differential properties of x mapsto x 2 t 1 we provide an extensive study of the differential properties of the functions x x2t 1 over bbf 2n for 1 t n we notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomials x2t bx2 b 1 x where b varies in bbf 2n we prove a strong relationship between the differential spectra of x x2t 1 and x x2s 1 for s n t 1 as a direct consequence this result enlightens a connection between the differential properties of the cube function and of the inverse function we also determine the complete differential spectra of x x7 by means of the value of some kloosterman sums and of x x2t 1 for t n 2 n 2 1 n 2

convergence of conditional expectations and strong laws of large numbers for multivalued random variables on etablit des lemmes de fatou et des theoremes de convergence de lebesgue pour des esperances conditionnelles multivaluees de variables aleatoires a valeurs dans des sous ensembles fermes d un espace de banach separable

on the attainable set of differential inclusions and control systems in several cases problems in optimal control mathematical economics and mechanics involving various types of ambiguity indeterminancy or shortage of information hysteresis stochastic problems dynamical price decentralization etc lead to the study of differential inclusions in a differential inclusion the tangent at each point is prescribed by a multifunction set valued function and the direction field is usually called an orientor field in this paper we consider the differential inclusion

on the convergence in distribution of measurable multifunctions random sets normal integrands stochastic processes and stochastic infima the concept of the distribution function of a closed valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi convergence in distribution of normal integrands and stochastic processes in particular various compactness criteria are exhibited the connections with the classical convergence theory for stochastic processes are analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of donsker s theorem brownian motion as a limit of normalized random walks we also suggest the potential application of the theory to the study of the convergence of stochastic infima

measurability and integrability of the weak upper limit of a sequence of multifunctions we provide several properties of the weak upper limit of a sequence of subsets of a separable banach space such as a criterion of non vacuity of closedness etc we also examine the measurability of the multifunction defined as the weak upper limit of a sequence of multifunctions at last applications to the existence of a measurable and bochner integrable selector for this multifunction are presented

distributional properties of correspondences on loeb spaces abstract we present some regularity properties for the set of distributions induced by the measurable selections of a correspondence over a loeb space which include closedness convexity compactness purification and semicontinuity we also note that all the properties reported in the main theorems are not satisfied by some correspondences on the unit lebesgue interval

permutations of finite fields with prescribed properties classes of permutations of finite fields with various specific properties are often needed for applications we use a recent classification of permutation polynomials using their carlitz rank with advantage to produce examples of classes of permutations of f p for odd p which for instance are random have low differential uniformity prescribed cycle structure high polynomial degree large weight and large dispersion they are also easy to implement we indicate applications in coding and cryptography

on the existence of solutions for random differential inclusions in a banach space abstract we prove two existence theorems for random differential inclusions defined in a separable banach space one is about differential inclusions defined on all of the banach space x and the other for differential inclusion defined on a closed convex subset k both theorems are proved through the use of analogous deterministic results which we also include and techniques from the theory of measurable multifunctions

a stochastic approach to stability in stochastic programming abstract a random optimization problem p 0 min x 0 0 x is approximated by a sequence of random surrogate problems p n n n with p n min x n n x p a given probability space we investigate the convergence almost surely and in probability of the optimal values and the solution sets the results can be regarded as random versions of well known stability statements of parametric programming semicontinuous convergence almost surely in probability of sequences of random functions is a crucial assumption in this framework and will be investigated in more detail

convergence of best approximations from unbounded sets given a metric space whose bounded sets are relatively compact i e have compact closures we show that a nearest point selection from a sequence of kuratowski converging sets converges to the nearest point in the limit set whenever the latter point is unique the result is extended to kuratowski limits of linear varieties in infinite dimensional hilbert spaces where this nearest point relative to the origin is necessarily unique finally we show that the kuratowski limit of hyperplanes must itself be a hyperplane and that a necessary and sufficient condition for the associated nearest points to the origin to converge as above is that the canonial points parametrizing the hyperplanes converge

pseudogroups and their etale groupoids abstract a pseudogroup is a complete infinitely distributive inverse monoid such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces the goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory c algebras and aperiodic tilings our starting point is an adjunction between a category of pseudogroups and a category of etale groupoids from which we are able to set up a duality between spatial pseudogroups and sober etale groupoids as a corollary to this duality we deduce a non commutative version of stone duality involving what we call boolean inverse semigroups and boolean etale groupoids as well as a generalization of this duality to distributive inverse semigroups non commutative stone duality has important applications in the theory of c algebras it is the basis for the construction of cuntz and cuntz krieger algebras and in the case of the cuntz algebras it can also be used to construct the thompson groups we then define coverages on inverse semigroups and the resulting presentations of pseudogroups as applications we show that paterson s universal groupoid is an example of a booleanization and reconcile exel s recent work on the theory of tight maps with the work of the second author

convergence of conditional expectations for unbounded random sets integrands and integral functionals given a sequence of unbounded convex random sets we study under which conditions fatou s lemma for the weak upper limit of their conditional expectations holds we also give multivalued versions of dominated and monotone convergence theorems and we discuss the special case of the integral finally applications to epigraphic convergence of integrands and to mosco convergence of certain integral functionals are provided

the status of costas arrays the definition the basic properties and all the currently known systematic constructions for costas arrays are presented as well as a table of the number c n of costas arrays of order n for 2 les n les 26 it is proved that lim supnrarrinfin c n infin and the conjecture liminfnrarrinfin c n 0 is discussed a costas array of order n is known to be equivalent to a permutation 1 2 n for which the difference triangle contains no repeated elements in any row a generalized costas array of order n q 1 or n q 2 is defined as a permutation of the nonzero elements or also excluding 1 of the q element field for which the difference triangle contains no repeated elements in any row two new constructions for these generalized costas arrays are described and illustrated

on a paper by khan and majumdar abstract in this paper we generalize two results of khan and majumdar on the weak sequential convergence of functions in the lebesgue bochner space l x 1 and on a fatou like convergence theorem

a groupoid approach to discrete inverse semigroup algebras abstract let k be a commutative ring with unit and s an inverse semigroup we show that the semigroup algebra ks can be described as a convolution algebra of functions on the universal etale groupoid associated to s by paterson this result is a simultaneous generalization of the author s earlier work on finite inverse semigroups and paterson s theorem for the universal c algebra it provides a convenient topological framework for understanding the structure of ks including the center and when it has a unit in this theory the role of gelfand duality is replaced by stone duality using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups generalizing the case of an inverse semigroup with finitely many idempotents more generally we describe the irreducible representations of an inverse semigroup s that can be induced from associated groups as precisely those satisfying a certain finiteness condition this finiteness condition is satisfied for instance by all representations of an inverse semigroup whose image contains a primitive idempotent

quasi uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions abstract we introduce several quasi uniformities on the hyperspace of a topological space in order to study convergence of nets of semicontinuous multifunctions the interplay between conjugate quasi uniformities yields results on both the topological structure of hyperspaces and the convergence of semicontinuous multifunctions

weak sequential convergence in l1 μ x and an exact version of fatouʼs lemma abstract the class of nonatomic finite measure spaces with the saturation property as developed in maharam 1942 and hoover keisler 1984 is characterized by the fatou and lebesgue property of a well dominated sequence of multifunctions taking values in a banach space with multifunctions reduced to functions this fatou characterization also extends to a variant of the closure property found in optimal control theory the results are developed through a considered overview of the relevant literature on the exact and approximate fatou lemma phrased in terms of bochner integration

