c0 is isometrically isomorphic to a subspace of cantor lebesgue functions abstract we prove what the title of this note states i e c 0 that is the set of all sequences converging to 0 is isometrically isomorphic to a subspace of continuous almost everywhere differentiable and non lipschitz functions on 0 1 having almost everywhere null derivative this in particular improves a recent result jimenez rodriguez et al 2012 9 on the lineability of this set of functions

algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

confidence regions for the multinomial parameter with small sample size consider the observation of n iid realizations of an experiment with d 2 possible outcomes which corresponds to a single observation of a multinomial distribution d n p where p is an unknown discrete distribution on 1 d in many applications the construction of a confidence region for p when n is small is crucial this challenging concrete problem has a long history it is well known that the confidence regions built from asymptotic statistics do not have good coverage when n is small on the other hand most available methods providing nonasymptotic regions with controlled coverage are limited to the binomial case d 2 here we propose a new method valid for any d 2 that provides confidence regions with controlled coverage and small volume the method involves inversion of the covering collection associated with level sets of the likelihood the behavior when d n tends to infinity remains an interesting open problem beyond the scope of this work

locating p poly optimally in the extended low hierarchy the low hierarchy in np and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes and almost all of them could be located optimally therein however until now the exact location of p poly remained open

riemann integrability versus weak continuity abstract in this paper we focus on the relation between riemann integrability and weak continuity a banach space x is said to have the weak lebesgue property if every riemann integrable function from 0 1 into x is weakly continuous almost everywhere we prove that the weak lebesgue property is stable under l 1 sums and obtain new examples of banach spaces with and without this property furthermore we characterize dunford pettis operators in terms of riemann integrability and provide a quantitative result about the size of the set of continuous nonriemann integrable functions with a locally convex topology weaker than the norm topology

a uniform approach to define complexity classes abstract complexity classes are usually defined by referring to computation models and by putting suitable restrictions on them following this approach many proofs of results are tightly bound to the characteristics of the computation model and of its restrictions and therefore they sometimes hide the essential properties which insure the obtained results in order to obtain more general results a uniform family of computation models which encompasses most of the complexity classes of interest is introduced as a first initial set of results derivable from the proposed approach we will give a sufficient and necessary condition for proving separations of relativized complexity classes a characterization of complexity classes with complete languages and a sufficient condition for proving strong separations of relativized complexity classes examples of applications of these results to some specific complexity classes are then given additional results related to separations by sparse oracles can be found in bovet 1991

spectral decomposition of the covariance matrix of a multinomial summary this paper gives information about the spectral decomposition of the covariance matrices of a multinomial it is a note to the paper of tanabe and sagae which gives fascinating information about the cholesky decomposition of the covariance matrices of multinomial distributions

on relativized exponential and probabilistic complexity classes an oracle x is constructed such that the exponential complexity class ep x 2 equals the probabilistic class r r x this shows that it will be difficult to prove that ep 2 is different from r r although it seems very unlikely that these two classes are equal the result subsumes several known results about relativized computations i the existence of relativized polynomial hierarchies extending two levels long t 1978 dissertation purdue univ lafayette ind heller h 1984 a siam j comput 13 717 725 heller h 1984 b math systems theory 17 71 84 ii the existence of an oracle x such that bpp x p x 2 stockmeyer l 1983 proc 15th stoc pp 118 126 iii the existence of an oracle x such that np x is polynomially turing reducible to a sparse set wilson c 1983 proc 24th focs pp 329 334 immerman n and mahaney s 1983 conference on computational complexity theory santa barbara march 21 25 the result shows possible inclusion relations for nonrelativized complexity classes and points out that certain results about probabilistic complexity classes and about polynomial size circuits cannot be improved unless methods are applied which do not relativize

nonseparable spaceability and strong algebrability of sets of continuous singular functions abstract let cbv denote the banach algebra of all continuous real valued functions of bounded variation defined in 0 1 we show that the set of strongly singular functions in cbv is nonseparably spaceable we also prove that certain families of singular functions constitute strongly c algebrable sets the argument is based on a criterion of strong c algebrability using compositions with exponential like functions

