critical exponents from power spectra we have simulated the two and three dimensional ising models at their respective critical points with a conventional monte carlo algorithm from the power spectrum of the magnetization autocorrelations we have determined the dynamic critical exponents and obtained the valuesz 2 16 2 19 andz 2 05 in agreement with the results quoted in the literature we have also studied the power spectrum for the kardar parisi zhang and sun guo grant equations describing interface dynamics arguments similar to what was recently used to conclude thatz 4 for model b in critical dynamics were applied to the sun guo grant growth model and the known exact values for the roughening and dynamic exponents were obtained from an analysis of the corresponding power spectrum in self organized critical sand models one obtains a recently proposed hyperscaling relation

time dependent critical properties of ising models by damage spreading abstract we use the method of damage spreading to study the time t taken for a spin at a distance r to be damaged for the first time in ising models with heat bath dynamics we report time dependent scaling exponents d t 2 0 1d 2 52 2d 2 26 3d and 2 0 4d if we assume that t r dt we also analyze our data assuming various forms of logarithmic corrections and we find that d t 2 25 or less 2d and 2 0 3d we argue that it is extremely plausible that d t is equal to z the dynamic critical exponent and thus our estimate of z in two dimensions is higher than the consensus value of 2 14 but we find good agreement in one three and four dimensions with the exact and expected values

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

generalized space forms spaces with radially symmetric curvature at base point p are shown to be diffeomorphic to space forms furthermore they are either isometric to r n or s n under a radially symmetric metric to rp n with riemannian universal covering of s n equipped with a radially symmetric metric or else have constant curvature outside a metric ball of radius equal to the injectivity radius at p

dimension gap under sobolev mappings abstract we prove an essentially sharp estimate in terms of generalized hausdorff measures for the images of boundaries of holder domains under continuous sobolev mappings satisfying suitable orlicz sobolev conditions this estimate marks a dimension gap which was first observed in 2 for conformal mappings

non uniform hyperbolicity in complex dynamics we say that a rational function f satisfies the summability condition with exponent if for every critical point c which belongs to the julia set j there exists a positive integer nc so that sum n 1 infty f n prime f n c c alpha infty and f has no parabolic periodic cycles let max be the maximal multiplicity of the critical points

a removability theorem for sobolev functions and detour sets we study the removability of compact sets for continuous sobolev functions in particular we focus on sets with infinitely many complementary components called detour sets which resemble the sierpinski gasket the main theorem is that if k subset mathbb r n is a detour set and its complementary components are sufficiently regular then k is w 1 p removable for p n several examples and constructions of sets where the theorem applies are given including the sierpinski gasket apollonian gaskets and julia sets

critical slowing down in ising model for creutz algorithm abstract dynamical critical exponents for the two dimensional ising model are computed on a cellular automaton from the relaxations of the time displaced correlation and auto correlation functions for the order parameter and the internal energy at the critical temperatures and from the nonlinear relaxation time for the order parameter near the critical temperature the analysis of the data within the frame of the dynamical finite size scaling hypothesis gives z m z e 2 20 and z a m 1 91 and z a e 0 19 for the linear dynamical exponents corresponding respectively to the relaxations of the correlation and the autocorrelation functions for the order parameter and the internal energy and nl m 2 07 for the nonlinear dynamical exponent for the order parameter these values verify the scaling relations between the dynamical exponents

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below ii we prove as our main theorem the finiteness of topological type of a complete open riemannian manifold m with a base point p m whose radial curvature at p is bounded from below by that of a non compact model surface of revolution m which admits a finite total curvature and has no pair of cut points in a sector here a sector is by definition a domain cut off by two meridians emanating from the base point p m notice that our model m does not always satisfy the diameter growth condition introduced by abresch and gromoll in order to prove the main theorem we need a new type of the toponogov comparison theorem as an application of the main theorem we present a partial answer to milnor s open conjecture on the fundamental group of complete open manifolds

rigidity theorems for circle domains a circle domain omega in the riemann sphere is conformally rigid if every conformal map from omega onto another circle domain is the restriction of a mobius transformation we show that circle domains satisfying a certain quasihyperbolic condition which was considered by jones and smirnov ark mat 38 263 279 2000 are conformally rigid in particular holder circle domains and john circle domains are all conformally rigid this provides new evidence for a conjecture of he and schramm relating rigidity and conformal removability

lifetimes of conditioned diffusions we investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non divergence form can be found the critical value of the parameter is found for each of the following classes of domains l p domains p n 1 uniformly regular twistedl p domains p n 1 and twisted holder domains 1 3 a related parabolic boundary harnack principle is proved

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

geometrical properties of swendsen wang clusters abstract several geometrical properties of swendsen wang clusters are studied numerically for the 2d ising and 3 state potts models at the critical temperature the fractal dimensions of the total mass backbone elastic backbone shortest path and red bonds are determined the results agree with some theoretical predictions

gradient flows on wasserstein spaces over compact alexandrov spaces we establish the existence of euclidean tangent cones on wasserstein spaces over compact alexandrov spaces of curvature bounded below by using this riemannian structure we formulate and construct gradient flows of functions on such spaces if the underlying space is a riemannian manifold of nonnegative sectional curvature then our gradient flow of the free energy produces a solution of the linear fokker planck equation

non removability of the sierpinski gasket we prove that the sierpinski gasket is non removable for quasiconformal maps thus answering a question of bishop nsf research proposal 2015 http www math stonybrook edu bishop vita nsf15 pdf the proof involves a new technique of constructing an exceptional homeomorphism from mathbb r 2 into some non planar surface s and then embedding this surface quasisymmetrically back into the plane by using the celebrated bonk kleiner theorem bonk and kleiner in invent math 150 1 127 183 2002 we also prove that all homeomorphic copies of the sierpinski gasket are non removable for continuous sobolev functions of the class w 1 p for 1 le p le 2 thus complementing and sharpening the results of the author s previous work ntalampekos in a removability theorem for sobolev functions and detour sets arxiv 1706 07687

quasiconformal and bi lipschitz homeomorphisms uniform domains and the quasihyperbolic metric let d be a proper subdomain of r and kd the quasihyperbolic metric defined by the conformal metric tensor ds2 dist x dd 2ds2 the geodesies for this and related metrics are shown by purely geometric methods to exist and have lipschitz continuous first derivatives this is sharp for kd we also obtain sharp estimates for the euclidean curvature of such geodesies we then use these results to prove a general decomposition theorem for uniform domains in r in terms of embeddings of bi lipschitz balls we also construct a counterexample to the higher dimensional analogue of the decomposition theorem of gehring and osgood the defining density dist x 3d 1 need not be differentiable we primarily are interested in studying the geometry of the quasihyperbolic meric and its geodesies we show that the geodesies of this metric are c11 i e the arclength parametrisation has lipschitz continuous derivatives we show that this is best possible and obtain a sharp result on the euclidean curvature of such geodesies these results are proved for a more general class of metrics in particular metrics defined by locally lipschitz densities we then use these results to prove a gener alisation of the decomposition theorem of gehring and osgood for uniform domains in ri space i wish to express my sincere thanks to f w gehring for suggesting many of these problems for many helpful ideas and simplifications throughout and for allowing me to present his proof of theorem 3 7 i also wish to thank him and d herron for carefully reading the manuscript

boundary harnack principle for symmetric stable processes in this paper we study potential theoretic properties of the symmetric stable processes 0 2 establishing the boundary harnack principle for ratios of harmonic functions on any open sets identifying the martin boundary with the euclidean boundary for open sets with a certain interior fatness property and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets

compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below let m be a complete open n manifold with a base point p at which the radial sectional curvature along every minimizing geodesic emanating from p is bounded below by the radial curvature function of a model surface we discuss the maximal diameter theorem for the compactification of m by attaching the ideal boundary under certain conditions we prove that p becomes a pole and that m is isometric to the n model