semigroups and ordered categories i the reduced case abstract we investigate the relationship which exists between certain classes of ordered small categories introduced by charles ehresmann in the course of his work on local structures and the class of u semiabundant semigroups first studied by el qallali and by de barros u semiabundant semigroups embrace both the abundant and the regular semigroups the former have been studied by fountain and his students in a number of papers the latter constitute a well known and wellinvestigated branch of semigroup theory one consequence of the ideas presented in this paper is that the work of nambooripad on inductive groupoids and regular semigroups should not be seen as sui generis but as belonging to the general framework of the theory of ordered small categories

convergence of multivalued mappings via approximable measurable selectors we investigate the convergence of measurable selectors for the limit of measurable multivalued maps the relationship between the convergence of measurable selectors and lower and upper limits of measurable multivalued mappings with closed images is also derived

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

fatou s lemma in reflexive banach spaces a version of an approximate fatou lemma for a uniformly integrable sequence of functions with values in a reflexive banach space is proved the usual assumption that this sequence is pointwisely dominated in norm by a real valued integrable function is omitted

a stability result for differential inclusions in banach spaces on etablit un resultat de stabilite pour les inclusions differentielles definies sur un espace de banach separable

general non commutative locally compact locally hausdorff stone duality abstract we extend the classical stone duality between zero dimensional compact hausdorff spaces and boolean algebras specifically we simultaneously remove the zero dimensionality restriction and extend to etale groupoids obtaining a duality with an elementary class of inverse semigroups

asymptotic stochastic programs consider a stochastic program with unique solution by the notion of epi convergence in distribution in local coordinates we define the asymptotic stochastic program associated to it stochastic programs may be classified according to the type of the pertaining asymptotic program in particular three such types are studied in detail the normal shift program the wiener type program and the poisson hyperplane program conditions for the convergence to each of the three types are given

on the nonlinearity of exponential welch costas functions we study the nonlinearity of the exponential welch costas functions using the fourier transform on z m these functions have been proposed for use in nonbinary cryptosystems high nonlinearity is required to ensure resistance to linear cryptanalysis we prove some properties of the nonlinearity of these functions and we suggest a plausible connection of the nonlinearity to the class number of a quadratic field

contributions to the theory of set valued functions and set valued measures measurable multifunctions and multimeasures with values in a banach space are studied we start by proving a variation of the known dunford theorem for weak compactness in l1 x with a similar technique we prove that the range of certain vector valued integrals that appear in applications is w compact and convex also we obtain dunford pettis type theorems for sequences of integrably bounded multifunctions some pointwise w compactness theorems are also obtained for certain families of measurable multifunctions then we prove a representation theorem for additive set valued operators defined on l1 x finally in the last section a detailed study of transition multimeasures is conducted and several representation theorems are proved

topological equivalence of tilings we introduce a notion of equivalence on tilings which is formulated in terms of their local structure we compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic

etale groupoids and their quantales abstract we establish close and previously unknown relations between quantales and groupoids in particular to each etale groupoid either localic or topological there is associated a unital involutive quantale we obtain a bijective correspondence between localic etale groupoids and their quantales which are given a rather simple characterization and here are called inverse quantal frames we show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids and as a consequence we obtain a non functorial correspondence between these and localic etale groupoids that generalizes more classical results concerning inverse semigroups and topological etale groupoids this generalization is entirely algebraic and it is valid in an arbitrary topos as a consequence of these results we see that a localic groupoid is etale if and only if its sublocale of units is open and its multiplication map is semiopen and an analogue of this holds for topological groupoids in practice we are provided with new tools for constructing localic and topological etale groupoids as well as inverse semigroups for instance via presentations of quantales by generators and relations the characterization of inverse quantal frames is to a large extent based on a new quantale operation here called a support whose properties are thoroughly investigated and which may be of independent interest

the local structure of tilings and their integer group of coinvariants the local structure of a tiling is described in terms of a multiplicative structure on its pattern classes the groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined this group furnishes part of the k0 group of the groupoid c algebra for tilings which reduce to decorations of the group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and v primitive this yields in particular the set of possible gap labels predicted by k theory for schrodinger operators describing the particle motion in such a tiling

asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems we study the asymptotic behavior of the statistical estimators that maximize a not necessarily dieren tiable criterion function possibly subject to side constraints equalities and inequalities the consistency results generalize those of wald and huber conditions are also given under which one is still able to obtain asymptotic normality the analysis brings to the fore the relationship between the problem of nding statistical estimators and that of nding the optimal solutions of stochastic optimization problems with partial information the last section is devoted to the properties of the saddle points of the associated lagrangians

restriction categories i categories of partial maps given a category with a stable system of monics one can form the corresponding category of partial maps to each map in this category there is on the domain of the map an associated idempotent which measures the degree of partiality this structure is captured abstractly by the notion of a restriction category in which every arrow is required to have such an associated idempotent categories with a stable system of monics functors preserving this structure and natural transformations which are cartesian with respect to the chosen monics form a 2 category which we call mcat the construction of categories of partial maps provides a 2 functor par mcat cat we show that par can be made into an equivalence of 2 categories between mcat and a 2 category of restriction categories the underlying ordinary functor par r0 mcat 0 ca t0 of the above 2 functor par turns out to be monadic and from this we deduce the completeness and cocompleteness of the 2 categories of m categories and of restriction categories we also consider the problem of how to turn a formal system of subobjects into an actual system of subobjects a formal system of subobjects is given by a functor into the category slat of semilattices this structure gives rise to a restriction category which via the above equivalence of 2 categories gives an m category this m category contains the universal realization of the given formal subobjects as actual subobjects

two new measures for permutations ambiguity and deficiency we introduce the concepts of weighted ambiguity and deficiency for a mapping between two finite abelian groups of the same size then we study the optimum lower bounds of these measures for permutations of an abelian group a construction of permutations by modifying some permutation functions over finite fields is given their ambiguity and deficiency is investigated most of these functions are apn permutations we show that when they are not optimal the mobius function in the multiplicative group of bbfq is closer to being optimal in ambiguity than the inverse function in the additive group of bbfq we note that the inverse function over bbf28 is used in aes finally we conclude that a twisted permutation polynomial of a finite field is again closer to being optimal in ambiguity than the apn function employed in the safer cryptosystem

convergence of set valued sub and supermartingales in the kuratowski mosco sense the purpose of this paper is to prove some convergence theorems of closed and convex set valued sub and supermartingales in the kuratowski mosco sense to get submartingale convergence theorems we give sufficient conditions for the kudo aumann integral and hiai umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables we also give an example of a bounded closed convex set valued random variable whose kudo aumann integral is not closed

on the conditional expectation and convergence properties of random sets in this paper we study random sets with values in a separable banach space first we establish several useful properties of the set valued conditional expectation and then prove some convergence theorems for set valued amarts and uniform amarts using the weak kuratowski mosco and hausdorff modes of set convergence

a perspective on non commutative frame theory abstract this paper extends the fundamental results of frame theory to a non commutative setting where the role of locales is taken over by etale localic categories this involves ideas from quantale theory and from semigroup theory specifically ehresmann semigroups restriction semigroups and inverse semigroups we prove several main results to start with we establish a duality between the category of complete restriction monoids and the category of etale localic categories the relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames this result builds on the work of pedro resende on the connection between pseudogroups and etale localic groupoids but in the process we both generalize and simplify for example we do not require involutions and in addition we render his result functorial a wider class of quantales called multiplicative ehresmann quantal frames is put into a correspondence with those localic categories where the multiplication structure map is semiopen and all the other structure maps are open we also project down to topological spaces and as a result extend the classical adjunction between locales and topological spaces to an adjunction between etale localic categories and etale topological categories in fact varying morphisms we obtain several adjunctions just as in the commutative case we restrict these adjunctions to spatial sober and coherent spectral equivalences the classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups consequently we deduce several dualities between distributive restriction semigroups and spectral etale topological categories we also specialize these dualities for the setting where the topological categories are cancellative or are groupoids our approach thus links unifies and extends the approaches taken in the work by lawson and lenz and by resende

on cournot nash equilibria in generalized qualitative games with a continuum of players in recent years the nash debreu theorem 28 8 of 1951 52 has been the subject of two particularly far reaching generalizations the first is a neglected contribution of ma 24 and is a result on the existence of cournot nash equilibria in games with an arbitrary set of players each of whom has strategy sets in an arbitrary hausdorff topological vector space the second is a result of toussaint 32 and it can be viewed as a generalization of ma s theorem to a setting where each player s pay off is generated by a nonordered relation both of these results are elegant and simple enough that they can be fully stated the following result was proved in 1969

fatou s lemma for multifunctions with unbounded values for multifunctions having unbounded sets as values we give fatou type inclusions for the kuratowski limes superior in finite as well as infinite dimensions these are derived from similar known fatou type inequalities for single valued multifunctions i e ordinary functions that is from balder s unifying fatou lemma in case the image set is finite dimensional and from an update of related results by balder in the infinite dimensional case for this extension from the single valued to the multivalued situation a lemma due to hess used to prove earlier fatou type inclusions is of critical importance also an asymptotic correction term introduced here plays an important role the two man results thus obtained subsume and extend an entire class of comparable fatou lemmas

on the convergence of closed valued measurable multifunctions in this paper we study the convergence almost everywhere and in measure of sequences of closed valued multifunctions we first give a number of criteria for the convergence of sequences of closed subsets these results are used to obtain various characterizations for the convergence of measurable multifunctions in particular we are interested in the convergence properties of measurable selections

on the convergence in probability of random sets measurable multifunctions it is shown that the convergence in probability of random sets introduced by salinetti and wets is consistent with the more standard definitions of convergence in probability and is metric invariant

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

subgroups of the group of homeomorphisms of the cantor space and a duality between a class of inverse monoids and a class of hausdorff etale groupoids abstract under non commutative stone duality there is a correspondence between second countable hausdorff etale groupoids which have a cantor space of identities and what we call tarski inverse monoids that is countable boolean inverse monoids with semilattices of idempotents which are countable and atomless tarski inverse monoids are therefore the algebraic counterparts of the etale groupoids studied by matui and provide a natural setting for many of his calculations under this duality we prove that natural properties of the etale groupoid correspond to natural algebraic properties of the tarski inverse monoid effective groupoids correspond to fundamental tarski inverse monoids and minimal groupoids correspond to 0 simplifying tarski inverse monoids particularly interesting are the principal groupoids which correspond to tarski inverse monoids where every element is a finite join of infinitesimals and idempotents here an infinitesimal is simply a non zero element with square zero the groups of units of fundamental tarski inverse monoids generalize the finite symmetric groups and include amongst their number the thompson groups g n 1 as well as the groups of units of af inverse monoids krieger s ample groups being examples

quantitative stability of variational systems i the epigraphical distance this paper proposes a global measure for the distance between the elements of a variational system parametrized families of optimization problems

on abstract conditional expectations i if c is a sub a field of 2 f o f o p a e and there is a nonnegative real valued function 40 e l q s t i f o 1 then lim ezf w e o this is just the lebesgue dominated convergence theorem for abstract conditional expectations see s ii if zm z are sub a fields of 2 v c z or fi i 2 c andfel q 1 pc co then ezn l e j furthermore if p 1 then eznf o ezf w p a e these are known as martingale convergence theorems see s and 14 theorem 51 together with the general theory of martingales have an increasingly important impact in banach space theory see 5 chap vi

convergence of set valued and fuzzy valued martingales the purpose of this paper is to prove the convergence theorems of set valued and fuzzy valued martingales in kuratowski mosco sense without assuming that their values are compact or of compact level sets