the asymmetric six vertex model the exact solution of the asymmetric six vertex model published nearly without derivation by sutherlandet al in 1967 is rederived in detail the transfer matrix method and the bethe ansatz solution for the free energy which can be calculated from an integral equation are discussed for some special cases zero or maximal polarization the integral equation can be solved exactly in addition an asymptotic analysis valid for small but nonzero polarization is carried out the analytical properties of the results and their relevance for the bcsos model are discussed

new constructive methods in classical ideal theory abstract the recent development of computer algebra allows us to take up problems of classical ideal theory for which constructive solutions by nearly unfeasible algorithms exist like the problem of deciding whether a polynomial belongs to a given ideal g hermann math ann 95 1926 736 788 even macaulay s methods by using h bases are not satisfactory at least when considering modules the concept of grobner bases introduced in computer algebra by buchberger allows us to find feasible algorithms for many problems of classical ideal theory by transforming them into some low dimensional systems of linear equations our aim is to demonstrate the effectivity of this new concept considering only polynomial rings p k x 1 x n k a field we generalize in part i the notion of grobner bases to modules present some equivalent properties characterizing these bases and show their good computational qualities in part ii we construct starting with grobner bases of ideals explicit free solutions where all the bases of submodules under consideration are also grobner bases and present a reduction strategy for obtaining shorter resolutions where all the bases have also favourable properties t bases part iii deals with applications we propose a method for constructing minimal resolutions starting with the resolutions of part ii and present a convenient way for computing the hilbert function

static and dynamic phase transitions in multidimensional voting models on continua a voting model or a generalization of the glauber model at zero temperature on a multidimensional lattice is defined as a system composed of a lattice each site of which is either empty or occupied by a single particle the reactions of the system are such that two adjacent sites one empty the other occupied may evolve to a state where both of these sites are either empty or occupied the continuum version of this model in a d dimensional region with a boundary is studied and two general behaviors of such systems are investigated the stationary behavior of the system and the dominant way of relaxation of the system toward its stationary state based on the first behavior a static phase transition discontinuous changes in the stationary profiles of the system is studied based on the second behavior a dynamical phase transition discontinuous changes in the relaxation times of the system is studied it is shown that the static phase transition is induced by the bulk reactions only while the dynamical phase transition is a result of both bulk reactions and boundary conditions

on polynomial ideals of finite codimension with applications to box spline theory we investigate the relation between an ideal i of finite codimension in the space of multivariate polynomials and ideals which are generated by lower order perturbations of some generators for i of particular interest are the codimension of these ideals and the local approximation order of their kernels the discussion stimulated by recent results in approximation theory allows us to provide a simple analysis of the polynomial and exponential spaces associated with box splines we describe their structure dimension and local approximation order and an algorithm for their construction the resulting theory is extended to subspaces of the above exponential polynomial spaces

real bargmann spaces fischer decompositions and sets of uniqueness for polyharmonic functions in this paper a positive answer is given to the following question of w k hayman if a polyharmonic entire function of order k vanishes on k distinct ellipsoids in the euclidean space r n then it vanishes everywhere moreover a characterization of ellipsoids is given in terms of an extension property of solutions of entire data functions for the dirichlet problem answering a question of d khavinson and h s shapiro these results are consequences from a more general result in the context of direct sum decom positions fischer decompositions of polynomials or functions in the algebra a br of all real analytic functions defined on the ball br of radius r and center 0 whose taylor series of homogeneous polynomials converges compactly in br the main result states that for a given elliptic polynomial p of degree 2k and suciently

random fields on model sets with localized dependency and their diffraction for a random field on a general discrete set we introduce a condition that the range of the correlation from each site is within a predefined compact set d for such a random field defined on the model set that satisfies a natural geometric condition we develop a method to calculate the diffraction measure of the random field the method partitions the random field into a finite number of random fields each being independent and admitting the law of large numbers the diffraction measure of consists almost surely of a pure point component and an absolutely continuous component the former is the diffraction measure of the expectation e while the inverse fourier transform of the absolutely continuous component of turns out to be a weighted dirac comb which satisfies a simple formula moreover the pure point component will be understood quantitatively in a simple exact formula if the weights are continuous over the internal space of then we provide a sufficient condition that the diffraction measure of a random field on a model set is still pure point

diffractive point sets with entropy after a brief historical survey this paper introduces the notion of entropic model sets cut and project sets and more generally the notion of diffractive point sets with entropy such sets may be thought of as generalizations of lattice gases we show that taking the site occupation of a model set stochastically results with probabilistic certainty in well defined diffractive properties augmented by a constant diffuse background we discuss both the case of independent but identically distributed i i d random variables and that of independent but different i e site dependent random variables several examples are shown