dynamical properties of the two and three dimensional ising models by damage spreading the method of damage spreading is used to measure the dynamic exponent z for both the two and three dimensional ising models with heat bath dynamics the authors also measure a new exponent alpha describing the rate at which a damage cloud grows they report values of z 2 24 or 0 04 and alpha 0 77 or 0 11 in two dimensions and z 2 202 or 0 03 and alpha 0 92 or 0 10 in three dimensions

non equilibrium relaxation and interface energy of the ising model abstract from the non equilibrium critical relaxation study of the two dimensional ising model the dynamical critical exponent z is estimated to be 2 165 0 010 for this model the relaxation in the ordered phase of this model is consistent with exp t behavior the interface energy of the three dimensional ising model is studied and the critical exponent of the correlation length and the critical amplitude of the surface tension 0 are estimated to be 0 6250 0 0025 and 1 42 0 04 respectively a dynamic monte carlo renormalization group method is successfully applied to the equilibrium properties of the three dimensional ising model

a density problem for sobolev spaces on gromov hyperbolic domains abstract we prove that for a bounded domain r n which is gromov hyperbolic with respect to the quasihyperbolic metric especially when is a finitely connected planar domain the sobolev space w 1 is dense in w 1 p for any 1 p moreover if is also jordan or quasiconvex then c r n is dense in w 1 p for 1 p

dynamic exponent for 2d ising model by power spectra method we consider power spectra for fluctuating quantities at a critical point p omega varies as omega phi where phi is determined by the dynamic exponent z and static exponents specific to the quantity considered from the magnetization spectra obtained with simulations z is found to be 2 13 or 0 03 for the two dimensional glauber kinetic ising model hence in accord with recent monte carlo relaxational studies but at odds with those obtained by series expansion and damage spreading methods excellent dynamic finite size scaling for system sizes l 10 to 128 supports our estimate

critical dynamics a consequence of a random stepwise growth of clusters critical dynamics of correlated particles here glauber dynamics of singly flipping ising spins is explained by a random stepwise growth and contraction of clusters as follows at equilibrium a cluster of size s is described by its length l a random walk like path connecting a sequence of neighbor spins the length scales as l s where constitutes a new static critical exponent we assume that on the average the random growth of a cluster from zero to size s and length ls requires a sequence of l2 spin flips this gives for dynamic critical exponent z 2 where and are the usual static exponents exact results at dimension d 1 and 4 and simulation results at d 2 and 3 support the theory

simulation of time exponent in 2d glauber kinetic ising model abstract simulations of the magnetization relaxation of large square lattices at the curie point give a dynamical exponent near 2 2 somewhat higher than expected

the caffarelli kohn nirenberg inequalities and manifolds with nonnegative weighted ricci curvature we prove that n dimensional n 3 complete and non compact metric measure spaces with non negative weighted ricci curvature in which some caffarelli kohn nirenberg type inequality holds are close to the model metric measure n space i e the euclidean metric n space

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below i we investigate the finiteness structure of a complete non compact n dimensional riemannian manifold m whose radial curvature at a base point of m is bounded from below by that of a non compact von mangoldt surface of revolution with its total curvature greater than we show as our main theorem that all busemann functions on m are exhaustions and that there exists a compact subset of m such that the compact set contains all critical points for any busemann function on m as corollaries by the main theorem m has finite topological type and the isometry group of m is compact

geometry and dynamics of randomly connected fractal clusters abstract the dynamics of critical ising droplets and suitable other clusters is argued to come from growth and decay in small steps not via aggregation of large clusters the resulting critical exponent z for metropolis glauber kinetics is related to the exponent for the chain lengths of the cluster branches simulations of 60960 2 and 1409 3 ising models confirm roughly the exponent z but throw doubts on the small step approach when interpreted with the help of the becker doring equation

a cluster model of critical dynamics abstract critical relaxation exponent z is described by a reversible growth of clusters particles of a cluster are ordered into connected sequences such that a stepwise variation of their length describes relaxation we derive z z where is a scaling exponent relating cluster size to connected length at equilibrium available from simulation of ising clusters furthermore a flory like equation gives and hence z z in d 2 and 3 theoretical z an simulation aided z agree reasonably well with each other and with direct simulations of z in d 4 e z agrees with the exact rg

