3 ) mm / s , and even if the larger lattice contribution to the electric field gradient at the cubic sites ( as determined at high temperature ) is considered , the difference remains . a possible explanation for this can be given if one interprets the fact that the ordering temperature of the cubic sites ( @xmath26 = 14 k ) is far lower than that of the hexagonal sites ( @xmath24 = 106 k ) as a consequence of the reduced strength of the exchange interactions at the cubic sites , which can be reflected in a lower degree of mixing of the @xmath44 = 7/2 excited multiplet into the ground one . the calculation of moon and koehler ( reference @xcite ) shows indeed that the ground state wavefunction of the cubic sites does not contain any mixing with the @xmath44 = 7/2 multiplet at all and that a larger 4__f _ _ magnetic moment is expected for these sites . by using the wavefunction which best fits their neutron data , we obtain values of @xmath1 270 t for the magnetic hyperfine field and @xmath1 - 1.5 mm / s for the 4__f _ _ contribution to the electric quadrupole interaction . although these values are lower than our measured ones , they qualitatively confirm that a reduction of the exchange interaction is in this case reflected in a decrease of the values of both magnetic hyperfine field and electric quadrupole interaction . we think that our set of data ( including the complete temperature dependence of @xmath31 and @xmath33@xmath34 ) in combination with the neutron data of koehler and moon ( reference @xcite ) might help evaluating better the effect of crystal field and exchange interactions on the ground state properties and especially on the value of the ordered magnetic moment of metallic sm at ambient pressure , which is still a subject of controversy @xcite . as pressure increases , the sm - type structure is expected to be stable up to @xmath13 @xmath1 7 gpa at 4.2 k. at @xmath13 = 3.4 gpa we observe indeed still the presence of hexagonal and cubic sites in the ratio 2:1 , with different hyperfine interactions , but in both cases slightly reduced with respect to the ambient pressure values . starting from @xmath13 = 9.4 gpa , in correspondence with the transition to the dhcp phase ( la - type structure , stable up to @xmath1 15 gpa at 4.2 k ) , the nfs spectra can be properly analyzed with a single set of hyperfine parameters at @xmath18 = 3 k. although in this structure two different crystallographic sites still exist for sm , with hexagonal and cubic coordination , respectively , and in the ratio 1:1 , they seem therefore to be characterized by the same values of their hyperfine parameters . this is in agreement with the resistivity studies of reference @xcite , where only one magnetic phase transition is observed for @xmath13 @xmath14 6 gpa . the dhcp structure of sm metal can be stabilized at ambient pressure , if samples are produced in the form of thick films on appropriate substrates . studies of these films with resonant magnetic x - ray scattering and neutron scattering @xcite reveal that , despite the presence of two crystallographically inequivalent sites , sm still orders antiferromagnetically , as in the sm - type structure , but only one ordering temperature ( @xmath59 = 25 k ) is observed for both hexagonal and cubic sites , owing probably to the enhanced coupling of layers with different coordinations and the consequent increase of the mutual exchange . our results are in perfect agreement with these findings , and the hyperfine interactions we determine from our measurements at 9.4 gpa have an intermediate value between those of the two sites in the sm - type phase ( see figure [ fig : seven ] ) . when pressure is increased further , the successive structural transitions from dhcp to fcc , from fcc to distorted fcc and from distorted fcc to primitive hexagonal are crossed at 15 , 20 and 39 gpa , respectively @xcite . at 39 gpa the coexistence of the two phases is responsible for the presence of two sets of hyperfine parameters as shown in figure [ fig : seven ] , however , at 46 gpa the pure primitive hexagonal phase is observed . from figure [ fig : seven ] it is evident that at 3 k the strengths of both magnetic hyperfine field and electric quadrupole interaction decrease considerably in the pressure range between 0 and 46 gpa , reaching the values of @xmath31 = 218(5 ) t and @xmath33@xmath34 = -0.53(3 ) mm / s at the highest pressure , corresponding to relative reductions of @xmath1 40% and @xmath1 70% with respect to the ambient pressure values , respectively . in the same pressure interval , the volume of the unit cell shrinks by almost 50% @xcite . the large reduction of the magnetic hyperfine field can be due to the variation with pressure of its different components . the largest contribution to the measured value is due to the 4__f _ _ electrons and is directly related to their magnetic moment , but other contributions arise from the polarization of core electrons and of conduction electrons . in sm compounds the core polarization has the same sign as the 4__f _ _ contribution but a considerably smaller value ( that can be estimated to amount to @xmath1 15 t ) , and it is generally considered as not dependent on pressure @xcite . the conduction electron polarization can , on the other hand , give a large contribution to the hyperfine field , with possibly also a large pressure dependence . at the present stage it is therefore not possible to state clearly whether the strong reduction of @xmath31 is due to a corresponding reduction of the 4__f _ _ magnetic moment or to a simple change in the balance between the 4__f _ _ and the conduction electron contributions . however , the decrease of @xmath31 is associated to an even larger decrease of the electric quadrupole interaction @xmath33@xmath34 , which can not be explained by the variation of its lattice component alone . we therefore believe that the large decrease of both @xmath31 and @xmath33@xmath34 at the same time can be taken as an indication that the 4__f _ _ magnetic moment of sm is reduced by pressure . moreover , although the decrease of the hyperfine parameters is mostly continuous , the entrance into the primitive hexagonal high pressure phase at 39 gpa seems to be accompanied by a small discontinuity ( downwards ) of both @xmath31 and @xmath33@xmath34 , which can be interpreted as a further sign of the incipient delocalization of the 4__f _ _ electrons , which are proposed to be involved in the chemical bonding in this phase @xcite . in conclusion we have studied the effect of external pressure on the magnetic and electronic properties of several sm compounds , which are nearly divalent ( sms ) , intermediate valent ( smb@xmath0 ) or trivalent ( sm metal ) at ambient pressure . in both sms and smb@xmath0 , where the valence is known to increase smoothly with pressure , we observe the onset of long - range magnetic order at a critical pressure which corresponds approximately to the entrance of these systems into the metallic state . the ordered state is very stable towards pressure and has the properties expected for trivalent compounds , although both systems are still intermediate valent when the magnetic order appears . for pressures just below that for the onset of long - range order we observe the presence at low temperatures of short - range magnetic correlations of dynamical origin . these correlations are found to persist in smb@xmath0 even at ambient pressure . in the case of metallic sm , pressure induces a strong decrease of the magnetic hyperfine field and the electric quadrupole interaction at the @xmath29sm nuclei . this can be related to a smooth decrease of the sm 4__f _ _ moments when pressure increases . the small discontinuity in the pressure dependence of the hyperfine field when sm undergoes the transition into the primitive hexagonal phase at 39 gpa might indicate the onset of the delocalization of the 4__f _ _ electrons . however , further studies are necessary at higher pressures in order to obtain the full delocalization .

we report on the study of the response to high pressures of the electronic and magnetic properties of several sm - based compounds , which span at ambient pressure the whole range of stable charge states between the divalent and the trivalent . our nuclear forward scattering of synchrotron radiation and specific heat investigations show that in both golden sms and smb@xmath0 the pressure - induced insulator to metal transitions ( at 2 and @xmath1 4 - 7 gpa , respectively ) are associated with the onset of long - range magnetic order , stable up to at least 19 and 26 gpa , respectively . this long - range magnetic order , which is characteristic of sm@xmath2 , appears already for a sm valence near 2.7 . contrary to these compounds , metallic sm , which is trivalent at ambient pressure , undergoes a series of pressure - induced structural phase transitions which are associated with a progressive decrease of the ordered 4__f _ _ moment .

resonant inelastic x - ray scattering ( rixs ) has emerged as a powerful technique to study momentum dependent magnetic- and charge - excitations in correlated materials.@xcite in particular a significant insight into the dynamics of high - temperature superconducting cuprates has been gained ( for a recent review see ref . ) . while cu @xmath11-edge rixs ( soft rixs ) is well suited for studying magnetic excitations in cuprates,@xcite cu @xmath0-edge rixs ( hard rixs ) is useful for understanding their charge dynamics.@xcite however in the case of iron based superconductors , experiments have been limited in both regimes . unlike cuprates , fe in pnictides / chalcogenides is in a tetrahedral environment , where on - site mixing between fe @xmath12 and @xmath13 orbitals is allowed . this mixing allows strong @xmath14 type fluorescence emission that dominates most rixs spectra,@xcite which is not the case in cuprates where cu is in octahedral environments . observation of charge or spin excitations in these fe compounds with rixs is thus very challenging , as the rixs features are usually much weaker than the fluorescence line . in their fe @xmath0-edge rixs experiment , jarrige and coworkers circumvented this problem by utilizing resonance at a much higher incident energy ; here the contribution from the fluorescence line was much smaller . comparison of their results with _ ab initio _ calculation supported a moderate coulomb repulsion of @xmath15 ev in the parent prfeaso compound.@xcite at the fe @xmath11-edge where spectra are also dominated by a strong fluorescence signal,@xcite a recent rixs experiment succeeded in observing paramagnons in ( ba@xmath16k@xmath17)fe@xmath3as@xmath3,@xcite in a similar fashion to a recent study on cuprates.@xcite although it was difficult to extract momentum dependence of the observed excitations , due to extremely low count rate , these results are encouraging . moreover they demonstrate that both fe @xmath0- and @xmath11-edge rixs can provide us with new insight into the physics of the iron based superconductors , just like for cuprates . among iron based superconductors , alkali metal iron selenides present an interesting case for rixs studies . these compounds , with a generic chemical composition a@xmath18fe@xmath19se@xmath3 ( a = k , cs and rb),@xcite are in many ways unique among isostructural iron pnictides.@xcite for instance , measurements of the fermi surfaces , by angle - resolved photoemission spectroscopy ( arpes),@xcite show that these systems exhibit no nesting properties of electron- and hole - like fermi surfaces , which is incompatible with the fermi surface driven spin - fluctuation mediated superconductivity.@xcite however , one of the unresolved issues in these materials is the intrinsic phase separation . there is not a consensus on whether the parent compound is insulating,@xcite semiconducting,@xcite or even metallic.@xcite a phase separation @xcite has been suggested to explain these observations . interestingly , the problem of phase separation , which exists in the superconducting materials ( insulating / magnetic and metallic domains),@xcite seems to disappear in the insulating samples which can be obtained for a particular value of @xmath20 ( e.g. k@xmath1fe@xmath2se@xmath3 ) . @xcite moreover , for that same value of @xmath20 , a novel blocked antiferromagnetic ( afm ) order with a large magnetic moment is found . @xcite these observations suggest that k@xmath1fe@xmath2se@xmath3 could give us an opportunity to study an insulating compound with fe @xmath0-edge rixs . in this work , we report an fe @xmath0-edge rixs study of insulating k@xmath1fe@xmath2se@xmath3 . we find a sharp @xmath21-excitation around 1 ev , whose spectral weight and line shape change as momentum transfer is varied . a broad fe @xmath12 interband transition is also observed at much higher energy of 3 - 7 ev , which is momentum - independent . calculations based on 70 band @xmath4 model , using a moderate @xmath5 ev , can capture the @xmath22 ev feature , which is found to have a dominant @xmath8 and @xmath9 orbital character , while a 102 band orbital model , which takes into account the fe @xmath12 states as well , is needed to describe the interband transition . our findings suggest that although k@xmath1fe@xmath2se@xmath3 is an insulator , it has a @xmath23 comparable to that of metallic iron pnictides . in addition , we discuss the observed behavior of @xmath21-excitations in k@xmath1fe@xmath2se@xmath3 in comparison with a similar excitation in prfeaso @xcite and cuprates . the rixs experiment was carried out at the advanced photon source using the 30id merix spectrometer . a spherical ( 1 m radius ) diced ge(620 ) analyzer was used and an overall energy resolution of 230 mev ( fwhm ) was obtained . the same experimental configuration was used in ref . . the energy calibration was based on the absorption spectrum through a thin fe foil . most of the measurements were carried out in a horizontal scattering geometry near * q * = ( 0 0 11 ) for which the scattering angle 2@xmath24 was close to 90@xmath25 , in order to minimize the elastic background intensity . the sample was freshly cleaved just before being mounted on a closed - cycle refrigerator . details of the growth and the characterization of single - crystal samples were reported earlier.@xcite in - plane dc resistivity and magnetic susceptibility data confirm that our sample is afm insulator . throughout this paper , we use for simplicity the tetragonal high temperature @xmath26 unit cell with two fe atoms per lattice point ( @xmath27 = @xmath28 = 3.8 @xmath29 and @xmath30 = 13.6 @xmath29 ) . in this notation the observed fe vacancy order and the blocked afm order , associated with the @xmath31 tetragonal @xmath32 unit cell , appear at * q*@xmath33 = ( 0.2 , 0.6 , 0 ) and * q*@xmath34 = ( 0.4 , 0.2 , 0 ) , respectively.@xcite the high x - ray energy used allows us to keep the rotation of the sample within 10@xmath25 when measuring over the whole brillouin zone , therefore minimizing any matrix element effect on our spectra . ( color online ) ( a ) fe @xmath0-edge pfy - xanes taken by monitoring the intensity of k@xmath35 emission line @xmath36 as a function of incident energy ( @xmath37 ) . ticks represent the incident energies used for our energy dependence in ( b ) and the blue triangle our resonance energy . ( b ) rixs spectra as a function of the incident energy . measurements were carried out at * q * = ( 0.5 0.5 10.75 ) at t = 15 k. the spectra have been shifted vertically for clarity ( horizontal ticks ) . ] the fe @xmath0-edge x - ray absorption near - edge spectra ( xanes ) taken in the partial fluorescence yield mode is shown in fig . [ fig01](a ) . the spectra were obtained by monitoring the intensity of k@xmath35 emission line @xmath36 as a function of incident energy ( @xmath37 ) . two distinct features are seen in the spectra , a sharp pre - edge peak around @xmath37 = 7.111 kev , corresponding to excitations of @xmath38 electrons into the empty fe @xmath39 states hybridized with se @xmath12 states , and the main edge around @xmath37 = 7.120 kev , corresponding to excitations into mostly empty fe @xmath12 states.@xcite the incident energy dependence of the rixs spectra for energies around the pre - edge ( as indicated by vertical lines in fig . [ fig01](a ) ) is plotted in fig . [ fig01](b ) . a broad and strong inelastic feature is seen in the @xmath403 - 7 ev range ( @xmath41 = @xmath42 with @xmath43 as the energy of the outgoing x - ray ) . we assign this feature to an fe @xmath12 interband transition . such a transition is possible since the absence of a center of symmetry at the tetrahedral fe site allows a significant on - site mixing between fe @xmath13 and @xmath12 orbitals to occur , pushing some of the fe @xmath12 states below the fermi level . therefore one can excite electrons from predominantly @xmath12 band to the hybridized @xmath13-@xmath12 band just above the fermi level . in between the fe @xmath12 interband transition and the elastic peak ( @xmath41 = 0 ) we observe a clear shoulder - like rixs feature around @xmath44 1 ev . their intensity and position dependence on incident energy , @xmath37 , was investigated . we fitted two spectral features using two gaussian functions of fixed widths and a linear background to account for the k@xmath35 emission line at higher energy loss . in fig . [ fig04 ] ( a ) we plot the integrated intensity of the 1 ev feature and the fe @xmath12 interband transition as a function of @xmath37 ; in both cases a large resonance enhancement near @xmath37 = 7.111 kev is observed . at higher incident energies the fe @xmath12 interband transition loses intensity and evolves into the k@xmath45 emission line ( includes @xmath46 transition ) . since emission occurs at fixed outgoing photon energy ( @xmath43 ) , the peak position as a function of energy transfer ( @xmath47 ) is proportional to @xmath37 , following a linear dashed line as shown in fig . [ fig04](b ) . the 1 ev peak is however only visible around the resonant incident energy . ( color online ) ( a ) the evolution of the integrated intensity of both the fe @xmath12 interband transition and the 1 ev features around the pre - edge . ( b ) peak position of the fe @xmath12 interband transition as a function of @xmath37 . numbers were derived from fit ( see text ) . ] ( color online ) ( a ) momentum dependence of the low energy rixs spectra of k@xmath1fe@xmath2se@xmath3 obtained at @xmath48 k. contribution from the elastic line has been subtracted . the spectra have been shifted vertically for clarity ( horizontal ticks ) and the solid lines are three - point smoothed spectra . superimposed as dashed red line is the smoothed spectrum taken at @xmath49-point . in ( b ) a schematic diagram of the ( h k 0 ) reciprocal space is shown . the brillouin zone ( bz ) corresponding to the tetragonal unit cell ( with two fe per lattice point ) is shown as a solid line . the filled circles are the points where rixs spectra in ( a ) are taken . ( c ) wide range rixs spectrum at @xmath49- , @xmath50- , and @xmath51-points . ] further insight into the nature of the 1 ev feature can be gained by measuring its momentum dependence . in fig . [ fig02 ] ( a ) we plot the low energy part of the rixs spectra of k@xmath1fe@xmath2se@xmath3 after subtracting contributions from the elastic line . the elastic line background was obtained by measuring the off - resonance spectrum at @xmath52 kev , as done previously in the study of two - magnon excitations in cuprates.@xcite each spectrum was normalized by the fe @xmath12 interband transition intensity , which does not show any variations with * q*. this is evident from fig . [ fig02 ] ( c ) where we compare spectra at @xmath49- , @xmath50- , and @xmath51-points . such a lack of momentum dependence allows us to use the fe @xmath12 interband transition to normalize each spectrum in fig . [ fig02 ] ( a ) . the momentum depedence of the 1 ev feature was measured along the high - symmetry directions shown in the brillouin zones ( bz ) in fig . [ fig02 ] ( b ) , where @xmath49 is * q * = ( 0 0 11.5 ) , @xmath50 is * q * = ( 0.5 0 11.25 ) , and @xmath51 is * q * = ( 0.5 0.5 11.3 ) . although overall intensity and peak position do not change drastically , one can clearly observe the difference between the @xmath49 point spectrum ( middle ) with the zone boundary spectra in either direction ( top or bottom ) . at the @xmath49 position , the spectral weight is at higher energy side , with maximum intensity occurring near 1.1 ev . as you move away from @xmath49 , the center of mass of the feature shifts to lower energy and the peak seems to become sharper with maximum intensity near 0.9 ev . since we only have data for a limited number of * q*-points and the spectral variation is small , extracting any dispersion relation is difficult . to emphasize the momentum dependence we have superimposed the @xmath49-point data as dashed red lines for each spectrum . we can clearly observe enhanced intensity around 0.9 ev for both zone - boundary @xmath51- and @xmath50-points ; scans along @xmath53 and @xmath54 exhibit similar variations . in order to understand the origin of the observed spectral features and their momentum dependence , we performed first - principles electronic structure calculations for @xmath55 fe vacancy ordered k@xmath56fe@xmath57se@xmath3 using the wien2k code.@xcite by fitting the band structure to a tight - binding model , using the wannier90 code,@xcite the rixs spectra could be explicitly calculated . short description on the calculation method is found in appendix [ app : rixstheory ] , for detailed description see refs . . we used two different tight - binding models . the first is a 102-orbital model , which includes fe @xmath13 , fe @xmath58 , fe @xmath12 , and se @xmath12 orbitals in order to reproduce the density of states ( dos ) near the fermi level ( @xmath59 ) as precisely as possible . the second , a 70-orbital model , inluding only fe @xmath13 and se @xmath12 orbitals , is used to calculate the low - energy rixs spectral weight , which has a strong contribution from the correlated fe @xmath13 electrons . we note that since the band structure calculations overestimate the bandwidth of the states in the energy range -2 ev @xmath60 1 ev , a renormalization factor of @xmath61 was needed . this factor is in fact consistent with the 2 - 2.5 bandwidth renormalization factor previously used to describe arpes results with lda calculations . @xcite in fig . [ fig03 ] ( a ) the fe 3@xmath7/4@xmath62 and se 4@xmath62 dos from the 102-orbital model is plotted . since the fe @xmath12 partial dos extends across @xmath59 , the fe @xmath12 states below @xmath59 should contribute significantly to the rixs spectra . the following excitation process is considered : a @xmath38 core - electron is promoted , through the absorption of a photon ( black arrow ) , into the empty fe @xmath12 states and subsequently an fe @xmath13 or @xmath12 electron below @xmath59 fills the @xmath38 core - hole ( blue arrows ) , leaving behind an excited state . in fig . [ fig03 ] ( b ) we plot this calculated rixs spectrum as obtained from the 102-orbital model at the @xmath49- and @xmath51-point . to account for the observed optical gap we have shifted the calculated spectrum by 350 mev towards higher energy loss . the overall agreement between the experimental results and our calculated rixs spectrum is very good . we can see that the two separate excitations at 1 ev and 3 - 7 ev are largely accounted for by the occupied density of states . we note that , since the fe 4@xmath62 states ( intermediate state ) have stronger hybridization with neighboring se or fe 4@xmath62 states than adjacent fe 3@xmath7 states , the 3 - 7 ev excitation becomes more pronounced than the 1 ev excitation . however , despite the good overall agreement , the observed momentum dependence of the 1 ev excitation can not be accounted for by the current 102-orbital model , which ignores electron correlation effect . since the low - energy rixs spectral features are expected to originate from fe @xmath13 and se @xmath12 , we carried out a separate calculation using the 70-orbital model which only focuses on these orbitals . this reduction in number of orbitals allows us to include the electron correlation , which is too computationally taxing for the 102-orbital model . in the 70-orbital calculation , processes to screen the @xmath38 core - hole are included and the random - phase approximation is used to account for fe @xmath13 electron correlations.@xcite the block - checkerboard afm ordering @xcite is described within the hartree - fock approximation . @xcite the on - site coulomb integrals are included as @xmath63 ( intra - orbital ) , @xmath64 ( inter - orbital ) , @xmath65 ( hund s coupling ) . for @xmath66 ev , an energy gap of about 540 mev at @xmath59 and an ordered moment of 3.5 @xmath67 are obtained , which is in agreement with neutron scattering@xcite and optical conductivity.@xcite recall that the bandwidth was normalized by a factor of @xmath61 in our band structure calculation to be consistent with arpes data . @xcite in order to keep the physically meaningful ratio @xmath68 unchanged , the experimentally relevant energy scale is @xmath69 . we start by investigating the effect of coulomb repulsion , @xmath10 , on our calculated rixs spectrum . in fig . [ fig05 ] we plot the contribution from fe @xmath13/se 4@xmath62 states to the rixs spectrum over wide energy range as derived from the 70-orbital model . calculations are shown for both @xmath49-point ( red ) and @xmath51-point ( blue ) for 1 ev @xmath70 3.5 ev . overall rixs spectrum changes significantly as @xmath10 is varied . as @xmath10 increases from 1 ev to 2.5 ev we notice that the spectrum above 1 ev , which arises from fe - se @xmath71 charge - transfer ( ct ) , is suppressed . in particular , the large difference between the @xmath49- and m - point spectra present for @xmath72 ev more or less disappears for @xmath73 ev . by increasing @xmath10 above 2.5 ev , the momentum dependence of the spectrum below 1 ev is also greatly suppressed , resulting in almost quenched momentum dependence in the entire energy range . based on these calculations and our results in fig . [ fig02](a ) , where variations with momentum were limited to the spectrum below 1 ev , we find that @xmath74 ev gives satisfying agreement with our results . we would like to point out that this @xmath10 value is in line with values obtained for a realistic blocked afm state in a five orbital hubbard model.@xcite this suggests that even in this insulating compound , a moderate coulomb repulsion similar to the iron pnictides is sufficient.@xcite in order to examine the momentum dependence in detail , the low - energy region of the calculated rixs spectra from the 70-orbital model with @xmath73 ev are plotted in fig . [ fig06](a)-(c ) . in fig . [ fig06](a ) , contributions from the fe @xmath13 and se @xmath12 states are highlighted . while these states have rather limited contributions below 1 ev at the @xmath49-point , a large increase of spectral weight below 1 ev is observed at the @xmath51-point . by looking at the orbital resolved contribution of the fe @xmath13 states at the @xmath51-point in fig . [ fig06 ] ( b ) we notice that the 1 ev feature originates mostly from the fe @xmath13 states ( lighter - shaded area ) , with the largest contribution from a transition involving the @xmath8 and @xmath9 orbitals ( darker - shaded area ) : @xmath75 , @xmath76 , @xmath77 , and @xmath78 . ( color online ) ( a ) contribution from fe @xmath13/se 4@xmath62 states at different transferred momentum . ( b ) orbital resolved contribution of the fe @xmath13 states at @xmath51-point . the lighter - shaded area represents the total contribution from the fe @xmath13 states while the darker - shaded area a transition into only the @xmath8 and @xmath9 orbitals . the plot demonstrates that the 0.9 ev feature consists mostly of excitation into the @xmath8 and @xmath9 orbitals . ( c ) difference between the calculated rixs spectra at momentum * q * and @xmath49-point . ( d ) difference between the experimental rixs spectra from fig . [ fig02](a ) and the @xmath49-point spectrum ( dashed red line in fig . [ fig02](a ) ) . ] in fig . [ fig06 ] ( c ) the low energy rixs spectrum , as obtained from the 70-orbital model , is shown for momentum transfers along the @xmath49-@xmath50 and @xmath49-@xmath51 directions . in order to emphasize the change with respect to the spectrum at @xmath49-point , the @xmath49-point spectrum has been subtracted from each spectrum . in fig . [ fig06 ] ( d ) , we plot the experimental data in the same manner . here we have subtracted the dashed red line ( @xmath49-point ) from the respective rixs spectra in fig . [ fig02 ] ( a ) . the calculation shows the spectral weight around 1 ev ( gray vertical bar ) increases significantly as we move away from @xmath49-point . similar momentum dependence , although weaker , is observed in the experimental data . therefore , we conclude that including moderate correlation energy @xmath10=2.5 ev in the 70-orbital calculation allows us to describe the observed momentum dependence of the 1 ev feature in our rixs data . to summarize our results , we find two spectral features , a sharp 1 ev peak and a broad feature around 3 - 7 ev , in our fe @xmath0-edge rixs investigation of k@xmath1fe@xmath2se@xmath3 , an insulating iron chalcogenide . the observed low - energy feature exhibits a weak * q*-dependence , while the broad high energy peak is * q*-independent . overall energy positions and intensities of these two excitations can be captured using the 102-orbital model , which include fe @xmath13 , @xmath58 , and @xmath12 , and se @xmath12 states . however , the 102-orbital model , which does not take into account electron correlation , can not describe the * q*-dependence of the 1 ev feature . by using instead the 70-orbital model , which focuses on the fe @xmath13 and se @xmath12 states and takes into account electron correlation , a satisfactory description of the 1 ev feature is achieved using @xmath73 ev . poor experimental statistics prevented us from extracting quantitative dispersion relation of the 1 ev feature . given that there seems to be multiple transitions contributing to this feature from our calculation , the momentum dependence could be due to changing spectral weight of these transitions . however , we point out that since the momentum dependent 1 ev feature consists mostly of excitations involving the @xmath8 and @xmath9 orbitals a sizable orbital dependent correlations could exist in the insulating k@xmath1fe@xmath2se@xmath3 . these are the same orbitals that are believed to give rise to the observed nematicity in the 122 feas.@xcite the @xmath79 orbital , which was found to go through metal - insulator ( mott ) transition in the superconducting 122 fese samples,@xcite does not make a significant contribution to the 1 ev feature . our observation seems to lend support to theoretical models that consider both orbital- and spin - fluctuation to contribute to the strong pairing interaction needed for superconductivity.@xcite we would like to mention that chen and coworkers observed a sharp double - peak structure below 1 ev in their optical conductivity measurements of k@xmath1fe@xmath2se@xmath3 , which were attributed to arising from the particular magnetic structure.@xcite each fe spin has two types of neighbors in this block antiferromagnetic ordered state : one aligned ferromagnetically and the other antiferromagnetically with the original spin . intersite @xmath21-excitations to these different spin arrangements have different energies , and the two peak structure in optical spectrum arises from these two transitions . however , one can not directly compare rixs data with these optical spectroscopy observations due to the difference in the response function these techniques are probing . rixs follows symmetry selection rules of raman scattering , and mostly sensitive to _ intrasite _ @xmath21-excitation , rather than _ intersite _ @xmath21-excitation . perhaps this is the reason why we observe only one peak in our rixs data , since local @xmath21-excitation should always satisfy the spin - selection rule . next , we compare the current results with previous fe @xmath0-edge rixs experiment on prfeaso.@xcite overall spectra bear many similarities , but there are some differences . first , in both systems a non - dispersive excitation at 3 - 5 ev was found . the fe @xmath12 interband transition energy is also higher in k@xmath1fe@xmath2se@xmath3 by about 600 mev , when compared to prfeaso.@xcite in addition , the higher incident energy used in the study of prfeaso seems to suppress the intensity of the fe @xmath12 interband transitions . as a result , the 3 - 5 ev feature in prfeaso was mostly associated with ct . in k@xmath1fe@xmath2se@xmath3 we also find in our 70-orbital model a non dispersive fe - se @xmath71 ct excitation in the same energy interval ( 3 - 7 ev ) as the fe @xmath12 interband transition . however , the ct excitation could not be resolved in our rixs spectrum and is instead hidden under the strong fe @xmath12 interband transition . secondly , although in both k@xmath1fe@xmath2se@xmath3 and prfeaso a low energy @xmath21-excitation with a dominant @xmath9 and @xmath8 orbital character is found , their behaviour with momentum are quite different . whereas in prfeaso the @xmath21-excitation is present at the @xmath49-point and disperses towards higher energy ( bandwidth @xmath80 ev ) with increasing in - plane momentum , the 1 ev feature in k@xmath1fe@xmath2se@xmath3 is much weaker at the @xmath49-point and exhibits very small dispersion when in - plane momentum is increased . since the dispersion of the @xmath21-excitation in prfeaso was associated with local magnetic correlations , reflecting the collinear antiferromagnetic order , we speculate that the lack of much momentum dependence of the 1 ev feature might be influenced by the different magnetic ordering pattern . finally , it is quite illuminating to compare the rixs spectra in fe based superconductors and cuprates . the most prominent difference between the rixs spectra of these two families of compounds is the difference in scattering cross - section . cuprate rixs spectra usually show prominent and well - defined charge - transfer excitations between 2 - 7 ev , while iron based materials do not show such a well defined feature . this contrast can be ascribed to the difference in the hybridization of the respective @xmath7-orbitals . in cuprates the spatial overlap between cu @xmath81 orbitals and oxygen @xmath82 orbitals is quite large , and the hybridization is sizable . in the cu @xmath0-edge process in cuprates the screening of the cu @xmath38 core hole is mainly provided by the charge transfer from the oxygen @xmath62 orbitals , resulting in salient features associated with the charge - transfer excitation in the spectra . on the other hand , in iron based superconductors fe @xmath7 orbitals are in tetrahedral environment of pnictogen or chalcogen atoms , the hybridization between orbitals from the two sites is thus much smaller . in addition , the tetrahedral environment allows on - site fe @xmath83 hybridization . therefore , there exist more channels to screen the @xmath38 core hole in the fe @xmath0-edge rixs experiment , since fe @xmath7 , @xmath62 , and se @xmath62 states are all available for the screening . as a result , the spectral features are all quite weak and perhaps there are more contributions than one dominant charge - transfer excitation . another consequence of the fe @xmath84 hybridization is the large fluorescence signal in iron compounds . since significant fe @xmath12 density of states are found near the fermi level , fluorescence occurs at low energy transfer , which overlaps with the rixs features . on the other hand , cu @xmath12 dos is far away from the fermi level in cuprates , and does not usually interfere with the rixs spectra . for these two reasons , rixs investigation of iron based superconductors are much more challenging . depending on the specific spectral feature one wants to focus on , a judicious choice of incident energy should be considered in the experimental design process . we have successfully measured charge excitations in the insulating k@xmath1fe@xmath2se@xmath3 using fe @xmath0-edge resonant inelastic x - ray scattering ( rixs ) . our key observation is the appearance of a sharp excitation around 1 ev when the incident energy is tuned to the pre - edge , as well as a broad spectral feature around 3 - 7 ev . this low energy @xmath21-excitation shows clear momentum dependence of the spectral weight and line shape , while the high energy peak is due to fe @xmath12 interband transitions . calculations based on 70 orbital model , using a moderate @xmath85 ev , indicate that the 1 ev feature originates from the correlated fe 3@xmath7 electrons with a dominant @xmath8 and @xmath9 orbital character , emphasizing the importance of those orbitals across different families of iron based superconductors . the momentum dependence of the @xmath21-excitation in k@xmath1fe@xmath2se@xmath3 is found to be quite different from a similar excitation in prfeaso,@xcite we speculate that this most likely originates from their dissimilar magnetic ordering . our results show that a moderate @xmath10 is in qualitative agreement with our rixs spectrum , suggesting that comparable correlations can be found in the insulating and metallic iron based superconductors . to discuss the rixs process microscopically , we consider the following form of hamiltonian : where @xmath87 and @xmath88 describe the inner - shell @xmath38 electrons and the dipole - transition by x - rays , respectively . @xmath89 describes the electrons ( fe-@xmath13 , se-@xmath12 , etc . ) near the fermi level ( for this part we use the 102-orbital or 70-orbital model ) . @xmath90 is the coulomb interaction between @xmath38 and @xmath13 electrons at fe sites . for @xmath38 electrons , we take completely localized @xmath38 orbitals at each fe site : where @xmath92 is the one - particle energy of the fe-@xmath38 state , @xmath93 and @xmath94 are the creation and annihilation operators of @xmath38 electrons with spin @xmath95 at fe site @xmath96 , respectively . @xmath88 describes resonant @xmath38-@xmath12 dipole transition induced by x - rays : where @xmath98 is the creation operator of fe @xmath99 electron ( @xmath100 ) at site @xmath96 with spin @xmath95 , and @xmath101 is the annihilation operator of a photon with momentum @xmath102 and polarization @xmath103 . the matrix elements of @xmath104 are given in the form : @xmath114 \delta ( \omega + e_{j_1}({\bf k})- e_{j_2}({\bf k } + { \bf q } ) ) \nonumber \\ & & \times \biggl| \sum_{\mu,\mu ' = x , y , z } \sum_i \sum_{\sigma } w_{\mu}({\bf r}_i ; { \bf q } , { \bf e } ) w_{\mu'}^*({\bf r}_i ; { \bf q } ' , { \bf e } ' ) \frac{u_{4p_{\mu}(i)\sigma , j_2}^*({\bf k}+{\bf q})u_{4p_{\mu'}(i)\sigma , j_1}({\bf k } ) } { \omega + \tilde{\varepsilon}_{1s}({\bf r}_i ) - e_{j_2}({\bf k}+{\bf q } ) } \biggr|^2 \nonumber\\ \label{eq : w4p}\end{aligned}\ ] ] @xmath115 \delta ( \omega + e_{j_1}({\bf k})- e_{j_2}({\bf k } + { \bf q } ) ) \nonumber \\ & & \times \biggl| \sum_{\mu = x , y , z } \sum_i w_{\mu}({\bf r}_i ; { \bf q } , { \bf e } ) w_{\mu}^*({\bf r}_i ; { \bf q } ' , { \bf e } ' ) \sum_{\ell_1\ell_2 } \sum_{\sigma_1\sigma_2 } \nonumber\\ & & v_{1s-3d}({\bf r}_i ) \lambda_{\ell_2\sigma_2 , \ell_1\sigma_1}({\bf r}_i ; q ) u_{\ell_2\sigma_2 , j_2}^*({\bf k}_1+{\bf q } ) u_{\ell_1\sigma_1 , j_1}({\bf k } ) \nonumber\\ & & \times \int_{e_f}^{\infty } d \varepsilon \frac { \rho_{4p_{\mu}(i)}(\varepsilon ) } { [ \omega + \tilde{\varepsilon}_{1s}({\bf r}_i ) - \varepsilon ] [ \omega ' + \tilde{\varepsilon}_{1s}({\bf r}_i ) - \varepsilon ] } \biggr|^2 \label{eq : w3d}\end{aligned}\ ] ] for fe @xmath38 core - hole screening processes @xcite , where @xmath116 is the energy of diagonalized band @xmath117 , @xmath118 is the electron occupation number of band @xmath117 at @xmath119 , @xmath120 are the diagonalization matrix elements , @xmath121 and @xmath122 are orbital and spin indices for fe @xmath13 electrons . @xmath123 and @xmath124 are the four - vectors of incoming and outgoing photons , respectively , and @xmath125 , where @xmath126 and @xmath127 are energy loss and momentum transfer , respectively . @xmath128 means the @xmath129 state at fe site @xmath112 with spin @xmath95 . @xmath130 is the fe @xmath12 density of states , and is calculated by the band structure calculation . @xmath131 is a vertex function , which is calculated within rpa to take account of fe @xmath13 electron correlations . @xmath132 , where @xmath133 is the damping rate of the @xmath38 core hole and set to 0.8 ev in the present study . summations in @xmath96 should be restricted only to eight fe sites in the unit cell . we calculate eq . ( [ eq : w4p ] ) using the 102-orbital model , and eq . ( [ eq : w3d ] ) using the 70-orbital model , starting from the identical first - principles band structure . the incident photon energy @xmath134 is set to the pre - edge peak .

we report an fe @xmath0-edge resonant inelastic x - ray scattering ( rixs ) study of k@xmath1fe@xmath2se@xmath3 . this material is an insulator , unlike many parent compounds of iron - based superconductors . we found a sharp excitation around 1 ev , which is resonantly enhanced when the incident photon energy is tuned near the pre - edge region of the absorption spectrum . the spectral weight and line shape of this excitation exhibit clear momentum dependence . in addition , we observe momentum - independent broad interband transitions at higher excitation energy of 3 - 7 ev . calculations based on a 70 band @xmath4 orbital model , using a moderate @xmath5 ev , indicate that the @xmath61 ev feature originates from the correlated fe 3@xmath7 electrons , with a dominant @xmath8 and @xmath9 orbital character . we find that a moderate @xmath10 yields a satisfying agreement with the experimental spectra , suggesting that the electron correlations in the insulating and metallic iron based superconductors are comparable .

in this paper , we confirm that the isomorphism conjecture is true for all baumslag - solitar groups . our main theorem is as follows the k- and l - theoretical isomorphism conjecture is true for every baumslag - solitar group with coefficients in any additive category . independently , g. gandini , s. meinert and h. rping proved the isomorphism conjecture with coefficients in an additive category for the fundamental group of any graph of abelian groups in @xcite , which includes all baumslag - solitar groups . recall that the baumslag - solitar group @xmath0 is defined by @xmath1 and all the solvable ones are isomorphic to @xmath2 . note that @xmath3 . when @xmath4 , @xmath5 are cat(0 ) groups , hence one can use @xcite and @xcite to conclude @xmath5 satisfies the isomorphism conjecture with coefficients in an additive category . since the construction of a @xmath6 space with universal cover a cat(0 ) space is enlightening for part of our proof , we construct it here . we start with two cylinders @xmath7 $ ] , @xmath8 $ ] with @xmath9 is the standard circle with radius @xmath10 , @xmath11 has radius @xmath12 . we will glue @xmath13 to @xmath14 by a @xmath10-fold ( orientation preserving ) local isometry covering map . and we do the same to @xmath15 and @xmath16 . what we get is a @xmath17 space with universal cover cat(0 ) . if we reverse orientation on the second gluing , we will get a @xmath18 space with universal cover cat(0 ) . in this paper , we will actually prove the isomorphism conjecture for @xmath0 with finite wreath products and coefficients in an additive category . the k- and l - theoretical isomorphism conjecture with finite wreath products and coefficients in an additive category is known for all solvable baumslag - solitar groups , see @xcite and @xcite . hence we only need to prove the case when @xmath19 . we will abbreviate the k- and l - theoretical isomorphism conjecture with finite wreath products and coefficients in an additive category by fjcw . the idea is to analyze the preimages of cyclic subgroups corresponding to a certain linear representation of @xmath0 by using bass - serre theory , and conclude they satisfy fjcw . our results relies on previous work on fjcw for cat(0 ) groups and virtually solvable groups by bartels and lck in @xcite , and wegner in @xcite , @xcite . note that fjcw implies the isomorphism conjecture with coefficients in an additive category . we list some inheritance properties and results on fjcw that we will need . for more information about fjcw we refer to @xcite section 2.3 . [ qut ] ( 1 ) if a group @xmath20 satisfies fjcw , then every subgroup @xmath21 satisfies fjcw . + ( 2 ) if @xmath22 and @xmath23 satisfy fjcw , then their direct product @xmath24 and free product @xmath25 satisfy fjcw . + ( 3 ) let @xmath26 be a directed system of groups ( with not necessarily injective structure maps ) . if each @xmath27 satisfies fjcw , then the direct limit @xmath28 satisfies fjcw . + ( 4 ) let @xmath29 be a group homomorphism . if @xmath30 and @xmath31 satisfy fjcw for every cyclic subgroup @xmath32 then g satisfies fjcw . + ( 5 ) cat(0 ) groups satisfy fjcw . + ( 6 ) virtually solvable groups satisfy fjcw . in particular @xmath33 } } \rtimes_{\frac{m}{n } } \mathbb{z}$ ] satisfies fjcw , where @xmath34 . + proof of ( 1 ) - ( 4 ) can be found for example in @xcite section 2.3 . ( 5 ) is the main result of @xcite and @xcite . ( 6 ) is proved in @xcite . in this section , we prove our main theorem . recall @xmath35 . we assume @xmath36 . let @xmath37 } } \rtimes_{\frac{m}{n } } \mathbb{z}$ ] . @xmath38 is a solvable linear group . there is a map @xmath39 mapping @xmath40 to @xmath41 and @xmath42 to @xmath43 . note that fjcw is known for @xmath38 by proposition [ qut ] ( 6 ) . hence by proposition [ qut ] ( 4 ) in order to prove fjcw for @xmath0 , we only need to prove fjcw holds for any subgroup @xmath31 of @xmath38 where @xmath44 is a cyclic subgroup of @xmath38 . viewing @xmath0 as an hnn extension , we have an associated oriented tree @xmath45 and @xmath0 acts on it without inversion , see for example @xcite , i.3.4 . let @xmath46 be the cyclic subgroup in @xmath0 generated by @xmath40 , @xmath47 be the subgroup generated by @xmath48 . then the vertices of @xmath45 are left cosets @xmath49 , and edges are the left cosets @xmath50 where @xmath51 . the edge @xmath50 connects from the tail vertex @xmath49 to the head vertex @xmath52 . the stabilizer of the vertex @xmath49 is the subgroup @xmath53 , and the stabilizer of the edge @xmath50 is @xmath54 . at each vertex , there are @xmath10 edges going out and @xmath55 edges going in . see figure [ tree2 - 3 ] for a picture of @xmath56 . we will show @xmath31 is a free group by showing it acts freely on the tree @xmath45 ( see for example @xcite , i.4.1 ) . for every vertex @xmath49 in @xmath45 , its stabilizer under the action of @xmath0 is the cyclic subgroup @xmath53 . let @xmath59 , then for any @xmath60 , @xmath61 note that @xmath44 is generated by @xmath57 with @xmath58 and the second coordinate of @xmath62 is @xmath63 . hence @xmath64 . note that @xmath65 restricted to @xmath46 is injective , hence it is injective when restricted to @xmath53 . therefore @xmath66 . we conclude that for every vertex @xmath49 , the stabilizer for the action of @xmath31 on @xmath45 is trivial . therefore @xmath31 acts freely on @xmath45 . now by proposition [ qut ] ( 5 ) , @xmath31 satisfies the fjcw . we now start the proof of proposition [ case2 ] . by equation ( 1 ) in the proof of proposition [ case1 ] , one sees that @xmath70 for any @xmath71 . hence @xmath72 , and @xmath73 . hence vertex or edge stabilizers for the action of @xmath68 on @xmath45 are the same as for the @xmath0 action . in particular , all the vertex and edge stabilizers are isomorphic to @xmath74 . moreover , for every edge @xmath50 , the stabilizer @xmath75 embeds into the tail vertex stabilizer @xmath76 by multiplying @xmath10 . correspondingly , @xmath75 embeds into the head vertex stabilizer @xmath77 by multiplying @xmath55 . note if we add one relation @xmath78 to the group @xmath0 then the new group is isomorphic to @xmath74 . hence there is a surjective homomorphism @xmath79 . if we denote the projection from @xmath33 } } \rtimes_{\frac{m}{n } } \mathbb{z}$ ] to @xmath74 by @xmath80 , then @xmath81 . and if @xmath82 , then @xmath83 . the action of @xmath84 on @xmath45 has the following key property . [ loo ] for any @xmath82 , the oriented geodesic connecting @xmath49 and @xmath85 contains an even number of edges , half of them are compatible with the orientation on @xmath45 , while the other half are oppositely oriented . we first define an algebraic distance " function @xmath86 . note the tree @xmath45 has a standard metric with edge length @xmath12 . for any two points @xmath87 , there is a unique oriented geodesic connecting @xmath88 to @xmath30 . @xmath89 is the distance from @xmath88 to @xmath30 counted with signs . in more detail , when the geodesic coincide with tree orientation , it contributes positively ; otherwise negatively . for example , in figure [ tree2 - 3 ] , @xmath90 , @xmath91 . note @xmath92 , @xmath93 for any @xmath94 . since @xmath0 acts on @xmath45 via orientation preserving isometries , @xmath95 , for any @xmath51 . in particular , @xmath96 since @xmath40 is in the stabilizer of the vertex @xmath46 . and @xmath97 , hence @xmath98 . note for any @xmath82 , @xmath83 . without loss of generality , we can write @xmath84 as a word @xmath99 , @xmath100 . then @xmath101 . therefore , @xmath102 @xmath103 + @xmath104 + @xmath105 + @xmath106 + @xmath107 + in general , @xmath108 since @xmath109 . the lemma now follows . to see this just observe that @xmath110 can be reduced to the oriented geodesic connecting @xmath49 and @xmath85 in a finite number of simple moves of the form @xmath111 where @xmath112 is an oriented edge and @xmath113 is the same edge with the opposite orientation . ( and @xmath114 , ets . means concatenation of paths . ) + summarize what we have so far , the quotient @xmath115 is an oriented graph of groups with the following properties : every vertex or edge stabilizer is isomorphic to @xmath74 , the edge stabilizer embeds into its tail vertex stabilizer by multiplying @xmath10 , and to its head vertex stabilizer by multiplying @xmath55 ; moreover , every oriented loop @xmath110 in @xmath115 has an even number of edges , where exactly half of them coincide with the orientation . this is a consequence of addendum [ add ] and the observation that @xmath110 lifts to an oriented path @xmath116 in @xmath45 with initial point and end point identified by the action of some element in @xmath68 . choosing a base vertex @xmath120 in @xmath117 , since its stabilizer is @xmath74 , we construct a cylinder @xmath121 $ ] with radius @xmath12 . for every edge and every other vertex , we will construct a cylinder @xmath121 $ ] with radius determined inductively by the following criteria : + _ _ + we now explicitly define these radius so that the above criteria are satisfied . we start by defining the radius @xmath122 of the cylinder corresponding to a vertex @xmath123 of @xmath117 as follows . let @xmath110 be an oriented path connecting @xmath120 to @xmath123 . define the integer @xmath124 to be the number of positive ( agreeing ) edges in @xmath110 minus the number of negative ( disagreeing ) edges . to see that @xmath124 is well defined , let @xmath125 be a second such path . then the concatenation @xmath126 is an oriented loop in @xmath115 which ( as observed above ) must have the same number of @xmath127 edges as @xmath128 edges . now let @xmath129 and then the radius @xmath130 of an edge @xmath112 is defined by @xmath131 where @xmath123 is the head vertex of @xmath112 . now we glue all these cylinders together by local isometries according to the graph @xmath117 . we glue them together using the following method . + therefore we have constructed a 2-complex which is a model for @xmath132 . it is now easy to check that its universal cover is a cat(0 ) space ; cf . @xcite , p502 , theorem i.2.7 . this completes the proof of proposition [ case2 ] .

in this paper , we prove the k- and l - theoretical isomorphism conjecture for baumslag - solitar groups with coefficients in an additive category .

we show in fig . [ fig : interwrtd ] the inter - band fock contribution to @xmath74 for three different densities @xmath101@xmath94 . especially for lower densities , it can be clearly seen that the inter - band contribution changes its sign close to the critical value @xmath8 nm . for higher densities , however , the crossover occurs at somewhat smaller values of @xmath7 . the reason for this tendency can be traced to the fact that , for larger densities , other terms besides the dominant hh - related contributions become important in the sum in eq . ( [ eq : fockself2 ] ) . inter - band exchange contribution to the inverse thermodynamic density of states @xmath74 as a function of the hgte quantum - well width @xmath7 for carrier sheet densities @xmath102@xmath94 ( blue solid curve ) , @xmath103@xmath94 ( red dashed curve ) , and @xmath104@xmath94 ( green dotted curve ) . the dashed vertical line indicates the value of the critical well width @xmath100nm.,width=226 ] spin - orbit coupling due to structural inversion asymmetry ( sia ) and bulk inversion asymmetry ( bia ) can be straightforwardly incorporated into the bhz hamiltonian . in the following , we discuss the influence of those types of spin - orbit coupling separately . _ sia_. the leading contribution due to sia arising from the presence of a perpendicular electric field @xmath105 is linear in the wave vector and given by @xcite @xmath106 0 & 0 & 0 & 0 \\[2 mm ] i r_0k_+ & 0 & 0 & 0 \\[2 mm ] 0 & 0 & 0 & 0 \end{pmatrix}~.\ ] ] typical values are @xmath107nm@xmath108 @xcite and @xmath109 @xcite . we define the sia - related dimensionless parameter @xmath110 ( @xmath111 in a typical hgte quantum well @xcite ) . figure [ fig : sia ] illustrates the effect of sia on @xmath74 in the inverted regime ( @xmath87 nm ) , demonstrating that sia reduces @xmath74 only slightly and that the reduction is larger for higher densities . however , the relative change amounts to a few percent only . for the normal case ( @xmath11 nm ) , the change is even less than 1% . dependence of @xmath112 on the sia magnitude measured in terms of @xmath113 . here @xmath0 ( @xmath114 ) is the thermodynamic density of states obtained with ( without ) sia for quantum - well width @xmath87 nm and @xmath115 ( red dashed curve ) and @xmath116 ( blue solid curve).,width=226 ] _ bia_. the effect of bia can be accounted for by augmenting the bhz hamiltonian , eq . ( [ eq : bhzham ] ) , by the term @xcite @xmath117 0 & 0 & \delta & 0 \\[2 mm ] 0 & \delta & 0 & 0 \\[2 mm ] - \delta & 0 & 0 & 0 \end{pmatrix}~.\ ] ] in fig . [ fig : bia ] , we plot the relative change of @xmath74 due to bia in the inverted regime . the figure shows that bia leads to an increase of @xmath118 by up to 15% , which further increases the difference between the magnitudes obtained for this quantity in the interacting and noninteracting cases . also , in contrast to sia , the effect of bia is larger for smaller densities . dependence of @xmath112 on the bia magnitude @xmath119 . here @xmath0 ( @xmath114 ) is the thermodynamic density of states obtained with ( without ) bia for quantum - well width @xmath87 nm and @xmath120 ( red dashed curve ) and @xmath116 ( blue solid curve ) . in actual samples , @xmath121mev @xcite.,width=226 ]

varying the quantum - well width in an hgte / cdte heterostructure allows to realize normal and inverted semiconducting band structures , making it a prototypical system to study two - dimensional ( 2d ) topological - insulator behavior . we have calculated the zero - temperature thermodynamic density of states @xmath0 for the electron - doped situation in both regimes , treating interactions within the hartree - fock approximation . a distinctively different behavior for the density dependence of @xmath0 is revealed in the inverted and normal cases , making it possible to detect the system s topological phase through measurement of macroscopic observables such as the quantum capacitance or electronic compressibility . our results establish the 2d electron system in hgte quantum wells as unique in terms of its collective electronic properties . _ introduction._capacitance measurements are a premier tool to elucidate the electronic properties of two - dimensional ( 2d ) electron systems @xcite . they fundamentally probe the thermodynamic density of states , @xmath1 where @xmath2 and @xmath3 denote the 2d system s electronic sheet density and chemical potential , respectively . more specifically , @xmath0 is related to the quantum capacitance per unit area @xmath4 and the electronic compressibility @xmath5 via @xmath6 the intriguing interplay between single - particle and coulomb - interaction contributions to @xmath0 has been intensely studied theoretically , both for conventional 2d electron systems realized in heterostructures @xcite and few - layer graphene @xcite . in particular , the tendency towards negative electronic compressibility in the low - density limit @xcite has attracted a lot of attention @xcite . here we show how the thermodynamic density of states of electrons in an hgte quantum well exhibits behavior different from any of the previously studied 2d electron systems , essentially because of the anomalous properties of an interaction - related inter - band contribution relevant for narrow - gap systems . our work provides new insight complementing the observation of unusual electric - transport properties in this system @xcite that relate to the existence of an unconventional , inverted , 2d electronic band structure when the quantum - well width @xmath7 is larger than a critical value @xmath8 nm @xcite . the deeper understanding derived from our results also enables novel characterization of topological phases @xcite in other 2d @xcite and bulk @xcite materials and extends the general knowledge about unusual collective properties of topological and dirac - semimetal systems @xcite . density dependence of the quantum capacitance per unit area for electrons in an hgte quantum well . the red solid ( blue dashed ) curve is obtained for a quantum - well width @xmath9 nm ( @xmath10 nm ) corresponding to the topological ( normal ) situation . clearly distinguishable opposite trends emerge in the low - density regime.,width=283 ] we calculate the thermodynamic density of states for electrons in hgte quantum wells , taking coulomb interactions into account within the hartree - fock approximation . to be specific , we focus on two experimentally feasible situations with quantum - well widths @xmath11 nm and @xmath12 nm , respectively , and present predictions for @xmath0 as a function of the 2d - system s fermi wave vector . in our calculations , crucial effects arising from the finite width of electronic bound states in the hgte / cdte heterostructure are included . quite generally , we find that interaction contributions significantly affect @xmath0 and , thus , observables such as the quantum capacitance and the electronic compressibility . see fig . [ fig : qcapac ] for a pertinent example . more specifically , it turns out that the inter - band exchange correction depends strongly on the quantum - well width and changes its sign for a value close to @xmath13 . we elucidate the underlying mechanisms such as the interplay of band - structure parameters that lead to this interesting behavior . _ model and formalism._the theoretical framework for our calculation of many - particle effects for electrons in an hgte quantum well is based on the bhz hamiltonian @xcite . the latter adequately describes the relevant single - particle states in the low - energy band structure , using basis functions @xmath14 , which are superposition of conduction - electron and light - hole ( lh ) states , and the heavy - hole ( hh ) states @xmath15 . within the representation defined by the basis - state vector ( @xmath16 ) , the bhz hamiltonian is block - diagonal and given by @xmath17 0&\mathcal{h}^{(- ) } \end{pmatrix } \quad , \label{eq : bhz}\end{aligned}\ ] ] with @xmath18 , @xmath19 and @xmath20 where @xmath21 are the pauli matrices . the quantum number @xmath22 distinguishes spin-1/2 projections parallel to the quantum - well growth direction , and the effective band - structure parameters @xmath23 are functions of the quantum - well width @xmath7 @xcite . for simplicity , we set the irrelevant overall energy shift @xmath24 to zero . the sign of the gap parameter @xmath25 distinguishes the ordinary and inverted - band situations : using the convention @xmath26 , the system is in the topological ( normal ) regime when @xmath27 ( @xmath28 ) . the energy eigenvalues of the bhz hamiltonian ( [ eq : bhzham ] ) are given by @xcite @xmath29 where @xmath30 distinguishes conduction and valence bands , both of which are doubly degenerate in @xmath31 . due to the inherent axial symmetry of the bhz model , the eigenvectors of the two @xmath32 matrices @xmath33 in eq . ( [ eq : bhzham ] ) can be expressed as @xmath34 in terms of the polar coordinates @xmath35 for wave vector @xmath36 , with @xmath37^{\frac{1}{2 } } \\[3 mm ] s \left[1 + \frac { \alpha(bk^2-m ) } { \sqrt{a^2 k^2 + ( bk^2-m)^2}}\right]^{\frac{1}{2 } } \end{pmatrix}\ ] ] and @xmath38 . _ quantum many - body effects._the single - particle band dispersions given in eq . ( [ eq : disper ] ) are renormalized by interaction effects . assuming that the electrostatic ( hartree ) terms are compensated by the influence of a neutralizing background charge density , we focus here on the exchange ( fock ) contributions . the fundamental quasi-2d character of the charge carriers is accounted for by retaining the full @xmath39 dependence of quantum - well bound states through the basis functions @xmath16 for the bhz hamiltonian . the fock self - energy of conduction - band electrons can then be written as @xmath40 \left[\psi^{(s)}_{{{\bm{\mathrm{k}}}}+}(z')^\dagger\cdot\psi^{(s)}_{{{\bm{\mathrm{k}}}}'\pm}(z')\right ] \quad , \ ] ] where @xmath41 measures the coulomb - interaction strength , @xmath42 is the fermi function , and the @xmath43 are six - dimensional spinor wave functions comprising the bands with @xmath44 and @xmath45 symmetry closest to the bulk - material s fundamental gap @xcite . intra-(inter-)band contributions to the fock self - energy are labeled by the subscript @xmath46 @xmath47 . note that terms with @xmath48 vanish for the block - diagonal bhz model given above because of the orthogonality of the associated basis states . however , such contributions do arise when spin - orbit - coupling effects are included . effects of the latter will be discussed briefly at the end of this paper . in the zero - temperature limit , which we consider in the following , the fermi functions in eq . ( [ eq : fockself ] ) reduce to @xmath49 for the fully occupied valence band and @xmath50 , where @xmath51 is the modulus of the fermi wave vector for electrons in the conduction band , and @xmath52 denoted the heaviside step function . to take into account both the in - plane dynamics described by the bhz hamiltonian as well as the nontrivial spinor structure of the bhz - model basis states , we employ subband @xmath53 theory @xcite to write the spinor wave functions @xmath43 as superpositions @xmath54 where the coefficients @xmath55 are the components of the corresponding eigenvectors , eq . ( [ eq : bhzeigenvec ] ) , of the bhz hamiltonian . the six - dimensional spinors @xmath56 are the bhz - model basis - state spinors for zero in - plane wave vector , which are determined by the solutions to a confined - particle problem for the hgte / cdte quantum - well heterostructure . their explicit expressions have been given in the supplemental information of ref . @xcite , where for instance @xmath57 and @xmath58 which are normalized , i.e. , @xmath59 , @xmath60 . as a result , we obtain for the intra- and inter - band contributions to the fock self - energy @xmath61 & & { } \hskip-0.9cm\times\sum_{i , j}\mathcal{f}_{ij}(\phi)~a_{k'\pm , i}^{(s)}a_{k'\pm , j}^{(s)}a_{k+,i}^{(s ) } a_{k+,j}^{(s)}|\psi_{0i}^{(s)}(z)|^2|\psi_{0j}^{(s)}(z')|^2,\nonumber\\\end{aligned}\ ] ] where the integration limits are @xmath62 ( intra - band ) and @xmath63 ( inter - band ) , with @xmath64 being an ultraviolet cutoff . in eq . ( [ eq : fockself2 ] ) , @xmath65 and @xmath66 , with @xmath67 and @xmath68 being the kronecker symbol . the inter - band contribution depends logarithmically on @xmath64 , which is typically chosen to be of the order of the inverse lattice constant @xcite . finally , with the chemical potential given in terms of @xmath51 as @xmath69 , and using the relation @xmath70 , the expression ( [ eq : thdydos ] ) for the thermodynamic density of states can be rewritten as @xmath71 . measuring wave vectors and energies in terms of the bhz - model scales @xmath72 and @xmath73 , the natural unit for @xmath74 is @xmath75 . the fine - structure constant that appears in the exchange - energy contributions to @xmath3 is given by @xmath76 when using @xmath77 as the dielectric constant of hgte . .[tab : input ] parameters of the bhz model applicable for two experimental realizations of hgte quantum wells @xcite having widths @xmath11 nm and @xmath12 nm , respectively . [ cols="<,^,^",options="header " , ] _ numerical results for @xmath0._we now present results obtained for the thermodynamic density of states in normal and topological hgte quantum wells . following the usual convention , @xmath78 is shown as a function of the fermi wave vector . we first consider an hgte quantum well with width @xmath11 nm , which is in the the normal ( non - inverted band - structure ) regime . the associated bhz parameters are given in table [ tab : input ] and correspond to an actual experimental realization @xcite . for the large - momentum cutoff of the inter - band contribution , we choose @xmath79 , with @xmath80 nm being the hgte bulk - material lattice constant . we show the result obtained for @xmath74 in fig . [ fig : compressnor ] , making also explicit the various contributions to @xmath74 . the purely kinetic ( i.e. , noninteracting ) part is given by a constant in the low - density regime , @xmath81 , \ ] ] which has the form expected for an ordinary 2d electron system ) can be expressed as @xmath82 , where @xmath83 is the effective band mass of 2d electrons obtained from the small-@xmath84 expansion of the dispersion ( [ eq : disper ] ) . ] . however , it exhibits a weak dependence on @xmath51 at larger carrier densities due to the hh - lh mixing of quantum - well bound states having finite in - plane wave vector . the intra - band interaction ( fock ) renormalization term is always negative and therefore reduces @xmath74 , thus leading to an enhancement of the electronic compressibility . at low - enough densities , the intra - band contribution drives @xmath74 to negative values . such a behavior is also reminiscent of that of an ordinary 2d electron system diverges @xmath85 in the low - density limit like the fock contribution for an ordinary 2d electron gas @xcite . in real samples , this divergence is cut off by image - charge effects @xcite . ] . in the normal regime ( except very close to the critical well width @xmath86 ) , the inter - band exchange contribution is also negative and thus reduces @xmath74 further . as a result , the crossover from positive to negative values of @xmath74 is shifted to higher densities . this behaviour has to be contrasted to that exhibited by single - layer graphene where the exchange renormalization of @xmath74 is positive @xcite . overall , from the results shown in fig . [ fig : compressnor ] , we see that the exchange contributions strongly influence the electronic compressibility . inverse thermodynamic density of states @xmath78 of an hgte quantum well in the normal regime ( well width @xmath11 nm ) . the red ( blue ) solid curve shows the result with ( without ) interactions . the magenta dashed ( green dot - dashed ) curve is the intra - band ( inter - band ) exchange contribution only . the black dotted curve is the sum of the noninteracting and intra - band exchange contributions.,width=283 ] inverse thermodynamic density of states @xmath78 of an hgte quantum well in the inverted regime ( well width @xmath87 nm ) . the red ( blue ) solid curve shows the result with ( without ) interactions . the magenta dashed ( green dot - dashed ) curve is the intra - band ( inter - band ) exchange contribution only . the black dotted curve is the sum of the noninteracting and intra - band exchange contributions . notice the opposite sign of the inter - band exchange contribution ( green dot - dashed curve ) , which shifts the crossover to negative compressibility to very low carrier densities.,width=283 ] we now consider the inverted regime of an hgte quantum well , which is realized for a well width @xmath88 nm . taking the bhz parameters of a feasible experimental situation corresponding to a well width @xmath87 nm ( see table [ tab : input ] ) , we again calculate the quantity @xmath74 . the result is shown in fig . [ fig : compresstop ] . the most salient feature is that the inter - band exchange contribution is now _ positive _ , like in single - layer graphene @xcite , and considerably larger in magnitude as compared to the situation in the normal regime . in contrast , the intra - band exchange term is of similar magnitude and has the same sign as in the normal case . the kinetic ( noninteracting ) contribution is much larger as compared to the @xmath89-nm case , which is mainly due to the smaller band gap in the present case this can be inferred from eq . ( [ eq : kineticpart ] ) . we see that the electronic compressibility is reduced by up to 35% due to exchange effects as compared with the noninteracting case . this trend is changed only at very low densities where the ( negative ) intra - band contribution becomes dominant . the striking difference observed between the inter - band interaction - renormalization contributions in the topological and normal regimes invites more detailed scrutiny . figure [ fig : compressintrainter ] illustrates the variation of intra - band and inter - band exchange terms as a function of the quantum - well width and @xmath90 are generally only weakly dependent on the hgte quantum - well width @xmath7 ( see table [ tab : input ] ) and , for this calculation , we use the @xmath91 and @xmath90 values for @xmath87 nm . the mass parameter @xmath25 , on the other hand , depends sensitively on @xmath7 , and we extract its functional dependence from the band - edge energies of the conduction and valence bands , which are obtained by appropriate matching conditions derived from the relevant confined - electron problem ( see the supplemental information of ref . ) . note that the bhz parameter @xmath92 does not enter the exchange corrections to @xmath74 . ] for a fixed carrier density @xmath93@xmath94 . the intra - band contribution is always negative and rather insensitive to a variation of @xmath7 . the inter - band contribution , however , depends strongly on the quantum - well width and changes its sign in the vicinity of the critical value @xmath8 nm . also around @xmath13 , due to the vanishing band gap , we can anticipate the onset of a divergence in the inter - band contribution for @xmath95 . figure s1 in the supplemental material shows this even more clearly . we can attribute the sign change in the inter - band exchange contribution to @xmath74 to a complex interplay of band - mixing effects ( due to the terms proportional to @xmath91 in the bhz hamiltonian ) and the change of the band characters when crossing over from @xmath28 to @xmath27 . to be more specific , we find that the heavy - hole term ( @xmath96 ) in eq . ( [ eq : fockself2 ] ) gives generally ( especially for low densities ) the largest contribution to @xmath97 ( as well as to @xmath98 ) , where for @xmath28 ( @xmath27 ) it is a monotonically decreasing ( increasing ) function of @xmath51 . intra - band ( dashed red curve ) and inter - band ( solid blue curve ) exchange contribution to the inverse thermodynamic density of states @xmath74 as a function of the hgte quantum - well width @xmath7 for carrier sheet density @xmath99@xmath94 . the dashed vertical line indicates the value of the critical well width @xmath100nm.,width=283 ] _ effect of spin - orbit coupling._we have extended our calculation of the thermodynamic density of states to the situation with bulk - inversion - asymmetry and structural - inversion - asymmetry spin - orbit coupling @xcite and find that , only for the largest expected magnitudes of the bulk - inversion - asymmetry energy scale of a few mev , results change quantitatively by upto 10% . however , our findings suggest that spin - orbit coupling affects the electronic compressibility of electrons in hgte quantum wells typically only at the percent level . see the supplemental material for more details . _ conclusions._we have presented results for the thermodynamic density of states for electrons in hgte quantum wells in experimentally feasible situations . interaction effects have been included within the hartree - fock approximation . we have also taken into account the finite width of the hgte / cdte quantum - well heterostructure , which is necessary to account for the attenuated coulomb repulsion in the transverse direction . markedly different behavior is exhibited for a well width of @xmath11 nm ( normal regime ) compared to one with @xmath87 nm ( topological regime ) . we have pinpointed the origin of this finding as the sizeable inter - band exchange correction whose sign differs in the topological and normal regimes . thus a measurement of the quantum capacitance of hgte quantum wells , e.g. using hgte double - quantum - well configurations @xcite , provides a useful way to determine the topological state of this system . the enhancement and eventual sign change of the compressibility found in the low - density limit of the non - topological phase is analogous to the behavior exhibited by ordinary 2d electron systems with parabolic dispersion @xcite . in contrast , the compressibility of the 2d electron system in the topological phase is strongly suppressed by coulomb interactions . additional contributions to the compressibility arising from image charges @xcite and disorder @xcite can be straightforwardly included to facilitate the description of real samples . 48ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/physrevb.32.2696 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.99649 [ * * , ( ) ] link:\doibase 10.1103/physrevb.42.3741 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.68.674 [ * * , ( ) ] link:\doibase 10.1103/physrevb.50.1760 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.77.3181 [ * * , ( ) ] link:\doibase 10.1103/physrevb.55.6715 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.84.4689 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.216407 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.82.041412 [ * * , ( ) ] link:\doibase 10.1126/science.1204168 [ * * , ( ) ] link:\doibase 10.1103/physrevb.85.235458 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.116.166802 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1103/physrevb.82.155111 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.201306 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.235407 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.99.226801 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.106805 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.155403 [ * * , ( ) ] link:\doibase 10.1103/physrevb.83.085429 [ * * , ( ) ] link:\doibase 10.1103/physrevb.39.5005 [ * * , ( ) ] link:\doibase 10.1126/science.1148047 [ * * , ( ) ] link:\doibase 10.1126/science.1174736 [ * * , ( ) ] @noop * * , ( ) @noop link:\doibase 10.1126/science.1133734 [ * * , ( ) ] @noop * * , ( ) \doibase http://dx.doi.org/10.1016/j.ssc.2012.09.002 [ * * , ( ) ] link:\doibase 10.1103/physrevb.91.081302 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.83.1057 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.236601 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.136603 [ * * , ( ) ] link:\doibase 10.1146/annurev - 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an amazing characteristic of some old fashion problems is their endurement . the projectile motion is one of them . being one of the main problems used to teach elementary physics , variations and not well known facts about it appear in the physics literature of the xxi century . a search in the web @xcite or in the _ science citation index _ gives an idea of this fact . some of the recent studies deal with the problem of air resistance in the projectile motion and its pedagogical character made of it an excellent example to introduce the lambert @xmath0 function , a special function.@xcite the lambert @xmath0 function is involved in many problems of interest for physicist and engineers , from the solution of the jet fuel problem to epidemics@xcite or , even , helium atom eigenfunctions.@xcite one of those problems is the solution for the range @xmath1 in the case that the air resistance has the from @xmath2.@xcite in this paper we analyze the not well known fact of the geometrical place formed by the maxima of all the projectile trajectories at launch angle @xmath3 and in the presence of a drag force proportional to the velocity , we shall denote this locus as @xmath4 . the resulting locus becomes a lambert @xmath0 function of the polar coordinate @xmath5 departing from the origin . this problem raises as a natural continuation from the nice fact that in the drag - free case such a locus is an ellipse@xcite with an universal eccentricity @xmath6.@xcite the paper is organized as follows . in section [ projectilemotion ] , the set of maxima for projectile trajectories moving under in the presence of air resistance is presented . in section [ solutionlambert ] we find a closed form , in polar coordinates , to express such a geometrical place , @xmath7 . in section [ param ] we present a numerical calculation of the curvature of @xmath7 using the polar angle and the launch angle as parameterizations . additionally , we demonstrate that the synchronous curve is a circle as in the drag - free case in section [ synchronous ] . in section [ conclusions ] we conclude . several approximations in order to consider the air resistance exist in the literature , the simplest is the linear case . in such a case the force is given by @xmath8 where @xmath9 is the mass of the projectile and @xmath10 is the drag coefficient . the units of @xmath10 are @xmath11 . the velocity components are labeled as @xmath12 , with @xmath13 and @xmath14 . the solutions for the position and velocity are obtained trough direct integration of eq.([eqnewton ] ) yielding rcl x(t ) & = & , [ timesolx ] rcl y(t ) & = & -g t / b , [ timesol ] for the coordinates , and @xmath15 for the speeds . we used the initial conditions @xmath16 and @xmath17 and @xmath18 . noticing that the terminal speed is @xmath19 in the @xmath20 axis . for the same initial speed @xmath21 these solutions are function of the launch angle @xmath3 and the locus formed by the apexes is obtained if time is eliminated between the solutions in time in eqs . ( [ timesolx ] ) and ( [ timesol ] ) , giving the equation @xmath22 and considering the value at the maximum , via @xmath23 . the corresponding solution is @xmath24 @xmath25 where we introduce the dimensionless perturbative parameter @xmath26 , the dimensionless length @xmath27 , and noticing that @xmath28 can be expressed as @xmath29 . an alternative procedure consists in set the derivative @xmath30 to zero to obtain the time of flight to the apex of the trajectory and , evaluate the coordinates at that time . the points @xmath31 conform the locus of apexes @xmath4 for all parabolic trajectories as a function of the launch angle @xmath3 . in fig . [ fig : geometricplace ] we plot @xmath4 described by eqs . ( [ xmeqno ] ) and ( [ ymeqno ] ) , for the drag - free case ( in dashed red line ) and for @xmath32 in continuous blue line . several projectile trajectories are plotted in thin black lines . the locus of apexes @xmath4 defined by @xmath31 of eqs . ( [ xmeqno ] ) and ( [ ymeqno ] ) is described parametrically by the launch angle @xmath3 and it changes for different values of @xmath26 . in the next section we shall find a description of @xmath4 in terms of polar coordinates and in a closed form using the lambert @xmath0 function . formed by the apexes of all the projectile trajectories ( continuous line in blue ) given by eqs . ( [ xmeqno ] ) and ( [ ymeqno ] ) in rectangular coordinates or by eq . ( [ w1 ] ) , the last one express @xmath7 in polar coordinates and in term of the lambert @xmath0 function . the dashed red line is the ellipse of eccentricity @xmath6 which represents the drag - free case , i.e. @xmath33 . the parameters are @xmath34 and @xmath32 . ] in order to obtain an analytical closed form of the locus we change the variables to polar ones , i.e. , @xmath35 and @xmath36 . the selection of a description departing from that origin instead of the center or the focus of the ellipse is because the resulting geometrical place is no longer symmetric and the only invariant point is just the launching origin . we substitute the polar forms of @xmath37 and @xmath38 into equations ( [ xmeqno ] ) and ( [ ymeqno ] ) and rearranging terms it must be expressed as @xmath39 the lhs depends on @xmath40 and @xmath41 meanwhile the rhs depends on @xmath3 , however the last angle is a function of @xmath41 and reads as @xmath42 by making @xmath43 from eqs . ( [ xmeqno ] ) and ( [ ymeqno ] ) . in order to obtain @xmath44 we set @xmath45 since eq . ( [ angleeq ] ) allows us to have , implicitly , @xmath46 . we shall return to this point later . hence , we can write eq . ( [ ymeqno2 ] ) as @xmath47 where we multiplied both sides of eq . ( [ ymeqno2 ] ) by @xmath48 . setting @xmath49 and @xmath50 in eq . ( [ pre1 ] ) , it shall have the familiar lambert @xmath0 function form , @xmath51 , from which we can obtain @xmath52 as @xmath53 it is important to note that the argument of the lambert function in this equation is negative for all the values @xmath54 . @xmath55 remains real in the range @xmath56 and have the branches denoted by @xmath57 and @xmath58.@xcite we select the principal branch , @xmath57 , since it is the bounded one , however , for values of @xmath59 there is a precision problem since the required argument values are near to @xmath60 . it is important to stress that in eq . ( [ w1 ] ) the independent variable is the angle @xmath61 and , it constitutes the parameterization of the curve @xmath7 . the polar expression of @xmath7 can also be written in terms of the tree function @xmath62 , giving @xmath63 we recover the drag - free result = 2 [ ellipse0 ] when @xmath64 . an explanation of this unfamiliar form of an ellipse is given in appendix [ appendix1 ] followed by a discussion about the @xmath64 limit of expression ( [ w1 ] ) in appendix [ appendix2 ] . formula ( [ w1 ] ) exhibits the deep relationship between the lambert @xmath0 function and the linear drag force projectile problem , since not only the range is given as this function @xcite . the problem open the opportunity to study the w function in polar coordinates , that , almost in the review of referencei @xcite , is absent . even when it is possible to write the locus in terms of @xmath65 this form does not shows the formal elegance of relation ( [ w1 ] ) . as a function of the launch angle @xmath3 for two different values of @xmath66 and their inverses . ] now we return to equation ( [ angleeq ] ) since we need to solve explicitly it in order to have the function @xmath67 . this task is not trivial since even when we approximate the rhs in expression ( [ angleeq ] ) up to first order in @xmath66 , @xmath68 the inversion is not easy . a way to do the inversion is to expand in a taylor series the rhs and then invert the series term by term.@xcite using _ mathematica _ to perform this procedure up to @xmath69 , we obtain as a result ll ( ) & ( 2 ) - ( 2 ) ^2 + ^2 ( 2 ) ^3 + & + ( 27 - 10 ^3 ) ( 2 ) ) ^4 + [ expantion ] the @xmath66-independent terms had been resumated to yield @xmath70 . however , the series does not converge for values in the argument larger than @xmath71 . the reason is the small convergence ratio for the taylor expansion of @xmath72 . an easier way to perform the inversion is to evaluate @xmath73 using eq . ( [ angleeq ] ) and plot the points @xmath74 , the result is shown in figure [ fig : inverse ] . the result is in agreement with the plot of eq . ( [ expantion ] ) up to its convergence ratio and it is not shown . notice that this method is exact in the sense that we can obtain as many pair of numbers as we need , a function is , finally , a relation one to one between two sets of real numbers . another result is to obtain the derivative @xmath75 , since it shall be needed in the following sections . to this end , we note that both functions increase monotonically and their derivatives are not zero , except at the interval end . hence , we can use the inverse function theorem in order to obtain = . the result is shown in fig . [ fig : invderiv](a ) as well as the second derivative in fig . [ fig : invderiv](b ) . the second derivative is calculated using an approximation to the slope to the function previously calculated and using @xmath76 points in the interval @xmath77 $ ] . a smaller number of points could be considered . as function of @xmath61 for various values of parameter @xmath66 . note that major changes occur for @xmath78 . ] in the drag - free situation , @xmath7 is an ellipse and its description is well know , however , in the presence of linear drag this is not the case . we do not expect that the locus could be a conic section and henceforth we need to characterize it . it is usual to consider curvature , radius of curvature or the length of arc in order to characterize a locus . in the present case we consider the curvature of @xmath7 in both parameterizations , first with the polar angle @xmath41 and secondly with the launch angle @xmath3 . we left the calculus of the length of arc to a posterior work , since the calculations became increasingly complex and the goal of the present section is to start the understanding of @xmath7 and to illustrate the way it can be done using the lambert w function . here and in the rest of the section we drop , for clearness , the subindex @xmath9 in @xmath79 and @xmath41 . the corresponding formula for the curvature @xmath80 for polar coordinates is @xcite @xmath81 in order to use eq . ( [ w1 ] ) . here @xmath61 corresponds a derivative respect to that variable . a direct calculation on the drag - free @xmath82 of eq.([ellipse0 ] ) yields to k_0 = , [ kappa0 ] which have a maximum at @xmath83 . a graph of this results appears in figure [ fig : kpolar](red line ) . note that this value is different from that we obtain if we evaluate @xmath84 , the launch angle of maximum range . the maximum curvature happens at a smaller angle than the angle of maximun range . it is interesting to note that the angle @xmath85 corresponds to a triangle which sides fulfill the relation @xmath86 , a pythagoras triple . using the numerical results for @xmath87 from the previous section , it is possible to carry on the calculation of @xmath80 ( see figure [ fig : kpolar ] ) performing the derivatives of @xmath52 from eq . ( [ w1 ] ) in a direct form and evaluating numerically the required values of @xmath3 and its derivatives . for the required derivatives of @xmath88 we used the expressions @xcite w(x ) = , and = . using this method , we obtained good results for values of @xmath66 up to @xmath89 but we require to calculate arguments of the lambert @xmath0 function near the limit @xmath90 for larger values of @xmath66 . the reliability of our numerical result was done comparing the first and second derivatives of @xmath91 with those corresponding to the ellipse . as can be seen in figure [ fig : kpolar ] , @xmath80 present a maximum in all the cases which can be calculated as well . we left to an ulterior work the analysis of the maxima distribution as a function of the pertubative parameter , that is not the case for the curvature with @xmath3 parameterization as we shall see in the next section . using the polar angle as the parameter from eq . ( [ kappa ] ) . in red line appears the corresponding result for the ellipse , eq . ( [ kappa0 ] ) . the maximum happens at @xmath92 , different from the value of maximum range @xmath93 . ] for the launch angle parameterization of @xmath7 we shall use expression @xcite @xmath94 for calculate the curvature from the rectangular form of equations ( [ xmeqno ] ) and ( [ ymeqno ] ) , where the primes denote derivative respect the parameterization variable , @xmath3 in this case . a direct calculation yields lll & = & ( 1+)^2 . + [ kappa_e ] with ll p_1 ( ) = & 16 + 6 ^ 2 - 8 ^ 22 + 2 ^ 2 4 + & + 30 - 3 + 5 , and ll p_2 ( ) = & 5 + 34 + 3 ^ 2 -4 ^ 2 2 + + & ^2 4 + 10- 53 + + & 5 . in the limit @xmath64 , we recover the drag - free curvature _ 0 = , [ kappa_0 ] which have a maximum at @xmath95 in the interval @xmath96 $ ] , as expected . a plot of @xmath97 for several values of @xmath66 beginning at zero and ending at @xmath98 appears in figure [ fig : angle](a ) . in red appears the drag - free case . note that both extremal values increase for increasing @xmath66 value as @xmath99 and @xmath100 . notice that for small @xmath66 , the @xmath101 crosses the drag - free curvature @xmath102 . for increasing values of parameter @xmath66 according to equation ( [ kappa_e ] ) . the values of @xmath66 are indicated in the inset . ( b ) several important angles as a function of dimensionless parameter @xmath66 are plotted . in lines and diamonds appears the angle , @xmath103 , at which the curvature is maximum . in red circles the angle at which the range is maximum according to exact solution , eq . ( [ exactangle ] ) , and in blue crosses the same angle according to eq . ( [ asympangle ] ) ( see text for discussion ) . in a dashed blue line the angle at which skewness is maximum is plotted . ] the angles @xmath103 at which @xmath101 attain their maxima are obtained in the usual way and requires to solve , numerical or graphically , the equation l 3 ( 1 + ^ * ) 2^ * _ 1(^ * ) q_2(^ * ) + + ^ * q_3(^ * ) q_4(^ * ) = 0 , with ll q_1 = & 4 ( 3 + ^2 ) 2 ^ * + ( -4 - 15^ * + + & 5 3^ * ) ; ll q_2 = & 16 + 6 ^2 - 8 ^2 2 ^ * + 2 ^2 4 ^ * + + & 30 ^ * - 3 ^ * + 5 ^ * ; ll q_3 = & 5 + 3 4 ^*+ 3 ^ 2 - 4 ^2 2 ^ * + + & ^2 4 ^ * + 10^ * - + & 53^ * + 5 ^ * ; and ll q_4 = & 70 + 36 ^2 - 16 ( 1 + 3 ^2 ) 2 ^ * + + & 2 ( 5 + 6 ^2 ) 4 ^ * + 154 ^ * - 31 3 ^ * + & + 7 5 ^*. in figure [ fig : angle](b ) the calculated values of @xmath103 as a function of @xmath66 appear . this angle is between the optimal angle for maximum range ( red circles and blue crosses ) and the angle for the greatest forward skew ( dashed line).@xcite _ skew = , where @xmath104 , valid for @xmath105.@xcite in figure [ fig : angle](b ) the optimal angles are drawn , in red circles the exact result in terms of lambert @xmath0 function@xcite _ max , s = , [ exactangle ] and the approximated result@xcite _ max , w = . [ asympangle ] both expressions are equivalent for large @xmath66 but differ at small @xmath66 , as expected . meanwhile the difference between these angles at small pertubative parameter is unimportant , at large @xmath66 the behavior of the corresponding trajectories is different . one reason is the large asymmetry in the locus formed by the set of apexes . in fig . [ fig : largeeps](a ) we plotted @xmath4 and the corresponding trajectories for the different launch angles for @xmath106 . the blue line corresponds to @xmath4 , note that the maximum height is @xmath107 in contrast to @xmath108 for the drag - free case , however , this can be the case of small friction parameter @xmath10 and large initial velocity @xmath21 giving a large @xmath66 value . in a black line appears the orbit launched at @xmath103 , in red line the corresponding to attain the maximum range and in blue dashed line the orbit with maximum skewness . for @xmath109 . see text for explanation . ] in macmillan s book@xcite the calculation of the synchronous curve was done for the drag - free case . this curve is formed if many projectiles were fired simultaneously from the same point , each one with at different launch angle and same initial speed @xmath21 . the locus will be a circle of radius @xmath110 and center in the point @xmath111 , i.e. @xmath112 here , we shall demonstrate that a circle is the synchronous curve in the linear drag case as well . following reference @xcite , we eliminate the launch angle @xmath113 from the position solutions , in the present case they are equations ( [ timesolx ] ) and ( [ timesol ] ) . we write down @xmath114 and @xmath115 and rearrange the terms to give , @xmath116 substituting these expressions in the identity @xmath117 we obtain @xmath118 with @xmath119 the center and @xmath120 the radius . in order to recover the case where @xmath121 , we consider a taylor expansion for the exponential up to second order in the exponential in eq . ( [ yc ] ) and up to first order in eq . ( [ rc ] ) . the fact that this circle exists in the presence of a drag force is remarkable . we obtained an explicit form for the locus @xmath7 composed by the set of maxima of all the trajectories of a projectile launched at an initial velocity @xmath122 , and in the presence of a linear drag force , @xmath123 , i.e. @xmath7 is the locus of the apexes . in polar coordinates , @xmath7 is written in terms of the principal branch of the lambert @xmath0 function for negative values . this represents the parameterization of the curve by the polar angle @xmath41 only and gives @xmath7 in a closed form and exhibits the deep relationship between the lambert @xmath0 function and the linear drag problem . the curvature of @xmath7 was calculated for different values of the dimensionless parameter @xmath124 in two parameterizations . the first one , the polar parameterization , shows a maximum that slightly departs from the drag - free case in @xmath92 . a wider exploration of the functional dependence respect to @xmath66 is pending due to numerical accuracy in the calculation of the lambert @xmath0 function near the limit at @xmath125 . in the case of a parameterization using the launch angle @xmath3 there is not such a restriction . in this case , the curvature was calculated for a wide range of the parameter @xmath66 yielding maximum at angle values larger than those corresponding to maximum range . comparison with the maximum skewness angle @xcite was also done and the difference is larger than the previous one . as an addendum , we demostrate that the synchronous curve , in this case , is a circle as in the drag - free case . this work was supported by promep 2115/35621 . hhs thanks to m. olivares - becerril for useful discussions and encouragement . ellipse canonical form or the polar form with the origin considered in one of the focus are standard knowledge . in the present case , however , we require to consider the origin of coordinates located in the _ bottom _ of the ellipse , since , in the presence of a drag force , the _ launching origin _ is the only invariant point when we change the drag force value . to obtain the ellipse form , we depart from the drag - free solutions at the locus of the apexes , x_m = , [ xmdf ] and y_m = ^2 , [ ymdf ] being @xmath126 . with the help of the trigonometric relations @xmath127 and @xmath128 we transform the upper equations into 2 & = & , + 2 & = & 1 - . taking the squares in both expressions , summing them and arranging terms , we arrive to ( ( 1 + 3 ^2 _ m ) - 2 _ m ) = 0 . where we used the polar coordinates @xmath35 and @xmath36 . the solutions are @xmath129 and r_m(_m ) = . the second one is the required form for the ellipse . in order to obtain the drag - free limit for the locus @xmath7 given in equation ( [ w1 ] ) , we note that = ( 1/2 ) [ tangent ] and that @xmath131 . the first expression is obtainable from the drag - free solutions eqs . ( [ xmdf ] ) and ( [ ymdf ] ) , and the second is obtained by setting @xmath132 in eq . ( [ fdtheta ] ) . where we used relation ( [ tangent ] ) in order to obtain the last line . using trigonometric identity @xmath133 and eq . ( [ tangent ] ) we obtain that = 1 + 3 ^2 _ m. using this result in the expression of @xmath130 we obtain the desired result , eq . ( [ ellipse0 ] ) . oo see for instance ` scholar.google.com ` . corless , g.h . gonnet , g.h . hare , d.e.g . jeffrey , and d.e . knuth , `` on the lambert @xmath0 function '' , adv . in comp . mathematics * 5 * , 329 - 359 ( 1996 ) . scott , a.lchow , d. bressanini , and j.d . morgan iii , `` the nodal surfaces of helium atom eigenfunctions '' , phys . a , * 75*. 060101 - 060104 ( 2007 ) . r.d.h . warburton and j. wang , `` analysis of asymptotic motion with air resistance using the lambert @xmath0 function '' , am . * 72 * , 1404 - 1407 ( 2004 ) . e. packel and d. yuen , `` projectile motion with resistance and the lambert function '' , coll . j. * 35*(5 ) , 337 - 350 ( 2004 ) . fernndez - chapou , a.l . salas - brito , and c.a . vargas , `` an elliptic property of parabolic trajectories '' , am . * 72*,1109 - 1109 ( 2004 ) . macmillan , _ theoretical mechanics : static and the dynamics of a particle _ ( mcgraw - hill , new york and london , 1927 ) . reprinted in ( dover , new york , 1958 ) , pp . 249 - 254 . thomas , m.b . weir , j. hass , f.r . _ calculus_. 11th ed . ( addisson - wesley , 2004 ) . _ mathematical methods for physicist_. 5th ed . ( academic press , 2000 ) . e. kreyzig . `` principal , normal , osculating circle '' in _ diffential geometry_. ( dover , n.y . , 1991 ) . e.w . wiesstein `` curvature '' from mathworld . ` http://mathworld.wolfram.com/curvature.html ` . steward , `` a little introductory and intermediate physics with the lambert function '' , proc . of the 16th biennial congress of the australian institute of physics . m. colla ed . pp 194 - 197 . australian institute of physics . parville , vic ( 2005 ) . steward , `` characteristics of the trajectory of a projectile in a linear resisting medium and the lambert @xmath0 function '' , australian inst . of physics . 17th . national congress 2006 . wc0035.(2006 ) . in ref . , the author comment that for @xmath134 we obtain the special value @xmath135 with @xmath136 , the golden ratio . however , the solution that appears in the article is not longer valid for @xmath134 . if the solution corresponds to another real root of the equation this special value corresponds to the case when the initial speed is equal to the limit speed @xmath137 . this makes much more intriguing this fact .

we present an analysis on the geometrical place formed by the set of maxima of the trajectories of a projectile launched in a media with linear drag . such a place , the locus of apexes , is written in term of the lambert @xmath0 function in polar coordinates , confirming the special role played by this function in the problem . in order to characterize the locus , a study of its curvature is presented in two parameterizations , in terms of the launch angle and in the polar one . the angles of maximum curvature are compared with other important angles in the projectile problem . as an addendum , we find that the synchronous curve in this problem is a circle as in the drag - free case .

the density matrix @xmath0 is a useful operator for quantum mechanical calculations . for a given system , one may be unsure about what is the state vector . if the possible state vectors and their associated probabilities are @xmath1 , one creates the _ _ proper__@xcite density matrix @xmath2 it is hermitian : @xmath3 . it is trace 1 : @xmath4 , where @xmath5 are an arbitrary complete orthonormal set of vectors . it is positive : @xmath6 for an arbitrary vector @xmath7 . all these properties can easily be verified from eq.([z1 ] ) . one can use the density matrix to conveniently calculate probabilities or mean values . if a measurement is set up to result in one of the eigenstates @xmath8 of an operator @xmath9 , that outcome s probability is @xmath10 and the mean eigenvalue of @xmath9 is @xmath11 . because each individual state vector evolves unitarily under the system hamiltonian * h * ( assumed for simplicity here to be time independent ) , @xmath12 , the density matrix in eq.([z1 ] ) satisfies the evolution equation @xmath13\implies{\bf\rho}(t)=e^{-i{\bf h}t}{\bf\rho}(0)e^{i{\bf h}t}.\ ] ] the operator acting on @xmath14 is often called a _ _ superoperator__@xcite since it describes a linear transformation on an operator : it operates on both sides of @xmath15 , so to speak . the case sometime arises where the system @xmath16 under consideration is a subsystem of a larger system @xmath17 , and @xmath18 is not measured . the pure ( so - called because it is formed from a single state vector ) density matrix for the joint system is @xmath19 where @xmath20 @xmath21 are orthonormal bases for @xmath16 , @xmath18 respectively , and @xmath22 . by taking the trace of @xmath23 with respect to @xmath18 , one arrives at an _ _ improper__@xcite density matrix for @xmath16 from which predictions can be extracted : @xmath24 one can easily see that this is hermitian , trace 1 and positive . however , while the density matrix of @xmath17 evolves unitarily , the density matrix of the subsystem @xmath16 evolving under the influence of @xmath18 generally does not evolve unitarily . nonetheless , sometimes @xmath25 can be written in terms of @xmath0 for a range of times earlier than @xmath26 . sometimes that range is short compared to the time scale of evolution of @xmath0 so that one may make an approximation whereby @xmath25 depends linearly just on @xmath27 . this is quite useful , and is what shall be considered in this paper . in this case , the evolution equation is highly constrained by the requirements on @xmath27 , to be satisfied at all times : hermiticity , trace 1 and positivity . ( the latter proves too general to simply implement , so a stronger requirement is imposed , called complete positivity see section [ secpos ] ) . ) the result , for an @xmath28-dimensional hilbert space , is the lindblad@xcite ( or lindblad - gorini - kossakowsky - sudarshan@xcite ) evolution equation for the density matrix : @xmath29\nonumber\\ & & \quad-\frac{1}{2}\sum_{\alpha=1}^{n^{2}-1}[{\bf l}^{\alpha\dagger}{\bf l}^{\alpha}{\bf\rho}(t ) + { \bf\rho}(t){\bf l}^{\alpha\dagger}{\bf l}^{\alpha}-2{\bf l}^{\alpha}{\bf\rho}(t){\bf l}^{\alpha\dagger } ] . \nonumber\\\end{aligned}\ ] ] in eq.([z3 ] ) , the hamiltonian @xmath30 is an arbitrary hermitian operator , but the lindblad operators @xmath31 are completely arbitrary operators . actually , as shall be shown , there need be no limitation on the number of terms in the sum in eq.([z3 ] ) , but this can always be reduced to a sum of @xmath32 terms . it is not a necessary condition , but if the equation is to be time - translation - invariant , the operators are time - independent . before deriving eq.([z3 ] ) , we give a few examples of the non - unitary evolutions it describes . since unitary evolution is well known , we shall let @xmath33 . it shall be seen that relaxation to some equilibrium ( constant ) density matrix is readily described . for simplicity , four of the five examples shall be in a @xmath34 hilbert space ( a restriction that is readily lifted ) . consider a state vector written in a basis whose phase factors undergo random walk . given an initial state vector @xmath35 ( @xmath36 ) , suppose at time @xmath26 , it has evolved to @xmath37 with probability @xmath38 the density matrix is @xmath39}\big[ab^{*}|\phi_{1}\rangle\langle\phi_{2}|+a^{*}b|\phi_{2}\rangle\langle\phi_{1}|\big ] . \nonumber\end{aligned}\ ] ] we see that the off - diagonal elements decay at a fixed rate while the diagonal elements remain constant . it satisfies @xmath40\big[{\bf \rho}(t)-\sum_{i=1}^{2}{\bf q}_{i}{\bf \rho}(t){\bf q}_{i}\big],\ ] ] where the projection operator @xmath41 . to see that this is a lindblad equation , note that @xmath42 and @xmath43 there are two lindblad operators : identify @xmath44 in eq.([z3 ] ) . suppose in time @xmath45 , a state vector @xmath46 has probability @xmath47 of changing to @xmath48 ( probability @xmath49 of being unchanged ) , where @xmath50 is a hermitian operator and @xmath51 . the density matrix at time @xmath52 is therefore @xmath53 so its evolution equation is @xmath54.\ ] ] this is of the lindblad form , with one lindblad operator @xmath55 . in the basis where @xmath50 is diagonal with elements @xmath56 , we get @xmath57 and @xmath58.\ ] ] so , again , its diagonal elements remain constant . its off - diagonal elements decay at the fixed rate @xmath59 $ ] and their phases change . suppose in time @xmath45 a state vector @xmath60 undergoes a unitary transformation to @xmath61 with probability @xmath62 the density matrix at @xmath52 ( neglecting terms of order higher than @xmath45 ) is given by @xmath63\bigg]\nonumber\\ & & = { \bf \rho}(t)-\frac{\lambda dt}{2}\big[{\bf g}^{2}{\bf \rho}(t)+{\bf \rho}(t){\bf g}^{2}-2{\bf g}{\bf \rho}(t){\bf g}\big]\nonumber\end{aligned}\ ] ] giving the lindblad equation @xmath64=-\frac{\lambda}{2}[{\bf g}[{\bf g } , { \bf \rho}(t ) ] \ ] ] with one lindblad operator @xmath65 . in the basis where @xmath50 is diagonal with elements @xmath56 , @xmath66 so , again , its diagonal elements remain constant but its off - diagonal elements decay at a rate determined by the difference in eigenvalues . suppose , in time @xmath45 , with probability @xmath47 , a state vector @xmath46 exchanges its basis states @xmath67 , becoming @xmath68 , where @xmath69 is the pauli matrix with diagonal elements 0 and off - diagonal elements 1 . it is easy to see that the density matrix evolution equation is @xmath70\ ] ] and is of the lindblad form , with one lindblad operator @xmath71 . the density matrix elements therefore satisfy @xmath72,\nonumber\\ \frac{d}{dt } \rho_{12}(t)&=&-\frac{d}{dt } \rho_{21}(t)=- \lambda[\rho_{12}(t)- \rho_{21}(t)].\nonumber\end{aligned}\ ] ] the diagonal density matrix elements change in this example , decaying to 1/2 . the off - diagonal matrix elements keep their real parts while the imaginary parts decay to 0 . here we consider arbitrary @xmath28 . suppose in time @xmath45 , a state vector @xmath46 makes a transition to state @xmath73 with probability @xmath74 ( @xmath75 , @xmath76 ) . the probability of all such transitions is @xmath77 , so the state vector is unchanged with probability @xmath49 . define @xmath78 . note that @xmath79 . the density matrix at time @xmath52 is @xmath80 so its evolution equation is @xmath81.\ ] ] this is of the lindblad form , with @xmath82 ( one more than the necessary maximum ! ) lindblad operators @xmath83 : the matrix elements of the density matrix obey @xmath84\nonumber\\ & = & -\lambda\big [ \rho_{rs}(t)-p_{r}\delta_{rs}\big].\nonumber\end{aligned}\ ] ] the off - diagonal elements decay at a uniform rate . the diagonal elements do not remain constant . they decay to predetermined values @xmath85 : @xmath86.\ ] ] this might be useful in modeling the approach to thermal equilibrium , where the states @xmath87 are energy eigenstates and @xmath85 is the boltzmann probability @xmath88 . we now turn to deriving the lindblad equation as the most general equation satisfying the constraints . while the hilbert space discussed here shall be assumed of dimension @xmath28 , @xmath28 may be allowed to go to to infinity and , also , the argument may readily be extended to a continuum basis . to make the argument easier to follow , examples of how its steps apply to a two - dimensional hilbert space shall occasionally be inserted . the _ markov _ constraint is that the density matrix @xmath89 at a later time @xmath90 , depends only upon the density matrix @xmath91 at an earlier time @xmath26 , not upon the density matrix over a range of earlier times . the _ linearity _ constraint , combined with the markov constraint , is that the matrix elements of @xmath92 can be written as the sum of constants multiplying the matrix elements of @xmath0 rather than , say , powers of the matrix elements of @xmath0 or any other kind of function of these matrix elements : @xmath93 here @xmath94 with @xmath95 some convenient orthonormal basis and , similarly , @xmath0 is expressed in the same basis . the constants @xmath96 can be functions of @xmath90 , @xmath26 . there are @xmath97 constants , and each can be complex , so there are @xmath98 real constants involved in eq.([x1 ] ) . the _ hermiticity _ constraint @xmath99 , applied to eq.([x1 ] ) , results in @xmath100\rho_{rs}.\ ] ] suppose we have the equation @xmath101 ( the @xmath102 are constants ) which holds for all possible density matrices . then , one can see @xmath103 as follows . first choose the density matrix @xmath104 , with all other elements vanishing : thus , @xmath105 . similarly , one shows @xmath106 . next , employ the density matrix @xmath107 , which results in @xmath108 finally , use the density matrix @xmath109 , @xmath110 , which results in @xmath111 and so @xmath112 . therefore , @xmath103 the four density matrices used here , @xmath113 ( written in terms of the pauli matrices ) we shall call _ the density matrix basis_. any @xmath114 matrix can be written as a linear sum with constant ( complex ) coefficients of these four matrices . more than that , they form a _ matrix basis for hermitian matrices _ , in that any hermitian matrix can be written as a linear sum with constant ( real ) coefficients of these four matrices . more than that , and this is the reason for their deployment here , they form a _ matrix basis for density matrices _ , in that any density matrix can be written as a linear sum with constant positive real coefficients of these four matrices such that the sum of the coefficients add up to 1 . this basis is to be distinguished from another basis , the pauli matrices plus the identity matrix , which we shall call the _ pauli+1 _ basis . this is also a matrix basis for hermitian matrices but it is _ not _ a density matrix basis . generalizing , if we have an equation @xmath115 for a matrix @xmath116 , which holds for all valid @xmath0 , then @xmath103 . this can be seen by using an @xmath117-size density matrix basis , ( generalizing the @xmath118-size density matrix basis of the previous section ) . first choose @xmath119 with all other elements vanishing , which implies @xmath120 . then for particular values of @xmath121 , choose @xmath122 with all other elements vanishing , from which one finds @xmath123 . finally , choose @xmath124 , from which one finds @xmath125 , so @xmath126 . letting @xmath127 , @xmath128 range over all possible pairs of indices results in @xmath103 . it therefore follows from eq.([x2 ] ) that @xmath129 where each index can take on the values 1 or 2 . a matrix @xmath116 for which @xmath130 is a hermitian matrix . therefore , @xmath131 is a hermitian matrix , where we regard the number pairs 11 , 12 , 21 , 22 as four different indices . that is , @xmath131 is a @xmath132 dimensional matrix . the most general @xmath132 hermitian matrix is characterized by 16 real numbers ( the four real diagonal matrix elements and the six complex matrix elements above the diagonal ) . since there are 32 real numbers which characterized the most general superoperator in a two dimensional space , the condition of hermiticity of the density matrix has cut that number in half . a hermitian matrix can be written in terms of its orthonormal eigenvectors and eigenvalues , and that decomposition shall prove very useful here . there are four real eigenvalues , @xmath133 , where @xmath134 . corresponding to each eigenvalue is an eigenvector @xmath135 in the four dimensional complex vector space . the four complex components of @xmath136 make 32 real numbers , but they are constrained . each eigenvector is normalized to 1 : @xmath137 provides 8 constraints , lowering the number of free components to 24 . the orthogonality of @xmath138 to the other three vectors provides 6 constraints , the orthogonality of @xmath139 to the remaining two vectors provides 4 constraints , and the orthogonality of @xmath140 to @xmath141 provides two constraints . thus , there are 12 constraints on the 24 free components , so the eigenvectors contain 12 free components . these , together with the four eigenvalues , comprise the 16 real numbers characterizing @xmath131 . an example of such an orthonormal basis is given by @xmath142 multiplying the pauli+1 basis . if we write the four components of @xmath135 as a four dimensional vector with components @xmath143 $ ] , then @xmath144 has components @xmath145 $ ] , @xmath146 has components @xmath147 $ ] , @xmath148 has components @xmath149 $ ] , @xmath150 has components @xmath151 $ ] , it is easy to verify that this is an orthonormal set of vectors . although each @xmath135 is a vector in a four dimensional space , with four components @xmath152 , @xmath135 can also be regarded as an operator in the two - dimensional hilbert space with four matrix elements @xmath152 . this leads to a neat way of writing the orthogonality relations for these eigenvectors . instead of @xmath153 , we can write @xmath154 where @xmath155 is the hermitian conjugate ( complex conjugate transpose ) of @xmath156 . it is easy to see how this works for the example where @xmath135 is @xmath157 the pauli+1 basis . the expression for the components of @xmath131 written in terms of its eigenvectors and eigenvalues is @xmath158 putting this into eq.([x1 ] ) results in the _ evolution equation _ now , lets impose the _ trace constraint _ , i.e. , @xmath160 . in terms of components this says @xmath161 writing @xmath162 , the trace constraint can be written as @xmath163\rho_{rs}=0\ ] ] or in matrix notation as @xmath164{\bf \rho}=0.\ ] ] where @xmath165 is the unit matrix . this must hold for arbitrary @xmath166 . we have seen how to handle such an expression . by successively putting in the four density basis matrices , we obtain the trace constraint @xmath167 the @xmath28-dimensional case works just like the two - dimensional case . it follows from eq.([x2 ] ) that @xmath131 can be viewed as an @xmath168 hermitian matrix . it has @xmath117 real eigenvalues . its @xmath117 complex eigenvectors @xmath136 satisfy the orthonormality conditions @xmath169 with * a * written in terms of its eigenvalues and eigenvectors , eq.([x1 ] ) becomes the evolution equation @xmath170 @xmath133 and @xmath135 depend upon @xmath171 , but we shall not write that dependence until it is needed . next , imposition of the trace constraint on eq.([x5 ] ) , with @xmath172 , gives @xmath173{\bf \rho } = 0.\ ] ] using the density matrix basis as in eq.([x3 ] ) et seq . , we obtain the trace constraint : @xmath174 by taking the trace of eq.([x6 ] ) and using eq.([x4a ] ) , we find the interesting relation @xmath175 the final constraint is _ positivity_. this says , given an arbitrary n - dimensional vector @xmath176 , that the expectation value of the density matrix @xmath177 is non - negative . this constraint , applied to eq.([x5 ] ) , is @xmath178 where we have defined @xmath179 . positivity of @xmath0 ensures @xmath180 . thus , we see from eq.([x7 ] ) , if all the @xmath133 s are non - negative , then @xmath177 will be positive too . however @xmath181 , while just shown to be _ sufficient _ for @xmath177 to be positive , is not _ necessary_. in the next section , we shall give an example where an eigenvalue is negative , yet @xmath92 is positive ! therefore , a stronger condition than positivity is necessary to ensure that @xmath181 . this condition , presented after the example , is _ complete positivity_. this example uses the pauli+1 eigenvectors @xmath182 and @xmath183 . ( note that the trace constraint ( [ x6 ] ) is satisfied , provided @xmath184 , since the square of each of the pauli+1 matrices is @xmath183 . ) choose @xmath185 : @xmath186 = \begin{bmatrix}\rho_{22}&\rho_{12}\\\rho_{21}&\rho_{11 } \end{bmatrix}.\ ] ] @xmath177 is just @xmath166 with its diagonal elements exchanged . thus , because @xmath166 is positive , then @xmath177 is positive . this is a particularly simple example of a more general case discussed in appendix [ a ] . it is not positivity but , rather , _ complete positivity _ that makes the non - negative eigenvalue condition necessary . here is what it means . add to our system a non - interacting and non - evolving additional system in its own @xmath28-dimensional hilbert space . the enlarged hilbert space is of dimension @xmath117 . the simplest state vector in the enlarged space is a direct product @xmath187 : @xmath8 is a vector from the original hilbert space , @xmath188 is a vector from the added system . the general state vector in the joint space is the sum of such products with c - number coefficients . form an arbitrary density matrix @xmath189 for the enlarged system . suppose it evolves according to eq.([x5 ] ) , where @xmath135 is replaced by @xmath190 ( i.e. , the evolution has no effect on the vectors of the added system . ) complete positivity says that the resulting density matrix @xmath191 must be positive . complete positivity says , given the evolution equation ( [ x5 ] ) , that @xmath192 for an arbitrary @xmath117 dimensional vector @xmath193 and for any initial density matrix @xmath194 in the enlarged hilbert space . we wish to prove that complete positivity implies the eigenvalues are non - negative . what we shall do is judiciously choose a single vector @xmath193 and four pure density matrices @xmath194 so that the expressions @xmath195 are @xmath196 , with a positive constant of proportionality . therefore , for complete positivity to hold , @xmath197 must be non - zero . here are choices that will do the job . we shall choose the maximally entangled vector @xmath198 ( @xmath199 , but it need not be normalized to 1 ) . we construct the state vectors @xmath200 and use them to make four pure density matrices @xmath201 . ( note that @xmath202 because of the orthogonality relation eq.([x4a ] ) ) . then , for one @xmath203 , @xmath204 putting this into eq.([x5 ] ) , the complete positivity condition is @xmath205 ( using the orthogonality relation ( [ x4a ] ) ) . thus , complete positivity implies @xmath206 . we follow the same procedure in the n - dimensional case . however , to be a bit more general , we shall use an arbitrary vector @xmath193 , and an arbitrary pure density matrix @xmath194 : @xmath207 where @xmath208 , @xmath209 are yet to be specified complex constants . the unit trace of @xmath0 in eq.([x8b ] ) requires @xmath210 . then , the complete positivity condition is @xmath211tr[{\bf e}^{\alpha\dagger}{\bf d}{\bf c}^{\dagger}].\end{aligned}\ ] ] now , choose @xmath212 , for any particular @xmath203 . this choice can be made in many ways . two are @xmath213 , @xmath214 ( the choice made in the two - dimensional example just discussed ) or @xmath215 , @xmath216 ( note , both choices respect @xmath210 ) . with this choice in eq.([x9 ] ) , and with use of the orthonormality conditions eq.([x4a ] ) , we obtain as the consequence of complete positivity : @xmath217 we have now applied all the constraints needed to obtain a valid density matrix @xmath92 at a later time @xmath90 from an earlier density matrix @xmath0 at time @xmath26 . this relation is eq.([x5 ] ) , supplemented by the orthonormality conditions ( [ x4a ] ) , the trace constraint ( [ x6 ] ) and the condition of non - negative eigenvalues ( [ x10 ] ) . it is customary to define @xmath218 , so that eqs.([x5 ] , [ x6 ] ) can be written in terms of @xmath219 alone : @xmath220 ( however , the orthonormality conditions , written in terms of @xmath221 , now depend upon @xmath133 ) . eq.([x11a ] ) is called the kraus representation and @xmath222 are called kraus operators@xcite . we have proved the necessity of the kraus representation , but it is also sufficient . that is , for _ any _ @xmath222 satisfying eqs.([x11a],[x11b ] ) , even for more than @xmath117 operators , also with no orthonormality conditions imposed , all the constraints on @xmath92 are satisfied . it is easy to see that hermiticity , trace 1 and positivity are satisfied . complete positivity requires a bit more work , and that is given in appendix [ b ] . this general statement of the kraus representation might seem to imply a larger class than we have derived as necessary , but that is not so . since the kraus representation is hermitian , trace 1 and completely positive , it may be written in the form eq.([x5 ] ) , as we have shown . now that we have satisfied all the constraints on the density matrix @xmath223 , we can let @xmath224 , and obtain the differential equation satisfied by @xmath27 . for the rest of this paper we shall only treat the @xmath28-dimensional case since the argument is precisely identical for the two - dimensional case , except that @xmath34 . first , lets see what we can say about the eigenvectors and eigenvalues when @xmath225 . then , eq.([x5 ] ) says @xmath226\rho_{rs}.\end{aligned}\ ] ] as we have done before , successive replacement of @xmath0 by the @xmath117 members of the density matrix basis results in @xmath227 multiply eq.([16 ] ) by @xmath228 and sum over @xmath229 . use of the orthonormality relation ( [ x4a ] ) gives @xmath230 if @xmath231 and @xmath232 , eq.([17 ] ) says that all the eigenvectors are @xmath233 . but only one of a set of orthogonal eigenvectors can be proportional to the identity . therefore , for the rest of the eigenvectors , @xmath234 and @xmath235 . call one eigenvector @xmath236 . from eq.([17 ] ) , we find the associated eigenvalue @xmath237 . for @xmath238 , the eigenvalues vanish . note that the condition @xmath239 says that these eigenvectors are orthogonal to @xmath240 . and , indeed , in this case , eq.([x5 ] ) becomes the identity @xmath241 when @xmath224 , the eigenvalues and eigenvectors change infinitesimally . accordingly we write @xmath242 , \medspace \lambda ^{\alpha}(dt)=c^{\alpha}dt \medspace(\alpha\neq n^{2}),\nonumber\\ & & { \bf e}^{n^{2}}(dt)=\frac{1}{\sqrt{n}}[{\bf1}+{\bf b}dt ] , \medspace { \bf e}^{\alpha}(dt)= { \bf k}^{\alpha } \medspace(\alpha\neq n^{2}),\nonumber\\\end{aligned}\ ] ] where the @xmath243 are constants . we do not include a term @xmath244 in the expression for @xmath245 since , because @xmath246 , it would contribute a negligible term @xmath247 to eqs.([x5],[x6 ] ) . because the eigenvalues must be positive , and because the eigenvalues sum to @xmath28 ( equation following eq.([x6 ] ) ) , we see that @xmath248 ( all @xmath249 ) . @xmath116 and @xmath250 are restricted by the orthonormality conditions , which we shall look at later . putting eqs.([19 ] ) into the evolution equation ( [ x5 ] ) gives @xmath251[{\bf 1}+{\bf b}dt]{\bf\rho}(t)[{\bf 1}+{\bf b^{\dagger}dt}]\nonumber\\ & & + dt\sum_{\alpha=1}^{n^{2}-1}c^{\alpha}{\bf k}^{\alpha}{\bf\rho}(t){\bf k}^{\alpha\dagger } , \medspace\hbox { or in the limit}\medspace dt\rightarrow 0,\nonumber \\ & & \frac{d}{dt}{\bf\rho}(t)=-c^{n^{2}}{\bf\rho}(t)+{\bf b}{\bf\rho}(t)+{\bf\rho}(t){\bf b^{\dagger}}+\sum_{\alpha=1}^{n^{2}-1}c^{\alpha}{\bf k}^{\alpha}{\bf\rho}(t){\bf k}^{\alpha\dagger}.\nonumber \\ \end{aligned}\ ] ] putting eqs.([19 ] ) into the trace constraint ( [ x6 ] ) gives @xmath252[{\bf 1}+{\bf b^{\dagger}dt][{\bf 1}+{\bf b}}dt]\nonumber\\ & & \qquad\qquad\qquad+dt\sum_{\alpha=1}^{n^{2}-1}c^{\alpha}{\bf k}^{\alpha\dagger}{\bf k}^{\alpha}={\bf 1},\medspace\hbox{or,}\nonumber\\ & & c^{n^{2}}{\bf 1}= { \bf b}+{\bf b}^{\dagger}+\sum_{\alpha=1}^{n^{2}-1}c^{\alpha}{\bf k}^{\alpha\dagger}{\bf k}^{\alpha}.\end{aligned}\ ] ] using ( [ 21 ] ) to replace @xmath253 in ( [ 20 ] ) ( specifically , @xmath254 $ ] ) results in @xmath255\nonumber\\ & & -\frac{1}{2}\sum_{\alpha=1}^{3}c^{\alpha}[{\bf k}^{\alpha}{\bf k}^{\alpha\dagger } { \bf\rho}(t)+ { \bf\rho}(t){\bf k}^{\alpha}{\bf k}^{\alpha\dagger}-{\bf k}^{\alpha}{\bf\rho}(t){\bf k}^{\alpha\dagger}].\nonumber\\ \end{aligned}\ ] ] if we define @xmath256 and @xmath257 , the evolution equation ( [ 22 ] ) becomes the lindblad equation ( [ z3 ] ) : @xmath258\nonumber\\ & & -\frac{1}{2}\sum_{\alpha=1}^{n^{2}-1}[{\bf l}^{\alpha}{\bf l}^{\alpha\dagger } { \bf\rho}(t)+ { \bf\rho}(t){\bf l}^{\alpha}{\bf l}^{\alpha\dagger}-2{\bf l}^{\alpha}{\bf\rho}(t){\bf l}^{\alpha\dagger}].\nonumber\\ \end{aligned}\ ] ] it is a consequence of this derivation that the lindblad operators @xmath259 in eq.([23 ] ) are not arbitrary operators , because they are restricted by the orthonormality conditions ( [ x4a ] ) . putting eqs.([19 ] ) into eq.([x4a ] ) constrains @xmath116 , @xmath250 . for what follows , we recall from the discussion in section ii that @xmath260 can be regarded in two ways . in one way , they are regarded as @xmath32 operators acting on vectors in an @xmath28 dimensional space , with matrix elements @xmath261 ( @xmath262 ) . in the other way , they are regarded as @xmath32 vectors in an @xmath117 dimensional space , each with components ( @xmath263 ) . in particular , the trace of two operators is the same as the scalar product of two vectors , as in eq.([x4a ] ) the orthonormality relation ( [ x4a ] ) , applied successively to @xmath264 , @xmath265 , @xmath266 , with use of eqs.([19 ] ) , are [ 24 ] @xmath267=0,\label{24a}\\ & & tr{\bf k}^{\alpha}=0,\label{24b } , \medspace(\alpha=1 , ... n^{2}-1)\\ & & tr { \bf k}^{\alpha}{\bf k}^{\beta\dagger}=\delta^{\alpha\beta } \medspace(\alpha , \beta=1 , ... n^{2}-1).\label{24c}\end{aligned}\ ] ] eq.([24a ] ) says that the hermitian part of @xmath116 vanishes . this provides no restriction at all on @xmath30 , which is the anti - hermitian part of @xmath116 . eq.([24c ] ) says that the vectors @xmath250 are orthonormal . eq.([24b ] ) says that @xmath268 , which implies that @xmath269 completes the orthonormal set . we shall now show that the lindblad equation ( [ 23 ] ) with _ arbitrary _ lindblad operators ( no constraints whatsoever ) can be transformed to new , _ constrained _ , lindblad operators of eq.([22 ] ) by adding a constant ( achieving the vanishing trace constraint ( [ 24b ] ) ) followed by a unitary transformation(achieving the orthogonality constraint ( [ 24c ] ) ) . first , we see that we can transform the arbitrary lindblad operators @xmath259 to lindblad operators @xmath270 which are traceless . define @xmath271 , where the @xmath272 are @xmath32 constants , and substitute that into the lindblad equation , obtaining : @xmath273\nonumber\\ & & -\frac{1}{2}\sum_{\alpha=1}^{3}[{\bf l}'^{\alpha}{\bf l}'^{\alpha\dagger } { \bf\rho}(t)+ { \bf\rho}(t){\bf l}'^{\alpha}{\bf l}'^{\alpha\dagger}-2{\bf l}'^{\alpha}{\bf\rho}(t){\bf l}'^{\alpha\dagger}].\nonumber \end{aligned}\ ] ] with a redefinition of @xmath30 , this is again the lindblad equation , expressed in terms of @xmath270 . by choosing @xmath274 , the new lindblad operators satisfy @xmath275 . now write the @xmath32 lindblad operators @xmath259 ( hereafter assumed traceless ) in terms of @xmath32 new operators @xmath276 ( @xmath249 , @xmath277 ) using the linear transformation @xmath278 we ask what the matrix @xmath279 must be in order that the lindblad form be unchanged ( @xmath280 replacing @xmath259 in eq.([23 ] ) ) . with arbitrary operators @xmath281 , @xmath282 , @xmath283 , @xmath284 ( the three terms in the lindblad equation have two of the @xmath285 while the third @xmath286 ) . this equals @xmath287 , leaving the lindblad equation unchanged in form , if and only if the matrix @xmath279 is unitary , @xmath288 . inverting eq.([25 ] ) , we see that @xmath276 is traceless . now , consider the matrix @xmath289 . it is hermitian , so its eigenvalues are real and its eigenvectors are orthogonal . it can be brought to diagonal form by properly choosing our unitary transformation , so we obtain @xmath290 this is almost the orthogonality constraint ( [ 24c ] ) . the eigenvalues @xmath291 are non - negative , since it follows from eq.([26 ] ) that @xmath292 thus , according to eq.([26 ] ) , the @xmath276 are orthogonal vectors , with squared norm @xmath291 . we can add one more vector @xmath293 to complete the set , orthogonal to the rest since @xmath294 we can define new operators @xmath295 which are orthonormal and traceless , by @xmath296 . in terms of these operators , the lindblad equation ( [ 23 ] ) written in terms of @xmath276 becomes @xmath297\nonumber\\ & & -\frac{1}{2}\sum_{\alpha=1}^{n^{2}-1}\tilde c^{\alpha}[\tilde{\bf k}^{\alpha}\tilde{\bf k}^{\alpha\dagger } { \bf\rho}(t)+ { \bf\rho}(t)\tilde{\bf k}^{\alpha}\tilde{\bf k}^{\alpha\dagger}-2\tilde{\bf k}^{\alpha}{\bf\rho}(t)\tilde{\bf k}^{\alpha\dagger}].\nonumber\\ \end{aligned}\ ] ] this is precisely eq.([22 ] ) with @xmath298 , @xmath299 , replacing @xmath250 , @xmath300 . moreover , the orthonormality constraints eqs.([24b],[24c ] ) on @xmath250 are satisfied by @xmath298 . we have shown that the constraints on the density matrix @xmath177 mandate its evolution equation ( [ 22 ] ) , where @xmath301 and @xmath302 satisfy the constraints ( [ 24 ] ) . we then showed that this is completely equivalent to the lindblad eq.([23 ] ) , with no constraints at all on the @xmath32 lindblad operators @xmath259 . however , there is no need to restrict the lindblad equation to no more than @xmath32 operators . we conclude our presentation by showing that the lindblad equation with any number of operators has all the required properties . ( of course , from what we have shown , such an equation may be reduced to one with no more than @xmath32 operators . ) looking at the lindblad equation in that case , @xmath303\nonumber\\ & & -\frac{1}{2}\sum_{\alpha}[{\bf l}^{\alpha}{\bf l}^{\alpha\dagger } { \bf\rho}(t)+ { \bf\rho}(t){\bf l}^{\alpha}{\bf l}^{\alpha\dagger}-2{\bf l}^{\alpha}{\bf\rho}(t){\bf l}^{\alpha\dagger } ] , \end{aligned}\ ] ] it is easy to see that hermiticity , and trace 1 ( in the form @xmath304 , with @xmath305 ) are satisfied . complete positivity requires a bit more work . we have shown in appendix [ b ] that complete positivity holds for the kraus form ( [ x11a ] ) subject to the trace constraint ( [ x11b ] ) , with an arbitrary number of operators . so , if eq.([28 ] ) can be written in the kraus form with the trace constraint , we have shown it is completely positive . accordingly , we write eq.([28 ] ) as @xmath306{\bf\rho}(t)\nonumber\\ & & \cdot\big[{\bf 1}-dt(-i{\bf h}+\frac{1}{2}\sum_{\alpha}{\bf l}^{\alpha\dagger}{\bf l}^{\alpha})\big]+dt\sum_{\alpha}{\bf l}^{\alpha}{\bf\rho}(t){\bf l}^{\alpha\dagger}.\nonumber\\\end{aligned}\ ] ] identifying the kraus operators as @xmath307 , { \bf m}^{\alpha\neq0}= \sqrt{dt}{\bf l}^{\alpha},\ ] ] gives the kraus form eq.([x11a ] ) , @xmath308 now , take the trace of eq.([31 ] ) . since eq.([28 ] ) implies @xmath309 , the result is @xmath310{\bf\rho}(t).\ ] ] as we have done before , by successively replacing @xmath27 by the members of the density matrix basis , we obtain the kraus trace constraint ( [ x11b ] ) : @xmath311 therefore , the lindblad form with an arbitrary number of lindblad operators , is completely positive . consider the pauli+1 matrices ( multiplied by @xmath142 ) as the eigenvectors @xmath156 . as noted in section iv , they satisfy the trace constraint ( [ x6 ] ) if @xmath184 . eq.([x5 ] ) becomes @xmath312\nonumber\\ & & \negmedspace\negmedspace\negmedspace\negmedspace=\frac{1}{2}\begin{bmatrix}(\lambda^{1}+\lambda^{2})\rho_{22}+(\lambda^{3}+\lambda^{4})\rho_{11},&(\lambda^{1}-\lambda^{2})\rho_{21}-(\lambda^{3}-\lambda^{4})\rho_{12 } \\(\lambda^{1}-\lambda^{2})\rho_{12}-(\lambda^{3}-\lambda^{4})\rho_{21},&(\lambda^{1}+\lambda^{2})\rho_{11}+(\lambda^{3}+\lambda^{4})\rho_{22 } \end{bmatrix}.\nonumber\\\end{aligned}\ ] ] now , we can successively replace @xmath166 by the four density basis matrices , and demand that the @xmath133 be chosen so @xmath177 is positive for all . since the sum of these four @xmath166 s with positive coefficients ( adding up to 1 ) is the most general two - dimensional density matrix , then the most general @xmath177 will be positive . the first density basis matrix has @xmath104 and the rest of the matrix elements vanishing . then , @xmath313 since the eigenvalues of the density matrix must lie between 1 and 0 , we obtain the two conditions : [ a3 ] @xmath314 the second density basis matrix , with @xmath315 and the rest of the matrix elements vanishing , gives the same results . the third density basis matrix is @xmath316 $ ] . using @xmath184 to simplify the result , we obtain : @xmath317 the eigenvalues of @xmath177 here are @xmath318 , @xmath319 , so the condition that they lie between 0 and 1 is @xmath320 the fourth density basis matrix is @xmath321 $ ] . using @xmath184 to simplify the result , we obtain : @xmath322 the eigenvalues of @xmath177 here are @xmath323 , @xmath324 , so the condition that they lie between 0 and 1 is @xmath325 so , we have obtained the result that @xmath177 will be positive if eqs.([a3a ] , [ a3b],[a5],[a7 ] ) and @xmath326 are satisfied . the sum of eqs.(a3b , [ a5 ] , [ a7]))minus ( [ a8 ] ) tells us that @xmath327 . eq.([a8 ] ) and the constraint boundaries are three - dimensional hyperplanes in the four dimensional @xmath328-space . their intersections delineate the allowed areas for the eigenvalues . we shall be content here to set @xmath329 ( in which case eqs.(a3a ) simplifies to @xmath330 ) . then , eq.([a8 ] ) describes a plane in @xmath331 space , and its intersection with the constraint boundary planes can be drawn . this is shown in fig . 1 . there are two regions where one of the eigenvalues is negative and the other two are positive : the points in the heavily outlined upper left triangle have @xmath332 , and the points in the heavily outlined lower right triangle have @xmath333 . allowed regions of eigenvalues for a positive density matrix @xmath177 . the two dark - outlined triangular regions are where an eigenvalue is negative . they abut an isosceles triangle , the restricted region of complete positivity , where all the eigenvalues are positive.,scaledwidth=50.0% ] the kraus form eq.([x11a ] ) and the kraus constraint eq.([x11b ] ) , generalized to _ any _ number of _ arbitrary _ operators @xmath219 , are respectively we want to show complete positivity . call any one of the @xmath335 . if we can show complete positivity for @xmath336 for an arbitrary @xmath337 , then eq.([b1a ] ) , which involves a sum of such terms , will be completely positive . and , it is only necessary to prove complete positivity for @xmath338 , where @xmath339 is any possible basis density matrix ( described in the paragraph following eq.([x3 ] ) ) in the added hilbert space , since the most general density matrix in the direct product hilbert space is the linear sum of such terms . we now calculate @xmath340 for arbitrary @xmath341 . there is no loss of generality if we pick the basis vectors @xmath342 in the added hilbert space any way we like . we shall pick them to be the eigenstates of @xmath343 . now , we note that each @xmath343 has one eigenvalue 1 and the remaining @xmath344 eigenvalues are 0 . call the eigenvector @xmath345 which corresponds to the eigenvalue 1 . then , @xmath346{\bf\rho } \big[\sum_{m ' = 1}^{n}d_{m1}{\bf m}^{\dagger}| \phi_{m}\rangle\big]\geq 0,\end{aligned}\ ] ] therefore , the kraus form with arbitrary @xmath219 s is completely positive .

the lindblad equation is an evolution equation for the density matrix in quantum theory . it is the general linear , markovian , form which ensures that the density matrix is hermitian , trace 1 , positive and completely positive . some elementary examples of the lindblad equation are given . the derivation of the lindblad equation presented here is simple " in that all it uses is the expression of a hermitian matrix in terms of its orthonormal eigenvectors and real eigenvalues . thus , it is appropriate for students who have learned the algebra of quantum theory . where helpful , arguments are first given in a two - dimensional hilbert space .

