higher order bernstein algebras given by symmetric bilinear forms let a be a kth order bernstein algebra and let n be the kernel of this article studies the structure of such algebras in which n2 has dimension one the algebras are of two types i and ii according as n2 u or n2 u a characterization of the algebras of type i is given power associative kth order bernstein algebras with dim n2 1 are then considered they turn out to be bernstein algebras of at most second order and multiplication tables for these algebras over the real field are given finally second order bernstein algebras of type ii are examined and a structure theorem for them is obtained

capturing complexity classes by fragments of second order logic abstract we investigate the expressive power of certain fragments of second order logic on finite structures the fragments are second order horn logic second order krom logic as well as a symmetric and a deterministic version of the latter it is shown that all these logics collapse to their existential fragments in the presence of successor relation they provide characterizations of polynomial time deterministic and nondeterministic logspace and of the complement of symmetric logspace without successor relation these logics still can express certain problems that are complete in the corresponding complexity classes but on the other hand they are strictly weaker than previously known logics for these classes and fail to express some very simple properties

biset transformations of tambara functors abstract if we are given an h g biset u for finite groups g and h then any mackey functor on g can be transformed by u into a mackey functor on h in this article we show that the biset transformation is also applicable to tambara functors when u is right free and in fact forms a functor between the category of tambara functors on g and h this biset transformation functor is compatible with some algebraic operations on tambara functors such as ideal quotients or fractions in the latter part we also construct the left adjoint of the biset transformation

the spectrum of the burnside tambara functor on a finite cyclic p group abstract for a finite group g a tambara functor on g is regarded as a g bivariant analog of a commutative ring in this analogy previously we have defined an ideal of a tambara functor in this article we calculate the prime spectrum of the burnside tambara functor when g is a finite cyclic p group for a prime integer p

tambarization of a mackey functor and its application to the witt burnside construction abstract for an arbitrary group g a semi mackey functor is a pair of covariant and contravariant functors from the category of g sets and is regarded as a g bivariant analog of a commutative semi group in this view a g bivariant analog of a semi ring should be a semi tambara functor a tambara functor is firstly defined by tambara which he called a tnr functor when g is finite as shown by brun a tambara functor plays a natural role in the witt burnside construction it will be a natural question if there exist sufficiently many examples of tambara functors compared to the wide range of mackey functors in the first part of this article we give a general construction of a tambara functor from any mackey functor on an arbitrary group g in fact we construct a functor from the category of semi mackey functors to the category of tambara functors this functor gives a left adjoint to the forgetful functor and can be regarded as a g bivariant analog of the monoid ring functor in the latter part when g is finite we investigate relations with other mackey functorial constructions crossed burnside ring elliott s ring of g strings jacobson s f burnside ring all these lead to the study of the witt burnside construction

existential second order logic over graphs charting the tractability frontier fagin s theorem the first important result of descriptive complexity asserts that a property of graphs is in np if and only if it is definable by an existential second order formula in this article we study the complexity of evaluating existential second order formulas that belong to prefix classses of existential second order logic where a prefix class is the collection of all existential second order formulas in prenex normal form such that the second order and the first order quantifiers obey a certain quantifier pattern we completely characterize the computational complexity of prefix classes of existential second order logic in three different contexts 1 over directed graphs 2 over undirected graphs with self loops and 3 over undirected graphs without self loops our main result is that in each of these three contexts a dichotomy holds that is to say each prefix class of existential second order logic either contains sentences that can express np complete problems or each of its sentences expresses a polynomial time solvable problem although the boundary of the dichotomy coincides for the first two cases it changes as one moves to undirected graphs without self loops the key difference is that a certain prefix class based on the well known ackermann class of first order logic contains sentences that can express np complete problems over graphs of the first two types but becomes tractable over undirected graphs without self loops moreover establishing the dichotomy over undirected graphs without self loops turns out to be a technically challenging problem that requires the use of sophisticated machinery from graph theory and combinatorics including results about graphs of bounded tree width and ramsey s theorem

trace forms of trinomials abstract in this paper we determine trace forms of field extensions defined by trinomials x n ax k b this gives an affirmative answer to a generalization of a question of conner and yui moreover we classify all such trace forms over algebraic number fields

ideals of tambara functors abstract for a finite group g a tambara functor on g is regarded as a g bivariant analog of a commutative ring in this article we consider a g bivariant analog of the ideal theory for tambara functors

rigidity for pseudo pretheories assume f is a homotopy invariant pseudo pretheory with torsion values and x is a smooth scheme of finite type over a field k we show for certain field extensions k k the map f x f xk is an isomorphism

unramified class field theory of arithmetical surfaces let k be an algebraic number field 9k its ring of integers and v a non empty open subscheme of spec ck let x be a projective smooth geometrically irreducible scheme over k and x a regular proper flat scheme over v such that x x vk x the purpose of this paper is to give an explicit description of the abelian fundamental groups ab x and 7t b x using some idele class groups attached to x and x by a k theoretical method its outline is as follows in general for a noetherian scheme z and i 0 or 1 we define the group ski z to be the cokernel of

regular languages defined with generalized quantifiers we study an extension of first order logic obtained by adjoining quantifiers that count with respect to an integer modulus it is shown that the languages definable in this framework are precisely the regular languages whose syntactic monoids contain only solvable groups we obtain an analogous result for regular languages and establish some connections with complexity theory for fixed depth families of circuits

replacing model categories with simplicial ones in this paper we show that model categories of a very broad class can be replaced up to quillen equivalence by simplicial model categories

a characterization of c fields via galois groups abstract it is known that those fields f which are of class c in the sense of r ware can be characterized by their witt ring structure as well as by the behavior of their binary quadratic forms and the structure of their galois 2 extensions in this paper we show that c fields which have at least 8 square classes are characterized by the absence of a certain group dc of order 16 as a galois group over them moreover we show that the level of such a field is determined by the presence or absence of the cyclic group of order 4 and the dihedral group of order 8 as galois groups over it these results provide an interesting continuation of the work of ware in the appendix we give an explicit construction of field extensions with galois group dc

multiplicativity of the idempotent splittings of the burnside ring and the g sphere spectrum abstract we provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the g equivariant sphere spectrum including their hill hopkins ravenel norms where g is any finite group our results describe explicitly how these structures depend on the subgroup lattice and conjugation in g algebraically our analysis characterizes the multiplicative transfers on the localization of the burnside ring of g at any idempotent element which is of independent interest to group theorists as an application we obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category

closure properties of locally finite ω languages locally finite omega languages were introduced by ressayre in journal of symbolic logic volume 53 no 4 p 1009 1026 they generalize omega languages accepted by finite automata or defined by monadic second order sentences we study here closure properties of the family loc omega of locally finite omega languages in particular we show that the class loc omega is neither closed under intersection nor under complementation giving an answer to a question of ressayre

on motivic cohomology with z l coefficients in this paper we prove the conjecture of bloch and kato which relates milnor s k theory of a field with its galois cohomology as well as the related comparisons results for motivic cohomology with finite coefficients in the nisnevich and etale topologies

topological hypercovers and 1 realizations we show that if u is a hypercover of a topological space x then the natural map hocolim u x is a weak equivalence this fact is used to construct topological realization functors for the 1 homotopy theory of schemes over real and complex fields in an appendix we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant

from algebraic cobordism to motivic cohomology let s be an essentially smooth scheme over a eld of characteristic exponent c we prove that there is a canonical equivalence of motivic spectra over s mgl a1 a2 1 c hz 1 c where hz is the motivic cohomology spectrum mgl is the algebraic cobordism spectrum and the elements an are generators of the lazard ring we discuss several applications including the computation of the slices of z 1 c local landweber exact motivic spectra and the convergence of the associated slice spectral sequences