pure point diffraction and cut and project schemes for measures the smooth case we present a cut and project formalism based on measures and continuous weight functions of sufficiently fast decay the emerging measures are strongly almost periodic the corresponding dynamical systems are compact groups and homomorphic images of the underlying torus in particular they are strictly ergodic with pure point spectrum and continuous eigenfunctions their diffraction can be calculated explicitly our results cover and extend corresponding earlier results on dense dirac combs and continuous weight functions with compact support they also mark a clear difference in terms of factor maps between the case of continuous and non continuous weight functions

non abelian symmetries of stochastic processes derivation of correlation functions for random vertex models and disordered interacting particle systems we consider systems of particles hopping stochastically on d dimensional lattices with space dependent probabilities we map the master equation onto an evolution equation in a fock space where the dynamics are given by a quantum hamiltonian continuous time or a transfer matrix discrete time using non abelian symmetries of these operators we derive duality relations expressing the time evolution of a given initial configuration in terms of correlation functions of simpler dual processes particularly simple results are obtained for the time evolution of the density profile as a special case we show that for any su 2 symmetric system the two point and three point density correlation functions in the n particle steady state can be computed from the probability distribution of a single particle moving in the same environment we apply our results to various models among them partial exclusion a simple diffusion reaction system and the two dimensional six vertex model with space dependent vertex weights for a random distribution of the vertex weights one obtains a version of the random barrier model describing diffusion of particles in disordered media we derive exact expressions for the averaged two point density correlation functions in the presence of weak correlated disorder

reaction diffusion processes of hard core particles we study a 12 parameter stochastic process involving particles with two site interaction and hard core repulsion on ad dimensional lattice in this model which includes the asymmetric exclusion process contact processes and other processes the stochastic variables are particle occupation numbers taking valuesn x 0 1 we show that on a ten parameter submanifold thek point equal time correlation functions n x1 n xk satisfy linear differential difference equations involving no higher correlators in particular the average density n x satisfies an integrable diffusion type equation these properties are explained in terms of dual processes and various duality relations are derived by defining the time evolution of the stochastic process in terms of a quantum hamiltonianh the model becomes equivalent to a lattice model in thermal equilibrium ind 1 dimensions we show that the spectrum ofh is identical to the spectrum of the quantum hamiltonian of ad dimensional anisotropic spin 1 2 heisenberg model in one dimension our results hint at some new algebraic structure behind the integrability of the system

multivariate refinable functions differences and ideals a simple tutorial this paper summarizes the algebraic quotient ideal approach to polynomial generation by refinable functions and connects it to strang fix conditions and factorization with respect to difference operators motivated by the latter one we also consider vector subdivision schemes with matrix valued coefficients and review some of their properties

similarity transformation in one dimensional reaction diffusion systems the voting model as an example the exact solution for a system with two particle annihilation and decoagulation has been studied the spectrum of the hamiltonian of the system is found it is shown that the steady state is twofold degenerate the average number density at each site ni t and the equal time two point functions ni t nj t are calculated any equal time correlation functions at large times ni nj are also calculated the relaxation behaviour of the system toward its final state is investigated and it is shown that generally it is exponential as expected for the special symmetric case the relaxation behaviour of the system is a power law for the asymmetric case it is shown that the profile of deviation from the final values is an exponential function of the position

on multiplicities in polynomial system solving this paper deals with the description of the solutions of zero dimensional systems of polynomial equations based on different models for describing solutions we consider suitable representations of a multiple root or more precisely suitable descriptions of the primary component of the system at a root we analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations

polynomial interpolation of minimal degree minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which lagrange interpolation with respect to the given points is uniquely solvable and degree reducing this is a generalization of the concept of least interpolation introduced by de boor and ron this paper investigates the behavior of lagrange interpolation with respect to these spaces giving a newton interpolation method and a remainder formula for the error of interpolation moreover a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view

multivariate polynomial equations with multiple zeros solved by matrix eigenproblems the eigenproblem method calculates the solutions of systems of polynomial equations f 1 x 1 ldots x s 0 ldots f m x 1 ldots x s 0 it consists in fixing a suitable polynomial f and in considering the matrix a f corresponding to the mapping p mapsto f cdot p where the equivalence classes are modulo the ideal generated by f 1 ldots f m the eigenspaces contain vectors from which all solutions of the system can be read off this access was investigated in 1 and 16 mainly for the case that is nonderogatory in the present paper we study the case where f 1 ldots f m have multiple zeros in common we establish a kind of jordan decomposition of a f reflecting the multiplicity structure and describe the conditions under which a f is nonderogatory the algorithmic analysis of the eigenproblem in the general case is indicated

on diffraction by aperiodic structures this paper gives a rigorous treatment of some aspects of diffraction by aperiodic structures such as quasicrystals it analyses diffraction in the limit of the infinite system through an appropriately defined autocorrelation the main results are a justification of the standard way of calculating the diffraction spectrum of tilings obtained by the projection method and a proof of a variation on a conjecture by bombieri and taylor

grobner bases h bases and interpolation the paper is concerned with a construction for h bases of polynomial ideals without relying on term orders the main ingredient is a homogeneous reduction algorithm which orthogonalizes leading terms instead of completely canceling them this allows for an extension of buchberger s algorithm to construct these h bases algorithmically in addition the close connection of this approach to minimal degree interpolation and in particular to the least interpolation scheme due to de boor and ron is pointed out

limit quasi periodic point sets as quasicrystals with p adic internal spaces model sets or cut and project sets provide a familiar and commonly used method of constructing and studying nonperiodic point sets here we extend this method to situations where the internal spaces are no longer euclidean but instead spaces with p adic topologies or even with mixed euclidean p adic topologies we show that a number of well known tilings precisely fit this form including the chair tiling and the robinson square tilings thus the scope of the cut and project formalism is considerably larger than is usually supposed applying the powerful consequences of model sets we derive the diffractive nature of these tilings

prony s method in several variables the paper gives an extension of prony s method to the multivariate case which is based on the relationship between polynomial interpolation normal forms modulo ideals and h bases though the approach is mainly of algebraic nature we also give an algorithm using techniques from numerical linear algebra to solve the problem in a fast and efficient way

on the singularity of multivariate hermite interpolation in this paper we study the singularity of multivariate hermite interpolation of type total degree we present two methods to judge the singularity of the interpolation schemes considered and by methods to be developed we show that all hermite interpolation of type total degree on m d k points in r d is singular if d 2k and then we solve the hermite interpolation problem on email protected d 3 nodes completely precisely all hermite interpolations of type total degree on email protected d 1 points with d 2 are singular only three cases for m d 2 and one case for m d 3 can produce regular hermite interpolation schemes respectively besides we also present a method to compute the interpolation space for hermite interpolation of type total degree

lagrange interpolation on subgrids of tensor product grids this note shows that a wide class of algebraically motivated constructions for lagrange interpolation polynomials always yields a tensor product interpolation space as long as the nodes form a tensor product grid or a lower subset thereof

the torus parametrization of quasiperiodic li classes the torus parametrization of quasiperiodic local isomorphism classes is introduced and used to determine the number of elements in such a class with special symmetries or inflation properties the method is explained in an illustrative fashion for some widely used tiling classes with golden mean rescaling namely for the fibonacci chain one dimensional the triangle and penrose patterns two dimensional and for kramer s and danzer s icosahedral tilings three dimensional we obtain a rather complete picture of the orbit structure within these classes and also discuss various general results

multivariate hermite interpolation by algebraic polynomials a survey abstract this is a survey of that theory of multivariate lagrange and hermite interpolation by algebraic polynomials which has been developed in the past 20 years its purpose is not to be encyclopedic but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples it takes classical hermite interpolation as a starting point but then successively broadens the assumptions so that finally interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed

equivalences between stochastic systems time dependent correlation functions of unstable particles undergoing biased or unbiased diffusion coagulation and annihilation are studied this is achieved by similarity transformations between different stochastic models and between stochastic and soluble nonstochastic models for special cases we obtain exact results which are in good agreement with experiments on one dimensional annihilation coagulation processes

weighted dirac combs with pure point diffraction a class of translation bounded complex measures which have the form of weighted dirac combs on locally compact abelian groups is investigated given such a dirac comb we are interested in its diffraction spectrum which emerges as the fourier transform of the autocorrelation measure we present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure simultaneously we establish a natural link to the cut and project formalism and to the theory of almost periodic measures our conditions are general enough to cover the known theory of model sets but also to include examples such as the visible lattice points

absence of singular continuous diffraction for discrete multi component particle models particle models with finitely many types of particles are considered both on d and on discrete point sets of finite local complexity such sets include many standard examples of aperiodic order such as model sets or certain substitution systems the particle gas is defined by an interaction potential and a corresponding gibbs measure under some reasonable conditions on the underlying point set and the potential we show that the corresponding diffraction measure almost surely exists and consists of a pure point part and an absolutely continuous part with continuous density in particular no singular continuous part is present

concentration inequalities for functions of gibbs fields with application to diffraction and random gibbs measures we derive useful general concentration inequalities for functions of gibbs fields in the uniqueness regime we also consider expectations of random gibbs measures that depend on an additional disorder field and prove concentration w r t the disorder field both fields are assumed to be in the uniqueness regime allowing in particular for non independent disorder fields the modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the dobrushin contraction coefficient and are explicitly computable on the basis of these inequalities we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in euclidean space restricted to a finite volume here we also allow for thermal dislocations of the scatterers around their equilibrium positions extending recent results for independent scatterers we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean the bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set

continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of bragg peaks we study uniquely ergodic dynamical systems over locally compact sigma compact abelian groups we characterize uniform convergence in wiener wintner type ergodic theorems in terms of continuity of the limit our results generalize and unify earlier results of robinson and assani respectively