we analyse the problem of controllability for linear finite - dimensional systems submitted to parametrised perturbations , depending on unknown parameters in a deterministic manner . in previous works we have analysed the property of averaged control looking for a control , independent of the values of these parameters , designed to perform well , in an averaged sense ( @xcite , @xcite ) . here we analyse the complementary issue of determining the most relevant values of the unknown parameters so to provide the best possible approximation of the set of parameter - depending controls . our analysis is based on previous work on greedy and weak greedy algorithms for parameter - depending pdes and abstract equations in banach spaces ( @xcite , @xcite ) , which we adapt to the present context . although the greedy control is applicable to more general control problems and systems , here we concentrate on controllability issues and , to better illustrate the main ideas of the new approach , we focus on linear finite - dimensional systems of parameter - dependent odes . infinite - dimensional systems , as a first attempt to later consider pde models , are discussed separately in section [ infinite ] , as well as in the conclusion section . consider the finite dimensional linear control system @xmath0 in ( [ eq1f - d ] ) the ( column ) vector valued function @xmath1 is the state of the system , @xmath2 is a @xmath3matrix governing its free dynamics and @xmath4 is a @xmath5-component control vector in @xmath6 , @xmath7 , entering and acting on the system through the control operator @xmath8 , a @xmath9 parameter - dependent matrix . in the sequel , to simplify the notation , @xmath10 will be simply denoted by @xmath11 . the matrices @xmath12 and @xmath13 are assumed to be lipschitz continuous with respect to the parameter @xmath14 , @xmath15 , @xmath16 being a compact set . however , some of our analytical results ( section [ infinite ] ) will additionally require analytic dependence conditions on @xmath17 . here , to simplify the presentation , we have assumed the initial datum @xmath18 to be controlled , to be independent of the parameter @xmath17 . despite of this , the matrices @xmath12 and @xmath13 being @xmath17-dependent , both the control and the solution will depend on @xmath17 . similar arguments allow to handle the case when @xmath19 also depends on the parameter @xmath17 , which will be discussed separately . we address the controllability of this system whose initial datum @xmath19 is given , known and fully determined . we assume that the system under consideration is controllable for all values of @xmath17 . this can be ensured to hold , for instance , assuming that the controllability condition is satisfied for some specific realisation @xmath20 and that the variations of @xmath2 and @xmath8 with respect to @xmath17 are small enough . in these circumstances , for each value of @xmath17 there is a control of minimal @xmath21^m$]-norm , @xmath22 . this defines a map , @xmath23^m$ ] , whose regularity is determined by that of the matrices entering in the system , @xmath2 and @xmath8 . here we are interested on the problem of determining the optimal selection of a finite number of realisations of the parameter @xmath17 so that all controls , for all possible values of @xmath17 , are optimally approximated . more precisely , the problem can be formulated as follows . * problem 1 * _ given a control time @xmath24 and arbitrary initial data @xmath19 and final target @xmath25 , we consider the set of controls of minimal @xmath21^m$]-norm , @xmath26 , corresponding to all possible values @xmath27 of the parameter satisfying the controllability condition : @xmath28 this set of controls is compact in @xmath21^m$ ] . _ given @xmath29 we aim at determining a family of parameters @xmath30 in @xmath16 , whose cardinal @xmath31 depends on @xmath32 , so that the corresponding controls , denoted by @xmath33 , are such that for every @xmath27 there exists @xmath34 steering the system in time @xmath35 within the @xmath32 distance from the target @xmath36 , i. e. such that @xmath37 here and in the sequel , in order to simplify the notation , we denote by @xmath38 the control @xmath22 , and similarly we use the simplified notation @xmath39 . note that , in practice , the controllability condition ( [ eq2f - d ] ) is relaxed to the approximate one ( [ eq2f - dap ] ) . this is so since , in practical applications , when performing numerical approximations , one is interested in achieving the targets within a given error . this fact is also intrinsic to the methods we employ and develop in this paper , and that can only yield optimal procedures to compute approximations of the exact control , which turn out to be approximate controls in the sense of ( [ eq2f - dap ] ) . this problem is motivated by the practical issue of avoiding the construction of a control function @xmath38 for each new parameter value @xmath17 which , for large systems , although theoretically feasible by the uniform controllability assumption , would be computationally expensive . by the contrary , the methods we develop try to exploit the advantages that a suitable choice of the most representative values of @xmath17 provides when computing rapidly the approximation of the control for any other value of @xmath17 , ensuring that the system is steered to the target within the given error ( [ eq2f - dap ] ) . of course , the compactness of the parameter set @xmath16 and the lipschitz - dependence assumption with respect to @xmath17 make the goal to be feasible . it would suffice , for instance , to apply a _ naive _ approach , by taking a fine enough uniform mesh on @xmath16 to achieve the goal . however , our aim is to minimise the number of spanning controls @xmath31 and to derive the most efficient approximation . the _ naive _ approach is not suitable in this respect . to achieve this goal we adapt to the present frame of finite - dimensional control , the theory developed in recent years based on greedy and weak - greedy algorithms for parameter dependent pdes or abstract equations in banach spaces , which optimise the dimension of the approximating space , as well as the number of steps required for its construction . the rest of this paper is organised as follows . in section [ control - prel ] we summarise the needed controllability results for finite - dimensional systems and reformulate problem 1 in terms of the corresponding gramian operator . section 3 is devoted to the review of ( weak ) greedy algorithms , while their application to the control problem under consideration and its solution is provided in the subsequent section . the computational cost of the greedy control approach is analysed in section 5 . section 6 contains a generalisation of the approach to infinite dimensional problems followed by a convergence analysis of the greedy approximation errors with respect to the dimension of the approximating space . section 7 contains numerical examples and experiments for finite - difference discretisations of 1-d wave and heat problems . the paper is closed pointing towards future development lines of the greedy control approach . in order to develop the analysis in this paper it is necessary to derive a convenient characterisation of the control of minimal norm @xmath38 , as a function of the parameter @xmath17 . this can be done in a straightforward manner in terms of the gramian operator . in this section we briefly summarise the most basic material on finite - dimensional systems that will be used along this article ( we refer to @xcite for more details ) . consider the finite - dimensional system of dimension @xmath40 : @xmath41 where @xmath42 is the @xmath40-dimensional state and @xmath43 is the @xmath5-dimensional control , with @xmath7 . this corresponds to a specific realisation of the system above for a given choice of the parameter @xmath17 . we omit however the presence of @xmath17 from the notation since we are now considering a generic linear finite - dimensional system . here @xmath12 is an @xmath44 matrix with constant real coefficients and @xmath13 is an @xmath45 matrix . the matrix @xmath12 determines the dynamics of the system and the matrix @xmath13 models the way @xmath5 controls act on it . in practice , it is desirable to control the @xmath40 components of the system with a low number of controls , the best possible case being the one of scalar controls : @xmath46 . recall that system ( [ fd ] ) is said to be _ controllable _ when every initial datum @xmath47 can be driven to any final datum @xmath36 in @xmath48 in time @xmath35 . this controllability property can be characterised by a necessary and sufficient condition , which is of purely algebraic nature , the so called _ kalman condition _ : system ( [ fd ] ) is controllable if and only if @xmath49=n.\ ] ] when this rank condition is fulfilled the system is controllable for all @xmath24 . there is a direct proof of this result which uses the representation of solutions of ( [ fd ] ) by means of the variations of constants formula . but for our purpose it is more convenient to use the point of view based on the dual problem of observability of the adjoint system that we discuss now . consider the _ adjoint system _ @xmath50 system ( [ fd ] ) is controllable in time @xmath35 if and only if the adjoint system ( [ afd ] ) is _ observable _ in time @xmath35 , i. e. if there exists a constant @xmath51 such that , for all solution @xmath52 of ( [ afd ] ) , latexmath:[\[\label{fdoi } hold in all time @xmath35 if and only if the kalman rank condition ( [ rc ] ) is satisfied . furthermore , the control of minimal @xmath21^m$]-norm can be built as a minimiser of a quadratic functional @xmath54 : @xmath55 more precisely , if @xmath56 is a minimiser for @xmath57 , then the control @xmath58 where @xmath59 is the corresponding solution of ( [ afd ] ) , is such that the state @xmath42 of ( [ fd ] ) satisfies the control requirement @xmath60 . note that the minimiser of @xmath54 exists since the functional @xmath57 is continuous , quadratic and coercive , in view of the observability inequality . this characterisation of controls ensuring controllability also yields explicit bounds on the controls . indeed , since the functional @xmath61 at the minimiser , and in view of the observability inequality ( [ fdoi ] ) , it follows that @xmath62^{1/2},\ ] ] @xmath63 being the same constant as in ( [ fdoi ] ) . therefore , we see that the square root of the observability constant is , up to a multiplicative factor , the norm of the control map associating to the data the control of minimal norm . summarising , the control of minimal @xmath21^m$]-norm is of the form @xmath64 where @xmath65 , @xmath56 being the minimiser of @xmath57 . let @xmath66 be the quadratic form , known as ( controllability ) gramian , associated to the pair @xmath67 , i.e. @xmath68 with @xmath69 being solutions to with the data @xmath70 and @xmath71 , respectively . then , under the rank condition , because of the observability inequality , this operator is coercive and symmetric and therefore invertible . its corresponding matrix , which we denote the same , is given by the relation @xmath72 the minimiser @xmath56 can be expressed as the solution to the linear system @xmath73 hereby , the left hand side , up to the free dynamics component , represents the solution of the control system ( [ fd ] ) with the control given by . as the solution is steered to the target @xmath36 , the last relation follows . in our context the adjoint system depends also on the parameter @xmath17 : @xmath74 we assume that the system under consideration is controllable for all values of @xmath17 . this can be ensured to hold assuming the following uniform controllability condition @xmath75 where @xmath76 are positive constants , while @xmath77 is the gramian of the system determined by @xmath78 . as we mentioned above , this assumption is fulfilled , in particular , as soon as the system is controllable for some specific value of the parameter @xmath79 , @xmath12 and @xmath13 depend continuously on @xmath17 , and @xmath17 is close enough to @xmath20 . but our discussion and presentation makes sense in the more general setting where ( [ g - bound ] ) is fulfilled . as we restrict the analysis to the set of minimal @xmath80^m$]-norm controls , each such control can be uniquely determined by the relation @xmath81 where @xmath82 is the unique minimiser of a quadratic functional @xmath83 given by , with @xmath13 replaced by @xmath84 , and @xmath85 by @xmath86 , the solution to . as explained above , the minimiser @xmath82 can be equivalently determined as the solution to the system @xmath87 where @xmath88 is the gramian associated to @xmath89 . according to the uniform controllability condition , this defines a mapping @xmath90 , whose smoothness is transferred from the mappings of @xmath91 and @xmath84 at all levels : lipschitz , analytic etc . having assumed these maps are lipschitz continuous and the parameter @xmath27 varies on a compact set , the set of minimisers @xmath92 constitutes also a compact set in @xmath48 . by using the 1 - 1 correspondence between the controls @xmath38 and the associated minimisers @xmath82 , the original problem formulation can be reduced to the following one . * problem 2 * _ given a control time @xmath24 , an arbitrary initial data @xmath19 , final target @xmath25 , and @xmath29 , and taking into account that the set of minimisers @xmath92 corresponding to all possible values @xmath27 of the parameter is compact in @xmath48 , we aim at determining a family of parameters @xmath93 in @xmath16 so that the corresponding minimisers , that we denote by @xmath94 , are such that for every @xmath27 there exists @xmath95 such that the control @xmath96 given by , with @xmath97 replaced by @xmath98 , steers the system to the state @xmath99 within the @xmath32 distance from the target @xmath36 . _ in this formulation the elements of the manifold we want to approximate are @xmath40-dimensional vectors ( instead of @xmath100^m$]-functions in the first formulation ) , uniquely determined as solutions to linear systems . this enables to adapt the ( weak ) greedy algorithms and reduced bases methods for parameter dependent problems , that we present in the following section . the approximate controls we obtain in this manner do not really belong to the space @xmath101 spanned by controls associated to selected parameter values , since the selection is done at the level of @xmath102 , the control being simply the natural one corresponding to the choice of @xmath103 . in this section we present a brief introduction and main results of the linear approximation theory of parametric problems based on the ( weak ) greedy algorithms , that we shall use in our application to controllability problems . a more exhaustive overview can be found in some recent papers , e.g. @xcite . the goal is to approximate a compact set @xmath104 in a banach space @xmath105 by a sequence of finite dimensional subspaces @xmath106 of dimension @xmath31 . by increasing @xmath31 one improves the accuracy of the approximation . determining _ offline _ an approximation subspace within a given error normally implies a high computational effort . however , this calculation is performed only once , resulting in a good subspace from which one can easily and computationally cheaply construct _ online _ approximations to every vector from @xmath104 . vectors @xmath107 spanning the space @xmath106 are called _ snapshots _ of @xmath104 . the goal of ( weak ) greedy algorithms is to construct a family of finite dimensional spaces @xmath108 that approximate the set @xmath104 in the best possible manner . the algorithm is structured as follows . * weak greedy algorithm * -3 mm fix a constant @xmath109 . in the first step choose @xmath110 such that @xmath111 at the general step , having found @xmath112 , denote @xmath113 and @xmath114 choose the next element @xmath115 such that @xmath116 the algorithm stops when @xmath117 becomes less than the given tolerance @xmath32 . + the algorithm produces a finite dimensional space @xmath106 that approximates the set @xmath104 within the given tolerance @xmath32 . the choice of a new element @xmath118 in each step is not unique , neither is the sequence of approximation rates @xmath119 . but every such a chosen sequence decays at the same rate , which under certain assumptions given below , is close to the optimal one . thus the algorithm optimises the number of steps required in order to satisfy the given tolerance , as well as the dimension of the final space @xmath106 . the pure greedy algorithm corresponds to the case @xmath120 . as we shall see below , the relaxation of the pure greedy method ( @xmath120 ) to a weak greedy one ( @xmath109 ) will not significantly reduce the efficiency of the algorithm , making it , by the contrary , much easier for implementation . when performing the ( weak ) greedy algorithm one has to chose the next element of the approximation space by exploring the distance for all possible values @xmath121 . such approach is faced with two crucial obstacles : * the set @xmath104 in general consists of infinitely many vectors . * in practical implementations the set @xmath104 is often unknown ( e.g. it represents the family of solutions to parameter dependent problems ) . the first problem is bypassed by performing a search over some finite discrete subset of @xmath104 . here we use the fact that @xmath104 , being a compact set , can be covered by a finite number of balls of an arbitrary small radius . as to deal with the second one , instead of considering the exact distance appearing in , one uses some _ surrogate _ , easier to compute . in order to estimate the efficiency of the weak greedy algorithm we compare its approximation rates @xmath119 with the best possible ones . the best choice of a approximating space @xmath106 is the one producing the smallest approximation error . this smallest error for a compact set @xmath104 is called the _ kolmogorov @xmath31-width of @xmath104 _ , and is defined as @xmath122 it measures how well @xmath104 can be approximated by a subspace in @xmath105 of a fixed dimension @xmath31 . in the sequel we want to compare @xmath119 with the kolmogorov width @xmath123 , which represents the best possible approximation of @xmath104 by a @xmath31 dimensional subspace of the referent banach space @xmath105 . a precise estimate in that direction was provided by @xcite in the hilbert space setting , and subsequently improved and extended to the case of a general banach space @xmath105 in @xcite . [ @xcite , corollary 3.3 ] [ greedy_rates ] for the weak greedy algorithm with constant @xmath124 in a hilbert space @xmath105 we have the following : if the compact set @xmath104 is such that , for some @xmath125 and @xmath126 @xmath127 then @xmath128 where @xmath129 . this theorem implies that the weak greedy algorithms preserve the polynomial decay rates of the approximation errors , and the result will be used in the convergence analysis of our method in section [ infinite ] . a similar estimate also holds for exponential decays ( cf . in addition , it is also remarkable that the constant @xmath124 effects the efficiency only up to a multiplicative constant , while leaving approximation rates unchanged . in this section we solve problem 2 implementing the ( weak ) greedy algorithm in the manifold @xmath130 consisting of minimisers determined by the relation . the * goal * is to choose @xmath31 parameters such that @xmath131 approximates the whole manifold @xmath92 within the given error @xmath32 . to this effect , as already stated in the introduction , we assume that the matrices @xmath2 and @xmath132 are lipschitz continuous with respect to the parameter . in turn , this implies that the mapping @xmath133 possesses the same regularity as well , with the lipschitz constant denoted by @xmath134 . the greedy selection of each new snapshot relies on the relation , which in this setting maximises the distance of elements of @xmath92 from the space spanned by already chosen snapshots . theoretically , this process requires that we solve for each value of @xmath17 . and this is exactly what we want to avoid . actually , here we face the obstacle _ ii ) _ from previous section , since one has to apply the greedy algorithm within a set whose elements are not given explicitly . the problem is managed by identifying an appropriate surrogate for the unknown distances @xmath136 . to this effect note that @xmath137 where @xmath138 , while @xmath139 denotes equivalence of terms resulting from the uniform controllability assumption . in such a way we replace the unknown @xmath82 by an easy computed term @xmath140 , combining the target @xmath36 and the solution of the free dynamics at time @xmath35 . as for the other term , note that @xmath141 represents the value at time @xmath35 of the solution to the system @xmath142 where @xmath143 is the _ control _ obtained by solving the corresponding adjoint problem ( for the parameter @xmath17 ) with initial datum @xmath144 . thus instead of dealing with @xmath136 we use the * surrogate * @xmath145 , obtained by projecting an easy obtainable vector @xmath140 to a linear space @xmath146 whose basis is obtained by solving the adjoint system plus the state one @xmath31 times . the surrogate measures the control performance of the snapshots @xmath147 when applied to the system associated to parameter @xmath17 ( figure 1 ) . namely , by relation the minimiser @xmath97 uniquely determines the control @xmath38 steering the system from the initial datum @xmath19 to the target @xmath36 . by replacing @xmath97 with @xmath144 in the system is driven to the state @xmath148 , whose distance from the target represents the surrogate value . ] if for every @xmath27 we can find a suitable linear combination of the above states close enough to the target , we deem that we have found a good approximation of the manifold @xmath92 . otherwise , we select as the next snapshot a value for which the already selected snapshots provide the worst performance . the precise description of the offline part of the algorithm is given below . -3 mm fix the approximation error @xmath149 . * step 1 ( discretisation ) * + choose a finite subset @xmath150 such that @xmath151 where @xmath152 is a constant to be determined later ( cf . , ) , in dependence on the problem under consideration and the tolerance @xmath32 . * step 2 ( choosing @xmath153 ) * + if @xmath154 stop the algorithm . else determine the first distinguished parameter value as @xmath155 and choose @xmath156 as the minimiser of @xmath157 corresponding to @xmath158 . * step 3 ( choosing @xmath159 ) * + having chosen @xmath160 calculate @xmath161 for each @xmath162 . if @xmath163 stop the algorithm . else @xmath164 and repeat step 3 . + the algorithm results in the approximating space @xmath165 , where @xmath31 is a number of chosen snapshots ( specially @xmath166 for @xmath167 ) . the value of the parameter @xmath153 is chosen by testing the performance of the null control as an initial guess for all @xmath168 . the selected value @xmath153 is the one for which this performance provides the worst approximation . the algorithm is stopped at this initial level only if the null control ensures the uniform control of all system realisations within the given tolerance . note that the set @xmath169 is linearly independent , as for vectors @xmath170 that linearly depend on already chosen ones , the corresponding surrogate distance ( 23 ) , which is the criterion for the choice of new snapshots , vanishes . thus the algorithm stops after , at most , @xmath171 iterations , and it fulfils the requirements of the weak greedy theory . more precisely the following result holds . [ main_result ] let the @xmath172 be lipschitz and such that the uniform controllability condition holds , and let @xmath134 be the lipschitz constant of the mapping @xmath133 determined by . for a given @xmath173 take the discretisation constant @xmath174 such that @xmath175 then the above algorithm provides a week greedy approximation of the manifold @xmath176 with the constant @xmath177 and the approximation error less than @xmath178 . the obtained approximation error @xmath178 of the family of minimisers @xmath176 is a consequence of the required @xmath32 approximation of the set @xmath179 ( as @xmath180 implies @xmath181 ) . the case @xmath167 , occurring when inequality holds , is a trivial one resulting in a null approximating space that we exclude from the proof . * proof : * in order to prove the theorem we have to show : * that the selected minimisers @xmath182 associated to parameter values determined by and satisfy @xmath183 for the constant @xmath124 given by ; * that the approximation error @xmath184 obtained at the end of the algorithm is less than @xmath178 . * a ) * to this effect note that for the first snapshot the following estimates hold : @xmath185 where we have used relation ( for @xmath158 ) and the criterion . in order to obtain an estimate including the whole set of parameters @xmath16 , we employ lipschitz regularity of the mapping @xmath133 . thus for an arbitrary @xmath27 , by taking @xmath186 such that @xmath187 , by means of and it follows @xmath188 having excluded the case , we have @xmath189 implying latexmath:[\[\label{gamma_1 } @xmath124 given by . for a general @xmath191-th iteration , and an arbitrary @xmath27 , by taking @xmath192 and using we get @xmath193 where the last two inequalities follow from the fact that the stopping criteria is not satisfied until @xmath194 and the definition of the next snapshot . combining the last relation with , estimate follows . * b ) * finally , having achieved inequality after @xmath31 iterations , for an arbitrary @xmath27 , taking @xmath195 as above we get @xmath196 thus obtaining the required approximation error of the set @xmath92 . @xmath197 besides a choice of the discretisation constant @xmath174 determined by , other choices are possible as well . actually , taken any @xmath152 , define @xmath198 . the implementation of the algorithm in that case would lead to the greedy constant @xmath199 and the approximation error of the set @xmath92 equal to @xmath200 . note , however , that no choice of @xmath174 can reduce the order of the error already obtained by . + having constructed an approximating space @xmath135 of dimension @xmath31 , we would like to exploit it for construction of an approximate control @xmath96 associated to an arbitrary given value @xmath201 . such a control is given by the relation @xmath202 where @xmath203 is appropriately chosen approximation of @xmath97 from @xmath204 . it steers the system to the state @xmath205 . comparing formula with the one fulfilled by the exact control , we note that the only difference lies in the replacement of the unknown minimiser @xmath97 by its approximation @xmath203 . thus @xmath206 is a suitable linear combination of vectors @xmath207 . as our goal is to steer the system to @xmath36 as close as possible , the best performance is obtained if @xmath208 is chosen as projection of @xmath209 to the space @xmath138 . for this reason we define approximation of @xmath97 as @xmath210 where the coefficients @xmath211 are chosen such that @xmath212 represents projection of the vector @xmath213 to the space @xmath214 . this choice of @xmath203 corresponds to the minimisation of the functional @xmath157 , determined by , over the space @xmath146 . in order to check performance estimate of the approximate control , note that for any value @xmath27 we have @xmath215 where we have used that @xmath216 is the orthogonal projection of @xmath217 to the space @xmath214 ( that also contains @xmath218 ) , while @xmath195 is taken from the set @xmath219 . taking into account the stopping criteria , the last term in is less than @xmath220 , while the preceding one equals @xmath221 where @xmath222 denotes the lipschitz constant of the mapping @xmath223 . thus in order to obtain a performance estimate less than @xmath32 , one possibly has to refine the discretisation used in theorem [ main_result ] and to take @xmath224 where , let it be repeated , @xmath134 denotes the lipschitz constant of the mapping @xmath133 . this finalises solving of problem 2 and leads us to the following result . [ thm - online ] let @xmath172 be a lipschitz mapping such that the uniform controllability condition holds . given @xmath173 , let @xmath131 be an approximating space constructed by the greedy control algorithm with the discretisation constant @xmath174 given by . then for any @xmath27 the approximate control @xmath96 given by and steers the control system to the state @xmath99 within the @xmath32 distance from the target @xmath36 . as already commented in section [ control - prel ] , the approximate control @xmath96 does not belong to the space @xmath225 spanned by controls associated to selected parameter values , as the control operator and system matrix entering correspond to the given value @xmath17 , while @xmath226 . the greedy control also applies if we additionally assume that initial datum @xmath19 , as well as target state @xmath36 depend on the parameter @xmath17 in a lipschitz manner . in that case , the greedy search is performed in the same manner as above , i.e. by exploring elements of the set @xmath227 in calculation of the surrogate distance , and the obtained results remain valid with the same constants . all the above results on greedy control remain valid if instead of lipschitz continuity we merely assume continuous dependence with respect to the parameter @xmath228 . namely , as @xmath104 is a compact set , the assumption directly implies uniform continuity , which suffices for the proof of theorems [ main_result ] and [ thm - online ] in which we need @xmath229 and @xmath230 to be close whenever @xmath17 and @xmath195 are . the only difference in that case is that the discretisation constant @xmath174 can not be given explicitly in terms of @xmath32 , unlike expressions and we finish the section describing the algorithm summarising the above procedure to construct the approximate control @xmath96 . hereby we suppose that , for fixed @xmath173 , the approximating space @xmath131 has been constructed by the weak greedy control algorithm with the choices of the constants @xmath124 and @xmath174 as in the statement of the theorem . -3 mm a parameter value @xmath27 is given . * step 1 * calculate @xmath209 . * step 2 * calculate @xmath231 , where @xmath232 * step 3 * project @xmath233 to @xmath138 . denote the projection by @xmath234 . * step 4 * solve the system @xmath235 for @xmath236 . * step 5 * the approximate control is given by @xmath237 where @xmath238 are already determined within step 2 . -3 mm for any @xmath27 the obtained approximate control steers the system within an @xmath32 distance from the target . note that for a parameter value @xmath195 that belongs to the discretisation set @xmath150 used in the construction of an approximating space @xmath135 steps 13 of the last algorithm can be skipped as the corresponding terms have already been calculated within the offline part of the algorithm . the offline part of the greedy control algorithm consists of two main ingredients . first , the search for distinguished parameter values @xmath239 by examining the surrogate value over the set @xmath240 , and , second , the calculation of the corresponding snapshots @xmath241 . this subsection is devoted to estimate its computational cost . * choosing @xmath153 * + in order to identify the first parameter value , one has to maximise the expression @xmath242 over the set @xmath243 , which represents the distance of the target @xmath36 from the free dynamics state . to this effect one has to solve the original system with zero control @xmath244 times . denoting the cost of solving the system by @xmath63 , the cost of choosing @xmath153 thus equals @xmath245 where the second term corresponds to calculation of the distance between two vectors in @xmath48 . in general , the cost @xmath63 differs depending on the type of matrices @xmath91 under consideration and the method chosen for solving the corresponding control system . for example , an implicit one step method consists on solving @xmath246 linear systems , where @xmath247 denotes the time discretisation step . however , as we consider time independent dynamics , all these systems have the same matrix , thus lu factorisation is required just ones . consequently , the cost @xmath63 can be estimated as @xmath248 where the first part corresponds to the @xmath249 factorisation , while the latter one is the cost of building and solving a system of a type @xmath250 . * calculating @xmath156 * + in order to determine the first snapshot one has to solve the system for the chosen parameter value @xmath153 . to this effect we construct ( the unknown ) gramian matrix @xmath251 by calculating @xmath252 for the vectors of the canonical basis @xmath253 . according to the results presented in section [ control - prel ] , for any @xmath254 the corresponding vector @xmath255 can be determined as the state of the control system at time @xmath35 with the control @xmath256 , with @xmath85 being the solution of the adjoint problem starting from @xmath257 . in such a way , the cost of composing the system for @xmath156 equals @xmath258 ( corresponding to @xmath40 control + adjoint problems starting from @xmath40 different data ) . as solving the system requires a number of operations of order @xmath259 , the cost of this part of the algorithm turns out as @xmath260 * choosing @xmath159 * + suppose we have determined the first @xmath191 snapshots and have constructed the approximating space @xmath261 . the next parameter value is chosen by maximising the distance of the vector @xmath262 , calculated already in the first iteration , from the space @xmath263 over the set @xmath264 ( the already chosen values can obviously be excluded from the search ) . the basis of the above space has been determined gradually throughout the previous iterations up to the last vector @xmath265 , whose calculation is performed at this level . as explained above , it requires @xmath266 operations , that have to be performed @xmath267 times . in addition , this basis is orthonormalised , thus enabling the efficient calculation of the distance in . this process is performed gradually throughout the algorithm as well , and in each iteration we just orthonormalise the last vector with respect to the the rest of the , already orthonormalised set . the corresponding cost for a single value @xmath268 is of order @xmath269 . similarly , the ( orthogonal ) projection of @xmath262 to @xmath263 takes into account its projection to @xmath270 used in the previous iteration , and adds just a projection to the last introduced vector . as its corresponding cost is of order @xmath271 , the total cost of this part of the algorithm equals @xmath272 where , as in , the last term corresponds to calculating the distance between two vectors in @xmath48 . * calculating @xmath273 * + in order to determine the next snapshot , we have to construct corresponding gramian matrix @xmath274 . as explained in the part related to the calculation of @xmath156 , this can be done by applying the gramian to some basis of @xmath48 . as in the previous part of the algorithm we have already calculated @xmath275 , it is enough to calculate @xmath276 , for vectors of canonical basis @xmath277 complementing the set @xmath278 to the basis of @xmath48 . thus building the matrix of the system requires @xmath279 operations . in addition , the complementation of the set @xmath278 takes advantage of the basis obtained as complementation of the set @xmath280 in the previous iteration . adding a new snapshot @xmath241 to that basis , one of vectors @xmath281 has to be removed such that the new set results in a basis again . identifying this vector requires solving a single @xmath44 system . finally , taking into account the cost of solving the system , the cost of calculating the next snapshot turns out to be @xmath282 * total cost * + summing the above costs for @xmath283 the total cost of the algorithm results in @xmath284 as the cost @xmath63 of solving the control system is @xmath285 , the most expensive part of the greedy control algorithm corresponds to the terms containing this cost . it is interesting to notice that , as the number of chosen snapshots @xmath31 approaches either the number of eligible parameter values @xmath286 , or the system dimension @xmath40 , this part converges to @xmath287 which corresponds , up to lower order terms , to the cost of calculating @xmath288 for all @xmath289 values @xmath290 . this demonstrates that the application of the greedy control algorithm is always cheaper than a naive approach that consists of calculating controls for all values of the parameter from an uniform mesh on @xmath16 , taken fine enough so to achieve the approximative control . let us note that in the above analysis we have assumed that in the each iteration we calculate the surrogate distance for all values of the set @xmath243 , except those already chosen by the greedy algorithm . and that we repeat the procedure until the residual is less than @xmath220 for all parameter values of @xmath243 . however , very likely , for some , or even many parameter values , the corresponding residual will be less than the given tolerance already before the last iteration @xmath31 . and once it occurs , these values do not have to be explored any more , neither we have to calculate the corresponding surrogates for subsequently chosen snapshots . therefore , the obtained cost estimates are rather conservative , and in practice we expect the real cost to be lower than the above one . further significant cost reductions are obtained under assumption of a system with parameter independent matrix @xmath12 and the control operator @xmath13 ( with dependence remaining only in initial datum and/or in the target ) . in that case it turns that the corresponding gramian matrix @xmath291 is also parameter independent . for this reason , maximising the distance appearing in requires solving of control + adjoint problem just once in each iteration . as a result , the most expensive term of the total cost is replaced by quite a moderate one @xmath292 * step 1 * + calculating @xmath233 corresponds to solving the control system once , whose computational cost equals @xmath63 . * step 2 * + calculating @xmath293 requires solving the loop of the adjoint+control system @xmath31 times , resulting in the cost of @xmath294 . * step 3 * + the most expensive part of this step consist in the gram schmidt orthonormalisation procedure of the set @xmath295 , whose cost equals @xmath296 . * step 4 * + solving the system with qr decomposition requires @xmath297 operations . * step 5 * + the cost of this step is negligible compared to the previous ones . * total cost * + thus we obtain that the total cost of finding an approximative control @xmath96 for a given parameter value equals @xmath298 as @xmath31 approaches the system dimension @xmath40 , its most expensive part @xmath299 converges , up to lower order terms , to the cost of calculating an exact control @xmath38 . consequently , the cost reduction obtained by choosing the approximative control @xmath96 obtained by the greedy control algorithm instead of the exact one depends linearly on the ratio between the number of used snapshots @xmath31 and the system dimension @xmath40 . the theory and the ( weak ) greedy control algorithm developed in the preceding section for finite dimensional linear control systems extend to odes in infinite dimensional spaces . they can be written exactly as in the form ( [ eq1f - d ] ) except for the fact that solutions @xmath300 for each value of the parameter @xmath17 and each time @xmath301 live in an infinite.dimensional hilbert space @xmath105 . the key assumption that distinguishes these infinite - dimensional odes from partial differential equations ( pde ) is that the operators @xmath302 entering in the system are assumed to be bounded . the controllability of such systems has been extensively elaborated during last few decades , expressed in terms of semigroups generated by a bounded linear operator , see , for instance,@xcite . in fact , the existing theory of controllability for linear partial differential equations ( pde ) can be applied in that context too . this allows to characterise controllability in terms of the observability of the corresponding adjoint systems . in this way , the uniform controllability conditions of parameter - dependent control problems can be recast in terms of the uniform observability of the corresponding adjoint systems . however , in here , we limit our analysis to the case of infinite - dimensional odes for which the evolution is generated by bounded linear operators , contrarily to the case of pdes in which the generator is systematically an unbounded operator . the greedy theory developed here is easier to implement in the context of infinite - dimensional odes since , in particular , the analytic dependence property of controls with respect to the parameters entering in the system can be more easily established . in fact , in the context of infinite - dimensional odes , most of the results in section 2 on finite - dimensional systems apply as well . in particular , the characterisation of the controls as in ( [ control_nu ] ) , in terms of minimisers of quadratic functionals of the form @xmath57 as in ( [ funct_j ] ) holds in this case too , together with ( [ gramian ] ) for the gramian operators . obviously , the kalman rank condition can not be extended to the infinite - dimensional case . but the open character of the property of controllability with respect to parametric perturbations remains true in the infinite - dimensional case too , i.e. if the system is controllable for a given value of @xmath17 , under the assumption that @xmath302 depends on @xmath17 in a continuous manner , it is also controllable for neighbouring values of @xmath17 . in this section we shall analyse convergence rates of the constructed greedy control algorithm as the dimension of the approximating space tends to infinity , an issue that only makes sense for infinite - dimensional systems . in particular , we are interested in the dimension of the approximating space @xmath92 required to provide an uniform control within the given tolerance @xmath32 . this problem can be stated in terms of the estimate : what is the number of the algorithm iterations @xmath31 we have to repeat until the estimate is fulfilled . in case of systems of a finite dimension @xmath40 , the algorithm constructs an @xmath31 dimensional approximating space of @xmath303 , and it certainly stops after , at most , @xmath171 iterations . for infinite - dimensional systems , however there is no such an obvious stopping criteria . in general we analyse the performance of the algorithm by comparing the greedy approximation errors @xmath304 with the kolmogorov widths @xmath305 , which represent the best possible approximation of @xmath92 by a @xmath31 dimensional subspace in @xmath48 . to this effect one could try to employ theorem [ greedy_rates ] which connects sequences @xmath306 and @xmath307 . however , we have to apply the theorem to the set @xmath92 ( instead of @xmath16 ) . but while kolmogorov width of a set of admissible parameters @xmath16 is usually easy to estimate , that is not the case for a corresponding set of solutions ( to a parametric dependent equation ) , or minimisers ( as studied in this manuscript ) . fortunately , a result in that direction has been provided recently for holomorphic mappings ( @xcite ) under the assumption of a polynomial decay of kolmogorov widths . [ greedy_rates2 ] for a pair of complex banach spaces @xmath105 and @xmath308 assume that @xmath43 is a holomorphic map from an open set @xmath309 into @xmath308 with uniform bound . if @xmath310 is a compact subset of @xmath105 then for any @xmath311 and @xmath126 @xmath312 for any @xmath313 and the constant @xmath314 depending on @xmath315 , @xmath316 and the mapping @xmath43 . the proof of the theorem provides an explicit estimate of the constant @xmath314 in dependence on @xmath315 , @xmath316 and the mapping @xmath43 . however , due to its rather complicated form we do not expose it here . going back to our problem the last theorem can be applied under the assumption that the mapping @xmath317 is analytic ( its image being embedded in the space of linear and bounded operators in @xmath105 ) , which implies , in view of the representation formula ( [ control_nu ] ) , that the mapping @xmath16 to @xmath92 is analytic as well . note that this issue is much more delicate in the pde setting , with the generator of the semigroup @xmath2 being an unbounded operator . in fact the property fails in the context of hyperbolic problems although it is essentially true for elliptic and parabolic equations . as we consider a finite number of parameters , lying in the set @xmath318 , the polynomial decay of @xmath319 can be achieved at any rate @xmath125 just by adjusting the corresponding constant @xmath320 in ( note that @xmath321 for @xmath322 ) . of course , the kolmogorov widths of the set @xmath92 do not have to vanish for @xmath31 large , but the last theorem ensures their polynomial decay at any rate . combining theorems [ greedy_rates ] , [ main_result ] and [ greedy_rates2 ] we thus obtain the following result . let the mapping @xmath172 , corresponding to the parameter - dependent infinite - dimensional odes , be analytic and such that the uniform controllability condition holds . then the greedy control algorithm ensures a polynomial decay of arbitrary order of the approximation rates . more precisely , for all @xmath323 there exists @xmath324 such that for any @xmath17 the minimiser @xmath82 determined by the relation can be approximated by linear combinations of the weak - greedy ones as follows : @xmath325 where @xmath326 can be determined by exploring constants appearing in , and . the last result provides us with a stopping criteria of the greedy control algorithm . for a given tolerance @xmath173 , the algorithm stops after choosing @xmath327 snapshots . it results with a @xmath31 dimensional space @xmath135 approximating the family of minimisers @xmath92 within the error @xmath178 , and providing , by means of formul and , a uniform control of our system within an @xmath32 tolerance . a similar result holds if @xmath16 is infinite - dimensional , provided its kolmogorov width decays polynomially . a typical example of such a set is represented by the so called affine model in which the parameter dependence is given by @xmath328 and/or similarly for @xmath329 . here it is assumed that @xmath330 belongs to the unit cube in @xmath331 , i.e. that @xmath332 for any @xmath333 , while the sequence of numbers @xmath334 belongs to @xmath335 for some @xmath336 . however , note that in this case the kolmogorov width of the set @xmath337 does not decay polynomially ( actually these are constants equal to 1 ) , but the polynomial decay is obtained for the set @xmath338 . indeed , rearranging the indices so that the sequence @xmath339 is decreasing , it follows @xmath340 where we have used that for a decreasing @xmath335 sequence we must have @xmath341 . thus one can consider the mapping @xmath342 and apply theorem [ greedy_rates2 ] to @xmath343 . furthermore , in the case of the affine model this theorem can be improved , implying the kolmogorov @xmath31-widths of sets @xmath344 and @xmath345 decay at the same rate @xmath346 ( e.g. @xcite ) . consequently , one obtains that the greedy approximation rates @xmath347 decay at the same rate as well . finally , let us mention that the cost of the greedy control is significantly reduced if one considers an affine model in which the control operator has finite representation of the form , while the system matrix @xmath12 is taken as parameter independent . in that case the corresponding gramian is of the form : @xmath348 where @xmath349 , while @xmath350 . here we consider a finite representation , but a more general one can be reduced to this one by truncation of the series . consequently , computing @xmath351 for a chosen snapshot does not require solving the loop of the adjoint+control system for each value @xmath352 . instead , it is enough to perform the loop @xmath353 times in order to obtain vectors @xmath354 , and express @xmath351 as their linear combination by means of . such approach can result in a lower cost of the greedy control algorithm compared to the one obtained in the previous section , with a precise reduction rate depending on the relation between the series length @xmath355 and the number of eligible parameters @xmath289 . we consider the control system whose governing matrix has the block form @xmath356 where @xmath357 is the identity matrix of dimension @xmath358 , while @xmath359 the control operator is assumed to be of the parameter - independent form @xmath360 the system corresponds to the semi - discretisation of the wave equation problem with the control on the right boundary : @xmath361 parameter @xmath17 represents the ( square of ) velocity of propagation , while @xmath358 corresponds to the number of inner points in the discretisation of the space domain . for this example we specify the following ingredients : @xmath362 the final target is set at @xmath363 and we assume @xmath364 the system satisfies the kalman s rank condition and accordingly the control exists for any value of @xmath17 . although the convergence of this direct approximation method , in which one computes the control for a finite - difference discretisation , hoping that it will lead to a good approximation of the continuous control , is false in general ( see @xcite ) , this is a natural way to proceed and it is interesting to test the efficiency of the greedy approximation for a given @xmath40 , which in practice corresponds to fixing the space - mesh for the discretisation . in any case , the time - horizon @xmath35 is chosen such that the geometrical control condition ( ensuring controllability of the continuous problem , see @xcite ) is satisfied for the continuous wave equation and all @xmath201 . the greedy control algorithm has been applied with @xmath365 and the uniform discretisation of @xmath16 in @xmath366 values . the algorithm stopped after 24 iterations , choosing 24 ( out of 100 ) parameter values in the following order : @xmath367 the corresponding minimisers @xmath144 have been calculated and stored , completing the offline part of the process . in the online part one explores the stored data in order to construct approximate controls for different parameter values ranging between 1 and 10 by means of formula . here we present the results obtained for @xmath368 . figure 2 displays the evolution of the last 5 components of the system , corresponding to the time derivative of the solution to at grid points , controlled by the approximate control @xmath96 . as required , their trajectories are driven ( close ) to the zero target . -4 cm . ] -4 cm the first 25 components of the system are plotted by a 3-d plot ( figure 3 ) depicting the evolution of the solution to a semi - discretised problem governed by the approximate control @xmath96 . the starting wave front , corresponding to the sinusoidal initial position , as well as the oscillations in time , gradually diminish and the solution is steered ( close ) to zero , as required . furthermore , the total distance of the solution from the target equals @xmath369 . -4 cm for @xmath368 . ] -4 cm the efficiency of the greedy control approach is checked by exploring the corresponding approximation rates @xmath370 , depicted by the red curve of figure 4 . the curve decreases rapidly below @xmath365 for @xmath371 , stopping the algorithm after 24 iterations . the bases of the approximation spaces @xmath135 are built iteratively in a _ greedy _ manner , by exploring the surrogate distance over the set @xmath240 . as explained in the previous section , this search is costly , but should produce optimal approximation rates in the sense of the kolmogorov widths . as opposed to that , one can explore some a cheaper approach , in which vectors spanning an approximation space are chosen arbitrarily , e.g. by taking vectors of the canonical basis . the blue curve of figure 4 plots approximation errors of spaces @xmath372 spanned by the first @xmath31 vectors of the canonical basis in @xmath48 . obviously the greedy approach wins over the latter one , in accordance with the theoretical results , ensuring optimal performance . eventually , both curves terminate at zero at the moment in which the size of the approximation space @xmath31 reaches the size of the ambient space @xmath48 . -4.5 cm -4.5 cm for @xmath373 , with @xmath374 given by , and the control operator @xmath375 the system corresponds to the space - discretisation of the heat equation problem with @xmath40 internal grid points and the control on the right boundary : @xmath376 the parameter @xmath17 represents the diffusion coefficient and is supposed to range within the set @xmath377 . the system satisfies the kalman s rank condition for any @xmath201 and any target time @xmath35 . we aim to control the system from the initial state @xmath378 to zero in time @xmath379 . the greedy control algorithm has been applied for the system of dimension @xmath380 with @xmath381 , and the uniform discretisation of @xmath16 in @xmath366 values . the algorithm stops after only three iterations , choosing parameter values ( out of 100 eligibles ) in the following order : @xmath382 the corresponding three minimisers @xmath144 are used for constructing approximate controls for all parameter values by means of formulas and . here we present the results obtained for @xmath383 . -4 cm for @xmath383 . ] -4 cm evolution of the solution , presented by a 3-d plot is given by figure 5 . the system is driven to the zero state within the error @xmath384 . the corresponding approximate control profile is depicted on figure 6 , exhibiting rather strong oscillations when approaching the target time an intrinsic feature for the heat equation . the choice of a rather tiny value of the target time @xmath379 is due to the strong dissipation effect of the heat equation , providing an exponential solution decay even in the absence of any control . for the same reason , the algorithm stopped after only three iterations although the precision was set rather high with @xmath381 ( for the wave problem 24 iterations were required in order to produce uniform approximation within the error 0.5 ) . -5 cm . ] -5 cm * _ exact controllability . _ in our analysis the exact control problem is relaxed to the approximate one by letting a tolerance @xmath32 on the target at time @xmath35 , what is realistic in applications . note however that this approximation of the final controllability condition is achieved by ensuring a sufficiently small error on the approximation of exact controls . thus , in practice , the methods we have built are of direct application to the problem of exact controllability as well as the tolerance @xmath32 tends to zero . * _ comparison of greedy and naive approach . _ as we have seen the weak greedy algorithm we have developed has various advantages with respect to the naive one , the latter consisting of taking a fine enough uniform mesh on the set of eligible parameters @xmath16 and calculating all corresponding controls . although the greedy approach requires an ( expensive ) search for distinguished parameter values , in the finite - dimensional case its computational cost is smaller than for the naive one . meanwhile , in the infinite - dimensional context it leads to algorithms with an optimal convergence rate . * _ model reduction : _ the computational cost of the greedy methods , as developed here , remains high . this is so because on the search of the new distinguished parameter values , to compute the surrogate , we are obliged to solve fully the adjoint system first and then the controlled one . it would be interesting to build cheaper ( from a computational viewpoint ) surrogates . this would require implementing model reduction techniques . but this , as far as we know , has non been successfully done in the context of controllability . model reduction methods have been implemented for elliptic and parabolic optimal control problems ( see @xcite , @xcite , and @xcite ) but , as far as we know , they have not been developed in the context of controllability . * _ parameter dependence . _ most existing practical realisations of ( weak ) greedy algorithms assume an affine parameter dependence presented at the end of section [ infinite ] ( @xcite ) . the assumption provides a cheaper computation of a surrogate and reduces the cost of the search for a next snapshot . + by the contrary , the greedy control algorithm presented in this paper allows for very general parameter dependence , still providing an efficient algorithm beating the naive approach . smooth dependence is needed in order to obtain sharp convergence rates . still the algorithms can be applied for models depending on the parameters in a rough manner . + the greedy control algorithm and corresponding approximation results of section 4 , as well as solution to our problem is derived under assumption of lipschitz continuity of the control system entries with respect to the parameter . however , additional analyticity requirement is imposed in section [ infinite ] in order to deduce convergence analysis of the algorithm . namely , transfer of kolmogorov width from the set of parameters into the set of solutions ( or controls ) is provided so far only by theorem [ greedy_rates2 ] , which requires analyticity assumption , as well as polynomial decay of kolmogorov widths . however , the lack of an analogous result under more general assumptions does not prevent possible applications of the greedy control algorithm . it can still be implemented , practically at no - risk . namely , in the worst case when dimension of approximation space reaches number of eligible parameters , one will calculate controls for all these parameter values , which corresponds to the naive approach . and this will cost the same as applying naive method directly from the beginning . but there are many reasons to believe the algorithm will stop before this ultimate point . * _ waves versus heat : _ in our numerical experiments we have observed that the greedy algorithm is more efficient for the semi - discrete heat equation than for the wave one , in the sense that for the first one it stops after @xmath385 iterations while for the second one it goes up to @xmath371 . this is a natural result to be expected . indeed , for the heat equation , because of the intrinsic dissipativity of the system , the high frequency components play a minor role when , as in our experiments , dealing with approximate controllability . thus , even if the algebraic dimension of both systems , waves and heat , are the same , in practice the relevant dimensions for the heat equation is much smaller , thus explaining the faster convergence of the greedy algorithm for heat like equations . * _ extension of greedy control to pde systems . _ as we have seen the methods and ideas developed in this paper can be easily adapted to infinite - dimensional odes . the same ideas and methods can be implemented without major changes on the pde setting . however , as we have seen above , in order to quantify the convergence rate of greedy algorithms , the analytic dependence of the controls with respect to parameters plays a key role . as far as we know , this issue has not been thoroughly addressed in the literature , i.e. that of whether or not controls for a given pde controllability problem depend analytically on the parameters entering in the system . and at this level , the type of pde under consideration can play a major role . indeed , for heat equations , even when the unknown parameters enter in the principal part of the diffusion operator , the analytic dependence of the controls can be expected . this is not the case for wave equations ( see @xcite ) . indeed , note that even for the simplest first order transport equation solutions fail to depend analytically on the velocity of propagation in a natural @xmath353 or sobolev energy setting . this issue requires significant further investigation . * robust control . _ the analysis and methods developed in this paper apply to control problems where the data to be controlled are prescribed a priori , either depending on the parameters or not . in that sense , our results are the analogue to what has been done for the greedy approximation of pde solutions with given data ( right hand side terms and boundary values ) as in @xcite , @xcite and @xcite , among others . + from the point of view of applications it would be interesting to develop greedy methods for control of potential application for all possible data to be controlled . this would require to establish a strategy for identifying the most relevant snapshots of the parameters @xmath17 for the approximation of the gramians @xmath386 , within the space of bounded linear operators . and this , in turn , requires identifying efficient surrogates . this is an interesting problem to be addressed . + actually , as far as we know , this has not been done even in the context of the solvability of elliptic problems . the work done so far , as mentioned above , addresses approximation issues for given specific data . but the problem of applying greedy algorithms to approximate the resolvent operators , in a robust manner and independently of the data entering in the pde , has not been addressed . , ( 2006 ) . _ controllability and observability of partial differential equations : some results and open problems_. _ handbook of differential equations : evolutionary equations , _ vol . 3 , c. m. dafermos and e. feireisl eds . , elsevier science , amsterdam , 527621 .

we analyse the problem of controllability for parameter - dependent linear finite - dimensional systems . the goal is to identify the most distinguished realisations of those parameters so to better describe or approximate the whole range of controls . we adapt recent results on greedy and weak greedy algorithms for parameter depending pdes or , more generally , abstract equations in banach spaces . our results lead to optimal approximation procedures that , in particular , perform better than simply sampling the parameter - space to compute the controls for each of the parameter values . we apply these results for the approximate control of finite - difference approximations of the heat and the wave equation . the numerical experiments confirm the efficiency of the methods and show that the number of weak - greedy samplings that are required is particularly low when dealing with heat - like equations , because of the intrinsic dissipativity that the model introduces for high frequencies . # 1 # 1 = = # 1h^#1 # 1#2 # 1 _ # 2 # 1 # 1 # 1#2#1,#2 # 1#2#1,#2 ] # 1(#1 ) # 1(#1 ) # 1#2[#1,#2 ] parametrised odes and pdes , greedy control , weak - greedy , heat equation , wave equation , finite - differences .

suppose the quarks and leptons are localized on a `` @xmath27-brane '' , i.e. , do not propagate in the extra dimensions ( the `` bulk '' ) in which the gauge bosons and higgs fields propagate . then translation invariance in the extra dimensions is broken by couplings of fields on the brane to fields in the bulk in this case , the couplings of quarks and leptons to gauge bosons . in momentum space , conservation of extra dimensional momentum is violated . thus , in the effective @xmath4 theory vertices with only one kk state , for example , tree - level couplings of kk gauge bosons to leptons and quarks , are allowed . these couplings result in a contribution to muon decay , atomic parity violation ( apv ) etc . at tree - level from feynman diagrams with exchange of kk @xmath12 s and @xmath13 s instead of ( zero - mode ) @xmath12 and @xmath13 as in the sm @xcite . thus , these contributions are @xmath28 compared to the sm . then , precision electroweak measurements ( @xmath29 etc . ) imply an upper limit on @xmath30 of about a few percent . and , in turn , @xmath31 ( see eq . ( [ amukk1 ] ) ) @xcite . suppose quarks and leptons are also in the ( same ) bulk the extra dimensions are `` universal '' . then , extra dimensional momentum is conserved , i.e. , in the effective @xmath4 theory , `` kk number '' is conserved . thus , there are no vertices with only one kk state ; in particular , couplings of ( zero - modes of ) quarks and leptons to kk gauge bosons are _ not _ allowed . this implies that there are _ no _ tree - level effects on muon decay , apv , @xmath32 etc . @xcite . however , there are constraints on @xmath26 from one loop contribution to @xmath0 from kk excitations of top quark , @xmath12 and higgs @xcite : @xmath33 - \left ( 0.25 + 0.034 \delta \right ) \sum _ n m_w^2 / \left ( n / r \right)^2 \nonumber \\ & & - 0.043 \sum _ n m_h^2 / \left ( n / r \right)^2 \left [ 1 + o \left ( m_h^2 / \left ( n / r \right)^2 \right ) \right ] \label{tuniv}\end{aligned}\ ] ] here , @xmath21 is the number of extra dimensions and @xmath34 is the higgs mass . we have assumed one higgs doublet as in the minimal sm . the contribution of kk states of top quark is positive and dominates over that of kk states of @xmath12 and higgs which is negative . thus , we get a constraint from the @xmath35 _ upper _ limit @xmath36 ( for @xmath37 gev ) . and @xmath13 and hence the @xmath35 limit on @xmath0 depends weakly ( logarithmically ) on @xmath34 . ] in the above equation , the contribution of kk states of @xmath12 and higgs at each level ( i.e. , for each value of @xmath38 ) depends on @xmath21 and @xmath34 . in addition , the sum over kk states @xmath39 depends on @xmath21 in fact , @xmath40 diverges for @xmath41 and so has to be regulated by a cut - off . thus , the constraint on @xmath26 depends on @xmath21 and @xmath34 . in any case , using @xmath42 and assuming @xmath43 and @xmath44 gev , we get the constraint @xmath45 ( if we neglect @xmath46 terms ) . the constraint is a little bit weaker if we allow larger @xmath21 and @xmath34 . nevertheless , this implies that @xmath47 . to relax the constraint from the @xmath0 parameter , we need to reduce the ( positive ) contribution of kk states of top quark to @xmath0 compared to the ( negative ) contribution from kk states of @xmath12 and higgs . this can be done by allowing the top quark to propagate in fewer extra dimensions than the electroweak particles . one possibility is to have only third generation quarks stuck on a @xmath27-brane . then , in the weak basis , only third generation quarks couple to kk gluons . thus , when we perform a unitary rotation on the quarks to go to mass basis , there is flavor violation at this kk gluon vertex @xcite . this results in contribution to , for example , @xmath48 mixing from tree - level exchange of kk gluons . if we assume ckm - like mixing among down - type quarks , then the coefficient of the @xmath49 operator is given by @xmath50 . then , the constraint that this coefficient should be smaller than that in the sm which is @xmath51 implies that @xmath52 . in turn , this shows that @xmath47 . also , if there are some extra dimensions in which gluons propagate but the first generation quarks do not , then there is a tree - level contribution to @xmath53 jets from exchange of kk excitations of gluons . this results in the constraint @xmath54 tev from cdf @xcite : such a large @xmath26 will result in @xmath31 . if higgs fields are on a @xmath27-brane ( or in general , propagate in a sub - space of the bulk in which electroweak gauge bosons propagate ) , then there is a mass - mixing term between zero - mode @xmath12 , @xmath13 and their kk excitations . this mixing is due to the coupling @xmath55 , where @xmath56 denotes a kk state of @xmath12 with mass @xmath57 and @xmath58 is the sm higgs field @xcite . due to this mixing , the physical @xmath12 and @xmath13 masses are shifted from the values given by the ( tree - level ) expression in the sm , i.e. , from @xmath59 and @xmath60 , where @xmath61 gev and @xmath62 , @xmath63 are the @xmath64 and @xmath65 gauge couplings . the relative shift in the masses is given by @xmath66 , whereas the tree - level expression for muon decay ( in terms of @xmath67 ) is the same as in the sm since only zero - mode of @xmath12 contributes in this model @xcite . as a result precision data on @xmath12 , @xmath13 masses give an upper limit on @xmath68 of about a few percent . from eq . ( [ amukk1 ] ) , we see that this constraint results in @xmath69 . so , to get @xmath22 , we have to assume that higgs fields propagate in the same extra dimensions as the electroweak gauge bosons . in order to evade all these constraints on @xmath26 , we assume that _ all _ quarks and gluons propagate in the _ same _ extra dimensional space , namely , @xmath70 extra dimensions . whereas , all leptons , electroweak gauge bosons _ and _ higgs doublet propagate in @xmath71 _ additional _ extra dimensions . as mentioned before , to evade the constraint from tree - level effects of kk states on electroweak observables involving leptons , the leptons and electroweak gauge bosons have to propagate in the same bulk . for simplicity , assume that all extra dimensions are compactified on circles of the same radius @xmath6 . a possible motivation for such a set - up is to explain @xmath72 at the cut - off ( and hence at the scale @xmath73 ) as we discuss below . the gauge coupling of the effective @xmath4 theory and the dimension_less _ gauge coupling of the fundamental @xmath74 theory , @xmath75 , are related at the cut - off @xmath76 by @xmath77 and thus if @xmath78 , then @xmath79 is reduced ( more ) by extra dimensional volume than @xmath80 . therefore , @xmath81 at the cut - off even though the _ fundamental _ ( dimension_less _ ) gauge coupling , @xmath75 , might be _ same _ for qcd and electroweak gauge groups . between the cut - off and the energy scale @xmath26 , @xmath80 and @xmath82 evolve with a power - law due to the contribution of the kk states @xcite : if @xmath78 , then this power - law evolution is more important for @xmath82 than for @xmath83 since there are fewer kk states with color . this evolution leads to an _ additional _ difference between @xmath84 and @xmath85 at the scale @xmath73 . this is to be contrasted to the `` standard '' picture with the gluon and electroweak gauge bosons propagating in the _ same _ bulk . in this picture , due to power - law evolution ( which is _ equally _ important for both gauge couplings ) , it is possible that the two _ effective _ @xmath4 couplings , i.e. , @xmath80 and @xmath82 , unify at the cut - off @xmath76 ( even though @xmath76 might not be much larger than @xmath26 ) @xcite . whereas , with @xmath78 , as discussed above , the _ fundamental _ dimension_less _ qcd and electroweak gauge couplings can `` unify '' at the cut - off , even though the _ effective _ @xmath4 gauge couplings do not . there are two cases within this type of model which we now discuss . in this case , there are no kk excitations of the top quark and hence eq . ( [ tuniv ] ) gives @xmath87 which is the @xmath35 lower limit on @xmath0 : this also depends weakly on @xmath34 . the range @xmath88 corresponds to varying @xmath21 and @xmath34 ( assuming @xmath43 and @xmath89 gev ) . this implies that @xmath90 so that @xmath22 is barely allowed by the constraint from @xmath0 ( see eq . ( [ amukk1 ] ) ) . however , there is a stronger constraint from inclusive hadronic decays of @xmath91 . the reason is that kk states of @xmath12 contribute at tree - level to hadronic weak decays , but not to semileptonic weak decays since ( zero - mode ) leptons do not couple to kk states of gauge bosons . again , this additional contribution to the amplitude is @xmath92 compared to the sm . the sm prediction for the ratio of the rates of hadronic and semileptonic inclusive @xmath91 decays , to be specific @xmath93 , agrees with experiment to within @xmath94 at @xmath95 ( combining theory and experiment errors ) @xcite . thus , we get @xmath96 ( the ratio of kk to sm @xmath12 _ amplitude _ ) is at most about @xmath97 so that @xmath47 in this case , the constraints from both @xmath0 and hadronic @xmath91 decays can be relaxed as we discuss below . using @xmath99 in eq . ( [ tuniv ] ) , we get @xmath100 , \label{t3b}\end{aligned}\ ] ] where ( and from now on ) @xmath101 denotes sum over kk states of a particle with momentum in only @xmath102 of the @xmath21 extra dimensions . as discussed before , the coefficient @xmath103 of the @xmath12 and higgs contribution to @xmath0 at each kk level depends on @xmath21 and @xmath34 we have allowed @xmath43 and @xmath44 gev . it is clear that there can be a cancelation between the contributions of top quark and @xmath104higgs , i.e. , @xmath105 is allowed at @xmath35 ( again depending weakly on @xmath34 ) , but for simplicity , we assume @xmath105 . ] if the following condition is satisfied : @xmath106 where the rhs depends on @xmath21 and @xmath34 . in other words , this case is intermediate between the case in section [ universal ] , where the contribution from top quark is dominant and the case in section [ subspace1 ] , where there is only the @xmath12 and higgs contribution . also , in this case , the additional contribution to hadronic @xmath91 decays ( relative to sm ) is given by @xmath107 since only kk excitations of @xmath12 with momentum in the @xmath108 extra dimensions in which quarks do not propagate contribute . from eq . ( [ amukk1 ] ) , to get @xmath22 , we require @xmath109 thus , even if eq . ( [ amuconstraint ] ) is satisfied , the contribution to hadronic @xmath91 decays can be smaller than @xmath110 in the amplitude ( as required by the agreement between theory and experiment ) provided the following condition is satisfied @xmath111 therefore , for given @xmath21 , it might be possible to evade the constraints from @xmath0 and hadronic decays ( while getting @xmath22 ) by an appropriate choice of @xmath70 and @xmath34 such that eqs . ( [ tconstraint ] ) and ( [ bconstraint ] ) are satisfied . next , we discuss the values of the parameters @xmath21 , @xmath70 , @xmath26 and the cut - off @xmath76 for which this is possible . we begin with direct bounds on @xmath26 from collider searches . in this case , there are kk excitations of quarks which appear as heavy stable quarks at hadron colliders and searches at cdf imply @xmath112 gev ( for @xmath113 ) @xcite . the direct lower bound on @xmath26 will be ( slightly ) larger for @xmath114 since there will be more kk modes at the lowest level , i.e. , with @xmath115 . the kk states of electroweak gauge bosons with momentum in the @xmath108 extra dimensions ( in which quarks do not propagate ) couple to zero - mode quarks ( with the same strength as in sm ) , but not to zero - mode leptons . thus , at hadron colliders , the signatures of these kk states will be similar to that of @xmath116 and @xmath117 decaying to dijets . the cdf search excludes hadronic decays of @xmath116 in the mass range @xmath118 gev at @xmath119 confidence level @xcite . for the kk states of @xmath12 , the branching ratio to quarks is slightly larger than that for @xmath116 since the leptonic decays are absent . also , since there are @xmath120 relevant states at the lowest kk level ( i.e. , with @xmath115 ) , the production cross - section can be larger than that for @xmath116 of same mass . we will assume the limit @xmath121 gev from a combination of the heavy stable quark and @xmath116 searches . in turn , this implies that @xmath122 to give @xmath22 ( see eq . ( [ amuconstraint ] ) ) . thus , @xmath123 will not suffice since @xmath124 for @xmath125 and is log - divergent ( but , @xmath126 ) for @xmath127 . the sum over kk states of a sm particle @xmath128 is power - divergent if @xmath129 . it can be approximated by an integral ( provided @xmath130 ) : @xmath131 , where the kk sum is cut - off at the mass scale @xmath76 ( i.e. , @xmath132 ) and @xmath133 $ ] ( the surface area of a unit - radius sphere in @xmath21 dimensions ) . thus , for @xmath129 , we get @xmath134 where @xmath135 is a factor of @xmath126 which includes two effects as follows . one effect is from the loop integral . the reason is that in the case of sm electroweak correction there is only one heavy particle in the loop ( @xmath12 ) whereas in the case of the kk contribution both the particles in the loop ( @xmath136 and @xmath137 ) are heavy . hence the ratio of the contribution from a kk state of mass @xmath138 to the sm electroweak contribution is not simply the ratio of heaviest masses in the loop , i.e. , it is @xmath139 _ only _ up to a factor of @xmath126 . in principle , @xmath135 can be computed in the effective @xmath4 theory . however , we see from eq . ( [ amukk2 ] ) that this factor can be absorbed into the definition of the cut - off @xmath76 . the other effect included in the factor @xmath135 is that @xmath140 includes the @xmath12 and @xmath13 loop contributions in the sm , whereas @xmath11 includes loop contributions from kk states of @xmath12 , @xmath13 _ and _ the photon . there is an upper limit on @xmath141 from the condition that the effective @xmath4 theory at the cut - off @xmath76 remain perturbative . the loop expansion parameter at the cut - off @xmath76 is given by @xmath142 . here @xmath143 is the number of `` colors '' and @xmath144 is the number of kk states lighter than @xmath76 , i.e , the total number of kk states in the effective @xmath4 theory . we can not trust perturbative calculations if this factor is larger than @xmath145 . therefore , we impose the constraint @xmath146 ( qcd effective coupling is smaller since number of kk states of gluons and quarks is smaller ) . thus , the upper limit on @xmath76 is not much larger than @xmath26 . nonetheless , we have checked if @xmath147 ( i.e. , @xmath76 is close to the upper limit of eq . ( [ pertconstraint ] ) ) , then the sum over kk states can _ still _ be approximated by an integral . we can rewrite eq . ( [ amukk2 ] ) in terms of @xmath148 ( using @xmath149 ) as follows : @xmath150 in fig . [ mmmplot ] , we plot @xmath151 as a function of @xmath148 for different values of @xmath21 and for @xmath152 gev . from eq . ( [ pertconstraint ] ) , the upper limit on @xmath148 is about @xmath153 . we have chosen @xmath154 for this plot ; as mentioned before , if @xmath155 , then it can be absorbed into the definition of the cut - off , or in other words , in the value of @xmath148 ( see eq . ( [ amukk3 ] ) ) . also , in this figure , we have not imposed the constraint from br@xmath156 ( see the discussion later ) . we see from this figure that it is possible to have @xmath22 for @xmath129 as long as @xmath157 . = 0.6 [ mmmplot ] we now discuss the values of @xmath70 for which the constraints from @xmath0 and hadronic @xmath91 decays can be satisfied . since @xmath158 for @xmath159 , it is clear from eq . ( [ b3b ] ) that if @xmath160 , then @xmath161 provided @xmath162 gev . thus , it is possible to ( barely ) satisfy the constraint from hadronic @xmath91 decays by choosing @xmath163 . for given values of @xmath21 and @xmath70 , in general , it should be possible to choose the value of @xmath34 such that @xmath164 ( see eq . ( [ t3b ] ) or eq . ( [ tconstraint ] ) ) . , then the results in fig . [ mmmplot ] really correspond to a `` redefined '' value of @xmath148 ( or in other words , the cut - off @xmath76 ) . then , the cut - off appearing in the sum over kk states in eq . ( [ t3b ] ) is , strictly speaking , different than that in fig . [ mmmplot ] . thus , for a given value of @xmath148 in fig . [ mmmplot ] the value of @xmath34 which gives @xmath164 is modified . however , the two values of cut - off and hence the two values of @xmath34 differ by a factor of at most @xmath126 . ] note that a small non - zero value of @xmath0 is allowed ( at @xmath35 level , @xmath165 as mentioned before ) and similarly the constraint from hadronic @xmath91 decays is also not very precise . thus , for given values of other parameters , there can be a _ range _ of values of @xmath70 and @xmath34 which is consistent with @xmath0 and with hadronic @xmath91 decays . next , we discuss the constraints from measurements of @xmath166 . the semileptonic decay is not affected by the kk states of @xmath12 since these states do not couple to ( zero - mode ) leptons . the amplitude for the process @xmath1 gets a contribution from kk states of @xmath12 @xmath167 ( the factor of @xmath168 reflects gim cancelation ) . this is to be compared to the sm contribution @xmath169 : the ratio is @xmath170 . thus , even if eq . ( [ amuconstraint ] ) is satisfied , i.e. , the kk contribution to @xmath171 is comparable to the sm electroweak correction , the @xmath136 contribution to @xmath1 is smaller than in the sm . this is due to ( a ) an additional suppression by a factor of @xmath172 ( since @xmath173 gev ) in the case of @xmath1 ( as compared to @xmath171 ) and ( b ) @xmath174 . however , there is a larger contribution to the amplitude for @xmath1 from loops with kk states of charged would - be - goldstone boson ( wgb ) as follows . with only one higgs doublet , the _ zero - mode _ charged wgb is unphysical since it becomes the longitudinal component of @xmath12 . however , the _ excited ( kk ) _ states of charged wgb _ are _ physical and have masses @xmath175 ( assuming @xmath176 ) . the coupling of the kk states of charged wgb to right - handed top quark is the same as for the zero - mode charged wgb ( i.e. , longitudinal @xmath12 ) and is @xmath177 . thus , the loop diagram with kk states of charged wgb and top quark gives an amplitude for @xmath1 given by @xmath178 . the ratio of this contribution to the sm amplitude is @xmath179 and is clearly larger than the @xmath136 contribution . moreover , if eq . ( [ amuconstraint ] ) is satisfied , i.e. , @xmath22 , then this additional amplitude for @xmath1 is also comparable to the sm amplitude . such a large contribution to the process @xmath1 is not allowed since the sm prediction for the rate agrees with experiment to within @xmath94 ( combining theory and experiment @xmath95 errors ) . thus , with one higgs doublet ( and assuming no other new physics contributions to @xmath1 ) , we see that the constraint from @xmath180 rules out the possibility that the contribution to the mmm from kk states of electroweak gauge bosons is comparable to the sm electroweak correction . however , if the higgs sector is _ non - minimal _ , then the contribution of kk states of charged higgs to the process @xmath1 is modified . for example , in the @xmath181-higgs - doublet model ( model - ii ) @xcite , a cancelation occurs between the contribution of kk states of _ physical _ charged higgs and the contribution of the kk states of the charged wgb mentioned above . this opens up the possibility that @xmath22 . note that the loop contribution to @xmath171 from kk states of charged higgs is negligible in this model since it is suppressed by additional powers of @xmath182 compared to the contribution due to kk states of @xmath12 . in summary , we have revisited the contributions to the anomalous magnetic moment of the muon ( mmm ) in models with flat extra dimensions of size @xmath2 tev@xmath3 accessible to the sm particles . in particular , we analyzed a model with colored particles ( quarks and gluon ) propagating in a sub - space of the bulk in which non - colored particles ( leptons , higgs and electroweak gauge bosons ) propagate . in such a model , the constraints on the size of the extra dimensions from tree - level processes and the @xmath0 parameter ( which have been analyzed previously ) can be evaded . thus , in this model the contribution to the mmm from the kk states of electroweak gauge bosons can be ( potentially ) comparable to the sm electroweak correction ( unlike in other types of models ) . however , with only one higgs doublet , once the constraint from contribution of kk states to the process @xmath1 ( which has _ not _ been considered before ) is imposed , we have shown that such a large contribution to the mmm is not possible . the @xmath1 constraint can be relaxed in models with more than one higgs doublet and then a large contribution to the mmm becomes possible . the first studies of possible effects of extra dimensions felt by sm particles were done in i. antoniadis , phys . b 246 ( 1990 ) 377 ; i. antoniadis , c. munoz and m. quiros , hep - ph/9211309 , nucl . phys . b 397 ( 1993 ) 515 ; i. antoniadis and k. benakli , hep - th/9310151 , phys . lett . b 326 ( 1994 ) 69 . k.r . dienes , e. dudas and t. gherghetta , hep - ph/9803466 , phys . b 436 ( 1998 ) 55 and hep - ph/9806292 , nucl . phys . b 537 ( 1999 ) 47 . n. arkani - hamed and m. schmaltz , hep - ph/9903417 , phys . rev . d 61 ( 2000 ) 033005 . arkani - hamed , h .- c . cheng , b.a . dobrescu and l.j . hall , hep - ph/0006238 , phys . d 62 ( 2000 ) 096006 . m. graesser , hep - ph/9902310 , phys . rev . d 61 ( 2000 ) 074019 . p. nath and m. yamaguchi , hep - ph/9903298 , phys . d 60 ( 1999 ) 116006 . r. casadio , a. gruppuso and g. venturi , hep - th/0010065 , phys . b 495 ( 2000 ) 378 . g.c . mclaughlin and j.n . ng , hep - ph/0008209 , phys . b 493 ( 2000 ) 88 . h. davoudiasl , j.l . hewett and t.g . rizzo , hep - ph/0006097 , phys . b 493 ( 2000 ) 135 . c.s . kim , j.d . kim and j. song , hep - ph/0103127 . s.c . park and h.s . song , hep - ph/0103072 . brown et al . ( muon @xmath183 collaboration ) , hep - ex/0102017 , phys . 86 ( 2001 ) 2227 . a. czarnecki and w.j . marciano , hep - ph/0102122 . p. nath and m. yamaguchi , hep - ph/9902323 , phys . d 60 ( 1999 ) 116004 . t. appelquist , h .- c . cheng and b.a . dobrescu , hep - ph/0012100 . carone , hep - ph/9907362 , phys . rev . d 61 ( 2000 ) 015008 . t.g . rizzo and j.d . wells , hep - ph/9906234 , phys . rev . d 61 ( 2000 ) 016007 . f. abe et al . ( cdf collaboration ) , hep - ex/9702004 , phys . d 55 ( 1997 ) 5263 . groom et al . ( particle data group ) , eur . j. c 15 ( 2000 ) 1 . b. grinstein and m.b . wise , phys . b 201 ( 1988 ) 274 ; w .- s . hou and r.s . willey , phys . b 202 ( 1988 ) 591 .

we investigate whether models with flat extra dimensions in which sm fields propagate can give a significant contribution to the anomalous magnetic moment of the muon ( mmm ) . in models with only sm gauge and higgs fields in the bulk , the contribution to the mmm from kaluza - klein ( kk ) excitations of gauge bosons is very small . this is due to the constraint on the size of the extra dimensions from _ tree - level _ effects of kk excitations of gauge bosons on precision electroweak observables such as fermi constant . if the quarks and leptons are also allowed to propagate in the ( same ) bulk ( `` universal '' extra dimensions ) , then there are _ no _ contributions to precision electroweak observables at tree - level . however , in this case , the constraint from _ one - loop _ contribution of kk excitations of ( mainly ) the top quark to @xmath0 parameter again implies that the contribution to the mmm is small . we show that in models with leptons , electroweak gauge and higgs fields propagating in the ( same ) bulk , but with quarks and gluon propagating in a _ sub - space _ of this bulk , _ both _ the above constraints can be relaxed . however , with only one higgs doublet , the constraint from the process @xmath1 requires the contribution to the mmm to be smaller than the sm electroweak correction . this constraint can be relaxed in models with more than one higgs doublet . epsf oits-702 + .05 in * can extra dimensions accessible to the sm explain the recent measurement of anomalous magnetic moment of the muon ? * .15 in k. agashe , n.g . deshpande , g .- h . wu .1 in _ institute of theoretical science + university of oregon + eugene or 97403 - 5203 _ .05 in theories with _ flat _ extra dimensions of size @xmath2 ( tev)@xmath3 in which sm gauge ( and in some cases quark and lepton ) fields propagate are motivated by susy breaking @xcite , gauge coupling unification @xcite , generation of fermion mass hierarchies @xcite and electroweak symmetry breaking by a composite higgs doublet @xcite . in the effective @xmath4 theory , the extra dimensions `` appear '' in the form of kaluza - klein ( kk ) excitations of sm gauge bosons ( and quarks and leptons ) . these kk states have masses quantized in units of @xmath5 , where @xmath6 is the size of an extra dimension . in this paper , we study the contributions to anomalous magnetic moment of the muon ( mmm ) from these kk excitations @xcite . ( tev)@xmath3 in which gravity and sm singlet fields ( for example , right - handed neutrino ) only propagate . in this case , the contribution to the mmm from kk excitations of graviton ( or right - handed neutrino ) is also important @xcite . contributions to the mmm from `` warped '' extra dimensions with and without sm gauge fields propagating in them have also been studied @xcite . ] to be specific , we investigate whether this contribution can be significant enough to be relevant for an explanation of the @xmath7 discrepancy between the sm prediction and the new world - average value of the mmm @xcite : @xmath8 . this discrepancy is ( roughly ) comparable to the electroweak correction to the mmm in the sm which is @xmath9 @xcite . of course , there are other new physics explanations of this discrepancy such as susy , non - minimal higgs sector etc . a priori , the one - loop contribution to the mmm from kk excitations of electroweak gauge bosons ( and the muon ) can be comparable to the sm electroweak correction to the mmm for two reasons . one is that the masses of these kk states ( the compactification scale ) need not be much larger than the electroweak scale . the other is the possible enhancement due to large number of kk states : in general , we get @xmath10 where @xmath11 is the one - loop contribution to the mmm from kk states of @xmath12 , @xmath13 and @xmath14 and @xmath15 is the one - loop electroweak correction in the sm . the mass of the kk state with momentum @xmath16 in the @xmath17 extra dimension is given by @xmath18 , where @xmath19 and @xmath20 is an integer ( assume @xmath21 extra dimensions compactified on circles of radius @xmath6 ) . thus , @xmath22 if @xmath23 few @xmath24 gev and/or @xmath25 . there are , of course , constraints on @xmath26 which depend on whether the sm quarks and leptons propagate in these extra dimensions or not .

cluster of galaxies are the largest gravitationally bound systems in the universe . they are excellent laboratories for studying the large - scale structure formation , structure mass assembly and galaxy evolution . numerical simulations show that massive clusters of galaxies form through hierarchical merging of smaller structures ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? clusters are complex systems , including a variety of interacting components such as galaxies , x - ray emitting gas and dark matter . optical and x - ray studies show that a large fraction of clusters contains sub - structures , revealing that clusters are indeed dynamically active structures , accreting galaxies and groups of galaxies from their neighborhoods ( e.g. * ? ? ? * ) . even though it is thought that rich clusters form at redshift 0.81.2 ( or as high as 3.0 , see * * ) , there are numerous evidences ( optical , x - ray ) that clusters are still accreting sub - structures at intermediate and low redshifts ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? we may witness the assembly of rich clusters by observing large groups or poor clusters which , in turn , would be the future core of rich clusters . the details of this process will depend in part on how these large groups / poor clusters relate to more nearby structures . most galaxies in the universe are concentrated in low - density environments ( groups and poor clusters ) . for intermediate redshifts , @xmath110.5 , while massive clusters of galaxies have been widely studied , the intermediate - mass systems , those between loose groups and rich clusters of galaxies , have received comparatively little attention , either in x - rays or in the optical . in the context of the hierarchical structure formation scenario , an intermediate - mass system is a fundamental player to understand the process involved in the assembly of massive clusters of galaxies . x - rays observations of intermediate - mass structures are particularly interesting , since the cluster x - ray faint - end luminosity function has eventually to turn over if the luminosity function of clusters is to meet that of single brightest elliptical galaxies with x - ray luminosities of a few @xmath12 ergs s@xmath5 . if this gap at intermediate luminosities could be closed , an x - ray luminosity function of all galactic systems could eventually be established , in analogy to that existing for the optical @xcite . in addition , the spatial distribution of low - mass clusters at intermediate - redshifts could be studied in order to map regions just entering the non - linear regime , i.e. , @xmath13 . in the optical , the intermediate - mass systems are also of great importance and have received little attention . many previous works have focused on the study of the galaxy population at intermediate redshifts but mostly in rich cluster of galaxies . these have established that the morphological content of galaxy clusters at intermediate redshift differs dramatically from that in nearby clusters ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? indeed , at @xmath110.5 there is an excess of spirals and a deficiency of lenticular galaxies in cluster cores when compared with the galaxy population in nearby clusters . it has been shown by these studies that the morphology - density relation is strong for concentrated , `` regular '' clusters , but nearly absent for clusters that are less concentrated and irregular , in contrast to the situation for low - redshift clusters , where a strong relation has been found for both . @xcite suggests that these observations indicate that the morphological segregation proceeds hierarchically along the time , i.e. irregular clusters at intermediate redshifts are not old enough to present segregation . however , nearby irregular clusters seem to be evolved enough to establish the correlation . taken together , these studies reveal that the morphological segregation has evolved significantly since @xmath14 , at least for regular clusters . however , it is not yet known at which degree , if any , morphological segregation evolves in the sparser environments of groups . a few poor clusters or groups at intermediate redshifts have been studied , either in x - ray and/or in the optical ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? one example is the work of @xcite , where it is presented the first spectroscopic survey of intrinsically low x - ray luminosity clusters ( @xmath15 ) at intermediate redshifts @xmath16 . the ten systems studied have velocity dispersions in the range 350850 km s@xmath5 , and are consistent with the local @xmath17@xmath18 correlation . they also find that the spectral and morphological properties of galaxies in these clusters are similar to those found in more massive systems at similar redshifts . more recently , @xcite described the properties of 6 intermediate redshift groups ( @xmath19 ) observed with xmm-_newton _ and concluded that they follow the same scaling relation observed in nearby groups . in this paper we analyze the properties of the low - luminosity x - ray cluster of galaxies rx j1117.4@xmath00743 hereafter [ vmf 98 ] 097 based on optical and x - ray data . the cluster was selected from the 160 square degree rosat cluster survey @xcite and is part of an ongoing project to study the cluster properties and the galaxy population of poor clusters in the redshift range @xmath20 . this paper is arranged as follows . in section 2 and 3 we describe the optical and x - ray data , respectively . section 4 shows the results based on the analysis of these data : the velocity distribution , the galaxy projected distribution , the cluster color - magnitude diagram , a weak - lensing analysis , and a study of the mass distribution , based on weak lensing and x - ray emission . in sect . 5 we discuss the evolutionary status of [ vmf 98 ] 097 and in section 6 we summarize our conclusions . throughout this paper we adopt when necessary a standard cosmological model : @xmath21 km s@xmath5 mpc@xmath5 , @xmath22 and @xmath23 . at @xmath24 , 1 corresponds to @xmath25kpc . this study is based on data collected with the gemini multi - object spectrograph ( hereafter gmos , * ? ? ? * ) at the gemini south telescope during the system verification process of the instrument . the cluster was imaged through the @xmath3 and @xmath2 sloan filters @xcite in 2003 march and may , using the detector array formed by three @xmath26 pixels eev ccds . with a pixel size of 13.5 microns and a scale of 0073 pixel@xmath5 , the detectors cover an area of 5.5 arcmin@xmath27 on the sky . a total of 12 images of 600 sec in @xmath3 and 7 images of 900 sec in @xmath2 were obtained , giving an effective exposure time of 7200 seconds and 6300 sec in both filters , respectively . we adopted a @xmath28 binning for the images ( 0146 pixel@xmath5 on the sky ) . offsets between exposures were used to take into account the gaps between the ccds ( 37 unbinned pixels ) and for cosmic ray removal . all images were observed under good transparency ( photometric ) and seeing conditions , with seeing median values of 07 and 08 in @xmath3 and @xmath2 , respectively . all observations were processed with the gemini iraf package v1.4 inside iraf . the images were bias / overscan - subtracted , trimmed and flat - fielded . the final processed images were registered to a common pixel position and then combined . the @xmath2 and @xmath3 magnitude zero - points were derived using @xcite standard stars observed immediately before and after the science exposures . the accuracy of the calibrations is of the order of 5% and 7% for @xmath3 and @xmath2 , respectively . we have used sextractor @xcite to detect objects in the images and to obtain their relevant photometric parameters . the combined @xmath3image was used to identify objects above a threshold of @xmath29 over the sky level ( 27.6 mag / arcsec@xmath30 ) and with at least 10 pixels ( 0.21 arcsec@xmath27 ) . the photometry in the @xmath2-band image was performed using the parameter assoc . this means that the photometric parameters in @xmath2 were obtained only for those objects detected in the @xmath3 image . the resulting catalog was then matched to obtain a final photometric catalog . we adopted the magnitude given by the parameter mag@xmath31auto as the value for the total magnitude of the objects . the colors of the objects were determined by measuring their magnitudes within a fixed 10 pixels diameter circular aperture ( 15 ) , corresponding to @xmath32 kpc at the cluster rest - frame . the sextractor `` stellarity '' index ( an indication of how certain an optical source is unresolved ) was used to separate stars from galaxies . objects with a stellarity index @xmath33 were selected as galaxies . this cut is in agreement with a separate classification done by plotting pair of object parameters , like central intensity _ vs. _ area , central intensity _ vs. _ size , and peak intensity _ vs. _ size , as well as by visual control . in all cases the classifications are consistent down to @xmath3=25.5 mag . the galaxy counts calculated using the objects classified as galaxies reach their maximum at @xmath3 = 25.8 mag . using this information and the uncertainties in the galaxy classification above @xmath3 = 25.5 mag , we have adopted this latter value as our limiting magnitude . the final catalog contains the total magnitudes , the colors and the structural parameters for 2698 objects classified as galaxies . of these , 1348 are brighter than 25.5 mag in @xmath3(@xmath34 at the distance of the cluster ) . the targets for spectroscopic follow up were selected based on their magnitudes only . no color selection was applied , meaning that the sample includes galaxies of different morphological types . all galaxies with apparent magnitudes brighter than @xmath3@xmath123 mag were selected for spectroscopy ( 31% of the total sample ) . of these , only 79 objects were observed spectroscopically . two masks were created : one for bright objects ( @xmath3 @xmath35 mag ) and another for faint objects ( @xmath36@xmath3 @xmath37 mag ) . the spectra of the galaxies were obtained with gmos in 2003 may 2930 , during dark time , with a good transparency , and with a seeing that varied between 08 and 09 . a total exposure times of 3600 seconds and 6000 seconds were used for masks containing bright and faint objects , respectively . small offsets of @xmath38 pixels in the spectral direction ( @xmath39 ) towards the blue and/or the red were applied between exposures to allow for the gaps between ccds and to avoid any lost of important emission / absorption lines present in the spectra . spectroscopic dome flats and comparison lamp ( cuar ) spectra were taken after each science exposure . all spectra were acquired using the 400 lines / mm ruling density grating ( r400 ) centered at 6700 , in order to maximize the wavelength coverage for galaxies at the cluster distance . all science exposures , comparison lamps and spectroscopic flats were bias subtracted and trimmed . spectroscopic flats were processed by removing the calibration unit plus gmos spectral response and the calibration unit uneven illumination , normalizing and leaving only the pixel - to - pixel variations and the fringing . the resulting 2-d spectra were then wavelength calibrated , corrected by s - shape distortions , sky - subtracted and extracted to an one - dimensional format using a fixed aperture of 13 . the residual values in the wavelength solution for 2030 points using a 4th or 5th - order chebyshev polynomial typically yielded _ rms _ values of @xmath400.20 . with the choice of a 075 slit width , the final spectra have a resolution of @xmath41 ( measured from the arc lines fwhm ) with a dispersion of @xmath42 pixel@xmath5 , covering a wavelength interval of @xmath439800 ( the wavelength coverage depends on the position of the slit in the gmos field - of - view ) . finally , the residuals of the 5577 , 5890 , and 6300 night - sky lines were removed from all spectra using a 10-th order cubic spline polynomial . beyond 7800 , the residuals of night - sky lines were simply masked . to obtain the galaxy radial velocities , we first inspected the spectra to search for obvious absorption and/or emission features characteristic of early- and late - type galaxy populations . for galaxies with clear emission lines , the routine rvidline in the iraf rv package was used employing a line - by - line gaussian fit to measure the radial velocity . the residual of the average velocity shifts of all measurements were used to estimate the errors . for early - type galaxies , the observed spectra were cross - correlated with high signal - to - noise templates using the fxcor program in the rv package inside iraf . the errors given by fxcor were estimated using the r statistic of tonry & davis ( 1979 ) : @xmath44 , where @xmath45 is the fwhm of the correlation peak and @xmath46 is the the ratio of the correlation peak height to the amplitude of the antisymmetric noise . the left panel in fig . [ spec ] shows the smoothed spectra of three galaxies identified as cluster members , corresponding to three different spectral types : early - type ( bottom ) , late - type ( top ) and intermediate - type ( middle ) . we were able to measure redshifts for 77 objects ( @xmath47% success rate ) . seventy five of them are galaxies and two are m - class stars . as expected , the fraction of emission - line galaxies is relatively high and constitute 33% of the total sample . however , the fraction of emission - line galaxies that are cluster members is lower and represent only @xmath48% of the cluster galaxy population . the emission - line galaxy fraction is in agreement with the results obtained by @xcite for 10 intrinsically low x - ray luminosity cluster ( @xmath49 10@xmath50 erg s@xmath5 ) at z@xmath510.25 ( see section 4.1 for more details ) . the measured redshifts , corrected to the heliocentric reference frame , and the corresponding errors are listed in table [ tab1 ] ( columns 6 and 7 , respectively ) . the galaxy identifications and their sky coordinates are given in the first 3 columns . the apparent total magnitudes in @xmath3-band and the @xmath2@xmath52 @xmath3 colors inside a fixed circular aperture of 15 are listed in columns 4 and 5 , respectively . the @xmath53 value @xcite listed in column 8 was used as a reliability factor of the quality of the measured velocity . for @xmath54 , the resulting velocity was that associated to the template which produced the lowest error . for galaxies with @xmath55 , we looked for absorption features like caii and g - band in the spectra , and performed a line - by - line gaussian fit using the package rvidline . the resulting values where then compared with the velocities given by cross - correlation . in all cases the agreement between the two procedures were good . the histogram of the redshift distribution is presented in the right panel of fig . the concentration of galaxies at @xmath56 ( shaded area ) indicates the position of the [ vmf 98 ] 097 cluster . the peak at @xmath57 corresponds to a group of galaxies , rixos f258_101 , located @xmath51 25 north of the cluster core ( see section [ sec : galprojdist ] ) . two other small peaks can be seen which are probably related to the groups reported by @xcite . the cluster [ vmf 98 ] 097 was serendipitously discovered in x - rays in a pointed rosat pspc observation of qso pg1115 + 080 @xcite . this object was observed by xmm-_newton _ in december 2002 ( obsid 082340101 ) and twice in june 2004 ( obsid 203560201 and 203560401 ) . [ vmf 98 ] 097 is found in the field of view of the mos1 in all exposures , but it was observed entirely by the pn detector only in the 2002 observation ( only half of the cluster appears in any mos2 field of view ) . this cluster was also observed by chandra acis - i3 in june 2000 ( p.i . garmire ) in a 26 ks exposure . however , it produced only @xmath58 net counts ( background corrected ) . therefore , the chandra observations are not used in the analysis . we have downloaded the odf files from xmm public archives and performed the mos and pn `` pipelines '' , which consist in the removal of bad pixels , electronic noise and correction for charge transfer losses with the program sas v6.5.0 . we have then applied the standard filters and removed the observation times with flares using the light - curve of the [ 8.014.0 kev ] energy band . the final exposure times after subtracting high particle background intervals of the cleaned event files for all observation are given in table [ tbl : resumoxmm ] together with the net count number ( i.e. , source minus background counts ) . for the spectral analysis we have selected a circular region of 11 centered at ra @xmath59@xmath60 , dec @xmath61@xmath62@xmath63 ( j2000 ) . the background was selected in the same observation . we have used a larger extraction region near the detector border ( this because [ vmf 98 ] 097 is itself near the border ) , without any visible sources . since about half of the cluster falls outside the mos2 field of view , we have used only mos1 and pn cameras of the 2002 observations , and only the mos1 of the 2004 observations . the total source counts , background subtracted , are also given in table [ tbl : resumoxmm ] . the left panel of figure [ xray_em ] shows the composite image made with mos1 and pn available data in the 0.38.0 kev band . the epic - mos has a fwhm @xmath64at the center of the detector . however the mos point spread function has somewhat extended wings and the half energy width ( hew ) is @xmath6514 . for off - axis sources there is a degradation in the resolution , which also depends on the energy @xcite . the cluster [ vmf 98 ] 097 is located @xmath65125 from the detector axis . therefore the resolution at 1.5 kev is 55 and 65 for the mos-1 and pn respectively . at 5 kev the resolution is 63 and 76 for the mos-1 and pn respectively . the effective exposure time is also affected , but it is taken into account by the redistribution matrix file ( rmf ) and the auxiliary response file ( arf ) @xcite . we also present in the right panel of fig . [ xray_em ] the smoothed x - ray emission plotted over the gmos @xmath3 image . the core of the cluster , about 1 south of the center of the image , is clearly detected in x - rays . the emission at the north of the image is associated with a foreground group at @xmath66 ( see section [ sec : galprojdist ] ) . two other x - ray emission features are worth mentioning since they are also present in the galaxy - density map and in the weak - lensing map discussed below : the feature at the east of the cluster core ( hereafter e - structure ) and another at the northeast , at the border of the optical image ( hereafter the ne - structure ) . the ancillary and redistribution files ( arf and rmf ) were created with the sas tasks ` arfgen ` and ` rmfgen ` , taking into account the extended nature of the source . the mos and pn spectra were fitted simultaneously , each spectrum with its own rmf , arf , and background files . the spectral fits were done with xspec 11.3 in the range [ 0.38.0 kev ] on the re - binned spectrum , with at least 12 counts per energy bin . we have used the mekal @xcite plasma model with a photoelectric absorption given by @xcite . for the hydrogen column density , we have adopted the galactic value at the position of [ vmf 98 ] 097 , @xmath67@xmath68 ( using the task ` nh ` from ftools , which is an interpolation from the * ? ? ? * galactic @xmath69 table ) . given the evidences presented in sect . [ sec : veldist ] that the velocity distribution has two peaks , we have tried the spectral fits with different redshifts but the results are virtually the same : we can not , with these spectra , obtain a redshift estimate of the source . we have also tried a two component mekal model , representing each source in the line - of - sight ; however the fit did not converge because of the low signal - to - noise ratio of the spectra . therefore , we give our results here as mean emission - weighted values , adopting a fixed mean redshift . the best - fit model is shown in the left panel of fig . [ xray_spec ] . fixing the redshift at @xmath70 ( average redshift obtained for the cluster , see below ) , we obtain the following temperature and metal abundance ( metallicity ) : @xmath71 kev and @xmath72 ( the errors are at the 90% confidence levels ) . the fit is fairly good with @xmath73/d.o.f . @xmath74 ( the null hypothesis probability is 0.54 ) . the metallicity is not well constrained since the fe - k line is not well detected . this is also shown in the temperature - metallicity correlation plot , in the right panel of fig . [ xray_spec ] . we have computed the unabsorbed x - ray flux and luminosity in different energy bands using the plasma model described in the previous section . table [ tbl : lx ] summarizes these results . with the determined temperature and bolometric luminosity , [ vmf 98 ] 097 is found to behave like a normal cluster , in agreement with the local @xmath75@xmath76 correlation @xcite and the correlation for intermediate redshift clusters ( @xmath9 , * ? ? ? the agreement with both local and intermediate relations comes from the intrinsic scatter in both relations and the error bars in our cluster . therefore , the x - ray emission of [ vmf 98 ] 097 is not affected ( at least significantly ) by the emission from the group / structure behind the cluster . a search with ned reveals that this cluster has a radio emission at 1.4 ghz associated to it . we have used this information to look for a radio image in the first survey . the radio image has a fwhm of 54 and the radio emission contours are shown in fig . [ fig : vlamosgmos ] . the peak of the radio emission coincides with galaxy 1098 ( tab . [ tab1 ] ) and the center of the x - ray emission . it has wide - angle tail morphology , which is often found in radio galaxies in the center of clusters @xcite . this radio morphology implies that the radio galaxy is moving with respect to the icm . this could be due to the bulk motion of the intra - cluster gas @xcite , for instance , because of a cluster or group merging . in the present case , based on the broadening of the radio emission , the direction of motion perpendicular to the line - of - sight is west east . the presence of an agn may contaminate the x - ray spectrum with a hard component , making the spectral determined temperature artificially higher . the radio emission associated with galaxy 1098 suggests such an agn . however , there is no sigh in either xmm or chandra data suggesting a point source or an excess x - ray emission at the spatial location of galaxy 1098 . since the x - ray surface brightness is quite flat in the cluster center , a bright x - ray agn would be detectable . in addition , there is no indication of an agn in the optical spectrum of the galaxy 1098 . thirty seven out of 75 galaxies with measured velocities are located in the redshift interval @xmath77 , corresponding to the prominent peak seen in the right panel of fig . the velocity distribution of these galaxies is shown in fig . [ hist_v ] . it is clear , from the figure , the complexity of the cluster . in order to investigate its structure , we use the kmm test @xcite , which is appropriate to detect the presence of two or more components in an observational data set . first we consider whether the data is consistent with a single component . the results of applying the test in the homoscedastic mode ( common covariance ) yields strong evidence that the redshift distribution of galaxies in the redshift interval above is at least bimodal , rejecting a single gaussian model at a confidence level of 97.7% ( p - value of 0.024 ) . the p - value is another way to express the statistical significance of the test , and is the probability that a likelihood test statistic would be at least as large as the observed value if the null hypothesis ( one component in this case ) were true . assuming two components , they are located at @xmath78 and @xmath79 , corresponding to the structures s1 and s2 in figure [ hist_v ] . they are separated by 3000 km s@xmath5 in the cluster rest frame . the histogram shows another gap of @xmath80 km s@xmath5 ( also in the cluster rest frame ) between s2 and s3 which is formed by 4 galaxies in the interval @xmath81 . if we assume that the velocity distribution in figure [ hist_v ] is indeed tri - modal , the kmm test rejects a single gaussian at a confidence level of 99% ( p - value of 0.010 ) . consequently , a model with three components is statistically more significant than a model with two components . in this case , the procedure assigns a mean value of @xmath78 , @xmath82 and @xmath83 with 23 ( 62% ) , 10 ( 27% ) and 4 ( 11% ) galaxies for each of the structures , respectively . we used the robust bi - weight estimators @xmath84 and @xmath85 of @xcite to calculate a reliable value for the average redshifts ( central location ) and the velocity dispersions ( scale ) of the two main velocity structures ( s1 and s2 ) present in the cluster . we used an iterative procedure by calculating the location and scale using the rostat program and applying a 3 @xmath18 clipping algorithm to the results . we repeated this procedure until the velocity dispersion converged to a constant value . the best estimates of the location and scale for s1 and s2 are shown in table [ tab : vel ] ( columns 5 and 6 respectively ) . the table also shows , for the velocity structure s1 , the virial radius @xmath86 and the virial mass ( column 8) , computed with the prescription of @xcite . the velocity structure s2 is not centrally concentrated and is probably not virialized ( see below ) . we then chose to not determine its @xmath86 and virial mass . the number of galaxies in the structure at @xmath87 ( s3 , see fig . [ hist_v ] ) is too small for a reliable determination of the velocity dispersion and other dynamical parameters @xcite . it is worth noting that the derived line - of - sight velocity dispersion for s1 of 592@xmath8882 km s@xmath5 agrees well ( inside the 68% confidence interval ) with the value inferred from the intra - cluster medium temperature . indeed , using the @xmath89@xmath18 relation from @xcite ( which is derived from a local sample ) the measured x - ray temperature of @xmath90 kev implies @xmath91 km s@xmath5 . this result suggests that the x - ray emission is associated to s1 . this seems to be the case since the x - ray emission is centered on the cluster core , which is associated with the velocity structure s1 . figure [ kernel_map ] shows an adaptive - kernel density map ( see * ? ? ? * ) based on a sample of 272 galaxies brighter than @xmath3@xmath92 mag . the area corresponds roughly to @xmath93 mpc@xmath30 at the rest - frame of the cluster . all structures shown in this map are above the 3 @xmath18 significance level . most of the structures identified in this figure are also present in the x - ray map ( figure [ xray_em ] ) . the [ vmf 98 ] 097 cluster is represented by the high density region located @xmath94 south from the center of the image . the second highest density region , located at the top of the figure , corresponds to a foreground group at @xmath57 ( rixos f258_101 ; * ? ? ? a third structure , located @xmath95 east from the cluster core , is the e - structure present in the x - ray map . most galaxies in the cluster core have velocities in the range of s1 ( squares ) , however the galaxies in s2 ( rhombi ) are mainly distributed , without any significant concentration , to the south of the cluster core . we have velocities for only four objects which overlap in space with the e - structure : one is in s1 , another in s2 and two others corresponding to a nearby and to a background object . the detection of two velocities at the cluster redshift , as well as the x - ray emission , suggests that the e - structure is probably dynamically associated with the cluster . figure [ kernel_map ] shows also two overdensities ne from the cluster center . the first one , at @xmath96 , may be associated to the cluster ( a substructure ) , since several velocities in the region are in the redshift of the cluster core . the second , at @xmath95 , is the ne - structure in the x - ray map , and may be either a substructure or a background cluster ( the galaxies there tend to be fainter than those in the cluster core ) . unfortunately we do not have any radial velocity in this region to confirm this point . as shown below , the weak - lensing analysis also detect most of the features present in figure [ kernel_map ] . galaxy colors provide valuable information about the stellar content of galaxies , allowing to identify passive and star - forming galaxies in clusters . [ cmd](a ) shows the color - magnitude diagram ( cmd ) for all galaxies detected in the images . colors and the total magnitudes have been corrected by galactic extinction from the reddening maps of @xcite and using the relations of @xcite ( @xmath97 mag and @xmath98 mag , respectively ) . an inspection of this cmd shows that the galaxy populations of the two main structures of the cluster are not the same , with s1 containing much more red galaxies ( at the so - called cluster red sequence in @xmath99 ) than s2 and s3 . this behavior is better seen in the color distribution histogram of galaxies with measured velocities . [ cmd](b ) indicates that s1 is dominated by a red galaxy population ( right shaded histogram ) , while s2 and s3 contains mostly blue galaxies ( left shaded histogram ) . it is important to note also that s2 and s3 are formed by galaxies that , in average , are @xmath100 fainter than those in s1 . in a cluster that is dinamically active , it is expected to find a high fraction of star - forming galaxies . the results shown above point into that direction . as we mentioned in section 2.2 , the average fraction of emission - line galaxies in the cluster is relatively small and constitute only 22% of the population . however , there are differences in the content of the galaxy population in the structures . in s1 , only 17% of the galaxies ( 4 out of 23 ) are emission - line objects . in the case of s2 and s3 , the fraction is much higher , and constitute 31% of the population ( 5 out of 13 ) . the results agree well with what we see in the color - magnitude diagram : the emission - line , star - forming ( blue ) galaxies are more numerous in s2 and s3 than in s1 . we investigated the median magnitudes and colors for all galaxies with @xmath3 @xmath37 in the two overdensities detected in x - ray , with the density map ( e - structure and ne - structure ) , and in the cluster core . for the analysis we select the galaxies inside a radius of 20 ( @xmath101 h@xmath10 mpc at the cluster distance ) from the center given by the maximum of the galaxy overdensity ( see fig . [ kernel_map ] ) . table [ tab : mags ] summarizes the median magnitudes and colors for the three overdensities . the galaxies in the e - structure are @xmath96 magnitude fainter and slightly bluer than the galaxies in the cluster core . the e - structure contains two galaxies with velocities at the cluster redshift . one of them is a member of s1 , with ( @xmath2@xmath52 @xmath3@xmath102 . the other is a member of s2 , with ( @xmath2@xmath52 @xmath3@xmath103 , similar to the median color value derived for this structure . the e - structure is also detected in x - rays and in the weak - lensing mass map . the galaxies in the ne - structure are fainter and bluer than in the cluster core . this structure is detected in x - ray ( fig . [ xray_em ] ) , and also by weak lensing ( see section [ sec : weaklensing ] ) . due to the lack of redshift information in this region , we can only speculate about the nature of this structure , i.e. , if it is a background or foreground cluster or even a sub - structure of [ vmf 98 ] 097 . if the ne - structure is a background cluster , then one would expect a much fainter galaxy population , but with much redder colors . this is not the case , since the galaxies in this region are faint , but bluer than in the core of [ vmf 98 ] 097 . another possibility is that it is indeed a background cluster of blue , star - forming galaxies , where the red sequence is not yet established . finally , this structure could be associated to the foreground group rixos f258_101 at @xmath57 ( the ne - structure is located at 17 from the center of this group ) . however this is very unlikely . the average magnitude of the galaxies in the group is @xmath104 mag with a median color of ( @xmath2@xmath52 @xmath3@xmath105 . although the median color obtained for the ne - structure is similar to the median color of the galaxies in rixos f258_101 , the galaxy population is much fainter ( 3.5 mag ) . gravitational lensing is a powerful tool for studying the matter distribution in galaxy clusters . in its weak regime gravitational lensing allows the reconstruction of the projected mass distribution through the analysis of the small morphological distortions induced by gravitational lensing of background sources ( weak shear field ) . this technique is completely independent of the dynamical state of the cluster . in this section we apply a weak - lensing analysis to the imaging data to estimate the mass distribution on the field of [ vmf 98 ] 097 . the determination of the shapes of faint , putative background galaxies , was performed using the method described in @xcite . in the following paragraphs we summarize the main steps of the procedure used in the analysis . we performed galaxy shape measurements , including the removal of seeing effects and psf anisotropies , using the algorithm im2shape @xcite . this program models an astronomical object as a sum of gaussian functions with an elliptical base and carries out the deconvolution of the object image with a local psf extracted from the image itself . while stars were modeled as one simple gaussian , galaxies are treated as a sum of two gaussians with same ellipticity and position angle . we use high signal - to - noise unsaturated stars ( @xmath106 ) to map the psf all over the frame . to make the final catalog , stellar objects with discrepant ellipticity or full width at half maximum ( fwhm ) were removed through a sigma - clipping procedure . in both images the psf showed to be nearly constant across the entire field , having an average ellipticity of 4.4% and 6.0% in the @xmath2 and @xmath3 images , respectively . to select background galaxies , which are the probes of the weak shear field , we need to rely on their magnitudes and colors to discriminate them from the cluster and/or foreground objects , given that we have not redshift information for the vast majority of them . ideally we would like to use just galaxies redder that the cluster red - sequence for those objects are supposed to be all behind the cluster ( e.g. * ? ? ? * ) but unfortunately , as we can see in fig . [ cmd ] , the red - sequence at @xmath107 is very red for this particular combination of filters , and the number of galaxies redder than the red - sequence is too small to provide an adequate sample . we opt therefore for a simple magnitude and signal - to - noise cut defining the weak lensing sample as all galaxies fainter than @xmath3@xmath123.0 mag ( @xmath108 at @xmath24 ) with ellipticities measured with precision greater than 0.2 . this criteria left us with a sample of 1001 ( 23 gal . arcmin@xmath109 ) and 1298 ( 30 gal . arcmin@xmath109 ) galaxies for the @xmath2 and @xmath3 images , respectively with an average magnitude of @xmath3@xmath124.9 mag . by using this criteria we expect some contamination by cluster or foreground galaxies to be present but it should not introduce any bias in the mass reconstruction , only increase the noise . the surface mass distribution of the cluster [ vmf 98 ] 097 has been recovered from the shear data using the second version of the lensent code @xcite . this algorithm takes the shape of every galaxy image as an independent estimator of the local reduced shear field . the reconstruction of the mass distribution incorporates an intrinsic smoothing whose characteristic scale is determined by bayesian methods , using a maximum entropy prior . this scale is chosen by maximizing the evidence , given the input data . using a gaussian function to smooth the data , we found that its optimal fwhm is 70 . figure [ massmaps ] shows the maps of the reconstructed surface mass density obtained by using data from each of the images . the maps are very consistent with each other , all showing basically the same features : a main structure clearly associated with the core of [ vmf 98 ] 097 , and two smaller structures at the eastern edge of the field , which can be associated with the e- and ne - structures discussed above . in the @xmath2 map there is a hint of a substructure between the cluster core and the ne - structure . the maps also suggest that there may exist a mass filament joining the e - structure with the main core . in general , it is actually impressive how x - ray emission , surface - mass and galaxy - number densities compare well in this field . figure [ massmaps ] also shows the mass center adopted for the radial analysis presented below ( dashed line ) . this center correspond to one of the brightest red galaxies on the cluster core which is close to the peak of both mass maps , particularly the one reconstructed with the @xmath2 image . we now address the measurement of the cluster mass , considering estimates based on weak - lensing of background galaxies and on the icm x - ray emission . for mass estimation through weak - lensing , we opt to use physically motivated mass - density models , to avoid the mass - sheet degeneracy bias @xcite . the two most widely adopted models for fitting shear data are the singular isothermal sphere ( sis ) and the nfw profile . the first is a solution of the hydrostatic equilibrium equation for an isothermal self - gravitating system , whereas the second provides a good fit to dark matter halos in numerical simulations @xcite . the sis profile has the advantage of having a single parameter , @xmath18 , which is associated with the line - of - sight velocity dispersion of the galaxies . this density profile is given by : @xmath110 the nfw profile is described by : @xmath111 where @xmath112 is the critical density , @xmath113 is a scale radius and @xmath114 is given by @xmath115 where @xmath116 is the concentration parameter . the approximate virial radius @xmath117 can be defined as @xmath118 . lensing formula for the sis and nfw profiles came from @xcite . these parametric models were fitted to the peak of the mass map corresponding to the cluster core using the procedure described in @xcite . we restricted the data to the region 15@xmath119 16 in relation to the mass center showed in fig . [ massmaps ] . data points closer to the center have been removed because they correspond to a region where the lensing effects are no longer linear ( strong lensing region ) and 16 is the distance to the closest image border . given these limits , the number of data points included in this analysis is 302 and 373 for the @xmath2 and @xmath3images , respectively . the reduced shear ( and the derived mass profile parameters ) depends on the mean redshift of the background galaxies through the mean value of the ratio @xmath120 of the angular diameter distances between the cluster and the sources and to the sources . we have estimated this quantity for our sample of background galaxies using a catalog of magnitudes and redshifts in the hubble deep field @xcite with both the same bright limit cutoff and the same average magnitude , obtaining in both cases @xmath121 . in fig . [ shear_profile ] we plot the binned data points of the galaxy ellipticities as well as the best fitted sis and nfw models . the best fitted parameters of these models are presented in table [ weak_results ] . our data poorly constrains the nfw concentration parameter because it controls the variation of the density slope in the very inner ( @xmath122 ) or outer ( @xmath123 ) regions , which we do not probe in our weak - lensing analysis . consequently , we decided to keep the value of @xmath116 constant , @xmath124 , and fit only @xmath117 . the results obtained for the @xmath3 and @xmath2 images are fully consistent within the errors . the same is not valid for the two models . the sis results tend to give smaller values for the cluster mass when compared with the results obtained with the nfw model , because within the restricted radial range we are considering here the sis profile is steeper than the nfw . it is worth mentioning that the values of @xmath117 obtained through weak - lensing are significantly above those from the virial analysis presented in sec . [ sec : veldist ] . we shall come back to this issue in sec . the gas - density profile is obtained from the radial x - ray surface - brightness profile . we assume that the gas has a number - density profile given by the @xmath125-model @xcite : @xmath126^{-3\beta/2 } \ , , \label{eq : gasbaeta}\ ] ] where @xmath127 is the core radius and @xmath128 the central number - density of electrons . then , the x - ray surface - brightness profile is : @xmath129^{-3\beta + 1/2 } \ , , \label{eq : sbx}\ ] ] assuming that the gas is isothermal and the core radius @xmath130 ( capital and lowercase symbols refer to 3d and projected quantities , respectively ) . the x - ray brightness profile of [ vmf 98 ] 097 was obtained using the stsdas / iraf task ellipse , with the sum of the mos1 images corresponding to observations with obsid 203560201 and 203560401 in the 0.38.0 kev energy band . each image was binned by a factor 128 so that 1 image pixel was 64 . the brightness profile was extracted up to 125 , corresponding to @xmath131kpc at the cluster redshift . figure [ x_surf ] shows the cluster core x - ray emission profile together with the best least - squares fitted @xmath125-model . we have obtained @xmath132 211@xmath8809 ( @xmath133kpc ) and @xmath134 . the central electronic density , @xmath128 , is estimated using the emission integral , @xmath135 ( see * ? ? ? * ) , which is related to the thermal spectrum normalization parameter given by xspec . using the thermal spectrum extracted within 66 , we have obtained @xmath136@xmath137 . band fitted by a @xmath125-model . the full line is the fit to the cluster plus background emission . the dashed line corresponds to the inferred x - ray emissivity of the cluster . ] the radial gas - mass profile can be simply obtained by integrating the density profile in concentric spherical shells . the dynamical ( total ) mass is computed assuming an isothermal gas in hydrostatic equilibrium . even summing all xmm observations , we have enough counts only to compute a single emission - weighted temperature . using the temperature previously determined , @xmath138kev , the computed dynamical mass is presented in fig . [ x_wl_mass ] . at @xmath139125 , the total mass inferred from x - rays is @xmath140 . the gas mass fraction , @xmath141 , is computed as the ratio between the gas mass and the total mass at a given radius . this ratio is related to the cluster baryon fraction as @xmath142 , where @xmath143 is the baryonic mass in the galaxy cluster members . the baryonic mass in galaxies may be estimated as @xmath144 @xcite . the bottom panel of fig . [ x_wl_mass ] shows the baryon fraction radial profile . at @xmath139125 , @xmath145 , with a rising trend . the x - ray observations are not deep enough to detect the point where @xmath141 flattens , as is observed in several clusters ( e.g. , * ? ? ? they are compared to the weak - lensing mass estimated with a sis ( short - dashed line ) and nfw ( long - dashed line ) profiles . the grey regions correspond to @xmath146 statistical errors . _ bottom panel : _ gas mass fraction profile derived using the x - ray data . ] at a radius of 0.5 h@xmath10 mpc , the inferred week - lensing masses from the @xmath3 image are @xmath147 and @xmath148 for the sis and nfw profiles , respectively . at the same radius , the x - ray mass is @xmath149 , i.e. the weak - lensing mass is 3.4 ( sis ) to 4.8 ( nfw ) times the value inferred through the x - ray emission . the possible causes for such discrepancy are given in the next section . our results strongly suggest that we are witnessing the mass assembly of a cluster at @xmath24 . there are several hints pointing towards this suggestion . the morphology of the cluster is complex , presenting at least two significant substructures . the x - ray emission , the galaxy distribution , and the surface - mass density map , all present the same overall features : the cluster core and the e and ne - structures . the e - structure and the cluster core have a projected distance less than 1 mpc ( at the cluster redshift ) and , as we argued in sections 4.2 and 4.3 , are probably at the same redshift . the substructure located between the cluster core and the ne - structure seem in the projected galaxy distribution ( figure [ kernel_map ] ) might be also real , since there is a feature present in our @xmath2 weak - lensing map close to it . the velocity distribution of the region is also complex , with multiple peaks in a small redshift range ( fig . [ hist_v ] ) . the statistical analysis presented in section 4.1 indicates , for instance , the presence of three velocity - substructures : s1 , s2 and s3 . additional evidence of dynamical activity in clusters can be obtained from its brightest galaxy . for local x - ray luminous groups and poor clusters , the most luminous galaxies ( bggs ) lie near the peak of the x - ray emission ( e.g. * ? ? ? . however , at intermediate redshift , this picture may be different . a recent study of x - ray groups and poor cluster at moderate redshif by @xcite suggest that the brightest galaxies in groups and poor clusters are still in the process of forming , as late as at @xmath1500.2 , in some systems . the indication is given by the offset between the bgg and the x - ray emission and offsets between the velocity of the bggs and the mean velocity of the system . this scenario is consistent with recent numerical simulations @xcite . in the case of [ vmf 98]097 , the x - ray emission is associated with the velocity structure s1 . this structure has two elliptical galaxies in the center with comparable luminosities : galaxy 947 with @xmath3@xmath120.26 mag , @xmath151 km s@xmath5 , and galaxy 1098 with @xmath3@xmath120.29 mag , @xmath152 km s@xmath5 . the peak of the x - ray emission is offset by @xmath153 from both galaxies . furthemore , the brightest elliptical galaxy ( number 947 ) is offset significantly in velocity from the mean velocity of the structure s1 ( @xmath154 1000 km s@xmath5 ) . these results provide additional evidences of an on - going dynamical activity in this cluster . the mass estimates obtaining using x - rays and weak - lensing are very discrepant , and this is usually interpreted as evidence of dynamical activity ( e.g. , * ? ? ? in fact , x - ray mass estimates are based on the assumption of hydrostatic equilibrium , which may not hold in the presence of mergers or strong tidal interactions . it also depends on the assumption that the @xmath125-model describes well the gas density radial profile . the accuracy of mass determination based on x - ray observations has been studied through hydrodynamical simulations . @xcite and @xcite find that , usually , x - ray estimates are good within 50% , with no systematic bias . in the case of an extreme non - equilibrium clusters , the mass deviation from the true value can be as high as a factor 2 . however , more recently , @xcite conclude that x - ray estimation of the total mass is biased towards lower values when using the @xmath125-model if the cluster is not in equilibrium . the typical estimated mass is around 40% of the true mass at about half the virial radius . another possibility is that the excess between lensing and x - ray masses ( a factor @xmath155 ) is due to the intervening mass along the line - of - sight of the cluster , which leads to an over - estimate of the weak - lensing mass of a few tens of percent @xcite , although this excess seems too high to be due to this effect only . on the other hand , the velocity dispersion in the region of the cluster core is consistent with equilibrium between gas and galaxies ( sect . [ sec : veldist ] ) . based on the color - magnitude analysis presented in sect.[sec : colmag ] , it is unlikely that the e - structure is a foreground group . besides , the e - structure contains two galaxies at the cluster redshift and is detected by weak - lensing mass reconstruction and by x - rays . it lies roughly at 1 @xmath156 mpc from the center of [ vmf 98 ] 097 and thus could be a substructure of the cluster . the ne - structure poses a more challenging problem , since we do not have any galaxy with measured redshift in this region . nevertheless , it is an overdensity detected in x - rays , in the galaxy projected number density map , and also by weak lensing . this suggests that it should be relatively massive . in order to probe its nature , we have compared their median magnitudes and the color ( @xmath2@xmath52@xmath3 ) with those of the cluster core and the e - structure . galaxies in the ne - structure are significantly fainter and bluer and , consequently , we suggest that it is indeed a background cluster , although spectroscopic information is necessary to verify such a claim . we have presented an optical and x - ray based study of [ vmf 98 ] 097 ( rx j1117.4@xmath00743 ) , an intermediate mass structure located at z@xmath10.485 . we demonstrate in this work that it is possible to obtain a good weak - lensing data for a distant cluster even with non - exceptional seeing conditions ( 0708 ) and with small field of view . our main results are summarized in the next paragraphs . the cluster shows a very complex structure . we find that velocity distribution of member galaxies is at least bimodal , with two well defined structures along the line - of - sight . the two main structures , named s1 with 23 galaxies and s2 with 9 galaxies , form the cluster core . these structures have a velocity dispersion of 592@xmath8882 km s@xmath5 and 391@xmath8885 km s@xmath5 respectively . using the projected density map of 272 galaxies brighter than @xmath3@xmath123 mag we were able to identify several structures in the neighborhood of [ vmf 98 ] 097 . these structure are also presented in x - ray and in the weak - lensing maps . the high density regions identified in the maps are : the cluster core formed by s1 and s2 , the foreground group rixos f258_101 ( located @xmath51 25 north from [ vmf 98 ] 097 ) , and two other overdensities , the e - structure and the ne - structure . we do not have redshift information of the galaxies belonging to these two structures , except for two galaxies at the cluster redshift in the e - structure . therefore , we have used the median magnitudes and colors of the galaxies inside a 20 radius from the peak of these overdensities to investigate the possibility if these structures are linked to [ vmf 98 ] 097 . based on this analysis and in the detection given by the weak - lensing and x - ray maps ( section 4.3 ) we conclude that the e - structure is a sub - structure associated to the cluster . using the same approach for ne - structure , we find that the galaxies in the region of this overdensity are significantly fainter and bluer than the galaxies in the cluster core and in the e - structure , suggesting that the ne - structure is a background cluster . however additional spectroscopic observations are necessary to verify this point . we have used the color - magnitude relation to analyze the galaxy contents in structures s1 , s2 and s3 . we find that the galaxy populations in s1 and s2@xmath0s3 differ in its content . most of the galaxies in s1 are redder than those presented in s2 and s3 , with an average color of @xmath2@xmath52@xmath3@xmath511.9 . they lie well inside the red sequence for passive galaxies ( see fig . 7(a ) ) . s2 and s3 are dominated by a population of blue , star - forming galaxies that , in average , are @xmath157mag fainter than the galaxies in s1 . further evidence is provided by the fraction of emission - line galaxies in each of the structure . we find that only 17% of the galaxies in s1 are emission - line galaxies , while the fraction of emission - line galaxies in s2 and s3 is of the order of 31% , in agreement with the galaxy contents derived for both structures from the color - magnitude relation analysis . we derived the x - ray temperature , the metal abundance and x - ray flux and luminosity in different energy bands using the plasma model described in section 3.1 . for the computed intra - cluster medium temperature of @xmath90 kev we find , using the @xmath158 relation from @xcite , a velocity dispersion of @xmath159 km s@xmath5 . this value agrees well with the derived line - of - sight velocity dispersion obtained for s1 ( 592@xmath8882 km s@xmath5 ) , suggesting that the x - ray emission is mainly associated to this structure . in addition , we find that the cluster is slightly hotter than expected for its luminosity ( @xmath160 @xmath161 erg s@xmath5 ) , compared to the local @xmath17@xmath89 correlation . however , its fall right on the sub - sample relation at @xmath162 of @xcite . we have used the weak - lensing analysis to map the mass distribution in the area of the cluster . we find a good agreement between the velocity dispersions derived from the @xmath2and @xmath3using the sis model profile . however , there is a disagreement with the values obtained from x - ray and from the kinematic of the member galaxies . we used two fit models , the sis and nfw , to compute the total mass of the cluster . we find that the total mass inferred from weak - lensing of 2.1 to 3.7 @xmath163 at @xmath164 mpc ( depending on the band and the model adopted ) is well in excess compared to the x - ray mass . however , given the several difficulties for an accurate gravitational lensing estimation in this field , particularly due to the contribution of the other mass clumps and the uncertainty on the average redshift of the background galaxies , these results should be taken with caution . the presence of several sub - structures in the x - ray , weak - lensing mass and galaxy density maps , the existence of a bridge of matter in the center of the cluster connecting different sub - structures ( detected in weak - lensing only ) and the complex velocity distribution of member galaxies reveal that this cluster is dynamical active . additional evidence of the dynamical activity is given by the offsets we see between the two brightest galaxies in the cluster core and the x - ray emission and the significantly offset between the velocity of the galaxy 947 ( the brigtest elliptical galaxy in s1 ) and the mean velocity of structure s1 . our main conclusion is that this poor cluster may be the core of a still forming rich cluster of galaxies . [ vmf 98 ] 097 is in an environment with other nearby substructures that , given their projected distance to the cluster , are probably gravitationally bound and will eventually merge to form a rich cluster . we would like to thank the anonymous referee for the useful comments and suggestions . erc acknowledges the hospitality of the departament of astronomy of the instituto de astronomia , geofsica e cincias atmosfricas , universidade de so paulo , where this work was partially done . erc also acknowledges the support from cnpq through the prosul project . gbln , lsj and cmo acknowledge support by the brazilian agencies fapesp and cnpq . we made use of the xmm - newton arquival data : the xmm - newton is an esa science mission with instruments and contribution direcly funded by esa member states and the usa ( nasa ) and the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the 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morris , s. l. , whitaker , r. j. , 2005 , mnras 358 , 71 west , m. 1991 , , 379 , 19 white , s.d.m . , navarro , j.f . , evrard , a.e . , & frenk , c.s . 1993 , nature , 366 , 429 xue , y.j . & wu , x.p . , 2000 , apj , 538 , 65 rcccccrrr 514 & 11 17 18.76 & @xmath007 42 42.3 & 20.49 & 1.98 & 183007 & 84 & 3.39 & + 2492 & 11 17 19.55 & @xmath007 46 01.0 & 22.63 & 1.28 & 143704 & 36 & 4.27 & + 1937 & 11 17 19.36 & @xmath007 45 07.7 & 19.41 & 0.94 & 109039 & 36 & & 8 + 202 & 11 17 19.35 & @xmath007 46 18.9 & 19.85 & 0.91 & 68022 & 10 & & 9 + 21 & 11 17 19.76 & @xmath007 47 03.2 & 19.11 & 1.46 & 107761 & 45 & 3.72 & + 1520 & 11 17 19.59 & @xmath007 44 30.5 & 19.12 & 1.58 & 99787 & 40 & 9.06 & + 2079 & 11 17 21.80 & @xmath007 45 18.4 & 22.56 & 0.98 & 144968 & 10 & & 8 + 2315 & 11 17 21.73 & @xmath007 45 41.8 & 22.42 & 0.71 & 70683 & 59 & & 10 + 676 & 11 17 22.40 & @xmath007 43 11.2 & 20.79 & 1.86 & 147672 & 61 & 4.60 & + 918 & 11 17 22.37 & @xmath007 43 22.4 & 20.35 & 1.90 & 144136 & 61 & 5.12 & + 2691 & 11 17 22.81 & @xmath007 46 23.4 & 21.25 & 1.46 & 170804 & 42 & 3.83 & + 1 & 11 17 22.69 & @xmath007 45 31.4 & 17.11 & 1.09 & 38300 & 34 & & 6 + 2572 & 11 17 23.04 & @xmath007 46 06.2 & 21.98 & 1.54 & 155596 & 35 & 4.81 & + 1570 & 11 17 23.02 & @xmath007 44 35.3 & 21.19 & 1.32 & 144939 & 48 & 2.57 & + 1576 & 11 17 23.00 & @xmath007 44 40.2 & 21.48 & 1.60 & 143740 & 46 & 3.70 & + 1568 & 11 17 23.29 & @xmath007 44 31.9 & 21.36 & 1.68 & 143192 & 45 & & 8 + 135 & 11 17 23.44 & @xmath007 46 36.5 & 20.73 & 1.75 & 202462 & 41 & 3.77 & + 2131 & 11 17 23.95 & @xmath007 45 23.3 & 21.57 & 1.42 & 141945 & 52 & 4.75 & + 1299 & 11 17 24.29 & @xmath007 44 10.6 & 19.65 & 1.72 & 147286 & 39 & 6.91 & + 190 & 11 17 24.14 & @xmath007 46 40.8 & 20.94 & 1.15 & 103336 & 23 & 8 & + 694 & 11 17 24.48 & @xmath007 42 58.2 & 21.53 & 1.94 & 144522 & 57 & 7.26 & + 3054 & 11 17 24.47 & @xmath007 42 13.7 & 22.16 & 1.82 & 148235 & 61 & 4.57 & + 339 & 11 17 24.80 & @xmath007 42 24.3 & 22.48 & 1.13 & 105922 & 13 & & 8 + 739 & 11 17 24.61 & @xmath007 43 04.5 & 21.83 & 0.86 & 93069 & 20 & & 13 + 908 & 11 17 25.05 & @xmath007 43 29.7 & 19.26 & 1.66 & 147157 & 48 & 4.23 & + 2973 & 11 17 25.22 & @xmath007 42 04.6 & 20.46 & 1.94 & 145243 & 42 & 7.25 & + 441 & 11 17 25.54 & @xmath007 42 34.0 & 22.37 & 0.91 & 146836 & 24 & & 8 + 910 & 11 17 25.92 & @xmath007 43 34.5 & 19.80 & 0.92 & 90311 & 12 & & 15 + 1085 & 11 17 25.86 & @xmath007 43 43.1 & 20.83 & 1.96 & 145337 & 56 & 5.35 & + 903 & 11 17 26.14 & @xmath007 43 53.3 & 20.64 & 1.89 & 146283 & 39 & 8.36 & + 2156 & 11 17 26.14 & @xmath007 45 29.7 & 20.59 & 1.88 & 145026 & 38 & 8.47 & + 1098 & 11 17 26.12 & @xmath007 43 41.0 & 20.29 & 1.84 & 144645 & 49 & 5.36 & + 947 & 11 17 26.09 & @xmath007 43 37.5 & 20.26 & 1.94 & 145882 & 47 & 6.11 & + 1205 & 11 17 26.48 & @xmath007 43 53.1 & 22.10 & 1.23 & 149339 & 30 & 5.16 & + 1269 & 11 17 26.45 & @xmath007 44 02.7 & 21.25 & 1.98 & 144528 & 24 & 7.79 & + 2346 & 11 17 26.28 & @xmath007 45 48.3 & 20.99 & 1.32 & 103397 & 72 & & 10 + 904 & 11 17 26.75 & @xmath007 43 24.0 & 18.90 & 0.94 & 48309 & 10 & & 10 + 1268 & 11 17 26.58 & @xmath007 44 07.6 & 22.21 & 1.14 & 103535 & 26 & & 6 + 798 & 11 17 26.52 & @xmath007 43 11.3 & 19.73 & 1.29 & 144724 & 44 & 7.50 & + 623 & 11 17 26.83 & @xmath007 42 51.6 & 20.77 & 1.96 & 147448 & 57 & 6.08 & + 1939 & 11 17 26.79 & @xmath007 45 08.1 & 20.88 & 1.77 & 170714 & 49 & 3.10 & + 1538 & 11 17 27.47 & @xmath007 44 28.5 & 21.48 & 1.81 & 143797 & 51 & 5.85 & + 1153 & 11 17 27.46 & @xmath007 43 50.1 & 21.18 & 1.86 & 183716 & 28 & 4.74 & + 10 & 11 17 27.45 & @xmath007 46 57.2 & 17.57 & 1.30 & 47253 & 45 & 9.84 & + 121 & 11 17 27.42 & @xmath007 46 47.7 & 19.15 & 1.27 & 47925 & 45 & 8.76 & + 1241 & 11 17 27.58 & @xmath007 44 00.0 & 21.54 & 1.45 & 145443 & 21 & 7.72 & + 545 & 11 17 27.76 & @xmath007 42 45.1 & 20.70 & 1.94 & 143211 & 50 & 9.12 & + 510 & 11 17 27.73 & @xmath007 42 38.4 & 22.57 & 1.16 & 148356 & 76 & 3.94 & + 3 & 11 17 28.36 & @xmath007 46 36.6 & 18.10 & 1.27 & 47375 & 29 & 11.94 & + 95 & 11 17 28.92 & @xmath007 46 55.2 & 20.58 & 0.59 & 65862 & 15 & & 12 + 1611 & 11 17 29.06 & @xmath007 44 37.3 & 21.46 & 1.12 & 149186 & 56 & & 8 + 1071 & 11 17 29.31 & @xmath007 43 37.6 & 20.68 & 1.90 & 144779 & 18 & 12.24 & + 2422 & 11 17 29.22 & @xmath007 45 52.1 & 21.09 & 1.89 & 149896 & 42 & 7.96 & + 2228 & 11 17 29.55 & @xmath007 45 33.9 & 21.62 & 1.14 & 143249 & 64 & 4.05 & + 7 & 11 17 29.68 & @xmath007 46 48.5 & 18.09 & 0.78 & 40722 & 10 & & 11 + 5 & 11 17 29.67 & @xmath007 46 23.7 & 16.65 & 1.26 & 47854 & 43 & 8.03 & + 6 & 11 17 30.08 & @xmath007 46 28.2 & 18.65 & 1.14 & 48516 & 26 & 8.75 & + 865 & 11 17 30.03 & @xmath007 43 16.2 & 22.12 & 0.82 & 65836 & 25 & & 12 + 24 & 11 17 30.01 & @xmath007 47 06.2 & 20.12 & 0.71 & 48649 & 18 & & 10 + 325 & 11 17 30.28 & @xmath007 42 13.1 & 22.43 & 0.99 & 146910 & 26 & 4.18 & + 2105 & 11 17 30.19 & @xmath007 45 23.3 & 20.27 & 1.28 & 149665 & 48 & 4.20 & + 1372 & 11 17 30.56 & @xmath007 44 13.4 & 22.12 & 1.18 & 157677 & 19 & & 9 + 864 & 11 17 30.50 & @xmath007 43 16.6 & 20.10 & 1.56 & 154346 & 40 & 6.53 & + 1099 & 11 17 30.42 & @xmath007 44 04.2 & 20.51 & 1.89 & 144382 & 37 & 7.69 & + 2571 & 11 17 31.07 & @xmath007 46 10.3 & 21.25 & 1.49 & 47531 & 56 & 3.29 & + 988 & 11 17 30.99 & @xmath007 43 29.6 & 19.70 & 1.04 & 107733 & 64 & & 8 + 1793 & 11 17 30.86 & @xmath007 44 52.4 & 21.89 & 0.56 & 300960 & 114 & & 7 + 2921 & 11 17 31.82 & @xmath007 41 57.9 & 21.21 & 2.01 & 153491 & 50 & 5.12 & + 1811 & 11 17 32.28 & @xmath007 44 55.3 & 21.14 & 1.80 & 157625 & 32 & 6.60 & + 1219 & 11 17 32.42 & @xmath007 43 58.0 & 21.66 & 1.94 & 145199 & 44 & 6.52 & + 663 & 11 17 33.43 & @xmath007 43 01.3 & 18.79 & 0.41 & 12677 & 16 & & 13 + 525 & 11 17 34.31 & @xmath007 42 40.0 & 22.30 & 1.92 & 184040 & 57 & 4.91 & + 359 & 11 17 35.53 & @xmath007 42 29.2 & 21.30 & 1.47 & 147726 & 60 & 3.62 & + 313 & 11 17 35.73 & @xmath007 42 52.6 & 19.99 & 1.98 & 143826 & 35 & 5.80 & + 1719 & 11 17 35.89 & @xmath007 44 59.7 & 19.33 & 0.68 & 33815 & 41 & & 12 + rccccccc s1 & 11 17 26.4 & @xmath007 44 01.3 & 23 & 0.48218@xmath880.00042 & 592@xmath8882 & 1.02@xmath170 & 1.05@xmath171 + s2 & 11 17 26.8 & @xmath007 42 52.4 & 9 & 0.49191@xmath880.00048 & 391@xmath8885 & & + lclc sis & @xmath2 & @xmath172 km s@xmath5 & @xmath173 + sis & @xmath3 & @xmath174 km s@xmath5 & @xmath175 + nfw(@xmath124 ) & @xmath2 & @xmath176 mpc & @xmath177 + nfw(@xmath124 ) & @xmath3 & @xmath178 mpc & @xmath179 +

we present a multiwavelength study of the poor cluster rx j1117.4@xmath00743 ( [ vmf 98 ] 097 ) at z@xmath10.485 , based on gmos / gemini south @xmath2 , @xmath3 photometry and spectroscopy , and xmm - newton observations . we examine its nature and surroundings by analyzing the projected galaxy distribution , the galaxy velocity distribution , the weak - lensing mass reconstruction , and the x - ray spectroscopy and imaging . the cluster shows a complex morphology . it is composed by at least two structures along the line - of - sight , with velocity dispersions of 592@xmath4 km s@xmath5 and 391@xmath6 km s@xmath5 respectively . other structures are also detected in x - ray , in the galaxy projected number density map , and by weak - lensing . one of these clumps , located east from the cluster center , could be gravitationally bound and associated to the main cluster . the derived temperature and bolometric x - ray luminosity reveal that [ vmf 98 ] 097 behave like a normal cluster , in agreement with @xmath7 correlation found for both local ( @xmath8 ) and moderate redshift ( @xmath9 ) clusters . we find that the mass determination inferred from weak - lensing is in average 3 to 4.8 times higher ( depending on the model assumed ) than the x - ray mass . we have two possible explanations for this discrepancy : _ i ) _ the cluster is in non - equilibrium , then the deviation of the x - ray estimated mass from the true value can be as high as a factor of two ; _ ii ) _ the intervening mass along the line - of - sight of the cluster is producing an over - estimation of the weak - lensing mass . based on the analysis presented , we conclude that [ vmf 98 ] 097 is a perturbed cluster with at least two substructures in velocity space and with other nearby structures at projected distances of about 1 h@xmath10mpc . this cluster is an example of a poor cluster caught in the process of accreting sub - structures to become a rich cluster .

