paley wiener spaces with vanishing conditions and painleve vi transcendents abstract we modify the classical paley wiener spaces pw x of entire functions of finite exponential type at most x 0 which are square integrable on the real line via the additional condition of vanishing at finitely many complex points z 1 z n we compute the reproducing kernels and relate their variations with respect to x to a krein differential system whose coefficient which we call the function and solutions have determinantal expressions arguments specific to the case where the trivial zeros z 1 z n are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non linear first order differential equations a computation having this non linear system at his start obtains quasi algebraic and among them rational painleve transcendents of the sixth kind as certain quotients of such functions

on the decomposition of tensor products of the representations of the classical groups by means of the universal characters on definit les caracteres universels d une representation rationnelle de gl n c et on donne les formules de decomposition des produits tensoriels entre eux en termes de coefficients de littlewood richardson on demontre les formules de decomposition du produit tensoriel de deux representations irreductibles de sp 2n c et de so n c

the sampling window corresp the concept of the sampling window is introduced for the central interpolation of finite energy band limited functions the sampling window does not increase the rate of convergence of the truncation error series as do various convergence factors but does significantly reduce the truncation error bound

an approach to the branching rule from sl2n c to sp2n c via littelmann s path model abstract we propose a conjecture describing the branching rule in terms of littelmann s path model from the special linear lie algebra sl 2 n c of type a 2 n 1 to the symplectic lie algebra of type c n embedded as the fixed point subalgebra of the diagram automorphism of sl 2 n c moreover we prove the conjecture in certain cases and also provide some supporting examples in addition we show that the branching coefficients can be obtained explicitly by using the inverse kostka matrix and path models for tensor products of symmetric powers of the defining or natural representation c 2 n of sl 2 n c

the truncation error in the application of sampling series to electromagnetic problems truncation error interpolation methods used in the evaluation and representation of scattered fields are discussed three main points are considered the choice of the sampling point lattice the choice of the type of sampling expansion and the determination of the number of terms of the sampling series that must be retained to insure a negligible truncation error the aim is to provide a clear reference frame for the applications which allow the optimal choice for the reconstruction algorithm the cases of bounded and square integrable band limited functions are separately dealt with in order to exploit the properties of each method a general analysis of the multidimensional truncation error is carried out no particular assumption on the sampling expansion is made except for that concerning the use of a rectangular sampling lattice simple and effective expressions relating the multidimensional error to the corresponding one dimensional bounds are derived the one dimensional error is then analyzed in detail with reference to the most relevant case of central interpolation by very simple methods error bounds for all the sampling expansions explicitly considered in the literature are derived in a systematic way expressions providing the values of the sampling rate and number of retained samples which minimize the overall computer time and memory requirement are derived some numerical examples are presented and briefly discussed

existence and stability of almost periodic solution for bam neural networks with delays by using the banach fixed point theorem and constructing suitable lyapunov function some sufficient conditions are obtained ensuring existence uniqueness and global stability of almost periodic solution of the bam neural networks with variable coefficients and delays these results are helpful to design global exponential stable bam networks and almost periodic oscillatory bam networks

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

on the averages of characteristic polynomials from classical groups we provide an elementary and self contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to weyl and littlewood

applications of minor summation formula i littlewood s formulas abstract the first and the third authors obtained a minor summation formula of pfaffian which expresses a weighted sum of minors of any rectangular matrix in terms of a single pfaffian in this paper as an application of this minor summation formula we give a new proof of littlewood s formulas with additional parameters and generalize them for the classical groups

the parafermion fock space and explicit mathfrak so 2n 1 representations the defining relations triple relations of n pairs of parafermion operators f j j 1 n are known to coincide with a set of defining relations for the lie algebra in terms of 2n generators with the common hermiticity conditions this means that the parafermions of order p correspond to a finite dimensional unitary irreducible representation w p of with highest weight although the dimension and the character of w p is known by classical formulae there is no explicit basis of w p available in which the parafermion operators have a natural action in this paper we construct an orthogonal basis for w p and present the explicit actions of the parafermion generators on these basis vectors we use group theoretical techniques in which the subalgebra of plays a crucial role a set of gelfand zetlin patterns of will be used to label the basis vectors of w p and also in the explicit action matrix elements certain clebsch gordan coefficients are essential

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

optimal reconstruction of bandlimited bounded signals a formula is presented for restoring bounded bandlimited signals from a finite number of samples the restoration is linear in the samples used and the error bound of the method is the theoretical minimum

the parafermion fock space and explicit so 2n 1 representations the defining relations triple relations of n pairs of parafermion operators f j j 1 n are known to coincide with a set of defining relations for the lie algebra so 2n 1 in terms of 2n generators with the common hermiticity conditions this means that the parafermions of order p correspond to a finite dimensional unitary irreducible representation w p of so 2n 1 with highest weight p 2 p 2 p 2 although the dimension and the character of w p is known by classical formulae there is no explicit basis of w p available in which the parafermion operators have a natural action in this paper we construct an orthogonal basis for w p and present the explicit actions of the parafermion generators on these basis vectors we use group theoretical techniques in which the u n subalgebra of so 2n 1 plays a crucial role a set of gelfand zetlin patterns of u n will be used to label the basis vectors of w p and also in the explicit action matrix elements certain u n clebsch gordan coefficients are essential

on the degrees of freedom of scattered fields starting from the observation that fields differing less than a prescribed error cannot be resolved as distinct entities the degrees of freedom of the scattered field are introduced and then computed the degrees of freedom are shown to be practically equal to the nyquist number appropriate to the effective spatial bandwidth of the scattered field and to the extension of the observation domain accordingly a finite number of elements of information can be used to determine the scattered field to a prescribed approximation error it is also shown that the field representation can be made in terms of field values and simple sampling functions provided that a marginal increase in the approximation error is tolerated the results not only completely justify the use of sampling interpolation for representing scattering fields but also demonstrate that such representation is practically an optimal one an algorithm for the reconstruction of scattered fields given the maximum allowed error is then produced

vicious walkers friendly walkers and young tableaux ii with a wall we derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of young tableaux and combinatorial descriptions of symmetric functions for the problem of n friendly walkers we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall

diagonally and antidiagonally symmetric alternating sign matrices of odd order we study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices dasasms of fixed odd order by introducing a case of the six vertex model whose configurations are in bijection with such matrices the model involves a grid graph on a triangle with bulk and boundary weights which satisfy the yang baxter and reflection equations we obtain a general expression for the partition function of this model as a sum of two determinantal terms and show that at a certain point each of these terms reduces to a schur function we are then able to prove a conjecture of robbins from the mid 1980 s that the total number of 2n 1 x 2n 1 dasasms is prod i 0 n 3i n i and a conjecture of stroganov from 2008 that the ratio between the numbers of 2n 1 x 2n 1 dasasms with central entry 1 and 1 is n n 1 among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980 s that for odd order dasasms is the last to have been proved

identities for classical group characters of nearly rectangular shape abstract we derive several identities that feature irreducible characters of the general linear the symplectic the orthogonal and the special orthogonal groups all of the identities feature characters that are indexed by shapes that are nearly rectangular by which we mean that the shapes are rectangles except for one row or column that might be shorter than the others as applications we prove new results in plane partitions and tableaux enumeration including new refinements of the bender knuth and macmahon ex conjectures

new schur function series this paper presents some new product identities for certain summations of schur functions these identities are generalizations of some famous identities known to littlewood and appearing in macdonald s book we refer to these identities as the littlewood type formulas in addition analogues for summations of characters of the other classical groups are also given the littlewood type formulas in this paper are separated into two classes the rational schur function series and the generalized schur function series an application of a rational schur function series to the infinite product representation of the elliptic theta functions is also given we prove these littlewood type formulas using the cauchy binet formula the cauchy binet formula is a basic but powerful tool applicable in the present context which can be derived from our pfaffian formula as we explain q 1998 academic press

fast and accurate near field far field transformation by sampling interpolation of plane polar measurements an optimal sampling interpolation algorithm which allows the accurate recovery of plane rectangular near field samples from the knowledge of the plane polar ones is developed this enables the standard near field far field nf ff transformation which takes full advantage of the fast fourier transform fft algorithm to be applied to plane polar scanning the maximum allowable sample spacing is also rigorously derived and it is shown that it can be significantly greater than lambda 2 as the measurement place moves away from the source this allows a remarkable reduction of both measurement time and memory storage requirements the sampling approach is compared with that based on the bivariate lagrange interpolation bli method the sampling reconstruction agrees with the exact results significantly better than the bli in spite of the significantly lower number of required measurements

an elliptic generalization of schur s pfaffian identity we present a pfaffian identity involving elliptic functions whose rational limit gives a generalization of schur s pfaffian identity for pf xj xi xj xi this identity can be regarded as the pfaffian counterpart of frobenius generalization of cauchy s determinant identity for det 1 xi xj