on barnes multiple zeta and gamma functions abstract we show how various known results concerning the barnes multiple zeta and gamma functions can be obtained as specializations of simple features shared by a quite extensive class of functions the pertinent functions involve laplace transforms and their asymptotics is obtained by exploiting this we also demonstrate how barnes multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic difference equations both of these new approaches to the barnes functions give rise to novel integral representations

an elliptic analogue of the multiple gamma function a hierarchy of functions including the elliptic gamma function is introduced it can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q analogue of the multiple gamma function some properties of the functions are considered

third neighbor correlators of a one dimensional spin 1 2 heisenberg antiferromagnet we exactly evaluate the third neighbor correlator s j z s j 3 z and all the possible nonzero correlators s j ensuremath alpha s j 1 ensuremath beta s j 2 ensuremath gamma s j 3 ensuremath delta of the one dimensional spin frac 1 2 heisenberg mathrm xxx antiferromagnet in the ground state without magnetic field all the correlators are expressed in terms of certain combinations of logarithm mathrm ln 2 the riemann zeta function ensuremath zeta 3 ensuremath zeta 5 with rational coefficients the results accurately coincide with the numerical ones obtained by the density matrix renormalization group method and the numerical diagonalization

adjusted estimates and wald statistics for the ar 1 model with constant abstract adjusted profile likelihood of cox and reid 1993 journal of the royal statistical society b 55 467 471 for the autoregressive coefficient of the ar 1 model with constant yields an estimator which can be more accurate than the maximum likelihood estimator in small samples whether the process is stationary or a random walk and in the latter case also asymptotically a related estimator is asymptotically distributed compactly and symmetrically around zero under a random walk but is not consistent in general the adjusted wald statistic can follow the asymptotic standard normal distribution much better than the non adjusted under stationarity or a drifted random walk

high temperature expansion of the emptiness formation probability for the isotropic heisenberg chain recently gohmann klumper and seel have derived novel integral formulae for the correlation functions of the spin 1 2 heisenberg chain at finite temperature we have found that the high temperature expansion hte technique can be effectively applied to evaluate these integral formulae actually as for the emptiness formation probability p n of the isotropic heisenberg chain we have found a general formula of the hte for p n with arbitrary up to o j t 4 if we fix a magnetic field to a certain value we can calculate the hte to much higher order for example the order up to o j t 42 has been achieved in the case of p 3 when h 0 we have compared these hte results with the data by quantum monte carlo simulations they exhibit excellent agreement

diagrammatics for su 2 invariant matrix product states we report on a systematic implementation of su 2 invariance for matrix product states mps with concrete computations cast in a diagrammatic language as an application we present a variational mps study of a spin 1 2 quantum chain for efficient computations we make systematic use of the su 2 symmetry at all steps of the calculations i the matrix space is set up as a direct sum of irreducible representations ii the local matrices with state valued entries are set up as superposition of su 2 singlet operators iii products of operators are evaluated algebraically by making use of identities for 3j and 6j symbols the remaining numerical computations like the diagonalization of the associated transfer matrix and the minimization of the energy expectation value are done in spaces free of symmetry degeneracies the energy expectation value is a strict upper bound of the true ground state energy and yields definite conclusions about the accuracy of density matrix renormalization group results reported in the literature furthermore we present explicit results with accuracy better than 10 4 for nearest and next nearest neighbour spin correlators and for general dimer dimer correlators in the thermodynamical limit of the spin heisenberg chain with frustration

subreflector shaping for antenna distortion compensation an efficient fourier jacobi expansion with go po analysis technological demands have brought a renewed interest in the application of large reflector antennas with steadily increasing operating frequencies and antenna dimensions the high surface accuracy of the main reflector required by these antennas can often not be achieved with available manufacturing technologies the utilization of a shaped subreflector for main reflector distortion compensation is considered an effective measure to enhance the overall radiation performance of an antenna system in the process of evaluating the suitability of the subreflector shaping however it is crucial to accurately assess the most suitable subreflector shape within a reasonable amount of computational time this is especially true for electrically large reflectors where simple analysis of the radiation characteristics already creates a serious computational burden moreover since reflector shaping is a synthesis process that necessitates repeated computation of the radiation characteristics in this paper the development of an efficient computational tool for subreflector shaping is presented the subreflector shaping is performed through a combination of geometrical optics go and physical optics po on the subreflector and the main reflector respectively to significantly limit the number of parameters subject to optimization the subreflector surface is parameterized by the coefficients of a global orthogonal fourier jacobi set related to zernike polynomials which allows us to accurately represent a surface with only a small number of coefficients the incorporation of this surface expansion into a go po synthesis technique is detailed representative results are given for a computationally challenging reflector configuration and the tolerances for the shaped subreflector surface are studied

on the theory of the synthesis of offset dual shaped reflectors case examples in an earlier paper by v galindo israel et al see ibid vol ap 35 p 887 96 august 1987 the geometrical optics go principles constraints and requirements of the dual and single offset shaped reflector synthesis problem were collected and developed into a set of nonlinear first order partial differential equations pdes methods of solving these pdes numerically were illustrated as were certain problems that may arise an extension of the methods by which solutions to the pdes can be obtained is presented together with several case examples these examples are independently analyzed by go and physical optics diffraction methods the starting point for the integration over each reflector can be taken on the outer rim at the center or at an intermediate point the intermediate starting point being the more general case the utility of the speed of this synthesis method is demonstrated this makes practical the incorporation of the synthesis into a search algorithm that can optimize one or more parameters of the reflector system as an example the optimization of the mapping equations for low cross polarization is discussed

the reconstruction of local quantum operators for the boundary xxz spin heisenberg chain the method of the reconstruction of local quantum operators in terms of the elements of the quantum monodromy matrix is applied to the xxz spin heisenberg chain with integrable boundary conditions

a q operator identity for the correlation functions of the infinite xxz spin chain an explicit construction for q operators of the finite xxz spin chain with twisted boundary conditions is presented the massless and the massive regimes are considered as well as the root of unity case it is explained how these results yield an alternative expression for the trace function employed in the description of the correlation functions of the inhomogeneous xxz model on the infinite lattice by boos jimbo miwa smirnov and takeyama

traces on the sklyanin algebra and correlation functions of the eight vertex model we propose a conjectural formula for correlation functions of the z invariant inhomogeneous eight vertex model we refer to this conjecture as ansatz it states that correlation functions are linear combinations of products of three transcendental functions with theta functions and derivatives as coefficients the transcendental functions are essentially logarithmic derivatives of the partition function per site the coefficients are given in terms of a linear functional tr on the sklyanin algebra which interpolates the usual trace on finite dimensional representations we establish the existence of tr and discuss the connection to the geometry of the classical limit we also conjecture that the ansatz satisfies the reduced qkz equation as a non trivial example of the ansatz we present a new formula for the next nearest neighbour correlation functions

dynamical correlation functions of the xxz model at finite temperature combining a lattice path integral formulation for thermodynamics with the solution of the quantum inverse scattering problem for local spin operators we derive a multiple integral representation for the time dependent longitudinal correlation function of the spin 1 2 heisenberg xxz chain at finite temperature and in an external magnetic field our formula reproduces the previous results in the following three limits the static the zero temperature and the xy limits

unitary representations of uq 2 ℝ the modular double and the multiparticle q deformed toda chain the paper deals with the analytic theory of the quantum q deformed toda chains the technique used combines the methods of representation theory and the quantum inverse scattering method the key phenomenon which is under scrutiny is the role of the modular duality concept first discovered by l faddeev in the representation theory of noncompact semisimple quantum groups explicit formulae for the whittaker vectors are presented in terms of the double sine functions and the wave functions of the n particle q deformed open toda chain are given as a multiple integral of the mellin barnes type for the periodic chain the two dual baxter equations are derived

on the physical part of the factorized correlation functions of the xxz chain it was recently shown by jimbo et al 2008 arxiv 0811 0439 that the correlation functions of a generalized xxz chain associated with an inhomogeneous six vertex model with a disorder parameter and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions and here we approach the description of the same correlation functions and in particular of the function from a different direction we start from a novel multiple integral representation for the density matrix of a finite chain segment of length m in the presence of a disorder field we explicitly factorize the integrals for m 2 based on this we present an alternative description of the function in terms of the solutions of certain linear and nonlinear integral equations we then prove directly that the two definitions of describe the same function the definition in the work of jimbo et al 2008 arxiv 0811 0439 was crucial for the proof of the factorization the definition given here together with the known description of in terms of the solutions of nonlinear integral equations is useful for performing e g the trotter limit in the finite temperature case or for obtaining numerical results for the correlation functions at short distances we also address the issue of the construction of an exponential form of the density matrix for finite

density matrix elements and entanglement entropy for the spin 1 2 xxz chain at δ 1 2 we have analytically obtained all the density matrix elements up to six lattice sites for the spin 1 2 heisenberg xxz chain at 1 2 we use the multiple integral formula of the correlation function for the massless xxz chain derived by jimbo and miwa as for the spin spin correlation functions we have newly obtained the fourth and fifth neighbour transverse correlation functions we have calculated all the eigenvalues of the density matrix and analyse the eigenvalue distribution using these results the exact values of the entanglement entropy for the reduced density matrix up to six lattice sites have been obtained we observe that our exact results agree quite well with the asymptotic formula predicted by the conformal field theory

the importance of being odd in this letter i mainly consider a finite xxz spin chain with periodic boundary conditions and an odd number of sites this system is described by the hamiltonian hxxz j 1n jx j 1x jy j 1y jz j 1z as it turns out the ground state energy is proportional to the number of sites e 3n 2 for a special value of the asymmetry parameter 1 2 the trigonometric polynomial q u the zeros of which are parameters of the ground state bethe eigenvector is explicitly constructed this polynomial of degree n n 1 2 satisfies the baxter t q equation using the second independent solution of this equation that corresponds to the same eigenvalue of the transfer matrix it is possible to find a derivative of the ground state energy w r t the asymmetry parameter this derivative is closely connected with the correlation function jz j 1z 1 2 3 2n2 this correlation function is related to the average number of spin strings for the ground state nstring n 1 n i would like to stress that all the above simple formulae are not applicable to the case of an even number of sites which is usually considered

critical correlations in a z invariant inhomogeneous ising model abstract we study the critical spin spin correlation function in the z invariant inhomogeneous ising model which includes the rectangular triangular honeycomb and checkerboard lattices as special cases using quadratic difference equations we derive various explicit representations for the correlation function confirming and simplifying also a miraculous result of baxter

string correlation functions of the spin 1 2 heisenberg xxz chain we calculate certain string correlation functions originally introduced as order parameters in integer spin chains for the spin 1 2 xxz heisenberg chain at zero temperature and in the thermodynamic limit for small distances we obtain exact results from bethe ansatz and exact diagonalization whereas in the large distance limit field theoretical arguments yield an asymptotic algebraic decay we also make contact with two point spin correlation functions in the asymptotic limit

projection matrix method for shaped beam synthesis in phased arrays and reflectors we present the projection matrix method for shaped beam synthesis in a general array of different size elements the method relies on orthogonal projection of the desired far field intensity vector onto the space spanned by the far field intensity vectors of the array elements it is found that for a uniform convergence of the solution the far field sample space must be extended beyond the coverage region otherwise the projection matrix becomes ill conditioned a general guideline for the far field sample space is provided the method with necessary amendments is then employed successfully for a reflector surface synthesis the method is found to be several times faster than the gradient search method commonly used for beam synthesis numerical results for array and shaped reflector syntheses are shown and the advantages discussed

factorization of the ising model form factors we present a general method for analytically factorizing the n fold form factor integrals f n n n t for the correlation functions of the ising model on the diagonal in terms of the hypergeometric functions 2f1 1 2 n 1 2 n 1 t which appear in the form factor f 1 n n t new quadratic recursion and quartic identities are obtained for the form factors for n 2 3 for n 2 3 4 explicit results are given for the form factors these factorizations are proved for all n for n 2 3 these results yield the emergence of palindromic polynomials canonically associated with elliptic curves as a consequence understanding the form factors amounts to describing and understanding an infinite set of palindromic polynomials canonically associated with elliptic curves from an analytical viewpoint the relation of these palindromic polynomials with hypergeometric functions associated with elliptic curves is made very explicitly and from a differential algebra viewpoint this corresponds to the emergence of direct sums of differential operators homomorphic to symmetric powers of a second order operator associated with elliptic curve

form factors and action of on cycles let be a sequence of skew symmetric polynomials in x1 xl satisfying degxjpn l n 1 whose coefficients are symmetric laurent polynomials in z1 zn we call p an cycle if holds for all n l these objects arise in integral representations for form factors of massive integrable field theory i e the su 2 invariant thirring model and the sine gordon model the variables a logxa are the integration variables and j logzj are the rapidity variables to each cycle there corresponds a form factor of the above models conjecturally all form factors are obtained from the cycles in this paper we define an action of on the space of cycles there are two sectors of cycles depending on whether n is even or odd using this action we show that the character of the space of even resp odd cycles which are polynomials in z1 zn is equal to the level 1 irreducible character of with lowest weight 0 resp 1 we also suggest a possible tensor product structure of the full space of cycles

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

a note on the high temperature expansion of the density matrix for the isotropic heisenberg chain abstract gohmann klumper and seel derived the multiple integral formula of the density matrix of the xxz heisenberg chain at finite temperatures we have applied the high temperature expansion hte method to isotropic case of their formula in a finite magnetic field and obtained coefficients for several short range correlation functions for example we have succeeded to obtain the coefficients of the hte of the third neighbor correlation function j z j 3 z for zero magnetic field up to the order of 25 these results expand our previous results on the emptiness formation probability z tsuboi m shiroishi j phys a math gen 38 2005 l363 l370 condmat 0502569 to more general correlation functions

a q operator for the twisted xxx model taking the isotropic limit 1 in a recent representation theoretic construction of baxter s q operators for the xxz model with quasi periodic boundary conditions we obtain new results for the xxx model we show that quasi periodic boundary conditions are needed to ensure convergence of the q operator construction and derive a quantum wronskian relation which implies two different sets of bethe ansatz equations one above the other below the equator of total spin sz 0 we discuss the limit to periodic boundary conditions at the end and explain how this construction relates to the trace functional introduced by boos et al in the context of correlation functions on the infinite lattice we also identify a special subclass of solutions to the quantum wronskian and numerically verify them up to spin chains of ten sites this special type of solutions might persist for longer chains

free boson representation of dyℏ gln k and dyℏ sln k we construct a realization of the yangian double dy h over bar gl n k and dy h over bar sl n k for an arbitrary level k in terms of free boson fields with a continuous parameter in the h over bar 0 limit this realization becomes the wakimoto realization of affine lie algebras gl n and sl n respectively the vertex operators and the screening currents are also constructed with the same spirits the screening currents commute with dy h over bar sl n module total difference c 1998 american institute of physics s0022 2488 98 00104 2

monte carlo methodology and the finite sample properties of instrumental variables statistics for testing nested and non nested hypotheses using monte carlo methodology this paper investigates the effect of dynamics and simultaneity on the finite sample properties of instrumental variables statistics for testing nested and nonnested hypotheses simple numerical analytical formulae response surfaces are obtained which closely approximate the statistics unknown size and power functions for a dynamic simultaneous equations model the analysis illustrates the value and limitations of asymptotic theory in interpreting finite sample properties two practical results arise the f form and the wald statistic is favored over its chi squared form and large sigma and small effective sample size strongly affect the test of over identifying restrictions and the cox type test copyright 1991 by the econometric society

free field representation for massive integrable models a new approach to massive integrable models is considered it allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form factors in thesu 2 thirring and sine gordon models

the critical an 1 1 chain we study the an 1 1 spin chain in the critical regime q 1 we give free boson realizations of the vertex operators and their duals using these free boson realizations we give integral representations for the correlation functions

a comparison between techniques for global surface interpolation in shaped reflector analysis usual geometrical optics based syntheses of shaped reflectors provide numerically defined surfaces which must be interpolated for subsequent evaluation of the antenna secondary field to check against design specifications the implementation of three techniques capable of performing global interpolations of numerically defined 3d surfaces namely the quintic pseudosplines polynomial fourier series and jacobi polynomial sinusoidal expansions is explored the relative superior stability of the first is demonstrated for the synthesis of shaped reflectors presenting critical behavior of their principal curvatures

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

difference equations for the correlation functions of the eight vertex model the authors propose that the correlation functions of the inhomogeneous eight vertex model in the anti ferroelectric regime satisfy a system of difference equations with respect to the spectral parameters solving the simplest difference equation they obtain the expression for the spontaneous staggered polarization conjectured by baxter and kelland 1974 they also discuss a related construction of vertex operators on the lattice

a general formula for soliton form factors in the quantum sine gordon model a general formula for soliton form factors of several local fields in the quantum sine gordon model is presented

finite temperature correlation functions from discrete functional equations we present a new approach to the static finite temperature correlation functions of the heisenberg chain based on functional equations an inhomogeneous generalization of the n site density operator is considered the lattice path integral formulation with a finite but arbitrary trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters we show that these equations yield a unique characterization of the density operator our functional equations are a discrete version of the reduced q knizhnik zamolodchikov equations which played a central role in the study of the zero temperature case as a natural result and independent of the arguments given by jimbo et al 2009 j phys a math theor 42 304018 we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature they are a sum of products of nearest neighbour correlators

a q operator for the quantum transfer matrix baxter s concept of a q operator is generalized to the quantum transfer matrix of the xxz spin chain by employing the representation theory of quantum groups the spectrum of this q operator is discussed and novel functional relations which describe the finite temperature regime of the xxz spin chain are derived for a non vanishing magnetic field the previously known bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies for vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six vertex transfer matrix at roots of unity the quantum classical crossover is also discussed in terms of the eigenvalues of the q operator for a few special examples

landau singularities and singularities of holonomic integrals of the ising class we consider families of multiple and simple integrals of the ising class and the linear ordinary differential equations with polynomial coefficients they are solutions of we compare the full set of singularities given by the roots of the head polynomial of these linear odes and the subset of singularities occurring in the integrals with the singularities obtained from the landau conditions for these ising class integrals we show that the landau conditions can be worked out either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral the singular behaviour of these integrals is obtained in the self dual variable w s 2 1 s2 with s sinh 2k where k j kt is the usual ising model coupling constant switching to the variable s we show that the singularities of the analytic continuation of series expansions of these integrals actually break the kramers wannier duality we revisit the singular behaviour zenine et al 2005 j phys a math gen 38 9439 74 of the third contribution to the magnetic susceptibility of ising model 3 at the points 1 3w 4w2 0 and show that 3 s is not singular at the corresponding points inside the unit circle s 1 while its analytical continuation in the variable s is actually singular at the corresponding points 2 s s2 0 outside the unit circle s 1

the exact moments of the least squares estimator for the autoregressive model corrections and extensions abstract for a first order autoregressive model with unknown intercept and normal errors the mean and variance of the finite sample distribution of the least squares estimator of the autoregressive parameter are numerically computed using formulae for evaluating the first and second moments of a ratio of two quadratic forms in normally distributed variables previous results in the literature are corrected and results for the case where the value of the autoregressive parameter is near unity in absolute value are presented

integrals of the ising class from an experimental mathematical perspective we analyze ising class integrals our experimental results involvedextreme precision multidimensional quadrature on intricate integrands thus highly parallel computation was required

reduced qkz equation and correlation functions of the xxz model correlation functions of the xxz model in the massive and massless regimes are known to satisfy a system of linear equations the main relations among them are the difference equations obtained from the qkz equation by specializing the variables 1 2n as 1 n n 1 1 1 we call it the reduced qkz equation in this article we construct a special family of solutions to this system they can be written as linear combinations of products of two transcendental functions open image in new window with coefficients being rational functions we show that correlation functions of the xxz model in the massive regime are given by these formulas with an appropriate choice of open image in new window we also present a conjectural formula in the massless regime

boundary spectrum in the sine gordon model with dirichlet boundary conditions we find the spectrum of boundary bound states for the sine gordon model with dirichlet boundary conditions closing the bootstrap and providing a complete description of all the poles in the boundary reflection factors the boundary coleman thun mechanism plays an important role in the analysis two basic lemmas are introduced which should hold for any 1 1 dimensional boundary field theory allowing the general method to be applied to other models

quiver text w _ boldsymbol epsilon _ mathbf 1 boldsymbol epsilon _ mathbf 2 algebras of 4d boldsymbol mathcal n mathbf mathbf 2 gauge theories we construct an e deformation of w algebras corresponding to the additive version of quiver w q t algebras which feature prominently in the 5d version of the bps cft correspondence and refined topological strings on toric calabi yau s this new type of algebras fill in the missing intermediate level between q deformed and ordinary w algebras we show that e deformed w algebras are spectral duals of conventional w algebras in particular the e deformed conformal blocks manifestly reproduce instanton partition functions of 4d n 2 quiver gauge theories in the full background and give dual integral representations of ordinary w conformal blocks

quantum spin chains and riemann zeta function with odd arguments the riemann zeta function is an important object of number theory we argue that it is related to the heisenberg spin 1 2 anti ferromagnet in the xxx spin chain we study the probability of formation of a ferromagnetic string in the anti ferromagnetic ground state in the thermodynamics limit we prove that for short strings the probability can be expressed in terms of the riemann zeta function with odd arguments

six vertex model with domain wall boundary conditions variable inhomogeneities we consider the six vertex model with domain wall boundary conditions we choose the inhomogeneities as solutions of the bethe ansatz equations these equations have many solutions so we can consider a wide variety of inhomogeneities for certain choices of the inhomogeneities we study arrow correlation functions on the horizontal line going through the centre in particular we obtain a multiple integral representation for the emptiness formation probability that generalizes the known formulae for xxz antiferromagnets

approximate bias correction in econometrics abstract this paper discusses methods for reducing the bias of consistent estimators that are biased in finite samples these methods are available whenever the bias function which relates the bias of the parameter estimates to the values of the parameters can be estimated by computer simulation or by some other method if so bias can be reduced by one full order in the sample size and in some cases that may not be unrealistic virtually eliminated unfortunately reducing bias may increase the variance or even the mean squared error of an estimator whether it does so depends on the shape of the bias function the results of the paper are illustrated by applying them to two problems estimating the autoregressive parameter in an ar 1 model with a constant term and estimating a logic model

holonomy of the ising model form factors we study the ising model two point diagonal correlation function c n n by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of wu mccoy tracy and barouch we extend this expansion weighting by powers of a variable the j particle contributions f j n n in the form factor expansion the corresponding extension of the two point diagonal correlation function c n n is shown for arbitrary to be a solution of the sigma form of the painleve vi equation introduced by jimbo and miwa in their isomonodromic approach to the ising model fuchsian linear differential equations for the form factors f j n n are obtained for j 9 and shown to have both a russian doll nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second order linear differential operator associated with the elliptic integral e from this we show that each f j n n is unexpectedly simple being expressed polynomially in terms of the elliptic integrals e and k in contrast we exhibit some mathematical objects built from these form factors f j n n which break the direct sum of symmetric powers decomposition with its associated polynomial expressions first we show that the scaling limit of these differential operators and form factors breaks the direct sum structure but not the russian doll structure secondly we show that the previous extension of two point diagonal correlation functions c n n is for singled out values cos m n m n integers also solutions of fuchsian linear differential equations these solutions of painleve vi are not polynomial in e and k but are actually algebraic functions being associated with modular curves

sensitivity analysis of reflector antennas and its application on shaped geo truss unfurlable antennas this paper presents the sensitivity of far field radiation performance with respect to reflector surface nodal displacements the reflector surface is divided into a series of triangular elements and in any deformation state the surface can be determined by the element nodal displacements the radiation integrals over each small element are summed to obtain the whole electromagnetic performance with the aid of shape functions in triangular elements the integration points inside each element are expressed as functions of element nodal displacements then the partial derivatives of electric far field and directivity with respect to each nodal displacement are derived the sensitivity contour plots of maximum directivity with respect to surface nodal displacements for axis symmetrical and offset parabolic reflectors are numerical simulated the sensitivity is applied to a shaped geo truss unfurlable antenna with accepted results

emptiness formation probability of the xxz spin heisenberg chain at δ using a multiple integral representation for the correlation functions we compute the emptiness formation probability of the xxz spin heisenberg chain at anisotropy we prove that it is expressed in terms of the number of alternating sign matrices

the finite sample effects of var dimensions on ols bias ols variance and minimum mse estimators vector autoregressions vars are important tools in time series analysis however relatively little is known about the finite sample behaviour of parameter estimators we address this issue by investigating ordinary least squares ols estimators given a data generating process that is a purely nonstationary first order var specifically we use monte carlo simulation and numerical optimisation to derive response surfaces for ols bias and variance in terms of var dimensions given correct specification and several types of over parameterisation of the model we include a constant and a constant and trend and introduce excess lags we then examine the correction factors that are required for the least squares estimator to attain the minimum mean squared error mse our results improve and extend one of the main finite sample multivariate analytical bias results of abadir hadri and tzavalis abadir k m hadri k tzavalis e 1999 the influence of var dimensions on estimator biases econometrica 67 163 181 generalise the univariate variance and mse findings of abadir abadir k m 1995 unbiased estimation as a solution to testing for random walks economics letters 47 263 268 to the multivariate setting and complement various asymptotic studies

the chiral space of local operators in su 2 invariant thirring model the space of local operators in the su 2 invariant thirring model su 2 itm is studied by the form factor bootstrap method by constructing sets of form factors explicitly we define a susbspace of operators which has the same character as the level one integrable highest weight representation of this makes a correspondence between this subspace and the chiral space of local operators in the underlying conformal field theory the su 2 wess zumino witten model at level one

relation between primes and nontrivial zeros in the riemann hypothesis legendre polynomials modified zeta function and schrodinger equation we study the dependence between prime numbers and the real and imaginary parts of the nontrivial zeros of the riemann zeta function the legendre polynomials and the partial derivatives of the riemann zeta function are used to investigate the above dependence along with the riemann hypothesis with physical interpretations a modified zeta function with finite terms is defined as a new implement for the study of the zeta function and its zeros

the central extension of the reflection equations and an analog of miki s formula two different types of centrally extended quantum reflection algebras are introduced realizations in terms of the elements of the central extension of the yang baxter algebra are exhibited a co action map is identified for the special case of a realization in terms of elements satisfying the zamolodchikov faddeev algebra a boundary analog of miki s formula is also proposed providing a free field realization of q onsager currents

elliptic algebra in the scaling limit the scaling limit of the elliptic algebra is investigated the limiting algebra is defined in terms of a continuous family of generators being fourier harmonics of gauss coordinates of the l operator the ding frenkel isomorphism between l operator s and current descriptions of the algebra is established and is identified with the riemann problem on a strip the representations coalgebraic structure and intertwining operators of the algebra are studied

cohomologies of affine hyperelliptic jacobi varieties and integrable systems we study the affine ring of the affine jacobi variety of a hyperelliptic curve the matrix construction of the affine hyperelliptic jacobi varieties due to mumford is used to calculate the character of the affine ring by decomposing the character we make several conjectures on the cohomology groups of the affine hyperelliptic jacobi varieties in the integrable system described by the familly of these affine hyperelliptic jacobi varieties the affine ring is closely related to the algebra of functions on the phase space classical observables we show that the affine ring is generated by the highest cohomology group over the action of the invariant vector fields on the jacobi variety

on the theory of the synthesis of single and dual offset shaped reflector antennas since kinber radio technika and engineering 1963 and galindo ieee trans antennas propagat 1963 1964 developed the solution to the circular symmetric dual shaped synthesis problem the question of existence and of uniqueness for offset dual or single shaped synthesis has been a point of controversy many researchers thought that the exact offset solutions may not exist later galindo israel and mittra ieee trans antennas propagat 1979 and others formulated the problem exactly and obtained excellent and numerically efficient but approximate solutions using a technique similar to that first developed by schruben for the single reflector problem journal of the optical society 1973 brickell and westcott proc institute of electrical engineering 1981 developed a monge ampere ma second order nonlinear partial differential equation for the dual reflector problem they solved an elliptic form of this equation by a technique introduced by rall 1979 which iterates by a newton method a finite difference linearized ma equation the elliptic character requires a set of finite difference equations to be developed and solved iteratively existence still remained in question although the second order ma equation developed by schruben is elliptic the first order equations from which the ma equation is derived can be integrated progressively e g as for an initial condition problem such as for hyperbolic equations a noniterative and usually more rapid type solution in this paper we have solved numerically the first order equations exact solutions are thus obtained by progressive integration furthermore we have concluded that not only does an exact solution exist but an infinite set of such solutions exists these conclusions are inferred in part from numerical results

hidden grassmann structure in the xxz model for the critical xxz model we consider the space mathcal w alpha of operators which are products of local operators with a disorder operator we introduce two anti commutative families of operators bf b zeta bf c zeta which act on mathcal w alpha these operators are constructed as traces over representations of the q oscillator algebra in close analogy with baxter s q operators we show that the vacuum expectation values of operators in mathcal w alpha can be expressed in terms of an exponential of a quadratic form of bf b zeta bf c zeta

quantum kz equation with q 1 and correlation functions of the xxz model in the gapless regime an integral solution to the quantum knizhnik zamolodchikov qkz equation with q 1 is presented upon specialization it leads to a conjectural formula for correlation functions of the xxz model in the gapless regime the validity of this conjecture is verified in special cases including the nearest neighbour correlator with an arbitrary coupling constant and general correlators in the xxx and xy limits

third neighbour and other four point correlation functions of spin 1 2 xxz chain the correlation functions of the spin 1 2 xxz chain in the ground state were expressed in the form of multiple integrals for 1 1 and 1 in particular adjacent four point correlation functions were given as certain four dimensional integrals we show that these integrals can be reduced to polynomials with respect to specific one dimensional integrals the results give the polynomial representation of the third neighbour correlation functions

new formulae for solutions of quantum knizhnik zamolodchikov equations on level 4 we present a new form of solution to the quantum knizhnik zamolodchikov equation qkz on level 4 in a special case corresponding to the heisenberg xxx spin chain our form is equivalent to the integral representation obtained by jimbo and miwa in 1996 7 an advantage of our form is that it is reduced to the product of single integrals this fact is deeply related to a cohomological nature of our formulae our approach is also based on the deformation of hyper elliptic integrals and their main property deformed riemann bilinear relation jimbo and miwa also suggested a nice conjecture which relates solution of the qkz on level 4 to any correlation function of the xxx model this conjecture together with our form of solution to the qkz makes it possible to prove a conjecture that any correlation function of the xxx model can be expressed in terms of the riemann function at odd arguments and rational coefficients suggested in 8 9 this issue will be discussed in our next publication

fifth neighbour spin spin correlator for the anti ferromagnetic heisenberg chain we study the generating function of the spin spin correlation functions in the ground state of the anti ferromagnetic spin 1 2 heisenberg chain without magnetic field we have found its fundamental functional relations from those for general correlation functions which originate in the quantum knizhnik zamolodchikov equation using these relations we have calculated the explicit form of the generating functions up to n 6 accordingly we can obtain the spin spin correlator szjszj k up to k 5

correlation functions of the higher spin xxx chains using the algebraic bethe ansatz we consider the correlation functions of the integrable higher spin chains we apply a method recently developed for the spin heisenberg chain based on the solution of the quantum inverse problem we construct a representation for the correlation functions on a finite chain for arbitrary spin then we show how the string solutions of the bethe equations can be considered in the framework of this approach in the thermodynamic limit finally a multiple integral representation for the spin 1 zero temperature correlation functions is obtained in the thermodynamic limit

interpolation methods for shaped reflector analysis the diffraction analysis of reflector surfaces which are described only at a discrete set of locations usually leads to the requirement of an interpolation to determine the surface characteristics over a continuum of locations two methods of interpolation the global and the local methods are presented the global interpolation representation is a closed form or series expression valid over the entire surface the coefficients of a series expression are found by an integration of all of the raw data since the number of coefficients used to describe the surface is much smaller than the number of raw data points the integration effectively provides a smoothing of the raw data the local interpolation provides a closed form expression for only a small area of the reflector surface the subreflector is divided into sectors each of which has constant discretized data each area segment is then locally described by a two dimensional quadratic surface the second derivative data give the desired smoothed values

density matrices for finite segments of heisenberg chains of arbitrary length we derive a multiple integral representing the ground state density matrix of a segment of length m of the xxz spin chain on l lattice sites which depends on l only parametrically this allows us to treat chains of arbitrary finite length specializing to the isotropic limit of the xxx chain we show for small m that the multiple integrals factorize we conjecture that this property holds for arbitrary m and suggest an exponential formula for the density matrix which involves only a double cauchy type integral in the exponent we demonstrate the efficiency of our formula by computing the next to nearest neighbour zz correlation function for chain lengths ranging from two to macroscopic numbers

integral representations for correlation functions of the xxz chain at finite temperature we derive a novel multiple integral representation for a generating function of the z z correlation functions of the spin xxz chain at finite temperature and finite longitudinal magnetic field our work combines algebraic bethe ansatz techniques for the calculation of matrix elements with the quantum transfer matrix approach to thermodynamics

form factors of boundary fields for the a2 affine toda field theory in this paper we carry out the boundary form factor program for the a2 affine toda field theory at the self dual point the latter is an integrable model consisting of a pair of particles which are conjugated to each other that is and possessing two bound states resulting from the scattering processes 1 1 2 and 2 2 1 we obtain solutions up to four particle form factors for two families of fields which can be identified with spinless and spin 1 fields of the bulk theory previously known as well as new bulk form factor solutions are obtained as a particular limit of ours minimal solutions of the boundary form factor equations for all an affine toda field theories are given which will serve as the starting point for a generalization of our results to higher rank algebras

the modular properties and the integral representations of the multiple elliptic gamma functions abstract we show the modular properties of the multiple elliptic gamma functions which are an extension of those of the theta function and the elliptic gamma function the modular property of the theta function is known as jacobi s transformation and that of the elliptic gamma function was provided by felder and varchenko in this paper we deal with the multiple sine functions since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite products we also derive integral representations of the multiple sine functions and the multiple elliptic gamma functions we introduce correspondences between the multiple elliptic gamma functions and the multiple sine functions

an approximation of pattern analysis for distorted reflector antennas using structural electromagnetic coupling model based on the concept of structural electromagnetic field coupling an approximation method is developed to analyze the radiation pattern for distorted reflectors in this method the integration point deformation is obtained by a local interpolation scheme which utilizes the same shape functions as that in the structural analysis for the radiation calculation the phase error caused by the distortion is added to the far field integral which is computed on the structural element similar to the assembly of element stiffness matrices in finite element method fem the far field electrical vector is derived as a function of structural nodal displacements in a matrix form by collecting the integrals on structural elements simulation is performed with good results the method can benefit the computation for a given reflector in the integrated structural electromagnetic optimization design procedure

q special functions with q 1 and their application to discrete integrable systems solutions for certain discrete integrable systems are constructed by using integral solutions for hypergeometric q difference systems with q 1 we apply a solution for lauricella s d type hypergeometric q difference system with q 1 to construct a solution for the discrete kp hierarchy furthermore an integral solution of a q difference analogue of bessel s equation is newly introduced and applied to construct a solution for a q difference analogue of the cylindrical toda equation with q 1

large distance asymptotic behavior of the emptiness formation probability of the xxz spin 1 2 heisenberg chain using its multiple integral representation we compute the large distance asymptotic behaviour of the emptiness formation probability of the xxz spin 1 2 heisenberg chain in the massless regime

asymptotic properties of multivariate nonstationary processes with applications to autoregressions asymptotic properties of multivariate time series with characteristic roots on the unit circle are considered for a vector autoregressive moving average process we derive the limiting distributions of certain statistics which are useful in understanding nonstationary processes these distributions are derived in a unified manner for all types of characteristic roots and are expressed in terms of stochastic integrals of brownian motions we use the limiting distributions to establish the consistency properties of the ordinary least squares estimates of various autoregressions of a vector process

higher level appell functions modular transformations and characters we study modular transformation properties of a class of indefinite theta series involved in characters of infinite dimensional lie superalgebras the level l appell functions open image in new window satisfy open quasiperiodicity relations with additive theta function terms emerging in translating by the period generalizing the well known interpretation of theta functions as sections of line bundles the open image in new window function enters the construction of a section of a rank l 1 bundle open image in new window we evaluate modular transformations of the open image in new window functions and construct the action of an sl 2 subgroup that leaves the section of open image in new window constructed from open image in new window invariant

reconstructing distortions on reflector antennas with the iterative field matrix method using near field observation data this work extends the mathematical formulation of the iterative field matrix method for observed data from the near field region of a perfect electric conductor the method is used as a diagnosis tool for reflector antennas to determine the positions and extent of distortions from their idealized shapes the new formulation is tested on a reflector antenna with several significant bumps and excellent results are achieved this work also presents an example where the method of moments is used to generate the synthetic data and the inversion is performed using physical optics such a configuration ensures that the forward model is unbiased with respect to the inversion model demonstrating that the new formulation is also robust for these realistic scenarios

new developments in the eight vertex model we demonstrate that the q matrix introduced in baxter s 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for q v which both contains the bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues

integral representation of the density matrix of the xxz chain at finite temperatures we present an integral formula for the density matrix of a finite segment of the infinitely long spin xxz chain this formula is valid for any temperature and any longitudinal magnetic field

bootstrap equations and correlation functions for the heisenberg xyz antiferromagnet we present two kinds of integral solutions to the quantum knizhnik zamolodchikov equations for the 2n point correlation functions of the heisenberg xyz antiferromagnet our first integral solution can be obtained from those for the cyclic sos model by using the vertex face correspondence by the construction the sum with respect to the local height variables k0 k1 k2n of the cyclic sos model remains other than the n fold integral in the first solution in order to perform these summations we improve this to find the second integral solution of r 1 n fold integral for r 1 where r is a parameter of the xyz model furthermore we discuss the relations among our formula lashkevich pugai s formula and that of shiraishi

matrix models with hard walls geometry and solutions we discuss various aspects of most general multisupport solutions to matrix models in the presence of hard walls i e in the case where the eigenvalue support is confined to subdomains of the real axis the structure of the solution at the leading order is described by semiclassical or generalized whitham krichever hierarchies as in the unrestricted case derivatives of tau functions for these solutions are associated with families of riemann surfaces with possible double points and satisfy the witten dijkgraaf verlinde verlinde equations we then develop the diagrammatic technique for finding free energy of this model in all orders of the t hooft expansion in the reciprocal matrix size generalizing the feynman diagrammatic technique for the hermitian one matrix model due to eynard