subspace arrangements over finite fields cohomological and enumerative aspects abstract the enumeration of points on or off the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function we discuss the basic relations between these two points of view counting points is also related to the l adic cohomology of the arrangement as a variety we describe the eigenvalues of the frobenius map acting on this cohomology which corresponds to a finer decomposition of the zeta function the l adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data finally it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases

text fi _ mathcal w fi w modules and constraints on classical weyl group characters in this paper we study the characters of sequences of representations of any of the three families of classical weyl groups mathcal w n the symmetric groups the signed permutation groups hyperoctahedral groups or the even signed permutation groups our results extend work of church et al duke math j 2012 arxiv 1204 4533 geom topol 18 5 2951 2984 2014 on the symmetric groups we use the concept of an text fi mathcal w module an algebraic object that encodes the data of a sequence of mathcal w n representations with maps between them defined in the author s recent work j algebra 420 269 332 2014 we show that if a sequence v n of mathcal w n representations has the structure of a finitely generated text fi mathcal w module then there are substantial constraints on the growth of the sequence and the structure of the characters for n large the dimension of v n is equal to a polynomial in n and the characters of v n are given by a character polynomial in signed cycle counting class functions independent of n we determine bounds the degrees of these polynomials we continue to develop the theory of text fi mathcal w modules and we apply this theory to obtain new results about a number of sequences associated to the classical weyl groups the cohomology of complements of classical coxeter hyperplane arrangements and the cohomology of the pure string motion groups the groups of symmetric automorphisms of the free group

grobner methods for representations of combinatorial categories given a category c of a combinatorial nature we study the following fundamental question how does the combinatorial behavior of c affect the algebraic behavior of representations of c we prove two general results the first gives a combinatorial criterion for representations of c to admit a theory of grobner bases from this we obtain a criterion for noetherianity of representations the second gives a combinatorial criterion for a general rationality result for hilbert series of representations of c this criterion connects to the theory of formal languages and makes essential use of results on the generating functions of languages such as the transfer matrix method and the chomsky schutzenberger theorem our work is motivated by recent work in the literature on representations of various specific categories our general criteria recover many of the results on these categories that had been proved by ad hoc means and often yield cleaner proofs and stronger statements for example we give a new more robust proof that fi modules originally introduced by church ellenberg farb and a family of natural generalizations are noetherian we give an easy proof of a generalization of the lannes schwartz artinian conjecture from the study of generic representation theory of finite fields we significantly improve the theory of modules introduced by snowden in connection to syzygies of segre embeddings and we establish fundamental properties of twisted commutative algebras in positive characteristic

categories of fi type a unified approach to generalizing representation stability and character polynomials abstract representation stability is a theory describing a way in which a sequence of representations of different groups is related and essentially contains a finite amount of information starting with church ellenberg farb s theory of fi modules describing sequences of representations of the symmetric groups we now have good theories for describing representations of other collections of groups such as finite general linear groups classical weyl groups and wreath products s n g for a fixed finite group g this paper attempts to uncover the mechanism that makes the various examples work and offers an axiomatic approach that generates the essentials of such a theory character polynomials and free modules that exhibit stabilization we give sufficient conditions on a category c to admit such structure via the notion of categories of fi type this class of categories includes the examples listed above and extends further to new types of categories such as the categorical power fi m whose modules encode sequences of representations of m fold products of symmetric groups the theory is applied in 9 to give homological and arithmetic stability theorems for various moduli spaces e g the moduli space of degree n rational maps p 1 p m

on hopf algebras with cocommutative coradicals 1 zf k is a right coideal subalgebra of h such that s k k h is faithfully flat as a right or left k module 2 zf x h n h is a quotient left h module coalgebra h is faithfully coflat as a right or left x h comodule 3 k h nk and 7 h h establish a 1 1 correspondence between the right coideal subalgebras k of h such that s k ko and the quotient left h module coalgebras n h n h 4 suppose k x in 3 there is a left x h comodule and right k module isomorphism h n x h q k if either a h is pointed or b k is finite dimensional over k

picard vessiot extensions of artinian simple module algebras abstract this paper pursues takeuchi s hopf algebraic approach m takeuchi a hopf algebraic approach to the picard vessiot theory j algebra 122 1989 481 509 to the picard vessiot pv theory for differential equations to involve the pv extensions of difference equations differential fields and total difference rings in the standard pv theory are unified here by artinian simple as module algebras over a cocommutative pointed smooth hopf algebra

geometric characterization of strongly normal extensions this paper continues previous work in which we developed the galois theory of strongly normal extensions using differential schemes in the present paper we derive two main results first we show that an extension is strongly normal if and only if a certain differential scheme splits i e is obtained by base extension of a scheme over constants this gives a geometric characterization to the notion of strongly normal second we show that picard vessiot extensions are characterized by their galois group being affine our proofs are elementary and do not use group chunks or cohomology we end by recalling some important results about strongly normal extensions with the hope of spurring future research

local contrast phase descriptor for fingerprint liveness detection we propose a new local descriptor for fingerprint liveness detection the input image is analyzed both in the spatial and in the frequency domain in order to extract information on the local amplitude contrast and on the local behavior of the image synthesized by considering the phase of some selected transform coefficients these two pieces of information are used to generate a bi dimensional contrast phase histogram used as feature vector associated with the image after an appropriate feature selection a trained linear kernel svm classifier makes the final live fake decision experiments on the publicly available livdet 2011 database comprising datasets collected from various sensors prove the proposed method to outperform the state of the art liveness detection techniques highlightswe propose a new local descriptor for fingerprint liveness detection it is based on the joint use of contrast and phase information image analysis is carried out in both the spatial and the transform domains we generate a bi dimensional contrast phase histogram used as feature vector a properly trained linear kernel svm classifier makes the final live fake decision

picard vessiot theory of differentially simple rings abstract in picard vessiot theory the galois theory for linear differential equations the picard vessiot ring plays an important role since it is the picard vessiot ring which is a torsor principal homogeneous space for the galois group scheme like fields are simple rings having only 0 and 1 as ideals the picard vessiot ring is a differentially simple ring i e a differential ring having only 0 and 1 as differential ideals having in mind that the classical galois theory is a theory of extensions of fields i e of simple rings it is quite natural to ask whether one can also set up a picard vessiot theory where the base is not a differential field but more generally a differentially simple ring it is the aim of this article to give a positive answer to this question i e to set up such a differential galois theory

a hopf algebraic approach to the picard vessiot theory abstract this paper shows what is really happening in picard vessiot theory our hopf algebraic approach simplifies and generalizes the theory among other things the generalization gives a picard vessiot theory for positive characteristic for fields with not necessarily commuting derivations and even for fields with a set of higher derivations

castelnuovo mumford regularity of representations of certain product categories abstract we show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite castelnuovo mumford regularity if and only if they are presented in finite degrees and hence the category consisting of them is abelian these results apply to examples such as the categories fi m and fi g m

coincidences between homological densities predicted by arithmetic abstract motivated by analogies with basic density theorems in analytic number theory we introduce a notion and variations of the homological density of one space in another we use weil s number field function field analogy to predict coincidences for limiting homological densities of various sequences z n d 1 d m x of spaces of 0 cycles on manifolds x the main theorem in this paper is that these topological predictions which seem strange from a purely topological viewpoint are indeed true one obstacle to proving such a theorem is the combinatorial complexity of all possible collisions of points this problem does not arise in the simplest and classical case m n 1 2 of configuration spaces to overcome this obstacle we apply the bjorner wachs theory of lexicographic shellability from algebraic combinatorics

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

the differential galois theory of strongly normal extensions differential galois theory the theory of strongly normal extensions has unfortunately languished this may be due to its reliance on kolchin s elegant but not widely adopted axiomatization of the theory of algebraic groups this paper attempts to revive the theory using a differential scheme in place of those axioms we also avoid using a universal differential field instead relying on a certain tensor product we identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product thus identifying the galois group with the closed points of an affine differential scheme moreover the tensor product has a natural coring structure which translates into the galois group operation composition of automorphisms this affine differential scheme splits i e is obtained by base extension from a not differential not necessarily affine group scheme as a consequence the galois group is canonically isomorphic to the closed or rational points of a group scheme defined over constants we obtain the fundamental theorem of differential galois theory giving a bijective correspondence between subgroup schemes and intermediate differential fields on the way to this result we study certain aspects of differential algebraic geometry e g closed immersions products local ringed space of constants and split differential schemes

no reference image quality assessment in the spatial domain we propose a natural scene statistic based distortion generic blind no reference nr image quality assessment iqa model that operates in the spatial domain the new model dubbed blind referenceless image spatial quality evaluator brisque does not compute distortion specific features such as ringing blur or blocking but instead uses scene statistics of locally normalized luminance coefficients to quantify possible losses of naturalness in the image due to the presence of distortions thereby leading to a holistic measure of quality the underlying features used derive from the empirical distribution of locally normalized luminances and products of locally normalized luminances under a spatial natural scene statistic model no transformation to another coordinate frame dct wavelet etc is required distinguishing it from prior nr iqa approaches despite its simplicity we are able to show that brisque is statistically better than the full reference peak signal to noise ratio and the structural similarity index and is highly competitive with respect to all present day distortion generic nr iqa algorithms brisque has very low computational complexity making it well suited for real time applications brisque features may be used for distortion identification as well to illustrate a new practical application of brisque we describe how a nonblind image denoising algorithm can be augmented with brisque in order to perform blind image denoising results show that brisque augmentation leads to performance improvements over state of the art methods a software release of brisque is available online http live ece utexas edu research quality brisque release zip for public use and evaluation

hopf module duality applied to x outer galois theory we work over a fixed ground field k in arbitrary characteristic let r be a prime algebra and let q denote the symmetric martindale quotient ring of r we know from 10 section 6 4 that to study group actions on r it is fundamental to consider the extended actions on q and those actions which are x outer or outer as automorphisms of q have nice properties which could be generalized for x outer in some appropriate sense hopf algebra actions to recall kharchenko s galois type correspondence theorem let g be a finite group of x outer automorphisms of r it was proved in 2 theorem 3 10 2 11 theorem b that

representation stability for families of linear subspace arrangements abstract church ellenberg farb 8 used the language of fi cha to identify certain sequences of hyperplane arrangements with s n actions that satisfy cohomological representation stability here we vastly extend their results and define when a collection of arrangements is finitely generated using this notion we get stability results to general linear subspace arrangements not necessarily of hyperplanes a wide class of group actions replacing fi by a general category c we show that the cohomology of such collections of arrangements satisfies a strong form of representation stability with many concrete applications for example this implies that their betti numbers are always given by certain polynomials for this purpose we use the theory of representation stability for quite general classes of groups developed in a 17 we apply this theory to get classical cohomological stability of quotients of linear subspace arrangements with coefficients in certain constructible sheaves

existence of parameterized picard vessiot extensions over fields with algebraically closed constants abstract the purpose of this short note is to establish the existence of parameterized picard vessiot extensions for systems of linear difference differential equations over difference differential fields with an algebraically closed field of constants

galois theory for iterative connections and nonreduced galois groups this article presents a theory of modules with iterative connection this theory is a generalisation of the theory of modules with connection in characteristic zero as given by n katz to modules over rings of arbitrary characteristic we show that these modules with iterative connection and also the modules with integrable iterative connection form a tannakian category assuming some nice properties for the underlying ring and we show how this generalises to modules over schemes stratifications on modules as introduced by a grothendieck generalise the notion of integrable ordinary connections in another way however over smooth rings we obtain an equivalence of stratifications and integrable iterative connections furthermore over a regular ring in positive characteristic we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by d gieseker in the second part of this article we set up a picard vessiot theory for fields of solutions for such a picard vessiot extension we obtain a galois correspondence which takes into account even nonreduced closed subgroup schemes of the galois group scheme on one hand and inseparable intermediate extensions of the picard vessiot extension on the other hand finally we compare our galois theory with the galois theory for purely inseparable field extensions given by s chase

difference galois theory of linear differential equations abstract we develop a galois theory for linear differential equations equipped with the action of an endomorphism this theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation the galois groups here are linear difference algebraic groups i e matrix groups defined by algebraic difference equations

histogram of gabor phase patterns hgpp a novel object representation approach for face recognition a novel object descriptor histogram of gabor phase pattern hgpp is proposed for robust face recognition in hgpp the quadrant bit codes are first extracted from faces based on the gabor transformation global gabor phase pattern ggpp and local gabor phase pattern lgpp are then proposed to encode the phase variations ggpp captures the variations derived from the orientation changing of gabor wavelet at a given scale frequency while lgpp encodes the local neighborhood variations by using a novel local xor pattern lxp operator they are both divided into the nonoverlapping rectangular regions from which spatial histograms are extracted and concatenated into an extended histogram feature to represent the original image finally the recognition is performed by using the nearest neighbor classifier with histogram intersection as the similarity measurement the features of hgpp lie in two aspects 1 hgpp can describe the general face images robustly without the training procedure 2 hgpp encodes the gabor phase information while most previous face recognition methods exploit the gabor magnitude information in addition fisher separation criterion is further used to improve the performance of hgpp by weighing the subregions of the image according to their discriminative powers the proposed methods are successfully applied to face recognition and the experiment results on the large scale feret and cas peal databases show that the proposed algorithms significantly outperform other well known systems in terms of recognition rate

homological stability for configuration spaces of manifolds let cn m be the configuration space of n distinct ordered points in m we prove that if m is any connected orientable manifold closed or open the homology groups hi cn m are representation stable in the sense of church and farb arxiv 1008 1368 applying this to the trivial representation we obtain as a corollary that the unordered configuration space bn m satisfies classical homological stability for each i hi bn m hi bn 1 m for n i this improves on results of mcduff segal and others for open manifolds applied to closed manifolds this provides natural examples where rational homological stability holds even though integral homological stability fails

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds