self similar transport in incomplete chaos particle chaotic dynamics along a stochastic web is studied for three dimensional hamiltonian flow with hexagonal symmetry in a plane two different classes of dynamical motion obtained by different values of a control parameter and corresponding to normal and anomalous diffusion have been considered and compared it is shown that the anomalous transport can be characterized by powerlike wings of the distribution function of displacement flights which are similar to l evy flights approximate trappings of orbits near the boundary layer of islands and anomalous behavior of the moments of a distribution function considered as a function of the number of the moment the main result is related to the self similar properties of different topological and dynamical characteristics of the particle motion this self similarity appears in the weierstrass like random walk process that is responsible for the anomalous transport exponent in the mean moment dependence on t this exponent can be expressed as a ratio of fractal dimensions of space and time sets in the weierstrass like process an explicit form for the expression of the anomalous transport exponent through the local topological properties of orbits has been given

volume inequalities for isotropic measures a direct approach to ball s simplex inequality is presented this approach which does not use the brascamp lieb inequality also gives barthe s characterization of the simplex for ball s inequality and extends it from discrete to arbitrary measures it also yields the dual inequality along with equality conditions and it does both for arbitrary measures

a petrov galerkin finite element method for variable coefficient fractional diffusion equations abstract fractional diffusion equations have found increasingly more applications in recent years but introduce new mathematical and numerical difficulties galerkin formulation which was proved to be coercive and well posed for fractional diffusion equations with a constant diffusivity coefficient may lose its coercivity for variable coefficient problems the corresponding finite element method fails to converge we utilize the discontinuous petrov galerkin dpg framework to develop a petrov galerkin finite element method for variable coefficient fractional diffusion equations we prove the well posedness and optimal order convergence of the petrov galerkin finite element method numerical examples are presented to verify the theoretical results

general affine surface areas abstract two families of general affine surface areas are introduced basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for l affine surface areas are established

the lp minkowski problem and the minkowski problem in centroaffine geometry abstract the l p minkowski problem introduced by lutwak is solved for p n 1 in the smooth category the relevant monge ampere equation 0 1 is solved for all p 1 the same equation for p 1 is also studied and solved for p n 1 1 when p n 1 the equation is interpreted as a minkowski problem in centroaffine geometry a kazdan warner type obstruction for this problem is obtained

the minkowski problem for polytopes abstract the traditional solution to the minkowski problem for polytopes involves two steps first the existence of a polytope satisfying given boundary data is demonstrated in the second step the uniqueness of that polytope up to translation is then shown to follow from the equality conditions of minkowski s inequality a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context in this article we adapt the classical argument to prove both the existence theorem of minkowski and his mixed volume inequality simultaneously thereby providing a new proof of minkowski s inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry

the logarithmic minkowski problem in analogy with the classical minkowski problem necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone volume measure of the unit ball of a finite dimensional banach space

small ball probability estimates ψ2 behavior and the hyperplane conjecture we introduce a method which leads to upper bounds for the isotropic constant we prove that a positive answer to the hyperplane conjecture is equivalent to some very strong small probability estimates for the euclidean norm on isotropic convex bodies as a consequence of our method we obtain an alternative proof of the result of j bourgain that every 2 body has bounded isotropic constant with a slightly better estimate if k is a symmetric convex body in rn such that q 2 for every sn 1 and every q 2 then lk c log where c 0 is an absolute constant

the even orlicz minkowski problem the classical minkowski problem leads to the lp minkowski problem and now to the orlicz minkowski problem existence is demonstrated for the even orlicz minkowski problem a byproduct is a new approach to the solution of the classical minkowski problem