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

moduleability algebraic structures and nonlinear properties we show that some pathological phenomena occur more often than one could expect existing large algebraic structures infinite dimensional vector spaces algebras positive cones or infinitely generated modules enjoying certain special properties in particular we construct infinite dimensional vector spaces of non integrable measurable functions completing some recent results shown in garcia pacheco et al 2009 13 garcia pacheco and seoane sepulveda 2006 15 munoz fernandez et al 2008 20 we prove as well the existence of dense and not barrelled spaces of sequences every non zero element of which has a finite number of zero coordinates giving partial answers to a problem originally posed by r m aron and v i gurariy in 2003

lineability spaceability and additivity cardinals for darboux like functions we introduce the concept of maximal lineability cardinal number ml m of a subset m of a topological vector space and study its relation to the cardinal numbers known as additivity a m homogeneous lineability hl m and lineability l m of m in particular we will describe in terms of l the lineability and spaceability of the families of the following darboux like functions on r n n 1 extendable jones and almost continuous functions

bounded queries approximations and the boolean hierarchy this paper investigates nondeterministic bounded query classes in relation to the complexity of np hard approximation problems and the boolean hierarchy nondeterministic bounded query classes turn out be rather suitable for describing the complexity of np hard approximation problems the results in this paper take advantage of this machine based model to prove that in many cases np approximation problems have the upward collapse property that is a reduction between np approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level for example if maxclique reduces to log n approximating maxclique using many one reductions then the traveling salesman problem tsp is equivalent to maxclique under many one reductions several upward collapse theorems are presented in this paper the proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes in fact these results depend on a surprising connection between the boolean hierarchy and nondeterministic bounded query classes

strong and robustly strong polynomial time reducibilities to sparse sets reducibility defined by oracle strong nondeterministic machines is studied two definitions of relativized strength are presented and separated the corresponding reduction classes of the sparse sets are characterized in terms of nonuniform complexity classes an oracle restricted positive relativization of the probabilistic class zpp is developed

nonuniform proof systems a new framework to describe nonuniform and probabilistic complexity classes the concept of non uniform proof systems is introduced this notion allows a uniform description of non uniform complexity classes 10 probabilistic classes e g bpp 8 15 27 32 am 2 3 and language classes defined by simultaneous non uniform and nondeterministic time bounds non uniform proof systems provide a better understanding of many results concerning these classes particularly their connections to uniform complexity measures we give an uniform approach to lowness results 19 20 for various complexity classes for instance we show that co np poly np is contained in the third level of the low hierarchy and that np c np co np poly implies that the polynomial time hierarchy collapses to its second level see also 1 finally some evidence is given that the low hierarchy cannot be extended beyond its third level by the current techniques

spaceability of the set of continuous injections from blp into lp with nowhere continuous inverses abstract let p 1 we show that in the banach space of all bounded continuous mappings from b l p the open unit ball in l p into l p the subset consisting of all injections with nowhere continuous inverses is isometrically l p spaceable i e it contains an isometric copy of l p the proof uses some ideas of creswell s example

lineability and integrability in the sense of riemann lebesgue denjoy and khintchine abstract in this paper we continue the ongoing research on lineability related questions on this occasion we shall consider among others the classes of integrable functions in the sense of riemann lebesgue denjoy and khintchine improving some already known results and expanding the study of lineability to other famous integrable classes never considered before

complexity classes without machines on complete languages for up this paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines often counting classes probabilistic classes and intersection classes lack such enumerations concentrating on the counting class up we show that there are relativizations for which upa has no complete languages and other relativizations for which pb upb npb and upb has complete languages among other results we show that p up if and only if there exists a set s in p of boolean formulas with at most one satisfying assignment such that s sat is not in p p up coup if and only if there exists a set s in p of uniquely satisfiable boolean formulas such that no polynomial time machine can compute the solutions for the formulas in s if up has complete languages then there exists a set r in p of boolean formulas with at most one satisfying assignment so that sat r is complete for up finally we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class bpp there is a relativized world where bppa has no complete languages if bpp has complete languages then it has a complete language of the form b majority where b p and majority f f is true for at least half of all assignments is the canonical pp complete set

the boolean hierarchy ii applications the boolean hierarchy i structural properties j cai et al siam j comput 17 1988 pp 1232 252 explores the structure of the boolean hierarchy the closure of np with respect to boolean

convolution functions that are nowhere differentiable abstract in 1951 v jarnik constructed two continuous functions whose volterra convolution is nowhere differentiable we generalize jarnik s results by proving that the set of such functions is maximal lineable this would shed some light on a question posed in 1973 on the structure of the set of continuous functions whose volterra convolution is nowhere differentiable

a generalization of resource bounded measure with application to the bpp vs exp problem we introduce resource bounded betting games and propose a generalization of lutz s resource bounded measure in which the choice of the next string to bet on is fully adaptive lutz s martingales are equivalent to betting games constrained to bet on strings in lexicographic order we show that if strong pseudorandom number generators exist then betting games are equivalent to martingales for measure on e and exp however we construct betting games that succeed on certain classes whose lutz measures are important open problems the class of polynomial time turing complete languages in exp and its superclass of polynomial time turing autoreducible languages if an exp martingale succeeds on either of these classes or if betting games have the finite union property possessed by lutz s measure one obtains the nonrelativizable consequence mbox bpp neq mbox exp we also show that if mbox exp neq mbox ma then the polynomial time truth table autoreducible languages have lutz measure zero whereas if mbox exp mbox bpp they have measure one

quantitative relativizations of complexity classes consider the following open problems i text p text np ii text np text co np iii text p text pspace iv text np text pspace in this paper we study these four problems from a particular point of view to illustrate our approach consider the first problem it is known that there exist recursive sets a and b such that text p a text np a and text p b ne text np b we study restrictions r on both the deterministic and also the nondeterministic polynomial time bounded oracle machines such that the following holds text p text np if and only if for every set a text p r a text np r a the restrictions are quantitative in the sense that the size of the set of strings queried by the oracle in computations of a machine on an input is bounded by a polynomial in the length of the input we study several different ways of specifying such quantitative restrictions each of which has the desire

lineability criteria with applications lineability is a property enjoyed by some subsets within a vector space x a subset a of x is called lineable whenever a contains except for zero an infinite dimensional vector subspace if additionally x is endowed with richer structures then the more stringent notions of dense lineability maximal dense lineability and spaceability arise naturally in this paper several lineability criteria are provided and applied to specific topological vector spaces mainly function spaces sometimes such criteria furnish unified proofs of a number of scattered results in the related literature families of strict order integrable functions hypercyclic vectors non extendable holomorphic mappings riemann non lebesgue integrable functions sequences not satisfying the lebesgue dominated convergence theorem nowhere analytic functions bounded variation functions entire functions with fast growth and peano curves among others are analyzed from the point of view of lineability

lineability of the set of bounded linear non absolutely summing operators in this note we solve except for extremely pathological cases a question posed by puglisi and seoane sepulveda on the lineability of the set of bounded linear non absolutely summing operators we also show how the idea of the proof can be adapted to several related situations

rolewicz type chaotic operators abstract in this article we introduce a new class of rolewicz type operators in l p 1 p we exhibit a collection f having the cardinality of the continuum consisting of operators of this type which are chaotic and remain so under almost all finite linear combinations provided that the linear combination has sufficiently large norm as a corollary to our main result we also obtain that there exists a countable collection of such operators whose all finite linear combinations are chaotic provided that they have sufficiently large norm

s linear and convex structures in function families abstract in this paper the notion of s lineability originally coined by vladimir i gurariy is introduced and developed in a general abstract setting this new notion is then applied to specific situations as for instance classes of differentiable nowhere monotone functions as well as families of vectors having dense orbit with respect to an operator large convex structures are also shown to exist inside the family of topologically mixing continuous selfmaps of a real compact interval

new collapse consequences of np having small circuits we show that if a self reducible set has polynomial size circuits then it is low for the probabilistic class zpp np as a consequence we get a deeper collapse of the polynomial time hierarchy ph to zpp np under the assumption that np has polynomial size circuits this improves on the well known result in karp and lipton proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 stating a collapse of ph to its second level sigmap 2 under the same assumption furthermore we derive new collapse consequences under the assumption that complexity classes like up fewp and c p have polynomial size circuits finally we investigate the circuit size complexity of several language classes in particular we show that for every fixed polynomial s there is a set in zpp np which does not have o s n size circuits

defying upward and downward separation upward and downward separation results link the collapse of small and large classes and are a standard tool in complexity theory we study the limitations of upward and downward separation pp we show that the exponential time limited nondeterminism hierarchy does not robustly possess downward separation we show that probabilistic classes do not robustly possess hartmanis immerman sewelson upward separation though np is known to robustly possess hartmanis immerman sewelson upward separation we show that np does not robustly possess hartmanis immerman sewelson upward separation with respect to strong immunity separation on the other hand we provide a structural sufficient condition for upward separation

basic sequences and spaceability in lp spaces let x be a sequence space and denote by z x the subset of x formed by sequences having only a finite number of zero coordinates we study algebraic properties of z x and show among other results that for p is an element of 1 infinity z l p does not contain infinite dimensional closed subspaces this solves an open question originally posed by r m aron and v i gurariy in 2003 on the linear structure of z l infinity in addition to this we also give a thorough analysis of the existing algebraic structures within the sets z x and x z x and their algebraic genericities

sparse sets in np p exptime versus nexptime this paper investigates the structural properties of sets in np p and shows that the computational difficulty of lower density sets in np depends explicitly on the relations between higher deterministic and nondeterministic time bounded complexity classes the paper exploits the recently discovered upward separation method which shows for example that there exist sparse sets in np p if and only if exptime nexptime in addition the paper uses relativization techniques to determine logical possibilities limitations of these proof techniques and exhibits one of the first natural structural differences between relativized np and conp

logarithmic advice classes abstract karp and lipton 1980 introduced the notion of nonuniform complexity classes where a certain amount of additional information the advice is given for free the advice only depends on the length of the input karp and lipton initiated the study of classes with either logarithmic or polynomial advice however later yap 1983 schoning 1984 balcazar 1987 and ko and schoning 1985 concentrated on the study of classes of the form c poly where c is p np or pspace and poly denotes a polynomial size advice this paper considers classes of the form c log as a main result it is shown that in the context of an np log computation log bounded advice is equivalent to a sparse oracle in np in contrast it has been shown that a poly bounded advice to an arbitrary sparse oracle set furthermore a general theorem is presented that generalizes karp and lipton s round robin tournament method

on the unique satisfiability problem unique sat is the problem of deciding whether a given boolean formula has exactly one satisfying truth assignment this problem is a typical moreover complete representative of a natural class of problems about unique solutions all these problems belong to the class dife l1 l2 l1 lz np studied by papadimitriou and yannakakis we consider the relationship between these two classes particularly whether unique sat is dife complete it is if np co np we construct an oracle relative to which unique sat is not complete for dif and another oracle relative to which unique sat is complete for dif e whereas np v co np

lineability and additivity in r r we give a condition for a family of functions to be lineable by means of its additivity the novelty presented here is that we solve the lineability problem using a technique that is not constructive as are most approaches to this problem we relate the notions of additivity and lineability and use this relation to give a general method to find the lineability of large families of functions we also study more examples of pathologically behaving functions in particular the class of jones functions which is a highly pathological subclass of perfectly everywhere surjective functions we work on the additivity lineability and main properties of this class

if np has polynomial size circuits then ma am abstract it is shown that the assumption of np having polynomial size circuits implies apart from a collapse of the polynomial time hierarchy as shown by karp and lipton that the classes am and ma of babai s arthur merlin hierarchy coincide this means that also a certain inner collapse of the remaining classes of the polynomial time hierarchy occurs

mathematical properties of the variance of the multinomial distribution in an interesting article tanabe and sagae 2 discuss in a wider context some properties of the variance of the multinomial distribution it is worthwhile to attempt a specific treatment of mathematical properties of this matrix a completely different approach will be used in this paper tanabe and sagae s results follow in an easy manner

p selectivity intersections and indices the p selective sets selman 1979 are those sets for which there is a polynomial time algorithm that given any two strings determines which is more likely to belong to the set if either of the strings is in the set the algorithm chooses one that is in the set we prove that for each k the k ary boolean connectives under which the p selective sets are closed are exactly those that are either completely degenerate or almost completely degenerate we determine the complexity of the index set of the r e p selective sets 30 complete

locally recurrent functions density topologies and algebraic genericity we study the algebraic and topological genericity of certain subsets of locally recurrent functions obtaining among other results algebrability and spaceability within these classes

non lipschitz functions with bounded gradient and related problems let e be a topological vector space and let us consider a property p we say that the subset m of e formed by the vectors in e which satisfy p is lineable for certain cardinal finite or infinite if m 0 contains an infinite dimensional linear space of dimension in this note we prove that there exist uncountably infinite dimensional linear spaces of functions enjoying the following properties 1 being continuous on 0 1 a e differentiable with a e bounded derivative and not lipschitz 2 differentiable in r2 r and not enjoying the mean value theorem 3 real valued differentiable on an open connected and non convex set having bounded gradient non lipschitz and therefore not verifying the mean value theorem

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

promise problems complete for complexity classes abstract a general framework is given to obtain hardness results for promise problems that derive from self reducible decision problems the principal theorem is that if a set a is d p equivalent to a disjunctive self reducible set in np then the natural promise problem associated with a is as hard to solve as it is to recognize a np hardness of the satisfiability promise problem follows and graph isomorphism hardness of a promise problem that derives from the graph isomorphism problem is proved

competing provers yield improved karp lipton collapse results via competing provers we show that if a language a is self reducible and has polynomial size circuits then s2a s2 building on this we strengthen the kamper afk theorem namely we prove that if np np conp poly then the polynomial hierarchy collapses to s2np conp we also strengthen yap s theorem namely we prove that if np conp poly then the polynomial hierarchy collapses to s2np under the same assumptions the best previously known collapses were to zppnp and zppnpnp respectively siam journal on computing 28 1 1998 311 journal of computer and system sciences 52 3 1996 421 building on proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 journal of computer and system sciences 39 1989 21 theoretical computer science 85 2 1991 305 theoretical computer science 26 3 1983 287 it is known that s2 zppnp proceedings of the 42nd ieee symposium on foundations of computer science ieee computer society press silver spring md 2001 pp 620 629 that result and its relativized version show that our new collapses indeed improve the previously known results the kamper afk theorem and yap s theorem are used in the literature as bridges in a variety of results ranging from the study of unique solutions to issues of approximation and so our results implicitly strengthen those results

robustness of probabilistic computational complexity classes under definitional perturbations various types of probabilistic algorithms play an increasingly important role in computer science especially in computational complexity theory probabilistic polynomial time complexity classes are defined and compared to each other emphasizing some structural relationships to the known complexity classes p np pspace the classes r and zpp corresponding to the so called las vegas polynomial time bounded algorithms are given special attention it is shown that many definitions that arise naturally from different types of algorithms are equivalent in defining the same class r or zpp resp these robustness results justify finally the tractability of the above probabilistic classes

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

counting classes are at least as hard as the polynomial time hierarchy in this paper it is shown that many natural counting classes such as pp c p and text mod k text p are at least as computationally hard as ph the polynomial time hierarchy in the following sense for each bf k of the counting classes above every set in bf k ph is polynomial time randomized many one reducible to a set in bf k with two sided exponentially small error probability as a consequence of the result it is seen that all the counting classes above are computationally harder than ph unless ph collapses to a finite level some other consequences are also shown

on lineability of sets of continuous functions we study the existence of vector spaces of dimension at least two of continuous functions on subsets of r every non zero element of which admits one and only one absolute maximum

on relativized polynomial and exponential computations the relationship between polynomial and exponential time bounded relativized computations is investigated it is shown that there exists an oracle x such that relative to x nondeterministic exponential computations yield only the languages in sigma 2 p x from the polynomial hierarchy more informally relative to that x two polynomial quantifiers are as powerful as one exponential quantifier

universality of sequences of operators related to taylor series abstract in this note the universality of a sequence of operators associated to the partial sums of the taylor series of a holomorphic function is investigated the emphasis is put on the fact that the taylor series are evaluated at a prescribed point and the variable is the center of the expansion the dynamics of the sequence of operators linked to the partial sums of a power series that is not generated by an entire function is also studied

geometric sets of low information content abstract in this paper we study language classes defined by nonuniform families of hyperplanes and halfspaces these are subclasses of p poly upto many one equivalence these classes are essentially the classes elt1 and lt1 of languages accepted by nonuniform families of polynomial size depth 1 circuits with a weighted exact threshold gate at the root respectively weighted threshold gate at the root we investigate the consequences of intractable sets being reducible to sets in elt1 and lt1 using the polynomial time algorithm for linear programming and ideas from the theory of convex polytopes we prove that every disjunctively self reducible bd cylinder that is many one reducible to a set in elt1 or lt1 is already in p as a consequence we derive that if a complete set for any class k np mod k p pp c p many one reduces to a set in elt1 or lt1 then k p for families of hyperplanes over finite fields we prove that every turing self reducible set that is many one reducible to a set defined by such a family is in p and every word decreasing self reducible set that is many one reducible to a set defined by such a family is in np co np furthermore as an interesting connection it turns out that several sparse and tally reduction classes are many one reducible to elt1 or lt1 thus our results concerning reductions to elt1 and lt1 subsume the strongest existing collapse results concerning reductions of any class k np mod k p pp c p to sparse sets for various reducibilities

computing solutions uniquely collapses the polynomial hierarchy is there an np function that when given a satisfiable formula as input outputs one satisfying assignment uniquely that is can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments we show that if there is such a nondeterministic function then the polynomial hierarchy collapses to zppnp and thus in particular to npnp as the existence of such a function is known to be equivalent to the statement every np function has an np refinement with unique outputs our result provides the strongest evidence yet that np functions cannot be refined we prove our result via a result of independent interest we say that a set a is npsv selective npmv selective if there is a 2 ary partial np function with unique values a 2 ary partial np function that decides which of its inputs if any is more likely to belong to a this is a nondeterministic analog of the recursion theoretic notion of the semi recursive sets and the extant complexity theoretic notion of p selectivity our hierarchy collapse result follows by combining foo the easy observation that every set in np is npmv selective with the following result if a in np is npsv selective then a in np cap conp poly relatedly we prove that if a in np is npsv selective then a is low 2 we prove that the polynomial hierarchy collapses even further namely to np if all conp sets are npmv selective this follows from a more general result we prove every self reducible npmv selective set is in np

a new derivation of eigenvalue inequalities for the multinomial distribution abstract we consider the covariance matrix of the multinomial distribution we suggest a new derivation of inequalities for the eigenvalues of this matrix using a classical result on the product of two positive semi definite matrices

complexity classes of equivalence problems revisited to determine if two lists of numbers are the same set we sort both lists and see if we get the same result the sorted list is a canonical form for the equivalence relation of set equality other canonical forms arise in graph isomorphism algorithms to determine if two graphs are cospectral have the same eigenvalues we compute their characteristic polynomials and see if they are equal the characteristic polynomial is a complete invariant for cospectrality finally an equivalence relation may be decidable in p without either a complete invariant or canonical form blass and gurevich 1984 asked whether these conditions on equivalence relations having an fp canonical form having an fp complete invariant and being in p are distinct they showed that this question requires non relativizing techniques to resolve we extend their results and give new connections to probabilistic and quantum computation

peano curves on topological vector spaces the starting point of this paper is the existence of peano curves that is continuous surjections mapping the unit interval onto the unit square from this fact one can easily construct of a continuous surjection from the real line r to any euclidean space rn the algebraic structure of the set of these functions as well as extensions to spaces with higher dimensions is analyzed from the modern point of view of lineability and large algebras are found within the families studied we also investigate topological vector spaces that are continuous image of the real line providing an optimal lineability result

nonlinear subsets of function spaces and spaceability abstract in this paper we study the existence of infinite dimensional closed linear subspaces of a rearrangement invariant space on 0 1 every nonzero element of which does not belong to any included rearrangement invariant space of the same class such that the inclusion operator is disjointly strictly singular we consider lorentz marcinkiewicz and orlicz spaces the answer is affirmative for marcinkiewicz spaces and negative for lorentz and orlicz spaces also the same problem is studied for nakano spaces assuming different hypothesis

an exact cholesky decomposition and the generalized inverse of the variance covariance matrix of the multinomial distribution with applications a symbolic formula is given for the square root free cholesky decomposition of the variance covariance matrix of the multinomial distribution the evaluation of the symbolic cholesky factors requires much fewer arithmetic operations than does the general cholesky algorithm since the symbolic formula is not affected by an ill conditioned matrix it is particularly useful when the elements of a probability vector are of quite different orders of magnitude a simpler formula is obtained for pederson s procedure of sampling from a multinomial population an explicit formula of the moore penrose inverse of the variance covariance matrix is given as well as a symmetric representation of a multinormal density approximation to the multinomial distribution these formulae facilitate symmetric manipulation of the matrix and are useful in statistical modelling and computation involving the logistic density transformation of the multinomial distribution and in computer simulations of dynamic models in population genetics each element of the cholesky factors is given interesting probabilistic interpretations

unambiguous computation boolean hierarchies and sparse turing complete sets this paper studies for up two topics that have been intensely studied for np boolean hierarchies and the consequences of the existence of sparse turing complete sets unfortunately as is often the case the results for np draw on special properties of np that do not seem to carry over straightforwardly to up for example it is known for np and more generally for any class containing sigma and emptyset and closed under union and intersection that the symmetric difference hierarchy the boolean hierarchy and the boolean closure all are equal we prove that closure under union is not needed for this claim for any class cal k that contains sigma and emptyset and is closed under intersection e g up us and fewp the symmetric difference hierarchy over cal k the boolean hierarchy over cal k and the boolean closure of cal k all are equal on the other hand we show that two hierarchies the hausdorff hierarchy and the nested difference hierarchy which in the np case are equal to the boolean closure fail to be equal for the up case in some relativized worlds regarding sparse turing complete sets for up we prove that if up has sparse turing complete sets then the levels of the unambiguous polynomial hierarchy are simpler than one would otherwise expect they collapse one level in terms of their location in the promise unambiguous polynomial hierarchy we obtain related results under the weaker assumption that up has sparse turing hard sets

spaceability of strict order integrability abstract in this paper sharp conditions on a measure space are provided in order that the subset of functions in the corresponding lebesgue space l p which are in no other l q contains except for zero an infinite dimensional closed subspace this completes earlier work of several authors the main result is derived from a new general criterion for spaceability on function spaces a further application to spaceability of the class of non absolutely continuous functions is supplied

relativizing complexity classes with sparse oracles baker gill and solovay constructed sparse sets a and b such that p a np a and np b co np b in contrast to their results we prove that p np if and only if for every tally language t p t np t and that np co np if and only if for every tally language t np t co np t we show that the polynomial hierarchy collapses if and only if there is a sparse set s such that the polynomial hierarchy relative to s collapses similar results are obtained for several other complexity classes

an undecidable case of lineability in rr abstract recently it has been proved that assuming that there is an almost disjoint family of cardinality 2 c in c which is assured for instance by either martin s axiom or the continuum hypothesis or even 2 c c one has that the set of sierpinski zygmund functions is 2 c strongly algebrable and thus 2 c lineable here we prove that these two statements are actually equivalent and moreover that they both are undecidable this would be the first time in which one encounters an undecidable proposition in the recently coined theory of lineability and spaceability

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6

p printable sets p printable sets arise naturally in the studies of generalized kolmogorov complexity and data compression as well as in other areas we present new characterizations of the p printable sets and present necessary and sufficient conditions for the existence of sparse sets in p that are not p printable as a corollary to one of our results we show that the class of sets of small generalized kolmogorov complexity is exactly the class of sets which are p isomorphic to a tally language

relativizing relativized computations abstract this paper introduces a technique of relativizing already relativized computations and gives two interesting applications the techniques developed here are simpler than the usual methods for constructing oracles that satisfy several requirements simultaneously the first application shows that a result of karp and lipton if sets in np are decidable with polynomial size circuits then p 2 p 2 cannot be strengthened in the presence of certain oracles this means that relativizable proof techniques cannot strengthen the conclusion to say p np such a stronger conclusion would be desirable as it would establish the equivalence of polynomial time programs and polynomial size circuits for solving np complete problems and would extend the known equivalence of polynomial time programs and programs that are allowed a single query to a polynomial size table the second application gives an oracle c for which p c np c conp c np c and np c conp c has complete sets under polynomial time many one reductions this complements a result of sipser in which an oracle b is constructed for which np b conp b has no complete sets these results suggest that current proof methods will not settle whether np conp has complete sets

separating complexity classes with tally oracles abstract long and selman 1986 proved that for most familiar pairs of complexity classes separating the classes with a tally oracle is no easier than truly separating the classes refuting a claim in the literature we prove for the first time that for many familiar pairs of complexity classes separating the classes with a tally oracle is easier than truly separating the classes

the complexity of promise problems with applications to public key cryptography a promise problem is a formulation of partial decision problem complexity issues about promise problems arise from considerations about cracking problems for public key cryptosystems using a notion of turing reducibility between promise problems this paper disproves a conjecture made by even and yacobi 1980 that would imply nonexistence of public key cryptosystems with np hard cracking problems in its place a new conjecture is raised having the same consequence in addition the new conjecture implies that np complete sets cannot be accepted by turing machines that have at most one accepting computation for each input word

almost overcomplete and almost overtotal sequences in banach spaces abstract we introduce the new concepts of almost overcomplete sequence in a banach space and almost overtotal sequence in a dual space we prove that any of such sequences is relatively norm compact and we present several applications of this fact

one way functions and the nonisomorphism of np complete sets abstract the one way conjecture states that the existence of nonisomorphic np complete sets implies the existence of one way functions this paper gives a relativized counterexample to the conjecture by constructing a relativized world that has nonisomorphic np complete sets but lacks one way functions

holomorphic functions on non runge domains and related problems let u be a domain in cn that is not a runge domain we study the topological and algebraic properties of the family of holomorphic functions on u which cannot be approximated by polynomials

infinite dimensional holomorphic non extendability and algebraic genericity abstract in this note the linear structure of the family h e g of holomorphic functions in a domain g of a complex banach space that are not holomorphically continuable beyond the boundary of g is analyzed more particularly we prove that h e g contains except for zero a closed and a dense vector space having maximal dimension as well as a maximally generated free algebra the results obtained complete a number of previous ones by several authors

testing for the supremacy of a multinomial cell probability tests for the supremacy of a multinomial cell probability are developed the tested null hypothesis states that a particular cell of interest is not more probable than all others rejection of this null leads to the conclusion that the cell of interest has a strictly greater probability than all other cells the null hypothesis constrains the multinomial probability vector to a nonconvex region that is a union of closed convex cones the likelihood ratio test for this problem is derived and shown to be equivalent to an intersection union test the least favorable configuration of the multinomial probability vector in the null parameter space is derived and the limiting null distribution of the test statistic that is stochastically greatest is shown to be a mixture of point mass at zero and a chi square distribution with a single degree of freedom asymptotic and valid finite sample testing procedures are proposed and examined via a simulation study and the analysis of two datasets the proposed procedure

hausdorff dimension and oracle constructions bennett and gill relative to a random oracle a pa npa co npa with probability 1 siam j comput 10 1981 96 113 proved that pa npa relative to a random oracle a or in other words that the set o p np a pa npa has lebesgue measure 0 in contrast we show that o p np has hausdorff dimension 1 this follows from a much more general theorem if there is a relativizable and paddable oracle construction for a complexity theoretic statement then the set of oracles relative to which holds has hausdorff dimension 1 we give several other applications including proofs that the polynomial time hierarchy is infinite relative to a hausdorff dimension 1 set of oracles and that pa npa conpa relative to a hausdorff dimension 1 set of oracles

some results and open questions on spaceability in function spaces a subset m of a topological vector space x is called lineable respectively spaceable in x if there exists an infinite dimensional linear space respectively an infinite dimensional closed linear space y subset of m boolean or 0 in this article we prove that for every infinite dimensional closed subspace x of c 0 1 the set of functions in x having infinitely many zeros in 0 1 is spaceable in x we discuss problems related to these concepts for certain subsets of some important classes of banach spaces such as c 0 1 or muntz spaces we also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of c 0 1 as well as oscillating and annulling properties of subspaces of c 0 1