unitary matrix models have many applications as a tool to study quantum systems , as well as an interesting mathematical structure @xcite . they make an appearance in the two - dimensional lattice yang - mills theory that was solved in the large n limit by gross and witten @xcite and they have a long history of applications to mean field models of lattice gauge theory @xcite and lattice models with induced qcd @xcite . they have interesting deformations which are related to integrable models @xcite and they have recently been of interest as effective field theories for the low energy degrees of freedom in yang - mills theory defined on certain compact spaces @xcite . in the latter context , with the appropriate effective action , the matrix integral that we shall be interested in , shown in equation ( [ loop ] ) below , computes the polyakov loop expectation value in an effective theory corresponding to finite temperature , large @xmath0 , four dimensional yang - mills theory where the three dimensional space is a sphere @xcite . in this paper , we wish to point out an interesting behavior of the expectation values of the character of the unitary matrix for representations with center charge of order n , in the large n limit . we focus on the completely symmetric representations , denoted @xmath6 , corresponding to a young tableaux with a single row of @xmath1 boxes . we will take @xmath1 large , @xmath7 , and typically in the region where @xmath1 is somewhat larger than @xmath0 . we strongly suspect that the behavior we find also occurs for other representations , but we shall not analyze them here . boxes on the length of a column in their young tableau , they do not display the behavior that we discuss here . ] the matrix integrals that we study have the generic form _ r u = e^-s[u ] _ r u e^-s[u ] , [ loop ] where @xmath8 is an @xmath9 unitary matrix , @xmath10 $ ] is the haar measure for unitary matrices , @xmath11 $ ] is a class function of @xmath8 ( in that it obeys @xmath11=s[vuv^\dagger]$ ] for any unitary matrix @xmath12 ) and @xmath13 is an irreducible representation of @xmath14 . @xmath11 $ ] is of order @xmath15 in the sense that @xmath16 \sim n^2 $ ] where @xmath17 is the unit matrix , when @xmath0 is large . for later reference , define @xmath18 by the relation _ s_k u = e^-n _ k . in the following , we shall consider a unitary matrix model with a generic action equipped with the properties that we have listed above , although in many instances , to be concrete , we will use the gross - witten model ( [ grosswitten ] ) since its eigenvalue density is known explicitly . in the gross - witten model @xcite , [ grosswitten ] s[u]=- . the matrix model in ( [ loop ] ) is what is referred to as an `` eigenvalue model '' since it is possible to use the symmetry of the action to write the integral as an integral over diagonal matrices . because it is a class function , @xmath11 $ ] is a function of the eigenvalues only , as is @xmath19 therefore , the above matrix integral can be written in the standard way as an integral over the @xmath0 eigenvalues of the matrix @xmath8 , @xmath20 , _ r u = _ i |(e^i_i)|^2 e^-s[_i ] _ r u _ i |(e^i_i)|^2 e^-s[_i ] , [ loop2 ] where @xmath21 is the vandermonde determinant . to evaluate this expression at large @xmath0 , several assumptions are usually made : 1 . * saddle point approximation * first there is the assumption that there is a good large n limit where the integral can be evaluated in a saddle point approximation . this includes the assumption that the effective potential @xmath22 - \ln |\delta(e^{i\phi_i})|^2 $ ] has isolated minima and the magnitude of the potential itself for generic values of @xmath23 is of order @xmath15 . then , for small enough representations @xmath13 , the expectation value in equation ( [ loop2 ] ) can be computed by the saddle point approximation . we will denote the location of the minimum of @xmath24 by @xmath25 and assume for simplicity that @xmath26 . with @xmath27 , we have simply [ saddle - point ] _ r u = _ r u + o(1/n^2 ) . * probe approximation * the saddle point approximation can still be applied to the integral at large @xmath0 where the representation @xmath13 is large , corresponding to a young tableau with @xmath1 boxes and @xmath1 is of order @xmath0 , that is , @xmath28 remains finite as @xmath29 @xcite . moreover , to compute the leading term , which is of order @xmath0 in @xmath30 , it is sufficient to assume that the position of the saddle point is not affected by the presence of an operator @xmath19 in the integrand . the upshot is that , for these large representations , ( [ saddle - point ] ) is replaced by [ probe0 ] _ r u = _ r u + o(1 ) . and has nontrivial content since @xmath31 is of order @xmath0 . * continuum approximation * an essential tool which is used to subsequently evaluate the matrix elements is the distribution function for the eigenvalues @xmath32 . for large @xmath0 , sums over eigenvalues are replaced by integrals over an eigenvalue density , [ eigenvaluedensity ] ( ) = _ i(-_i ) which is assumed to approach a piece - wise smooth function of @xmath33 in the limit where @xmath0 is large . it is then employed to compute traces , for example , u^k = _ j e^ik_j n d ( ) e^ik . in the following , we shall examine the reliability of the large @xmath0 expansion for computing expectation values of the forms @xmath34 and @xmath35 when @xmath7 . we shall see that , for good reasons , the saddle point computation of @xmath34 fails once @xmath36 is large enough . on the other hand , the expectation value of the character , @xmath35 , is much better behaved . for it , we shall find that points 1 and 2 , which are essentially the saddle point and probe approximations , hold up to the largest values of @xmath28 that we can study , and that @xmath35 has a well - defined large @xmath0 limit in the regime where @xmath36 is of order one . however , the assumption of a continuous eigenvalue distribution fails , leading to intriguing behaviour . to illustrate the latter point , let us assume that all aspects of the large @xmath0 expansion are valid and use them to compute @xmath37 in the large @xmath0 limit . in that limit , we assume that the eigenvalues are classical , given by those values of @xmath23 which minimize the effective potential , which we denote by @xmath38 , and we can simply evaluate the expectation value by substitution : @xmath39 where @xmath40 is the classical diagonal matrix . for concreteness , let us consider the strong coupling phase of the gross - witten model which has action @xmath11 $ ] given in ( [ grosswitten ] ) . it is known to be solved by the eigenvalue density ( ) = 12 ( 1 + 2p ( ) ) , where @xmath41 when @xmath42 . for the sake of this argument , we will assume that this eigenvalue density can be used to compute @xmath43 . using this density , traces are given by u^k = _ j=1^n e^ik_j \ { ll n & k=0 , + n p & |k| = 1 , + 0 & |k| > 1 . . [ grosswittentraces]the trace of a matrix in the irrep @xmath44 can be written in terms of the eigenvalues as _ s_k u = _ j_1 j_k ( _ a=1^k i_j_a ) , [ s ] which is equivalent , due to the frobenius formula , to [ frobenius ] _ s_k u = 1k!_s_k ( _ j=1^p u^l_j ) , where @xmath45 is the symmetric group , the sum is over all its elements @xmath46 , and where @xmath47 , @xmath48 , @xmath49 , @xmath50 are the lengths of the cycles in the decomposition into cycles of the element @xmath46 . we can now combine equations ( [ grosswittentraces ] ) and ( [ frobenius ] ) to obtain _ s_k ( np)^k ( ( n / k)pe)^kor _ s_k u e^- k . [ saddle - point - strong - coupling ] several things are interesting about this expression : as expected , since @xmath7 , @xmath51 is of order @xmath0 . for sufficiently large @xmath2 , @xmath52 , it is negative . this crossing from a scenario where @xmath53 is an effectively infinite exponential of a positive quantity of order @xmath0 to one which is exponentially small in @xmath0 was interpreted as a phase transition in references @xcite . to test the adequacy of this expression , we have performed several numerical computations . in one , we treat @xmath8 as a random variable , approximating the integrals in equation ( [ loop2 ] ) by gaussian integrals near the saddle point ( the precise methodology is described in section [ sec : gaussian ] ) . in another , we rely on both the saddle point approximation and the probe approximation ( but not the continuum approximation ) , treating @xmath8 as a strictly classical variable but maintaining discreteness of the eigenvalue distribution . the results , at @xmath54 , are given in figure [ fa ] . three things are evident in this figure : * there is a good match between the results of the saddle point computation plus continuum distribution ( [ saddle - point - strong - coupling ] ) and the numerics for smaller values of @xmath28 . this match ends abruptly at @xmath55 . * for larger values of @xmath28 , the numerical computations agree with each other rather well , but disagree with ( [ saddle - point - strong - coupling ] ) . agreement of the numerical computations suggests that the existence of a saddle point and use of the classical matrix @xmath40 are still valid , apparently for the whole range of @xmath28 that we consider . however , the continuum approximation which led to the result ( [ saddle - point - strong - coupling ] ) must fail for values at and above @xmath56 . * treating @xmath8 as a random variable and treating it as a classical variable with the operator @xmath57 in the probe approximation results in differences of sub - leading order , consistent with equation ( [ probe0 ] ) . is plotted on the vertical axis as a function of @xmath2 which is plotted on the horizontal axis , in the strongly coupled gross - witten model with @xmath54 and @xmath58 . crosses represent a computation where @xmath8 is treated as a random variable , open circles represent a computation assuming the saddle point approximation and the probe approximation , _ i.e. _ treating @xmath8 as a classical variable , but retaining discreteness of the eigenvalue distribution . the solid black line represents the result due to the continuum approximation , equation ( [ saddle - point - strong - coupling]).,title="fig : " ] is plotted on the vertical axis as a function of @xmath2 which is plotted on the horizontal axis , in the strongly coupled gross - witten model with @xmath54 and @xmath58 . crosses represent a computation where @xmath8 is treated as a random variable , open circles represent a computation assuming the saddle point approximation and the probe approximation , _ i.e. _ treating @xmath8 as a classical variable , but retaining discreteness of the eigenvalue distribution . the solid black line represents the result due to the continuum approximation , equation ( [ saddle - point - strong - coupling]).,title="fig : " ] thus the saddle point approximation and the probe approximation appear to be valid for the computation of @xmath59 , but the continuum approximation does not . nonetheless , validity of the saddle point and probe approximations allow us to write , using equation ( [ frobenius ] ) , _ s_k u = _ s_k ( _ j=1^p u^l_j ) 1k!_s_k ( _ j=1^p u^l_j ) . [ s - frobenius ] in the last equality , it is legitimate to substitute the classical discrete eigenvalues . we now draw the reader s attention to the strange feature of @xmath60 as a function of @xmath1 , namely the recurrence , or approximate periodicity , which is seen in figure [ fa ] . the remainder of this introduction will discuss the origin and interpretation of this feature , assuming the validity of the formula ( [ s - frobenius ] ) above . formula ( [ s - frobenius ] ) implies that _ for the purpose of computing _ @xmath60 , we can treat @xmath8 as a classical variable , frozen at the saddle point value @xmath40 . to study the implications of this formula , we need to examine the properties of @xmath61 for large @xmath1 and large @xmath0 , in the regime where @xmath62 . we illustrate these properties in figure [ f00 ] . for @xmath63 , figure [ f00](a ) , the results shown agree with the continuum approximation result ( [ grosswittentraces ] ) up to @xmath64 . above this point , @xmath65 is no longer zero , as was predicted by equation ( [ grosswittentraces ] ) , and is instead of order @xmath0 ( we should recall here that since @xmath40 is a unitary matrix , @xmath66 , so above @xmath64 , @xmath65 is quite large , reaching as much as half of its maximum value . ) . and ( a ) @xmath63 and ( b ) @xmath54 . @xmath65 is plotted on the vertical axis as a function of @xmath2 on the horizontal axis.,title="fig : " ] and ( a ) @xmath63 and ( b ) @xmath54 . @xmath65 is plotted on the vertical axis as a function of @xmath2 on the horizontal axis.,title="fig : " ] ( a ) and ( a ) @xmath63 and ( b ) @xmath54 . @xmath65 is plotted on the vertical axis as a function of @xmath2 on the horizontal axis.,title="fig : " ] and ( a ) @xmath63 and ( b ) @xmath54 . @xmath65 is plotted on the vertical axis as a function of @xmath2 on the horizontal axis.,title="fig : " ] ( b ) a reasoning for why this is so comes from the fact that , at any finite @xmath0 , trace relations allow us to compute the traces of a single classical matrix @xmath43 for @xmath67 from the first @xmath68 traces . it is therefore not possible for just the first trace to be arbitrary and non - zero , and the rest zero , as the continuum approximation would demand . similarly , for general ( but assumed smooth ) eigenvalue distributions with a infinite number of frequency components , @xmath43 would have to decay to zero with @xmath69 , which again does not appear compatible with trace relations . thus , we might expect @xmath43 to become noisy at some point in the vicinity of @xmath70 , and this is indeed what we see for the case of @xmath63 in figure [ f00](a ) . notice , in figure [ f00](b ) , that at @xmath54 which is close to the maximum value that @xmath71 can reach in the strong coupling phase of the gross - witten model ( before the phase transition to the weak coupling phase with a gapped distribution ) , the agreement with the continuum approximation is even weaker . more intuitively , consider the eigenvalues @xmath72 and the spacing between consecutive eigenvalues @xmath73 . at large @xmath0 , this spacing is approximately @xmath74 where @xmath75 is the eigenvalue density . when summing over the phases @xmath76 , we are probing the structure of the distribution of eigenvalues at wave - number @xmath1 , and we can not replace the discrete sum with an integral unless @xmath77spacing@xmath78 , or @xmath79 . the continuum approximation to the density will thus hold for a finite @xmath2 , as long as it is not too large , the cutoff being of order the inverse height of @xmath75 . the narrower the eigenvalue distribution , the larger the density @xmath75 and therefore the continuum approximation will hold for a larger range of @xmath2 . only for the delta - function eigenvalue distribution will the continuum approximation hold for any @xmath2 of order @xmath80 . returning to the strongly coupled gross - witten model , the continuum approximation result ( [ saddle - point - strong - coupling ] ) , which is based on equation ( [ grosswittentraces ] ) , implies that @xmath60 becomes very small once @xmath2 is large enough . this is a generic feature of @xmath60 which appears when @xmath43 goes to zero fast enough as @xmath81 . define @xmath82 by @xmath83 in such a scenario . we can think of @xmath82 as @xmath18 as obtained in the continuum approximation and pictured with a solid line in figure [ fa ] . as we have just discussed , at any finite @xmath0 , @xmath43 can not in fact go to zero for large @xmath1 , due to restrictions placed on it by the trace relations . we will examine the consequences of this fact by assuming that @xmath43 fails to be zero at precisely one large value of @xmath1 , which we will denote with @xmath84 , @xmath85 . we will assume that @xmath86 . this is motivated by figure [ f00 ] , where we see that @xmath43 takes exceptionally large values at isolated points , the smallest of which is @xmath87 ( for @xmath54 , figure [ f00](b ) ) . what we are doing is replacing the situation in figure [ f00 ] with a caricature shown in figure [ f000 ] . ( to be clear , @xmath82 could be computed by assuming that the only non - zero trace is @xmath88 whereas @xmath18 is computed assuming that the only nonzero traces are @xmath88 , @xmath89 , though the argument is somewhat more general than that , as it relies on the large @xmath1 behaviour of @xmath82 , and not the details of @xmath90 for small @xmath1 . ) .,title="fig : " ] .,title="fig : " ] equation ( [ frobenius ] ) now implies that , for @xmath91 of order @xmath80 and @xmath92 , e^-n_k+nm=_s_k+nm u = k ! ( k+nm ) ! . assuming that @xmath84 is large enough that @xmath93 , the @xmath94 term dominates the sum and we have _ s_k+nm u e^-n _ k = 1n ! ( s m / n)^n _ s_k u , [ periodicity ] or _ k+nm = _ k + o((n ) ) . [ periodic ] we discover that the order @xmath0 part of @xmath18 is periodic in @xmath1 , with the period given by @xmath84 . this simple calculation reproduces the behavior seen in figure [ fa ] quite well . in particular , we see that the period of the recurrence in figure [ fa ] is about @xmath95 which matches really well anomalously large value of @xmath96 seen at @xmath97 in figure [ f00](b ) . going back to figure [ fa ] we see that @xmath43 is also large at @xmath98 and @xmath99 ( and probably at higher multiples ) . how does that affect our computation ? let s say that @xmath100 . then , equation ( [ periodicity ] ) , for @xmath101 , becomes _ s_k+2 m u ( 12 ! ( s m / n)^2 + 11 ! ( s m / n)^1 ) _ s_k u , which gives @xmath102 as before . thus , the presence of further points where @xmath43 is nonzero does not affect our conclusions as long as these appear at @xmath1 equal to integer multiples of @xmath84 . this turns out to be a generic feature of simple eigenvalue distributions , and we will discuss it further in section [ sec : gw ] . we will also examine a more complicated example with multiple periods in section [ sec : discussion ] . before moving on , let us briefly discuss the origin of the values of @xmath1 for which @xmath43 is anomalously large . they can occur when there is a region in the interval @xmath103 $ ] where a large number of the classical eigenvalues are almost equally spaced . this would damp the destructive interference between the individual terms in the sum over @xmath104 of the phases @xmath76 . it would occur for values of the wave - vector @xmath1 which are @xmath105(spacing ) . eigenvalues are almost equally spaced when the derivative of the eigenvalue distribution is zero , that is , at an extremum of the distribution . we will return to this issue in the following section and see a high degree of correlation between the period of recurrences and the maxima of the eigenvalue distribution . this would also explain why the anomalously large @xmath96 seen in figure [ f00 ] apparently occur for values of @xmath1 which are integer multiples of a basic number . so far we have focused on the properties of @xmath43 . what about @xmath106 ? figure [ f0 ] addresses this question . it shows @xmath106 in the gross - witten model with @xmath63 computed by treating @xmath8 as a random variable , correcting the classical limit by integrating over quadratic fluctuations around the saddle point ( filled circles ) , as well the classical limit itself , @xmath107 ( from figure [ f00](a ) ) for comparison . and @xmath58 . @xmath106 is plotted on the vertical axis as a function of @xmath2 on the horizontal axis . open circles are obtained by finding @xmath38 that minimize the effective action and evaluating @xmath107 using those eigenvalues . this is identical to figure [ f00](a ) . filled circles include the leading correction to the large @xmath0 result , arising from integrating the fluctuations about @xmath38 . the precise methodology is given in section [ sec : gaussian ] . , title="fig : " ] and @xmath58 . @xmath106 is plotted on the vertical axis as a function of @xmath2 on the horizontal axis . open circles are obtained by finding @xmath38 that minimize the effective action and evaluating @xmath107 using those eigenvalues . this is identical to figure [ f00](a ) . filled circles include the leading correction to the large @xmath0 result , arising from integrating the fluctuations about @xmath38 . the precise methodology is given in section [ sec : gaussian ] . , title="fig : " ] curiously , the filled circles do agree with the large @xmath0 result ( [ grosswittentraces ] ) over the entire range . we note that this integration is done for @xmath14 , rather than @xmath108 by imposing the constraint that the average value of the eigenvalues must be zero . if we use the unconstrained integral over @xmath108 , there is a soft mode which changes the result considerably . we take the upshot of this discussion as evidence that the saddle point approximation can not be trusted for the computation of @xmath109 when @xmath1 is bigger than some fraction of @xmath0 . the fluctuations are apparently at least as important as the classical eigenvalues in that regime . this is consistent with equations ( [ saddle - point ] ) and ( [ probe0 ] ) , since @xmath109 is bounded by @xmath0 and therefore its logarithm can not be large . the failure of the saddle point approximation arises from the variance in the eigenvalues @xmath23 being of the same order as the spacing between @xmath32s in the classical limit . thus integrating over the eigenvalues ` washes ' out the effect of a discrete spectrum and seems to restore the validity of the continuum approximation for @xmath109 . another way to view this is that re - introducing some randomness to the matrix so that it is no longer strictly classical relaxes the trace relations . in fact , we see that this works rather well in this case , in that fluctuations allow the traces to agree with what one would obtain from the eigenvalue density in the large @xmath0 limit over the entire range of @xmath28 that we explore . the curious ( and central to our argument ) fact is that this is _ not _ the case with @xmath60 . if we were to use @xmath60 instead of @xmath107 in equation ( [ s - frobenius ] ) , we would not obtain the correct answer . this can potentially be explained by a failure of factorisability ( it is unlikely that @xmath110 if @xmath109 can not be accurately computed in the saddle point approximation ) , but we do not explore this any further here . in contrast to @xmath109 , @xmath60 seems to be computable in the saddle point approximation ( as long as we do not use the continuum approximation as well ) and to have a well defined large @xmath0 limit , which we will explore further in section [ sec : gw ] . the periodic behavior of equation ( [ periodic ] ) has an interesting implication for some physical applications of this matrix model . for example , when the character is the expectation value of the polyakov loop in yang - mills theory on the sphere , @xmath18 is interpreted as the free energy of a heavy quark with center charge @xmath1 and in the totally symmetric representation . if one thinks of this heavy quark as being composed of @xmath1 partons , it appears that @xmath84 of these partons can combine together to form a ( ` bound ' ) state of low free energy . thus , if @xmath111 ( @xmath112 ) , the free energy of @xmath1 partons receives contributions mainly from the @xmath113 partons that are ` free ' , while the contribution of the @xmath91 ` bound ' states of @xmath84 partons is negligible . we will argue that this periodicity always exists , since it is related to maxima of the eigenvalue distribution . the rest of the paper is organized as follows . in section [ sec : gw ] , we study further the character and its large @xmath0 limit in the gross - witten model , both in strong and in weak coupling . we find that the character is indeed quasi - periodic and link the period to the maximum value taken by the eigenvalue density . in section [ sec : gaussian ] , we numerically compute the integrals in equation ( [ loop2 ] ) to test approximations 1 and 2 , and to see whether our results from section [ sec : gw ] are robust . finally , in section [ sec : discussion ] , we compute @xmath18 for more complicated eigenvalue densities and discuss generic behaviour . on the vertical axis is plotted as a function of @xmath2 on the horizontal axis for p=0.25 , 0.40 , 0.75 , 0.90 ( from the shortest recurrence period to the longest ) . red dots represent numerical results at n=200 and the solid black line is the result in the continuum approximation.,title="fig : " ] on the vertical axis is plotted as a function of @xmath2 on the horizontal axis for p=0.25 , 0.40 , 0.75 , 0.90 ( from the shortest recurrence period to the longest ) . red dots represent numerical results at n=200 and the solid black line is the result in the continuum approximation.,title="fig : " ] in this section , we shall examine the difference between using a discrete and continuous eigenvalue distribution in the limit where @xmath0 is large . we will assume that the large @xmath0 limit localizes the matrix integral onto eigenvalues @xmath38 and that to a first approximation , these eigenvalues are not disturbed by the presence of the character in the integral . to aid numerical computation at large @xmath0 , will use an approximation to the eigenvalues @xmath38 which we obtain be re - discretizing the continuum distribution . we will show that this is in fact a good approximation in the next section . here , it will prove sufficient to demonstrate our point about the consequences of the failure of the continuum approximation . to be concrete , we will study the expectation value of a character in the gross - witten model @xcite , both below and above the third - order phase transition . at finite @xmath0 , we approximate the actual positions of the eigenvalues , @xmath32 , by approximate positions @xmath114 computed from the large-@xmath0 density , via the formula which ( at infinite @xmath0 ) defines the density , [ eigenvalues ] _ -^_k ( ) = k-12n k=1 if @xmath115 , the @xmath116 simply leads to a symmetric distribution of eigenvalues . @xmath114s differ from @xmath32s by corrections which are sub - leading in the large-@xmath0 limit . we will demonstrate that they are a good enough approximation to @xmath38 in the next section . we will find it convenient to use the following generating function for the character : [ genfun ] _ k=0^t^k _ s_k u = _ j=1^n 11-t e^i _ j = ( - ( 1-tu ) ) . then , @xmath117 defined by [ sigma ] _ k=0^t^k e^n_k = ( -n _ j=1^n ( 1-t e^-_j ) ) is approximately equal to @xmath118 . we will see in the next section that the difference between @xmath117 and @xmath118 is of order @xmath119 . in the ungapped phase of the gross - witten model , the eigenvalue density is ( ) = 12 ( 1 + 2p ) for @xmath120 , while in the gapped phase , the eigenvalue density is ( ) = ( 2 - 2p ) for @xmath121 . here , in both cases , we have parameterized the distributions by the expectation value @xmath122 . with these explicit eigenvalue densities , at finite fixed @xmath0 , we solve equation ( [ eigenvalues ] ) ( numerically ) for @xmath114 at different values of @xmath71 . this allows us to write down the generating function ( [ sigma ] ) as a function of @xmath123 , which we then expand ( using maple ) in a taylor series for small @xmath123 to obtain @xmath117 as a function of @xmath1 . the results , at n=200 , are displayed in figure [ f1 ] . since both axes of the figure are scaled so that the plots are independent of @xmath0 ( to leading order ) , finite @xmath0 results should approximate infinite @xmath0 results . for comparison , solid lines show the answer at infinite @xmath0 in the continuum approximation , as computed in the introduction and in @xcite . the approximate periodicity of @xmath117 as a function of @xmath2 , predicted in the introduction , is clearly visible in the figure . one might worry that the recurrences shown in figure [ f1 ] are simply a consequence of working at finite @xmath0 and do not accurately represent the large @xmath0 limit of the theory . to show that this is not the case , figure [ f2 ] contains the results of a computation at @xmath124 , @xmath125 and @xmath126 . finite @xmath0 results seem to converge to a well - defined answer with a stable recurrence period . . @xmath117 on the vertical axis is plotted as a function of @xmath2 on the horizontal axis for n=100 ( green ) , n=200 ( red ) and n=400 ( blue ) . the solid black line is the result in the continuum approximation.,title="fig : " ] . @xmath117 on the vertical axis is plotted as a function of @xmath2 on the horizontal axis for n=100 ( green ) , n=200 ( red ) and n=400 ( blue ) . the solid black line is the result in the continuum approximation.,title="fig : " ] we can understand the period of the recurrences as follows . in the introduction , we argued that the recurrences are due to a single large trace , @xmath127 at some @xmath84 . since @xmath128 , @xmath127 is large if the spacing between consecutive @xmath129 is an integer multiple of @xmath130 . of course , the spacing is not constant , so what we want is that a large fraction of the eigenvalues be spaced at approximately @xmath130 . the eigenvalue spacing , equal to @xmath131 varies slowest where the derivative of @xmath75 is zero . since there are more eigenvalues near the point where @xmath75 attains its maximum than near the point where it attains its minimum , we conjecture that @xmath84 is given by @xmath132 , where @xmath133 is given by _ max = \ { ll 1 + 2p 2 & 0<p<1/2 , + 2(1-p ) & 1/2 < p<1 . . the recurrence period for @xmath117 as a function of @xmath2 should then be simply @xmath134 . this conjecture is clearly supported by figure [ f3 ] . we will see in section [ sec : discussion ] that for more complicated eigenvalue densities with multiple maxima , there will be several competing recurrence periods . nonetheless , the periods will be related to the local maxima of the eigenvalue density in the way described above . . the crosses are data based on our numerical results and the solid line is our prediction that recurrence period = @xmath135.,title="fig : " ] . the crosses are data based on our numerical results and the solid line is our prediction that recurrence period = @xmath135.,title="fig : " ] of course , the toy picture painted in the introduction , where there @xmath136 is large for only one isolated value of @xmath84 is not accurate even in the case of the relatively simple eigenvalue distributions discussed in this section . this was already illustrated in figure [ f00 ] . here we complement that figure with a computation in the weak coupling regime . figure [ f4 ] shows @xmath137 for @xmath138 and @xmath139 . several things are apparent in this plot : @xmath140 is large , while @xmath137 is small at first , but grows with @xmath1 . it also appears nearly random with a distribution whose mean is zero . since @xmath137 oscillates , the effect described in the introduction from each individual @xmath137 roughly cancels between different values of @xmath1 . however , at @xmath141 something interesting happens : @xmath137 is of order @xmath0 and negative for several values of @xmath1 in a row . thus , the effect for several values of @xmath1 in a row can reinforce , which should lead to a recurrence . we see in figure [ f2 ] that indeed , at @xmath138 , the recurrence period is about 2 . not surprisingly , a similar reinforcement will occur for @xmath142 . in the gross - witten model for @xmath138 and @xmath139.,title="fig : " ] in the gross - witten model for @xmath138 and @xmath139.,title="fig : " ] in this section we provide the methodology behind figures [ fa ] and [ f0 ] in the introduction and further support the validity of our approximation in section [ sec : gw ] . while we have shown already that the recurrence pattern is stable in the large @xmath0 limit , one could wonder whether it is also stable under small ( of order @xmath119 ) corrections to the eigenvalue density . we have checked the impact of such order @xmath119 changes as removing a single eigenvalue and adding a small high frequency component to the eigenvalue density , and these had no impact on the overall pattern beyond @xmath119 corrections . another possibility is that the since the eigenvalues are random variables with a variance of order @xmath119 , the discreteness of the eigenvalue distribution might be washed out when the integrals in equation ( [ loop2 ] ) are evaluated . to test for this possibility , in this section we evaluate these integrals by approximating the potential near the saddle point ( to the leading , quadratic order ) . while this computation is not exact , it should provide us with the next - to - leading order corrections to @xmath117 and allow us to see whether the discreteness of the eigenvalue density still matters when the eigenvalues are random variables . the expectation value we are interested in can be written as e^- ( 1-tu ) = _ j=1^n dx_j e^-v_(x_j ) ( -_j ( 1-te^ix_j ) ) _ j=1^n dx_j e^-v_(x_j ) , [ veff ] where the effective potential for the eigenvalues is v _ = -_i < j - 2 p n _ i ( x_i ) . the first term in the potential , which comes from the vandermonde determinant , causes eigenvalues to repel , while the second term attracts them towards @xmath143 . the second term comes from the gross - witten action ( [ grosswitten ] ) where @xmath144 for @xmath145 and @xmath146 for @xmath147 . this action is known@xcite to produce the eigenvalue distributions discussed in the section [ sec : gw ] . we will denote by @xmath148 the positions of the eigenvalues for which @xmath149 attains its minimum . @xmath148 are , up to corrections of order @xmath119 , the same as @xmath114 in equation ( [ eigenvalues ] ) . the minimum is unique up to a permutation of the eigenvalues @xmath150 , and satisfies @xmath151 . given @xmath148 , we can expand @xmath149 around its minimum v_(x_1 , , x_n ) & & v_(x_1 , , x_n ) + & + & _ a , b=1^n ^2 x_a x_b v _ ( x_1 , , x_n ) |_x_k = x_k ( x_a - x_a ) ( x_b - x_b ) . we also expand the inserted operator @xmath152 around the point @xmath153 , o ( x_1 , , x_n ) & & o ( x_1 , , x_n ) + _ a=1^n x_a o ( x_1 , , x_n ) |_x_k = x_k ( x_a - x_a ) + & + & _ a , b=1^n ^2 x_a x_b o ( x_1 , , x_n ) |_x_k = x_k ( x_a - x_a ) ( x_b - x_b ) . the second term on the first line does not contribute to the integrals , by symmetry . evaluating the appropriate gaussian integrals we obtain e^- ( 1-tu ) = o ( x_1 , , x_n ) + , [ gaussian - result ] where we have arranged the second - order derivatives into @xmath154 matrices in the natural way . @xmath148 , and the matrices of second - order derivatives can be obtained in maple . the resulting expectation value is a function of @xmath123 , and can be expanded at small @xmath123 as before . the first term in the above equation is simply the saddle point result we have obtained before ( but with @xmath155 instead of @xmath156 ) , while the second term represents @xmath119 corrections due to variance in the eigenvalues . as can be seen in figure [ f5 ] ( and in figure [ fa ] ) , the general pattern we observed in the previous section persists . the difference between the saddle point results at @xmath58 ( open circles ) and at @xmath139 ( points ) is of order @xmath119 , as is the difference between the result of a gaussian integral ( crosses ) and just the saddle point ( open circles ) . this again justifies approximations 1 and 2 described in the introduction , demonstrates that our methodology in section [ sec : gw ] is sufficient and shows that the recurrences we have seen appear in the exact computation as well . in the gross - witten model as a function of @xmath2 for @xmath138 . open circles represent the saddle point result ( the first term in equation ( [ gaussian - result ] ) ) , while crosses include @xmath119 corrections ( both terms in equation ( [ gaussian - result ] ) ) ; n=45 . the black points show the result of an approximate saddle point computation with n=200 as discussed in section [ sec : gw ] , for comparison . the three plots differ by @xmath119 corrections . , title="fig : " ] in the gross - witten model as a function of @xmath2 for @xmath138 . open circles represent the saddle point result ( the first term in equation ( [ gaussian - result ] ) ) , while crosses include @xmath119 corrections ( both terms in equation ( [ gaussian - result ] ) ) ; n=45 . the black points show the result of an approximate saddle point computation with n=200 as discussed in section [ sec : gw ] , for comparison . the three plots differ by @xmath119 corrections . , title="fig : " ] to compute the results shown in figure [ f0 ] , we proceed in a similar fashion , except that we do not expand the operator , @xmath157 around @xmath158 , as it is not necessary to do that to obtain the integrals . also , we impose a constraint @xmath159 in our integrals , corresponding to the @xmath14 model , as the @xmath108 model has a soft mode which leads to @xmath160 even at small @xmath1 ( including @xmath161 ) . the results presented in section [ sec : gw ] are quite generic if one focuses on eigenvalue distributions with a single maximum ( either ungapped or one - cut gapped distributions ) . when multiple maxima are present , however , the behaviour can be much richer , as multiple maxima can lead to multiple recurrence periods . as an example , consider the following two - cut distribution , consisting of two rescaled semicircle pieces , one centered around @xmath162 and one around @xmath163 ( ) = \ { ll h_1 & || < w_1 , + h_2 & |-| < w_2 , + . [ two - semi ] where @xmath164 , @xmath165 , @xmath166 and @xmath167 . the eigenvalue distribution together with the corresponding @xmath117 as a function of @xmath2 are shown in figure [ f6 ] . two different recurrence periods are clearly visible , each related to one of the two maxima by the formula ( recurrence period ) = @xmath134 . the eigenvalue distribution was carefully chosen to make the amplitude of the two series of recurrences approximately equal so that they would both be visible . generically , even if the distribution has multiple maxima , one of the recurrence sequences dominates the others . given in equation ( [ two - semi ] ) and the corresponding @xmath117 as a function of @xmath2 , for @xmath139.,title="fig : " ] given in equation ( [ two - semi ] ) and the corresponding @xmath117 as a function of @xmath2 , for @xmath139.,title="fig : " ] given in equation ( [ two - semi ] ) and the corresponding @xmath117 as a function of @xmath2 , for @xmath139.,title="fig : " ] given in equation ( [ two - semi ] ) and the corresponding @xmath117 as a function of @xmath2 , for @xmath139.,title="fig : " ] other phenomena are possible as well . for example , in figure [ f7 ] , with an ungapped distribution given by ( ) = 12 ( -0.3 ( ) + 0.15(2 ) + 0.25(3 ) ) , [ two - peak ] we see that the peaks can broaden and eventually merge as they reoccur . given in equation ( [ two - peak ] ) and the corresponding @xmath117 as a function of @xmath2 , for @xmath139.,title="fig : " ] given in equation ( [ two - peak ] ) and the corresponding @xmath117 as a function of @xmath2 , for @xmath139.,title="fig : " ] given in equation ( [ two - peak ] ) and the corresponding @xmath117 as a function of @xmath2 , for @xmath139.,title="fig : " ] given in equation ( [ two - peak ] ) and the corresponding @xmath117 as a function of @xmath2 , for @xmath139.,title="fig : " ] the recurrences as a function of @xmath2 seem to be a fairly generic feature of @xmath18 at large @xmath0 . this is interesting partly because for many theories of interest , such as @xmath168 sym , the eigenvalue density is not known . therefore , while we can not compute @xmath18 in @xmath168 sym , we can make a conjecture that it will exhibit some kind of recurrences . notice further that @xmath18 seems also to generically experience intriguing first - order phase transitions ( kinks ) as a function of @xmath2 . in the introduction , we have already noted the qualitative difference between the behaviour of @xmath60 and @xmath169 . not only is the former exponentially larger than the latter , but it exhibits a very clean large @xmath0 limit which can be obtained from the saddle point approximation . in contrast , @xmath169 can not be obtain this way . this suggests that if one is interested in studying the moments of the eigenvalue distribution , @xmath60 is a better object to compute than @xmath169 . these two objects contain in principle the same information , but organized in a different way . in particular , through exponentiation , @xmath60 magnifies certain features of the eigenvalue distribution @xmath170 ( such as the value of @xmath133 ) making them easier to extract . since @xmath60 has a good large - n limit , it is also the natural candidate for computations using the ads - cft duality ( the same is true of the expectation value of a character in the rank @xmath1 totally anti - symmetric tensor representation , as was shown by similar methods in @xcite ) . it would be most interesting to find a stringy counterpart of the recurrences conjectured in this paper . finally , the potentially complex recurrence pattern means that computing @xmath60 in finite temperature @xmath168 sym on a sphere ( for example , through the ads - cft duality ) might teach us about more than just the eigenvalue distribution . if the picture we presented in the introduction , in which the recurrences are due to presence of bound states of partons , is correct , the recurrence pattern directly carries information about the dynamics of quarks in sym . this work is supported by nserc of canada . gws acknowledges the aspen center for physics , kitp santa barbara , galileo galilei institute and nordita , where parts of this work were completed . work done at kitp is supported in part by the national science foundation under grant no . nsfphy05 - 51164 and in part by darpa under grant no . hr0011 - 09 - 1 - 0015 and by the national science foundation under grant no . phy05 - 51164 . a. y. a. morozov , `` unitary integrals and related matrix models , '' theor . math . phys . * 162 * , 1 ( 2010 ) [ teor . mat . fiz . * 161 * , 3 ( 2010 ) ] [ arxiv:0906.3518 [ hep - th ] ] . d. j. gross , e. witten , `` possible third order phase transition in the large n lattice gauge theory , '' phys . * d21 * , 446 - 453 ( 1980 ) . m. makeenko and s. b. khokhlachev , `` symmetry of gauge group center and the problem of quark confinement in quantum chromodynamics . ( in russian ) , '' sov . jetp * 53 * , 228 ( 1981 ) [ zh . fiz . * 80 * , 448 ( 1981 ) ] . v. a. kazakov and a. a. migdal , `` induced qcd at large n , '' nucl . phys . b * 397 * ( 1993 ) 214 [ arxiv : hep - th/9206015 ] . i. i. kogan , g. w. semenoff and n. weiss , `` induced qcd and hidden local z(n ) symmetry , '' phys . lett . * 69 * , 3435 ( 1992 ) [ arxiv : hep - th/9206095 ] . i. i. kogan , a. morozov , g. w. semenoff and n. weiss , `` area law and continuum limit in induced qcd , '' nucl . b * 395 * , 547 ( 1993 ) [ arxiv : hep - th/9208012 ] . i. i. kogan , a. morozov , g. w. semenoff and n. weiss , continuum limits of induced qcd : lessons of the gaussian model at d = 1 int . j. mod . phys . a * 8 * , 1411 ( 1993 ) [ arxiv : hep - th/9208054 ] . m. bergere , b. eynard , `` some properties of angular integrals , '' j. phys . a * a42 * , 265201 ( 2009 ) . [ arxiv:0805.4482 [ math - ph ] ] . m. i. dobroliubov , a. morozov , g. w. semenoff and n. weiss , evaluation of observables in the gaussian n = infinity kazakov - migdal int . j. mod . phys . a * 9 * , 5033 ( 1994 ) [ arxiv : hep - th/9312145 ] . r. brower , p. rossi and c. i. tan , `` the external field problem for qcd , '' nucl . b * 190 * , 699 ( 1981 ) . k. n. anagnostopoulos , m. j. bowick , a. s. schwarz , `` the solution space of the unitary matrix model string equation and the sato grassmannian , '' commun . phys . * 148 * , 469 - 486 ( 1992 ) . [ hep - th/9112066 ] . a. mironov , a. morozov , g. w. semenoff , `` unitary matrix integrals in the framework of generalized kontsevich model . 1 . brezin - gross - witten model , '' int . j. mod . phys . * a11 * , 5031 - 5080 ( 1996 ) . [ hep - th/9404005 ] . a. morozov , in _ particles and fields _ , proceedings , cap - crm summer school , banff , canada , august 16 - 24 , 1994 . semenoff , l. vinet eds . , new york , usa : springer ( 1999 ) ; `` matrix models as integrable systems , '' arxiv : hep - th/9502091 . o. aharony , j. marsano , s. minwalla , k. papadodimas and m. van raamsdonk , the hagedorn - deconfinement phase transition in weakly coupled large n adv . theor . math . * 8 * , 603 ( 2004 ) [ arxiv : hep - th/0310285 ] . o. aharony , j. marsano , s. minwalla , k. papadodimas and m. van raamsdonk , a first order deconfinement transition in large n yang - mills theory on a phys . rev . d * 71 * , 125018 ( 2005 ) [ arxiv : hep - th/0502149 ] . g. grignani , j. l. karczmarek and g. w. semenoff , `` hot giant loop holography , '' phys . d * 82 * , 027901 ( 2010 ) [ arxiv:0904.3750 [ hep - th ] ] . g. w. semenoff , g. grignani and j. karczmarek , `` large representation polyakov loop in hot yang - mills theory , '' pos * qcd - tnt09 * , 041 ( 2009 ) . j. l. karczmarek , g. w. semenoff and s. yang , `` comments on k - strings at large n , '' jhep * 1103 * , 075 ( 2011 ) [ arxiv:1012.5875 [ hep - lat ] ] .

in a random unitary matrix model at large @xmath0 , we study the properties of the expectation value of the character of the unitary matrix in the rank @xmath1 symmetric tensor representation . we address the problem of whether the standard semiclassical technique for solving the model in the large @xmath0 limit can be applied when the representation is very large , with @xmath1 of order @xmath0 . we find that the eigenvalues do indeed localize on an extremum of the effective potential ; however , for finite but sufficiently large @xmath2 , it is not possible to replace the discrete eigenvalue density with a continuous one . nonetheless , the expectation value of the character has a well - defined large @xmath0 limit , and when the discreteness of the eigenvalues is properly accounted for , it shows an intriguing approximate periodicity as a function of @xmath2 . nsf - kitp-11 - 182 0.5 in joanna l. karczmarek@xmath3 and gordon w. semenoff@xmath4 0.3 in @xmath3 _ department of physics and astronomy + university of british columbia vancouver , canada v6 t 1z1_0.3 in @xmath5 _ kavli institute for theoretical physics , university of california , + santa barbara , california 93106 - 4030 _ 0.5 in

low mass x ray binaries are binary systems in which a @xmath7 star transfers matter to a neutron star or a black hole . a large fraction of the low mass x ray binaries are transients ; these are called soft x ray transients ( sxts ) . although several of the neutron star systems were already detected in quiescence with einstein , exosat , asca , and rosat ( e.g. @xcite ; ; ; @xcite ; @xcite ) , detailed studies of these faint quiescent counterparts to neutron star transients have only become possible with _ chandra _ and xmm / newton ( e.g. @xcite , @xcite ) . several mechanisms have been proposed to explain the observed x ray luminosity and spectra of quiescent neutron star sxts . accretion may be ongoing at a low level producing a soft spectrum ( @xcite ) . several authors have pointed out that the presence of a @xmath8 gauss magnetic field would have a large influence on the accretion flow . the onset of a propeller or pulsar wind mechanism (; @xcite ; @xcite ) has been proposed as an explanation of the hard power law spectral component . although detailed theoretical model calculations predicting the spectral shape are absent so far , @xcite argue that in the propeller phase the spectrum will be hard . in addition to these two apparently mutually exclusive models , it is thought that a soft , thermal spectral component with a luminosity of typically @xmath9 will be generated due to the fact that the neutron star crust and core are heated via pycnonuclear reactions in the crust during the accretion phase . the crust will thermally radiate in ( soft ) x rays , cooling the neutron star ( e.g. @xcite ; @xcite ; @xcite ) . in this model the observed quiescent luminosity can differ from outburst to outburst by a factor of 23 since the fraction of hydrogen and helium left in the atmosphere after an outburst will vary from outburst to outburst . this fraction influences the heat flux that flows from the core to the surface ( @xcite ) . the quiescent luminosity is also likely to vary somewhat from source to source since the neutron star core and crust temperature depend on the mass accretion history of the source and the cooling rate may depend among other things on the neutron star mass . using hydrogen neutron star atmosphere models to fit the soft part of the quiescent spectrum , neutron star radii and temperatures can be determined . the observed values are in the range expected from neutron star theory . if it can be established that the quiescent emission is indeed due to the hot neutron star surface or core , these systems could provide a way to determine neutron star radii . together with information about the neutron star spin ( e.g. obtained through burst oscillations [ e.g. @xcite ] and/or pulsations observed during outburst [ e.g. @xcite ] ) and mass ( see @xcite ) this provides important information about the behaviour of matter under physical conditions that are unattainable on earth . recently , using the _ chandra _ satellite we followed the neutron star sxt rx j170930.2263927 , also called xte j1709267 , towards quiescence ( @xcite ) . xte j1709267 was detected for the first time using rosat all sky survey observations performed in 1990 ( ) . the source was also detected by rosat in 1992 ( see ; @xcite ) . since then , the source has been detected with rxte three times ; in 1997 ( @xcite ) , in 2002 ( @xcite ) , and in 2004 ( @xcite ) . during the 1997 outburst @xcite found type i bursts using the wide field cameras onboard the _ bepposax _ satellite . since the source is located only 910 arcminutes away from the core of the globular cluster ngc 6293 it has been speculated that xte j1709267 is associated with ngc 6293 (; @xcite ) . in this paper we present x ray and optical observations of xte j1709267 in quiescence and outburst . a preliminary announcement of the optical observations was already made in @xcite . we have observed the neutron star sxt xte j1709267 during quiescence using the acis i ccds operated in the very faint mode on board the _ chandra _ satellite ( @xcite ) for @xmath225 ksec on may 12 , 2003 ( observation i d 3507 ) . due to the short deadtime introduced by reading out the ccds the effective on source time was 22.7 ksec . the x ray data were processed by the _ chandra _ x ray centre but we reprocessed the data starting with the level 1 products in order to take full advantage of the newest available calibrations . we used the _ ciao _ software to reduce the data ( version 3.0.2 and caldb version 2.26 ) . events with asca grades of 1 , 5 , 7 , cosmic rays , hot pixels , and events close to ccd node boundaries were rejected . we searched the data for periods of enhanced background radiation but none was present . hence , all the data were used in our analysis . we offset pointed the satellite with respect to the known accurate coordinates of xte j1709267 ( see @xcite ) in order to put some of the globular cluster ngc 6293 in the field of view . we detected @xmath1015 sources , but the coordinates of only one were consistent with those of xte j1709267 ( the analysis of the other sources will be presented elsewhere ) . we detected 166 source counts in 22.7 ksec . the spectrum of xte j1709267 was extracted from a circular region with a 5 arc second radius centred on the source whereas the background spectrum was extracted from a circular region with a radius of 5 arc seconds located 50 arc seconds east of the source ( 3 background counts were detected in this region ) . we rebinned the spectrum such that each bin contained at least 5 counts per bin . because of this low number of counts we used the cash statistic method in our spectral fitting to estimate the errors on the fitting parameters ( @xcite ) . we only include photons with energies above 0.3 and below 10 kev in our spectral analysis since the acis timed exposure mode spectral response is not well calibrated outside that range . in order to validate the cash statistics we did not subtract the background photons . these background photons all have energies above 3 kev . we fit the spectra using the xspec package ( version 11.3.0 ; @xcite ) . we fit the spectrum with an absorbed black body model and with an absorbed neutron star atmosphere model ( nsa ; @xcite ; ) . we kept the absorption fixed at the value we found during outburst ( @xmath11 ; @xcite ) . the galactic absorption , the nsa normalisation , the mass , and the radius of the neutron star , were held fixed during the fit at @xmath12 @xmath4 , @xmath13 pc@xmath14 ( for the distance , d in pc , we took the value of the globular cluster 8.8@xmath15 pc , see section 2.2 ) , and 1.4 m@xmath16 , 10 km , respectively . the best fitting parameters are presented in table [ fitpars ] . the reddening to the globular cluster ngc 6293 of @xmath17 implies a column density of @xmath18 assuming @xmath19 and using the relation between @xmath20 and @xmath21 of . we also fitted a nsa and a black body to the data with the @xmath21 fixed at @xmath22 ( see table [ fitpars ] ) . the best fitting parameters are consistent within the 90 per cent confidence contours with those obtained using @xmath23 . the flux in the last two bins is underestimated for both the black body ( see figure [ bbfit ] ) and the nsa model fits . this can ( partially ) be explained by the fact that we did not subtract the background . had we subtracted the background as defined above it would have reduced the count rate above 3 kev ( 3 out of the 10 photons detected above 3 kev would have been labelled background photons ) . additionally , a hard ( power law ) spectral could be present . a power law spectral component with index 2 contributes less than 20 per cent to the unabsorbed flux in the 0.510 kev band . the absorbed 0.510 kev source flux for both models and both @xmath21 values considered above was consistent with @xmath24 erg @xmath4 s@xmath5 , whereas the unabsorbed flux was @xmath25 erg @xmath4 s@xmath5 . here the error is determined from the range in fluxes derived from the various models . for all models we performed a monte carlo simulation ( using the goodness command in xspec ) . we simulated 10@xmath26 spectra based on a gaussian distribution of parameters centred on the best fit model parameters with a gaussian width set by the 1@xmath27 errors on the fit parameters . the percentage of these simulations with the fit statistic less than that for the data is more than the fiducial 50 per cent mark for 3 of the 4 cases considered , but again this may be partially explained by the presence of background photons in the last two bins . the goodness percentages are given in table [ fitpars ] . [ cols="<,^,^,^,^ " , ] @xmath28 value in outburst , corrected for the contribution of the nearby interloper we observed the neutron star sxt xte j1709267 on several occasions in outburst and quiescence in x ray and optical . the brightening of a star from @xmath0=22.24@xmath10.03 in quiescence to @xmath0=20.5@xmath10.1 in outburst and its positional coincidence with the chandra position for xte j1709267 provides convincing evidence that we have discovered the optical counterpart ( @xcite ) . obviously , there is a small chance that the star found in quiescence is an interloper and not the true counterpart ; spectroscopic observations of the proposed quiescent counterpart will test this . if we assume , however , that we also detected the optical counterpart in quiescence , the absolute magnitude , @xmath29 , for the counterpart would be @xmath30 where we have assumed an @xmath0band interstellar absorption , @xmath31 , of 0.921.84 magnitudes , respectively [ for a distance of 8.8 kpc , from @xmath32 , and from the relations between @xmath21 , @xmath20 , and @xmath31 given by @xcite and ] . these @xmath29values are consistent with xte j1709267 being a low mass x ray binary with a late type ( k ) dwarf companion . the unabsorbed luminosity in the 0.510 kev x ray band we find using _ chandra _ observations of the source in quiescence is only slightly lower than the lowest luminosity measured by @xcite ( approximately a factor 2 ; in that paper we gave the unabsorbed flux in the 0.110 kev and not the 0.510 kev band ) . so , the decay in luminosity in about 14 months is small ( approximately factor of 2 ) . this is in contrast with the decay rate of the quasi persistent source mxb 165929 . @xcite found that for that source the bolometric luminosity decreased by a factor 79 in 18 months . such a difference could be explained by the fact that mxb 165929 had been accreting steadily for several years before returning to quiescence . this extended period of steady accretion may have heated the neutron star crust to temperatures higher than that of the neutron star core and after the outburst the crust cools down ( @xcite ) . however , this difference could also reflect a difference between the quiescent mass accretion rates , although it is unclear why in some sources the mass accretion rate hits a minimum close after the outburst whereas in other neutron star stxs like mxb 165929 the mass accretion rate keeps decreasing gradually . recently , we found ( @xcite ) that there seems to be an anti correlation between the fractional power law contribution to the 0.510 kev luminosity and the source luminosity in quiescence for quiescent luminosities lower than @xmath6 erg s@xmath5 and a correlation between these two parameters for luminosities above this luminosity . in figure [ relation ] we plot the power law fractional contribution to the total 0.510 kev unabsorbed quiescent luminosity for several neutron star sxts for which the distance is accurately known . to obtain the quiescent luminosities we use distances quoted in @xcite for the systems where photospheric radius expansion bursts have been observed . since the photospheric radius expansion burst luminosity is thought to be close to 2.0 or 3.8@xmath33 ( see ) we use a range in quiescent luminosities to account for this ambiguity in source distance . in the case of xte j1709267 we took a distance range of 812 kpc . for the neutron star sxts in the globular clusters terzan 5 and ngc 6440 we took a distance of 8.7@xmath34 kpc and 8.5@xmath35 kpc , respectively ( @xcite ; ) . finally for cen x4 we use a distance of 1.2 kpc ( @xcite ; ; @xcite ) . we do not take into account errors on the source quiescent luminosities due to errors on the measured source flux since these are typically smaller than the uncertainty in the burst chemical composition . when the error on the power law contribution to the flux was not given in the literature we assumed an error of 10 per cent . finally , we included in the plot globular cluster sources which are thought to be quiescent neutron star sxts on the basis of their soft spectrum ( e.g. @xcite ) . we used the sources and limits on the power law component in the spectrum as found by @xcite.0.3 kpc for the globular cluster @xmath36 cen ( @xcite ) , 10.3@xmath10.8 kpc for m 80 ( ) , 5.2@xmath10.3 kpc for 47 tuc , 3.6@xmath10.3 kpc for ngc 6397 , and 9.5@xmath10.9 kpc for m 30 ( all from @xcite ) . ] to the high luminosity side , the trend of increasing power law fraction with luminosity is dominated by the data points of xte j1709267 which was followed by _ chandra _ during its decay to quiescence after an outburst ( jonker et al . 2003 ; see also table [ refit ] ) . we found that the power law contribution to the 0.510 kev x ray luminosity decreased from 72 per cent on mjd 52365.018 , to 48 per cent on mjd 52374.727 , to less than 19 per cent using the combined data from observation 3492 and 3507 . the detailed study of aql x1 confirms the observed trend ( rutledge et al . however , the quiescent properties of the counterpart to the neutron star sxt exo 1745248 in the dense globular cluster core of terzan 5 seemingly do not fit the correlation ( wijnands et al . perhaps the identified source is an interloper or perhaps the distance to terzan 5 is much smaller than what is assumed ( derived a distance of 5.6 kpc ) . alternatively , the apparent correlation is spurious and terzan 5 is the first source to fill in the gap . if the apparent smooth change in power law contribution to the quiescent luminosity is real it could mean that the nature of the power law spectral component at high and low source luminosities is different . since we observed the power law contribution in xte j1709267 to decrease when the source returned to quiescence after an outburst it is conceivable that the power law component at luminosities above @xmath6 erg s@xmath5 finds its origin in residual accretion . it has been proposed that neutron star sxts enter a propeller regime when the outburst decay rate steepens impeding most if not all accretion ( @xcite ) . however , pulsations were still detected in sax j1808.43658 after the alleged onset of the propeller mechanism ( @xcite ) . furthermore , steepening of the decay is also found in black hole candidate sxts ( e.g. @xcite ) . finally , the work of shows that there is a class of burst sources which likely accrete at a low level . deep observations a few hours after the detection of a type i burst in sax j2224 + 5421 did not reveal a persistent source with a 210 kev upper limit of 1.3@xmath37 erg @xmath4 s@xmath5 ( ) , which for the distance of sax j2224 + 5421 leads to an upper limit on the luminosity in that band of 7.4@xmath38 erg s@xmath5 . therefore , we think it is more likely that the origin of the power law at relatively high quiescent luminosities lies in residual accretion . if so this can help explain the observed short term neutron star temperature changes in aql x1 ( @xcite ) . these short term temperature changes pose a problem for the cooling neutron star core / crust model . however , if residual accretion is ongoing the observed changes can , for instance , be explained as being the result of a different hydrogen and helium content in the atmosphere during the two observations caused by the residual accretion . hence , the fact that this power law component can be explained at least qualitatively in terms of residual accretion helps the cooling neutron star model as an explanation for the thermal component . the origin of the power law on the low quiescent luminosities side of @xmath39 erg s@xmath5 is still unclear . since sax j1808.43658 is known to have a sizable magnetic field the power law component could be explained as being due to a pulsar wind mechanism ( e.g. @xcite ; ) . however , this can not explain the strong power law components in the non pulsating sources cen x4 , sax j1810.82609 , and xte j2123058 although one must bear in mind that cen x4 and sax j1810.82609 have not been observed with _ rxte_/pca when the sources were in outburst ( whereas xte j2123058 _ has _ been observed with _ rxte_/pca in outburst see @xcite and @xcite ) . a possible explanation for the apparent correlation between the fractional power law contribution and the total 0.510 kev luminosity at low quiescent luminosities is that there is a power law spectral component with a luminosity of @xmath40 . this power law luminosity would then need to be approximately the same for all sources ; the luminosity of the black body can differ between sources . in the cooling neutron star model the luminosity of the thermal component depends on the time averaged mass accretion rate and the neutron star mass ( @xcite ; @xcite ) . a low black body luminosity could point at a low time averaged mass accretion rate and/or a large neutron star mass allowing for enhanced core cooling ( cf . the nature of the power law and why it would have a luminosity close to @xmath40 in the 0.510 kev band is unclear . finally , we found a precursor to several type i x ray bursts of xte j1709267 . a similar burst with a precursor was found for the bursting atoll source 4u 163653 . a precursor to the main burst event in relatively long bursts has been associated with photospheric radius expansion bursts ( see @xcite ) . however , from a comparison of the burst properties of photospheric radius expansion bursts in 4u 163653 ( galloway et al . in preparation ) with the properties of the burst with a precursor in 4u 163653 we conclude that these bursts with precursors in 4u 163653 and hence also in xte j1709267 are not photospheric radius expansion bursts . perhaps these precursor events are related to bursts with multiple peaks observed in 4u 163653 ( @xcite ; ) . support for this work was provided by nasa through chandra postdoctoral fellowship grant number pf340027 awarded by the chandra x ray center , which is operated by the smithsonian astrophysical observatory for nasa under contract nas839073 . mrg acknowledges support from ltsa grant nag510889 and contract nas839073 to the chandra x ray center . pgj would like to thank sergio campana and albert kong for comments on an earlier version of the manuscript .

in this paper we report on the discovery of the optical counterpart to the neutron star soft x ray transient ( sxt ) xte j1709267 at an @xmath0band magnitude of @xmath0=20.5@xmath10.1 and 22.24@xmath10.03 , in outburst and quiescence , respectively . we further report the detection of type i x ray bursts in _ rxte _ data obtained during an outburst of the source in 2002 . these bursts show a precursor before the onset of the main burst event , reminiscent of photospheric radius expansion bursts . sifting through the archival _ rxte _ data for the burster 4u 163653 we found a nearly identical burst with precursor in 4u 163653 . a comparison of this burst to true photospheric radius expansion bursts in 4u 163653 leads us to conclude that these bursts with precursor do not reach the eddington limit . nevertheless , from the burst properties we can derive that the distance to xte j1709267 is consistent with the distance of the globular cluster ngc 6293 . we further report on the analysis of a 22.7 ksec observation of xte j1709267 obtained with the _ chandra _ satellite when the source was in quiescence . we found that the source has a soft quiescent spectrum which can be fit well by an absorbed black body or neutron star atmosphere model . a power law contributes less than @xmath220 per cent to the 0.510 kev unabsorbed flux of @xmath3 erg @xmath4 s@xmath5 . this flux is only slightly lower than the flux measured right after the outburst in 2002 . this is in contrast to the recent findings for mxb 165929 , where the quiescent source flux decreased gradually by a factor of @xmath279 over a period of 18 months . finally , we compared the fractional power law contribution to the unabsorbed 0.510 kev luminosity for neutron star sxts in quiescence for which the distance is well known . we find that the power law contribution is low only when the source quiescent luminosity is close to @xmath6 erg s@xmath5 . both at higher and lower values the power law contribution to the 0.510 kev luminosity increases . we discuss how models for the quiescent x ray emission can explain these trends . stars : individual : ( xte j1709267 , 4u 163653 ) accretion : accretion discs binaries : general stars : neutron stars x - rays : binaries

it is now rather well established that the quantum energy levels of classically chaotic billiards have wigner dyson correlations @xcite . these universal correlations are closely associated with the concept of `` quantum chaos '' and have first been obtained in random matrix theory @xcite ( rmt ) in a different context ( statistical description of the spectral properties of complex nuclei and small metallic particles ) . futhermore , many works@xcite have considered the case where the system hamiltonian @xmath12 depends on an external parameter @xmath13 . instead of a single matrix ensemble defined by a certain probability density , a continuous family of ensembles , defined by a @xmath13dependent probability density , is considered . the universality of the level correlations can then be extended to a broader domain , which involves not only ( small ) level separations at a given @xmath13 , but also ( small ) external parameter separation @xmath14 . this broader universality has been numerically checked in very different physical systems : disordered conductors@xcite , quantum billiards@xcite , the hydrogen atom in a magnetic field@xcite and correlated electron systems@xcite . analytical derivations use perturbation theory@xcite or the non linear @xmath11-model @xcite . another fruitful way to calculate the universal expressions of those parametric level correlations is based on brownian motion ensembles of random matrices , first introduced by dyson . the idea is to assume that the @xmath15 hamiltonian matrix diffuses when one varies the parameter @xmath13 in the available manifold given its symmetries . the @xmath13-dependence of the level distribution is then determined by a fokker planck equation@xcite , which can be solved in the large @xmath1-limit for arbitrary symmetries and for finite @xmath1 in the unitary case ( sutherland method ) . for quantum transport in the mesoscopic regime , the system is not closed , but open to electron reservoirs , and becomes a @xmath1-channel scatterer with transmission eigenvalues @xmath0 . many transport properties@xcite ( conductance , quantum shot noise ... ) are linear statistics of the @xmath1 eigenvalues @xmath0 , which are the appropriate levels for a scattering problem . the scattering matrix @xmath2 or transfer matrix @xmath8 , suitably parametrized by the @xmath1 transmission eigenvalues @xmath0 and certain auxiliary unitary matrices , are the relevant matrices for which rmt approaches have been formulated . for instance , if the unitary @xmath2-matrix is distributed according to one of dyson s classical circular ensembles @xcite , the corresponding transmission eigenvalue distributions have been recently obtained @xcite , exhibiting the same logaritmic pairwise repulsion than for the energy levels , and hence giving essentially the same wigner dyson correlations for the transmission eigenvalues . this rmt description is mainly relevant for quantum transport through ballistic chaotic cavities , which is a recent field of significant experimental investigations . such cavities can be made with semiconductor nanostructures known as quantum dots @xcite with a few channel contacts to two electron reservoirs . the validity of this rmt description is confirmed by microscopic semiclassical approaches@xcite and by numerical quantum calculations@xcite . the universality of the transmission eigenvalue correlations are established for a given @xmath13 , and therefore we want to know if it can be extended to their parametric dependence , too , as for the energy levels . futhermore , we want to see if these parametric correlations obey a similar behavior than that of the energy levels . this is one of the issues which we address in this work , using two brownian motion ensembles for the transmission eigenvalues , which result from two different assumptions concerning the @xmath13dependence of @xmath2 ( @xmath2-brownian motion ensemble ) or of @xmath8 ( @xmath8-brownian motion ensemble ) . this issue has been recently considered also in refs . @xcite . the @xmath2-brownian motion ensemble is simply the original ensemble introduced by dyson@xcite for the scattering matrix @xmath2 @xmath16 the idea is to assume that @xmath2 _ diffuses _ in the @xmath2-matrix space with respect to a fictitious time @xmath13 of the brownian motion , and converges in the limit @xmath17 to a stationary probability distribution given by one of the well known circular ensembles : circular unitary ensemble ( cue , @xmath18 ) when there is no time reversal symmetry ( applied magnetic field ) ; circular orthogonal ensemble ( coe , @xmath19 ) when there are time reversal and spin rotation symmetries ( no spin orbit coupling or applied magnetic field ) and circular symplectic ensemble ( cse , @xmath20 ) otherwise . choosing an initial probability distribution at @xmath21 of higher symmetry than the stationary limit , e.g. two independent circular ensembles diffusing towards a single one , one can perform with the parameter @xmath13 the crossover from the initial ensemble to the stationary ensemble . actually , the brownian motion ensembles coe @xmath5 cue , cse @xmath5 cue , and @xmath4cue @xmath5 cue have been investigated in ref . @xcite in terms of the scattering phase shifts @xmath22 . semiclassical and numerical justifications of these models are given in ref . the _ decisive _ difference between our approach and refs . @xcite is that we do not consider the usual eigenvector eigenvalue parametrization of @xmath2 , but the parametrization which explicitly uses the @xmath1 transmission eigenvalues @xmath0 of @xmath23 as introduced in ref . @xcite . since there exists neither a simple mathematical method nor an intuitive way to relate the @xmath24 eigenvalues @xmath25 of @xmath2 with the @xmath1 transmission eigenvalues @xmath0 , it is justified to reconsider the brownian motion in terms of the latter . this parametrization is suitable to determine the `` time '' dependence of the @xmath0 , by a fokker planck equation which we map , using a transformation introduced by sutherland , onto a schrdinger equation with imaginary time . in the case of the stationary cue limit , one has a problem of @xmath1 non interacting fermions which we solve exactly for arbitrary @xmath1 . one obtains for the corresponding parametric correlations of the transmission eigenvalues the same universal form than for the energy levels , after an appropriate rescaling and _ within certain limits_. a physical application of such a parametric ensemble can for example be realized by two weakly coupled ballistic ( irregular ) cavities , each of them connected to one electron reservoir ( fig . the crossover between two cue towards a single cue is performed by changing the strength of the coupling from nearly uncoupled cavities ( two cue ) to idealy coupled cavities ( one cue ) , when a small constant magnetic field is applied . the transmission eigenvalue parametrization is very well adapted to this case since the initial condition is easily realized by the requirement @xmath26 . another application of the dyson brownian motion ensemble , concerning the crossover coe @xmath5 cue in terms of the @xmath0 is presented elsewhere @xcite . outside the problem of two weakly coupled quantum dots , we are also motivated by numerical calculations @xcite showing that the scattering matrix of a quasi one - dimensional disordered wire exhibits also some properties of the circular ensembles , in the sense that the correlation of the scattering phase shifts @xmath22 were found to be approximately described by the universal correlations functions of these ensembles . for the disordered wire , a crossover from one circular ensemble towards two independent circular ensembles was observed as the length of the wire exceeds its localization length . the appropriate statistical description @xcite of the transmission eigenvalues of disordered wires is given by a different brownian motion ensemble than the original dyson model , and results from the multiplicative combination law of the transfer matrix @xmath8 . this is what we call the @xmath8-brownian motion ensemble which yields another fokker - planck equation taking into account the quasi one - dimensional structure . this fokker - planck equation has been solved by beenakker and rejaei@xcite in the unitary case , using the sutherland transformation . a remarkable result which they found is that the transmission eigenvalue pairwise interaction is universal , in the sense that it does not contain any adjustable parameter . it coincides with the standard logaritmic rmt repulsion for small eigenvalue separations , and it is halved for larger separations . the physical interpretation of this halving is unclear , but must be somewhat related to the statistical decoupling of the reflection properties of the two opposite system edges . if it is right , the dyson brownian motion ensemble must also exhibit in the crossover regime @xmath4cue @xmath27cue a somewhat similar pairwise interaction for weak transmission . it is therefore an interesting idea to compare those two different brownian motion ensembles for the transmission eigenvalues . let us mention the obvious difference between them : their initial and stationary limits are inverse of each other . for the @xmath2matrix model , we start from no transmission towards a good cue transmission . for the @xmath8matrix model on the contrary , the initial condition corresponds to perfect transmission , and the system evolves with the time @xmath13 ( here to be identified with conductor length @xmath9 ) to the zero transmission localized limit . this paper is composed of four main sections . first , we introduce the transmission eigenvalue parametrization of @xmath2 and define the @xmath2brownian motion ensemble ( section [ section:2a ] ) . the corresponding fokker - planck equation is derived in appendix a. in section [ section:2b ] , the fokker - planck equation is solved for an arbitrary initial condition in terms of fermionic one particle green s functions , using the method introduced by beenakker and rejaei for the @xmath8-brownian motion ensemble . in section [ section:2e ] , we show how to recover for the transmission eigenvalues the same universal parametric correlations than for the energy levels , after rescaling and within certain limits . in section [ section:2add ] , we treat completely the system of two coupled ballistic cavities by the @xmath2brownian motion ensemble for the unitary case . the time of the brownian motion is given in terms of the numbers of channels in the contacts to the electron reservoirs ( @xmath1 ) and in the contact between the two cavities ( @xmath3 ) . in section [ section:2c ] , the crossover @xmath4cue @xmath5 cue is considered . the method of orthogonal polynomials , well known in the framework of random matrix theory @xcite , is extended to a more general crossover situation and yields the correlation functions as determinants ( eq . ( [ corresult ] ) ) containing a function @xmath28 given as a sum of legendre polynomials ( eq . ( [ knexp ] ) ) . this section contains with eqs . ( [ gaverage]-[gautocorr2 ] ) also exact results for the average and the fluctuations of the conductance . in section [ section:2d ] , we show that the transmission eigenvalue interaction deviates from the pure rmt logarithmic interaction , in the limit of weakly coupled cavities . one can note some interesting similarities with the pairwise interaction of the @xmath8-brownian motion model given in ref . @xcite . the third part of this paper ( section [ section:3 ] ) reconsiders the @xmath8brownian motion ensemble of ref . @xcite for disordered wires in the unitary case . we use the exact expression of the joint probability distribution found in ref . @xcite and calculate the corresponding correlation functions , using a method very similar to the one used in section [ section:2c ] . this is possible because , _ from a mathematical point of view _ , the brownian motion models for @xmath2 and @xmath8 are very similar . we get an expression of the @xmath29-point correlations as a determinant ( eq . ( [ mellocorrfunc ] ) ) with a function @xmath30 given as a sum and an integral with legendre polynomials and legendre functions ( eq . ( [ melloknexplizit ] ) ) . this allows us to prove the equivalence between the @xmath8brownian motion ensemble and a more microscopic approach , based on a non linear @xmath11-model formulation and supersymmetry , as far as the behavior of the first two moments of the conductance @xmath31 and @xmath32 is concerned ( section [ section:3b ] ) . we emphasize that the solution of the brownian motion ensemble is now complete , contrary to the sigma model approach where one only knows @xmath31 and @xmath32 in the large @xmath1-limit . in section [ section:3c ] , some further simplifications valid in the localized limit lead to asymptotic expressions for the density and the two point function for the logarithm of the transmission eigenvalues . in section [ section:4 ] , we review some essential differences between the transmission properties characterizing two weakly coupled ballistic chaotic cavities ( @xmath2brownian motion ensemble ) on one side and an homogeneous disordered wire ( @xmath8brownian motion ensemble ) on the other side . we give some concluding remarks in section [ section:5 ] , notably on the mathematical similarities between the @xmath2 and @xmath8 brownian motion ensembles . the scattering matrix @xmath2 of a system connected to two electron reservoirs by two @xmath1-channel contacts can be described in the transmission eigenvalue parametrization by @xmath33 where @xmath34 , @xmath35 are @xmath36 unitary matrices and @xmath37 is a diagonal @xmath36-matrix with real entries @xmath38 . the transmission matrix is @xmath13 , but we mean by `` transmission eigenvalues '' @xmath39 the eigenvalues of @xmath40 . this parametrization has recently @xcite been used to calculate the conductance properties of a ballistic cavity , which is known to be suitably described by one of dyson s circular ensembles @xcite . in the unitary case ( @xmath18 ) all unitary matrices @xmath34 , @xmath35 are independent whereas for the orthogonal ( @xmath19 ) and symplectic ( @xmath20 ) symmetry classes , one has @xmath41 and @xmath42 ( with @xmath43 in the orthogonal case and @xmath44 , @xmath45 , in the symplectic case ) . in the following , the transmission eigenvalues @xmath39 are described by angles @xmath46 $ ] via @xmath47 and therefore @xmath48 where @xmath49 is a diagonal matrix with entries @xmath50 . the invariant measure for @xmath2 in the parametrization ( [ par ] ) and the change of variables ( [ rep ] ) is given by @xcite @xmath51 with @xmath52 in the cases @xmath53 only two products @xmath54 appear . the brownian motion ensemble which we consider ( cp . dyson @xcite ) describes a unitary random matrix which depends on a fictitious time . a small change on the time variable @xmath55 yields ( in the unitary case ) for @xmath2 the change : @xmath56 where @xmath57 is an infinitesimal hermitian random matrix with averages @xmath58 and that is independently distributed of @xmath59 . the diffusion constant @xmath60 will be precised afterwards , depending on the considered physical problem . we will only consider the unitary case . the brownian motion ensembles for the other cases are more involved and their precise definition can be found in ref . let @xmath61 be the joint probability density for the angles @xmath50 of @xmath2 at the time @xmath13 , obtained after integration over the unitary matrices @xmath34 , @xmath35 . this brownian motion is then characterized by the fokker - planck equation : @xmath62 with @xmath63 given by eq . ( [ fbeta ] ) . in the appendix this fokker - planck equation is derived for the unitary case @xmath18 which is our main concern . the details of the derivation in the appendix show that the fokker - planck equation ( [ fp1 ] ) remains valid even if the statistics of the matrix @xmath57 is arbitrarily chosen ( including the case of a constant matrix @xmath57 ) instead of ( [ x_statistic ] ) . for this to be true , one then needs instead of eq . ( [ x_statistic ] ) the non - trivial assumption that the unitary matrices @xmath64 and @xmath65 that appear in the parametrization ( [ par ] ) of @xmath59 are for each @xmath13 uniformly distributed on the space of @xmath36 unitary matrices ( i.e. they have always two independent @xmath36 cue distributions ) . the time step @xmath14 is then determined by eq . ( [ dm_inform ] ) of the appendix . it is interesting to note that this assumption concerning @xmath65 and @xmath64 is fulfilled at least at @xmath21 for the two applications concerning the crossovers @xmath4cue @xmath5 cue or coe @xmath5 cue which are treated here or in ref . @xcite respectively . we assume that this holds also for arbitrary @xmath66 and the brownian motion model is applicable to rather general physical situations . we want now to solve the fokker - planck equation ( [ fp1 ] ) for an arbitrary initial condition @xmath67 and for @xmath18 . we set @xmath68 for convenience . we proceed in a similar way as in ref . @xcite and apply the so - called sutherland transformation @xcite : @xmath69 the function @xmath70 fullfils a schrdinger equation with imaginary time : @xmath71 where @xmath72 is a many particle hamilton operator @xmath73 with a potential @xmath74 having the form @xmath75 and the constant @xmath76 one can see that the unitary case @xmath18 is apparently much more simpler , since the complicated many particle hamilton operator reduces in this case to a sum of independent one particle hamiltonians , i.e. @xmath77 with @xmath78 the eigenvalue problem of this operator can be solved using legendre polynomials . the properly normalized eigenfunctions @xmath79 with eigenvalues @xmath80 , @xmath81 , are given by @xmath82 the solution of the fokker - planck equation can be expressed by the many particle fermionic green s function @xmath83 , as already done in ref . @xcite via @xmath84 where @xmath85 eq . ( [ green1 ] ) describes the one particle green s function , defined by @xmath86 for technical reasons , we will now switch to new coordinates @xmath87 $ ] which are just the arguments of the legendre polynomials . let @xmath88 be the probability density in terms of the @xmath89 taking into account the jacobian of this transformation , i.e. @xmath90 . ( [ sol1]-[green1 ] ) are then replaced by @xmath91 we have used in ( [ xsol1]-[xgreen1 ] ) the same symbols @xmath92 , @xmath93 , @xmath94 , @xmath95 and @xmath96 for objects which are related with the corresponding objects of ( [ sol1]-[green1 ] ) via suitable transformations including the corresponding jacobians , and some factors @xmath97 . one can easily check that @xmath88 in the limit @xmath17 corresponds to the cue distribution . for this , we need to consider only the first @xmath1 contributions , i.e. @xmath98 , in the sum of ( [ xgreen1 ] ) . the matrix @xmath99 is then the matrix product of two matrices @xmath100 , @xmath101 and a diagonal matrix with entries @xmath102 . then the determinant of eq . ( [ xgreen0 ] ) becomes proportional to the two vandermond determinants @xmath103 , @xmath104 and to the factor @xmath105 which is just canceled by the contribution @xmath106 in eq . ( [ xgreen0 ] ) . this gives : @xmath107 which corresponds to a stationary solution equal to @xmath108 , once the jacobian of the transformation @xmath109 has been taken into account . the first correction to the limit ( [ limit ] ) is proportional to @xmath110 , giving the typical time scale @xmath111 for the brownian motion to reach the stationary solution . the universality of the energy level parametric correlation has recently attracted a considerable interest . in mesoscopic quantum physics , this remarkable property obtained after an appropriate rescaling of the variables , was proven by diagrammatic methods in ref . an alternative derivation of this result can be directly applied to our problem : beenakker and rejaei have indeed shown @xcite how to recover this universal behavior from a brownian motion model for the hamiltonian ( @xmath12-brownian motion ensemble ) . this derivation can be easily adapted to the @xmath2-brownian motion ensemble and to the transmission eigenvalue parametric correlation . as in refs . @xcite , we do not consider here a crossover regime ( i.e : @xmath4 cue @xmath5 cue or coe @xmath5 cue for instance ) but just a cue @xmath5 cue parametric dependence , which reduces to the large @xmath13-behavior of the crossover of two decoupled @xmath15 cue towards a single @xmath112 cue considered in section [ section:2add ] . our discussion follows very closely the derivation of ref . @xcite , where a parametric dependence on a quantity @xmath113 was considered , related to the brownian motion time through the relation @xmath114 . we measure @xmath13 in units of @xmath115 in order to have a stationary distribution proportional to @xmath116 , and we use again the variables @xmath49 .. ] usually , one considers @xcite two types of correlation functions which are the density correlation function @xmath117 and the current correlation function @xmath118 for an arbitrary linear statistic ( e. g. conductance , shot noise power , ... ) of the form @xmath119 we can calculate the correlator via @xmath120 the two correlation functions ( [ ss_corr_def ] ) and ( [ cc_corr_def ] ) are related @xcite through @xmath121 in subsection [ section:2c ] , we calculate the probability density @xmath122 to find the eigenvalue @xmath123 at time @xmath13 and the eigenvalue @xmath124 at time @xmath125 . for a given @xmath126 and in the limit @xmath17 , the particular information of the initial condition will be lost and we have the density correlation function for the cue @xmath5 cue parametric dependence : @xmath127 the parameter @xmath113 is now related with @xmath126 through @xmath128 and the sinus - prefactors are just the jacobians due the variable transformation @xmath129 . the subscript means that @xmath1 can have arbitrary values and that the limit @xmath10 has still not been taken . using the results ( [ corr11res ] ) , ( [ kn2def ] ) and ( [ hkdef ] ) derived in the next section , we get directly the exact expression : @xmath130 where @xmath131 is the eigenvalue of the one particle hamiltonian @xmath132 given by ( [ ham1 ] ) . in another work@xcite , we have calculated the function @xmath133 for the crossover coe @xmath5 cue which differs at small @xmath13 from the result ( [ corr11res ] ) we have used here . however , in the limit @xmath17 both functions become identical and give the expression ( [ sn_corr_result ] ) which has in fact the same structure than in eq . ( 5.27 ) of ref . one has just to replace the harmonic oscillator eigenfunctions used in @xcite by the legendre polynomials appearing in the @xmath2-brownian motion . first , we consider the limit @xmath10 and differences @xmath134 of the order @xmath135 , in order to use the large @xmath3 expansion of the legendre polynomials : @xmath136 and to replace the sums by integrals . after some smoothing over the fast oscillating terms , the correlation function depends only on the difference @xmath137 , i.e. @xmath138 where @xmath139 is the average density of the variable @xmath49 and @xmath140 . the expression ( [ sunend_result ] ) , which is valid in the large @xmath1- and small @xmath141-limit , coincides exactly with the result of ref . @xcite given by a brownian motion for the hamiltonian @xmath12 and also with a supersymmetric calculation for the unitary case @xcite . the calculations of the current correlation function ( [ cc_corr_def ] ) and of the correlator ( [ lin_corr ] ) are then straightforward and can be found in ref . we note that ( [ sunend_result ] ) yields an algebraic decay for the correlator of an arbitrary linear statistic as @xmath142 @xcite @xmath143 which is also found @xcite in a simplified brownian motion for the transmission eigenvalues . however , we can not directly use the expression ( [ sunend_result ] ) to evaluate the integrals in ( [ lin_corr ] ) , due to the restriction @xmath144 . in fact , we will see in the next section that the conductance correlator decays exponentially in the @xmath2-brownian motion ensemble . in order to understand this behavior , we consider again the limit @xmath10 , but now without any restriction on @xmath49 and @xmath145 . we replace in the exact expression ( [ sn_corr_result ] ) again the legendre polynomials by the asymptotic formula ( [ legendre_asym ] ) but now we keep the discrete sums instead of replacing them by integrals . one can see that this step is very crucial : the variable @xmath49 varies between @xmath146 and @xmath147 and the correlation function should rather be expanded in a discrete fourier - series instead of a continuous fourier - integral , being responsible @xcite for the algebraic decay ( [ corr_decay_alg ] ) . in the limit @xmath10 at fixed ratio @xmath148 , the main contributions comes from the terms with @xmath149 and @xmath150 so that we can linearize the energy difference @xmath151 . in addition , all fast oscillating terms of the type @xmath152 are omited . then , one of the sums is easily done and we obtain for the density correlation function the expression @xmath153 where the function @xmath154 is given as a fourier - series @xmath155 with fourier - coefficients @xmath156 our derivation accounts of course only for the unitary case @xmath18 but eq . ( [ ss_coeff ] ) is valid for the orthogonal ( @xmath19 ) and symplectic case ( @xmath20 ) too @xcite . the derivation for all @xmath157 can be done by an asymptotic expansion of the fokker - planck equation @xcite similarly as in section 4 of ref . @xcite and eq . ( [ ss_coeff ] ) is indeed the discrete version of eq . ( 4.14 ) in ref . @xcite . we are now able to calculate the correlator for the linear statistics ( [ lin_stat ] ) . if we expand the function @xmath158 in a fourier - series @xmath159 we find @xmath160 in ( [ aa_four ] ) , we have extended @xmath158 to an even function on the interval @xmath161 so that the fourier - coefficients are even in @xmath3 ( one should imagine that @xmath162 is a well defined function of @xmath129 . ) . as an example we consider the conductance @xmath163 with @xmath164 , i.e. @xmath165 , @xmath166 and @xmath167 if @xmath168 , @xmath169 which is ( in the limit @xmath17 ) consistent with eq . ( [ gautocorr2 ] ) of the next section . the current correlation function is calculated from ( [ corr_relation ] ) and the assumption that @xmath2 and @xmath170 depend only on the difference @xmath171 . we obtain @xmath172 where @xmath173 has the fourier - coefficients @xmath174 the coefficient @xmath175 is not determined and not needed due to ( [ cc_better ] ) . the modified sign in the first contribution in ( [ cc_better ] ) is important because the current correlation function is odd as a function of @xmath49 ( or @xmath145 ) if @xmath145 ( or @xmath49 ) is fixed . in summary , we have found that the parametric correlations of the @xmath2-brownian motion are only _ locally _ ( on a range @xmath176 ) identical with the universal correlations found by a @xmath12-brownian motion @xcite or by a microscopic approach in the framework of the zero - dimensional supersymmetric non linear @xmath11-model @xcite . this agreement does not hold if we consider the whole range of @xmath49 , @xmath145 . in particular , quantities _ integrated over the whole spectrum _ like the correlator of a linear statistic must be treated with more care and indeed the conductance correlator shows an exponential instead of an algebraic decay , _ in the @xmath2-brownian motion ensemble_. the @xmath2-brownian motion ensemble with cue stationary limit can be applied to a system made by two quantum dots in series under an applied magnetic field , for describing how its transmission eigenvalue distribution varies as a function of the strength of their coupling . we assume that the two cavities are coupled between them by a wire with @xmath3 conducting channels ( see fig . 1 ) , and each of them to one electron reservoir through @xmath177 channels . the underlying classical dynamics is supposed to be fully chaotic in each dots . for this particular application , we need to express the fictitious time @xmath13 of the brownian motion , which has been introduced in a rather abstract mathematical way , in terms of the physical parameters @xmath1 and @xmath3 . in ref . @xcite , it was shown that a closed system where the classical motion remains ( nearly ) in several separated chaotic regions of the classical phase space can be modelled by a random matrix approach similar to dyson s gaussian brownian motion ensemble discussed in ref . the time parameter of the latter can then be related with the particle flux between the nearly separated regions of phase space . in principle , one can relate @xcite the @xmath12-brownian motion ( for a closed system ) with the @xmath2-brownian motion ( for an open system ) by some microscopic assumptions about the @xmath178-dependence and then apply the results of ref . @xcite in order to obtain the physical meaning of the @xmath2-brownian motion time . for the scope of this paper , we will only give a simple heuristic derivation . therefore , we assume that the @xmath179-dimensional @xmath2-matrix of each cavity has a cue distribution when an external parameter is varied ( fermi energy , magnetic field , shape of the dot ... ) and has the form shown in eq . ( [ smatrix ] ) , excepted that @xmath180 is @xmath36-matrix whereas the coupling gives rise to a @xmath181-matrix @xmath13 . when one electron is injected from a reservoir into one of the cavities by the @xmath182-th of the @xmath1 incoming channels of the corresponding lead , one can assume that the probability that it goes directly back to reservoir is just @xmath183 and that the probability to go through the @xmath3 channel wire into the other cavity is @xmath184 this results from the assumption that the chaotic motion inside the dot uniformly explores the dot boundaries . furthermore , one can assume that the typical time for the electron to stay in one cavity is very long compared to the time needed to cross the connection , i.e. the electron looses completely the memory from which reservoir it was injected to the actual cavity . in this case , the total probability @xmath185 for the electron to be transmitted from one reservoir to the other is given by @xmath186 , a relation which accounts for all type of processes where the electron can move between the cavities an arbitrary number of times before it leaves the system into the other reservoir . the solution of this equation is given by @xmath187 then , the average conductance can be estimated by @xmath188 which leads with the exact result eq . ( [ gaverage ] ) of section [ section:2c ] to the identification @xmath189 where @xmath13 the brownian motion time . the prefactor in ( [ timemeaning ] ) accounts for the typical time scale @xmath190 which is found in the end of section [ section:2b ] . in the weak coupling limit @xmath191 , we have the simple proportionality @xmath192 . in the parametrization ( [ par ] ) , the case of two independent cues of dimension @xmath1 corresponds to the requirement that all @xmath193 or @xmath194 respectively . we have therefore @xmath195 for the initial condition . the evaluation of ( [ xsol1 ] ) leads to the limit @xmath196 . we note that the legendre polynomials are particular hypergeometric functions , i.e. @xmath197 which can be expanded as @xmath198 with the polynomials @xmath199 of degree @xmath200 . we now expand the one particle propagator @xmath99 with respect to @xmath201 up to terms @xmath202 . again we can identify a matrix product in the determinant of eq . ( [ xsol1 ] ) which yields a factor @xmath104 just cancelling the inverse contribution in ( [ xsol1 ] ) . for the further calculation we use the notation @xmath203 and find from ( [ xsol1 ] ) , ( [ xgreen0 ] ) @xmath204 with @xmath205 because of the determinant in eq . ( [ prob0 ] ) , we note that one can replace in eq . ( [ gkdef ] ) the polynomials @xmath206 , @xmath207 by an _ arbitrary _ set of linear independent polynomials of ( maximal ) degree @xmath208 . such a substitution affects only the normalization constant . in the following , we will use two different substitutions for two different purposes : first to calculate exactly all kind of correlation functions and second to determine the effective _ interaction _ between the @xmath89 ( or @xmath50 ) variables . we begin with the calculation of the correlation functions defined by @xmath209 we put @xmath210 , @xmath211 and replace the polynomials @xmath206 by the lagrangian interpolation polynomials @xmath212 for the @xmath213 , i.e. @xmath214 where @xmath215 . we find now for the joint probability density the result @xmath216 with @xmath217 the normalization in ( [ prob1 ] ) follows directly from the obvious property @xmath218 actually , this `` orthogonality '' relation enables us to calculate all correlation functions ( [ corfunc0 ] ) . we define the function @xmath219 and find the following three properties @xmath220 from these properties and a well known theorem in the theory of random matrices @xcite ( cp . theorem 5.2.1 . of ref . @xcite ) , we find directly the result for the correlation functions @xmath221 for the cases @xmath222 , we have @xmath223 for practical applications , we give the more explicit expression for ( [ kndef ] ) @xmath224 with coefficients @xmath225 one might also consider correlation functions at two different times , i.e. @xmath226 without going into technical details , we mention that one can calculate the function @xmath227 with the technic of functional derivatives @xcite . the result is @xmath228 where @xmath229 is a generalization of ( [ kndef ] ) for different times . the limit @xmath230 is given by @xmath231 . the knowledge of the correlations functions ( [ corr1 ] , [ corr11res ] ) enables us to calculate exact expressions for the average of the conductance @xmath232 and the auto correlation of its variance @xmath233 . using the orthogonality of the legendre polynomials and their recursion relation we find @xmath234 of particular interest is the large @xmath1 limit . in order to obtain a non trivial crossover , we have to measure the time variable in units of the typical crossover scale @xmath111 and to keep the ratio @xmath235 fixed in the limit @xmath236 . we find then up to corrections of order @xmath135 @xmath237 we want evaluate ( [ prob0 ] ) in a suitable way for having the effective pairwise interaction between the @xmath89- or @xmath50-variables . we now replace in eq . ( [ gkdef ] ) the polynomials @xmath238 by @xmath239 and find @xmath240 with @xmath241 the quantity @xmath242 can be related to the heat kernel on the unit sphere @xmath243 with initial condition at the origin , which yields a useful approximation in the limit @xmath244 @xmath245 from the expansion @xmath246 ( here @xmath247 denote the laguerre polynomials ) and ( [ gapprox ] ) , ( [ gkdef2 ] ) we find @xmath248 this results leads to a probability density ( now for the variables @xmath50 ) @xmath249 where @xmath250 is a normalization constant which depends on @xmath13 . if we write ( [ prob4 ] ) under the form of a `` gibbs factor '' @xmath251 $ ] , we find that the fictitious corresponding hamiltonian has the pairwise interaction and the one body confining potential : @xmath252 this can be simplified when @xmath13 is very small , i.e. @xmath253 since @xmath254 in this limit . one gets a joint probability distribution ( [ prob4 ] ) identical to the so - called `` laguerre ensemble '' of ref . @xcite , with a logarithmic repulsion between the @xmath255 ( see also ref . @xcite ) . for @xmath236 , the density reduces to a quarter circle law : @xmath256 if the argument of the square root is not negative ( otherwise @xmath257 ) . using @xmath258 , one gets @xmath259 for the variable @xmath260 $ ] . one can note that the exact pairwise interaction for the @xmath50 variables ( [ interact ] ) differs for @xmath261 from the standard rmt interaction given by ( [ fbeta ] ) for @xmath18 . however , this concerns essentially the quantitative difference between @xmath262 and @xmath263 whereas for the quasi one - dimensional disordered wire considered in ref . @xcite this effect was much more stronger due to the appearance of @xmath264 instead of @xmath265 . in the sections [ section:4 ] , [ section:5 ] , we will return to this point and compare the results eqs . ( [ corr1 ] ) , ( [ semicircle2 ] ) with the corresponding expressions obtained for disordered lines , assuming the @xmath8-brownian motion ensemble . the @xmath2-brownian motion ensemble was built on the evolution law @xmath266 where @xmath57 was an infinitesimal hermitian random matrix ( eq . ( [ s_dynamics ] ) ) . a fokker - planck equation was deduced , giving the @xmath13-dependence of the transmission eigenvalues . a different fokker - planck equation is more natural when one builds a quasi - one dimensional wire , adding in series small diffusive blocs @xcite . the evolution law of the matrices @xmath267 and @xmath268 ( `` angular part '' ) and of the matrix @xmath37 ( `` radial part '' ) defined in eq . ( [ par ] ) can indeed be deduced from the _ multiplicative _ transfer matrix @xmath8 , expressed @xcite in the same parametrization . the law of @xmath8 describing the addition in series of the scatterers is now : @xmath269 the brownian motion time @xmath13 is in this case related to the number of scatterers in series ( i. e. : the length @xmath9 of the wire ) . @xmath8 can be parametrized with the same coordinates used for @xmath2 in eq . [ par ] , as detailed in ref . @xcite for instance . assuming that the scatterers are isotropically distributed , the length dependence of the transmission eigenvalues for the @xmath8-brownian motion ensemble is given by a fokker - planck equation @xcite , which presents interesting analogies and differences with eq . ( [ fp1 ] ) . the comparison of those two brownian motions , assuming two different evolution laws , will enable us to describe in details the similarities and the differences between quantum transmission through two weakly coupled cavities ( @xmath2-ensemble ) and through a disordered wire ( @xmath8-ensemble ) . before this , having used the beenakker - rejaei method for solving the @xmath2-brownian motion ensemble , we have been able to improve this method and we show in return how to complete the solution of the transmission problem through a disordered wire . indeed , we explain how to calculate exactly the length dependence of @xmath3 order correlation functions of the transmission eigenvalues , for arbitrary values of @xmath3 . this allowes us to establish an important equivalence between the @xmath8-brownian motion ensemble and a more microscopic approach using the non linear @xmath11-model . we start with the exact result given in ref . @xcite for the joint probability distribution of the variables @xmath270 which satisfies the fokker - planck equation of the @xmath8-brownian motion ensemble : @xmath271 with @xmath272 where @xmath273 is just given by ( [ constcn ] ) for @xmath18 . we underline that the changed sign in the exponential factor of ( [ probmello ] ) if compared with ( [ xgreen0 ] ) is correct . the variable @xmath13 is now @xmath274 ( @xmath275 is the elastic mean free path , @xmath9 is the length of the conductor , and @xmath276 is the quasi one - dimensional localization length for the unitary case ) . the index @xmath29 in ( [ mellogndef ] ) takes the values @xmath277 . for later use we mention that the generalized legendre function can be expressed as a hypergeometric function via @xmath278 this function is an eigenfunctions of the differential operator @xmath279 with eigenvalue @xmath280 . the operator @xmath60 is of course related with the one particle hamiltonian @xmath281 used in ref . @xcite by a suitable transformation . we can now state the following properties of @xmath282 @xmath283 eqs . ( [ gncalc ] ) , ( [ g0calc ] ) follow directly from ( [ mellogndef ] ) . eq . ( [ g0delta ] ) reflects the initial condition for the one particle green s function @xcite and was already given in @xcite , where the expression ( [ probmello ] ) was found for the one channel case . we are now interested in integrals of the form @xmath284 where @xmath285 again denotes the legendre polynomial of degree @xmath3 . we consider first the case @xmath286 . from ( [ g0delta ] ) we find @xmath287 and from ( [ g0calc ] ) we get the differential equation @xmath288 which has the solution @xmath289 . in eq . ( [ andiffeq ] ) we have used that @xmath290 is an eigenfunction of @xmath60 with eigenvalue @xmath291 , a property following directly from legendre s differential equation . from eqs . ( [ gncalc ] ) , ( [ coeffdef ] ) we get by a similar calculation the general result @xmath292 as in sec . [ section:2c ] we can replace the exponential factors @xmath293 in eq . ( [ mellogndef ] ) by an arbitrary set of linear independent polynomials of ( maximal ) degree @xmath208 in the variable @xmath280 because the determinant in eq . ( [ probmello ] ) yields only a constant factor after such a transformation . we now choose the replacement @xmath294 where @xmath295 are just the lagrangian interpolation polynomials already given by eq . ( [ lagdef ] ) . we introduce the following notations @xmath296 and @xmath297 from eq . ( [ anresult ] ) we get directly for @xmath298 @xmath299 the joint probability distribution ( [ probmello ] ) can then be written in the form @xmath300 with @xmath301 similarly as in sec . [ section:2c ] we find from ( [ melloorth ] ) @xmath302 and @xmath303 these properties lead as in eq . ( [ corresult ] ) to the @xmath29-point correlation functions @xmath304 where @xmath30 can also be expressed by the more explicit formula @xmath305 as a first application we calculate an exact expression for the average conductance , when the length of the wire increases . this expression is valid from the disordered conductor towards the anderson insulator . since the conductance is a linear statistic of the @xmath306 , @xmath307 , we have to the evaluate the corresponding integral over the density : @xmath308 since @xmath309 is a polynomial of degree @xmath29 in the variable @xmath310 , we can write @xmath311 where @xmath312 is a polynomial of degree @xmath313 which does not contribute in the integral ( [ melloavcond ] ) due to the quasi orthogonality relation ( [ melloorth ] ) . we find therefore @xmath314 where @xmath315 stands for the integral @xmath316 we use the transformation formula 15.3.4 of ref . @xcite for the hypergeometric function in ( [ ikdef ] ) and expand afterwards the latter in its power series . the integral over @xmath317 can then be done and gives the sum @xmath318 where @xmath319 denotes the generalized binomial coefficient which is defined for arbitrary complex values of @xmath162 . the evaluation of the sum ( [ iksum ] ) yields @xmath320 we now use the expression ( [ lagdef ] ) for the polynominals @xmath321 and find for the averaged conductance the expression @xmath322 where we have introduced the coefficient @xmath323 the result ( [ melloavcond3 ] ) is exact for all values of @xmath1 and @xmath324 . in the limit @xmath325 the coefficient @xmath326 is just @xmath327 and the expression ( [ melloavcond3 ] ) becomes identical with the microscopic result of zirnbauer et al @xcite based on the supersymmetric non linear sigma model , in the unitary case . a similar calculation yields the second moment of @xmath95 @xmath328 we omit the details , since this result can also be derived in a more direct way by the general identity @xmath329 which follows directly from the original fokker - planck equation @xcite for the case @xmath18 . this gives the proof that the microscopic supersymmetric approach and the @xmath8-brownian motion ensemble of ref . @xcite are equivalent in the limit @xmath10 , for arbitrary values of @xmath9 , at least as far as the average and the variance of the conductance are concerned . in the localized limit @xmath331 the integral in eq . ( [ mellohndef ] ) can be evaluated by a saddle point approximation . in order to do this , we have to consider the oscillatory behavior of the legendre function for large values of the variable @xmath317 . we use now the transformation formula 15.3.8 of @xcite and express the @xmath200-integration measure as a product of gamma functions . instead of eq . ( [ mellohndef ] ) we get then the modified expression @xmath332 with the abbreviation @xmath333 it is more convenient to replace @xmath317 by the new variable @xmath334 . a simple substitution then leads to the expression @xmath335 which is still exact . we mention that this expression is rather useful for an efficient numerical evaluation since the difficult oscillatory behavior of the legendre functions has been taken into account by a shift of the integration path into the complex plane . in the limits @xmath331 , @xmath336 , the integral can be done and we obtain up to corrections of the order @xmath337 or @xmath338 respectively @xmath339 the corresponding limit for the @xmath309 in eq . ( [ melloqndef ] ) is @xmath340 the density @xmath341 and the two point function @xmath342 for the variable @xmath343 can be obtained from eq . ( [ mellocorrfunc ] ) and eq . ( [ mellokndef ] ) . the result ( containing additional factors @xmath344 or @xmath345 due to the jacobian of the variable transformation @xmath346 ) is @xmath347 where we have used the abbreviation @xmath348 for the gaussian distribution . we mention that for ( [ rz1result ] ) we have replaced the argument @xmath349 of the gamma functions and of the polynomial @xmath321 in eq . ( [ hmapprox ] ) by the most probable values @xmath350 or @xmath351 due to the gaussian factors . we recover the simple sum of gaussian distributions with mean values @xmath352 and a variance @xmath353 already obtained by a direct simplification of the fokker planck equation valid in the localized limit@xcite . the above calculation shows that the general expressions ( [ mellocorrfunc ] ) , ( [ melloknexplizit ] ) of sec . [ section:3a ] are indeed consistent with known results concerning the quasi-@xmath330 insulators . we describe now the differences between transmission through two ballistic cavities coupled by a narrow constriction ( section [ section:2add ] ) and transmission through an homogeneously disordered wire of constant transverse section ( section [ section:3 ] ) . this can be of practical interest for many purposes which are not restricted only to mesoscopic electrical conductances , but which could be relevant for other systems as wave guide communication lines . one can imagine that the signal from an antenna to a receiver is anomalously weak and that the radio - engineering problem is to know if the weakness of the signal is due to an homogeneous deterioration of the transmission line or only to a local deterioration resulting from an accidental narrow constriction . characteristic behaviors of the @xmath2-brownian motion ensemble are first given , characterizing the coupled ballistic dots under a small applied magnetic field . in fig . 2 , the density of the variable @xmath354 is shown for the one channel case ( @xmath355 ) at four different values of @xmath356 ( @xmath111 is the typical time scale introduced in section [ section:2b ] ) . the variables @xmath357 are the radial parameters of the transfer matrix , which are usual in random transfer matrix theory @xcite . the curves are calculated from eqs . ( [ corr1]),([knexp ] ) . the figure 3 gives this density for the same values of @xmath13 but for @xmath358 channels . in addition for @xmath359 the asymptotic expression ( [ semicircle2 ] ) that arises from the laguerre approximation is shown , too . the latter is valid if @xmath236 and @xmath360 . in both figures one can see that the typical values for @xmath361 increase ( on a logarithmic scale ) if @xmath13 decrease . one can identify a rather sharp minimum value @xmath362 for @xmath361 if @xmath360 . a simple estimation from eq . ( [ semicircle1 ] ) yields @xmath363 for values @xmath364 the distribution is rather broad and the overall form is ( nearly ) independent of @xmath1 ( apart from the case @xmath365 ) . a reduction of the time leads essentially to a translation of the density . for finite values of @xmath1 ( see @xmath358 ) , one can see typical but rather small oscillations around the limiting distribution for @xmath10 . the value @xmath366 corresponds within the accuracy of the plots to the stationary limit @xmath367 , i.e. the single cue behavior . for this case the position of the first maximum decreases and becomes sharper when @xmath1 increases . this gives rise to a conductance @xmath368 ( cp . ( [ gaverage ] ) ) . the corresponding behaviors of the @xmath8-brownian motion ensemble exhibit essential differences from the previous case . 4 shows the @xmath361-density for the disordered wire with @xmath358 channels in three different regimes , i.e. the localized regime ( @xmath369 ) , the crossover regime ( @xmath370 ) and the metallic regime ( @xmath371 ) . 5a - c contain the same density curves in a more readable scale than in fig . 4 . the gaussian approximation ( [ rz1result ] ) ( dotted line ) which fits very well the exact density at @xmath369 is indicated in fig . 5c .. used in section [ section:3c ] is only approximately equal to @xmath361 ( i.e. @xmath372 when @xmath373 ) ] the results shown in the figs . 4 , 5 were obtained by a numerical evaluation of the integral ( [ hfuncsub ] ) except for the limits @xmath244 , @xmath374 where the integral ( [ mellohndef ] ) is better suited . the substitution @xmath375 was of course exactly considered . one can see that the @xmath361-density of the disordered line differs essentially from the density shown in fig . 3 for the two weakly coupled cavities . for small transmission ( i.e. the localized limit ) , the @xmath361-density is mainly concentrated around maximal values @xmath376 ( @xmath277 ) with a relative variance @xmath377 . between the maxima , the density nearly vanishes . this effect can be seen in fig . 5c and is still increased if the ratio @xmath378 is increased . but in the crossover regime @xmath379 ( fig . 5b ) , the minima are no longer negligible compared to the maxima . in fig . 5a ( metallic regime ) , the amplitude of the finite @xmath1 oscillations are small but still clearly observable . further calculations for different values of @xmath1 show that the amplitude of the oscillations remains unchanged if @xmath1 is increased for a fixed ratio @xmath378 . in order to reach a metallic regime where the density is really a smooth function of @xmath361 , one has to increase @xmath1 to very high values and at the same time to decrease @xmath378 ( within the limit @xmath380 ) . finally , we compare @xmath6 and of @xmath7 between the two cases in the limit of small transmission , i.e. @xmath360 for the @xmath4cue @xmath5 cue crossover or @xmath381 for the disordered line . for the two weakly coupled cavities , we get in the lowest order of @xmath235 from eqs . ( [ gaverage ] ) , ( [ gautocorr ] ) @xmath382 where @xmath383 is the effective number of coupling channels introduced in section [ section:1 ] . these results lead to the limit @xmath384 for the relative variance of @xmath95 . the corresponding expressions for the quasi one - dimensional line in the localized limit @xmath381 can be obtained from eqs . ( [ melloavcond3 ] ) , ( [ gderiv ] ) by an evaluation of the @xmath200-integral in a saddle point approximation @xmath385 @xmath386 the relative variance of @xmath95 is now infinite and the infinite length limit is characterized by @xmath387 the comparison with ( [ glimit1 ] ) shows again a qualitative difference in the statistics of the conductance between the two cases . ( [ gslok ] ) and ( [ deltaglok ] ) illustrate the well known fact that the average of the conductance does not significantly characterize the conductance properties in the localized limit . on the other hand , the limit ( [ glimit1 ] ) for the two weakly coupled ballistic cavities is a small quantity for a large number of channels . the @xmath2 and @xmath8 brownian motion ensembles which we have used are characterized by two fokker - planck equations expressing isotropic diffusion equations on the compact space of unitary matrices ( @xmath2 ) or on a non compact symmetric space of pseudo - unitary transfer matrices @xcite respectively . the differential operators which appear in the fokker - planck equations are just the radial part of the laplace - beltrami operators of the corresponding spaces , describing free isotropic diffusion on those curved spaces . in the case of the @xmath8-brownian motion ensemble , the variable @xmath361 is the good coordinate since it measures in some way the geodesic distance on the space of pseudo - unitary matrices ( cp . @xcite ) . fig . 4 shows very clearly the free diffusion of @xmath361 . since the range of @xmath361 is non compact the rather strong level repulsion gives well separated maxima . for the compact space of unitary matrices , it is the variable @xmath49 used in the sections [ section:2a ] and [ section:2b ] which measures the geodesic distance . the range for @xmath49 is compact and diffusion leads after a long time to an essentially uniform distribution . the level repulsion only causes some small oscillations around this distribution . apart from the ( important ) differences which we have reviewed , we can also notice nice mathematical similarities between the two fokker - planck approaches . many formulas and results of section [ section:2 ] coincide with the results of ref . @xcite and section [ section:3 ] , after replacing the variables @xmath50 by the @xmath388 ( @xmath389 in @xcite ) and @xmath390 by @xmath391 . this is partly related to the similarity of the parametrization ( [ par ] ) with the corresponding parametrization of the transfer matrix@xcite ( with the substitution @xmath392 ) . this occurs only for the unitary case ( @xmath18 ) . for the orthogonal and unitary cases ( @xmath53 ) this similarity does not exist , since the unitary matrices @xmath35 , @xmath34 in ( [ par ] ) are related in a different way than for the transfer matrix @xcite by time reversal symmetry . our results ( eqs . ( [ gsmallt])-([glimit2 ] ) are valid for an arbitrary number @xmath393 of channels . in the case of a single channel , the quotient of the gamma functions takes the value @xmath394 and eq . ( [ gslok ] ) becomes identical with the result of ref . @xcite obtained using a supervector model for the one - dimensional white noise potential at small disorder . indeed , it was known that the one - dimensional white noise model for small disorder leads @xcite to the same fokker - planck equation as the one of refs . @xcite for one channel . in the opposite limit @xmath10 , the gamma function quotient reduces to one and we recover the corresponding expression of ref . @xcite obtained by the supersymmetric non linear @xmath11-model : the expressions ( [ melloavcond3 ] ) , ( [ mellofluct ] ) for @xmath6 and @xmath395 coinciding exactly for all length scales @xmath396 with those of ref . @xcite . the non linear sigma model for quasi one - dimensional disordered conductors can at least be derived assuming three different microscopic hamiltonians , i.e. the white noise potential @xcite , wegner s @xmath1 orbital model @xcite , and a random banded matrix hamiltonian @xcite . it is then very remarkable that in the unitary case , these microscopic approaches become equivalent to the @xmath8-brownian motion ensemble of ref . this behavior is apparently not true for the symplectic case , since the non - linear sigma model predicts @xcite finite values of @xmath6 and @xmath7 in the localized regime , while the fokker - planck equation gives an exponential decrease of the typical conductance for each @xmath397@xcite . in principle , it could be still possible that in the large @xmath1 limit the average @xmath398 takes a non vanishing value because of some very subtle effects concerning the tails of the normal distribution of @xmath399 . but even this can not happen , as it can be easily seen from the generalization @xcite of eq . ( [ gderiv ] ) for arbitrary @xmath157 @xmath400 here @xmath401 denotes the localization length and @xmath402 is just the trace of the squared transmission matrix @xmath403 , i.e. @xmath404 . from the inequality @xmath405 one finds directly that in the limit @xmath406 the behavior @xmath407 ( as predicted in ref . @xcite for @xmath20 ) is not possible since @xmath395 must vanish , which yields directly @xmath408 . this contradiction when @xmath20 between the two approaches is surely worth to be considered in future work . though the two fokker - planck approaches are very similar from a mathematical point of view , they nevertheless describe two different physical situations where the fokker - planck time controls the crossover between systems with a high and a low conductance . for both approaches we have found ( rather ) closed expressions ( valid for a finite channel number ) for any @xmath29-point correlation functions of the transmission eigenvalues , allowing us to calculate the first and second moment of the conductance . this complete analytical solution is essentially restricted to the unitary case where both fokker planck approaches can be solved in terms of fermionic one particle green s functions . at the moment , the extension of this solution for arbitrary @xmath1 to the orthogonal and symplectic cases looks difficult , since the hamilton operator obtained after the sutherland transformation does not reduces to the sum of one particle operators . we thank b. l. altshuler and c. w. j. beenakker for helpful discussions . in particular , our understanding of the universality of the transmission eigenvalue correlation functions has been improved thank to stimulating remarks by c. w. j. beenakker . this work was supported in part by eec , contract no . scc - ct90 - 0020 . klaus frahm acknowledges the d.f.g . for a post - doctoral fellowship in this appendix , we derive the fokker - planck equation ( [ fp1 ] ) for the case @xmath18 . in order to do this , we consider the matrix @xmath409 where @xmath410 the matrix @xmath411 then has the form @xmath412 on the other hand the evolution equation @xmath413 implies @xmath414 with @xmath415+{\cal o}(\delta x^3)\quad.\ ] ] the matrix @xmath416 has the form @xmath417 where @xmath418 is an arbitrary complex @xmath36-random matrix with @xmath419 the matrix @xmath59 in ( [ pert1 ] ) contains still the unitary matrices @xmath64 and @xmath65 which can be taken into account by the replacement @xmath420 , this gives for @xmath421 the same statistics than those described by eq . ( [ stat ] ) . at this point , we can make an important remark concerning the case where the matrix @xmath57 obeys a more general distribution than that of eq . ( [ x_statistic ] ) . in this case , the property ( [ stat ] ) for the matrix @xmath418 is of course no longer valid . but let us assume that the two unitary matrices @xmath65 and @xmath64 are independently distributed according to the invariant measure for unitary @xmath36-matrices , i.e. @xmath65 and @xmath64 are described by two independent @xmath36 cues . then the average over these two unitary matrices yields just the distribution ( [ stat ] ) for the transformed matrix @xmath422 . the only information that we need from @xmath418 is therefore the invariant quantity @xmath423 which relates the time step @xmath14 with the perturbation @xmath57 or @xmath418 respectively . this proves that the fokker - planck equation derived below is also valid for a more general perturbation @xmath57 , provided that the two unitary matrices @xmath65 and @xmath64 are for each time @xmath13 independently cue distributed . the latter assumption is of course of crucial importance . in the case of an initial condition of one @xmath424 coe or two @xmath36 cues for @xmath425 , this assumption is at least valid at @xmath21 and we assume that it remains correct for arbitrary @xmath66 . we conclude that the fokker - planck equation derived for a perturbation @xmath57 obeying ( [ x_statistic ] ) , still holds if one can relate the fokker - planck time to the perturbation via ( [ dm_inform ] ) . since we are interested in the statistics of the @xmath426 in ( [ pert0 ] ) , we have to express the @xmath427 in terms of @xmath421 . this corresponds to an eigenvalue problem after the transformation @xmath428 which implies @xmath429 the matrix @xmath430 which appears in ( [ a10 ] ) and ( [ a11 ] ) is just the unitary matrix @xmath431 where @xmath432 is defined in ( [ uu_def ] ) . the matrix @xmath430 results only in a similarity transformation which does not change the eigenvalues . ( [ pert0 ] ) implies that @xmath433 has just the eigenvalues @xmath434 and @xmath435 , which can be obtained by second order perturbation theory @xmath436 the perturbing matrix @xmath437 in eq . ( [ pert2 ] ) is just @xmath438 eq . ( [ pert2 ] ) leads after some algebra to @xmath439 we insert eq . ( [ dmdef ] ) in eq . ( [ pert3 ] ) and apply the average ( [ stat ] ) for the matrix @xmath421 that results in @xmath440 with @xmath441 the corresponding fokker - planck equation has the form @xmath442 which is just eq . ( [ fp1 ] ) for @xmath18 because @xmath443 with @xmath108 given by eq . ( [ fbeta ] ) . b. d. simons and b. l. altshuler , phys . lett . * 70 * , 4063 ( 1993 ) ; b. d. simons , a. szafer and b. l. altshuler , pisma zh . eksp * 57 * , 268 ( 1993 ) [ jetp lett . * 57 * , 276 ( 1993 ) ] ; b. d. simons , p. a. lee and b. l. altshuler , nucl . phys . * b 409 * , 487 ( 1993 ) . c. m. marcus , a. j. rimberg , r. m. westervelt , p. f. hopkins , and a. c. gossard , phys . * 69 * , 506 ( 1992 ) ; c. m. marcus , r. m. westervelt , p. f. hopkins , and a. c. gossard , phys . b * 48 * , 2460 ( 1993 ) . 1 . two chaotic ballistic cavities coupled with a @xmath3 channel contact and each of them connected to an electron reservoir through a @xmath1 channel contact . 2 . density of the variable @xmath354 for the crossover @xmath4cue @xmath5 cue with @xmath355 at different values of @xmath444 . the curves are calculated from eqs . ( [ corr1 ] ) and ( [ knexp ] ) . 3 . density of the variable @xmath354 for the crossover @xmath4cue @xmath5 cue with @xmath358 at different values of @xmath444 . the curves are calculated from eqs . ( [ corr1 ] ) and ( [ knexp ] ) . in addition , for @xmath359 the laguerre approximation ( [ semicircle2 ] ) is included . 4 . density of the variable @xmath354 for the disordered line at three different length scales , @xmath445 . the curves are obtained from a numerical evaluation of eq . ( [ hfuncsub ] ) . same as fig . 4 for @xmath371 but with a modified scale . same as fig . 4 for @xmath370 but with a modified scale . same as fig . 4 for @xmath369 but with a modified scale . the gaussian approximation ( eq . ( [ rz1result ] ) , dotted line ) is included for comparison

the parametric correlations of the transmission eigenvalues @xmath0 of a @xmath1-channel quantum scatterer are calculated assuming two different brownian motion ensembles . the first one is the original ensemble introduced by dyson and assumes an isotropic diffusion for the @xmath2-matrix . we derive the corresponding fokker - planck equation for the transmission eigenvalues , which can be mapped for the unitary case onto an exactly solvable problem of @xmath1 non - interacting fermions in one dimension with imaginary time . we recover for the @xmath0 the same universal parametric correlation than the ones recently obtained for the energy levels , within certain limits . as an application , we consider transmission through two chaotic cavities weakly coupled by a @xmath3-channel point contact when a magnetic field is applied . the @xmath2-matrix of each chaotic cavity is assumed to belong to the dyson circular unitary ensemble ( cue ) and one has a @xmath4 cue @xmath5 one cue crossover when @xmath3 increases . we calculate all types of correlation functions for the transmission eigenvalues @xmath0 and we get exact finite @xmath1 results for the averaged conductance @xmath6 and its variance @xmath7 , as a function of the parameter @xmath3 . the second brownian motion ensemble assumes for the transfer matrix @xmath8 an isotropic diffusion yielded by a multiplicative combination law . this model is known to describe a disordered wire of length @xmath9 and gives another fokker - planck equation which describes the @xmath9-dependence of the @xmath0 . an exact solution of this equation in the unitary case has recently been obtained by beenakker and rejaei , which gives their @xmath9-dependent joint probability distribution . using this result , we show how to calculate all types of correlation functions , for arbitrary @xmath9 and @xmath1 . this allows us to get an integral expression for the average conductance which coincides in the limit @xmath10 with the microscopic non linear @xmath11-model results obtained by zirnbauer et al , establishing the equivalence of the two approaches . we review the qualitative differences between transmission through two weakly coupled quantum dots and through a disordered line and we discuss the mathematical analogies between the fokker - planck equations of the two brownian motion models . # 1

pair - density - wave ( pdw ) superconducting order has emerged as a realistic candidate for order in the charge - ordered region of the pseudogap phase of the cuprates near one - eighth filling . it naturally accounts for both superconducting ( sc ) correlations and for static quasi - long - range charge - density - wave ( cdw ) order observed near this hole doping and at temperatures below approximately 150 k @xcite , and it can explain observed signatures of broken time - reversal symmetry @xcite . moreover , pdw can lead to the quantum oscillations seen in the cuprates @xcite and can also explain anomalous quasi - particle properties observed by angle - resolved photoemission ( arpes ) measurements @xcite . in addition , numerical simulations of theories of a doped mott insulator reveal pdw order to be a competitive ground - state to @xmath0-wave superconductivity @xcite . it is therefore important to find experiments that can identify pdw order in the cuprates . motivated by the observation of checkerboard cdw order inside @xmath0-wave vortex cores by scanning tunneling microscopy ( stm ) @xcite and by nuclear magnetic resonance ( nmr ) @xcite , we examine the competition between @xmath0-wave and pdw superconductivity in applied magnetic fields . previous theoretical studies of competing orders in a magnetic field have emphasized competing spin - density - wave ( sdw ) order @xcite , cdw order @xcite , and staggered flux phases @xcite with @xmath0-wave superconductivity . competing pdw and @xmath0-wave order has not been extensively studied ( note that superconducting phase disordered pdw competing with @xmath0-wave order has been examined @xcite ) . here , we find that inside the vortex cores of @xmath0-wave superconductivity , pdw order drives the observed checkerboard cdw order and , in conjunction with @xmath0-wave superconductivity , it also drives an additional cdw order that appears in a ring - like region outside the vortex cores . this additional cdw order has twice the period of the observed checkerboard cdw order and serves as a smoking gun for pdw order . to in the following , we develop a phenomenological theory for competing pdw and @xmath0-wave superconductivity , sketched in fig . [ fig : htdiagram ] . we assume that in zero field , only @xmath0-wave superconductivity appears at the expense of the pdw order . the pdw order can only appear when the @xmath0-wave order is weakened by the external field . this is followed by an analysis of the core structure of a single @xmath0-wave vortex , where we show that pdw order appears inside these cores , without any phase winding , generating the cdw order discussed above . finally , we examine the behavior of this competing system as the field is further increased and identify a transition at which pdw order develops phase coherence and forms a vortex phase . at the mean - field level , pdw order simultaneously breaks gauge invariance and translational symmetry . fluctuations can lead to two separate transitions : one for which gauge symmetry is broken and one for which translational symmetry is broken @xcite . we argue that at high fields , the superconducting order is removed by phase fluctuations , leaving behind the cdw order seen through nmr experiments . to investigate the physics resulting from the @xmath1-@xmath2 phase diagram shown in fig . [ fig : htdiagram ] , we consider a model with competing @xmath0-wave and pdw superconductivity . the pdw order parameter is represented by a four component complex vector @xmath3 , defined as @xmath4 and the @xmath0-wave by one complex ( scalar ) field @xmath5 . for an external applied field @xmath6 , which we will take to be along the @xmath7-axis , @xmath8 , the ginzburg - landau free - energy density is @xmath9 where @xmath10 is the magnetic field and @xmath11 its vector potential . @xmath12 describes the pair - density - wave @xmath3 , and @xmath13 its coupling to the @xmath0-wave order that obeys @xmath14 with @xmath15 . symmetry arguments dictate that the free energy of the pdw has the following structure @xcite : @xmath16 here , we neglect variations along the @xmath7-axis , thus @xmath17 is the spatial index , while @xmath18 is a wave - vector index : @xmath19 . in the following , another convenient index @xmath20 , will also be used . the coefficients @xmath21 of the kinetic term satisfy the following relation @xmath22 and @xmath23 , and @xmath24 measures the anisotropy of the system . has very little influence on the physics we describe here . we verified that indeed for @xmath25 , moderate anisotropies do not qualitatively change the physical properties we discuss . thus in the rest of the paper , we consider only the isotropic case @xmath26 . ] here @xmath27 represents the wavevector @xmath28 , @xmath29 represents @xmath30 , and @xmath31 represents the gap associated with the pairing between the fermion states @xmath32 and @xmath33 , where @xmath34 is the momentum and @xmath35 , @xmath36 denote the spin - states . our choice of the wavevectors and model for the pdw order is motivated by the recent proposal of amperean pairing by p.a . lee @xcite , for which it has been shown that pdw order can account for both the anomalous quasi - particle properties observed by arpes and the cdw order ( at momenta 2@xmath37 and 2@xmath38 ) observed in the pseudogap phase of bi@xmath39sr@xmath40la@xmath41cuo@xmath42 ( bi2201 ) . depending on the parameters @xmath43 , the free energy of the pdw sector allows five possible distinct ground - states @xcite . we choose parameters such that , in the non - competing case , the pdw ground - state has the form @xmath44 . this pdw ground - state is the same as that proposed in ref . and is also found to be a ground - state in the spin - fermion model @xcite . both @xmath5 and @xmath3 interact with the magnetic field ( through the kinetic terms ) and are therefore indirectly coupled . they also directly interact through @xmath13 : @xmath45{\delta_d}^2 + c.c.\right)\,.\end{aligned}\ ] ] the first term in is a bi - quadratic coupling between the @xmath0-wave and the pair - density - wave @xmath46 . the coexistence of both order parameters is penalized for positive values @xmath47 , and when strong enough , only one of the condensates supports a nonzero ground - state density . our choice of parameters is such that when @xmath48 , @xmath5 has lower condensation energy and @xmath3 is completely suppressed , because of the interaction terms . moreover , as cdw order emerges at high field , we require @xmath3 to have a higher second critical field ( @xmath49 ) than @xmath5 ( @xmath50 ) . these conditions lead to fig . [ fig : htdiagram ] . we note that in principle , the existence of the competing pdw order can allow for the pdw driven cdw order to appear in zero field in the vicinity of inhomogeneities or due to fluctuations in some materials . indeed cdw order has been observed in yba@xmath39cu@xmath51o@xmath52 in zero field through high - energy x - ray diffraction @xcite ( this cdw order is enhanced by magnetic fields ) . we take cdw order to be denoted by @xmath53 ( note that @xmath54 ) . the coupling between @xmath55 ( with @xmath20 ) and pdw order is given by @xcite : @xmath56 assuming that the cdw order is induced by the pdw order , we find that @xmath57 the cdw order given by @xmath55 corresponds to that observed in the pseudogap phase in zero field and to the checkerboard order observed inside the @xmath0-wave vortex cores . an important feature of this work is that the interplay between @xmath0-wave and pdw orders gives rise to an additional contribution to the cdw order . in particular , this coupling is given by @xcite @xmath58 \nonumber\\ & + \rho_{-q}[\delta_{q}\delta_d^*+\delta_{-q}^*\delta_d ] \big ) \,.\end{aligned}\ ] ] differentiation with respect to @xmath59 and @xmath60 yields the relations ( this also assumes the cdw order is purely induced ) : @xmath61 the contributions @xmath62 and @xmath63 to the cdw are reconstructed according to @xmath64 which shows the @xmath65th - order contribution to the cdw . the cdw order @xmath62 has twice the periodicity of @xmath63 and is not an induced order of the pure @xmath3 : it only appears when both @xmath5 and @xmath3 coexist . consequently , @xmath62 is a signature of the appearance of @xmath3 in a @xmath0-wave superconductor . note that the existence of @xmath62 requires superconducting phase coherence for both the pdw and @xmath0-wave orders ( strictly speaking , coherence in the phase difference between these two orders will suffice ) . we note that an observation of @xmath62 has been reported @xcite , and below we make predictions about the structure of @xmath62 around a vortex in @xmath5 . in order to investigate the interplay of @xmath3 and @xmath5 , within the framework sketched in fig . [ fig : htdiagram ] , we numerically minimize the free energy both for single vortices and for a finite sample in external field . the theory is discretized within a finite element formulation @xcite and minimized using a nonlinear conjugate gradient algorithm ( for detailed discussion on the numerical methods , see , for example , @xcite ) . to typical single vortex solutions ( see fig . [ fig : vortex ] ) clearly show that the components of the pdw order acquire small , yet nonzero density at the center of the @xmath0-wave vortex core . as a result , the cdw order is also nonzero at the vortex core . far from the vortex , the @xmath3 decays to zero , and the induced cdw is suppressed as well . [ fig : checkerboard ] shows the magnitude of the total cdw order as well as the contributions from different orders in @xmath66 . here , we used the values @xmath67 and @xmath68 , where @xmath69 is the cu - cu distance in cuprates and , in qualitative accordance with experimental data @xcite , we take the @xmath0-wave coherence length to be @xmath70 . @xmath63 forms a checkerboard pattern that extends significantly outside the vortex core , and this is consistent with the observations . to in addition to this checkerboard order , we also find that @xmath62 , which varies at twice the wave - length of @xmath63 , is nonzero and also has a non - trivial structure . more precisely , at the singularity in the @xmath0-wave , @xmath71 , and when @xmath72 becomes nonzero , @xmath62 also becomes nonzero . since @xmath3 exhibits no phase winding , @xmath73 inherits the phase winding of @xmath5 . a phase winding in @xmath62 implies a dislocation in the corresponding real - space order @xcite . consequently , the cdw order associated with @xmath62 has a dislocation at the vortex core . since @xmath62 is suppressed in vortex cores , the checkerboard pattern that appears there , is essentially due to @xmath63 . the contribution of @xmath62 to the cdw becomes important at distances larger than @xmath74 . moreover , as it varies with a doubled wave - length , every other charge peak is magnified in a region outside the core . note that away from the vortex , @xmath62 is suppressed at a much slower rate than @xmath63 . furthermore , if @xmath62 is observable at all , then it should vanish at @xmath50 , while @xmath63 , will persist to much higher fields . to investigate the evolution of the pdw and @xmath0-wave orders in external field @xmath6 , for parameters corresponding to fig . [ fig : htdiagram ] , we minimize the free energy , while imposing @xmath75 at the ( insulating ) boundary of the domain . we follow the vertical line sketched in fig . [ fig : htdiagram ] . that is , starting from @xmath48 , the field is sequentially increased after the solution for the current value of @xmath1 is found . typical results illustrating such a simulation are shown in fig . [ fig : magnetization ] . to in low fields , only @xmath5 has a nonzero ground - state density and , as a result of the competition with @xmath5 in the interacting terms , @xmath3 is fully suppressed ( or vanishingly small ) . above the first critical field , vortices in @xmath5 , carrying a small amount @xmath3 in their core , start entering the system . the averaged pdw over the whole sample @xmath76 is still vanishingly small . with increasing field , the density of vortices increases and they start to overlap . that is , @xmath77 and @xmath78 do not have enough room " to recover their ground - state values . at this point , the lumps of @xmath3 , previously isolated in vortex cores , interconnect and @xmath3 acquires a phase coherence globally . this behavior was also found to occur in a similar system with competing orders @xcite . at this phase transition , not only does @xmath76 become nonzero , but the induced cdw @xmath79 and @xmath80 , also become nonzero on average ( see fig . [ fig : magnetization ] ) . we conjecture that this phase transition is related to that seen though nmr @xcite . when the pdw order is on average nonzero , energetic considerations dictate that it should acquire phase winding as well . indeed , when two condensates have nonzero density , the energy of configurations that has winding in only one condensate diverges ( at least logarithmically ) with the system size . as a result vortices in @xmath81 are created when @xmath82 @xcite . note that as it is still beneficial to have nonzero @xmath3 inside the vortex cores of @xmath5 , the singularities that are formed due to the winding in @xmath81 do not overlap with those of @xmath5 [ and they do not overlap with each other due to the terms @xmath43 in , which favor core splitting ] . thus , the cdw order still appears within the vortex cores of @xmath5 . since all the vortices that are created do not overlap with each other , the magnetic induction is smeared out and is much more spatially uniform than in usual vortex phases . for fields above the second critical field of @xmath5 , only the pdw order survives . as a result , the contribution @xmath73 to the induced cdw also vanishes and the observed cdw order above @xmath50 is solely that induced by the pdw ( that is @xmath63 ) . in this state , at the mean - field level , the vortices in @xmath81 do not overlap , as the terms with @xmath43 in favor vortex core splitting . in principle , the parameters @xmath43 can also be chosen so that the @xmath81 cores coincide for some or all pdw components . this will not change the qualitative physics associated with the competition between @xmath5 and @xmath3 . however , it will affect the resulting high - field regime . in either case , we expect superconducting phase fluctuations to play an important role in the high - field phase . in particular , it is known that for type - ii superconductors , high magnetic fields significantly enhance the role of fluctuations @xcite . phase fluctuations will remove the superconducting long - range order of the pdw state , but the cdw order can still survive @xcite . a related mechanism was also considered in a different but related model of superconductivity @xcite . we have considered a model of competing pair - density - wave and @xmath0-wave superconductivity . the superconducting state in the meissner phase is purely @xmath0-wave . with increasing external field , vortices in the @xmath0-wave superconductor are formed and they carry pdw and induced cdw order in their core . when these vortices significantly interact , the lumps of pdw order acquire global phase coherence and both pdw and @xmath0-wave superconductivity coexist . in the regions where both pdw and @xmath0-wave order exist , the induced cdw order features a @xmath73 contribution that exists at twice the periodicity of the cdw order observed in the pseudogap phase at zero fields . the observation of @xmath73 can serve to identify the existence of pdw order in the pseudogap phase . we thank egor babaev , andrey chubukov , marc - henri julien , manoj kashyap , patrick lee , and yuxuan wang for fruitful discussions . dfa acknowledges support from nsf grant no . jg was supported by national science foundation under the career award dmr-0955902 and by the swedish research council grants 642 - 2013 - 7837 , 325 - 2009 - 7664 . the computations were performed on resources provided by the swedish national infrastructure for computing ( snic ) at the national supercomputer center at linkping , sweden . 41ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1126/science.1223532 [ * * , ( ) ] link:\doibase 10.1126/science.1242996 [ * * , ( ) ] link:\doibase 10.1126/science.1243479 [ * * , ( ) ] @noop link:\doibase 10.1038/nphys999 [ * * , ( ) ] link:\doibase 10.1088/1367 - 2630/11/11/115004 [ * * , ( ) ] link:\doibase 10.1103/physrevx.4.031017 [ * * , ( ) ] link:\doibase 10.1088/1742 - 6596/449/1/012012 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.127002 [ * * , ( ) ] link:\doibase 10.1126/science.1198415 [ * * , ( ) ] link:\doibase 10.1038/416610a [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.047003 [ * * , ( ) ] link:\doibase 10.1103/physrevb.91.054502 [ * * , ( ) ] link:\doibase 10.1103/physrevb.86.104507 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.113.046402 [ * * , ( ) ] link:\doibase 10.1126/science.1066974 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.95.257005 [ * * , ( ) ] link:\doibase 10.1038/nature10345 [ * * , ( ) ] http://dx.doi.org/10.1038/ncomms3113 [ * * , ( ) ] link:\doibase 10.1103/physrevb.66.094501 [ * * , ( ) ] link:\doibase 10.1103/physrevb.69.144504 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.054511 [ * * , ( ) ] link:\doibase 10.1103/physrevb.63.224517 [ * * , ( ) ] link:\doibase 10.1103/physrevb.74.104506 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.93.187002 [ * * , ( ) ] link:\doibase 10.1038/nphys1389 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.035149 [ * * , ( ) ] link:\doibase 10.1103/physrevb.91.115103 [ * * , ( ) ] link:\doibase 10.1038/nphys2456 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/87/37005 [ * * , ( ) ] link:\doibase 10.1515/jnum-2012 - 0013 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.064509 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.79.353 [ * * , ( ) ] http://www.cambridge.org/se/academic/subjects/physics/condensed-matter-physics-nanoscience-and-mesoscopic-physics/principles-condensed-matter-physics[__ ] ( , ) link:\doibase 10.1103/physrevb.91.014510 [ * * , ( ) ] link:\doibase 10.1038/nphys2502 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.28.1025 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.66.1125 [ * * , ( ) ] link:\doibase 10.1016/j.nuclphysb.2004.02.021 [ * * , ( ) ]

we consider competing pair - density - wave ( pdw ) and @xmath0-wave superconducting states in a magnetic field . we show that pdw order appears in the cores of @xmath0-wave vortices , driving checkerboard charge - density - wave ( cdw ) order in the vortex cores , which is consistent with experimental observations . furthermore , we find an additional cdw order that appears on a ring outside the vortex cores . this cdw order varies with a period that is twice that of the checkerboard cdw and it only appears where both pdw and @xmath0-wave order co - exist . the observation of this additional cdw order would provide strong evidence for pdw order in the pseudogap phase of the cuprates . we further argue that the cdw seen by nuclear magnetic resonance at high fields is due to a pdw state that emerges when a magnetic field is applied .

