distribution free consistent independence tests via center outward ranks and signs this article investigates the problem of testing independence of two random vectors of general dimensions for this we give for the first time a distribution free consistent test our approach com

fourier transforms and frobenius eigenvalues for finite coxeter groups abstract lusztig s classification of the unipotent characters of a finite chevalley or steinberg group involves a certain non abelian fourier transformation we construct analogous transformations for the suzuki and ree groups based on a set of axioms derived from lusztig s theory of character sheaves we also determine fourier matrices for the spetses in the sense of broue michel and the second author associated with twisted dihedral groups this completes the determination of fourier matrices for all spetses associated with finite coxeter groups we end by collecting common properties of these fourier matrices and the eigenvalues of frobenius of character sheaves and unipotent characters

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

categorification of induced cell modules and the rough structure of generalised verma modules this paper presents categorifications of right cell modules and induced cell modules for hecke algebras of finite weyl groups fit type a we show that these categorifications depend only on the isomorphism class of the cell module not on the cell itself our main application is multiplicity formulas for parabolically induced modules over a reductive lie algebra of type a which finally determines the so called rough structure of generalised verma modules on the way we present several categorification results and give a positive answer to kostant s problem from a joseph kostant s problem goldie rank and the gelfand kirillov conjecture invent math 56 3 1980 191 213 in many cases we also present a general setup of decategorification precategorification and categorification

minimal length elements in twisted conjugacy classes of finite coxeter groups let w be a finite coxeter group and let f be an automorphism of w that leaves the set of generators of w invariant we establish certain properties of elements of minimal length in the f conjugacy classes of w that allow us to define character tables for the corresponding twisted iwahori hecke algebras these results are extensions of results obtained by geck and pfeiffer in the case where f is trivial

consistent distribution free k sample and independence tests for univariate random variables a popular approach for testing if two univariate random variables are statistically independent consists of partitioning the sample space into bins and evaluating a test statistic on the binned data the partition size matters and the optimal partition size is data dependent while for detecting simple relationships coarse partitions may be best for detecting complex relationships a great gain in power can be achieved by considering finer partitions we suggest novel consistent distribution free tests that are based on summation or maximization aggregation of scores over all partitions of a fixed size we show that our test statistics based on summation can serve as good estimators of the mutual information moreover we suggest regularized tests that aggregate over all partition sizes and prove those are consistent too we provide polynomial time algorithms which are critical for computing the suggested test statistics efficiently we show that the power of the regularized tests is excellent compared to existing tests and almost as powerful as the tests based on the optimal yet unknown in practice partition size in simulations as well as on a real data example

measuring and testing dependence by correlation of distances distance correlation is a new measure of dependence between random vectors distance covariance and distance correlation are analogous to product moment covariance and correlation but unlike the classical definition of correlation distance correlation is zero only if the random vectors are independent the empirical distance dependence measures are based on certain euclidean distances between sample elements rather than sample moments yet have a compact representation analogous to the classical covariance and correlation asymptotic properties and applications in testing independence are discussed implementation of the test and monte carlo results are also presented

the exotic robinson schensted correspondence abstract we study the action of the symplectic group on pairs of a vector and a flag considering the irreducible components of the conormal variety we obtain an exotic analogue of the robinson schensted correspondence conjecturally the resulting cells are related to exotic character sheaves

minimal and maximal operator spaces and operator systems in entanglement theory abstract we examine k minimal and k maximal operator spaces and operator systems and investigate their relationships with the separability problem in quantum information theory we show that the matrix norms that define the k minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k positive linear maps and bound entanglement similarly we investigate the k super minimal and k super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k block positive operators and unnormalized states with schmidt number no greater than k respectively we characterize a class of norms on the k super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the schmidt number of a quantum state that generalizes the recently developed separability criterion based on trace contractive maps

minimal length elements of coxeter groups let w be a coxeter group and wj a finite parabolic subgroup we present here a geometric new proof of x he s result on minimal length elements of an orbit of wj which acts on w in a certain way

cones of positive maps and their duality relations the structure of cones of positive and k positive maps acting on a finite dimensional hilbert space is investigated special emphasis is given to their duality relations to the sets of superpositive and k superpositive maps we characterize k positive and k superpositive maps with regard to their properties under taking compositions a number of results obtained for maps are also rephrased for the corresponding cones of block positive k block positive separable and k entangled operators due to the jamiolkowski choi isomorphism generalizations to a situation where no such simple isomorphism is available are also made employing the idea of mapping cones as a side result to our discussion we show that extreme entanglement witnesses which are optimal should be of special interest in entanglement studies

