critical phenomena and universal exponents in statistical physics on dyson s hierarchical model on etudie le comportement des etats de gibbs a la temperature critique ou a proximite on etudie en details le modele hierarchique de dyson et on presente les plus importants resultats

monotone matrix functions of two variables abstract monotone operator functions were studied by koranyi who generalised lowner s theorem on monotone matrix functions of arbitrary high order n to two variables an alternate proof of koranyi s representation of monotone operator functions is provided

a space time finite element method for the nonlinear schrodinger equation the continuous galerkin method the convergence of a class of continuous galerkin methods for the nonlinear cubic schrodinger equation is analyzed in this paper these methods allow variable temporal stepsizes as well as changing of the spatial grid from one time level to the next we show the existence of the resulting approximations and prove optimal order error estimates in l infty l 2 and in l infty h 1 these estimates are valid under weak restrictions on the space time mesh these restrictions are milder if the elliptic projection is used at every time step instead of the l2 projection we also give superconvergence results at the temporal nodes tn

discontinuous polynomial approximations in the theory of one step hybrid and multistep methods for nonlinear ordinary differential equations this paper studies the approximation of the solution of nonlinear ordinary differen tial equations by discontinuous piecewise polynomials of degree k and traces at the nodes of discretization a mesh dependent variational framework underlying this discontinuous ap proximation is derived several families of one step hybrid and multistep schemes are obtained it is shown that the convergence rate in the l2 norm is k 1 the nodal conver gence rate can go up to 2 k 2 depending on the particular scheme under consideration the mesh dependent variational framework introduced here is of special interest in the approxi mation of the solution of optimal control problems governed by differential equations 1 introduction the object of this paper is the study of discontinuous piecewise polynomial approximations to the solution of systems of nonlinear ordinary dif

hp discontinuous galerkin time stepping for parabolic problems abstract we consider the hp version of the discontinuous galerkin dg time stepping method for linear parabolic problems with non symmetric elliptic spatial operators we derive new analyticity estimates for the exact solutions by means of semigroup techniques these estimates allow us to show that the hp dg time stepping method can resolve start up singularities at exponential rates of convergence

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

matrix convexity of functions of two variables abstract let i j be intervals such that 0 i j let mm be the algebra of all m m complex matrices and let mm i be the set of all hermitian members of mm whose spectrum is contained in i let m be the set of all hermitian projections in mm we give a complete characterization of the set of functions f i j r which satisfy the inequality p q f pap qbq p q p q f a b p q for all a m m i b mn j p m q n we also generalize some results on matrix convexity of functions of one variable to the case of functions of two variables

a hierarchical model of quantum anharmonic oscillators critical point convergence a hierarchical model of interacting quantum particles performing anharmonic oscillations is studied in the euclidean approach in which the local gibbs states are constructed as measures on infinite dimensional spaces the local states restricted to the subalgebra generated by fluctuations of displacements of particles are in the center of the study they are described by means of the corresponding temperature green matsubara functions the result of the paper is a theorem which describes the critical point convergence of such matsubara functions in the thermodynamic limit

optimal order a posteriori error estimates for a class of runge kutta and galerkin methods we derive a posteriori error estimates which exhibit optimal global order for a class of time stepping methods of any order that include runge kutta collocation rk c methods and the continuous galerkin cg method for linear and nonlinear stiff odes and parabolic pdes the key ingredients in deriving these bounds are appropriate one degree higher continuous reconstructions of the approximate solutions and pointwise error representations the reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time step they do not hinge on asymptotics

ergodicity ofl2 semigroups and extremality of gibbs states we extend the classical results of holley stroock on the characterization of extreme gibbs states for the ising model in terms of the irreducibility resp ergodicity of the corresponding glauber dynamics to the case of lattice systems with unbounded linear spin spaces we first develop a general framework to discuss questions of this type using classical dirichlet forms on infinite dimensional state spaces and their associated diffusions we then describe concrete applications to lattice models with polynomial interactions i e the discretep d models of euclidean quantum field theory in addition we prove the equivalence of extremality and shift ergodicity for tempered gibbs states of these models and also discuss this question in the general framework

non gaussian fixed points of the block spin transformation hierarchical model approximation with the use of analyticity techniques recently developed by the authors the e and 1 n expansion type arguments are turned into a rigorous control of the non gaussian fixed point of the hierarchical model renormalization group the present approach should extend beyond the hierarchical approximation and result in mathematical theory of the critical point of statistical mechanics or quantum field theory in three dimensions for small e or largen