the inset zooms in the region @xmath97 , revealing the existence of the two boundary transtions at b1 and b2 . ] the boundary transitions inside the ( ih , p ) phase are peculiar in 1d , since edge spins are separated by a macroscopic distance , and the only way to communicate between them is through the bulk from which they effectively decouple . to prove that we are dealing with a boundary phenomenon we compare the excitation gaps for open and periodic boundary conditions . one can clearly see from fig . [ fig : gaps ] that low - lying states below the bulk modes develop for open boundary conditions . another illustration of this boundary transition is provided by fig . [ fig : edgespins ] , which shows the behavior of the first excited states in the @xmath98 and @xmath99 sector . to describe the physics of this transition at the qualitative level , it is instructive to consider the limit of large @xmath26 . in the strong @xmath26 limit one can integrate out orbital degrees of freedom to obtain an effective spin-@xmath0 model . in the leading order in @xmath100 , its hamiltonian has the form of a @xmath101-@xmath102 model with modified first and last nearest - neighbor links : @xmath103 where @xmath104 one can see that with increasing @xmath25 , the boundary link strength @xmath105 goes through zero at some point and changes its sign to a ferromagnetic coupling . this effectively creates impurity spins attached ferromagnetically at the ends of the spin-@xmath0 chain . interaction between the end spins is mediated by the bulk . for an even number of sites the effective interaction is antiferromagnetic , whereas for odd number of sites the effective interaction between the end spins is ferromagnetic . the second boundary transition , @xmath106 , is similar in nature to the first one , but now the last two spins decouple from the bulk creating an effective spin-@xmath107 localized at each boundary and ferromagnetically attached to the anitferromagnetic spin-@xmath0 chain . the interaction between the spin-1 edge impurities is antiferromagnetic for even number of sites and ferromagnetic for odd number of sites . the lowest excitations are boundary excitations : a boundary triplet with total spin @xmath108 and a little higher boundary quintet with spin @xmath109 . and @xmath99 sectors , at two points of the @xmath59 phase diagram , along the @xmath49 line : ( a ) at a point between the b1 and b2 boundary transition lines , the @xmath98 excitation is localized at the system edges , while the @xmath99 excitation belongs to the bulk ; ( b ) at another point between the b2 line and the boundary of the fsol phase , both @xmath98 and @xmath99 excitations are localized at the edges.,title="fig : " ] and @xmath99 sectors , at two points of the @xmath59 phase diagram , along the @xmath49 line : ( a ) at a point between the b1 and b2 boundary transition lines , the @xmath98 excitation is localized at the system edges , while the @xmath99 excitation belongs to the bulk ; ( b ) at another point between the b2 line and the boundary of the fsol phase , both @xmath98 and @xmath99 excitations are localized at the edges.,title="fig : " ]

two - component dipolar fermions in zigzag optical lattices allow for the engineering of spin - orbital models . we show that dipolar lattice fermions permit the exploration of a regime typically unavailable in solid - state compounds that is characterized by a novel spin - liquid phase with a finite magnetization and spontaneously broken su(2 ) symmetry . this peculiar spin liquid may be understood as a luttinger liquid of composite particles consisting of bound states of spin waves and orbital domain walls moving in an unsaturated ferromagnetic background . in addition , we show that the system exhibits a boundary phase transitions involving non - local entanglement of edge spins . _ introduction. _ frustrated spin systems provide a wealth of novel phenomena , both at the classical and quantum levels @xcite . frustration becomes particularly important in low - dimensional systems , where quantum and thermal fluctuations are strongly enhanced and long - range order is suppressed . one of the most interesting frustration - inducing mechanisms involves the interaction of spins with orbital degrees of freedom @xcite , which may result in spin - liquid states that lack long - range magnetic order @xcite . however , in solid state systems controlling the strength of spin - orbital interactions is hardly possible , limiting the exploration of spin - liquid phases . several recent works @xcite have shown that ultra - cold atomic gases in optical lattices can serve as quantum simulators of spin - orbital models , providing the required freedom of controlling the effective interactions by tweaking the optical lattice or by using feshbach resonances . moreover , rapidly - developing experimental techniques make possible to study the physics of higher energy bands , and to exploit orbital degeneracy @xcite . in particular , it has been recently shown @xcite that spin - orbital models of the kugel - khomskii type @xcite , relevant in transition metal oxides @xcite , can be realized in systems of dipolar spin-@xmath0 fermions loaded in doubly - degenerate @xmath1-bands of optical zigzag lattices . for comparable on - site intra - orbital repulsion @xmath2 and inter - orbital repulsion @xmath3 , which is the typical situation in solid - state scenarios @xcite , it was shown that dipolar fermions have a rich ground state phase diagram containing states with ferromagnetic ( fm ) , antiferromagnetic ( af ) , dimerized and quadrumerized spin order @xcite . spin liquid phases are however absent in this regime . interestingly , contrary to the usual case in solid - state systems , a large ratio @xmath4 may be attained for the case of dipolar fermions in zig - zag lattices by properly controlling the ratio between dipolar and contact interactions . in this letter we show that for @xmath5 the ground state diagram contains a novel spin - orbital liquid phase with a finite magnetization . this phase has a spontaneously broken su(2 ) spin symmetry and algebraically decaying longitudinal spin correlations , while the orbital correlations decay exponentially . the mechanism driving the transition into this phase is given by the softening of composite excitations formed by bound states of spin waves and orbital domain walls . we support our analytical arguments by numerical results obtained by means of the density matrix renormalization group ( dmrg ) technique @xcite . in addition , for open boundary conditions we observe peculiar boundary phase transitions that involve the formation of edge spins that decouple from the bulk and get non - locally entangled . ) ; ( b ) phase diagram of the model for @xmath6 ; ( c ) sketch of the kink - magnon bound state in the ( f , af ) phase , where @xmath7 denotes the effective spin exchange on the corresponding link . a magnon binds only to the orbital kink , but not to the antikink . ] _ spin - orbital model. _ we consider two - component ( pseudo - spin-@xmath8 ) fermions loaded in doubly degenerate @xmath1-bands of a zig - zag lattice ( see fig . [ fig : pathkink](a ) , details on the experimental implementation of this system can be found in ref . @xcite ) . the energy scales that determine the system are the nearest - neighbor ( nn ) hopping @xmath9 between equal orbitals , the average on - site repulsion energies @xmath2 ( @xmath3 ) between same ( different ) orbitals , the hund coupling @xmath10 , and an in - plane deformation of the optical lattice , distorting the xy rotational symmetry of a single - site potential that mixes the orbitals within the same site with an amplitude @xmath11 . in the mott insulator regime ( one fermion per site and strong coupling @xmath12 ) , the system is described by an effective spin - orbital hamiltonian ( for details of the derivation we refer to ref . @xcite ) : @xmath13 where @xmath14 are spin-@xmath0 operators acting on the lattice site @xmath15 , and @xmath16 are the pauli matrices describing the orbitals . the parameters of the model , in the leading order in @xmath17 , are given by @xmath18 , @xmath19 , @xmath20 , and the hamiltonian ( [ ham-2d ] ) has overall units of @xmath21 . the dipole - cipole coupling is crucial because for purely contact interaction there is no repulsion between fermions in the spin - triplet state , and hence the mott phase with one particle per site could not stabilize . _ analytical estimates. _ figure [ fig : pathkink](b ) shows the phase diaram for the case @xmath6 , for which the orbitals are classical . the @xmath22 region is dominated by a phase with fm spin order and af orbital order , which we label ( f , af ) following the notation of ref . @xcite . a smaller @xmath23 favors the spontaneously dimerized ( d , f ) state , with the spin sector described by a product of singlets on even ( odd ) nn bonds and ferromagnetic orbitals . on the line @xmath24 , @xmath6 , @xmath22 the spins are fully decoupled , whereas adding infinitesimally small @xmath25 ( @xmath26 ) favors fm ( af ) spin exchange . this competition between @xmath26 and @xmath25 leads to a first - order transition from the ( f , af ) phase to the ( ih , af ) phase where the spin sector behaves as an isotropic heisenberg antiferromagnet , and the orbitals retain af order . for @xmath27 this transition line can be easily estimated by computing the leading order correction in @xmath26 to the energy @xmath28 of a magnon in the ( f , af ) state . for small momenta @xmath29 , one obtains @xmath30 , collapsing at @xmath31 . a further increase of @xmath26 at fixed small @xmath25 eventually leads to an ising transition in the orbital sector , bringing the system into the ( ih , p ) phase with paramagnetic orbitals . the ( f , af ) ground state factorizes into a product of spin and orbital wave functions , so there is a purely orbital ising - type transition from the ( f , af ) phase to the ( f , p ) phase where orbitals are paramagnetic ( disordered ) and spins remain fully polarized ; the transition line thus can be obtained exactly as @xmath32 . however , there is another , previously unknown , instability of the ( f , af ) phase which is of crucial interest here . this instability can be traced down to the fact that in the ( f , af ) phase magnons tend to bind to kinks in the orbital order ( see fig . [ fig : pathkink](c ) ) . if a kink - antikink pair is excited on top of the ( f , af ) state , on the link at the kink position the effective exchange @xmath7 changes from ferromagnetic ( @xmath33 in zeroth order in @xmath26 ) to antiferromagnetic ( @xmath34 ) , acting as an impurity which can bind a magnon . there is another impurity link with @xmath35 at the antikink position , but it does not support bound states . to the leading order in @xmath26 , the energy of the kink - antikink pair with a magnon bound to the kink is @xmath36\cos p-\cos k \big\},\end{aligned}\ ] ] where @xmath1 and @xmath29 are the kink and antikink momenta , respectively . the lower edge of this continuum is achieved at @xmath37 , @xmath38 , i.e. , when the magnon is essentially a propagating singlet dimer . this excitation softens at @xmath39 . hence , for @xmath40 a novel phase is expected with a finite density of composite kink - dimer particles `` floating '' in the ferromagnetic background @xcite . an infinitesimal density of moving kinks and antikinks immediately suppresses the orbital order , so the orbital af order parameter experiences a jump at the transition . indeed , the ( f , af ) product wave function remains an exact eigenstate all the way up to @xmath41 , and @xmath42 remains smaller than the ising transition value @xmath43 in a finite range of @xmath25 . the ferromagnetic order in spins is retained , but the magnetization is no more fully saturated . . symbols denote numerical results ( solid and dotted lines are a guide to the eye ) , whereas dashed lines correspond to the analytical estimates @xmath44 and @xmath45 . curves b1 and b2 mark the boundary phase transitions that involve non - local entanglement between edge spins @xmath46 and @xmath47 . the inset shows the ground - state correlation between edge spins @xmath48 as a function of @xmath25 for @xmath49.,scaledwidth=48.0% ] in the novel phase mentioned above , the su(2 ) symmetry in the spin sector remains spontaneously broken , exactly as in the ( f , af ) phase , but the ground state belongs to a degenerate multiplet with some spin @xmath50 , where @xmath51 is the total number of particles ( @xmath52 at unit filling considered here ) . this phase is expected to have _ two _ branches of gapless excitations , one with a quadratic dispersion at small momenta ( `` spin '' mode , ferromagnetic magnons ) , and the other with a linear dispersion ( `` charge '' mode , sound waves in the luttinger liquid of kink - dimer particles ) . this resembles the situation found in spin-@xmath0 bose gas , where such a spin - charge separation has been found both in the 1d @xcite and 2d @xcite cases . since the longitudinal spin correlator is related to the kink - dimer density fluctuations , it must decay algebraically on top of the long - range order . this highly unusual phase can be called a ferromagnetic spin - orbital liquid ( fsol ) @xcite . and @xmath49 : ( a ) magnetization curve for different system sizes @xmath53 ; ( b ) magnon density ( circles ) and orbital domain wall density ( triangles ) in the ground state at @xmath54 , @xmath55 , @xmath56 ; finite - size scaling of the particle - hole ( c ) and magnon ( d ) excitation gaps at @xmath57 ; ( e ) ground state correlators for @xmath54 , @xmath56 . see text for details . ] _ numerical results. _ our dmrg results confirm the analytical arguments given above . figure [ fig : phd - alpha3 ] shows our numerical results for the @xmath58 phase diagram of the 1d version of the model ( [ ham-2d ] ) at @xmath59 . we considered open systems consisting of up to @xmath60 sites , monitoring different correlation functions , total spin of the ground state and fidelity susceptibility @xcite to detect phase boundaries . in addition , we have checked our data on systems of up to @xmath56 sites with periodic boundary conditions . we have typically kept up to @xmath61 states ( within a subspace with fixed @xmath62 ) in our dmrg calculations . we indeed observe the fsol phase in a wide region of @xmath25 and @xmath26 . as shown in fig . [ fig : allferri](a ) , spontaneous magnetization in the fsol phase changes smoothly , confirming that there is no gap for single - particle excitations . in accordance with the composite - particle transition mechanism outlined above , there is a clear correlation between the peaks in the densities of orbital domain walls and magnons ( see fig . [ fig : allferri](b ) ) . we have checked that such a correlation persists at all magnetization values , and that the number of peaks in the domain wall density is always equal to the number of magnons in the ground state . moreover , the energy of the lowest excitation in the same @xmath62 sector as the ground state ( the particle - hole gap ) scales as @xmath63 with the system size @xmath53 , while that in the sector corresponding to adding a magnon scales as @xmath64 , as shown in figs . [ fig : allferri](c ) and ( d ) . thus these are two gapless excitation branches with linear and quadratic dispersion , respectively . finally , in the fsol phase the correlators @xmath65 and @xmath66 are very close to each other , despite the fact that @xmath67 can be significantly lower than one , see fig . [ fig : allferri](e ) ( in fact , for the parameters presented in fig . [ fig : allferri](e ) @xmath68 is larger in absolute value than @xmath69 even though @xmath70 ) . one can straightforwardly check that this follows from the fact that the wave function of the bound state is very close to a singlet bond across the orbital domain wall , so that the operator @xmath71 nearly anihilates the ground state . _ boundary transitions. _ in addition to the existence of the fsol phase , the spin - orbital model with @xmath22 is characterized by the appearance of peculiar boundary phase transitions within the ( ih , p ) phase ( curves b1 and b2 in fig . [ fig : phd - alpha3 ] ) at which the behavior of the edge spins in open chains changes drastically . when increasing @xmath25 at fixed @xmath26 , localized and strongly correlated @xmath72 edge spins emerge when crossing the b1 curve . further increasing @xmath25 leads to a second transition at the b2 line , where the value of the boundary spin changes from @xmath72 to @xmath73 . this effect is illustrated in the inset of fig . [ fig : phd - alpha3 ] , where the correlation between edge spins is depicted as a function of @xmath25 for @xmath49 @xcite . we refer to the supplemental material @xcite for further details . _ realization. _ dipolar spinor fermions may be realized using polar molecules ( see ref . @xcite for a detailed discussion ) or employing atoms with large magnetic dipole moments , such as chromium @xcite , dysprosium @xcite , or erbium @xcite . for the particular case of @xmath74cr , in a strong magnetic field , any two of the four lowest energy states @xmath75 , @xmath76 , @xmath77 , and @xmath78 , can be chosen to simulate the @xmath79 and @xmath80 pseudospin-@xmath0 states . those states have approximately the same large magnetic moments , given by the electronic spin projection @xmath81 , differing only by their nuclear moment . the total interparticle potential is of the form @xmath82 , where @xmath83 characterizes the contact interactions , where @xmath84 is the @xmath85-wave scattering length , @xmath86 is the atomic mass , @xmath87 is the vacuum permitivity and @xmath88 is the magnetic dipole moment . the interaction , resulting from the electronic degrees of freedom , is pseudospin - independent , providing the desired su(2 ) spin symmetry of the problem . for @xmath74cr the natural value of ratio @xmath4 , where @xcite @xmath89 ( here @xmath90 and @xmath91 are the orbital wave functions centered at the same site ) is in the regime @xmath92 considered in this letter . _ summary. _ we have shown that dipolar two - component fermions loaded in the @xmath1-bands of a zigzag optical lattice may allow for the realization of a novel , unaccessible in solid - state systems , spin - orbital liquid phase characterized by a finite but unsaturated magnetization . this phase , like ferromagnet , has spontaneously broken su(2 ) symmetry , but , unlike a ferromagnet , it has algebraically decaying longitudinal spin correlations . this phase can be viewed as a luttinger liquid of bound composites of singlet spin dimers and orbital domain walls on top of a fully polarized ferromagnetic phase . _ acknowledgments. _ we thank h. frahm , t. osborne , and h .- j . mikeska for helpful discussions . this work has been supported by quest ( center for quantum engineering and space - time research ) and dfg research training group ( graduiertenkolleg ) 1729 . 10 _ introduction to frustrated magnetism : materials , experiments , theory _ , ed . by c. lacroix , p. mendels , and f. mila ( springer series in solid - state sciences * 164 * , 2011 ) . y. tokura and n. nagaosa , science * 288 * , 462 ( 2000 ) . e. dagotto , science * 309 * , 257 ( 2005 ) . a. m. ole , j. phys . : condens . matter * 24 * , 313201 ( 2012 ) . l.f . feiner , a.m. ole , and j. zaanen , phys . rev . lett . * 78 * , 2799 ( 1997 ) . g. khaliullin and s. maekawa , phys . rev . lett . * 85 * , 3950 ( 2000 ) . f. wang and a. vishwanath , phys . rev . b * 80 * , 064413 ( 2009 ) . j. chaloupka , g. jackeli , and g. khaliullin , phys . rev . lett . * 105 * , 027204 ( 2010 ) . p. corboz , a. m. luchli , k. penc , m.troyer , and f. mila , phys . rev . lett . * 107 * , 215301 ( 2011 ) . c. wu , d. bergman , l. balents , and s. das sarma , phys . rev . lett . * 99 * , 070401 ( 2007 ) . m. hermele , v. gurarie , and a.m. rey , phys . rev . lett . * 103 * , 135301 ( 2009 ) . a.v . gorshkov , m. hermele , v. gurarie , c. xu , p.s . julienne , j. ye , p. zoller , e. demler , m. d. lukin and a. m. rey , nature phys . * 6 * , 289 ( 2010 ) . g. sun , g. jackeli , l. santos , and t. vekua , phys . rev . b * 86 * , 155159 ( 2012 ) . g. wirth , m. lschlger and a. hemmerich , nature phys . * 7 * , 147 ( 2010 ) . k.i . kugel and d.i . khomskii , sov . phys . usp . * 25 * , 231 ( 1982 ) ( uspekhi fiz . nauk * 136 * , 621 ( 1982 ) [ in russian ] ) . s. di matteo , g. jackeli , c. lacroix , and n. b. perkins , phys . rev . lett . * 93 * , 077208 ( 2004 ) . s. r. white , phys . rev . lett . * 69 * , 2863 ( 1992 ) ; phys . rev . b * 48 * , 10345 ( 1993 ) . u. schollwck , rev . mod . phys . * 77 * , 259 ( 2005 ) . the expression for @xmath45 , obtained from the perturbation theory in @xmath26 , is formally valid when @xmath25 and @xmath93 are both small . with increasing @xmath26 , the first instability of the ( f , af ) phase is determined by @xmath44 at very small @xmath25 , but starting from @xmath94 it is described by @xmath45 . j. n. fuchs , d. m. gangardt , t. keilmann , and g. v. shlyapnikov , phys . rev . lett . * 95 * , 150402 ( 2005 ) . m. t. batchelor , m. bortz , x. w. guan , and n. oelkers , j. stat . mech . p03016 ( 2006 ) . m. b. zvonarev , v. v. cheianov , and t. giamarchi , phys . rev . lett . * 99 * , 240404 ( 2007 ) . a. kleine , c. kollath , i. p. mcculloch , t. giamarchi , and u. schollwck , new j. phys . * 10 * , 045025 ( 2008 ) . m .- c . chung and a. b. bhattacherjee , phys . rev . lett . * 101 * , 070402 ( 2008 ) . it should be mentioned that a phase with unsaturated spontaneous magnetization with @xmath95 , has been numerically observed in a 1d purely spin model with exchange interactions up to the @xmath96-th neighbor ( see t. shimokawa and h. nakano , j. phys . . jpn . * 80 * , 043703 ( 2011 ) ; h. nakano and m. takahashi , j. phys . . jpn . * 66 * , 228 ( 1997 ) ) . the mechanism of its formation remains unknown . in any case , the mechanism is different in our model , since the transition into the fsol phase is essentially driven by the orbital degrees of freedom . see , e.g. , shi - jian gu , int . j. mod . phys . b * 24 * , 4371 ( 2010 ) for a review . we define the edge spins as the first three and the last three spins of the chain . this grouping is in principle arbitrary and is just chosen to account for the fact that the edge spins have some finite localization length ( see the supplemental material @xcite ) . see supplemental material at [ url will be inserted by publisher ] for the details concerning the boundary transitions in the ( ih , p ) phase . t. lahaye , t. koch , b. frhlich , m. fattori , j. metz , a. griesmaier , s. giovanazzi , and t. pfau , nature * 448 * , 672 ( 2007 ) . m. lu , n. q. burdick , s. h. youn , and b. l. lev , phys . rev . lett . * 107 * , 190401 ( 2011 ) . a. frisch , k. aikawa , m. mark , f. ferlaino , e. berseneva , and s. kotochigova , e - print arxiv:1304.3326 ( unpublished ) .

one geometrical method used to obtain remarkable fractal sets of extreme beauty and complexity having the property of being self - similar ( i.e. conformally equal to itself at infinite small scales ) , is by considering the limit sets of schottky groups , consisting on finitely generated groups of reflections on codimension one round spheres . as a testimony of such a beauty and complexity , one can consult the wonderful book _ indra s pearl : the vision of felix klein _ written by d. mumdord , c. series and d. wright @xcite . + the purpose of the present paper is to construct , in the spirit of indra s pearls book , an example of a wildly embedded 2-sphere in @xmath0 ( i.e. a wild 2-knot in @xmath0 ) obtained as limit set of a kleinian group . + in section 2 , we present the preliminary definitions and results in knot theory and kleinian groups that we will use in this paper . in section 3 , we describe the geometric ideas involved to construct a wild 2-knot , and we give an explicit example of such a group . in section 4 , we prove that the limit set obtained in section 3 , is a wild 2-knot in @xmath0 . in sections 5 , 6 and 7 we give very interesting topological properties in the case where the original arc ( see section 2 ) fibers over the circle . we show that the wild 2-knot also fibers over the circle and we determine its monodromy . in section 8 , we lift the action of this kleinian group to the twistor space of @xmath0 , obtaining a dynamically defined @xmath1 wildly embedded in the twistor space @xmath2 . + i want to thank prof . alberto verjovsky for all the discussions and very important suggestions . i also want to thank prof . aubin arroyo for drawing the beautiful figures 10 and 11 . in 1925 emil artin described two methods for constructing knotted spheres of dimension two in @xmath0 from knots in @xmath3 . the first of them is called _ suspension_. roughly speaking , this method consists of taking the suspension of @xmath4 , where @xmath5 is a tame knot , to obtain a 2-knot @xmath6 in @xmath0 . by construction , we have that the fundamental group of @xmath6 is easily computable . + the second method is called _ spinning _ and uses the rotation process . a way to visualize it is the following . we can consider @xmath7 as an @xmath8-family of half - equators ( meridians=@xmath9 ) such that the respective points of their boundaries are identified to obtain the poles . then the formula @xmath10 means to send homeomorphically the unit interval @xmath9 to a meridian of @xmath7 such that @xmath11 is mapped to the poles and , multiply the interior of @xmath9 by @xmath8 . in other words , one spins the meridian with respect to the poles to obtain @xmath7 . similarly , consider @xmath12 as an @xmath8-family of half - equators ( @xmath13 ) where boundaries are respectively identified , hence @xmath14 means to send homeomorphically @xmath13 to a meridian of @xmath12 and keeping @xmath15 fixed , multiply the interior of @xmath13 by @xmath8 . in particular @xmath16 . + it is this second method that we will use to construct a 2-sphere wildly embedded in @xmath0 , so we will give a more detailed description of it . + consider in @xmath17 the half - space @xmath18 whose boundary is the plane @xmath19 we can spin each point @xmath20 of @xmath21 with respect to @xmath22 according to the formula @xmath23 we define @xmath24 of a set @xmath25 , as @xmath26 to obtain a knot in @xmath17 , we choose a tame arc @xmath27 in @xmath21 with its end - points in @xmath22 and its interior in @xmath28 . then @xmath29 is a 2-sphere in @xmath17 called a _ spun knot_. + we can think of @xmath27 as the image of an embedding @xmath30 with @xmath31 , and which will be denoted by the same letter . then we will say that an arc @xmath32 is a _ spinnable arc _ if it is smooth in every point with contact of infinite order with respect to the normal on its end - points . + it can be proved that the fundamental group of @xmath29 is isomorphic to @xmath33 and by the seifert - van kampen theorem s , this is isomorphic to the fundamental group of @xmath34 in @xmath35 , where @xmath36 is an unknotted segment joining the end - points of @xmath27 ( see @xcite , @xcite ) . + our goal is to obtain a wild 2-sphere as the limit set of a conformal kleinian group . we will give briefly some basic definitions about kleinian groups . + let @xmath37 denote the group of mbius transformations of the n - sphere @xmath38 , i.e. conformal diffeomorphisms of @xmath39 with respect to the standard metric . for a discrete group @xmath40 _ the discontinuity set _ @xmath41 is defined as follows @xmath42 @xmath43 .3 cm the complement @xmath44 is called the _ limit set _ ( see @xcite ) . + both @xmath41 and @xmath45 are @xmath46-invariant , @xmath41 is open , hence @xmath45 is compact . + a subgroup @xmath40 is called _ kleinian _ if @xmath47 is not empty . we will be concerned with very specific kleinian groups of schottky type . + we recall that a conformal map @xmath48 on @xmath39 can be extended in a natural way to the hyperbolic space @xmath49 , such that @xmath50 is an orientation - preserving isometry with respect to the poincar metric . hence we can identify the group @xmath37 with the group of orientation preserving isometries of hyperbolic @xmath51-space @xmath49 . this allows us to define the limit set of a kleinian group through sequences . + a point @xmath52 is a _ limit point _ for the kleinian group @xmath46 , if there exist a point @xmath53 and a sequence @xmath54 of _ distinct elements _ of @xmath46 , with @xmath55 . the set of limit points is @xmath56 ( see @xcite section ii.d ) . + one way to illustrate the action of a kleinian group @xmath46 is to draw a picture of @xmath57 . for this purpose a fundamental domain is very helpful . roughly speaking , it contains one point from each equivalence class in @xmath41 ( see @xcite pages 78 - 79 , @xcite pages 29 - 30 ) . + a fundamental domain @xmath58 for a kleinian group @xmath46 is a codimension- zero piecewise - smooth submanifold ( subpolyhedron ) of @xmath41 satisfying the following 1 . @xmath59 ( @xmath60 denotes closure ) . @xmath61 for all @xmath62 ( @xmath63 denotes the interior ) . 3 . the boundary of @xmath58 in @xmath47 is a piecewise - smooth ( polyhedron ) submanifold in @xmath47 , divided into a union of smooth submanifolds ( convex polygons ) which are called faces . for each face @xmath64 , there is a corresponding face @xmath65 and an element @xmath66 such that @xmath67 ( @xmath68 is called a face - pairing transformation ) ; @xmath69 . only finitely many translates of @xmath58 meet any compact subset of @xmath47 . ( @xcite , @xcite ) let @xmath70 denote the orbit space with the quotient topology . then @xmath71 is homeomorphic to @xmath72 . the main idea of this construction is to use the symmetry of the spinning process to find a `` packing '' ( i.e. a cover ) of an embedded @xmath7 in @xmath0 consisting of closed round balls of dimension 4 , such that the group @xmath73 generated by inversions in their boundaries ( spheres of dimension 3 ) is kleinian , and its limit set is a wild sphere of dimension two . + let @xmath74 . we will say that @xmath75 is a _ packing _ for @xmath76 if this is contained in the interior of @xmath77 , where @xmath78 is a closed ball of dimension @xmath79 for @xmath80 . let @xmath27 be a spinnable knotted arc in @xmath21 . a _ semi - pearl solid necklace _ subordinate to it , is a collection of consecutive closed round 4-balls @xmath81 such that 1 . the end - points of @xmath27 are the centers of @xmath82 and @xmath83 respectively . 2 . the arc @xmath27 is totally contained in @xmath84 . 3 . two consecutive balls are orthogonal ; otherwise @xmath85 , @xmath86 . the segment of @xmath27 lying in the interior of each ball is unknotted . each ball @xmath87 is called a _ solid pearl_. its boundary is a 3-sphere @xmath88 called a _ pearl_. a _ semi - pearl necklace _ subordinate to @xmath27 is @xmath89 . next , we will define a pearl - necklace @xmath90 subordinate to the 2-knot @xmath29 , with all the requirements needed for the group , generated by reflections on each pearl , to be kleinian . + let @xmath32 be a spinnable knotted arc . a pearl - necklace @xmath90 subordinate to the tame knot @xmath91 is constructed in the following way 1 . let @xmath92 be a semi - pearl necklace subordinate to @xmath27 consisting of the pearls @xmath93 . consider the subset @xmath94 . now , in @xmath29 we will select six isometric copies @xmath95 of @xmath27 , called @xmath96 , @xmath97 , in such a way that each @xmath96 has subordinate a isometric copy @xmath98 of @xmath99 , denoted by @xmath100 , @xmath97 . we will require that @xmath101 is orthogonal to the corresponding @xmath102 . 2 . at each pole of the knot @xmath29 we set a pearl @xmath103 , @xmath104 , orthogonal to the pearls of the next and previous levels , @xmath105 and @xmath106 , @xmath97 , respectively ; such that @xmath107 and @xmath108 3 . at each intersection point ( see proposition 3.4 ) of two consecutive pearls @xmath109 , @xmath110 in @xmath100 and the corresponding @xmath111 , @xmath112 in @xmath113 , we set a pearl @xmath114 which is orthogonal to these four pearls and does not intersect any other , i.e. we require that @xmath115 for @xmath116 , @xmath117 and @xmath118 , for @xmath104 . the intersection @xmath119 is an unknotted disk , where @xmath120 is the solid pearl whose boundary is @xmath109 . let @xmath121 be a pearl - necklace . we define the _ filling _ of @xmath90 as @xmath122 , where @xmath78 is the round closed 4-ball whose boundary @xmath123 is the pearl @xmath124 . + geometrically , the above definition means that when we rotate @xmath27 and @xmath99 with respect to @xmath22 , we obtain an infinite number of pearls covering @xmath29 . we shall select a finite number of them keeping a `` symmetry '' , i.e. we will choose @xmath99 such that when we spin a pearl @xmath125 , we can select six of @xmath126 in such a way that their centers form a regular hexagon and adjacent pearls are orthogonal ( two pearls are orthogonal if the square of the distance between their centers is equal to the sum of the squares of their radii ) . in other words , we will choose six @xmath21 ( six pages of the open book decomposition of @xmath17 ) . as a consequence , in @xmath29 we will have six preferential meridians ( one for each page ) . each meridian @xmath96 ( @xmath127 ) , is a copy of @xmath27 and has a semi pearl - necklace @xmath100 ( @xmath127 ) subordinate to it which is an isometric copy of @xmath99 . the pearls belong to @xmath100 will be denoted by @xmath109 , where the superscript @xmath128 , indicates its `` latitude '' and the subscript @xmath97 , indicates its `` meridian '' ( see figure 2 and compare @xcite page 208 ) . + at each pole of the knot @xmath29 we will set a pearl @xmath103 ( @xmath104 ) orthogonal to the other six of the previous or next level respectively , such that there is no hole among them ( see figure 3 ) , i.e. @xmath129 ( @xmath130 ) , for the 3-tuples of indexes @xmath131 and @xmath132 . by standard arguments of euclidean geometry this sphere always exists . + = 1.5 in at this point , we have chosen a finite number of pearls . however , we have not proved that @xmath29 is totally covered by them . + the pearls @xmath109 , @xmath128 , @xmath97 and the two pearls @xmath103 , ( @xmath104 ) at the poles , totally cover a knot isotopic to @xmath29 . * _ proof . * + firstly , we will verify that the intersection of the pearls @xmath133 , @xmath134 and the corresponding @xmath135 , @xmath136 is not empty . + let @xmath137 be the radius of the pearls @xmath133 and @xmath135 with centers @xmath138 and @xmath139 respectively . let @xmath140 be the radius of the pearls @xmath141 and @xmath142 with centers @xmath143 and @xmath144 respectively ( see figure 4 ) . we know that two pearls are orthogonal if the square of the distance between their centers is equal to the sum of the squares of their radii . hence @xmath145 @xmath146 @xmath147 this implies that @xmath148 so that the intersection of these four pearls is a point . + = 1.2 in now consider the filling of each @xmath100 . we shall prove that there exists a knot isotopic to @xmath29 which is totally contained in @xmath149 , where @xmath150 is the closed 4-ball whose boundary is the pearl @xmath103 . + let @xmath151 be a parametrization of the arc @xmath27 by @xmath152 $ ] . let @xmath153 be an affine plane parallel to the @xmath154-plane . notice that @xmath155 is a circle for @xmath156 and a point for @xmath157 . + let @xmath158 be the smallest @xmath159 for which @xmath160 ( remember that @xmath120 is the 4-ball such that @xmath161 ) . let @xmath162 be the smallest @xmath159 that satisfies @xmath163 . for @xmath164 $ ] , we have that @xmath165 for some index @xmath166 . then @xmath167 is a union of six disks with the property that adjacent disks are either overlapped or tangent . observe that @xmath168 may be not contained in @xmath169 ( see figures 5 and 6 ) . + = 1.5 in = 1.5 in by an isotopy of @xmath170 , we can send @xmath168 to a circle @xmath171 , which passes through either the middle point of each chord joining the two intersection points of each overlap or the points of tangency of adjacent circles ( see figures 5 and 6 ) . indeed , this isotopy @xmath172 can be constructed radially from a function @xmath173 whose graph appears in figure 7 ( both cases ) . thus @xmath174 is a stable isotopy , i.e. is the identity in the complement of a closed set . + = 3.5 in for the pearls at the poles , we have that @xmath175 for @xmath176 $ ] and @xmath177 for @xmath178 $ ] . we can transform , by an isotopy , @xmath179 @xmath176\cup [ \epsilon_{2},1]$ ] in two arcs contained in the interior of the respective balls with the condition that their end - points coincide with @xmath180 , @xmath181 and @xmath182 , @xmath183 , respectively . + by the above , we can define a function such that in each level @xmath179 is the previous isotopy . this function depends of the parameter of the isotopy on each level and @xmath159 . since it is continuous with respect to each variable , it is continuous . notice that on each level @xmath179 , we have that the corresponding isotopy is the identity in the complement of some disk . hence we can conclude that this function is the identity in the complement of a closed ball . + we can extend this function to an isotopy defined on @xmath0 ( see @xcite ) that sends @xmath168 to @xmath171 . + therefore , an isotopic knot to @xmath29 is totally covered by the pearls @xmath109 , @xmath128 , @xmath97 and the two pearls at the poles . @xmath184 + the intersection of four pearls @xmath133 , @xmath134 and @xmath135 , @xmath136 is a single point that will be denoted by @xmath185 ( see figure 8) . we centered at @xmath185 a pearl @xmath114 orthogonal to these four pearls such that it does not overlap to any other . notice that this sphere always exists and its construction uses standard euclidean geometry . + = 1.5 in hence , @xmath90 consists of the pearls @xmath109 , @xmath128 , @xmath97 , the two pearls @xmath103 , @xmath104 , at the poles and the pearls @xmath114 . we will say that @xmath29 is the _ template _ of @xmath90 . + consider the group @xmath73 generated by reflections through each pearl . to guarantee that the group @xmath73 is kleinian we will use the poincar polyhedron theorem . this theorem establishes conditions for the group to be discrete . in practice these conditions are very hard to be verify , but in our case all of them are satisfied automatically from the construction ( see @xcite , @xcite , @xcite ) . + this theorem also gives us a presentation for the group @xmath73 . suppose that the pearl - necklace @xmath90 is formed by the pearls @xmath186 , @xmath187 and we denote by @xmath188 the reflection with respect to @xmath186 . since the dihedral angles between the faces @xmath189 , @xmath190 are @xmath191 , where @xmath192 is either @xmath193 if the faces are adjacent or @xmath194 in other case . therefore , we have the following presentation of @xmath73 @xmath195 the group @xmath73 generated by reflections through each pearl , is kleinian . * _ proof . _ * by the poincar polyhedron theorem , we have that @xmath73 is discrete and its fundamental domain is @xmath196 . therefore it is kleinian . @xmath184 + the first question to appear is if there exists a pearl - necklace @xmath90 for some knot @xmath29 . in the next theorem we will exhibit a semi - necklace @xmath99 subordinate to an embedded of the trefoil arc @xmath27 , satisfying all the requirements of the definition 3.3 . + there exists an embedding of the trefoil arc @xmath27 in @xmath21 that admits a semi - necklace satisfying all the requirements of the definition 3.3 . * _ * by proposition 3.4 it follows that if we have constructed a pearl - necklace @xmath90 subordinate to the knot @xmath29 , it is always possible to find a knot isotopic to @xmath29 such that it is totally contained in the interior of @xmath90 . the group @xmath73 is defined through the pearl - necklace , this means that the pearl - necklace is more fundamental for our purpose than the knot itself . this allows us to consider the trefoil arc @xmath27 as a polygonal arc ( see figure 9 ) obtained joining the centers @xmath197 of the pearls @xmath198 whose coordinates appear in the next table . + observe that the pearl @xmath109 and the corresponding rotated @xmath111 are orthogonal if and only if their radii are equal to the @xmath199-coordinate divided by @xmath200 . + [ cols="<,<,<",options="header " , ] the coordinates of the centers of @xmath201 , @xmath202 and @xmath203 are rational numbers and their radii are equal to @xmath204 . we obtained the rest of centers and radii using the equations @xmath205 and @xmath206 hence , we conclude that all centers and radii of the pearls belong to a finite algebraic extension of the rational numbers . let @xmath73 be the group generated by reflections @xmath188 , through the pearl @xmath186 ( @xmath207 ) of the necklace @xmath90 formed by @xmath79 pearls . then @xmath73 is a conformal kleinian group . let @xmath27 be a spinnable knotted arc in @xmath3 . consider the 2-knot @xmath91 and take a pearl - necklace @xmath90 subordinate to @xmath29 consisting on @xmath79 pearls . + let @xmath73 be the group generated by reflections @xmath188 through the pearl @xmath208 . the natural question is : what is its limit set ? recall that to find the limit set of @xmath73 , we need to find all the accumulation points of orbits . to do that we are going to consider all the possible sequences of elements of @xmath73 . we will do this in steps : 1 . first step : reflecting with respect to each @xmath186 ( @xmath209 ) , a copy of the exterior of @xmath90 is mapped within it . at the end we obtain a new knot @xmath210 , which is in turn isotopic to the connected sum of @xmath211 copies of @xmath29 and it is totally covered by @xmath212 pearls ( packing ) called @xmath213 . + notice that there exists an isotopy of @xmath0 such that the knot @xmath214 is sent to @xmath215 . actually , this remains true for the connected sum of any couple of knotted arcs . therefore @xmath210 is isotopic to spin of the connected sum of @xmath211 copies of @xmath27 . + we have that @xmath216 . in fact , each pearl of @xmath90 lies in @xmath213 . hence @xmath217 is a closed neighbourhood of @xmath29 . to clarify the above , see figure 10 for a simpler case , i.e. for an unknotted necklace . + + + @xmath218 is isotopic to a closed tubular neighbourhood of @xmath29 . + _ proof . _ let @xmath219 be a closed tubular neighbourhood of @xmath29 with the condition that @xmath220 . since @xmath27 is a spinnable arc , it follows that @xmath29 is smooth . given @xmath221 , consider the tangent plane @xmath222 of @xmath29 at @xmath223 . let @xmath224 be a 2-sphere totally geodesic with respect to the spherical metric ( i.e. radius 1 ) that passes through @xmath223 and intersects transversally @xmath222 . thus @xmath225 cuts each solid pearl @xmath226 that contains @xmath223 in a disk @xmath227 . then @xmath228 is a star - shaped set with respect to @xmath223 . this neighbourhood is contained in a closed disk @xmath229@xmath230 , where the radius is @xmath231 . notice that @xmath232 ( see figure 11 ) . + + = 1.8 in + hence for each point @xmath221 , we have found a neighbourhood @xmath233 of it which is star - shaped with respect to @xmath223 and retracts onto @xmath234 . this retraction can be constructed in the following way . one draws a ray @xmath235 going from @xmath223 with angle @xmath236 . let @xmath237 be the intersection point of @xmath235 with @xmath234 and let @xmath238 be the intersection point of @xmath235 with @xmath239 ( see figure 13 ) . we can send the segment @xmath238 to the segment @xmath237 through the radial isotopy @xmath240 , where @xmath241 is the unique polygonal function whose graph appears in figure 12 . observe that this function is the identity beyond a distance @xmath242 from @xmath223 . + + = 1.8 in + therefore we have an isotopy defined on @xmath225 that transforms @xmath239 to @xmath234 and is the identity outside of some closed disk @xmath229@xmath230 . since @xmath225 depends continuously of @xmath223 , we have an isotopy that sends @xmath243 to @xmath219 . by @xcite we can extend this isotopy to @xmath0 . this isotopy is the identity outside of a closed tubular neighborhood @xmath244 , where the radius of @xmath8 is @xmath245 . @xmath184 + 2 . second step : if we consider the action of elements of @xmath73 on @xmath213 , we obtain a new knot @xmath246 totally covered by a packing consisting of @xmath247 pearls , called @xmath248 . the knot @xmath246 is isotopic to the connected sum of @xmath249 copies of @xmath29 . by the above observation , it follows that it is also isotopic to the spin of the connected sum of @xmath249 copies of @xmath27 . + let @xmath250 . then @xmath251 is connected and is a closed neighbourhood of the 2-knot @xmath252 , which is in turn isotopic to the connected sum of @xmath253 copies of @xmath29 . by the above claim , @xmath251 is isotopic to a closed tubular neighbourhood of @xmath252 . notice that @xmath254 ( see figure 13 ) . @xmath255-step : the action of elements of @xmath73 on @xmath256 determines a tame knot @xmath257 , which is in turn isotopic to the connected sum of @xmath258 + 1 $ ] copies of @xmath29 and is also isotopic to the spin of the connected sum of @xmath258 + 1 $ ] copies of @xmath27 . + let @xmath259 . thus @xmath260 is connected and is a closed neighbourhood of the knot @xmath261 which is in turn isotopic to the connected sum of @xmath262+n(n-3)^{k-2}+1 $ ] . this neighbourhood consists of @xmath263 pearls and is isotopic to a closed tubular neighbourhood of @xmath261 . by construction , @xmath264 . let @xmath265 . we shall prove that @xmath52 is a limit point . indeed , there exists a sequence of closed balls @xmath266 with @xmath267 such that @xmath268 for each @xmath269 . we can find a @xmath270 and a sequence @xmath271 of distinct elements of @xmath73 , such that @xmath272 . since @xmath273 it follows that @xmath274 converges to @xmath52 . the other inclusion clearly holds . therefore , the limit set is given by @xmath275 the limit set @xmath276 is isotopic to @xmath277 , where @xmath278 is a wild arc in the sense of @xcite , @xcite , and is contained in each page ( @xmath21 ) of the open book decomposition of @xmath17 . * _ proof . _ * let @xmath27 be a spinnable knotted arc . construct the 2-knot @xmath29 and take the necklace of @xmath79-pearls @xmath90 subordinate to @xmath29 . + now consider a semi - pearl necklace @xmath279 consisting of @xmath79 consecutive orthogonal round 2-spheres that cover completely to @xmath27 , in which its end - points are the centers of the first pearl , @xmath280 , and the last one , @xmath281 . construct @xmath282 ( see section 2 ) . + @xmath283 is isotopic to @xmath284 . indeed , we have already proved that @xmath283 is isotopic to a closed tubular neighbourhood of the knot @xmath29 . by the same argument , @xmath284 is isotopic to a closed tubular neighbourhood of @xmath29 . now two closed tubular neighbourhoods of @xmath29 are isotopic ( @xcite ) . this proves the claim . @xmath184 + in the first step of the reflecting process applied to @xmath90 , we get a packing @xmath213 , of @xmath210 formed by pearls . now , for the case of the semi - necklace @xmath279 , we join the end - points of @xmath27 by an unknotted curve @xmath285 obtaining a knot @xmath286 ( see figure 14 ) . + = 1.5 in we complete the semi - pearl necklace @xmath279 for the knot @xmath286 , with pearls @xmath287 , @xmath288 , keeping the same conditions on consecutive pearls . this new necklace is called @xmath289 ( see figure 15 ) . + = 1.5 in now , we reflect only with respect to each pearl @xmath124 , @xmath290 , of @xmath279 . then we obtain a new knot @xmath291 isotopic to the connected sum of @xmath211 copies of @xmath286 . to return @xmath291 to an arc , we remove the unknotted curve joining the image of the end - points of @xmath27 under the corresponding reflections . this new arc is called @xmath292 and is totally covered by a set of pearls @xmath293 . observe that @xmath294 ( see figure 16 ) . + = 1.5 in notice that @xmath295 is isotopic to @xmath210 and @xmath296 is a packing for it . thus , @xmath297 ( where @xmath298 is defined as @xmath283 ) is a closed neighbourhood of @xmath29 . + in the second step for the necklace @xmath90 , we get a packing @xmath248 of the knot @xmath246 . for the semi - necklace @xmath279 , we join again the end - points of the arc @xmath292 by an unknotted curve , @xmath299 , forming again the knot @xmath291 in such a way that when we complete the semi - pearl necklace @xmath293 , we add the pearls @xmath287 @xmath288 , obtaining the necklace @xmath289 . we can assume that the end - points of the arc @xmath292 coincide with the centers of the pearls @xmath280 and @xmath300 , respectively ( see figure 17 ) . + = 1.5 in now reflecting only with respect to each pearl of the semi - necklace @xmath279 , i.e. with respect to the pearls @xmath124 @xmath290 , we get as in the previous step , the packing @xmath301 of the new arc @xmath302 , which is in turn isotopic to the connected sum of @xmath253 copies of @xmath286 minus an unknotted curve @xmath303 . when we spin @xmath301 and @xmath302 , we obtain the packing @xmath304 of @xmath305 . define @xmath306 . then @xmath307 is a closed neighbourhood of a 2-knot @xmath308 , which is isotopic to the connected sum of @xmath253 copies of @xmath29 . so @xmath251 and @xmath307 are closed neighbourhoods of isotopic 2-knots . using the same arguments of claim 4.1 and the standard fact that any locally flat embedding of @xmath7 in @xmath0 has trivial normal bundle , it follows that two closed tubular neighbourhoods of isotopic knots are isotopic , hence @xmath251 is isotopic to @xmath307 and the following diagram commutes @xmath309 where the row maps are inclusions . notice that this isotopy is stable , i.e. is the identity on some open in @xmath0 , and is orientation - preserving ( see @xcite ) . + inductively , for the @xmath255-step we obtain the packings @xmath310 of @xmath257 and @xmath311 of @xmath312 . where @xmath257 is obtained through the reflecting process previously described . the arc @xmath313 is formed applying the reflecting process to @xmath314 subordinate to the knot @xmath315 and removing an unknotted curve @xmath316 . then @xmath317 and @xmath318 are closed neighborhoods of the knots @xmath319 and @xmath320 respectively , which are in turn isotopic to the connected sum of @xmath258+n(n-3)^{k-1}+1 $ ] copies of @xmath29 . hence , @xmath321 is isotopic to @xmath322 and the following diagram commutes @xmath323 observe that this isotopy is stable and orientation - preserving . summarizing , we have the commutative diagram @xmath324 where the row maps are inclusions and the vertical arrows are orientation - preserving stable isotopies . + the inverse limit in the first row of the above diagram is @xmath325 and the inverse limit in the second row is @xmath326 . but @xmath327 is a wild arc denoted by @xmath328 ( see @xcite , @xcite ) , i.e. the inverse limit in the second row is @xmath329 . + by the universal property of the inverse limit , there exists a homeomorphism of @xmath0 to @xmath0 which sends @xmath330 to @xmath331 . this homeomorphism is stable because it coincides with a stable homeomorphism on some open set ( see @xcite ) and is orientation - preserving . this implies that it is isotopic to the identity ( see @xcite ) . + therefore , the knots @xmath331 and @xmath330 are isotopic . this proves the theorem.@xmath184 the limit set @xmath276 is homeomorphic to @xmath7 * _ proof . _ * by the above theorem , we have that @xmath332 let @xmath90 be a pearl - necklace subordinate to the non - trivial tame knot @xmath29 . then @xmath276 is wildly embedded in @xmath0 . * _ _ * the fundamental group of @xmath333 is isomorphic to the fundamental group of the knot obtained joining the end - points of the arc @xmath278 by an unknotted curve ( see @xcite ) . it is well - known that this fundamental group has no finite representation ( see @xcite , @xcite ) . @xmath184 let @xmath90 be a pearl - necklace subordinate to @xmath29 where @xmath27 is the trefoil arc , @xmath334 . then @xmath335 hence @xmath336 @xmath337 @xmath338 is infinitely generated with a infinite number of relations . the action of @xmath73 can be extended to the hyperbolic space @xmath339 and in this case @xmath73 is a subgroup of isom @xmath339 , which acts properly and discontinuously on @xmath340 . its fundamental polyhedron is @xmath341 , where @xmath342 is the natural extension of the pearl - necklace to @xmath339 . it is a convex subset and has a finite number of sides , hence @xmath73 is geometrically finite ( see @xcite ) . + the group @xmath73 acts properly and discontinuously on @xmath343 , then the quotient @xmath344 ( see theorem 2.2 ) is a compact orbifold such that its interior is a non - compact hyperbolic manifold of infinite volume and its compactification as a subset of @xmath345 has boundary which possesses a conformally flat structure given by the action . + for the kleinian group @xmath73 acting on the pearl - necklace @xmath90 , its fundamental domain is @xmath346 . the group @xmath73 acts properly and discontinuously on @xmath347 , hence @xmath348 is an orientable , compact , conformally flat 4-orbifold with boundary . its fundamental group coincides with the fundamental group of the template of @xmath90 . + in the next section , we will describe @xmath349 under the restriction that @xmath29 is a fibered knot . + consider now the index - two subgroup @xmath350 consisting of even words , i.e. @xmath351 is the orientation preserving index two subgroup of @xmath73 . its fundamental polyhedron is @xmath352 , where tilde means the natural extensions to the hyperbolic space of both the pearl - necklace and the corresponding reflection map . since @xmath353 is a convex subset and has a finite number of sides , it follows that @xmath351 is geometrically finite . + since @xmath351 acts freely on its domain of discontinuity , then the quotient space @xmath354 is a compact , orientable manifold , such that @xmath355 is a non - compact , orientable hyperbolic manifold of infinite volume . this space as a subset of @xmath356 , has a boundary which possesses a natural conformally flat structure given by the action . + for the kleinian group @xmath351 acting on @xmath0 , its fundamental domain is @xmath357 . since @xmath351 acts freely on @xmath358 , we have that @xmath359 is a compact , orientable , conformally flat 4-manifold with boundary . its fundamental group is the fundamental group of the knot @xmath214 . we recall that a mapping @xmath361 is said to be a locally trivial fibration with fiber @xmath65 if each point of @xmath362 has a neighbourhood @xmath363 and a `` trivializing '' homeomorphism @xmath364 for which the following diagram commutes @xmath365_f\ar[r]^h & u\times f\ar[dl]^-{\txt{projection}}\\ u & } \ ] ] @xmath77 and @xmath362 are known as the _ total _ and _ base _ spaces , respectively . each set @xmath366 is called a _ fiber _ and is homeomorphic to @xmath65 . we will be concerned with fibrations with base space @xmath8 . + a knot or link @xmath285 in @xmath3 is _ fibered _ if there exists a locally trivial fibration @xmath367 . we require that @xmath368 be well - behaved near @xmath285 . that is , each component @xmath369 is to have a neighbourhood framed as @xmath370 , with @xmath371 , in such a way that the restriction of @xmath368 to @xmath372 is the map into @xmath8 given by @xmath373 . + it follows that each @xmath374 , @xmath375 , is a 2-manifold with boundary @xmath285 : in fact a seifert surface for @xmath285 ( see @xcite , page 323 ) . + let @xmath376 be the 3-sphere centered at the origin of radius @xmath377 . let @xmath378 . then @xmath379 is the right - handed trefoil knot and the map @xmath380 given by @xmath381 is a locally trivial fibration with fiber the punctured torus ( see @xcite section 1 , @xcite pages 327 - 333 ) . .3 cm ( @xcite)let @xmath27 be a spinnable knotted arc . suppose that the knot @xmath286 , obtained from @xmath27 joining its end - points by an unknotted curve , fibers over the circle with fiber the surface @xmath64 . then @xmath29 fibers over the circle with fiber @xmath382 , an @xmath8-family of surfaces @xmath383 all glued onto a single meridian of @xmath384 with longitude @xmath236 . the interior of @xmath383 is @xmath64 and its boundary is a meridian of @xmath384 ( see figure 18 ) . * _ proof . _ * the fibering of the complement of @xmath286 induces a fibering of @xmath385 by surfaces @xmath386 , @xmath387 . the interior of @xmath386 is @xmath64 and its boundary is @xmath388=@xmath389 the meridian of @xmath384 with longitude @xmath236 ( see figure 18 ) . + = 1.5 in recall that in the spinning process we multiply the interior of @xmath390 by @xmath8 and @xmath384 stays fixed . hence , we get a fibering of @xmath391 by an @xmath8-family of surfaces @xmath383 all glued onto the single meridian @xmath389 of @xmath392 with longitude @xmath236 . .5 cm henceforth , a fibered arc will mean that the knot obtained from it joining its end - points by an unknotted curve , fibers over the circle . + let @xmath27 be a spinnable fibered arc with fiber the surface @xmath64 . let @xmath90 be an @xmath79-pearl necklace subordinate to the tame knot @xmath29 . let @xmath276 be the limit set . then @xmath393 fibers over the circle with fiber @xmath394 , the closure of the surface @xmath382 of the previous lemma . * _ proof . _ * let @xmath395 be the given fibration with fiber the 3-manifold @xmath382 . observe that @xmath396 is a fibration with fiber @xmath382 . + as we know @xmath397 . in our case @xmath398 , which fibers over the circle with fiber the closure of @xmath382.@xmath184 + by the above , to describe @xmath399 when the original knot is fibered , we just need to determine its monodromy . it coincides with the knot s monodromy . hence we have a complete description of @xmath399 . + let @xmath27 be a spinnable fibered arc with fiber the surface @xmath64 . let @xmath90 be an @xmath79-pearl necklace subordinate to the tame knot @xmath29 . let @xmath351 be the orientation preserving index two subgroup of @xmath73 . let @xmath400 be the limit set . then @xmath401 fibers over the circle with fiber @xmath402 , which is homeomorphic to the connected sum along the boundary of the 3-manifold @xmath382 with itself . * _ proof . _ * we can assume , up to isotopy , that the fiber @xmath64 cuts each pearl of the semi - necklace corresponding to @xmath27 , in arcs going from one intersection point to another . hence , we can assume that the fiber @xmath382 cuts each pearl @xmath403 in disks @xmath404 , whose boundary is the intersection of @xmath124 with the adjacent pearls . + when we reflect with respect to @xmath124 a copy of @xmath382 , called @xmath405 , is mapped to the interior of @xmath124 and it is joined to @xmath382 along the disk @xmath404 . @xmath184 + since @xmath351 is a normal subgroup of @xmath73 , it follows by lemma 8.1.3 in @xcite that @xmath351 has the same limit set as @xmath73 . therefore @xmath406 . let @xmath27 be a non - trivial spinnable fibered arc . let @xmath90 be a pearl - necklace subordinate to the fibered knot @xmath29 . let @xmath73 be the group generated by reflections through the pearls and let @xmath351 be the orientation preserving index two subgroup of @xmath73 . let @xmath407 be the corresponding limit set . then : 1 . there exists a locally trivial fibration @xmath408 , where the fiber @xmath409 is an @xmath8-family of surfaces @xmath410 all glued onto a meridian @xmath236 , of @xmath384 ( see lemma 6.3 ) . where @xmath410 is an orientable infinite genus surface with one end . 2 . @xmath411 . * _ proof . _ * we know that @xmath412 is an infinite - fold covering . by the previous lemma , there exists a locally trivial fibration @xmath413 with fiber @xmath402 . + then @xmath414 is a locally trivial fibration . the fiber is @xmath415 , i.e. the orbit of the fiber . + we now give another proof . as we know from theorem 4.1 , the knot @xmath276 is isotopic to the knot @xmath331 , where @xmath328 is a wild arc . since @xmath27 is fibered , so is @xmath328 . in this case the fiber , @xmath410 , is an orientable infinite genus surface with one end . hence @xmath331 fibers over the circle with fiber @xmath416 , an @xmath8-family of surfaces @xmath410 all glued onto a meridian @xmath236 , of @xmath384 ( see lemma 6.3 ) . + the first part of the theorem has been proved . for the second part , observe that the closure of the fiber is the closure of the @xmath8-family of surfaces @xmath410 , i.e. is the closure of an @xmath8-family of ends . as we can see in the figure 20 , each end has as boundary the wild arc @xmath328 . hence the closure of the fiber is exactly the limit set . therefore @xmath411 . @xmath184 1 . this theorem can be generalized to fibered links . 2 . this theorem gives an open book decomposition of @xmath333 , where the `` binding '' is the wild knot @xmath276 , and each `` page '' , @xmath417 , is a 3-manifold which fibers over @xmath8 and is the @xmath8-family of surfaces @xmath410 all glued onto a meridian @xmath236 , of @xmath390 ( see lemma 6.3 ) . here @xmath410 is an orientable infinite genus surface with one end . + indeed , this decomposition can be viewed in the following way . for the above theorem , @xmath333 is @xmath418 $ ] modulo the identification of the top with the bottom through an identifying homeomorphism . consider @xmath419 $ ] and identify the top with the bottom . this is equivalent to keep @xmath420 fixs and to spin @xmath421 with respect to @xmath420 until glue it with @xmath422 . removing @xmath420 we obtain the open book decomposition . let @xmath29 be a non - trivial fibered tame knot and let @xmath64 be the fiber . since @xmath391 fibers over the circle , we know that @xmath391 is a mapping torus equal to @xmath423 $ ] modulo an identifying homeomorphism @xmath424 that glues @xmath425 to @xmath426 . this homeomorphism induces a homomorphism @xmath427 called _ the monodromy of the fibration_. + another way to understand the monodromy is through the _ first return poincar map _ , defined as follows . let @xmath428 be connected , compact manifold and let @xmath429 be a flow that possesses a transversal section @xmath430 . it follows that if @xmath431 then there exists a continuous function @xmath432 such that @xmath433 . we may define the first return poincar map @xmath434 as @xmath435 . this map is a diffeomorphism and induces a homomorphism of @xmath436 called _ the monodromy _ ( see @xcite , chapter 5 ) . + for the manifold @xmath391 , the flow that defines the first return poincar map @xmath437 is the flow that cuts transversally each page of its open book decomposition . + consider a pearl - necklace @xmath90 subordinate to @xmath29 . as we have observed during the reflecting process , @xmath29 and @xmath64 are copied in each reflection . so the flow @xmath437 is also copied . hence , the poincar map can be extended in each step , giving us in the end a homeomorphism @xmath438 that identifies @xmath439 with @xmath440 , and induces the monodromy of the wild knot . + from the above , if we know the monodromy of the knot @xmath29 then we know the monodromy of the wild knot @xmath441 . + by the long exact sequence associated to a fibration , we have @xmath442 which has a homomorphism section @xmath443 . therefore ( 1 ) splits . as a consequence @xmath444 is the semi - direct product of @xmath445 with @xmath446 . + let @xmath27 be the trefoil arc . consider the knot @xmath29 . then the fiber @xmath416 @xmath447 is an @xmath8-family of punctured torus all glued onto a single meridian ( see previous section ) . the fundamental group of @xmath416 is the free group in two generators , @xmath448 and @xmath449 . since @xmath450trefoil knot ) , it follows that the monodromy maps , in both cases , coincide . that is , @xmath451 sends @xmath452 and @xmath453 . its order is six up to an outer automorphism ( see @xcite pages 330 - 333 ) . + the monodromy in the limit @xmath454 is given by @xmath455 and @xmath456 , where @xmath457 . so @xmath458 let @xmath459 ; + @xmath460 this gives another method for computing the fundamental group of a wild 2-knot whose complement fibers over the circle . let @xmath90 be a pearl - necklace whose template is a non - trivial tame fibered knot @xmath29 . then @xmath461 . in this section we will lift the action of the group @xmath73 on @xmath0 to its twistorial space , which is complex projective 3-space @xmath2 . we refer to @xcite and @xcite for details . + let us now recall briefly the _ twistor fibration _ of @xmath0 , also known as _ the calabi - penrose fibration _ @xmath462 ( see @xcite ) . there are several equivalent ways to construct this fibration . a geometric way to describe it is by thinking of @xmath0 as being the quaternionic projective line @xmath463 , of _ right _ quaternionic lines in the quaternionic plane @xmath464 ( regarded as a 2-dimensional right @xmath465-module ) . that is , for @xmath466 ( @xmath467 ) the right quaternionic line passing through @xmath468 is the linear space @xmath469 . we can identify @xmath464 with @xmath470 via the @xmath471-linear map given by @xmath472 , where @xmath473*j*=@xmath474*i*+@xmath475*j*+@xmath476*k * and @xmath477*j*=@xmath478*i*+@xmath479*j*+@xmath480*k*. in this notation * i * , * j * , * k * denote the standard quaternionic units , @xmath481*i * , @xmath482*i * , @xmath483*i * and @xmath484*i*. + under this identification each right quaternionic line is invariant under _ right _ multiplication by * i*. hence such a line is canonically isomorphic to @xmath485 . if we think of @xmath2 as being the space of complex lines in @xmath470 , then there is an obvious map @xmath462 , whose fiber over a point @xmath486 is the space of complex lines in the given right quaternionic line @xmath487 ; thus the fiber is @xmath488 . + the group @xmath489 of orientation preserving conformal automorphisms of @xmath0 is isomorphic to @xmath490 , the projectivization of the group @xmath491 , invertible , quaternionic matrices . this is naturally a subgroup of @xmath492 , since every quaternion corresponds to a couple of complex numbers . hence @xmath489 has a canonical lifting to a group of holomorphic transformations of @xmath2 , carrying twistor lines into twistor lines . ( @xcite ) by a twistor kleinian group we mean a discrete subgroup @xmath46 of @xmath493 of holomorphic automorhisms , which acts on @xmath2 with non - empty region of discontinuity @xmath494 and which is a lifting of a conformal kleinian group acting on @xmath0 . there is no `` good '' general definition of the discontinuity set @xmath495 for general groups , hence an appropiate definition must be given in each case ( see @xcite . we are considering the definition 1.4 of @xcite , in which @xmath41 is an open @xmath46-invariant set and @xmath46 acts properly and discontinuously on @xmath41 . the space @xmath57 has the quotient topology , and the map @xmath496 is continuous and open . it has been proved in @xcite that if @xmath497 is a discrete subgroup acting on @xmath0 with limit set @xmath278 , then its canonical lifting @xmath498 acts on @xmath2 with limit set @xmath499 , thus @xmath500 is a fibered bundle over @xmath278 with fiber @xmath7 . in @xcite , is also proved that if we restrict the twistor bundle to a proper subset of @xmath0 . + we consider the kleinian group @xmath73 such that its limit set is @xmath7 wildly embedded on @xmath0 . then 99 e. artin , _ zur isotopie zweidimensionalen flchen i m @xmath503 _ abh . hamburg ( 1926 ) , 174 - 177 . _ geometrical finiteness for hyperbolic groups_. journal of functional analysis 113 ( 1993 ) , 245 - 317 . j. dugundji . _ topology_. allyn and bacon , inc . d. b. a. epstein , c. petronio . _ an exposition of poincare s polyhedron theorem_. enseignement mathematique 40 , 1994 , 113 - 170 . r. h. fox . _ a quick trip through knot theory_. topology of 3-manifolds and related topics . prentice - hall , inc . , 1962 . m. gromov , h. b. lawson , w. thurston . _ hyperbolic 4-manifolds and conformally flat 3-manifolds_. publ . i.h.e.s . vol . 68 ( 1988 ) , 27 - 45 . w. goldman . _ conformally flat manifolds with nilpotent holonomy and the uniformization problem for 3-manifolds_. transactions of the american mathematical society vol . 278 no . 2 , 573 - 583 . w. hirsch . _ smooth regular neighbourhoods_. annals of mathematics vol . 76 , no.3 ( 1962 ) , 524 - 530 . m. kapovich . _ topological aspects of kleinian groups in several dimensions_. preprint ( 1988 ) . m. kapovich . _ hyperbolic manifolds and discrete groups_. progress in mathematics , birkhauser , 2001 . _ stable homeomorphisms and the annulus conjecture_. ann of math ( 2 ) 89 , 1969 , 575 - 582 . r. s. kulkarni . _ groups with domains of discontinuity_. math . 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( 1965 ) , 471 - 495 .

the purpose of this paper is to construct an example of a 2-knot wildly embedded in @xmath0 as the limit set of a kleinian group . we find that this type of wild 2-knots has very interesting topological properties .

in recent years , the rapidly growing field of diluted magnetic semiconductors ( dms ) @xcite has attracted a considerable interest owing to their potential for spintronic devices . one of the main goals is to combine , the traditional electronic functionality ( charge ) and the spin degree of freedom of the electrons / holes . this requires optimal candidates that exhibit room temperature ferromagnetism . recent progress in growth processes of tm - doped iii - v semiconductors has boosted the interest for such novel materials . among iii - v dms , mn doped gaas that could be considered as the prototype is certainly the most widely studied ( both transport and magnetic properties ) . however , the understanding of the fundamental physical properties in these doped compounds involves theoretical speculations which are subject to controversy . the quest for a model able to capture quantitatively the physics and identify the key physical parameters that control both magnetic and transport properties was a clear open issue over the last decade . till recently dms based theoretical studies could be split into two main distinct types : ( i ) first principle based approaches @xcite and ( ii ) zener mean field type theories ( zmf ) @xcite . the first kind is based on density functional theory ( dft ) such as local spin density approximation ( lsda ) or generalized gradient approximation ( gga ) for instance . they require no adjustable parameters and are essentially material specific . the second type is a model approach that includes a realistic description of the host band structure within a 6 bands or 8 bands kohn - luttinger hamiltonian @xcite and a local p - d exchange between itinerant holes and localized impurity spins . in zener mean field theory the p - d coupling is treated pertubatively and the dilution effects at the lowest order , also known as virtual crystal approximation ( vca ) . as a consequence , the fermi level lies inside the unperturbed valence band ( vb ) leading to the so called valence band scenario ( see fig . [ fig1](a ) ) . regarding the specific case of ( ga , mn)as , the perturbative vb picture is inconsistent with first principle based studies . indeed , density functional calculations clearly predict the existence of a well defined preformed impurity band ( see fig . [ fig1](b ) ) that for a sufficiently large concentration of mn ( beyond 1% ) overlaps with the valence band @xcite . it is found that the fermi level lies in the resonant impurity band . thus , ab initio studies clearly support the so called `` impurity band picture '' ( ib ) . it is worth noticing that both optical conductivity measurements @xcite and proximity of mn doped gaas to the metal - insulator transition @xcite fully support the ib - picture . in spite of all that the issue of vb versus ib scenario is still controversial on the other hand , the vb scenario remains suitable to describe the physics in ii - vi materials such as mn doped znte , cdte , znse for instance . the reason for this is the absence of hybridized p - d states in the vicinity of the top of the valence band in these alloys . in other words , treating the substitution by mn as a perturbation remains a good approximation in ii - vi materials . in the following , we present a two step approach that allows to describe both magnetic and/or transport properties of a wide range of diluted magnetic semiconductors . concerning the magnetic properties that are our main concern in this paper the two steps are described as follows . the first one consists in calculating the magnetic couplings between localized spins randomly distributed in the semiconductor host . to this end one can use first principle calculations or suitable model approaches . the purpose is to build the effective heisenberg hamiltonian of the problem . this spin hamiltonian is diagonalized during the second step within the self consistent local random phase approximation @xcite ( sc - lrpa ) procedure that is described in the next section . the hamiltonian that describes n@xmath0 interacting spins @xmath1 ( classical or quantum ) randomly distributed in the host lattice is the dilute / disordered heisenberg model , @xmath2 @xmath3 are the spin - spin couplings , the sum runs over all sites and the random variable p@xmath4 is 1 if the site is occupied by an impurity , otherwise 0 . in the case of mn doped iii - v dms , the localized spin s=5/2 , thus it can be treated classically . in what follows the dilute heisenberg hamiltonian is diagonalized using the self - consistent local random phase approximation ( sc - lrpa ) . it is a semi - analytical method based on finite temperature green s functions that describe the spin fluctuations . this powerful approach offers several advantages . compared to standard classical monte carlo simulations ( mc ) , ( i ) it allows calculations on large system sizes , ( ii ) the cpu and memory cost are relatively low , ( iii ) the critical temperature is given by a semi - analytical expression ( no need to calculate binder cumulants as in mc ) , ( iv ) the t - dependent local magnetizations , susceptibility and magnetic excitation spectrum can be calculated directly as well . in order to calculate the magnetic properties we define the following retarded green s function @xmath5 , where @xmath6\rangle$ ] , ( @xmath7 denotes the thermal average ) . g@xmath8 describes the transverse and thermal spin fluctuations . after tyablicov ( or rpa ) decoupling of the higher - order green s functions that appear in the equation of motion of @xmath9 one finds , @xmath10 for convenience we have introduced the reduced variable @xmath11 , @xmath12 being the averaged magnetization in the sample ( @xmath13 ) , and @xmath14 . for each configuration of the disorder ( random positions of the impurities ) , the local t - dependent magnetizations @xmath15 are calculated self - consistently . @xmath16 is an effective n@xmath17n@xmath0 non hermitian , bi - orthogonal matrix @xcite whose matrix elements are , @xmath18 the property of bi - orthogonality implies a set of right and left eigenvectors @xmath19 and @xmath20 , each pair associated with the same eigenvalue @xmath21 ( @xmath22=1,2, .. n@xmath0 ) . the retarded green s function can be re - written , @xmath23 the sc - lrpa has been used in several studies and was proven accurate and reliable . it properly treats thermal fluctuations and disorder effects such as localization and percolation physics . one can for instance find detailed discussions in refs . a summary of the sc - lrpa procedure is illustrated in fig . [ fig2 ] . as mentioned previously , one of the great advantages of the sc - lrpa is to allow a direct calculation of the critical temperature . indeed , one can derive a semi - analytical expression for t@xmath24 that reads @xcite , @xmath25 where @xmath26 and @xmath27 f@xmath28 s depend on the reduced local magnetizations @xmath29 that are calculated self - consistently . in the limit of vanishing magnetization one finds the following set of equations , @xmath30 the nature of the magnon modes ( localized / extended ) is explicitly taken into account in the expression of t@xmath31 ( eq.([tc1 ] ) and ( [ tc2 ] ) ) . the accuracy and reliability of the sc - lrpa to handle both thermal fluctuations and disorder ( localization , percolation ) has been often addressed by direct comparison with monte carlo calculations @xcite . the agreement was systematically very good , the critical temperatures calculated by both methods usually differ by less than 10@xmath32 . for instance , in the case of 5@xmath32 doped gaas , using the same couplings , monte carlo calculations have lead respectively to 137 k @xcite and 110 k @xcite ( error bars were not given but should be at least of the order of 10@xmath32 ) , whilst the sc - lrpa value is 132 @xmath33 k. the use of the two step approach combining first principle calculations of the magnetic couplings and sc - lrpa treatment of the effective diluted heisenberg hamiltonian is a tool of choice . however , due to the complexity and material specific nature of first principle calculations , it remains difficult to discern the relevant parameters that govern the physical features in various diluted magnetic semiconductors . a suitable minimal model allowing a coherent and consistent picture of the physics in these materials is needed . in the next section we introduce such a model , the v - j hamiltonian . it captures most of the relevant features in dilute magnets . it continuously shows how the couplings change from rkky nature in ii - vi mn doped compounds such as ( zn , mn)te or ( cd , mn)te to double exchange type in ( ga , mn)n . we will see that ( ga , mn)as is located near the metal to insulator transition and the resonant effects due to the position of the mn acceptor level is responsible for the highest curie temperature observed in this family of materials . the aim of this section is to compare ( i ) ab initio , ( ii ) model based calculations and ( iii ) experimental data . it will be seen that the non perturbative v - j model @xcite provides naturally a coherent and unified picture of the physics ( magnetism and transport ) in dms . as mentioned in the introduction part , in zener mean field theory the focus is put on the realistic description of the host band structure ( kohn - luttinger hamiltonian ) whilst the coupling between holes and localized spins ( s=5/2 ) of mn@xmath35 is treated perturbatively . in contrast , the key point of the v - j model ( defined below ) lies in the non - perturbative treatment of the impurity - hole coupling , while the details in the host band structure are ignored . as will be shown , it appears that the details of the band structure indeed play a secondary role . note , that the v - j model has been recently extended by including a more realistic description of the host band structure . this study has given further support to the validity of the model@xcite . denotes the impurity density and @xmath36 is the carrier concentration . , width=480 ] the v - j model is defined as follows@xcite , @xmath37 where the hopping term t@xmath38 for i and j nearest neighbours , otherwise zero . @xmath39 ( @xmath40 ) is the creation ( annihilation ) operator of a hole of spin @xmath41 at site i. j@xmath28=j if the site is occupied by mn otherwise it is zero . j is the p - d coupling between the localized mn spin @xmath1 ( s=5/2 ) and the itinerant hole quantum spin @xmath42 . the on - site potential v@xmath28 results from the substitution of the host cation by the magnetic impurity . thus j@xmath28=p@xmath28j and v@xmath28=p@xmath28v where p@xmath28=1 if the site is occupied by an impurity , otherwise zero . the one band model contains 3 parameters only ( t , j , v ) . the hopping term has been set to t=0.7 ev ( to reproduce the typical vb bandwidth in iii - v / ii - vi semiconductors ) . j is of the order of 1 ev in both mn doped ii - vi and iii - v dms , thus , j has been set to 1.2 ev ( widely accepted value for mn doped gaas ) . in this way , the remaining last parameter v fully characterizes a given mn doped compound . it is chosen in order to reproduce the specific position of the acceptor hybridized p - d state@xcite . we will see that the on - site scattering potential v , missing in previous theories @xcite , is the key parameter . this is discussed in what follows . [ fig3 ] shows an illustration of the two steps procedure used to calculate both transport and magnetic properties within the v - j model . during the first step , the hamiltonian is diagonalized exactly for each disorder configuration ( random positions of the magnetic impurities ) in both spin sectors . then , from the itinerant carrier green s functions @xmath43 , the magnetic couplings between localized impurity spins are calculated for all distances ( see ref . for further details ) . note that transport properties can be computed at this stage . next , the calculated magnetic couplings enter the heisenberg hamiltonian which is treated in the second step in the framework of the sc - lrpa . in this paper , we have chosen to restrict ourselves to magnetic properties only . moreover , we will not mention here the effects of compensation defects . this issue has been addressed in several papers@xcite . transport properties in the framework of the v - j model ( metal - insulator phase transition and optical conductivity ) have been discussed in ref . . we would like to stress that a good quantitative agreement between this theory and experiments has been found for transport properties as well . ( ev ) as a function of @xmath44 in mn doped gaas ( ab initio and v - j model ) . @xmath45 denotes the pertubative value . the calculations are extracted from ref .. ( right ) measured and calculated ( ab initio and v - j model ) critical temperatures in ga@xmath46mn@xmath47as as a function of @xmath44 . the calculations are performed for well annealed compounds ( no compensation defects , e.g. 1 hole / mn ) . the experimental data are from ( a ) edmonds et al . @xcite , ( b ) chiba et al . @xcite , ( c ) edmonds et al . @xcite , ( d ) jungwirth et al . @xcite , ( e ) stone et al . , width=480 ] in fig . [ fig4 ] we focus first on the case of mn doped gaas . the parameter v has been set to 1.8 t=1.26 ev in order to reproduce the position of the measured p - d acceptor level ( 110 mev ) . we clearly observe in fig . [ fig4]-left a very good agreement between v - j model and ab initio calculated spin splitting of the valence band up to 12@xmath32 doped samples . on the other hand , the perturbative estimate ( zener mean field value ) of the spin splitting ( @xmath48js ) is found much smaller than that obtained from ab initio studies ( about 4 times smaller for the 5@xmath32 doped sample ) . it is also worth mentioning that in the absence of v ( v=0 ) , @xmath49 is well approximated by @xmath45 . this regime corresponds to that of mn doped ii - vi compounds such as zn@xmath46mn@xmath47te , cd@xmath46mn@xmath47se for instance . in fig . [ fig4]-right , we compare the theoretical values of the curie temperature ( ab initio and v - j model ) to available experimental measurements . first , it is seen that ab initio and v - j model calculations agree surprisingly well with each other . in addition , we clearly find an overall very good agreement between theory and experiments in the whole concentration range . note that much more experimental data can be found in the literature for t@xmath24 . we have only chosen some values corresponding to annealed samples . because of the presence of compensating defects , as grown samples usually exhibit much smaller critical temperatures . this issue has been theoretically addressed in other papers @xcite . thus , an excellent quantitative agreement is found between v - j model and ab initio on one side and with the experimental data on the other side . this clearly supports the fact ( i ) that the v - j model captures the relevant and essential physics and ( ii ) that the key physical parameter is indeed the acceptor level position . the curie temperatures result from a systematic average over a large number of disorder configurations . the calculations are systematically performed on large systems ( negligible finite size effects ) . the concentration of mn is fixed @xmath50 and the hole concentration is @xmath51 ( well annealed systems ) . the open circles are obtained with the v - j model and the other symbols ( ( a)-(e ) ) to t@xmath24 computed with ab initio exchange integrals ( see ref . ) . the dashed vertical line correspond to the calculated metal - insulator transition . , width=480 ] we now proceed further and show that the agreement between v - j model and ab initio is not limited to the case of mn doped gaas only . for that purpose we have calculated t@xmath24 as a function of the acceptor level e@xmath52 ( by tuning v ) and compared it with existing ab initio calculations . the concentration of mn is set to @xmath44=0.05 and we have considered the case of optimally annealed systems , thus the hole density is set to @xmath51 . the results are depicted in fig . regarding the ab initio results , the @xmath44-coordinate is the measured or calculated mn acceptor level in the compound . we observe that the v - j calculated t@xmath24 increases rapidly with increasing e@xmath52 till it reaches a maximum and then decreases . note that the rapid increase of t@xmath24 occurs on a very short energy scale . after the maximum , we observe two regimes : first t@xmath24 decreases rapidly till e@xmath52 @xmath53 and then the decay slope becomes significantly reduced . the maximum of the curie temperature which is about 125@xmath54 is reached for e@xmath52 @xmath55 . remarkably , this acceptor level energy coincides almost exactly with that of mn doped gaas ( measured value is 110 mev ) . thus the v - j model explains for the first time why among ii - vi and iii - v magnetic impurity doped semiconductors the highest measured t@xmath24 is that of mn doped gaas . in this figure , we also see the very good quantitative agreement between the v - j calculated critical temperatures and that obtained from ab initio based calculations . the reason of this maximum is the fact that the couplings are optimal ( resonant effects ) when the acceptor level is not too far and not too close to the top of the valence band . when the acceptor level is too small or vanishes ( e@xmath52 @xmath56 0 ) the couplings are rkky like ( case of mn doped ii - vi ) this leads to small t@xmath24 ( 1 - 2 k ) or eventually spin glass phase because of the frustration effects . as we increase e@xmath52 the couplings rapidly loose their oscillating character and become more and more ferromagnetic , thus t@xmath24 increases until it reaches its maximum . afterwards , when the acceptor level becomes larger and larger the ferromagnetic couplings becomes shorter range ( double exchange regime ) , thus leading to a suppression of ferromagnetism . ( see text ) in the ( @xmath57,@xmath58 ) plane for a 3@xmath32 mn doped gaas compound and a hole density @xmath51 ( see ref . ) . the couplings used have been obtained from lsda calculations . ( ( b ) right ) spin stiffness d ( mev @xmath59 ) in ga@xmath46mn@xmath47as as a function of @xmath44 ( see ref . ) . the hole concentration is set to @xmath51 ( well annealed compounds ) . squares correspond to the v - j model calculations @xcite ) , hexagons to ab initio values @xcite , ( a ) and ( b ) circles to experimental measurements extracted from ref . and . the circles with a cross correspond to annealed samples , width=624 ] we now discuss the low energy spin excitation spectrum in iii - v doped systems . we will focus on the case of mn doped gaas since , to our knowledge , no experimental data are available for the other compounds . for this purpose we have calculated the dynamical spectral function that provides deeper insight into the underlying spin dynamics . this physical quantity can be directly and accurately probed by inelastic neutron scattering ( ins ) experiments . the averaged dynamical spectral function is defined as follows @xmath60 where @xmath61 means average over disorder . for a given configuration the spectral function is given by , @xmath62 where , @xmath63 in fig . [ fig6]-left we have plotted @xmath64 in the ( @xmath57,@xmath58 ) plane . the calculations are performed for a mn concentration of 3@xmath32 in gaas . the mn - mn couplings are those obtained from local spin density approximation calculations ( see ref . ) . in contrast to what one usually observes in weakly disordered magnetic systems , in the dilute mn doped gaas , well - defined excitations exist only in a restricted region of the brillouin zone centered around the @xmath65-point ( q = ( 0 , 0 , 0 ) ) . as we move away from the center of the brillouin zone , the width of the excitation increases rapidly . beyond a momentum cut - off q@xmath66 no well defined excitations exists anymore . this is a consequence of the short range nature of the exchange integrals . a recent study @xcite of the magnetic properties of mn doped zro@xmath67 has revealed that @xmath68 , where @xmath69 is the percolation threshold . that should also hold in the present case . note that @xmath69 is about 0.0075 in mn doped gaas ( see fig . [ fig4 ] ) . in the region of well defined excitations we find the expected quadratic magnon dispersion @xmath58(q)=d@xmath70q@xmath59 , where d is the so called spin stiffness . in fig . [ fig6]-right we have plotted d as a function of @xmath44 . the theoretical values , obtained both via v - j model and ab initio , are shown together with available experimental data . details concerning the calculations are given in refs . and . first , we observe , for the whole concentration range a very good agreement between the v - j and ab initio based calculations . we have also found a good quantitative agreement with the experimental data for both 3@xmath32 and 5@xmath32 doped samples . regarding the 6@xmath32 doped case , one of the annealed sample agree very well whilst the other has a lower value . in the latter case the average experimental spin stiffness is 90 @xmath71 20 mev @xmath72 . on the other hand the v - j model and ab initio calculations give respectively 120 @xmath71 30 mev@xmath72 and 130@xmath71 30 mev@xmath72 . note that , the uncertainty in the theoretical values result from the sensitivity to the magnetic couplings at relatively high distances between localized spins . thus the agreement is still reasonably good for this concentration as well . we have demonstrated that we can describe quantitatively the physics of dms , both transport and magnetism , within a coherent picture . the minimal v - j hamiltonian is the missing link that bridges the gap between complex and material specific first principle studies and model approaches . it has been shown that the physics is essentially controlled by the position of the p - d acceptor level with respect to the top of the valence band . this model approach continuously explain the change in the nature of the couplings . the two extreme regimes , rkky in mn doped ii - vi such as ( zn , mn)te and double exchange like in ( ga , mn)n , are described within the same picture . the agreement between ab initio , v - j model and experimental data are impressive ( curie temperatures , low energy magnetic excitation spectrum ) . the v - j model clearly explains the reason why mn doped gaas exibits the highest critical temperature among both ii - vi and iii - v compounds and its proximity to metal - insulator transition . hence , this model provides an efficient tool to find other pathways towards room temperature ferromagnetism , such as the influence of correlated disorder and nanostructuration of the materials for instance . 99 t. jungwirth et al . , rev . * 78 , * 809 ( 2006 ) . for a review see k. sato et al . * 82 , * 1633 ( 2010 ) . c. timm , j. phys . matter * 15 , * r1865 ( 2003 ) . t. dietl et al . science . * 287 , * 1019 ( 2000 ) . j. m. luttinger and w. kohn , phys . rev . * 97 , * 869 ( 1955 ) . e. o. kane , j. phys . solids * 1 , * 249 ( 1957 ) . more details on bi - orthogonality can be found in j. dieudonn , mich . j. * 2 , * 7 ( 1953 ) . g. bouzerar et al . * 85 , * 4941 ( 2004 ) ; g. bouzerar et al . * 69 , * 812 ( 2005 ) . g. bouzerar , eur . * 79 , * 57007 ( 2007 ) . a. chakraborty and g. bouzerar , phys . rev . b * 81 , * 172406 ( 2010 ) . g. bouzerar et al . , phys . rev . b * 72 , * 125207 ( 2005 ) . r. bouzerar et al . b * 82 , * 035207 ( 2010 ) . g. bouzerar and o. cpas phys . b * 76 , * 020401 ( 2007 ) . g. bouzerar , r. bouzerar and o. cpas phys . b * 76 , * 144419 ( 2007 ) . l. bergqvist et al . , * 93 , * 137202 ( 2004 ) . k. sato et al . , phys . b * 70 , * 201202 ( 2004 ) . r. bouzerar et al . , * 78 , * 67003 ( 2007 ) . r. bouzerar and g. bouzerar , europhys . * 92 , * 47006 ( 2010 ) . s. barthel et al . , eur . phys . j. b * 86 , * 11 ( 2013 ) . r. bouzerar and g. bouzerar , new journal of physics * 13 , * 023002 ( 2011 ) . edmonds et al . appl . phys . lett * 81 , * 4991 ( 2002 ) . k. w. edmonds et al . * 92 , * 03201 ( 2004 ) . d. chiba et al . applied physics letters * 82 , * , 3020 ( 2003 ) . t. jungwirth et al . b * 72 , * 165204 ( 2005 ) . p. r. stone et al . * 101 , * 087203 ( 2008 ) .

after more than a decade of intensive research in the field of diluted magnetic semiconductors ( dms ) , the nature and origin of ferromagnetism , especially in iii - v compounds is still controversial . many questions and open issues are under intensive debates . why after so many years of investigations mn doped gaas remains the candidate with the highest curie temperature among the broad family of iii - v materials doped with transition metal ( tm ) impurities ? how can one understand that these temperatures are almost two orders of magnitude larger than that of hole doped ( zn , mn)te or ( cd , mn)se ? is there any intrinsic limitation or is there any hope to reach in the dilute regime room temperature ferromagnetism ? how can one explain the proximity of ( ga , mn)as to the metal - insulator transition and the change from ruderman - kittel - kasuya - yosida ( rkky ) couplings in ii - vi compounds to double exchange type in ( ga , mn)n ? in spite of the great success of density functional theory based studies to provide accurately the critical temperatures in various compounds , till very lately a theory that provides a coherent picture and understanding of the underlying physics was still missing . recently , within a minimal model it has been possible to show that among the physical parameters , the key one is the position of the tm acceptor level . by tuning the value of that parameter , one is able to explain quantitatively both magnetic and transport properties in a broad family of dms . we will see that this minimal model explains in particular the rkky nature of the exchange in ( zn , mn)te/(cd , mn)te and the double exchange type in ( ga , mn)n and simultaneously the reason why ( ga , mn)as exhibits the highest critical temperature among both ii - vi and iii - v dms s . aprs plus dune dcennie de recherches intensives dans le domaine des semi - conducteurs magntiques dilus ( dms ) , la nature et lorigine du ferromagntisme , en particulier dans les composs iii - v , restent controverses . de nombreuses questions et problmes ouverts sont toujours sujets dintenses dbats . pourquoi parmi la grande famille des matriaux iii - v , et pour une concentration donne en mtal de transition , le compos ( ga , mn)as reste - t - il le candidat prsentant encore la temprature critique la plus leve ? comment peut - on comprendre que ces tempratures soient presque deux ordres de grandeur suprieures celles observes dans ( zn , mn)te dop en trous ou ( cd , mn)se ? subsiste - t - il pour ces matriaux dilus un espoir dobserver un ordre ferromagntique au del de la temprature ambiante ou est - il fatalement ananti par des limitations physiques intrinsques ? comment expliquer que ( ga , mn)as soit si proche de la transition mtal - isolant ? comment comprendre la nature des couplages magntiques passant typiquement de rkky dans les composs ii - vi double change dans ( ga , mn)n ? des tudes , bases sur la thorie de la fonctionnelle de la densit , ont pu fournir avec prcision les tempratures critiques dans divers composs . cependant un modle thorique en mesure de fournir une vision unifie et une comprhension de la physique sous - jacente manquait toujours . trs rcemment , dans le cadre dun modle minimal , il a t possible de montrer que , parmi les paramtres physiques , la cl rside dans la position du niveau accepteur de limpuret magntique . en adaptant ce dernier , il devient en effet possible dapprhender la diversit des proprits magntiques et aussi de transport dans une large famille de dms . nous verrons alors que le modle minimal explique non seulement la nature rkky des couplages magntiques dans ( zn , mn)te/(cd , mn)te ou leur caractre double change dans ( ga , mn)n , mais aussi la raison pour laquelle ( ga , mn)as prsente les tempratures de curie les plus leves parmi les dms ii - vi et iii - v .

the decomposition of fields into eigenmodes is a well established technique to solve various problems within physical sciences . the most prominent example refers to schrdinger s equation , within the field of quantum mechanics , where energy spectra of atoms are determined via the eigenvalue spectra and associate wavefunctions of the hamiltonian operator . indeed , electron orbits are eigenmodes of the energy , angular momentum , and spin operators @xcite and as such they deliver fundamental insights into the physics of atoms . within classical mechanics , modes of vibration of music instruments give , for example , their resonant frequencies while their spectrum is associated with the shape of the instrument @xcite . in the optical domain , mode decomposition is used in order to describe light propagation within waveguides @xcite , photonic crystals @xcite , and optical cavities @xcite . in the case of waveguides and photonic crystals , for example , eigenmodes describe electromagnetic fields that are invariant in their intensity profile as they propagate along the fibre or crystal . additionally , these modes are orthogonal and as such light coupled to one of these modes remains , in theory , in this mode forever . this optical mode decomposition can be expanded to include additional operators such as orbital and spin angular momentum @xcite . in this paper , we report a novel method which we term `` quadratic measure eigenmodes ( qme ) '' which represents a generalization of the powerful concept of eigenmode decomposition within the field of optics . crucially , it is shown that eigenmode decomposition is applicable to the case of any quadratic measure which is defined as a function of the electromagnetic field . prominent examples of optical quadratic measures include the energy density and the energy flux of electromagnetic fields . the qme method makes it possible to describe an optical system and its response to incident electromagnetic fields as a simple mode coupling problem and to determine the optimal `` excitation '' for the given measure considered . intuitively , a superposition of initial fields is optimized in a manner that the minimum / maximum measure is achieved . for instance , the transmission through a pinhole is optimized by maximizing the energy flux through the pinhole . from a theoretical perspective , the qme optimization method is mathematically rigorous and may be distinguished from the multiple techniques currently employed ranging from genetic algorithms @xcite and random search methods @xcite to direct search methods @xcite . the major challenge encountered in any such approximate optimization and engineering of optical properties is the fact that electromagnetic waves interfere . as such the interference pattern not only makes the search for an optimum beam problematic but crucially renders the superposition found unreliable , as the different algorithms may converge on different local minima which are unstable with respect to the different initial parameters in the problem . in contrast , our proposed qme method yields a unique solution to the problem and directly determines the optimum ( maximal / minimal ) measure possible . in the first part of the paper , we introduce the background of the qme theory and show its properties in a general context of optimizing the quadratic measures of interfering waves . in the second part , we apply the qme formalism to maximize the transmission through apertures and to minimize the focal spot size . for these applications , we describe , respectively , the electromagnetic field as a superposition of scalar laguerre - gaussian beams and vectorial bessel beams . in the final part of the paper , we report a particular experimental implementation of the qme method using computer controlled spatial light modulators to squeeze the spot size of a superposition of bessel beams . the paper concludes with a discussion of the particular results obtained and with general comments on the versatility of the qme method to a wide range of problems . the qme method is based upon two fundamental properties of the electromagnetic field and its interactions . firstly , the approach relies on the linearity of the electromagnetic fields , _ i.e. _ , the sum of two solutions of maxwell s equations is itself a solution of them . as we consider free space propagation , this criteria is satisfied . the second property relates to the interaction of the electromagnetic field with its environment . all such interactions can be written in the form of quadratic expressions with respect to the electric and magnetic fields . examples include the energy density , the energy flow , and maxwell s stress tensor . this allows us to designate appropriate qme to various parameters ( _ e.g. _ spot size ) and subsequently ascertain the optimal eigenvalue which , in the case of a spot size operator , yields a sub - diffraction optical spot . in this section , we present the details of the theory underpinning our approach . [ [ electromagnetic - waves ] ] electromagnetic waves + + + + + + + + + + + + + + + + + + + + + to demonstrate our method , we consider monochromatic solutions of the free space maxwell s equations : @xmath0 where @xmath1 and @xmath2 are the spatial part of the electric and magnetic vector fields and where @xmath3 and @xmath4 denote the vacuum permittivity and permeability . the time dependent carrier wave is given by @xmath5 . these monochromatic solutions of maxwell s equations can be written in an integral form linking the electromagnetic fields on the surface @xmath6 with the fields at any position @xmath7 , @xmath8 where @xmath9 is a shorthand for the two electromagnetic fields having six scalar components @xmath10 . the integration kernel @xmath11 corresponds to a propagation operator giving rise to different vector diffraction integrals such as huygens , kirchhoff @xcite , and stratton - chu @xcite . [ [ quadratic - measures ] ] quadratic measures + + + + + + + + + + + + + + + + + + crucially all `` linear '' and measurable properties of the electromagnetic field can be expressed as quadratic forms of the local vector fields and are therefore termed _ quadratic measures_. for instance , the time averaged energy density of the field is proportional to @xmath12 while the energy flux to @xmath13 . the asterisk @xmath14 stands for the complex conjugate . integrating the first quantity over a volume determines the total electromagnetic energy in this volume , and integrating the normal energy flux across a surface yields the intensity of the light field incident on this surface . all the quadratic measures @xmath15 can be represented in a compact way by considering the integral @xmath16 where the kernel @xmath17 is hermitian with @xmath18 the adjoint operator including boundary effects for finite volumes . table [ tab ] enumerates some operators associated to common quadratic measures . the integrand part of most of these quadratic measures corresponds to the conserving densities , which together with the associated currents are lorentz invariant @xcite . the volume , over which the integral is taken , does not need to be the whole space and can be a region of space , a surface , a curve , or simply multiple points . to account for this general integration volume , we broadly term it the region of interest ( roi ) in the following . .common quadratic measure operators including the energy operator ( eo ) , intensity operator ( io ) , spot size operator ( sso ) , linear momentum operator ( lmo ) , orbital angular momentum operator ( oamo ) , and circular spin operator ( cso ) . the vector operators include the subscript @xmath19 indicating the different coordinates and @xmath20 the associated unit vectors . [ cols="^,^",options="header " , ] [ [ quadratic - measures - eigenmodes ] ] quadratic measures eigenmodes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + finally , using the general definition ( [ qm : genl ] ) of the quadratic measure it is possible to define a hilbert sub - space , over the solutions of maxwell s equations , with the energy operator defining the inner product . furthermore , any general quadratic measure defined by ( [ qm : genl ] ) can be represented in this hilbert space by means of its spectrum of eigenvalues and eigenfunctions defined by : @xmath21 depending upon the kernel @xmath22 or operator @xmath23 , the eigenvalues @xmath24 form a continuous or discrete real valued spectrum which can be ordered . this gives direct access to the solution of maxwell s equations with the largest or smallest measure . the eigenfunctions are orthogonal to each other ensuring simultaneous linearity in both field and measure . in the following , we study the case of different quadratic measure operators and their spectral decomposition into the qme . the convention for operator labeling we adopt is to use the shorthand qme followed by a colon and a shorthand of the operator name . in the following , we discuss some examples of different quadratic measure operators . [ [ qme - intensity - operator - qmeio ] ] qme : intensity operator ( qme : io ) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the quadratic measure corresponding to the qme : io measures the electromagnetic energy flow across a surface roi : @xmath25 where @xmath26 is chosen such that it corresponds to the projection of the poynting vector on the normal @xmath27 to the surface . the eigenvector decomposition of this operator can be used to maximize the optical throughput through a pinhole or to minimize the intensity in dark spots . considering a closed surface roi surrounding an absorbing particle , the qme approach gives access to the field that either maximizes or minimizes the absorption of this particle . [ [ qme - spot - size - operator - qmesso ] ] qme : spot size operator ( qme : sso ) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + one way to define the spot size of a laser beam is by measuring the second order intensity moment ( soim ) @xmath28 @xcite . @xmath28 can be expressed as @xmath29 with @xmath30 as defined in eq . and the qme : sso defined by @xmath31 where @xmath7 is the position vector and @xmath32 the centre of the beam . accordingly , the square root of the qme : sso eigenvalues multiplied by two gives direct access to the respective beam size provided that the intensity is normalized to one within the roi , i.e. , @xmath33 ; [ [ qme - optical - force - operator - qmeofo ] ] qme : optical force operator ( qme : ofo ) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the optical force acting on a scattering object can be calculated by considering the momentum flux , given by maxwell s stress tensor , across a surface roi , surrounding the scattering particle . there are three quadratic measures associated with the optical force acting on the particle , one for each orthogonal direction : @xmath34 where @xmath35 corresponds to the measure of the force . the eigenvector decomposition of this operator can be used to maximize the optical scattering and optical trapping force on microparticles . this approach can be extended through the use of the angular momentum operator , defined locally by @xmath36 . in this section , we put the qme concept into practice and provide a couple of examples which demonstrate the striking applicability of the qme formalism to answer different questions within the field of optics . one such question is determining the largest intensity that may be transmitted through a given optical structure such as metallic apertures @xcite . another question is what is the smallest optical spot one may achieve . before we discuss these questions , we show how the operator formalism presented in section [ s : fundcon ] is rendered into a practical formalism which we have also applied in the experiments described in section [ s : sec4 ] . for practical purposes , in particular in terms of the experimental realization of the qme concept , the optimization procedure is spatially separated ; that is we consider 1 ) an initial plane at the propagation distance @xmath37 where we can superpose a set of @xmath38 fields @xmath39 and @xmath40 ( @xmath41 ) shaped both in amplitude and phase and 2 ) a target plane at the propagation distance @xmath42 where we have the roi within which the optimization is actually carried out . due to linearity a superposition of fields in the initial plane is rendered into a superposition in the target plane both featuring the same set of superposition coefficients : @xmath43 based on this , the qme : io defined in can be rewritten as @xmath44 the matrix @xmath45 is a @xmath46 matrix with the elements given by the overlap integrals @xmath47 this matrix is equivalent to the qme : io on the hilbert subspace defined by the fields @xmath48 . @xmath45 is hermitian and positive - definite which implies that its eigenvalues @xmath49 ( @xmath50 ) are real and positive and the eigenvectors @xmath51 are mutually orthogonal . accordingly , the largest eigenvalue @xmath52 and the associated eigenvector @xmath53 will deliver the initial plane ( @xmath37 ) or target plane ( @xmath42 ) superposition @xmath54 which maximizes the intensity within the roi . we remark , that the case of a superposition of scalar fields @xmath55 , the matrix operator takes on the simpler form @xmath56 an approach used in section [ s : sec32 ] to maximize the transmission through a pinhole . similar to the qme : io , the qme : sso defined in eq . can be rewritten as @xmath57 where @xmath58 and @xmath59 must be represented in the intensity normalised base @xmath60 in order to fulfill the requirement of unity intensity within the roi ( @xmath61 , see paragraph `` qme : spot size operator ( qme : sso ) '' in section [ sec : theoreticalbackground ] ) . @xmath62 is a @xmath46 matrix with the elements given by @xmath63 after transforming back to the original base we denote the eigenvalues of @xmath58 as @xmath64 and the eigenvectors as @xmath65 . then , the eigenvector @xmath66 associated with the smallest eigenvalue @xmath67 corresponds to the smallest spot achievable within the roi . the respective superposition is @xmath68 for @xmath37 and @xmath42 . we employed this vectorial definition to minimize the size of a focused spot in section [ s : sec34 ] using a superposition of vector bessel beams . the spot size minimization performed both in section [ s : sec33 ] using a superposition of lg beams and in the experimental section [ s : sec4 ] is based on the simpler scalar version ot the qme : sso matrix defined as @xmath69 . ( a ) transversal and ( b ) longitudinal cross section of a gaussian beam ( @xmath70 and @xmath71 ) . ( c ) and ( d ) qme superposition delivering the highest intensity in the roi ( @xmath72).,width=491 ] and for different numbers @xmath73 of lg modes considered . ( b ) relative intensity @xmath74 of the lg modes ( @xmath75 ) decomposing the qme intensity optimized beam.,width=491 ] in this subsection , we consider a superposition of lg beams propagating in the @xmath76-direction and modulating the carrier wave @xmath77 . in cylindrical coordinates , lg beams are defined by @xcite : @xmath78 with @xmath79 , @xmath80 , and @xmath81 where @xmath82 , @xmath83 , @xmath84 are the gaussian beam waist , vacuum wave vector , and optical frequency , respectively . this beam profile is a solution of the paraxial equation for integer values of @xmath85 and @xmath86 corresponding respectively to the radial and azimuthal index of the beam . within the paraxial approximation , the intensity of the beam transmitted through a planar roi defined by a disk centered on @xmath87 is proportional to @xmath88 where @xmath89 is the radius of the disk . the coefficient @xmath90 is the normalization factor such that the total intensity of the beam , for an infinite roi , is unity . the transmission is maximized using the practical implementation of the qme concept described above in section [ s : sec31 ] ; that is the qme : io was assembled according to eq . using the respective representation in cylindrical coordinates . we only considered the radial family of lg beams ( @xmath70 ) and performed the optimization in the plane at @xmath91 . figure [ fig1 ] shows the final superposition @xmath92 in the case of a roi with a radius equal to the waist of the gaussian envelope . in fig . [ fig2 ] , we observe that the maximal transmission achievable via the qme intensity optimized beams depends on the number of lg beams considered in the superposition and on the size of the roi . indeed , the qme for a smaller roi needs a larger number of lg modes to achieve 100% transmission . ) considering 25 lg modes . @xmath93 is the relative soim measured according to eq . . the strehl ratio in ( a ) is @xmath94.,width=491 ] fulfilling the intensity criteria for the @xmath95 case . the arrows indicate the corresponding scales . ( b ) ratio between the roi intensity of the smallest spot size eigenmode and the largest intensity achievable in the roi ( strehl ratio).,width=491 ] using a superposition of lg beams we have also minimized the size of a focal spot using the representation of the qme : sso in cylindrical coordinates . it is important to note at this point that we only retain the intensity eigenmodes whose eigenvalues are within a chosen fraction of total intensity . this is equivalent to considering only beams that have a significant intensity contribution in the roi . intuitively , the optimization procedure may be performing so well that a spot of size zero is finally obtained if no intensity threshold is applied . figure [ fig3 ] shows the smallest spot superposition where we observe the appearance of sidebands just outside the roi . these sidebands are a secondary effect of squeezing the light below its diffraction limit . it is these sidebands that decrease the efficiency of the squeezed spot with respect to the maximal possible intensity in the roi as calculated via the qme : io . using the ratio between these two intensities we can define the intensity strehl ratio @xcite for the qme : sso ( see fig . [ fig4]b ) . we remark that both , the spot size and the strehl ratio , show resonances as a function of the roi size . this can be explained considering the number of intensity eigenmodes used for the spot size operator . indeed , as the roi size decreases , so does the number of significant intensity eigenmodes . each time one of these modes disappears ( step in fig . [ fig4 ] ) , we have a sudden increase in the minimum spot size achievable accompanied with an enhanced strehl ratio as we drop the most intensity inefficient mode . overall , the strehl ratios determined in our studies predominantly exceeded values of @xmath96 even when spots were tightly squeezed . therefore , the observed decrease of intensity is not to severe in terms of potential applications of squeezed beams for optical manipulation and imaging . , ( c ) @xmath97.,title="fig:",width=453 ] , ( c ) @xmath97.,title="fig:",width=453 ] analogously to the approach to calculate the smallest spot size in a planar roi , we can determine the qme : sso for a volume . in this case , the sidebands appear in the outside of the roi in both the lateral and longitudinal directions . the different volumes considered in fig . [ fig5 ] suggest that there is an intrinsic link between squeezing light in the lateral direction and in the longitudinal direction . on a final note , we remark that squeezing light below its diffraction limit may be associated with the effect of super - oscillations @xcite . this refers specifically to the ability to have a local @xmath19-vector ( gradient of the phase ) larger than the spectral bandwidth of the original field . to visualize this effect , in the case of qme spot size optimized beams , we have calculated the spectral density of the radial wave - vector for the smallest planar spot @xcite . as shown in fig . [ fig6 ] , this spectral density clearly identifies a spectral bandwidth ( white background in fig . [ fig6 ] ) . regions of the beam which exhibit locally larger wave - vectors than the ones supported by this spectral band width correspond to super - oscillating regions . the local wave vector is defined as @xmath98 where @xmath99 defines the phase of the analytical signal @xmath100 . we observe , that super - oscillations occur only in the dark region of the beam . additionally , when the roi is large compared to the gaussian beam waist @xmath82 , there are no super - oscillating regions . these only appear when the beam starts to be squeezed . . the yellow dashed circle shows the position of the smallest zero - intensity circle taken as the roi inside which the spot size is calculated . the spot size is normalized to the spot size of the reference bessel beam . ( b ) reference bessel beam corresponding to the largest cone angle @xmath101 . the soim of the reference bessel beam is denoted as @xmath102 . ( c ) qme spot size optimized beam for a superposition of bessel beams ( @xmath103 $ ] ) for a large roi highlighted by the dashed yellow circle . strehl ratio : @xmath104 . ( d ) qme spot size optimized beam for a small roi . strehl ratio : 0.@xmath104 . the gray - scaled region shows the sidebands while the color range the roi . notice that the two scales are different.,width=453 ] of the bessel beam superposition as a function of the relative roi radius @xmath105 . the soim @xmath102 and the roi radius @xmath106 are associated with the reference bessel beam shown in fig . [ fig7](b ) , where the roi is indicated as dashed circle . for comparison , the red dot indicates the location of the reference beam in the @xmath107 vs. @xmath105 plot . ( b ) strehl ratio vs relative roi radius @xmath105.,width=453 ] the paraxial approximation employed above in the case of lg beams can be used to describe sub - diffracting beams but breaks down when beams are tightly focused . as a consequence we must consider full vectorial solutions of maxwell s equations . here , we have chosen bessel beams as a base - set and determined the superposition of bessel beams which minimized the spot size in a planar finite roi . note that the problem of the finite intensity of bessel beams @xcite is easily circumvented here due to the finite roi size considered . the monochromatic electric vector field of the vectorial bessel beam may explicitly be expressed as @xcite @xmath108 where @xmath109 and @xmath110 are the transversal and longitudinal wave vectors with @xmath111 the characteristic cone angle of the bessel beam . @xmath112 , @xmath113 and @xmath114 are the unit vectors in the cartesian coordinate system . the parameter @xmath86 corresponds to the azimuthal topological charge of the beam while @xmath115 and @xmath116 are associated with the polarization state of the beam . the magnetic field @xmath2 was deduced according to maxwell s equations and the qme : io and qme : sso operators are assembled according to the expressions and , respectively . figure [ fig7 ] shows a comparison between airy disk , bessel beam , and qme optimized spot considering a numerical aperture of na@xmath117 . as in the case of the lg beams , squeezing the focal spot is accompanied by side bands and a loss in efficiency shown by the strehl ratio ( see fig . [ fig8 ] ) . , @xmath118 , @xmath119 , @xmath120 . laser : jds uniphase hene laser , @xmath121 , @xmath122 , slm : holoeye heo 1080 p dual display system , @xmath123 , @xmath124 . ccd camera : basler pilot pia640 - 210gm , @xmath125 , @xmath126.,scaledwidth=90.0% ] according to the theoretical foundations of the qme concept , the successful experimental implementation within the field of optics requires a _ linear _ optical system along with the ability to shape laser fields in both amplitude and phase . we have achieved this by using the experimental setup shown in fig . [ fig : setup ] . a @xmath127 laser beam is expanded and subsequently amplitude modulated by a spatial light modulator ( slm ) display operating in conjunction with a pair of crossed polarizers . analogously to a liquid crystal display on a computer or laptop monitor , the liquid crystal slm display rotates the polarization of the incident light by an angle depending upon the voltage applied to the display pixels . the amplitude modulated beam is then imaged onto a second slm display through a pair of lenses . this second slm display along with a subsequent fourier lens and aperture served to modulate the phase of the laser beam in the standard first order configuration @xcite . the field modulations of interest were encoded as rgb images where the blue channel represented the amplitude and the green channel the phase modulation . the slm controller extracted these information and applied the two channels to the respective panel . we have performed calibration measurements to ensure that both the amplitude and phase modulation exhibited a linear dependence on the applied 8-bit color value between 0 and 255 . a ccd camera allowed us to record images of laser fields in the fourier plane of lens 5 . to conform this experimental section to the conventions introduced in section [ s : sec31 ] we remark that we shaped a set of test fields both in amplitude and phase in the initial plane at @xmath37 which coincided with the two slm panels and subsequently minimized the size of a focal spot in the target plane at @xmath42 which coincided with the ccd camera chip . for our proof - of - principle experiments we ignored polarization effects and considered a set of scalar fields @xmath128 ( @xmath38 ) where the ccd camera detected the associated intensity @xmath129 . both the qme : io and the qme : sso were assembled from these fields according to the scalar expressions and , respectively . the amplitudes @xmath130 were determined by simply recording an associated intensity image @xmath131 and subsequently taking the square root . we used the three - step phase retrieval algorithm described in ref . @xcite to retrieve the phase modulations @xmath132 . this algorithm is based on interference with a reference field @xmath133 . the reference field s intensity was distributed uniformly over the rois considered in our experiments by adding a square phase @xmath134 to the reference field using the slm phase panel . moreover , a constant phase term @xmath135 was added for @xmath136 , and the superimposed fields @xmath137 were then encoded onto the slm . the associated intensity distributions explicitly read @xmath138 where the spatial coordinates @xmath139 were omitted for brevity . these three intensity distributions represent a 3-dimensional equation system with three unknowns @xmath140 , @xmath141 and @xmath142 . the latter is explicitly obtained as @xmath143 where @xmath144 is the two argument arctangent function corresponding to the argument of the complex number @xmath145 . note that the reference phase @xmath146 cancels out when calculating the operator elements due to multiplication of a complex conjugate field @xmath147 with a complex field @xmath148 . therefore eq . yields the adequate phase modulation required to assemble the qme operators . during the course of our experiments we verified the linearity of our optical system by performing a comparison between what we term the `` experimental superposition ( exp - s ) '' and the `` numerical superposition ( num - s ) '' . the exp - s refers to the case where the set of qme optimized superposition coefficients @xmath149 is used to encode the optimized superimposed field onto the slm . the ccd camera then detected the intensity @xmath150 corresponding to this encoded optimized field . the num - s utilizes the fields @xmath151 , which were individually measured to assemble the qme operators , in order to _ numerically _ determine the intensity distribution as @xmath152 . crucially , linearity is verified if @xmath153 . this is indeed observed in our experiments as demonstrated in the following subsection which features a comparison of experimental and numerical intensity distributions . of the bessel beam s central core . @xmath28 in ( d ) is the soim of the airy disk.,scaledwidth=80.0% ] in our experiments , we used @xmath154 non overlapping amplitude ring masks with a constant phase modulation as fields of interest @xmath155 . after propagation through the fourier lens 5 ( see fig . [ fig : setup ] ) the resulting fields @xmath151 form a set of bessel beams . figure [ fig : bessels](a ) shows the largest ring modulation encoded onto the slm with the resulting bessel beam shown in fig . as this particular bessel beam comes along with the highest na compared to the bessel beams created with smaller ring modulations , the beam shown in fig . ( b ) exhibits the smallest central spot of all beams realized in our experiments . the soim of the bessel beam featuring the smallest core is denoted as @xmath156 and used as reference for the measurements presented below . for comparison figure [ fig : bessels](c ) depicts a circular aperture which is encoded onto the slm in order to observe the airy disk ( see fig . the soim of the airy disk is approximately @xmath157 times larger than the core of the reference bessel beam as expected @xcite . for different roi radii in pixel as indicated in the top left corner of all graphs shown . the roi is exemplary indicated as a dashed ring in the left hand side intensity distribution . the number in the bottom left corner represents the soim @xmath28 in units of the reference soim @xmath102 . _ central row _ : optimized experimental distribution as rgb encoded onto the slm . _ bottom row _ : intensity distributions @xmath150 . the relative soim @xmath158 is indicated in the lower left corner.,scaledwidth=80.0% ] the results of the performed qme spot size minimization are shown in fig . [ fig : spots ] for different sizes of the roi . to begin with , the comparison of the num - s intensity distribution @xmath159 ( top row ) and the exp - s intensity distributions @xmath150 ( bottom row ) clearly reveals good agreement and thus verifies the linearity of our optical system as elucidated above . for completeness , the central row shows the exp - s superposition in rgb format as encoded onto the slm . the color code features a blue channel representing the amplitude modulation from 0 ( black ) to 1 ( blue ) and a green channel corresponding to phase modulations from 0 ( black ) to @xmath160 ( green ) . next , we conclude from the measured relative soim @xmath158 that the spot size decreases if the roi size is reduced . the reduced spot size is achieved at the expense of the spot intensity which is redistributed to a ring outside of the roi similar to the theoretical results presented in section [ s : sec34 ] and fig . [ fig7 ] . referring to the exp - s data , for @xmath161 the spot size is reduced to @xmath162 of the size of the reference bessel beam s core and even further to @xmath163 for @xmath164 . the latter result is somewhat vague , though , due to the low spot intensity which may be truncated by the sensitivity threshold of the ccd detector and thus may appear smaller . however , our experimental results overall clearly verify the qme concept applied to spot size minimization . moreover , the results strongly suggest that the qme optimization may indeed squeeze spots to the subdiffractive regime since the optimal superposition of bessel beams not only beats the airy disk but also the reference bessel beam . within the roi depending on the roi radius @xmath89 in pixel for numerical superposition of the fields @xmath151 in the ccd plane.,scaledwidth=60.0% ] we have performed an additional comparison of the experimental results to the theoretically predicted ones : the relative soim @xmath158 was evaluated for the num - s for roi radii ranging from @xmath165 to @xmath166 in steps of one pixel ( see fig . [ fig : somvsr ] ) . in comparison to the corresponding graph shown in fig . [ fig8 ] we observe agreement in terms of both the general decrease of @xmath158 with decreasing @xmath89 and the peaks occurring periodically along the @xmath89-axis . surprisingly , for @xmath167 , we do not observe values of @xmath168 ; that is the qme spot minimization does not deliver , as one might naively expect , the reference bessel beam featuring the smallest soim @xmath156 of all bessel beams considered . this is due to the ring structure of bessel beams which significantly adds to the soim and therefore to the spot size in the case of large rois . as a consequence , the qme optimization aims to superimpose the set of bessel beams in a manner that the rings are destructively interfered within the roi , only retaining the central core . given that the ring structure is essential for the reduced core size of bessel beams compared to the diffraction limited airy disk we observe values of @xmath28 which are larger than @xmath102 . indeed , the values of @xmath28 are fairly close to the size of the airy disk which was determined as @xmath169 . this strongly suggests that for large rois the qme spot minimization yields a superposition of bessel beams which closely resembles the diffraction limited airy disk . finally , we remark that for @xmath170 the relative soim @xmath158 becomes very small and has to be taken with care due to the possible truncation of the spot mediated by the ccd detector s sensitivity threshold as mentioned above . we have theoretically and experimentally demonstrated a novel approach based on quadratic measures eigenmodes that enables the optimization of different optical measures . the theory that we employ is rigorous and based on considering the light - matter interaction as a quadratic measure originating from the fields described by maxwell s equations . excitingly , we can define many quadratic measure operators to which our approach is applicable ( see table [ tab ] ) . the method is thus very powerful and the generic nature of our approach means that it may be applied for example to optimize the size and contrast of optical dark vortices , the raman scattering or fluorescence of any samples , the optical dipole force , and the angular / linear momentum transfer in optical manipulation . in the present paper we have verified the rigor of the method by demonstrating experimental spot size operator and intensity operator optimization using laguerre - gaussian and bessel light modes using a dual slm approach to implement the technique . we envisage the qme approach as providing a powerful and versatile theoretical and practical toolbox . our generic approach is applicable to all linear physical phenomena where generalized fields interfere to give rise to quadratic measures . we thank the epsrc nanoscope basic technology consortium for funding . mark dennis is acknowledged for the introduction to super - oscillations . kd is a royal society - wolfson merit award holder .

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in three - dimensional einstein gravity , there are no dynamical degrees of freedom , i.e. , no gravitons that mediate the interactions between massive objects in the vacuum @xcite , with or without the cosmological constant , unless some higher derivative terms , like the ( gravitational ) chern - simons term , are introduced @xcite . although spacetime outside of matter is locally of constant curvature , some non - trivial spacetime solutions to the field equations exist that have either black hole @xcite or cosmological event horizons @xcite when the cosmological constant is nonzero . their metrics have the general form @xmath6 with the lapse ( squared ) and shift functions @xmath7 respectively . here , @xmath8 and @xmath9 are constant parameters whose signs ( and values ) depend on whether we are considering the black hole solution in ads space or the cosmological solution in ds space . the adm mass and angular momentum of this class of spacetimes , with a cosmological constant @xmath10 , are given by @xmath11 for the ( btz ) black hole solution ( where @xmath12 @xcite and @xmath13 for the ( kds@xmath14 ) cosmological solution ( where @xmath15 @xcite , respectively , where the three - dimensional newton constant @xmath16 is assumed positive . for the black hole case of eq . ( [ btz ] ) , we have @xmath17 in order that there is no naked conical singularity and @xmath18 denotes the _ outer / inner _ horizon . on the other hand , for the cosmological solution , there is no constraint on @xmath19 and @xmath20 in order that the horizon exists , unless @xmath21 is considered @xcite : even @xmath22 is allowed also . in this case , @xmath23 denotes the ( cosmological ) event horizon @xcite but @xmath24 does not signify an inner horizon ; the real parameter @xmath24 is introduced just for convenience @xcite . over the years black hole and cosmological solutions in three dimensions have fascinated theorists because of the potential insights they afford into quantum gravity . amongst the most intriguing applications @xcite have been black hole formation and the associated issues of gravitational collapse and the cosmic censorship conjecture @xcite . while there are numerous examples of initial conditions that form a naked singularity in the context of general relativity , none of them are generic as required by the terms of cosmic censorship conjecture . however it has been proposed that three dimensions might be an exception @xcite . a recent study of collapsing _ shells _ in three dimensions with _ no _ angular momentum showed that a naked singularity and a cauchy horizon can form as the final result of shell collapse for a _ broad _ variety of initial data @xcite . this feature is quite generic since the results are independent of the types of collapsing shell matter such as pressureless dust , polytropic matter , and a generalized chaplygin gas ( gcg ) . furthermore , similar behaviour also appeared in previous studies of collapsing dust in three dimensions @xcite and the formation of topological black holes in four dimensions @xcite . ( see refs . @xcite for higher - dimensional extensions . ) the formation of a naked singularity is of great importance in association with the ads / cft correspondence @xcite since one might ask whether or not the quantum correlation function at the ads boundary is properly defined in the presence of a bulk singularity . this in turn raises additional issues associated with the collapse to a naked singularity , particularly the proper inclusion of quantum ( gravity ) effects in the bulk @xcite . thus far the possible scenarios for the emergence of a naked singularity have been explicitly analyzed for collapse to final states ( black holes or otherwise ) with no angular momentum @xcite . indeed , relatively little is explicitly known about the gravitational collapse of matter with nonzero angular momentum in any dimension . it is therefore natural to take into account an extension of gravitational collapse to exterior black holes with rotation ( or other possible configurations ) and to investigate how rotational effects alter previous results @xcite concerning cosmic censorship violation . while this problem is technically formidable in higher dimensions , a study of gravitational collapse of shells in three dimensions is considerably more tractable . so we shall consider this problem in our paper . to this end , we introduce a co - rotating coordinate system on the shell , which simplifies the matching procedure . the angular momentum produces a potential barrier around the origin , preventing the shell from contracting to zero size . we analyze the possible collapse scenarios by investigating the effective potential for all types of equation of state . for a pressureless dust shell , we find that its angular momentum prevents the creation of a singularity at the origin , unlike the non - rotating case @xcite . previous work on rotating dust shell collapse was carried out in the hamiltonian formalism , an alternative to the darmois - israel formalism @xcite . for a more restricted class of scenarios , they found that singularities were also avoided . our work goes beyond this insofar as we consider shells with pressure . in this case , curvature singularities of finite spatial extent are formed before they meet the barrier , and the effective potential and surface stress - energy tensor diverge . therefore under rather generic conditions it is possible for a naked singularity to form , violating cosmic censorship as in the non - rotating collapse scenario @xcite . the outline of our paper is as follows . in sec . [ sec : coll ] , we present a set - up for the two - dimensional hypersurfaces with arbitrary rotations and a spherical symmetry . considering a co - rotating frame where the co - moving observer on the shell sees only the radial motion yields a simplified equation of motion for the shell . in sec . [ sec : dust ] , we consider the evolution of a pressureless dust shell and compare it to the results for non - rotating shell collapse @xcite . in sec . [ sec : pressure ] , we consider shells with pressure whose equations of state are the polytrope type , which includes the perfect fluid and the gcg . finally , we shall summarize and discuss our results in sec . [ sec : discussion ] . in this section , we consider rotating shells in three - dimensional einstein gravity with a cosmological constant @xmath25 . the bulk einstein equation is given by @xmath26 with a bulk stress - tensor @xmath27 . if we introduce a two - dimensional hypersurface with surface stress - energy tensor denoted by @xmath28 , then the three - dimensional manifold is divided into three parts the interior space @xmath29 , the exterior space @xmath30 , and the thin - shell hypersurface @xmath31 . the metric away from the shell decomposes as @xmath32 , where @xmath33 is a geodesic coordinate and @xmath34 is the heaviside step function . is equal to @xmath35 if @xmath36 , @xmath37 if @xmath38 , and indeterminate if @xmath39 . it has the following properties : @xmath40 , @xmath41 , and @xmath42 , where @xmath43 is the dirac delta function . ] we shall use the coordinate system @xmath44 in the interior and exterior spaces while we use the co - moving coordinate system @xmath45 on the shell . then , the evolution of the shell is obtained by the darmois - israel matching conditions between metrics and the corresponding extrinsic curvatures in the interior and the exterior geometries @xcite , @xmath46=0,\\ & & { 8 \pi g } { \mathcal s}_{ij}=-([k_{ij}]-g_{ij}[k ] ) \label{eq : jc2}\end{aligned}\ ] ] where @xmath47\equiv \lim_{\sigma\to 0 } ( { a_{+}}-{a_{-}})$ ] with the subscripts ` @xmath48 ' and ` @xmath49 ' denoting exterior and interior spacetimes , respectively . greek letters @xmath50 denote three - dimensional spacetime indices , whereas roman letters @xmath51 denote the two - dimensional indices on the shell . the combination of the metric junction condition and the induced einstein s equation on the shell will describe the effective motion of the shell . if we take a co - rotating frame on the shell by introducing @xmath52 , then the metric ( [ eq : metbtz ] ) becomes @xmath53 ^ 2\label{eq : indmet}\ ] ] and each metric in both regions is simply expressed as @xmath54 ^ 2 , \label{eq : indmet2}\ ] ] with @xmath55 and @xmath56 . here @xmath57 and @xmath58 depend on the choice of spacetime . the black hole and cosmological solutions are respectively given by @xmath59 and @xmath60 , as explained in sec . 1 . note that @xmath61 is a time ( space ) -like coordinate and @xmath62 is a space ( time ) -like coordinate in the exterior ( interior ) of an outer black hole horizon or interior ( exterior ) of a cosmological horizon . moreover , the horizonless geometry with point masses , where @xmath61 is time - like and @xmath62 is space - like always , corresponds to setting @xmath63 with @xmath64 for ads space and @xmath65 for flat space @xcite . on the shell s surface @xmath31 with @xmath66 and @xmath67 , the metric reduces to for both interior and exterior spacetimes . for the different parameters @xmath68 , the analysis is the same with the rescaled interior and exterior coordinates @xmath69 , without changing the physical parameters @xmath70 and @xmath71 . ] @xmath72 which yields ( from the first junction condition in eq . ( [ eq : jc ] ) ) @xmath73 the induced basis vectors and the normal vectors on @xmath31 are @xmath74 and @xmath75 respectively . then , the non - vanishing components of the extrinsic curvature defined by @xmath76 are computed to be @xmath77 and its trace is @xmath78 note that @xmath79 does not vanish even though there is no @xmath80 component of the metric on the shell where @xmath81 , i.e. , @xmath82 . this is basically because @xmath83 has a term @xmath84 which does not vanish on the shell . we shall assume that the surface stress - energy tensor of the shell is that of a perfect fluid @xmath85 where @xmath86 is an energy density , @xmath87 is a pressure , and @xmath88 is the shell s two - velocity . then , from the second junction condition ( [ eq : jc2 ] ) , the surface stress - energy tensor with three - velocity @xmath89 is straightforwardly evaluated to be @xmath90 where @xmath91 . in addition , we have @xmath92 since @xmath93 vanishes on the shell in the co - rotating frame . hence @xmath94=0 $ ] , which gives @xmath95 implying @xmath96 on the shell . combining both equations in ( [ eq : eqns ] ) yields @xmath97 the first equation states that the ( relativistic ) energy of the shell , @xmath98 is balanced by the respective difference of energies @xmath99 , as measured from the interior and exterior spacetimes with given gravitational backgrounds . the second equation occurs because the shell is a closed , adiabatic system , with the loss of shell energy @xmath100 under expansion occurring at the expense of the work @xmath101 done by the shell . for a pressureless ( @xmath102 ) dust shell , eq . ( [ eq : eqn2 ] ) yields @xmath103 , where @xmath104 is an initial rest mass of the shell and is assumed to be a non - vanishing positive constant . inserting this into eq . ( [ eq : eqn1 ] ) gives the equation of motion for the shell @xmath105 or alternatively @xmath106 where the effective potential is @xmath107.\ ] ] note that this equation describes the one - dimensional newtonian motion of a point particle with zero energy in the potential @xmath108 insofar as surfaces of constant @xmath109 are spacelike are spacelike , the system corresponds to a non - conservative system with a time - dependent potential @xmath110 , due to absence of a time - like killing vector . this situation is analogous to that of the @xmath111-brane geometry @xcite . for a related analysis in a @xmath111-brane geometry , see ref . @xcite . the behaviour of the shell will depend on the number of roots of the effective potential in @xmath112 @xcite . now if we define the dimensionless parameters ( ` @xmath49 ' for ads and ` @xmath48 ' for ds ) @xmath113 with the overdot denoting @xmath114 , then eq . ( [ eq : eqx ] ) becomes @xmath115 where the effective potential is @xmath116 and we employ henceforth the convention @xmath117 for convenience , unless otherwise stated . ] . the coefficients are @xmath118\nonumber\\ & & a_{4 } = - \left[m_{0}^2/4 - ( \alpha_{o}^{+}k_{i}^{+}+\alpha_{i}^{+}k_{o}^{+}+\alpha_{i}^{-}k_{o}^{-}+\alpha_{o}^{-}k_{i}^{-})\right]^2 \nonumber\\ & & \qquad\qquad + 4(\alpha_{o}^{+}\alpha_{o}^{-}k_{i}^{+}k_{i}^{-}+\alpha_{i}^{+}\alpha_{i}^{-}k_{o}^{+}k_{o}^{- } + \alpha_{o}^{+}\alpha_{i}^{-}k_{i}^{+}k_{o}^{-}+\alpha_{o}^{-}\alpha_{i}^{+}k_{o}^{+}k_{i}^{-}),\nonumber\\ & & a_{2 } = k_o^{+}k_{i}^{+ } m_{0}^2= k_{o}^{-}k_{i}^{-}m_{0}^2\nonumber\end{aligned}\ ] ] with the condition @xmath119 ( @xmath120 ) on the shell from eq . ( [ eq : jpjm ] ) . note that a non - vanishing coefficient @xmath121 implies the interior and exterior geometries differ ; for example exterior ads and interior ds spacetimes . the coefficient @xmath122 , which is positive , is peculiar to geometries with rotation ; if there is no rotation then @xmath122 vanishes ( @xmath123 ) , and the effective potential agrees with that for the non - rotating case @xcite . the effective potential can be classified by the values of its coefficients . for a non - vanishing @xmath121 ( different geometries ) , the effective potential behaves as @xmath124 as @xmath125 corresponding to a centrifugal barrier around the origin at @xmath126 . hence the shell can not collapse to zero size . we also have the asymptotic behaviour @xmath127 as @xmath128 since @xmath129 . the shape of the effective potential is one of four types , depending on the values of the parameters ( as illustrated in fig . [ fig : efpa8 ] ) and its numerator is a cubic polynomial in @xmath130 . for vanishing @xmath121 , i.e. , the same interior and exterior geometries , the numerator of the effective potential reduces to a quadratic polynomial in @xmath130 . the coefficient of the highest order term @xmath131 can have both positive and negative values , depending on the values of parameters , and the shapes of the effective potential are depicted in fig . [ fig : effp ] . the crucial point concerning the different effective potentials depicted in figs . [ fig : efpa8 ] and [ fig : effp ] is the centrifugal barrier that appears around @xmath126 . it is this barrier that prevents the shell from contracting toward zero size , in turn preventing the formation of a curvature singularity at @xmath126 unlike the non - rotating case @xcite . we now turn to a discussion of exact solutions . we first consider the case of interior ds and exterior ads spaces by setting @xmath133 , @xmath134 , @xmath135 , @xmath136 , @xmath137 , then we get @xmath138 we can also reverse the above case by setting @xmath139 , @xmath140 , @xmath141 , @xmath142 , @xmath143 , and easily find that the coefficients are obtained by switching ` @xmath48'@xmath144 ` @xmath49 ' , as expected due to the corresponding symmetry in the configuration . next consider an interior ds and exterior flat space . this corresponds to setting @xmath145 , @xmath134 , @xmath146 , @xmath147 , @xmath148 , which reduces the coefficients to @xmath149,\nonumber\\ a_4&= & -\left[\frac{m_{0}^2}{4}-\frac{1}{4 } ( m_+ -2 ) ^2 + m_- \right]^2 - ( m_+ -2 ) ^2 m_- , \nonumber \\ a_2&= & \frac{1}{16 \ell^2 } ( m_+- 2 ) ^2 j_+^2 m_{0}^2=\frac{j_-^2 m_{0}^2}{4 \ell^2}.\end{aligned}\ ] ] in the last line , we have used the junction condition @xmath150 and this shows that it is not the angular momentum @xmath151 itself but the combination @xmath152 ( with @xmath153 the mass parameter of the exterior locally flat space ) that is continuous across the shell and matches with the angular momentum @xmath154 of the interior ds space . note that the case of interior ads and exterior flat spaces can be similarly obtained by changing @xmath155 in the above formula with @xmath141 . reversing the interior and exterior spaces can be obtained by switching ` @xmath48 ' @xmath144 ` @xmath49 ' . ( [ eq : eqnmot ] ) is not a particularly convenient form for obtaining an exact solution . rather , by introducing @xmath156 , we find that ( [ eq : eqnmot ] ) can be rewritten as @xmath157 where @xmath158 since the effective potential can be rewritten in the form @xmath159 @xmath160 where @xmath161 , @xmath162 , @xmath163 , and @xmath164 , one finds that the differential equation has an exact solution in terms of the jacobi elliptic function @xmath165\nonumber\\ & = & \frac{m_{0}}{\sqrt{a(d - b)}}{\rm ellipticf}\left[\sqrt{\frac{x^2-b}{c - b}},\sqrt{\frac{c - b}{d - b}}\right],\end{aligned}\ ] ] which can be rewritten as @xmath166^{1/2}.\ ] ] the integration constant @xmath167 is @xmath168 \nonumber \\ & = & \frac{m_{0}}{\sqrt{a(d - b)}}{\rm ellipticf}\left[\sqrt{\frac{x_0 ^ 2-b}{c - b}},\sqrt{\frac{c - b}{d - b}}\right]\end{aligned}\ ] ] determined by setting @xmath169 at @xmath170 . the three roots @xmath171 , @xmath172 , @xmath173 can be rewritten as @xmath174 where @xmath175{r \pm i \sqrt{q^3-r^2}}$ ] with @xmath176 and here the choice of identification of @xmath177 with @xmath178 is arbitrary . for @xmath179 , the number of `` real '' roots depends on the discriminant @xmath180 if @xmath181 , there are three distinct real roots and if @xmath182 , ( at least ) two roots coincide , while if @xmath183 , there is only one real root with a pair of complex conjugate roots ( fig . 1 ) . plugging eq . ( [ a : kds / btz ] ) into eq . ( [ eq : delta ] ) yields @xmath184 with @xmath185 , @xmath186 , and @xmath187 . thus we conclude that collapse scenarios with different interior and exterior geometries ( see fig . [ fig : efpa8 ] ) are described by the exact solution of eq . ( [ eq : exsolution ] ) , with parameters appropriately chosen . note that for the non - rotating case ( @xmath188 and @xmath189 ) @xmath190 always ( fig . [ fig : efpa8 ] ( a ) ) . the crucial difference between the rotating case and the non - rotating case @xcite is the centrifugal barrier around the origin that prevents the shell from collapsing to zero size , forbidding the formation of a curvature singularity for the former . this is manifest in eq . ( [ deltaeqn ] ) , where we can see that the effect of rotation gives _ positive _ contributions for @xmath191 , and so can render @xmath192 non - negative ( fig . [ fig : efpa8 ] ( b ) , ( c ) , and ( d ) ) . for the same interior and exterior geometries we have @xmath194 . the solutions become simpler in that they can be expressed by trigonometric or exponential functions . to see this , we first consider ads spaces in both regions by setting @xmath195 . then we get @xmath196 ^ 2 + 4(k_{o}^{+}+k_{i}^{+})(k_{o}^{-}+k_{i}^{-}),\nonumber \\ & & a_{2}=k_{o}^{+}k_{i}^{+}m_{0}^2\end{aligned}\ ] ] with the condition @xmath197 . alternatively , if we define the mass and angular momentum parameters @xmath198 and @xmath199 , then the black hole and the ads point mass spacetimes can be described by @xmath200 and @xmath201 , respectively , and one finds @xmath202 where @xmath203 is positive definite when at least one of the interior / exterior geometries is a black hole geometry . for example for btz black holes in both regions @xmath204 , we have @xmath205 . likewise for interior ads point mass and exterior btz black hole case @xmath206 we have @xmath207 . however for ads point masses in both regions , @xmath208 can not be positive always . the shape of the effective potential is given in fig . [ fig : effp ] ( a ) since @xmath209 . so there are two positive roots at @xmath210 , where @xmath211 and the shell moves between them as illustrated in fig . [ fig : effp ] ( a ) . an exact solution can be found by a straightforward computation : @xmath212.\ ] ] alternatively we can write @xmath213^{1/2}\ ] ] or @xmath214^{1/2},\ ] ] where the integration constant @xmath167 is determined by an initial condition on the position of the shell @xmath215 , @xmath216.\ ] ] it is clear that a ( real ) solution exists only for @xmath217 : for ads point masses in both regions , i.e. , @xmath218 , we need particular initial configurations satisfying @xmath219 or @xmath220 in order to have real solutions . note that the minimum bound at @xmath221 arises due to the angular momentum parameter @xmath222 , which implies that the shell will not shrink to zero size . the intrinsic ricci scalar on the shell computed from the induced metric ( [ eq : indmet ] ) , @xmath223 = \frac{2}{x(t)}\frac{d^2x(t)}{dt^2 } = - \frac{1}{x}\frac{d v_{{\rm eff}}}{dx } , \label{eq : ricci}\ ] ] has no singularity since the shell can not reach the origin at @xmath126 due to the angular momentum barrier . this implies that rotational effects prevent the curvature singularity at @xmath126 from being formed . formation of a btz black hole will take place if certain initial conditions hold . first the inequalities @xmath224 must hold , so that the shell s initial location is outside of the putative horizon @xmath225 of the exterior spacetime , which in turn must be located between @xmath226 and @xmath227 . a btz black hole can form for the exterior observer before the shell moving inward either collapses _ onto _ the interior point mass , collapses _ into _ a point mass if the interior is pure ads , or is absorbed into the outer horizon of an interior btz black hole . to see this , consider eq . ( [ junction1 ] ) , which can be written as @xmath228 and note that everything in the square - root terms is the same except for @xmath229 and @xmath230 . if there is a black hole in the interior spacetime , i.e. , @xmath231 , then a black hole in the exterior spacetime , i.e. , @xmath232 , will always form for a positive @xmath233 , for any initial point @xmath234 : the shell moving inward forms the ( outer ) black hole horizon before being absorbed by the outer horizon of the interior black hole , i.e. , @xmath235 which is equivalent to @xmath236 . if the interior is pure ads or has a point mass , i.e. , @xmath237 , then a black hole in the exterior spacetime ( @xmath232 ) forms if @xmath238 with @xmath239 , for any initial point @xmath234 . for @xmath240 then @xmath241 and the shell collapses into ( or onto ) a point mass , which is similar to the condition for dust _ cloud _ collapse in the non - rotating case @xcite furthermore , for @xmath242 we have @xmath243 ; the deficit angle , which is defined by @xmath244 @xcite , of the point mass outside can not be smaller than that of the point mass inside . note that the physics governing the endstate of collapse is basically the same regardless of the values of @xmath245 and @xmath222 . on the other hand , by squaring ( [ junction1 ] ) one find that the shell s gravitational mass @xmath246 , which becomes the black hole mass after its formation , is given by @xmath247 ( reinstating newton s constant @xmath16 ) where the first and second terms correspond to the shell s relativistic kinetic energy and binding energy , respectively @xcite . note that the binding energy is negative only for _ positive _ @xmath16 , as in the conventional higher dimensional black holes @xcite with @xmath248 the four - dimensional newton s constant . ] ; this is unique to ads spacetimes and this could provide a physical explanation of why the black hole solution can exist only in this case , but not in ds or flat spacetimes . next we consider ds spaces in both regions by setting @xmath249 and @xmath250 . then we get @xmath251 ^ 2 + 4(k_{o}^{+}-k_{i}^{+})(k_{o}^{-}-k_{i}^{-}),\nonumber \\ & & a_{2}=k_{o}^{+}k_{i}^{+}m_{0}^2\end{aligned}\ ] ] with the condition @xmath197 . the effective potential is ( @xmath252 ) @xmath253 in this case , the shape of the effective potential is fig . 2(b ) since @xmath254 with a single root at @xmath255 , where @xmath256 provided @xmath257 ; for vanishing angular momentum @xmath258 , i.e. , non - rotating ds spaces , there is no @xmath259 ( the lower dotted line in fig . the solution for the shell s edge is @xmath260 ^ 2},\ ] ] where the integration constant @xmath167 is @xmath261.\ ] ] as with black hole spacetimes , when some appropriate conditions are imposed , a ( cosmological ) event horizon @xmath262 can form from the perspective of the interior observer . if the cosmological horizon @xmath262 of the interior spacetime is located larger than @xmath259 and if the initial location of the dust shell is in between @xmath259 and @xmath263 , the expanding ( or collapsing and later expanding ) shell will form a cosmological horizon for the interior ( kds@xmath14 ) observer . to see this explicitly , we note that , as in the black hole case , the cosmological horizons are located always larger than @xmath264 ) which is always satisfied when the shell s location coincides with one of the horizons , i.e. , @xmath265 or @xmath266 . ] , i.e. , @xmath267 , from @xmath268 for any non - zero @xmath269 . regardless of the sign of @xmath270 this condition is trivially satisfied since there are no real values of @xmath269 satisfying @xmath271 . consider again eq . ( [ junction1 ] ) , which we write as @xmath272 note that the contributions of the mass terms @xmath273 are opposite to those of the black hole spacetime due to our definition of mass @xcite . then the expanding ( or collapsing and later expanding ) shell will form a cosmological horizon for the interior ( kds@xmath14 ) observer for a _ negative _ @xmath233 : if @xmath274 , then @xmath275 , which is equivalent to @xmath276 , and the shell will be absorbed by the cosmological horizon of the exterior spacetime , before forming the cosmological horizon in the interior spacetime . finally we consider flat spaces in both regions by setting @xmath277 and @xmath249 . we have @xmath278 ^ 2 + 4k_{i}^{+}k_{i}^{-},\nonumber \\ & & a_{2}=k_{o}^{+}k_{i}^{+}m_{0}^2\end{aligned}\ ] ] with the condition @xmath197 . if we define the mass and angular momentum parameters @xmath279 and @xmath280 of the point masses to be @xmath281 , @xmath282 , then one finds the effective potential becomes @xmath283 where @xmath284 ^ 2 -4 k^+_i k^-_i$ ] . its shape is shown in fig . [ fig : effp](c ) , since @xmath285 with a single root at @xmath255 , where @xmath286 here we note that the angular momenta @xmath287 are not separately continuous across the shell , but only the combinations @xmath288 are . the exact solution is found to be @xmath289 where the integration constant @xmath167 is @xmath290 to summarize this section , we find that rotating dust shells can collapse , but only down to a minimal size @xmath259 ( after which they expand ) due to their angular momentum . no curvature singularity is formed during the gravitational collapse process , unlike the non - rotating case @xcite . the alternative collapse endpoint is a btz black hole or a kds@xmath14 spacetime . in this section , we shall consider shells with pressure determined by somewhat generalized equations of state . the effect of pressure is to produce a varying shell energy @xmath291 from eq . ( [ eq : eqn2 ] ) , i.e. , deviations from the inverse radius dependence of @xmath292 . specifically we consider shells with the polytropic - type equation of state @xmath293 that encompasses many sorts of known fluids by choosing a specific equation of state parameter @xmath294 and polytropic index @xmath295 . for instance , we have constant energy density ( @xmath296 ) , non - relativistic degenerate fermions ( @xmath297 ) , non - relativistic matter or radiation pressure ( @xmath298 ) , and linear ( perfect ) fluid ( @xmath299 ) , respectively @xcite . moreover , the equation of state in eq . ( [ eq : poly ] ) can describe a chaplygin gas by choosing @xmath300 . plugging the equation of state ( [ eq : poly ] ) into eq . ( [ eq : eqn2 ] ) yields @xmath301 where @xmath302 is an integration constant . similarly , for the linear fluid ( @xmath303 ) , we have @xmath304 from these we obtain the equations @xmath305 for finite @xmath295 , and @xmath306 for the linear fluid , where @xmath307 so that @xmath308 when @xmath309 . for finite @xmath295 and the linear fluid , the equations of motion can be expressed in the alternate form @xmath310 where the effective potential is @xmath311 with the junction condition @xmath197 . note that the only change to the effective potential compared to pressureless dust shells in eq . ( [ veff1 ] ) is the replacement of @xmath312 in eq . ( [ veff1 ] ) . up to now , the preceding results are valid for any @xmath295 . but the behaviour of the effective potential differs for @xmath313 and @xmath314 , which need separate consideration . for ordinary fluids with finite and positive definite @xmath295 and @xmath294 , the effective potential and the intrinsic ricci scalar also _ negatively _ diverge at @xmath316 , where the density @xmath86 also diverges . so the shell will not collapse to a point but rather to a ring of finite size @xmath317 in a finite proper time . the shape of the effective potential is depicted in figs . [ fig : numpics ] and [ fig : numpics2 ] . on the other hand , for the linear fluid ( @xmath303 ) there is no ring singularity since @xmath86 is finite for any finite @xmath318 . rather , from the effective potential near @xmath126 , @xmath319 due to @xmath320 , a shell with @xmath321 will collapse to zero size shows similar behaviour for the effective potential and @xmath322 is a marginal case . we shall not consider these possibilities , i.e. , @xmath323 with the usual values of @xmath313 , since their physical relevance is not quite clear . ] , _ regardless of _ the initial values of @xmath104 and @xmath324 , in a finite proper time where the effective potential and the energy density ( or pressure ) diverge ( figs . [ fig : numpics6 ] and [ fig : numpics7 ] ) . provided the exterior geometry has no black hole ( event ) horizon such as the kds@xmath14 space or the ads / flat spaces with point masses , there is nothing to prevent the collapsing shell from developing a curvature singularity . the resultant singularity is naked and we have a violation of the cosmic censorship for `` generic '' initial data . these are qualitatively the same behaviours as those of non - rotating shell collapse @xcite , which implies that their angular momentum is not large enough to overcome forming the curvature singularity for most fluids whose equations of state are of the form ( [ eq : poly ] ) . the centrifugal barrier that prevents collapse in the pressureless dust case still occurs at @xmath126 , but is always dominated by the negatively divergent effective potential at the finite value of @xmath318 . however , for the linear fluid with @xmath325 , angular momentum effects again dominate , ensuring that cosmic censorship is upheld regardless of the relative values of @xmath104 and @xmath222 ( figs . [ fig : numpics8 ] and [ fig : numpics9 ] ) . note that this is the case where there is a similarity with the pressureless dust shells in sec . 3 : actually , figs . [ fig : numpics8 ] and [ fig : numpics9 ] ( and fig . [ fig : numpics5 ] as well ) show all the possible cases in figs . [ fig : efpa8 ] and [ fig : effp ] . the case @xmath326 is a marginal case that crucially depends on the initial data : one has an infinite well for @xmath327 ( fig . [ fig : numpics3 ] ) but an infinite barrier for @xmath328 ( fig . [ fig : numpics5 ] ) ; for @xmath329 , the point @xmath126 is naked when @xmath330 ( fig . [ fig : numpics4 ] ) . for the chaplygin - type gas shells ( @xmath332 ) , by setting @xmath333 one can conveniently rewrite the equation of state as @xmath334 and @xmath335 , in agreement with more standard conventions @xcite . then , for finite @xmath295 , i.e. , @xmath336 one finds @xmath337^{1/(\lambda+1)},\ ] ] from eq . ( [ rho : finite_n ] ) . the shape of the effective potential depends on several parameters @xmath338 , @xmath339 , and @xmath340 , as shown in fig . [ fig : numpics10 ] . near @xmath126 , the effective potential behaves as @xmath341 due to @xmath342 for @xmath343 , from eq . ( [ rho : finite_-n ] ) . again there is a centrifugal barrier near the origin . the asymptotic behaviour as @xmath128 is given by @xmath344 x^2\end{aligned}\ ] ] and this depends on the values of @xmath338 and @xmath339 , similarly to fig . [ fig : effp ] . in addition , as can be seen from a consideration of the effective potential ( [ v_eff : p ] ) , it will not diverge for any finite value of @xmath318 when @xmath345 , while it will _ negatively _ diverge due to the vanishing energy density , i.e. , @xmath346 at @xmath347^{1/(\lambda+1)}$ ] when @xmath348 for all @xmath349 . on the other hand , for @xmath350 , we have a uniform density @xmath351 from eq . ( [ rho : finite_-n ] ) also and the ( full ) effective potential is given by @xmath352 near @xmath126 , this behaves as @xmath353 and it depends on initial data : one has an infinite well for @xmath354 ( this is what has been plotted in fig . [ fig : numpics10 ] ) , and an infinite barrier for @xmath355 . the case @xmath356 is a marginal case that has a finite well and a finite intrinsic curvature ; the intrinsic curvature is finite even at the point @xmath126 , from eq . ( [ eq : ricci ] ) term ( omitted in ( [ v : bounce ] ) ) , which cancels the @xmath357 factor in eq . ( [ eq : ricci ] ) . ] , and so @xmath126 is a bounce point . as @xmath358 , the effective potential behaves as @xmath359 x^2\end{aligned}\ ] ] which depends on the initial data also . as an explicit example , we consider @xmath360 , describing a conventional chaplygin gas shell @xcite . then the effective potential becomes @xmath361 where @xmath362 , \nonumber\\ & & a_{6}= -m_{0}^4 a ( 1-a)/8 + \left[(\alpha_{o}^{+ } k_{i}^{+ } + \alpha_{i}^{+ } k_{o}^{+ } + \alpha_{i}^{-}k_{o}^{-}+\alpha_{o}^{-}k_{i}^{-})a + ( \alpha_{o}^{+}\alpha_{i}^{+}+\alpha_{o}^{-}\alpha_{i}^{-})(1-a ) \right ] m_{0}^2/2\nonumber\\ & & \qquad -2(\alpha_{o}^{+}\alpha_{i}^{+}-\alpha_{o}^{-}\alpha_{i}^{- } ) ( k_{i}^{+}\alpha_{o}^{+}+\alpha_{i}^{+}k_{o}^{+ } - \alpha_{i}^{-}k_{o}^{-}-\alpha_{o}^{-}k_{i}^{- } ) , \nonumber\\ & & a_{4 } = -m_{0}^4 ( 1-a)^2/16 + \left[(\alpha_{o}^{+}k_{i}^{+ } + \alpha_{i}^{+}k_{o}^{+ } + \alpha_{o}^{-}k_{i}^{-}+\alpha_{i}^{-}k_{o}^{- } ) ( 1-a ) + 2k_{o}^{-}k_{i}^{-}a\right ] m_{0}^2/2\nonumber \\ & & \qquad -(\alpha_{o}^{+ } k_{i } + \alpha_{i}^{+}k_{o}^{+ } + \alpha_{i}^{-}k_{o}^{-}+\alpha_{o}^{-}k_{i}^{- } ) ^2\nonumber\\ & & \qquad+ 4(\alpha_{o}^{+}\alpha_{o}^{-}k_{i}^{+}k_{i}^{- } + \alpha_{i}^{+}\alpha_{i}^{-}k_{o}^{+}k_{o}^{- } + \alpha_{o}^{+}\alpha_{i}^{-}k_{i}^{+}k_{o}^{- } + \alpha_{o}^{-}\alpha_{i}^{+}k_{o}^{+}k_{i}^{-}),\nonumber\\ & & a_{2}=m_{0}^2 k_{o}^{-}k_{i}^{- } ( 1-a).\nonumber\end{aligned}\ ] ] for a black hole spacetime outside the shell , the collapsing shell may form a black hole within a finite time for @xmath363 . but this is not always the case for @xmath348 since there exists a singular point at @xmath364 . if the horizon @xmath365 is located at @xmath366 , then it will form a black hole , while if @xmath367 , it will form a finite - sized ring singularity unless the numerator vanishes at the point @xmath368 , @xmath369 . apart from a contrivance of very restrictive conditions on the parameters @xmath370 , and @xmath338 , this is still a somewhat singular configuration : even though the intrinsic ricci scalar of the shell is finite and its energy density vanishes , the pressure diverges at the point @xmath368 . this suggests a bounce solution . comparing to the non - rotating case @xcite , we again find that angular momenta does not in general prevent the emergence of a naked ring singularity at a finite position of @xmath371 . violation of cosmic censorship occurs from the gravitational shell collapse , regardless of the rotation and initial data . our investigation of the gravitational collapse of rotating shells in three dimensions has uncovered a number of interesting features . we have studied whether or not angular momentum can significantly change the collapse scenario and its resulting cosmic censorship violations in the non - rotating cases in the literature . for asymptotically ads boundary conditions , we find that the rotating shell collapses to either a black hole or a minimum value and then expands out to infinity . for shells composed of pressureless dust these are the only scenarios ; the centrifugal barrier forbids a naked singularity from forming . however for shells with pressure we have another scenario in which a naked singular ring can form , violating cosmic censorship . when the exterior spacetime is taken to be a geometry with a point mass , one might expect that the collapsing shell could form a curvature singularity within finite time , since there is no event horizon as with the non - rotating system in ref . however we have shown that a naked singularity never forms in the collapse of a rotating dust shell due to a centrifugal barrier in the effective potential experienced by the shell . collapse scenarios for shells with pressure show that a naked ring singularity of finite size can be formed , where the effective potential and the surface stress - energy tensor diverge . for asymptotically ds boundary conditions a collapsing shell with pressure can form a naked singularity , also . if the interior spacetime is ( k)ds , then a cosmological horizon can form from an expanding shell . which of these scenarios occurs depends on the choice of parameters and initial conditions . for asymptotically flat boundary conditions the qualitative behaviours are similar , i.e. _ , a rotating collapsing dust shell does not form a naked singularity , but a shell with pressure can form a naked singularity . for a polytropic shell this will be a naked singular ring of finite size . the qualitative behaviour is more or less intermediate between the asymptotically ads and ds cases as implied by the choice of @xmath57 and illustrated in figs . [ fig : numpics2 ] and [ fig : numpics8 ] , though there are some anomalous regions that do not reveal this simple trend . the most intriguing lesson of this paper is that the centrifugal barrier in the effective potential governing the time evolution of a rotating dust shell can prevent formation of a naked singularity . however if the shell has pressure , a ring singularity may form for a typical class of equations of state , i.e. , @xmath321 , where the pressure dominates the angular momentum , with quite general initial data . so if the exterior spacetime is assumed to be a geometry without a ( covered ) black hole event horizon , then the singularity may be naked and the violation of cosmic censorship is possible . we have found that a collapsing shell can form either a black hole / cosmological horizon or a naked singularity or bounce to infinity , depending on the initial data . this suggests a set of phase transitions @xcite along with accompanying critical phenomena , similar to that discovered for scalar matter @xcite . it would be of great interest to study these phenomena explicitly as they could reveal universal features of the critical exponents in three dimensions . since we have confined our study to three dimensions ( where there is no gravitational radiation ) , some caution is warranted in applying our results to higher dimensions . we have found that the angular momentum does not in general prevent violation of cosmic censorship in three dimensions , where local gravitational interactions vanish outside of matter . an interesting extension of our work would be to include higher derivative terms , whose effect is to produce such interactions . another interesting extension is the inclusion of quantum effects , since they might be expected to prevent singularity formation , thereby sidestepping the cosmic censorship issue . what impact these modifications have on our results remains to be investigated . + we would like to thank sang pyo kim and shin nakamura for exciting discussions and the headquarter of apctp for warm hospitality during the apctp - tpi joint focus program and workshop . j. j. oh would like to thank wontae kim , seungjoon hyun , hongbin kim , jaehoon jeong , hyeong chan kim , gungwon kang , inyong cho , constantinos papageorgakis , andrew strominger , and chi - ok hwang for useful and helpful discussions . r. b. mann was supported by the natural sciences and engineering research council of canada . j. j. oh was supported by the korea research council of fundamental science & technology ( krcf ) . park was supported by the korea research foundation grant funded by korea government(moehrd ) ( krf-2007 - 359-c00011 ) . s. deser , r. jackiw , and s. templeton , ann . 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we study the gravitational collapse problem of rotating shells in three - dimensional einstein gravity with and without a cosmological constant . taking the exterior and interior metrics to be those of stationary metrics with asymptotically constant curvature , we solve the equations of motion for the shells from the darmois - israel junction conditions in the _ co - rotating _ frame . we study various collapse scenarios with _ arbitrary _ angular momentum for a variety of geometric configurations , including anti - de sitter , de sitter , and flat spaces . we find that the collapsing shells can form a btz black hole , a three - dimensional kerr - ds spacetime , and an horizonless geometry of point masses under certain initial conditions . for pressureless dust shells , the curvature singularity is _ not _ formed due to the angular momentum barrier near the origin . however when the shell pressure is nonvanishing , we find that for all types of shells with polytropic - type equations of state ( including the perfect fluid and the generalized chaplygin gas ) , collapse to a naked singularity is _ possible _ under generic initial conditions . we conclude that in three dimensions angular momentum does not in general guard against violation of cosmic censorship . arxiv:0812.2297v3 [ hep - th ] + + + robert b. mann@xmath0 , john j. oh@xmath1 , and mu - in park@xmath2 + _ @xmath3 department of physics and astronomy , university of waterloo , + waterloo , ontario , n2l 3g1 , canada + @xmath4 division of interdisciplinary mathematics , national institute for mathematical sciences , + daejeon , 304 - 350 , korea + @xmath5 research institute of physics and chemistry , chonbuk national university , + chonju , 561 - 756 , korea + _ pacs numbers : 04.20.dw , 04.40.-b , 04.60.kz , 04.70.bw

in 1952 lee and yang @xcite proved that the zeroes of the partition function of a ferromagnetic ising model ( or equivalently , a lattice gas with single - site hard cores ) as a function of complex magnetic field @xmath0 ( or chemical potential @xmath1 ) are confined to the imaginary @xmath0 axis for real temperatures @xmath2 . yang and lee argued @xcite that for a system above its critical temperature @xmath3 , the partition function must be nonzero throughout some neighborhood of the real axis in the complex reduced magnetic - field plane , @xmath4 thus for @xmath5 , a gap free of zeros will be found on the imaginary @xmath6 axis with edges at , say , @xmath7 . equivalently , in fluid - language , for @xmath8 , there will be a gap in the complex activity plane with edges at @xmath9 as illustrated in fig . 1 . here we define the reduced activity in @xmath10 spatial dimensions by @xmath11 where @xmath12 , while @xmath13 is a microscopic reference volume ( taken as the cell volume for a lattice gas ) and @xmath14 is the thermal de broglie wavelength . if one defines the density of yang - lee zeros , @xmath15 , so that when @xmath16 , the number of spins ( or lattice sites ) becomes infinite , @xmath17 approaches the number of zeros between @xmath18 and @xmath19 on the imaginary @xmath6 axis , one must have @xmath20 . kortman and griffiths @xcite pointed out that the density of zeros beyond the gap should be expected to exhibit a power law singularity the yang - lee edge singularity @xcite of the form @xmath21 this singularity proves analogous to an ordinary critical point obeying scaling laws and exponent relations although there is only one relevant scaling field [ 3 , 4 ] . thus the basic exponent @xmath22 in ( 4 ) satisfies @xmath23@xcite . in a general renormalization group analysis , it was shown @xcite that the field theory controlling the yang - lee fixed point is described by a pure imaginary @xmath24 coupling . this leads to a critical dimension @xmath25 above which the classical , mean - field value @xcite @xmath26 applies . to first order in @xmath27 one has @xmath28 and @xmath29 . expansions of @xmath30 to order @xmath31 are known @xcite and one has @xmath32 @xcite , @xmath33 , and finds , numerically , @xmath34 @xcite : see lai and fisher @xcite who review previously known relationships of the yang - lee edge singularity to a number of different problems , specifically , isotropic branched polymers or , equivalently , undirected lattice animals with or without loops allowed , in @xmath35 dimensions [ 8 , 9 ] ; anderson localization @xcite ; and _ directed _ branched polymers ( or directed , loop - free lattice animals ) in @xmath36 dimensions @xcite . by contrast , an apparently quite different type of singularity arises in fluid systems when the particle interactions have repulsive cores . if the pair interaction potentials are purely repulsive ( i.e. , positive ) , the ( reduced ) cluster integrals , @xmath37 , in the activity or fugacity series for the ( reduced ) pressure , namely , @xmath38 are known @xcite to alternate in sign : this implies a dominant singularity on the _ negative _ @xmath39-axis that determines the radius of convergence , @xmath40 , of the series : see fig . 1 . in the vicinity of this singularity , say at @xmath41 , the reduced pressure can be written as @xmath42 + \cdots , \end{aligned}\ ] ] where the exponents @xmath43 and @xmath44 are anticipated to be nonintegral . indeed , in 1984 , poland @xcite studied a variety of lattice models and a continuum fluid of hard squares and proposed that this repulsive - core singularity is characterized by a _ universal _ exponent @xmath45 . subsequent confirmation came from baram and luban @xcite who investigated further models including dimers on lattices , parallel hypercubes in continuum space and the _ soft_-core single - component gaussian - molecule model . more recently , lai and fisher @xcite found similar behavior in a _ gaussian - molecule mixture using very long series expansions for @xmath46 . in that case the singularity at @xmath47 was drawn out into a continuous locus ; but the estimates for @xmath48 supported the universality hypothesis . ( precise estimates , for the leading correction - to - scaling " exponent @xmath49 , in ( 6 ) for all @xmath10 , including @xmath50 , @xmath51 , and @xmath52 , were also generated @xcite . ) however , lai and fisher @xcite noticed in particular that , when compared with previous knowledge about the yang - lee edge exponent @xmath53 , the exact results for @xmath54 for @xmath55 and @xmath56 @xcite and for @xmath57 @xcite , and the various numerical estimates @xcite for @xmath58 , strongly suggested the identification @xmath59 hence they proposed @xcite that the _ universal repulsive - core singularity belongs to the yang - lee edge critical universality class_. in analytical support of this identification ( for general @xmath10 ) they appealed to earlier work by kurtze and fisher @xcite who had proved that when @xmath60 the yang - lee edge singularity in ferromagnetic ising models precisely describes the dominant singularity on the negative @xmath39 axis of a fluid of _ hard dimers _ on the same lattice , each dimer occupying one bond and two adjacent lattice sites . this correspondence with a dimer gas was carried further by shapir @xcite who used field - theoretic arguments to demonstrate the identity generically for all @xmath61 . however , dimers are rather special objects with orientational degrees of freedom ( although these seem , _ post facto _ , to have no effect on @xmath48 @xcite ) . furthermore , shapir s field - theoretic approach was rather special and did not seem extendable to more general particles with soft repulsive cores , with the additional presence of attractive forces , or to systems also displaying critical behavior for positive @xmath39 @xcite ( as illustrated in fig . 1 ) . in this article we repair this gap in the theory . specifically , we consider a general single - component fluid with a pair interaction potential , @xmath62 , which contains both repulsive _ and _ attractive parts : of course , repulsive terms are always essential to ensure thermodynamic stability . to be concrete and explicit we analyze lattice systems in which multiple occupancy of a site is forbidden ; however , it must be stressed that the repulsive interactions we consider are _ not _ confined to such trivial single - site hard cores . on the contrary , essentially we suppose only that the potential @xmath63 is of finite range and may be decomposed according to @xmath64 where the sum runs over lattice sites @xmath65 while the repulsive and attractive parts , @xmath66 and @xmath67 , respectively , are both positive ( if they do not vanish identically ) . more specifically we will use @xmath68 @xmath69 so that @xmath70 and @xmath71 represent the interaction ranges of the repulsive and attractive components . the real - space potentials acting between sites @xmath72 and @xmath73 ( @xmath74 ) , namely , @xmath75 and @xmath76 , follow by fourier inversion . for @xmath77 one may imagine yukawa forms : @xmath78 and @xmath79 with @xmath80 ; but that is certainly not essential . likewise , the leading lattice isotropy assumed for convenience in ( 9 ) and ( 10 ) is not necessary . our treatment extends straightforwardly to multicomponent fluids @xcite and , at least formally , generalizes readily to continuum systems . on the basis of ( 8)-(10 ) , we develop a field - theoretic analysis and show that there is , in general , a repulsive - core singularity at some @xmath81 on the negative activity axis ( see fig . 1 ) that is described within an lgw renormalization group framework by a fixed point hamiltonian with a purely imaginary cubic coupling , @xmath82 , and , hence , lies in the same universality class as yang - lee edge criticality . a more or less novel aspect of our treatment is that _ separate_sine - gordan @xcite and kac - hubbard - stratonovich ( khs ) @xcite transformations are used for handling the repulsive and attractive parts of the interaction potential : compare with refs . 23 and 24 . for a lattice of volume @xmath83 with sites labeled @xmath84 , @xmath85 , let @xmath86 or @xmath87 according as site @xmath72 is or is not occupied by a particle . then , recalling ( 3 ) and ( 8)-(10 ) _ et seq_. , the grand partition function for the lattice system is @xmath88 \right\ } .\ ] ] now , utilizing the positivity of @xmath89 and @xmath90 , we may apply a sine - gordon transformation @xcite to the repulsive terms , @xmath91 , and a khs transformation @xcite to the attractive terms @xmath92 ( where , as usual , it is most convenient to utilize periodic lattice boundary conditions ) . neglecting an unimportant constant factor , this yields @xmath93 \\ & & \times { \rm tr } _ { \rho } ^ { \tiny { \cal n } } \left\ { z^ { { \tiny \sum_j } { \rho}_j } \exp [ \sum_j ( - i { \varphi}_j + { \chi}_j ) { \rho}_j ] \right\ } , \end{aligned}\ ] ] where @xmath94 = [ w_{jk}^{-1 } ] ^{-1 } \hspace{1em}$ ] and similarly for @xmath95 . performing the trace over the @xmath96 then yields the transformed reduced hamiltonian @xmath97 & = & { \textstyle \frac{1}{2 } } \sum_{j , k } ( { \varphi}_j w_{jk}^{-1 } { \varphi}_k + { \chi}_j v_{jk}^{-1 } { \chi}_k ) \\ \nonumber & & - \sum_j \ln ( 1 + z e^ { - i { \varphi}_j + { \chi}_j } ) . \end{aligned}\ ] ] now , in seeking to understand the possible singularities in the reduced pressure @xmath98 , we will , initially , neglect fluctuations and study the saddle point(s ) which extremize the integrand @xmath99 \ } $ ] in ( 12 ) . we expect spatially uniform solutions @xmath100 , @xmath101 ( all @xmath72 ) to suffice : from @xmath102 one thus finds @xmath103 for the repulsive terms . on premultiplying by the matrix @xmath104 $ ] ( see , e.g. , [ 25 ] ) and using the fourier representation ( 9 ) , one obtains the simpler form @xmath105 similarly from @xmath106 we find @xmath107 for the attractive terms . before analyzing these coupled saddle - point equations , note that if one puts @xmath108 the two equations combine simply and can be rewritten as @xmath109 where the total strength of the pair potential is measured by @xmath110 furthermore , by substitution of any solution of ( 18 ) on the right hand sides of ( 15 ) and ( 16 ) one obtains the separate solutions @xmath111 , which may evidently be imaginary , and @xmath112 . we remark , first , that following the pioneering study of hubbard and schofield @xcite , all authors interested in ordinary gas - liquid criticality have treated the repulsive interactions in a fluid by use of a _ reference system _ : for a recent example , see brilliantov @xcite . unless this reference fluid is essentially trivial , as for single - site hard cores on a lattice , this entails increasingly detailed knowledge of the correlation functions of the repulsive - core system @xcite . nevertheless , it will be helpful for us to make contact with this approach by , initially , neglecting the nontrivial repulsive interactions embodied in @xmath113 . if the attractions are _ also _ neglected the grand partition function ( 11 ) reduces simply to @xmath114 . this predicts a repulsive - core singularity at @xmath115 ( all @xmath2 ) with an exponent @xmath116(log ) @xcite . evidently , the actual dimensionality , @xmath10 , plays no role . now , following traditional treatments ( e.g.,@xcite ) let us introduce attractive terms with @xmath117 . the khs transformation then leads to the saddle - point equation ( 18 ) in which , now , @xmath118 is negative and we can identify @xmath119 directly with @xmath120 ( since @xmath121 and @xmath122 etc . , can be ignored ) . fig . 2 then provides a graphical representation of ( 18 ) which can be readily analyzed : note that the bold curve depicts @xmath123 which has a point of inflection at @xmath124 and a corresponding tangent that intersects the axis at @xmath125 : see the dashed line . by inspection , one then sees that when @xmath39 increases from @xmath126 ( along the real axis ) for temperatures such that @xmath127 there is always a single saddle - point solution , @xmath128 , which varies analytically with @xmath2 and @xmath39 : see , e.g. , the lines labelled ( a ) and ( b ) in fig . conversely , for lower @xmath2 , when @xmath129 , there is a single analytic solution , @xmath130 , for small @xmath39 [ as on the line ( c ) ] but for an intermediate range of @xmath39 three distinct solutions arise , as illustrated by ( d ) ; finally , for larger @xmath39 only a single solution remains : line ( e ) . evidently the three solutions merge at a bifurcation point determined by the inflection point at @xmath124 : this leads to @xmath131 and @xmath132 . expanding @xmath133 $ ] in powers of @xmath134 , @xmath135 and @xmath136 , all taken as real variables , shows that this saddle - point bifurcation simply represents the anticipated classical or mean - field gas - liquid critical point at @xmath137 . as usual , the corresponding lgw hamiltonian can be used as a starting point in a field - theoretic renormalization - group ( rg ) treatment which then leads to all the standard results . on the other hand , for @xmath138 @xmath139 , i.e. , @xmath2 @xmath140 , one can follow ref . @xcite and discover two yang - lee edge singularities at complex @xmath39 ( with small imaginary parts when @xmath2 is near @xmath3 ) : at the saddle - point level these have @xmath141 ; but the fixed - point hamiltonian is controlled by an @xmath142 coupling leading , as explained above , to an rg @xmath143 expansion @xcite . if , next , _ small _ repulsive terms , @xmath144 , are introduced , the overall interaction parameter @xmath145 , in ( 18 ) remains negative and the arguments proceed in essentially the same manner ( although , in due course , the field @xmath146 for the repulsive terms would normally be integrated out ) . as expected , neither the usual gas - liquid nor the yang - lee edge singularities undergo any change in character . however , if @xmath147 becomes sufficiently large , @xmath148 becomes _ positive _ and then , clearly , the previous analysis fails ! in particular , as seen in fig . 3 [ lines ( a ) and ( b ) ] , for positive ( real ) @xmath39 , there is always only a single , smoothly varying saddle - point solution , @xmath149 . of course , this does _ not _ ( necessarily ) mean that the usual gas - liquid and yang - lee edge singularities are lost . rather , the form of the hamiltonian , @xmath150 $ ] , represents an inadequate starting point for a perturbative rg approach : instead , it becomes necessary to integrate out ( at least to some degree ) the repulsive terms , i.e. , to perform some @xmath151 integrals and , thereby , make contact with the reference - fluid treatments @xcite . on the other hand , when the repulsions dominate , so that @xmath152 ( as we will assume hereon ) , one sees from fig . 3 [ e.g. , lines ( c ) and ( d ) ] that for a real _ negative _ @xmath39 there is a unique positive saddle - point solution , say @xmath153 , that vanishes when @xmath154 . however , when @xmath39 approaches @xmath155 ( from above ) a bifurcation point is reached at which the saddle - point must become complex : see the broken line in fig . 3 that corresponds to @xmath156 . ( at @xmath157 a second , larger saddle - point solution , @xmath158 , merges with @xmath159 . for @xmath39 much larger , additional real _ negative _ saddle - point solutions , @xmath160 , appear but these are not relevant to the dominant repulsive - core singularity . ) as evident from fig . 3 , the bifurcation point is located by the tangent passing through @xmath161 : this leads to @xmath162^{1/2 } - 1 \ } < 1 , \ ] ] and @xmath163 \exp [ - { \psi}_0 ( t ) ] , \nonumber \\ & { \approx } & - 1 / e \beta u_0 [ 1 + { \cal o } ( 1 / \beta u_0 ) ] , \end{aligned}\ ] ] the last result applying for strong repulsions ( @xmath164 ) . clearly , this saddle - point bifurcation represents the repulsive - core singularity . incidentally , if following hauge and hemmer @xcite , we had treated the ( @xmath165)-dimensional continuum hard - rod gas with additional infinite - range , infinitely weak repulsive kac potentials , we would , at this point , have found @xmath166 in precise accord with the exact ( limiting ) calculations @xcite . but , of course , our saddle - point treatment is not restricted to @xmath165 even though in the kac limit it will also become exact . to complete the analysis we may now follow standard procedures by expanding about the saddle point values @xmath167 and @xmath112 [ following from ( 15)-(17 ) ] . in terms of @xmath168 , @xmath169 , and @xmath170 , the hamiltonian truncated at fourth order becomes @xmath171 & = & { \textstyle \frac{1}{2 } } \sum_{j , k } ( { \varphi}_j w_{jk}^{-1 } { \varphi}_k + { \chi}_j v_{jk}^{-1 } { \chi}_k ) \\ \nonumber & & + \sum_j [ { \textstyle \frac{1}{2 } } r_0 ( \delta { \psi}_j ) ^2 -g_0 ( \delta { \psi}_j ) ^3 + u_0 ( \delta { \psi}_j ) ^4 ] , \end{aligned}\ ] ] where it is convenient to put @xmath172 ( which is real and positive for @xmath173 when @xmath174 ) so that @xmath175 and @xmath176 in the usual continuum approximation this becomes @xmath177 , \end{aligned}\ ] ] with @xmath178 and @xmath179 and @xmath180 before proceeding generally , let us suppose that only repulsive terms act , i.e. , @xmath181 . then we may drop the integrals @xmath182 and see , by ( 17 ) , that @xmath183 ; but note that @xmath184 remains real , positive and less than unity . the continuum hamiltonian then reduces to @xmath185= \int d^d x [ { \textstyle \frac{1}{2 } } t_0 ( \delta \varphi ) ^2 + { \textstyle \frac{1}{2 } } c _ { \varphi } ( \nabla \delta \varphi ) ^2 - i g_0 { \delta \varphi } ^3 + u_0 { \delta \varphi } ^4 ] , \ ] ] where the controlling coefficient is @xmath186 } { [ 1 + z e^ { { \psi}_0 ( t ) } ] ^2 } , \ ] ] which is positive for small @xmath39 , decreases as @xmath187 increases in magnitude , and vanishes linearly when @xmath188 . the fourth order coefficient , @xmath189 , is positive and approaches @xmath190 when @xmath191 , while the cubic term is purely imaginary with a coefficient approaching @xmath192 at this point we have thus reached the same stage as in the original treatment @xcite of the yang - lee edge singularities . the imaginary term @xmath193 dominates the behavior . a momentum - shell rg analysis reveals an upper critical dimension @xmath194 , an exponent @xmath195 for @xmath196 , and , using a unit cutoff in momentum space , a fixed point value of @xmath197 , where @xmath198 is the area of the unit sphere at @xmath199 dimensions . all other results follow as previously [ 4,7 ] . returning now to the general case of ( 25 ) , in which attractive interactions are present ( i.e. , @xmath200 ) but @xmath201 remains positive , we anticipate that only one linear combination of @xmath202 and @xmath203 will become critical . to check this at the saddle - point level , it suffices to put @xmath204 the quadratic ( nongradient ) terms in ( 25 ) then become @xmath205 the coefficient @xmath206 remains positive for all negative @xmath39 but one finds @xmath207 } , \ ] ] for @xmath208 : thus by ( 28 ) , @xmath209 vanishes when @xmath210 . integration over the noncritical field @xmath211 is hence justified and may be carried out perturbatively . apart from additive terms that are nonsingular near @xmath81 , this finally leads to a renormalized lgw hamiltonian @xmath212= \int d^d { x } & & [ { \textstyle \frac{1}{2 } } t_r ( \delta \varphi ) ^2 + { \textstyle \frac{1}{2 } } a_r^2 ( \nabla \delta \varphi ) ^2 \\ & & - i h_r \delta \varphi - i g_r ( \delta \varphi ) ^3 + \cdots ] , \end{aligned}\ ] ] where the fourth and higher order terms have been dropped . for @xmath213 sufficiently small , the renormalized couplings @xmath214 and @xmath215 will differ little from @xmath209 and @xmath216 and , in particular , @xmath217 and @xmath215 will be real , so that the @xmath218 term once again dominates . stability of the saddle point requires that the renormalized repulsive range , @xmath219 , in ( 34 ) be real ( and positive ) . roughly , we expect @xmath220 : the positivity of this factor then represents a previously unstated restriction on the initial potentials . in practice , however , if this ( or the more precise ) condition fails , it may be sufficient , as explained in the discussion of gas - liquid criticality , to perform further , nonperturbative renormalizations to dampen the attractive interactions for @xmath221 and so expose again a saddle - point representation of the repulsive - core singularity at @xmath81 . certainly , we must expect the form ( 34 ) to represent the singularity whenever it is actually realized in a system ; and , then , clearly it must belong to the yang - lee edge universality class . in summary , by using separate field - theoretic transformations for the repulsive and attractive parts of the pair interactions in a fluid , we have demonstrated in general the presence of a universal repulsive - core singularity on the negative axis at a value @xmath222 : see fig . the behavior of the pressure in the vicinity of @xmath223 is stated in ( 6 ) . the singularity belongs to the same universality class as yang - lee edge criticality , as proposed by lai and fisher @xcite : see eqs . ( 1)-(4 ) and fig . the basic critical exponents are related via ( 7 ) ( see also @xcite ) ; the borderline dimensionality is @xmath25 ; and an @xmath224 coupling characterizes the lgw fixed - point hamiltonian @xcite . y. p thanks the institute for physical science and technology at the university of maryland for hospitality . her work has been supported in part by the ministry of education of korea through the basic science research institute of seoul national university . m.e.f . acknowledges support from the u. s. national science foundation ( through grant che 96 - 14495 ) . the interest of youngchan kim has been appreciated . 99 t. d. lee and c. n. yang , phys . rev . * 87 * , 410 ( 1952 ) . c. n. yang and t. d. lee , phys . rev . * 87 * , 404 ( 1952 ) . p. j. kortman and r. b. griffiths , phys . * 27 * , 1439 ( 1971 ) . m. e. fisher , phys . lett . * 40 * , 1610 ( 1978 ) . o. f. de alcantara bonfim , j. e. kirkham , and a. j. mckane , j. phys . a * 13 * , l247 ( 1980 ) . m. e. fisher , prog . phys . suppl . * 69 * , 14 ( 1980 ) . s .- lai and m. e. fisher , j. chem . phys . * 103 * , 8144 ( 1995 ) . g. parisi and n. sourlas , phys . lett . * 49 * , 871 ( 1981 ) . a. r. day and t. c. lubensky , j. phys . a * 15 * , l285 ( 1982 ) . t. c. lubensky and a. j. mckane , j. phys . ( paris ) * 42 * , l331 ( 1981 ) . j. l. cardy , j. phys . a * 15 * , l593 ( 1982 ) . j. groenevold , phys . lett . * 3 * , 50 ( 1962 ) . d. poland , j. stat . phys . * 35 * , 341 ( 1984 ) . a. baram and m. luban , phys a * 36 * , 760 ( 1987 ) . e. hiis hauge and p. c. hemmer , physica * 29 * , 1338 ( 1963 ) . r. j. baxter , j. phys . a * 13 * , l61 ( 1980 ) has obtained exact results for the hard - hexagon lattice gas . d. a. kurtze and m. e. fisher , phys . b * 20 * , 2785 ( 1979 ) . y. shapir , j. phys . a * 15 * , l433 ( 1982 ) . j. glimm and a. jaffe , _ quantum physics : a functional point of view _ , 2nd ed . ( springer verlag , berlin , 1987 ) . m. kac , phys . fluids * 2 * , 8 ( 1959 ) . j. hubbard , phys . lett . * 3 * , 77 ( 1954 ) . r. l. stratonovich , doklady akad . nauk sssr * 115 * , 1097 ( 1957 ) [ sov . doklady * 2 * , 416 ( 1957 ) ] . a. g. moreira , m. m. telo da gama , and m. e. fisher , j. chem . phys . * 110 * , 10058 ( 1999 ) . r. r. netz and h. orland , _ one and two - component hard - core plasmas _ ( cond - mat/9902220 , 16 feb 1999 ) . j. w. negele and h. orland , _ quantum many particle systems _ ( addison wesley , 1988 ) p. 199 . j.hubbard and p. schofield , 40a * , 245 ( 1972 ) . n. v. brilliantov , phys . e * 58 * , 2628 ( 1998 ) .

lattice and continuum fluid models with repulsive - core interactions typically display a dominant , critical - type singularity on the real , _ negative _ activity axis . lai and fisher recently suggested , mainly on numerical grounds , that this repulsive - core singularity is universal and in the same class as the yang - lee edge singularities , which arise above criticality at complex activities with _ positive _ real part . a general analytic demonstration of this identification is presented here using a field - theory approach with separate representation of the repulsive and attractive parts of the pair interactions . = 10000 2

the run of the lhc delivered about of integrated luminosity to both the atlas and cms experiments . among the many important results coming from these data , the properties of the top - quark have been measured with unprecedented precision . at the same time , theoretical calculations of top - quark related observables have seen significant advancements in the last few years . in particular , very recently the next - to - next - to - leading order ( nnlo ) qcd corrections to differential cross sections in top - quark pair ( @xmath1 ) production have been calculated @xcite . in @xcite , the cms collaboration performed a comprehensive comparison between their measurements @xcite of the differential cross sections and various theoretical predictions , including those from the nnlo calculation and those from monte carlo event generators with next - to - leading order ( nlo ) accuracy matched to parton showers . the overall agreement between theory and data is truly remarkable , which adds to the success of the standard model ( sm ) as an effective description of nature . however , a persistent issue in the 8 tev results is that the transverse momentum ( @xmath2 ) distribution of the top quark is softer in the data than in theoretical predictions , i.e. , the experimentally measured differential cross section at high @xmath2 is lower than predictions from event generators or from nlo fixed - order calculations @xcite . while the nnlo corrections bring the fixed - order predictions into better agreement with the cms data , as noted in @xcite and @xcite , there is still some discrepancy in the high-@xmath2 bins where @xmath3 . given the importance of the @xmath1 production process as a standard candle for validating the sm and as an essential background for new physics searches , it would be disconcerting if this feature were to persist at higher @xmath2 values and with more data . it is therefore important to assess the effects of qcd corrections even beyond nnlo , in order to see whether the gap between theory and data at high @xmath2 can be bridged . for boosted top - quark pairs with high @xmath2 there are two classes of potentially large contributions . the first is the sudakov - type double logarithms arising from soft gluon emissions . the second comes from gluons emitted nearly parallel to the top quarks , resulting in large logarithms of the form @xmath4 , where @xmath5 is the top quark mass , and @xmath6 is the transverse mass of the top quark . in @xcite , some of the authors of the current work developed a formalism for the simultaneous resummation of both type of logarithms to all orders in the strong coupling constant @xmath7 . in this letter , we report the first phenomenological applications of that formalism , giving predictions for the top - quark @xmath2 and the @xmath1 invariant mass distributions at the lhc , and comparing with experimental measurements as well as the nnlo calculations when possible . with an eye to the future , we also present predictions for the lhc , where nnlo results are not yet available . our main finding is that the higher - order effects contained in our resummation formalism significantly alter the high - energy tails of the @xmath2 and @xmath1 invariant mass distributions , softening that of the @xmath2 distribution but enhancing that of the @xmath1 invariant mass distribution . these effects bring our results into better agreement with the experimental data compared to pure nlo fixed - order calculations . interestingly , for the case of the @xmath2 distribution , this softening of the spectrum is slightly stronger than the similar effect displayed in recent nnlo results , and leads to a better modeling of the @xmath8 gev portion of the cms data @xcite . we comment further on this fact in the conclusions . our predictions are based on the factorization and resummation formula derived in @xcite . the technical details will be given in a forthcoming article , although the main elements have already been sketched out in @xcite . in the kinematic situation where the top quarks are highly boosted and the events are dominated by soft gluon emissions , the resummed partonic differential cross section in mellin space can be written as @xmath9 \\ & \hspace{-8em } \times \widetilde{u}_d^2(\mu_f,\mu_{dh},\mu_{ds } ) \ , c_d^2(m_t,\mu_{dh } ) \ , \widetilde{s}_d^2 \biggl ( \ln\frac{m_t}{\bar{n}\mu_{ds } } , \mu_{ds } \biggr ) \ , , \nonumber\end{aligned}\ ] ] where for simplicity , we have suppressed some variables in the functional arguments which are unnecessary for the explanations below . in the above formula , @xmath10 is the invariant mass of the @xmath1 pair ( which can be related to the @xmath2 of the top quark in the soft limit through a change of variables ) , @xmath11 is the mellin moment variable , and @xmath12 with @xmath13 the euler constant . the soft limit corresponds to @xmath14 in mellin space . the four coefficient functions @xmath15 , @xmath16 , @xmath17 and @xmath18 encode contributions from four widely - separated energy scales @xmath10 , @xmath19 , @xmath5 and @xmath20 , respectively . the presence of the four scales leads to the two types of large logarithms discussed in the introduction . in correspondence with these four physical scales , there are four unphysical renormalization scales @xmath21 , @xmath22 , @xmath23 and @xmath24 , one for each coefficient function . the philosophy of resummation is to choose the four unphysical scales to be around their corresponding physical scales , so that the four coefficient functions are free of large logarithms and are well - behaved in fixed - order perturbation theory . one can then use renormalization group ( rg ) equations to evolve these functions to the factorization scale @xmath25 in order to convolute with the parton distribution functions ( pdfs ) and obtain the hadronic cross sections . the effects of the rg running are encoded in the two evolution factors @xmath26 ( for @xmath15 and @xmath16 ) and @xmath27 ( for @xmath17 and @xmath18 ) , which resum all the large logarithms to all orders in @xmath7 in an exponential form . at the moment , the four coefficient functions are known to nnlo @xcite , while the two evolution factors are known to next - to - next - to - leading logarithmic ( nnll ) accuracy @xcite . such a level of accuracy is usually referred to as nnll@xmath0 in the literature , and we adopt that nomenclature here . while the formula ( [ eq : boosted - resummed ] ) is only applicable in the boosted soft limit , we can extend its domain of validity by combining it with information from nnll soft gluon resummation derived in @xcite ( recast into mellin space ) as well as the nlo fixed - order result calculated in @xcite and implemented in mcfm @xcite . the precise matching formula can be found in @xcite . after such a matching procedure , we denote the final accuracy of our predictions , which are valid throughout phase space , as nlo+nnll@xmath0 . it would be desirable to match with the recent nnlo results in @xcite to achieve nnlo+nnll@xmath0 accuracy . however , at the moment nnlo results are only available for fixed ( i.e. , kinematics - independent ) factorization and renormalization scales @xmath28 , whereas for the study of differential distributions over large ranges of phase space we consider it important to follow common practice and use dynamical ( i.e. , kinematics - dependent ) scale choices . therefore , such an improvement over our result is not currently possible , and we leave it for the future . in the following we present nlo+nnll@xmath0 predictions for the @xmath10 and @xmath2 distributions at the lhc . in all our numerics we choose @xmath29 and use mstw2008nnlo pdfs @xcite . for @xmath2 distributions , the default values for the factorization scale and the four renormalization scales are chosen as @xmath30 , @xmath31 , @xmath32 , @xmath33 and @xmath34 . for @xmath10 distributions , the only difference is @xmath35 . we estimate scale uncertainties by varying the five scales around their default values by factors of two and combining the resulting variations of differential cross sections in quadrature ; we do not consider uncertainties from pdfs and @xmath7 in this letter . the hadronic differential cross sections are first evaluated in mellin space at a given point in phase space , and we then perform the inverse mellin transform numerically using the minimal prescription @xcite . this procedure relies on an efficient construction of mellin - transformed parton luminosities , for which we use methods outlined in @xcite . the differential cross sections considered below span several orders of magnitude when going from low to high values of @xmath2 or @xmath10 . in order to better display the relative sizes of various results , we show in the lower panel of each plot the differential cross sections normalized to our default prediction , i.e. , the ratio defined by @xmath36 resummed prediction ( blue band ) for the normalized top - quark @xmath2 distribution at the 8 tev lhc compared with cms data ( red crosses ) @xcite and the nnlo result ( magenta band ) @xcite . the lower panel shows results normalized to the default nlo+nnll@xmath0 prediction . ] [ fig : ptt_cms ] compares our nlo+nnll@xmath0 resummed prediction for the normalized top - quark @xmath2 distribution to the cms measurement @xcite in the lepton+jet channel at the lhc with a center - of - mass energy @xmath37 . also shown is the nnlo result from @xcite , which adopted by default the renormalization and factorization scales @xmath38 , and also used a slightly different top - quark mass , @xmath39 . at low @xmath2 , it is clear that both the nlo+nnll@xmath0 and the nnlo results describe the data fairly well . with the increase of @xmath2 , it appears that the nnlo prediction systematically overestimates the data , although there is still agreement within errors . on the other hand , with the simultaneous resummation of the soft gluon logarithms and the mass logarithms and also with the dynamical scale choices , our nlo+nnll@xmath0 resummed formula produces a softer spectrum which agrees well with the data . resummed prediction ( blue band ) for the absolute @xmath2 distribution at the lhc in the boosted region compared with the atlas data ( red crosses ) @xcite and the nlo result ( magenta band ) . ] resummed prediction ( blue band ) for the absolute @xmath10 distribution at the lhc compared with atlas data ( red crosses ) @xcite and the nlo result ( magenta band ) . ] in @xcite , the atlas collaboration carried out a measurement of the top - quark @xmath2 spectrum in the highly - boosted region using fat - jet techniques . although the experimental uncertainty is rather large due to limited statistics , it is interesting to compare it with the theoretical predictions here , since it is expected that the soft and small - mass logarithms become more relevant at higher energies . in fig . [ fig : ptt_atlas_boosted ] we show such a comparison . the nnlo result for such high @xmath2 values is not yet available , so we compare instead with the nlo result computed using mcfm with mstw2008nlo pdfs and dynamical renormalization and factorization scales , whose default values are @xmath40 . scale uncertainties of the nlo results are estimated through variations of @xmath41 by a factor of two around the default value . from the plot one can see that the nlo result calculated in this way does a good job in estimating the residual uncertainty from higher order corrections , as the resummed band lies almost inside the nlo one up to @xmath42 . on the other hand , the inclusion of the higher - order logarithms in the nlo+nnll@xmath0 result significantly reduces the theoretical uncertainty , which is crucial for future high precision experiments at the lhc . resummed predictions ( blue bands ) for the @xmath2 and @xmath10 distributions at the lhc compared with the nlo results ( magenta bands).,title="fig : " ] + resummed predictions ( blue bands ) for the @xmath2 and @xmath10 distributions at the lhc compared with the nlo results ( magenta bands).,title="fig : " ] our formalism is flexible and can be applied to other differential distributions as well . to demonstrate this fact , in fig . [ fig : mtt_atlas ] we show the nlo+nnll@xmath0 resummed prediction for the top - quark pair invariant mass distribution along with a measurement from the atlas collaboration @xcite at the lhc . since the nnlo result in @xcite for this distribution has an incompatible binning , it is currently not possible to include it in the plot , so we show instead the nlo result computed with the same input as in fig . [ fig : ptt_atlas_boosted ] , but this time with the default scale choice @xmath43 . one can see from the plot that the nlo result with this scale choice is consistently lower than the experimental data . the resummation effects significantly enhance the differential cross sections , especially at high @xmath10 . as a result , the nlo+nnll@xmath0 prediction agrees with data quite well . we have found that choosing the default renormalization and factorization scales to be half the invariant mass increases the fixed - order cross section and therefore mimics to some extent the resummation effects . in fact , this procedure has been extensively employed in the literature for processes such as higgs production @xcite , where higher - order corrections are also large . consequently , it may be advisable to employ a renormalization and factorization scale of the order of @xmath44 in fixed - order calculations ( and monte carlo event generators ) , and we shall use this choice when studying the @xmath45 distribution at the 13 tev lhc below . relative sizes of the corrections at approximate nnlo ( blue ) and beyond ( black ) , with respect to nlo . ( [ rat - defs ] ) and the explanations there for precise definitions . ] the lhc has started the run in 2015 . so far there are only two cms measurements @xcite of differential cross sections for @xmath1 production , based on just of data . the resulting experimental uncertainties are therefore quite large and it is not yet possible to probe higher @xmath2 or @xmath10 values . nevertheless , in the near future there will be a large amount of high - energy data , which will enable high - precision measurements of @xmath1 kinematic distributions , also in the boosted regime . in fig . [ fig : lhc13 ] we show our predictions for the @xmath2 and @xmath10 spectrum up to @xmath46 and @xmath47 , contrasted with the nlo results . note that for the @xmath10 distribution , we have changed the default @xmath25 to a lower value @xmath44 for the reasons explained above . the plots exhibit similar patterns as observed at @xmath48 , namely that the higher - order resummation effects serve to soften the tail of the @xmath2 distribution but enhance that of the @xmath10 distribution compared to a pure nlo calculation . as mentioned before , we would like to match our calculations with the nnlo results when they become available in the future . we end this section by discussing the expected effects of such a matching , by estimating the size of resummation corrections beyond nnlo . we do this in fig . [ fig : lhc13-radcors ] , where the relative sizes of the beyond - nnlo corrections generated through the resummation formula are displayed as a function of @xmath10 or @xmath2 with the default scale choices . the exact nnlo results for these scale choices are not yet available , so we show in comparison the relative sizes of the approximate nnlo ( annlo ) corrections obtained by expanding and truncating our resummation formula to that order . more precisely , the blue and black curves in fig . [ fig : lhc13-radcors ] correspond to @xmath49 where @xmath50 refers to the approximate nnlo result . the figure clearly shows that corrections beyond nnlo are significant in the tails of the distributions , especially in the case of the @xmath51 distribution . in this letter we have presented new resummation predictions for differential cross sections in @xmath1 production at the lhc . the predictions include the simultaneous resummation to nnll@xmath0 accuracy of both soft and small - mass logarithms , which endanger the convergence of the fixed - order perturbative series in the boosted regime where the partonic center - of - mass energy is much larger than the mass of the top quark . this resummation is matched with both standard soft - gluon resummation at nnll accuracy and fixed - order nlo calculations , so that our results are applicable in the whole phase space . such predictions for @xmath1 differential distributions at the lhc are not only the first to be calculated in mellin space , but also represent the highest resummation accuracy achieved to date , namely nlo+nnll@xmath0 . our results are thus a major step forward in the modeling of high - energy tails of distributions , which is of great importance for new physics searches . the agreement of nlo+nnll@xmath0 predictions with data indicates the value of including resummation effects and using dynamical scale settings correlated with @xmath2 or @xmath10 when studying differential distributions . interestingly , in the case of normalized @xmath2 distribution measured by the cms collaboration @xcite , the nlo+nnll@xmath0 calculation produces a slightly softer spectrum than recent nnlo predictions ( which use a fixed scale setting where @xmath52 by default ) , thus achieving a better agreement with the data . however , we emphasize that the optimal use of resummation is to supplement nnlo calculations , not to replace them . with this in mind , we have studied the size of corrections beyond nnlo encoded in our resummation formula , and found that their effects are significant in the high - energy tails of distributions , especially for the @xmath1 invariant mass distribution where they enhance the differential cross section . it will therefore be an essential and informative exercise to produce nnlo+nnll@xmath0 predictions once nnlo calculations are available with dynamical scale settings . _ acknowledgments : _ we would like to thank alexander mitov for providing us the results of the nnlo calculations in @xcite . we are grateful to andrea ferroglia for collaboration on many related works . x. wang and l. l. yang are supported in part by the national natural science foundation of china under grant no . d. j. scott is supported by an stfc postgraduate studentship . a. ferroglia , b. d. pecjak and l. l. yang , jhep * 1210 * , 180 ( 2012 ) [ arxiv:1207.4798 [ hep - ph ] ] . v. ahrens , a. ferroglia , m. neubert , b. d. pecjak , l. l. yang , jhep * 1009 * , 097 ( 2010 ) [ arxiv:1003.5827 [ hep - ph ] ] . p. nason , s. dawson and r. k. ellis , nucl . b * 327 * , 49 ( 1989 ) ; m. l. mangano , p. nason and g. ridolfi , nucl . b * 373 * , 295 ( 1992 ) ; s. frixione , m. l. mangano , p. nason and g. ridolfi , phys . b * 351 * , 555 ( 1995 ) [ hep - ph/9503213 ] . a. d. martin , w. j. stirling , r. s. thorne and g. watt , eur . j. c * 63 * , 189 ( 2009 ) [ arxiv:0901.0002 [ hep - ph ] ] . s. catani , m. l. mangano , p. nason and l. trentadue , nucl . b * 478 * , 273 ( 1996 ) [ hep - ph/9604351 ] . m. bonvini and s. marzani , jhep * 1409 * , 007 ( 2014 ) [ arxiv:1405.3654 [ hep - ph ] ] . m. bonvini , arxiv:1212.0480 [ hep - ph ] . g. aad _ et al . _ [ atlas collaboration ] , arxiv:1511.04716 [ hep - ex ] .

we present state of the art resummation predictions for differential cross sections in top - quark pair production at the lhc . they are derived from a formalism which allows the simultaneous resummation of both soft and small - mass logarithms , which endanger the convergence of fixed - order perturbative series in the boosted regime , where the partonic center - of - mass energy is much larger than the mass to the top quark . we combine such a double resummation at nnll@xmath0 accuracy with standard soft - gluon resummation at nnll accuracy and with nlo calculations , so that our results are applicable throughout the whole phase space . we find that the resummation effects on the differential distributions are significant , bringing theoretical predictions into better agreement with experimental data compared to fixed - order calculations . moreover , such effects are not well described by the nnlo approximation of the resummation formula , especially in the high - energy tails of the distributions , highlighting the importance of all - orders resummation in dedicated studies of boosted top production .

stochastic modelling is often the most effective tool available in order to describe complex systems in physical , biological and social networks @xcite . in particular , since natural noise sources are mostly gaussian , stationary and non - stationary gaussian processes are often used to model the response of a system exposed to environmental noise . in view of the increasing interest towards complex systems , a question thus naturally arises on whether an effective characterization of gaussian processes is achievable . in this paper we address the characterization of classical random fields and focus attention on fractional gaussian processes . the reason is twofold : on the one hand , most of of the noise sources in nature are gaussian and the same is true for the linear response of systems exposed to environmental noise @xcite . on the other hand , fractional processes have recently received large attention since they are suitable to describe noise processes leading to complex trajectories , e.g. irregular time series characterized by a haussdorff fractal dimension in the range @xmath0 . in particular , in order to maintain the discussion reasonably self contained , we focus on systems exposed to fractional brownian noise @xcite ( fbn ) @xmath1 , which is a paradigmatic nonstationary gaussian stochastic process with zero mean @xmath2_b=0 $ ] and covariance @xcite @xmath3_b & \equiv k(t , s ) \notag \\ & = \frac12 v_h\left(|t|^{2h } + \label{bh}\end{aligned}\ ] ] where @xmath4 @xmath5 being the euler gamma function . in the above formulas @xmath6 is a real parameter @xmath7 $ ] , usually referred to as the hurst parameter @xcite . the hurst parameter is directly linked to the fractal dimension @xmath8 of the trajectories of the particles exposed to the fractional noise . the notation @xmath9_b$ ] denotes expectation values taken over the values of the process and represents a shorthand for the functional integral @xmath10_b&=\int\ ! { \cal d}[b_h(t)]\ , { \cal p}[b_h(t)]\ , f(t ) \notag\\ & 1 = \int\ ! { \cal d}[b_h(t)]\ , { \cal p}[b_h(t ) ] \ , , \notag\end{aligned}\ ] ] performed over all the possible realizations of the process @xmath1 , each one occurring with probability @xmath11 $ ] . we remind that fbn is a self - similar gaussian process , i.e. @xmath12 , and that it is suitable to describe anomalous diffusion processes with diffusion coefficients proportional to @xmath13 , corresponding to ( generalized ) noise spectra with a powerlaw dependence @xmath14 on frequency @xcite . the characterization of fbn amounts to the determination of the fractal dimension of the resulting trajectories , i.e. the determination of the parameter @xmath6 . in the following , in order to simplify notation and formulas , we will employ the complementary hurst parameter @xmath15 $ ] and upon replacing @xmath16 in eq . ( [ bh ] ) , we will denote the fbn process by @xmath17 the purpose of this paper is to address in some details the characterization of fbn , i.e. the determination of the parameter @xmath18 , using _ quantum probes_. this means that we consider a system , say a particle , subject to fbn , and assume that its motional degree of freedom , regarded to be classical , is coupled to a quantum degree of freedom of the same system , e.g. its spin . we then ignore the noisy classical part and exploit quantum limited measurements on the spin part to extract information about the fbn . notice , however , that our approach and our results are also valid to assess the performances of quantum limited measurements for any two - level system subject to dephasing perturbations described by fractional brownian noise , i , e . without the need of referring to a qubit coupled to the motion of a particle . we will address both _ estimation _ and _ discrimination _ problems for the fractal dimension of the fbn , i.e. situations where the goal is to estimate the unknown values of the parameter @xmath19 $ ] , and cases where we know in advance that only two possible values @xmath20 and @xmath21 are admissible and want to discriminate between them @xcite . several techniques have been suggested for the estimation of the hurst parameter in the time or in the frequency domain @xcite , or using wavelets @xcite . among them we mention range scale estimators @xcite , maximum likelihood @xcite , karhunen - loeve expansion @xcite , p - variation @xcite , periodograms @xcite , weigthed functional @xcite , and linear bayesian models @xcite . compared to existing techniques , quantum probes offers the advantage of requiring measurements performed at a fixed single ( optimized ) instant of time , without the need of observing the system for a long time in order to collect a time series , and thus avoiding any issue related to poor sampling @xcite . as we will see , quantum probes may be effectively employed to characterize fractional gaussian process when the the system - environment coupling is weak , provided that a long interaction time is achievable , or when the coupling is strong and the quantum probe may be observed shortly after that the interaction has been switched on . overall , and together with results obtained for the characterization of stationary process @xcite , our results indicate that quantum probes may represent a valid alternative to other techniques to characterize classical noise . the paper is structured as follows : in section [ s : mod ] we introduce the physical model and discuss the dynamics of the quantum probe . in section [ s : qest ] we briefly review the basic notions of quantum information geometry and evaluate the figures of merit that are relevant to our problems . in section [ s : qp ] we discuss optimization of the interaction time , and evaluate the ultimate bounds to the above figures of merit that are achievable using quantum probes . section [ s : out ] closes the paper with some concluding remarks . we consider a spin @xmath22 particle in a situation where its motion is subject to environmental fbn noise and may be described classically . we assume that the motional degree of freedom of the particle is coupled to its spin , such that the effects of noise influence also the dynamics of the spin part . we also assume that the noise spectrum of the fbn contain frequencies that are far away from the natural frequency @xmath23 of the spin part . when the spectrum contains frequencies that are _ smaller _ than @xmath23 than the fluctuation induced by the fbn are likely to produce decoherence of the spin part , rather the damping , such that the time - dependent interaction hamiltonian between the motional and the spin degrees of freedom may be written as @xmath24 where @xmath25 denotes a pauli matrix and @xmath26 denotes the coupling between the spin part and its classical environment . we do not refer to any specific interaction model between the motional degree of freedom and the spin part and assume that eq . ( [ hi ] ) describes the overall effect of the coupling . the full hamiltonian of the spin part is given by @xmath27 and may be easily treated in the interaction picture . upon denoting by @xmath28 the initial state of the spin part , the state at a subsequent time @xmath29 is given by @xmath30_b$ ] , where @xmath31 upon substituting the above expression of @xmath32 in @xmath33 we arrive at @xmath34_b\ , \rho_0 + e[\sin^2\varphi(t)]_b\ , \sigma_z\rho_0\sigma_z \notag \\ & - i e[\sin\varphi(t)\cos\varphi(t)]_b\,[\sigma_z,\rho_0 ] \notag \\ = & p_\gamma ( t,\lambda)\ , \rho_0 + [ 1-p_\gamma(t,\lambda ) ] \,\sigma_z\rho_0\sigma_z\,.\label{evol}\end{aligned}\ ] ] in writing the last equality , we have already employed the averages over the realizations of the fractional process @xmath35_b & = \frac12 \left[1+\exp\left\{-\frac{\lambda\ , t^{2\gamma } v_\gamma}{\gamma}\right\}\right ] \notag \\ e[\cos\varphi(t)\sin\varphi(t)]_b & = 0\notag\,,\end{aligned}\ ] ] which have been evaluated taking into account that @xmath17 is a gaussian process with zero mean and covariance @xmath36 , i.e. by using the generating function @xmath37_b = \notag\\ & \exp\left\{-\frac12\int_0^t\!\int_0^t\ ! ds ds^\prime f(s ) \,k(s , s^\prime)\ , f(s^\prime)\right\}\,,\end{aligned}\ ] ] which leads to @xmath38_b & = e\left [ \exp\left\{-i m \int_0^t\!\ ! ds\ , b_\gamma(s)\right\ } \right]_b \notag \\ & = \exp\left\{-\frac12 m^2 \beta(t ) \right\ } \quad \forall m\in { \mathbbm z}\ , , \notag \end{aligned}\ ] ] where @xmath39 in the complementary case , i.e. when the noise spectrum of the fbn contains frequencies that are larger than the natural frequency of the spin part , the dominant process induced by the environmental noise is damping , such that the overall hamiltonian may be written as @xmath40 . due to the presence of the transverse field in the time - dependent stochastic hamiltonian there is no exact ( close ) solution for the unitary evolution , which involves time ordering . when the quantity @xmath41 in the characteristic function is small @xcite , e.g. in the limit of slowly varying @xmath17 we may write the quasi static unitary evolution , which reads as follows @xmath42 where the last equality is valid if @xmath43 , i.e. assuming @xmath44 , @xmath45 . in this limit , the damping evolution operator in eq . ( [ ud ] ) is just a rotated version of the decoherence one in eq . ( [ udc ] ) . in general in the following we limit ourselves to estimation and discrimination problems involving a fbn inducing nondissipative decoherence , i.e. with noise spectrum containing frequencies smaller than @xmath23 and leading to an evolution operator of the form ( [ udc ] ) . the characterization of fbn by quantum probes amount to distinguish quantum states in the class @xmath46 , i.e. states originating from a common initial state @xmath28 and evolving in different noisy fbn channels , each one characterized by a different hurst parameter , and thus inducing trajectorie with different fractal dimension . distinguishability of quantum states is generally quantified by a distance in the hilbert space . however , depending on the nature of the estimation / discrimination problem at hand , different distances are involved to capture the relevant notion of distinguishability @xcite . in situations where we want to estimate the unknown value of @xmath47 $ ] the problem is to discriminate a quantum state within the continuous family @xmath46 . in this case , the relevant quantity is the so - called bures infinitesimal distance between nearby point in the parameter space @xcite @xmath48 , where the _ bures metric _ @xmath49 is given by @xmath50 @xmath51 being the eigenvectors of @xmath52 in terms of the fidelity @xmath53\right)^2 $ ] . the relevance of the bures metric in estimation problems comes from the fact that @xmath54 where @xmath55 is the quantum fisher information of the considered statistical model @xmath33 @xcite . in order to appreciate this fact , let us remind that any estimation problem consists in inferring the value of a parameter @xmath18 , which is not directly accessible , by measuring a related quantity @xmath56 . the solution of the problem amounts to find an estimator @xmath57 , _ i.e. _ a real function of the measurements outcomes @xmath58 to the parameters space . classically , the variance @xmath59 of any unbiased estimator satisfies the cramer - rao bound @xmath60 , which establishes a lower bound on variance in terms of the number of independent measurements @xmath61 and the fisher information @xmath62 ^ 2 $ ] , @xmath63 being the conditional probability of obtaining the value @xmath64 when the parameter has the value @xmath18 . when quantum systems are involved , we have @xmath65 $ ] , @xmath66 being the probability operator - valued measure ( povm ) describing the measurement . a quantum estimation problem thus corresponds to a quantum statistical model , _ i.e. _ a set of quantum states @xmath33 labeled by the parameter of interest , with the mapping @xmath67 providing a coordinate system . upon introducing the symmetric logarithmic derivative ( sld ) @xmath68 as operator satisfying the equation @xmath69 $ ] one can prove @xcite that @xmath70 is upper bounded by the quantum fisher information @xmath71 $ ] . in turn , the ultimate limit to precision is given by the quantum cramer - rao theorem ( qcr ) @xmath72 which provides a measurement - independent lower bound for the variance which is attainable upon measuring a povm built with the eigenprojectors of the sld . in fact , quantum estimation theory has been successfully employed for the estimation of static noise parameters @xcite and in several other scenarios , as for example quantum thermometry @xcite . for quantum systems with a bidimensional hilbert space , as those we are investigating in this paper , the optimal measurement is a projective one @xcite . besides , using eqs . ( [ evol ] ) and ( [ gb ] ) , it is straightforward to show that starting from a generic pure initial state @xmath73 the maximum of @xmath49 is achieved for @xmath74 . in this case , the evolved state @xmath46 is a mixed state with eigenvectors independent on @xmath18 . in other words , the dependence on @xmath18 is only in the eigenvalues , and thus eq . ( [ gb ] ) reduces to @xmath75 ^ 2}{p_\gamma(t,\lambda)[1-p_\gamma(t,\lambda ) ] } \notag \\ = & \frac{t^{4 \gamma}\,\lambda^2}{\gamma^4 } \left[\gamma\ , \partial_\gamma v_\gamma - ( 1 - 2\gamma \log t)\,v_\gamma \right]^2 \notag \\ & \times \left(e^{\frac{2\lambda\,t^{2\gamma}}{\gamma}v_\gamma}-1\right)^{-1}\ , , \label{gbg}\end{aligned}\ ] ] where @xmath76\,,\ ] ] @xmath77 being the the log - derivative of the euler gamma function . the quantum cramer - rao theorem implies that the optimal conditions to estimate @xmath18 by quantum probes correspond to the maxima of @xmath49 . as mentioned above , the optimization over the initial state is trivial and correspond to prepare the spin of the particle in the superposition @xmath78 , whereas the maximization over the time evolution will be discussed in the next section . let us now consider situations where we have to discriminate between two fixed and known values of @xmath18 , e.g. the null hypothesis @xmath79 and the alternative @xmath80 corresponding to a non trivial fractal dimension . the corresponding states @xmath81 and @xmath82 are assumed to be known , as well as the _ a priori _ probabilities @xmath83 and @xmath84 , but we do nt know which state is actually received at the end of propagation . the simplest case occurs when the _ a priori _ probabilities are equal @xmath85 . any strategy for the discrimination between the two states amounts to define a two - outcomes povm @xmath86 on the system and establish the inference rule that after observing the outcome @xmath87 the observer infers that the state of the system is @xmath88 @xcite . the probability of inferring @xmath89 when the true value is @xmath90 is thus given by @xmath91 $ ] and the optimal povm for the discrimination problem is the one minimizing the overall probability of a misidentification i.e. @xmath92 . for the simplest case of equiprobable hypotheses ( @xmath93 ) we have @xmath94\right)$ ] where @xmath95 . @xmath96 is minimized by choosing @xmath97 as the projector over the positive subspace of @xmath98 . then we have @xmath99 = { \mathop{\text{tr}}\nolimits}|\lambda| $ ] and @xmath100 where @xmath101 . this is usually referred to as the helstrom bound , and represent the ultimate quantum bound to the error probability in a binary discrimination problem . in our case , @xmath96 is minimized when the two output states commute , i.e. for @xmath74 leading to @xmath102 where @xmath103 is given in section [ s : mod ] . the minimization over the interaction time will be discussed in the next section . we notice , however , that any single - copy discrimination strategy based on quantum probes is inherently inefficient since eq . ( [ peqp ] ) imposes an error probability larger than @xmath104 at any time . one is therefore led to consider different strategies , as those involving several copies of the quantum probes . indeed , let us now suppose that @xmath105 copies of both states are available for the discrimination . the problem may be addressed using the above formulas upon replacing @xmath106 with @xmath107 . we thus need to analyze the quantity @xmath108 . the evaluation of the trace distance for increasing @xmath105 may be difficult and for this reason , one usually resort to the quantum chernoff bound , which gives an upper bound to the probability of error @xcite @xmath109 where @xmath110= \inf_{0\leq s\leq 1 } { \mathop{\text{tr}}\nolimits}\left[\rho_{\gamma_1 } ^s\ : \rho_{\gamma_2}^{1-s}\right]\,.\end{aligned}\ ] ] the bound may be attained in the asymptotic limit of large @xmath105 . notice that while the trace distance is capturing the notion of distinguishability for single copy discrimination this is not the case for multiple copies strategies , where the quantity @xmath111 represent the proper figure of merit . also in eq . ( [ q ] ) we omitted the explicit dependence on the interaction time . for nearby states the relevant distance is the so - called infinitesimal quantum chernoff bound ( qcb ) distance @xmath112 , where the qcb metric @xmath113 is given by @xmath114 the qcb introduces a measure of distinguishability for density operators which acquires an operational meaning in the asymptotic limit . the larger is the qcb distance , the smaller is the asymptotic error probability of discriminating a given state from its neighbors . on the other hand , for a fixed probability of error @xmath96 , the smaller is @xmath111 , the smaller the number of copies of @xmath81 and @xmath82 we will need in order to distinguish them . also the quantity @xmath111 is minimized when the two output states commute , i.e. for @xmath74 and , in this case we have @xmath115^s \,[1-p_{\gamma_2}(t,\lambda)]^{1-s } \big\}\ , . \notag\end{aligned}\ ] ] the minimization over the parameter @xmath116 and the interaction time will be discussed in the next section . concerning the qcb metric , we have the general relation @xmath117 . in our case , since the maximum is achieved when only the eigenvalues of @xmath118 depends on @xmath18 , the only non zero terms in eqs . ( [ gb ] ) and ( [ gqcb ] ) are those with @xmath119 . as a consequence the first inequality above is saturated and we have @xmath120 , @xmath121 . the working conditions to optimize the estimation or the discrimination of nearby states are thus the same . in this section we discuss optimization of the estimation / discrimination strategies for fbn over the possible values of the interaction time . more explicitly , we maximize the bures metric and minimize the helstrom and qcb bound to error probability , as a function of the interaction time . in this way , we individuate the optimal working conditions , maximizing the performances of quantum probes , and establish a benchmark to assess any strategy based on non optimal measurements . for the estimation of the complementary hurst parameter @xmath18 as a function of @xmath18 and of the interaction time , for different values of the coupling @xmath26 . the contour plots correspond , from top left to bottom right , to @xmath122 , and @xmath123 , respectively . whiter regions correspond to larger values of the bures metric . ] upon inspecting the functional dependence of the bures metric on the quantities @xmath29 , @xmath26 and @xmath18 in eq . ( [ gbg ] ) one sees that @xmath49 is somehow a function of the quantity @xmath124 and thus maxima are expected , loosely speaking , for small @xmath29 and large @xmath26 or viceversa . on the other hand , this scaling is not exact and thus a richer structure is expected . this is illustrated in fig . [ f : f1 ] , where we show contour plots of @xmath125 as a function of @xmath18 and of the interaction time for different values of the coupling @xmath26 . as it is apparent from the plots , for any value of the coupling there are two maxima located in different regions ( notice the different ranges for the interaction time ) . the global maximum moves from one region to the other depending on the values of the coupling ( see below ) . in fig . [ f : f2 ] we show the results obtained from the numerical maximization of the bures metric @xmath126 over the interaction time . the upper left panel is a log - log - plot of the maximized bures metric as a function of the coupling for randomly chosen values of @xmath127 $ ] and @xmath128 $ ] ( gray points ) . we also report some curves at fixed values of @xmath18 , showing that for any value of the complementary hurst parameter , except those close to the limiting values @xmath129 and @xmath130 , a threshold value @xmath131 on the coupling , i.e. on the intensity of the noise , naturally emerges . the bures metric is large , i.e. estimation may achieve high precision , in the weak and in the strong coupling limit , that is , when @xmath132 or @xmath133 . on the other hand , for intermediate values of the coupling @xmath134 the estimation of the fractal dimension is inherently inefficient . this behavior is further illustrated in the lower left panel , where we report the same random points as a function of @xmath18 , also showing curves at fixed values of the coupling . values of @xmath18 close to @xmath129 or @xmath130 may be precisely estimated for any value of the coupling whereas intermediate values needs a tuning of @xmath26 , in order to be placed in the corresponding weak ( or strong ) coupling limit . the threshold value @xmath135 increases with @xmath18 and does not appear for @xmath136 or @xmath137 . for those values high precision measurements are achievable only in the strong coupling limit ( for @xmath136 , i.e. fractal dimension close to @xmath138 ) or the weak coupling limit ( @xmath137 , i.e. negligible fractal dimension @xmath139 ) . [ cols="^,^ " , ] the plots confirm the overall symmetry of the helstrom bound @xmath140 at fixed @xmath26 . another feature that emerges from fig . [ f : f3 ] is that , say , the pairs @xmath141 and @xmath142 or @xmath143 and @xmath144 have different discriminability despite the fact that for both pairs we have @xmath145 , i.e. the helstrom bound is not uniform . the plots also confirm the overall picture obtained in discussing estimation problems : for each pair of values @xmath146 , two regimes of strong or weak coupling may be individuated , where discrimination may be performed with reduced error probability , whereas for intermediate values of the coupling performances are degraded . the only exception regards values close to the limiting values @xmath129 or @xmath130 , where no threshold appears . we also notice that by increasing the coupling one enlarges the region in the @xmath20-@xmath21 plane where discrimination may be performed with reduce error probability . this is illustrated in the right panels of fig . [ f : f3 ] , where we show a density plot of the minimized helstrom bound as a function of both the values @xmath20 and @xmath21 for two different values of the coupling : @xmath147 ( top panel ) and @xmath148 ( bottom panel ) of values of the complementary hurst parameter by quantum probes . in the left panel we report the maximized chernoff bound as a function of the coupling with the environment for pair of values @xmath149 with @xmath18 not too close to the limiting values @xmath129 or @xmath130 . from left to right we have , @xmath150 ( blue squares ) , @xmath151 ( green triangles ) , @xmath152 ( red circles ) , @xmath153 ( magenta stars ) , @xmath154 ( gray squares ) , @xmath155 ( gray circles ) . in the right panel we show the same quantity for pair of values @xmath156 close to the boundaries @xmath129 and @xmath130 . the increasing curves correspond to @xmath157 ( blue circles ) , @xmath158 ( blue stars ) , @xmath159 ( blue triangles ) , whereas the decreasing ones are for @xmath160 ( black circles ) , @xmath161 ( black stars ) , @xmath162 ( black triangles ) . , title="fig : " ] of values of the complementary hurst parameter by quantum probes . in the left panel we report the maximized chernoff bound as a function of the coupling with the environment for pair of values @xmath149 with @xmath18 not too close to the limiting values @xmath129 or @xmath130 . from left to right we have , @xmath150 ( blue squares ) , @xmath151 ( green triangles ) , @xmath152 ( red circles ) , @xmath153 ( magenta stars ) , @xmath154 ( gray squares ) , @xmath155 ( gray circles ) . in the right panel we show the same quantity for pair of values @xmath156 close to the boundaries @xmath129 and @xmath130 . the increasing curves correspond to @xmath157 ( blue circles ) , @xmath158 ( blue stars ) , @xmath159 ( blue triangles ) , whereas the decreasing ones are for @xmath160 ( black circles ) , @xmath161 ( black stars ) , @xmath162 ( black triangles ) . , title="fig : " ] as mentioned in section [ s : qest ] , the helstrom bound to the single - shot error probability by quantum probes is bounded from below by the value @xmath163 , making these kind discrimination schemes of little interest for applications . we are thus naturally led to consider multiple - copy discrimination . in fig . [ f : f4 ] we report the results of the optimization of the chernoff bound of eq . ( [ q ] ) over the parameter @xmath116 and the interaction time . in the left panel we show the quantity @xmath164 , minimized over the interaction time , as a function of the coupling with the environment for different pairs of values @xmath20 and @xmath21 not too close to the limiting values @xmath129 and @xmath130 . also in this case , the plot also confirms that better performances are obtained in the regimes of weak and strong coupling , whereas for intermediate values no measurements are able to effectively extract information from the quantum probe . the threshold to define the two regimes increases with the value of the @xmath18 s themselves . when the values of the hurst parameter are approaching the limiting values @xmath129 and @xmath130 no threshold appears . in these two limiting cases discrimination may be reliably performed in the weak coupling limit ( for negligible fractal dimension ) or in the strong coupling one ( fractal dimension closer to its maximum value ) . this behavior is illustrated in the right panel of fig . [ f : f4 ] , where we show the minimized @xmath164 as a function of the coupling for pairs of values @xmath20 and @xmath21 close to @xmath129 or @xmath130 . for both , single- and multiple - copy discrimination , the behavior of the optimal interaction time is analogue to that observed in the discussion of estimation problem . we have addressed estimation and discrimination problems involving the fractal dimension of fractional brownian noise . upon assuming that the noise induces a dephasing dynamics on a qubit , we have analyzed in details the performances of inferences strategies based on quantum limited measurements . in particular , in order to assess the performances of quantum probes , we have evaluated the bures metric , the helstrom bound and the chernoff bound , and have optimized their values over the interaction time . our results show that quantum probes provide an effective mean to characterize fractional process in two complementary regimes : either when the the system - environment coupling is weak , provided that a long interaction time is achievable , or when the coupling is strong and the quantum probe may be observed shortly after that the interaction has been switched on . the two regimes of weak and strong coupling are defined in terms of a threshold value of the coupling , which itself increases with the fractional dimension . our results overall indicate that quantum probes may represent a valid alternative to characterize classical noise . this work is dedicated to the memory of r. f. antoni . the author acknowledges support by miur project firb lichis - rbfr10yq3h ) . 99 p. sibani , j. h. jensen , _ stochastic dynamics of complex systems _ ( world scientific , new york , 2013 ) . d. j. wilkinson , nat . * 10 * , 122 ( 2009 ) . d. most , d. keles , eur . j. op . res . * 207 * , 543 ( 2010 ) . p. e. smouse , s. focardi , p. r. moorcroft , j. g. kie , j. d. forester , j. m. morales , phyl . b * 365 * , 2201 ( 2010 ) . r. f. fox , phys . lett . * 48 * , 179 ( 1978 ) . b. b. mandelbrot , j. w. van ness , siam rev . * 10 * , 432 ( 1968 ) . b. b. mandelbrot , j. r. wallis , water resour . res . * 4 * , 909 ( 1969 ) . m. s. taqqu , stat . * 28 * , 131 ( 2013 ) . r. j. barton , h. v. poor , ieee trans . th . * 34 * , 943 ( 1988 ) . h. e. hurst , trans . eng . * 116 * , 770 ( 1951 ) . p. flandrin , iee trans . th * 35 * , 197 ( 1989 ) . r. b. davies , d. s. harte , biometrika * 74 * , 95 ( 1987 ) . h. d. jeong , j. s. lee , d. mcnickle , and k. pawlikowski , simul . theory * 15 * , 1173 ( 2007 ) . j. barunik , l. kristoufek , physica a * 389 * , 3844 ( 2010 ) . g. w. wornell , a. v. oppenheim , ieee trans . signal pro- cess . * 40 * , 611 ( 1992 ) . l. zunino , d. g. peez , m. t. martn , a. plastino , m. garavaglia , o. a. rosso , phys . e * 75 * , 021115 ( 2007 ) . c. m. kendziorski , j. b. bassingthwaighte , p. j. tonellato , physica a * 273 * , 439 ( 1999 ) . l. a. salomon , j. c. fort , j. stat . comp . simul . * 83 * , 542 ( 2013 ) . m. magdziarz , j. k. slezak , j. wjcik , j. phys . a * 46 * , 325003 ( 2013 ) . m. s. taqqu , v. teverovsky , w. willinger , fractals * 3 * , 785 ( 1995 ) . y. liu , y. liu , k. wang , t. jiang , l. yang , phys . rev . e * 80 * , 066207 ( 2009 ) . d. boyer , d. s. dean , c meja - monasterio , g. oshanin , phys . e * 87 * , 030103(r ) ( 2013 ) . n. makarava , s. benmehdi , m. holschneider , phys . e * 84 * , 021109 ( 2011 ) . j. schmittbuhl , j .- vilotte , s. roux , phys . e * 51 * , 131 ( 1995 ) . a. mehrabi , h. rassamdana , m. sahimi , phys . e * 56 * , 712 ( 1997 ) . c. castelnovo , a. podest , p. piseri , p. milani , phys rev . e * 65 * , 021601 ( 2002 ) . c. benedetti , f. buscemi , p. bordone , m. g. a. paris , phys , rev . a * 89 * , 032114 ( 2014 ) . c. benedetti , m. g. a. paris , int . j. quantum inf . * 12 * , 1461004 ( 2014 ) . j. ehek , m. g. a. paris ( eds ) _ quantum state estimation _ , lect . not * 649 * ( springer , berlin , 2004 ) i. bengtsson , k. zyczkowski , _ geometry of quantum states _ , ( cambridge university press , 2006 ) . d. bures , trans . am . math . soc . * 135 * , 199 ( 1969 ) . a. uhlmann , rep . math . phys . * 9 * , 273 ( 1976 ) . w. k. wootters phys . d * 23 * , 357 ( 1981 ) . r. josza , j. mod . opt . * 41 * , 2314 ( 1994 ) . sommers , k. zyczkowski , j. phys . a * 36 * , 10083 ( 2003 ) . s. braunstein and c. caves , phys . lett . * 72 * , 3439 ( 1994 ) . s. braunstein , c. caves , and g. milburn , ann . 247 * , 135 ( 1996 ) . d. c. brody , l. p. hughston , proc . a * 454 * , 2445 ( 1998 ) ; 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we address the characterization of classical fractional random noise via quantum probes . in particular , we focus on estimation and discrimination problems involving the fractal dimension of the trajectories of a system subject to fractional brownian noise . we assume that the classical degree of freedom exposed to the environmental noise is coupled to a quantum degree of freedom of the same system , e.g. its spin , and exploit quantum limited measurements on the spin part to characterize the classical fractional noise . more generally , our approach may be applied to any two - level system subject to dephasing perturbations described by fractional brownian noise , in order to assess the precision of quantum limited measurements in the characterization of the external noise . in order to assess the performances of quantum probes we evaluate the bures metric , as well as the helstrom and the chernoff bound , and optimize their values over the interaction time . we find that quantum probes may be successfully employed to obtain a reliable characterization of fractional gaussian process when the coupling with the environment is weak or strong . in the first case decoherence is not much detrimental and for long interaction times the probe acquires information about the environmental parameters without being too much mixed . conversely , for strong coupling information is quickly impinged on the quantum probe and can effectively retrieved by measurements performed in the early stage of the evolution . in the intermediate situation , none of the two above effects take place : information is flowing from the environment to the probe too slowly compared to decoherence , and no measurements can be effectively employed to extract it from the quantum probe . the two regimes of weak and strong coupling are defined in terms of a threshold value of the coupling , which itself increases with the fractional dimension .

several chiral liquid crystal ( lc ) compounds have found to exhibit intermediate subphases , smc@xmath0 ( also called smc@xmath1 ) , smc@xmath2 ( also called af ) , and other ferrielectric phases ( fi@xmath3 , fi@xmath4 and fi ) between ferroelectric smc@xmath5 and antiferroelectric smc@xmath6 phases @xcite . all these phases are tilted smectic phases and they have approximately the same tilt angle from the layer normal . each phase consists of a helical stack of unit - set - of - layers ( unit cells ) ; the helix pitch is typically on the order of several @xmath7 @xcite . structures of smc@xmath8 , smc@xmath0 and smc@xmath2 among those are well - known @xcite : the unit cells characterizing smc@xmath5 , smc@xmath8 , smc@xmath0 , and smc@xmath2 consist of 1 , 2 , and 3 , 4 layers , respectively . the early experiments @xcite have identified that the @xmath9-directors ( the projection of a director onto layer - plane ) of neighboring layers are parallel or antiparallel . this experimental result indicates that the system can be described with an ising model . in fact , it has shown that an axial next nearest neighbor ising ( annni ) model , which has competing interactions , exhibits infinitely many subphases and can reproduce the transition sequence shown in experiments @xcite . however , the assumption of the ising symmetry on these lc systems is unnatural . in practice , the @xmath9-directors of layers are proved to deviate slightly from the parallel - antiparallel configuration by later high - resolution experiments @xcite . in this context , various phenomenological theories have been carried out to clarify the transition phenomena of the intermediate phases @xcite . in these models a continuous rotational freedom of tilt direction is taken into account . to a greater or less extent , these models are generalization of annni model and the models are called axial next - nearest - neighbor xy ( annnxy ) models . although the annnxy model exhibits infinitely many subphases , the system does not undergo discrete transitions between these subphases but continuously changes . by introducing an external aligning field on annnxy model , the transitions between subphases are shown to be recovered @xcite . however such external field is artificial and its origin is ambiguous . as a more realistic model applicable to the antiferroelectric materials , the annnxy model with a biquadratic interaction has been introduced and introductory remarks on the crossover from xy character to ising character are given @xcite . in the present paper , we investigate the annnxy model with the biquadratic interaction and disclose general properties of this model . following ref . @xcite , we introduce a hamiltonian of the annnxy model with a biquadratic interaction as @xmath10 @xmath11 , \label{eq : hamiltonian}\end{gathered}\ ] ] where the @xmath12 denotes the azimuthal angle of a molecule @xmath13 in @xmath14-th layer . the parameter @xmath15 is the interaction parameter between molecules belonging to the same layer . the summation in the first term of eq . is done over all neighboring pairs in the @xmath14-th layer . the parameters @xmath16 and @xmath17 are the interaction parameters between molecules in the nearest neighboring layers and next nearest neighboring layers , respectively . let @xmath16 be positive without loss of generality since the transformation @xmath18 followed by @xmath19 leaves the above hamiltonian invariant . in the following , we set @xmath17 to be negative . the last term of eq . is the biquadratic interaction with an interaction parameter @xmath20 . this biquadratic interaction term stabilizes the parallel and antiparallel configuration of @xmath9-directors . this hamiltonian can be viewed as a truncated fourier approximation of an exact hamiltonian . our model turns to the ordinary annnxy model for vanishing value of @xmath21 . in the limit of @xmath22 , it turns to the annni model . thus for sufficiently large @xmath21 the ordered phase has parallel and antiparallel @xmath9-directors ; we call such phases ising symmetric phases . we call the other phases xy symmetric phases . in the present paper , we ignore the helical structure of the chiral smectic phases , and assume these phases as simple periodic repetitions of unit cells . thus a smectic phase is characterized by a set of order parameters in the unit cell , and may be labeled with its wavenumber @xmath23 . we note that the transformation @xmath24 results in @xmath25 . thus , e.g. , @xmath26 ( smc@xmath0 ) and @xmath27 ( smc@xmath28 ) phases under negative @xmath16 correspond , respectively , to @xmath29 and @xmath30 phases under positive @xmath16 . at zero temperature , there are three stable phases @xcite : @xmath30 phase , @xmath31 phase , and a phase q in which the wave number changes continuously in the range @xmath32 . the phase diagram for temperature @xmath33 is shown in the inset of fig . [ fig : k0phasediagram ] . in the phase @xmath31 , irrespective of the value @xmath21 , the azimuthal angles in a unit cell can be written as @xmath34 and @xmath35 , i.e. , this phase has ising symmetry . the stable distribution of azimuthal angle in phase q is @xmath36 , and the stable wave number is determined by interaction parameters as @xmath37 . in this section , we investigate the stability of phases at finite temperature within the mean field approximation . we choose the order parameters of @xmath14-th layer as @xmath38 where the angular brackets indicate a thermal average over all molecules in a single layer . we can assume , without loss of generality , that in ising symmetric phases the @xmath9-directors are parallel to @xmath39-axis and thus @xmath40 . the mean fields due to these order parameters are @xmath41 , \\ \eta_{l } & = \beta \left[jzs_{l } + j_{1}(s_{l-1 } + s_{l+1 } ) + j_{2}(s_{l-2 } + s_{l+2})\right],\\ \mu_{l } & = \beta k(c_{l-1 } + c_{l+1}),\\ \nu_{l } & = \beta k(s_{l-1 } + s_{l+1 } ) , \end{split } \label{eq : meanfield}\ ] ] where we assumed that each molecule interacts @xmath42 molecules in the same layer , one molecule in each of the two nearest neighboring layers , and one molecule in each of the two next nearest neighboring layers . we can derive the mean field free energy per layer @xcite using the above order parameters and mean fields as @xmath43 , \label{eq : freeenergy}\end{gathered}\ ] ] where @xmath44 is a period ( number of layers in a unit cell ) . the function @xmath45 in the last term of eq . is the one - molecule partition function defined as @xmath46 from the derivatives of the free energy with respect to the mean fields , we obtain a set of self - consistent equations : @xmath47 where @xmath48 , @xmath49 , @xmath50 , and @xmath51 in the right - hand sides are functions of order parameters defined in eqs .. one of the solutions of eqs . which minimizes the free energy gives a set of order parameters of the thermodynamically stable state . the transition temperature between ordered and disordered phases is @xmath52 where @xmath53 and @xmath54 are , respectively , @xmath55 with the @xmath56-th order modified bessel functions @xmath57 . the variable @xmath58 is a solution of @xmath59 thus @xmath60 is given by solving eqs . and self - consistently . the above critical temperature is derived @xcite , under the assumption that the transition is second - order , by expanding the free energy @xmath61 around @xmath62 , @xmath63 and @xmath64 , and letting the quadratic terms of the expansion vanish at critical temperature . the order - disorder transition temperature deviates from eq . if the transition is first - order . indeed , as we will see below , the transition temperature is slightly higher than @xmath60 when @xmath21 is comparable to @xmath60 . in the following , we restrict our consideration to a set of _ major _ ordered phases with wavenumbers @xmath30 , @xmath65 , @xmath66 , @xmath67 , @xmath68 , and a disordered phase . since we discuss the relative stability of these phases rather than infinitely many phases , the resulting phase diagrams should be viewed as rough approximations of correct phase diagrams . however , we can expect , from the analogy of annni model , that other than the above six phases will occupy relatively small regions on the phase diagrams , and that we can safely ignore these minor phases . in order to reduce the number of order parameters , we make a reasonable assumption that the ordered phases are highly symmetric . in precise , when the period @xmath44 is odd ( i.e. , @xmath69 in this paper ) , we assume that a set of @xmath9-directors of a unit cell has mirror symmetry with respect to a plane perpendicular to the smectic layers ; when @xmath44 is even , we assume a two - fold rotational symmetry of the set of @xmath9-directors , in addition to the mirror symmetry . in the following calculation , we set @xmath70 and @xmath71 . we take @xmath72 as the unit of energy . each ordered phase has a set of ( reduced number of ) self - consistent equations . we solved the sets of equations numerically , and determined the thermodynamic stable phase by comparing the free energy per molecule . at @xmath73 , our model is reduced to an ordinary annnxy model . then the system transits through the phases of @xmath74 continuously @xcite and thus the successive phase transitions are never observed . we show the phase diagram for @xmath73 in fig . [ fig : k0phasediagram ] , in which we considered the relative stability of @xmath30 , @xmath65 , @xmath66 , @xmath67 , and @xmath68 phases . -@xmath16 plane for @xmath73 . each fractional number indicates the wavenumber @xmath23 of relatively stable phase , and d denotes a disordered phase . the shaded areas indicate the phases with xy symmetry , where non - vanishing @xmath75 and @xmath76 are allowed ; while the unshaded areas with @xmath30 and @xmath68 indicate the phases with ising symmetry , where @xmath40 . the inset is the phase diagram in the @xmath21-@xmath16 plane for zero temperature . the wavenumber of the phase q continuously changes from 0 to 1/4 . ] the phases with @xmath77 , @xmath66 , and @xmath67 have xy symmetry , while @xmath30 and @xmath68 phases have ising symmetry . it seems there is a contradiction between @xmath21-@xmath16 diagram ( inset ) and @xmath78-@xmath16 diagram at @xmath73 and @xmath33 , i.e. , stable regions of @xmath30 and @xmath68 phases are overestimated in @xmath78-@xmath16 diagram ; such a contradiction , which can be observed in all the following phase diagrams , is due to our approximation in which infinitely many minor phases are ignored . since the effect of the biquadratic interaction stabilizes the ising symmetric phases , @xmath30 and @xmath68 phases dominate larger regions as @xmath21 increases . the phase diagram for @xmath79 , fig . [ fig : k02phasediagram ] , shows that the ising symmetric phases spread from the lower - temperature region . but for @xmath79 . ] the phases with @xmath77 , @xmath29 , and @xmath80 still have xy symmetry . when @xmath81 exceeds @xmath82 , the phases with @xmath32 vanish for any @xmath16 at zero - temperature ( see the inset of fig . [ fig : k0phasediagram ] ) . the phases with @xmath77 , @xmath66 , and @xmath67 change qualitatively at this stage . in these phases , ising symmetric phases appear at lower temperature region and the ising symmetric regions expand as @xmath21 increases . figure [ fig : k10phasediagram ] is a phase diagram for @xmath83 . but for @xmath83 . the phases with @xmath29 and @xmath67 exhibit ising symmetric phases . the broken line indicates the boundary between xy symmetric phases and ising symmetric phases . ( inset ) : the order parameter ratio @xmath84 of @xmath29 phase as a function of temperature at @xmath85 around the xy - ising transition temperature . ] in this phase diagram , we can clearly observe the cross - over from xy symmetric phases to ising symmetric phases . figure [ fig : k30phasediagram ] shows the phase diagram for @xmath86 . but for @xmath86 . the xy symmetric phase remains only in @xmath77 phase . the dotted line around @xmath87 and @xmath88 is @xmath60 defined in eq .. the order(@xmath31)-disorder transition temperature ( indicated by solid line ) is slightly higher than the @xmath60 and the transition is first - order in the region where @xmath21 and @xmath89 are comparable . ] the ising symmetric phases are spread out almost the whole region . this figure also shows a difference from figs . [ fig : k0phasediagram ] , [ fig : k02phasediagram ] and [ fig : k10phasediagram ] in the order - disorder transition temperature . the temperature @xmath60 ( the order - disorder transition temperature under the assumption of the second - order transition ) deviates from @xmath53 when @xmath21 exceeds unity ( see eqs . and ) ; the deviation arises from the region where @xmath90 . however the @xmath60 does not give correct order - disorder transition temperature . in fact , the actual transition temperature is slightly higher than the calculated @xmath60 in the region where @xmath21 and @xmath89 are comparable , as shown in fig . [ fig : k30phasediagram ] . this fact indicates that the order - disorder transition is first - order , rather than second - order . indeed , we observed that the order parameters changes discontinuously at the region where the order - disorder transition temperature is different from @xmath60 . we studied the transition behavior of annnxy model with biquadratic interaction which can be applicable to the successive phase transitions of antiferroelectric smectics ; the biquadratic interaction is interpreted as the second fourier component of a pair directional interaction . the crossover from the xy symmetry to the ising one is identified for several representative phases . the ising symmetric phase has been suggested to appear from the low temperature region as the strength of biquadratic interaction is increased @xcite . at the xy symmetric phase with period @xmath44 , the difference of tilt directions of directors in successive layers , @xmath91 , is shown to be @xmath92 , which indicates a simple helical structure . our results show that the annnxy model with biquadratic interaction reproduce the phase sequence observed in the experiments . however , we may not expect the quantitative agreement with experiments . as an example , let us consider the lc molecules in @xmath29 phase , which is essentially equivalent to @xmath26 phase for negative @xmath16 , and thus this phase corresponds to the 3-layer smc@xmath0 . ellipsometric experiments have shown that @xmath9-directors of this phase do approximately but not exactly lie in a co - plain @xcite . in precise , the azimuthal angles of molecules in a period are @xmath93 , @xmath94 , and @xmath95 , with @xmath96rad ( @xmath97 ) . in terms of our model , this fact corresponds to the situation where @xmath98 . the inset of fig . [ fig : k10phasediagram ] shows a typical behavior of @xmath99 in @xmath29 phase at the xy - ising transition temperature . as shown in fig . [ fig : k10phasediagram ] , the ratio @xmath99 is rather large(@xmath100 ) at high temperature and approximately zero at low temperature ; we can not find any region where @xmath101 is comparable to the experimental results . for quantitative agreement of the present results with experimental ones , some additional modification to the present model is required . we may expect to obtain quantitatively reliable results by introducing a chiral interaction together with long range interactions @xcite , as done by olson _ et al . _ @xcite to explain the @xmath94 in the framework of phenomenological free energy . the profile of phase diagram for ising symmetric phases in fig . [ fig : k30phasediagram ] suggests apparently a character of devil s staircase @xcite and thus the system undergoes successive phase transitions . for phases of xy symmetry in figs . [ fig : k02phasediagram ] and [ fig : k10phasediagram ] , however , it is not clear from our present analysis whether the system exhibits the successive phase transitions or not . analysis of the stability of the soliton excitation can be a powerful tool to certify the successive transitions between xy symmetric phases . in the annnxy model under the two - fold external field , the character of devil s staircase is certified even in the ground state @xcite . however , in the present model , the phase changes continuously at the ground state for small biquadratic interaction as shown in the inset of fig . 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an axial next - nearest - neighbor xy model is studied as a model of chiral liquid crystals which exhibit many ferro- , ferri- and antiferroelectric tilted smectic phases . depending on the values of interaction parameters , this model exhibits ising symmetric ( i.e. , the tilt directions of directors are parallel or anti parallel ) phases or xy symmetric phases . phases with each type - of - symmetry show the character of devil s staircase , which has been observed in experiments .

precision studies of electroweak phenomena provide very important tests of the @xmath4 standard electroweak model . the measurement of the parity nonconserving ( pnc ) components of the atomic transitions belongs to this class . it offers a unique opportunity for testing the electroweak radiative corrections at the one loop level , and , possibly , to search for new physics beyond the standard model @xcite . the pnc effects in atoms are caused by the @xmath5 interference in the electron nucleus interaction . the dominant contribution comes from the coupling of the axial electronic current to the vector nuclear current . ( the interaction of the electronic vector current with the nuclear axial current is weaker in heavy atoms , and can be eliminated by summing over the pnc effects in the resolved hyperfine components of the atomic transitions . the hyperfine dependent effect , which also includes the nuclear anapole moment , is of interest in its own right @xcite , but is not considered hereafter . ) since the vector current is conserved , atomic pnc essentially measures the electroweak coupling of the elementary quarks . at the present time , pnc measurements in stable @xmath6cs atoms have @xmath72% experimental uncertainty @xcite . ( an earlier experiment in cs was performed by bouchiat _ _ @xcite ; the studies of pnc effects in atoms have been reviewed by commins @xcite and telegdi @xcite . ) however , improvement by an order of magnitude in the experimental accuracy is anticipated and a possibility of measuring pnc effects in unstable cesium and francium isotopes has been discussed @xcite . at this level , two issues must be resolved before an interpretation of the pnc data in terms of the fundamental electroweak couplings is possible . the atomic theory , even in its presently most sophisticated form @xcite , introduces about @xmath71% uncertainty . moreover , the small but non negligible effects of nuclear size @xcite must be addressed . this latter problem is the main topic of the present work . atomic pnc is governed by the effective bound electron nucleus interaction ( when taking only the part that remains after averaging over the hyperfine components ) of the form @xmath8 \times \psi_e^{\dagger } \gamma_5 \psi_e d^3r ~,\ ] ] where the proton and neutron densities @xmath9 are normalized to unity , and we have assumed the standard model nucleon couplings @xmath10 @xmath11 the electron part in eq . ( 1 ) can be parametrized as @xcite @xmath12 where @xmath13 contains all atomic structure effects for a point nucleus , @xmath14 is a precisely calculable normalization factor , and @xmath15 describes the spatial variation ( normalized such that @xmath16 ) . it is the integrals @xmath17 that determine the effect of the proton and neutron distributions on the pnc observables . the formfactors @xmath15 can be calculated to the order @xmath18 for a sharp nuclear surface of radius @xmath19 @xcite , @xmath20 ~. \label{ff}\ ] ] for a diffuse nuclear surface numerical evaluation of @xmath15 is necessary ( see below ) . however , the coefficients at @xmath21 and @xmath22 remain numerically of the order @xmath18 and depend only weakly on the exact shape of @xmath23 . in addition , since the electric potential near the nucleus is very strong , one can safely neglect atomic binding energies in the evaluation of @xmath15 . below we will separate the effects of the finite nuclear size ( i.e. , effects related to the deviations of @xmath24 from unity ) ; these terms will be represented by a nuclear structure correction to the weak charge . taking the matrix element of @xmath25 , one obtains @xmath26 ~,\ ] ] where @xmath27 , the quantity of primary interest from the point of view of testing the standard model , is the `` weak charge . '' in the standard model , with couplings ( 2 ) and ( 3 ) , the weak charge is @xmath28 the nuclear structure correction @xmath29 describes the part of the pnc effect that is caused by the finite nuclear size . in the same approximation as eq . ( 8) above @xmath30 where @xmath24 are the integrals of @xmath15 defined above . ( nuclear structure also affects the normalization factor @xmath14 , which is , however , determined by the known nuclear charge distribution @xcite . ) in a measurement that involves several isotopes of the same element , ratios of the pnc effects depend essentially only on the ratio of the weak charges and the corresponding nuclear structure corrections @xmath31 . ( the dependence @xmath14 on the neutron number @xmath32 will not be considered here . ) the ratios of the nuclear structure corrected weak charges , in turn , depend , to a good approximation , only on the _ differences _ @xmath33 of the neutron distributions in the corresponding isotopes . the uncertainties in these quantities , or equivalently , in the differences of the neutron mean square radii @xmath34 , then ultimately limit the accuracy with which the fundamental parameters , such as @xmath35 , can be determined . it is the purpose of this work to evaluate quantities @xmath24 for a number of cesium isotopes , which might be used in future high precision pnc experiments @xcite . moreover , we estimate the uncertainty in these quantities , respectively in their differences , since they represent the ultimate limitations for the interpretation of the pnc measurements . in section ii we describe the nuclear hartree fock calculations that we performed . in section iii we compare the calculated binding energies , ground state spins and charge radii with the experiment . there we also discuss how corrections for the zero point vibrational motion can be estimated and added . from the spread between the results obtained with two different successful effective skyrme forces , and from the pattern of deviations between the calculated and measured isotope shifts in the charge radii , we then estimate the uncertainties in the corresponding differences of the neutron radii . finally , in section iv , we calculate the nuclear structure corrections to the weak charges @xmath36 and their uncertainties and discuss the corresponding limiting uncertainties in the determination of the fundamental parameters of the standard model . ( our notation follows that of ref . others , e.g. , ref . @xcite do not explicitly separate the nuclear structure dependent effects . we believe that such a separation is very useful , since , as stated above , @xmath15 in eq . ( [ ff ] ) and hence also @xmath24 , eq . ( [ qf ] ) , are essentially independent of atomic structure . ) as demonstrated by numerous calculations , the microscopic description of nuclear ground state properties by means of the hartree fock ( hf ) method with an effective skyrme force like interaction is remarkably successful @xcite . the few adjustable parameters in the skyrme force are chosen to fit the various bulk properties ( energy per nucleon , compressibility modulus , symmetry energy , etc . ) , and properties of several doubly magic nuclei ( binding energies , charge radii , etc . ) the two most popular sets of skyrme parameters , namely skyrme iii and skyrmem@xmath37 have been successfully employed to describe the properties of nuclei in several regions of the periodic table @xcite . below we show only a few formulae essential to the basic understanding of the numerical calculation that we performed ; details can be found in the quoted references . the generalized skyrme force ( including all possible spin exchange terms and zero range density dependent interaction ) can be written as , @xmath38 where @xmath39 and @xmath40 are the adjustable parameters , and @xmath41 . because we are dealing with odd a nuclei , the unpaired nucleon introduces terms that break time reversal symmetry in the hf functional . when the spin degrees of freedom are taken into account , the breaking of time reversal symmetry leads to a rather complicated functional @xcite . the total energy @xmath42 , which is minimized in the hf method , can be written as a space integral of a local energy density @xmath43 with @xmath44 for complete expressions of the coulomb energy @xmath45 and the coefficients @xmath46 see ref . @xcite , where the dependence on skyrme force parameters in eq . ( [ sf ] ) is given . the mass densities @xmath47 , kinetic density @xmath48 , current density @xmath49 , spin orbit density @xmath50 and vector density @xmath51 in eq . ( [ hfd ] ) can , in turn , be expressed in terms of the single particle wave functions @xmath52 . the variation of @xmath42 with respect to @xmath53 defines the one body hartree fock hamiltonian @xmath54 @xcite . in the following we will use the mass densities @xmath47 , which can be expressed as @xmath55 here @xmath56 denotes the component of the @xmath57th single nucleon wave function with spin @xmath58 along the @xmath59 direction , and @xmath60 are the bcs occupation factors ( see below ) . the expressions for the other densities are again given in ref . @xcite . the mean square proton and neutron radii are given by the usual formulae @xmath61 in this work , two discrete symmetries , namely parity and @xmath59signature , are imposed on the wave functions @xcite . the complete description of a wave function requires four real functions corresponding to the real , imaginary , spin up and spin down parts of @xmath52 @xcite . the numerical approximation to the hf energy @xmath42 is obtained by a discretization of the configuration space on a three dimensional rectangular mesh . the mesh size @xmath62 is the same in the three directions and the abscissae of the mesh points are @xmath63 . in this work , @xmath62 is @xmath64 , and the mesh size is @xmath65 . the numerical procedure is described in detail in ref . @xcite . pairing correlations need to be included in a realistic description of medium and heavy nuclei . we choose to describe pairing between identical nucleons within the bcs formalism using a constant strength seniority force @xcite . in the usual bcs scheme , the paired states are assumed to be the two time reversed orbitals @xmath52 and @xmath66 . although time reversal symmetry is broken in our calculations of odd a nuclei , the time reversal breaking terms in the functional generated by the unpaired odd nucleon are very small compared to the time reversal conserving terms so that the time reversal symmetry is still approximately good . in our calculation we define the pairing partner @xmath67 of state @xmath52 to be the eigenstate of @xmath54 whose overlap with @xmath68 is maximal ( @xmath69 is the time reversal operator ) . because the single particle orbital occupied by the unpaired nucleon and its signature partner do not contribute to the pairing energy , we introduce blocking in our code to prevent these two orbitals from participating in pairing and force their bcs occupation numbers to be @xmath70 and @xmath71 , respectively . as some of the cesium isotopes considered here are deformed , it is very important to take the deformation degrees of freedom into account . the method of solving the hf+bcs equations by discretization of the wave functions on a rectangular mesh allows any type of even multipole deformation . the deformation energy curves are obtained by a constraint on the mass quadrupole tensor @xmath72 . the two discrete symmetries of the wave functions @xmath52 ensure that the principal axes of inertia lie along the coordinate axes . the quadrupole tensor is , therefore , diagonal and its principal values @xmath73 can be expressed in terms of two quantities @xmath74 and @xmath75 as @xmath76 where @xmath74 and @xmath75 satisfy the inequalities @xmath77 the values of the three constraints @xmath73 were computed from the desired values of @xmath74 and @xmath75 and inserted in a quadratic constraint functional added to the variational energy , according to the method described in ref . @xcite . in the calculations described below , we constrain the nuclear shape to be axially symmetric ( @xmath78 ) . in fig . 1 we show the potential energy curves for @xmath0cs @xmath1cs . according to our calculations with skyrmeiii ( skmiii ) and skyrmem * ( skm * ) forces the lighter cesium isotopes @xmath79 are deformed . for skmiii such an assignment is able to explain the observed ground state spins of @xmath80 for @xmath81 and @xmath82 for @xmath83 . for skm * the mean field proton states @xmath84 and @xmath85 are interchanged and therefore the ground state spin assignments for the deformed cesium isotopes are not correct . ( this turns out not to be a very crucial problem . ) binding energies and shifts @xmath86 and @xmath87 calculated with the skm * and skmiii interactions are shown in tables i and ii . the binding energies agree in both cases with the experimental values with largest deviation of 4 mev out of about 1000 mev of total binding energy . the comparison between the measured and calculated isotope shifts is illustrated in figs . 2 and 3 as a series of successively better approximations . first , the crosses , connected by dashed lines to guide eyes , show the isotope shifts for spherical nuclei . the agreement with experiment is not very good even though the spherical calculation correctly predicts that the slope of the dependence @xmath88 is about half of the slope expected from the simple relation @xmath89 . this means that , on average , the neutron proton interaction we use has the correct magnitude . next , the equilibrium deformation for the lighter cesium isotopes is included ( open squares ) , leading to a much better agreement . further improvement is achieved when the effect of zero point quadrupole vibrational motion is taken into account . it is well known that the mean square radius of a vibrating nucleus is increased by @xcite @xmath90 we include this effect of the shape fluctuations using the quantities @xmath91 extracted from the measured transition matrix elements @xmath92 and the relation @xmath93^{-2 } ~.\ ] ] we take the average @xmath94 of the corresponding xe and ba isotopes with neutron numbers @xmath95 and correct the radii of @xmath6cs@xmath1cs accordingly , as shown in figs . 2 and 3 . thus , further improvement in the comparison with the measured isotope shifts results . ( for @xmath96 the @xmath94 values are not known . we use instead the empirical relation between the energy of the lowest @xmath97 state and the deformation parameter @xmath94 @xcite . ) this correction results in changes in @xmath98 of 0.2124 @xmath99 in @xmath6cs , 0.1325 @xmath99 in @xmath100cs , 0.0724 @xmath99 in @xmath101cs , and 0.1263 @xmath99 in @xmath1cs . in a fully consistent calculation , one should make a similar correction for the deformed cesium isotopes as well . since the corresponding @xmath94 values for the vibrational states are not known , and the corrections are expected to be small , we do not make them . instead , we somewhat arbitrarily assume that the zero point motion correction is the same as in the semimagic @xmath101cs . we believe this explains the somewhat poorer agreement in the deformed cesium isotopes . even though the quadrupole @xmath97 states contribute most to the mean square radius via eq . ( [ rb ] ) , other vibrational states , e.g. , the octupole @xmath102 and the giant resonances , contribute as well ; however , all such states not only have smaller collective amplitudes but , even more importantly , vary more smoothly with the atomic mass ( or neutron number ) than the @xmath97 states , and hence their contribution to the shifts @xmath103 should be correspondingly smaller . altogether , the error in the shift @xmath104 is at most 0.2 @xmath99 , and appears to be independent of the change in the neutron number @xmath105 . thus , for the following considerations we assign an uncertainty in the relative value of @xmath104 of 0.2 @xmath99 . very little is known experimentally about the moments @xmath106 . quite conservatively , we assume that the uncertainty in @xmath107 is @xmath108 5 @xmath109 . before turning our attention to the neutron radii , it is worthwhile to make a brief comment about the comparison with @xmath110 values of @xmath111 and @xmath112 . experimentally , muonic x ray energies for the stable @xmath6cs have been fitted to the fermi distribution with the halfway radius @xmath113 = 5.85 fm , surface thickness @xmath114 = 1.82 fm @xcite , and @xmath111 = 23.04 @xmath99 . such a fermi distribution gives @xmath112 = 673 @xmath109 . our hf calculation corrected for zero point vibrational motion with @xmath115 = 0.024 , as described above , gives @xmath116 = 23.27 @xmath99 for skmiii and 22.69 @xmath99 for skm * interaction , both quite close to the experimental value . the calculated @xmath112 moments ( not corrected for the zero point motion ) are 671(skmiii ) and 652(skm * ) @xmath109 . we see , therefore , that the calculation is quite successful in the absolute radii ( and even surface thicknesses ) , in particular for the skmiii interaction ( which gives also the correct ground state spin ) . the calculated shifts in the neutron radii @xmath117 and @xmath118 are listed in tables i ( skm * ) and ii ( skmiii ) and the quantities @xmath117 corrected for the effect of zero point vibrational motion are displayed in fig . several comments about these are in order . first , the slope of the dependence of @xmath119 for spherical configurations is correspondingly steeper than the slope following from @xmath89 . that is obviously a correct result ; the combination of a smaller slope in the proton radii and a larger slope in the neutron radii when neutrons are added is necessary to maintain on average the @xmath89 relation . second , the hf calculations imply that the proton and neutron distributions have essentially identical deformations . this agrees with the general conclusion about the isoscalar character of low frequency collective modes in nuclei ( see , e.g. , ref . thus , we accept this result and do not assign any additional uncertainty to the possible difference in the deformation of protons and neutrons . finally , for the same reason , we use the same @xmath94 values , and the @xmath91 extracted from them , to correct the neutron radii using eq . ( [ rb ] ) . assuming all of the above , we assign _ identical _ uncertainties to the neutron shifts @xmath117 and the proton shifts @xmath104 , and similarly to the fourth moments @xmath120 . very little reliable experimental information on neutron distribution in nuclei is available . in ref . @xcite , data from pionic atoms are analyzed . the corresponding best fit for neutron mean square radii agrees very well with the hf results quoted there . the nearest nucleus to cesium in ref . @xcite is @xmath121ce . scaling it with @xmath122 one arrives at @xmath123 = 24.7 @xmath99 for @xmath6cs , somewhat larger than our calculated values 23.7 and 24.0 for skm * and skmiii , respectively . in ref . @xcite , the theoretical neutron density of brack _ et al . _ @xcite with @xmath123 = 23.5 @xmath99 , was used . that value , presumably obtained by interpolation from the values obtained by the hf method using the skm * interaction , is , not surprisingly , quite close to our calculated values . this limited comparison suggests that the absolute radii @xmath123 have uncertainties of about 1 @xmath99 . the uncertainty in the shifts @xmath117 should be substantially smaller , and our estimated error of 0.2 @xmath99 does not seem unreasonable . in ref . @xcite the uncertainty in the integrals @xmath24 was estimated from the spread of the calculated values with a wide variety of interactions . some of the interactions employed in @xcite give better agreement for known quantities ( charge radii , binding energies , etc . ) than others . we chose to use only the two most successful interactions . the spread in the calculated shifts @xmath86 for these two interactions is less than our postulated error of 0.2 @xmath99 . pollock _ et al . _ @xcite also argue that the isovector surface term @xmath124 in the skyrme lagrangian is poorly determined and may affect the neutron skin significantly , without affecting most bulk nuclear properties . we tested this claim by modifying simultaneously the coefficients @xmath125 and @xmath126 in eq . ( [ hfd ] ) . we find that when we vary @xmath127 ( i.e. , the relative strength of the isovector surface term ) from + 0.3 to -0.3 the proton radius @xmath111 changes indeed very little ( about 0.06 @xmath99 ) and the neutron radius changes somewhat more ( by about 0.1 @xmath99 , still less than our estimated error ) . however , the binding energy changes by about 5 mev , more than the largest discrepancy between the theory and experiment . thus , we do not think that the uncertainty in this particular coefficient of the skyrme force alters our conclusions . the nuclear structure effects are governed by the coefficients @xmath24 , eq . ( [ qf ] ) , which in turn involve integrals of the formfactors @xmath15 , eq . ( [ ff ] ) . the function @xmath15 is slowly varying over the nuclear volume , and may be accurately approximated by a power series @xmath128 and , therefore , @xmath129 for a sharp nuclear surface density distribution the only relevant parameter is the nuclear radius @xmath19 and @xmath130 . using the experimental @xmath131 = 23.04 @xmath99 for @xmath6cs @xcite , we find from eq . ( [ ff ] ) @xmath132 where the distance is measured in fermis . if , instead , we solve numerically the dirac equation for the @xmath133 and @xmath134 bound electron states in the field of the finite size diffuse surface nucleus , we obtain the coefficients @xmath135 of @xmath136 when we use the standard surface thickness parameter @xmath137 fm , and @xmath138 when we use the surface thickness adjusted so that the nuclear density parametrized by the two parameter fermi distribution resembles as closely as possible the hartree fock charge density in @xmath6cs . the expansion coefficients @xmath139 depend , primarily , on the mean square charge radius . to take this dependence into account , we use for @xmath6cs the @xmath140 and @xmath141 above , and for the other isotopes , we use the same surface thickness parameter @xmath142 and adjust the halfway radius in such a way that the experimental @xmath143 are correctly reproduced . it is easy now to evaluate the uncertainty in the factors @xmath24 given the coefficients @xmath139 and our estimates of the uncertainties in @xmath131 and @xmath22 . substituting the corresponding values , we find that the uncertainty is @xmath144 = 4.6@xmath14510@xmath146 , caused almost entirely by the uncertainty in the mean square radii @xmath147 . this uncertainty represents about 1% of the deviations of @xmath24 values from unity . before evaluating the nuclear structure corrections @xmath29 we have to consider the effect of the intrinsic nucleon structure . following @xcite we use @xmath148 where @xmath149 are the nucleon weak radii , and @xmath150 are the nucleon weak charges . neglecting the `` strangeness radius '' of the nucleon , and using the fitted two parameter fermi density distribution , we find @xmath151 very close to the sharp nuclear surface values of pollock _ et al . the above intrinsic nucleon structure corrections are small , but not negligible . more importantly , they are independent of the nuclear structure , and cancel out in the differences @xmath152 . the quantities 100@xmath145(@xmath153 - 1 ) and 100@xmath145(@xmath154 - 1 ) are listed in table iii for all cesium isotopes and for the two skyrme interactions we consider . one can see that they vary by about 4% for neutrons and are essentially constant for protons when the neutron number increases from @xmath32 = 70 to 84 . the variation with @xmath32 is essentially identical for the two forces , while the small difference between the @xmath24 values calculated with the two forces reflects the difference in the _ absolute _ values of radii for the two interactions . the weak charges @xmath27 and the nuclear structure corrections @xmath29 in table iii are radiatively corrected . thus , instead of the formulae ( [ qweak ] ) , ( [ qnuc ] ) we use @xmath155 ~,~ \bar{x } = 0.2323 + 0.00365s ~,\ ] ] following @xcite . here @xmath156 is the parameter characterizing the isospin conserving `` new '' quantum loop corrections @xcite . also , @xmath157 ~.\ ] ] these quantities , evaluated for @xmath158 , are shown in table iii . the assumed uncertainty in the shifts of the mean square radii , and consequently in the changes in factors @xmath24 results in the relative uncertainty @xmath159 of 5@xmath14510@xmath146 . that uncertainty , therefore , represents the `` ultimate '' nuclear structure limitation on the tests of the standard model in the atomic pnc experiments involving several isotopes . in the atomic pnc experiments involving a _ isotope , the uncertainty in the neutron mean square radius is larger , and 1 @xmath99 appears to be a reasonable choice . thus , from nuclear structure alone , the weak charge in a single isotope has relative uncertainty of about 2.5@xmath14510@xmath160 , perhaps comparable to the best envisioned measurements , but considerably smaller then the present uncertainty associated with the _ atomic _ structure . suppose now that in an experiment involving several cesium isotopes one is able to determine the ratio @xmath161 with some relative uncertainty @xmath162 . to a ( reasonable ) first approximation @xmath163 ~.\ ] ] thus , we see that nuclear structure contributes to the uncertainty of @xmath19 at the level of roughly 7@xmath14510@xmath146 , where we added the individual errors in quadrature . this uncertainty is much smaller than the anticipated experimental error . in such a measurement , therefore , the uncertainty in @xmath164 will be @xmath165 ( see also @xcite ) where the last factor is evaluated for @xmath166 = 70 , 84 . the above equation illustrates the obvious advantage of using isotopes with large @xmath105 . also , by performing the measurement with several isotope pairs , one can further decrease the uncertainty @xmath167 . on the other hand , the uncertainty in the important parameter @xmath156 is determined from the relation @xmath168 , and thus @xmath169 in conclusion , we have evaluated the nuclear structure corrections to the weak charges for a series of cesium isotopes , and estimated their uncertainties . we concluded that the imperfect knowledge of the neutron distribution in cesium isotopes does not represent in the foreseeable future a limitation on the accuracy with which the standard model could be tested in the atomic pnc experiments . we would like to thank c. wieman and d. vieira whose discussion of the proposed experiments inspired the work described here . this research was performed in part using the intel touchstone delta system operated by caltech on behalf of the concurrent supercomputing consortium . this work was supported in part by the u.s . department of energy under contract # de f60388er40397 , and by the national science foundation , grant no . phy9013248 . p. langacker , m .- x . luo , and a. k. mann , rev . 64 * , 87 ( 1992 ) . w. marciano and j. rosner , phys . lett . * 65 * , 2963 ( 1990 ) . v. v. flambaum , i. b. khriplovich , and o. p. sushkov , phys . * 146b * , 367 ( 1984 ) . w. c. haxton , e. m. henley , and m. j. musolf , phys . 63 * , 949 ( 1989 ) . m. c. noecker , b. p. masterson , and c. wieman , phys . * 61 * , 310 ( 1988 ) . m. a. bouchiat _ _ , phys . lett . * 117b * , 358 ( 1982 ) ; * 134b * , 463 ( 1984 ) ; j. phys . 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the interpretation of future precise experiments on atomic parity violation in terms of parameters of the standard model could be hampered by uncertainties in the atomic and nuclear structure . while the former can be overcome by measurement in a series of isotopes , the nuclear structure requires knowledge of the neutron density . we use the nuclear hartree fock method , which includes deformation effects , to calculate the proton and neutron densities in @xmath0cs @xmath1cs . we argue that the good agreement with the experimental charge radii , binding energies , and ground state spins signifies that the phenomenological nuclear force and the method of calculation that we use is adequate . based on this agreement , and on calculations involving different effective interactions , we estimate the uncertainties in the differences of the neutron radii @xmath2 and conclude that they cause uncertainties in the ratio of weak charges , the quantities determined in the atomic parity nonconservation experiments , of less than @xmath3 . such an uncertainty is smaller than the anticipated experimental error . atomic parity nonconservation and neutron radii in cesium isotopes w. k. kellogg radiation laboratory , 10638 + california institute of technology , pasadena , ca 91125 norman bridge laboratory of physics , 16133 + california institute of technology , pasadena , ca 91125

the distribution of baryons in the universe remains one of the puzzling issues in modern cosmology . at the present epoch , the total amount of baryons inferred from a census of hi absorption , gas and stars in galaxies , and x - ray emission from hot gas in galaxy clusters is far smaller than that predicted by nucleosynthesis calculations ( persic & salucci 1992 ; fukugita , hogan & peebles 1998 ) and by measurements of the cosmic microwave background radiation ( spergel et al . hence , it is believed that a large fraction of the baryons lies in an as yet undiscovered _ dark _ state . three dimensional hydrodynamic simulations of cosmic structure formation ( e.g. , cen & ostriker 1999 ; dav et al . 2001 ; croft et al . 2002 ) predict that about 3040% of the baryons reside in the so - called warm - hot intergalactic medium ( whim ) at the present epoch . this gas is mostly shock - heated to a temperature of @xmath3@xmath4 k during large scale structure formation , and this relatively low temperature makes its thermal emission difficult to detect with conventional x - ray probes . a variety of observational approaches have been considered for studying the whim , including both ultraviolet ( e.g. , furlanetto et al . 2003 , 2004 ) and x - ray lines . with respect to the latter possibility , there have been several tentative claims that the whim has been detected locally in absorption ( e.g. , nicastro et al . 2002 ) and in emission ( finoguenov et al . 2003 ) . moreover , yoshikawa et al . ( 2004 ) have proposed using ovii / oviii lines to examine the whim in detail with high spectral resolution x - ray detectors . they argue that , under a variety of assumptions , about half of the whim ( by mass ) can be detected via oxygen line emission ( see also fang et al . while the proposed missions appear promising , the detectability depends crucially on the assumed metallicity and the relative abundances of particular ions . previous theoretical studies used a single - temperature model for the whim , under the assumption that equipartition is fully achieved . however , the outskirts of galaxy clusters may actually have a two - temperature structure ( e.g. , fox & loeb 1997 ; takizawa 1998 ; courty & alimi 2004 ) . near galaxy clusters , infalling gas shock - heats to roughly the virial temperature as kinetic energy is converted into thermal energy . because the infalling ions carry nearly all of the bulk kinetic energy ( exceeding that contained in electrons by a factor @xmath5 ) , the post - shock electrons can gain thermal energy only via collisions with ions . if collisional relaxation takes a time comparable to or longer than the age of the universe , the two temperatures , i.e. those of the electrons and ions , remain different . one can easily show that the relaxation time is rather long for diffuse warm - hot gas around clusters and in large - scale filamentary structures . since the relative abundance of metal ions and their emissivities are sensitive to the electron temperature , this has important implications for future observations of the whim , particularly for the hotter component . in this _ letter _ , we study the thermal evolution of the intergalactic medium ( igm ) using a large hydrodynamic simulation of structure formation . we show that a considerable fraction of the whim indeed has a two - temperature structure , particularly in and around rich clusters . in these regions , the electron temperature is typically smaller than the ion temperature by a factor of a few , substantially modifying estimates of the abundances of ions and their emissivities . for our simulations , we use the parallel tree - pm / sph code gadget2 ( v. springel , in preparation ) , which employs a fully conservative scheme for integrating the equations of motion ( springel & hernquist 2002 ) . we implement non - equipartition processes between ions and electrons following fox & loeb ( 1997 ) and takizawa ( 1998 ) . equilibrium in an electron - proton plasma is achieved in the following manner . after passing through a shock , electrons and ions thermalize into ( separate ) maxwellian distributions on equilibration timescales @xmath6 , with @xmath7 ( spitzer 1962 ) . here , @xmath8 and @xmath9 are the electron and ion masses and we have assumed protons dominate the ionic component . equipartition between protons and electrons is achieved on an even longer timescale @xmath10 . hence , we assume that electrons and ions quickly achieve maxwellian distributions with temperatures @xmath11 and @xmath12 , respectively , and consider only the non - equipartition effect . spitzer ( 1962 ) showed that the appropriate timescale is given by @xmath13 where @xmath14 is the electric charge , @xmath15 is the charge of an ion , @xmath16 is the ion number density , and @xmath17 is the coulomb logarithm , which is given by @xmath18 for a hot , fully ionized plasma , @xmath19 and this timescale can be comparable to the hubble time in regions with @xmath20 k and overdensities @xmath21 with respect to the cosmic mean . the evolution of the electron temperature is given by @xmath22 we do not consider heat conduction on the assumption that it will be suppressed by tangled magnetic fields . some other electron heating processes driven by plasma instabilities have also been proposed ( laming 2000 ) , but detailed calculations in the context of supernova remnants suggest that these mechanisms convert less than @xmath23% of the bulk kinetic energy to electron thermal energy ( e.g. , cargill & papadopoulos 1988 ) , so we neglect these effects . our treatment therefore maximizes the possible offset between @xmath11 and @xmath12 . we work with a flat @xmath24-dominated cold dark matter cosmology with matter density @xmath25 , cosmological constant @xmath26 and expansion rate at the present time @xmath27 km s@xmath28 mpc@xmath28 . we set the baryon density @xmath29 and the normalization parameter @xmath30 . our simulation employs @xmath31 cold dark matter particles and the same number of non - radiative gas particles in a cosmological volume 100 @xmath32 mpc on a side . the mass per gas particle is @xmath33 , and the nominal gas mass resolution is about @xmath34 for our choice of the number of sph neighbors , @xmath35 . we note that the cdm initial conditions and the large simulation volume allow us to correctly model the merging process during the formation of filamentary structures and the assembly of massive halos which contribute to heating of the igm as well as the intracluster medium ( icm ; e.g. , keshet et al . figure 1 shows the baryon distribution at @xmath36 in a slice 20 @xmath32 mpc thick ( top ) , the hot component with temperature @xmath37 k ( middle ) , and gas in which @xmath38 . the latter two components are quite similar , suggesting that much of the hotter whim has a two - temperature structure . the distribution of the two - temperature gas appears quite complex even in clusters , indicating that shocks induced by mergers during hierarchical assembly propagate and leave two - temperature gas within clusters . we have checked the time variation of the specific entropy of the gas elements and verified that the two - temperature regions have recently been shock - heated . in the gas around large - scale structure , _ continuous _ shock - heating disturbs equipartition sufficiently close to the shock , while more distant gas is often relaxed . figure 2 shows the electron temperature @xmath11 plotted against the mean temperature @xmath39 , which is the temperature we would obtain in a single temperature model . interestingly , the two - temperature deviation is largest in the whim regime ( indicated by the rectangular box ) . note that there are some gas elements with @xmath40 . this is because our simulation does not include electron heating processes other than coulomb collisions . figure 3 shows the radial distribution of @xmath11 ( solid line ) and @xmath41 ( dashed line ) for one of the most massive halos in our simulation . we used a friends - of - friends group - finder to locate dark matter halos and define the virial radius such that the mean matter density within this radius is 200 times the critical density . the halo we consider here has a virial mass of @xmath42 and a virial radius of @xmath43 mpc . in figure 3 , the difference between the two temperatures is large at @xmath44 mpc and appreciable even at @xmath45 mpc . the overall profile looks similar to those found in simulations of single clusters by takizawa ( 1998 ) and chieze et al . owing to the difference in relaxation time ( see eq . [ 3 ] ) in the dense central part and in the low density outer regions , the electron temperature profile appears steeper than that of the mean temperature at @xmath46 mpc . the bottom panel of figure 3 shows the baryonic mass fraction as a function of both @xmath11 and @xmath41 ( thick and thin histograms , respectively ) . we selected gas particles within 5 @xmath32 mpc of the cluster center . the distribution of @xmath47 is skewed significantly to smaller values . the total gas mass contained in this region ( @xmath48 mpc ) is @xmath49 , and nearly 50% of it has @xmath50 kev ; i.e. , the mass of the warm component surrounding the cluster is greater than the gas mass within the cluster itself . the two - temperature structure of the whim and icm has many important implications . for example , the relative fractions of metal ions are of considerable interest for whim observations . figure 4 shows the fraction of ovi , ovii , and oviii ions as a function of temperature assuming collisional ionization equilibrium . clearly the abundances have a strong dependence on _ electron _ temperature , because the populations are primarily determined by electron collisions ( photo - ionization is unimportant for these high - level ions unless @xmath51k or the incident x - ray intensity is high ) . the bottom panel indicates the possible errors caused by assuming a single temperature for the whim . a factor of two systematic shift in temperature , typical of the offsets we find between @xmath41 and @xmath11 near shocks , can lead to significant over / under - estimates of the abundances . we further quantify the importance of non - equipartition with a model oxygen line emission map . we put the super - cluster region indicated in figure 1 at @xmath52 and compute the surface brightness of ovii emission following the procedure of yoshikawa et al . we adopt a simple relation between the gas metallicity and the local gas density of the form @xmath53 , where @xmath54 is the mean baryon density . we then compute the surface brightness for both @xmath11 and @xmath41 . figure 5 shows the derived profiles . while the overall results appear similar in the two cases , there are substantial local deviations . in particular , around some bright spots , the emissivity is greater by up to an order of magnitude when computed using the electron temperature . the ratio of the surface brightness @xmath55 is shown in the bottom panel of figure 5 . there , we mark the regions with @xmath56 , which is the nominal detection limit of the proposed dios mission ( yoshikawa et al . 2003 ) . similar considerations may also apply to efforts to study the whim using uv observations ( e.g. , furlanetto et al . 2004 ) , although the steep temperature dependence of @xmath57 implies that equipartition is more accurate in this regime ( see fig . 2 ) . in principle , one can measure the degree of equilibration using various line intensities from far uv to soft x - rays ( e.g. , ghavamian et al . 2001 ; raymond et al . 2003 ) as well as the metal line doppler widths . measuring the age of the whim from the degree of equilibration will put a stringent constraint on theoretical models of large - scale _ baryonic _ structure formation . although our simulation does not include radiative cooling , we note that the lower electron temperature can have a significant impact on the radiative cooling efficiency in dense regions . the gas cooling rate has a very steep temperature dependence at @xmath58k regardless of the metallicity ( sutherland & dopita 1993 ) . as shown in figure 2 , recently shock - heated gas with @xmath59k may have an electron temperature @xmath60k . such gas should cool rapidly and in a highly non - equilibrium manner . this warrants further study of the effect of the two - temperature structure on the igm using a simulation with radiative cooling . overall , our simulation suggests that theoretical studies of the whim and correct interpretation of the observational data from current and future x - ray missions requires explicit consideration of these relaxation processes .

