three friendly walkers more than 15 years ago guttmann and voge 2002 j stat plan inference 101 107 introduced a model of friendly walkers since then it has remained unsolved in this paper we provide the exact solution to a closely allied model which essentially only differs in the boundary conditions the exact solution is expressed in terms of the reciprocal of the generating function for vicious walkers which is a d finite function however ratios of d finite functions are inherently not d finite and in this case we prove that the friendly walkers generating function is the solution to a non linear differential equation with polynomial coefficients it is in other words d algebraic we find using numerically exact calculations a conjectured expression for the generating function of the original model as a ratio of a d finite function and the generating function for vicious walkers we obtain an expression for this d finite function in terms of a hypergeometric function with a rational pullback and its first and second derivatives

stationary half space last passage percolation in this paper we study stationary last passage percolation with exponential weights and in half space geometry we determine the limiting distribution of the last passage time in a critical window close to the origin the result is a new two parameter family of distributions one parameter for the strength of the diagonal bounding the half space strength of the source at the origin in the equivalent tasep language and the other for the distance of the point of observation from the origin it should be compared with the one parameter family giving the baik rains distributions for full space geometry we finally show that far enough away from the characteristic line our distributions indeed converge to the baik rains family we derive our results using a related inhomogeneous integrable model having pfaffian correlations together with careful analytic continuation and steepest descent analysis

exit problems associated with affine reflection groups we obtain a formula for the distribution of the first exit time of brownian motion from the alcove of an affine weyl group in most cases the formula is expressed compactly in terms of pfaffians expected exit times are derived in the type widetilde a case the results extend to other markov processes we also give formulas for the real eigenfunctions of the dirichlet and neumann laplacians on alcoves observing that the hot spots conjecture of j rauch is true for alcoves

flattening and analytic continuation of affinoid morphisms remarks on a paper of gardener and schoutens we give an example of an affinoid curve without analytic continuation we use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow ups thus the global rigid analogue of hironaka s complex analytic flattening theorem given by t gardener and h schoutens in theorem 2 3 of flattening and subanalytic sets in rigid analytic geometry proc london math soc 3 83 2001 681 707 is not true since this is a key step in the proof of the affinoid elimination theorem loc cit theorem 3 12 that proof contains a serious gap we also give an example of an affinoid subset of the plane that is not the image under a proper rigid analytic map of a set that is globally semianalytic in the domain of that map this clarifies the relationship among several natural categories of rigid subanalytic sets

symmetrized models of last passage percolation and non intersecting lattice paths it has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups u l sp 2l and o l we present a theory of such results based on non intersecting lattice paths and integration techniques familiar from the theory of random matrices detailed derivations of probabilities relating to two further symmetrizations are also given

pfaffian formulas and schur q function identities abstract we establish pfaffian analogues of the cauchy binet formula and the ishikawa wakayama minor summation formula each of these pfaffian analogues expresses a sum of products of subpfaffians of two skew symmetric matrices in terms of a single pfaffian by using these pfaffian formulas we give transparent proofs to several identities for schur q functions

entropy and boundary conditions in random rhombus tilings the tilings of rhombi in two dimensions and of rhomboedra in three dimensions are studied when they are constrained by fixed boundary conditions we establish a link between those conditions and free or periodic boundary ones the entropy is written as a functional integral which is treated via a saddle point method we can exhibit the dominant states of the statistical ensemble of tilings and show that they can display a strong structural inhomogeneity caused by the boundary this inhomogeneity is responsible for a difference of entropy between the studied fixed boundary tilings and free boundary ones this method uses a representation of tilings by membranes embedded in a higher dimensional hypercubic lattice it is illustrated in the case of 60 degree rhombus tilings

vapnik chervonenkis density in some theories without the independence property ii we study the vapnik chervonenkis vc density of denable families in certain stable rst order theories in particular we obtain uniform bounds on vc density of denable families in nite u rank theories without the nite cover property and we characterize those abelian groups for which there exist uniform bounds on the vc density of denable families

algebraic properties of separated power series we study the commutative algebra of rings of separated power series over a ring e and that of their extensions rings of separated and more specifically convergent power series from a field k with a separated e analytic structure both of these collections of rings already play an important role in the model theory of non archimedean valued fields and we establish their algebraic properties this will make a study of the analytic geometry over such fields through the classical methods of algebraic geometry possible

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

on the evaluation of r chapman s evil determinant abstract we prove a conjecture of r chapman asserting that for any prime p 3 mod 4 the determinant of the p 1 2 p 1 2 matrix c c ij with c ji j i p is always 1

fluctuation properties of the tasep with periodic initial configuration we consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process tasep they are expressed as fredholm determinants with a kernel defining a signed determinantal point process we then consider certain periodic initial conditions and determine the kernel in the scaling limit this result has been announced first in a letter by one of us sasamoto in j phys a 38 l549 l556 2005 and here we provide a self contained derivation connections to last passage directed percolation and random matrices are also briefly discussed

tropicalization of the moduli space of stable maps let x be an algebraic variety and let s be a tropical variety associated to x we study the tropicalization map from the moduli space of stable maps into x to the moduli space of tropical curves in s we prove that it is a continuous map and that its image is compact and polyhedral loosely speaking when we deform algebraic curves in x the associated tropical curves in s deform continuously moreover the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral our main tools are berkovich spaces formal models balancing conditions vanishing cycles and quantifier elimination for rigid subanalytic sets

young tableaux gelfand patterns and branching rules for classical groups abstract young tableaux and gelfand patterns which combinatorially describe characters of irreducible finite dimensional representations of complex classical lie groups are presented some branching rules and or character recursions are given explicit combinatorial descriptions these recursions are derived from manipulations of determinantal character formulas

polynuclear growth on a flat substrate and edge scaling of goe eigenvalues we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints through the robinson schensted knuth rsk construction one obtains the multilayer png model which consists of a stack of non intersecting lines the top one being the png height the statistics of the lines is translation invariant and at a fixed position the lines define a point process we prove that for large times the edge of this point process suitably scaled has a limit this limit is a pfaffian point process and identical to the one obtained from the edge scaling of the gaussian orthogonal ensemble goe of random matrices our results give further insight to the universality structure within the kpz class of 1 1 dimensional growth models

holographic algorithms without matchgates abstract the theory of holographic algorithms which are polynomial time algorithms for certain combinatorial counting problems surprised the complexity community by showing certain problems very similar to p complete problems were in fact in the class p in particular the theory produces algebraic tests for a problem to be in p in this article we describe the geometric basis of these algorithms by i replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix and ii replacing the computation of weighted perfect matchings of an auxiliary graph by computing the pfaffian of the directly constructed skew symmetric matrix this procedure indicates a more geometric approach to complexity classes it also leads to more general constructions where one replaces the grassmann plucker identities which test for admissibility by other algebraic tests natural problems treatable by these methods have been previously considered in a different context and we present one such example

multiobjective design of water distribution systems under uncertainty 1 the water distribution system wds design problem is defined here as a multiobjective optimization problem under uncertainty the two objectives are 1 minimize the total wds design cost and 2 maximize wds robustness the wds robustness is defined as the probability of simultaneously satisfying minimum pressure head constraints at all nodes in the network decision variables are the alternative design options for each pipe in the network the sources of uncertainty are future water consumption and pipe roughness coefficients uncertain variables are modeled using probability density functions pdfs assigned in the problem formulation phase the corresponding pdfs of the analyzed nodal heads are calculated using the latin hypercube sampling technique the optimal design problem is solved using the newly developed rnsgaii method based on the nondominated sorting genetic algorithm ii nsgaii in rnsgaii a small number of samples are used for each fitness evaluation leading to significant computational savings when compared to the full sampling approach chromosome fitness is defined here in the same way as in the nsgaii optimization methodology the new methodology is tested on several cases all based on the new york tunnels reinforcement problem the results obtained demonstrate that the new methodology is capable of identifying robust pareto optimal solutions despite significantly reduced computational effort

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

robust optimization for water distribution systems least cost design 1 the objective of the least cost design problem of a water distribution system is to find its minimum cost with discrete diameters as decision variables and hydraulic controls as constraints the goal of a robust least cost design is to find solutions which guarantee its feasibility independent of the data i e under model uncertainty a robust counterpart approach for linear uncertain problems is adopted in this study which represents the uncertain stochastic problem as its deterministic equivalent robustness is controlled by a single parameter providing a trade off between the probability of constraint violation and the objective cost two principal models are developed uncorrelated uncertainty model with implicit design reliability and correlated uncertainty model with explicit design reliability the models are tested on three example applications and compared for uncertainty in consumers demands the main contribution of this study is the inclusion of the ability to explicitly account for different correlations between water distribution system demand nodes in particular it is shown that including correlation information in the design phase has a substantial advantage in seeking more efficient robust solutions

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

correlation functions for β 1 ensembles of matrices of odd size using the method of tracy and widom we rederive the correlation functions for 1 hermitian and real asymmetric ensembles of n n matrices with n odd

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

some identities for determinants of structured matrices abstract in this paper we establish several relations between the determinants of the following structured matrices hankel matrices symmetric toeplitz hankel matrices and toeplitz matrices using known results for the asymptotic behavior of toeplitz determinants these identities are used in order to obtain fisher hartwig type results on the asymptotics of certain skew symmetric toeplitz determinants and certain hankel determinants

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

on schur multiple zeta functions a combinatoric generalization of multiple zeta functions abstract we introduce schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta star functions of the euler zagier type combinatorially we first study their basic properties including a region of absolute convergence and the case where all variables are the same then under an assumption on variables some determinant formulas coming from theory of schur functions such as the jacobi trudi giambelli and dual cauchy formula are established with the help of macdonald s ninth variation of schur functions moreover we investigate the quasi symmetric functions corresponding to the schur multiple zeta functions we obtain the similar results as above for them and furthermore describe the images of them by the antipode of the hopf algebra of quasi symmetric functions explicitly finally we establish iterated integral representations of the schur multiple zeta values of ribbon type which yield a duality for them in some cases

ising model observables and non backtracking walks this paper presents an alternative proof of the connection between the partition function of the ising model on a finite graph g and the set of non backtracking walks on g the techniques used also give formulas for spin spin correlation functions in terms of non backtracking walks the main tools used are viennot s theory of heaps of pieces and turning numbers on surfaces

configurational entropy of codimension one tilings and directed membranes the calculation of random tiling configurational entropy amounts to an enumeration of partitions a geometrical description of the configuration space is given in terms of integral points in a high dimensional space and the entropy is deduced from the integral volume of a convex polytope in some cases the latter volume can be expressed in a compact multiplicative formula and in all cases in terms of binomial series the origin of which is given a geometrical meaning our results mainly concern codimension one tilings but can also be extended to higher codimension tilings we also discuss the link between freeboundary and fixed boundary condition problems

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

a combinatorial proof of the giambelli identity for schur functions the giambelli identity provides a formula for expressing an arbitrary schur function of shape as a determinant of schur functions of hook shapes l 1k we give the first complete combinatorial proof of the giambelli identity we show how to derive various hook formulas from the giambelli identity and show how to extend our methods to derive extensions of the giambelli identity and the hook formula for the number of standard tableaux to certain skew shapes

fluctuations of the one dimensional polynuclear growth model in half space we consider the multi point equal time height fluctuations of the one dimensional polynuclear growth model in half space for special values of the nucleation rate at the origin the multi layer version of the model is reduced to a process with a determinantal weight for which the asymptotics can be analyzed in the scaling limit the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal symplectic to unitary transition ensemble at soft edge in random matrix theory

vicious walkers and young tableaux i without walls we rederive previously known results for the number of star and watermelon configurations by showing that these follow immediately from standard results in the theory of young tableaux and integer partitions in this way we provide a proof of a result previously only conjectured for the total number of stars

odd symplectic groups on definit des groupes symplectiques sur des espaces vectoriels de dimension impaire ces groupes de lie ne sont ni simples ni reductifs on obtient une formule de caracteres pour certaines representations tensorielles de ces groupes