existential second order logic over strings existential second order logic eso and monadic second order logic mso have attracted much interest in logic and computer science eso is a much expressive logic over successor structures than mso however little was known about the relationship between msoand syntatic fragments of eso we shed light on this issue by completely characterizing this relationship for the prefix classes of eso over strings i e finite successor structures moreover we determine the complexity of model checking over strings for all eso prefix classes let eso q denote the prefix class containing all sentences of the shape r q 4 where r is a list of predicate variables q is a first order predicate qualifier from the prefix set q and 4 is quantifier free we show that eso and eso are the maximal standard eso prefix classes contained in mso thus expressing only regular languages we further prove the following dichotomy theorem an eso prefix class either expresses only regular languages and is thus in mso or it expresses some np complete languages we also give a precise characterization of those eso prefix classes that are equivalent to mso over strings and of the eso prefix classes which are closed under complementation on strings

on the andre quillen homology of tambara functors abstract we lift to equivariant algebra three closely related classical algebraic concepts abelian group objects in augmented commutative algebras derivations and kahler differentials we define mackey functor objects in the category of tambara functors augmented to a fixed tambara functor r and we show that the usual square zero extension gives an equivalence of categories between these mackey functor objects and ordinary modules over r we then describe the natural generalization to tambara functors of a derivation building on the intuition that a tambara functor has products twisted by arbitrary finite g sets and we connect this to square zero extensions in the expected way finally we show that there is an appropriate form of kahler differentials which satisfy the classical relation that derivations out of r are the same as maps out of the kahler differentials

the crossed burnside rings iii the dress construction for a tambara functor abstract we discuss the dress construction for a tambara functor associated to a finite commutative monoid with action of a finite group

on very effective hermitian k theory we argue that the very effective cover of hermitian k theory in the sense of motivic homotopy theory is a convenient algebro geometric generalization of the connective real topological k theory spectrum this means the very effective cover acquires the correct betti realization its motivic cohomology has the desired structure as a module over the motivic steenrod algebra and that its motivic adams and slice spectral sequences are amenable to calculations

power associative algebras that are train algebras we investigate the structure of power associative algebras that are train algebras we flrst show the existence of idempotents which are all principal and absolutely primitive we then study the train equation involving the peirce decomposition when the algebra is flnite dimensional it turns out that the dimensions of the peirce components are invariant and that the upper bounds for their nil indexes are reached for some idempotent further locally train algebras are shown to be train algebras we then get a complete description of the set of idempotents by giving their explicit formulas including several illustrative examples some attention is paid to the jordan case where we discuss conditions forcing power associative train algebras to be jordan algebras it is also shown that flnitely generated jordan train algebras are flnite dimensional for a n th order bernstein algebra of period p we prove that power associativity necessitates p 1 in this case there are 2 ni1 possible train equations which are explicitly described 2000 mathematics subject classiflcation primary 17d92 17a05

vanishing theorems for real algebraic cycles we establish the analogue of the friedlander mazur conjecture for teh s reduced lawson homology groups of real varieties which says that the reduced lawson homology of a real quasi projective variety x vanishes in homological degrees larger than the dimension of x in all weights as an application we obtain a vanishing of homotopy groups of the mod 2 topological groups of averaged cycles and a characterization in a range of indices of dos santos real lawson homology as the homotopy groups of the topological group of averaged cycles we also establish an equivariant poincare duality between equivariant friedlander walker real morphic cohomology and dos santos real lawson homology we use this together with an equivariant extension of the mod 2 beilinson lichtenbaum conjecture to compute some real lawson homology groups in terms of bredon cohomology

some computational aspects of circumscription the effects of circumscribing first order formulas are explored from a computational standpoint first extending work of v lifschitz it is shown that the circumscription of any existential first order formula is equivalent to a first order formula after this it is established that a set of universal horn clauses has a first order circumscription if and only if it is bounded when considered as a logic program thus it is undecidable to tell whether such formulas have first order circumscription finally it is shown that there arefirst order formulas whode circumscription has a conp complete model checking problem