we study the temperature structure of the intergalactic medium ( igm ) using a large cosmological @xmath0-body / sph simulation . we employ a two - temperature model for the thermal evolution of the ionized gas , in which we include explicitly the relaxation process between electrons and ions . in the diffuse , hot igm , the relaxation time is comparable to the age of the universe and hence the electron temperature in post - shock regions remains significantly smaller than the ion temperature . we show that , at the present epoch , a large fraction of the warm / hot intergalactic medium ( whim ) has a well - developed two - temperature structure , with typical temperature differences of order a few . consequently , the fraction of metals in various ionization states such as ovi , ovii , and oviii , as well as their line emissivities , can differ locally by more than an order of magnitude from those computed with a single - temperature model , especially in gas with @xmath1 k. it is thus necessary to follow the evolution of the electron temperature explicitly to determine absorption and emission by the whim . although equipartition is nearly achieved in the denser intracluster medium ( icm ) , we find an appreciable systematic deviation between the gas - mass weighted electron temperature and the mean temperature even at half the virial radii of clusters . there is thus a reservoir of warm ( @xmath2 kev ) gas in and around massive clusters . our results imply that relaxation processes need to be considered in describing and interpreting observational data from existing x - ray telescopes as well as from future missions designed to detect the whim , such as the _ diffuse intergalactic oxygen surveyor _ and the _ missing baryon explorer_.

revealing the magnetic coupling mechanism is often a critical step towards understanding the role of magnetism in intriguing phenomena such as colossal magnetoresistance ( cmr ) , high @xmath8 superconductivity , multiferroicity or frustration in correlated electron materials @xcite . by way of example , the indirect double- and super - exchange interactions were successfully elaborated in qualitatively explaining the cmr effect and associated magnetic orders based only on the spin and charge degrees of freedom @xcite . in 4__f__-based insulators , the indirect oscillating interaction @xcite between pairs of localized 4__f _ _ moments via the intermediary of valence electrons is blocked . therefore , possible super- , dipole - dipole and multipolar , and dzyaloshinsky - moriya ( dm ) exchange interactions are primarily responsible for potential magnetic ordering @xcite . without detailed knowledge of the structural and magnetic parameters , it is hard to uniquely determine which interaction acts as the major exchange mechanism @xcite . in this case , the origins of the related incommensurable spin structures become elusive @xcite . in addition , the competition between spin - orbital coupling and crystal electric field ( cef ) at low temperatures largely affects the highly - degenerate hund s rule ground state , and besides the anisotropic dipolar and dm interactions , determine the magnitude of the magnetic anisotropy @xcite . this anisotropy strongly influences the degree of magnetic frustration . sometimes , it may disorder or even quench potential magnetic moments , leading to a virtually nonmagnetic ground state @xcite . magnetic frustration can lead to novel quantum states such as spin liquid , spin ice , cooperative paramagnetism or the magnetic coulomb phase based on magnetic monopole excitations , providing an excellent testing ground for theories @xcite . a monte carlo simulation indicates that the observed diffuse scattering in srer@xmath1o@xmath2 originates from a ladder of er triangles @xcite . a computation of the crystal - field levels demonstrates site - dependent anisotropic single - ion magnetism in the compounds of srho@xmath1o@xmath2 and srdy@xmath1o@xmath2 @xcite . lanthanide - based magnetic compounds , e.g. edge - sharing tetrahedra , corner - sharing spinels , or triangular kagom@xmath0 and pyrochlore lattices , often show anomalous magnetic properties due to geometric frustration @xcite . the family of sr__re__@xmath1o@xmath2 ( _ re _ = y , gd , ho , yb ) compounds was first synthesized in 1967 @xcite . recently , a study on polycrystalline sr__re__@xmath1o@xmath2 ( _ re _ = gd , dy , ho , er , tm , yb ) samples demonstrates that they adopt the orthorhombic structure @xcite with a geometric frustration for the magnetic ions revealed by the existence of magnetic short - range orders down to @xmath3 1.5 k @xcite . subsequently , single crystals of sr__re__@xmath1o@xmath2 ( _ re _ = y , lu , dy , ho , er ) were successfully grown @xcite . single - crystal neutron - scattering studies on sr__re__@xmath1o@xmath2 ( _ re _ = ho , er , yb ) compounds with respective antiferromagnetic ( afm ) transition temperatures at 0.62 , 0.73 , and 0.9 k were reported @xcite , generally confirming that there exists a coexistence of long- and short - range magnetic orders . it is pointed out that for the case of srho@xmath1o@xmath2 , young , et al . @xcite observed only a short - range spin order inconsistent with other reports @xcite . further experimental tests would be necessary to address this discrepancy . since the adopted orthorhombic structure accommodates two _ re _ sites ( _ re1 _ and _ re2 _ ) , it is hard to derive the crystallographic origins of the two types of spin ordering . in addition , single - crystal srdy@xmath1o@xmath2 displays only weak diffuse magnetic scattering which persists down to @xmath3 20 mk @xcite . the low transition temperatures of the magnetic orders , to some extent , prevent a complete understanding of the nature of magnetic interactions and frustrations in the family . to overcome these problems and address the relevant interesting physics necessitate a search in the sr__re__@xmath1o@xmath2 ( _ re _ = rare earth ) family for a new compound that displays a higher n@xmath9el temperature , thus permitting a technically easier study of the two coupling mechanisms . in this study , we report on a new frustrated member to the family of sr__re__@xmath1o@xmath2 , namely srtb@xmath1o@xmath2 , which has not been studied yet by neutron scattering . the single - crystal srtb@xmath1o@xmath2 displays a long - range magnetic order relative to the underlying lattice . the noncollinear incommensurate afm structure forms at @xmath4 = 4.28(2 ) k upon cooling . the synthesis of srtb@xmath1o@xmath2 with the highest n@xmath9el temperature in the family opens up an easier route to elucidate the magnetic coupling and frustrating mechanisms . by polarized and unpolarized neutron scattering we uniquely determine the detailed structural and magnetic parameters to understand the magnetism in srtb@xmath1o@xmath2 . polycrystalline samples of srtb@xmath1o@xmath2 were synthesized from stoichiometric mixtures of srco@xmath10 ( 99.99% ) and tb@xmath2o@xmath11 ( 99.99% ) compounds by standard solid - state reaction @xcite . both raw materials were preheated at 800@xmath7 for 12 h and weighted at @xmath3 200@xmath7 . the mixed and milled raw materials were calcined twice at 1473 and 1573 k for 48 h each in air in order to perform decarbonization and prereaction . the resulting powder was pressed into cylindrical rods with an isostatical pressure of @xmath3 78 mpa . the rods were sintered two times at 1573 and 1673 k for 48 h at each temperature in air . after each round of the isostatic pressing and subsequent firing , the product was reground and ball - remilled , which results in a dense and homogenous sample and ensures a complete chemical reaction . the single crystal of srtb@xmath1o@xmath2 was grown by optical floating - zone method with an atmosphere of @xmath3 98% ar and @xmath3 2% o@xmath1 . the growing speed is @xmath3 4 mm / h with rotations of the feed and seed rods at + 32 and -28 rpm , respectively . the phase purity of the polycrystalline and single - crystalline samples was checked by in - house x - ray powder diffraction . the electrical resistivity of a bar - shaped single crystal by standard dc four - probe technique was measured on a commercial physical property measurement system . high - resolution neutron powder diffraction ( npd ) patterns were collected with a pulverized srtb@xmath1o@xmath2 single crystal ( @xmath3 5 g ) mounted in a @xmath12he insert on the structure powder diffractometer ( spodi ) @xcite with constant wavelength @xmath13 = 2.54008(2 ) at the frm - ii research reactor in garching , germany . the srtb@xmath1o@xmath2 single crystal ( @xmath3 2.2 g ) for the neutron - scattering studies was oriented in the ( _ h _ , _ k _ , 0 ) scattering plane with the neutron laue diffractometer orientexpress @xcite and the in3 thermal triple - axis spectrometer at the institut laue - langevin ( ill ) , grenoble , france . the mosaic of this single crystal is 0.494(5)@xmath14 full width at half maximum ( fwhm ) for the nuclear ( 2 , 0 , 0 ) bragg reflection at 1.5 k. longitudinal _ xyz _ neutron polarization analysis was carried out on the d7 ( ill ) diffractometer @xcite with a dilution fridge and @xmath13 = 4.8 . unpolarized elastic neutron - scattering studies were performed at the two - axis d23 diffractometer ( ill ) with incident wavelength 1.277 and the in12 ( ill ) cold triple - axis spectrometer with fixed final energy of 5.3 mev and the beam collimation set as open-40@xmath15-sample-60@xmath15-open . here the wave vector * * q**@xmath16 ( @xmath17 ) = ( @xmath18 , @xmath19 , @xmath20 ) is defined through ( _ h _ , _ k _ , _ l _ ) = ( @xmath21 , @xmath22 , @xmath23 ) quoted in units of r.l.u . , where _ a _ , _ b _ , and _ c _ are relevant lattice constants referring to the orthorhombic @xcite unit cell . * figure [ fig1 ] * shows the neutron polarization analysis in the spin - flip ( sf , i.e. , z - flipper on ) and non - spin - flip ( nsf , i.e. , z - flipper off ) channels . compared with the maps at 300 k ( * figure [ fig1]a * ) , it is clear that extra fourfold bragg peaks around ( @xmath241.6 , @xmath241 , 0 ) appear symmetrically in both sf and nsf reciprocal space maps at 50 mk ( * figure [ fig1]b * ) due to a long - range magnetic transition . polarized neutron magnetic scattering depends on the direction of the neutron polarization @xmath25 with respect to the scattering vector @xmath26 , and also the direction of the ordered - moments @xmath27 . in our case , @xmath25 ( z - component ) @xmath28 _ c_-axis @xcite , and the magnetic bragg reflections are observed in the ( _ h _ , _ k _ , 0 ) plane , i.e. , @xmath25 @xmath29 @xmath26 . in this case , the neutron - scattering cross sections of the nsf and sf channels are @xmath30{1.00,1.00,1.00}{ab } } { \propto } \text{\textcolor[rgb]{1.00,1.00,1.00}{ab } } { \langle}{\hat{\mu}}\parallel\hat{\textbf{p}}{\rangle}^2 \text { , and } \\ * \label{eq2 } & & \mathlarger{\mathlarger{\mathlarger{(}}}\frac{d\sigma}{d\omega}\mathlarger{\mathlarger{\mathlarger{)}}}^{^{\texttt{sf}}}_{_{\texttt{z - on } } } \text{\textcolor[rgb]{1.00,1.00,1.00 } { , } } = \frac{1}{2}\mathlarger{\mathlarger{\mathlarger{(}}}\frac{d\sigma}{d\omega}\mathlarger{\mathlarger{\mathlarger{)}}}_{_{\texttt{mag } } } + \frac{2}{3}\mathlarger{\mathlarger{\mathlarger{(}}}\frac{d\sigma}{d\omega}\mathlarger{\mathlarger{\mathlarger{)}}}_{_{\texttt{si } } } \text { , } \nonumber \\ * & & \mathlarger{\mathlarger{\mathlarger{(}}}\frac{d\sigma}{d\omega}\mathlarger{\mathlarger{\mathlarger{)}}}^{^{\texttt{sf}}}_{_{\texttt{mag } } } \text{\textcolor[rgb]{1.00,1.00,1.00}{ab } } { \propto } \text{\textcolor[rgb]{1.00,1.00,1.00}{ab } } { \langle}{\hat{\mu}}\perp\hat{\textbf{p } } \times \hat{\textbf{q}}{\rangle}^2 \text{,}\end{aligned}\ ] ] respectively . the first and the second terms in each equation refer to the magnetic and spin - incoherent scatterings , respectively . the third term in eq . ( [ eq1 ] ) denotes nuclear and isotope incoherent contributions @xcite . the presence of the incommensurable afm bragg peaks in the nsf channel ( * figure [ fig1]b * ) indicates that one component of @xmath27 is parallel to the _ c _ axis , while their appearances in the sf channel imply a @xmath27 component lying in the _ ab _ plane . we observe the magnetic bragg peak only at 0.5 k in our npd study ( * figure [ fig2 ] * ) . we thereby refine the afm wave vector exactly as * * q**@xmath31 = ( 0.5924(1 ) , 0.0059(1 ) , 0 ) by the profile - matching mode @xcite and a total moment @xmath32 = 1.92(6 ) @xmath33 at the maximum amplitude for the tb1 ions only with the _ b_- and _ c_-components equalling to + 1.88(8 ) and + 0.40(23 ) @xmath33 ( * table 1 * ) , respectively . the moment size of the tb2 site is negligible . * figure [ fig3 ] * schematically shows the resulting crystal and magnetic structures as well as the structural parameters for the bent tb@xmath34 honeycombs . the temperature dependence of the afm ( 1.6 , 1 , 0 ) bragg peak is shown in * figure [ fig4]*. the extracted integrated intensity ( _ i _ ) was fit to a power law @xmath35 , which produces a n@xmath9el temperature @xmath36 4.28(2 ) k , and a critical exponent @xmath37 0.55(2 ) probably indicative of a second - order type phase transition and possible three - dimensional heisenberg - like spin interactions @xcite . we record a reciprocal space map ( * figure [ fig5]a * ) around the afm ( 1.6 , 1 , 0 ) bragg peak at 1.7 k using d23 , and the central scans along the @xmath38 and @xmath39 directions ( * figure [ fig5]b * ) were measured at in12 . in both * figures * , the fwhm of the magnetic bragg peak along the @xmath38 and @xmath39 directions is sharply different . both magnetic bragg peaks are broader than the nuclear bragg ( 2 , 0 , 0 ) reflection in the reciprocal space as shown in * figure [ fig5]b * , which indicates that the observed magnetic bragg peaks are beyond the instrument resolution . therefore , * figure [ fig5]b * shows a real in - plane magnetic anisotropy . rccccccccc + _ t _ ( k ) & & & + @xmath40 ( ) & 10.0842(1 ) & 11.9920(2 ) & 3.4523(1 ) & 10.0844(1 ) & 11.9918(1 ) & 3.4522(1 ) & 10.0852(1 ) & 11.9922(1 ) & 3.4525(1 ) + atom & _ x _ & _ y _ & _ b _ ( @xmath41 ) & _ x _ & _ y _ & _ b _ ( @xmath41 ) & _ x _ & _ y _ & _ b _ ( @xmath41 ) + sr & 0.7497(1 ) & 0.6487(1 ) & 0.85(5 ) & 0.7493(2 ) & 0.6485(2 ) & 0.87(7 ) & 0.7498(3 ) & 0.6491(3 ) & 1.04(8 ) + tb1 & 0.4243(2 ) & 0.1126(1 ) & 0.27(4 ) & 0.4241(2 ) & 0.1124(2 ) & 0.25(5 ) & 0.4250(3 ) & 0.1123(2 ) & 0.56(6 ) + tb2 & 0.4182(2 ) & 0.6116(1 ) & 0.47(4 ) & 0.4180(2 ) & 0.6116(2 ) & 0.50(5 ) & 0.4178(3 ) & 0.6114(2 ) & 0.53(7 ) + o1 & 0.2133(2 ) & 0.1799(1 ) & 0.64(5 ) & 0.2138(3 ) & 0.1798(2 ) & 0.75(7 ) & 0.2125(3 ) & 0.1796(2 ) & 0.63(9 ) + o2 & 0.1293(2 ) & 0.4818(1 ) & 0.16(5 ) & 0.1295(2 ) & 0.4819(2 ) & 0.21(7 ) & 0.1288(3 ) & 0.4824(2 ) & 0.43(9 ) + o3 & 0.5092(2 ) & 0.7859(2 ) & 0.56(4 ) & 0.5095(2 ) & 0.7857(2 ) & 0.48(6 ) & 0.5095(3 ) & 0.7859(3 ) & 0.69(8 ) + o4 & 0.4273(2 ) & 0.4216(1 ) & 0.55(5 ) & 0.4271(3 ) & 0.4218(2 ) & 0.49(7 ) & 0.4270(4 ) & 0.4217(2 ) & 0.74(9 ) + @xmath27(tb1 ) @xmath42 & & & & & & & + @xmath43tb1-o2-tb1 ( @xmath7 ) & & & + @xmath43tb1-o3-tb1 ( @xmath7 ) & & & + @xmath43tb1-o1-tb2 ( @xmath7 ) & & & + @xmath43tb1-o3-tb2 ( @xmath7 ) & & & + @xmath43tb2-o1-tb2 ( @xmath7 ) & & & + @xmath43tb2-o4-tb2 ( @xmath7 ) & & & + @xmath43o1-tb1-o2 ( @xmath7 ) & & & + @xmath43o2-tb1-o3 ( @xmath7 ) & & & + @xmath43o3-tb2-o4 ( @xmath7 ) & & & + @xmath43o1-tb2-o4 ( @xmath7 ) & & & + @xmath44tb1-o1,2,3@xmath45 ( ) & & & + @xmath44tb2-o1,3,4@xmath45 ( ) & & & + @xmath46 ( @xmath47 ) & & & + @xmath48 & & & + to quantitatively estimate the in - plane anisotropy , we take the fwhm of the nuclear bragg ( 2 , 0 , 0 ) peak as the detecting accuracy which is convoluted in fitting the magnetic peaks by a gaussian function shown as the solid lines in * figure [ fig5]b*. this results in fwhm = 0.0183(1 ) and 0.0492(2 ) @xmath17 along the @xmath38 and @xmath39 directions , respectively , implying highly anisotropic in - plane spin correlations consistent with the observation that strong magnetic frustration exists in srtb@xmath1o@xmath2 . we roughly estimate the spin - correlation length ( @xmath49 ) by @xmath50 , i.e. , @xmath51 and @xmath52 . therefore , @xmath53 . similar in - plane anisotropic magnetic correlations were also observed in the iron - based superconductors @xcite that are highly frustrated , too , where its microscopic origin , from the ellipticity of the electron pockets or the competing exchange interactions associated with the local - moment magnetism , is still being strongly argued @xcite . it is undoubted that the observed in - plane magnetic anisotropy in srtb@xmath1o@xmath2 indicates an appearance of the competing spin exchanges and is certainly associated with a description of the purely - localized magnetism of ionic tb@xmath54 ions . a deeper understanding of the insulating state necessitates theoretical band structure calculations . we tentatively estimate the compatibility between ordered magnetic and nuclear crystalline domains based on the non - deconvoluted fwhm ( @xmath55 ) of the bragg ( 1.6 , 1 , 0 ) ( @xmath56 = 0.0300(7 ) @xmath17 ) and ( 2 , 0 , 0 ) ( @xmath57 = 0.0238(2 ) @xmath17 ) peaks , i.e. , @xmath57/@xmath56 = 79(2)% , which implies that the incommensurate afm structure orders with a long - range fashion relative to the underlying lattice of the single crystal . we further analyze the spin - correlation length with our npd data ( * figure [ fig2]c * ) . firstly , it is pointed out that the positive and negative momenta can not technically be differentiated in a npd study . as shown in * figure [ fig6]a * , taking into account the corresponding spodi instrument resolution ( dashed line ) @xcite , a gaussian fit ( solid line ) to the afm bragg ( 1.5924 , 1.0059 , 0 ) peak ( squares ) results in an average @xmath58 in real space . this indicates that the afm ordering observed in srtb@xmath1o@xmath2 is indeed of long range in character in comparison with the reported extremely - broad magnetic diffuse scattering which was attributed to the presence of short - ranged magnetic ordering in polycrystalline sr@xmath59o@xmath2 ( _ re _ = ho , er , dy ) samples in the study of reference @xcite . with the same method utilized in the analysis of the data as shown in * figure [ fig6]a * , we also analyze the npd peak of the nuclear bragg ( 2 , 0 , 0 ) reflection as shown in * figure [ fig6]b * and extract that @xmath60 . this indicates that @xmath61 = 66(3)% basically in accord with the compatibility between ordered magnetic and nuclear crystalline domains extracted with our single - crystal neutron - scattering data . since our npd data were collected from a pulverized srtb@xmath1o@xmath2 single crystal , that @xmath62 is @xmath3 2.5 times larger than @xmath63 may indicate that there have strong magnetic and crystalline domain effects in single - crystal srtb@xmath1o@xmath2 , or a large part of spins are blocked probably due to a pining effect by strains accumulated during single crystal growth . in any case , this difference between single - crystalline and polycrystalline samples in turn supports the fact that there is a strong magnetic frustration in single - crystal srtb@xmath1o@xmath2 . further studies with high pressures would be of great interest . in most cases , the strength of the indirect magnetic interactions such as conventional double- or super - exchange @xcite can be influenced more or less by the value of the revelent bond angle @xcite , e.g. the @xmath43tb - o - tb bond angles in srtb@xmath1o@xmath2 as listed in * table 1 * ( see also * figure [ fig7 ] * ) . however , the respective values of @xmath43tb - o - tb display no appreciable difference within accuracy between 0.5 and 10 k ( * table 1 * ) , below and above the @xmath64 , respectively , which may indicate an invalidity of the two conventional magnetic coupling mechanisms ( double- or super - exchange ) in srtb@xmath1o@xmath2 . this is consistent with the study of srtm@xmath1o@xmath2 @xcite and in excellent agreement with our transport study , where any attempts to measure possible resistivity in srtb@xmath1o@xmath2 from 2 to 300 k were fruitless . we estimate that the resistance of the single crystal measured is beyond at least 10@xmath65 ohm . we thus conclude that srtb@xmath1o@xmath2 is a robust insulator , and the electrons responsible for the incommensurable antiferromagnetism are mainly from the localized 4@xmath66 shell of the ionic tb@xmath54 ions . in this localized picture , the interionic exchange interactions dominate for the formation of the magnetic structure @xcite . the nearest tb neighbours are stacked linearly along the _ c _ axis ( * figure [ fig3]b * ) . the shortest tb1-tb1 and tb2-tb2 have the same bond length . however , the nn tb1 ions have a ferromagnetic ( fm ) arrangement . by contrast , the interaction between the nn tb2 ions is blocked unexpectedly ( * figure [ fig3]b * ) . there is no appreciable difference in the nn tb - tb bond length , i.e. , the _ c _ lattice constant , between 0.5 and 10 k ( * table 1 * ) , which probably rules out the potential direct exchange interaction consistent with the fact that unpaired 4__f _ _ electrons are deeply embedded under the @xmath67 shells and also indicates that the prevalent dipole - dipole interaction is subjected to some condition , i.e. , the octahedral distortion as discussed below , in agreement with the study of srtm@xmath1o@xmath2 @xcite . as a non - kramers ion , tb@xmath54 ( @xmath68 ) in principle keeps the time reversal symmetry and does nt show any energy degeneracy in the presence of the purely - localized electric field . however , we refine two kinds of octahedra as shown in * figure [ fig7 ] * : tb1o@xmath34 and tb2o@xmath34 , corresponding to the partially - ordered and totally - frozen tb1 and tb2 ions , respectively . the average octahedral distortion @xcite can be quantitatively measured by the parameter @xmath46 defined as : @xmath69}}^2 $ ] , where @xmath70 and @xmath71 are the six tb - o bond lengths along the six crossed directions ( * figure [ fig7 ] * ) and the mean tb - o bond length ( * table 1 * ) , respectively . it is noteworthy that the @xmath46 values of the tb1 and tb2 ions are in the same magnitudes as those of the mn@xmath54 kramers and mn@xmath72 non - kramers ions , respectively , in the jahn - teller ( jt ) distorted regime of single - crystal la@xmath73sr@xmath74mno@xmath10 @xcite . this sharp contrast implies that the tb1 ions are strongly distorted , while the tb2 ions behave normally within the non - kramers scheme . therefore , the @xmath46 magnitude that reflects the ion local symmetry and thus the strength of the surrounding cef directly determines the existence of the magnetic ordering , which is supported by the observation that below @xmath75 the respective @xmath46 values of the tb1 and tb2 ions change oppositely with temperature ( * figure [ fig8 ] * ) . we therefore infer that one possible reason for the formation of the incommensurable magnetic structure is the modulated distribution of the @xmath76 valence electrons which modify the surrounding environment experienced by the localized unpaired 4__f _ _ electrons . the corresponding modulation of the local symmetry may plausibly be attributed to the spatial zigzag - type tb arrangements along the _ a _ and _ b _ axes in the process of forming the crystallographic domains . this is supported by the fact that the honeycomb columns run straightly along the _ c _ axis , and there is no spin modulation at all in that direction . based on the refined tb - o bond lengths , we deduce two distortion modes for the tb1o@xmath34 and tb2o@xmath34 octahedra ( * figure [ fig7 ] * ) , respectively . the possible product of the tb1 subjected stress - vectors ( small arrows ) should point qualitatively to the direction of the tb1 moment , implying a strong single - ion anisotropy . this jt - like distortion mode leads to the large @xmath46 value of the tb1 ions , and possibly lifts further the degenerate multiplets . by contrast , the tb2 ions are subjected to opposing stresses in all the three pair - directions . in this case , the octahedral distortion strongly depends on their competing strengths . this mode makes the small @xmath46 value of the tb2 ions and their potential total magnetic moments quenched vitally . the maximum tb1 moment size is mere 1.92(6 ) @xmath33 , 21.3(7)% of the theoretical saturation value ( @xmath77 9 @xmath33 ) . it is of particular interest to explore the frustrating mechanism . the virtual non - kramers state of the tb2 site reduces the total moment size per molar formula by 50% . the tb1 moment fluctuates like a wave defined as @xmath27 = @xmath78 , where @xmath79 is a spin coordinate along the _ a _ axis and @xmath80 is a phase parameter . the existence of the strong single - ion anisotropy indicates a large cef effect which should be comparable to the energy scale of the magnetic interactions . we have shown the clear evidence for a large magnetic exchange anisotropy ( * figure [ fig5 ] * ) , which is ascribed to the anisotropic dipole - dipole interaction . the nn magnetic arrangement is fm ( * figure [ fig3]b * ) , implying no possibility for a magnetic frustration . the nnn magnetic configurations display a dual character , i.e. , fm and afm for the equivalent tb11 and tb12 sites , respectively . this sharp difference may frustrate the heisenberg - exchange coupled nnn spins . to summarize , we have synthesized large enough srtb@xmath1o@xmath2 single crystals suitable for neutron scattering studies and revealed a modulated spin structure in srtb@xmath1o@xmath2 with the highest afm transition temperature at @xmath81 4.28(2 ) k in the sr@xmath59o@xmath2 family , which provides a technically friendly platform to explore the related magnetic coupling and frustrating mechanisms . our studies show that the localized tb1 moments lie in the _ bc _ plane with the fm chains along both the _ b _ and _ c _ directions and the afm modulation mainly along the _ a _ axis . we have found two distinct octahedra for the non - kramers tb@xmath54 ions : tb1o@xmath34 being strongly distorted , corresponding to the partially - ordered moments ; tb2o@xmath34 being frustrated entirely in the non - kramers state . therefore , the octahedral distortion has a decisive influence on the hund s rule magnetic ground state @xmath82 and the related frustrations . the magnetocrystalline anisotropy is crucial in determining the direction of the ordered moments . the direct nn interaction results in a fm arrangement for the tb1 ions along the _ c _ axis , and the different nnn tb configurations ( fm and afm ) further lift the magnetic frustration . the present results make srtb@xmath1o@xmath2 a particularly significant compound in the family for theoretical and further experimental studies . inelastic neutron - scattering studies to determine the detailed crystal - field and magnetic - interaction parameters would be of great interest . the factors that influence the value of the afm transition temperature would be further explored in combination with theoretical calculations . this work at rwth aachen university and j@xmath83lich centre for neutron science jcns outstation at ill was funded by the bmbf under contract no . 05k10pa3 . h.f.l thanks the sample environment teams at ill and frm - 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the n@xmath0el temperature of the new frustrated family of sr__re__@xmath1o@xmath2 ( _ re _ = rare earth ) compounds is yet limited to @xmath3 0.9 k , which more or less hampers a complete understanding of the relevant magnetic frustrations and spin interactions . here we report on a new frustrated member to the family , srtb@xmath1o@xmath2 with a record @xmath4 = 4.28(2 ) k , and an experimental study of the magnetic interacting and frustrating mechanisms by polarized and unpolarized neutron scattering . the compound srtb@xmath1o@xmath2 displays an incommensurate antiferromagnetic ( afm ) order with a transverse wave vector * * q**@xmath5 = ( 0.5924(1 ) , 0.0059(1 ) , 0 ) albeit with partially - ordered moments , 1.92(6 ) @xmath6 at 0.5 k , stemming from only one of the two inequivalent tb sites mainly by virtue of their different octahedral distortions . the localized moments are confined to the _ bc _ plane , 11.9(66)@xmath7 away from the _ b _ axis probably by single - ion anisotropy . we reveal that this afm order is dominated mainly by dipole - dipole interactions and disclose that the octahedral distortion , nearest - neighbour ( nn ) ferromagnetic ( fm ) arrangement , different next nn fm and afm configurations , and in - plane anisotropic spin correlations are vital to the magnetic structure and associated multiple frustrations . the discovery of the thus far highest afm transition temperature renders srtb@xmath1o@xmath2 a new friendly frustrated platform in the family for exploring the nature of magnetic interactions and frustrations .

in recent years spintronics and magneto - electronic devices have been a major research topic , biased in part by the prospect of new technological applications in information processing based on the simultaneous use both the electron s spin and charge degree of freedom @xcite . all spintronic devices depend on driven spin currents between different subsystems , which is why a deep understanding of these spin - currents is essential for this field of research . a naive identification of total transfer of angular momentum with the spin - polarized charge current passed through a junction , @xmath0 , neglects the possibility of spin transfer due to exchange couplings . the aim of this paper is to derive an expression for the spin current through a tunnel junction , that is suitable to describe interacting electron systems , such as quantum dots , and that explicitly demonstrates the two different contributions to the spin current . we start our discussion with the calculation of the spin current between a ferromagnetic lead @xmath1 and an island @xmath2 via a tunnel contact . in this first section , we do not specify the electronic structure of the island yet , i.e. , any kind of many - body effects or couplings of the island to other leads are included . the hamiltonian of such a tunnel system is given by @xmath3 where @xmath4 are the fermion creation operators in the lead and @xmath5 are the corresponding operator for the island . in the lead ( island ) we label the momentum states of the electrons with @xmath6 ( @xmath7 ) and the spin with @xmath8 and @xmath9 ( @xmath10 and @xmath11 ) . starting from this hamiltonian we calculate the spin current in close analogy to the derivation of the charge current by meir and wingreen for interacting electron systems @xcite . if the spin is a conserved quantity , the time derivative of the total spin in the lead equals the spin current through the tunnel barrier @xmath12 . in the heisenberg picture , the time evolution of the spin operator @xmath13 is given by @xmath14 $ ] , which yields @xmath15 where we introduced the keldysh green s functions @xmath16 . by use of a dyson equation @xcite we can replace the latter with ( free ) green s functions @xmath17 of the lead and green s functions @xmath18 of the island . by choosing the magnetization direction @xmath19 as spin - quantization axis , the lead greens functions @xmath20 are diagonal in the spin index . the lead green s functions are then @xmath21 , @xmath22 , @xmath23 , and @xmath24 . there @xmath25 stands for the fermi distribution function in the lead @xmath26 and @xmath27 . assuming , furthermore , that tunnel events conserve the spin of the electrons , we substitute the tunnel matrix elements by @xmath28 , and define the spin - dependent transition rates @xmath29 . after a lengthy but straightforward calculation , the spin current can be written as @xmath30\nonumber\\ & & + { \bm\sigma}^{}_{\gamma\delta}(\gamma^\gamma_{q , p}-\gamma^\delta_{q , p})\left [ f^+_{\rm l}(\omega)\,(g^{\rm ret}_{q\delta , p\gamma}+g^{\rm adv}_{q\delta , p\gamma})+\frac{1}{i\pi}{\int}^\prime \!\!de\,\frac{g_{q\delta , p\gamma}^<(e)}{e-\omega}\,\right]\ , .\end{aligned}\ ] ] this is the central most general result of our calculation . since we did not specify the green s functions @xmath31 of the island yet , the expression for the spin current can be used for many situation , including strongly - correlated systems such as quantum dots @xcite . by comparison with the expression of the charge current derived by meir and wingreen @xcite for nonmagnetic systems , we identify the first line of eq . ( [ final ] ) with a spin - polarized charge current . the origin of the spin - current contribution in the second line of eq . ( [ final ] ) is the exchange interaction between lead and island . if the island does posses a magnetic moment , due to spin accumulation or ferromagnetic order , this moment couples to the magnetization of the lead and both precess around each other @xcite . this exchange coupling changes the average spin on each side of the tunnel junction , and therefore it must also appear as contribution to the spin current crossing the tunnel barrier . in the latter case , the transfered angular moment is perpendicular to the magnetic moments of lead and island , which is sometimes described by `` spin mixing conductances '' @xcite . for concreteness , we restrict the following discussion to the special case , that the island is an itinerant ferromagnet . thereby the direction of magnetization @xmath32 of the island encloses a finite angle @xmath33 with the lead magnetization direction . we further assume , that the island is large , to be also described as a reservoir in thermal equilibrium . due to the non - collinear magnetization directions , the green s function of the island is non - diagonal in the spin space @xmath34 with the su(2 ) rotations @xmath35 . to simplify the result further , we assume that the absolute value of the tunnel matrix elements @xmath36 is independent of the momentum index . then we can replace the transition rates @xmath37 by the spin - resolved density of states @xmath38 and @xmath39 . after performing all spin summations , we get in lowest - order in the tunnel coupling @xmath40\chi_{\rm l}(\omega)\rho_{\rm i}(\omega)\nonumber \\ & & \lambda_2(\omega)=[f^-_{\rm l}(\omega)f^+_{\rm i}(\omega)-f^+_{\rm l}(\omega)f^-_{\rm i}(\omega)]\rho_{\rm l}(\omega)\chi_{\rm i}(\omega ) \nonumber \\ & & \lambda_3(\omega)=\frac{1}{\pi}{\int}^\prime \!\!de\,\ , \frac{f^+_{\rm l}(\omega)-f^+_{\rm i}(e)}{\omega - e } \chi_{\rm i}(e)\chi_{\rm l}(\omega)\end{aligned}\ ] ] with the full density of states @xmath41 and the spin - polarization density @xmath42 , and analogue definitions for the lead @xmath1 . in the first and second term we can recognize the spin current contribution of the charge transfer between the two reservoirs . with the cross product @xmath43 the third term shows the typical structure for a precession movement . in the approach of spin dependent circuit theory @xcite this spin current contribution corresponds to the imaginary part of the spin - mixing conductance . a very pronounced effect of a spin current is current - induced magnetization reversal @xcite . however , there it is difficult to selectively address the spin - current contribution arising from the exchange interaction . the latter goal can be achieved , e.g. , in measuring the charge current through a single - level quantum dot connected to two ferromagnetic leads . a current forced through such a quantum - dot spin valve will accumulate a non - equilibrium spin on the dot . this spin is sensitive to the exchange field generated by the leads . its precession is predicted to be visible in the magneto - resistance of the device @xcite . s. a. wolf , d. d. awschalom , r. a. buhrman , j. m. daughton , s. von molnr , m. l. roukes , a. y. chtchelkanova , d. m. treger , science , * 294 * , 1488 - 1495 ( 2001 ) ; i. zutic , j. fabian , and s. das sarma , rev . phys . * 76 * , 323 ( 2004 ) .

we derive an expression for the spin - current through a tunnel barrier in terms of many - body green s functions . the spin current has two contributions . one can be associated with angular - momentum transfer by spin - polarized charge currents crossing the junction . if there are magnetic moments on both sides of the tunnel junction , due to spin accumulation or ferromagnetic ordering , then there is a second contribution related to the exchange coupling between the moments . 72.25.mk , 72.25.pn , 72.25.-b , 73.23.hk , 85.75.-d

the numerous ly@xmath0 absorption lines seen in quasar spectra can be considered a very deep window on the nature of the young universe . all recent results obtained with high resolution spectroscopy indicate that the associated gas clouds represent most likely a consistent fraction of the dark side of the baryonic matter physically connected with primeval galaxies . clustering properties have been studied basically using the two point correlation function . a positive signal was detected in high resolution spectra only for small scales ( up to a few hundred , webb 1987 , rauch et al . 1992 , chernomordik 1995 , cristiani et al . 1996 ) . the absorption lines can be fitted by voigt profiles in order to obtain hi column densities , doppler widths and redshifts . at high redshift ( @xmath1 ) the redshift evolution and hi column density distribution is well reproduced by a double power law : @xmath2 where @xmath3 and @xmath4 . the important features of this kind of non gaussian distribution are the index of the scaling laws and not the amplitudes at every scale and we analyse it using a mathematical tool which is most suitable for their determination . = 11.6 cm = 5.44 cm = 5.6 cm in fig . [ f1 ] we show the visual similarity between the distribution of two different physical quantities : the redshift distribution of hi column densities in a quasar spectrum and the kinetic energy dissipation in a fully developed turbulent fluid . the energy transfer , underlying the phenomenon shown in fig . [ f1]b , can be described by means of a self similar cascade with an associated multiplicative process : a break down of large scale structures into small scale ones , each receiving a fraction of energy . an analogous mechanism ( for instance in a cdm scenario with an associated inverse cascade " of gravitationally confined structures ) can leads to the phenomenon shown in fig . the result is an intermittent process , where rare events ( the localized peaks , or singularities , in the distribution ) have a higher probability to occur with respect to a gaussian process . every scaling process , like turbulence and hi column density distribution in the early universe , can be treated in the context of the fractal formalism or , more generally , the multifractal formalism . a multifractal is a scale - invariant distribution described by a local exponent @xmath0 @xmath5 where @xmath6 is the probability measure in the @xmath7th interval of size @xmath8 around the point @xmath9 . in a multifractal , @xmath0 depends on the position @xmath9 . to investigate the multifractal structure of the measure , we define the generalized partition function of the box counting method ( paladin & vulpiani , 1987 ) : @xmath10^q~,\ ] ] where the sum is extended to all the subsets @xmath7 at a given scale @xmath8 . the information relative to the multifractal structure can be recognized by calculating the generalized rnyi dimensions @xmath11 from the scaling law : @xmath12^{(q-1)d_q}\ ] ] for the way the partition function is defined , big values of @xmath13 emphasize the scaling properties of overdense regions , while small values those of underdense regions . a multifractal structure is a non homogeneous fractal ( with different @xmath11 at every @xmath13 ) where the presence of clusters is enhanced by positive values of @xmath13 and that of voids by negative values . the two point correlation function describes clustering at the first order only , while the infinite set of singularities , each being characterized by a different fractal dimension @xmath11 , describes clustering at every scale and has the important property to show ( in principle ) the _ entire hierarchy of clusters _ ( if any ) . we have used in a previous paper ( carbone & savaglio , 1996 , hereafter cs96 ) the high resolution spectrum of q@xmath14 ( @xmath15 , cristiani et al . 1995 ) in order to test the multifractality of the forest . for each scale @xmath16 ( between two redshifts @xmath17 and @xmath18 ) we define a probability measure by dividing the redshift range into disjoint subsets @xmath7 . this measure @xmath19 is defined as the total hi column density in the @xmath7th subset characterized by a velocity separation @xmath16 , normalized to the total column density in the spectrum . this can be related to the probability of occurrence of a certain amount of gas in the @xmath7-th box at a certain scale @xmath16 . the results for @xmath20 are shown in fig . it is evident the presence of two regimes with a linear relation and two different @xmath21 values . the separation between the two regimes occurs at @xmath22 , which at a mean redshift of @xmath23 corresponds to a comoving scale of about 8 @xmath24 mpc . similar features are visible for higher values of @xmath13 up to @xmath25 ( cs96 ) . for comparison with the distribution of galaxies in the local universe , one can see martinez et al . 1990 , coleman & pietronero 1992 , borgani et al . 1994 , martinez & coles 1994 and garrido et al . 1996 . in the multifractal analysis of the qdot redshift survey of 2086 iras galaxies ( dominated by spiral galaxies ) , martinez & coles ( 1994 ) found multifractality and observed two regimes for different values of @xmath13 . for @xmath20 , at small scales ( @xmath26 mpc ) they found scaling properties with correlation dimension @xmath27 , which in one dimension corresponds to a fractal dimension of 0.25 ( @xmath28 ) . for large scales , the iras galaxies reach homogeneity . we notice that we are comparing the distribution of galaxies of the local universe with that of clouds at high redshift . if the change of regime occurs at similar comoving scales , we conclude that clouds have undergone a faster clustering evolution compared to iras galaxies . however multifractality in galaxies is matter of controversy and it is strongly dependent on the galaxy morphology . the statistics of the forest is poor in comparison with galaxy surveys . one of the main problems is thus to test the significance of the results . a first check has been presented in cs96 . the observed distribution has been compared with a set of 2000 `` fake '' distributions . in a following work ( savaglio & carbone , in preparation ) , new tests will be presented and sets of different simulations will be compared to the observed distributions . as very preliminary and crude results , we have seen that in 100 simulations of the q@xmath14 sight line , 53% of the cases shows no scaling law , 37% one single scaling law with @xmath29 . in 10% a very weak double scaling law , with @xmath21 which goes from about 0.6 at small scales to about 0.8 at large scales and a correlation length much smaller than the observed one . even if a genuine multifractality in the clouds distribution is evident , a richer sample of lines and comparison with simulations would help to clarify this situation . we applied the box counting method to a sample of qso spectra . it represents part of the sample used by cristiani et al . ( 1996 ) , with a total of 2412 lines for 12 sight lines . the redshift coverage is @xmath30 and the resolution better than 15 , the best available for these sources . the lower limit for the redshift is imposed by the lack of statistics of hst high resolution data . we restricted our analysis to high resolution spectroscopy in order to minimize the problem of line blanketing ( the confusion of lines ) which at high redshift can dramatically affect the analysis . = 8.8 cm = 8.5 cm = 6.5 cm the partition functions with @xmath20 for the 12 forests are shown in fig . we confirm the presence of two regimes in almost all of them , except for one object ( @xmath31 ) , where we do not see any clear scaling law in the distribution . for two objects ( @xmath32 and @xmath33 ) the determination of the two scales is particularly difficult . for the remaining 9 objects , two slopes are clearly visible . in the plot of @xmath21 as function of the mean redshift for the two different scales ( fig . [ f4 ] ) there is no correlation for large scales , with a mean value of 0.8 . this value is an indication that homogeneity has been reached in the sample . for small scales there is a clear deviation from homogeneity . a weak correlation with redshift of @xmath21 , being smaller for lower redshifts , can also be noticed . a more clear redshift evolution is shown by the distribution of the correlation lengths ( fig . a simple fit with a power law gives : @xmath34 these results suggest a picture where an initial homogeneous distribution of gas clouds or mass in the universe is broken by process of fragmentation ( in a cold dark matter scenario ) or of aggregation of matter around some singularities ( in the hot dark matter scenario ) . the ultimate fate of both the processes is a highly intermittent distribution like that shown in fig . the work presented here is in progress . a more extended analysis , with a full description of different methods testing the stability of the results , will be presented elsewhere . even if the main problem is the lack of statistics , we can firmly conclude that multi - scaling analysis of forests is a very promising approach to the study of the large scale structure of the universe at high redshift and its evolution . this has to be regarded as a parallel and complementary point of view with respect to the study of the galaxy distribution . the two point correlation function analysis can be replaced by different statistics which are more suitable to describe highly inhomogeneous distributions . clouds have shown scaling laws for much larger scales with respect to previous analysis , around 10 @xmath35 mpc in comoving distance at redshift of about @xmath36 . for larger scales , we have no indication against a homogeneous distribution . the multifractal properties of a sample of 11 quasars show evolution with redshift . in particular both the amplitude and the strength of the multifractality decrease with redshift , which is what one expects to see in a universe where gravitational clustering gives rise to larger , correlated structures . we are grateful to l. amendola for helpful discussions . we also thank c. meneveau for kindly providing us with the postscript file of fig . [ f1]b , and s. cristiani , s. dodorico , v. dodorico , a. fontana and e. giallongo for making the data of the eso key programme available . borgani s. , martinez v. j. , prez m. a. & valdarnini r. 1994 , apj , 435 , 37 carbone v. , savaglio s. 1996 , mnras , _ in press _ , cs96 chernomordik v. v. 1995 , apj , 440 , 431 coleman p. h. & pietronero , l. 1992 , phys . , 213 , 311 cristiani s. , dodorico s. , dodorico v.,fontana a. , giallongo e. , savaglio s. 1996 , mnras , _ in press _ cristiani s. , dodorico s. , fontana a. , giallongo e. & savaglio , s. 1995 , mnras , 273 , 1016 garrido p. , lovejoy s. , schertzer d. 1996 , physica a , 225 , 294 martinez v. j. & coles p. 1994 , apj , 437 , 550 martinez v. j. , jones b. j. t. , dominguez tenreiro r. & van de weygaert r. 1990 , apj , 357 , 50 meneveau c. & sreenivasan k. r. 1991 , j. fluid mech . , 224 , 429 paladin g. , & vulpiani a. 1987 , phys . rep . , 156 , 147 rauch m. , carswell r. f. , chaffee f. h. , foltz c. b. , webb j. k. , weymann r. j. , bechtold j. , green r. f. , 1992 , apj , 390 , 387 webb j. k. , 1987 , in : _ iau symposium 124 , observational cosmology _ , ed . a. hewett , g. burbidge , l. z. fang , p803

we present some statistical features of the large number of absorption lines detected in high redshift quasar spectra , obtained by using the multifractal approach . in the analysed sample of 12 qso sight lines , 11 show scaling behaviour with a crossover between two distinct regimes : a non - homogeneous regime at small scales and a homogeneous regime at large scales . the correlation length shows a redshift dependence , suggesting that the forest can be an intermediate phenomenon between a strongly inhomogeneous galaxy distribution in the local universe and a homogeneous initial mass distribution .

the immense complex of the taurus , perseus , and auriga ( tpa ) clouds is known to be one of the largest local associations of dark nebula . according to @xcite , the total masses of the whole complex is @xmath3 solar mass . the tpa complex contains star - forming cloudlets ( tmc1 , tmc2 , ngc 1333 , and ic 348 , for example ) , a bright emission nebula ( ngc 1499 , also known as california nebula ) , and stellar associations ( per ob2 , for example ) . low - mass star formation processes are believed to be taking place in the eastern section of the complex ( taurus and auriga ) , where mostly low - mass t tauri type stars are observed , while both low- and high - mass stars are being formed in the western section of the perseus complex ( the per ob2 association , the open cluster ngc 1333 , and so on ) . the whole complex may be physically associated with the per ob2 stellar association , centered at ( l , b)=(160.5 , -14.7 ) with a diameter of @xmath4 , which is the second closest ob association to the sun with a distance of @xmath0300 pc on average @xcite . recent supernova explosions in the per ob2 association are believed to have created an expanding hi supershell that is moving into the surrounding interstellar medium @xcite . the tpa complex may consist of multiple layers of molecular gas located at various distances . the taurus cloud is believed to spread between 100 pc and 200 pc @xcite , with a large part of it blocked by the cloud itself @xcite . the most distant part of the taurus molecular cloud complex may be interacting with a supershell blown by the per ob2 association @xcite . @xcite argued that the cloud in the region of ngc 1333 , one of the star forming regions located in and around the perseus cloud , consists of two layers , and estimated their distances to be @xmath0170 pc and @xmath0230 pc based on an extinction analysis . the auriga dark cloud , located at a distance similar to that of the taurus cloud , is usually regarded as part of the taurus - auriga complex , and was suggested to overlap with the california cloud at its farthest side @xcite . the california cloud seems to be the most distant cloud in this region : @xcite argued that the cloud is located at @xmath0450 pc from the sun . the interstellar radiation field ( isrf ) , composed of radiations originating directly from stellar sources as well as the diffuse background , plays an important role in the physical and chemical processes of the interstellar medium ( ism ) . for example , far - ultraviolet ( fuv ) isrf photons may ionize atoms , dissociate molecules , or heat gases by ejecting electrons from dust grains or by directly exciting atoms and molecules . the isrf itself is also affected by interactions with the ism . in fact , most of the diffuse fuv background radiation is believed to originate from the scattering of starlight by interstellar dust @xcite . the effect of dust scattering is well observed in reflection nebulae and dark clouds as well as in the diffuse galactic light ( dgl ) . hence , the observations of these objects have been used not only to trace interstellar dust , but also to obtain the optical properties of dust itself , such as the albedo ( a ) and the phase function asymmetry factor ( g ) . in the case of the dgl , dust scattering was observed to be moderately reflective in the forward direction @xcite , with the estimated values of a and g in general agreement with those of the theoretical model for carbonaceous - silicate grains , i.e. , 0.40 - 0.60 and 0.55 - 0.65 , respectively , at uv wavelengths @xcite . for reflection nebulae @xcite and individual clouds @xcite , however , the results were varied and dependent on the individual targets , probably due to uncertainties in the scattering geometry of the target clouds as well as the star systems that were the dominant sources of photons @xcite . @xcite determined a and g to be 0.7 - 0.8 and @xmath00.75 around 1600 , respectively , for dust toward ic 43 , while @xcite estimated a and g to be @xmath00.22 and @xmath00.74 , respectively , for wavelengths below 2200 @xmath5 for the pleiades nebula . for the ophiuchus cloud , @xcite found a and g to be @xmath00.40 and @xmath00.55 , respectively , at @xmath01100 , while @xcite obtained @xmath00.36 and @xmath00.52 for a and g , respectively , for 1370 1670 . [ cols="^,^ " , ] we have estimated the locations of the four prominent clouds in the tpa region by comparing the observed fuv images with the results of the monte carlo simulations of dust scattering . for the taurus cloud , we obtained a distance to the front face and a thickness of the cloud of @xmath6 pc and @xmath7 pc , respectively . there have been many reports on the physical structure of the taurus cloud . @xcite estimated the mean distance to the taurus complex to be @xmath0135 pc , based on the distances to the stars associated with the taurus cloud . @xcite suggested that the cloud spreads between 100 pc and 200 pc , based on the results of star counting , the distances of the reflection nebulae , and the color excess of the field stars . @xcite also reported that extinction begins to increase sharply at 130 - 180 pc . the present result agrees well with all these observations . for the perseus cloud , the present simulation estimates a distance to the front face and a thickness of @xmath8 pc and @xmath9 pc , respectively . the scatter plot of the simulated vs. the observed fuv intensity in figure [ fig : simnobs_scatt](b ) shows severely scattered data points for the perseus cloud , indicating that the model cloud does not adequately describe its actual structure . in fact , the data points corresponding to the observation are spread over a wide range of intensity , which we believe comes from the assumption of a single layer for the cloud . hence , we have divided the perseus cloud into two sub - regions and fit them separately . we name per a as the western part of the cloud and per b as the eastern part , as shown in figure [ fig : multiwave](d ) , and these include the two regions ngc 1333 and ic 348 , respectively . with the same fixed values of the albedo ( a = 0.42 ) and the asymmetry factor ( g = 0.47 ) as used in the previous simulations , we find a distance to the front face and a thickness of the per a region of @xmath10 pc and @xmath11 pc , respectively , and a distance to the front face and a thickness of the per b region of @xmath0320 pc and @xmath030 pc within @xmath12 pc error range , respectively . the scatter plot is now improved considerably , as shown in figure 9 . for the western portion of the perseus cloud , @xcite obtained @xmath0170 pc and @xmath0230 pc for the distances of the two layers in this region , while @xcite , using the parallax observation of the 22 ghz h@xmath13o maser , directly measured the distance of ngc 1333 to be @xmath0240 pc . the rather thick cloud obtained in the present simulation for per a may reflect the multi - layer nature of the cloud suggested by @xcite . the distance to the eastern portion of the perseus cloud containing ic 348 was estimated to be larger than 300 pc @xcite . it is interesting that the thickness was estimated to be rather small for per b in the present simulation . it was seen that the intensity of the scattered emission was very high when the cloud was placed at a distance below 300 pc . we believe the emission observed in the per b region comes from backward scattering reflected by the cloud itself . + for the auriga cloud , the distance to its front face and its thickness are estimated in the present study to be @xmath14 pc and @xmath15 pc , respectively , and for the california cloud , they are @xmath16 pc and @xmath17 pc , respectively . @xcite suggested that the auriga cloud has a two - layered structure near the eastern end of the california cloud , with the layers located at @xmath0115 - 135 pc and @xmath0270 - 380 pc , based on an extinction study . @xcite determined a distance to the california cloud of @xmath0350 pc while @xcite suggested that the cloud is located at @xmath0450 pc from the sun . @xcite argued that the california cloud region is two - layered , with the first layer at @xmath0160 pc and the second layer at @xmath0350 pc . with the present single - slab model , it is impossible to reproduce such multi - layer structures . instead , we obtained single thick clouds , encompassing two layers of the clouds . we note that the estimated value of @xmath0330 pc to the center of the california cloud is larger than the distance to the per ob2 association ( @xmath0300 pc on average , @xcite ) . hence , the california cloud , being located behind the per ob2 association which is a candidate source for the bright fuv emission seen in this region , does not block the emission from the association . figure [ fig : summary ] shows a summary of our distance estimations for the four clouds , along with previous observations for the corresponding clouds . we have plotted in figure [ fig : dameco ] the velocity - clipped co emission maps for the ranges given in the local standard of rest ( lsr ) frame : ( a ) from -21.8 km / s to + 30.2 km / s , ( b ) from -10.7 km / s to -0.3 km / s , ( c ) from + 0.3 km / s to + 2.9 km / s , and ( d ) from + 4.2 km / s to + 10.7 km / s . these maps are reproduced from the co sky survey data by @xcite . while the differences in the lsr velocities do not necessarily imply their relative distances , they may give insightful clues to the physical associations of the corresponding clouds . for example , the california cloud is seen in figure [ fig : dameco](b ) and no other clouds are seen in that velocity range . in figure [ fig : dameco](d ) , however , we can see indications of the taurus , auriga , and perseus clouds , but not of the california cloud . such a grouping may indicate that the california cloud is physically not connected to other clouds , while the rest of the clouds system may be more or less associated . furthermore , parts of the auriga and perseus clouds are also seen in figure [ fig : dameco](c ) . hence , the fact that these clouds are seen in a wide range of the lsr velocities may be consistent with them having multi - layered structures . @xcite also noted that the large velocity gradient observed in the perseus cloud may be due to the nature of multi - components superimposed along the line of sight . + the error ranges of the estimated optical properties are a little large , especially in the value of the asymmetry factor , where the reason was attributed earlier to the statistical fluctuations of the fims data with low exposure time . nevertheless , the best fit values of the distances to the respective front face and the thicknesses of the clouds do not seem to change much when even the upper or lower limits of these error ranges are used for fitting . for example , the changes are mostly @xmath010 pc for the distances to the front faces and @xmath030 pc for the thicknesses of the clouds . the most sensitive parameters for fitting are instead the locations of the field stars as the photon sources relative to the clouds , not the optical properties of dust . we have also made a number of assumptions in the present study to obtain the optical properties of dust and the geometrical structures of the clouds in the tpa region . first , we used a conversion formula for the fuv continuum intensity of 60 cu for 1 rayleigh intensity of h@xmath18 to estimate the two - photon effects , which holds only for the case of a hot diffuse ionized gas whose temperature is @xmath08000 k @xcite . to see the effect of the choice of the conversion factor , we ran the same monte carlo simulation by but instead subtracting the two - photon effect with a conversion factor of 30 cu for 1 rayleigh of h@xmath18 in one case and even with no subtraction in another case . the results were a = 0.43 , g = 0.49 for the former case , and a = 0.44 , g = 0.52 for the latter case . all these values are within the 1-sigma range of the nominal case , a = @xmath1 and g = @xmath2 , implying that the choice of the conversion factor does not alter the conclusions made in the present study . next , we subtracted 300 cu as an isotropic background of the fuv emission , which was obtained for high galactic regions . when we subtracted 500 cu instead , only the thicknesses of the clouds changed while the distances to the respective front face remained the same . the thickness decreased by @xmath010 % for the auriga and the california clouds as well as per a , and @xmath030 % for the taurus cloud region , which is darkest of all and thus most affected . the reason for the decrease in thickness is that reduction in fuv intensity was accomplished by increasing the shielding effect in the case of these clouds . on the other hand , the thickness of per b increased by @xmath020 pc , reflecting the nature of backward scattering in the case of per b. nevertheless , these new values are all within the 1-sigma range of the results obtained for the nominal case . we have constructed a fuv continuum map of the tpa complex using fims and galex data . the morphological features seen in the complex are well understood in terms of dust scattering , as demonstrated by monte carlo simulations . the following are the main findings of the present study : 1 . the diffuse fuv emission seen in the complex originates mostly from scattering of stellar photons by dust grains . the fuv intensity of 1000 cu is observed at a very high extinction level , which we regard as diffuse background and attribute to scattered fuv photons located in the foreground to the thick clouds . 3 . molecular hydrogen fluorescent emission constitutes @xmath010% of the total fuv intensity throughout the region . we have derived the following scattering parameters for this region based on the monte carlo radiative transfer ( mcrt ) simulations : albedo ( a ) = @xmath1 and asymmetry factor ( g ) = @xmath2 . these values agree well with those obtained previously for the orion - eridanus superbubble region as well as theoretical estimations . we have estimated the distances to the four prominent clouds in this complex , which are in good agreement with those estimated observationally using other methods . the thickness of each cloud was also reasonably determined when it has a simple structure such as the taurus cloud . the present single slab model , however , was not able to reproduce multi - layered clouds , but gives rather thick clouds , instead . the geometrical structures of the clouds are less sensitive to the exact values of the albedo and the asymmetry factor . instead , the locations of the field stars relative to the clouds are the main factors that constrain the distance and thickness of the clouds . fims / spear is a joint project of kaist and kasi ( korea ) and uc berkeley ( usa ) , funded by the korea most and nasa grant nag5 - 5355 . this research was supported by basic science research program ( 2010 - 0023909 ) and national space laboratory program ( 2008 - 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we have constructed a far - ultraviolet ( fuv ) continuum map of the taurus - auriga - perseus complex , one of the largest local associations of dark clouds , by merging the two data sets of galex and fims , which made observations at similar wavelengths . the fuv intensity varies significantly across the whole region , but the diffuse fuv continuum is dominated by dust scattering of stellar photons . a diffuse fuv background of @xmath01000 cu is observed , part of which may be attributable to the scattered photons of foreground fuv light , located in front of the thick clouds . the fluorescent emission of molecular hydrogen constitutes @xmath010 % of the total fuv intensity throughout the region , generally proportional to the local continuum level . we have developed a monte carlo radiative transfer code and applied it to the present clouds complex to obtain the optical properties of dust grains and the geometrical structures of the clouds . the albedo and the phase function asymmetry factor were estimated to be @xmath1 , and @xmath2 , respectively , in accordance with theoretical estimations as well as recent observations . the distance and thickness of the four prominent clouds in this complex were estimated using a single slab model applied individually to each cloud . the results obtained were in good agreement with those from other observations in the case of the taurus cloud , as its geometrical structure is rather simple . for other clouds , which were observed to have multiple components , the results gave distances and thicknesses encompassing all components of each cloud . the distance and thickness estimations were not crucially sensitive to the exact values of the albedo and the phase function asymmetry factor , while the locations of the bright field stars relative to the clouds as initial photon sources seem to be the most important factor in the process of fitting .

random matrices are useful for a wide range of physical problems . in particular , by the means of feynman rules , random matrices can be interpreted in term of two - dimensional surfaces , which themselves are related to two - dimensional quantum gravity @xcite . 2d quantum gravity models can be coupled to matter fields , with a non - zero central charge @xmath1 . in term of matrices , this leads to consider multi - matrix models , which , studied near their critical points , in the scaling limit , allow to recover a continuous theory . some of the random matrices models have been solved exactly @xcite and the continuous limit one recovers in the neighborhood of their critical points has been related to @xmath2 quantum gravity models . although the behaviour of @xmath3 models is well understood , exact methods have not been able to solve @xmath4 models yet . there is a `` @xmath5 barrier '' , which prevents us from using @xmath4 matrix models as a tool to understand @xmath4 two - dimensional quantum gravity . whereas exact technics are unable to deal with @xmath4 models , an approximate method is likely to succeed better . in 1992 , brezin and zinn - justin @xcite introduced a new method to solve random matrices models : the large @xmath0 renormalization group ( rg ) , where the rescaling parameter @xmath0 is the size of the matrices . integrating over part of the matrices to reduce @xmath0 one obtains a rg flow in the space of actions . fixed points should correspond to critical points of matrix models in the large @xmath0 limit and the scaling dimensions of operators give the corresponding critical exponents . for instance , the scaling dimension @xmath6 of the most relevant operator is related to the `` string exponent '' @xmath7 by the relation @xmath8 while the large-@xmath0 renormalization group method was introduced to study directly @xmath4 models , its application to already solved ( by exact methods ) @xmath3 models is also useful . indeed , in @xcite it was argued by one of the authors that the understanding of the behaviour of the flows for @xmath3 models would throw light on what happens at @xmath4 : taking into account `` branching interactions '' in matrix models , it is expected that in addition to the gravity fixed point there is another fixed point ( corresponding to the branching " transition between 2d gravity and branched polymer behavior ) . the value of the critical exponents when @xmath9 ( known from the exact solutions and kpz scaling ) support the conjecture of @xcite that at @xmath5 these two fixed points would merge , and that there would be generically no gravity fixed point for @xmath4 models . in this paper we develop the large @xmath0 rg method in order to study precisely matrix models for 2d gravity + branched polymers . we shall describe first in sec . [ rgmm ] the renormalization group method as it was introduced by brezin and zinn - justin , and the improvements made by higuchi et al . @xcite , using the equation of motions . then we explain why these methods are not sufficient to study our class of models . we show in sec . [ om ] that the linear approximation of @xcite is not sufficient and we propose a method to go further , and we apply it to the one matrix model . the method still requires numerical analysis , which is performed in sec . [ na ] , where the resulting rg flow equations are analysed and the results of the method are compared to the previous results of @xcite and to the exact results . finally in sec . [ isingsection ] we shall consider the generalizations of our method , in particular to the ising plus branched polymer two - matrix model . we shall consider a matrix model for gravity plus branched polymers . the partition function is @xmath10 where @xmath11 \label{vandtn}\ ] ] in eq . ( [ vandtn ] ) @xmath12 is the gravity coupling constant and @xmath13 is the branched polymer coupling constant . this model can be solved exactly , thus it should be a good test of renormalization group methods . when @xmath13 is equal to zero , it is the pure gravity model that was considered by brezin and zinn - justin in @xcite when they first tested their renormalization group ( rg ) method . the general idea of rg is to start from the potential @xmath14 for a @xmath15 matrix @xmath16 with @xmath17 , and to integrate over the last row and the last column of the matrix , leading to an effective action @xmath18 for the @xmath19 remaining matrix @xmath20 . performing a linear rescaling of @xmath20 ( wave - function renormalization ) to keep the @xmath21 term unchanged one obtains the renormalized action @xmath22 and the rg flow equation in the space of action @xmath23 . in @xcite it was shown that at order @xmath24 the action ( [ vandtn ] ) ( with @xmath25 ) stayed closed ( up to a additive shift coresponding to a @xmath26 operator ) , i.e. no new operators were generated by the rg . the corresponding beta - function for @xmath12 was found to have a zero for @xmath27 , corresponding to the critical coupling where pure gravity scaling is recovered . however at that order the first numerical result of @xcite were only very qualitative , since the error on the critical coupling itself is of order 100% . this method was then developped further by several authors @xcite . shortly after , higuchi , itoi , nishigaki and sakai @xcite improved the rg method by using the equations of motion of the matrix model ( the so - called loop equations ) to eliminate some of the new operators which appear in the rg transformation at higher orders in @xmath12 . they first considered the pure gravity model with cubic action @xmath28 and showed that in the planar limit , the rg equation for the free energy @xmath29 can be written as @xmath30 where @xmath31 is a non - linear function of @xmath12 , the gravity coupling constant , and of @xmath32 . this is a non linear differential equation , at variance with the standard rg equations which should be linear and of the generic form given in @xcite @xmath33 ( where @xmath34 and @xmath35 are regular functions in @xmath12 ) . in order to obtain a standard renormalization group equation the authors of @xcite truncated @xmath36 to the first order in @xmath37 . for this problem this approximation is in fact quite good . the results they obtained for the cubic action ( [ iguchiv ] ) are : a @xmath38 error for the value of the critical coupling and a @xmath39 error for the eigenvalue corresponding to the string critical exponent @xmath40 . moreover , they have generalized this method to a one - matrix model with two couplings : @xmath41 and also to the ising two - matrix model , with in both cases results quite close to the exact results . in particular for the quartic action ( @xmath42 ) the relative errors on the coupling and the eigenvalue are @xmath43 and @xmath44 . nevertheless , the linear approximation method has only been applied up to now to gravity models without branching interactions . the case of gravity plus branched polymers , which are the models one has to consider if one wants to verify the scenario of @xcite , would be more difficult to treat . one would have to introduce @xmath45 , partial derivative of @xmath46 with respect to @xmath13 , and linearly develop the equations in @xmath45 and @xmath47 . moreover , the success of the linear approximation method of @xcite may be attributed to the fact that the value of the critical coupling for pure gravity , @xmath48 , is in fact quite small , since the corrections to the linear approximation turn out to be of order @xmath24 , and are therefore small . for branching interactions , we expect that this will not be the case . already for the pure branched polymer model ( @xmath49 , @xmath50 ) the critical point is @xmath51 , thus we can guess that the estimates for the general critical points would be far less precise than those for the pure gravity critical point . since in this paper we shall also use the equations of motion to write rg equations for matrix models with branching interactions , let us briefly recall the general idea . if one starts from a simple action @xmath52 such as ( [ vandtn ] ) ( which depends only on the two operators @xmath53 and @xmath54 ) one obtains a variation @xmath55 which is a function of all the @xmath56 for any @xmath57 . to take all these terms into account , one can try to write the action of the renormalization group on the most general partition function . this leads to expressions with an infinite number of terms which must be truncated in some way . however we know no general and natural truncation scheme , and the simplest truncations lead to a very slow convergence for the value of critical points , while the estimates for the critical exponents do not converge at all ! this can be shown explicitely on the simple example of the pure branched polymer model with action @xmath58 ( which is in fact a vector model ) as discussed in appendix a. nevertheless one must realize that all the operators are not independent . one is free ( in perturbation theory ) to include in the rg transformation a non - linear reparametrization of the field variable ( i.e. the matrix @xmath20 ) in the partition function , of the form @xmath59\phi^k \label{}\ ] ] this induces an additional variation in the effective action of the form @xmath60 this variation does not change the physical content of the rg equations , since @xmath61 where @xmath62 denotes the `` vacuum expectation value '' ( v.e.v . ) @xmath63 ; these are the quantum equations of motion of the model , which express the relations between the v.e.v . of the traces . this ( unphysical ) arbitrariness in the rg equations is known as the problem of the `` redundant operators''@xcite . the idea is that one should use the equation of motion to simplify the expression of the rg equations and to reduce the number of operators and of coupling constants involved in the rg flows . this is what has been done in @xcite to reduce the rg equation to the form of eq . ( [ linearapprox ] ) . in appendix [ vectorappendix ] we show how the method applies to the simple branched polymer model . we are now going to introduce our method , which aims at calculating the renormalization flows of gravity plus branched polymer models . we are now going to explain our method on the example of the one matrix model : @xmath64 ( more general cases as the ising plus branched polymer model will be considered in section [ isingsection ] and we shall discuss then the generalization of our method ) . as the use of the equations of motion does not allow us to put the renormalized action exactly in the same form as the action we would like to study , we are going to consider a slightly more general model : @xmath65 where @xmath66 is of the form : @xmath67 with @xmath68 and @xmath69 . we are going to show that when the renormalization group acts on this model , the renormalized action can be put in the same form , with a renormalized @xmath66 and a renormalized @xmath12 , and @xmath70 always equal to one . in the following we denote @xmath71 the derivative of @xmath66 @xmath72 we start from @xmath73 a @xmath74 hermitian matrix , @xmath20 is a @xmath75 hermitian matrix , @xmath76 is a vertical vector , and @xmath77 a real number . @xmath78 \label{znplus1}\ ] ] can be rewritten , to the first order in @xmath79 , separating the variables @xmath20 , @xmath76 , @xmath80 and @xmath77 , as : @xmath81 \qquad \makebox{with } \qquad \nonumber\\ \delta v= g t_4 + 2 \psi - t_2 u + g { \alpha^4 \over 4 } + u { \alpha^2 \over 2 } + v^{\ast}(u + g \alpha^2 + g \alpha \phi + g \phi^2 ) v + g { ( v^{\ast } v)^2 \over 2 } \end{aligned}\ ] ] let us introduce first the auxiliary field @xmath82 and rewrite the part of the integral containing the variables @xmath76 and @xmath83 as : @xmath84 \label{i}\ ] ] by integrating over @xmath76 and @xmath80 , we obtain , @xmath85 \right ) ] \label{ibis}\ ] ] we then can use a saddle point method and minimize this expression with respect to @xmath82 . we find an implicit equation for the value @xmath86 of @xmath82 : @xmath87 \label{sigma}\ ] ] we now have to integrate over @xmath77 . this can also be done by using the saddle point method . the @xmath88 terms disappear thanks to eq . ( [ sigma ] ) , and the saddle point @xmath89 verifies : @xmath90 \ = \ 0 \label{alpha}\ ] ] then we can rewrite , where @xmath91 means that the average is done over @xmath20 , @xmath92\rangle \label{moy}\ ] ] with @xmath93\ ] ] and use the factorization property which is valid in the large @xmath0 limit @xmath94 to express the variation of @xmath52 : @xmath95 \ -\ g { { \langle\sigma_s\rangle}^2 \over 2}\rangle \label{reneq}\end{aligned}\ ] ] from the saddle point equation on @xmath89 , and the parity of the action which leads to @xmath96 we immediately see that @xmath97 is solution of the averaged saddle point equation . thus : @xmath98 \ -\ g { \sigma_s^2 \over 2}\rangle \label{reneqbis}\ ] ] @xmath99 as discussed in sec . 2.2 , one should use the equation of motion and the freedom to add redundant operators to the rg transformation to simplify the flows . we now express the equations of motion . the change in variables : @xmath100 , @xmath101 leads to the equation of motion : @xmath102 where the @xmath103 are defined as in eq . ( [ vandtn ] ) @xmath104 we introduce the function @xmath105 which is the generating function of the @xmath106 , @xmath107 ( this function @xmath108 is almost the resolvent of the model ) . using eq . ( [ eqmvt1 ] ) @xmath105 can be rewritten : @xmath109 noting that @xmath110\rangle = { 1 \over ( u+g \sigma_s ) } f\left({-g \over u+g \sigma_s}\right ) \label{vevsigma}\ ] ] we immediately find that the solution of the above equation is @xmath111 integrating @xmath108 , we also have @xmath112 \rangle \ = \ \langle\ln(u + g \sigma_s)\ + \ \int_0^{-{g \over u + g \sigma_s } } { f(z ) - 1 \over z}\ dz\rangle \label{tralint}\ ] ] finally , denoting @xmath113 we obtain @xmath114 this expression , if expanded in powers of @xmath53 , contains a linear term in @xmath53 : @xmath115 ( we recall that @xmath116 ) . we would like to keep @xmath70 equal to one . this is possible if one substracts the coefficient of @xmath53 , multiplicated by the expression appearing in the first equation of motion : @xmath117 . then we finally obtain a renormalized action exactly in the same form as the original action . beyond this point , we shall suppress , for simplicity reasons , the @xmath91 in all our expressions . indeed , eq . ( [ equafin ] ) means that , if we replace @xmath118 in @xmath119 by the right - hand part of our equation , the result will be the same . we now have obtained a renormalization group equation containing only linear terms in @xmath54 and powers of @xmath53 . so , starting from the action @xmath120 ( with @xmath121 ) , we can write ( at least in principle ) the equations for @xmath122 and @xmath123 : we just have to expand the above expression in powers of @xmath53 . this method , however , leads to very complicated expressions with many non - trivial integrals depending on the parameter @xmath12 . to be able to treat our expression easily , it is better to expand it first in powers of @xmath124 ( @xmath124 is of order @xmath12 , as @xmath125 is of order one ) . the last integral in eq . ( [ equafin ] ) is then also expanded in powers of @xmath126 , and it remains to expand @xmath127 in powers of @xmath53 , which is easy . one should notice that the expansion in @xmath124 is _ not _ trivial : one can not simply expand the integrand and the bounds of integration , as this would lead to divergent expressions . in fact , the integral can be expressed as an expansion in @xmath128 and @xmath129 . what one has to do to expand the last integral properly is to treat separately the integral between @xmath130 and @xmath131 , and between @xmath131 and @xmath132 . let us briefly describe these operations : the last integral can be expressed as a sum of two integrals : @xmath133 the first one given by @xmath134 @xmath135 and the second one given by @xmath136 @xmath137 before expanding the integrand in powers of @xmath124 , up to the order @xmath138 , we have to notice that , when @xmath124 tends to zero , the bounds of integration in @xmath139 and @xmath140 tend respectively to @xmath141 and @xmath130 . let us take @xmath140 as an example . for @xmath142 large , the terms in the expansion of the integrand are of the form @xmath143 . this implies that the integration leads to a term of order @xmath128 . thus , the terms in the expression are really of increasing powers of @xmath124 , which justifies our expansion , keeping in mind that expanding the integrand up to order @xmath144 amounts to expand the integral up to order @xmath128 . moreover , when expanding the integrand @xmath145 , we obtain expressions of the form : @xmath146 where @xmath147 is a polynomial . the primitives of such terms lead to logarithmic expressions , and we finally obtain a @xmath128 and @xmath129 expansion , that is to say , in term of the coupling @xmath12 , we have an expansion in @xmath148 _ and _ @xmath149 . we would not have realized the existence of these logarithmic terms if we had used a finite number of equations of motion ; they appear here because , by the use of the @xmath105 function , we have used an infinite number of equations of motion , and expanded _ properly _ only after . this phenomenon should not come as a surprise , it was already observed in @xcite . once the expansion in @xmath12 has been done , we expand @xmath71 in powers of @xmath53 and , finally , we obtain a quite simple expression with two orders of expansion : @xmath150 in @xmath12 and @xmath151 in @xmath53 . after development in @xmath124 at order @xmath152 in @xmath124 , for example , the expression of the integral @xmath153 is : @xmath154 inserting the expansion for @xmath155 in eq . ( [ equafin ] ) we obtain the rg flow equations for the couplings @xmath12 and @xmath156 ( @xmath157 ) . this flow equations can easily be integrated numerically . in figure [ flot1mfig ] we show the results obtained at orders @xmath158 and @xmath159 . the axes represented correspond to the two couplings that interest us : the gravity coupling @xmath12 and the branched polymer coupling @xmath160 . of course , there are @xmath161 other couplings which are not represented there . the rg flows represented here correspond to initial condition @xmath162 , for all the @xmath163 , i.e. we study the evolution of the model with inital action@xmath164 as in eq . ( [ vandtn ] ) . it is easy to find the critical line in the ( @xmath165 ) plane : it separates the domain where one flows towards the gaussian fixed point ( @xmath166 ) from the domain where @xmath12 and @xmath13 diverge . we have chosen here to show only the flows for @xmath167 near the critical line , on both sides of it . since under the rg flows the @xmath156 s becomes non - zero , and since we project the flows on the @xmath167 plane the flows may seem to cross , of course this is unphysical and a projection of what happens in higher dimension . first one recovers the correct phase diagram and critical line ( with an average @xmath44 relative error at this order ) . the critical line is separated into two parts by a multicritical point @xmath168 . starting from the rightmost part of the critical line the flows are driven towards a fixed point * b * which lies on the @xmath49 plane . starting from the leftmost part of the critical line the flows are driven towards a fixed point * a * ( with @xmath12 , @xmath13 , @xmath169 , @xmath170 @xmath171 ) . we have represented on fig [ flot1mfig ] the pull - back " * a * of * a * , i.e. the point on the critical line which flows in the fastest way towards * a * ( this is equivalent to say that the leading subdominant corrections to scaling vanish at * a * ) . both fixed points * b * and * a * have one unstable direction . thus * b * corresponds to branched polymers and * a * is the fixed point corresponding to pure gravity . starting from the point * c * one flows towards a fixed point * c * ( also with with @xmath12 , @xmath13 , @xmath169 , @xmath170 @xmath171 ) with two unstable directions , which should therefore correspond to the branching transition " between pure gravity and branched polymer scaling behaviour . these features of the rg flows are in full agreement with the picture which was proposed in @xcite . this agreement is confirmed by studying more precisely the numerical results . we note immediately that we recover the good critical line ( with @xmath172 error at this order ) , the bicritical point c ( which is a totally repulsive fixed point of the renormalization group ) . we also note the existence of the fixed point a of reference * ? ? ? * : for this particular model , the shape of the flows fits the conjecture . for @xmath173 , the flows are attracted by the gaussian fixed point @xmath174 , and beyond the critical line , they diverge . all these qualitative facts show that , for this particular model , we have proven reference @xcite s conjecture is right . of course , to verify the conjecture further , we have to study the evolution of the two fixed points ( a and c ) when the central charge tends to @xmath175 . at this order of approximation ( @xmath158 , @xmath176 ) , we obtain for the critical point @xmath177 located on the axis @xmath25 a precision which is smaller than the one obtained by higuchi et al .. our computations , however , can be performed for several values of @xmath150 and @xmath151 , and the results can be extrapolated to obtain a very good precision . for example , for @xmath177 , critical value of @xmath12 for @xmath25 , we have : + [ cols="^,^,^,^,^,^",options="header " , ] [ extrapolgc ] we then extrapolate these results by a polynom of degree three in @xmath178 . fig [ extrapolgc ] shows that this extrapolation behaves almost as a straight line : the value of @xmath177 converges as @xmath179 . the extrapolated value for @xmath180 is @xmath181 , while the exact value of @xmath177 is @xmath182 . we have thus obtained a @xmath183 relative error , to be compared with the @xmath184 relative error of @xcite . + by the same method , we can obtain the whole critical line in the plane @xmath167 , which was not obtained in ref.@xcite . on the pure branched polymer line ( @xmath49 ) , we obtain for the branched polymer critical point @xmath185 instead of the exact value @xmath51 i.e. a @xmath186 relative error . for the position of the multicritical point * c * we obtain respectively for @xmath187 and @xmath177 a @xmath188 and @xmath189 relative errors . linearizing the rg flow equations in the vicinity of the fixed points , we obtain also good results for the critical exponents . for instance , studying the multicritical fixed point * c * , we obtain for the largest eigenvalue @xmath190 , i.e. a @xmath191 relative error when compared to the exact result @xmath192 ! it is not surprising that the critical exponents converge less rapidly than the positions of the critical points . better but more tedious calculations could be done , for we can go in principle to any orders @xmath150 and @xmath151 , and the limit @xmath193 and @xmath194 should give the exact result , but this is not our purpose here . our method can be generalized to a two - matrix model which describes the ising plus branched polymer model : @xmath195 @xmath196 and @xmath197 are two @xmath75 hermitian matrices , @xmath12 is the gravity coupling constant , @xmath13 the branched polymer coupling constant and @xmath198 corresponds to the temperature of the ising model . for simplicity reasons , we have not introduced a external field , which would have broken the symmetry in @xmath196 and @xmath197 . as previously , we are led to study a more general model : if we note @xmath199 , and @xmath200 , the model we are going to study can be expressed as : @xmath201 where @xmath66 is , as in the case of one matrix models , a function of the quadratic terms in @xmath20 . if @xmath202 and @xmath203 , we note @xmath204 and @xmath205 . here @xmath198 is a series in @xmath206 and @xmath53 , with constant coefficient @xmath207 . the renormalization group equation reads : @xmath208 where @xmath209 . to simplify this expression , we have to use the equations of motion . similarly as what was done in @xcite , we can express , by introducing @xmath210 changes in variables , @xmath211^{-1}\rangle $ ] as the solution of a quartic polynomial : @xmath212 we have then to integrate : @xmath213 without going through all the details of the procedure , let us note that , to expand @xmath214 , we have to be cautious . indeed , as for the one - matrix model , the integral will have to be cut into two parts , as @xmath215 can be large ( @xmath216 ) or small ( @xmath217 ) . this will lead to two different changes of variables in the integral and , once more , to logarithmic terms in @xmath12 _ and _ in @xmath218 . let us also stress that one has to expand the integral both in powers of @xmath12 and of @xmath207 . the easiest way to do so is to expand simultaneously @xmath155 in powers of @xmath12 and @xmath207 , with @xmath219 . then , we can compute approximate flows for this model . figure 3 shows approximate rg flows , at @xmath220 , at order @xmath221 in @xmath12 ( i.e. three in @xmath207 ) and @xmath222 in @xmath53 , where @xmath12 is in abcissa and @xmath207 in ordinate . here , we recover the pure gravity ( @xmath223 , @xmath220 ) critical point at @xmath224 , i.e. with @xmath225 error ( the exact critical point is @xmath226 ) . by extrapolating the two first results at @xmath227 and @xmath228 , however , we obtain an extrapolated @xmath229 that is to say only @xmath230 error ! moreover , we can see on figure @xmath231 that we recover the ising bicritical point * c * at @xmath232 ( the exact value is @xmath233 ) , and the shape of the critical line . all these results show we can compute good approximations of the flows , not only for one - matrix models , but also for multimatrix models . we can theoretically compute that way the flows of an open chain of @xmath234 matrices with nearest neighbour coupling , for @xmath235 , @xmath236(ising ) , @xmath237 this series of models is all the more interesting as we know that when @xmath238 , the central charge of the model @xmath9 . these models could thus allow us to verify the evolution of the flows with @xmath1 and their shape when @xmath1 is equal to one . this leads us , however , to a practical problem : though the initial model ( the open chain ) has the same coupling constants for all the matrices of the chain , the action of the renormalization group leads to almost as many coupling constants as matrices . this comes from the fact the roles of the matrices of the open chain are not symmetric . thus , we do not know if it is practically manageable to study , for example , a @xmath239 matrices open chain , or if it leads to too long computations . a solution is to study a symmetric problem : the closed chain , for example ( we do not know its exact solution ) , or the @xmath234-matrices potts model , where all matrices are coupled . the study of the latter model could be indeed very interesting : when @xmath240 , then @xmath241 and we would thus enter the @xmath242 domain . in this paper we have developped the large @xmath0 renormalisation group method to study matrix models containing interaction terms corresponding to branched polymer interactions . we have shown the analytical basis and the successes of our method . it can deal with models containing branched polymers , it gives us the shape of the flows , and also good approximations of the position of the critical points and critical exponents of the models . we have applied our method to the case of the pure gravity plus branched polymer one - matrix model , and to the case of the ising two - matrix model . our method is an approximation method : the exact expressions must be truncated to a certain order to be numerically manageable ( the ideal case of the infinite chain being the exact solution ) . however , the extrapolation of the first orders gives good results without taking high computation times . but , when studying models with a growing complexity ( @xmath234-matrices open chains ) , we may reach for big @xmath234 the practical limits of the method , and thus it would be a good thing to find more technical simplifications . we also plan to study @xmath234-matrices potts models , which are very symmetric models ( so technically simpler ) but which would allow to cross the @xmath4 barrier for @xmath243 , and are thus theoretically more complex models . * * we thank j. zinn - justin for his interest and his careful reading of the manuscript . this work has partial support from european contract tmr erbfmrxct960012 . the large @xmath0 rg method has already been applied to the ( very ) simple case of the vector model in @xcite . this model corresponds to the limit @xmath49 of the matrix model with action ( [ vt4 ] ) , since then the u(@xmath0 ) matrix model becomes a o(@xmath244 ) vector model ) . the authors of @xcite found that if one does not use the equations of motion ( i.e. if no non - linear reparametrization of the fields is performed ) the rg flow equation seems to lead to strange results , with the apparent existence of a one - parameter family of fixed points . however they showed that when using the equations of motion one can reduce the rg flows to much simpler flows in a finite dimensional space of couplings . moreover in that case the flows can be easily calculated exactly and it is found that one recovers the exact critical points and critical exponents . the purpose of this appendix is just to present the results of a simple exercise : how do the results of the rg method change if one does not use the equations of motion , or if one uses the equations of motion in a approximate way ? we start from the action ( [ vt4 ] ) with @xmath49 for a @xmath19 hermitean matrix @xmath245 since we can rewrite @xmath246 where @xmath247 is the @xmath244-dimensional real vector whose components are the real and imaginary part of the matrix elements @xmath248 of @xmath20 , the model reduces to the vector model studied in @xcite . the rg flow equation for the potential @xmath66 can be written exactly , either by integrating explicitely over some components of @xmath247 , or by computing exactly the effective potential at large @xmath0 and studying its variation with @xmath0 . both methods yield , to the first order in @xmath79 , @xmath249 \label{vpsiflot}\ ] ] this simple rg equation becomes even simpler if we consider instead of @xmath66 its derivative @xmath250 and if we invert the function @xmath251 and consider instead the function @xmath252 defined as @xmath253 ( this is at least possible for @xmath53 small ) . the rg flow equation becomes linear for the function @xmath150 and is trivial to solve . if we start at @xmath255 from the initial potential @xmath256 , i.e. from the initial function @xmath257 with @xmath258 we get at @xmath259 @xmath260 and inverting again @xmath253 we obtain the renormalized derivative of the potential @xmath261 . performing a linear rescaling @xmath262 to keep the coefficient @xmath70 of the @xmath53 term fixed amounts to changing @xmath263 in eq . ( [ vxflot ] ) , with @xmath264 a linear rescaling factor fine tuned such that the constraint @xmath265 ( i.e. @xmath266 ) is kept for @xmath267 . first let us study the exact flows if one starts from the quartic potential @xmath268 , i.e. from the initial function @xmath269 . for @xmath270 small , the function @xmath71 remains analytic around the origin , but it develops a square root singularity at a finite @xmath271 , which starts for @xmath272 at the zero @xmath273 of @xmath274 , i.e. at the critical point of @xmath256 . of course the flow can be studied analytically but they are better depicted graphically ( see fig . [ coloredflows ] ) * if @xmath275 , the singularity @xmath276 goes to infinity as @xmath277 and the function @xmath278 tends towards the constant function @xmath279 , which corresponds to the gaussian potential @xmath280 ( gaussian fixed point ) ( fig . [ coloredflows].a ) . * at the critical value @xmath281 the singularity @xmath282 tends toward a _ finite value _ @xmath283 , so that the function @xmath71 tends toward a non - analytic fixed point ( fig . [ coloredflows].b ) @xmath284 * for @xmath285 , the singularity @xmath276 reaches the origin @xmath286 in a _ finite _ rg time @xmath287 , the potential @xmath66 becomes singular at the origin and the rg flow thus diverges in a finite time and reaches no fixed point ( fig . [ coloredflows].c ) . this analysis can be extended to general initial potentials . we thus recover ( without using the equations of motion ) a sensible picture of the rg flows : the attraction domain of the gaussian fixed point , ( @xmath288 ) is separated from the domain where rg flows diverge by a critical ( unstable ) manifold where one is driven towards the non - trivial fixed point @xmath289 ( which corresponds to branched polymers ) . the only subtle point is that the non trivial fixed point is non - analytic and can not be distinguished by a local analysis around the origin ( @xmath290 ) from the gaussian fixed point . this explains the apparent paradoxes of @xcite . let us mention that if instead of using the potential @xmath291 one uses the inverse function @xmath292 , the rg flot is very simple : @xmath293 . the structure of the flow is described by the expansion of the function @xmath294 around the largest zero @xmath289 such that @xmath295 , and the ( only ) relevant scaling field @xmath296 corresponds to the first derivative @xmath297 of @xmath151 ( since the critical manifold corresponds to @xmath298 ) . this @xmath296 scales with @xmath299 as @xmath300 , therefore has scaling dimension @xmath236 . however the mapping @xmath301 is highly non - linear , and becomes singular along the critical manifold @xmath298 . therefore this does not contradict the fact that the real scaling dimension of the relevant operator is @xmath302 . one can try to truncate the rg flow equation eq . ( [ vpsiflot ] ) in the most naive way : we keep only the couplings @xmath156 with dimension @xmath303 in @xmath66 , then expand the @xmath304 $ ] in powers of @xmath53 and _ truncate this expansion at order k in @xmath305_. we thus obtain for fixed @xmath306 approximate rg flow equations for the @xmath306 couplings @xmath156 , which are of the standard form @xmath307 $ ] , with the @xmath308 functions polynomials of order @xmath306 in the @xmath156 s . at a given truncation order @xmath306 the explicit form of these functions is not especially illuminating and will not be given here . using computing software the approximate fixed points can be found exactly and the structure of the rg flows studied . we find indeed a non - trivial fixed point @xmath309 with one unstable direction , which should correspond to the branched polymer fixed point . the derivative @xmath310 is depicted on fig . [ coloredflows ] as a function of the order of truncation @xmath306 ( @xmath71 is a polynomial of degree @xmath311 . one sees that as @xmath306 increase the approximated fixed points converge towards the exact ( but singular ) fixed point . however a more precise analysis ( that we do not reproduce here ) shows that the convergence is very slow , typically as @xmath312 . the finite @xmath306 estimates for the critical coupling @xmath187 ( if all higher order couplings @xmath156 , @xmath163 are set to zero ) converge towards the exact value @xmath313 at the same rate . this very slow convergence is insufficient to obtain good estimates for the critical exponents . it turns out that within this approximation scheme , the scaling dimension @xmath314 of the scaling fields at the approximate fixed point are independent of the trucation order @xmath306 ! indeed they are found to be integers @xmath315 . the @xmath316 dimension should correspond to the dimension @xmath6 of the relevant perturbation , which is known to be @xmath302 . thus , although the estimates for the critical points converge towards the correct result as @xmath317 , the estimates for the scaling exponents do not ! a procedure to accelerate the convergence is required . this is precisely what the equation of motions are doing .

we develop a method to obtain the large @xmath0 renormalization group flows for matrix models of 2 dimensional gravity plus branched polymers . this method gives precise results for the critical points and exponents for one matrix models . we show that it can be generalized to two matrices models and we recover the ising critical points . + saclay t98/123 + revised version + + * renormalization group for matrix models + with branching interactions * + gabrielle * * bonnet * * and franois * * david * * + cea / saclay , service de physique thorique + f-91191 gif - sur - yvette cedex , france

in the milky way , there is a clear separation between known open clusters ( which have diffuse structure , generally have low masses and ages , and belong to the disk ) and globular clusters ( which have a more concentrated structure , higher masses and ages , and in many cases belong to the halo ) . when the only well - studied globular cluster system was that of the milky way , it was generally thought that this separation was because globular clusters were fundamentally different from other star clusters , perhaps because of conditions in the early universe @xcite . however , it is possible to produce this apparent bimodality from clusters formed in a single process , with the same cluster initial mass function . in this picture , cluster disruption mechanisms , which are more effective at destroying low - mass clusters in particular because of two - body relaxation @xcite , would remove almost all of the low - mass older clusters . if all clusters were born with similar cluster mass functions , then we would expect to see the occasional high - mass young cluster . in fact , we do see these in other galaxies . the `` populous blue clusters '' of the lmc @xcite have been suggested as examples of young objects which will evolve into globular clusters . m33 also has a few likely massive young clusters @xcite , and such clusters have been found in a number of normal isolated spirals @xcite . it is possible that the seeming absence of such objects in the milky way is merely an observational selection effect ; recently , there have been discoveries of heavily reddened open clusters such as westerlund 1 , which likely has mass in excess of @xmath3m@xmath4 @xcite . what of m31 s clusters ? while its clusters have been studied since the 1960s ( eg * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and it was noted even then that some of these clusters had colors indicating young populations , their nature is still not entirely clear . @xcite called them open clusters , while @xcite adopted the simple convention of calling any cluster projected on m31 s disk a `` disk cluster '' until proved otherwise by kinematics . this avoids the question of whether young clusters are fundamentally different from globular clusters in structure , formation , etc . our detailed study of m31 young clusters , incorporating kinematics , should cast some more light on these questions . it is only recently that detailed constraints on the mass , kinematics , age and structure of cluster populations in m31 have been obtained , particularly for clusters projected on the inner disk and bulge . multi - fiber spectroscopy and hst imaging have played an important role here . a number of m31 globular cluster catalogs have been created over the years , giving a very heterogeneous result , with significant contamination by both background galaxies , foreground objects and even non - clusters in m31 itself . while the work of @xcite , @xcite , @xcite , @xcite , and @xcite has gone a long way towards cleaning up the catalogs and winnowing out the non - clusters , still more work is needed for both young and old clusters . here we present a new catalog of m31 star clusters which were originally classified as globular clusters , all with updated high - quality coordinates . we have observed a large number of these clusters with the mmt and the hectospec multi - fiber system . in this paper we study more than 100 young m31 clusters in detail . in subsequent papers we will address the kinematics , ages and metallicities of the older clusters . the m31 young clusters have been studied both by authors aiming to study its open clusters ( eg * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and also by others who have aimed to study its globular clusters . for example , @xcite pointed out the existence of young clusters in m31 , estimated masses between @xmath5 and @xmath6 , and drew attention to their similarity to the populous blue clusters in the lmc . @xcite noted the existence of 8 clusters with strong balmer lines in their spectra , which they tentatively classified as young globular clusters . @xcite ( confirmed with a later sample by * ? ? ? * ) added more clusters , and commented that hst imaging ( then unavailable ) was needed to distinguish between structure typical of open and globular clusters . @xcite added more new young clusters , bringing the total to 19 , and @xcite increased the sample to 67 . in general , authors have associated these young clusters with m31 s disk , although @xcite invoke an accretion of an lmc - sized galaxy by m31 . observations are particularly challenging for clusters projected on m31 s disk : many of the early velocities had large errors , and there were issues with background subtraction . here we discuss high - quality spectroscopic measurements of kinematics and age for the young clusters , supplemented with hst imaging to delineate the structural , spatial and kinematical properties of these young clusters . we find that while they are almost all kinematically associated with m31 s young disk , and their age distribution will allow us to test suggestions that m32 has had a recent passage through m31 s disk @xcite . the young clusters have structure and masses which range all the way from the low mass milky way open clusters to higher mass , more concentrated globular clusters , although they are dominated by the lower concentration clusters . we will also discuss the likelihood that these clusters will survive . the starting point for our cluster catalog was the revised bologna catalog ( rbc ) @xcite , itself a compilation of many previous catalogs . coordinates from this catalog were used to inspect images from the local group survey ( lgs ) images @xcite , which cover a region out to 18 kpc radius on the major axis and 2.8 kpc away from the major axis , hst archival wfpc2 and acs images , and the dss for the outermost clusters . the lgs images have median seeing @xmath7 1 . we also added in some new clusters , found visually by ourselves on the lgs images and on hst images ( discussed below ) . even a casual inspection of the lgs images , particularly the i band images , reveals the presence of a vast number of uncataloged faint disk clusters , presumably similar to the galactic open clusters ( * ? ? ? * estimate 10000 such clusters from hst images ) . we have elected not to take on the enormous task of cataloging and measuring those clusters in this paper ; rather we choose to deal with the more massive clusters for which some information , however fragmentary , already exists . at a later stage in the project , the catalogs of @xcite , extracted from kpno 0.9 m images , and @xcite , from hst images became available , from which we collected objects not already in our own catalog and subjected them to the same editing as we now describe . the archival images were thus used to answer the following questions from the catalog we created from the rbc and our own additions . first , did the catalog coordinates correspond to a unique object ? in cases where the identification of the cluster on the local group survey was ambiguous or unclear , we consulted the original papers and finding charts where these were available @xcite . in some cases , there is no clear object that can be associated with the published coordinates , or the nearest object in fact already had a different i d . the large number of hectospec fibers meant that we were able to verify classifications of many low - probability candidates . the cataloging and observation parts of this program occurred in a feedback fashion , allowing some target names and/or coordinates to be changed for the succeeding observations . as a result , we had some objects whose identifications were incorrect ; to these we add an `` x '' to their original name in our tables below . second , were the existing coordinates accurate ? in general , the answer was no . our final catalog contains 1200 objects from the rbc , without considering the newer candidates of @xcite and @xcite . 830 of those required coordinate corrections larger than 0.5to place them on the fk5 system used in the dss and the lgs images . the median error in coordinates is 0.8 , with the largest error being of order 10 ; at which point the identification of the actual object becomes uncertain . similarly , there are 379 objects in the @xcite catalog found within 2of an lgs object . for these , the median error is 0.8offset , where the largest error is 1.9 . many of the discrepant velocities between us and the rbc or @xcite tabulations reported here are likely due to inaccurate coordinates used in previous spectroscopic work . the coordinates newly derived from the lgs images are accurate to 0.2 rms . third , were the targets really clusters ? the lgs v and i band images , and wfpc2 or acs images taken with non - uv filters were used to confirm the cluster nature of the objects , by visual inspection as well as the automated image classifier contained in the sextractor code . a number of cluster candidates were stellar on the lgs images ; all of these were later confirmed as stars in our spectra , from either the spectral characteristics or the velocities , in regions of m31 where there is no confusion between the local m31 velocity field and the velocity distribution of foreground galactic stars . we found that about 90 of the @xcite candidates listed as new , probable and possible ( indicating that they appeared non - stellar in their kpno 0.9 m images ) appeared stellar on the lgs or hst images . objects for which we have no new classification data are kept in the catalog , but noted as still in question in our table . the large majority of the misclassified objects are stars ( foreground galactic or m31 members ) ; more than 130 objects considered to be clusters as recently as 2007 by @xcite are in fact stars . quite a number of cataloged objects are background galaxies , and a few are either unidentifiable , or are accidental clumpings of galactic or m31 member stars . @xcite have recently stated that four clusters that were classified as disk clusters by @xcite are `` asterisms '' . they note that in their adaptive optics ( ao ) images there was no cluster visible generally there were only a few bright stars . however , for young clusters , red supergiants would dominate the light at infrared wavelengths and the hotter mainsequence stars would appear much fainter . we would need to use multi - wavelength data to classify these objects correctly . we show below that the optical spectra of those four clusters are indeed consistent with clusters of massive , main - sequence stars , and although the magnitudes , and hence masses , of these few objects were certainly overestimated , the objects will still be considered as clusters in our catalog , at least until hst images show otherwise . figure [ cohen ] shows this complication for one cluster , by comparing the high resolution , but long wavelength ao image with the lgs i band image . a star cluster is clearly visible in the i band , and even more prominent at bluer wavelengths - the object is indeed a young star cluster , though certainly not a globular and not very massive . our own hst images do reveal two cataloged clusters as asterisms : these are comprised of a small number of ob stars and late supergiants , resulting in a distinctive integrated spectrum with strong balmer and he i lines in the blue , and tio bands in the red . even if these two are real clusters , the derived masses are small enough to exclude them from a list of massive clusters . some cataloged objects have no real object even within a generous radius . in a few cases , a nearby background galaxy had also led to confusion in previous papers ( though not in the most recent version of the rbc ) , whereby an actual cluster was labeled as background . thus , while for the most part we have removed objects from the list of clusters , we have also restored a few objects to the cluster list . table [ main ] lists all objects believed to be clusters . for object names , we use the naming convention of @xcite where possible , where the name consists of a prefix with the rbc number followed by the number of the object from the next most significant catalog . objects for which we have no new information other than improved coordinates , and which have not been convincingly shown to be clusters by previous workers , are italicized . objects in the rbc which we did not observe and for which our coordinates are within 0.5of the rbc coordinates are not listed here , nor are the @xcite or @xcite cluster candidates that we did not observe . some objects that we did not observe could of course still be background even though they have non - stellar profiles , but these , few in number , are still listed here . a rough classification based on the spectra is included in this table , for objects with good quality spectra . `` young '' clusters are those with ages less than 1 gyr , `` interm '' refers to those with ages between 1 and 2 gyr , and `` old '' refers to clusters older than that . a subsequent paper will provide a detailed analysis and evidence that few if any of these `` old '' clusters have ages less than 10 gyr . `` hii '' indicates the spectrum is emission - line dominated . `` na '' appears for objects known to be clusters from an hst image , but for which we have no spectrum , or cases where the spectrum is too poor to determine the age , even in a coarse manner . the v magnitudes come from this paper , using the aperture size listed ( see [ s : phot ] ) or otherwise as indicated . column c describes what information was used to classify the target as a cluster . the possibilities are `` s '' , where our spectrum clearly indicates a star cluster , `` l '' , where the lgs image is non - stellar , and/or `` h '' , where an hst image indicates a star cluster . table [ young ] gives a list of those clusters that have ages less than 2 gyr . ( sections [ s : acs ] , [ s : ages ] and [ s : masses ] of this paper will discuss the measurement of ages and masses for these clusters . ) table [ stars ] lists objects from previous cluster catalogs that are in fact stars . many of these had also been classified as stars by previous workers . asterisms are also listed here . table [ maybestars ] gives a list of possible stars . in these cases , the local group survey imaging indicates a stellar profile , but either we have no spectrum , or the spectrum is ambiguous . note that some of the stars in tables [ stars ] and [ maybestars ] are certainly members of m31 . objects thought to be clusters in the @xcite catalog but which have stellar profiles in the lgs images are listed in [ maybestars ] , with coordinates derived from the lgs . table [ gals ] lists background galaxies . table [ nobodyhome ] lists cataloged objects where there was no obvious object within a reasonable distance of the previously published coordinates , which are listed here again . as a commentary on the difficulty experienced by all of those who have endeavored to collect true m31 star clusters ( including us ) , here is a brief summary of the contents of the most recent version of the rbc , excluding the additions of @xcite and @xcite , but including the lists compiled by other astronomers starting with edwin hubble . the rbc , restricted as just mentioned , contains 1170 entries . we here , and others ( particularly * ? ? ? * ) , have provided classifications for 991 of these . only 620 entries are actually star clusters , while 20 more could be considered clusters though the large amount of ionized gas present indicates the clusters may still be forming . 270 entries are stars , mostly foreground , and 224 entries are background galaxies or agn . at least 2 objects are chance arrangements of luminous m31 stars , together which appear as clusters from the ground . in the @xcite catalogs , there 113 , 258 and 234 `` new '' , `` probable '' and `` possible '' clusters , respectively . the lgs survey contains images of 94 , 152 and 105 members of those catalogs , respectively . of those subsets , 79 , 106 , and 129 , respectively have non - stellar profiles in the lgs , some of which were observed with hectospec . the hectospec multi - fiber positioner and spectrograph is ideally suited for this project , in that its usable field is 1 degree in diameter , and the instrument itself is mounted on the 6.5 m mmt telescope . we obtained data in observing runs in the years 2004 to 2007 , and now have high - quality spectral observations of over 400 confirmed clusters in m31 , and lower quality spectra of 50 more . we used the 270 gpm grating ( except for a small number of objects taken with a 600 gpm grating ) , which gave spectral coverage from 36509200 and a resolution of @xmath75 . the normal operating procedure with hectospec , and other multi - fiber spectrographs , is to assign a number of fibers to blank sky areas in the focal plane , and then combine those in some fashion to allow sky subtraction of the target spectra . for instance , the 4 - 5 fibers nearest on the sky to the target may be combined . these methods are satisfactory for our outer m31 fields , but not for the central areas where the local background is high relative to the cluster targets . for those fields , we alternated exposures on - target and off - target to allow local background subtraction to be performed . many of the discrepancies between our bulge cluster velocities and those of previous workers might be explained by their lack of proper background subtraction , and/or inaccurate target coordinates . we obtained exposures for 25 fields , with total exposure times varying between 1800s and 4800s . the signal / noise ratios for the 500 objects we classified as clusters have a median of 60 at 5200 and 30 at 4000 , with more than 100 clusters having a ratio at 5200 better than 100 . figure [ hectospec ] shows the locations and types of objects observed in all of these fields . the multifiber spectra were reduced in a uniform manner . for each field , or configuration , the separate exposures were debiased and flat fielded , and then compared before extraction to allow identification and elimination of cosmic rays through interpolation . spectra were then extracted , combined and wavelength calibrated . each fiber has a distinct wavelength dependence in throughput , which can be estimated using exposures of a continuum source or the twilight sky . the object spectra were thus corrected for this dependence next , followed by a correction to put all the spectra on the same exposure level . the latter correction was estimated by the strength of several night sky emission lines . sky subtraction was performed , using object - free spectra as near as possible to each target . for the targets where local m31 background was high , the method was reversed , such that only sky spectra far from the disk of m31 were used . an offset exposure for such fields , taken concurrently , was reduced in a similar way ( thus contemporaneous sky subtraction was performed for on- and off - target exposures ) , and then the off - target spectra were subtracted from the on - target . this process increased the resultant noise of course , but we deemed it essential for targets in the bulge and disk of m31 . the off - target exposures have the additional advantage of giving measurements of the unresolved light in over 800 locations over the entire disk of m31 . velocities were measured using the sao xcsao software . given the wide variety of spectra in this study , it was deemed necessary to develop new velocity templates , from the spectra themselves . the procedure was to derive an initial velocity of all spectra using library templates ( typically a k giant star ) . the spectra were shifted to zero velocity , and sorted into three different spectral types , a , f and g type spectra . the best spectra in each group were combined to make new templates , and the procedure was repeated now using the new templates . by using these templates we have assured that all the m31 targets are on the same velocity system . they are tied to an external velocity in the initial step , whose accuracy depended on the accuracy of the initial set of templates used . a good test of the internal accuracy was provided by repeat measurements of clusters . we have 386 repeat measurements ( on different nights ) for 224 clusters . the median difference in velocity for these repeats was 0.5 , with an implied median single measurement error of 11 ( smaller than our formal errors listed in the tables ) . we will present external comparisons in a subsequent paper , but note that the cluster velocities agree very well with the hi rotation curve ( see [ s : kin ] ) . velocities for the young clusters , stars and galaxies are presented in tables [ young ] , [ stars ] , [ maybestars ] , and [ gals ] . velocities for the old clusters will presented in a subsequent paper . the spectra were corrected to relative flux values , using observations of standard stars ( the mmt f/5 optics system employs an atmospheric dispersion compensator , adc ) . the flux correction has been determined to be very stable over several years . thus , observations of the same targets taken in different seasons can be combined where available . @xcite highlighted the heterogeneous quality of the m31 cluster catalogs when they claimed , using ao techniques at k , that four out of six observed young clusters were in fact asterisms . figure [ blue_spectra ] shows , from top to bottom , spectra of the four disputed clusters , the average of three genuine young clusters verified by acs images , and the average spectra of single supergiant stars ( these were verified to be stars from the lgs images , and members of m31 from their velocities ) . if the disputed clusters were in fact merely a few stars , those stars would have to be supergiant stars , whose absorption line widths would be as narrow as seen in our blue supergiant spectra . this is not the case for the four disputed clusters ( note in particular the h@xmath8 and h@xmath9 widths , narrow in the stars and wide in the other spectra ) , and we conclude that those objects are true clusters and not asterisms . to be sure , these particular clusters are not globular clusters either , and , additionally , are perhaps not massive enough to be considered young , populous clusters . high spatial resolution imaging can both check for asterisms and also explore the clusters spatial structure : is their concentration low , like typical milky way open clusters , or high , like globular clusters ? there are acs or wfpc2 images available for 25 of the clusters with ages less than 2 gyr . two of these show no evidence of an underlying cluster , but the remaining 23 are clearly not asterisms . figure [ acs_images ] shows the range of structure seen in these young clusters . while many of them show the low - concentration structure typical of milky way open clusters , a number of them , such as b374-g306 and b018-g071 , are quite centrally concentrated , like the majority of the milky way globular clusters . @xcite have measured the structure of m31 clusters with available hst imaging at the time of publication . there are 70 clusters in their sample which we have classified as old , and 7 of our young clusters . it is straightforward to compare the structure of the clusters they study with the milky way globulars , because they use the same technique as @xcite , who have produced a careful summary of the structure of the milky way globular clusters . however , it should be noted that their fitting technique ( fitting ellipses to cluster isophotes ) is not well - suited to very low - concentration clusters , and in fact one of our young clusters , b081d , is omitted from their analysis because of its low density . figure [ structure ] compares their results for old clusters from our sample with the structure of milky way globulars ( from the work of * ? ? ? it can be seen that the concentration parameter = log(@xmath10 ) for king model fits , @xcite p. 307 ] for the old clusters in m31 has a similar distribution to the milky way globulars . we note that although there are no old clusters in our sample with concentration greater than 2.2 , such clusters are definitely present in the @xcite sample so this absence is unlikely to be significant . the similarity in structure is interesting , because at first sight it would seem that the m31 clusters with high concentration would be preferentially discovered in surveys . perhaps the m31 globular cluster surveys ( which , as we have seen above , include a large number of non - globular clusters , as well as the low - concentration young clusters ) are now sufficiently thorough that they are not strongly affected by this bias . although only seven of our young m31 clusters were analyzed by @xcite , it can be seen from figure [ structure ] ( where they are marked by asterisks and five - pointed stars in the top panel ) that their concentrations cover the whole range of the older clusters in m31 and in the milky way . ( the five - pointed star represents hubble v from ngc 205 . ) our observational selection biases may over - emphasize high - concentration clusters , but it is still interesting to see that three of the young clusters have quite high concentration parameters . how does their structure compare with the milky way open clusters ? it is quite difficult to answer this question because the available samples of milky way open clusters are severely incomplete , and it is a challenging task to fit king models to the known open clusters , because cluster membership is hard to determine . the 2mass database @xcite has been used by @xcite to measure the structure of 21 open clusters . they used a cmd - fitting technique to remove contamination from disk field stars . we have also used data from @xcite , who used radial velocity to decide membership . because of the high background in all these cases , it is possible that the limiting radius " given by the authors is in fact smaller than the tidal radius , in which case the cluster concentrations would be smaller than those plotted . it can be seen in the middle panel of figure [ structure ] that all these open clusters have quite low concentrations . however , the sample of clusters with concentration measurements is quite small , and it is quite possible that there are a few open clusters in the milky way which are yet to be discovered and have high concentrations , like the two m31 young clusters . in summary , the young clusters in m31 show a range of structure . most have the low concentration typical of milky way open clusters , but there are a few which have high concentrations , like most milky way globulars . we note that any survey of m31 clusters will preferentially discover ones with high concentrations . in addition , the incompleteness of milky way open cluster samples and the difficulty of measuring cluster concentration in crowded fields means that we can not rule out the existence of such clusters in the milky way . in this section , we describe methods for determining ages from the spectra and color magnitude diagrams for the verified clusters . since the emphasis in this paper is on the younger clusters , and more specifically , on their m / l ratios , our task is first to distinguish young from old clusters , and then to obtain accurate age measurements among the younger clusters . a more refined age determination ( for clusters older than 2 gyr ) is postponed for a later paper . the methods for obtaining ages for young stellar populations from their integrated spectra are similar to those used for older stellar populations , except that instead of employing empirical stellar libraries ( e.g. , * ? ? ? * ) , modelers focussing on younger stellar populations have used synthetic spectra , partly due to a lack of empirical spectra of young stars over a range of metallicities . here we have made use of the starburst99 stellar population modeling program ( sb99 , * ? ? ? to distinguish young from old clusters , we compared our cluster spectra with two external sets of spectra , which served as population templates . one set was the sample of 41 mw globular spectra obtained by @xcite , covering the abundance range of @xmath11[fe / h]@xmath12 . these spectra have a wavelength coverage from 3500 to 6300 , a dispersion of 1 / pixel and a resolution of about 4.5 , and served as our old population templates . our young population templates were created from the sb99 program using the padova z=0.05 interior models and z=0.04 stellar atmospheres , since it seemed likely the young clusters have supersolar abundances . using a solar abundance set of isochrones in the models has the net effect of increasing the derived ages for clusters older than 0.1 gyr , in a logarithmic scaling such that a 1 gyr supersolar model and a 2.5 gyr solar abundance model have nearly the same resultant spectra . a grid of 40 high resolution spectra was created for ages from zero to 2 gyr , with a logarithmic time step of 0.07 between models . our approach then was to simplify the old populations by only using a set of mw globular cluster templates , and to simplify the young populations by restricting ourselves to a single metal - rich chemical composition . the evolution of integrated spectra is essentially logarithmic with time ; for example , the difference between a 5 and 12 gyr population is relatively small compared to the large changes which occur over the first 1 gyr . thus our simplification seems warranted , particularly since in this paper we are concerned only with identifying the younger clusters and then studying them in detail . the major issue , as seen below , in separating old from young clusters , is the potential degeneracy between very metal - poor old populations and young populations , both of which are dominated by balmer lines in their integrated spectra . both sets of template spectra were rebinned to the same resolution and dispersion as the m31 set , and then each template spectrum and m31 spectrum had a low order fit subtracted . the significance of this step was that we did not use the continuum shape to help determine the best matching template . a scaling factor was then determined between each template and all the available m31 cluster spectra , and the reduced @xmath13 calculated over the spectral ranges of 3750 - 4500 , 4750 - 5000 , and 5080 - 5360 . these ranges were chosen to exclude spectral regions where there are few lines , as well as where the mw cluster spectra have no data due to bad columns in the ccd used for that set . specifically included were the balmer lines from h@xmath8 to h@xmath14 , the mg b lines , ca ii h&k , and the he i lines at 4009 and 4026 , the last being prominent in ob stars and thus strong in the youngest clusters . the noise used in the reduced @xmath13 calculations was that calculated from the spectra themselves . a few logic decisions had to be made in analyzing the resultant list of reduced @xmath13 values . for most cases , the lowest reduced @xmath13 occurred clearly either in the sb99 set or the mw cluster set , thus allowing a particular cluster to be identified as young or old . for the young clusters , the age chosen was the average of ages where the reduced @xmath13 was within 10% of the lowest value . about 10% of the clusters were equally well fit by a mw cluster , typically a metal - poor one with [ fe / h ] @xmath15 , and a metal - rich sb99 model . nearly all of these are low s / n , and thus the poor fits were not surprising . visual inspection of the spectrum clarified 11 of these as being very young ( with very blue continuum shapes ) , and the remaining 48 we grouped as old . figure [ young_spectra ] compares the data and best - fitting models for a range of determined ages . we estimate the errors in determined ages for the young clusters to be about a factor of two , which leads to m / l uncertainties of 50% . the model m / l values for each chosen age then allowed masses to be estimated from the cluster integrated v band photometry , which is described in [ s : phot ] . to allow a comparison of young clusters to be made , we also calculated spectroscopic m / l values for the old m31 clusters from the models in @xcite , by obtaining estimates of [ fe / h ] for each cluster via fe and mg line indices and assuming an age of 12 gyr for each . the detailed analysis of the old cluster spectra will be presented in future paper . many readers may be more familiar with extracting ages from diagrams that plot a largely age - sensitive index versus a largely metallicity - sensitive index . to help demonstrate the efficacy of the @xmath13 approach we show two such diagrams . figure [ indices ] is a plot of a balmer line index versus a metal line index . we have defined m@xmath16=(fe5270+mgb)/2 and h@xmath16=(h@xmath17 h@xmath18h@xmath8)/3 , all lick indices @xcite . these indices are equivalent widths : units are . for clarity in these diagrams , a signal - to - noise ratio cutoff was made , which eliminated 20% of the clusters from being plotted . different symbols represent mw globular clusters , and four age bins for m31 clusters , which were determined by the @xmath13 method : very young ( @xmath19 0.1 gyr ) , young ( 0.1 @xmath19 age @xmath19 1 gyr ) , intermediate ( 1 @xmath19 age @xmath19 2 gyr ) , and old . clearly , there is a sequence of m31 clusters that closely matches the sequence of mw globular clusters , and a large spray of clusters that fall mostly in regions of higher balmer equivalent width . clusters with younger ages will of course have stronger balmer absorption , until the very youngest ages are reached , at which point the balmer strength declines again . the second diagram ( figure [ indices2 ] ) uses indices defined in @xcite , namely the ratio of residual light in the h@xmath9 line to the nearby fe4045 line , and the ratio of the line at 3969 which contains both h@xmath20 and caii h , to the caii k line . the indices are unitless . this diagram also shows the old cluster metallicity sequence and again distinguishes them from the young clusters , again excepting for the extremely young clusters . the shortcoming of using such diagrams is that occasionally , due to the nature of real data , one index will be bad , causing the cluster to look young or old in one diagram , and the opposite in another diagram . we are thus more confident in a fitting procedure that uses many diagnostic lines , but are gratified that in the vast majority of cases , these two diagrams verify the ages assigned by the @xmath13 method . @xcite estimated ages of many young disk clusters in m31 from wfpc2 color - magnitude diagrams ( cmd ) and isochrone fitting to the main sequence or to luminous evolved stars . four of their clusters are bright enough to be in our spectroscopic study . their ages agree quite well with ours ( table [ ages ] ) . additionally , as part of hst go proposal 10407 , we obtained acs images of several young clusters , three of which we report on here . the multidrizzle package @xcite was used to combine the 3 individual exposures taken in the f435w and f606w filters ( corresponding to b and v , respectively ) . stellar photometry was obtained using the daophot package of @xcite , modeling the spatially variable psfs for each of the combined images separately , using only stars on those images . psfs were constructed using 5 - 10 bright stars which had no pixels above a level of 20,000 counts , the point at which an ostensible non - linearity set in and the psf no longer matched those of fainter stars . aperture corrections were also measured using these stars , to determine any photometric offset between the psf photometry and the aperture magnitude within 0.5 . @xcite have provided aperture corrections from that aperture size to infinity , in all acs filters . generally , two passes of photometry were run . first , a star list was made and entered into allstar , which aside of the photometry , produces a star - subtracted image . stars missed in the first round were located in the subtracted image and added to the original list . the original frame was then measured again by allstar . the photometry was then placed on the standard johnson / kron - cousins vi system using the aperture corrections and synthetic transformations provided in @xcite . to lessen the severe problems with crowding in these clusters , only stars that fall in an annulus with radii of 15 and 50 pixels ( 0.75 and 2.5 ) are shown in the color - magnitude diagrams ( figure [ cmd ] ) . the background field shown has the same area as the cluster fields , and refers to an annulus around b049-g112 with inner radius of 60 pixels . isochrones with super solar abundances from the padova group @xcite have been placed in the diagrams to allow age determination , using a distance modulus of m31 of 24.43 and the reddenings determined above for these two clusters ( 0.25 for both ) . these cmds and that of a third we have worked on ( b367-g292 ) give ages in reasonable agreement with those from the spectroscopic analysis ( table [ ages ] ) . multicolor photometry for most of the clusters in this project is already collated in @xcite , but enough clusters are missing to warrant remeasuring all the clusters , to allow the photometry to be used with the spectroscopic m / l values to obtain masses . the lgs survey of m31 consisted of 10 separate but overlapping fields . stellar photometry from these fields using psf - fitting has been reported in @xcite , but the aperture photometry needed for resolved star clusters has not yet been reported . to limit the scope of the work , we elected to measure objects in our catalogs only in the v band . targets from our entire catalog were located on the images , and photometry for 12 separate apertures ranging from 0.7 to 16(spaced logarithmically ) was collected using daophot . growth curves from these apertures were constructed , and used in an automatic fashion to estimate the aperture which enclosed the total light of the cluster . these apertures were then inspected on the images and increased , if the clusters were in fact larger , or decreased for cases where the apertures included substantial light from objects clearly not part of the clusters . the local background was measured in annuli with inner radii 1 pixel larger than the outer radius of the aperture for the object . the apertures used are listed in table [ main ] . extraneous stars remaining in the apertures were accounted for by measuring their magnitudes separately , and subtracting their contribution to the cluster aperture magnitudes . the resultant instrumental magnitudes were then placed on the standard v system using stars from the @xcite tables which we were measured in the same way as the clusters . the color term in the v mag transformation ( described in * ? ? ? * ) was ignored , as it was smaller than the errors we report . tables [ main ] , [ stars ] , and [ maybestars ] list the results from this work . the formal errors in the standardized photometry were less than 0.03 mag , set by the uncertainties in the transformations . however , since our goal was actually cluster total magnitudes , our calculated uncertainties refer to the uncertainty in setting the proper apertures and correcting for extraneous objects within those apertures . specifically , the uncertainties were set to be equal to the difference in the magnitude of the aperture chosen as best representing the limiting radius of the cluster , and that of the next larger aperture in our logarithmic spacing of aperture sizes . in practice this means that clusters in crowded fields have larger uncertainties than those in less dense areas . the tables also list photometry from other sources for the objects that are outside of the lgs images ; we do not list the errors in such cases . comparing our aperture magnitudes with those collected from various sources and listed in the rbc , we find excellent agreement over the magnitude range of 14 to 18 , with an rms in the differences of 0.18 mag in the set of 200 objects in common ; this in spite of the fact that no effort was made to insure that the apertures used in the two data sets were the same . between v=18 and 20 , our photometry tended to be fainter by 0.2 mag and the scatter increased to 0.5 mag , some of which was likely due to differences in object identification . a further comparison was made with the 58 v magnitudes measured for m31 clusters from archival hst images in @xcite . aside of one cluster whose v magnitude appears to be a typo ( b151-g205 ) , the rms of the differences of that set with the magnitudes presented here is 0.28 , with no apparent systematics . @xcite misclassified 15 of the young clusters as old disk globular clusters ( 17% of their disk globulars ) . this was pointed out by @xcite . in some cases this was due to the low s / n of the wyffos spectrum , in others the problem was misinterpretation of the spectra . our new study of the clusters has resolved most of this confusion and changed the classification of a number of clusters from old and massive to younger , not very massive . however , we still find clusters with significant masses , above @xmath21m@xmath22 , and with ages less than 150 myr ( see table [ young ] ) . of the 10 clusters with those physical characteristics , the hst or lgs images of three confirm them as clusters similar in appearance to the populous clusters of the lmc ( these are b315-g038 , b318-g042 , and vdb0 , the latter still the most massive , young cluster known in m31 , @xcite ) . five appear more like ob associations , and thus may not survive as bound clusters ( b319-g044 , b327-g053 , b442-d033 , b106d and bh05 ) . the case for the other two ( b040-g102 , b043-g106 ) is not as clear , but their lgs images are more similar to the cases like b315-g038 than to the ob associations . figure [ mips ] shows a spitzer / mips 24@xmath23 mosaic of m31 @xcite with the positions of the young clusters overlaid . clusters younger than 0.1 gyr , between 0.1 and 0.32 gyr , and 0.32 and 2 gyr are shown in different colors . the latter two groupings divide the clusters older than 0.1 gyr into two equal parts . it can be seen that the spatial distribution of the young clusters is well correlated with the star - forming regions in m31 , with the majority associated with the 10 kpc `` ring of fire '' . the comparison of these young clusters and the warm dust emission is distinct from the comparison of the latter with the location of hii regions , as we have excluded the clusters embedded in hii regions from our sample . @xcite and @xcite use the curious appearance of the mid - ir ring - the split near the location of m32 , creating the appearance of a `` hole '' in between , and the possible offset of the ring from the nucleus - to suggest a recent encounter of m32 and m31 . both groups suggest that the split is caused by m32 s passage through the disk , and their models also produce rings offset from m31 s center , albeit not as extreme as is observed ( * ? ? ? * produce an offset of 1@xmath24 , not 6@xmath24 ) . an examination of our figure [ mips ] shows that the cluster distribution favors the outer parts of the hole , and is generally quite symmetric about m31 s nucleus . most models of an offset ring assume that the inner part of the split in the observed ring is the one which should be traced by star formation . however , this is not where most of the clusters are found . ] @xcite need a very recent interaction between m32 and the disk ( their model has the disk passage occurring @xmath25 years ago ) because the passage of m32 through the disk in their model results in a burst of star formation that propagates outward through the disk . however , we do not see any radial trends with cluster age , which might be expected with a propagating ring of star formation ( figure [ radial_age ] ) . @xcite prefer a collision about @xmath26 years ago , which triggers expanding density waves . our young cluster ages range from 0.04 gyr to 1 gyr , but most are between 10@xmath27 and 10@xmath28 years . if the ring of fire " was produced by a single event , as modeled by the above authors , we might expect the age distribution of clusters associated with it to be more peaked . the ages of the younger clusters presented in @xcite range from around @xmath29 to @xmath26 years , and there is little evidence of a peak in star formation in this age range either . these results seem to suggest that star formation has been fairly high in this region of m31 for 1 gyr or more . in summary , we see no evidence of enhancement in star formation rate or any spatial age separation , as we might expect from the m32 disk passage . do the kinematics of the young clusters bear out the disk origin suggested by their close association ( in projection ) with star forming regions in m31 s disk ? m31 s inner disk kinematics are more complex than originally supposed , due to m31 s bar @xcite . the velocities from our many sky fibers , taken both as a part of regular observing and also from entire exposures devoted to offset sky in the crowded inner regions , give us a new way of quantifying disk rotation throughout the inner regions where the young clusters are found . ( we plan to use these data in a study of bar kinematics , athanassoula et al . in preparation ) . figure [ spider_diag ] shows the disk mean velocity field . we also use kinematics of hii regions that we observed as fillers in the hectospec fields to give us an indication of the kinematics of young disk objects . these data will be published in athanassoula et al . ( in preparation ) . figure [ dist_vel ] shows the kinematics of the mean disk light , the young clusters , and hii regions , versus major axis distance x , in kpc . we have split the sample into objects which are close to the major axis ( @xmath19 1 kpc ) and those projected from @xmath30 kpc from the major axis , because the projection of a circular orbit looks different in these two cases . objects on the major axis in circular orbits have all of their velocity projected on the line of sight ; as we move further away from the major axis , less of the circular velocity is projected on the line of sight and so the tilt of the distance - velocity line becomes smaller . this can be seen clearly in figure [ dist_vel ] for all types of objects . the curious flattening of the mean velocities from absorption line spectra for major axis distances between 3 and 10 kpc is likely to be caused by the bar . it can be seen that the young clusters follow the disk mean velocity curve from absorption spectra quite well , and show an even better correlation with the kinematics of the youngest objects : hii regions and sky fibers showing emission spectra . this kinematic analysis confirms our spatial association of the young clusters with the star forming young disk in m31 . the m / l values obtained from the spectroscopic age estimates can be combined with the v band photometry to derive masses of all the observed m31 clusters , young and old . reddening values are of course also needed , and a large number of e(b v ) values were derived from photometry in @xcite . they and we consider only the total reddenings , foreground and internal to m31 . the methodology used in @xcite meant that the reddenings would only be valid for old clusters , and indeed few of the clusters we have identified here as young were included in their study , thus reddenings for those objects are needed . therefore , we elected to rederive the reddenings for all of the clusters in our study , young and old . in the case of the young clusters we compared the fluxed spectra with the sb99 model spectra of the appropriate age . as described above , the ages were obtained by matching spectral line features in the observed and model spectra , not by comparing the continua shapes . once the ages have been found in this way , differences in the continua shapes may be assumed to be due to reddening , except for a few cases where a late - type star , whether member or not , clearly dominates the redder wavelengths as evidenced by the presence of tio bands . for those cases , we use the mean reddening for the young clusters of 0.28 . for the old clusters we did not use models , but rather the sample of spectra themselves . initial values of reddenings were obtained from @xcite , and were used to deredden the spectra of those clusters with e(b@xmath31v)@xmath32 , about 190 in number ( there are about 350 old clusters in our spectroscopic sample ) . these spectra were ordered in metallicity , which was estimated from the spectral line indices as mentioned above , rebinned to a coarse grid in wavelength , and normalized to have the same intensity at the arbitrary wavelength of 5000 . interpolation formulae were developed from these spectra , via a least squares method to avoid bad spectra , for intensity as a function of both wavelength and metallicity . as a result , a cluster spectrum of arbitrary metallicity could be created , dereddened to the accuracy of the @xcite reddenings . the individual spectra in this low reddening sample were then compared with the appropriate interpolated spectra , and reddenings were adjusted as needed to bring their continua shapes closer to that of the expected template shape . the method is thus similar to methods that use the metal abundance to predict the intrinsic broad band colors , and then require the derived reddening to reproduce the observed colors . the overall goal in working with the low reddening sample was to retain the mean value of the reddenings found in @xcite , but to correct those that varied significantly . after cleaning up those reddenings , the interpolation formulae were then used to derive reddenings for the 150 clusters for which we have spectra and whose reddenings were not measured in @xcite . thus while we have not improved upon the absolute levels of the m31 old cluster reddenings , we believe we have improved the precision of the values in a relative sense , and as well have nearly doubled the number of reddenings available . about 10% of the spectra were taken during nights when the adc was not operating properly , thus we ca nt use the continuum shape to estimate reddenings . for objects whose only spectra were taken on those nights , we assume the average reddening of 0.28 . a comparison of our derived e(b@xmath31v ) values ( which range up to 1.4 mag ) and those in @xcite results in a scatter of 0.17 mag rms , which is good enough for our overall goal of comparing the m31 cluster system in bulk with that of other galaxies . interestingly , both the young cluster and old cluster groups have clusters with e(b@xmath31v)@xmath33 , though the highest measured value ( e(b@xmath31v)=1.4 ) is still found in the old cluster b037-v327 , probably a selection effect since that cluster also has the highest luminosity in all of m31 . the young cluster reddenings are listed in table [ young ] ; those of the old clusters will be presented in a subsequent paper . by using the position of blue - plume stars in the color - magnitude diagram , @xcite estimated the average reddening for young stars in m31 to be 0.13 mag , significantly lower than the mean of the clusters younger than 100 myr presented here , which may place a constraint on the accuracy of the values presented here . the mass histogram for the all of the young clusters is shown in figure [ massdist ] . we have also shown the mass distribution of milky way open clusters within 600 pc of the sun . this is based on the sample of @xcite , with mass calculations by @xcite . the kharchenko catalog is the most homogeneous and complete catalog of open clusters in the solar neighborhood currently available , and is based on a stellar catalog complete to v=11.5 . the cluster masses were estimated by counting the number of cluster members brighter than the limiting magnitude , then correcting for the stars fainter than this using a salpeter mass function and a lower mass limit of 0.15 m@xmath4 . this catalog does not include the most massive clusters in the galaxy because of its relatively small sample size ; for example , there have been recent discoveries of more distant young clusters which may have masses as high as @xmath3m@xmath4 ( e.g. @xcite ) , and we add westerlund 1 to the histogram as an example . the milky way globular and lmc young massive cluster histograms are shown in the bottom two panels ( from * ? ? ? * ) . these mass estimates are based on king model fits . obviously , m31 clusters with masses less than @xmath34m@xmath35 and ages greater than a few @xmath36 years are too faint to be part of this study , and await a future study . @xcite estimate over 10000 such clusters in the disk of m31 ; these would form the low mass tail in the mass distribution of figure [ massdist ] . nonetheless , there is a trend in cluster mass , with the milky way open clusters having the lowest median mass , the milky way and m31 globulars the highest , and the lmc young massive clusters and the m31 young clusters in between . this trend is consistent with a single cluster imf plus disruption , taking into account the small size of the volume searched for clusters in the milky way . would we expect these young m31 clusters to survive as they age , or to disrupt ? one of the main processes that leads to cluster disruption is 2-body relaxation enhanced by an external tidal field @xcite . the lower - mass clusters suffer more strongly from relaxation effects . another property of the cluster itself which will affect its survival is its density lower - density clusters will disrupt more quickly @xcite . thus we would expect that massive , concentrated clusters such as b018 and bh05 would be more likely to survive . @xcite derive an empirical expression for the disruption of clusters as a function of their mass , studying cluster populations in the solar neighborhood , the smc , m33 and m51 . @xcite point out that observational selection effects could mimic the decrease in the number of clusters with age which boutloukos et al . ascribe to cluster disruption . however , this is almost certainly not true of the solar neighborhood open clusters studied by @xcite using a similar analysis . we show the age - mass diagram for the young m31 clusters in figure [ agemass ] . while our sample is clearly very incomplete below @xmath37 years , the diagram shows some similarity to the lmc cluster age - mass diagram of @xcite in the age range we cover . unfortunately , we do not expect our catalog to be complete enough to permit an analysis using the techniques of boutloukos et al . ] environmental effects also control the tidal stripping of the cluster . for stars whose orbits are mostly confined to the disk , encounters with giant molecular clouds and spiral arms contribute to their disruption @xcite . for clusters whose orbits are not confined to the disk , bulge and disk shocking are more important @xcite . the similarity of the m31 young cluster kinematics to that of other young disk objects suggests strongly that these clusters are confined to m31 s disk plane , so giant molecular clouds should be the relevant external disruptor . @xcite show that disruption times for clusters in galaxies ranging in size from m51 to the smc , scale with molecular gas density in the expected way . s molecular gas density is highest near the ring of fire " where many of our clusters are found @xcite . this density is similar to the molecular gas density in the solar neighborhood @xcite . thus we would expect the survival due to giant molecular cloud interactions of the m31 young clusters to be similar to that of the solar neighborhood open clusters . we expect that most of these young clusters will be disrupted in the next gyr or so ( * ? ? ? * derive a disruption time of 1.3 gyr for a cluster of mass 10@xmath38 in the solar neighborhood ) . however , some of the more massive and concentrated of the young clusters will likely survive for longer . we present a new catalog of 670 m31 clusters , with accurate coordinates . in this paper we focus on the 140 clusters ( many originally classified in the literature as globular clusters ) which have ages less than 2 gyr : most have ages between @xmath37 and @xmath39 years . using high - quality mmt / hectospec spectra , excellent ground based images , and in some cases , hst images , we explore the nature of these clusters . with the exception of ngc 205 s young cluster , they have spatial and kinematical properties consistent with formation in the star - forming disk of m31 . many are located close to the 10 kpc `` ring of fire '' which shows active star formation . the age distribution of our clusters , plus that of the younger clusters of @xcite , shows no evidence for a peak in star formation there between @xmath25 and @xmath39 years ago , which we might expect if the ring was created by a recent passage of m32 through the disk , as suggested by @xcite and @xcite . we have estimated their masses using spectroscopic ages and m / l ratios , ( in some cases ) acs color - magnitude diagrams , and new photometry from the local group survey . the clusters have masses ranging from 250 to 150,000 @xmath40 . these reach to higher values than the known milky way open clusters , but it must be remembered that our sample of open clusters in the milky way is far from complete . the most massive of our young clusters overlap the mass distributions of m31 s old clusters and the milky way globulars . interestingly , although most of the young clusters show the low - concentration structure typical of the milky way open clusters , a few have the high concentrations typical of the milky way globulars and the old m31 clusters . we estimate that most of these young clusters will disrupt in @xmath30 gyr , but the massive , concentrated clusters may well survive longer . we would like to thank dan fabricant for leading the effort to design & build the hectospec fiber positioner and spectrograph , perry berlind & mike calkins for help with the observations , john roll , maureen conroy & bill joye for their many contributions to the hectospec software development project , and phil massey , pauline barmby & jay strader for comments and data tables on m31 . hlm was supported by nsf grant ast-0607518 , and would like to thank dean mclaughlin for helpful conversations . work on this project has also been supported by hst grant go10407 . baade , w. , & arp , h. 1964 , , 139 , 1027 barmby , p. et al . 2000 , aj , 119 , 727 barmby , p. , & huchra , j. p. 2001 , , 122 , 2458 barmby , p. , mclaughlin , d. e. , harris , w. e. , harris , g. l. h. , & forbes , d. a. 2007 , , 133 , 2764 battistini , p. , bonoli , f. , braccesi , a. , fusi - 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we present an updated catalog of 1300 objects in the field of m31 , including 670 likely star clusters of various types , the rest being stars or background galaxies once thought to be clusters . the coordinates in the catalog are accurate to 0.2 , and are based on images from the local group survey ( lgs , * ? ? ? * ) or from the dss . archival hst images and the lgs were inspected to confirm cluster classifications where possible , but most of the classifications are based on spectra taken of @xmath0 objects with the hectospec fiber positioner and spectrograph on the 6.5 m mmt . the spectra and images of young clusters are analyzed in detail in this paper ; analysis of older clusters will appear in a later paper . ages and reddenings of 140 young clusters are derived by comparing the observed spectra with model spectra . seven of these clusters also have ages derived from hst color - magnitude diagrams ( two of which we present here ) ; these agree well with the spectroscopically determined ages . combining new v band photometry with the m / l values that correspond to the derived cluster ages , we derive masses for the young clusters , finding them to have masses as great as @xmath1 with a median of @xmath2 , and a median age of 0.25 gyr . in comparison therefore , milky way open clusters have the lowest median mass , the milky way and m31 globulars the highest , and the lmc young massive clusters and the m31 young clusters are in between . the young clusters in m31 show a range of structure . most have the low concentration typical of milky way open clusters , but there are a few which have high concentrations . we expect that most of these young clusters will be disrupted in the next gyr or so , however , some of the more massive and concentrated of the young clusters will likely survive for longer . the spatial distribution of the young clusters is well correlated with the star - forming regions as mapped out by mid - ir emission . a kinematic analysis likewise confirms the spatial association of the young clusters with the star forming young disk in m31 .

in metallic ferromagnet - nonferromagnet - ferromagnet multilayers ( see fig . 1 ) magnetic structure oscillates between ferromagnetic and antiferromagnetic orientations of the ferromagnets magnetizations as a function of thickness of nonmagnetic metal @xmath0 with a period of order of the fermi wave length @xcite@xcite @xcite@xcite@xcite@xcite . the explanation of this phenomenon is based on the fact that the interlayer coupling is due to ruderman - kittel interaction between electron spins in different ferromagnets . further investigations discovered structures with perpendicular orientations of the ferromagnets magnetizations ( see for review @xcite ) . often phenomenological coupling between magnetizations of ferromagnetic layers can be represented as sum of bilinear and biquadratic contributions @xmath1 here @xmath2 is angle between directions of magnetizations of ferromagnetic films . bilinear constant @xmath3 oscillates as function of interlayer distance @xmath4 in case of large positive biquadratic constant @xmath5 minimum of @xmath6 corresponds to @xmath7 . as explained by slonzevskii large positive biquadratic coupling might be result of spatial fluctuations of bilinear coupling @xmath3 due to ferromagnet - nonferromagnet surface roughness @xcite . in disordered system , when @xmath0 is larger than electron mean free path @xmath8 , rkky interaction @xmath9 , averaged over realizations of scattering potential exponentially decreases @xcite . at the same time fluctuations of local exchange become much larger than @xmath10 @xcite giving rise to biquadratic contribution @xmath11 @xcite . here we propose mechanism of coupling in disordered multilayers based on interaction between electrons in nonmagnetic layer . spin fluctuations in system of interacting electrons give rise to contribution to thermodynamic potential@xcite , which depends on magnetic field or , in our case , on relative orientation of magnetizations in ferromagnetic layers . here we show that in magnetic multilayer this mechanism describes transition between ferromagnetic and noncolinear ordering with increasing distance between ferromagnetic layers or value of ferromagnetic splitting of conducting electrons . we assume that magnetic multilayer can be described by hamiltonian @xmath12 here @xmath13 is hamiltonian of free electrons in random field . second term describes exchange field in ferromagnetic layers . @xmath14 is ferromagnetic splitting of conducting electrons . @xmath15 is unit vector of direction of magnetization of ferromagnetic layers . @xmath16at @xmath17 and @xmath18at @xmath19 as it is shown on figure 1 . @xmath20 and @xmath21 are creation and annihilation operators , @xmath22 are pauli matrixes . integration in second term is over ferromagnetic layers . the last term @xmath23 describes coulomb interaction between electrons in nonmagnetic layer . we assume that interaction in ferromagnetic layers is taken into account self consistently in @xmath14 . details of calculation are given in the last part of the paper . here we present the main results . characteristic energies in the problem are ferromagnetic splitting of conducting electrons @xmath14 and thouless energy @xmath24 . @xmath25 is diffusion constant of conduction electrons . we assume that it is the same in nonmagnetic and ferromagnetic layers . in case of small thickness when @xmath26 coupling between ferromagnetic layers has bilinear form and coupling energy per unit area is @xmath27 here @xmath28 is characteristic constant of interaction in diffusion channel@xcite . it is positive for coulomb repulsion between electrons . let us note that in this regime coupling ( 3 ) does not depend on @xmath0 . minimum of ( 3 ) corresponds to ferromagnetic orientation of magnetizations in multilayer @xmath29 . note that result is obtained in limit when @xmath30 , or @xmath31 smaller that inverse mean free time @xmath32 . at larger distance @xmath0 when @xmath33 coupling has biquadratic form and coupling energy per unit area is @xmath34 this quantity decreases as @xmath35 with increasing distance . minimum of coupling energy corresponds to noncollinear state @xmath7 . both expressions are given for the case of infinite thickness of ferromagnetic layers . calculation show that in case @xmath36 coupling weakly depends on @xmath37 . results ( 3 ) and ( 4 ) are also valid provided @xmath38 . in opposite case coupling energy decreases exponentially as @xmath39 . let us compare results ( 2 ) and ( 3 ) with biquadratic contribution due to mesoscopic fluctuations of rkky interaction @xcite , which is @xmath40 at @xmath33 and @xmath41 at @xmath42 . here @xmath43 is an intralayer ferromagnetic stiffness and thickness @xmath44 . the quantity @xmath45 decreases with @xmath0 much faster than ( 4 ) . also for @xmath46 , @xmath47 coupling energy given by expressions ( 3 ) and ( 4 ) is larger than biquadratic contribution due to mesoscopic fluctuations of rkky in whole range of distances . in this case with increasing distance @xmath48 system is undergo transition between ferromagnetic and perpendicular @xmath7 ordering . such transition was observed in @xcite . correction to thermodynamic potential which depends on @xmath49 is given by expression @xcite @xmath50 here constant @xmath28 describes screened coulomb interaction in diffusion channel . @xmath51 is matsubara frequency . @xmath52 is electron mean free time . diffusion ladder satisfies equation @xmath53 it is convenient to present solution of equation ( 6 ) at @xmath54 in the form @xmath55 here we introduce @xmath56 . @xmath57 is matrix of relative rotation of magnetizations of ferromagnetic layers . in case when direction of magnetization in ferromagnetic layer @xmath17 is directed along @xmath58 axes @xmath59 and at @xmath19 @xmath60direction is @xmath61 , it is matrix of rotation along @xmath62 axes @xmath63 . for simplicity we consider limit of semiinfinite ferromagnetic layers . more detail consideration shows that at @xmath44 results weakly depend on thickness of ferromagnetic layers . it is convenient to introduce boundary conditions for diffusion ladder at ferromagnet - nonferromagnet interfaces taking into account that according to equation ( 6 ) in coordinate system where spins are directed along magnetization , components of ladder with antiparallel spins decreases as @xmath64 and @xmath65 at @xmath66 , where @xmath67 . components of ladder with parallel spins decreases as @xmath68 at @xmath66 . at @xmath69 where @xmath70 boundary conditions are @xmath71 here we introduce projectors of spins on @xmath58-axes @xmath72 . the same kind of boundary conditions can be introduced for rotated diffusion ladder @xmath73 at @xmath74 . solving system of equations ( 6,8 ) we obtain @xmath75 } { % \left [ 1-\left ( % tcimacro{\func{re}}% % beginexpansion \mathop{\rm re}% % endexpansion \lambda \right ) ^{2}+2\left ( \left| \lambda \right| ^{2}-\left ( % tcimacro{\func{re}}% % beginexpansion \mathop{\rm re}% % endexpansion \lambda \right ) ^{2}\right ) \cos \varphi + \left ( \left| \lambda \right| ^{4}-\left ( % tcimacro{\func{re}}% % beginexpansion \mathop{\rm re}% % endexpansion \lambda \right ) ^{2}\right ) \cos ^{2}\varphi \right ] } \end{array}\ ] ] here @xmath76 . expression ( 9 ) contains divergent terms , which do not depend @xmath2 and must be subtracted . in limit of large exchange splitting when @xmath77 parameter @xmath78 is real . in this case energy is function of @xmath79 . subtracting in expression ( 9 ) terms which do no depend on angle we obtain @xmath80 main contribution in expression ( 10 ) is from region where @xmath81 , denominator therefore gives only small correction . neglecting it we obtain expression ( 4 ) . in opposite limit of small exchange splitting @xmath82 , @xmath83 and to the order @xmath84 coupling energy is proportional to @xmath85 @xmath86 calculating ( 11 ) at zero temperature we obtain ( 3 ) . transition between limits ( 10 ) and ( 11 ) occurs at @xmath87 . this work is supported by russian fund for fundamental research grant number 01 - 02 - 17794 .

we consider mechamism of exchange coupling based on interaction between electrons in nonmagnetic layer . depending on ratio of inverse time of diffusion of electrons between ferromagnetic layers and ferromagnetic splitting of conducting electrons this mechanism describes transition from ferromagnetic to concollinear ordering of magnetizations of ferromagnetic layers . [ theorem]acknowledgement [ theorem]algorithm [ theorem]axiom [ theorem]claim [ theorem]conclusion [ theorem]condition [ theorem]conjecture [ theorem]corollary [ theorem]criterion [ theorem]definition [ theorem]example [ theorem]exercise [ theorem]lemma [ theorem]notation [ theorem]problem [ theorem]proposition [ theorem]remark [ theorem]solution [ theorem]summary

1.0truecm with the publication of the new results of the sno solar neutrino experiment @xcite ( see also @xcite ) on i ) the measured rates of the charged current ( cc ) and neutral current ( nc ) reactions , @xmath12 and @xmath13 , ii ) on the day - night ( d - n ) asymmetries in the cc and nc reaction rates , and iii ) on the day and night event energy spectra , further strong evidences for oscillations or transitions of the solar @xmath14 into active neutrinos @xmath15 ( and/or antineutrinos @xmath16 ) , taking place when the solar @xmath14 travel from the central region of the sun to the earth , have been obtained . the evidences for oscillations ( or transitions ) of the solar @xmath14 become even stronger when the sno data are combined with the data obtained in the other solar neutrino experiments , homestake , kamiokande , sage , gallex / gno and super - kamiokande @xcite . global analysis of the solar neutrino data @xcite , including the latest sno results , in terms of the hypothesis of oscillations of the solar @xmath14 into active neutrinos , @xmath17 , show @xcite that the data favor the large mixing angle ( lma ) msw solution with @xmath1 , where @xmath18 is the angle which controls the solar neutrino transitions . the low solution of the solar neutrino problem with transitions into active neutrinos is only allowed at approximately 99.73% c.l . @xcite ; there do not exist other solutions at the indicated confidence level . in the case of the lma solution , the range of values of the neutrino mass - squared difference @xmath19 , characterizing the solar neutrino transitions , found in @xcite at 99.73% c.l . reads : @xmath20 the best fit value of @xmath21 obtained in @xcite is @xmath22 . the mixing angle @xmath18 was found in the case of the lma solution to lie in an interval which at 99.73% c.l . is determined by @xcite @xmath23 the best fit value of @xmath24 in the lma solution region is given by @xmath25 . strong evidences for oscillations of atmospheric neutrinos have been obtained in the super - kamiokande experiment @xcite . as is well known , the atmospheric neutrino data is best described in terms of dominant @xmath26 ( @xmath27 ) oscillations . the explanation of the solar and atmospheric neutrino data in terms of neutrino oscillations requires the existence of 3-neutrino mixing in the weak charged lepton current ( see , e.g. , @xcite ) . assuming 3-@xmath0 mixing and massive majorana neutrinos , we analyze the implications of the latest results of the sno experiment for the predictions of the effective majorana mass in neutrinoless double beta ( - ) decay ( see , e.g. , @xcite ) : @xmath28 here @xmath29 are the masses of 3 majorana neutrinos with definite mass @xmath30 , @xmath31 are elements of the lepton mixing matrix @xmath32 - the pontecorvo - maki - nakagawa - sakata ( pmns ) mixing matrix @xcite , and @xmath33 and @xmath34 are two majorana cp - violating phases satisfy the majorana condition : @xmath35 , where @xmath36 is the charge conjugation matrix . ] @xcite . if cp - invariance holds , one has @xcite @xmath37 , @xmath38 , @xmath39 , and @xmath40 represent the relative cp - parities of the neutrinos @xmath41 and @xmath42 , and @xmath41 and @xmath43 , respectively . the experiments searching for -decay test the underlying symmetries of particle interactions ( see , e.g. , @xcite ) . they can answer the fundamental question about the nature of massive neutrinos , which can be dirac or majorana fermions . if the massive neutrinos are majorana particles , the observation of -decay ev ( 95% c.l . ) are claimed to have been obtained in @xcite . the results announced in @xcite have been criticized in @xcite . ] can provide unique information on the type of the neutrino mass spectrum and on the lightest neutrino mass @xcite . combined with data from the @xmath44 @xmath45-decay neutrino mass experiment katrin @xcite , it can give also unique information on the cp - violation in the lepton sector induced by the majorana cp - violating phases , and if cp - invariance holds - on the relative cp - parities of the massive majorana neutrinos @xcite . rather stringent upper bounds on have been obtained in the @xmath46ge experiments by the heidelberg - moscow collaboration @xcite , @xmath47 ( @xmath48c.l . ) , and by the igex collaboration @xcite , @xmath49 ( @xmath48c.l . ) . taking into account a factor of 3 uncertainty in the calculated value of the corresponding nuclear matrix element , we get for the upper limit found in @xcite : @xmath50 ev . considerably higher sensitivity to the value of @xmath51 is planned to be reached in several @xmath52decay experiments of a new generation . the nemo3 experiment @xcite , which will begin to take data in july of 2002 , and the cryogenic detector cuore @xcite , are expected to reach a sensitivity to values of @xmath53ev . an order of magnitude better sensitivity , i.e. , to @xmath54ev , is planned to be achieved in the genius experiment @xcite utilizing one ton of enriched @xmath46ge , and in the exo experiment @xcite , which will search for @xmath52decay of @xmath55xe . two more detectors , majorana @xcite and moon @xcite , are planned to have sensitivity to in the range of @xmath56 ev . in what regards the @xmath44 @xmath45-decay experiments , the currently existing most stringent upper bounds on the electron ( anti-)neutrino mass @xmath57 were obtained in the troitzk @xcite and mainz @xcite experiments and read @xmath58 . the katrin @xmath44 @xmath45-decay experiment @xcite is planned to reach a sensitivity to @xmath59 ev . the fact that the solar neutrino data implies a relatively large lower limit on the value of @xmath24 , eq . ( [ thlma ] ) , has important implications for the predictions of the effective majorana mass parameter in -decay @xcite and in the present article we investigate these implications . 1.0truecm according to the analysis performed in @xcite , the solar neutrino data , including the latest sno results , strongly favor the lma solution of the solar neutrino problem with @xmath60 . we take into account these new development to update the predictions for the effective majorana mass , derived in @xcite , and the analysis of the implications of the measurement of , or obtaining a more stringent upper limit on , performed in @xcite . the predicted value of depends in the case of 3-neutrino mixing of interest on ( see e.g. @xcite ) : i ) the value of the lightest neutrino mass @xmath8 , ii ) @xmath21 and @xmath18 , iii ) the neutrino mass - squared difference which characterizes the atmospheric @xmath61 ( @xmath62 ) oscillations , @xmath63 , and iv ) the lepton mixing angle @xmath64 which is limited by the chooz and palo verde experiments @xcite . the ranges of allowed values of @xmath21 and @xmath18 are determined in @xcite , while those of @xmath63 and @xmath64 are taken from @xcite ( we use the best fit values and the 99% c.l . results from @xcite ) . given the indicated parameters , the value of depends strongly @xcite on the type of the neutrino mass spectrum , as well as on the values of the two majorana cp - violating phases , @xmath33 and @xmath34 ( see eq . ( [ effmass2 ] ) ) , present in the lepton mixing matrix . let us note that if @xmath63 lies in the interval @xmath65 , as is suggested by the current atmospheric neutrino data @xcite , its value will be determined with a high precision ( @xmath66 uncertainty ) by the minos experiment @xcite . similarly , if @xmath67 , which is favored by the solar neutrino data , the kamland experiment will be able to measure @xmath21 with an uncertainty of @xmath68 ( 99.73% c.l . ) . combining the data from the solar neutrino experiments and from kamland would permit to determine @xmath24 with a high precision as well ( @xmath69 uncertainty at 99.73% c.l . , see , e.g. , @xcite ) . somewhat better limits on @xmath70 than the existing one can be obtained in the minos experiment . we number the massive neutrinos ( without loss of generality ) in such a way that @xmath71 . in the analysis which follows we consider neutrino mass spectra with normal mass hierarchy , with inverted hierarchy and of quasi - degenerate type @xcite . in the case of neutrino mass spectrum with normal mass hierarchy ( @xmath72 ) we have @xmath73 and @xmath74 , while in the case of spectrum with inverted hierarchy ( @xmath75 ) one finds @xmath76 and @xmath77 . in both cases one can choose @xmath78 . it should be noted that for @xmath79 , the neutrino mass spectrum is of the quasi - degenerate type , @xmath80 , and the two possibilities , @xmath73 and @xmath76 , lead to the same predictions for . 1.0truecm if @xmath81 , the effective majorana mass parameter is given in terms of the oscillation parameters , , @xmath82 and @xmath83 which is constrained by the chooz data , as follows @xcite : @xmath84 the effective majorana mass can lie anywhere between 0 and the present upper limits , as fig . 1 ( left panels ) shows . and the cp violating phases which enter the effective majorana mass are not constrained . ] this conclusion does not change even under the most favorable conditions for the determination of , namely , even when , , @xmath18 and @xmath64 are known with negligible uncertainty @xcite . our further conclusions for the case of the lma solution of the solar neutrino problem @xcite are illustrated in fig . 1 ( left panels ) and are summarized below . * case a : @xmath85 , @xmath86 . * taking into account the new constraints on the solar neutrino oscillation parameters following from the sno data @xcite does not change qualitatively the conclusions reached in ref . the maximal value of , @xmath87 , for given @xmath8 reads : @xmath88 where @xmath89 and @xmath90 are the values corresponding to @xmath91 , and @xmath92 is the maximal value of allowed for the @xmath93 @xcite . for the values of and @xmath94 from the lma solution region @xcite , eqs . ( [ dmsollma ] ) and ( [ thlma ] ) , we get for @xmath95 : @xmath96 . using the best fit values of the oscillation parameters found in refs . @xcite , one obtains : @xmath97 . the maximal value of corresponds to the case of cp - conservation and @xmath30 having identical cp - parities , @xmath98 . there is no significant lower bound on because of the possibility of mutual compensations between the terms contributing to and corresponding to the exchange of different virtual massive majorana neutrinos . furthermore , the uncertainties in the oscillation parameters do not allow to identify a `` just - cp violation '' region of values of @xcite ( a value of in this region would unambiguously signal the existence of cp - violation in the lepton sector , caused by majorana cp - violating phases ) . however , if the neutrinoless double beta - decay will be observed , the measured value of , combined with information on @xmath8 and a better determination of the relevant neutrino oscillation parameters , might allow to determine whether the cp - symmetry is violated due to majorana cp - violating phases , or to identify which are the allowed patterns of the massive neutrino cp - parities in the case of cp - conservation ( for a detailed discussion see ref . @xcite ) . * case b : neutrino mass spectrum with partial hierarchy ( @xmath99 ) * for @xmath100 there exists a lower bound on the possible values of ( fig . 1 , left panels ) . using the 99.73% c.l . allowed values of and @xmath24 from @xcite , we find that this lower bound is significant , i.e. , @xmath101 ev , for @xmath102 ev . for the best fit values of the oscillation parameters obtained in @xcite , one has @xmath101 ev for @xmath103 ev . for a fixed @xmath100 , the minimal value of , @xmath104 , is given by @xmath105 where again @xmath92 is the maximal allowed value of for the @xmath93 @xcite . the upper bound on , which corresponds to cp - conservation and @xmath106 ( @xmath30 possessing identical cp - parities ) , can be found for given @xmath8 by using eq . ( [ meffmaxhierlma ] ) . for the allowed values of @xmath8 , @xmath107 , we have @xmath108 . 1.0 cm if @xmath109 , the effective majorana mass is given in terms of the oscillation parameters , , @xmath82 and @xmath110 which is constrained by the chooz data @xcite : @xmath111 the new predictions for differ substantially from those obtained before the appearance of the latest sno data due to the existence of a significant lower bound on for every value of @xmath8 : even in the case of @xmath112 ( i.e. , even if @xmath113 ev ) , we get @xmath114 ( see fig . 1 , right panels ) . given the neutrino oscillation parameters , the minimal allowed value of depends on the values of the cp violating phases @xmath33 and @xmath34 . * case a : @xmath85 , @xmath115 . * 1.0 cm the effective majorana mass can be considerably larger than in the case of a hierarchical neutrino mass spectrum @xcite . the maximal value of corresponds to cp - conservation and @xmath116 , and for given @xmath8 reads : @xmath117 where @xmath89 and @xmath90 are the values corresponding to @xmath91 , and @xmath118 is the minimal allowed value of @xmath119 for the @xmath92 . for the allowed ranges - eqs . ( 1 ) and ( 2 ) for and @xmath94 , and the best fit values of the neutrino oscillation parameters , found in @xcite , we get @xmath120 and @xmath121 , respectively . there exists a non - trivial lower bound on in the case of the lma solution for which @xmath122 is found to be significantly different from zero . for the 99.73% c.l . allowed values of and @xmath122 @xcite , this lower bound reads : @xmath123 . using the best fit values of the oscillation parameters @xcite , we find : @xmath124 the lower bound is present even for @xmath125 : in this case @xmath126 . the minimal value of , @xmath127 , is reached in the case of cp - invariance and @xmath128 , and is determined by : @xmath129 where @xmath89 and @xmath90 are the values corresponding to @xmath91 , and is the maximal allowed value of @xmath130 for the . in the two other cp conserving cases of @xmath131 , the lower bound on depends weakly on the allowed values of @xmath82 and reads @xmath132 . if the neutrino mass spectrum is of the inverted hierarchy type , a sufficiently precise determination of , @xmath82 and ( or a better upper limit on ) , combined with a measurement of in the current or future -decay experiments , could allow one to get information on the difference of the majorana cp - violating phases @xmath133 @xcite . the value of @xmath134 is related to the experimentally measurable quantities as follows @xcite : @xmath135 ( @xmath85 ) . the constraints on @xmath134 one could derive on the basis of eq . ( [ alpha2131 ] ) are illustrated is not constrained in the case under discussion . even if it is found that @xmath136 , @xmath33 can be a source of cp - violation in @xmath137 processes other than -decay . ] in fig . 11 of ref . obtaining an experimental upper limit on of the order of 0.03 ev would permit , in particular , to get a lower bound on the value of @xmath134 and possibly exclude the cp conserving case corresponding to @xmath138 ( i.e. , @xmath139 ) . * case b : spectrum with partial inverted hierarchy ( @xmath140 ) . * 1.0 cm the discussion and conclusions in the case of the spectrum with partial inverted hierarchy are identical to those in the same case for the neutrino mass spectrum with normal hierarchy given in sub - section 2.1 , case b , except for the maximal and minimal values of , @xmath87 and @xmath141 , which for a fixed @xmath8 are determined by : @xmath142 @xmath118 ( @xmath143 ) in eq . ( [ maxmefflmainva ] ) ( in eq . ( [ maxmefflmainvb ] ) ) being the minimal ( maximal ) allowed value of @xmath119 given the maximal ( minimal ) value @xmath92 ( @xmath144 ) . for any @xmath100 , the lower bound on reads : @xmath145 . using the best fit values of the neutrino oscillation parameters , obtained in @xcite , one finds : @xmath146 . 1.0 cm the new element in the predictions for in the case of quasi - degenerate neutrino mass spectrum , @xmath147 , is the existence of a lower bound on the possible values of ( fig . the lower limit on is reached in the case of cp - conservation and @xmath149 . one finds a significant lower limit , @xmath145 , if @xmath150 more specifically , using the best fit value , and the @xmath151 c.l . and the @xmath152 c.l . allowed values , of @xmath24 from @xcite , we obtain , respectively : @xmath153 , @xmath154 and @xmath155 ( figs . 1 and 3 ) . these values of @xmath156 are in the range of sensitivity of the current and future -decay experiments . the upper bound on , which corresponds to cp - conservation and @xmath157 ( @xmath30 possessing the same cp - parities ) , can be found for a given @xmath8 by using eq . ( [ meffmaxhierlma ] ) . for the allowed values of @xmath147 ( which is limited from above by the @xmath158h @xmath11decay data @xcite , @xmath159 ) , @xmath87 is limited by the upper bounds obtained in the -decay experiments : @xmath160 @xcite and @xmath161 @xcite . in the case of cp conservation and @xmath162 , is constrained to lie in the interval @xcite @xmath163 . an upper limit on would lead to an upper limit on @xmath164 which is more stringent than the one obtained in the present @xmath44 @xmath45-decay experiments : for @xmath165 we have @xmath166 . furthermore , the upper limit @xmath167 would permit to exclude the cp - parity pattern @xmath162 for the quasi - degenerate neutrino mass spectrum . if the cp - symmetry holds and @xmath168 , there are both an upper and a lower limits on , @xmath169 . using eq . ( [ thlma ] ) and the results on @xmath170 from ref . @xcite , one finds @xmath171 . given the allowed values of @xmath24 , eq . ( [ thlma ] ) , the observation of the -decay in the present and/or future -decay experiments , combined with a sufficiently stringent upper bound on @xmath172 from the tritium beta - decay experiments , @xmath173 , would allow one , in particular , to exclude the case of cp - conservation with @xmath174 ( fig . 2 ) . for values of , which are in the range of sensitivity of the future -decay experiments , there exists a `` just - cp - violation '' region . this is illustrated in fig . 2 , where we show @xmath175 for the case of quasi - degenerate neutrino mass spectrum , @xmath147 , @xmath176 , as a function of @xmath122 . the `` just - cp - violation '' interval of values of @xmath177 is determined by @xmath178 taking into account eq . ( [ thlma ] ) and the existing limits on @xmath179 , this gives @xmath180 . information about the masses @xmath181 can be obtained in the katrin experiment @xcite . a rather precise determination of , @xmath182 , @xmath18 and @xmath83 would imply an interdependent constraint on the two cp - violating phases @xmath33 and @xmath34 @xcite ( see fig . 16 in @xcite ) . for @xmath183 ev , the cp - violating phase @xmath33 could be tightly constrained if @xmath170 is sufficiently small and the term in containing it can be neglected , as is suggested by the current limits on @xmath170 : @xmath184 the term which depends on the cp - violating phase @xmath34 in the expression for , is suppressed by the factor @xmath170 . therefore the constraint one could possibly obtain on @xmath185 is trivial ( fig . 16 in @xcite ) , unless @xmath186 . 1.0 cm the existence of a lower bound on in the cases of inverted mass hierarchy ( @xmath109 ) and quasi - degenerate neutrino mass spectrum , eqs . ( [ meffminimh ] ) and ( [ meffminqds ] ) , implies that the future -decay experiments might allow to determine the type of the neutrino mass spectrum ( under the general assumptions of 3-neutrino mixing and massive majorana neutrinos , -decay generated only by the ( v - a ) charged current weak interaction via the exchange of the three majorana neutrinos , neutrino oscillation explanation of the solar and atmospheric neutrino data ) . this conclusion is valid not only under the assumption that the -decay will be observed in these experiments and will be measured , but also in the case only a sufficiently stringent upper limit on will be derived . more specifically , as is illustrated in fig . 3 , the following statements can be made : 1 . a measurement of @xmath187 , would imply that the neutrino mass spectrum is of the quasi - degenerate type ( @xmath188 ) and that there are both a lower and an upper limit on @xmath8 , @xmath189 . the values of @xmath190 and @xmath191 are fixed respectively by the equalities @xmath192 and @xmath193 , where @xmath194 and @xmath195 are given by eqs . ( [ minmefflma ] ) and ( [ meffmaxhierlma ] ) ; 2 . if is measured and is found to lie in the interval @xmath196 , one could conclude that either + \i ) @xmath73 and the spectrum is of the quasi - degenerate type ( @xmath197 ) or with partial hierarchy ( @xmath140 ) , with @xmath198 , where the maximal and minimal values of @xmath8 are determined as in the _ case 1 _ ; + or that ii ) @xmath76 and the spectrum is quasi - degenerate ( @xmath199 ) or with partial inverted hierarchy ( @xmath107 ) , with @xmath200 and @xmath201 , where @xmath190 and @xmath191 are given by the equalities @xmath192 and @xmath193 , and @xmath194 and @xmath195 are determined by eqs . ( [ maxmefflmainvb ] ) and ( [ maxmefflmainva ] ) ; 3 . a measured value of satisfying @xmath202 , would imply that ( see fig . 3 ) either + \i ) @xmath73 and the spectrum is of quasi - degenerate type ( @xmath199 ) , with @xmath203 , or with partial hierarchy ( @xmath140 ) , + or that ii ) @xmath76 and the spectrum is quasi - degenerate ( @xmath199 ) , or with partial inverted hierarchy ( @xmath140 ) , or with inverted hierarchy ( @xmath85 ) , with only a significant upper bound on @xmath8 , @xmath204 , @xmath205 , where @xmath190 is determined by the equation @xmath192 , with @xmath194 given by eq . ( [ maxmefflmainvb ] ) ; 4 . a measurement or an upper limit on , @xmath206 , would lead to the conclusion that the neutrino mass spectrum is of the normal mass hierarchy type , @xmath73 , and that @xmath8 is limited from above by @xmath207 , where @xmath190 is determined by the condition @xmath192 , with @xmath194 given by eq . ( [ minmefflma ] ) . for the allowed values of the oscillation parameters ( at a given confidence level , fig . 3 ) , an upper bound on , @xmath208 , would imply an upper limit on @xmath8 , @xmath209 - fig . 3 , middle panel , and @xmath210 - fig . 3 , lower panel . for the best fit values of , , @xmath82 and @xmath64 , the bound @xmath211 would lead to a rather narrow interval of possible values of @xmath8 , @xmath212 ( fig . 3 , upper panel ) . thus , a measured value of ( or an upper limit on ) the effective majorana mass @xmath213 would disfavor ( if not rule out ) the quasi - degenerate mass spectrum , while a value of @xmath214 would rule out the quasi - degenerate mass spectrum , disfavor the spectrum with inverted mass hierarchy and favor the hierarchical neutrino mass spectrum . using the best fit values of @xmath215 , @xmath216 from @xcite and of @xmath63 and @xmath217 from @xcite , we have found that ( fig . 3 , upper panel ) : i ) @xmath218 in the case of neutrino mass spectrum with normal hierarchy , ii ) @xmath219 if the spectrum is with inverted hierarchy , and iii ) @xmath220 for the quasi - degenerate mass spectrum . therefore , if @xmath63 , @xmath215 and @xmath216 will be determined with a high precision ( @xmath221 uncertainty ) using the data from the minos , kamland and the solar neutrino experiments and their best fit values will not change substantially with respect to those used in the present analysis . ] , a measurement of @xmath132 would rule out a hierarchical neutrino mass spectrum ( @xmath86 ) even if there exists a factor of @xmath222 ( or smaller ) uncertainty in the value of due to a poor knowledge of the corresponding nuclear matrix element(s ) . an experimental upper limit of @xmath223 suffering from the same factor of @xmath222 ( or smaller ) uncertainty would rule out the quasi - degenerate mass spectrum , while if the uncertainty under discussion is only by a factor which is not bigger than @xmath224 , the spectrum with inverted hierarchy would be strongly disfavored ( if not ruled out ) . if the minimal value of @xmath216 inferred from the solar neutrino data , is somewhat smaller than that in eq . ( [ thlma ] ) , the upper bound on in the case of neutrino mass spectrum with normal hierarchy ( @xmath73 , @xmath113 ev ) might turn out to be larger than the lower bound on in the case of spectrum with inverted mass hierarchy ( @xmath76 , @xmath113 ev ) . thus , there will be an overlap between the regions of allowed values of in the two cases of neutrino mass spectrum at @xmath95 . the minimal value of @xmath122 for which _ the two regions do not overlap _ is determined by the condition : @xmath225 where we have neglected terms of order @xmath226 . for the values of the neutrino oscillation parameters used in the present analysis this `` border '' value turns out to be @xmath227 . let us note that @xcite if the -decay is not observed , a measured value of @xmath57 in @xmath44 @xmath45-decay experiments , @xmath228 ev , which is larger than @xmath190 , @xmath229 , where @xmath190 is determined as in the _ case 1 _ ( i.e. , from the upper limit on , @xmath192 , with @xmath194 given in eq . ( [ minmefflma ] ) ) , might imply that the massive neutrinos are dirac particles . if the -decay has been observed and measured , the inequality @xmath230 , would lead to the conclusion that there exist contribution(s ) to the -decay rate other than due to the light majorana neutrino exchange which partially cancel the contribution due to the majorana neutrino exchange . a measured value of , @xmath231 , and a measured value of @xmath57 or an upper bound on @xmath57 , such that @xmath232 , where @xmath191 is determined by the condition @xmath193 , with @xmath195 given by eq . ( [ maxmefflmainva ] ) , would imply that @xcite there are contributions to the -decay rate in addition to the ones due to the light majorana neutrino exchange ( see , e.g. , @xcite ) , which enhance the -decay rate . this would signal the existence of new @xmath137 processes beyond those induced by the light majorana neutrino exchange in the case of left - handed charged current weak interaction . 1.0truecm assuming 3-@xmath0 mixing and massive majorana neutrinos , we have analyzed the implications of the results of the solar neutrino experiments , including the latest sno data , which favor the lma msw solution of the solar neutrino problem with @xmath1 , for the predictions of the effective majorana mass in -decay . neutrino mass spectra with normal mass hierarchy , with inverted hierarchy and of quasi - degenerate type are considered . for @xmath233 , which follows ( at 99.73% c.l . ) from the analysis of the solar neutrino data performed in @xcite , we find significant lower limits on in the cases of quasi - degenerate and inverted hierarchy neutrino mass spectrum , @xmath234 ev and @xmath4 ev , respectively . if the neutrino mass spectrum is hierarchical ( with inverted hierarchy ) , the upper limit holds @xmath235 ev . correspondingly , not only a measured value of @xmath6 , but even an experimental upper limit on of the order of @xmath236 ev can provide information on the type of the neutrino mass spectrum ; it can provide also a significant upper limit on the mass of the lightest neutrino @xmath8 . further reduction of the lma solution region due to data , e.g. , from the experiments sno , kamland and borexino , leading , in particular , to an increase ( a decreasing ) of the current lower ( upper ) bound of @xmath237 can strengthen further the above conclusions . using the best fit values of @xmath215 , @xmath216 from @xcite and of @xmath63 and @xmath217 from @xcite , we have found that ( fig . 3 , upper panel ) : i ) @xmath218 in the case of neutrino mass spectrum with normal hierarchy , ii ) @xmath219 if the spectrum is with inverted hierarchy , and iii ) @xmath220 for the quasi - degenerate neutrino mass spectrum . therefore , if @xmath63 , @xmath215 and @xmath216 will be determined with a high precision ( @xmath221 uncertainty ) using the data from the minos , kamland and the solar neutrino experiments and their best fit values will not change substantially with respect to those used in the present analysis , a measurement of @xmath132 would rule out a hierarchical neutrino mass spectrum ( @xmath86 ) even if there exists a factor of @xmath222 uncertainty in the value of due to a poor knowledge of the corresponding nuclear matrix element(s ) . an experimental upper limit of @xmath223 suffering from the same factor of @xmath222 ( or smaller ) uncertainty would rule out the quasi - degenerate neutrino mass spectrum , while if the uncertainty under discussion is by a factor not bigger than @xmath224 , the spectrum with inverted hierarchy would be strongly disfavored ( if not ruled out ) . finally , a measured value of @xmath9 ev , which would imply a quasi - degenerate neutrino mass spectrum , combined with data on neutrino masses from the @xmath10h @xmath11decay experiment katrin ( an upper limit or a measured value ) , might allow to establish whether the cp - symmetry is violated in the lepton sector . after the work on the present study was essentially completed , few new global analyses of the solar neutrino data have appeared @xcite . the results obtained in @xcite do not differ substantially from those derived in @xcite ; in particular , the ( 99.73% c.l . ) minimal allowed values of @xmath237 in the lma solution region found in @xcite and in @xcite practically coincide . the best fit values of and @xmath237 found in @xcite also practically coincide , with @xmath238 lying in the interval ( 0.41 - 0.50 ) and @xmath239 . the authors of @xcite find a similar @xmath238 , but a somewhat larger @xmath240 . according to @xcite , @xcite and @xcite , the lower limit @xmath241 holds approximately at 94% c.l . , 90% c.l . and 81% c.l . , respectively . larger maximal allowed values of than that given in eq . ( [ dmsollma ] ) - of the order of @xmath242 ( 99.73% c.l . ) , have been obtained in the analyses performed in @xcite . the authors of @xcite used the full sno data on the day and night event spectra @xcite in their analyses , while the authors of @xcite did not use at all or used only part of these data . cleveland et al . , _ astrophys . j. _ * 496 * ( 1998 ) 505 ; y. fukuda _ et al . _ , _ phys . * 77 * ( 1996 ) 1683 ; v. gavrin , _ nucl . _ * 91 * ( 2001 ) 36 ; w. hampel _ et al . _ , _ phys . _ * b447 * ( 1999 ) 127 ; m. altmann _ et al . _ , _ phys . lett . _ * b490 * ( 2000 ) 16 . v. barger and k. whisnant , _ phys . _ * b456 * ( 1999 ) 194 ; h. minakata and o. yasuda , _ nucl . _ * b523 * ( 1998 ) 597 ; t. fukuyama _ _ , _ phys . rev . _ * d57 * ( 1998 ) 5844 and hep - ph/0204254 ; p. osland and g. vigdel , _ phys . * b520 * ( 2001 ) 128 ; d. falcone and f. tramontano , _ phys . rev . _ * d64 * ( 2001 ) 077302 ; t. hambye , hep - ph/0201307 . f. vissani , _ jhep _ * 06 * ( 1999 ) 022 ; m. czakon _ * b465 * ( 1999 ) 211 , hep - ph/0003161 and _ phys . rev . _ * d65 * ( 2002 ) 053008 ; h. v. klapdor - kleingrothaus , h. pas and a. yu . smirnov , _ phys . _ * d63 * ( 2001 ) 073005 ; h. minakata and h. sugiyama , _ phys . _ * b526 * ( 2002 ) 335 ; z. xing , _ phys . _ * d65 * ( 2002 ) 077302 ; n. haba , n. nakamura and t. suzuki , hep - ph/0205141 .

assuming 3-@xmath0 mixing and massive majorana neutrinos , we analyze the implications of the results of the solar neutrino experiments , including the latest sno data , which favor the lma msw solution of the solar neutrino problem with @xmath1 , for the predictions of the effective majorana mass in neutrinoless double beta - decay . neutrino mass spectra with normal mass hierarchy , with inverted hierarchy and of quasi - degenerate type are considered . for @xmath2 , which follows ( at 99.73% c.l . ) from the sno analysis of the solar neutrino data , we find significant lower limits on in the cases of quasi - degenerate and inverted hierarchy neutrino mass spectrum , @xmath3 ev and @xmath4 ev , respectively . if the spectrum is hierarchical the upper limit holds @xmath5 ev . correspondingly , not only a measured value of @xmath6 , but even an experimental upper limit on of the order of @xmath7 ev can provide information on the type of the neutrino mass spectrum ; it can provide also a significant upper limit on the mass of the lightest neutrino @xmath8 . a measured value of @xmath9 ev , combined with data on neutrino masses from the @xmath10h @xmath11decay experiment katrin , might allow to establish whether the cp - symmetry is violated in the lepton sector . * the sno solar neutrino data , neutrinoless double beta - decay and neutrino mass spectrum * s. pascoli and s. t. petcov _ scuola internazionale superiore di studi avanzati , i-34014 trieste , italy + _

the polarization of gluons in @xmath0 annihilation @xcite , in deep inelastic scattering @xcite and in quarkonium decays @xcite has been studied in a series of papers dating back to the early 80 s . several proposals have been put forward to measure the polarization of the gluon among which is the proposal to measure angular correlation effects in the splitting process of a polarized gluon into a pair of gluons or quarks @xcite . latter proposal has led to a beautiful confirmation of the presence of the three - gluon vertex in the @xmath0 data @xcite ( see also @xcite ) . the earlier calculations of the gluon s polarization in @xmath0 annihilations had been done for massless fermions which was quite sufficient for the purposes of that period @xcite . in the meantime the situation has changed in so far as the heavy top quark has been discovered whose production properties in @xmath0 annihilations will be studied in the proposed next - linear - collider ( nlc ) . typical running energies of the nlc would extend from @xmath1-threshold at about @xmath2 to maximal energies of about @xmath3 . it is quite clear that top mass effects can not be neglected in this energy range even at the highest c.m . energies . it is therefore timely to redo the calculations of @xcite for heavy quarks and to investigate the influence of heavy quark mass effects on the polarization observables of the gluon . as is usual we shall represent the two - by - two differential density matrix @xmath4 of the gluon with gluon helicities @xmath5 in terms of its components along the unit matrix and the three pauli matrices . accordingly one has @xmath6 where @xmath7 is the unpolarized differential rate and @xmath8 are the three components of the ( unnormalized ) differential stokes vector . specifying to @xmath9 we perform an azimuthal and polar averaging over the relative beam - event orientation . after azimuthal averaging the @xmath10-component of the stokes vektor @xmath11 drops out @xcite . one retains only the @xmath12- and @xmath13-components of the stokes vector which are referred to as the gluon s linear polarization in the event plane and the circular polarization of the gluon , respectively . in this report we study the differential energy distribution of the polarization of the gluon , differential with regard to the scaled gluon energy @xmath14 . after having integrated over the quark ( or antiquark ) energy the circular polarization of the gluon averages to zero due to @xmath15-invariance . the differential unpolarized and polarized rates ( with @xmath16 ) are then given by @xmath17 the notation @xmath18 stands for either @xmath7 or @xmath19 , and the same for @xmath20 ( the index @xmath21 is explained later on ) . this notation closely follows the one in @xcite where the nomenclature @xmath22 has been used to denote the total rate ( @xmath23 : unpolarized transverse , @xmath24 : longitudinal ) . the electro - weak cross section is written in modular form in terms of two building blocks . the first building block specifies the electro - weak model dependence through the parameters @xmath25 ( @xmath26 ) . they are given by @xmath27 where , in the standard model , @xmath28 , with @xmath29 and @xmath30 the mass and width of the @xmath31 and @xmath32 . @xmath33 are the charges of the final state quarks to which the electro - weak currents directly couple ; @xmath34 and @xmath35 , @xmath36 and @xmath37 are the electro - weak vector and axial vector coupling constants . for example , in the weinberg - salam model , one has @xmath38 , @xmath39 for leptons , @xmath40 , @xmath41 for up - type quarks ( @xmath42 ) , and @xmath43 , @xmath44 for down - type quarks ( @xmath45 ) . in this paper we use standard model couplings with @xmath46 . the second building block is determined by the hadron dynamics , i.e. by the current - induced production of a heavy quark pair with subsequent gluon emission . we shall work in terms of the components of the polarized and unpolarized hadronic tensor @xmath47 which are related to the differential rate by @xmath48 the index @xmath49 specifies the current composition in terms of the two parity - conserving products of the vector and the axial vector currents according to ( we drop all further indices on the hadron tensor ) @xmath50 eq . ( [ eqn2 ] ) gives the differential cross section for unpolarized beams . the case of longitudinally polarized beams can easily be included and leads to the replacement @xmath51 in the unpolarized and linearly polarized differential rates @xmath20 where the electroweak coefficients @xmath52 are given in eqs . ( [ eqn3 ] ) and where @xmath53 ( @xmath54 ) denote the longitudinal polarization of the electron ( positron ) . the various pieces of the hadronic tensor can be calculated from the relevant feynman diagrams . after integration over the quark ( or antiquark ) energy one obtains @xmath55,\nonumber\\ \tilde h_{u+l}^2(x)&=&\xi n\bigg[-6\left(\frac1x-1\right)\frac1xw_+(x ) + \left(3\frac{2-\xi}x-6-x\right)t_{\ell+}(x)\bigg],\nonumber\\ \tilde h_{u+l}^{1x}(x)&=&(4-\xi)n\bigg[-2\left(\frac1x-1\right)\frac1xw_+(x ) + \left(\frac{2-\xi}x-2\right)t_{\ell+}(x)\bigg],\nonumber\\ \tilde h_{u+l}^{2x}(x)&=&3\xi n\bigg[-2\left(\frac1x-1\right)\frac1xw_+(x ) + \left(\frac{2-\xi}x-2\right)t_{\ell+}(x)\bigg]\end{aligned}\ ] ] where @xmath56 , @xmath57 , @xmath58 with @xmath59 and @xmath60 , @xmath61 and @xmath62 the normalized linear polarization @xmath67 of the gluon is given by the normalized stokes vector components . one has @xmath68 in fig . 1(a ) we plot the linear polarization of the gluon as a function of the gluon s fractional energy @xmath69 for the top and charm quark cases ( @xmath70 ) . we use a c.m . energy of @xmath71 . at both ends of the spectrum the linear polarization of the gluon is fixed by general and model independent considerations . at the soft gluon end it is well - known that the linear polarization is 100% while at the hard end of the spectrum the linear polarization has to go to zero for the simple reason that one can no longer define a hadronic plane in this collinear configuration . these limits can be easily verified by taking the corresponding @xmath72 and @xmath73 limits in eqs . ( [ eqn4 ] ) . we have chosen to compare the polarization of the gluon in the top and charm quark cases at the same fractional energy @xmath74 . for a given fractional energy @xmath69 the linear polarization of the gluon is always higher in the top quark case than in the charm quark or mass zero case . contrary to this one finds a higher degree of polarization in the charm quark case than in the top quark case when comparing the linear polarization at _ fixed gluon energies_. however , a comparison at a fixed fractional energy of the gluon is more appropiate from a physics point of view in particular if one is interested in the average linear polarization of the gluon to be discussed later on . the linear polarization of the gluon remains above 50% for 85% of the avaliable energy range in the top quark case . as fig . 1(b ) shows , the rate for top quark production is strongly weighted towards smaller gluon energies where the linear polarization is large . we anticipate a large average linear polarization of the top quark . in the charm quark case the linear gluon polarization is already quite close to the zero mass case which , according to eq . ( [ eqn5 ] ) , is given by . ] @xmath75 the good quality of the zero mass formula eq . ( [ eqn6 ] ) when applied to the charm quark case ( @xmath76 and @xmath77 ) must be judged against the fact that the dominating logarithmic term @xmath78 is not yet overly large . the linear polarization is flavour independent ( and beam polarization independent ) in the zero mass limit since the flavour dependent @xmath79 factor drops out in the ratio eq . this is different in the massive case where flavour dependence comes in through the nonvanishing of the hadron tensor component @xmath80 . it can , however , be checked that the dependence on the electro - weak parameters is also quite weak in the massive quark case . the reason is two - fold . first the contributions of @xmath81 and @xmath82 are somewhat suppressed even for top quark pair production . secondly the rate is dominated by one photon exchange at the energy @xmath76 leading again to a near cancellation of the electro - weak model dependence . the last step is the integration over the second phase - space parameter @xmath12 . it is clear that we have to introduce a gluon energy cut - off at the soft end of the gluon spectrum in order to keep the rate finite . denoting the cut - off energy by @xmath83 the integration extends from @xmath84 to @xmath85 . we obtain @xmath86 \nonumber\\&&\qquad\qquad-4{{\cal g}}(1)+(4+\xi){{\cal g}_\ell}(1)\bigg],\nonumber\\ h_{u+l}^2&=&\xi n\bigg[-3\big[2{{\cal g}}(-1)-2{{\cal g}}(0)-(2-\xi){{\cal g}_\ell}(-1)+2{{\cal g}_\ell}(0)\big ] -{{\cal g}_\ell}(1)\bigg],\nonumber\\ h_{u+l}^{1x}&=&-(4-\xi)n\bigg[2{{\cal g}}(-1)-2{{\cal g}}(0)-(2-\xi){{\cal g}_\ell}(-1 ) + 2{{\cal g}_\ell}(0)\bigg],\nonumber\\ h_{u+l}^{2x}&=&-3\xi n\bigg[2{{\cal g}}(-1)-2{{\cal g}}(0)-(2-\xi){{\cal g}_\ell}(-1 ) + 2{{\cal g}_\ell}(0)\bigg]\end{aligned}\ ] ] using the integrals @xmath87 { { \cal g}}(-1)&=&-\ln\left(\frac{1+a}{1-a}\right ) -v\ln\left(\frac{v - a}{v+a}\right),\\[12pt ] { { \cal g}}(0)&=&\frac{\xi a}{1-a^2}-\frac\xi2\ln\left(\frac{1+a}{1-a}\right),\\[12pt ] { { \cal g}}(1)&=&\frac{\xi a}{4(1-a^2)^2}(4-\xi-(4+\xi)a^2 ) -\frac\xi8(4-\xi)\ln\left(\frac{1+a}{1-a}\right),\\[12pt ] { { \cal g}_\ell}(m)&:=&\int_{2\lambda}^{1-\xi}x^mt_{\ell+}(x)dx,\\[12pt ] { { \cal g}_\ell}(-1)&=&\frac12\ln\left(\frac{1-a^2}4\right)\ln\left(\frac{1+a}{1-a}\right ) -\ln\left(\frac{1+v}{1-v}\right)\ln\left(\frac{v - a}{v+a}\right)\nonumber\\ & & + { { \rm li}_2}\left(\frac{1+a}{2}\right)-{{\rm li}_2}\left(\frac{1-a}2\right ) + { { \rm li}_2}\left(-\frac{v+a}{1-v}\right)\\ & & -{{\rm li}_2}\left(-\frac{v - a}{1-v}\right)+{{\rm li}_2}\left(\frac{v - a}{1+v}\right ) -{{\rm li}_2}\left(\frac{v+a}{1+v}\right),\nonumber\\[12pt ] { { \cal g}_\ell}(0)&=&\frac\xi2\left(\frac{1+a^2}{1-a^2}\right ) \ln\left(\frac{1+a}{1-a}\right)-\frac{\xi a}{1-a^2},\\[12pt ] { { \cal g}_\ell}(-1)&=&\frac\xi{16(1-a^2)^2}(8 - 5\xi-6\xi a^2-(8 - 3\xi)a^4 ) \ln\left(\frac{1+a}{1-a}\right)\nonumber\\ & & + \frac{\xi a}{8(1-a^2)^2}(-8 + 5\xi+(8 - 3\xi)a^2)\end{aligned}\ ] ] where @xmath88 in fig . 2 we show a plot of the average linear polarization of the gluon as a function of the c.m . energy @xmath89 for three different cut - off values @xmath90 , @xmath91 and @xmath92 . gluon energies of this magnitude are sufficient to make the corresponding gluon jets detectable . because of the `` dead cone '' effect in the massive case , the gluon jet would be pointing away from the original top or antitop direction . the average linear polarization of the gluon rises steeply from threshold and quickly attains very high values around @xmath93 for the top quark case . in the charm quark case the average linear polarization is also large but is somewhat smaller than in the top quark case . the linear polarization @xmath94 becomes larger for smaller values of @xmath83 and tends to one as @xmath95 goes to zero . the approach to the asymptotic value @xmath96 is , however , rather slow . in the leading log aproximation as @xmath97 the linear polarization formula considerably simplifies . the leading log contributions can be easily identified in the terms @xmath98 and @xmath99 . they are obtained by setting @xmath100 in the other terms one can savely set @xmath101 as @xmath97 . we mention that the leading log representation of the linear polarization gives very accuarate numerical results for the above range of cut - off values except for energies close to theshold . for example , for top production and for @xmath102 the leading log result is @xmath103 below the full result at @xmath76 and @xmath104 at @xmath105 . in conclusion we have computed gluon polarization effects in the process @xmath106 . compared to the zero quark mass case the average linear polarization of the gluon is somewhat enhanced through quark mass effects . if one aims to study gluon polarization effects in the splitting process @xmath107 the present on - shell calcuation should be sufficient to identify and discuss the leading effects of gluon polarization without that one has to perform a full @xmath108 calculation of @xmath109 and @xmath110 @xcite . * acknowledgements : * this work is partially supported by the bmbf , frg , under contract no . 06mz865 , and by hucam , eu , under contract no . chrx - ct94 - 0579 . s.g . acknowledges financial support by the dfg , frg . the work of j.a.l . is supported by the daad , frg . olsen , p. osland and i. verb , + phys . lett . * 89 b * ( 1980 ) 221 ; nucl . * b192 * ( 1981 ) 33 j.g . krner and d.h . schiller , + desy preprint desy-81 - 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we calculate the linear polarization of gluons radiated off top quarks produced in @xmath0 annihilations . for typical top pair production energies at the next - linear - collider ( nlc ) the degree of linear polarization remains close to its soft gluon value of 100% over almost the whole energy spectrum of the gluon . the massive quark results are compared with the corresponding results for the massless quark case . mz - th/96 - 34 + hep - ph/9704416 + july 1997 + * gluon polarization in * + + institut fr physik , johannes - gutenberg - universitt , + staudinger weg 7 , d-55099 mainz , germany +

* statement 1 . * _ if conditions ( [ 1.1 ] ) are satisfied , then , for fixed external momenta @xmath72 and @xmath170 , the equality @xmath171 holds while the expressions appearing in ( [ a1 ] ) exist and the integral over @xmath69 on the right - hand side is absolutely convergent . it is assumed that the momenta of lines @xmath37 are expressed in terms of loop momenta @xmath38 , @xmath133 is the domain corresponding to the presence of full lines , type-1 lines , and type-2 lines ( the definitions are given following formula ( [ 22 ] ) ) , @xmath39 is the sphere of radius @xmath40 , where @xmath172 , and @xmath173 is a number depending on the diagram structure . _ let us prove the statement . for each type-1 line in ( [ a1 ] ) , we perform the following partitioning : @xmath174 . } \ ] ] then both sides of eq . ( [ a1 ] ) become the sum of expressions of the same form in which , however , the domain @xmath133 corresponds to the presence of only full and type-2 lines . it is clear that by proving the statement for this @xmath133 : ( which is done below ) , we prove the original statement as well . let @xmath175 be a domain such that the surfaces on which are not tangent to the boundary @xmath175 . first , we prove that in the expression @xmath176 the integral over @xmath69 is absolutely convergent ( here the integral over @xmath70 is finite because @xmath177 , @xmath178 ) . this becomes obvious ( considering conditions ( [ 1.1 ] ) and the fact that , in type-2 lines , the momentum @xmath60 is separated from zero ) if the contours of the integration over @xmath70 can be deformed in such a way that the momenta @xmath60 of the full lines are separated from zero by a finite quantity ( within the domain @xmath179 ) . in this case , we can repeat the well - known weinberg reasoning @xcite . what can prevent deformation is either a `` clamping '' of the contour or the point @xmath180 falling on the integration boundary . let us investigate the first alternative . we divide the domain of integration over @xmath47 into sectors such that the momenta of all full lines @xmath87 have a constant sign within one sector . let us examine one sector . we take a set of full lines whose @xmath60 may simultaneously vanish . in the vicinity of the point where @xmath60 from this set vanish simultaneously , we bend the contours of the integration over @xmath70 such that these contours pass through the points @xmath181 and the momenta @xmath60 of the type-2 lines do not change . let @xmath182 be such that @xmath183 for the lines from the set ( for @xmath184 from the sector under consideration ) . it is easy to check that this bending is possible . ( since the contours of integration over @xmath41 are bent and @xmath185 are expressed in terms of @xmath41 , one should only check that such @xmath186 exist , where the necessary @xmath182 are expressed in the same way , i.e. , that @xmath182 obey the conservation laws and flow only along the full lines ) . with this bending , rather small in relation to the deviation and the size of the deviation region , the contours do not pass through the poles because , for the denominator of each line from the set in question , we have @xmath187 and for the other denominators , the bending takes place in a region separated from the point where the corresponding momenta @xmath60 are equal to @xmath188 . repeating the reasoning for all sets , we can see that there is no contour `` clamping '' . the other alternative is excluded by the above condition for @xmath175 . to make this clear , one should introduce such coordinates @xmath189 in the @xmath38-space that the boundary of the domain @xmath190 is determined by the equation @xmath191 and then argue as above for the coordinates @xmath189 with @xmath192 . after bending the contours , integral ( [ a2 ] ) is absolutely convergent in @xmath47 , @xmath41 if tlie integration in @xmath47 is carried out within the sector under consideration . on pointing out that the result , of internal integration in ( [ a2 ] ) does not depend on the bending , we add the integrals over all sectors and conclude that ( [ a2 ] ) converges in @xmath69 absolutely . now let us prove that if @xmath193 is a quite small , finite vicinity of the point @xmath194 that lies outside the sphere @xmath39 , then expression ( [ a2 ] ) is equal to zero . we consider the momentum @xmath60 of one line . flowing along the diagram , it can ramify or it can merge with other momenta . clearly , two situations are possible : either it flows away completely through external lines , or , probably , after long wandering , part of it , @xmath195 , makes a complete loop . the former situation is possible only if @xmath196 , where all external momenta leaving the diagram ( but not entering it ) are summed . obviously , @xmath173 can be chosen such that for @xmath70 from @xmath175 , a line exists whose momentum violates this condition . the latter situation results in the existence of a loop , where the inequality @xmath197 holds for all momenta of its lines and the positive direction of the momenta is along the loop . then the integral over @xmath47 of the loop in question can be interchanged with the integrals over @xmath70 ( because it is absolutely and uniformly convergent for all @xmath41 ) and the residue formula can be used to perform this integration . since , for the loop in question , the momenta @xmath60 of the lines of this loop are separated from zero and are of the same sign , the result is zero . this has a simple physical meaning . if we pass to stationary noncovariant perturbation theory , we find that only quanta with positive @xmath9 can exist . in this case , external particles with positive @xmath198 are incoming and those with negative @xmath198 are outgoing . then , the momentum conservation law favors the occurrence of the first situation . the entire outside space for @xmath175 can be composed of the above domains @xmath39 ( everything converges well at infinity due to the factor @xmath199 ) . thus , on the left - hand side of ( [ a1 ] ) , one can substitute the integration domain @xmath200 for @xmath133 , set the limit in @xmath53 under the sign of integration over @xmath69 because of its absolute convergence , and also set the limit in @xmath54 under the integration sign because the domain of the integration over @xmath70 is bounded . thus , we obtain the right - hand side . the statement is proved . * statement 2 . * _ if @xmath133 corresponds to the presence of type-2 lines alone , then , under the same conditions as in statement 1 , the equality @xmath201 is valid . _ the proof of this statement is analogous to the second part of the proof of statement 1 . * statement . * _ if conditions ( [ 1.1 ] ) are satisfied , the limits in @xmath53 and @xmath54 in ( [ 18 ] ) can be interchanged ( in turn ) with the sign of the integral over @xmath55 and then with @xmath202 . _ to prove this , we define the vectors , , and , where the vector @xmath203 is built only from external momenta and @xmath204 is an @xmath205 matrix of rank @xmath206 , @xmath207 , @xmath208 . next , we introduce the following notation : @xmath209 then it follows from ( [ 19 ] ) that @xmath210 the function @xmath78 is a polynomial and we consider each of its terms separately . up to a factor , each term has the form @xmath211 . these derivatives act on @xmath212 and @xmath213 . the action on @xmath212 results in the constant factor @xmath214 , the action on @xmath213 results in the factor @xmath215 or @xmath216 ( the latter is the result of the action of two derivatives ; @xmath217 , @xmath218 , and @xmath219 are constant vectors ) . it is necessary to prove the correctness of the following three procedures : ( i ) setting the limit in @xmath53 under the integral sign for fixed @xmath178 ; ( ii ) setting the limit in @xmath54 for @xmath220 ; ( iii ) setting the limits in @xmath53 and @xmath54 under the signs of differentiation with respect to @xmath221 . in cases ( i ) and ( ii ) , one must obtain the bounds @xmath222 @xmath223 where @xmath224 and @xmath225 are functions integrable ( for @xmath224 if @xmath178 ) in any finite domain over @xmath226 , with @xmath227 . then , for case ( i ) , we have @xmath228 i.e. , a limit on the integrated function arises , and , thus , the limit in @xmath53 can be put under the integral sign . the situation is similar for case ( ii ) . it is evident from ( [ b4 ] ) that the function @xmath229 can be singular only if the eigenvalues of matrix @xmath230 become zero . on finding the lower bound of these eigenvalues , one can prove through rather long reasoning that bounds ( [ b5 ] ) , ( [ b6 ] ) exist if condition ( [ 1.1 ] ) is satisfied . after the limits in @xmath53 and @xmath54 are put under the integral sign , it is not difficult to interchange them with the differentiation with respect to @xmath221 . one need do it only for @xmath231 ( for each @xmath232 ) and , in this case , one can show that the eigenvalues of the matrix @xmath230 are nonzero and @xmath229 is not singular . soldati r. the mandelstam - leibbrandt prescription and the discretized light front quantization . in the book : theory of hadrons and light - front qcd , ed . by s.d . world scientific .

the relationship between the perturbation theory in light - front coordinates and lorentz - covariant perturbation theory is investigated . a method for finding the difference between separate terms of the corresponding series without their explicit evaluation is proposed . a procedure of constructing additional counter - terms to the canonical hamiltonian that compensate this difference at any finite order is proposed . for the yukawa model , the light - front hamiltonian with all of these counter - terms is obtained in a closed form . possible application of this approach to gauge theories is discussed . @xmath0 ) = -10 mm published in theoretical and mathematical physics , vol . 112 , no . 3 , 1997 . translated from teoreticheskaya i maternaticheskaya fizika . vol . 112 , no . 3 , + pp . 399 - 416 , september , 1997 . [ vvede ] startsectionsection10pt 3.5ex plus 1ex minus .2ex2.3ex plus .2ex***@xmath1 . introduction @xmath1 the hamiltonian approach to quantum field theory in light - front coordinates ( lf ) , , , is attractive as a possible method of solving strong interaction problems . in this approach , the formal triviality of the physical vacuum allows one to seek bound states without prior investigation of the complex vacuum structure . however , as is already known , canonical quantization in lf , i.e. , on the @xmath2 hypersurface , can result in a theory not quite equivalent to the lorentz - invariant theory ( i.e. , to the standard feynman formalism ) . this is due , first of all , to strong singularities at zero values of the `` light - like '' momentum variables . to restore the equivalence with a lorentz - covariant theory , one has to add unusual counter - terms to the formal canonical hamiltonian for the lf , @xmath3 ( the operator of a shift along the @xmath4-axis ) . these counter - terms can be found by comparing the perturbation theory based on the canonical lf formalism with lorentz - covariant perturbation theory . this is done in the present paper . the light - front hamiltonian thus obtained can then be used in nonperturbative calculations . it is possible , however , that perturbation theory does not provide all of the necessary additions to the canonical hamiltonian , as some of these additions can be nonperturbative . in spite of this , it seems necessary to examine this problem within the framework of perturbation theory first . for practical purposes a stationary noncovariant light - front perturbation theory , which is similar to the one applied in nonrelativistic quantum mechanics , is widely used . it was found @xcite that the `` light - front '' dyson formalism allows this theory to be transformed into an equivalent light - front feynman theory ( under an appropriate regularization ) . then , by re - summing the integrands of the feynman integrals , one can recast their form so that they become the same as in the lorentz - covariant theory . ( this is not the case for diagrams without external lines , which we do not consider here . ) then , the difference between the light - front and lorentz - covariant approaches that persists is only due to the different regularizations and different methods of calculating the feynman integrals ( which is important because of the possible absence of their absolute convergence in pseudo - euclidean space ) . in the present paper , we concentrate on the analysis of this difference . a light - front theory needs not only the standard uv regularization , but also a special regularization of the singularities @xmath5 . in our approach , this regularization eliminates the creation operators @xmath6 and annihilation operators @xmath7 with @xmath8 from the fourier expansion of the field operators in the field representation . as a result , the integration w.r.t . the corresponding momentum @xmath9 over the range @xmath10 is associated with each line before removing the @xmath11-functions . different propagators are regularized independently , which allows the described re - arrangement of the perturbation theory series . on the other hand , this regularization is convenient for further nonperturbative numerical calculations with the light - front hamiltonian , to which the necessary counter - terms are added ( the `` effective '' hamiltonian ) . we require that this hamiltonian generate a theory equivalent to the lorentz - covariant theory when the regularization is removed . note that lorentz - invariant methods of regularization ( e.g. , pauli - villars regularization ) are far less convenient for numerical calculations and we shall only briefly mention them . the specific properties of the light - front feynman formalism manifest themselves only in the integration over the variables @xmath12 , while integration over the transverse momenta @xmath13 is the same in the light - front and the lorentz coordinates ( though it might be nontrivial because it requires regularization and renormalization ) . therefore , we concentrate on a comparison of diagrams for fixed transverse momenta ( which is equivalent to a two - dimensional problem ) . in the present paper , we propose a method that allows one to find the difference ( in the limit @xmath14 ) between any light - front feynman integral and the corresponding lorentz - covariant integral without having to calculate them completely . based on this method , a procedure is elaborated for constructing an effective hamiltonian in lf in any order of perturbation theory . the procedure can be applied to all nongauge field theories , as well as to abelian and non - abelian gauge theories in the gauge @xmath15 with the vector meson propagator chosen according to the mandelstam - leibbrandt prescription @xcite . the question of whether the additional components of the hamiltonian that arise can be combined into a finite number of counter - terms must be dealt with separately in each particular case . application of this formalism to the yukawa model makes it possible to obtain the effective light - front hamiltonian in a closed form . the result agrees with the conclusions of the work @xcite , where a comparison was made of the light - front and lorentz - covariant methods via calculating self - energy diagrams in all orders of perturbation theory and other diagrams in lowest orders . conversely , for gauge theories ( both abelian and non - abelian ) , it was found that counter - terms of arbitrarily high order in field operators must be added to the effective hamiltonian . this result may turn out to be wrong if the contributions to the counter - terms are mutually canceled . this calls for further investigation , but such possibility appears to be very unlikely . what we have said above does not depreciate the light - front formalism as applied to gauge theories . this is because the only requirement concerning the light - front hamiltonian is that it correctly reproduces all gauge - invariant quantities rather than the off - mass - shell feynman integrals in a given gauge . however , renormalization of the light - front hamiltonian turns out to be a difficult problem and it requires new approaches . we do not examine the possibilities of changing the light - front hamiltonian by introducing new nonphysical fields by a method different from the pauli - villars regularization @xcite or the possibilities of using gauges more general than @xmath15 with the mandelstam - leibbrandt propagator . these points also need to be investigated further . [ integ ] startsectionsection10pt 3.5ex plus 1ex minus .2ex2.3ex plus .2ex***@xmath1 . reduction of light - front and lorentz - covariant @xmath1 + @xmath1feynman integrals to a form convenient for comparison @xmath1 let us examine an arbitrary ipi feynman diagram . we fix all external momenta and all transverse momenta of integration , and integrate only over @xmath16 and @xmath9 : @xmath17 we assume that all vertices are polynomial and that the propagator has the form @xmath18 where @xmath19 is a polynomial . a propagator of the second type in ( 2 ) arises in gauge theories in the gauge @xmath15 if the mandelstam - leibbrandt formalism @xcite with the vector boson propagator @xmath20 is used . in eq . ( [ 1 ] ) either @xmath21 , where @xmath22 is the particle mass , or @xmath23 . the function @xmath24 involves the numerators of all propagators and all vertices with the necessary @xmath11-functions , that include the external momenta @xmath25 ( the same expression without the @xmath11-functions is a polynomial , which we denote by @xmath26 ) . we assume for the diagram @xmath27 and for all of its subdiagrams that the conditions @xmath28 hold , where @xmath29 is the index of divergence w.r.t . @xmath16 at @xmath30 , and @xmath31 is the index of divergence in @xmath16 and @xmath9 ( simultaneously ) ; . the diagrams that do not meet these conditions should be examined separately for each particular theory ( their number is usually finite ) . we seek the difference between the value of integral ( [ 1 ] ) obtained by the lorentz - covariant calculation and its value calculated in light - front coordinates ( light - front calculation ) . in the light - front calculation , one introduces and then removes the light - front cutoff @xmath32 : @xmath33 where @xmath34 . here ( and in the diagram configurations to be defined below ) we take the limit w.r.t . @xmath35 , but , generally speaking , this limit may not exist . in this case , we assume that we do not take the limit , but take the sum of all nonpositive power terms of the laurent series in @xmath35 at the zero point . if conditions ( [ 1.1 ] ) are satisfied , statement 2 from appendix i can be used . this results in the equality @xmath36 from here on , the momenta of the lines @xmath37 are assumed to be expressed in terms of the loop momenta @xmath38 , @xmath39 is a sphere of a radius @xmath40 in the @xmath41-space , and @xmath40 depends on the external momenta . now , using statement 2 from appendix i , we obtain @xmath42 to reduce the covariant feynman integral to a form similar to ( [ 5 ] ) , we introduce a quantity @xmath43 : @xmath44 let us prove that this quantity coincides with the result of the lorentz - covariant calculation @xmath45 . to this end , we introduce the in the minkowski space of the propagator @xmath46 then we substitute ( [ 9 ] ) into ( [ 8 ] ) . due to the exponentials that cut off @xmath47 , @xmath41 and @xmath48 the integral over these variables is absolutely convergent . therefore , one can interchange the integrations over @xmath47 , @xmath41 and @xmath48 . as a result , we obtain the equality @xmath49 where @xmath50 } \bigr|_{y_i=0}. { \hfill ( \theform ) \cr}}\ ] ] for the lorentz - covariant calculation in the satisfying conditions ( [ 1.1 ] ) , there is a known expression @xcite @xmath51 where @xmath52 } \bigr|_{y_i=0}. { \hfill ( \theform ) \cr}}\ ] ] in appendix 2 , it is shown that in ( [ 18 ] ) the limits in @xmath53 and @xmath54 can be interchanged , in turn , with the integration over @xmath55 , and then with @xmath56 . after that , a comparison of relations ( [ 18 ] ) , ( [ 19 ] ) and ( [ 17 ] ) , ( [ 16 ] ) , clearly shows that @xmath57 . considering ( [ 8 ] ) and using statement 1 from appendix 1 , we obtain the equality @xmath58 expression ( [ 20 ] ) differs from ( [ 5 ] ) only by the range of the integration over @xmath41 . [ polos ] startsectionsection10pt 3.5ex plus 1ex minus .2ex2.3ex plus .2ex***@xmath1 . reduction of the difference between the light - front and@xmath1 + @xmath1lorentz - covariant feynman integrals to a sum of configurations @xmath1 let us introduce a partition for each line , @xmath59 . { \hfill ( \theform ) \cr}}\ ] ] we call a line with integration w.r.t . the momentum @xmath60 in the range @xmath61 ( before removing @xmath11-functions ) a type-1 line , a line with integration in the range @xmath10 a type-2 line , and a line with integration over the whole range @xmath62 a full line . in the diagrams , they are denoted as shown in figs . la , b , and c , respectively . fig1.pic let us substitute partition ( [ 22 ] ) into expression ( [ 5 ] ) for @xmath63 and open the brackets . among the resulting terms , there is @xmath64 ( expression ( [ 20 ] ) ) . we call the remaining terms `` diagram configurations '' and denote them by @xmath65 . then we arrive at the relation @xmath66 , where @xmath67 and @xmath68 is the region corresponding to the arrangement of full lines and type-1 lines in the given configuration . note that before taking the limit in @xmath35 , eqs . ( [ 20 ] ) and ( [ 24 ] ) can be used successfully : first , they are applied to a subdiagram and , then , are substituted into the formula for the entire diagram . this is admissible because , after the deformation of the contours described in the proof of statement 1 from appendix 1 , the integral over the loop momenta @xmath69 of the subdiagram converges ( after integration over the variables @xmath70 of this subdiagram ) absolutely and uniformly with respect to the remaining loop momenta @xmath71 . therefore , one can interchange the integrals over @xmath69 and @xmath71 . thus , the difference between the light - front and lorentz - covariant calculations of the diagram is given by the sum of all of its configurations . a configuration of a diagram is the same diagram , but where each line is labeled as a full or type-1 line , provided that at least one type-1 line exists . [ epsil ] startsectionsection10pt 3.5ex plus 1ex minus .2ex2.3ex plus .2ex***@xmath1 . behavior of the configuration as @xmath14 @xmath1 we assume that all external momenta @xmath72 are fixed for the diagram in question and @xmath73 where the summation is taken over any subset of external momenta ; all of these momenta are assumed to be directed inward . let us consider an arbitrary configuration . we apply the term `` @xmath35-line '' to all type-1 lines and those full lines for which integration over @xmath9 actually does not expand outside the domain @xmath74 , where @xmath75 is a finite number ( below , we explain when these lines appear ) . the remaining full lines are called @xmath76-lines . in the diagrams , the @xmath35-lines and @xmath76-lines are denoted as shown in figs . 1d and e , respectively . note that the diagram can be drawn with lines `` a '' and `` c '' from fig . 1 ( this defines the configuration unambiguously ) , or with lines `` d '' and `` e '' ( then the configuration is not uniquely defined ) . if among the lines arriving at the vertex only one is full and the others are type-1 lines , this full line is an @xmath35-line by virtue of the momentum conservation at the vertex . the remaining full lines form a subdiagram ( probably unconnected ) . by virtue of conditions ( [ d0 ] ) , there is a connected part to which all of the external lines are attached . all of the external lines of the remaining connected parts are @xmath35-lines . consequently , using statement 1 from appendix 1 , we can see that integration over the internal momenta of these connected parts can be carried out in a domain of order @xmath35 in size , i.e. , all of their internal lines are @xmath35-lines . thus , an arbitrary configuration can be drawn as in fig . 2 and integral ( [ 24 ] ) , with the corresponding integration domain , is associated with it . fig2.pic let us investigate the behavior of the configuration as @xmath14 . from here on , it is convenient to represent the propagator as @xmath77 rather than as ( [ 1.2 ] ) . then , in ( [ 1 ] ) , @xmath21 and the function @xmath78 is no longer a polynomial . if the numerator of the integrand consists of several terms , we consider each term separately ( except when the terms arise from expressing the propagator momentum @xmath60 in terms of loop and external momenta ) . we denote the loop momenta of subdiagram @xmath76 in fig . 2 by @xmath79 and the others by @xmath80 . we make following change of integration variables in ( [ 24 ] ) : @xmath81 then , the integration over @xmath82 goes within finite limits independent of @xmath35 . we denote the power of @xmath35 in the common factor by @xmath83 ( it stems from the volume elements and the numerators when the transformation ( [ d10 ] ) is made ) . the contribution to @xmath83 from the expression @xmath84 ( eq . ( [ d1 ] ) ) , which is related to the @xmath35-line , is equal to -1 . we divide the domain of integration over @xmath85 and @xmath86 into sectors such that the momenta of all full lines @xmath87 have the same sign within one sector . in statement 1 of appendix 1 , it is shown that for each sector , the contours of integration over @xmath88 and @xmath82 can be bent in such a way that absolute convergence in @xmath86 , @xmath85 , @xmath88 and @xmath82 takes place . since , in this case , the momenta @xmath60 of @xmath76-lines are separated from zero by an @xmath35-independent constant , the corresponding @xmath76-line - related propagators and factors from the vertices can be expanded in a series in @xmath35 . this expansion commutes with integration . it is also clear that the denominators of the propagators allow the following estimates under an infinite increase in @xmath89 : @xmath90 here @xmath91 and @xmath92 are @xmath35-independent constants . note that for fixed finite @xmath16 , the estimated expressions are bounded as @xmath93 . after transformation ( [ d10 ] ) and release of the factor @xmath94 ( in accordance with what was said about the contribution to @xmath83 ) , the @xmath35-line - related expression from ( [ d1 ] ) becomes @xmath95 where a @xmath16-independent quantity was used for the estimate ( this quantity is meaningful and does not depend on @xmath35 because the value of @xmath9 is separated from zero by an @xmath35-independent constant ) . we integrate first over @xmath86 , @xmath85 within one sector and then over @xmath88 , @xmath82 ( the latter integral converges uniformly in @xmath35 ) . let us examine the convergence of the integral over @xmath86 , @xmath85 with canceled denominators of the @xmath35-lines ( which is equivalent to estimating expressions ( [ d12 ] ) by a constant ) . if it converges , then the initial integral is obviously independent of @xmath35 and the contribution from this sector to the configuration is proportional to @xmath96 . let us show that if it diverges with a degree of divergence @xmath97 , the contribution to the initial integral is proportional to @xmath98 up to logarithmic corrections . to this end , we divide the domain of integration over @xmath86 , @xmath85 into two regions : @xmath99 , which lies inside a sphere of radius @xmath100 ( @xmath101 is fixed ) , and @xmath102 , which lies outside this sphere ( recall that in our reasoning , we deal with each sector separately ) . now we estimate ( [ d11 ] ) ( like ( [ d12 ] ) ) in terms of @xmath103 ( which is admissible ) and change the integration variables as follows : @xmath104 after @xmath35 is factored out of the numerator and the volume element , the integrand becomes independent of @xmath35 . thus , the integral converges . one can choose such @xmath101 ( independent of @xmath35 ) that the contribution from the domain @xmath102 is smaller in absolute value than the contribution from the domain @xmath99 . consequently , the whole integral can he estimated via the integral over the finite domain @xmath99 . now we make an inverse replacement in ( [ d13 ] ) and estimate ( [ d12 ] ) by a constant ( as above ) . since the size of the integration domain is @xmath100 and the degree of divergence is @xmath97 , the integral behaves as @xmath105 ( up to logarithmic corrections ) , q.e.d . this reasoning is valid for each sector and , thus , for the configuration as a whole . obviously , @xmath106 where @xmath107 is the subdiagram divergence index and the maximum is taken over all subdiagrams @xmath108 ( including unconnected subdiagrams for which @xmath107 is the sum of the divergence indices of their connected parts ) . in the case under consideration , @xmath109 , where @xmath110 is the number of internal @xmath35-lines in the subdiagram @xmath108 . the quantities @xmath111 are the uv divergence indices of the subdiagram @xmath108 w.r.t . @xmath112 . above , we introduced a quantity @xmath83 , which is equal to the power of @xmath35 that stems from the numerators and volume elements of the entire configuration . we can write @xmath113 , where @xmath114 is the index of the uv divergence in @xmath9 of a smaller subdiagram ( probably , a tree subdiagram or a nonconnected one ) consisting of @xmath76-lines entering @xmath108 . the term @xmath115 is the power of @xmath35 in the common factor , which , during transformation ( [ d10 ] ) , stems from the volume elements and numerators of the lines that did not enter @xmath108 . ( it is implied that the integration momenta are chosen in the same way as when calculating the divergence indices of @xmath108 . ) then , up to logarithmic corrections , we have @xmath116 consequently , for @xmath14 , the configuration is equal to zero if @xmath117 . relation ( [ 13.4 ] ) allows all essential configurations to be distinguished . [ ispra ] startsectionsection10pt 3.5ex plus 1ex minus .2ex2.3ex plus .2ex***@xmath1 . correction procedure and analysis of counter - terms @xmath1 we want to build a corrected light - front hamiltonian @xmath118 with the cutoff @xmath119 , which would generate green s functions that coincide in the limit @xmath120 with covariant green s functions within the perturbation theory . we begin with a usual canonical hamiltonian in the light - front coordinates @xmath121 with the cutoff @xmath119 . we imply that the integrands of the feynman diagrams derived from this light - front hamiltonian coincide with the covariant integrands after some resummation @xcite . however , a difference may arise due to the various methods of doing the integration , e.g. , due to different auxiliary regularizations . as shown in sec . [ polos ] , this difference ( in the limit @xmath120 ) is equal to the sum of all properly arranged configurations of the diagram . one should add such correcting counter - terms to @xmath121 , which generates additional `` counter - term '' diagrams , that reproduce nonzero ( after taking limit w.r.t . @xmath35 ) configurations of all of the diagrams . were we able to do this , we would obtain the desired @xmath122 . in fact , we can only show how to seek the @xmath122 that generates the green s functions coinciding with the covariant ones everywhere except the null set in the external momentum space ( defined by condition ( [ d0 ] ) ) . however , this restriction is not essential because this possible difference does not affect the physical results . our correction procedure is similar to the renormalization procedure . we assume that the perturbation theory parameter is the number of loops . we carry out the correction by steps : first , we find the counterterms to the hamiltonian that generate all nonzero configurations of the diagrams up to the given order and , then , pass to the next order . we take into account that this step involves the counter - term diagrams that arose from the counter - terms added to the hamiltonian for lower orders . thus , at each step , we introduce new correcting counter - terms that generate the difference remaining in this order . let us show how to successfully look for the correcting counter - terms . we call a configuration nonzero if it does not vanish as @xmath14 . we call a nonzero configuration `` primary '' if @xmath76 is a tree subdiagram in it ( see fig . 2 ) . note that for this configuration , breaking any @xmath76-line results in a violation of conditions ( [ d0 ] ) ; then , the resulting diagram is not a configuration . we say that the configuration is changed if all of the @xmath76-lines in the related integral ( [ 24 ] ) are expanded in series in @xmath35 ( see the reasoning above eq . ( [ d11 ] ) in sec . [ epsil ] ) and only those terms that do not vanish in the limit @xmath14 after the integration are retained . as mentioned above , developing this series and integration are interchangeable operations . thus , in the limit @xmath120 , the changed and unchanged configurations coincide . therefore , we always require that the hamiltonian counter - terms generate changed configurations , as this simplifies the form of the counter - terms . using additional terms in the hamiltonian , one can generate only counter - term diagrams , which are equal to zero for external momenta meeting the condition @xmath123 , because with the cutoff used ( see the introduction ) , the external lines of the diagrams do not carry momenta with @xmath123 . we bear this in mind in what follows . we seek counter - terms by the induction method . it is clear that , in the first order in the number of loops , all nonzero configurations are primary . we add the counter - terms that generate them to the hamiltonian . now , we examine an arbitrary order of perturbation theory . we assume that in lower orders , all nonzero configurations that can be derived from the counter - terms , accounting for the above comment , have already been generated by the hamiltonian . let us proceed to the order in question . first , we examine nonzero configurations with only one loop momentum @xmath124 and a number of momenta @xmath125 ( see the notation above eq . ( [ d10 ] ) ) . we break the configuration lines one by one without touching the other lines ( so that the ends of the broken lines become external lines ) . the line break may result in a structure that is not a configuration ( if conditions ( [ d0 ] ) are violated ) ; a line break may also result in a zero configuration or in a nonzero configuration . if the first case is realized for each broken line , then the initial configuration is primary and it must be generated by the counter - terms of the hamiltonian in the order under consideration . if breaking of each line results in either the first or the second case , we call the initial configuration real and it must be also generated in this order . assume that breaking a line results in the third case . this means that the resulting configuration stems from counter - terms in the lower orders . then , after restoration of the broken line ( i.e. , after the appropriate integration ) , it turns out that the counter - terms of the lower orders have generated the initial configuration ( we take into account the comment on successive application of eq . ( [ 24 ] ) ; see the end of sec . [ polos ] ) with the following distinctions : ( i ) the broken line ( and , probably , some others , if a nonsimply connected diagram arises after breaking the line ) is not a @xmath76-line but a type-2 line , due to the conditions @xmath126 ; ( ii ) if , after restoration of the broken line , the behavior at small @xmath35 becomes worse ( i.e. , @xmath127 decreased ) , then fewer terms than are necessary for the initial configuration were considered in the above - mentioned series in @xmath35 . we expand these arising type-2 lines by formula ( [ 22 ] ) and obtain a term where all of these lines are replaced by @xmath76-lines or other terms where some ( or all ) of these lines have become type-1 lines . in the latter case , one of the momenta @xmath125 becomes the momentum @xmath124 . we call these terms `` repeated parts of the configuration '' and analyze them together with the configurations that have two momenta @xmath124 . in the former case , we obtain the initial configuration up to distinction ( ii ) . we add a counter - term to the hamiltonian that compensates this distinction ( the counter - term diagrams generated by it are called the compensating diagrams ) . if there is only one line for which the third case is realized , it turns out that , in the given order , it is not necessary to generate the initial configuration by the counter - terms , except for the compensating addition and the repeated part that is considered at the next step . if there are several lines for which the third case is realized , the initial configuration is generated in lower orders more than once . for compensation , it should be generated ( with the corresponding numerical coefficient and the opposite sign ) by the hamiltonian counter - terms in the given order . we call this configuration a secondary one . next , we proceed to examine configurations with two momenta @xmath124 and so on up to configurations with all momenta @xmath124 , which are primary configurations . thus , the configurations to be generated by the hamiltonian counter - terms can be primary ( not only the initial primary configurations but also the repeated parts analogous to them , called primary - like ) , real , compensating , and secondary . if the theory does not produce either the loop consisting only of lines with @xmath16 in the numerator ( accounting for contributions from the vertices ) or a line with @xmath128 in the numerator for @xmath129 , then real configurations are absent because a line without @xmath16 in the numerator can always be broken without increasing @xmath127 ( see eq . ( [ 13.4 ] ) ) . it is not difficult to demonstrate that if each appearing primary , real , and compensating configuration has only two external line , then there are no secondary configurations at all . the dependence of the primary configuration on external momenta becomes trivial if its degree of divergence @xmath97 is positive , the maximum in formula ( [ d13.1 ] ) is reached on the diagram itself , and @xmath130 . then , only the first term is taken into account in the above - mentioned series . thus , not all of the @xmath76-line - related propagators and vertex factors depend on @xmath82 and they can be pulled out of the sign of the integral w.r.t . @xmath131 in ( [ 24 ] ) . we then obtain @xmath132 where @xmath133 is a domain of order @xmath35 in size . let us carry out transformations ( [ d10 ] ) and ( [ d13 ] ) . for the denominator of the @xmath76-line , we obtain @xmath134 here we neglect terms of order @xmath35 in the denominator because the singularity at @xmath135 is integrable under the given conditions for @xmath97 and everything can be calculated in zero order in @xmath35 at @xmath130 . thus , the dependence on external momenta can be completely collected into an easily obtained common factor . [ jukav ] startsectionsection10pt 3.5ex plus 1ex minus .2ex2.3ex plus .2ex***@xmath1 . application to the yukawa model @xmath1 the yukawa model involves diagrams that do not satisfy condition ( [ 1.1 ] ) . these are displayed in figs . 3a and b. we have @xmath136 for diagram `` a '' and @xmath137 for diagram `` b '' . fig3.pic nevertheless , these diagrams can be easily included in the general scheme of reasoning . to this end , one should subtract the divergent part , independent of external momenta , in the integrand of the logarithmically divergent ( in two - dimensional space , with fixed internal transverse momenta ) diagram `` a '' . we obtain an expression with @xmath138 ( i.e. , which converges in two - dimensional space ) and @xmath137 , as in diagram `` b '' . this means that the integral over @xmath139 converges only in the sense of the principal value ( and it is this value of the integral that should be taken in the light - front coordinates to ensure agreement with the stationary noncovariant perturbation theory ) . this value can be obtained by distinguishing the @xmath139-even part of the integrand . two approaches are possible . one is to introduce an appropriate regularization in transverse momenta and to imply integration over them ; then , it is convenient to distinguish the part that is even in four - dimensional momenta @xmath125 . the other is to keep all transverse momenta fixed ; then , the part that is even in longitudinal momenta @xmath140 can be released . for the yukawa theory , we use the first approach . for the transverse regularization , we use a `` smearing '' of vertices , which is equivalent to dividing each propagator by @xmath141 . in four - dimensional space , diagram `` a '' diverges quadratically . under introduction and subsequent removal of the transverse regularization , the divergent part , which was previously subtracted from this diagram , acquires the form @xmath142 . after separating the even part of the regularized expression , we fix all of the transverse momenta again . then it turns out that diagrams `` a '' and `` b '' in fig . 3 meet conditions ( [ 1.1 ] ) and one can show that after all of the operations mentioned , the exponent @xmath127 ( see ( [ 13.4 ] ) ) does not decrease for any of their configurations . hence , they can be included in the general scheme without any additional corrections . let us first analyze the primary configurations ( see the definition in sec . [ ispra ] ) . in the numerators , @xmath143 appears only in the zero or one power and there are no loops where the numerators of all of the lines contain @xmath143 . consequently , one always has @xmath144 , @xmath145 , and @xmath146 ( see the definitions in sec . [ epsil ] ) . analyzing the properties of the expression @xmath147 for the yukawa model diagrams , we conclude from ( [ 13.4 ] ) that @xmath148 always holds . the general form of the nonzero primary configurations with @xmath130 is depicted in fig . 4 . note that they are all configurations with two external line . fig4.pic further , it is clear that there are no nonzero real configurations ( see the comment at the end of sec . [ ispra ] ) , and it can be shown by induction that there are no nonzero compensating or secondary configurations either ( the definitions are given in sec . [ ispra ] also ) . thus , only primary or primary - like configurations can be nonzero and all of them have the form shown in fig . 4 . it can be shown that their degree of divergence @xmath97 is positive and the maximum in formula ( [ d13.1 ] ) is reached for the diagram itself . thus , the reasoning above and below formula ( [ d14 ] ) applies to them . then , denoting the configurations displayed in figs . 4a - d by , we arrive at the equalities , , and , where the expressions depend only on the masses and transverse momenta , but not on the external longitudinal momenta , and have a finite limit as @xmath14 . now we assume that @xmath149 @xmath150 are not single configurations but are the sums of all configurations of the same form and that integration over the internal transverse momenta has already been carried out , ( with the above - described regularization ) . in four - dimensional space , the diagrams @xmath149 and @xmath151 diverge linearly while @xmath152 and @xmath150 diverge quadratically . therefore , because of the transverse regularization , the coefficients @xmath153 and @xmath154 in the limit for removing this regularization take the form @xmath155 , where @xmath156 and @xmath157 do not depend on the external momenta ( neither do @xmath158 , @xmath159 ) ) . thus , to generate all nonzero configurations by the light - front hamiltonian , only the expression @xmath160 should be added , where @xmath161 and @xmath162 are the boson and fermion fields , respectively , and @xmath163 , are the constant coefficients . comparing ( [ u1 ] ) with the initial canonical light - front hamiltonian , one can easily see that the found counter - terms are reduced to a renormalization of various terms of the hamiltonian ( in particular the boson mass squared and the fermion mass squared without changing the fermion mass itself ) . the explicit lorentz invariance is absent , which compensates the violation of the lorentz invariance inherent , in the light - front formalism . note that in the framework of the second approach , mentioned at the beginning of this section , one can obtain the same results . the only difference is that in two - dimensional space , the contributions from the configurations displayed in fig . 3 would additionally depend on external transverse momenta however , this dependence disappears after integration over internal transverse momenta with the introduction and subsequent removal of an appropriate regularization . in the pauli villars regularization , it is easy to verify that the expression @xmath164 from ( [ 13.4 ] ) increases . this is because the number of terms in the numerators of the propagator increases . then , the contribution from the @xmath35-lines does not change , while the @xmath76-lines belonging to @xmath108 make zero contribution to @xmath147 and @xmath115 , but @xmath165 contribution to @xmath114 . since @xmath144 , this regularization makes it possible to meet the condition @xmath117 for the configurations that were nonzero ( one additional boson field and one additional fermion field are enough ) . then it turns out that the canonical light - front hamiltonian can not be corrected at all . [ kalib ] startsectionsection10pt 3.5ex plus 1ex minus .2ex2.3ex plus .2ex***@xmath1 . application to gauge theories @xmath1 let us consider a gauge theory ( e.g. , qed or qcd ) in the gauge @xmath15 . the boson propagator in the mandelstam - leibbrandt prescription has the form @xmath166 all of the above reasoning was organized such that it could be applied to a theory like this ( with fixed transverse momenta @xmath167 ) . it turns out that there are nonzero configurations with arbitrarily large numbers of external lines . an example of such a configuration is given in fig . 5 . fig5.pic indeed , using formula ( [ 13.4 ] ) , we can see that for the configuration in fig . 5 , @xmath168 and , thus , @xmath169 , i.e. , this is a nonzero configuration . it is also clear that introduction of the pauli - villars regularization does not improve the situation because it does not affect @xmath83 . thus , within the framework of the above - described method for correcting the canonical light - front hamiltonian of the gauge theory , an infinite number of counter - terms must be added to the hamiltonian . note , however , that the formulated conditions for the vanishing of the configuration are sufficient , but , generally speaking , not necessary . because of this and because of the possible cancellation of different configurations after integration w.r.t . transverse momenta , the number of necessary counter - terms may be smaller . the authors are thankful to e. v. prokhvatilov for the discussion of the paper and for the valuable comments . this investigation was supported by the russian foundation for basic research , grant no . 92 - 02 - 05520-a .

fix an isocrystal and the corresponding newton vector . mazur s inequality ( see @xcite , pg.658 , and the group - theoretic generalization in @xcite , theorem 4.2 ) states that , given a lattice in our isocrystal , the hodge vector associated to it is greater than or equal to the newton vector . conversely , any vector satisfying mazur s inequality is the hodge vector for some lattice ( see e.g. @xcite , proposition 4.2 , and @xcite , theorem 2 ) . the last result can be regarded as a statement for @xmath0 , but , in @xcite , 4 , r. kottwitz and m. rapoport formulated an analogous version for other groups and they proved it for @xmath2 and @xmath0 . they also formulated a combinatorial statement involving respective root systems that would imply the converse to mazur s inequality in each case ( see also sections 4.3 and 4.4 in @xcite , for a detailed version ) . c. lucarelli in @xcite , theorem 0.2 , showed that this combinatorial statement is true ( see proposition [ cathy ] below ) , and therefore a converse to mazur s inequality was deduced , for all split classical groups . we prove this combinatorial statement for @xmath1 ( theorem a ) and hence deduce a converse to mazur s inequality for @xmath1 . we also generalize the combinatorial result for @xmath0 ( theorem b ) , which in particular gives a new and simple proof of the converse to mazur s inequality for @xmath0 . the results obtained here are proved using the theory of toric varieties . in fact , part of our aim is to show how the combinatorial result mentioned above can be naturally treated as a vanishing result for higher cohomology groups of certain line bundles on toric varieties associated to root systems ( theorems c and d on section [ resultssection ] ) , which improves the previously known results for these varieties . this raises the question of new vanishing results for toric varieties in general ( see @xcite for work in this direction concerning toric varieties associated with root systems ) . it is worth mentioning that these toric varieties , where the fan of the variety is the weyl fan , appear naturally in the work of group compactifications ( see e.g. @xcite , pp.187206 , and references therein ) . to give a flavor of the type of problem that we deal with here , it is very instructive to use a picture ( see below ) to illustrate the statement of theorem e for n=3 . consider a toric variety @xmath3 with initial lattice ( i.e. , lattice of co - characters ) @xmath4 and the fan consisting of six maximal cones as pictured below in dotted lines . the dual lattice ( of characters ) is @xmath5 let @xmath6 be a globally - generated torus - equivariant line bundle on @xmath7 . then @xmath6 is completely determined by a set of characters ( one for each maximal cone in the fan ) , such that if @xmath8 and @xmath9 are two `` neighboring '' maximal cones , and @xmath10 and @xmath11 are their corresponding characters , then @xmath12 , with respect to the canonical pairing of @xmath13 and @xmath14 , is perpendicular to the common face of @xmath8 and @xmath9 , non - negative on @xmath8 , and non - positive on @xmath9 . this way @xmath6 determines a hexagon @xmath15 where the vertices of @xmath15 are the characters corresponding to @xmath6 , one for each maximal cone of the fan . ( in our picture all these characters are distinct and we have a true hexagon , but in general we may get some degenerate version of @xmath15 , where some of the vertices coincide . ) then @xmath16 is given by the number of @xmath13-points in the convex hull @xmath17 of @xmath15 . note that @xmath17 lies in @xmath18 . this should explain why we are using dotted lines for the cones of our initial fan they lie in the initial world , not the dual one . fix a root @xmath19 as in the picture . then the line - segment " @xmath20 corresponds to a ( non - torus - invariant ) divisor on @xmath7 . if we denote by @xmath21 the embedding @xmath22 , then @xmath23 equals the number of points that are both on @xmath20 and that are projections along the root @xmath19 of points in @xmath13 . our question is as follows : for any point @xmath24 on the line segment @xmath20 that is obtained by the projection along @xmath19 of some point @xmath25 ( @xmath26 is not necessarily in @xmath17 ) , can we always find a point @xmath27 that is in both @xmath13 and @xmath17 , and which maps to the initial point @xmath24 ? the answer is yes " and , because of the long - exact sequence @xmath28 where @xmath29 stands for the ideal sheaf of @xmath20 , it is equivalent to the statement that @xmath30 we should mention here that the above long - exact sequence has trivial entries to the right of @xmath31 , because higher cohomology groups vanish for globally - generated line bundles on complete toric varieties , and @xmath6 and @xmath32 are globally - generated on ( the complete toric varieties ) @xmath7 and @xmath20 , respectively . also , note that the map @xmath33 is induced by projecting along @xmath19 . in this paper we precisely formulate the above question , using co - characters and co - roots , for a split connected reductive group @xmath34 over a finite extension of @xmath35 , or equivalently , by passing to the ( langlands ) complex dual world , using characters and roots , for the corresponding group @xmath36 over @xmath37 . then we reformulate the question using certain toric varieties associated to the root system of @xmath36 ( where the fan of our variety is the weyl fan and the dual lattice ( i.e. , lattice of characters ) is that of characters for @xmath36 ) . we answer the question in the affirmative for @xmath38 and for a weaker version ( where only line bundles @xmath6 arising from weyl orbits are considered ) for @xmath39 . as mentioned at the beginning , this type of combinatorial question , without reference to toric varieties , was investigated by kottwitz and rapoport ( and others ) . if we use the language of toric varieties to describe what they did , then they considered only globally generated line bundles @xmath6 arising from weyl - orbits , i.e. , where the corresponding polygon @xmath15 is a weyl - orbit for some element in the character lattice . let us now describe how our paper is organized . in the first section we introduce notation , set up the problem and state our main results ( theorems a through e ) . we also show that theorem c is equivalent to theorem a. sections 2 and 3 are devoted to some useful results in computing cohomology groups for line bundles on our toric varieties using the so - called piece - wise linear continuous functions on the support of the respective fans . we use these results in the next section , where we prove theorems b , d , and e , and where we show how e implies d and d implies b. finally , the last section is devoted to the proof of theorem c. [ [ acknowledgments ] ] * acknowledgments : * + + + + + + + + + + + + + + + + + + the author is greatly indebted to his thesis adviser , professor robert kottwitz , for suggesting that he work on the problems dealt with in this paper , for numerous helpful comments , and for continuous encouragement and support . prior to @xcite , a converse to mazur s inequality for @xmath40 and @xmath2 was proved in @xcite , theorem 4.11 ( see also @xcite , theorem 2 , for the @xmath40 case ) , whose notation we follow . once a notation is introduced , it will be fixed for the rest of the paper . let @xmath41 be a finite extension of @xmath35 . denote by @xmath42 the ring of integers of @xmath41 . suppose @xmath34 is a split connected reductive group , @xmath43 a borel subgroup and @xmath44 a maximal torus in @xmath43 , all defined over @xmath42 . let @xmath45 be a parabolic subgroup of @xmath34 which contains @xmath43 , where @xmath46 is the unique levi subgroup of @xmath15 containing @xmath44 . we write @xmath47 for the set of co - characters @xmath48 . then @xmath49 and @xmath50 will stand for the quotient of @xmath47 by the co - root lattice for @xmath34 and @xmath46 , respectively . also , we let @xmath51 and @xmath52 denote the respective natural projection maps . let @xmath53 be @xmath34-dominant and let @xmath54 be the weyl group of @xmath44 in @xmath34 . the group @xmath54 acts on @xmath47 and so we consider @xmath55 and the convex hull of @xmath56 in @xmath57 , which we denote by @xmath58 . define @xmath59 let @xmath60 and write @xmath61 for the natural projection induced by @xmath62 . note that although @xmath63 is a quotient of @xmath47 , after tensoring with @xmath64 any possible torsion is lost and we can therefore consider @xmath65 as a subspace of @xmath66 ; we shall do so throughout the paper . the aim is to generalize the proposition below for @xmath38 and prove an analogous result for @xmath39 . kottwitz in @xcite ( sections 4.3 and 4.4 ) explains how , in each case , a converse to mazur s inequality follows from the result below and therefore from our results . [ cathy]_(cf . theorem 0.2 in @xcite ) _ let @xmath34 be a split connected reductive group over @xmath41 with every irreducible component of its dynkin diagram of type @xmath67 , or @xmath68 . with the notation as above , we have @xmath69 @xmath70 we are first going to phrase our results using arthur s notion of @xmath71-orthogonal sets ( see e.g. @xcite , pg . 217 ) . for more on these sets see for example @xcite , pp.441447 , whose notation we follow . a family of points @xmath72 in @xmath47 , one for every borel subgroup @xmath43 of @xmath34 that contains @xmath44 , is called a _ @xmath71-orthogonal set _ in @xmath47 if for every two adjacent borel subgroups @xmath73 , there exists an integer @xmath74 such that @xmath75 where @xmath76 is the unique co - root for @xmath44 that is positive for @xmath43 and negative for @xmath77 . similarly , if we have a point @xmath78 for each parabolic subgroup @xmath15 of @xmath34 that admits @xmath46 as a levi component , then the family @xmath79 is called a _ _ in @xmath63 ( see e.g. @xcite , pg.442 ) if for every two adjacent parabolic subgroups @xmath81 that admit @xmath46 as a levi component , there exists an integer @xmath82 so that @xmath83 here @xmath84 is the unique element in @xmath85 with the property that all the other elements of @xmath85 are positive multiples of it ; @xmath85 consists of the images in @xmath63 of the co - roots @xmath76 where @xmath19 is a root occurring in lie@xmath86 lie@xmath87 , where @xmath45 and @xmath88 , with @xmath89 being the parabolic subgroup containing @xmath46 opposite @xmath15 . if all the numbers @xmath74 ( resp . @xmath82 ) above are non - negative then we say that @xmath90 is a _ positive _ @xmath71-orthogonal set ( resp . @xmath79 is a _ positive _ @xmath80-orthogonal set ) . an important example of a positive @xmath71-orthogonal set arises from weyl orbits @xmath56 ( see e.g. @xcite , pg.443 ) . let @xmath91 , then we say that @xmath92 is dominant with respect to a borel group @xmath93 if for every root @xmath19 in lie@xmath94 we have that @xmath95 we get a positive @xmath71-orthogonal set by associating to every borel group @xmath43 the unique element @xmath96 that is both dominant with respect to @xmath43 and lies in @xmath56 . if @xmath90 is a @xmath71-orthogonal set in @xmath47 , then we get a @xmath80-orthogonal set as follows . the points @xmath90 , where @xmath43 is a borel subgroup containing @xmath44 and @xmath97 , form an @xmath98-orthogonal set in @xmath47 , and we get a point @xmath99 as the common image in @xmath63 of all the points in @xmath100 . the set of all such points @xmath101 , where @xmath15 is a parabolic that contains @xmath43 and admits @xmath46 as a levi component , is a @xmath80-orthogonal set in @xmath63 . moreover , if @xmath90 is positive , then @xmath79 obtained in this way is positive as well . we can now state our main results . let @xmath39 . with notation as above we have that @xmath102 for every @xmath71-orthogonal set @xmath90 that arises from a weyl orbit and its corresponding @xmath80-orthogonal set @xmath79 . let @xmath38 . with notation as above we have that @xmath102 for every positive @xmath71-orthogonal set @xmath90 and its corresponding @xmath80-orthogonal set @xmath79 . it is clear that the @xmath0 case of proposition [ cathy ] becomes a special case of theorem b when @xmath90 arises from a weyl orbit . also note that in both theorems a and b , the left - hand side is obviously contained in the right - hand side . the non - trivial part is to show the other containment . it is important to mention that , while we believe that theorem a is also true for the rest of the exceptional groups , theorem b is probably only true for a split connected reductive group with irreducible components of dynkin diagram of type @xmath103 . more precisely , it is easy to construct counterexamples to theorem b for other classical groups and for the @xmath1 case . at the end of section 5 we do so explicitly , but only for @xmath1 , since the construction for classical groups is completely analogous . we believe that the same construction should also provide counterexamples to theorem b for the other exceptional groups . the proof of these two theorems will involve the theory of toric varieties and we will freely use standard terminology and basic facts from this theory which appear in @xcite . let us first make clear the correspondence between @xmath71-orthogonal sets and line bundles on a certain projective nonsingular toric variety @xmath104 , which we now define ; we follow the exposition in @xcite , 23 , and keep the same notation as above . let @xmath36 and @xmath105 be the ( langlands ) complex dual group for @xmath34 and @xmath44 , respectively . let @xmath106 be the center of @xmath36 . then the fan of our toric variety @xmath104 is the weyl fan in @xmath107 and the torus is @xmath108 . ( we would like to remark that the toric variety @xmath104 appears naturally in the theory of group compactifications see e.g. @xcite , pp.187206 . ) clearly @xmath108 acts on @xmath104 , but we are interested in the action of @xmath105 on @xmath104 . the latter action is obtained using the canonical surjection @xmath109 and the action of @xmath108 on @xmath104 . as with any toric variety , there is a one - to - one correspondence between @xmath105-orbits in @xmath104 and cones in the fan of @xmath104 . in our case , because the fan is the weyl fan , this means that we have a one - to - one correspondence between @xmath105-orbits in @xmath104 and parabolic subgroups of @xmath34 that contain @xmath44 . according to this identification , if @xmath15 is a parabolic subgroup of @xmath34 that contains @xmath44 , then the corresponding @xmath105-orbit is given by @xmath110 , where @xmath46 is the unique levi component of @xmath15 containing @xmath44 and @xmath111 is the corresponding levi subgroup of @xmath36 containing @xmath105 . again , as with any toric variety , there is a one - to - one correspondence between maximal cones in the fan of @xmath104 and @xmath105-fixed points in @xmath104 . in our case , since maximal cones correspond to borel subgroups of @xmath34 that contain @xmath44 , we get a one - to - one correspondence between borel subgroups of @xmath34 containing @xmath44 and @xmath105-fixed points in @xmath104 . it is very important for us to note that there is a one - to - one correspondence between isomorphism classes of @xmath105-equivariant line bundles on @xmath104 and @xmath71-orthogonal sets in @xmath47 ( see e.g. @xcite , 23.1 ) . we describe the map that gives this correspondence . let @xmath6 be a @xmath105-equivariant line bundle on @xmath104 . then at each @xmath105-fixed point @xmath112 in @xmath104 the torus @xmath105 acts by a character , say , @xmath72 on the line in @xmath6 at @xmath112 , where @xmath43 is the borel subgroup corresponding to the fixed point @xmath112 . in this way , for every borel subgroup @xmath43 ( containing @xmath44 ) we get a character @xmath113 , i.e. , a co - character @xmath114 . the fact that @xmath90 is a @xmath71-orthogonal set comes from the fact that the characters defining @xmath6 must agree on the overlaps . a very useful remark is that positive @xmath71-orthogonal sets correspond to line bundles on @xmath104 which are generated by their sections , and hence their higher cohomology groups vanish . this last result follows from the more general fact that if the support of the fan ( i.e. , the union of all the cones in the fan ) of a toric variety is a convex set , then the higher cohomology groups vanish for line bundles ( on this variety ) which are generated by their sections ( see e.g. @xcite , pg.74 ) . remember that we are trying to reformulate theorems a and b in terms of toric varieties ; we have already described the toric analog of @xmath71-orthogonal sets ; the map @xmath115 corresponds to the map @xmath116 where @xmath117 stands for the root lattice for @xmath111 ( note that the codomain of the last map is just @xmath118 ) ; now we need to describe the toric analog of @xmath80-orthogonal sets . we continue to have the same notation . then for @xmath46 as above ( a levi subgroup containing @xmath44 ) , we will need a toric variety @xmath119 for the torus @xmath120 ( see e.g. @xcite , 23.2 ) . first assume that the group @xmath36 is adjoint ; we can do this since the center of @xmath121 in @xmath122 is equal to @xmath120 ) . then @xmath123 is a subtorus of @xmath105 and so @xmath124 is a subgroup of @xmath125 . the collection of cones from the weyl fan inside @xmath126 that lie in the subspace @xmath127 gives a fan . this is the fan for the complete , nonsingular , projective toric variety @xmath119 . now we go back to the general case where we no longer assume that @xmath36 is adjoint . denote by @xmath128 the simply connected cover of the derived group of @xmath34 and by @xmath129 the levi subgroup in @xmath128 that is obtained as the inverse image of @xmath46 under the map @xmath130 . then we have that @xmath131 , @xmath132 , and @xmath133 . from the adjoint case , we can define the toric variety @xmath134 for @xmath133 . we can therefore view @xmath134 as a space on which @xmath123 acts using the canonical map @xmath135 . it is this toric variety , which sits inside @xmath104 as a closed @xmath123-stable subspace , that we denote by @xmath119 . as with the toric variety @xmath104 , it is easily seen that @xmath123-fixed points in @xmath119 are in one - to - one correspondence with parabolic subgroups of @xmath34 that admit @xmath46 as a levi component . also , there is a one - to - one correspondence between isomorphism classes of @xmath123-equivariant line bundles on @xmath119 and @xmath80-orthogonal sets in @xmath63 . this correspondence is defined is the same way as the one for @xmath104 ( see e.g. @xcite , 23.4 ) . we note that ( see e.g. @xcite , 23.4 ) restricting a @xmath105-equivariant line bundle @xmath6 on @xmath104 to @xmath119 corresponds the procedure described earlier in this section of obtaining a @xmath80-orthogonal set ( the one corresponding to the restriction @xmath136 ) from a @xmath71-orthogonal set ( the one corresponding to @xmath6 ) . let us now assume that our parabolic subgroup @xmath15 is of semisimple rank 1 . this implies that the root lattice @xmath117 is just @xmath137 for a unique , up to a sign , root @xmath19 of @xmath36 , and that the toric variety @xmath119 , which we now denote by @xmath20 , is a non - torus - invariant divisor in @xmath104 . the map @xmath115 will now be denoted by @xmath138 . by tensoring with @xmath64 we get a map from @xmath138 , which we still denote by @xmath139 since tensoring with @xmath64 will loose any possible torsion , we can identify the codomain of the last map with the co - root hyperplane @xmath140 : = \{x \in x^*(\hat{t})\otimes_{\mathbb{z}}\mathbb{r } : \langle \alpha^{\vee},x \rangle = 0 \ } , \ ] ] where @xmath141 is the canonical pairing between co - characters and characters , and @xmath76 stands for the co - root of @xmath36 corresponding to @xmath19 . we will therefore say that the map @xmath138 is the projection along @xmath19 onto the co - root hyperplane @xmath142 $ ] . also , the fan of @xmath20 is contained in the root hyperplane @xmath143 : = \ { x \in x_*(\hat{t } ) \otimes_{\mathbb{z}}\mathbb{r } : \langle x , \alpha \rangle = 0 \}.\ ] ] now let @xmath6 be a @xmath105- line bundle on @xmath104 that is generated by its sections . then we have a short - exact sequence of sheaves on @xmath104 : @xmath144 where @xmath29 is the ideal sheaf of @xmath20 and @xmath21 is the inclusion map @xmath145 . note that @xmath146=0 , for all @xmath147 , since @xmath6 is generated by its sections and @xmath104 is projective , and also @xmath148 for all @xmath147 , since @xmath32 is generated by its sections and @xmath20 is a projective toric variety . therefore the short - exact sequence gives rise to the long - exact sequence @xmath149 so , we see that surjectivity of the map @xmath33 is equivalent to @xmath150 let @xmath39 . we claim that the following result is equivalent to theorem a. with notation as above , we have that @xmath151 whenever @xmath6 arises from a weyl orbit . theorem c is true for split classical groups . in fact , in the same way that we will show the equivalence of theorems a and c one can show that theorem c for a split classical group is equivalent to proposition [ cathy ] for the case of parabolic subgroups @xmath15 of semisimple rank 1 . let us demonstrate the equivalence between theorems a and c. put @xmath39 . suppose that @xmath6 corresponds to the weyl - orbit orthogonal set @xmath90 in @xmath47 , i.e. in @xmath152 , and , as before , denote by @xmath79 the corresponding @xmath80-orthogonal set in @xmath63 , i.e. in @xmath118 . write @xmath153 for the intersection of @xmath152 with the convex hull @xmath154 of @xmath90 . then we have ( see e.g. @xcite , pg.66 ) @xmath155 where the section @xmath156 is an eigenvector corresponding to the character @xmath157 . ( recall that @xmath105 acts on @xmath158 and so @xmath158 decomposes according to the characters of @xmath105 . ) assume that @xmath19 is the root of @xmath36 , up to a sign , that corresponds to @xmath111 , in the sense that the projection map @xmath159 $ ] along the root @xmath19 corresponds to the projection map @xmath115 . then we write @xmath160 for the intersection of @xmath161 with @xmath162 and we get @xmath163 the map @xmath164 from the long - exact sequence above is given by @xmath165 . this explains why theorem c is equivalent to theorem a , since @xmath138 corresponds to @xmath115 . now let @xmath38 . theorem b will follow from the following result . with notation as above , we have that @xmath166 whenever @xmath6 is generated by its sections . theorem d itself will be deduced ( in section 4.5 ) from the result for @xmath167 : with notation as above , we have that @xmath168 whenever @xmath6 is generated by its sections . we note that while theorems a and c were seen to be equivalent , to show that theorems b and d imply each other will take a little more work ( because we are no longer working with a two - dimensional space ! ) ; however , the same argument above for @xmath39 applies in our case to show that theorem b for the case of parabolic subgroups @xmath15 of semisimple rank 1 is equivalent to theorem d. but , we will deduce our theorem b _ from _ theorem d , where the latter will be proved using toric geometry . we explain in section [ proof of theorem b ] how the general case is handled , but here let us just show , using a similar method used there , how theorem d , i.e. surjectivity of @xmath33 , implies theorem b for all @xmath169 , where @xmath170 and only one of @xmath171 s is equal to 2 and the rest are equal to 1 . assume that @xmath172 for some @xmath173 and that @xmath174 for all @xmath21 such that @xmath175 and @xmath176 . keeping the same notation as above , we have in this case @xmath177 and @xmath178 where , as mentioned earlier , we have identified @xmath65 as a subspace of @xmath66 . also , the map @xmath115 , which in general is given by averaging over each of the @xmath179 `` batches '' , now agrees with @xmath138 where @xmath180 ( with 1 in the @xmath181-th place and -1 in the @xmath182-st place ) : @xmath183 ( in general @xmath115 does not correspond to one @xmath138 , rather , it is a composition of a number of @xmath138 s , for distinct roots @xmath19 . ) as in the @xmath1 case , we find that @xmath155 @xmath184 and @xmath185 . then surjectivity of @xmath33 clearly implies theorem b for this particular case . as we mentioned in the introduction , the next sections are devoted to proving our results . it is now safe for the reader to assume that , for the rest of the paper , we are working over the complex numbers we will be in the toric geometry setting . also , since we will be working in the complex dual world , when we use terms root , co - root , character , and co - character , they will always refer to those terms for the complex dual groups @xmath36 and @xmath105 . before we end this section it is worth mentioning that vanishing results like the ones in theorems c and d ( and e ) are also of independent interest just from a toric - variety point of view . a very important vanishing result for toric varieties has been proved by musta ( see e.g. @xcite , theorem 0.1 ) and for the particular toric varieties arising from @xmath0 and @xmath1 theorems c and d ( and e ) give vanishings of higher cohomology groups for more line bundles on these varieties . ( see also @xcite for more vanishing results for toric varieties associated with root systems . ) we keep the same notation as in the previous section . let @xmath186 be the non - singular , projective toric variety corresponding to @xmath34 as described in section [ resultssection ] and let @xmath187 be the fan of @xmath7 . each element of @xmath187 corresponds to a unique torus orbit in @xmath7 , and under this identification , if we denote by @xmath188 the set of rays in @xmath187 and if @xmath189 , we write @xmath190 for the closure of the orbit of @xmath191 , a @xmath105-cartier divisor ( cf . @xcite,3.1 and 3.3 ) . in fact , @xmath105-cartier divisors on @xmath7 can be written as a linear combination of @xmath190 s . from now on let us agree to refer to @xmath105-cartier divisors simply as divisors . ( no confusion should arise from this agreement even though our divisor @xmath20 is not a @xmath105-divisor ; in fact , thanks to lemma [ dalpha2 ] below , when computing cohomology groups , we will deal with a @xmath105-cartier divisor instead of @xmath20 . ) let us mention another way of characterizing divisors on toric varieties ( see e.g. @xcite , 3.4 , but be aware of a sign difference ) . a divisor @xmath192 on a toric variety @xmath7 , with fan @xmath187 , is completely determined by a set of characters @xmath10 , one for every element @xmath193 , such that they `` agree '' on the overlaps , i.e. , such that they form a @xmath71-orthogonal set ( we will usually call these just orthogonal sets ; here and throughout this paper , @xmath194 stands for the set of maximal cones in @xmath187 ) . this orthogonal set in turn gives a continuous ( integral ) piece - wise linear function @xmath195 on the _ support _ @xmath196 of the fan , where @xmath197 , for all @xmath198 . and conversely , given a continuous ( integral ) piece - wise linear function @xmath199 on @xmath200 , i.e. , a continuous function that is linear and given by an element of the character lattice on each cone of @xmath187 , we get a divisor by considering the characters @xmath10 , @xmath193 , which define @xmath199 on the maximal affine pieces of @xmath7 . ( the continuity of @xmath199 guarantees that the characters defining it agree on the overlaps and hence give a divisor . ) from what we said , it is easy to see that if @xmath201 , then the corresponding piece - wise linear function is given by @xmath202 @xmath203 where , for each @xmath8 , the character @xmath10 is found by solving the system of equations @xmath204 where @xmath191 varies through the rays in the cone @xmath8 and @xmath205 stands for the primitive lattice element in the ray @xmath191 . let us mention that @xmath206 is called _ convex _ or _ lower convex _ if for any @xmath207 and @xmath208 , we have that @xmath209 . the following result is going to be very useful . [ fultontheorem ] * _ ( cf . @xcite , pg . 68 ) _ let @xmath192 be a cartier divisor on @xmath7 . then @xmath210 is generated by its sections if and only if @xmath206 is ( lower ) convex . * _ ( cf . @xcite , pg.74 ) _ for every @xmath211 there exist canonical isomorphisms @xmath212 and @xmath213 where @xmath214 denotes the @xmath157-eigenspace that is obtained by the action of @xmath105 on @xmath215 and @xmath216 . keep the same notation as above and let @xmath19 be a root of @xmath36 . the following result was suggested to us by r. kottwitz . [ dalpha ] for the toric variety @xmath186 , we have that @xmath20 and @xmath217 are linearly equivalent divisors . this follows from considering the rational function @xmath218 on @xmath104 and computing the corresponding principal divisor . note that the divisor of zeros corresponds to @xmath20 . since we will be interested in computing the cohomology group @xmath219 , the last lemma allows us to focus instead on @xmath220 the lemma below describes the orthogonal set corresponding to the @xmath105-equivariant divisor @xmath221 , which , due to the previous lemma , will be referred to as the orthogonal set corresponding to @xmath20 . [ dalpha2 ] let @xmath19 be a root of @xmath36 and consider @xmath104 . the corresponding orthogonal set for @xmath20 is given by @xmath222 , where @xmath223 if @xmath224 on @xmath8 , and @xmath225 if @xmath226 on @xmath8 . the result follows at once from lemma [ dalpha ] and the fact that , as we discussed earlier , if @xmath201 , then @xmath10 s can be found by solving the system of equations @xmath227 , where @xmath189 varies through the rays of @xmath8 . [ importantlemma ] consider @xmath104 and let @xmath19 be a root of @xmath36 . suppose @xmath228 is generated by its sections . if one of the conditions _ ( i ) _ or _ ( ii ) _ below is satisfied , then the 0-th eigenspace @xmath229 * @xmath230 * @xmath231 and + @xmath232 first note that using proposition [ fultontheorem ] , part b ) , it is sufficient ( and necessary ) to show that @xmath233 where @xmath234 looking at the long - exact sequence we get from the pair @xmath235 , since @xmath236 is a vector space ( @xmath104 is complete ) , we only need to show that @xmath237 is path - wise connected . let us do that . according to lemma [ dalpha2 ] , @xmath237 is equal to the union of sets @xmath238 = \left\ { v\in |\delta| : \psi_{d}(v ) < 0\ , \emph{and } \left\langle \alpha , v\right\rangle \leq 0 \right\}\ ] ] and @xmath239 = \left\ { v\in |\delta| : \psi_{d - d_{\alpha}}(v ) < 0\ , \emph{and } \left\langle \alpha , v\right\rangle \geq 0 \right\}.\ ] ] here we have written @xmath240 $ ] and @xmath241 $ ] for the sets @xmath242 and @xmath243 , respectively . proposition [ fultontheorem ] , part a ) , implies that sets ( [ eqn : psid ] ) and ( [ eqn : psidalpha ] ) are convex . let us see how this works in the case of ( [ eqn : psidalpha ] ) . let @xmath112 and @xmath244 be in ( [ eqn : psidalpha ] ) and let @xmath245 . then using lemma [ dalpha2 ] . we have @xmath246 which , since @xmath206 is convex , is not greater than @xmath247 and the last expression is less than zero since @xmath248 , @xmath249 and @xmath245 . clearly , from our assumptions , we also have that @xmath250 $ ] , so ( [ eqn : psidalpha ] ) is a convex set , as desired . if condition ( i ) of the lemma is satisfied , then @xmath237 equals the set ( [ eqn : psidalpha ] ) and hence ( [ eqn : vanish ] ) holds , because the set ( [ eqn : psidalpha ] ) is convex . suppose that ( ii ) is satisfied . since @xmath251 is convex , this implies that there exists a ( non - empty , convex ) subset @xmath252 in @xmath253 \cap [ \alpha \geq 0 ] $ ] that is contained in both sets appearing in ( ii ) . we claim that @xmath252 is then contained in both sets ( [ eqn : psid ] ) and ( [ eqn : psidalpha ] ) . this claim is sufficient to prove that @xmath237 is path - wise connected ( and hence ( [ eqn : vanish ] ) holds ) since we already saw that ( [ eqn : psid ] ) and ( [ eqn : psidalpha ] ) are convex . first , @xmath252 is contained in ( [ eqn : psid ] ) by our assumption that it is contained in both sets appearing in ( ii ) , and in particular the second one . second , since @xmath252 is contained in the first set appearing in ( ii ) , we see that @xmath254 and @xmath255 , for all @xmath256 . therefore , using lemma [ dalpha2 ] , we see that @xmath257 thus @xmath252 is contained in ( [ eqn : psidalpha ] ) as well , which ends the proof of the claim and hence of the lemma . note that if we knew that ( [ eqn : vanish ] ) is true for _ all _ globally generated divisors @xmath192 , then we could conclude that for those divisors we have @xmath258 for _ all _ @xmath157 . indeed , to prove this , consider the ( globally generated ) divisor @xmath259 associated to the orthogonal set @xmath260 , where @xmath261 corresponds to @xmath192 . the result follows by using ( [ eqn : vanish ] ) ( which is assumed to be true for all globally generated divisors ) , where instead of @xmath237 we have @xmath262 . using proposition [ fultontheorem ] , part b ) , we deduce that if ( [ eqn : vanish ] ) holds for all globally generated divisors @xmath192 , then @xmath263 . an important consequence of this is that theorem d needs to be proved only in the case when the set ( [ eqn : psid ] ) is non - empty and @xmath264 = \varnothing.\ ] ] more concretely , it suffices to show that , under these conditions , @xmath265 = \varnothing , \ ] ] because then we get that @xmath237 is equal to the set ( [ eqn : psid ] ) , which we know is a convex set , and we can apply the same reasoning as in lemma [ importantlemma ] to conclude that ( [ eqn : vanish ] ) is true , and by what we just wrote , @xmath266 is true as well . consider @xmath104 , where @xmath267 ( and so @xmath268 ) . we want to concretely describe the fan @xmath187 of @xmath104 . but first , the initial lattice ( i.e. lattice of characters ) for @xmath104 is @xmath269 and so the dual one is @xmath270 for every @xmath271 we put @xmath272 for ( the element represented by ) @xmath273 , where 1 appears only on the @xmath21-th place . identify the rays in @xmath187 with their corresponding minimal lattice points . then @xmath274 consists of the sums of the form @xmath275 , where @xmath276 s are distinct elements from @xmath277 and @xmath278 . the maximal cones in @xmath104 are the n - dimensional cones whose rays are of the type @xmath279 where @xmath280 are distinct elements from @xmath281 . let @xmath19 be a root of @xmath36 . we can assume that @xmath282 is simple ( for a choice of a borel subgroup ) and , with a change of coordinates , we can always ensure that @xmath19 is written as @xmath283 . therefore , without loss of generality , throughout this section we let @xmath284 . the proof of theorem e will be proved by induction , although we begin by giving the proof for the first two nontrivial cases , @xmath285 and @xmath286 . this may seem a little unusual , but we do so because in order to apply the induction process we will need many " root hyperplanes and for @xmath286 we do nt have enough " of them ( see case 2 ) of the proof in section 4.3 where we use @xmath287 ) . in this case the set of maximal cones @xmath194 in the fan @xmath187 of @xmath104 consists of six cones @xmath288 , generated respectively by @xmath289 and @xmath290 ; @xmath290 and @xmath291 ; @xmath291 and @xmath292 ; @xmath292 and @xmath293 ; @xmath293 and @xmath294 ; and @xmath294 and @xmath289 . let @xmath192 be a divisor on @xmath104 such that @xmath228 is generated by its sections . if @xmath295 form the orthogonal set corresponding to @xmath192 , where @xmath296 is assigned to the maximal cone @xmath297 , then , according to lemma [ dalpha2 ] , the orthogonal set corresponding to the divisor @xmath298 consists of @xmath299 for @xmath300 and @xmath296 for @xmath301 . from our discussion in section [ lemmasection ] , we only need to prove that ( [ eqn : psidelta5 ] ) is true , under the assumption that the set ( [ eqn : psid ] ) is non - empty and ( [ eqn : psidelta4 ] ) is true . suppose , for a contradiction , that under these assumptions there exists @xmath302 such that @xmath303 and @xmath304 . since @xmath206 is continuous and piecewise linear , there are only two cases we need to consider : ( i ) @xmath305 , and ( ii ) @xmath306 . we can use the symmetry @xmath307 of the root system , which preserves the half - space @xmath240 $ ] , to see that we only need to consider the case ( i ) . let @xmath308 . suppose that @xmath309 , where @xmath310 . since @xmath311 on @xmath240 $ ] , we find that @xmath312 and @xmath313 . therefore @xmath314 and @xmath315 . but , by assumption @xmath316 , hence @xmath317 . so @xmath318 and @xmath319 . now , since @xmath228 is generated by its sections , @xmath296 s form a _ positive _ orthogonal set and therefore there exists a non - negative number @xmath320 such that @xmath321 . we get @xmath322 . for the same reasons , there exists a non - negative integer @xmath82 such that @xmath323 , and therefore @xmath324 . thus , since @xmath206 is continuous and piece - wise linear , @xmath206 takes non - negative values on all of @xmath241 $ ] , and this contradicts the non - emptiness of ( [ eqn : psid ] ) . theorem e follows for @xmath285 . [ inductionremarkn=3 ] if we are working with @xmath325 , then we can consider sub - fans of the initial fan @xmath187 that are contained in the root hyperplanes . for example , if our given root is @xmath326 , then the sub - fan corresponding to the root hyperplane @xmath327 $ ] has rays @xmath328 and the maximal - dimensional cones are the 2-dimensional cones in @xmath187 obtained from these rays . this is just a `` copy '' of the fan of @xmath329 and indeed we can carry out the same calculations as above to see that theorem e holds for this `` copy '' of @xmath329 inside @xmath325 . this remark is important for our induction process and is part of a more general story which we tell in section [ inductionsection ] . note that we are still assuming that @xmath284 . to simplify notation , write @xmath330 for the maximal cone whose rays are @xmath331 , where @xmath332 are distinct elements of @xmath333 . let @xmath192 be a divisor on @xmath104 with @xmath228 generated by its sections . denote by @xmath334 the character corresponding to @xmath192 for the cone @xmath330 . then these characters form an orthogonal set and , according to lemma [ dalpha2 ] , the orthogonal set corresponding to @xmath298 consists of @xmath335 for the maximal cones @xmath330 contained in the half - space @xmath240 $ ] and @xmath334 for the ones contained in @xmath241 $ ] . following the discussion in section [ lemmasection ] , we assume that the set ( [ eqn : psid ] ) is non - empty and that ( [ eqn : psidelta4 ] ) is true . we want to prove ( [ eqn : psidelta5 ] ) and , for a contradiction , suppose that there exists a point @xmath336 in the half - space @xmath240 $ ] such that @xmath337 . since @xmath206 is continuous and piece - wise linear on @xmath236 , there are only four cases we need to consider , corresponding to the rays whose primitive lattice points are contained in @xmath338 $ ] : ( i ) @xmath305 , ( ii ) @xmath339 , ( iii ) @xmath340 , and ( iv ) @xmath341 . but , we can use the symmetry @xmath342 of the root system , which preserves the half - space @xmath240 $ ] , to see that we only need to consider cases ( i ) and ( ii ) . ( i ) : our aim is to show that @xmath206 takes non - negative values at @xmath343 and @xmath344 , which would contradict our assumption that the set ( [ eqn : psid ] ) is non - empty . first , if we look at the hyperplane corresponding to the root @xmath345 , as mentioned in remark [ inductionremarkn=3 ] , we get a copy of @xmath329 , where the rays of the sub - fan are @xmath346 . these are precisely the rays of @xmath187 contained in our hyperplane . apply theorem e , case n=3 , to see that @xmath347 and @xmath348 . similarly , by looking at the hyperplanes corresponding to @xmath349 and @xmath350 and applying theorem e , for n=3 , we get that @xmath351 and @xmath352 respectively , which is what we intended to prove . _ case _ ( ii ) : completely similarly as in case ( i ) we see that @xmath351 and @xmath352 ( use the root @xmath353 ) , and @xmath348 ( use the root @xmath350 ) . the only non - trivial case is when we want to show that @xmath347 , because there is no root - hyperplane which contains both @xmath354 and @xmath355 . we proceed as follows . from our assumptions we already know that @xmath356 , @xmath357 , @xmath358 and @xmath359 therefore if @xmath360 where @xmath361 , then @xmath362 , @xmath363 , @xmath364 and @xmath365 , i.e. , @xmath366 , @xmath364 and @xmath367 . but since @xmath334 s form a _ positive _ orthogonal set , we can find a non - negative integer @xmath82 so that @xmath368 . thus @xmath369 , i.e. , @xmath347 , and this concludes the proof . we use induction to prove the theorem in the general case . assume that the theorem is true for all @xmath370 ( as well as for the root - hyperplane copies of @xmath370 inside @xmath371 ; again see section [ inductionsection ] ) , and we want to prove it for @xmath371 . let @xmath192 be a divisor on @xmath371 such that @xmath228 is generated by its sections . once more we consider @xmath298 , where we have taken without loss of generality @xmath284 . following the discussion in section [ lemmasection ] , we assume that the set ( [ eqn : psid ] ) is non - empty and that ( [ eqn : psidelta4 ] ) is true . we want to prove that ( [ eqn : psidelta5 ] ) holds . for a contradiction , suppose that there exists a point @xmath336 in the half - space @xmath240 $ ] such that @xmath337 . since the function @xmath206 is continuous and piece - wise linear , we would get a contradiction if we showed that @xmath206 takes non - negative values on all the primitive lattice points along the rays of the fan of @xmath371 which are contained in the half - space @xmath241 $ ] . moreover , due to the same properties of @xmath206 , we may assume that @xmath372 where @xmath276 s are distinct elements of @xmath373 . we want to prove that for all the primitive lattice points of the form @xmath374 where @xmath375 s are distinct elements of @xmath373 , we have @xmath376 for this , it suffices to show that there is a root @xmath377 of @xmath36 such that both @xmath378 and @xmath379 are contained in the root hyperplane corresponding to @xmath377 , because we can then apply the induction hypothesis to conclude that ( [ eqn : psipq ] ) is true . if there exists an @xmath276 from ( [ eqn : ij ] ) which is not equal to any of @xmath375 s appearing in ( [ eqn : pq ] ) , then put @xmath380 . if there exists a @xmath375 from ( [ eqn : pq ] ) which is not equal to any of @xmath276 s from ( [ eqn : ij ] ) , then we put @xmath381 . in both cases , ( [ eqn : psipq ] ) follows at once . therefore we are only left with the possibility that , up to permutation , @xmath276 s are equal to @xmath375 s . if this is the case , then we distinguish two cases : case 1 ) . the number of @xmath382 s is greater than one , and so if , say , @xmath383 and @xmath384 appear in ( [ eqn : ij ] ) we put @xmath385 to get ( [ eqn : psipq ] ) . case 2 ) . the number of @xmath382 s is less than two , and so , because @xmath287 , we can find @xmath386 and @xmath387 which do nt appear in ( [ eqn : ij ] ) and therefore in ( [ eqn : pq ] ) either . put @xmath388 to see that @xmath336 and @xmath379 belong to the root hyperplane corresponding to @xmath377 . so we get ( [ eqn : psipq ] ) . this proves theorem e for all @xmath74 . let us justify the induction process we used in the proof of theorem e. suppose @xmath389 and consider @xmath104 , with the initial lattice @xmath390 and with dual lattice @xmath391 . take without loss of generality @xmath284 . then the fan of @xmath20 consists of the @xmath392-dimensional cones of the fan @xmath187 of @xmath371 which lie inside the hyperplane @xmath393=\{v\in { \mathbb{r}}^{n+1 } : \langle \alpha , v \rangle = 0 \}.$ ] the dual lattice for @xmath20 is @xmath162 . more concretely , the dual lattice for @xmath20 is @xmath394 because the map @xmath138 is given by @xmath395 . the claim is that we can treat @xmath20 as a copy of @xmath396 inside @xmath371 . for ease of notation , let @xmath397 and @xmath398 be the dual lattice of @xmath396 . note that we have a bijection @xmath399 , given by @xmath400 . now let @xmath401 be a root of @xmath36 . we only need to consider two cases : i ) @xmath402 ; ii ) @xmath326 , to prove that we can treat @xmath20 as a copy of @xmath396 . \i ) let @xmath402 . define @xmath403 , the projection using @xmath377 , in the same way as @xmath138 . we project @xmath404 via @xmath403 to get the lattice @xmath405 if we project @xmath398 via the corresponding @xmath403 , we get the lattice @xmath406 and we also have a natural bijection @xmath407 , given by @xmath408 further , if we have an orthogonal set @xmath409 in @xmath396 , then we get an orthogonal set @xmath410 in @xmath20 by applying @xmath411 to the former . and conversely , we apply @xmath412 to an orthogonal set in @xmath20 in order to get an orthogonal set in @xmath396 . also , points in @xmath413 are identified with points in @xmath414 via the map @xmath415 and , under the same map , points in @xmath416 are identified with those in @xmath417 . the proof of the following lemma then solely involves unravelling the definitions of the maps involved and the definition of a convex hull . incidentally , it proves , as we wanted , that we can treat @xmath20 as a copy of @xmath396 inside @xmath371 , for which theorem e holds . for the corresponding orthogonal sets @xmath409 and @xmath410 we have that @xmath418 if and only if @xmath419 . \ii ) let @xmath326 . this is handled analogously to i ) and the lemma above holds in this case , too . let us mention that in this case we have @xmath420 and @xmath421 let @xmath38 and , as earlier , take without loss of generality @xmath284 . we have that @xmath422 . suppose @xmath192 is a divisor on @xmath104 such that @xmath228 is globally generated and let @xmath423 be the corresponding orthogonal set , where for each @xmath424 , @xmath425 corresponds to the maximal cone @xmath297 of the fan @xmath187 of @xmath104 . then , the fact that @xmath426 is an orthogonal set means in this case that there exists @xmath427 such that @xmath428 . if we denote by @xmath429 the intersection of @xmath152 with the convex hull of the orthogonal set @xmath423 , then we have that ( see e.g. @xcite , pg.66 ) @xmath430 and @xmath431 similarly , we get that @xmath432 where @xmath433 , with @xmath434 standing for the image of the convex hull of the orthogonal set @xmath423 under the map @xmath138 . ( recall that the map @xmath138 was defined in section [ resultssection ] ) . using the long - exact sequence ( * ) from section [ resultssection ] , theorem d can be re - written in the following way : @xmath435 is surjective , for all globally generated line bundles @xmath228 on @xmath436 , where @xmath437 . let us prove theorem d using theorem e. suppose @xmath438 . it suffices to find a point @xmath26 in @xmath429 such that @xmath439 . first `` shift '' the orthogonal set @xmath423 by @xmath440 , i.e. , consider the orthogonal set @xmath441 , which gives a new divisor @xmath259 on @xmath104 . then , if we write @xmath442 , we see that @xmath443 . so @xmath441 gives an orthogonal set in @xmath396 . note that @xmath444 lies in the intersection of the image of the set @xmath445 under the map @xmath138 and the set @xmath446 . using theorem e , we can then conclude that there exists @xmath447 such that @xmath448 , where @xmath449 one can easily check that @xmath450 and @xmath451 . this means that we found @xmath452 in @xmath429 such that @xmath439 , which proves theorem d. now it is time to explain how theorem b follows from theorem d. we keep the same notation as in section [ resultssection ] , with the difference that we consider @xmath453 instead of @xmath38 . then @xmath46 must be of the form @xmath169 where @xmath454 . we can also see that @xmath455 , @xmath456 @xmath457 and @xmath115 is given by averaging over each of the @xmath179 `` batches '' . ( we remind the reader that here , as before , we are considering @xmath65 as a subspace of @xmath66 . ) the simplest case is when only one of @xmath171 s is equal to 2 and the rest are equal to 1 , but we saw in section [ resultssection ] how theorem b is deduced from theorem d in that case . now assume that only two of the @xmath171 s are equal to 2 and the rest equal 1 . we can assume without loss of generality that @xmath458 and @xmath459 , with @xmath460 . then @xmath461 and so @xmath115 is just the composition @xmath462 . therefore when we apply theorem d to @xmath463 , where we consider @xmath463 as a copy of @xmath464 and @xmath465 , we find that theorem b follows in this case , too . the next case would be to consider , without loss of generality , @xmath466 and @xmath467 , where @xmath468 . now we see that @xmath115 is the composition @xmath469 and we get our desired conclusion again using theorem d. in general we write @xmath115 as a composition of a finite number of @xmath138 s for distinct roots @xmath19 and then apply theorem d as many times as there are roots @xmath19 appearing in that composition . if we want to be very specific , then , for @xmath169 we can see that @xmath115 agrees with the composition of maps @xmath470 where @xmath471 with the convention @xmath472 . this concludes the proof of theorem b. consider @xmath104 for @xmath473 . just as in the case of @xmath474 , we would like to describe the fan @xmath187 of @xmath7 . but first , the initial lattice is @xmath475 and so the dual one is @xmath476 . identify the rays in @xmath187 with their corresponding minimal lattice points . then @xmath274 consists of twelve points : @xmath477 and @xmath478 , for @xmath479 . there are twelve maximal cones : @xmath297 , @xmath480 , where the rays of @xmath297 are @xmath481 and @xmath482 ( with @xmath483 ) . let @xmath192 be a @xmath105-cartier divisor on @xmath7 corresponding to a weyl orbit ( see section [ resultssection ] for the precise definition ) and @xmath19 a root of @xmath36 . for every maximal cone @xmath297 denote by @xmath296 the character corresponding to the divisor @xmath192 . obviously , @xmath296 s form a positive orthogonal set . we want to prove that @xmath484 this is true if the orthogonal set corresponding to @xmath192 is _ strictly positive _ , i.e. , if @xmath296 s are distinct and they form a positive orthogonal set . indeed , use lemma [ dalpha2 ] and the definition of orthogonal sets to conclude that the orthogonal set corresponding to @xmath298 is still positive ( but not necessarily strictly positive ) , i.e. , @xmath485 is globally generated . it is also obvious that if all @xmath296 s are equal to each other , then we get a degenerate case for which the result is true as well . therefore the only problem might arise if only some but not all of the @xmath296 s are the same . using symmetries of the current root system , there are only four cases we need to consider , two for a short root and two for a long root : * @xmath486 and there exists a natural number @xmath74 so that @xmath487 @xmath488 @xmath489 * @xmath490 and there exists a natural number @xmath74 so that @xmath487 @xmath488 @xmath489 * @xmath486 and there exists a natural number @xmath74 so that @xmath491 @xmath492 @xmath493 * @xmath490 and there exists a natural number @xmath74 so that @xmath491 @xmath492 @xmath494 let us mention again that we want to prove that @xmath263 . as we explained in section [ resultssection ] , we have a long - exact sequence @xmath495 where terms on the right vanish because @xmath496 is generated by its sections . therefore it suffices to prove that @xmath33 is surjective in each of the cases above . case ( a ) : we have that ( cf . @xcite , pg.66 ) @xmath497 where @xmath498 . also , @xmath499 where @xmath500\cap { \mathbb{z}}\}$ ] . the map @xmath33 is given by @xmath501 , where @xmath502 . it is clear that if @xmath181 is even , we have @xmath503 and @xmath504 . if @xmath181 is odd , then we have @xmath505 . therefore to prove that @xmath33 is surjective , it is enough to show that @xmath506 for all odd integers @xmath181 between @xmath507 and @xmath508 . but this is easily seen to be true by checking that @xmath509 , where @xmath510 s are the respective primitive lattice points along the rays of our fan @xmath187 , written explicitly at the beginning of this section . case ( b ) : keeping the same notation as in case ( a ) we now have that @xmath511 \cap { \mathbb{z}}\}$ ] . @xmath33 is given by @xmath512 , where @xmath513 . if @xmath181 is even then @xmath514 and @xmath33 fixes @xmath515 . if @xmath181 is odd then we have that @xmath516 it is an easy calculation to see that for all odd @xmath181 between @xmath517 and @xmath518 we get @xmath519 and therefore @xmath33 is surjective , which we wanted to prove . case ( c ) : this time we have @xmath520 \cap { \mathbb{z}}\}$ ] and @xmath33 is the same as in case ( a ) . clearly @xmath521 is in @xmath429 and @xmath522 is fixed by @xmath33 , whenever @xmath181 is even . if @xmath181 is odd then consider @xmath523 , which is easily seen to lie in @xmath429 . @xmath33 is surjective because it maps @xmath524 to @xmath522 . case ( d ) : in our final case , we have @xmath525 \cap { \mathbb{z}}\}$ ] . @xmath33 is the same as in case ( b ) and it fixes @xmath515 for @xmath181 even . clearly @xmath514 and it is not difficult to see that if @xmath181 is odd then @xmath526 as well . moreover , still for @xmath181 odd , @xmath527 hence @xmath33 is surjective . we would like to note that the proof of theorem d was different from that of theorem c because in the former we used _ all _ positive orthogonal sets whereas in the latter we only used some and not all positive orthogonal sets , namely only the weyl - orbit ones . more concretely , in section 3 , after lemma [ importantlemma ] , we saw that the vanishing of the first cohomology group there was implied by the vanishing of its corresponding 0-eigenspace ( under the action of our torus ) . this is not true for @xmath39 . the crucial step was to note that by shifting " a positive orthogonal set , say @xmath261 , by a character , say , @xmath157 , we get a positive orthogonal set @xmath260 ; however , it is clear that a shift of a weyl - orbit is in general not a weyl - orbit . at the very end , we give an example to show that theorem b fails in the case of @xmath1 . consider @xmath528 and let @xmath529 . let @xmath192 be the positive orthogonal set ( i.e. , the globally generated torus - equivariant divisor on @xmath528 ) given by @xmath530 for @xmath531 and @xmath532 for @xmath533 . following the same notation as before , we see that in our case @xmath534 and @xmath535 therefore @xmath536 , but @xmath537 . hence the map @xmath538 , induced by the projection @xmath138 along @xmath19 , is not surjective , and theorem b is no longer true if we put @xmath1 instead of @xmath0 . 99 j. arthur , _ the characters of discrete series as orbital integrals _ , invent . * 32 * 1976 , no.3 , 205261 . m. brion and s. kumar , _ frobenius splitting methods in geometry and representation theory _ , birkhuser boston ( 2004 ) . fontaine and m. rapoport , _ existence de filtrations admissible sur des isocristaux _ , bull . france * 133 * ( 2005 ) , no.1 , 7386 . w. fulton , _ introduction to toric varieties _ , ann . of math . 131 , princeton univ . press , princeton , nj , 1993 . q. r. gashi , _ vanishing results for toric varieties associated with root systems _ , in preparation . r. kottwitz , _ on the hodge - newton decomposition for split groups _ , int . res . not . * 2003 , * no.26 , 14331447 . r. kottwitz , _ harmonic analysis on reductive @xmath539-adic groups and lie algebras _ , in _ harmonic analysis , the trace formula , and shimura varieties _ , 393522 , clay math . proc . , * 4 , * amer . soc . , providence , ri , 2005 . r. kottwitz and m. rapoport , _ on the existence of f - isocrystals _ , comment . * 78 * ( 2003 ) , 153184 . c. lucarelli , _ a converse to mazur s inequality for split classical groups _ , journal of the inst . jussieu ( 2004 ) * 3 * ( 2 ) , 165183 . b. mazur , _ frobenius and the hodge filtration _ , bull . * 78 * ( 1972 ) 653667 . m. musta , _ vanishing theorems on toric varieties _ , tohoku math . j. ( 2 ) * 54 * ( 2002 ) , no.3 , 451470 . m. rapoport , _ a positivity property of the satake isomorphism _ , manuscripta math . * 101 * ( 2000 ) , no.2 , 153166 . m. rapoport and m. richartz , _ on the classification and specialization of f - isocrystals with additional structure _ , composito math . * 103 * ( 1996 ) , no.2 , 153181

toric varieties associated with root systems appeared very naturally in the theory of group compactifications . here they are considered in a very different context . we prove the vanishing of higher cohomology groups for certain line bundles on toric varieties associated to @xmath0 and @xmath1 . this can be considered of general interest and it improves the previously known results for these varieties . we also show how these results give a simple proof of a converse to mazur s inequality for @xmath0 and @xmath1 respectively . it is known that the latter imply the non - emptiness of some affine deligne - lusztig varieties . dedicated to scarlett mccgwire and dr . christian duhamel

in the theory of gravitational radiation , it remains true what r. sachs stressed in his 1963 les houches lectures , i.e. that we understand the following three main features @xcite : 0.3 cm ( i ) we can give a description of radiation at large distances from its sources in an asymptotically flat universe ; this description is geometrically elegant and sufficiently detailed to analyze all conceptual experiments concerning the behaviour of test particles or test absorbers in the far field . 0.3 cm ( ii ) how the exact theory relates the far field to the near field and the sources or non - gravitational fields . 0.3 cm ( iii ) we have approximation methods that make it possible to obtain numerical results for the amount of radiation emitted in a particular situation , or for scattering cross - sections , etc . ; these approximation methods do not have a profound geometrical nature , but are very important in comparing theory with the experiment , in case the latter succeeds in finding observational evidence in favour of gravitational waves . 0.3 cm within this framework , it is often desirable to use green - function methods , since the construction of suitable inverses of differential operators lies still at the very heart of many profound properties in classical and quantum field theory . for example , the theory of small disturbances in local field theory can only be built if suitable invertible operators are considered @xcite . in a functional - integral formulation , these correspond to the gauge - field and ghost operators , respectively @xcite . moreover , the peierls bracket on the space of physical observables , which is a poisson bracket preserving the invariance under the full infinite - dimensional symmetry group of the theory , is obtained from the advanced and retarded green functions of the theory via the supercommutator function @xcite and leads possibly to a deeper approach to quantization . last , but not least , a perturbation approach to classical general relativity relies heavily on a careful construction of green functions of operators of hyperbolic @xcite and elliptic @xcite type . in particular , following @xcite , we shall be concerned with the axisymmetric collision of two black holes travelling at the speed of light , each described in the centre - of - mass frame before the collision by an impulsive plane - fronted shock wave with energy @xmath0 . one then passes to a new frame to which a large lorentz boost is applied . there the energy @xmath1 of the incoming shock @xmath2 obeys @xmath3 , where @xmath4 is the energy of the incoming shock @xmath5 and @xmath6 ( @xmath7 being the usual relativistic parameter ) . in the boosted frame , to the future of the strong shock @xmath2 , the metric can be expanded in the form @xcite @xmath8 , \label{(1.1)}\ ] ] where @xmath9 is the standard notation for the minkowski metric . the task of solving the einstein field equations becomes then a problem in singular perturbation theory , having to find @xmath10 by solving the linearized field equations at first , second , ... order respectively in @xmath11 , once that characteristic initial data are given just to the future of the strong shock @xmath2 . the perturbation series ( 1.1 ) is physically relevant because , on boosting back to the centre - of - mass frame , it is found to give an accurate description of space - time geometry where gravitational radiation propagates at small angles away from the forward symmetry axis @xmath12 . the news function @xmath13 ( see appendix ) , which describes gravitational radiation arriving at future null infinity in the centre - of - mass frame , is expected to have the convergent series expansion @xcite @xmath14 with @xmath15 a suitable retarded time coordinate , and @xmath0 the energy of each incoming black hole in the centre - of - mass frame . in @xcite a very useful analytic expression of @xmath16 was derived , exploiting the property that perturbative field equations may all be reduced to equations in only two independent variables , by virtue of a remarkable conformal symmetry at each order in perturbation theory . the green function for perturbative field equations was then found by reduction from the retarded flat - space green function in four dimensions . however , a _ direct _ approach to the evaluation of green functions appears both desirable and helpful in general , and it has been our aim to pursue such a line of investigation . for this purpose , following hereafter our work in @xcite , reduction to two dimensions with the associated hyperbolic operator is studied again in section 2 . section 3 performs reduction to canonical form with the associated riemann function . equations for the goursat problem obeyed by the riemann function are derived in section 4 , while the corresponding numerical algorithm is discussed in section 5 . some backgound material is described in the appendix . as is well known from the work in @xcite and @xcite , the field equations for the first - order correction @xmath17 in the expansion ( 1.1 ) are particular cases of the general system given by the flat - space wave equation ( here @xmath18 ) @xmath19 supplemented by the boundary condition @xmath20 , \label{(2.2)}\ ] ] @xmath21 moreover , @xmath22 should be of the form @xmath23 for @xmath24 , where @xmath25 @xmath26 for the homogeneous wave equation ( 2.1 ) there is no advantage in eliminating @xmath27 and @xmath28 from the differential equation . however , the higher - order metric perturbations turn out to obey inhomogeneous flat - space wave equations of the form @xmath29 where @xmath30 is a source term equal to @xmath31 . this leads to the following equation for @xmath32 : @xmath33 where @xmath34 is an hyperbolic operator in the independent variables @xmath35 and @xmath36 , and takes the form @xcite @xmath37 the proof of hyperbolicity of @xmath34 , with the associated normal hyperbolic form , can be found in section 3 of @xcite , and in @xcite . the advantage of studying eq . ( 2.7 ) is twofold : to evaluate the solution at some space - time point one has simply to integrate the product of @xmath38 and the green function @xmath39 of @xmath34 : @xmath40 and the resulting numerical calculation of the solution is now feasible @xcite . if one defines the variables @xmath41 the operator @xmath34 is turned into @xmath42 the operator @xmath43 is the ` sum ' of an elliptic operator in the @xmath44 variable and a two - dimensional wave operator ` weighted ' with the exponential @xmath45 , which is the main source of technical complications in these variables . it is therefore more convenient , in our general analysis , to reduce first eq . ( 2.7 ) to canonical form , and then find an integral representation of the solution . reduction to canonical form means that new coordinates @xmath46 and @xmath47 are introduced such that the coefficients of @xmath48 and @xmath49 vanish . as is shown in @xcite , this is achieved if @xmath50 @xmath51 @xmath52 the resulting formulae are considerably simplified if one defines @xmath53 the dependence of @xmath54 on @xmath55 and @xmath56 is obtained implicitly by solving the system @xcite @xmath57 @xmath58 this leads to the equation @xmath59 which can be cast in the form @xmath60 this suggests defining @xmath61 so that one first has to solve the transcendental equation @xmath62 to obtain @xmath63 , from which one gets @xmath64 on denoting by @xmath65 the left - hand side of eq . ( 3.10 ) , one finds that , in the plane @xmath66 , the right - hand side of eq . ( 3.10 ) is a line parallel to the @xmath67-axis , which intersects @xmath65 at no more than one point for each value of @xmath68 . for example , when @xmath69 , @xmath65 intersects the line taking the constant value @xmath2 , for which @xmath70 . the function @xmath71 is asymmetric and has the limiting behaviour described by @xmath72 @xmath73 thus , in the lower half - plane , @xmath74 has an horizontal asymptote given by the @xmath67-axis , and a vertical asymptote given by the line @xmath75 , while it has no asymptotes in the upper half - plane , since @xmath76 in addition to ( 3.13 ) . the first derivative of @xmath74 reads as @xmath77 one therefore has @xmath78 for all @xmath79 , and @xmath80 for all @xmath81 , and @xmath74 is monotonically decreasing for negative @xmath67 and monotonically increasing for positive @xmath67 . the point @xmath82 , at which @xmath83 vanishes , is neither a maximum nor a minimum point , because @xmath84 @xmath85 these formulae imply that @xmath86 but @xmath87 , and hence @xmath82 yields a flex of @xmath65 ( see fig . 1 ) . in the @xmath88 variables , the operator @xmath34 therefore reads @xmath89 where , exploiting the formulae @xmath90 @xmath91 one finds @xmath92 , \label{(3.20)}\end{aligned}\ ] ] @xmath93 , \label{(3.21)}\ ] ] @xmath94 . \label{(3.22)}\ ] ] the resulting canonical form of eq . ( 2.7 ) is @xmath95&= & \left({\partial^{2}\over \partial x \partial y } + a(x , y){\partial \over \partial x } + b(x , y){\partial \over \partial y}+c(x , y ) \right)\chi(x , y ) \nonumber \\ & = & { \widetilde h}(x , y ) \label{(3.23)}\end{aligned}\ ] ] where @xmath96 @xmath97 @xmath98 @xmath99 note that @xmath100 . for an hyperbolic equation in the form ( 3.23 ) , we can use the riemann integral representation of the solution . for this purpose , recall from @xcite that , on denoting by @xmath101 the adjoint of the operator @xmath102 in ( 3.23 ) , which acts according to @xmath103=\chi_{xy}-(a\chi)_{x}-(b \chi)_{y}+c \chi , \label{(3.28)}\ ] ] one has to find a ` function ' @xmath104 ( actually a kernel ) subject to the following conditions ( @xmath105 being the coordinates of a point @xmath106 such that characteristics through it intersect a curve @xmath107 at points @xmath108 and @xmath109 , @xmath110 being a segment with constant @xmath56 , and @xmath111 being a segment with constant @xmath55 , as is shown in fig . 2 ) : 0.3 cm ( i ) as a function of @xmath55 and @xmath56 , @xmath112 satisfies the adjoint equation @xmath113=0 , \label{(3.29)}\ ] ] ( ii ) @xmath114 on @xmath110 , i.e. @xmath115 and @xmath116 on @xmath111 , i.e. @xmath117 ( iii ) @xmath112 equals @xmath2 at @xmath106 , i.e. @xmath118 it is then possible to express the solution of eq . ( 3.23 ) in the form @xmath119 + \int_{ab}\biggr ( \left[{r\over 2}\chi_{x } + \left(br-{1\over 2}r_{x}\right)\chi \right]dx \nonumber \\ & - & \left[{r\over 2}\chi_{y } + \left(ar-{1\over 2}r_{y}\right)\chi \right]dy \biggr ) + \int \int_{\omega}r(x , y;\xi,\eta){\widetilde h}(x , y)dx dy , \label{(3.33)}\end{aligned}\ ] ] where @xmath120 is a domain with boundary . note that eqs . ( 3.30 ) and ( 3.31 ) are ordinary differential equations for the riemann function @xmath104 along the characteristics parallel to the coordinate axes . by virtue of ( 3.32 ) , their integration yields @xmath121 @xmath122 which are the values of @xmath112 along the characteristics through @xmath106 . equation ( 3.33 ) yields instead the solution of eq . ( 3.23 ) for arbitrary initial values given along an arbitrary non - characteristic curve @xmath107 , by means of a solution @xmath112 of the adjoint equation ( 3.29 ) which depends on @xmath123 and two parameters @xmath124 . unlike @xmath125 , the riemann function @xmath112 solves a characteristic initial - value problem . by fully exploiting the reduction to canonical form of eq . ( 2.7 ) we have considered novel features with respect to the analysis in @xcite , because the riemann formula ( 3.33 ) also contains the integral along the piece of curve @xmath107 from @xmath108 to @xmath109 , and the term @xmath126 $ ] . this representation of the solution might be more appropriate for the numerical purposes considered in @xcite , but the task of finding the riemann function @xmath112 remains extremely difficult . one can however use approximate methods for solving eq . ( 3.29 ) . for this purpose , we first point out that , by virtue of eq . ( 3.28 ) , eq . ( 3.29 ) is a canonical hyperbolic equation of the form @xmath127 where @xmath128 @xmath129 @xmath130 thus , on defining @xmath131 @xmath132 the equation ( 4.1 ) for the riemann function is equivalent to the hyperbolic canonical system @xcite @xmath133 @xmath134 where @xmath135 @xmath136 @xmath137 @xmath138 for the system described by eqs . ( 4.7 ) and ( 4.8 ) with boundary data ( 3.34 ) and ( 3.35 ) an existence and uniqueness theorem holds ( see @xcite for the lipschitz conditions on boundary data ) , and we can therefore exploit the finite differences method to find approximate solutions for the riemann function @xmath104 , and eventually @xmath139 with the help of the integral representation ( 3.33 ) . the inverses of hyperbolic operators @xcite and the cauchy problem for hyperbolic equations with polynomial coefficients @xcite have always been the object of intensive investigation in the mathematical literature . we have here considered the application of such issues to axisymmetric black hole collisions at the speed of light , relying on the work in @xcite . we have pointed out that , for the inhomogeneous equations ( 2.7 ) occurring in the perturbative analysis , the task of inverting the operator ( 2.8 ) can be accomplished with the help of the riemann integral representation ( 3.33 ) , after solving eq . ( 4.1 ) for the riemann function . one has then to solve a characteristic initial - value problem for a homogeneous hyperbolic equation in canonical form in two independent variables , for which we have developed formulae to be used for the numerical solution with the help of a finite differences scheme . for this purpose one studies the canonical system ( cf ( 4.7 ) and ( 4.8 ) ) @xmath140 @xmath141 in the rectangle @xmath142 , y \in [ y_{0},y_{0}+b ] \right \}$ ] with known values of @xmath143 on the vertical side @xmath144 where @xmath145 , and known values of @xmath146 on the horizontal side @xmath147 where @xmath148 . the segments @xmath147 and @xmath144 are then divided into @xmath149 and @xmath150 equal parts , respectively . on setting @xmath151 and @xmath152 , the original differential equations become equations relating values of @xmath143 and @xmath146 at three intersection points of the resulting lattice , i.e. @xmath153 @xmath154 it is now convenient to set @xmath155 , so that these equations read as @xmath156 @xmath157 thus , if both @xmath143 and @xmath146 are known at @xmath158 , one can evaluate @xmath143 at @xmath159 and @xmath146 at @xmath160 . the evaluation at subsequent intersection points of the lattice goes on along horizontal or vertical segments . in the former case , the resulting algorithm is @xmath161 @xmath162 while in the latter case one obtains the algorithm expressed by the equations @xmath163 @xmath164 stability of such solutions is closely linked with the geometry of the associated characteristics , and the criteria to be fulfilled are studied in section 13.2 of @xcite ( stability depends crucially on whether or not @xmath165 ) . to sum up , one solves numerically eq . ( 3.10 ) for @xmath166 , from which one gets @xmath167 with the help of ( 3.11 ) , which is a fractional linear transformation . this yields @xmath168 and @xmath169 as functions of @xmath88 according to ( 3.24)(3.27 ) , and hence @xmath170 and @xmath107 in the equation for the riemann function are obtained according to ( 4.2)(4.4 ) , where derivatives with respect to @xmath55 and @xmath56 are evaluated numerically . eventually , the system given by ( 4.7 ) and ( 4.8 ) is solved according to the finite - differences scheme of the present section , with @xmath171 @xmath172 once the riemann function @xmath173 is obtained with the desired accuracy , numerical evaluation of the integral ( 3.33 ) yields @xmath139 , and @xmath174 is obtained upon using eqs . ( 3.5 ) and ( 3.6 ) for the characteristic coordinates . our steps are conceptually desirable since they rely on well established techniques for the solution of hyperbolic equations in two independent variables @xcite , and provide a viable alternative to the numerical analysis performed in @xcite , because all functions should be evaluated numerically . our method is not obviously more powerful than the one used in @xcite , but is well suited for a systematic and lengthy numerical analysis , while its analytic side provides an interesting alternative for the evaluation of green functions both in black hole physics and in other problems where hyperbolic operators with variable coefficients might occur . this task remains very important because a strong production of gravitational radiation is mainly expected in the extreme events studied in @xcite and which motivated our paper . any viable way of looking at mathematical and numerical aspects of the problem is therefore of physical interest for research planned in the years to come @xcite . in our expository article , we find it appropriate to include some background material , following , for example , the presentation in section iv of @xcite . we therefore consider some four - dimensional region of space - time and choose in it a set of null hypersurfaces @xmath175 ; the corresponding ray congruence with tangent vector @xmath176 is assumed to have expansion @xmath177 , which can always be arranged in a space - time patch , whereas outside of some patch the rays start to cross and hence our construction breaks down globally . on completing the @xmath178 direction to a quasi - normal tetrad @xmath179 , one finds the following split of the vacuum einstein equations with einstein tensor @xmath180 : main equations ( 6 equations ) @xmath181 trivial equation @xmath182 and 3 supplementary conditions @xmath183 where a single complex equation has been counted as two real equations . remarkably , if the main equations hold everywhere , then the trivial equation holds everywhere and the supplementary conditions hold everywhere if they hold at one point on each ray . the fulfillment of the trivial equation is proved by writing , from the vacuum einstein equations , that @xmath184 and then exploiting the main equations ( a1 ) jointly with the split of @xmath185 as given in @xcite : @xmath186 hence one gets @xmath187 by hypothesis the expansion @xmath27 does not vanish , so that the trivial equation is , indeed , identically satisfied . the fulfillment of ( a3 ) everywhere is proved along similar lines . thus , one can again integrate the main equations ( a1 ) first and worry about the supplementary conditions ( a3 ) later . choose now the coordinate @xmath188 as the retarded time : @xmath189 . let @xmath190 be a luminosity distance along the rays ; let @xmath191 ( with @xmath192 ) be any other pair of coordinates constant along the rays . the line element in these coordinates takes therefore the form ( no confusion should arise with the @xmath7 of section 1 ) @xmath193 where @xmath194 depend on the @xmath191 coordinates . since @xmath36 is a luminosity distance , the determinant of @xmath195 is independent of @xmath36 . bearing in mind that the luminosity distance is defined only up to a factor constant along each ray , one can demand without loss of generality that ( here @xmath196 ) @xmath197 the metric corresponding to the line element ( a7 ) contains only six unknown functions of four variables , and our coordinate system is ` rigid ' enough for our purposes @xcite . one can either analyze the field in the neighbourhood of some point , or the field near infinity in an asymptotically flat space - time . indeed , if in minkowski space - time one uses a retarded time @xmath198 and spherical coordinates @xmath199 one finds for the line element @xmath200 hence one is led to require that , if asymptotic flatness holds , @xmath201 where all limits are taken as @xmath36 approaches infinity with @xmath202 fixed . the second requirement in ( a10 ) , i.e. that all quantities of interest admit a power - series expansion in @xmath203 , e.g. @xmath204 is indeed restrictive . such a requirement can be drastically weakened but not fully eliminated ; moreover , it is closely related to an outgoing radiation condition of the sommerfeld type . it should be stressed that all these requirements no longer hold when @xmath36 becomes small to the extent that rays start to cross each other . in the axially- and reflection - symmetric case considered by bondi et al . @xcite , one has @xmath205 and the @xmath28-direction is a hypersurface - orthogonal killing direction . the line element acquires the simpler form @xmath206 , \label{(a13)}\ ] ] where the peculiar form of the first coefficient is chosen to simplify the resulting calculations . interestingly , two main equations are found to be identically satisfied by virtue of axial symmetry , whereas the other four turn out to be linear combinations of @xcite @xmath207 @xmath208}_{1 } -2r^{2}\biggr(-\gamma_{12}+2\gamma_{1}\gamma_{2}-2\gamma_{1 } \cot \theta + \beta_{12}-2{\beta_{2}\over r}\biggr)=0 , \label{(a15)}\ ] ] @xmath209=0 , \label{(a16)}\end{aligned}\ ] ] @xmath210=0 . \label{(a17)}\end{aligned}\ ] ] equations ( a14)(a16 ) are called _ hypersurface equations _ because they contain no @xmath211 derivatives , while eq . ( a17 ) is called the _ standard equation_. now if @xmath212 is given for one value of @xmath211 , eq . ( a14 ) and the boundary conditions ( a10 ) determine @xmath7 uniquely . ( a15 ) and the boundary conditions determine @xmath143 up to a function of integration @xmath213 that can be added to @xmath214 . equation ( a16 ) determines @xmath146 up to the additive function @xmath215 ; last , eq . ( a17 ) determines @xmath216 up to an additive function @xmath217 . one can then differentiate eqs . ( a14)(a17 ) with respect to @xmath211 and repeat the whole procedure . to sum up , given @xmath212 at one moment the main equations determine the future or past up to the three integration functions just mentioned . in the general case , the results are completely similar . one has to assign at one value of @xmath211 the two functions @xmath212 and @xmath218 . the future is then determined up to five integration functions : a term @xmath219 to be added to @xmath220 ; two functions @xmath221 which occur in the @xmath222 term for @xmath143 ; and two ` news functions ' @xmath223 , where the complex function @xmath224 is given by eq . ( a11 ) . as far as the supplementary conditions are concerned , the lemma just given makes it clear that they should only involve the functions @xmath225 and @xmath13 , while a long calculation yields @xcite @xmath226 \right \ } , \label{(a18)}\ ] ] @xmath227 { \overline c}_{0 } , \label{(a19)}\ ] ] where @xmath228 . the desired @xmath229 and @xmath230 can be determined once that @xmath231 and some initial values are given . in the axially symmetric case , eqs . ( a18 ) and ( a19 ) take the simpler form @xcite @xmath232 @xmath233 the functions @xmath212 and @xmath218 given on the initial hypersurface @xmath175 , jointly with the two news functions @xmath13 given at @xmath234 describe the two transverse degrees of freedom . moreover , one should specify @xmath229 and @xmath235 at the initial ( or final ) retarded time , and these three functions of two variables must be related to the longitudinal - timelike degrees of freedom of the gravitational field . in the characteristic value problem for general relativity , the independent data appear therefore in a very explicit form @xcite . the work of g. esposito has been partially supported by prin _ sintesi_. he is grateful to decio cocolicchio and sorin dragomir for warm encouragement . iop publishing has kindly granted permission to republish material from the author s paper in @xcite . sachs , in _ relativity , groups and topology _ , edited by c. dewitt and b.s . dewitt ( gordon & breach , new york , 1964 ) . dewitt , _ dynamical theory of groups and fields _ ( gordon and breach , new york , 1965 ) . dewitt , in _ relativity , groups and topology ii _ , edited by b.s . dewitt and r. stora ( north holland , amsterdam , 1984 ) . g. esposito , _ quantum gravity in four dimensions _ ( nova science , new york , 2001 ) . p. cartier and c. dewitt morette , _ j. math . phys . _ * 41 * , 4154 ( 2000 ) . death and p.n . payne , _ phys . _ d * 46 * , 658 ( 1992 ) . death and p.n . payne , _ phys . _ d * 46 * , 675 ( 1992 ) . death and p.n . payne , _ phys . _ d * 46 * , 694 ( 1992 ) . death , _ black holes : gravitational interactions _ ( clarendon press , oxford , 1996 ) . g. esposito and c. stornaiolo , _ class . quantum grav . _ * 17 * , 1989 ( 2000 ) . g. esposito and c. stornaiolo , _ found . * 13 * , 279 ( 2000 ) . g. esposito , _ class . quantum grav . _ * 18 * , 1997 ( 2001 ) . r. courant and d. hilbert , _ methods of mathematical physics . ii . partial differential equations _ ( interscience , new york , 1961 ) . j. leray , _ hyperbolic differential equations _ ( princeton university press , princeton , 1953 ) . j. leray , _ c. r. acad . paris _ * 242 * , 953 ( 1956 ) . garabedian , _ partial differential equations _ ( chelsea , new york , 1964 ) . b. allen and a. ottewill , _ gen . grav . _ * 32 * , 385 ( 2000 ) . h. bondi , m.g.j . van der burg , a.w.k . metzner , _ proc . lond . _ * a 269 * , 21 ( 1962 ) .

previous work in the literature has studied gravitational radiation in black - hole collisions at the speed of light . in particular , it had been proved that the perturbative field equations may all be reduced to equations in only two independent variables , by virtue of a conformal symmetry at each order in perturbation theory . the green function for the perturbative field equations is here analyzed by studying the corresponding second - order hyperbolic operator with variable coefficients , instead of using the reduction method from the retarded flat - space green function in four dimensions . after reduction to canonical form of this hyperbolic operator , the integral representation of the solution in terms of the riemann function is obtained . the riemann function solves a characteristic initial - value problem for which analytic formulae leading to the numerical solution are derived . # 1

the quantization of non - abelian gauge theory is complicated by the presence of the so - called gribov problem@xcite@xcite@xcite[4 ] . the gribov problem in general suggests that the coulomb gauge can not completely fix the gauge due to the presence of more than one gauge configurations which satisfy the coulomb gauge condition or , in certain circumstances in compactified space - time , it even suggests the absence of the gauge configuration which satisfies the coulomb gauge condition [ 2 ] . in euclidean formulation of gauge theory , the landau gauge also suffers from the same complications . although the full details of the gribov problem are not understood yet , a working prescription is figured out if one regards the gribov problem as the appearance of several gauge copies ( i.e. , if one assumes that one can always find at least one gauge configuration which satisfies the coulomb gauge condition ) . in fact , it was noted some time ago@xcite that if the functional correspondence between @xmath0 and the gauge parameter @xmath1 defined by @xmath7 is `` globally single valued '' , the path integral with brst symmetry@xcite is defined by summing over gribov copies in a very specific way . for a more general gauge fixing , ( 1.1 ) may be replaced by @xmath8 where @xmath9 specifies the gauge condition . in ( 1.1 ) or ( 1.2 ) @xmath10 stands for the gauge field obtained from @xmath11 by a gauge transformation specified by the gauge orbit parameter @xmath12 . for an infinitesimal @xmath12 , one has @xmath13 one may regard ( 1.1 ) or ( 1.2 ) as a functional correspondence between @xmath0 and @xmath1 parametrized by @xmath14 . the globally single valued correspondence between @xmath0 and @xmath1 , which is explained in more detail in section 2 , is required for any value of @xmath14 to write a simple path integral formula . [ if one restricts oneself to only one of the gribov copies by some means@xcite , one can also incorporate the idea of brst symmetry . but a simple prescription which selectively picks up only one copy appears to be misssing at this moment . ] recently a very detailed analysis of the gribov problem was performed by friedberg , lee , pang and ren@xcite on the basis of a soluble gauge model which exhibits gribov - type copies . one of the main conclusions in @xcite is that the singularity associated with the so - called gribov horizon is immaterial at least in their soluble model and one may incorporate all the gribov copies in a very specific way . they showed how this prescription works in the soluble model proposed by them . the present work is motivated by the fact that the model and the gauge choice in ref.@xcite satisfy our criterion in ( 1.1 ) or ( 1.2 ) . one can thus formulate a brst invariant path integral for the model proposed in [ 8 ] by summing over gribov - type copies ; the physical quantities thus calculated agree with those in ref.@xcite . note that the brst symmetry deals with an extended hilbert space which contains indefinite metric in general , although the physical sector specified by brst cohomology contains only positive metric . since the brst symmetry plays a fundamental role in modern gauge theory , we here present a detailed brst analysis of the soluble model proposed in @xcite . in this section we recapitulate the essence of the argument presented in @xcite . we start with the faddeev - popov formulation of the feynman - type gauge condition @xcite@xcite . the vacuum - to - vacuum transition amplitude is defined by @xmath15 where @xmath16 stands for the action invariant under the yang - mills local gauge transformation . the positive constant @xmath17 is a gauge fixing parameter which specifies the feynman - type gauge condition . [ the equations in this section are written in the minkowski metric , but they should really be interpreted in the euclidean metric to render the functional integral and the gribov problem well - defined . ] in the following we often suppress the internal symmetry indices , and instead we write the gauge parameter explicitly : @xmath18 indicates the gauge field which is obtained from @xmath14 by a gauge transformation specified by @xmath19 . the determinant factor @xmath20 is defined by@xcite @xmath21|^{-1}\}\end{aligned}\ ] ] where the summation runs over all the gauge equivalent configurations satisfying @xmath22 , which were found by gribov@xcite and others@xcite@xcite@xcite . equation(2.2 ) is valid only for sufficiently small @xmath17 , since the parameter @xmath23 defined by @xmath24 has a complicated branch structure for large @xmath25 in the presence of gribov ambiguities . obviously the feynman - type gauge formulation becomes even more involved than the landau - type gauge condition . it was suggested in @xcite to replace equation ( 2.1 ) by @xmath26 exp\{is(a_{\mu}^{\omega } ) - \frac{i}{2\alpha}{\int}c(x)^{2}dx\}\ ] ] the crucial difference between ( 2.1 ) and ( 2.4 ) is that ( 2.4 ) is local in the gauge space @xmath19 ( i.e. , the gauge fixing factor and the compensating factor are defined at the identical @xmath1 ) , whereas @xmath20 in ( 2.1 ) is gauge independent and involves a non - local factor in @xmath1 as is shown in ( 2.2 ) . as the determinant in ( 2.4 ) depends on @xmath18 , the entire integrand in ( 2.4 ) is in general no more degenerate with respect to gauge equivalent configurations even if the gauge fixing term itself may be degenerate for certain configurations . another important point is that one takes the absolute values of determinant factors in ( 2.2 ) thanks to the definition of the @xmath27-function , whereas just the determinant which can be negative as well as positive appears in ( 2.4 ) . it is easy to see that ( 2.4 ) can be rewritten as @xmath28 where @xmath29 with @xmath30 the lagrangian multiplier field , and @xmath31 and @xmath32 the ( hermitian ) faddeev - popov ghost fields ; @xmath33 is the structure constant of the gauge group and @xmath34 is the gauge coupling constant . if one imposes the hermiticity of @xmath31 and @xmath32 , the phase factor of the determinant in ( 2.4 ) can not be removed . the normalization constant @xmath35 in ( 2.5 ) includes the effect of gaussian integral over @xmath36 in addition to @xmath37 in ( 2.4 ) , and in fact @xmath35 is independent of @xmath17 . see eq . ( 2.10 ) . this @xmath38 as well as the starting gauge invariant lagrangian are invariant under the brst transformation defined by @xmath39\nonumber\\ \delta_{\theta } c^{a } & = & i\theta ( g/2)f^{abc}c^{b}c^{c}\nonumber\\ \delta_{\theta } \bar{c}^{a } & = & \theta b^{a}\nonumber\\ \delta_{\theta } b^{a } & = & 0\end{aligned}\ ] ] where @xmath40 and the ghost variables @xmath41 and @xmath42 are elements of the grassmann algebra , i.e. , @xmath43 . this transformation can be confirmed to be nil - potent @xmath44 , for example , @xmath45 one can also confirm that the path integral measure in ( 2.5 ) is invariant under ( 2.7 ) . note that the transformation ( 2.7 ) is `` local '' in the @xmath1 parameter;precisely for this property , the prescription in ( 2.4 ) was chosen . to interprete the path integral measure in ( 2.4 ) as the path integral over all the gauge field configurations divided by the gauge volume , namely @xmath46 one needs to define the normalization factor in ( 2.4 ) by @xmath47 exp\{- \frac{i}{2\alpha}{\int}c(x)^{2}dx\}\nonumber\\ & = & { \int}{\cal d}\omega det[\frac{\partial}{\partial\omega}\partial^{\mu}a_{\mu}^{\omega } ] exp\{- \frac{i}{2\alpha}{\int}(\partial^{\mu}a_{\mu}^{\omega})^{2}dx\}\nonumber\\ & = & { \int}{\cal d}\tau\ exp\{- \frac{i}{2\alpha}{\int}\tau ( x)^{2}dx\}\end{aligned}\ ] ] where the function @xmath0 is defined by @xmath48 and the determinant factor is regarded as a jacobian for the change of variables from @xmath12 to @xmath49 . although we use the feynman - type gauge fixing ( 2.11 ) as a typical example in this section , one may replace ( 2.11 ) by @xmath50 to deal with a more general gauge condition @xmath51 it is crucial to establish that the normalization factor in ( 2.10 ) is independent of @xmath14 . only in this case , ( 2.4 ) defines an acceptable vacuum transition amplitude . the gribov ambiguity in the present case appears as a non - unique correspondence between @xmath49 and @xmath12 in ( 2.11 ) , as is schematically shown in fig . 1 which includes 3 gribov copies . the path integral in ( 2.10 ) is performed along the contour in fig . 1 . as the gaussian function is regular at any finite point , the complicated contour in fig . 1 gives rise to the same result in ( 2.10 ) as a contour corresponding to @xmath52 . in the present path integral formulation , the evaluation of the normalization factor in ( 2.10 ) is the only place where we explicitly encounter the multiple solutions of gauge fixing condition . [ if the normalization factor @xmath37 should depend on gauge field @xmath14 , the factor @xmath37 , which is gauge independent in the sense that we integrated over entire gauge orbit , needs to be taken inside the path integral in ( 2.4 ) . in this case one looses the simplicity of the formula ( 2.4 ) . ] the basic assumption we have to make is therefore that ( 2.11 ) in the context of the path integral ( 2.10 ) is `` globally single - valued '' , in the sense that the asymptotic functional correspondence between @xmath1 and @xmath0 is little affected by a fixed @xmath14 with @xmath53@xcite ; fig.1 satisfies this requirement . this assumption appears to be physically reasonable if the second derivative term of the gauge orbit parameter dominates the functional correspondence in ( 2.11 ) , though it has not been established mathematically . to define the functional correspondence between @xmath1 and @xmath0 in ( 2.11 ) , one needs in general some notion of norm such as @xmath54-norm for which the coulomb gauge vacuum is unique [ 3][4 ] . the functional configurations which are square integrable however have zero measure in the path integral[11 ] , and this makes the precise analysis of ( 2.11 ) very complicated : at least what we need to do is to start with an expansion of a generic field variable in terms of some complete orthonormal basis set ( which means that the field is inside the @xmath54-space ) and then let each expansion coefficient vary from @xmath55 to @xmath56(which means that the field is outside the @xmath54-space ) . if the gribov problem simply means the situation as is schematically shown in fig . 1 , the prescription in ( 2.4 ) may be justified . the indefinite signature of the determinant factor in ( 2.4 ) is not a difficulty in the framework of indefinite metric field theory@xcite@xcite since the determinant factor is associated with the faddeev - popov ghost fields and the brst cohomology selects the positive definite physical space . on the other hand , the gribov problem may aslo suggest that one can not bring the relation(2.11 ) with fixed @xmath14 to @xmath57 by any gauge transformation@xcite . if this is the case , the asymptotic behavior of the mapping ( 2.11 ) is in general modified by @xmath14 and our prescription can not be justified . consequently , the prescription ( 2.4 ) may be valid to the extent that one can always achieve the condition @xmath53 by means of suitable ( but not necessarily unique ) gauge transformations . we also note that the limit @xmath58 in ( 2.5 ) and ( 2.10 ) corresponds to a very loose gauge fixing . * 3.1 , a soluble gauge model * the soluble gauge model of friedberg , lee , pang and ren@xcite is defined by @xmath59^{2 } + [ \dot{y}(t ) - g\xi(t)x(t)]^{2 } + [ \dot{z}(t ) - \xi(t)]^{2}\ } - u(x(t)^{2 } + y(t)^{2})\ ] ] where @xmath60 , for example , means the time derivative of @xmath61 , and the potential @xmath62 depends only on the combination @xmath63 . this lagrangian is invariant under a local gauge transformation parametrized by @xmath64 , @xmath65 the gauge condition ( an analogue of @xmath66 gauge ) @xmath67 or ( an analogue of @xmath68 gauge ) @xmath69 is well - defined without suffering from gribov - type copies . however , it was shown in @xcite that the gauge condition ( an analogue of the coulomb gauge ) @xmath70 with a constant @xmath71 suffers from the gribov- type complications . this is seen by using the notation in ( 3.2 ) as @xmath72 = 0\end{aligned}\ ] ] where we used the relation(3.5 ) and @xmath73 from a view point of gauge fixing , @xmath74 is a solution of ( 3.6 ) if ( 3.5 ) is satisfied . by analyzing the crossing points of two graphs in @xmath75 plane defined by @xmath76 one can confirm that eq.(3.6 ) in general has more than one solutions for @xmath1 . from a view point of general gauge fixing procedure , we here regard the algebraic gauge fixing such as ( 3.3 ) and ( 3.4 ) well - defined ; in the analysis of the gribov problem in ref.[2 ] , the algebraic gauge fixing [ 3 ] is excluded . the authors in ref.@xcite started with the hamiltonian formulated in terms of the well - defined gauge @xmath77 in ( 3.3 ) and then faithfully rewrote the hamiltonian in terms of the variables defined by the `` coulomb gauge '' in ( 3.5 ) . by this way , the authors in @xcite analyzed in detail the problem related to the gribov copies and the so - called gribov horizons where the faddeev - popov determinant vanishes . they thus arrived at a prescription which sums over all the gribov - type copies in a very specific way . as is clear from their derivation , their specification satisfies the unitarity and gauge independence . in the context of brst invariant path integral discussed in section 2 , the crucial relation ( 2.11 ) becomes @xmath78 in the present model . for @xmath79 , the functional correspondence between @xmath1 and @xmath0 is one - to - one and monotonous for any fixed value of @xmath80 . when one varies @xmath81 and @xmath82 continuously , one deforms this monotonous curve continuously . but the asymptotic correspondence between @xmath64 and @xmath83 at @xmath84 for each value of @xmath80 is still kept preserved , at least for any fixed @xmath81 and @xmath82 . this correspondence between @xmath64 and @xmath83 thus satisfies our criterion discussed in connection with ( 2.11 ) . the absence of terms which contain the derivatives of @xmath64 in ( 3.9 ) makes the functional correspondence in ( 3.9 ) well - defined and transparent . from a view point of gauge fixing in ( 3.6 ) , this `` globally single - valued '' correspondence between @xmath1 and @xmath0 means that one always obtains an _ odd _ number of solutions for ( 3.6 ) . the prescription in @xcite is then viewed as a sum of all these solutions with signature factors specified by the signature of the faddeev - popov determinant @xmath85\ } = det\{[1 + \lambda g y^{\omega}(t ) ] \delta(t - t^{\prime})\}\ ] ] evaluated at the point of solutions , @xmath86 , of ( 3.6 ) . the row and column indices of the matrix in ( 3.10 ) are specified by @xmath80 and @xmath87 , respectively . in the context of brst invariant formulation , a pair - wise cancellation of gribov - type copies takes place , except for one solution , in the calculation of the normalization factor in ( 2.10 ) or ( 3.21 ) below . * 3.2 , brst invariant path integral * the relation ( 3.9 ) satisfies our criterion discussed in connection with ( 2.11 ) . we can thus define an analogue of ( 2.5 ) for the lagrangian ( 3.1 ) by @xmath88 where @xmath89 in terms of the lagrangian @xmath90 in ( 3.1 ) . the gauge fixing part of ( 3.11 ) is defined by @xmath91 where @xmath92 and @xmath71 are numerical constants , and @xmath93 and @xmath94 are ( hermitian ) faddeev - popov ghost fields . @xmath36 is a lagrangian multiplier field . note that @xmath38 is hermitian . the integral measure in ( 3.11 ) is given by @xmath95 the lagrangians @xmath90 and @xmath38 and the path integral measure ( 3.14 ) are invariant under the brst transformation defined by @xmath96 where the parameter @xmath40 is a grassmann number , @xmath43 . note that @xmath40 and ghost variables anti - commute . in ( 3.15 ) we used a brst superfield notation : in this notation , the second component of a superfield proportional to @xmath40 stands for the brst transformed field of the first component . the second component is invariant under brst transformation which ensures the nil - potency of the brst charge . in the operator notation to be defined later , one can write , for example , @xmath97 with a nil - potent brst charge @xmath98 , @xmath99 . namely , the brst transformation is a translation in @xmath40-space , and @xmath100 is analogous to momentum operator . in ( 3.11)@xmath101 ( 3.15 ) , we explicitly wrote the gauge parameter @xmath1 to emphasize that brst transformation is `` local '' in the @xmath1-space . for @xmath38 in ( 3.13 ) , the relation ( 3.9 ) is replaced by ( an analogue of the landau gauge ) @xmath102 the functional correspondence between @xmath1 and @xmath0 is monotonous and one - to - one for weak @xmath103 and @xmath104 fields ; this is understood if one rewrites the relation ( 3.17 ) for euclidean time @xmath105 by neglecting weak fields as @xmath106 the fourier transform of this relation gives a one - to - one monotonous correspondence between the fourier coefficients of @xmath0 and @xmath1 for non - negative @xmath107 . the asymptotic functional correspondence between @xmath1 and @xmath0 for weak field cases is preserved even for any fixed strong fields @xmath103 and @xmath104 for non - negative @xmath107 to the extent that the term linear in @xmath64 dominates the cosine and sine terms . the correspondence between @xmath0 and @xmath1 in ( 3.17 ) is quite complicated for finite @xmath64 due to the presence of the derivatives of @xmath108 . thus ( 3.17 ) satisfies our criterion of brst invariant path integral for any non - negative @xmath107 . the relation ( 3.9 ) is recovered if one sets @xmath109 in ( 3.17 ) ; the non - zero parameter @xmath110 however renders a canonical structure of the theory better - defined . for example , the kinetic term for ghost fields in ( 3.13 ) disappears for @xmath109 . in this respect the gauge ( 3.5 ) is also analogous to the unitary gauge . in the following we set @xmath111 in ( 3.13 ) , @xmath112 and let @xmath113 later . in the limit @xmath114 , one recovers the gauge condition ( 3.5 ) defined in ref.@xcite . this procedure is analogous to @xmath4-gauge ( or the @xmath115-limiting process of lee and yang@xcite ) @xcite , where the ( singular ) unitary gauge is defined in the vanishing limit of the gauge parameter , @xmath116 : in ( 3.18 ) the parameter @xmath17 plays the role of @xmath115 in @xmath4- gauge . in @xmath4-gauge one can keep @xmath117 without spoiling gauge invariance , and the advantage of this approach with @xmath117 is that one can avoid the appearance of ( operator ordering ) correction terms @xcite when one moves from the hamiltonian formalism to the lagrangian formalism and vice versa . this point will be discussed in detail when we analyze perturbative corrections to the ground state energy in section 4 . by using the brst invariance , one can show the @xmath71- independence of ( 3.11 ) as follows : @xmath118 \ exp\{i { \int}({\cal l}+{\cal l}_{g})dt\}\ ] ] where we perturbatively expanded in the variation of @xmath38 for a change of the parameter @xmath119 , @xmath120\ ] ] this expansion is justified since the normalization factor defined by ( see eq.(2.10 ) ) @xmath121 is independent of @xmath71 provided that the global single - valuedness in ( 3.17 ) is satisfied . [ see also the discussion related to eq.(5.1 ) . ] as was noted before , this path integral for @xmath37 , which depends on @xmath17 , is the only place where we explicitly encounter the gribov - type copies in the present approach . by denoting the brst transformed variables by prime , for example , @xmath122 we have a brst identity ( or slavnov - taylor identity [ 16 ] ) @xmath123 \ exp\{i { \int}({\cal l}+{\cal l}_{g})dt\}\end{aligned}\ ] ] where the first equality holds since the path integral is independent of the naming of integration variables provided that the asymptotic behavior and the boundary conditions are not modified by the change of variables . the second equality in ( 3.23 ) holds due to the brst invariance of the measure and the action @xmath124 but @xmath125\ ] ] from ( 3.23 ) one concludes @xmath126 \ exp\{i { \int}({\cal l}+{\cal l}_{g})dt\ } = 0\ ] ] and thus @xmath127 in ( 3.19 ) . this relation shows that the ground state energy is independent of the parameter @xmath71 ; in particular one can choose @xmath128 in evaluating the ground state energy , which leads to the gauge condition ( 3.4 ) without gribov complications . in the path integral ( 3.11 ) , one may impose _ periodic _ boundary conditions in time @xmath80 on all the integration variables and let the time interval @xmath129 later so that brst transformation ( 3.15 ) be consistent with the boundary conditions . the analysis in this sub - section is general but formal . in the next sub - section we convert the path integral ( 3.11 ) to an operator hamiltonian formalism and analyze in detail the structure of the ground state and the gauge independence of physical energy spectrum . * 3.3 , brst analysis : operator hamiltonian formalism * + we start with the brst invariant effective lagrangian @xmath130^{2 } + [ \dot{y}^{\omega}(t ) - g\xi^{\omega}(t)x^{\omega}(t)]^{2 } + [ \dot{z}^{\omega}(t ) - \xi^{\omega}(t)]^{2}\}\nonumber\\ & & - u[(x^{\omega}(t))^{2 } + ( y^{\omega}(t))^{2}]\nonumber\\ & & -\alpha \dot{b}(t ) \xi^{\omega}(t ) + b(t)(z^{\omega}(t ) - \lambda x^{\omega}(t ) ) + \alpha i \dot{\bar{c}}(t)\dot{c}(t)\nonumber\\ & & -i\bar{c}(t ) [ 1 + g\lambda y^{\omega}(t)]c(t ) + \frac{\alpha}{2}b(t)^{2}\end{aligned}\ ] ] obtained from ( 3.1 ) and ( 3.18 ) . a justification of ( 3.28 ) , in particular its treatment of gribov - type copies , rests on the path integral representation ( 3.11 ) . in the following we suppress the suffix @xmath1 , which emphasizes that the brst transformation is local in @xmath1-space . one can construct a hamiltonian from ( 3.28 ) as @xmath131 + u(x^{2 } + y^{2})\nonumber\\ & & + \xi g - b(z-\lambda x ) + i\frac{1}{\alpha}p_{c}p_{\bar{c } } + i\bar{c } ( 1+\lambda gy)c - \frac{1}{2}\alpha b^{2}\end{aligned}\ ] ] where @xmath132 and the gauss operator is given by @xmath133 we note that @xmath134 and @xmath135 from ( 3.30 ) . the quantization condition @xmath136 = \frac{1}{i}\ ] ] implies @xmath137 = \frac{i}{\alpha}\ ] ] and thus one may take the representation @xmath138 which is used later . we also note the quantization conditions @xmath139 the brst charge is obtained from @xmath140 ( 3.28 ) via the noether current as @xmath141 the brst charge @xmath98 is hermitian @xmath142 and nil - potent @xmath143 by noting @xmath144 . the brst transformation ( 3.15 ) is generated by @xmath98 , for example , @xmath145\nonumber\\ & = & x(t ) - i\theta g c(t)y(t),\nonumber\\ e^{-\theta q}\bar{c}(t)e^{\theta q } & = & \bar{c}(t ) - [ \theta q , \bar{c}(t)]\nonumber\\ & = & \bar{c}(t ) + \theta b(t)\end{aligned}\ ] ] by noting @xmath43 . some of the brst invariant operators are given by @xmath146 the hamiltonian in ( 3.29 ) is rewritten by using the brst charge as @xmath147 with @xmath148 + u ( x^{2 } + y^{2 } ) \ ] ] one defines a physical state @xmath149 as an element of brst cohomology @xmath150 namely @xmath151 but @xmath149 is _ not _ written in a form @xmath152 with a non - vanishing @xmath153 . the time development of @xmath149 is dictated by schroedinger equation @xmath154 and thus @xmath155 in the sense of brst cohomology by noting ( 3.40 ) and @xmath156 . note that the hamiltonian is brst invariant @xmath157 = [ q , h ] = 0\ ] ] if one solves the time independent schroedinger equation @xmath158 with @xmath156 , one obtains @xmath159 by noting @xmath160 . the eigen - value equation ( 3.47 ) is gauge independent and thus @xmath161 is formally gauge independent , but the value of @xmath161 is constrained by @xmath162 and thus we need a more detailed analysis . the basic task in the present brst approach is to construct physical states @xmath149 satisfying ( 3.42 ) . we construct such physical states , in particular the ground state , as fock states . for an explicit construction of physical states , we use the harmonic potential considered in ref.@xcite @xmath163 and we write @xmath164 in ( 3.29 ) as @xmath165 + \frac{\omega^{2}}{2 } ( x^{2 } + y^{2 } ) + \frac{1}{2}(gl_{z})^{2}\nonumber\\ & & + \frac{1}{2}(\tilde{p_{z } } + \tilde{\xi})^{2}+ \frac{1}{2\alpha}z^{2 } - \frac{1}{2}\tilde{\xi}^{2 } - \frac{1}{2\alpha}(\alpha b + z)^{2}\nonumber\\ & & + \frac{i}{\alpha}p_{c}p_{\bar{c } } + i\bar{c}c -\lambda\ { q , \bar{c}x\}_{+}\end{aligned}\ ] ] with@xmath166 eq.(3.50 ) indicates that the freedom associated with @xmath167 has positive norm but the freedom @xmath168 has negative norm . we thus define @xmath169,\nonumber\\ \tilde{\xi } & = & \xi - gl_{z } = ( -i)\sqrt{\frac{\nu}{2 } } ( b - b^{\dagger } ) , \nonumber\\ \alpha b + z & = & \frac{1}{\sqrt{2\nu } } ( b + b^{\dagger } ) , \nonumber\\c & = & \frac{1}{\sqrt{2}\mu } ( \hat{c } + \hat{c}^{\dagger } ) , \nonumber\\ p_{\bar{c } } & = & \frac{\mu}{\sqrt{2 } } ( \hat{c } - \hat{c}^{\dagger } ) , \nonumber\\ \bar{c } & = & \frac{1}{\sqrt{2}\mu } ( \hat{\bar{c } } + \hat{\bar{c}}^{\dagger } ) , \nonumber\\ p_{c } & = & \frac{\mu}{\sqrt{2 } } ( -\hat{\bar{c } } + \hat{\bar{c}}^{\dagger } ) \end{aligned}\ ] ] with @xmath170 in ( 3.52 ) the operator expansion of @xmath171 , and @xmath93 is the standard one , but the expansion of @xmath172 and @xmath173 is somewhat unconventional . the canonical commutators are satisfied by postulating @xmath174 = 1,\nonumber\ ] ] @xmath175 = 1,\nonumber\ ] ] @xmath176 = 1,\nonumber\ ] ] @xmath177 & = & -1,\nonumber\\ \ { \hat{c } , \hat{\bar{c}}^{\dagger}\}_{+ } & = & -i,\nonumber\\ \ { \hat{c}^{\dagger } , \hat{\bar{c}}\}_{+ } & = & i\end{aligned}\ ] ] and all other commutators are vanishing . here we defined @xmath178 so that @xmath179,\ ] ] @xmath180 @xmath181 the operator @xmath182 also satisfies @xmath183 = -1\ ] ] the variables @xmath184 in ( 3.54 ) and @xmath182 in ( 3.56 ) carry negative norm . the hamiltonian ( 3.50 ) is then written as @xmath185 + i\nu[\hat{\bar{c}}^{\dagger}\hat{c } - \hat{c}^{\dagger}\hat{\bar{c } } + i]\nonumber\\ & & + \lambda\{q , \bar{c}x\}_{+}\end{aligned}\ ] ] with the brst charge @xmath186\nonumber\\ & = & i\nu[\hat{c}^{\dagger}(\bar{b } - d ) - \hat{c}(\bar{b } - d)^{\dagger } ] + \sqrt{\frac{\nu}{2}}(\hat{c}^{\dagger } + \hat{c})gl_{z}\end{aligned}\ ] ] and @xmath187 the brst charge in ( 3.58 ) is nil - potent @xmath188 = 0\ ] ] by noting ( 3.54 ) and ( 3.56 ) . we thus define the ( physical ) ground state at @xmath189 by @xmath190 which ensures @xmath191 the zero - point energy of @xmath167 and @xmath115 and the zero - point energy of @xmath94 and @xmath93 in ( 3.57 ) cancel each other for the state @xmath192 . when one defines a unitary transformation [ 8 ] @xmath193 we can write the physical part of @xmath164 in ( 3.57 ) as @xmath194 if one recalls the relation @xmath195\nonumber\\ b - d & = & \frac{1}{\sqrt{2\nu } } [ -i(p_{z } + gl_{z } ) + \nu\frac{\partial}{i\partial \xi}]\end{aligned}\ ] ] where we used @xmath196 in ( 3.34 ) , the state @xmath192 in ( 3.61 ) is required to satisfy @xmath197 eq.(3.66 ) has a solution @xmath198 which depends on @xmath199 ; @xmath200 is a normalization constant . the inner product of this state needs to be defined by means of a @xmath201 degree rotation in the variable @xmath115 @xmath202 which reflects the fact that the @xmath115-variable carries negative norm . if one projects the state @xmath203 to the one with @xmath204 by fourier transformation , which is the general procedure of dirac in his treatment of singular lagrangian [ 17 ] , one obtains @xmath205 this is the physical ground state naively expected for @xmath206 system for a given @xmath199 on the basis of invariance under the gauss operator in ( 3.31 ) , and it is independent of the gauge parameter @xmath207 . other sectors of the ground state @xmath208 are constructed in a standard manner @xmath209 and the entire ground state in ( 3.62 ) is written as @xmath210 where @xmath211 is replaced by a c - number by acting @xmath199 on @xmath208 . in the present case , @xmath212 . the ground state thus defined has a time dependence described by @xmath164 in ( 3.57 ) @xmath213 in the sense of brst cohomology by noting ( 3.62 ) . thus we can _ represent _ the vacuum ( or ground ) state in @xmath214 by @xmath192 in ( 3.71 ) for any time , and we have @xmath215 excited physical states are represented by @xmath216 one obtains the eigenvalue of @xmath164 in ( 3.57 ) and ( 3.64 ) as @xmath217 \frac{1}{\sqrt{n_{1}!n_{2}!}}(\tilde{a}_{x}^{\dagger})^{n_{1 } } ( \tilde{a}_{y}^{\dagger})^{n_{2}}|0\rangle \nonumber\\ & & + \lambda q\bar{c}x \frac{1}{\sqrt{n_{1}!n_{2}!}}(\tilde{a}_{x}^{\dagger})^{n_{1 } } ( \tilde{a}_{y}^{\dagger})^{n_{2}}|0\rangle\nonumber\\ & \simeq & [ \omega(n_{1 } + n_{2 } + 1 ) + \frac{g^{2}}{2}(n_{1 } - n_{2})^{2 } ] \frac{1}{\sqrt{n_{1}!n_{2}!}}(\tilde{a}_{x}^{\dagger})^{n_{1 } } ( \tilde{a}_{y}^{\dagger})^{n_{2}}|0\rangle \nonumber\\\end{aligned}\ ] ] in the sense of brst cohomology , since @xmath218 this ( 3.76 ) is confirmed by using the expression of @xmath98 in ( 3.58 ) as @xmath219 where we used @xmath220 the second relation in ( 3.78 ) is regarded as a constraint on the ground state of @xmath184 for a given value of @xmath199,which is a manifestation of the gauss constraint in the present formulation . the states which include @xmath221,and @xmath222 excitations become unphysical and are removed by brst cohomology . for example , the state @xmath223 is brst invariant and has energy ( for @xmath224 ) @xmath225 but it is obviously excluded by brst cohomology . the unphysical excitations @xmath221,and @xmath222 form the components of _ non - trivial _ brst superfields @xmath226 and @xmath227 in ( 3.15 ) , @xmath228\end{aligned}\ ] ] a characteristic property of these non - trivial superfields is that the second components of the superfields , which are brst transform of the first components , contain terms @xmath229 in the elementary field . the basic theorem of brst symmetry is that any brst invariant state which contains those unphysical degrees of freedom , @xmath230 and @xmath222 , is written in the brst exact form such as in ( 3.79 ) and thus it is removed by brst cohomology . see ref.[13 ] . the present brst analysis is in accord with an explicit construction of physical states in ref.@xcite . one can safely take the limit @xmath113 ( or @xmath231 ) in the physical sector , though unphysical excitations such as in ( 3.80 ) acquire infinite excitation energy in this limit just like unphysical excitations in gauge theory defined by @xmath4-gauge@xcite . it has been shown in @xcite that the correction terms arising from operator ordering plays a crucial role in the evaluation of perturbative corrections to ground state energy in lagrangian path integral formula . this problem is often treated casually in conventional perturbative calculations ; a general belief ( and hope ) is that lorentz invariance and brst invariance somehow takes care of the operator ordering problem . in the following , we show that brst invariance and @xmath232-product prescription reproduce the correct result of ref.@xcite provided that one uses a canonically well - defined gauge such as @xmath4-gauge with @xmath117 in ( 3.18 ) . this check is important to establish the equivalence of ( 3.11 ) to the path integral formula in ref.@xcite . see also ref.@xcite . if one starts with @xmath114 from the on - set , one needs correction terms calculated in ref.@xcite . to be precise , what we want to evaluate is eq.(3.11 ) , namely @xmath233 we define the path integral for a sufficiently large time interval @xmath234\ ] ] and let @xmath235 later . in the actual calculation , there appear two important aspects which need to be taken into account : + ( i ) we impose periodic boundary conditions on all the variables so that brst transformation ( 3.15 ) is well - defined including the boundary conditions . + ( ii ) in the actual evaluation of the path integral as well as feynman diagrams , we may apply the wick rotation and perform euclidean calculations . + the exact ground state energy of ( 4.1 ) is given by eq.(3.73 ) as @xmath236 namely , we have no correction depending on the gauge parameter @xmath71 and the coupling constant @xmath34 . as was already shown in ( 3.27 ) , the absence of @xmath71 dependence is a result of brst symmetry . this property is thus more general and , in fact , it holds for all the energy spectrum of physical states;this can be shown by using the schwinger s action principle [ 18 ] and the definition of physical states in ( 3.42 ) . the perturbative check of @xmath71-independence or slavnov - taylor identities in general is carried out in the standard manner . on the other hand , the absence of @xmath34- dependence is an effect of more dynamical origin . in this section we concentrate on the evaluation of @xmath34-dependence by taking a view that the @xmath71- independence has been generally established in ( 3.27 ) . we first perform the gaussian path integral over @xmath36-variable by noting @xmath237^{2 } - \frac{1}{2\alpha}[\alpha\dot{\xi } + z - \lambda x]^{2}\nonumber\\ & \rightarrow & - \frac{1}{2\alpha}[\alpha\dot{\xi } + z - \lambda x]^{2}\end{aligned}\ ] ] we thus consider the path integral @xmath238 where @xmath37 is the original normalization constant in ( 3.21 ) , and @xmath239^{2 } + [ \dot{y}(t ) - g\xi(t)x(t)]^{2}\ } - \frac{\omega^{2}}{2}[{x(t)}^{2 } + { y(t)}^{2 } ] \nonumber\\ & & + \frac{1}{2}{\dot{z}(t)}^{2 } - \frac{1}{2\alpha}[z(t ) - \lambda x(t)]^{2 } - \frac{\alpha}{2}{\dot{\xi}(t)}^{2 } + \frac{1}{2}{\xi ( t)}^{2}\nonumber\\ & & + \lambda\dot{\xi}(t)x(t ) + \alpha i \dot{\bar{c}}(t)\dot{c}(t ) -i\bar{c}(t ) ( 1 + g\lambda y(t))c(t ) \nonumber\\ & \equiv & { \cal l}_{0 } + { \cal l}_{i}\end{aligned}\ ] ] with @xmath240 - \frac{\omega^{2}}{2}[{x^(t)}^{2 } + { y(t)}^{2 } ] \nonumber\\ & & + \frac{1}{2}{\dot{z}(t)}^{2 } - \frac{1}{2\alpha}z(t)^{2 } - \frac{\alpha}{2}{\dot{\xi}(t)}^{2 } + \frac{1}{2}{\xi ( t)}^{2}\nonumber\\ & & + \alpha i \dot{\bar{c}}(t)\dot{c}(t ) -i\bar{c}(t)c(t),\\ { \cal l}_{i } & \equiv & g\xi ( t)[\dot{x}(t)y(t ) - \dot{y}(t)x(t ) ] + \frac{1}{2}g^{2}{\xi ( t)}^{2}[x(t)^{2 } + y(t)^{2}]\end{aligned}\ ] ] where we set @xmath224 in the final expressions in ( 4.7 ) and ( 4.8 ) . the propagators are defined by @xmath241 in ( 4.7 ) in the standard manner as @xmath242 where we took @xmath243 limit in the evaluation of those propagators ; this procedure is justified for the evaluation of corrections to the ground state energy since all the momentum integrations in ( 4.11 ) below are well - convergent . up to the second order of perturbation in @xmath244,we have @xmath245\rangle \nonumber\\ & + & \frac{(i)^{2}}{2!}\int dt_{1}dt_{2}\langle t^{\star}g^{2}\xi ( t_{1})\xi ( t_{2})[\dot{x}(t_{1})y(t_{1 } ) - \dot{y}(t_{1})x(t_{1})]\nonumber\\ & & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times [ \dot{x}(t_{2})y(t_{2 } ) - \dot{y}(t_{2})x(t_{2})]\rangle \}\nonumber\\ & = & \frac{1}{n}{\int}d\mu \ e^{i{\int}{\cal l}_{0}dt}\ { 1 + ig^{2}\int dt \langle t^{\star}\xi ( t)\xi ( t)\rangle \langle t^{\star}x(t)x(t)\rangle \nonumber\\ & + & ( i)^{2}g^{2}\int dt_{1}dt_{2}\langle t^{\star}\xi ( t_{1})\xi ( t_{2})\rangle \langle t^{\star}\dot{x}(t_{1})\dot{x}(t_{2})\rangle \langle t^{\star}y(t_{1})y(t_{2})\rangle \nonumber\\ & - & ( i)^{2}g^{2}\int dt_{1}dt_{2}\langle t^{\star}\xi ( t_{1})\xi ( t_{2})\rangle \langle t^{\star}\dot{x}(t_{1})x(t_{2})\rangle \langle t^{\star}y(t_{1})\dot{y}(t_{2})\rangle \}\end{aligned}\ ] ] by taking the symmetry in @xmath246 and @xmath247 into account . the terms of order @xmath248 in ( 4.10 ) are written by using the propagators in ( 4.9 ) as @xmath249 after the wick rotation . we here note that the @xmath232-product prescription is crucial in obtaining ( 4.11 ) ; the @xmath232-product commutes with the time derivative operation , which is intuitively understood from the fact that the basic path integration variables are field variables ( and not their time derivatives ) in the lagrangian path integral formula [ 19 ] . in this approach , the conventional @xmath250-product is defined from @xmath232-product via the bjorken - johnson - low prescription [ 20 ] . the first two terms in ( 4.11 ) cancel each other . the last two terms in ( 4.11 ) can be evaluated as follows : in the third term in ( 4.11 ) , one can rewrite the integrand as @xmath251 since the poles displaced in the same side of the real axis do not contribute to the integral over @xmath252 ; the contour can be shrunk to zero without encircling poles . similarly , in the last term in ( 4.11 ) , @xmath253 the last two terms in ( 4.11 ) thus cancel each other , and we obtain only the lowest order contribution in ( 4.11 ) @xmath254^{2}}\frac{1}{n}\frac{det [ \alpha\partial_{t}^{2 } + 1]}{\sqrt{det [ \partial_{t}^{2 } + 1/\alpha ] det [ \alpha\partial_{t}^{2 } + 1]}}\nonumber\\ & = & const \times \frac{1}{[2i\sin ( \omega t/2)]^{2 } } \ \ \ for\ t\rightarrow \infty,\end{aligned}\ ] ] where the determinant factors coming from @xmath255 and @xmath94 integration combined with the normalization constant @xmath37 cancel completely among themselves . the last expression in ( 4.14 ) is a standard path integral of harmonic oscillators @xmath246 and @xmath247 with periodic boundary conditions [ 21 ] , and it may be expanded as @xmath256^{2 } } = const \times e^{-i\omega t}(1 + \sum_{n_{1 } , n_{2 } = 0}^{\infty } e^{-i\omega t(n_{1 } + n_{2})})\ ] ] for @xmath243 , where the summation over the non - negative integers @xmath257 and @xmath258 excludes the case @xmath259 . to be precise , the @xmath250-dependence of the normalization constant @xmath37 needs to be taken into account to obtain the last expression of ( 4.14 ) [ 21 ] . the ground state energy is then obtained from @xmath260}|0\rangle \nonumber\\ & = & \lim_{t\rightarrow \infty } const \times e^{-i\omega t}\end{aligned}\ ] ] which is justified for @xmath261 and @xmath262 in euclidean theory . we thus obtain the ground state energy @xmath236 to be consistent with ( 3.73 ) . the brst symmetry plays a central role in modern gauge theory , and the brst invariant path integral can be formulated by summing over all the gribov - type copies in a very specific manner provided that the crucial correspondence in ( 2.11 ) or ( 3.17 ) is globally single valued[5 ] . this criterion is satisfied by the soluble gauge model proposed in ref.[8 ] , and it is encouraging that the brst invariant prescription is in accord with the canonical analysis of the soluble gauge model in ref.@xcite . the detailed explicit analysis in ref.@xcite and the present somewhat formal brst analysis are complementary to each other . in ref.@xcite , the problem related to the so - called gribov horizon , in particular the possible singularity associated with it , has been analyzed in greater detail ; this is crucial for the analysis of more general situation . on the other hand , an advantage of the brst analysis is that one can clearly see the gauge independence of physical quantities such as the energy spectrum as a result of brst identity . the brst approach allows a transparent treatment of general class of gauge conditions implemented by ( 3.18 ) . this gauge condition with @xmath117 renders the canonical structure better - defined , and it allows simpler perturbative treatments of the problems such as the corrections to the ground state energy . our calculation vis - a - vis the explicit canonical analysis in ref.@xcite may provide a ( partial ) justification of conventional covariant perturbation theory in gauge theory , which is based on lorentz invariance ( or @xmath232-product ) and brst invariance without the operator ordering terms . motivated by the observation in ref.[8 ] to the effect that the gribov horizons are not really singular in quantum mechanical sense , which is in accord with our path integral in ( 3.21 ) , we would like to make a speculative comment on the role of gribov copies in qcd . first of all , the topological phenomena related to instantons for which the coulomb gauge is generally singular may be analyzed in the temporal gauge @xmath66 ; this gauge is relatively free from the gribov complications[3 ] . a semi - classical treatment of instantons with small quantum fluctuations around them will presumably give qualitatively reliable estimates of topological effects . as for other non - perturbative effects such as quark confinement and hadronic spectrum , one may follow the argument of witten [ 22 ] on the basis of @xmath5 expansion in qcd[23 ] ; he argues that the @xmath5 expansion scheme comes closer to the real qcd than the instanton analysis in the study of hadron spectrum . if this is the case , one can analyze the qualitative aspects of hadron spectrum on the basis of a sum of ( an infinite number of ) feynman diagrams . this diagramatic approach or an analytical treatment equivalent to it in the feynman - type gauge deals with topologically trivial gauge fields but may still suffer from the gribov copies , as is suggested by the analysis in ref.[4 ] : if one assumes that the vacuum is unique in this case as is the case in @xmath54-space , the global single - valuedness in ( 2.11 ) in the context of path integral ( 2.10 ) will be preserved for infinitesimally small fields @xmath14 . by a continuity argument , a smooth deformation of @xmath14 in @xmath54-space ( or its extension as explained in section 2 ) will presumably keep the integral ( 2.10 ) unchanged . if this argument should be valid , our path integral formula in ( 2.4 ) would be justified . if our speculation is correct , the formal path integral formula ( 2.5 ) will provide a basis for the analysis of some non - perturbative aspects of qcd . on the other hand , the gribov problem may also suggest the presence of some field configurations which do not satisfy any given gauge condition in four dimensional non - abelian gauge theory [ 2 ] . for example , one may not be able to find any gauge parameter @xmath12 which satisfies @xmath263 for some fields @xmath14 . although the measure of such field configurations in path integral is not known , the presence of such filed configurations would certainly modify the asymptotic correspondence in ( 2.11 ) . in the context of brst symmetry , the gribov problem may then induce complicated phenomena such as the dynamical instability of brst symmetry[24 ] . if the dynamical instability of brst symmetry should take place , the relation corresponding to ( 3.27 ) , which is a result of the brst invariance of the vacuum , would no longer be derived . in the framework of path integral , this failure of ( 3.27 ) would be recognized as the failure of the expansion ( 3.19 ) since the normalization factor @xmath37 in ( 3.21 ) would generally depend on not only field variables but also @xmath71 if the global single - valuedness in ( 3.17 ) should be violated . finally , we note that the lattice gauge theory [ 25 ] , which is based on compactified field variables , is expected to change the scope and character of the gribov problem completely . the gribov problem is intricately related to the difficult issue of the non - perturbative continuum limit of lattice gauge theory . [ note that the perturbative continuum limit is not quite relevant in the present context , since we do not worry about the gribov problem much in weak field perturbation theory ] . v. n. gribov , nucl . i. m. singer , comm . phys . * 60*(1978)7 . r. jackiw , i. muzinich and c. rebbi , phys . rev . * t. maskawa and h. nakajima , prog . phys . * 60*(1978)1526 . k. fujikawa , prog . . phys . * 61*(1979)627 . + p. hirschfeld , nucl . phys . * b157*(1979)37 . c. becchi , a. rouet and r. stora , comm . * 42*(1975)127 ; ann . + m. henneaux , phys . reports * 126*(1985)1 . + l. baulieu , phys . reports * 129*(1985)1 . d. zwanziger , nucl . phys . * b412*(1994)657 and references therein . + l. j. carson , nucl . phys . * + h. yabuki , ann . phys . * 209*(1991)231 . + f. g. scholtz and g. b. tupper , phys . rev . * r. friedberg , t. d. lee , y. pang and h. c. ren , columbia report , cu - tp-689 ; ru-95 - 3-b . l. d. faddeev and v. n. popov , phys . lett . * 25b*(1967)29 . + g. t hooft , nucl . phys . * b33*(1971)173 . l. d. faddeev , theor . math . phys . * 1*(1970)1 . s. coleman , the uses of instantons in _ aspects of symmetry _ ( cambridge univ . press , cambridge,1985 ) . t. kugo and i. ojima , phys . lett . * 73b*(1978)459 . k. fujikawa , prog . phys.*63*(1980)1364 ; ibid , * 59*(1978)2045 t. d. lee and c. n. yang , phys . rev . * 128 * ( 1962)885 . k. fujikawa , b. w. lee and a. i. sanda , phys . rev . * d6*(1972)2923 . e. s. fradkin and i. v. tyutin , phys . rev * d2*(1970)2841 . + a. a. slavnov , theor . phys . * 10*(1972)99 . + j. c. taylor , nucl . phys . * b33*(1971)436 . p. a. m. dirac , _ lectures on quantum field theory_(yeshiva univ . , new york , 1966 ) . c. s. lam , nuovo cim . * 38*(1965 ) 1755 . y. nambu , prog . . phys . * 7*(1952)131 . j. d. bjorken , phys . * 148*(1966)1467 . + k. johnson and f. e. low , prog . suppl.*37 - 38*(1966)74 . r. p. feynman and a. r. hibbs , _ quantum mechanics and path integrals _ ( new york , mcgraw - hill,1965 ) . e. witten , nucl . phys . * b149*(1979)285 . g. t hooft , nucl . phys . * b72*(1974)461 . k. fujikawa , nucl . phys . * b223*(1983)218 . k. g. wilson , phys .

a path integral with brst symmetry can be formulated by summing the gribov - type copies in a very specific way if the functional correspondence between @xmath0 and the gauge parameter @xmath1 defined by @xmath2 is `` globally single valued '' , where @xmath3 specifies the gauge condition . a soluble gauge model with gribov - type copies recently analyzed by friedberg , lee , pang and ren satisfies this criterion . a detailed brst analysis of the soluble model proposed by the above authors is presented . the brst symmetry , if it is consistently implemented , ensures the gauge independence of physical quantities . in particular , the vacuum ( ground ) state and the perturbative corrections to the ground state energy in the above model are analysed from a view point of brst symmetry and @xmath4-gauge . implications of the present analysis on some aspects of the gribov problem in non - abelian gauge theory , such as the @xmath5 expansion in qcd and also the dynamical instability of brst symmetry , are briefly discussed . 9.3 in 6.3 in = -0.4 in = -0.8 in @xmath6 ut-722 , 1995 1.5 truecm .75 truecm * kazuo fujikawa * .4 truecm _ department of physics , university of tokyo _ _ bunkyo - ku , tokyo 113,japan _ \1 . truecm addtoresetequationsection \1 . truecm

we gratefully acknowledge support by a rgc grant from the sar government of hong kong under grant number hku 7113/02p and from nserc of canada and fcar of quebec ( h.g ) . m. julliere , phys . lett . a * 54 * , 225 ( 1975 ) ; r. meservey and p.m. tedrow , phys . rep . * 238 * , 173 ( 1994 ) ; j.s . moodera et al , phys . lett . * 74 * , 3273 ( 1995 ) ; g.a . prinz , science , * 282 * , 1660 ( 1998 ) ; h. mehrez et al , phys . lett . * 84 * , 2682 ( 2000 ) ; a. wolf et al , science * 294 * , 1488 ( 2001 ) .

we investigate a non - adiabatic parametric quantum pump consists of a nonmagnetic scattering region connected by two ferromagnetic leads . the presence of ferromagnetic leads allows electrons with different spins to experience different potential landscape . using this effect we propose a quantum spin pump that drives spin - up electrons to flow in one direction and spin - down electrons to flow in opposite direction . as a result , the spin pump can deliver a spin current with vanishing charge current . a parametric quantum pump generates a dc electric current by a cyclic variation of system parameters while keeping the leads at a constant chemical potential@xcite . considerable effort has been devoted to understand the physics of parametric pumping@xcite . it was found that the pumped current is rather sensitive to various parameters of the system such as potential of the pump , frequency of the driving force , and fermi energy of the leads . as parameters vary , the pumped current can even change directions : this has been predicted for charge pumping in resonant tunneling diodes@xcite , nanotube quantum pumps@xcite , finite frequency pumping process@xcite , and quantization of the pumped charge@xcite . to fully exploit this behavior , in this paper we investigate parametric pumping in the presence of ferromagnetic leads . when ferromagnetic leads are present , electrons with different spins experience different potentials . under certain conditions the pumped current for electron with different spin may flow in opposite directions . as a result , the total pumped charge current can be zero while a pure spin current is delivered . such a quantum spin pump may have potential applications in the fascinating field of spintronics@xcite . recently , several different spin pumps have been proposed . in the uni - pole spin battery studied in refs . , a spin current is generated by either a rotating magnetic moment or rotating external magnetic field . in these uni - pole pumps , spin current is not conserved due to spin flips@xcite . in contrast , the spin pump considered in this work satisfies spin current conservation when magnetizations of the two leads are parallel or antiparallel . in another direction , an adiabatic quantum pump was proposed@xcite such that a spin polarized current is generated in a chaotic quantum dot in the presence of an in - plane magnetic field . in this work we go beyond the adiabatic regime and examine the frequency dependence of the pumped spin current . our results show that a pure spin current can be achieved by varying the pumping frequency . the system we investigate is a spin - valve which consists of a non - magnetic scattering region connected by two ferromagnetic electrodes to the reservoir . the magnetic moment @xmath0 of the left electrode is pointing to the @xmath1-direction , the electric current is flowing in the @xmath2-direction , while the moment of the right electrode is at an angle @xmath3 to the @xmath1-axis in the @xmath4 plane . for simplicity of discussion , we assume that the value of molecular field @xmath5 is the same for the two electrodes , thus a standard spin - valve effect is obtained@xcite by varying the angle @xmath3 . essentially , @xmath5 mimics the difference of density of states ( dos ) between spin - up and down electrons@xcite in the electrodes . the pumped current for this system can be calculated at finite frequency and up to the second order in pumping amplitude using the perturbation theory based on nonequilibrium green s functions@xcite . the pumped particle current due to spin component @xmath6 through lead @xmath7 is found to be ( @xmath8)@xcite @xmath9 { \bf g}^a \right]_{\beta \beta } \label{eq8}\end{aligned}\ ] ] where @xmath10 , @xmath11 is the pumping potential profile@xcite , @xmath12 , and @xmath13 is a @xmath14 matrix representing the equilibrium retarded green s function @xmath15 with @xmath16 the hamiltonian in the absence of pumping potential . here @xmath17 is the self energy and @xmath18 $ ] is the linewidth function . the self - energies are given@xcite @xmath19 with the rotational matrix @xmath20 for electrode @xmath7 defined as @xmath21 here angle @xmath22 is defined as @xmath23 and @xmath24 and @xmath25 is the usual self energy@xcite . in this paper , we will study two special cases : @xmath26 or @xmath27 for spin current . from eq.([eq8 ] ) , we observe that up to the second order in pumping amplitude , the particle can absorb or emit a photon during the pumping process . the contribution due to these two photon assisted processes have different sign and tend to cancel each other . as the result of this competition , the pumped particle current can reverse its direction upon varying system parameters . the pumped charge current @xmath28 is given by @xmath29 and the pumped spin current @xmath30 is ( we have set @xmath8 ) @xmath31 now we examine the conservation law for pumped current . when @xmath32 , the @xmath14 green s function @xmath33 and self energy are diagonal . as a result , we have @xmath34_{\beta \beta } \label{eq9}\end{aligned}\ ] ] where we have moved @xmath35 to the beginning of the trace . this can be done only if the green s function and self energy are diagonal . using the fact that @xmath36 and changing variable from @xmath37 to @xmath38 in the second term of eq.([eq9 ] ) , it is straightforward to show the @xmath39 . this means that both the pumped electric current and pumped spin current are conserved in this device . now we use eqs.([eq8 ] ) , ( [ ele ] ) , and ( [ spin ] ) to calculate pumped electric current and spin current . the system we studied is a symmetric double @xmath40 barrier structure modeled by potential @xmath41 . for this system the green s function @xmath42 can be calculated exactly@xcite . the pumping potential is chosen to be sinusoidal @xmath43 $ ] . we will calculate the pumped electric and spin current from the left lead at zero temperature and set @xmath44 . we set @xmath45 and @xmath46 . a gate voltage @xmath47 is applied in the double barrier structure to control the resonant electron level . we assume that the fermi level of the leads is in line with the resonant level at @xmath48 . finally the unit is set by @xmath49 . for the system of fe / ge / fe with @xmath50 , the energy unit is @xmath51 which corresponds to @xmath52 ghz . the unit for pumped current is @xmath53a . the unit for spin current is @xmath54 . in fig.1 , we show the pumped electric current ( solid line ) and spin current ( dotted line ) as a function pumping frequency when @xmath26 for fixed fermi energy and magnetization @xmath5 . as the frequency increases , the pumped electric current increases and reaches a maximum value at @xmath55 . as the frequency increases further , the pumped electric current decrease and becomes negative at large frequency . in the presence of magnetic electrodes , the pumped electric current is spin polarized with non - zero spin current . the behavior of the pumped spin current is similar to that of the pumped electric current . it is positive at small pumping frequency and becomes negative at larger frequencies . at @xmath56 ( thin vertical line in fig.1 ) , the pumped electric current is zero and a pure spin current is achieved . the physics behind this is the following . during parametric pumping , the system pumps out spin - up and spin - down electrons . for electron with a given fermi energy , the potential of the ferromagnetic lead for spin - up electron is @xmath57 while for spin - down electron it is @xmath5 . as a result , the pumped electric current for different spins can have different sign , _ i.e. _ spin - up electron pumps out from the right lead to the left whereas spin - down electron flows to the right lead during the pumping . this way a spin current is generated . at certain value of the frequency , a complete cancellation of electric current occurs and a pure spin current is delivered . in fig.2 we plot the pumped electric current and spin current versus pumping frequency for @xmath58 . we observe that the pumped electric current for @xmath26 has the same order of magnitude as that at @xmath58 . the pumped spin current , however , is quite different from that at @xmath26 . it displays a linear dependency on the pumping frequency . the frequency at which a pure spin current occurs is also different from the case when @xmath26 . comparing with the case of @xmath26 , the spin current reverses its direction when @xmath58 . our study shows that the pure spin current is a generic property and can occur at a wide range of parameters . in fig.1 , we see that at @xmath59 and @xmath56 the system pumps out pure spin current . as we vary @xmath5 , the frequency @xmath60 at which the pure spin current occurs also changes . the trajectory or the `` phase diagram '' is depicted in fig.3 where pumping frequency versus magnetization of the leads for @xmath26 is shown ( solid line ) along with the magnitude of pumped spin current . in summary , we have proposed a non - adiabatic quantum spin pump which can generate a pure spin current with zero electric current during the parametric pumping . the device consists of a non - magnetic system connected by two ferromagnetic leads . since electrons with different spin experience different potential landscape , the corresponding pumped current can be quite different . at certain condition , the pumped electric current can be zero while pure spin current is produced . our numerical results show that this can be easily achieved by varying system parameters such as the pumping frequency .

considerable effort has recently been directed towards high spatial resolution imaging of the nucleus of ngc 1068 from uv to radio wavelengths ( uv & visible : macchetto et al . 1994 ; capetti et al . 1995 , 1997 ; near - ir : chelli et al . 1987 ; gallais , 1991 ; young et al . 1996 ; marco et al . 1997 ; mid - ir : braatz et al . 1993 ; cameron et al . 1993 ; radio : wilson & ulvestad 1987 ; planesas et al . 1991 ; blietz et al . 1994 , tacconi et al . 1994 ; muxlow et al . 1996 ; gallimore et al . 1996a , b , 1997 ; greenhill & gwinn 1996 ) . indeed , as the closest seyfert 2 nucleus , ngc 1068 deserves such efforts since it is a key object in investigating models of active galactic nuclei ( agn ) . the popular `` unified '' model ( e.g. , antonucci 1993 ) includes a parsec - scale torus of dust and molecular gas around the central engine . models of the infrared emission from the torus ( krolik & begelman 1986 ; pier & krolik 1993 ; efstathiou & rowan - robinson 1994 ; granato , danese & franceschini 1997 ) explore torus sizes from 1 to 100 pc . recently , gallimore et al . ( 1997 ) detected a 1 pc elongated distribution of ionized gas at 8 ghz with the vlba which they interpreted as the hot zone " of obscuring material surrounding the agn . however , direct images showing the torus , or any elongated structure , are still lacking in the near - ir where it is expected to be most conspicuous . to achieve this goal , one requires both high angular resolution ( 1= 72 pc at the distance of ngc 1068 ) and high contrast , the light from the central region being a complex mixture of starlight , synchrotron emission and excited gas emission , all scattered and absorbed by a clumpy dust component . near - ir imaging at high angular resolution offers potential advantages in the study of agn because : _ i ) _ the wavelength range lies between two domains carrying complementary pieces of information the visible ( excited gas in the nlr ) , and the thermal ir - radio range ( cool dust and synchrotron emission from electrons ) , _ ii ) _ the extinction is reduced , _ iii ) _ diffraction - limited images are now possible thanks to adaptive optics ( ao hereafter ) . here , we present new results obtained with _ pueo _ , the ao system recently commissioned on the 3.6 m canada - france - hawaii telescope . the cfht ao system , based on a concept by roddier et al . ( 1991 ) , uses a 19 zone bimorph mirror controlled by a curvature wavefront sensor ( wfs ) to produce diffraction - limited observations ( fwhm @xmath4012 ) in the near - ir ( rigaut et al . , 1994 , lai et al . , 1996 ) . the monica infrared camera , a facility instrument of the universit de montral ( nadeau al . 1994 ) , was mounted at the output focus of _ pueo _ , itself installed at the cassegrain focus of cfht . special optics in monica give a scale on the nicmos-3 array of 00344 per pixel . the observations of ngc 1068 , and a nearby blank sky reference position , were obtained on feb 14 , 16 and 18 1997 . a dither pattern , with relative offsets of @xmath408 , was used in order to reduce the effect of bad pixels and to improve the flat - fielding , which was initially derived from dome flats . the ukirt faint standard stars fs8 and fs7 were used to provide flux and psf calibrations . the ao system was servoed on the nucleus itself ; its rather high brightness and the quality of the seeing ( ranging from 038 in k to 07 in j ) allowed good correction of the turbulence and strehl ratios of over 60% were obtained at k. the true point spread function ( psf ) in each band was recovered using wfs information ( vran et al . more specifically , it can be shown that departure of a long exposure psf from a perfect airy pattern is contributed by : _ i ) _ the non - corrected static aberrations , due for instance to the ir camera optics : these can be measured using an artificial source ; _ ii ) _ the non - perfect compensation by the adaptive mirror of the wf deformations : at low spatial frequencies they are actually measured by the wfs but can not be completely accomodated by the mirror , while at high spatial frequencies , they are not measured by the wfs , but can be estimated using a kolmogorov model of the atmosphere , scaled on the actual seeing , itself evaluated from the amplitude of the correction ( vran et al . vran ( 1997 ) showed that this method for retrieving the real - time psf is valid , the source being point - like or extended . similarly , one may wonder if the exactness of the ao correction is maintained , using a slightly extended reference as in the case of ngc 1068 . indeed , it has been shown already that the use of an extended source instead of a point - like one to perform the curvature - sensing and ao correction only reduces the efficiency of the wf sensing , but not its accuracy ( rousset , 1994 ) . moreover , the source extension in the case of ngc 1068 is smaller than the seeing value for all images : in such conditions , even the efficiency loss is small ( equ . 13 in rousset , 1994 ) . in conclusion , the non point - like shape of the reference should not introduce systematic errors in the structure of the ao corrected image , nor on the reconstructed psf . this is illustrated by the comparison between the recovered psf at k and the image of a nearby star ( fig . 1e , upper left and lower left respectively ) . image processing proceeded as follows : _ i ) _ bad pixel correction ; _ ii ) _ sky subtraction , using a median - averaged sky estimate ; _ iii ) _ flat - field correction ; _ iv ) _ re - centering of the different exposures through cross - correlation techniques ; _ v ) _ adjustment of the sky level among the overlapping regions to produce a homogeneous background ; _ vi ) _ co - addition of the overlapping regions , rejecting deviant pixels ( clipped mean ) . the resulting images were then deconvolved using the classical lucy - richardson algorithm ( 60 iterations ) while following the recipe recently proposed by magain et al . ( 1997 ) to constrain the final psf . the resulting k image is presented in fig . [ f - raw]-a , on a magnitude ( log ) scale , chosen because ao provides a high dynamic range ( typically 1.3 10@xmath5 at k ) and significant details are seen at all flux levels . in fig . [ f - raw]-e , we show a set of psf images : the psfs recovered using the wfs information ( vran et al . 1997 ) and used in the deconvolution procedure are shown for the 3 bands and , for comparison , the psf observed in k on a nearby star is also shown . we show in figs . [ f - raw]-b and -c the h and k deconvolved images . within the 1 inner region the following features are visible , especially on the deconvolved k 1300 above the background ( measured at 6 away from the core and showing a mean s / n of 10 ) . this core was known already ( marco et al . 1997 ) , although we can place a more stringent upper limit on its size . b ) _ an ese - wnw elongated structure at pa @xmath1 102@xmath6 , roughly perpendicular to the axis of the inner ionizing cone ( p.a . = 15@xmath6 , as originally derived by evans et al . , 1991 ) . this structure starts to show up at a radius 020 with a contrast , i.e. the ratio of the flux excess in the structure at the relevant radius , with respect to the mean inter - structure brightness measured at the same radius . ] of 0.28 in the ese quadrant and of 0.19 in the wnw quadrant . the mean brightness per pixel of this structure is a factor 340 above the background . c ) _ along the ns direction , a s - shaped feature extends over 03 on each side of the unresolved core , with a brightness per pixel 200 times the background level . the contrast of the structure at a radius of 02 is 0.29 to the n and 0.75 to the s. moreover , within the s - shaped structure , there is an elongation in the direction of the axis of the ionizing cone . we hereafter refer to it as the bar - like structure . in order to ascertain the reality of the structures we see at faint levels , we have built maps of ngc 1068 , of the observed psf ( star ) and of the recovered psf , normalizing the flux at each position ( @xmath7 ) to the flux averaged along an annulus of the same radius ( @xmath8 ) . these maps reveal , especially at faint brightness levels ( less than 2% of the peak ) , a few radial structures that are residual aberrations not compensated by ao ( mostly due to the camera optics ) . in particular , a faint feature at p.a . @xmath9 is present on all maps that is clearly such an artifact . however , none of the features in ngc 1068 mentioned above has such a spurious counterpart in the psf map , and their brightness is well above those of the artifacts . owing to the complexity of the central region , we have discarded a simple fit to the stellar distribution in terms of elliptical isophotes . instead , we assume that the j band emission is a satisfactory representation of the underlying stellar component and we have derived [ j@xmath0k ] and [ j@xmath0h ] color images in order to minimize effects of the stellar component . the [ j@xmath0k ] isophotes ( see fig . [ f - raw]-d ) delineate particularly well the three features already mentioned , i.e. , the core , the p.a . @xmath1 102 structure and the s - shaped ns structure . similar information is also available at 1.65 @xmath3 m in the [ j@xmath0h ] color image ( not shown ) . neither of the two extended features ( b ) and ( c ) is apparent on the j image . regarding the core observed in j , h and k , integration of the k flux within a 02 diameter circular diaphragm on the un - deconvolved image gives k = 9.3 mag . psf fitting with fwhm = 012 gives a result in excellent agreement , confirming that indeed the core is unresolved at 2.2 @xmath3 m . in j and h , the contribution of the underlying extended component must first be subtracted . to accomplish this , the j and h profiles at p.a . = 125 , a direction free of the small scale structures discussed previously , were extracted . beyond r = 015 , these profiles are quite well fitted by two exponential disks with characteristic radii of 31 and 026 ( see fig . 1-f and the discussion in section 3 ) . once the contribution of the extended component is removed , aperture photometry of the 02 core gives j = 16.3 mag and h = 13.1 mag . assuming that the core centers are coincident in j , h and k , this leads to the extremely red colors [ j@xmath0k ] = 7.0 and [ h - k ] = 3.8 . for comparison , we find a mean value of [ j@xmath0k ] = 3.5 for the regions situated at r = 02 either along the p.a . = 102 elongated structure or along the bar - like elongation within the s - shaped feature at p.a . we compare the inner s - shaped feature with other bar / spiral structures in ngc 1068 . from the archived f547 m wfpc / hst image in the continuum around 547 nm , one can distinguish ( _ i _ ) an outer barred spiral structure , with the bar at p.a . = 43@xmath6 , extending over 16 in diameter , and ( _ ii _ ) an intermediate barred spiral structure , with the bar at p.a . = 26@xmath6 , extending over 33 in diameter . finally , on the current near - ir images , we find that the inner s - shaped feature shows a bar - like central elongation at p.a . = 4@xmath6 which extends over a scale of 05 . the existence of interwoven spiral / barred structures in galaxies has been suggested by simulations ( friedli & martinet , 1993 ; heller & shlosman 1994 ; combes 1994a ) . they are thought to build a fueling channel for the active nucleus in agn . ngc 1068 , in which we detect overlapping spiral structures on three different scales ( 1.15 kpc , 240 pc , 36 pc ) at a p.a . rotating from 43@xmath6 to 4@xmath6 inward , appears to be a good test case for detailed modeling of this effect . this is deferred to a future paper . in order to interpret the new data in the perspective of the agn modeling , it is necessary to accurately register the near - ir peak with respect to the visible one . one way to achieve this is through simultaneous visible / ir observations , as carried out by marco et al . ( 1997 ) , who found the k peak to be offset by 028@xmath100.05 s and 008@xmath100.05 w from the optical ( i band ) peak . here , we have to rely on observations of a nearby star interlaced with the ngc 1068 measurements . this star provides a positional reference both on the wfs and on the near - ir camera . moreover , because the nearby stellar light source is coincident in the visible and near - ir , any relative offset in the near - ir between the nearby stellar peak and the ngc 1068 peak reflects an intrinsic separation between the visible and the near - ir sources within ngc 1068 . however , since the wfs bandpass is rather wide and the central region of ngc 1068 quite complex , the image of ngc 1068 as seen by the wfs has to be generated . such a composite image was synthesized from archival f502n , f547 m , f658n and f791n wfpc2/hst images , kindly made available to us in their fully reduced and precisely aligned form by z. tsetanov . these four images were properly scaled , weighted by the wfs response and summed to provide the required image . because the f547 m intensity peak is widely used as the positional reference for the visible peak " ( lynds et al . 1991 ) for ngc 1068 research , we have compared the _ pueo _ wfs composite image , just obtained above , to the f547 m image . we find that the center of gravity of the pueo wfs image is 0011 n and 0007 e of the visible peak . finally we have derived the offset between the unresolved core on the k image of ngc 1068 and its visible peak to be 0180@xmath11 s and 0153@xmath11 w. this result broadly agrees with that cited above by marco et al . ( 1997 ) and that derived by thatte et al . ( 1997 ) ( 0216@xmath100.100 s and 0095@xmath10 0.100 w , as deduced from their fig . 6 , although no indication is given about which image of ngc 1068 was used for the wfs ) . within the remaining uncertainty in the location of the k peak in ngc 1068 , we are led to conclude that the unresolved core in k ( size less than 8 pc ) is coincident with the radio component s1 ( gallimore et al . 1996a ) , with the 12.4 @xmath3 m source observed by braatz et al . ( 1993 ) , and with the center of symmetry of the uv / optical polarization map ( capetti et al . 1995 ) . as a result , we consider that the k unresolved core can be identified with the immediate surroundings of the central engine , most probably with hot dust close to sublimation . assuming classical dust grains at 1500 k , with associated [ j - k ] = 2.76 and [ h - k ] = 1.18 , the extreme red colors of the 02 diameter core , [ j - k ] = 7.0 [ h - k ] = 3.8 provide an extinction a@xmath12 of 25 and 41 mag , respectively ( rieke & lebofsky 1985 ) . the discrepancy between the two a@xmath12 figures obtained remains to be elucidated , and might be related to the nature of the dust grains . we also notice , given the 012 resolution of the _ pueo _ data set , that there is some evidence from the profile analysis that the core is resolved in h and j , while this is not the case in k. this result suggests some contribution from scattered light in j and h. how could we interpret the two extended structures also detected in the k and h bands ? the ese - wnw belt at p.a.=102@xmath6 extends over about 20 pc on either side of the core . we argue that it _ is very probably the trace of the warm dust within the molecular / dusty torus _ invoked in the unified model of agn since : a ) its direction is perpendicular to the axis of the inner ionization cone and to the direction of the radio jet originating from s1 ; b ) its direction follows within 1 that of the small scale ( 1pc ) radio disk recently discovered by gallimore et al . ( 1997 ) with the vlba interpreted by these authors as the ionized outer envelope of the torus ; c ) the observed k flux ratio of the core to the region at r = 15 pc , is the same as the ratio of black body emission at 1500 k and 600 k , respectively ; such a set of temperatures and radius is consistent with a simple model of grains heated by a central source ( t@xmath13r@xmath14 ) , the hot dust ( 1500 k ) being located at a radius of 1.5 pc and the warm dust ( 600 k ) at a radius of 15 pc . concerning the elongated s - shaped feature ( @xmath10 20 pc on each side of the unresolved core ) , a first question arises about the emission mechanism at 2.2 @xmath3 m . we envisaged that the 2.2 @xmath3 m emission is free - free radiation from ionized gas stripped off the inner edges of the molecular / dusty torus and driven away along the radio axis / inner ionization cone . under this scenario , the related radio emission at 6 cm , proportional to the k emission , would be @xmath1 300 mjy , a value in rough agreement with that obtained by ulvestad et al . ( 1987 ) . however , this interpretation would be in conflict with the finding by gallimore et al . ( 1996a ) that the radio emission along the jet ( from s1 and up to @xmath1 04 n ) has an increasingly steep spectrum , fully consistent with synchrotron emission . we are left with the assumption that the emission at 2.2 @xmath3 m is from stellar photospheres or from warm dust . it is true that the observed [ j - k ] color , at @xmath15 , both along the ese - wnw belt at p.a . = 102and along the pseudo - bar within the s - shaped feature , remains very similar , in the range 3.4 to 3.6 . as we have seen above , the ese - wnw belt emission at 2.2 @xmath16 m is consistent with warm dust emission ( 600k ) : therefore , we might envisage that the 2.2 @xmath16 m along the pseudo - bar within the s - shaped feature originates from warm dust at the back of nlr clouds shaded from the uv photons in the ionizing cone of the agn . as a very rough estimate , if we assume a dust temperature of 600k and a black - body emission , then the measured flux of 90 mjy in the 02 @xmath17 0 3 area of the pseudo - bar would correspond to a bolometric luminosity of the dust of 1.8 10@xmath18 l@xmath19 , i.e. one tenth of the bolometric luminosity of ngc 1068 . a lower temperature would lead to much larger luminosities that must be ruled out : for instance 6 10@xmath20 l@xmath19 if t@xmath21 = 300k . starlight is totally unable to produce so large a dust temperature pc of a b2 star , a distance one thousand times smaller than the mean distance between stars if the luminosity of 1.8 10@xmath18 l@xmath19 was accounted for by a cluster of b2 stars within the 0 2 @xmath17 0 3 ] and another dust heating mechanism should be invoked : one is direct exposure to x - rays emitted by the central engine , another could be shocks , either through direct dust heating within the shock or through secondary heating by uv photons from the shocked gas . the central pseudo - bar may correspond to dust in direct view of the agn , but then the s - shape would not be explained in this scheme . shocks at different locations may overcome this difficulty ; in terms of energy balance , since models of fast , radiative shocks are indeed able to account for the total number of ionizing photons produced in a seyfert nucleus ( e.g. dopita & sutherland , 1995 ) , we can assume that a significant fraction of this uv flux can efficiently heat the dust . on the basis of this sole data set , it is difficult to go beyond this stage in the interpretation of the origin of the 2.2 @xmath3 m photons . else , the s - shape is intriguing and might tell us about the origin of the 2.2 @xmath3 m emission . as in the case of ngc 2110 ( mulchaey et al . , 1994 ) , we might contemplate an interpretation involving the ejection of optical - emitting gas and radio - emitting gas along the same axis although with different velocities . but then , the emission mechanism at 2.2 @xmath3 m would be dominated by free - free emission which seems to be implausible . we have considered an alternative scenario , involving the presence of a spiral and bar structure made of a mixture of stellar and gaseous components . this later scenario seems promising as it provides as well clues on the central engine feeding . indeed , the bar within bar mechanism has been studied in details through n - body simulations of a stellar and gaseous component mixture ( athanassoula , 1994 ) . as gas is pushed inward along the bar , it forms a bar - unstable disc on a smaller scale , which will generate in turn a smaller bar . this mechanism could repeat itself on scales which are smaller and smaller . the dynamical behavior of such a system has been modeled with various configurations ( combes , 1994b ; friedli , 1996 ) , the prediction being that the innermost disc is in relation with the presence of a bar about twice as large ( combes , private communication ) . in the case of ngc 1068 , we noticed that each of the radial brightness distributions in j and h ( fig . [ f - raw]-f ) is quite well fitted by two exponential disks with characteristic radii of r@xmath22 = 220 pc , r@xmath23 = 19 pc . considering that the j band is mainly a tracer of stars , then the scale - length of the inner exponential disk , 19 pc , is about what would be obtained after a few rotations of a 40 pc bar , the size of the elongated ns feature . under this scenario , the 2.2 @xmath3 m emission would be related as well to the stellar component , although the presence of gas and dust taking part in the dynamics is not unlikely . detailed information on the kinematics of the innermost regions are required to distinguish between the two possibilities , calling for the obtainment of 2d - spectroscopic data sets at high angular resolution . yet , the presence of two other bar / spiral systems on larger scales in ngc 1068 and the natural explanation for inward mass transfer , lead us to conclude that the s - shaped structure may represent the third level of an interwoven system of bars , spiral and ring structures , bringing material inward to build up and feed the black - hole / accretion - disk system ( schlosman et al . this constitutes the first direct suggestion of the existence of a micro - barred / spiral structure in an agn . we are gratefully indebted to z. tsvetanov for making available to us the fully calibrated and aligned hst / wfpc2 images and to j.p . vran for reconstructing psfs with his powerful method . we warmly thank the cfht team who allowed successful _ pueo _ observations . thanks are extended to the referee , j. gallimore , who made several useful suggestions and to c. and g. robichez for pertinent comments that helped improving the manuscript .

we present diffraction - limited near - ir images in j , h and k of the nucleus of ngc 1068 , obtained with the adaptive optics system _ pueo _ at cfht . the achieved resolution ( 012 ) reveals several components , particularly prominent on the [ j@xmath0k ] 9 pc ) ; b ) an elongated structure at p.a . @xmath1 102 , beginning to show up at radius @xmath1 15 pc ; c ) a s - shaped structure with radial extent @xmath1 20 pc , including a bar - like central elongation at p.a . @xmath1 15 and two short spiral arms . a precise registration of the ir peak was carried out relative to the hst i - band peak . the k unresolved core is found to be close to the location of the putative central engine ( radio source s1 ) . consistent with the unified model of agn , the near - ir core is likely the emission from the hot inner walls of the dust / molecular torus . the extremely red colors of the 02 diameter core , [ j - k]=7.0 , [ h - k]=3.8 , lead to an intrinsic extinction a@xmath225 , assuming classical dust grains at 1500 k. the elongated structure at p.a . @xmath1 102 may trace the presence of cooler dust within and around the torus . this interpretation is supported by two facts at least : a ) the elongated structure is perpendicular to the local radio jet originating at s1 ; b ) its direction follows exactly that of the disk of ionized gas recently found with the vlba . regarding the s - shaped feature , the near - ir flux of the bar - like central elongation at p.a.= @xmath1 4 , if interpreted in terms of free - free emission from ionized gas , is roughly consistent with the level of 5 ghz emission . however , the radio spectrum behaviour is indicative of synchrotron emission and we rather interpret the 2.2 @xmath3 m emission as originating from warm dust in the shaded part of nlr clouds or in stellar photospheres . the shape itself suggests an extremely compact barred spiral structure , that would be the innermost of a series of nested spiral structures , as predicted by models and simulations . this is supported by the inner stellar distribution deduced from the j image which clearly follows an exponential disk with a 19 pc scale - length , precisely that expected from the rotation of a bar twice this size .