sample based polynomial approximation of rational bezier curves we present an iteration method for the polynomial approximation of rational bezier curves starting with an initial bezier curve we adjust its control points gradually by the scheme of weighted progressive iteration approximations the l p error calculated by the trapezoidal rule using sampled points is used to guide the iteration approximation we reduce the l p error by a predefined factor at every iteration so as to obtain the best approximation with a minimum error numerical examples demonstrate the fast convergence of our method and indicate that results obtained using the l 1 error criterion are better than those obtained using the l 2 error and l error criteria

asymptotic couplings by reflection and applications for nonlinear monotone spdes abstract asymptotic couplings by reflection are constructed for a class of nonlinear monotone spdes stochastic partial differential equations as applications gradient holder estimates as well as exponential convergence are derived for the associated markov semigroup the main results are illustrated by stochastic generalized porous media equations stochastic p laplacian equations and stochastic generalized fast diffusion equations we emphasize that the gradient estimate is studied for the first time for these equations

preconditioned progressive iterative approximation for triangular bezier patches and its application abstract in this paper we propose a preconditioned progressive iterative approximation for triangular bezier patches firstly the diagonally compensated reduction is used in designing the preconditioner some properties of the preconditioner are shown the validity of the presented iterative method by using lu factorization of the preconditioner is discussed then the preconditioned progressive iterative approximation is used to approximate rational triangular bezier patches the given numerical examples show that the proposed method achieves a great improvement on the convergence rate compared to the existing progressive iterative approximation

robust dimension free isoperimetry in gaussian space we prove the first robust dimension free isoperimetric result for the standard gaussian measure n and the corresponding boundary measure n in rn the main result in the theory of gaussian isoperimetry proven in the 1970s by sudakov and tsirelson and independently by borell states that if n a 1 2 then the surface area of a is bounded by the surface area of a half space with the same measure n a 2 1 2 our results imply in particular that if a rn satisfies n a 1 2 and n a 2 1 2 then there exists a half space b rn such that n a b clog 1 2 1 for an absolute constant c since the gaussian isoperimetric result was established only recently a robust version of the gaussian isoperimetric result was obtained by cianchi et al who showed that n a b c n for some function c n with no effective bounds compared to the results of cianchi et al our results have optimal i e no dependence on the dimension but worse dependence on

singular stochastic equations on hilbert spaces harnack inequalities for their transition semigroups we consider stochastic equations in hilbert spaces with singular drift in the framework of g da prato m rockner singular dissipative stochastic equations in hilbert spaces probab theory related fields 124 2 2002 261 303 we prove a harnack inequality in the sense of f y wang logarithmic sobolev inequalities on noncompact riemannian manifolds probab theory related fields 109 1997 417 424 for its transition semigroup and exploit its consequences in particular we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding kolmogorov operator has at most one infinitesimally invariant measure satisfying some mild integrability conditions finally we prove existence of such a measure for noncontinuous drifts

frobenius morphisms and representations of algebras by introducing frobenius morphisms f on algebras a and their modules over the algebraic closure f q of the finite field f q of q elements we establish a relation between the representation theory of a over f q and that of the f fixed point algebra a f over f q more precisely we prove that the category mod a f of finite dimensional a f modules is equivalent to the subcategory of finite dimensional f stable a modules and when a is finite dimensional we establish a bijection between the isoclasses of indecomposable a f modules and the f orbits of the isoclasses of indecomposable a modules applying the theory to representations of quivers with automorphisms we show that representations of a modulated quiver or a species over f q can be interpreted as f stable representations of the corresponding quiver over f q we further prove that every finite dimensional hereditary algebra over f q is morita equivalent to some a f where a is the path algebra of a quiver q over f q and f is induced from a certain automorphism of q a close relation between the auslander reiten theories for a and a f is established in particular we prove that the auslander reiten modulated quiver of a f is obtained by folding the auslander reiten quiver of a finally by taking frobenius fixed points we are able to count the number of indecomposable representations of a modulated quiver over f q with a given dimension vector and to generalize kac s theorem for all modulated quivers and their associated kac moody algebras defined by symmetrizable generalized cartan matrices

liouville theorems for non local operators abstract the paper characterizes some classes of pseudo differential operators for which there are or there are not non constant bounded harmonic functions non local perturbations of ornstein uhlenbeck operators and operators with dissipative coefficients are considered the methods used are probabilistic and based on the concept of absorption function and on a new extension of the bismut elworthy li formula the probabilistic interpretation of the liouville theorem by means of absorption functions for general markov processes is given as well

an isoperimetric inequality for uniformly log concave measures and uniformly convex bodies abstract we prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log concave measures that we introduce these inequalities imply up to universal constants the log sobolev inequalities proved by bobkov ledoux s g bobkov m ledoux from brunn minkowski to brascamp lieb and to logarithmic sobolev inequalities geom funct anal 10 5 2000 1028 1052 and the isoperimetric inequalities due to bakry ledoux d bakry m ledoux levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator invent math 123 2 1996 259 281 and bobkov zegarlinski s g bobkov b zegarlinski entropy bounds and isoperimetry mem amer math soc 176 829 2005 x 69 we also recover a concentration inequality for uniformly convex bodies similar to that proved by gromov milman m gromov v d milman generalization of the spherical isoperimetric inequality to uniformly convex banach spaces compos math 62 3 1987 263 282

integrability conditions for sdes and semilinear spdes by using the local dimension free harnack inequality established on incomplete riemannian manifolds integrability conditions on the coefficients are presented for sdes to imply the nonexplosion of solutions as well as the existence uniqueness and regularity estimates of invariant probability measures these conditions include a class of drifts unbounded on compact domains such that the usual lyapunov conditions cannot be verified the main results are extended to second order differential operators on hilbert spaces and semilinear spdes

isoperimetric constants for product probability measures a dimension free lower bound is found for isoperimetric constants of product probability measures from this some analytic inequalities are derived

analysis on wiener spaces ii differential forms abstract the author gives the notions of p forms and p forms and shows some properties of them in particular a vanishing theorem of de rham cohomology of poincare type also he shows the hypoellipticity of certain elliptic operators on wiener space and a theorem of poinare inequality type

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

a parametrix for the neumann problem on pseudoconvex domains of finite type abstract we construct a parametrix for the neumann problem on any pseudoconvex domain of finite type which reduces completely the study of isotropic lp sobolev and holder estimates to those for b and b we also establish the maximal nonisotropic estimates in the case of d q domains and provide a precise comparison of the bergman and szego projections and of the canonical solutions to with respect to these two operators the argument is substantially simpler than the known ones for the cases of strongly pseudoconvex domains and pseudoconvex domains of finite type in c 2

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

logarithmic sobolev inequalities on noncompact riemannian manifolds this paper presents a dimension free harnack type inequality for heat semigroups on manifolds from which a dimension free lower bound is obtained for the logarithmic sobolev constant on compact manifolds and a new criterion is proved for the logarithmic sobolev inequalities abbrev lsi on noncompact manifolds as a result it is shown that lsi may hold even though the curvature of the operator is negative everywhere

levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator we establish by simple semigroup arguments a levy gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the gaussian measure this produces in particular a new proof of the gaussian isoperimetric inequality this isoperimetric inequality strengthens the classical logarithmic sobolev inequality in this context a local version for the heat kernel measures is also proved which may then be extended into an isoperimetric inequality for the wiener measure on the paths of a riemannian manifold with bounded ricci curvature

on gradient bounds for the heat kernel on the heisenberg group it is known that the couple formed by the two dimensional brownian motion and its levy area leads to the heat kernel on the heisenberg group which is one of the simplest sub riemannian space the associated diffusion operator is hypoelliptic but not elliptic which makes difficult the derivation of functional inequalities for the heat kernel however driver and melcher and more recently h q li have obtained useful gradient bounds for the heat kernel on the heisenberg group we provide in this paper simple proofs of these bounds and explore their consequences in terms of functional inequalities including cheeger and bobkov type isoperimetric inequalities for the heat kernel

bv functions in abstract wiener spaces abstract functions of bounded variation in an abstract wiener space i e an infinite dimensional banach space endowed with a gaussian measure and a related differential structure have been introduced by m fukushima and m hino using dirichlet forms and their properties have been studied with tools from analysis and stochastics in this paper we reformulate in an integral geometric vein and with purely analytical tools the definition and the main properties of bv functions and investigate further properties

hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n sphere a logarithmic sobolev inequality with respect to the probability measure a 1 x2 12 dx on 1 1 is proved for all 12 from this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n sphere

on the application of two symmetric gauss legendre quadrature rules for composite numerical integration over a triangular surface this paper first presents a gauss legendre quadrature rule for the evaluation of i t f x y d x d y where f x y is an analytic function in x y and t is the standard triangular surface x y 0 x y 1 x y 1 in the two space x y we transform this integral into an equivalent integral s f x y frac x y d d where s is the 2 square in space 1 1 we then apply the one dimensional gauss legendre quadrature rules in and variables to arrive at an efficient quadrature rules with new weight coefficients and new sampling points then a second gauss legendre quadrature rule of composite type is obtained this rule is derived by discretising t into three new triangles tic i 1 2 3 of equal size which are obtained by joining centroid of t c 1 3 1 3 to the three vertices of t by use of affine transformations defined over each tic and the linearity property of integrals leads to the result i underover i 1 3 tic f x y d x d y frac 1 3 t g x y d x d y where g x y i 1n n f xic x y yic x y and x xic x y and y yic x y refer to affine transformations which map each tic into t the standard triangular surface we then write t g x y d x d y s g x y frac x y d d and a composite rule of integration is thus obtained we next propose the discretisation of the standard triangular surface t into n2 right isosceles triangular surfaces ti i 1 1 n2 each of which has an area equal to 1 2 n2 units we have again shown that the use of affine transformation over each ti and the use of linearity property of integrals lead to the result t f x y d x d y underover i 1 n n ti f x y d x d y frac 1 n2 t h x y d x d y where h x y i 1n n f xi x y yi x y and x xi x y y yi x y refer to affine transformations which map each ti in x y space into t a standard triangular surface t in the x y space we can now apply the two rules earlier derived to the integral t h x y d x d y this amounts to application of composite numerical integration of t into n2 and 3n2 triangles of equal sizes respectively we can now apply the rules which are derived earlier to the evaluation of the integral t f x y d x d y and each of these procedures converges to the exact value of the integral t f x y d x d y for sufficiently large value of n and the convergence is much faster for higher order rules we have demonstrated this aspect by applying the above composite integration method to some typical integrals 2007 elsevier inc all rights reserved

harnack and functional inequalities for generalized mehler semigroups abstract harnack inequalities are established for a class of generalized mehler semigroups which in particular imply upper bound estimates for the transition density moreover poincare and log sobolev inequalities are proved in terms of estimates for the square field operators furthermore under a condition well known in the gaussian case we prove that generalized mehler semigroups are strong feller the results are illustrated by concrete examples in particular we show that a generalized mehler semigroup with an stable part is not hyperbounded but exponentially ergodic and that the log sobolev constant obtained by our method in the special gaussian case can be sharper than the one following from the usual curvature condition moreover a harnack inequality is established for the generalized mehler semigroup associated with the dirichlet heat semigroup on 0 1 we also prove that this semigroup is not hyperbounded

a harnack type inequality for non symmetric markov semigroups abstract for a strong feller and irreducible markov semigroup on a locally compact polish space the harnack type inequality 1 1 holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive moreover new sufficient conditions for this inequality to hold as well as upper bound estimates of the underlying constant are presented for diffusion semigroups on riemannian manifolds

representations of finite lie algebras abstract we develop an approach to investigate representations of finite lie algebras g f over a finite field f q through representations of lie algebras g with frobenius morphisms f over the algebraic closure k f q as an application we first show that frobenius morphisms on classical simple lie algebras can be used to determine easily their f q forms and hence reobtain a classical result given in g b seligman modular lie algebras springer verlag berlin 1967 we then investigate representations of finite restricted lie algebras g f regarded as the fixed point algebra of a restricted lie algebra g with restricted frobenius morphism f by introducing the f orbital reduced enveloping algebras u g associated with a reduced enveloping algebra u g we partition simple g f modules via f orbits of their p characters we further investigate certain relations between the categories of g modules with p character g f modules with p character and g f modules with p character for an automorphism of g finally we illustrate the theory with the example of sl 2 f q

a functional form of the isoperimetric inequality for the gaussian measure abstract letgbe a smooth function on r nwith values in 0 1 using the isoperimetric property of the gaussian measure itis proved that 1 eg e 1 g e g conversely this inequality implies the isoperimetric property of the gaussian measure

harnack inequality and applications for stochastic generalized porous media equations by using coupling and girsanov transformations the dimension free harnack inequality and the strong feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations as applications explicit upper bounds of the l p norm of the density as well as hypercontractivity ultracontractivity and compactness of the corresponding semigroup are derived

harnack inequality and strong feller property for stochastic fast diffusion equations abstract as a continuation to f y wang harnack inequality and applications for stochastic generalized porous media equations ann probab 35 2007 1333 1350 where the harnack inequality and the strong feller property are studied for a class of stochastic generalized porous media equations this paper presents analogous results for stochastic fast diffusion equations since the fast diffusion equation possesses weaker dissipativity than the porous medium one does some technical difficulties appear in the study as a compensation to the weaker dissipativity condition a sobolev nash inequality is assumed for the underlying self adjoint operator in applications some concrete examples are constructed to illustrate the main results

log harnack inequality for stochastic burgers equations and applications abstract by proving an l 2 gradient estimate for the corresponding galerkin approximations the log harnack inequality is established for the semigroup associated to a class of stochastic burgers equations as applications we derive the strong feller property of the semigroup the irreducibility of the solution the entropy cost inequality for the adjoint semigroup and entropy upper bounds of the transition density

a new approach to kac s theorem on representations of valued quivers by using the ringel hall algebra approach we first present a new proof of kac s theorem for species over a finite field the method used here is quite different from earlier techniques in sv sevenhant and van den bergh have discovered an interesting relation between a conjecture of kac on representations of quivers and the structure of the ringel hall algebras in some special cases we second show that this relation still holds for species

an iterative algorithm for polynomial approximation of rational triangular bezier surfaces this paper presents the weighted progressive iteration approximation wpia property for the triangular bernstein basis over a triangle domain with uniform parameters which is extended from the pia property for triangular bernstein basis proposed by chen and wang in j chen g j wang progressive iterative approximation for triangular bezier surfaces computer aided design 43 2011 889 895 we also provide how to choose an optimal value of the weight to own the fastest convergence rate for triangular bernstein basis furthermore a new and efficient iterative method is proposed for polynomial approximation of rational triangular bezier surfaces the algorithm is reiterated until a halting condition about approximation error is satisfied and the approximation error in l p norm p 1 2 is calculated by the symmetric gauss legendre quadrature rule for composite numerical integration over a triangular surface finally several numerical examples are presented to validate the effectiveness of this method

structural properties of semilinear spdes driven by cylindrical stable processes we consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise we investigate structural properties of the solutions like markov irreducibility stochastic continuity feller and strong feller properties and study integrability of trajectories the obtained results are applied to semilinear stochastic heat equations with dirichlet boundary conditions and bounded and lipschitz nonlinearities

generalized mehler semigroups and applications we construct and study generalized mehler semigroups p t t 0 and their associated markov processesm the construction methods for p t t 0 are based on some new purely functional analytic results implying in particular that any strongly continuous semigroup on a hilbert spaceh can be extended to some larger hilbert spacee with the embeddingh e being hilbert schmidt the same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedx t c dw t ax t dt with possibly unbounded linear operatorsa andc onh on a suitably chosen larger spacee for gaussian generalized mehler semigroups p t t 0 with corresponding markov processm the associated non symmetric dirichlet forms ed e are explicitly calculated and a necessary and sufficient condition for path regularity ofm in terms of e d e is proved then using dirichlet form methods it is shown thatm weakly solves the above stochastic differential equation if the state spacee is chosen appropriately finally we discuss the differences between these two methods yielding strong resp weak solutions

tight monomials and the monomial basis property abstract we generalize a criterion for tight monomials of quantum enveloping algebras associated with symmetric generalized cartan matrices and a monomial basis property of those associated with symmetric classical cartan matrices to their respective symmetrizable case we then link the two by establishing that a tight monomial is necessarily a monomial defined by a weakly distinguished word as an application we develop an algorithm to compute all tight monomials in the rank 2 dynkin case

convergence in variation of solutions of nonlinear fokker planck kolmogorov equations to stationary measures abstract we study convergence in variation of probability solutions of nonlinear fokker planck kolmogorov equations to stationary solutions we obtain sufficient conditions for the exponential convergence of solutions to the stationary solution in case of coefficients that can have an arbitrary growth at infinity and depend on the solutions through convolutions with unbounded discontinuous kernels in addition we study a more difficult case where the nonlinear equation has several stationary solutions and convergence to a stationary solution depends on initial data finally we obtain sufficient conditions for solvability of nonlinear fokker planck kolmogorov equations

logarithmic sobolev inequalities for pinned loop groups abstract letgbe a connected compact type lie group equipped with anadg invariant inner product on the lie algebra g ofg given this data there is a well known left invariant h1 riemannian structure on l l g the infinite dimensional group of continuous based loops ing using this riemannian structure we define and construct a heat kernel t g0 associated to the laplace beltrami operator on l g heret 0 g0 l g and t g0 is a certain probability measure on l g for fixedg0 l g andt 0 we use the measure t g0 and the riemannian structure on l g to construct a classical pre dirichlet form the main theorem of this paper asserts that this pre dirichlet form admits a logarithmic sobolev inequality

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

approximating rational triangular bezier surfaces by polynomial triangular bezier surfaces an attractive method for approximating rational triangular bezier surfaces by polynomial triangular bezier surfaces is introduced the main result is that the arbitrary given order derived vectors of a polynomial triangular surface converge uniformly to those of the approximated rational triangular bezier surface as the elevated degree tends to infinity the polynomial triangular surface is constructed as follows firstly we elevate the degree of the approximated rational triangular bezier surface then a polynomial triangular bezier surface is produced which has the same order and new control points of the degree elevated rational surface the approximation method has theoretical significance and application value it solves two shortcomings fussy expression and uninsured convergence of the approximation of hybrid algorithms for rational polynomial curves and surfaces approximation

an isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in gauss space we prove an isoperimetric inequality on the discrete cube which is the precise analog of a logarithmic inequality due to talagrand as a consequence the gaussian isoperimetric inequality is derived

pbw bases of the twisted generic composition algebras of affine valued quivers abstract pbw type bases of the twisted generic composition algebras of the affine valued quivers are constructed by using the frobenius morphism

harnack inequalities for stochastic equations driven by levy noise abstract by using coupling argument and regularization approximations of the underlying subordinator dimension free harnack inequalities are established for a class of stochastic equations driven by a levy noise containing a subordinate brownian motion the harnack inequalities are new even for linear equations driven by levy noise and the gradient estimate implied by our log harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for levy processes or linear equations driven by levy noise the main results are also extended to semilinear stochastic equations in hilbert spaces

totally positive bases and progressive iteration approximation in this paper we study the progressive iteration approximation property of a curve tensor product surface generated by blending a given data point set and a set of basis functions the curve tensor product surface has the progressive iteration approximation property as long as the basis is totally positive and the corresponding collocation matrix is nonsingular thus the b spline and nurbs curve surface have the progressive iteration approximation property and bezier curve surface also has the property if the corresponding collocation matrix is nonsingular

levy ornstein uhlenbeck transition semigroup as second quantized operator abstract let be an invariant measure for the transition semigroup p t of the markov family defined by the ornstein uhlenbeck type equation d x a x d t d l on a hilbert space e driven by a levy process l it is shown that for any t 0 p t considered on l 2 is a second quantized operator on a poisson fock space of e at from this representation it follows that the transition semigroup corresponding to the equation on e r driven by an stable noise l 0 2 is neither compact nor symmetric

minimal generating system of ringel hall algebras of affine valued quivers abstract let f q be a finite field with q elements and h f q q the ringel hall algebra of affine valued quiver q all indecomposable representations of q which can be generated inside h f q q by smaller dimension are classified and minimal homogeneous generating systems of h f q q are given explicitly

isoperimetric rigidity and distributions of 1 lipschitz functions abstract we prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal then the space is foliated by minimal geodesics where the observable variance is defined to be the supremum of the variance of 1 lipschitz functions on the space our result can be considered as a variant of cheeger gromoll s splitting theorem and also of cheng s maximal diameter theorem as an application we obtain a new isometric splitting theorem for a complete weighted riemannian manifold with a positive bakry emery ricci curvature

stability of the bakry emery theorem on rn abstract we establish quantitative stability estimates for the bakry emery bound on logarithmic sobolev and poincare constants of uniformly log concave measures more specifically we show that if a 1 uniformly log concave measure has almost the same logarithmic sobolev or poincare constant as the standard gaussian measure then it almost splits off a gaussian factor our results are dimension free leading to dimension free stability estimates for gaussian concentration of lipschitz functions the proofs are based on stein s method optimal transport and an approximate integration by parts identity relating measures and near extremals in the associated functional inequality

on the stochastic p laplace equation the p laplace equation with random perturbation is studied for the singular case 1 p 2 in this paper some properties of the invariant measure and transition semigroups are obtained the main tool is the dimension free harnack inequality which is established by using the coupling argument as consequences some ergodicity compactness and contractive properties are derived for the associated transition semigroups the main results are applied to stochastic reaction diffusion equations and the stochastic p laplace equation in hilbert space

a realization of elliptic lie algebras of type f 4 2 2 by the ringel hall approach abstract this article is to study relations between an elliptic lie algebra g of type f 4 2 2 and the f fixed point algebra a f of a tubular algebra a of type t 3 3 3 under a frobenius morphism f using the explicit structure of the root category of the f fixed point algebra a f we prove that the elliptic lie algebra g of type f 4 2 2 is isomorphic to the ringel hall lie algebra of the root category of the f fixed point algebra a f