generalized burnside rings and group cohomology we define the cohomological burnside ring b n g m of a finite group g with coefficients in a zgmodule m as the grothendieck ring of the isomorphism classes of pairs x u where x is a g set and u is a cohomology class in a cohomology group h n x g m the cohomology groups h x g m are defined in such a way that h x g m i h hi m when x is the disjoint union of transitive g sets g hi i fa is an abelian group with trivial action then b 1 g a is the same as the monomial burnside ring over a and when m is taken as a g monoid then b 0 g m is equal to the crossed burnside ring b c g m we discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological burnside rings we also give an interpretation of b 2 g m in terms of twisted

witt vectors and tambara functors abstract we give a combinatorial description of the ring of g witt vectors on a polynomial algebra over the integers for every finite group g using this description we show that the functor which takes a commutative ring with trivial action of g to its ring of witt vectors coincides with the left adjoint of the algebraic functor from the category of g tambara functors to the category of commutative rings with an action of g

comparison of stable homotopy categories and a generalized suslin voevodsky theorem abstract let k be an algebraically closed field of exponential characteristic p given any prime l p we construct a stable etale realization functor et l s pt k pro s pt h z l from the stable category of motivic p 1 spectra over k to the stable category of h z l local pro spectra see section 3 for the definition this is induced by the etale topological realization functor a la friedlander the constant presheaf functor naturally induces the functor sh 1 p sh k 1 p where k and p are as above and sh and sh k are the classical and motivic stable homotopy categories respectively we use the stable etale realization functor to show that this functor is fully faithful furthermore we conclude with a homotopy theoretic generalization of the etale version of the suslin voevodsky theorem

on the computation of the trace form of some galois extensions abstract we investigate the trace form trl k l k x trl kx2of a finite galois extensionl k in particular we study 2 extensions of degree 16 using some reduction theorems these results yield a classification of nearly all trace forms of galois extensions of degree 31 finally we study the trace form of a cyclotomic extension and of its maximal real subfield

quantifiers and approximation we investigate the relationship between logical expressibility of np optimization problems and their approximation properties first such attempt was made by papadimitrou and yannakakis 1988 who defined the class of npo problems max np we show that many important optimization problems do not belong to max np and that in fact there are problems in p which are not in max np the problems that we consider fit naturally in a new complexity class that we call max 1 we prove that several natural optimization problems are complete for max 1 under approximation preserving reductions all these complete problems are not approximable unless p np this motivates the definition of subclasses of max 1 that only contain problems which are presumably eaiser with respect to approximation in particular the class that we call rmax 2 contains approximable problems and problems like max clique that are not known to be nonapproximable we prove the max clique and several other optimization problems are complete for rmax 2 all the complete problems in rmax 2 share the interesting property that they either are nonapproximable or are approximable to any degree of accuracy

the homotopy limit problem for hermitian k theory equivariant motivic homotopy theory and motivic real cobordism abstract the homotopy limit problem for karoubi s hermitian k theory karoubi 1980 26 was posed by thomason 1983 44 there is a canonical map from algebraic hermitian k theory to the z 2 homotopy fixed points of algebraic k theory the problem asks roughly how close this map is to being an isomorphism specifically after completion at 2 in this paper we solve this problem completely for fields of characteristic 0 theorems 16 20 we show that the 2 completed map is an isomorphism for fields f of characteristic 0 which satisfy cd 2 f i but not in general

sur les algebres de bernstein d ordre 2 abstract we show that the duplicate of an n th order bernstein algebra is a bernstein algebra of order n 1 and we characterize the set of generalized idempotents in such an algebra we also characterize those bernstein algebra of order 2 which are jordan or are power associative the results are quite different from what is true for bernstein algebras of order 1

induced orthogonal representations of galois groups abstract we prove a result showing the connection between induced orthogonal representations and corestrictions of group extensions derived from their clifford groups by the means of the corestriction map we then obtain new obstructions to the p embedding problems given by the group extensions of the modular p group and to one 2 embedding problem given by a group extension of the dihedral group

on a class of baric algebras abstract it is known that baric algebras satisfying the identity x 2 2 w x x 3 have idempotent elements and every linear form w a k is a multiplicative map we prove that these algebras are jordan bernstein of order 2 and special train algebras moreover as a corollary we obtain that the train equation of these algebras is x 4 w x x 3 0 and we give examples of baric algebras satisfying x 4 w x x 3 0 but not satisfying x 2 2 w x x 3

