quadratic pencils and least squares piecewise polynomial approximation for a partition o o m denotes the space of piecewise polynomials of order k and of smoothness m 1 this space can be represented as the graph of the appropriate linear operator between two finite dimensional hilbert spaces it gives an approach to the c de boor problem 1972 on uniform boundedness with respect to in the l norm of the orthogonal projections onto stm and we give the detailed analysis of a quadratic pencil matrix valued polynomial of the second degree which appears in the case of a geometric mesh if 2m k the explicit calculations and estimates of zeros of the characteristic polynomial show that in the case sbx x the geometric mesh with the parameter x 0 m 0 and any partition ax n of the unit interval 0 1 01 0 1 0 1 n n i 1 we define the piecewise polynomial subspace 0 2 s t km n c m1 0 11 where p fel 01vfia k i pk e f ej l il f ia is a polnom1ial caxi of order k 0 s a s n each interior breakpoint a 1 a n generates m continuity conditions 0 3 ia c af ia ya i 0 in to make a functionf e pkt be an element of scm let q qs be the orthogonal projection onto s in l2 0 1 i e with respect to the inner product f g f x g x dx we are interested in q as a map in c 0 1 or l 0 1 and we would like to get the estimates of its norm iiqiioo sup iiqfli lflfloo f where ilf 11k0 esssupo x if x i f e l o 1 received september 29 1980 1980 mathematics subject classification primary 41a1 5 secondary 47a68 this paper was partly supported by nsf grant 7906079 1983 american mathematical society 0025 571 8 82 0000 0750 05 25

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

cutoff for the ising model on the lattice introduced in 1963 glauber dynamics is one of the most practiced and extensively studied methods for sampling the ising model on lattices it is well known that at high temperatures the time it takes this chain to mix in l 1 on a system of size n is o logn whether in this regime there is cutoff i e a sharp transition in the l 1 convergence to equilibrium is a fundamental open problem if so as conjectured by peres it would imply that mixing occurs abruptly at c o 1 logn for some fixed c 0 thus providing a rigorous stopping rule for this mcmc sampler however obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple markov chains already for the one dimensional ising model showing cutoff is a longstanding open problem

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

on the layering transition of an sos surface interacting with a wall ii the glauber dynamics we continue our study of the statistical mechanics of a 2d surface above a fixed wall and attracted towards it by means of a very weak positive magnetic fieldh in the solid on solid sos approximation when the inverse temperature is very large in particular we consider a glauber dynamics for the above model and study the rate of approach to equilibrium in a large cube with arbitrary boundary conditions using the results proved in the first paper of this series we show that for allh h k 1 h k h k being the critical values of the magnetic field found in the previous paper the gap in the spectrum of the generator of the dynamics is bounded away from zero uniformly in the size of the box and in the boundary conditions on the contrary forh h k and free boundary conditions we show that the gap in a cube of sidel is bounded from above and from below by a negative exponential ofl our results provide a strong indication that contrary to what happens in two dimensions for the three dimensional dynamical ising model in a finite cube at low temperature and very small positive external field with boundary conditions that are opposite to the field on one face of the cube and are absent free on the remaining faces the rate of exponential convergence to equilibrium which is positive in infinite volume may go to zero exponentially fast in the side of the cube

the applications of partial integro differential equations related to adaptive wavelet collocation methods for viscosity solutions to jump diffusion models this paper presents adaptive wavelet collocation methods for the numerical solutions to partial integro differential equations pides arising from option pricing in a market driven by jump diffusion process the first contribution of this paper lies in the formulation of the wavelet collocation schemes the integral and differential operators are formulated in the collocation setting exactly and efficiently in both adaptive and non adaptive wavelet settings the wavelet compression technique is employed to replace the full matrix corresponding to the nonlocal integral term by a sparse matrix an adaptive algorithm is developed which automatically obtains the solution on a near optimal grid the second contribution of this paper is the theoretical analysis of the wavelet collocation schemes due to the possible degeneracy of the parabolic operators classical solutions of the jump diffusion models may not exist in this paper we first prove the convergence and stability of the proposed numerical schemes under the framework of viscosity solution theory and then the numerical experiments demonstrate the accuracy and computational efficiency of the methods we developed

cutoff phenomena for random walks on random regular graphs the cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite markov chains to equilibrium many natural families of chains are believed to exhibit cutoff and yet establishing this fact is often extremely challenging an important such family of chains is the random walk on g n d a random d regular graph on n vertices it is well known that almost every such graph for d 3 is an expander and even essentially ramanujan implying a mixing time of o logn according to a conjecture of peres the simple random walk on g n d for such d should then exhibit cutoff with high probability whp as a special case of this durrett conjectured that the mixing time of the lazy random walk on a random 3 regular graph is whp 6 o 1 log2n in this work we confirm the above conjectures and establish cutoff in total variation its location and its optimal window both for simple and for non backtracking random walks on g n d namely for any fixed d 3 the simple random walk on g n d whp has cutoff at d d 2 logd 1n with window order logn surprisingly the non backtracking random walk on g n d whp has cutoff already at logd 1n with constant window order we further extend these results to g n d for any d no 1 that grows with n beyond which the mixing time is o 1 where we establish concentration of the mixing time on one of two consecutive integers

using the refinement equation for evaluating integrals of wavelets the wavelet galerkin method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets this paper studies the problem from the point of view of stationary subdivision schemes one of the main results is to identify these integrals as components of the unique solution of a certain eigenvector moment problem associated with the coefficients of the refinement equation asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

uniformity of the uncovered set of random walk and cutoff for lamplighter chains we show that the measure on markings of znd d 3 with elements of 0 1 given by i i d fair coin flips on the range mathcal r of a random walk x run until time t and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold t tcov znd as a consequence of our methods we show that the total variation mixing time of the random walk on the lamplighter graph z2 znd d 3 has a cutoff with threshold tcov znd we give a general criterion under which both of these results hold other examples for which this applies include bounded degree expander families the intersection of an infinite supercritical percolation cluster with an increasing family of balls the hypercube and the caley graph of the symmetric group generated by transpositions the proof also yields precise asymptotics for the decay of correlation in the uncovered set

glauber dynamics for the mean field ising model cut off critical power law and metastability we study the glauber dynamics for the ising model on the complete graph also known as the curie weiss model for 1 we study metastability in particular we show that the glauber dynamics restricted to states of non negative magnetization has mixing time o n log n

the equivalence of the logarithmic sobolev inequality and the dobrushin shlosman mixing condition given a finite range lattice gas with a compact continuous spin space it is shown cf theorem 1 2 that a uniform logarithmic sobolev inequality cf 1 4 holds if and only if the dobrushin shlosman mixing condition cf 1 5 holds as a consequence of our considerations we also show cf theorems 3 2 and 3 6 that these conditions are equivalent to a statement about the uniform rate at which the associated glauber dynamics tends to equilibrium in this same direction we show cf theorem 3 19 that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the glauber dynamics

universality of cutoff for the ising model on any locally finite geometry the stochastic ising model is known to be contractive when the inverse temperature beta is small enough via classical results of dobrushin and of holley in the 1970s by a general principle proposed by peres the dynamics is then expected to exhibit cutoff however so far cutoff for the ising model has been confirmed mainly for lattices heavily relying on amenability and log sobolev inequalities without these cutoff was unknown at any fixed beta 0 no matter how small even in basic examples such as the ising model on a binary tree or a random regular graph we use the new framework of information percolation to show that in any geometry there is cutoff for the ising model at high enough temperatures precisely on any sequence of graphs with maximum degree d the ising model has cutoff provided that beta kappa d for some absolute constant kappa a result which up to the value of kappa is best possible moreover the cutoff location is established as the time at which the sum of squared magnetizations drops to 1 and the cutoff window is o 1 just as when beta 0 finally the mixing time from almost every initial state is not more than a factor of 1 varepsilon beta faster then the worst one with varepsilon beta to0 as beta to0 whereas the uniform starting state is at least 2 varepsilon beta times faster

information percolation and cutoff for the stochastic ising model we introduce a new framework for analyzing glauber dynamics for the ising model the traditional approach for obtaining sharp mixing results has been to appeal to estimates on spatial properties of the stationary measure from within a multi scale analysis of the dynamics here we propose to study these simultaneously by examining information percolation clusters in the space time slab using this framework we obtain new results for the ising model on z nz throughout the high temperature regime total variation mixing exhibits cutoff with an o 1 window around the time at which the magnetization is the square root of the volume previously cutoff in the full high temperature regime was only known in dimensions d 2 and only with an o log log n window furthermore the new framework opens the door to understanding the effect of the initial state on the mixing time we demonstrate this on the 1d ising model showing that starting from the uniform disordered initial distribution asymptotically halves the mixing time whereas almost every deterministic starting state is asymptotically as bad as starting from the ordered all plus state

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

l boundedness of l 2 projections on splines for a multiple geometric mesh this paper concerns the l2 projectors from loo to the normed linear space of polynomial splines it is shown that for the multiple geometric meshes the loo norms of the corresponding l2 projectors are bounded independently of the mesh ratio 0 introduction let us begin with some notation let k be a positive integer and x x i z a real nondecreasing knot sequence with xi xi k all i set x 00 lim xi i x o lim xji i x oo x00 i 0 i o00o fori e z and x e i define mi k x k xi xi k x i ni k x xi k xi mi k x k where po pr f denotes the rth divided difference of the function f at the points po 0 pr then ni k is called the i th b spline of order k on the knot sequence x for a e rz the rule f x a i niak x defines a function on i if we take the sum to be pointwise every such function is called a polynomial spline of order k with the knot sequence x and their collection is denoted by skx further let denote the normed linear space of bounded polynomial splines of order k with the knot sequence x and norm fll ii sup i f x ix xei we shall be concerned with p the orthogonal projector onto with respect to the ordinary inner product f g ff x g x dx but restricted to l i we want to bound its norm 11 p ii sup iipsf ii 1 iii o fel j received march 26 1984 revised december 16 1985 1980 mathematics subject classification primary 41a15 15a60 this is a part of the author s thesis which was submitted to the graduate school university of wisconsin madison in partial fullfillment of the requirement for ph d degree present address department of mathematics zhejiang university hangzhou zhejiang the people s republic of china 1987 american mathematical society 0025 5718 87 1 00 25 per page 675 this content downloaded from 207 46 13 178 on wed 15 jun 2016 07 00 20 utc all use subject to http about jstor org terms

fast initial conditions for glauber dynamics in the study of markov chain mixing times analysis has centered on the performance from a worst case starting state here in the context of glauber dynamics for the one dimensional ising model we show how new ideas from information percolation can be used to establish mixing times from other starting states at high temperatures we show that the alternating initial condition is asymptotically the fastest one and surprisingly its mixing time is faster than at infinite temperature accelerating as the inverse temperature beta ranges from 0 to beta 0 frac 1 2 mathrm arctanh frac 1 3 moreover the dominant test function depends on the temperature at beta beta 0 it is autocorrelation whereas at beta beta 0 it is the hamiltonian

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc