the p t degrees of the recursive sets lattice embeddings extensions of embeddings and the two quantifier theory abstract ambos spies 1984a showed that the two basic nondistributive lattices can be embedded in rp t the polynomial time turing degrees of the recursive sets we introduce more general techniques to extend his results to show that every recursive lattice can be embedded in rp t in addition to lattice theoretic representation theorems we use the scheme of priority style arguments coupled with looking back techniques presented in shinoda and slaman 1988 1990 we also generalize the density type results of ladner 1975 and many others to settle the full extension of the embedding problem for rp t combined with the logical analysis of sentences with one alternation of quantifiers shore 1978 lerman 1983 these results suffice to decide the full theory of rp t they also give a strong nonhomogeneity result the p time degrees of the sets recursive in and if desired p time above two distinct sets a and b are almost never isomorphic the situation for the p time many one degrees is quite different we decide the extension of the embedding problem differently than for rp t but not the theory

a uniform approach to obtain diagonal sets in complexity classes abstract a uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given we obtain some known results as well as some new ones as corollaries of our main theorem the new results concern the complexity classes p np co np pspace apt almost polynomial time r random polynomial time and the polynomial hierarchy

flat ideals and stability in integral domains abstract we introduce the concept of quasi stable ideal in an integral domain d a nonzero fractional ideal i of d is quasi stable if it is flat in its endomorphism ring i i and study properties of domains in which each nonzero fractional ideal is quasi stable we investigate some questions about flatness that were raised by s glaz and w v vasconcelos in their 1977 paper 17

the radical trace property and primary ideals let r be an integral domain and let m be an r module then the trace 4 of m is the ideal generated by the set fm f hom m r and m m r for an ideal i of r the trace is simply the product of i and i 1 we call an ideal j a trace ideal if j is the trace of some r module an elementary result which will be used freely throughout this paper is that if j is a trace 1 1 ideal then jj j i e j equals the ring j j 6 proposition 7 2 thus the trace ideals of r are precisely those ideals j for which j 1 equals j j such ideals are also referred to as being strong see for example 3 if r is a valuation domain and m is an r module then the trace of m is either r or a prime ideal of r 14 proposition 2 1 extracting the conclusion of this result fontana et al give the following definition a domain r is said to satisfy the trace property or to be a tp domain if for each r module m the trace of m is equal to either r or a prime ideal of r 14 p 169 see also 1 theorem 2 8 theorem 3 5 of 14 gives a characterization of noetherian tp domains namely for a noetherian domain r r is a tp domain if and only if r is onedimensional has at most one non invertible maximal ideal m and if such 1 a maximal ideal exists then m equals the integral closure of r or 1 equivalently m m m is a dedekind domain in section 2 of 17 gabelli shows that by replacing integral closure with complete integral closure the same list of conditions characterizes the class of mori domains which satisfy the trace property recall that a mori domain is an integral domain which satisfies the ascending chain condition on divisorial ideals

minimal degrees for polynomial reducibilities the existence of minimal degrees is investigated for several polynomial reducibilities it is shown that no set has minimal degree with respect to polynomial many one or turing reducibility this extends a result of ladner in which only recursive sets are considered a polynomial reducibility h t is defined this reducibility is a strengthening of polynomial turing reducibility and its properties relate to the p np question for this new reducibility a set of minimal degree is constructed under the assumption that p np however the set constructed is nonrecursive and it is shown that no recursive set is of minimal h t degree

the radical trace property or ii e spec r 4 prop 2 41 using this formulation of tp fontana huckaba and papick gave characterizations for noetherian tp domains and a special class of priifer tp domains 4 it is our goal to extend this work by studying rtp radical trace property domains i e domains satisfying the condition that zz

conducive integral domains this article introduces the concept of a conducive domain that is an integral domain each of whose overrings apart from the quotient field has nonzero conductor the principal examples of conducive domains are all d m constructions and all pseudo valuation domains several characterizations are obtained notably that a domain r is conducive if and only if r has a valuation overring with nonzero conductor it is proved that if r is a conducive domain but not a field then spec r is pinched at a prime p such that the set of primes within p is linearly ordered the converse is shown for r a prufer domain in which case p prp consequences include pullback characterizations of the seminormal resp prufer conducive domains special attention is paid to the class of conducive g domains with attendant interplay between conducive and the property of having a maximum overring

