equilibrium fluctuations for a system of harmonic oscillators with conservative noise we investigate the harmonic chain forced by a multiplicative noise the evolution is given by an infinite system of stochastic differential equations total energy and deformation are preserved the conservation of momentum is destroyed by the noise gaussian product measures are the extremal stationary states of this model equilibrium fluctuations of the conserved fields at a diffusive scaling are described by a couple of generalized ornstein uhlenbeck processes

logarithmic sobolev inequality for some models of random walks we determine the logarithmic sobolev constant for the bernoulli laplace model and the time to stationarity for the symmetric simple exclusion model up to the leading order our method for proving the logarithmic sobolev inequality is based on a martingale approach and is applied to the random transposition model as well the proof for the time to stationarity is based on a general observation relating the time to stationarity to the hydrodynamical limit

conditional lie backlund symmetries and functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations abstract the functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations are studied by applying the conditional lie backlund symmetry method a complete list of canonical forms for such equations are presented exact solutions to the resulting equations are constructed the asymptotic behaviors and blow up properties of some solutions are also discussed

hydrodynamical limit for a hamiltonian system with weak noise starting from a general hamiltonian system with superstable pairwise potential we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles we prolve that in the scaling limit the time conserved quantities energy momenta and density satisfy the euler equation of conservation laws up to a fixed timet provided that the euler equation has a smooth solution with a given initial data up to timet the strength of the noise term is chosen to be very small but nonvanishing so that it disappears in the scaling limit

linearizing mappings for certain nonlinear diffusion equations in this paper we consider the nonlinear diffusion equations of the type it is shown that linearizing point transformations do not exist this equation can be equivalently written as a system of two and three equations respectively linearizing point transformations are sought for these two auxiliary systems and the complete list is presented these in turn may be employed to construct contact transformations which map the nonlinear diffusion equation to a linear partial differential equation such linearizing point transformations exist only if n 2 m and n 2 m

derivation of the leroux system as the hydrodynamic limit of a two component lattice gas the long time behavior of a couple of interacting asymmetric exclusion pro cesses of opposite velocities is investigated in one space dimension we do not allow two particles at the same site and a collision effect exchange takes place when particles of opposite velocities meet at neighboring sites there are two conserved quantities and the model admits hyperbolic euler scaling the hydrodynamic limit results in the classical leroux system of conservation laws even beyond the appearance of shocks actually we prove convergence to the set of entropy solutions the question of unique ness is left open to control rapid oscillations of lax entropies via logarithmic sobolev inequality estimates the symmetric part of the process is speeded up in a suitable way thus a slowly vanishing viscosity is obtained at the macroscopic level following 4 5 the stochastic version of tartar murat theory of compensated compactness is extended to two component stochastic models

conditional lie backlund symmetries and differential constraints for inhomogeneous nonlinear diffusion equations due to linear determining equations abstract the method of linear determining equations is developed to study conditional lie backlund symmetries for evolution equations which is more general than the classical determining equations for lie s generators as an application of this approach the complete classification is presented for the inhomogeneous nonlinear diffusion equations which admit the second order and third order conditional lie backlund symmetries several examples are given to illustrate the corresponding symmetry reductions due to the compatibility of the invariant conditions and the governing equations

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

exact similarity solutions to some nonlinear diffusion equations the author gives new closed form similarity solutions to n dimensional radially symmetric nonlinear diffusion equations with power law and exponential diffusivities these solutions are generalisations of known similarity solutions

hydrodynamic limit of condensing two species zero range processes with sub critical initial profiles two species condensing zero range processes zrps are interacting particle systems with two species of particles and zero range interaction exhibiting phase separation outside a domain of sub critical densities we prove the hydrodynamic limit of nearest neighbour mean zero two species condensing zrp with bounded local jump rate for sub critical initial profiles i e for initial profiles whose image is contained in the region of sub critical densities the proof is based on h t yau s relative entropy method which relies on the existence of sufficiently regular solutions to the hydrodynamic equation in the particular case of the species blind zrp we prove that the solutions of the hydrodynamic equation exist globally in time and thus the hydrodynamic limit is valid for all times

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

on point transformations of generalized nonlinear diffusion equations this paper classifies all finite point transformations of a general class between generalized diffusion equations of the form ut x1 m xn 1f u ux x these transformations may be divided into three cases depending on the functional form of f u i f arbitrary ii f un and iii f eu in particular these transformations include all the invariant infinitesimal transformations and in addition they include a number of point transformations which relate different equations of the above form many exact solutions are already known and the transformations which are derived may be used to obtain new solutions from these

entropy dissipation estimates in a zero range dynamics we study the exponential decay of relative entropy functionals for zero range processes on the complete graph for the standard model with rates increasing at infinity we prove entropy dissipation estimates uniformly over the number of particles and the number of vertices