laguerre entire functions and the lee yang property laguerre entire functions are the polynomials of a single complex variable possessing real nonpositive zeros only or their limits on compact subsets of c these functions are employed to establish a property of isotropic i e o n invariant probability measures on r n email protected n it is called the lee yang property since in the case n 1 it corresponds to the property of the partition function of certain models of statistical physics first established by t d lee and c n yang a class of measures possessing this property is described certain connections of the lee yang property with other aspects of the analytic theory of probability measures are discussed

a space time finite element method for the nonlinear schro dinger equation the discontinuous galerkin method the convergence of the discontinuous galerkin method for the nonlinear cubic schrodinger equation is analyzed in this paper we show the existence of the resulting approximations and prove optimal order error estimates in l l 2 these estimates are valid under weak restrictions on the space time mesh

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

a posteriori error bounds and global error control for approximation of ordinary differential equations the author analyzes a finite element method for the integration of initial value problems in ordinary differential equations general and contractive problems are treated and quasi optimal a priori and a posteriori error bounds obtained in each case in particular good results are obtained for a class of stiff dissipative problems these results are used to construct a rigorous and robust theory of global error control the author also derives an asymptotic error estimate that is used in a discussion of the behavior of the error in conclusion the properties of the error control are exhibited in a series of numerical experiments

continuous and discontinuous galerkin time stepping methods for nonlinear initial value problems with application to finite time blow up we consider continuous and discontinuous galerkin time stepping methods of arbitrary order as applied to first order initial value ordinary differential equation problems in real hilbert spaces our only assumption is that the nonlinearities are continuous in particular we include the case of unbounded nonlinear operators specifically we develop new techniques to prove general peano type existence results for discrete solutions in particular our results show that the existence of solutions is independent of the local approximation order and only requires the local time steps to be sufficiently small independent of the polynomial degree the uniqueness of local solutions is addressed as well in addition our theory is applied to finite time blow up problems with nonlinearities of algebraic growth for such problems we develop a time step selection algorithm for the purpose of numerically computing the blow up time and provide a convergence result

renormalization of dyson s hierarchical vector valued φ 4 model at low temperatures we investigate dyson s hierarchical vector valued 4 model at low temperatures the case 2 c sqrt 2 is considered the pure phase is constructed and the existence of its large scale limit is proved the limit is gaussian but an unusual normalization has to be chosen in the direction of the spontaneous magnetization one has to divide by the square root of the volume but in the orthogonal direction one has to divide by a different power of the volume for all low temperatures

a nonlinear dynamical system on the set of laguerre entire functions a nonlinear modification of the cauchy problem dtf t z dzf t z zd 2 zf t z t ir 0 z c 0 f 0 z g z l is considered the set l consists of laguerre entire functions which one obtains as a closure of the set of polynomials having real nonpositive zeros only in the topology of uniform convergence on compact subsets of c the modification means that the time half line ir is divided onto the intervals in n 1 n n in 0 and on each in the evolution is to be described by the above equation but at the endpoints the function f t z is changed f n z

convergence of a continuous galerkin method with mesh modification for nonlinear wave equations we consider space time continuous galerkin methods with mesh modification in time for semilinear second order hyperbolic equations we show a priori estimates in the energy norm without mesh conditions under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates estimates of the jump in the riesz projection in two successive time steps are also derived

exponential convergence in h 1 of hp fem for gevrey regularity with isotropic singularities for functions u in h 1 omega in a bounded polytope omega subset mathbb r d d 1 2 3 with plane sides for d 2 3 which are gevrey regular in overline omega backslash mathscr s with point singularities concentrated at a set mathscr s subset overline omega consisting of a finite number of points in overline omega we prove exponential rates of convergence of hp version continuous galerkin finite element methods on affine families of regular simplicial meshes in omega the simplicial meshes are geometrically refined towards mathscr s but are otherwise unstructured

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

hp discontinuous galerkin time stepping for volterra integrodifferential equations we present an hp error analysis of the discontinuous galerkin time stepping method for volterra integrodifferential equations with weakly singular kernels we derive new error bounds that are explicit in the time steps the degrees of the approximating polynomials and the regularity properties of the exact solution it is then shown that start up singularities can be resolved at exponential rates of convergence by using geometrically graded time steps our theoretical results are confirmed in a series of numerical tests