sur les algebres de bernstein iv the extension to bernstein algebras of higher order of known results for one order is not trivial in this paper we give some results in this direction

galois equivariance and stable motivic homotopy theory for a finite galois extension of fields l k with galois group g we study a functor from the g equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence we show that after completing at a prime and the motivic hopf map this results in a full and faithful embedding whenever k is real closed and l k i it is a full and faithful embedding after completion if a motivic version of serre s finiteness theorem is valid we produce strong necessary conditions on the field extension l k for this functor to be full and faithful along the way we produce several results on the stable c2 equivariant betti realization functor and prove convergence theorems for the p primary c2 equivariant adams spectral sequence

rational isomorphisms between k theories and cohomology theories the well known isomorphism relating the rational algebraic k theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map the segre map of infinite loop spaces moreover the associated chern character map on rational homotopy groups is shown to be a ring isomorphism a technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi projective complex varieties to induce a homotopy equivalence of semi topological singular complexes since semi topological k theory and morphic cohomology can be formulated as the semi topological singular complexes associated to algebraic k theory and motivic cohomology this criterion provides a rational isomorphism between the semi topological k theory groups and the morphic cohomology groups of a smooth complex variety consequences include a riemann roch theorem for the chern character on semi topological k theory and an interpretation of the topological filtration on singular cohomology groups in k theoretic terms

relational queries computable in polynomial time we characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe we also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation this proves that the fixed point query hierarchy suggested by chandra and harel collapses at the first fixed point level it is also a general result showing that in finite model theory one application of fixed point suffices

a generalization of the dress construction for a tambara functor and its relation to polynomial tambara functors abstract for a finite group g semi mackey functors and semi tambara functors are regarded as g bivariant analogs of semi groups and semi rings respectively in fact if g is trivial they agree with the ordinary semi groups and semi rings and many naive algebraic properties concerning rings and groups have been extended to these g bivariant analogous notions in this article we investigate a g bivariant analog of the semi group rings with coefficients just as a coefficient ring r and a monoid q yield the semi group ring r q our construction enables us to make a tambara functor t m out of a semi mackey functor m and a coefficient tambara functor t this construction is a composant of the tambarization and the dress construction as expected this construction is the one uniquely determined by the righteous adjoint property besides in analogy with the trivial group case if m is a mackey functor then t m is equipped with a natural hopf structure moreover as an application of the above construction we also obtain some g bivariant analogs of the polynomial rings

picard groups grothendieck rings and burnside rings of categories abstract we discuss the picard group the grothendieck ring and the burnside ring of a symmetric monoidal category and we consider examples from algebra homological algebra topology and algebraic geometry

the tambara structure of the trace ideal for cyclic extensions abstract this paper explores the tambara functor structure of the trace ideal of a galois extension in the case of a pro cyclic extension we are able to explicitly determine the generators of the ideal furthermore we show that the absolute trace ideal of a cyclic group is strongly principal when viewed as an ideal of the burnside tambara functor applying our results we calculate the trace ideal for extensions of finite fields the appendix determines a formula for the norm of a quadratic form over an arbitrary finite extension of a finite field

on the fractions of semi mackey and tambara functors for a finite group g a semi mackey resp tambara functor is regarded as a g bivariant analog of a commutative monoid resp ring as such some naive algebraic constructions are generalized to this g bivariant setting in this article as a g bivariant analog of the fraction of a ring we consider fraction of a tambara and a semi mackey functor by a multiplicative semi mackey subfunctor

defect theory for maximal ideals and simple functors keywords green functor source green correspondence maximal ideal finite group fixed point algebra defect theory for g algebras defect group restriction transfer conjugation defect pair functorial ideal simple g functors reference ctg article 1991 002doi 10 1016 0021 8693 91 90168 8 record created on 2008 12 16 modified on 2017 05 12