sublattices of the polynomial time degrees we show that any countable distributive lattice can be embedded in any interval of polynomial time degrees furthermore the embeddings can be chosen to preserve the least or the greatest element this holds for both polynomial time bounded many one and turing reducibilities as well as for all of the common intermediate reducibilities

core of ideals in integral domains abstract this paper uses objects and techniques from multiplicative ideal theory to develop explicit formulas for the core of ideals in various classes of integral domains not necessarily noetherian we also investigate the existence of minimal reductions originally established by rees and sally for local noetherian rings all results are illustrated by original examples in noetherian and non noetherian settings where we explicitly compute the core and validate some open questions recently raised in the literature

on mockor s question abstract for certain classes of prufer domains a we study the completion a t of a with respect to the supremum topology t sup t w w where is the family of nontrivial valuations on the quotient field which are nonnegative on a and t w is a topology induced by a valuation w it is shown that the concepts sft prufer domain and generalized dedekind domain are the same we show that if e is the ring of entire functions then e t is a bezout ring which is not a t prufer ring and if a is an sft prufer domain then a t is a prufer ring under a certain condition we also show that under the same conditions as above a t is a t prufer ring if and only if the number of independent valuation overrings of a is finite in particular if a is a dedekind domain resp h local prufer domain then a t is a t prufer ring if and only if a has only finitely many prime ideals resp maximal ideals these provide an answer to mockor s question

a counterpart to nagata idealization abstract idealization of a module k over a commutative ring s produces a ring having k as an ideal all of whose elements are nilpotent we develop a method that under suitable field theoretic conditions produces from an s module k a subring r of s that behaves like the idealization of k but is such that when s is a domain so is r the ring s is contained in the normalization of r but is finite over r only when r s we determine conditions under which r is noetherian cohen macaulay gorenstein a complete intersection or a hypersurface when r is local then its m adic completion is the idealization of the m adic completions of s and k

two generated ideals and representations of abelian groups over valuation rings abstract let r be a commutative ring with identity we give some general results on non noetherian commutative rings with the property that each finitely generated ideal can be generated by n elements and characterize the quasi local reduced group rings r g and closely related group rings which have this property for n 2 it is then shown that finitely generated torsionfree r g modules are direct sums of ideals if r g is a reduced quasi local group ring with this 2 generator property the group rings r g which have only finitely many isomorphism classes of finitely generated torsionfree modules are also determined where the coefficient ring r is as in the above mentioned characterization of when r g has the 2 generator property these results depend on a determination of when a simple ring extension of the form r x p r x is a valuation domain for a prime power p r where p i x x p i 1 p 1 x p i 1 1 and some related results which are given in section 3 the relationship between the 2 generator property and stability of finitely generated regular ideals is also considered

numbers of generators of ideals in a group ring of an elementary abelian p group abstract let i denote the minimal number of generators of an ideal i of a local ring r m the dilworth number d r max i i an ideal of r and sperner number sp r max mi i 0 are determined in the case that r a g where g z pz k is an elementary abelian p group a za is a principal local ring with a za of characteristic p 0 and zn 0 for small values of n

index sets and presentations of complexity classes this paper draws close connections between the ease of presenting a given complexity class b and the position of the index sets ib lcub i l mi b rcub and jb lcub i miis total l mi b rcub in the arithmetical hierarchy for virtually all classes b studied in the literature the lowest levels attainable are ib 03 and jb 02 the first holds iff b is 02 presentable and the second iff b is recursively presentable a general kind of priority argument is formulated and it is shown that every property enforcible by it is not recursively presentable it follows that the classes of p immune and p biimmune languages in exponential time are not recursively presentable it is shown that for all b with ib 03 many members of b do not provably in true 2 arithmetic belong to b a class h is exhibited such that ih 03 is open and ih 03 implies that the polynomial hierarchy is infinite

integral domains of finite t character abstract an integral domain d is said to be of finite t character if each nonzero nonunit of d is contained in only finitely many maximal t ideals of d for example noetherian domains and krull domains are of finite t character in this paper we study several properties of integral domains of finite t character we also show when the ring d s d x d s x is of finite t character where x is an indeterminate over d and s is a multiplicative subset of d

core of ideals in one dimensional noetherian domains abstract this paper contributes to the study of the core of ideals in noetherian rings we use techniques and objects from multiplicative ideal theory to develop explicit formulas for the core in various classes of one dimensional noetherian domains all results are illustrated with original examples where we explicitly compute the core

on the relative complexity of hard problems for complexity classes without complete problems abstract we show that any recursive sequence of recursive sets which is ascending with respect to the standard polynomial time reducibility notions has no minimal upper bound as a consequence any complexity class with certain natural closure properties possesses either complete problems or no easiest hard problems a further corollary is that assuming p np the partial ordering of the polynomial time degrees of np sets is not complete and that there are no degree invariant approximations to np complete problems

ideal theory in prufer domains an unconventional approach abstract in prufer domains of finite character ideals are represented as finite intersections of special ideals which are proper generalizations of the classical primary ideals we show that representations of ideals as shortest intersections of primal or quasi primary ideals exist and are unique moreover every non zero ideal is the product of uniquely determined pairwise comaximal quasi primary ideals semigroups of primal and quasi primary ideals with fixed associated primes are also investigated in arbitrary prufer domains their structures can be described in terms of the value groups of localizations

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

structural properties of bounded relations with an application to np optimization problems we introduce the notion bounded relation which comprises most resource bounded reducibilities which can be found in the literature including non uniform bounded reducibilities such as tp poly we state conditions on bounded relations which are again satisfied for most bounded reducibilities and which imply that every countable partial ordering can be embedded into every proper interval of the recursive degrees as corollaries we obtain that every countable partial ordering can be embedded into every proper interval of rec tp poly as well as into every proper interval between either two maximization or two minimization problems in the structures npo e and npo l for the two latter structures we show further that the result about embeddings of partial orderings extends to embeddings of arbitrary countable distributive lattices where in addition the least or the greatest element of the lattice can be preserved among other corollaries we obtain that for both structures every non trivial np optimization problem bounds a minimal pair in connection with embeddings into npo we introduce a representation of maximization or minimization problems by polynomial time computable functions we consider decidability issues w r t this representation and show for example that there is no effective procedure which decides for a subrecursive index of a polynomial time computable function whether the corresponding maximization problem is approximable within a constant factor

minimal pairs and complete problems two sets are said to form a minimal pair for polynomial many one reductions if neither set is in p and the only sets which are polynomially reducible via polynomial many one reductions to both sets are in p we consider the question of which complete sets can be the top that is the supremum of a minimal pair our main result is that any elementary recursive set which is hard for deterministic exponential time cannot be the top of a minimal pair hence in particular any complete set for an elementary recursive class containing exponential time cannot be such a supremum for np complete sets the problem remains open we show here that it is oracle dependent this is the first example of an oracle dependent property which is completely degree theoretic

constituent groups of clifford semigroups arising from t closure the t class semigroup of an integral domain r denoted st r is the semigroup of fractional t ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t multiplication we recently proved that if r is a krull type domain m griffin rings of krull type j reine angew math 229 1968 1 27 then st r is a clifford semigroup s kabbaj a mimouni t class semigroups of integral domains j reine angew math 612 2007 213 229 in this paper we aim to describe the idempotents of st r and the structure of their associated groups we extend and recover well known results on class semigroups of valuation domains and prufer domains of finite character

reductions on np and p selective sets abstract p selective sets are used to distinguish polynomial time bounded reducibilities on np in particular we consider different kinds of positive reductions these preserve membership in np and are not a priori closed under complements we show that the class of all sets which are both p selective and have positive reductions to their complements is p this is used to show that if dext next then there exists a set in np p that is not positive reducible to its complement various similar results are obtained we also show that p is the class of all sets which are both p selective and positive truth table self reducible from this result it follows that various naturally defined apparently intractible problems are not p selective unless p np

bazzoni s conjecture a conjecture posed 11 years ago by s bazzoni is solved by showing that a prufer domain with the property that every locally finitely generated ideal is finitely generated is in fact a prufer domain of finite character

extension of ideal theoretic properties of a domain to submodules of its quotient field abstract we examine when multiplicative properties of ideals extend to submodules of the quotient field of an integral domain an integral domain r is stable if each non zero ideal of r is invertible as an ideal over its ring of endomorphisms we show that an integral domain r is stable if and only if an analogue of this invertibility property extends to submodules of the quotient field of r by contrast the class of integral domains for which every non zero ideal is locally free over its ring of endomorphisms is shown to properly contain the class of domains r for which each submodule of the quotient field is locally free over its ring of endomorphisms and we give complete characterizations of both classes of domains

diagonalization uniformity and fixed point theorems abstract we derive new fixed point theorems for subrecursive classes together with a theorem on the uniformity of certain reductions from a general formulation of the technique of delayed diagonalization this formulation extends the main theorem of u schoning theoret comput sci 18 1982 95 103 to cases which involve infinitely many diagonal classes c k and which allow each c k to contain uncountably many members the main technical work ties the familiar concept of a witness function directly to the often studied cantor set topology on languages and provides a delay construction which refines those due to schoning s breidtbart and d schmidt our a e fixed point theorems do not require that the programming system for the subrecursive class in question be well behaved we compare them to results which do the other theorem is similar to the uniform boundedness theorem of classical analysis and extends work of j grollmann and a selman

modules over dedekind like rings abstract all finitely generated modules are described over a class of rings that includes the integral group ring zgn of the cyclic group of square free order n some rings of algebraic integers that are not integrally closed in their field of quotients and many subrings of z z z special attention is paid to behavior of these modules in direct sums

diagonalizations over polynomial time computable sets abstract a formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets the class of polynomial time computable functions respectively like e g p immunity it is shown that there are sets called p generic which have all properties enforceable by such diagonalizations we study the behaviour and the complexity of p generic sets in particular we show that the existence of p generic sets in np is oracle dependent even if we assume p np

class semigroups of integral domains this paper seeks ring theoretic conditions of an integral domain r that reflect in the clifford property or boolean property of its class semigroup s r that is the semigroup of the isomorphy classes of the nonzero integral ideals of r with the operation induced by multiplication precisely in section 3 we characterize integrally closed domains with boolean class semigroup in this case s r identifies with the boolean semigroup formed of all fractional overrings of r in section 4 we investigate noetherian like settings where the clifford and boolean properties of s r coincide with lipman and sally vasconcelos stability conditions a main feature is that the clifford property forces t locally noetherian domains to be one dimensional noetherian domains section 5 studies the transfer of the clifford and boolean properties to various pullback constructions our results lead to new families of integral domains with clifford or boolean class semigroup moving therefore beyond the contexts of integrally closed domains or noetherian domains

almost bezout domains an integral domain r is said to be an almost bezout domain respectively almost gcd domain if for x y r 0 there exists an n with xn yn respectively xn yn v principal in this paper we continue the investigation of agcd domains begun by the second author and introduce the notion of an almost bezout domain we show that r is an almost bezout domain if and only if r the integral closure of r is a prufer domain with torsion class group and for every x r there exists an n with xsn r

on an optimal propositional proof system and the structure of easy subsets of taut in this paper we develop a connection between optimal propositional proof systems and structural complexity theory specifically there exists an optimal propositional proof system if and only if there is a suitable recursive presentation of the class of all easy polynomial time recognizable subsets of taut as a corollary we obtain the result that if there does not exist an optimal propositional proof system then for every theory t there exists an easy subset of taut which is not t provably easy

the density and complexity of polynomial cores for intractable sets let a be a recursive problem not in p lynch has shown that a then has an infinite recursive polynomial complexity core this is a collection c of instances of a such that every algorithm deciding a needs more than polynomial time almost everywhere on c we investigate the complexity of recognizing the instances in such a core and show that every recursive problem a not in p has an infinite core recognizable in subexponential time we further study how dense the core sets for a can be under various assumptions about the structure of a our main results in this direction are that if p np then np complete problems have polynomially nonsparse cores recognizable in subexponential time and that exptime complete problems have cores of exponential density recognizable in exponential time

minimal pairs for p abstract we prove a general minimal pair theorem which yields as corollaries many results about minimal pairs for p obtained previously by various authors furthermore it yields the new counterintuitive result that there exist arbitrarily complex minimal pairs we also investigate the question of whether minimal pairs in np are low

ideal class semigroups of overrings abstract let r be an integral domain an let t be an overring of r there is a canonical semigroup homomorphism between the ideal class semigroup of r and the ideal class semigroup of t we investigate conditions under which this semigroup homomorphism is surjective and we apply the results we obtain to the study of overrings of clifford regular domains we recover some known results of bazzoni and we prove in certain more general situations that the clifford regular property is inherited by an overring in particular we prove that if r is a clifford regular domain such that the integral closure of r is a fractional overring then every overring of r is clifford regular we also characterize among clifford regular domains the ones that are stable

on the classification of stable domains abstract in the second half of a two part study of stable domains we explore the extent to which the stability of an integral domain is determined by the stability of its prime and finitely generated ideals this yields pullback theorems for stable domains a method of constructing nonstandard examples of stable domains and a complete classification of coherent stable domains a non coherent stable domain is constructed

commutative ideal theory without finiteness conditions primal ideals our goal is to establish an efficient decomposition of an ideal a of a commutative ring r as an intersection of primal ideals we prove the existence of a canonical primal decomposition a p xa a p where the a p are isolated components of a that are primal ideals having distinct and incomparable adjoint primes p for this purpose we define the set ass a of associated primes of the ideal a to be those defined and studied by krull we determine conditions for the canonical primal decomposition to be irredundant or residually maximal or the unique representation of a as an irredundant intersection of isolated components of a using our canonical primal decomposition we obtain an affirmative answer to a question raised by fuchs and also prove for p spec r that an ideal a c p is an intersection of p primal ideals if and only if the elements of r p are prime to a we prove that the following conditions are equivalent i the ring r is arithmetical ii every primal ideal of r is irreducible iii each proper ideal of r is an intersection of its irreducible isolated components we classify the rings for which the canonical primal decomposition of each proper ideal is an irredundant decomposition of irreducible ideals as precisely the arithmetical rings with noetherian maximal spectrum in particular the integral domains having these equivalent properties are the prufer domains possessing a certain property

a note on the usefulness of superkernels in density estimation we consider the akaike parzen rosenblatt density estimate f nh based upon any superkernel l i e an absolutely integrable function with int l 1 whose characteristic function is 1 on lbrack 1 1 rbrack and compare it with a kernel estimate g nh based upon an arbitrary kernel k we show that for a given subclass of analytic densities inf l sup k lim sup n rightarrow infty frac inf h mathbb e int f nh f inf h mathbb e int g nh f 1 where h 0 is the smoothing factor thus asymptotically the class of superkernels is as good as any other class of kernels when certain analytic densities are estimated we also obtain exact asymptotic expressions for the expected l 1 error of the kernel estimate when superkernels are used

domains satisfying the trace property e h om m r when no confusion will result we will suppress the r and write y m in section 2 we prove that if r is a valuation domain and m a unital r module then y m either equals r or a prime ideal of r it is this result that motivates our work and gives rise to the following definition a domain r is said to satisfy the

well centered overrings of an integral domain abstract let a be an integral domain with field of fractions k we investigate the structure of the overrings b k of a that are well centered on a in the sense that each principal ideal of b is generated by an element of a we consider the relation of well centeredness to the properties of flatness localization and sublocalization for b over a if b a b is a simple extension of a we prove that b is a localization of a if and only if b is flat and well centered over a if the integral closure of a is a krull domain in particular if a is noetherian we prove that every finitely generated flat well centered overring of a is a localization of a we present examples of non finitely generated flat well centered overrings of a dedekind domain that are not localizations

on some classes of integral domains defined by krullʼs a b operations let d be an integral domain with quotient field k the b operation that associates to each nonzero d submodule e of k eb ev v valuation overring of d is a semistar operation that plays an important role in many questions of ring theory e g if i is a nonzero ideal in d ib coincides with its integral closure in a first part of the paper we study the integral domains that are b noetherian i e such that for each nonzero ideal i of d ib jb for some a finitely generated ideal j of d for instance we prove that a b noetherian domain has noetherian spectrum and if it is integrally closed is a mori domain but integrally closed mori domains with noetherian spectra are not necessarily b noetherian we also characterize several distinguished classes of b noetherian domains in a second part of the paper we study more generally the e a b semistar operation of finite type a canonically associated to a given semistar operation for instance the b operation is the e a b semistar operation of finite type canonically associated to the identity operation these operations introduced and studied by krull jaffard gilmer and halter koch play a very important role in the recent generalizations of the kronecker function ring in particular in the present paper we classify several classes of integral domains having some of the fundamental operations d t w and v equal to some of the canonically associated e a b operations b ta wa and va

one dimensional stable rings a commutative ring r is stable provided every ideal of r containing a nonzerodivisor is projective as a module over its ring of endomorphisms the class of stable rings includes the one dimensional local cohen macaulay rings of multiplicity at most 2 as well as certain rings of higher multiplicity necessarily analytically ramified the former are important in the study of modules over gorenstein rings while the latter arise in a natural way from generic formal fibers and derivations we characterize one dimensional stable local rings in several ways the characterizations involve the integral closure r of r and the completion of r in a relevant ideal adic topology for example we show if r is a reduced stable ring then there are exactly two possibilities for r 1 r is a bass ring that is r is a reduced noetherian local ring such that r is finitely generated over r and every ideal of r is generated by two elements or 2 r is a bad stable domain that is r is a one dimensional stable local domain such that r is not a finitely generated r module

a note on deterministic and nondeterministic time complexity it is proved that there exists for every set b e np co np a set a np co np such that a and b have the same deterministic time complexity note that b is easier nondeterministically than both a and a in that b e np while a np co np we also prove there exists for each b np co np a set a np co np such that the deterministic time complexity of a is less than that of b on some infinite set in this case then a is easier than b deterministically while b is easier than a nondeterministically

clifford regular domains abstract the class semigroup of a commutative integral domain r is the semigroup s r of the isomorphism classes of the nonzero ideals of r with operation induced by multiplication a domain r is said to be clifford regular if s r is a clifford semigroup i e s r is the disjoint union of the subgroups associated to the idempotent elements in this paper we characterize the noetherian and the integrally closed clifford regular domains and find some properties of an arbitrary clifford regular domain

constructing np intermediate problems by blowing holes with parameters of various properties the search for natural np intermediate problems is one of the holy grails within computational complexity ladner s original diagonalization technique for generating np intermediate problems blowing holes has a serious shortcoming it creates problems with a highly artificial structure by arbitrarily removing certain problem instances in this article we limit this problem by generalizing ladner s method to use parameters with various characteristics this allows one to define more fine grained parameters resulting in np intermediate problems where we only blow holes in a controlled subset of the problem we begin by fully characterizing the problems that admit np intermediate subproblems for a broad and natural class of parameterizations and extend the result further such that structural csp restrictions based on parameters that are hard to compute such as tree width are covered thereby generalizing a result by grohe for studying certain classes of problems including csps parameterized by constraint languages we consider more powerful parameterizations first we identify a new method for obtaining constraint languages such that csp are np intermediate the sets can have very different properties compared to previous constructions by for instance bodirsky grohe and provides insights into the algebraic approach for studying the complexity of infinite domain csps second we prove that the propositional abduction problem parameterized by constraint languages admits np intermediate problems this settles an open question posed by nordh zanuttini we provide a framework to obtain np intermediate problems based on diagonalization we categorize different types of parameters that are usable in our framework we construct np intermediate constraint satisfaction problems we resolve the open question whether boolean abduction admits intermediate problems

lattice embeddings for abstract bounded reducibilities we give an abstract account of resource bounded reducibilities as exemplified by the polynomially time or logarithmically space bounded reducibilities of turing truth table and many one type we introduce a small set of axioms that are satisfied for most of the specific resource bounded reducibilities appearing in the literature some of the axioms are of a more algebraic nature such as the requirement that the reducibility under consideration is a reflexive relation while others are formulated in terms of recursion theory and for example are related to delayed computations of arbitrary recursive sets the main technical result shown is that for any reducibility that satisfies these axioms every countable distributive lattice can be embedded into any proper interval of the structure induced on the recursive sets this extends a corresponding result for polynomially time bounded reducibilities due to ambos spies inform and control 65 1985 pp 63 84 as well as a result on embeddings of partial orderings for axiomatically described reducibilities due to mehlhorn j comput system sci 12 1976 pp 147 178

robust simulations and significant separations abstract we define a new notion of robust simulations between complexity classes which is intermediate between the traditional notions of infinitely often and almost everywhere as well as a corresponding notion of significant separations a language l has a robust simulation in a complexity class c if there is a language in c which agrees with l on arbitrarily large polynomial stretches of input lengths we show that various implications in complexity theory such as the collapse of ph if np p and the karp lipton theorem have analogues for robust simulations we then use these results to prove that most known separations in complexity theory can be strengthened to significant separations though in each case an almost everywhere separation is unknown proving our results requires several new ideas including a completely different proof of the hierarchy theorem for non deterministic polynomial time than the ones previously known

a universally acceptable smoothing factor for kernel density estimates we define a minimum distance estimate of the smoothing factor for kernel density estimates based on a methodology first developed by yatracos it is shown that if f nh denotes the kernel density estimate on r d for an i i d sample of size n drawn from an unknown density f where h is the smoothing factor and if f n is the kernel estimate with the same kernel and with the proposed new data based smoothing factor then under a regularity condition on the kernel k formula math this is the first published smoothing factor that can be proven to have this property

class semigroups of prufer domains abstract the class semigroup of a commutative integral domainris the semigroup s r of the isomorphism classes of the nonzero ideals ofrwith the operation induced by multiplication the aim of this paper is to characterize the prufer domainsrsuch that the semigroup s r is a clifford semigroup namely a disjoint union of groups each one associated to an idempotent of the semigroup we find a connection between this problem and the following local invertibility property an idealiofris invertible if and only if every localization ofiat a maximal ideal ofris invertible we consider the property introduced in 1967 for prufer domainsr stating that if 1and 2are two distinct sets of maximal ideals ofr then rm m 1 rm m 2 let c be the class of prufer domains satisfying the separation property or with the property that each localization at a maximal ideal if finite dimensional we prove that ifrbelongs to c then the local invertibility property holds onrif and only if every nonzero element ofris contained only in a finite number of maximal ideals ofr moreover ifrbelongs to c then s r is a clifford semigroup if and only if every nonzero element ofris contained only in a finite number of maximal ideals ofr

w divisorial domains abstract we study the class of domains in which each w ideal is divisorial extending several properties of divisorial and totally divisorial domains to a much wider class of domains in particular we consider p v md s and mori domains

uniformly hard languages ladner j assoc comput mach 22 1975 155 showed that there are no minimal recursive sets under polynomial time reductions given any recursive set a ladner constructs a set b such that b strictly reduces to a but b does not lie in p the set b does have very long sequences of input lengths of easily computable instances we examine whether ladner s results hold if we restrict ourselves to uniformly hard languages which have no long sequences of easily computable instances under a hard to disprove assumption we show that there exists a minimal recursive uniformly hard set under honest many one polynomial time reductions

the recursion theoretic structure of complexity classes abstract we prove easy recursion theoretic results which have as corollaries generalizations of existing diagonalization theorems on complexity classes roughly speaking almost no reasonable time space or even abstract complexity class can be expressed as the non trivial union of two recursively presentable classes which are closed under finite variations e g unless np p np p np complete languages and consequently the non trivial complement of one complexity class in another e g npp provided np p is almost never recursively presentable