flattening and subanalytic sets in rigid analytic geometry let k be an algebraically closed field endowed with a complete non archimedean norm with valuation ring r let f colon y to x be a map of k affinoid varieties in this paper we study the analytic structure of the image f y subset x such an image is a typical example of a subanalytic set we show that the subanalytic sets are precisely the mathbf d semianalytic sets where mathbf d is the truncated division function first introduced by denef and van den dries this result is most conveniently stated as a quantifier elimination result for the valuation ring r in an analytic expansion of the language of valued rings to prove this we establish a flattening theorem for affinoid varieties in the style of hironaka which allows a reduction to the study of subanalytic sets arising from flat maps that is we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups the case of a flat map is then dealt with by a small extension of a result of raynaud and gruson showing that the image of a flat map of affinoid varieties is open in the grothendieck topology using embedded resolution of singularities we derive in the zero characteristic case a uniformization theorem for subanalytic sets a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centres as a corollary we obtain the fact that any subanalytic set in the plane r2 is semianalytic 2000 mathematical subject classification 32p05 32b20 13c11 12j25 03c10

a two phase decomposition method for optimal design of looped water distribution networks a two phase decomposition method is proposed for the optimal design of new looped water distribution networks as well as for the parallel expansion of existing ones the main feature of the method is that it generates a sequence of improving local optimal solutions the first phase of the method takes a gradient approach with the flow distribution and pumping heads as decision variables and is an extension of the linear programming gradient method proposed by alperovits and shamir 1977 for nonlinear modeling the technique is iterative and produces a local optimal solution in the second phase the link head losses of this local optimal solution are fixed and the resulting concave program is solved for the link flows and pumping heads these then serve to restart the first phase to obtain an improved local optimal solution the whole procedure continues until no further improvement can be achieved some applications and extensions of the method are also discussed

broken translation invariance in quasifree fermionic correlations out of equilibrium abstract using the c algebraic scattering approach to study quasifree fermionic systems out of equilibrium in quantum statistical mechanics we construct the nonequilibrium steady state in the isotropic xy chain whose translation invariance has been broken by a local magnetization and analyze the asymptotic behavior of the expectation value for a class of spatial correlation observables in this state the effect of the breaking of translation invariance is twofold mathematically the finite rank perturbation not only regularizes the scalar symbol of the invertible toeplitz operator generating the leading order exponential decay but also gives rise to an additional trace class hankel operator in the correlation determinant physically in its decay rate the nonequilibrium steady state exhibits a left mover right mover structure affected by the scattering at the impurity

rigid subanalytic subsets of the line and the plane working over an algebraically closed complete non archimedean non trivially valued field we show that any subanalytic subset of the line is semialgebraic and that any subanalytic subset of the plane is semianalytic

dimer problem on the cylinder and torus institute of mathematics academia sinica nsfc 10771086 10671162 fujian province university nsc nsc96 2115 m 001 005

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

hyperdeterminantal expressions for jack functions of rectangular shapes abstract we derive a jacobi trudi type formula for jack functions of rectangular shapes in this formula we make use of a hyperdeterminant which is cayley s simple generalization of the determinant in addition after developing the general theory of hyperdeterminants we give summation formulas for schur functions involving hyperdeterminants and evaluate toeplitz type hyperdeterminants by using jack function theory

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

on some identities of schur q functions abstract we give a proof of the jozefiak pragacz identity using the representation theory of affine lie algebras new results of this type are also obtained

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

tropical analytic geometry newton polygons and tropical intersections abstract in this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results we use tropical methods to prove a theorem about the newton polygon for convergent power series in several variables if f 1 f n are n convergent power series in n variables with coefficients in a non archimedean field k we give a formula for the valuations and multiplicities of the common zeros of f 1 f n we use rigid analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper these results are naturally formulated and proved using the theory of tropicalizations of rigid analytic spaces as introduced by einsiedler kapranov and lind 2006 14 and gubler 2007 20 we have written this paper to be as readable as possible both to tropical and arithmetic geometers

coupling functions for domino tilings of aztec diamonds the inverse kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics in this paper we consider three different weightings of domino tilings of the aztec diamond and show using recurrence relations that we can compute the inverse kasteleyn matrix these weights are the one periodic weighting where the horizontal edges have one weight and the vertical edges have another weight the qvol weighting which corresponds to multiplying the product of tile weights by q if we add a box to the height function and the two periodic weighting which exhibits a flat region with defects in the center

the combinatorics of a class of representation functions abstract in the study of the irreducible representations of the unitary group u n one encounters a class of polynomials defined on n 2 indeterminates z ij 1 i j n which may be arranged into an n n matrix array z z ij these polynomials are indexed by double gelfand patterns or equivalently by pairs of column strict young tableaux of the same shape using the double labeling property one may define a square matrix d z whose elements are the double indexed polynomials these matrices possess the remarkable group multiplication property d xy d x d y for arbitrary matrices x and y even though these matrices may be singular for z u u n these matrices give irreducible unitary representations of u n these results are known but not always fully proved from the extensive physics literature on representation of the unitary groups where they are often formulated in terms of the boson calculus and the multiplication property is unrecognized the generality of the multiplication property is the key to understanding group representation theory from the purview of combinatorics the combinatorial structure of the general polynomials is expected to be intricate and in this paper we take the first step to explore the combinatorial aspects of a special class which can be defined in terms of the set of integral matrices with given row and column sums these special polynomials are denoted by l z where and are integral vectors representing the row sums and column sums of a class of integral matrices we present a combinatorial interpretation of the multiplicative properties of these polynomials we also point out the connections with macmahon s master theorem and schwinger s inner product formula which is essentially equivalent to macmahon s master theorem finally we give a formula for the double pfaffian which is crucial in the studies of the generating function of the 3 n j coefficients in angular momentum theory we also review the background of the general polynomials and give some of their properties

domino tilings and the six vertex model at its free fermion point at the free fermion point the six vertex model with domain wall boundary conditions dwbc can be related to the aztec diamond a domino tiling problem we study the mapping on the level of complete statistics for general domains and boundary conditions this is obtained by associating with both models a set of non intersecting lines in the lindstrom gessel viennot lgv scheme one of the consequences for dwbc is that the boundaries of the ordered phases are described by the airy process in the thermodynamic limit

symmetric functions and wavefunctions of xxz type six vertex models and elliptic felderhof models by izergin korepin analysis we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an elliptic extension of the trigonometric felderhof model we show that the wavefunctions can be expressed as one parameter deformations of an elliptic analog of the vandermonde determinant and elliptic symmetric functions we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an

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

a formula for the number of tilings of an octagon by rhombi we propose the first algebraic determinantal formula to enumerate tilings of a centrosymmetric octagon of any size by rhombi this result uses the gessel viennot technique and generalizes to any octagon a formula given by elnitsky in a special case

a dual of macmahon s theorem on plane partitions a classical theorem of macmahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula in this paper we present a counterpart of this formula corresponding to the exterior of a concave hexagon obtained by turning 120 after drawing each side macmahon s hexagon is obtained by turning 60 after each step

uniform properties of rigid subanalytic sets in the context of rigid analytic spaces over a non archimedean valued field a rigid subanalytic set is a boolean combination of images of rigid analytic maps we give an analytic quantifier elimination theorem for complete algebraically closed valued fields that is independent of the field in particular the analytic quantifier elimination is independent of the valued field s characteristic residue field and value group in close analogy to the algebraic case this provides uniformity results about rigid subanalytic sets we obtain uniform versions of smooth stratification for subanalytic sets and the lojasiewicz inequalities as well as a unfiorm description of the closure of a rigid semianalytic set

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

excited young diagrams and equivariant schubert calculus we describe the torus equivariant cohomology ring of isotropic grassmannians by using a localization map to the torus fixed points we present two types of formulas for equivariant schubert classes of these homogeneous spaces the first formula involves combinatorial objects which we call excited young diagrams and the second one is written in terms of factorial schur q or p functions as an application we give a giambelli type formula for the equivariant schubert classes we also give combinatorial and pfaffian formulas for the multiplicity of a singular point in a schubert variety

the ginibre ensemble of real random matrices and its scaling limits we give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the pfaffian of an antisymmetric matrix formed from a 2 2 matrix kernel associated to the ensemble we apply this result to the real ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large

advanced determinant calculus a complement abstract this is a complement to my previous article advanced determinant calculus c krattenthaler advanced determinant calculus seminaire lotharingien combin 42 1999 the andrews festschrift article b42q 67 pp in the present article i share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory g almkvist c krattenthaler j petersson some new formulas for experiment math 12 2003 441 456 moreover i add a list of determinant evaluations which i consider as interesting which have been found since the appearance of the previous article or which i failed to mention there including several conjectures and open problems

fuzzy probabilistic design of water distribution networks 1 the primary aim of this paper is to present a fuzzy probabilistic approach for optimal design and rehabilitation of water distribution systems combining aleatoric and epistemic uncertainties in a unified framework the randomness and imprecision in future water consumption are characterized using fuzzy random variables whose realizations are not real but fuzzy numbers and the nodal head requirements are represented by fuzzy sets reflecting the imprecision in customers requirements the optimal design problem is formulated as a two objective optimization problem with minimization of total design cost and maximization of system performance as objectives the system performance is measured by the fuzzy random reliability defined as the probability that the fuzzy head requirements are satisfied across all network nodes the satisfactory degree is represented by necessity measure or belief measure in the sense of the dempster shafer theory of evidence an efficient algorithm is proposed within a monte carlo procedure to calculate the fuzzy random system reliability and is effectively combined with the nondominated sorting genetic algorithm ii nsgaii to derive the pareto optimal design solutions the newly proposed methodology is demonstrated with two case studies the new york tunnels network and hanoi network the results from both cases indicate that the new methodology can effectively accommodate and handle various aleatoric and epistemic uncertainty sources arising from the design process and can provide optimal design solutions that are not only cost effective but also have higher reliability to cope with severe future uncertainties

a triangular gap of side 2 in a sea of dimers in a 60 angle we consider a triangular gap of side 2 in a 60 angle on the triangular lattice whose sides are zigzag lines we study the interaction of the gap with the corner as the rest of the angle being completely filled with lozenges we show that the resulting correlation is governed by the product of the distances between the gap and its five images in the sides of the angle this provides a new aspect of the parallel between the correlation of gaps in dimer packings and electrostatics developed by the first author in previous work this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

vicious walkers friendly walkers and young tableaux iii between two walls we derive exact and asymptotic results for the number of star and watermelon configurations of vicious walkers confined to lie between two impenetrable walls as well as corresponding results for the analogous problem of friendly walkers our proofs make use of results from symmetric function theory and the theory of basic hypergeometric series

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

loop erased walks and random matrices it is well known that there are close connections between non intersecting processes in one dimension and random matrices based on the reflection principle there is a generalisation of the reflection principle for more general e g planar processes due to fomin in which the non intersection condition is replaced by a condition involving loop erased paths in the context of independent brownian motions in suitable planar domains this also has close connections to random matrices an example of this was first observed by sato and katori phys rev e 83 041127 2011 we present further examples which give rise to various cauchy type ensembles we also extend fomin s identity to the affine setting and show that in this case by considering independent brownian motions in an annulus one obtains a novel interpretation of the circular orthogonal ensemble

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

shifted tableaux and the projective representations of symmetric groups les groupes de representation de s n classes de conjugaison algebres de clifford et la representation spin de base produits induits pour des representations spin une application caracteristique pour des caracteres de spin tableaux decales et q fonctions de schur les caracteres de spin irreductibles de s n et a n une analogue decalee de la regle lr produits tensoriels interieurs

closure of rigid semianalytic sets abstract letkbe an algebraically closed field of characteristic zero endowed with a complete nonarchimedean norm letxbe ak rigid analytic variety and a semianalytic subset ofx then the closure of inxwith respect to the canonical topology is again semianalytic the proof uses embedded resolution of singularities