the lipschitz continuity of the distance function to the cut locus let n be a closed submanifold of a complete smooth riemannian manifold m and uv the total space of the unit normal bundle of n for each v c uv let p v denote the distance from n to the cut point of n on the geodesic y with the velocity vector tyv 0 v the continuity of the function p on uv is well known in this paper we prove that p is locally lipschitz on which p is bounded in particular if m and n are compact then p is globally lipschitz on uy therefore the canonical interior metric 6 may be introduced on each connected component of the cut locus of n and this metric space becomes a locally compact and complete length space let n be an immersed submanifold of a complete cc riemannian manifold m and 7 uv n the unit normal bundle of n for each positive integer k and vector v c uv let a number k v denote the parameter value of fy where 7yv denotes the geodesic for which the velocity vector is v at t 0 such that yv ak v is the k th focal point conjugate point for the case in which n is a point of n along y counted with focal or conjugate multiplicities a unit speed geodesic segment 0 a m emanating from n is called an n segment if t d n y t on 0 a by the first variation formula an n segment is orthogonal to n a point 7y to on an n segment 7y v e uv is called a cut point of n if there is no n segment properly containing y o to for each v c uv let p v denote the distance from n to the cut point on y of n whitehead 27 investigated the structure of the conjugate locus and the cut locus of a point on a real analytic finsler manifold he determined the structure of the conjugate locus around a conjugate point for which the conjugate multiplicity is locally constant on its neighborhood cf also 25 and proved the continuity of the function p in compact symmetric spaces t sakai 19 and m takeuchi 23 determined the detailed structure of the cut locus of a point the detailed structure of the cut locus of a point in a 2 dimensional riemannian manifold has been investigated by poincare myers and others 7 11 13 hartman first tried to show the absolute continuity of the function p when m is 2 dimensional he proved in 8 that if p is of bounded variation then p is absolutely continuous recently hebda 11 and the first named author 13 independently proved ambrose s problem by showing that p is absolutely continuous on a closed arc on which p is bounded when n is a point in a 2 dimensional riemannian manifold therefore the cut locus of a point in a compact 2 dimensional received by the editors october 14 1998 and in revised form april 13 1999 2000 mathematics subject classification primary 53c22 secondary 28a78 supported in part by a grant in aid for scientific research from the ministry of education science sports and culture japan 2000 american mathematical society

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

weighted sobolev spaces and embedding theorems in the present paper we study embedding operators for weighted sobolev spaces whose weights satisfy the well known muckenhoupt a p condition sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities the proposed method is based on the concept of generalized quasiconformal homeomorphisms homeomorphisms with bounded mean distortion the choice of the homeomorphism type depends on the choice of the corresponding weighted sobolev space such classes of homeomorphisms induce bounded composition operators for weighted sobolev spaces with the help of these homeomorphism classes the embedding problem for non smooth domains is reduced to the corresponding classical embedding problem for smooth domains examples of domains with anisotropic holder singularities demonstrate the sharpness of our machinery comparatively with known results

a boundary harnack principle in twisted holder domains the boundary harnack principle for the ratio of positive harmonic functions is shown to hold in twisted holder domains of order a for a e 1 2 1 for each a e 0 1 2 there exists a twisted h6lder domain of order a for which the boundary harnack principle fails extensions are given to l harmonic

description of critical dynamics by static geometry of clusters abstract critical slowing down of relaxation exponent z is modelled by clusters which grow and recede at the end of connected sequences of particles consequently the relaxation time of a cluster of size s is proportional to the number of back and forth steps needed to travel its typical connected length ls this and an assumption on mutual screening of ends lead to z 2 5 0 25 where is a static geometric exponent in ls s ising exponents measured elsewhere with the help of simulation give z in addition a flory like approximation is derived here it relates to and analytically giving z in d 2 3 and 4 respectively we findz 2 14 2 05 and 2 275 1 while z 2 17 4 and 2 04 4 both sets of values are in fair agreement with the literature