hamiltonian reduction and nearby cycles for mirabolic d modules abstract we study holonomic d modules on sl n c c n called mirabolic modules analogous to lusztig s character sheaves we describe the supports of simple mirabolic modules we show that a mirabolic module is killed by the functor of hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus we introduce an analogue of verdier s specialization functor for representations of cherednik algebras which agrees on category o with the restriction functor of bezrukavnikov and etingof in type a we also consider a verdier specialization functor on mirabolic d modules we show that hamiltonian reduction intertwines specialization functors on mirabolic d modules with the corresponding functors on representations of the cherednik algebra this allows us to apply known hodge theoretic purity results for nearby cycles in the setting considered by bezrukavnikov and etingof

cells in affine weyl groups and tensor categories abstract in this paper we construct a tensor or monoidal category for any two sided cell in a finite or affine weyl group the isomorphism classes of simple objects of this category correspond to the elements of the two sided cell the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold which imitates geometrically the definition of the ring j given in g lusztig j algebra 109 1987 536 548 similarly we construct for any left cell a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse a right cell this contains as a special case the category of equivariant perverse sheaves on an affine grassmannian which as a consequence of g lusztig asterisque 101 102 1983 208 209 is a tensor category for convolution without truncation closely related to the tensor category of representations of the group dual in the sense of langlands to the group which gives rise to the affine grassmannian but our construction can be specialized in other ways and it gives a conjectural realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the langlands dual group

generalized springer correspondence and green functions for type b c graded hecke algebras this paper presents a combinatorial approach to the tempered representation theory of a graded hecke algebra h of classical type b or c with arbitrary parameters we present various combinatorial results which together give a uniform combinatorial description of what becomes the springer correspondence in the classical situation of equal parameters more precisely by using a general version of lusztig s symbols which describe the classical springer correspondence we associate to a discrete series representation of h with central character w0c a set w0c of w0 characters where w0 is the weyl group this set w0c is shown to parametrize the central characters of the generic algebra which specialize into w0c using the parabolic classification of the central characters of h t r on the one hand and a truncated induction of weyl group characters on the other hand we define a set w0c for any central character w0c of h t r and show that this property is preserved we show that in the equal parameter situation we retrieve the classical springer correspondence by considering a set u of partitions which replaces the unipotent classes of so2n 1 c and sp2n c and a bijection between u and the central characters of h t r we end with a conjecture which basically states that our generalized springer correspondence determines h t r exactly as the classical springer correspondence does in the equal label case in particular we conjecture that w0c indexes the modules in h t r with central character w0c in the following way a module m in h t r has a natural grading for the action of w0 and the w0 representation m in its top degree is irreducible when m runs through the modules in h t r with central character w0c m runs through w0c moreover still in analogy with the equal parameter case we conjecture that the w0 structure of the modules in h t r can be computed using generalized green functions

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

trivial intersection of blocks and nilpotent subgroups abstract let p q be different primes and suppose that the principal p and the principal q block of a finite group have only one irreducible complex character in common namely the trivial one we conjecture that this condition implies the existence of a nilpotent hall p q subgroup and prove that a minimal counter example must be an almost simple group where pq divides the order of its simple nonabelian normal subgroup as an immediate consequence we obtain that the conjecture holds true for p solvable or q solvable groups furthermore we prove the conjecture in case 2 p q using the classification theorem of finite simple groups finally we consider the situation that the intersection of an arbitrary p block with an arbitrary q block contains only one irreducible character

bounds on the distance between a unital quantum channel and the convex hull of unitary channels motivated by the recent resolution of asymptotic quantum birkhoff conjecture aqbc we attempt to estimate the distance between a given unital quantum channel and the convex hull of unitary channels we provide two lower bounds on this distance by employing techniques from quantum information and operator algebras respectively we then show how to apply these results to construct some explicit counterexamples to aqbc we also point out an interesting connection between the grothendieck s inequality and aqbc

the character sheaves on the group compactification abstract we give a definition of character sheaves on the group compactification which is equivalent to lusztig s definition in g lusztig parabolic character sheaves ii mosc math j 4 4 2004 869 896 we also prove some properties of the character sheaves on the group compactification

characterizations of character sheaves for complex reductive algebraic groups character sheaves are geometric objects associated to a reductive algebraic group they are closely related to irreducible characters of finite chevalley groups the existence of character sheaves was conjectured by g lusztig in the book characters of reductive groups over finite fields 1984 there he described what one should expect character sheaves to be for complex reductive groups a definition of character sheaves for an arbitrary characteristic quite different from this description was given by him later 1985 it is shown in this thesis in particular that the original 1984 lusztig s suggestion for describing character sheaves is equivalent to the later definition we use alternative characterizations of character sheaves for a connected complex reductive algebraic group g that are obtained in this thesis and also the one conjectured by g lusztig and g laumon that was obtained in 1988 by v ginsburg and independently by i mirkovic and k vilonen these characterizations involve microsupport and microlocalization of a sheaf a microsupport ss cal f of a complex cal f of sheaves on g is a closed conic subset of t sp g g times bf g it tells us about singularities of cal f a microlocalization mu sb m cal f of cal f along a submanifold m of g is a complex of sheaves on the conormal bundle t sbsp m g which allows us to analyze cal f on the infinitesimally small neighbourhood of m theorem let g be a complex connected reductive algebraic group cal f be a g equivariant for conjugation irreducible perverse sheaf on g let cal n be the nilpotent cone of the lie algebra g of g s may denote either g or the set of all semisimple elements of g the following are equivalent i cal f is a character sheaf ii ss cal f vert sb s subset s times cal n iii supp mu sb cal o cal f subset cal o times cal n where cal o subset s is a conjugacy class iv supp mu sb x cal f subset cal n x in s characterization ii when s g is the one given by ginsburg mirkovic and vilonen characterization iii with s being the set of semisimple elements was conjectured by j bernstein for arbitrary characteristic this kind of characterizations essentially simplifies the development and our understanding of the theory currently there is still no such description for positive characteristic an important intermediate result is the description of the microsupport of a doubc constructible sheaf cal f on a manifold x in terms of microlocalizations to the points of this manifold ss cal f overline bigcup limits sb x in x rm supp mu sb x cal f

a remark on the orthogonality relations for green functions let g be a connected linear algebraic group defined over the finite field f of characteristic p we denote by k an algebraic closure of f and by f the frobenius automorphism q h ay of k over if then f operates on g and its group g of fixed points is the finite group of 5 rational points of g now assume g to be reductive and let ti be the if variety of unipotent elements of g the green functions of gf are class functions q7 c defined on the set j of unipotent elements of g parametrized by a maximal torus t of g defined over f and depending only on the g iconjugacy class of t they are of importance for the representation theory of g in his fundamental paper 7 on the character theory of the finite groups gl f i a green first introduced such functions in a combinatorial guise subsequently various definitions of green functions for general reductive groups were given in 131 elaborated in 141 in 6 and most recently in 9 conjecturally all definitions should be equivalent at least if the characteristic p is good in the sense of 141 as far as i know this has not yet been established and the equivalence of the various definitions is known only under more severe restrictions on p a basic property of the green functions are their orthogonality relations which can be written as

remarks on computing irreducible characters 0 1 let g be a connected reductive algebraic group defined over a finite field fq and let g fq be the finite group of all fq rational points of g we would like to present here a strategy for computing the character table of g fq under the assumption that p the characteristic of fq is sufficiently large we can assume that g has a simply connected derived group indeed in the general case we can find a connected reductive group g with simply connected derived group and a surjective homomorphism f g g whose kernel is a central torus such that g and f are defined over fq then the character table

extremal generalized quantum measurements a measurement on a section k of the set of states of a finite dimensional c algebra is defined as an affine map from k to a probability simplex special cases of such sections are used in description of quantum networks in particular quantum channels measurements on a section correspond to equivalence classes of the so called generalized povms which are called quantum testers in the case of networks we find extremality conditions for measurements on k and characterize generalized povms such that the corresponding measurement is extremal these results are applied to the set of channels we find explicit extremality conditions for two outcome measurements on qubit channels and give an example of an extremal qubit 1 tester such that the corresponding measurement is not extremal

conditions for optimal input states for discrimination of quantum channels we find optimality conditions for testers in discrimination of quantum channels these conditions are obtained using semidefinite programming and are similar to optimality conditions for discrimination of quantum states we get a simple condition for existence of an optimal tester with any given input state with maximal schmidt rank in particular with a maximally entangled input state in the case when maximally entangled state is not optimal an upper bound on the optimal success probability is obtained the results for discrimination of two channels are applied to covariant channels qubit channels unitary channels and simple projective measurements

galois group action and jordan decomposition of characters of finite reductive groups with connected center abstract let g be a connected reductive group with connected center defined over f q with frobenius morphism f given an irreducible complex character of g f with its jordan decomposition and a galois automorphism gal q q we give the jordan decomposition of the image of under the action of on its character values

green polynomials of weyl groups elliptic pairings and the extended dirac index abstract in this paper we give a uniform construction of irreducible genuine characters of the pin cover w of a weyl group w and put them into the context of theory of springer representations in the process we provide a direct connection between springer theory via green polynomials the irreducible representations of w and an extended dirac operator for graded hecke algebras we also introduce a q elliptic pairing for w with respect to the reflection representation v these constructions are of independent interest the q elliptic pairing is a generalization of the elliptic pairing of w introduced by reeder and it is also related to s kato s notion of graded kostka systems for the semidirect product a w c w s v

fast computing for distance covariance distance covariance and distance correlation have been widely adopted in measuring dependence of a pair of random variables or random vectors if the computation of distance covariance and distance correlation is implemented directly accordingly to its definition then its computational complexity is o n2 which is a disadvantage compared to other faster methods in this article we show that the computation of distance covariance and distance correlation of real valued random variables can be implemented by an o nlog n algorithm and this is comparable to other computationally efficient algorithms the new formula we derive for an unbiased estimator for squared distance covariance turns out to be a u statistic this fact implies some nice asymptotic properties that were derived before via more complex methods we apply the fast computing algorithm to some synthetic data our work will make distance correlation applicable to a much wider class of problems a supplementary file to this article available on

on the number of irreducible real valued characters of a finite group abstract we prove that there exists an integer valued function f on positive integers such that if a finite group g has at most k real valued irreducible characters then g sol g f k where sol g denotes the largest solvable normal subgroup of g in the case k 5 we further classify g sol g this partly answers a question of iwasaki 15 on the relationship between the structure of a finite group and its number of real valued irreducible characters

local cohomology of modules of covariants abstract let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let w u be two finite dimensional representations of g in this paper we compute the local cohomology of u sw g provided a certain relatively weak technical condition is true

topological generation of exceptional algebraic groups abstract let g be a simple algebraic group over an algebraically closed field k and let c 1 c t be non central conjugacy classes in g in this paper we consider the problem of determining whether there exist g i c i such that g 1 g t is zariski dense in g first we establish a general result which shows that if is an irreducible subvariety of g t then the set of tuples in generating a dense subgroup of g is either empty or dense in in the special case c 1 c t by considering the dimensions of fixed point spaces we prove that this set is dense when g is an exceptional algebraic group and t 5 assuming k is not algebraic over a finite field in fact for g g 2 we only need t 4 and both of these bounds are best possible as an application we show that many faithful representations of exceptional algebraic groups are generically free we also establish new results on the topological generation of exceptional groups in the special case t 2 which have applications to random generation of finite exceptional groups of lie type in particular we prove a conjecture of liebeck and shalev on the random r s generation of exceptional groups

fisher exact scanning for dependency abstractwe introduce a method called fisher exact scanning fes for testing and identifying variable dependency that generalizes fisher s exact test on 2 2 contingency tables to r c contingency tables and continuous sample spaces fes proceeds through scanning over the sample space using windows in the form of 2 2 tables of various sizes and on each window completing a fisher s exact test based on a factorization of fisher s multivariate hypergeometric mhg likelihood into the product of the univariate hypergeometric likelihoods we show that there exists a coarse to fine sequential generative representation for the mhg model in the form of a bayesian network which in turn implies the mutual independence up to deviation due to discreteness among the fisher s exact tests completed under fes this allows an exact characterization of the joint null distribution of the p values and gives rise to an effective inference recipe through simple multiple testing procedures such as sidak and bonferroni

generalized green functions associated to complex reflection groups abstract in this paper we consider the set of r symbols in a full generality we construct hall littlewood functions and kostka functions associated to those r symbols we also discuss a multi parameter version of those functions we show that there exists a general algorithm of computing the multi parameter kostka functions as an application we show that the generalized green functions of symplectic groups can be described combinatorially in terms of our one parameter kostka functions

a consistent multivariate test of association based on ranks of distances summary we consider the problem of detecting associations between random vectors of any dimension few tests of independence exist that are consistent against all dependent alternatives we propose a powerful test that is applicable in all dimensions and consistent against all alternatives the test has a simple form is easy to implement and has good power

dynamical entanglement unlike the entanglement of quantum states very little is known about the entanglement of bipartite channels called dynamical entanglement here we work with the partial transpose of a superchannel and use it to define computable measures of dynamical entanglement such as the negativity we show that a version of it the max logarithmic negativity represents the exact asymptotic dynamical entanglement cost we discover a family of dynamical entanglement measures that provide necessary and sufficient conditions for bipartite channel simulation under local operations and classical communication and under operations with positive partial transpose

equitability mutual information and the maximal information coefficient how should one quantify the strength of association between two random variables without bias for relationships of a specific form despite its conceptual simplicity this notion of statistical equitability has yet to receive a definitive mathematical formalization here we argue that equitability is properly formalized by a self consistency condition closely related to data processing inequality mutual information a fundamental quantity in information theory is shown to satisfy this equitability criterion these findings are at odds with the recent work of reshef et al reshef dn et al 2011 science 334 6062 1518 1524 which proposed an alternative definition of equitability and introduced a new statistic the maximal information coefficient mic said to satisfy equitability in contradistinction to mutual information these conclusions however were supported only with limited simulation evidence not with mathematical arguments upon revisiting these claims we prove that the mathematical definition of equitability proposed by reshef et al cannot be satisfied by any nontrivial dependence measure we also identify artifacts in the reported simulation evidence when these artifacts are removed estimates of mutual information are found to be more equitable than estimates of mic mutual information is also observed to have consistently higher statistical power than mic we conclude that estimating mutual information provides a natural and often practical way to equitably quantify statistical associations in large datasets

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

nilpotent orbits in classical lie algebras over f2n and the springer correspondence we give the number of nilpotent orbits in the lie algebras of orthogonal groups under the adjoint action of the groups over f2n let g be an adjoint algebraic group of type b c or d defined over an algebraically closed field of characteristic 2 we construct the springer correspondence for the nilpotent variety in the lie algebra of g

character d modules via drinfeld center of harish chandra bimodules the category of character d modules is realized as drinfeld center of the abelian monoidal category of harish chandra bimodules tensor product of harish chandra bimodules is related to convolution of d modules via the long intertwining functor radon transform by a result of beilinson and ginzburg represent theory 3 1 31 1999 exactness property of the long intertwining functor on a cell subquotient of the harish chandra bimodules category shows that the truncated convolution category of lusztig adv math 129 85 98 1997 can be realized as a subquotient of the category of harish chandra bimodules together with the description of the truncated convolution category bezrukavnikov et al in isr j math 170 207 234 2009 this allows us to derive under a mild technical assumption a classification of irreducible character sheaves over obtained by lusztig by a different method

modular generalized springer correspondence iii exceptional groups we complete the construction of the modular generalized springer correspondence for an arbitrary connected reductive group with a uniform proof of the disjointness of induction series that avoids the case by case arguments for classical groups used in previous papers in the series we show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the sylow subgroups of the weyl group under some assumptions we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character we also provide tables and other information on the modular generalized springer correspondence for quasi simple groups of exceptional type including a complete classification of cuspidal pairs in the case of good characteristic and a full determination of the correspondence in type g 2

on a measure of distance for quantum strategies the present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem a full formal proof of strong duality is presented for the semidefinite optimization problem in question this norm and its properties are employed to generalize a state discrimination result of gutoski and watrous in proceedings of the 22nd symposium on theoretical aspects of computer science stacs 05 lecture notes in computer science vol 3404 springer 2005 pp 605 616 the generalized result states that for any two convex sets s0 s1 of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of s0 s0 from any choice of s1 s1 with bias proportional to the minim

multiscale likelihood analysis and complexity penalized estimation we describe here a framework for a certain class of multiscale likelihood factorizations wherein in analogy to a wavelet decomposition of an l 2 function a given likelihood function has an alternative representation as a product of conditional densities reflecting information in both the data and the parameter vector localized in position and scale the framework is developed as a set of sufficient conditions for the existence of such factorizations formulated in analogy to those underlying a standard multiresolution analysis for wavelets and hence can be viewed as a multiresolution analysis for likelihoods we then consider the use of these factorizations in the task of nonparametric complexity penalized likelihood estimation we study the risk properties of certain thresholding and partitioning estimators and demonstrate their adaptivity and near optimality in a minimax sense over a broad range of function spaces based on squared hellinger distance as a loss function in particular our results provide an illustration of how properties of classical wavelet based estimators can be obtained in a single unified framework that includes models for continuous count and categorical data types

optimal and fast detection of spatial clusters with scan statistics we consider the detection of multivariate spatial clusters in the bernoulli model with n locations where the design distribution has weakly dependent marginals the locations are scanned with a rectangular window with sides parallel to the axes and with varying sizes and aspect ratios multivariate scan statistics pose a statistical problem due to the multiple testing over many scan windows as well as a computational problem because statistics have to be evaluated on many windows this paper introduces methodology that leads to both statistically optimal inference and computationally efficient algorithms the main difference to the traditional calibration of scan statistics is the concept of grouping scan windows according to their sizes and then applying different critical values to different groups it is shown that this calibration of the scan statistic results in optimal inference for spatial clusters on both small scales and on large scales as well as in the case where the cluster lives on one of the marginals methodology is introduced that allows for an efficient approximation of the set of all rectangles while still guaranteeing the statistical optimality results described above it is shown that the resulting scan statistic has a computational complexity that is almost linear in n

process tomography for unitary quantum channels we study the number of measurements required for quantum process tomography under prior information such as a promise that the unknown channel is unitary we introduce the notion of an interactive observable and we show that any unitary channel acting on a d level quantum system can be uniquely identified among all other channels unitary or otherwise with only o d2 interactive observables as opposed to the o d4 required for tomography of arbitrary channels this result generalizes to the problem of identifying channels with at most q kraus operators and slight improvements can be obtained if we wish to identify such a channel only among unital channels or among other channels with q kraus operators these results are proven via explicit construction of large subspaces of hermitian matrices with various conditions on rank eigenvalues and partial trace our constructions are built upon various forms of totally nonsingular matrices

sp2n fq2 invariants in irreducible unipotent representations of sp4n fq abstract we show that for any irreducible representation of sp 4 n f q the subspace of all its sp 2 n f q 2 invariants is at most one dimensional in terms of lusztig symbols we give a complete list of irreducible unipotent representations of sp 4 n f q which have a non zero sp 2 n f q 2 invariant and in particular we prove that every irreducible unipotent cuspidal representation has a one dimensional subspace of sp 2 n f q 2 invariants as an application we give an elementary proof of the fact that the unipotent cuspidal representation is defined over q which was proved by lusztig in 12

legendrian knots and constructible sheaves we study the unwrapped fukaya category of lagrangian branes ending on a legendrian knot our knots live at contact infinity in the cotangent bundle of a surface the fukaya category of which is equivalent to the category of constructible sheaves on the surface itself consequently our category can be described as constructible sheaves with singular support controlled by the front projection of the knot we use a theorem of guillermou kashiwara schapira to show that the resulting category is invariant under legendrian isotopies a subsequent article establishes its equivalence to a category of representations of the chekanov eliashberg differential graded algebra we also find two connections to topological knot theory first drawing a positive braid closure on the annulus the moduli space of rank n objects maps to the space of local systems on a circle the second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the colored by n triply graded khovanov rozansky homology second drawing a positive braid closure in the plane the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in a of the homfly polynomial of the braid closure

how to compute the frobenius schur indicator of a unipotent character of a finite coxeter system abstract for each finite irreducible coxeter system w s lusztig has associated a set of unipotent characters uch w there is also a notion of a fourier transform on the space of functions uch w r due to lusztig for weyl groups and to broue lusztig and malle in the remaining cases this paper concerns a certain w representation w in the vector space generated by the involutions of w our main result is to show that the irreducible multiplicities of w are given by the fourier transform of a unique function uch w 1 0 1 which for various reasons serves naturally as a heuristic definition of the frobenius schur indicator on uch w the formula we obtain for extends prior work of casselman kottwitz lusztig and vogan addressing the case in which w is a weyl group we include in addition a succinct description of the irreducible decomposition of w derived by kottwitz when w s is classical and prove that w defines a gelfand model if and only if w s has type a n h 3 or i 2 m with m odd we show finally that a conjecture of kottwitz connecting the decomposition of w to the left cells of w holds in all non crystallographic types and observe that a weaker form of kottwitz s conjecture holds in general in giving these results we carefully survey the construction and notable properties of the set uch w and its attached fourier transform

base norms and discrimination of generalized quantum channels we introduce and study norms in the space of hermitian matrices obtained from base norms in positively generated subspaces these norms are closely related to discrimination of so called generalized quantum channels including quantum states channels and networks we further introduce generalized quantum decision problems and show that the maximal average payoffs of decision procedures are again given by these norms we also study optimality of decision procedures in particular we obtain a necessary and sufficient condition under which an optimal 1 tester for discrimination of quantum channels exists such that the input state is maximally entangled