construction of modular branching functions from bethe s equations in the 3 state potts chain in the present work the authors use the single particle excitation energies and the completeness rules of the 3 state antiferromagnetic potts chain which have been obtained from bethe s equation to compute the modular invariant partition function this provides a fermionic construction for the branching functions of the d sub 4 representation of z sub 4 parafermions which complements the bosonic constructions it is found that there are oscillations in some of the correlations and a new connection with the field theory of the lee yang edge is presented 47 refs 9 tabs

design of experiments for estimating inverse quadratic polynomial responses the precision of the parameter estimates from nonlinear response curves depends on the number and positioning of the levels of the stimulus variable and the parameter values for the four parameter inverse quadratic polynomial we show that the optimal designs have four geometrically spaced levels the efficiencies of equally and geometrically spaced designs with more than four levels are compared and it is shown that it is important to extend the range of experimentation as far as possible and that designs with six levels over a wide range with a geometric spacing factor of 2 are robust to a wide range of parameter values

continued fractions with applications i introductory examples ii more basics iii convergence criteria iv continued fractions and three term recurrence relations v correspondence of continued fractions vi hypergeometric functions vii moments and orthogonality viii pade approximants ix some applications in number theory x zero free regions xi digital filters and continued fractions xii applications to some differential equations appendix some continued fraction expansions references index

optimal designs for rational models and weighted polynomial regression sion model of degree p with efficiency function 1 x are presented interest in these designs stems from the fact that they are equivalent to locally d optimal designs for inverse quadratic polynomial models for the unrestricted design space and p n the d optimal designs put equal masses on p 1 points which coincide with the zeros of an ultraspherical polynomial while for p n they are equivalent to d optimal designs for certain trigonometric regression models and exhibit all the curious and interesting features of those designs for the restricted design space 1 1 sufficient but not necessary conditions for the d optimal designs to be based on p 1 points are developed in this case the problem of constructing p 1 point d optimal designs is equivalent to an eigenvalue problem and the designs can be found numerically for n 1 and 2 the problem is solved analytically and specifically the d optimal designs put equal masses at the points 1 and at the p 1 zeros of a sum of n 1 ultraspherical polynomials a conjecture which extends these analytical results to cases with n an integer greater than 2 is given and is examined empirically 1 introduction weighted polynomial regression models with variance functions which depend on the explanatory variable have played and continue to play an important role in the development of classical optimal design theory the reasons for this are essentially twofold first there is a wealth of elegant mathematics associated with the construction of d optimal designs for many of these models in particular for certain classes of variance functions it is possible to show that the d optimal designs put equal masses on p 1 points of support where p is the degree of the polynomial embedded in the model and furthermore it is possible to use tools from the theory of differential equations and of canonical moments in order to establish that these support points coincide with the zeros of classical orthogonal polynomi

stable trace formula cuspidal tempered terms consider a connected reductive group g over a number field f for technical reasons we assume that the derived group of g is simply connected see l1 in l3 langlands partially stabilizes the trace formula for g after making certain assumptions he writes the elliptic regular part of the trace formula for g as a linear combination of the elliptic g regular parts of the stable trace formulas for the elliptic endoscopic groups h of g the function f used in the stable trace formula forh is obtained from the function f used in the trace formula for g by transferring orbital integrals

on the de la garza phenomenon deriving optimal designs for nonlinear models is in general challenging one crucial step is to determine the number of support points needed current tools handle this on a case by case basis each combination of model optimality criterion and objective requires its own proof the celebrated de la garza phenomenon states that under a p 1 th degree polynomial regression model any optimal design can be based on at most p design points the minimum number of support points such that all parameters are estimable does this conclusion also hold for nonlinear models if the answer is yes it would be relatively easy to derive any optimal design analytically or numerically in this paper a novel approach is developed to address this question using this new approach it can be easily shown that the de la garza phenomenon exists for many commonly studied nonlinear models such as the emax model exponential model three and four parameter log linear models emax pk1 model as well as many classical polynomial regression models the proposed approach unifies and extends many well known results in the optimal design literature it has four advantages over current tools i it can be applied to many forms of nonlinear models to continuous or discrete data to data with homogeneous or nonhomogeneous errors ii it can be applied to any design region iii it can be applied to multiple stage optimal design and iv it can be easily implemented

fusion products kostka polynomials and fermionic characters of su r 1 k using a form factor approach we define and compute the character of the fusion product of rectangular representations of su r 1 this character decomposes into a sum of characters of irreducible representations but with q dependent coefficients we identify these coefficients as generalized kostka polynomials using this result we obtain a formula for the characters of arbitrary integrable highest weight representations of su r 1 in terms of the fermionic characters of the rectangular highest weight representations

two dimensional current algebras and affine fusion product in this paper we study a family of commutative algebras generated by two infinite sets of generators these algebras are parametrized by young diagrams we explain a connection of these algebras with the fusion product of integrable irreducible representations of the affine sl2 lie algebra as an application we derive a fermionic formula for the character of the affine fusion product of two modules these fusion products can be considered as a simplest example of the double affine demazure modules

a stable trace formula iii proof of the main theorems this paper is the last of three articles designed to stabilize the trace formula our goal is to stabilize the global trace formula for a general connected group subject to a condition on the fundamental lemma that has been established in some special cases in the first article i we laid out the foundations of the process we also stated a series of local and global theorems which together amount to a stabilization of each of the terms in the trace formula in the second paper ii we established a key reduction in the proof of one of the global theorems in this paper we shall complete the proof of the theorems we shall combine the global reduction of ii with the expansions that were established in section 10 of i we refer the reader to the introduction of i for a general discussion of the problem of stabilization the introduction of ii contains further discussion of the trace formula with emphasis on the elliptic coefficients agll s these objects are basic ingredients of the geometric side of the trace formula

stability and accuracy of differential quadrature method in solving dynamic problems in this paper the differential quadrature method is used to solve dynamic problems governed by second order ordinary differential equations in time the legendre radau chebyshev chebyshev gauss lobatto and uniformly spaced sampling grid points are considered besides two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated the stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices it is found that higher order accurate solutions can be obtained at the end of a time step if the gauss and radau sampling grid points are used however the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used unfortunately the commonly used chebyshev gauss lobatto sampling grid points would not generate unconditionally stable algorithms

spaces of coinvariants and fusion product i from equivalence theorem to kostka polynomials the fusion rule gives the dimensions of spaces of conformal blocks in wess zumino witten wzw theory we prove a dimension formula similar to the fusion rule for spaces of coinvariants of affine lie algebras widehat mathfrak g an equivalence of filtered spaces is established between spaces of coinvariants of two objects highest weight widehat mathfrak g modules and tensor products of finite dimensional evaluation representations of mathfrak g otimes mathbb c t in the widehat mathfrak sl 2 case we prove that their associated graded spaces are isomorphic to the spaces of coinvariants of fusion products and that their hilbert polynomials are the level restricted kostka polynomials

e optimal designs for rational models in this paper experimental designs for a rational model y p x q x e are investigated where p x o 1 x p x p and q x 1 p 1 x p q x q are polynomials and e is a random error two approaches bayesian d optimal and bayesian optimal design for extrapolation are examined the first criterion maximizes the expected increase in shannon information provided by the experiment asymptotically and the second minimizes the asymptotic variance of the maximum likelihood estimator mle of the mean response at an extrapolation point x e corresponding locally optimal designs are also discussed conditions are derived under which a p q 1 point design is a locally d optimal design the bayesian d optimal design is shown to be independent of the parameters in p x and to be equally weighted at each support point if the number of support points is the same as the number of parameters in the model the existence and uniqueness of the locally optimal design for extrapolation are proven the number of support points for the locally optimal design for extrapolation is exactly p q 1 these p q 1 design points are proved to be independent of the extrapolation point x e and the parameters in p x the corresponding weights are also independent of the parameters in p x but depend on x e and are not equal

optimal designs for a class of nonlinear regression models for a broad class of nonlinear regression models we investigate the local e and c optimal design problem it is demonstrated that in many cases the optimal designs with respect to these optimality criteria are supported at the chebyshev points which are the local extrema of the equi oscillating best approximation of the function f 0 0 by a normalized linear combination of the regression functions in the corresponding linearized model the class of models includes rational logistic and exponential models and for the rational regression models the e and c optimal design problem is solved explicitly in many cases

the l2 lefschetz numbers of hecke operators introduction 5 1 hecke operators and l2 cohomology 52 the spectral decomposition of cohomology 6 3 application of the trace formula 54 the functions pa t 5 5 proof of theorem 5 1 6 the main formula appendix bibliography