gauge theories labelled by three manifolds we propose a dictionary between geometry of triangulated 3 manifolds and physics of three dimensional n 2 gauge theories under this duality standard operations on triangulated 3 manifolds and various invariants thereof classical as well as quantum find a natural interpretation in field theory for example independence of the sl 2 chern simons partition function on the choice of triangulation translates to a statement that s 3 b partition functions of two mirror 3d n 2 gauge theories are equal three dimensional n 2 field theories associated to 3 manifolds can be thought of as theories that describe boundary conditions and duality walls in four dimensional n 2 scfts thus making the whole construction functorial with respect to cobordisms and gluing

quantization of models with non compact quantum group symmetry modular xxz magnet and lattice sinh gordon model we define and study certain integrable lattice models with non compact quantum group symmetry the modular double of including an integrable lattice regularization of the sinh gordon model and a non compact version of the xxz model their fundamental r matrices are constructed in terms of the non compact quantum dilogarithm our choice of the quantum group representations naturally ensures self adjointness of the hamiltonian and the higher integrals of motion these models are studied with the help of the separation of variables method we show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations baxter equation and quantum wronskian equation which is characterized by suitable analytic and asymptotic properties a key technical tool is the so called operator for which we give an explicit construction our results allow us to establish some connections to related results and conjectures on the sinh gordon theory in continuous spacetime our approach also sheds some light on the relations between massive and massless models in particular the sinh gordon and liouville theories from the point of view of their integrable structures

elliptic u 2 quantum group and elliptic hypergeometric series we investigate an elliptic quantum group introduced by felder and varchenko which is constructed from the r matrix of the andrews baxter forrester model containing both spectral and dynamical parameter we explicitly compute the matrix elements of certain corepresentations and obtain orthogonality relations for these elements using dynamical representations these orthogonality relations give discrete bi orthogonality relations for terminating very well poised balanced elliptic hypergeometric series previously obtained by frenkel and turaev and by spiridonov and zhedanov in different contexts

a commutative algebra on degenerate cp1 and macdonald polynomials we introduce a unital associative algebra a associated with degenerate cp1 we show that a is a commutative algebra and whose poincare series is given by the number of partitions thereby we can regard a as a smooth degeneration limit of the elliptic algebra introduced by feigin and odesskii int math res notices 11 531 1997 then we study the commutative family of the macdonald difference operators acting on the space of symmetric functions a canonical basis is proposed for this family by using a and the heisenberg representation of the commutative family studied by shiraishi commun math phys 263 439 2006 it is found that the ding iohara algebra lett math phys 41 183 1997 provides us with an algebraic framework for the free field construction an elliptic deformation of our construction is discussed showing connections with the drinfeld quasi hopf twisting leningrad math j 1 1419 1990 in the sence of babelon bernard billey phys lett b 375 89 1996 the ruijsenaar

elliptic hypergeometry of supersymmetric dualities ii orthogonal groups knots and vortices we consider seiberg electric magnetic dualities for 4d mathcal n 1 sym theories with so n gauge group for all such known theories we construct superconformal indices scis in terms of elliptic hypergeometric integrals equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals in particular we describe a number of new elliptic beta integrals associated with the s confining theories with the spinor matter fields reductions of some dualities from sp 2n to so 2n or so 2n 1 gauge groups are described interrelation of scis and the witten anomaly is briefly discussed possible applications of the elliptic hypergeometric integrals to a two parameter deformation of 2d conformal field theory and related matrix models are indicated connections of the reduced scis with the state integrals of knot theory generalized agt duality for 3 3 d theories and a 2d vortex partition function are described

comment on star star relations in statistical mechanics and elliptic gamma function identities we prove a recently conjectured star star relation which plays the role of an integrability condition for a class of 2d ising type models with multicomponent continuous spin variables namely we reduce this relation to an identity for elliptic gamma functions previously obtained by rains

abcd of instantons we solve open image in new window 2 supersymmetric yang mills theories for an arbitrary classical gauge group i e su n so n sp n in particular we derive the prepotential of the low energy effective theory and the corresponding seiberg witten curves we manage to do this without resolving singularities of the compactified instanton moduli spaces

properties of generalized univariate hypergeometric functions based on spiridonov s analysis of elliptic generalizations of the gauss hypergeometric function we develop a common framework for 7 parameter families of generalized elliptic hyperbolic and trigonometric univariate hypergeometric functions in each case we derive the symmetries of the generalized hypergeometric function under the weyl group of type e 7 elliptic hyperbolic and of type e 6 trigonometric using the appropriate versions of the nassrallah rahman beta integral and we derive contiguous relations using fundamental addition formulas for theta and sine functions the top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with ruijsenaars relativistic hypergeometric function and the askey wilson function respectively we show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions we also describe an intimate connection between the hyperbolic and trigonometric theory which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions

from weak to strong coupling in abjm theory the partition function of mathcal n 6 supersymmetric chern simons matter theory known as abjm theory on mathbb s 3 as well as certain wilson loop observables are captured by a zero dimensional super matrix model this super matrix model is closely related to a matrix model describing topological chern simons theory on a lens space we explore further these recent observations and extract more exact results in abjm theory from the matrix model in particular we calculate the planar free energy which matches at strong coupling the classical iia supergravity action on rm ads 4 times mathbb c mathbb p 3 and gives the correct n3 2 scaling for the number of degrees of freedom of the m2 brane theory furthermore we find contributions coming from world sheet instanton corrections in mathbb c mathbb p 3 we also calculate non planar corrections both to the free energy and to the wilson loop expectation values this matrix model appears also in the study of topological strings on a toric calabi yau manifold and an intriguing connection arises between the space of couplings of the planar abjm theory and the moduli space of this calabi yau in particular it suggests that in addition to the usual perturbative and strong coupling ads expansions a third natural expansion locus is the line where one of the two t hooft couplings vanishes and the other is finite this is the conifold locus of the calabi yau and leads to an expansion around topological chern simons theory we present some explicit results for the partition function and wilson loop observables around this locus

the supersymmetric index in four dimensions we review the calculation and properties of the supersymmetric index for four dimensional mathcal n 1 theories illustrating its physical significance in several examples

a generalized hypergeometric function satisfying four analytic difference equations of askey wilson type the hypergeometric function 2 f 1 can be written in terms of a contour integral involving gamma functions we generalize this barnes representation by using a certain generalized gamma function as a building block in this way we obtain a new 2 f 1 generalization with various symmetry features we determine the analyticity properties of the r function in all of its eight arguments and show that it is a joint eigenfunction of four distinct askey wilson type difference operators two acting on v and two on the askey wilson polynomials can be obtained by a suitable discretization of v or

a generalized hypergeometric function ii asymptotics and d 4 symmetry in previous work we introduced and studied a function r a a bf c v hat v that generalizes the hypergeometric function in this paper we focus on a similarity transformed function mathcal e a a gamma v hat v with parameters 4 related to the couplings c 4 by a shift depending on a a we show that the function is invariant under all maps w with w in the weyl group of type d 4 choosing a a positive and bf gamma hat v real we obtain detailed information on the re v asymptotics of the function in particular we explicitly determine the leading asymptotics in terms of plane waves and the c function that implements the similarity r

hellinger reissner principles for plate and shell finite elements the hellinger reissner generalized variational principle is used to deduce further principles directed especially towards formulation of high integrity finite elements for plates and curved shells in polynomial parametric representation assumed stresses in these principles are derived from displacements and supplemented with stresses derived from stress functions

rarefied elliptic hypergeometric functions abstract two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved they generalize evaluations of the type i and ii elliptic beta integrals attached to the root system c n in a special n 1 case the simplest p 0 limit is shown to lead to a new class of q hypergeometric identities symmetries of a rarefied elliptic analogue of the euler gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed some extensions of the latter function to c n and a n root systems and corresponding symmetry transformations are considered an application of the rarefied type ii c n elliptic hypergeometric function to some eigenvalue problems is briefly discussed

elliptic hypergeometry of supersymmetric dualities we give a full list of known mathcal n 1 supersymmetric quantum field theories related by the seiberg electric magnetic duality conjectures for su n sp 2n and g 2 gauge groups many of the presented dualities are new not considered earlier in the literature for all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals this gives a systematic extension of the related romelsberger and dolan osborn results equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals about half of them were proven earlier and another half represents new challenging conjectures in particular we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author

hyperbolic beta integrals abstract hyperbolic beta integrals are analogues of euler s beta integral in which the role of euler s gamma function is taken over by ruijsenaars hyperbolic gamma function they may be viewed as q q bibasic analogues of the beta integral in which the two bases q and q are interrelated by modular inversion and they entail q analogues of the beta integral for q 1 the integrals under consideration are the hyperbolic analogues of the ramanujan integral the askey wilson integral and the nassrallah rahman integral we show that the hyperbolic nassrallah rahman integral is a formal limit case of spiridonov s elliptic nassrallah rahman integral

gauge theories and macdonald polynomials we study the mathcal n 2 four dimensional superconformal index in various interesting limits such that only states annihilated by more than one supercharge contribute extrapolating from the su 2 generalized quivers which have a lagrangian description we conjecture explicit formulae for all a type quivers of class mathcal s which in general do not have one we test our proposals against several expected dualities the index can always be interpreted as a correlator in a two dimensional topological theory which we identify in each limit as a certain deformation of two dimensional yang mills theory the structure constants of the topological algebra are diagonal in the basis of macdonald polynomials of the holonomies

edgeworth expansion for u statistics under minimal conditions berry esseen bounds for u statistics under the optimal moment conditions were derived by koroljuk and borovskich and friedrich under the same optimal moment assumptions with an additional nonlattice condition we establish a one term edgeworth expansion with remainder o n 1 2 for u statistics

alternative ways for formulation of hybrid stress elements an element stiffness matrix can be derived by the conventional potential energy principle and indirectly also by generalized variational principles such as the hu washizu principle and the hellinger reissner principle the present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the hellinger reissner principle it is found that the resulting scheme is equivalent to that considered by tong 1982 for the construction of hybrid stress elements in tong s scheme the inversion of a large flexibility matrix can be avoided it is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the hellinger reissner principle and the hu washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method

partition functions of n 2 2 gauge theories on s 2 and vortices we apply localization techniques to compute the partition function of a two dimensional mathcal n 2 2 r symmetric theory of vector and chiral multiplets on s2 the path integral reduces to a sum over topological sectors of a matrix integral over the cartan subalgebra of the gauge group for gauge theories which would be completely higgsed in the presence of a fayet iliopoulos term in flat space the path integral alternatively reduces to the product of a vortex times an antivortex partition functions weighted by semiclassical factors and summed over isolated points on the higgs branch for applications we evaluate the partition function for some u n gauge theories showing equality of the path integrals for theories conjectured to be dual by hori and tong and deriving new expressions for vortex partition functions

physical shape functions in finite element analysis of moderately thick plates a new rectangular finite element for moderately thick plates is presented the element is based on the concept of physical shape functions i e functions which contain in themselves physical and geometrical properties of the element the analytic formulae of stiffness geometric stiffness and mass matrices are presented for isotropic material the element is free from locking and zero energy modes not corresponding to rigid body motions several examples are presented for static initial stability and free vibration problems

transformations of elliptic hypergeometric integrals we prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions corresponding to the root systems bc n and a n as a special case we recover some integral identities conjectured by van diejen and spiridonov for bc n we also consider their type ii integral their proof of that integral together with our transformation gives rise to pairs of adjoint integral operators a different proof gives rise to pairs of adjoint difference operators these allow us to construct a family of biorthogonal abelian functions generalizing the koornwinder polynomials and satisfying the analogues of the macdonald conjectures finally we discuss some transformations of type ii style integrals in particular we find that adding two parameters to the type ii integral gives an integral invariant under an appropriate action of the weyl group e 7

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

lower estimates of the convergence rate for u statistics recent results on the berry esseen bound for u statistics assumed the following conditions suppose a u statistic of degree 2 is nondegenerate then the rate of convergence in the clt is of the order o n 1 2 provided that ee h x 1 x 2 x 1 3 0 similar results hold for von mises statistics

elliptic modular double and 4d partition functions we consider 4d supersymmetric special unitary gamma quiver gauge theories on compact manifolds which are t 2 fibrations over s 2 we show that their partition functions are correlators of vertex

quantum zonal spherical functions and macdonald polynomials abstract a unified theory of quantum symmetric pairs is applied to q special functions previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions here a distinguished family of such functions invariant under the weyl group associated to the restricted roots is shown to be a family of macdonald polynomials as conjectured by koornwinder and macdonald our results place earlier work for lie algebras of classical type in a general context and extend to the exceptional cases

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

an edgeworth expansion for symmetric statistics we consider asymptotically normal statistics which are symmetric functions of n i i d random variables for these statistics we prove the validity of an edgeworth expansion with remainder o n 1 under cramer s condition on the linear part of the statistic and moment assumptions for all parts of the statistic by means of a counterexample we show that it is generally not possible to obtain an edgeworth expansion with remainder o n 1 without imposing additional assumptions on the structure of the nonlinear part of the statistic

a generalized hypergeometric function iii associated hilbert space transform for generic parameters a a c 0 2 4 we associate a hilbert space transform to the relativistic hypergeometric function r a a bf c v hat v studied in previous papers restricting the couplings c to a certain polytope we show that the renormalized r function kernel gives rise to an isometry from the even subspace of l 2 mathbb r hat w hat v d hat v to the even subspace of l2 w v dv where hat w hat v and w v are positive and even weight functions we prove that the orthogonal complement of the range of this isometry is spanned by n pairwise orthogonal functions the latter are in essence askey wilson polynomials arising from the r function by choosing hat v i kappa n with kappa 0 ldots kappa n 1 distinct negative numbers the two commuting analytic difference operators acting on the variable v for which r is a joint eigenfunction give rise to two commuting self adjoint hamiltonians on the even subspace of l2 w v dv we explicitly determine the relation of the time dependent scattering theory for these dynamics to their joint spectral transform

spectral transformation chains and some new biorthogonal rational functions a discrete time chain associated with the generalized eigenvalue problem for two jacobi matrices is derived various discrete and continuous symmetries of this integrable equation are revealed a class of its rational elementary and elliptic functions solutions appearing from a similarity reduction are constructed the latter lead to large families of biorthogonal rational functions based upon the very well poised balanced hypergeometric series of three types the standard hypergeometric series 9 f 8 basic series 10 9 and its elliptic analogue 10 e 9 for an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly

the berry esseen bound for studentized statistics we derive berry esseen bounds for a class of studentized statistics the results are applied to studentized ustatistics studentized l statistics and studentized functions of the sample mean to give the berry esseen bounds under conditions weaker than those obtained by alternative methods

extended global symmetries for 4d mathcal n 1 sqcd theories in spiridonov and vartanov 2010 nucl phys b 824 192 the authors using the superconformal index technique found that four dimensional sqcd theory with su 2 gauge group and four flavors has 72 dual representations recently in dimofte and gaiotto 2012 j high energy phys jhep10 2012 129 the authors showed that these dual theories when coupled to five dimensional 5d hypermultiplets with specific boundary conditions have an extended e7 global symmetry in this work we find that for a reduced theory with three flavors the explicit su 6 global symmetry is enhanced to an e6 symmetry in the presence of 5d hypermultiplets we also show connections between indices of different theories in three and four dimensions

inextensional bending of a shell triangular element in quadratic parametric representation abstract exact closed form inextensional bending solutions are developed for a class of shell triangular elements in quadratic parametric representation these elements may have positive zero or negative gaussian curvature previous exact closed form inextensional solutions of the equations of thin shell theory are known for non zero gaussian curvature only for spherical surfaces the polynomial solutions are relevant to finite element design and provide inextensional bending behaviours with slowly varying curvature changes ready for use in patch test validation of any shell finite element

quantum integrability of the generalized elliptic ruijsenaars models the quantum integrability of the generalized elliptic ruijsenaars models is shown these models are mathematically related to the macdonald operator and the macdonald koornwinder operator which appeared in the q orthogonal polynomial theories we construct these integrable families by using the yang baxter equation and the reflection equation

symmetry groups of bailey s transformations for 10 φ 9 series although most of the symmetry groups or invariance groups associated with two term transformations between basic hypergeometric series have been studied and identified this is not the case for the most general transformation formulae in the theory of basic hypergeometric series namely bailey s transformations for f910 series first we show that the invariance group for both bailey s two term transformations for terminating f910 series and bailey s four term transformations for non terminating f910 series rewritten as a two term transformation of a so called f series is isomorphic to the weyl group of type e 6 we continue our recent research concerning the group structure underlying three term transformations s lievens j van der jeugt invariance groups of three term transformations for basic hypergeometric series j comput appl math 197 2006 1 14 and demonstrate that the group associated with a three term transformation between these f series each admitting bailey s two term transformation is the weyl group of type e 7 we do this by giving a description of the root system of type e 7 that allows to find a transformation between equivalent three term identities in an easy way a computation shows that there are five essentially different three term transformations between these f series we give an explicit form of each of these five transformations in an elegant way to our knowledge only one of these transformations has appeared in the literature

ruijsenaars hypergeometric function and the modular double of u q sl 2 c abstract simultaneous eigenfunctions of two askey wilson second order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized universal enveloping algebra u q sl 2 c these eigenfunctions are shown to be equal to ruijsenaars hypergeometric function under a proper parameter correspondence

a family of hybrid plate elements hybrid stress elements are known to provide accurate results for analyses of plate bending in particular for the prediction of moment distribution the construction of hybrid stiffness matrix requires numerical inversion of a moment matrix evaluation of some relatively complicated boundary integrals and several matrix transformations each of these operations can be time consuming and the matrix inversion can result in a loss of numerical accuracy this paper devises methods to explicitly invert the moment matrices we find that for triangular elements the inverse is independent of the element shape and is only inversely proportional to its size we also use a novel set of displacement variables which greatly simplifies the boundary integration the displacement variables are chosen in a hierarchical form so that lower order elements can be determined by straightforward reduction of excess terms in a higher order element except for the nodal displacements at the vertices the present approach involves only variables at the midpoints of the sides of an element

an a r threesome matrix models 2d conformal field theories and 4d n 2 gauge theories we explore the connections between three classes of theories a quiver matrix models d 2 conformal a r toda field theories and d 4 n 2 supersymmetric conformal a r quiver gauge theories in particular we analyze the quiver matrix models recently introduced by dijkgraaf and vafa unpublished and make detailed comparisons with the corresponding quantities in the toda field theories and the n 2 quiver gauge theories we also make a speculative proposal for how the matrix models should be modified in order for them to reproduce the instanton partition functions in quiver gauge theories in five dimensions c 2010 american institute of physics doi 10 1063 1 3449328

study of trial functions in shell triangular finite elements of quadratic parametric representation abstract a study is made of trial functions in a curved triangular finite element for the solution of problems in first approximation shell theory the element geometry is in quadratic parametric representation of the real shell surface while the displacement and stress function trial functions are in cubic parametric representation special attention is given to inextensional bending and homogeneous membrane states numerical examples are provided for specimen elements with positive zero and negative gaussian curvature vector equations are listed for the underlying shell theory together with fortran subroutines for the element geometry kinematics and statics

on normal approximations to u statistics let x1 xn be i i d random observations taking their values in a measurable space consider a u statistic of order k says s x1 xn based on the sample x1 xn assume that s can be represented in the form s l t with el et 0 where l is a linear statistic and t is a stochastically smaller statistic assume further that the linear statistic l is approximately normal and that we have a bound for error of such approximation our objective is to show that a similar bound holds for the distribution ofs provided that we correct it by adding vart i ln 2 3 1 vart where vart et 2 is the variance of t it occurs that one has to correct the normal approximation as well replacing it by some asymptotically as n 1 equivalent distribution furthermore we show that the class of u statistics up to negligible error terms is closed under studentization or self normalization for example using the jackknife estimators of variances this leads to error bounds of normal approximations for non deterministically normalized u statistics our results extend reflne and yield a number of related known results the bounds have a simple and convenient form for applications and analysis since they involve only the variance of the noise termt in the case k 3 of higher order statistics it remains open whether one can improve the bounds removing the logarithmic factor ln 2 3 1 vart we provide as well an optimal bound under a lower moment assumption ejtj fi 1 fi 2