mixing of the symmetric exclusion processes in terms of the corresponding single particle random walk we prove an upper bound for the e mixing time of the symmetric exclusion process on any graph g with any feasible number of particles our estimate is proportional to trw g ln v e where v is the number of vertices in g and trw g is the 1 4 mixing time of the corresponding single particle random walk this bound implies new results for symmetric exclusion on expanders percolation clusters the giant component of the erdos renyi random graph and poisson point processes in rd our technical tools include a variant of morris s chameleon process

regression adjustment in completely randomized experiments with a diverging number of covariates extending r a fisher and d a freedman s results on the analysis of covariance lin 2013 proposed an ordinary least squares adjusted estimator of the average treatment effect in completely randomized experiments we further study its statistical properties under the potential outcomes model in the asymptotic regimes allowing for a diverging number of covariates we show that lin 2013 s estimator is consistent when kappa log p rightarrow 0 and asymptotically normal when kappa p rightarrow 0 under mild moment conditions where kappa is the maximum leverage score of the covariate matrix in the favorable case where leverage scores are all close together his estimator is consistent when p o n log n and is asymptotically normal when p o n 1 2 in addition we propose a bias corrected estimator that is consistent when kappa log p rightarrow 0 and is asymptotically normal with the same variance in the fixed p regime when kappa 2 p log p rightarrow 0 in the favorable case the latter condition reduces to p o n 2 3 log n 1 3 similar to lin 2013 our results hold for non random potential outcomes and covariates without any model specification our analysis requires novel analytic tools for sampling without replacement which complement and potentially enrich the theory in other areas such as survey sampling matrix sketching and transductive learning

the logarithmic sobolev constant of the lamplighter we give estimates on the logarithmic sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base also we give examples of application

perturbation of singular equilibria of hyperbolic two component systems a universal hydrodynamic limit we consider one dimensional locally finite interacting particle systems with two conservation laws which under the eulerian hydrodynamic limit lead to two by two systems of conservation laws t x u 0 t u x u 0

hydrodynamics and hydrostatics for a class of asymmetric particle systems with open boundaries we consider attractive particle systems in mathbb z d with product invariant measures we prove that when particles are restricted to a subset of mathbb z d with birth and death dynamics at the boundaries the hydrodynamic limit is given by the unique entropy solution of a conservation law with boundary conditions in the sense of bardos et al comm part diff equ 4 1017 1034 1979 for the hydrostatic limit between parallel hyperplanes we prove a multidimensional version of the phase diagram conjectured in popkov and schutz europhys lett 48 257 263 1999 and show that it is robust with respect to perturbations of the boundaries

onsager relations and eulerian hydrodynamic limit for systems with several conservation laws we present the derivation of the hydrodynamic limit under eulerian scaling for a general class of one dimensional interacting particle systems with two or more conservation laws following yau s relative entropy method it turns out that in case of more than one conservation laws in order that the system exhibit hydrodynamic behaviour some particular identities reminiscent of onsager s reciprocity relations must hold we check validity of these identities whenever a stationary measure with product structure exists it also follows that as a general rule the equilibrium thermodynamic entropy as function of the densities of the conserved variables is a globally convex lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit as concrete examples we also present a number of models modeling deposition or domain growth phenomena the onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel the fact that equilibrium thermodynamic entropy is lax entropy for the arising euler equations was noticed earlier in the context of hamiltonian systems with weak noise see ref 7

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

logarithmic sobolev inequality for generalized simple exclusion processes let be a probability measure on the set 0 1 r for some r and l a cube of width l in d denote by gc l the grand canonical product measure on the configuration space on l with as the marginal measure here the superscript indicates the grand canonical ensemble the canonical ensemble denoted by c l n is defined by conditioning gc l given the total number of particles to be n consider the exclusion dynamics where each particle performs random walk with rates depending only on the number of particles at the same site the rates are chosen such that for every n and l fixed the measure c l n is reversible we prove the logarithmic sobolev inequality in the sense that flogfd c l n for any probability density f with respect to c l n here the constant is independent of n or l and d denotes the dirichlet form of the dynamics the dependence on l is optimal

symmetries and form preserving transformations of generalised inhomogeneous nonlinear diffusion equations we consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form f x ut g x unux x we present a complete classification of lie symmetries and form preserving point transformations in the case where f x 1 which is equivalent to the original equation we also introduce certain nonlocal transformations when f x xp and g x xq we have the most known form of this class of equations if certain conditions are satisfied then this latter equation can be transformed into a constant coefficient equation it is also proved that the only equations from this class of partial differential equations that admit lie backlund symmetries is the well known nonlinear equation ut u 2ux x and an equivalent equation finally two examples of new exact solutions are given

on point transformations of evolution equations two evolution equations are considered of the form ut hi x t u ux uxx i 1 2 which are related by the general point transformation x p x t u t q x t u u r x t u it is shown that the time transformation must necessarily be of the form t q t and that if hi are polynomials in the spatial derivatives of u then x p x t

random walk on upper triangular matrices mixes rapidly we present an upper bound o n 2 for the mixing time of a simple random walk on upper triangular matrices we show that this bound is sharp up to a constant and find tight bounds on the eigenvalue gap we conclude by applying our results to indicate that the asymmetric exclusion process on a circle indeed mixes more rapidly than the corresponding symmetric process

lattice gases large deviations and the incompressible navier stokes equations we study the incompressible limit for a class of stochastic particle systems on the cubic lattice zd d 3 for initial distributions corresponding to arbitrary macroscopic l2 initial data the distributions of the evolving empirical momentum densities are shown to have a weak limit supported entirely on global weak solutions of the incompressible navier stokes equations furthermore explicit exponential rates for the convergence large deviations are obtained the probability to violate the divergence free condition decays at rate at least exp e d l while the probability to violate the momentum conservation equation decays at rate exp fe d 2 with an explicit rate function given by an hi1 norm

on symmetries of radially symmetric nonlinear diffusion equations in this paper symmetries are sought for radially symmetric nonlinear diffusion equations of the type ut r1 n rn 1f u ur r the nonlinear forms of this equation which admit lie classical symmetries are completely classified and are shown to exist when f un eu also a trivial symmetry exists for arbitrary f lie backlund symmetries are also considered when f un and shown to exist when n 0 linear case and when n 2 and n 1

microscopic theory of isothermal elastodynamics this paper examines random perturbations of the anharmonic chain of coupled oscillators the microscopic system has two conservation laws and its hyperbolic scaling limit results in the quasi linear wave equation p system of isothermal isentropic elasticity in the shock regime the compensated compactness method is used lastly results from j w shearer and d serre are applied

exact solutions to a nonlinear diffusion equation similarity solutions are derived for the nonlinear diffusion equation ct del c del c in one dimension and with cylindrical symmetry some applications are indicated

on form preserving point transformations of partial differential equations new identities are presented relating arbitrary order partial derivatives of and for the general point transformation these identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 1 partial differential equations these results facilitate the search for point symmetries both discrete and continuous lie and assist the search for point transformations which reduce equations to canonical but similar form a simple test for the existence of hodograph type transformations between equations of similar form is given

similarity solutions to three dimensional nonlinear diffusion equations the similarity transformation method has been used to solve three dimensional nonlinear diffusion equations the general lie group is calculated exact solutions to the cylindrical and spherical symmetry cases are found and their relations to some real physical processes are discussed

classical lie point symmetry analysis of nonlinear diffusion equations describing thermal energy storage in this paper we employed the linear transformation group approach to time dependent nonlinear diffusion equations describing thermal energy storage problem symmetry analysis of the governing equation resulted in admitted large lie symmetry algebras for some special cases of the arbitrary constants and the source term some transformations that lead to equations with fewer arbitrary parameters are applied and classical lie point symmetry methods are employed to analyze the transformed equations some symmetry reductions are performed and wherever possible the reduced ordinary differential equations are completely solved subject to realistic boundary conditions

spin modules in characteristic 2 abstract some properties of spin modules for orthogonal groups are developed particularly with regard to parabolic subgroups an application is given to a problem in the structure theory of finite simple groups

irreducible modules and parabolic subgroups summary it seems desirable from the point of view of finite group theorists to develop further the representation theory of finite chevalley groups in the characteristic of their definition say p by focusing attention on the full array of p local subgroups that is the parabolics rather than the usual single borel or cartan subgroup the following observation may be regarded as a generalization to arbitrary parabolic subgroups of the standard result 3 theorem 39 d for a borel subgroup namely that in an irreducible module the subspace fixed by a maximal unipotent subgroup is 1 dimensional and so affords an irreducible module for a levi complement which is an abelian cartan subgroup it seems natural to state the result first for algebraic groups

the cutoff profile for the simple exclusion process on the circle in this paper we give a very accurate description of the way the simple exclusion process relaxes to equilibrium let ptpt denote the semi group associated the exclusion on the circle with 2n2n sites and nn particles for any initial condition and for any t 4n29 2lognt 4n29 2log n we show that the probability density pt pt is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system as 4n29 2logn4n29 2log n is smaller than the mixing time which is n22 2lognn22 2log n this allows to give a sharp description of the cutoff profile if dn t dn t denote the total variation distance starting from the worse initial condition we have limn dn n22 2logn n2 2s erf 2 e s limn dn n22 2log n n2 2s erf 2 e s where erferf is the gauss error function

sheaf homology and complete reducibility abstract in the study of finite simple groups by means of the geometries provided by their local subgroups problems of structure often reduce to questions about modular representations in finite characteristic of particular interest are modules spanned by fixed points of sylow groups including high weight modules for chevalley groups for which the homology methods of ronan and smith 6 can be very useful this note presents an elementary sufficient condition for splitting of some reducible modules of this type among the applications we generalize a number of results appearing in recent work of timmesfeld 11 which provided the original inspiration for this analysis some of the ideas had appeared in a special case in 10

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant