the sharp hardy uncertainty principle for schrodinger evolutions we give a new proof of hardy s uncertainty principle up to the end point case which is only based on calculus the method allows us to ex tend hardy s uncertainty principle to schrodinger equations with non constant coefficients we also deduce optimal gaussian decay bounds for solutions to these schrodinger equations

matrices characters and descents abstract a new family of asymmetric matrices of walsh hadamard type is introduced we study their properties and in particular compute their determinants and discuss their eigenvalues the invertibility of these matrices implies that certain character formulas are invertible yielding expressions for the cardinalities of sets of combinatorial objects with prescribed descent sets in terms of character values of the symmetric group

primitive ideals and orbital integrals in complex exceptional groups in 2 two related problems were studied for complex classical groups determination of the primitive spectrum of the enveloping algebra as a set and fourier inversion of unipotent orbital integrals the first of these was solved completely extending work of joseph and others in sl n c and small groups although our techniques presumably solve the second as well up to determination of normalizing constants we carried out the calculations only for special unipotent classes see 181 we had of course tried to apply the same methods to the exceptional groups but they seem to be inadequate since that work was done however brylinski and kashiwara and beilinson and bernstein have established the kazhdan lusztig conjecture giving character formulas for irreducible highest weight modules 3 71 because of a conjecture of joseph proved in 24 this determines in principle the primitive ideals with a fixed regular integral infinitesimal character they are in one to one correspondence with what kazhdan and lusztig call left cells in the weyl group however the algorithm given by kazhdan and lusztig to compute these cells is enormously complicated requiring one to compute roughly 1 i polynomials of degrees on the order of half the number of positive roots this is very unsatisfactory however it

uniqueness properties of solutions of schrodinger equations under suitable assumptions on the potentials v and a we prove that if u c 0 1 h1 is a solution of the linear schrodinger equation i t x u vu a xu on rd 0 1 and if u 0 in x r 0 1 for some r 0 then u 0 in rd 0 1 as a consequence we obtain uniqueness properties of solutions of nonlinear schrodinger equations of the form i t x u g x t u u xu xu on rd 0 1 where g is a suitable nonlinear term the main ingredient in our proof is a carleman inequality of the form e x1 v lx2lt2 e x1 xv bx 2lt2 c e x1 i t x v bx1 2lt2 for any v c r h1 with v t 0 for t 0 1 in this inequality bx 2 and bx1 2 are banach spaces of functions on rd and e x1 is a suitable weight

hardy uncertainty principle and unique continuation properties of covariant schrodinger flows abstract we prove a logarithmic convexity result for exponentially weighted l 2 norms of solutions to electromagnetic schrodinger equation without needing to assume smallness of the magnetic potential as a consequence we can prove a unique continuation result in the style of the hardy uncertainty principle which generalizes the analogous theorems which have been recently proved by escauriaza kenig ponce and vega

induction and restriction of kazhdan lusztig cells abstract barbasch and vogan gave a beautiful rule for restricting and inducing kazhdan lusztig representations of weyl groups in this paper we show that this rule implies and generalizes the littlewood richardson rule for decomposing outer products of representations of the symmetric groups a new recursive rule for computing characters of arbitrary coxeter groups follows another application is a generalization of garsia remmel s algorithm for decomposing certain tensor products of symmetric groups representations

fourier series and the special values of l functions let k be a number field with ring of integers a let x be an idele class character for k which is trivial at each infinite place the l series l s x c x b nb is defined by the sum over all non trivial ideals b of a which are prime to the conductor f of x this sum is absolutely convergent for re s 1 when x is non trivial l s x extends analytically to a neighborhood of s 1 and jc 1 x is o although this value is of great arithmetic interest truly satisfactory formulas for l 1 x have only been obtained in the cases when k q and when k is imaginary quadratic where they are due to dirichlet and kronecker respectively these number fields are precisely the one with a unique archimedean completion k in this case a is a discrete co compact subring of a locally compact field k an analogous situation exists for global function fields k over finite fields let oc be a place of k and let a denote the ring of functions which are regular outside co then a is a discrete co compact subring of the completion k which is a locally compact field if x is an idele class character for k which is trivial at cc and we define l s x as above then for x 1 the series is regular and non zero at s 1 formulas for l 1 x were obtained by galovich and rosen in the case when a is isomorphic to the ring of polynomials over a finite field and by hayes in general in this paper we will reprove all the limit formulas for l 1 x in a uniform manner using fourier analysis on the locally compact field k we shall also obtain the functional equation of l s x following the

on unique continuation for the generalized schrodinger equations abstract in this paper we study the unique continuation properties of solutions of the generalized schrodinger equations we prove the carleman inequalities as well as the exponential decay weighted estimates as a consequence we obtain sufficient conditions on the behavior of the solution u at two different times t 0 0 and t 1 1 which guarantee that u 0 is the unique solution

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

decentralized dimensionality reduction for distributed tensor data across sensor networks this paper develops a novel decentralized dimensionality reduction algorithm for the distributed tensor data across sensor networks the main contributions of this paper are as follows first conventional centralized methods which utilize entire data to simultaneously determine all the vectors of the projection matrix along each tensor mode are not suitable for the network environment here we relax the simultaneous processing manner into the one vector by one vector ovbov manner i e determining the projection vectors pvs related to each tensor mode one by one second we prove that in the ovbov manner each pv can be determined without modifying any tensor data which simplifies corresponding computations third we cast the decentralized pv determination problem as a set of subproblems with consensus constraints so that it can be solved in the network environment only by local computations and information communications among neighboring nodes fourth we introduce the null space and transform the pv determination problem with complex orthogonality constraints into an equivalent hidden convex one without any orthogonality constraint which can be solved by the lagrange multiplier method finally experimental results are given to show that the proposed algorithm is an effective dimensionality reduction scheme for the distributed tensor data across the sensor networks

image fusion using higher order singular value decomposition a novel higher order singular value decomposition hosvd based image fusion algorithm is proposed the key points are given as follows 1 since image fusion depends on local information of source images the proposed algorithm picks out informative image patches of source images to constitute the fused image by processing the divided subtensors rather than the whole tensor 2 the sum of absolute values of the coefficients savc from hosvd of subtensors is employed for activity level measurement to evaluate the quality of the related image patch and 3 a novel sigmoid function like coefficient combining scheme is applied to construct the fused result experimental results show that the proposed algorithm is an alternative image fusion approach

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

an algorithm for characters of hecke algebras hn q of type an 1 recently the characters of irreducible representations of the hecke algebra hn q of type an 1 were identified with the transition coefficients relating q generalized power sum symmetric functions to schur functions king and wybourne 1990 using this result one obtains an easy combinatorial algorithm for calculating the characters

unique continuation properties of the higher order nonlinear schrodinger equations in one dimension abstract in this paper we study the unique continuation properties of the higher order nonlinear schrodinger equations in one dimension and show exponential decay weighted estimates as well as a l p type carleman estimate based on the littlewood paley theory as a consequence we obtain that if u is a solution of the schrodinger equation such that there exists x 0 with supp u 0 supp u 1 x 0 or x 0 then u 0 in r 0 1

hecke algebra actions on the coinvariant algebra abstract two actions of the hecke algebra of type a on the corresponding polynomial ring are studied both are deformations of the natural action of the symmetric group on polynomials and keep symmetric functions invariant we give an explicit description of these actions and deduce a combinatorial formula for the resulting graded characters on the coinvariant algebra

gaussian decay of harmonic oscillators and related models abstract we prove that the decay of the eigenfunctions of harmonic oscillators uniform electric or magnetic fields is not stable under 0 order complex perturbations even if bounded of these hamiltonians in the sense that we can produce solutions to the evolutionary schrodinger flows associated to the hamiltonians with a stronger gaussian decay at two distinct times we then characterize in a quantitative way the sharpest possible gaussian decay of solutions as a function of the oscillation frequency or the strength of the field depending on the hamiltonian which is considered this is connected to the hardy s uncertainty principle for free schrodinger evolutions

the ivp for a higher dimensional version of the benjamin ono equation in weighted sobolev spaces abstract we study the initial value problem associated to a higher dimensional version of the benjamin ono equation our purpose is to establish local well posedness results in weighted sobolev spaces and to determinate according to them some sharp unique continuation properties of the solution flow in consequence optimal decay rate for this model is determined a key ingredient is the deduction of a new commutator estimate involving riesz transforms

