generalized bibasic hypergeometric series and theiru n extensions we survey summation theorems for generalized bibasic hypergeometric series found recently by chu and perfected by macdonald these contain arbitrary sequences of parameters and generalize bibasic summation theorems found by gosper gasper and rahman they also contain a host of matrix inversions for infinite lower triangular matrices including those found by gould hsu carlitz and krattenthaler further special cases of krattenthaler s theorem are andrews matrix formulation of the bailey transform and bressoud s and gasper s inversions we extend the telescoping arguments used for bibasic summation theorems to provideu n 1 extensions of gosper s and gasper s bibasic summation theorems special cases includeu n 1 extensions of carlitz s and gasper s matrix inversions further specializations correspond to theu n 1 bailey transform andu n 1 bressoud matrix inversion found previously by milne applications includeu n 1 transformation and expansion formulas and au n 1 extension of aq analogue of abel s binomial theorem thisq analogue is different from those found by jackson and johnson

from canonical tensor operators of su 3 and 3 to bi orthogonal coupling coefficients explicit expansion expansion of the matrix elements of su 3 and canonical tensor operators in terms of the bi orthogonal coupling coefficients and overlaps of the draayer akiyama construction are considered special bi orthogonal extremal isoscalar factors with subscripts as multiplicity labels and proportional to q racah q 6j coefficients or the generalized wilson rahman rational bi orthogonal functions in terms of balanced or hypergeometric series are used as a natural basis for extremal matrix elements of the highest weight component of the canonical tensor operators of su 3 in the generating function approach of biedenharn lohe and louck the expansion that is obtained triple sum together with previously derived asymmetric seed isofactors and elementary overlaps gives the explicit overlap coefficients and can be used to derive su 3 and canonical tensor operators as well as new explicit normalized seed isofactors with regge type symmetry specified for the minimal null space case

orthogonalization coefficients for the elliott draayer states of the two parametric irreps of su n so n explicit algebraic polynomial expressions for the direct and inverse orthogonalization coefficients of the projected elliott draayer basis states of su n so n are constructed in the case of two parametric irreducible representation irreps they cover the general su 3 so 3 case as well as the two parametric covariant irreps of su 4 restricted to the su 2 su 2 subgroup of spin and isospin the obtained orthonormal bases are equivalent to those constructed by means of the ordered gram schmidt process the orthogonalization coefficients are expressed up to explicitly given elementary factors in terms of the numerator and denominator polynomials related with those that appeared in the orthogonalization coefficients of the paracanonical coupling coefficients isofactors of su 3 and with a functions of louck biedenharn and lohe

basic hypergeometric series very well poised in u n abstract we introduce and study a u n multiple series generalization of classical very well poised basic hypergeometric series of one variable two new multiple series generalizations of the terminating 6 5 summation theorem are derived one of these extends to a u n generalization of the nonterminating 6 5 summation theorem multiple series generalizations of the summation theorems of kummer and dixon appear as special cases we obtain the corresponding theorems for ordinary series by taking the limit as q 1 of all these results

computer model calibration using high dimensional output this work focuses on combining observations from field experiments with detailed computer simulations of a physical process to carry out statistical inference of particular interest here is determining uncertainty in resulting predictions this typically involves calibration of parameters in the computer simulator as well as accounting for inadequate physics in the simulator the problem is complicated by the fact that simulation code is sufficiently demanding that only a limited number of simulations can be carried out we consider applications in characterizing material properties for which the field data and the simulator output are highly multivariate for example the experimental data and simulation output may be an image or may describe the shape of a physical object we make use of the basic framework of kennedy and o hagan however the size and multivariate nature of the data lead to computational challenges in implementing the framework to overcome these challenges we make use of basis repre

schur functions good s identity and hypergeometric series well poised in su n abstract a simple direct proof is given of a fundamental identity involving schur functions which contains as special cases the identity responsible for good s proof of the dyson conjecture and the summation theorem of biedenharn and louck that appears frequently in dealing with the explicit matrix elements which arise in the unitary groups by using the weyl character formula a general identity is obtained which implies our result involving schur functions when a root system of type a n 1 is considered as a further application of our general identity explicit analogs of good s identity are given corresponding to the root systems of types b n c n and d n in addition methods to obtain q analogs of all of these results are briefly described

generalized factorial schur functions biorthogonal rational functions and the rii chain generalized factorial schur functions are introduced through a newton type interpolation they are applied to construct a class of biorthogonal rational functions and a special solution to its spectral transformation chain the rii chain a two parameter deformation of the schur type functions is also discussed

sharp l 1 a posteriori error analysis for nonlinear convection diffusion problems we derive sharp l l 1 a posteriori error estimates for initial boundary value problems of nonlinear convection diffusion equations of the form u t div f u a u g under the nondegeneracy assumption a s 0 for any s r the problem displays both parabolic and hyperbolic behavior in a way that depends on the solution itself it is discretized implicitly in time via the method of characteristic and in space via continuous piecewise linear finite elements the analysis is based on the kruzkov doubling of variables device and the recently introduced boundary layer sequence technique to derive the entropy error inequality on bounded domains the derived a posteriori error estimators have the correct convergence order in the region where the solution is smooth and recover the standard a posteriori error estimators known for parabolic equations with strong diffusions

a q analog of the 5f4 1 summation theorem for hypergeometric series well poised in su n abstract as an application of a general q difference equation for basic hypergeometric series well poised in su n an elementary proof is given of a q analog of holman s su n generalization of the terminating 5 f 4 1 summation theorem this provides an su n generalization of the terminating 6 5 summation theorem for classical basic hypergeometric series

a q analog of the gauss summation theorem for hypergeometric series in u n in this paper we derive a q analog of holman s u n generalization of the gauss summation theorem this result is obtained by iterating our mutivariable generalization of the gauss reduction formula for basic hypergeometric series in u n multiple series generalizations of both classical q analogs of the vandermonde summation theorem appear as special cases we obtain the corresponding theorems for ordinary series by taking the limit as q 1 of all these results

on the projected bases for sp 4 u 2 and the orthogonalization problem the mutual expansion and overlaps of the projected smirnov tolstoy and szpikowski berej bases for the irreducible representations irreps of sp 4 restricted to u 2 are considered the equivalence relation connecting the overlaps of both st and sb bases after the definite substitution of parameters up to an elementary factor is presented the overlaps of the szpikowski berej basis states are rearranged to double sums restricted by the parameters characterizing the multiplicities of the repeating irreps of subgroup and thus representable in polynomial forms the regge type symmetry of these new expressions allows the proof of the earlier conjectured symmetry of the orthogonalization coefficients oc for the family of the biorthogonal bases of sp 4 u 2 su 4 su 2 su 2 and su n so n for two parametric irreps the structure of the orthogonalization coefficients for the szpikowski berej basis is also considered the symmetry of the b c abde functions as the conjectured numerator polynomials

towards the canonical tensor operators of uq 3 ii the denominator function problem the explicit denominator normalization function of the canonical tensor operators of the quantum algebra uq 3 corresponding to the maximal null space case is derived ab initio in terms of double basic hypergeometric series which cannot be obtained as any q extension of the su 3 denominator polynomial gb 1 x in terms of multiple double or triple balanced hypergeometric series introduced by biedenharn louck and their collaborators although their q 1 versions are shown being equivalent the corresponding orthonormal seed isoscalar factors of the coupling wigner clebsch gordan coefficients of uq 3 and su 3 with multiple irreducible representations are presented conjectured expression of the q polynomials which ratios appear in the uq 3 and new su 3 denominator functions for an arbitrary value of the canonical multiplicity label t of the repeating irreducible representations in terms of multiple partition dependent q series extension of the maximal and minimal null space versions i

on the denominator function for canonical su 3 tensor operators the definition of a canonical unit su 3 tensor operator is given in terms of its characteristic null space as determined by group theoretic properties of the intertwining number this definition is shown to imply the canonical splitting conditions used in earlier work for the explicit and unique up to phases construction of all su 3 wcg coefficients wigner clebsch gordan using this construction an explicit su 3 invariant denominator function characterizing completely the canonically defined wcg coefficients is obtained it is shown that this denominator function squared is a product of linear factors which may be obtained explicitly from the characteristic null space times a ratio of polynomials these polynomials denoted gtq are defined over three shift parameters and three barycentric coordinates the properties of these polynomials hence of the corresponding invariant denominator function are developed in detail these include a derivation of their degree symmetries and zeros the

inhomogeneous basis set of symmetric polynomials defined by tableaux abstract a basis set of inhomogeneous symmetric polynomials denoted by tlambda z where z z1 zn and lambda lambda1 lambdan is a partition is defined in terms of young weyl standard tableaux or equivalently in terms of gel fand weyl patterns a number of significant properties of these polynomials are given together with outlines of proofs and compared with properties of the well known basis set of schur functions which are homogeneous polynomials the basis of the ring of symmetric polynomials defined here is shown to be natural for the expansion of inhomogeneous symmetric functions constructed from rising factorials

adaptive galerkin methods with error control for a dynamical ginzburg landau model in superconductivity the time dependent ginzburg landau model which describes the phase transitions taking place in superconductors is a coupled system of nonlinear parabolic equations it is discretized semi implicitly in time and in space via continuous piecewise linear finite elements a posteriori error estimates are derived for the l infty l 2 norm by studying a dual problem of the linearization of the original system other than the dual of error equations numerical simulations are included which illustrate the reliability of the estimators and the flexibility of the proposed adaptive method

an adaptive finite element method for the h ψ formulation of time dependent eddy current problems in this paper we develop an adaptive finite element method based on reliable and efficient a posteriori error estimates for the h formulation of eddy current problems with multiply connected conductors multiply connected domains are considered by making cuts the competitive performance of the method is demonstrated by an engineering benchmark problem team workshop problem 7 and a singular problem with analytic solution

u n very well posed 10 p 9 transformations abstract in this paper we derive multivariable generalizations of bailey s classical terminating very well poised 10 o 9 transformation we work in the setting of multiple basic hypergeometric series very well poised on unitary groups u n 1 we obtain these u n 1 10 o 9 transformations from an alternate form of our general u n 1 classical bailey transform and the u n 1 generalizations of jackson s q analog of dougall s terminating 7 f 6 1 summation one of these 10 o 9 transformations contains the u n 1 10 o 9 transformation of denis and gustafson 1992 as a speci we also deduce a u n 1 extension of one of sears 4 o 3 transformations the classical case of all this work corresponds to u 2 or equivalently a 1

bayesian calibration of inexact computer models bayesian calibration is used to study computer models in the presence of both a calibration parameter and model bias the parameter in the predominant methodology is left undefined this results in an issue where the posterior of the parameter is suboptimally broad there has been no generally accepted alternatives to date this article proposes using bayesian calibration where the prior distribution on the bias is orthogonal to the gradient of the computer model problems associated with bayesian calibration are shown to be mitigated through analytic results in addition to examples supplementary materials for this article are available online

efficient calibration for imperfect computer models many computer models contain unknown parameters which need to be estimated using physical observations tuo and wu 2014 show that the calibration method based on gaussian process models proposed by kennedy and o hagan j r stat soc ser b stat methodol 63 2001 425 464 may lead to an unreasonable estimate for imperfect computer models in this work we extend their study to calibration problems with stochastic physical data we propose a novel method called the l 2 calibration and show its semiparametric efficiency the conventional method of the ordinary least squares is also studied theoretical analysis shows that it is consistent but not efficient numerical examples show that the proposed method outperforms the existing ones

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

balanced3ϕ2summation theorems foru n basic hypergeometric series abstract in this paper we begin the theory and application of the u n 1 generalization of the classical bailey transform and bailey lemma we work in the setting of multiple basic hypergeometric series very well poised on unitary groups u n 1 the u n 1 bailey transform is obtained from a suitably modified u n 1 terminating very well poised 4 3 summation theorem and termwise transformations it is then interpreted as a matrix inversion result for two infinite lower triangular matrices this provides a higher dimensional generalization of andrews matrix inversion formulation of the bailey transform as in the classical case the concept of a u n 1 bailey pair is introduced and then inverted this u n 1 inversion enables us to derive u n 1 terminating balanced 3 2 summations directly from suitable u n 1 terminating very well poised 6 5 summations and vice versa these pairs of u n 1 dual identities extend the classical one variable case of andrews special limiting cases of the u n 1 terminating balanced 3 2 summation yield u n 1 q gauss summations several u n 1 q chu vandermonde summations u n 1 q binomial theorems and a u n 1 cauchy identity many other consequences of the u n 1 6 2 and 3 2 summations and the u n 1 bailey transform are further developed in our subsequent papers these include a derivation of the u n 1 bailey lemma several terminating u n 1 q whipple transformations u n 1 10 9 transformations and nonterminating u n 1 q whipple transformations the classical case of all this work corresponding to a 1 or equivalently u 2 contains a substantial amount of the theory and application of one variable basic hypergeometric series

coupling coefficients of so n and integrals involving jacobi and gegenbauer polynomials the expressions for the coupling coefficients 3j symbols for most degenerate symmetric representations of orthogonal groups so n in a canonical basis with so n restricted to so n 1 and different semicanonical or tree bases with so n restricted to so n so n n n n are considered respectively in context of integrals involving triplets of the gegenbauer and the jacobi polynomials since the directly derived triple hypergeometric series do not reveal the apparent triangle conditions of the 3j symbols they are rearranged using their relation with semistretched isofactors of the second kind for the complementary chain sp 4 su 2 su 2 and analogy with the stretched 9j coefficients of su 2 into formulae with more rich limits for summation intervals and obvious triangle conditions the isofactors of class one representations of orthogonal groups or class two representations of unitary groups and of course the related integrals involving triplets of the gegenbauer and the jacobi polynomials turn into double sums in the cases of canonical so n so n 1 or u n u n 1 and semicanonical so n so n 2 so 2 chains as well as into the 4f3 1 series under more specific conditions some ambiguities of the phase choice of the complementary group approach are adjusted as well as problems with an alternating sign parameter of so 2 representations in the so 3 so 2 and so n so n 2 so 2 chains

adaptive finite element frequency domain method for eddy current problems in the present paper adaptive finite element methods are studied for time harmonic eddy current problems in the case of three dimensional isotropic and linear materials we adopt the formulation based on the magnetic field and a magnetic scalar potential in this paper since it needs the least number of unknowns reliable and efficient a posteriori error estimates are obtained and the efficiency of the adaptive algorithm is demonstrated by numerical and engineering experiments

generalizations of cauchy s summation theorem for schur functions cauchy s summation theorem for schur functions is generalized and a number of related results are given the result is applied to a com binatorial problem involving products of pairs of permuations by appeal to properties of the group algebra of the symmetric group in 3 this summation theorem is used to derive a generalization of gessel s 3 result on counting 3 rowed latin rectangles with respect to the number of cycles in each row considered as permuations we also derive proposition 3 2 an explicit expression for a useful character sum associated with the symmetric group 4 contains a number of results for symmetric functions that are equivalent to theorem 2 1 these include a generalization of cauchy s summation theorem for

an adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems an efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator an adaptive algorithm with variable time step sizes and space meshes is proposed and studied which at each time step delays the mesh coarsening until the final iteration of the adaptive procedure allowing only mesh and time step size refinements before it is proved that at each time step the adaptive algorithm is able to reduce the error indicators and thus the error below any given tolerance within a finite number of iteration steps the key ingredient in the analysis is a new coarsening strategy numerical results are presented to show the competitive behavior of the proposed adaptive algorithm

a q analog of transposition symmetry for invariant g functions abstract we give an algebraic construction of milne s v g m n z functions in terms of the matrix coefficients of certain linear operators b z we show that the v g m n z functions all satisfy transposition symmetry if and only if z t t c n unless v g 1 n z is identically zero this transposition symmetry is a q analog of that for the ordinary v g m n polynomials of biedenharn gustafson and milne we also define a q analog of the elementary reduced wigner coefficients for u n and give both a combinatorial and representation theoretic interpretation of this definition

a new symmetry for biedenharn s g functions and classical hypergeometric series abstract we show that the general bisymmetric polynomials m g n q 1 n 1 m are a limiting case of the bisymmetric invariant polynomials n g n q 1 n 1 n which characterize u n tensor operators p q q 0 0 by taking suitable limits of a pair of difference equations for n g n q we then deduce transposition symmetry for m g n q from the same symmetry for n g n q as an application of transposition symmetry for m g n q we derive an elegant new contiguous relation for classical well poised hypergeometric series and also prove an identity between these series and multiple hypergeometric series well poised in su n

a q analog of hypergeometric series well poised in tsu n and invariant g functions abstract we introduce natural q analogs of hypergeometric series well poised in su n the related hypergeometric series in u n and invariant g functions we prove that both the su n multiple q series and the invariant g functions satisfy general q difference equations both the su n and u n q series are new multivariable generalizations of classical basic hypergeometric series of one variable we prove an identity which expresses our u n multiple q series as a finite sum of finite products of classical basic hypergeometric series these u n q series also satisfy an elegant reduction formula which is analogous to the inclusion lemma for classical invariant g functions

factorial type schur functions and the discrete kp equation a class of factorial type schur functions is introduced from a viewpoint of a newton type interpolation their basic properties are presented it is shown that the factorial type schur functions can be used to describe a particular solution to a nonautonomous version of the discrete kp equation

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

an a posteriori error estimator for anisotropic refinement besides an algorithm for local refinement an a posteriori error estimator is the basic tool of every adaptive finite element method using information generated by such an error estimator the refinement of the grid is controlled for 2nd order elliptic problems we present an error estimator for anisotropically refined grids like n d cuboidal and 3 d prismatic grids that gives correct information about the size of the error additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way numerical examples are presented for 2 d rectangular and 3 d prismatic grids

a u n 1 bailey lattice abstract multiple basic hypergeometric series associated with the unitary group u n 1 have been investigated by various authors the purpose of this paper is to obtain a u n 1 analogue of the agarwal andrews bressoud bailey lattice and to give some applications

on a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics this paper deals with the weak formulation of a free moving boundary problem arising in theoretical glaciology considering shallow ice sheet flow we present the mathematical analysis and the numerical solution of the second order nonlinear degenerate parabolic equation modelling in the isothermal case the ice sheet non newtonian dynamics an obstacle problem is then deduced and analyzed the existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method then the solutions are numerically computed with a method of characteristics and a duality algorithm to deal with the resulting variational inequalities the weak framework we introduce and its analysis both qualitative and numerical are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension they can be successfully applied to more realistic and sophisticated models related to other geophysical settings

barnes type degenerate euler polynomials in this paper we consider the barnes type degenerate euler polynomials we present several explicit formulas and recurrence relations for these polynomials also we establish a connection between our polynomials and several known families of polynomials

operator identities involving the bivariate rogers szego polynomials and their applications to the multiple q series identities abstract in this paper we first give several operator identities involving the bivariate rogers szego polynomials by applying the technique of parameter augmentation to the multiple q binomial theorems given by milne s c milne balanced 2 3 summation theorems for u n basic hypergeometric series adv math 131 1997 93 187 we obtain several new multiple q series identities involving the bivariate rogers szego polynomials these include multiple extensions of mehler s formula and rogers s formula our u n 1 generalizations are quite natural as they are also a direct and immediate consequence of their often classical known one variable cases and milne s fundamental theorem for a n or u n 1 basic hypergeometric series in theorem 1 49 of s c milne an elementary proof of the macdonald identities for a l 1 adv math 57 1985 34 70 as rewritten in lemma 7 3 on p 163 of s c milne balanced 2 3 summation theorems for u n basic hypergeometric series adv math 131 1997 93 187 or corollary 4 4 on pp 768 769 of s c milne m schlosser a new a n extension of ramanujan s 1 1 summation with applications to multilateral a n series rocky mountain j math 32 2002 759 792

the normal law under linear restrictions simulation and estimation via minimax tilting summary simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing and is typically only feasible by using approximate markov chain monte carlo sampling we propose a minimax tilting method for exact independently and identically distributed data simulation from the truncated multivariate normal distribution the new methodology provides both a method for simulation and an efficient estimator to hitherto intractable gaussian integrals we prove that the estimator has a rare vanishing relative error asymptotic property numerical experiments suggest that the scheme proposed is accurate in a wide range of set ups for which competing estimation schemes fail we give an application to exact independently and identically distributed data simulation from the bayesian posterior of the probit regression model

a new symmetry related to su n for classical basic hypergeometric series abstract a direct proof is given of an elegant new contiguous relation for classical well poised basic hypergeometric series which preserves the well poised condition the proof involves elementary series manipulations and does not depend upon the transposition symmetry of the general bisymmetric polynomials m g n q 1 n 1 m which was used to establish the ordinary or q 1 case of the identity the new continguous relation can be considered a generalization of the 6 5 summation theorem

equivariant littlewood richardson skew tableaux we give a positive equivariant littlewood richardson rule also discovered independently by molev our proof generalizes a proof by stembridge of the classical littlewood richardson rule we describe a weight preserving bijection between our indexing tableaux and trapezoid puzzles which restricts to a bijection between positive indexing tableaux and knutson tao puzzles

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

degenerate poly cauchy polynomials in this paper we study several properties for the degenerate poly cauchy polynomials we present several explicit formulas and recurrence relations for these polynomials also we establish a connection between our polynomials and several known families of polynomials

a new tableau representation for supersymmetric schur functions abstract we give a new tableau definition for supersymmetric skew schur functions and obtain a number of properties of these functions as easy corollaries

irreducibility of induced supermodules for general linear supergroups abstract in this note we determine when an induced supermodule h g 0 corresponding to a dominant integral highest weight of the general linear supergroup g g l m n is irreducible using the contravariant duality given by the supertrace we obtain a characterization of irreducibility of weyl supermodules v this extends the result of kac 12 13 who proved that for ground fields of characteristic zero v is irreducible if and only if is typical

gaussian predictive process models for large spatial data sets with scientific data available at geocoded locations investigators are increasingly turning to spatial process models for carrying out statistical inference over the last decade hierarchical models implemented through markov chain monte carlo methods have become especially popular for spatial modelling given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics however fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations rendering such models infeasible for large spatial data sets this computational burden is exacerbated in multivariate settings with several spatially dependent response variables it is also aggravated when data are collected at frequent time points and spatiotemporal process models are used with regard to this challenge our contribution is to work with what we call predictive process models for spatial and spatiotemporal data every spatial or spatiotemporal process induces a predictive process model in fact arbitrarily many of them the latter models project process realizations of the former to a lower dimensional subspace thereby reducing the computational burden hence we achieve the flexibility to accommodate non stationary non gaussian possibly multivariate possibly spatiotemporal processes in the context of large data sets we discuss attractive theoretical properties of these predictive processes we also provide a computational template encompassing these diverse settings finally we illustrate the approach with simulated and real data sets

on the denominator function for canonical su 3 tensor operators ii explicit polynomial form the canonical resolution of the multiplicity problem for tensor operators in su 3 is equivalent to the map the denominator mapping from the set of all su 3 unit tensor operators to su 3 invariant functions the denominator functions the denominator function vanishes precisely on that characteristic null space that specifies each operator uniquely since for su 3 the characteristic null spaces are known to be simply ordered each denominator function can be expressed up to explicitly known multiplicative factors as a ratio of two successive polynomials in the set gtq t 0 1 q 1 q 0 1 by obtaining explicitly the set of all polynomials gtq this paper completes the construction of all su 3 denominator functions

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

umbral calculus and sheffer sequences of polynomials in this paper we investigate some properties of sheffer sequences of polynomials arising from umbral calculus from these properties we derive new and interesting identities between sheffer sequences of polynomials an application to normal ordering is presented

the factorial schur function the application of the divided difference of a function to the inhomogeneous symmetric functions factorial schur functions of biedenharn and louck is shown to lead to new relations and simplified proofs of their properties these results include determinantal definitions and the factorial jacobi trudi identities with extensions to skew versions similar properties of a second class of symmetric functions depending on an arbitrary parameter and of importance for generalized hypergeometric functions and series are shown also to be derivable from the divided difference notion slightly extended

an umbral calculus for polynomials characterizing u n tensor operators after the change of variables i i i and xi i 1 i i 1 we show that the invariant polynomials g n q i xi i 1 characterizing u n tensor operators p q q 0 0 become an integral linear combination of schur functions s in the symbol where denotes the difference of the two sets of variables 1 n and 1 n we obtain a similar result for the yet more general bisymmetric polynomials m g n q 1 n 1 m making use of properties of skew schur functions s and s we put together an umbral calculus for m g n q that is working entirely with polynomials we uniquely determine m g n q from m g n q 1 and combinatorial rules involving ferrers diagrams i e partitions provided that n 1 q this restriction does not interfere with writing the general case of m g n q as a linear combination of s as an application we deduce conjugation symmetry for n g n q from transposition symmetry by showing that these two symmetries are equivalent

a u n generalization of ramanujan s 1ψ1 summation abstract in this paper we derive a u n generalization of ramanujan s 1 1 summation directly from a recent u n multiple series refinement of the q binomial theorem we also obtain a new generalization of the macdonald identities for a 1 l

the bal and bcl bailey transform and lemma we announce a higher dimensional generalization of the bailey transform bailey lemma and iterative bailey chain concept in the setting of basic hypergeometric series very well poised on unitary a or symplectic c groups the classical case corresponding to a 1 or equivalently u 2 contains an immense amount of the theory and application of one variable basic hypergeometric series including elegant proofs of the rogers ramanujan schur identities

factorial type schur functions orthogonal rational functions and discrete dressing chains a relationship between orthogonal rational functions and discrete integrable systems is studied by an approach based on schur type symmetric functions a system of orthogonal rational functions is constructed using a multiparameter deformation of the schur functions spectral equations for the orthogonal rational functions are derived by using properties of the schur type symmetric functions the compatibility condition of the spectral equations induces a discrete dressing chain which is a toda type discrete integrable system describing dressing transformations for orthogonal rational functions

weight 2 zeros of 3j coefficients and the pell equation all weight 2 zeros of the wigner 3j coefficients may be obtained from the quadratic diophantine equation known as pell s equation these zeros may then be classified by the orbits of a discrete infinite order subgroup of the lorentz group so 1 1 this is carried out by transforming the polynomial part of a weight 2 3j coefficient to pellian form and obtaining the fundamental zeros numerically the relation of this polynomial to a family of binary quadratic forms is also given together with a discussion of the invariance group

towards the canonical tensor operators of uq 3 i the maximal null space case generalizing the su 3 canonical tensor operator concept biedenharn and louck to the quantum algebra uq 3 the wigner clebsch gordan coefficients of uq 3 with repeating irreducible representations are considered extremal projectors of the quantum algebra uq 3 in terms of the ordered generator polynomials are used for evaluation of the bilinear combinations of the uq 3 canonical isoscalar factors explicit expressions of the uq 3 isofactors corresponding to the maximal null space case of the uq 3 unit canonical tensor operators and their normalization factors denominator functions are presented the transposition and conjugation phase factors for the su 3 and uq 3 canonical isofactors are correlated with phases and zeros of boundary isofactors invariance of the canonical isofactors or absence of such invariance under interchange of the tensor operator and the initial or final state parameters is correlated with the existence and invariance or numerical degeneracy of the usual splitting

littlewood richardson polynomials abstract we introduce a family of rings of symmetric functions depending on an infinite sequence of parameters a distinguished basis of such a ring is comprised by analogues of the schur functions the corresponding structure coefficients are polynomials in the parameters which we call the littlewood richardson polynomials we give a combinatorial rule for their calculation by modifying an earlier result of b sagan and the author the new rule provides a formula for these polynomials which is positive in the sense of w graham we apply this formula for the calculation of the product of equivariant schubert classes on grassmannians which implies a stability property of the structure coefficients the first manifestly positive formula for such an expansion was given by a knutson and t tao by using combinatorics of puzzles while the stability property was not apparent from that formula we also use the littlewood richardson polynomials to describe the multiplication rule in the algebra of the casimir elements for the general linear lie algebra in the basis of the quantum immanants constructed by a okounkov and g olshanski

a littlewood richardson rule for factorial schur functions we give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial schur functions it is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial schur function applications to capelli operators and quantum immanants are also given

explicit orthogonalization of some biorthogonal bases for su n so n and sp 4 u 2 mutually related explicit algebraic polynomial expressions of the orthogonalization coefficients are proposed for the biorthogonal polynomial bargmann moshinsky stretched antistretched quasistretched and their dual bases of the two parametric mixed tensor and covariant irreducible representations irreps of su n restricted to so n as well as for the projected smirnov tolstoy and dual bases of the five dimensional quasispin the orthogonalization coefficients of the essentially simplified gram schmidt process are expressed up to explicitly given elementary factors in terms of the numerator and denominator polynomials represented as compositions of the generalized hypergeometric coefficients 3f2 and 1f0 and equivalent in the diagonal denominator and boundary cases to the a cabde functions of biedenharn and louck the distribution of zeros and the symmetry properties of the introduced polynomials are crucial for the conjectured solutions

a characteristic galerkin method with adaptive error control for the continuous casting problem the continuous casting problem is a convection dominated nonlinearly degenerate diffusion problem it is discretized implicitly in time via the method of characteristics and in space via continuous piecewise linear finite elements a posteriori error estimates are derived for the l1l1 norm of temperature which exhibit a mild explicit dependence on velocity the analysis is based on special properties of a linear dual problem in non divergence form with vanishing diffusion and strong advection several simulations with realistic physical parameters illustrate the reliability of the estimators and the flexibility of the proposed adaptive method

stanley s zrank conjecture on skew partitions we present an affirmative answer to stanley s zrank conjecture namely the zrank and the rank are equal for any skew partition we show that certain classes of restricted cauchy matrices are nonsingular and furthermore the signs are determined by the number of zero entries we also give a characterization of the rank in terms of the giambelli type matrices of the corresponding skew schur functions our approach also applies to the factorial cauchy matrices and the inverse binomial coefficient matrices

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

model calibration with censored data abstractthe purpose of model calibration is to make the model predictions closer to reality the classical kennedy o hagan approach is widely used for model calibration which can account for the inadequacy of the computer model while simultaneously estimating the unknown calibration parameters in many applications the phenomenon of censoring occurs when the exact outcome of the physical experiment is not observed but is only known to fall within a certain region in such cases the kennedy o hagan approach cannot be used directly and we propose a method to incorporate the censoring information when performing model calibration the method is applied to study the compression phenomenon of liquid inside a bottle the results show significant improvement over the traditional calibration methods especially when the number of censored observations is large

bayesian computation using design of experiments based interpolation technique in this article a new deterministic approximation method for bayesian computation known as design of experiments based interpolation technique doit is proposed the method works by sampling points from the parameter space using an experimental design and then fitting a kriging model to interpolate the unnormalized posterior the approximated posterior density is a weighted average of normal densities and therefore most of the posterior quantities can be easily computed doit is a general computing technique that is easy to implement and can be applied to many complex bayesian problems moreover it does not suffer from the curse of dimensionality as much as some quadrature methods it can work using fewer posterior evaluations which is a great advantage over the monte carlo and markov chain monte carlo methods especially when dealing with computationally expensive posteriors this article has supplementary material that is available online

explicit orthonormal clebsch gordan coefficients of su 3 the construction of the explicit algebraic polynomial expressions for the nonmultiplicity free orthonormal clebsch gordan wigner coefficients of su 3 u 2 is completed in the case of the paracanonical coupling scheme related with the explicit minimal biorthogonal systems by means of the hecht or gram schmidt process the direct and inverse orthogonalization coefficients the first of them being equivalent to the boundary orthonormal isofactors are expressed up to explicitly given multiplicative factors in terms of the numerator and denominator polynomials related with the auxiliary a function of louck biedenharn and lohe that appears as a fragment of the denominator g functions of canonical su 3 tensor operators

new infinite families of exact sums of squares formulas jacobi elliptic functions and ramanujan s tau function in this paper we give two infinite families of explicit exact formulas that generalize jacobi s 1829 4 and 8 squares identities to 4n2 or 4n n 1 squares respectively without using cusp forms our 24 squares identity leads to a different formula for ramanujan s tau function n when n is odd these results arise in the setting of jacobi elliptic functions jacobi continued fractions hankel or turanian determinants fourier series lambert series inclusion exclusion laplace expansion formula for determinants and schur functions we have also obtained many additional infinite families of identities in this same setting that are analogous to the function identities in appendix i of macdonald s work macdonald i g 1972 invent math 15 91 143 a special case of our methods yields a proof of the two conjectured kac v g and wakimoto m 1994 in progress in mathematics eds brylinski j l brylinski r guillemin v kac v birkhauser boston boston ma vol 123 pp 415 456 identities involving representing a positive integer by sums of 4n2 or 4n n 1 triangular numbers respectively our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series gustafson s cl nonterminating 6 5 summation theorem and andrews basic hypergeometric series proof of jacobi s 4 and 8 squares identities we have elsewhere applied symmetry and schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n n 1 squares respectively our sums of more than 8 squares identities are not the same as the formulas of mathews 1895 glaisher 1907 ramanujan 1916 mordell 1917 1919 hardy 1918 1920 kac and wakimoto and many others

overlaps of the biorthogonal 3 coupling coefficients and related basic hypergeometric and other q factorial series the intermediate expansion technique is developed for the overlap coefficients of different biorthogonal coupled states and coupling coefficients of the quantum algebra and group su 3 with repeating irreducible representations and the role of separate basic and classical hypergeometric functions is demonstrated some expansion coefficients such as new prime overlap functions equivalent to the bilinear combinations of the different boundary isofactors are expressed in terms of the balanced saalschutzian basic or classical hypergeometric series these new overlap functions their analytical inversion some known triangular expansion matrices transition matrices related to definite racah coefficients the well poised or series and compositions of q factorial series resembling the well poised and series or equivalent to and series provide themselves as generators for mutual expansion of different non orthonormal systems of and su 3 isofactors which may be orthonormalized to the paracanonical or with a definite caution canonical version

space time adaptive algorithm for the mixed parabolic problem in this paper we present an a posteriori error estimator for the mixed formulation of a linear parabolic problem used for designing an efficient adaptive algorithm our space time discretization consists of lowest order raviart thomas finite element over graded meshes and discontinuous galerkin method with variable time step finally several examples show that the proposed method is efficient and reliable

schubert classes in the equivariant cohomology of the lagrangian grassmannian abstract let lg n denote the lagrangian grassmannian parametrizing maximal isotropic lagrangian subspaces of a fixed symplectic vector space of dimension 2n for each strict partition 1 k with 1 n there is a schubert variety x let t denote a maximal torus of the symplectic group acting on lg n consider the t equivariant cohomology of lg n and the t equivariant fundamental class of x the main result of the present paper is an explicit formula for the restriction of the class to any torus fixed point the formula is written in terms of factorial analogue of the schur q function introduced by ivanov as a corollary to the restriction formula we obtain an equivariant version of the giambelli type formula for lg n as another consequence of the main result we obtained a presentation of the ring h t lg n

anisotropic mesh adaptation for the solution of the stefan problem a new finite element formulation for the solution of the classical stefan problem is introduced it is obtained by a slight modification of the phase field formulation used for dendritic growth where both the phase field and the temperature t are discretized an anisotropic mesh adaptation strategy is also presented to further increase the accuracy of the method numerical results for two dimensional examples illustrating the performance and accuracy of the proposed method are presented

multivariable wilson polynomials a multivariable biorthogonal generalization of the wilson polynomials is presented these are four distinct families which in a special case occur in two complex conjugate pairs that satisfy four biorthogonality relations among them an interesting limit case is the multivariable continuous dual hahn polynomials

orthogonalization of the projected states and isofactors the explicit direct and inverse orthogonalization coefficients ocs are derived for the projected basis states of the two parametric covariant irreducible representations of including elliott s n 3 case and draayer with the alternative choices of the gram schmidt processes began respectively from the lowest vergados or highest absolute value of the intrinsic parameter k the different direct and inverse ocs the latter equivalent to the definite boundary isofactors or the boundary resubducing coefficients are found to be connected by the definite analytical continuation contraction procedures with mutually related ocs of the biorthogonal states and the boundary orthonormal pseudocanonical and paracanonical isofactors of the orthogonalization coefficients considered are presented as definite algebraic polynomial structures under the square root with the linear factors and numerator denominator polynomials in terms of the partition dependent functions of biedenharn and louck

multivariable biorthogonal continuous discrete wilson and racah polynomials several families of multivariable biorthogonal partly continuous and partly discrete wilson polynomials are presented these yield limit cases that are purely continuous in some of the variables and purely discrete in the others or purely discrete in all the variables the latter are referred to as the multivariable biorthogonal racah polynomials interesting further limit cases include the multivariable biorthogonal hahn and dual hahn polynomials

an elementary proof of the macdonald identities for a 1 l abstract in this paper it is shown that the macdonald identities for a 1 l are a natural consequence of the recent multivariable generalization of classical basic hypergeometric series known as basic hypergeometric series in u n more precisely a u n multiple series generalization of the q binomial theorem is derived and used to generalize cauchy s elegant proof of jacobi s triple product identity and to give a direct elementary proof of the macdonald identities for a 1 l

giambelli formulae for the equivariant quantum cohomology of the grassmannian we find presentations by generators and relations for the equivariant quantum cohomology of the grassmannian for these presentations we also find determinantal formulae for the equivariant quantum schubert classes to prove this we use the theory of factorial schur functions and a characterization of the equivariant quantum cohomology ring

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

a posteriori error estimation and adaptivity for degenerate parabolic problems two explicit error representation formulas are derived for degenerate parabolic pdes which are based on evaluating a parabolic residual in negative norms the resulting upper bounds are valid for any numerical method and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient they are applied to a practical space time discretization consisting of c 0 piecewise linear finite elements over highly graded unstructured meshes and backward finite differences with varying time steps two rigorous a posteriori error estimates are derived for this scheme and used in designing an efficient adaptive algorithm which equidistributes space and time discretization errors via refinement coarsening a simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach and hints at the potentials and reliability of the proposed method