entropic multipliers method for langevin diffusion and weighted log sobolev inequalities in his work about hypocercivity villani 18 considers in particular convergence to equilibrium for the kinetic langevin process while his convergence results in l 2 are given in a quite general setting convergence in entropy requires some boundedness condition on the hessian of the hamiltonian we will show here how to get rid of this assumption in the study of the hypocoercive entropic relaxation to equilibrium for the langevin diffusion our method relies on a generalization to entropy of the multipliers method and an adequate functional inequality as a byproduct we also give tractable conditions for this functional inequality which is a particular instance of a weighted logarithmic sobolev inequality to hold

isotropic hypoellipticity and trend to equilibrium for the fokker planck equation with a high degree potential we consider the fokker planck equation with a confining or anti confining potential which behaves at infinity like a possibly high degree homogeneous function hypoellipticity techniques provide the well posedness of the weak cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding witten laplacian with detailed applications

semi martingale inequalities via the garsia rodemich rumsey lemma and applications to local times abstract with the help of the garsia rodemich rumsey lemma some improvements of the burkholder davis gundy inequalities for continuous martingales are obtained as well as certain estimates of the l p norm of the supremum of a the local times of a continuous semi martingale and b the difference of the local times of two continuous martingales

a finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh a finite volume fv procedure is described for solving the elastic solid mechanics equations in three dimensions on an unstructured mesh for bodies undergoing thermal or mechanical loads the fv procedure is developed in parallel with the conventional fe galerkin procedure so that the differences in each approach may be clearly distinguished the matrix form of the fv procedure is described and is implemented in parallel with the fe procedure both for two dimensional quadrilateral and three dimensional brick meshes the fv and fe procedures are then compared against a range of benchmark problems that test the basic capability of the fv technique it is shown to be approximately as accurate as the fe procedure on similar meshes though its system matrix set up time is twice as long for a node by node set up procedure

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

a finite volume method for analysing the bending deformation of thick and thin plates the formulation of a novel finite volume based procedure for determining deformations and moment distributions within transversely loaded plates is described in detail the formulation employs the mindlin theory of plate deformation transverse displacement and bending moment distributions are determined for a number of test problems the effect of mesh refinement upon the accuracy of the displacement and moment fields is investigated the finite volume procedure exhibits a number of interesting features it is conservative in the sense that equilibrium is satisfied locally within each cell and across cell boundaries it has the capacity to represent the behaviour of thick and thin plates and it is geometrically versatile because the cell shapes can take the form of multifaceted polygons

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

grids and the arithmetics of jordan pairs abstract we prove that every 2 finitedimensional covering standard division grid of a jordan pair v over a henselian field canonically determines a norm on v this is used to classify maximal orders which contain a covering division grid of v we show that weakly separable orders always satisfy this condition and link their existence to ramification properties of v

degenerate fokker planck equations bismut formula gradient estimate and harnack inequality abstract by constructing successful couplings for degenerate diffusion processes explicit derivative formula and harnack type inequalities are presented for solutions to a class of degenerate fokker planck equations on r m r d the main results are also applied to the study of gradient estimate entropy transportation cost inequality and heat kernel inequalities

multilevel monte carlo method for ergodic sdes without contractivity abstract this paper proposes a new multilevel monte carlo mlmc method for the ergodic sdes which do not satisfy the contractivity condition by introducing the change of measure technique we simulate the path with contractivity and add the radon nikodym derivative to the estimator we can show the strong error of the path is uniformly bounded with respect to t moreover the variance of the new level estimators increase linearly in t which is a great reduction compared with the exponential increase in standard mlmc then the total computational cost is reduced to o e 2 log e 2 from o e 3 log e of the standard monte carlo method numerical experiments support our analysis

quantitative harris type theorems for diffusions and mckean vlasov processes we consider d valued diffusion processes of type dxt b xt dt dbt assuming a geometric drift condition we establish contractions of the transitions kernels in kantorovich l1 wasserstein distances with explicit constants our results are in the spirit of hairer and mattingly s extension of harris theorem in particular they do not rely on a small set condition instead we combine lyapunov functions with reflection coupling and concave distance functions we retrieve constants that are explicit in parameters which can be computed with little effort from one sided lipschitz conditions for the drift coefficient and the growth of a chosen lyapunov function consequences include exponential convergence in weighted total variation norms gradient bounds bounds for ergodic averages and kantorovich contractions for nonlinear mckean vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities we also establish quantitative bounds for sub geometric ergodicity assuming a sub geometric drift condition

a large strain finite volume method for orthotropic bodies with general material orientations abstract this paper describes a finite volume method for orthotropic bodies with general principal material directions undergoing large strains and large rotations the governing and constitutive relations are presented and the employed updated lagrangian mathematical model is outlined in order to maintain equivalence with large strain total lagrangian methods the constitutive stiffness tensor is updated transforming the principal material directions to the deformed configuration discretisation is performed using the cell centred finite volume method for unstructured convex polyhedral meshes the current methodology is successfully verified by numerically examining two separate test cases a circular hole in an orthotropic plate subjected to a traction and a rotating orthotropic plate containing a hole subjected to a pressure the numerical predictions have been shown to agree closely with the available analytical solutions in addition a 3 d composite component is examined to demonstrate the capabilities of the developed methodology in terms of a variable material orientation and parallel processing

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

couplings and quantitative contraction rates for langevin dynamics we introduce a new probabilistic approach to quantify convergence to equilibrium for kinetic langevin processes in contrast to previous analytic approaches that focus on the associated kinetic fokker planck equation our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the langevin equation it yields contractions in a particular wasserstein distance and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime in particular we are able to recover kinetic behavior in terms of explicit lower bounds for the contraction rate for example for a rescaled double well potential with local minima at distance a we obtain a lower bound for the contraction rate of order a 1 provided the friction coefficient is of order a 1

convergence of numerical time averaging and stationary measures via poisson equations numerical approximation of the long time behavior of a stochastic differential equation sde is considered error estimates for time averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the sde the error analysis is based on using an associated poisson equation for the underlying sde the main advantages of this approach are its simplicity and universality it works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables and for hypoelliptic sdes to simplify the exposition we consider only the case where the state space of the sde is a torus and we study only smooth test functions however we anticipate that the approach can be applied more widely an analogy between our approach and stein s method is indicated some practical implications of the results are discussed

numerical method for coupled fluid flow heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology abstract a numerical method is presented that can be used for both solid body stress analysis and fluid flow predictions independently as well as in a coupled manner the method uses an integral form of equations governing mass energy and momentum balance for an arbitrary control volume a detailed description is provided of a novel second order accurate spatial discretisation technique which can accommodate unstructured moving meshes with cells of arbitrary topology this is accompanied by a fully implicit temporal discretisation which makes the method stable for any time step size the resulting set of coupled non linear algebraic equations is solved by employing a segregated approach leading to a decoupled set of linear algebraic equations for each dependent variable with a sparse diagonally dominant coefficient matrix these equations are solved by an iterative conjugate gradient solver which retains the sparsity of the coefficient matrix thus achieving a very efficient use of computer resources the method has been tested on a number of fluid flow and stress analysis problems and numerical calculations show favourable agreement with analytical and or experimental results in order to demonstrate the full capabilities of the present method an example of numerically complex coupled fluid flow stress analysis and heat transfer calculations is presented

hitting times functional inequalities lyapunov conditions and uniform ergodicity the use of lyapunov conditions for proving functional inequalities was initiated in 5 it was shown in 4 30 that there is an equivalence between a poincare inequality the existence of some lyapunov function and the exponential integrability of hitting times in the present paper we close the scheme of the interplay between lyapunov conditions and functional inequalities by showing that strong functional inequalities are equivalent to lyapunov type conditions showing that these lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times we also give some complement concerning the link between lyapunov conditions and in tegrability property of the invariant probability measure and as such transportation inequalities and we show that some unbounded lyapunov conditions can lead to uniform ergodicity and coming down from infinity property

computing ergodic limits for langevin equations abstract to evaluate averages with respect to the invariant law for langevin type equations one has to integrate a system over long time intervals especially when dissipation is small this is a challenging problem from the computational point of view since coefficients of the langevin equations are typically not globally lipschitz the difficulties are redoubled in the nonglobal lipschitz case we observe an exploding behavior of some approximate trajectories due to many usual numerical methods in g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 we propose and justify a concept which in principle allows us to apply any numerical method to stochastic differential equations with nonglobally lipschitz coefficients here quasi symplectic integrators from g n milstein m v tretyakov quasi symplectic methods for langevin type equations ima j numer anal 23 2003 593 626 they are the most appropriate methods for solving langevin type equations especially when damping is small and the concept of g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 are applied for calculation of ergodic limits special attention is paid to the case when the invariant measure is known e g when this measure is gibbsian some computer experiments with three model problems van der pol s equation with additive noise one dimensional arrays of oscillators in thermal equilibrium and a physical pendulum with linear friction and additive noise are presented

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

finite volume method for thermo elasto plastic stress analysis abstract in this paper a new numerical method for prediction of stresses and displacements in thermo elasto plastic material is presented the method is based on the direct solution of the governing equations for thermal energy and momentum balance expressed in terms of temperature and displacements as dependent variables via corresponding constitutive relations and discretized by the finite volume technique as a result the unsteady fields of temperature displacements and stresses as well as residual stresses in a solid body subjected to a non uniform temperature and or external loads are obtained the method has been applied to a number of test cases and the results are compared with the analytical solution calculations obtained by another numerical method or experimental data results show that the method is accurate stable and efficient and that it can be used as a useful tool for predicting thermo mechanical deformation processes in a solid body

idempotents and peirce gradings of jordan algebras abstract in this paper we find a necessary and sufficient condition for a peirce grading of a jordan algebra j to come from a peirce decomposition with respect to an idempotent of a jordan algebra j containing j as a subalgebra we also show that the above condition holds automatically when j is nondegenerate

hypocoercivity for linear kinetic equations conserving mass we develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law local mass conservation is assumed at the level of the collision kernel while transport involves a confining potential so that the solution relaxes towards a unique equilibrium state our goal is to evaluate in an appropriately weighted l 2 norm the exponential rate of convergence to the equilibrium the method covers various models ranging from diffusive kinetic equations like vlasov fokker planck equations to scattering models like the linear boltzmann equation or models with time relaxation collision kernels corresponding to polytropic gibbs equilibria including the case of the linear boltzmann model in this last case and in the case of vlasov fokker planck equations any linear or superlinear growth of the potential is allowed