a generalization of the littlewood richardson rule for the general linear group gz the littlewood richardson rule see 18 193 gives a method to calculate the decomposition of a tensor product of two irreducible g representations the aim of this article is to give a generalization of this rule for all simple simply connected algebraic groups of type a b c d g2 and e we obtain also partial results for g of type f4 e and es the restrictions in the last three cases come from the fact that for the formulation of the decomposition rules we need the notion of a standard young tableau such a notion has been developed by seshadri lakshmibai musili and rajeswari in a series of articles see ll 13 14 16 but not yet for all representations of the last three exceptional groups the advantage of the notion of a young tableau developed by seshadri et al is that it is independent of the type of the group for the convenience of the reader not used to this notion we give first a seperate proof for g sz using the classical notion of a standard young tableau of course for applications the classical notion is much more appropriate in the appendix we give a translation of the notion of a standard young tableau in the sense of seshadri et al into the classical notion of a young tableau for g sp2 and spin in 3 8 we give also such a translation for g g for the other exceptional groups such a translation is not known to the author let p be a dominant weight for g and denote by i the corresponding simple g module we assume that the character of vp is given by ct eycr where the sum has to be taken over all standard young tableaux f of shape p p and v 5 is the weight of the tableau y now by the

a combinatorial formula for kazhdan lusztig polynomials in their fundamental paper 14 kazhdan and lusztig defined for every coxeter group w a family of polynomials indexed by pairs of elements of w which have become known as the kazhdan lusztig polynomials of w see e g 12 chap 7 these polynomials are intimately related to the bruhat order of w and to the algebraic geometry of schubert varieties and have proven to be of fundamental importance in representation theory our aim in this work is to give a simple nonrecursive combinatorial formula for any kazhdan lusztig polynomial of any coxeter group theorem 4 1 and to study some consequences of it the main idea involved in the proof and statement of this formula is that of extending the concept of the rpolynomial see e g 12 w to any finite multichain of w so that for multichains of length 1 one obtains apart from sign the usual r polynomials once this has been done then the kazhdan lusztig polynomial of a pair u v turns out to be just the sum over all multichains from u to v of the corresponding generalized r polynomials the r polynomial of a multichain can be readily defined and computed from the ordinary r polynomials see 9 10 and 11 since several combinatorial formulas and interpretations are known for these polynomials see e g 5 10 and for the case of symmetric groups 3 and simple recurrences exist for them we feel that this formula is a significant step forward in the understanding of the kazhdanlusztig polynomials though combinatorial formulas for kazhdan lusztig polynomials have appeared before in the literature see e g 16 21 8 6 none of them hold in complete generality

a frobenius formula for the characters of the hecke algebras this paper uses the theory of quantum groups and the quantum yang baxter equation as a guide in order to produce a method of computing the irreducible characters of the hecke algebra this approach is motivated by an observation of m jimbo giving a representation of the hecke algebra on tensor space which generates the full centralizer of a tensor power of the standard representation of the quantum group u q mathfrak s l n by rewriting the solutions of the quantum yang baxter equation for u q mathfrak s l n in a different form one can avoid the quantum group completely and produce a frobenius formula for the characters of the hecke algebra by elementary methods using this formula we derive a combinatorial rule for computing the irreducible characters of the hecke algebra this combinatorial rule is aq extension of the murnaghan nakayama for computing the irreducible characters of the symmetric group along the way one finds connections apparently unexplored between the irreducible characters of the hecke algebra and hall littlewood symmetric functions and kronecker products of symmetric groups

combinatorial gelfand models abstract a combinatorial construction of a gelfand model for the symmetric group and its iwahori hecke algebra is presented

unique continuation for schrodinger evolutions with applications to profiles of concentration and traveling waves we prove unique continuation properties for solutions of the evolution schrodinger equation with time dependent potentials as an application of our method we also obtain results concerning the possible concentration profiles of blow up solutions and the possible profiles of the traveling waves solutions of semi linear schrodinger equations

uniqueness properties for discrete equations and carleman estimates abstract using carleman estimates we give a lower bound for solutions to the discrete schrodinger equation in both dynamic and stationary settings that allows us to prove uniqueness results under some assumptions on the decay of the solutions

uniqueness properties of solutions to schrodinger equations nx j 1 2 u x 2 x 0 harmonic functions in the unit ball x r n x 2 since harmonic functions are still real analytic in x r n x 1 in fact it is well known that if p x d is a linear elliptic differential operator with real analytic coefficients andp x d u 0 in a open set r n then u is real analytic in hence the s u c p also holds for such solutions through the work of hadamard 28 on the uniqueness of the cauchy problem which is closely related to the strong unique continuation property discussed earlier it became clear for applications in nonlinear problems that it would be desirable to establish the strong unique continuation property for operators whose coefficients are not necessarily real analytic oreven c the first results in this direction were found in the pioneering work of carleman 9 when

hardy uncertainty principle convexity and parabolic evolutions we give a new proof of the l2 version of hardy s uncertainty principle based on calculus and on its dynamical version for the heat equation the reasonings rely on new log convexity properties and the derivation of optimal gaussian decay bounds for solutions to the heat equation with gaussian decay at a future time we extend the result to heat equations with lower order variable coefficient

representations and traces of the hecke algebras hn q of type an 1 the notion of the connectivity class of minimal words in the algebra hn q is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented guided by these results the connection between the ocneanu trace on hn q and schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of hn q they appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to schur functions their evaluation involves the use of the littlewood richardson rule which is proved to apply to hn q just as it does to sn both representation matrices and characters are tabulated

a recursive rule for kazhdan lusztig characters abstract a formula for computing kazhdan lusztig characters in hecke algebras of coxeter groups is given this formula holds for products of coxeter elements of parabolic subgroups in an arbitrary coxeter group where the parabolic subgroups are of type a and the kazhdan lusztig representations are of finite dimension the formula yields new combinatorial interpretations of murnaghan nakayama type to these characters and to other related characters

the ivp for the benjamin ono equation in weighted sobolev spaces abstract in this work we continue our study initiated in fonseca and ponce 2011 11 on the uniqueness properties of real solutions to the ivp associated to the benjamin ono bo equation in particular we shall show that the uniqueness results established in fonseca and ponce 2011 11 do not extend to any pair of non vanishing solutions of the bo equation also we shall prove that the uniqueness result established in fonseca and ponce 2011 11 under a hypothesis involving information of the solution at three different times cannot be relaxed to two different times

distributed dictionary learning for sparse representation in sensor networks this paper develops a distributed dictionary learning algorithm for sparse representation of the data distributed across nodes of sensor networks where the sensitive or private data are stored or there is no fusion center or there exists a big data application the main contributions of this paper are 1 we decouple the combined dictionary atom update and nonzero coefficient revision procedure into two stage operations to facilitate distributed computations first updating the dictionary atom in terms of the eigenvalue decomposition of the sum of the residual correlation matrices across the nodes then implementing a local projection operation to obtain the related representation coefficients for each node 2 we cast the aforementioned atom update problem as a set of decentralized optimization subproblems with consensus constraints then we simplify the multiplier update for the symmetry undirected graphs in sensor networks and minimize the separable subproblems to attain the consistent estimates iteratively and 3 dictionary atoms are typically constrained to be of unit norm in order to avoid the scaling ambiguity we efficiently solve the resultant hidden convex subproblems by determining the optimal lagrange multiplier some experiments are given to show that the proposed algorithm is an alternative distributed dictionary learning approach and is suitable for the sensor network environment

murnaghan nakayama rules for characters of iwahori hecke algebras of classical type in this paper we give murnaghan nakayama type formulas for computing the irreducible characters of the iwahori hecke algebras of types an 1 bn and dn our method is a generalization of a derivation of the murnaghan nakayama formula for the irreducible characters of the symmetric group given by curtis greene greene s approach is to sum up the diagonal entries of the matrices of certain cycle permutations in young s seminormal representations the analogues of the young seminormal representations for the iwahori hecke algebras of types an 1 bn and dn were given by hoefsmit