partition function zeros for the ising model on complete graphs and on annealed scale free networks we analyse the partition function of the ising model on graphs of two different types complete graphs wherein all nodes are mutually linked and annealed scale free networks for which the degree distribution decays as p k k we are interested in zeros of the partition function in the cases of complex temperature or complex external field fisher and lee yang zeros respectively for the model on an annealed scale free network we find an integral representation for the partition function which in the case 5 reproduces the zeros for the ising model on a complete graph for 3 5 we derive the dependent angle at which the fisher zeros impact onto the real temperature axis this in turn gives access to the dependent universal values of the critical exponents and critical amplitudes ratios our analysis of the lee yang zeros reveals a difference in their behaviour for the ising model on a complete graph and on an annealed scale free network when 3 5 whereas in the former case the zeros are purely imaginary they have a non zero real part in latter case so that the celebrated lee yang circle theorem is violated

some convex and monotone matrix functions we prove some matrix monotonicity and matrix convexity properties for functions derived from operator connections as a consequence we prove that a certain product of matrix monotone functions is a matrix monotone function similar results are proved for matrix convex functions these results include some known results for matrix monotone and matrix convex functions

hierarchical ferromagnetic vector spin model possessing the lee yang property thermodynamic limit at the critical point and above the hierarchical ferromagneticn dimensional vector spin model as a sequence of probability measures onrn is considered the starting element of this sequence is chosen to belong to the lee yang class of measures that is defined in the paper and includes most known examples 4 measures gaussian measures and so on for this model we probe two thermodynamic limit theorems one of them is just the classical central limit theorem for weakly dependent random vectors it describes the convergence of classically normed sums of spins when temperature is sufficiently high the other theorem describes the convergence of more than normally normed sums that holds for some fixed temperature it corresponds to the strong dependence of spins which appears at the critical point of the model

dobrushin s uniqueness for quantum lattice systems with nonlocal interaction based on dobrushin s fundamental criterion we prove uniqueness of euclidean gibbs states for a certain class of quantum lattice systems with unbounded spins nonharmonic pair potentials and infinite radius of interaction the necessary estimates on dobrushin s coefficients are obtained from the log sobolev inequality which holds for the one point conditional distributions on the infinite dimensional single spin loop spaces

on an operator inequality abstract jensen s operator inequality characterizes operator convex functions of two variables f hansen proc amer math soc 125 1997 2093 2102 we give a simplified proof of this theorem formulated for matrices

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

euclidean gibbs measures of interacting quantum anharmonic oscillators a rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting dimensional quantum anharmonic oscillators is given the oscillators are indexed by the elements of a countable set mathbb l subset mathbb r d possibly irregular the anharmonic potentials vary from site to site and the interaction has infinite range the description is based on the representation of the gibbs states in terms of path measures the so called euclidean gibbs measures it is proven that a the set of such measures mathcal g rm t is non void and compact b every mu in mathcal g rm t obeys an exponential integrability estimate the same for the whole set mathcal g rm t c every mu in mathcal g rm t has a lebowitz presutti type support d mathcal g rm t is a singleton at high temperatures the case of attractive interaction and nu 1 is studied in more detail we prove that a mathcal g rm t 1 at low temperatures b mathcal g rm t 1 due to quantum effects and at a nonzero external field thereby a qualitative theory of phase transitions and quantum effects which interprets most important experimental data known for the corresponding physical objects is developed

a posteriori error estimation for hp version time stepping methods for parabolic partial differential equations the aim of this paper is to develop an hp version a posteriori error analysis for the time discretization of parabolic problems by the continuous galerkin cg and the discontinuous galerkin dg time stepping methods respectively the resulting error estimators are fully explicit with respect to the local time steps and approximation orders their performance within an hp adaptive refinement procedure is illustrated with a series of numerical experiments

a quantum crystal with multidimensional anharmonic oscillators the quantum anharmonic crystal is made up of a large number of multidimensional anharmonic oscillators arranged in a periodic spatial lattice with a nearest neighbor coupling if the coupling coefficient is sufficiently small then there is a convergent expansion for the ground state of the crystal the estimates on the convergence are independent of the size of the crystal the proof uses the path integral representation of the ground state in terms of diffusion processes the convergence of the cluster expansion depends on the ergodicity properties of these processes

phase transitions and algebra of fluctuation operators in an exactly soluble model of a quantum anharmonic crystal a complete description of the fluctuation operator algebra is given for a quantum crystal showing displacement structural phase transitions in the one phase region the fluctuations are normal and its algebra is non abelian in the two phase region and on the critical line t c 0 the momentum fluctuation is normal the displacement is critical and the algebra is abelian att c 0 quantum phase transition this algebra is non abelian with abnormal displacement and supernormal squeezed momentum fluctuation operators both being dimension dependent