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

new orlicz affine isoperimetric inequalities abstract the orlicz brunn minkowski theory received considerable attention recently and many results in the l p brunn minkowski theory have been extended to their orlicz counterparts the aim of this paper is to develop orlicz l affine and geominimal surface areas for a single convex body as well as for multiple convex bodies which generalize the l p mixed affine and geominimal surface areas fundamental concepts in the l p brunn minkowski theory our extensions are different from the general affine surface areas by ludwig in 21 moreover our definitions for orlicz l affine and geominimal surface areas reveal that these affine invariants are essentially the infimum supremum of v k l the orlicz mixed volume of k and the polar body of l where l runs over all star bodies and all convex bodies respectively with volume of l equal to the volume of the unit euclidean ball b 2 n properties for the orlicz l affine and geominimal surface areas such as affine invariance and monotonicity are proved related orlicz affine isoperimetric inequalities are also established

a stability result for mean width of lp centroid bodies we give a different proof of a recent result of klartag b klartag a central limit theorem for convex sets invent math 168 1 2007 91 131 concerning the concentration of the volume of a convex body within a thin euclidean shell and proving a conjecture of anttila ball and perissinaki m anttila k ball i perissinaki the central limit problem for convex bodies trans amer math soc 355 12 2003 4723 4735 it is based on the study of the lp centroid bodies we prove an almost isometric reverse holder inequality for their mean width and a refined form of a stability result

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

on solvability of two dimensional lp minkowski problem abstract given p 0 and a positive continuous function g with g x t g x for some 0 0 the functional f u 0 t u x 2 u 2 x dx c 0 t g x u p x dx 2 p has a minimizer in the class of positive functions u c1 r for which u x t u x for all x r this minimizer is used to prove the existence of a positive periodic solution y c2 r of two dimensional lp minkowski problem y1 p x y x y x g x where p 0 2

the discrete planar l0 minkowski problem abstract in the discrete setting the l 0 minkowski problem extends the question posed and answered by the classical minkowski s existence theorem for polytopes in particular the planar extension which we address in this paper concerns the existence of a convex polygonal body which contains the origin whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area

the cramer rao inequality for star bodies associated with each body k in euclidean n space mathbb r sp n is an ellipsoid gamma sb 2k called the legendre ellipsoid of k it can be defined as the unique ellipsoid centered at the body s center of mass such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of the body in an earlier paper the authors showed that corresponding to each convex body k subset mathbb r sp n is a new ellipsoid gamma sb 2 k that is in some sense dual to the legendre ellipsoid the legendre ellipsoid is an object of the dual brunn minkowski theory while the new ellipsoid gamma sb 2 k is the corresponding object of the brunn minkowski theory the present paper has two aims the first is to show that the domain of gamma sb 2 can be extended to star shaped sets the second is to prove that the following relationship exists between the two ellipsoids if k is a star shaped set then gamma sb 2 k subset gamma sb 2k with equality if and only if k is an ellipsoid centered at the origin this inclusion is the geometric analogue of one of the basic inequalities of information theory the cramer rao inequality

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

a unified petrov galerkin spectral method for fractional pdes existing numerical methods for fractional pdes suffer from low accuracy and inefficiency in dealing with three dimensional problems or with long time integrations we develop a unified and spectrally accurate petrov galerkin pg spectral method for a weak formulation of the general linear fractional partial differential equations fpdes of the form 0dt2 u j 1dcj ajdxj2 ju u f where 2 2 j 0 1 j 0 1 in a 1 d1 d dimensional space time domain subject to dirichlet initial and boundary conditions we perform the stability analysis in 1 d and the corresponding convergence study of the scheme in multi d the unified pg spectral method applies to the entire family of linear hyperbolichyperbolic parabolicparabolic and ellipticelliptic like equations we develop the pg method based on a new spectral theory for fractional sturm liouville problems fslps recently introduced in zayernouri and karniadakis 2013 specifically we employ the eigenfunctions of the fslp of first kind fslp i called jacobi poly fractonomials as temporal spatial bases next we construct a different space for test functions from poly fractonomial eigenfunctions of the fslp of second kind fslp ii besides the high order spatial accuracy of the pg method we demonstrate its efficiency and spectral accuracy in time integration schemes for solving time dependent fpdes as well rather than employing algebraically accurate traditional methods especially when 2 12 1 finally we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices which reduces the computational cost drastically we demonstrate that this framework can reduce to hyperbolic fpdes such as time and space fractional advection tsfa parabolic fpdes such as time and space fractional diffusion tsfd model and elliptic fpdes such as fractional helmholtz poisson equations with the same ease and cost several numerical tests confirm the efficiency and spectral convergence of the unified pg spectral method for the aforementioned families of fpdes moreover we demonstrate the computational efficiency of the new approach in higher dimensions e g 1 3 1 5 and 1 9 dimensional problems

new lp affine isoperimetric inequalities we prove new lp affine isoperimetric inequalities for all p 8 1 we establish for all p n a duality formula which shows that lp affine surface area of a convex body k equals affine surface area of the polar body k

spectral analysis and structure preserving preconditioners for fractional diffusion equations fractional partial order diffusion equations are a generalization of classical partial differential equations used to model anomalous diffusion phenomena when using the implicit euler formula and the shifted grunwald formula it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a toeplitz like structure in this paper we focus our attention on the case of variable diffusion coefficients under appropriate conditions we show that the sequence of the coefficient matrices belongs to the generalized locally toeplitz class and we compute the symbol describing its asymptotic eigenvalue singular value distribution as the matrix size diverges we employ the spectral information for analyzing known methods of preconditioned krylov and multigrid type with both positive and negative results and with a look forward to the multidimensional setting we also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system with krylov methods such as cgnr and gmres a number of numerical examples show that our proposal is more effective than recently used circulant preconditioners

a hybrid perturbation galerkin technique that combines multiple expansions a two step hybrid perturbation galerkin method for the solution of a variety of differential equations type problems is found to give better results when multiple perturbation expansions are employed the method assumes that there is a parameter in the problem formulation and that a perturbation method can be used to construct one or more expansions in this parameter an approximate solution is constructed in the form of a sum of perturbation coefficient functions multiplied by computed amplitudes in step one regular and or singular perturbation methods are used to determine the perturbation coefficient functions the results of step one are in the form of one or more expansions each expressed as a sum of perturbation coefficient functions multiplied by a priori known gauge functions in step two the classical bubnov galerkin method uses the perturbation coefficient functions computed in step one to determine a set of amplitudes that replace and improve upon the gauge functions the hybrid method has

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation in this article we adopt the saddle point theoretical framework to analyze the conservative fractional diffusion equations by introducing a fractional order flux as auxiliary variable we establish the well posedness of the saddle point variational formulation based on the formulation we propose a locally conservative expanded mixed finite element procedure to approximate the unknown its derivative and the fractional flux directly and prove the existence and uniqueness of the mixed finite element solution by proving that the negative fractional derivative spaces are equivalent to the negative fractional order sobolev spaces we derive the approximation capability of the standard projection operator in fractional sobolev spaces for non smooth functions this leads an optimal order error estimates in terms of the right hand side both for sufficiently smooth solution and non smooth solution numerical experiments are included to confirm our theoretical findings a locally conservative expanded mixed finite element scheme is developed based on the saddle point variational formulation the well posedness of the saddle point variational formulation and the expanded mixed formulation is established the equivalence of the negative fractional spaces and the negative sobolev spaces is established optimal order error estimates are derived for sufficiently smooth solution and non smooth solution

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

the orlicz brunn minkowski theory a general framework additions and inequalities the orlicz brunn minkowski theory introduced by lutwak yang and zhang is a new extension of the classical brunn minkowski theory it represents a generalization of the lp brunn minkowski theory analogous to the way that orlicz spaces generalize lp spaces for appropriate convex functions 0 1 m 0 1 a new way of combining arbitrary sets in r n is introduced this operation called orlicz addition and denoted by has several desirable properties but is not associative unless it reduces to lp addition a general framework is introduced for the orlicz brunn minkowski theory that includes both the new addition and previously introduced concepts and makes clear for the rst time the relation to orlicz spaces and norms it is also shown that orlicz addition is intimately related to a natural and fundamental generalization of minkowski addition called m addition the results obtained show roughly speaking that the orlicz brunn minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the lp brunn minkowski theory inequalities of the brunn minkowski type are obtained both for m addition and orlicz addition the new orlicz brunn minkowski inequality implies the lp brunn minkowski in equality new orlicz minkowski inequalities are obtained that generalize the lp minkowski inequality one of these has connections with the conjectured log brunn minkowski inequal ity of lutwak yang and zhang and in fact these two inequalities together are shown to split the classical brunn minkowski inequality

dual orlicz brunn minkowski theory abstract in this paper a dual orlicz brunn minkowski theory is presented an orlicz radial sum and dual orlicz mixed volumes are introduced the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality are established the variational formula for the volume with respect to the orlicz radial sum is proved the equivalence between the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality is demonstrated orlicz intersection bodies are defined and the orlicz busemann petty problem is posed

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

spectral approximation of a variable coefficient fractional diffusion equation in one space dimension abstract in this article we consider the approximation of a variable coefficient two sided fractional diffusion equation fde having unknown u by introducing an intermediate unknown q the variable coefficient fde is rewritten as a lower order constant coefficient fde a spectral approximation scheme using jacobi polynomials is presented for the approximation of q q n the approximate solution to u u n is obtained by post processing q n an a priori error analysis is given for q q n and u u n two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

orlicz centroid bodies the sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body from below by the volume of the star body itself is the busemann petty centroid inequality a decade ago the lp analogue of the classical busemannpetty centroid inequality was proved here the definition of the centroid body is extended to an orlicz centroid body of a star body and the corresponding analogue of the busemann petty centroid inequality is established for convex bodies the centroids of the intersections of an origin symmetric body with half spaces form the surface of a convex body this centroid body is a concept that dates back at least to dupin the classical affine isoperimetric inequality that relates the volume of a convex body with that of its centroid body was conjectured by blaschke and established in a landmark work by petty 53 petty s inequality became known as the busemann petty centroid inequality because in establishing his inequality petty not only made critical use of busemann s random simplex inequality but as petty stated he reinterpreted it see e g the books by gardner 12 leichtweiss 25 schneider 55 and thompson 58 for reference the concept of a centroid body had a natural extension in what became known as the lp brunn minkowski theory and its dual this theory had its origins in the early 1960s when firey see e g schneider 55 introduced his concept of lp compositions of convex bodies three decades later in 34 and 35 these firey minkowski lp combinations were shown to lead to an embryonic lp brunn minkowski theory this theory and its dual has witnessed a rapid growth see e g 1 9 17 23 26 32 34 44 46 48 54 56 57 59 62 the lp analogues of centroid bodies became a central focus within the lp brunn minkowski theory and its dual and establishing the lpanalogue of the busemann petty centroid inequality became a major goal this was accomplished by the authors of the present paper in 37 with an independent approach presented by campi and gronchi 3 the lp centroid bodies quickly became objects of interest in asymptotic mathematics subject classification mr 52a40 research supported in part by nsf grant dms 0706859 proof copy 1 not for distribution 2 erwin lutwak deane yang and gaoyong zhang geometric analysis see e g 10 11 24 49 50 51 52 and even the theory of stable distributions see e g 48 using concepts introduced by ludwig 29 haberl and schuster 21 were led to establish asymmetric versions of the lp busemann petty centroid inequality that for bodies that are not origin symmetric are stronger than the lp busemann petty centroid inequality presented in 37 and 3 the asymmetric inequalities obtained by haberl and schuster are ideally suited for non symmetric bodies this can be seen by looking at the haberl schuster version of the lp analogue of the classical blaschke santalo inequality that was presented in 46 while for origin symmetric bodies the lp extension of 46 does recover the original blaschke santalo inequality as p for arbitrary bodies only the haberl schuster version does so the works of haberl and schuster 21 see also 22 ludwig and reitzner 32 and ludwig 31 have demonstrated the clear need to move beyond the lp brunn minkowski theory to what we are calling an orlicz brunn minkowski theory this need is not only motivated by compelling geometric considerations such as those presented in ludwig and reitzner 32 but also by the desire to obtain sobolev bounds see 22 of a far more general nature this paper is the second in a series intended to develop a few of the elements of an orlicz brunn minkowski theory and its dual here we define the orlicz centroid body establish some of its basic properties and most importantly establish what we call the orlicz busemannpetty centroid inequality for orlicz centroid bodies we consider convex r 0 such that 0 0 this means that must be decreasing on 0 and increasing on 0 we require that one of these is happening strictly so i e is either strictly decreasing on 0 or strictly increasing on 0 the class of such will be denoted by c if k is a star body see section 1 for precise definitions with respect to the origin in rn with volume k and c then we define the orlicz centroid body k of k as the convex body whose support function at x rn is given by h k x inf 0 1 k k x y dy 1 where x y denotes the standard inner product of x and y in rn and the integration is with respect to lebesgue measure in rn when p t t p with p 1 then

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

a simple finite element method for boundary value problems with a riemann liouville derivative we consider a boundary value problem involving a riemann liouville fractional derivative of order 3 2 2 on the unit interval 0 1 the standard galerkin finite element approximation converges slowly due to the presence of singularity term x 1 in the solution representation in this work we develop a simple technique by transforming it into a second order two point boundary value problem with nonlocal low order terms whose solution can reconstruct directly the solution to the original problem the stability of the variational formulation and the optimal regularity pickup of the solution are analyzed a novel galerkin finite element method with piecewise linear or quadratic finite elements is developed and l 2 d error estimates are provided the approach is then applied to the corresponding fractional sturm liouville problem and error estimates of the eigenvalue approximations are given extensive numerical results fully confirm our theoretical study

valuations on polytopes containing the origin in their interiors we give a classification of non negative or borel measurable sl d invariant homogeneous valuations on the space of d dimensional convex polytopes containing the origin in their interiors the only examples are volume volume of the polar body and the euler characteristic

the orlicz brunn minkowski inequality abstract the orlicz brunn minkowski theory originated with the work of lutwak yang and zhang in 2010 in this paper we first introduce the orlicz addition of convex bodies containing the origin in their interiors and then extend the l p brunn minkowski inequality to the orlicz brunn minkowski inequality furthermore we extend the l p minkowski mixed volume inequality to the orlicz mixed volume inequality by using the orlicz brunn minkowski inequality

on the number of solutions to the discrete two dimensional l0 minkowski problem abstract our main result shows the uniqueness of planar convex polygons of given outer orientations to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices if they exist such polygons are solutions to the discrete l 0 minkowski problem in the plane proposed in a larger generality by lutwak in his extension of the brunn minkowski theory the method exploits the equivalence between the l 0 polygons and the homothetic solutions to appropriate anisotropic crystalline flows hence as a by product we obtain a positive answer to jean taylor s conjecture on the uniqueness of polygonal shapes which are deformed homothetically under the flow by crystalline curvature without any symmetry assumptions

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

positions of convex bodies associated to extremal problems and isotropic measures abstract we show that there are close relations between extremal problems in dual brunn minkowski theory and isotropic type properties for some borel measures on the sphere the methods we use allow us to obtain similar results in the context of firey brunn minkowski theory we also study reverse inequalities for dual mixed volumes which are related with classical positions such as l position or isotropic position

the brunn minkowski inequality for p capacity of convex bodies in this paper we prove the brunn minkowski inequality for the p capacity of convex bodies i e convex compact sets with non empty interior in r n for every p 1 n moreover we prove that the equality holds in such inequality if and only if the involved bodies coincide up to a translation and a dilatation

the lp busemann petty centroid inequality abstract the ratio between the volume of the p centroid body of a convex body k in r n and the volume of k attains its minimum value if and only if k is an origin symmetric ellipsoid this result the lp busemann petty centroid inequality was recently proved by lutwak yang and zhang in this paper we show that all the intrinsic volumes of the p centroid body of k are convex functions of a time like parameter when k is moved by shifting all the chords parallel to a fixed direction the lp busemann petty centroid inequality is a consequence of this general fact

on the l p monge ampere equation abstract for the case where 0 p 1 we prove the existence of solutions of the l p monge ampere equation for arbitrary measures and show that the solution is in general not unique

the hadamard variational formula and the minkowski problem for p capacity abstract a hadamard variational formula for p capacity of convex bodies in r n is established when 1 p n the formula is applied to solve the minkowski problem for p capacity which involves a degenerate monge ampere type equation uniqueness for the minkowski problem for p capacity is established when 1 p n and existence and regularity when 1 p 2 these results are non linear extensions of the now classical solution of jerison of the minkowski problem for electrostatic capacity p 2

the lp minkowski problem for torsion abstract in the present paper we introduce the so called l p torsion measure l p mixed torsion a hadamard variational formula for torsion under the p sum is established as an application the existence of solutions to a l p minkowski problem for torsion is proved for any fixed p 1 and p n 2 which extends a result of colesanti and fimiani 24 a l p minkowski inequality for torsion is also obtained which implies the uniqueness of the solutions for the l p minkowski problem mentioned above

spectral analysis and multigrid preconditioners for two dimensional space fractional diffusion equations abstract fractional diffusion equations fdes are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance in this paper we focus on a two dimensional space fde problem discretized by means of a second order finite difference scheme obtained as combination of the crank nicolson scheme and the so called weighted and shifted grunwald formula by fully exploiting the toeplitz like structure of the resulting linear system we provide a detailed spectral analysis of the coefficient matrix at each time step both in the case of constant and variable diffusion coefficients such a spectral analysis has a very crucial role since it can be used for designing fast and robust iterative solvers in particular we employ the obtained spectral information to define a galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped jacobi as smoother and to prove the linear convergence rate of the corresponding two grid method the theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the v cycle method and the geometric multigrid on this basis we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for krylov methods several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate

the orlicz minkowski problem for torsional rigidity abstract in this paper we prove a variational formula of torsional rigidity under the orlicz sum and then introduce the orlicz mixed torsional rigidity which yields the orlicz brunn minkowski inequality and orlicz minkowski inequality for torsional rigidity as an application we prove the existence of solution to the orlicz minkowski problem for discrete measure and general measure associated with torsional rigidity

analysis of a forced strongly non linear two degree of freedom system by means of the power series method abstract a power series solution is presented for the periodic motion of an undamped strongly non linear two degree of freedom system excited by harmonic forces the analysis is facilitated by transforming the time variable into an harmonically oscillating time the frequency of the new time variable is determined by observing the equality between the rate of change of dynamical energy and the power delivered by the forces the results show good agreement with the modified lindstedt poincare method and the incremental harmonic balance method

lp minkowski problem with not necessarily positive data abstract the l p minkowski problem in the plane has been the object of a number of recent investigations here we present various existence results for 0 p 2 in the case 0 p 2 a necessary and sufficient condition for the solvability of the problem is given

a spectral method of exponential convergence for singular solutions of the diffusion equation with general two sided fractional derivative an open problem in the numerical solution of fractional partial differential equations fpdes is how to obtain high order accuracy for singular solutions even for smooth right hand sides solutions of fpdes are singular here we consider the one dimensional diffusion equation with general two sided fractional derivative characterized by a parameter p in 0 1 for p 1 2 we recover the riesz fractional derivative while for p 1 0 we obtain the one sided fractional derivative we employ a petrov galerkin projection in a properly weighted sobolev space with two sided jacobi polyfracnomials as basis and test functions in particular we derive these two sided jacobi polyfractonomials as eigenfunctions of a sturm liouville problem with weights uniquely determined by the parameter p we provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term i e for smooth data corresponding to singular solutions we obtain exponential convergence wh

the dual orlicz brunn minkowski theory this paper introduces the dual orlicz brunn minkowski theory for star sets a radial orlicz addition of two or more star sets is proposed and a corresponding dual orlicz brunn minkowski inequality is established based on a radial orlicz linear combination of two star sets a formula for the dual orlicz mixed volume is derived and a corresponding dual orlicz minkowski inequality proved the inequalities proved yield as special cases the precise duals of the conjectured log brunn minkowski and log minkowski inequalities of boroczky lutwak yang and zhang a new addition of star sets called radial m addition is also introduced and shown to relate to the radial orlicz addition

wellposedness of variable coefficient conservative fractional elliptic differential equations fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second order differential equations there are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second order differential equations these are i fractional differential operators are nonlocal operators and ii the adjoint of a fractional differential operator is not the negative of itself the wellposedness of a galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously many subsequent works were carried out to extend the analysis to other numerical methods a constant diffusivity coefficient has been assume

a superfast preconditioned iterative method for steady state space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second order diffusion equations because of the nonlocal property of fractional differential operators the corresponding numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 for a problem of size n we develop a superfast preconditioned conjugate gradient squared method for the efficient solution of steady state space fractional diffusion equations the method reduces the computational work from o n 2 to o nlogn per iteration and reduces the memory requirement from o n 2 to o n furthermore the method significantly reduces the number of iterations to be mesh size independent preliminary numerical experiments for a one dimensional steady state diffusion equation with 2 1 3 nodes show that the fast method reduces the overall cpu time from 3h and 27min for the gaussian elimination to 0 39s for the fast method while retaining the accuracy of gaussian elimination in contrast the regular conjugate gradient squared method diverges after 2days of simulations and more than 20 000 iterations

inhomogeneous dirichlet boundary value problems of space fractional diffusion equations and their finite element approximations we prove the wellposedness of the galerkin weak formulation and petrov galerkin weak formulation for inhomogeneous dirichlet boundary value problems of constant or variable coefficient conservative caputo space fractional diffusion equations we also show that the weak solutions to their riemann liouville analogues do not exist in general in addition we develop an indirect finite element method for the dirichlet boundary value problems of caputo fractional differential equations which reduces the computational work for the numerical solution of variable coefficient fractional diffusion equations from o n 3 to o n and the memory requirement from o n 2 to o n on any quasiuniform space partition we further prove a nearly sharp error estimate for the method which is expressed in terms of the smoothness of the prescribed data of the problem only we carry out numerical experiments to investigate the performance of the method in comparison with the galerkin finite element method

efficient spectral galerkin methods for fractional partial differential equations with variable coefficients efficient spectral galerkin algorithms are developed to solve multi dimensional fractional elliptic equations with variable coefficients in conserved form as well as non conserved form these algorithms are extensions of the spectral galerkin algorithms for usual elliptic pdes developed in 24 more precisely for separable fpdes we construct a direct method by using a matrix diagonalization approach while for non separable fpdes we employ a preconditioned bicgstab method with a suitable separable fpde with constant coefficients as preconditioner the cost of these algorithms is of o n d 1 flops where d is the space dimension we derive rigorous weighted error estimates which provide more precise convergence rate for problems with singularities at boundaries we also present ample numerical results to validate the algorithms and error estimates

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

orlicz projection bodies abstract minkowski s projection bodies have evolved into l p projection bodies and their asymmetric analogs these all turn out to be part of a far larger class of orlicz projection bodies the analog of the classical petty projection inequality is established for the new orlicz projection bodies

brunn minkowski inequalities for variational functionals and related problems abstract recently several inequalities of brunn minkowski type have been proved for well known functionals in the calculus of variations e g the first eigenvalue of the laplacian the newton capacity the torsional rigidity and generalizations of these examples in this paper we add new contributions to this topic in particular we establish equality conditions in the case of the first eigenvalue of the laplacian and of the torsional rigidity and we prove a brunn minkowski inequality for another class of variational functionals moreover we describe the links between brunn minkowski type inequalities and the resolution of minkowski type problems

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

a high accuracy preserving spectral galerkin method for the dirichlet boundary value problem of variable coefficient conservative fractional diffusion equations fractional diffusion equations were shown to provide an adequate and accurate description of transport processes exhibiting anomalous diffusion behavior recently spectral galerkin methods were developed for space fractional diffusion equations aiming at achieving exponential convergence an optimal order error estimate in the fractional energy norm was proved under the assumption that the true solution to the fractional diffusion equation has the desired regularity an optimal order error estimate in the l 2 norm was proved via the well known nitsche lifting technique under the assumption that the true solution to the corresponding boundary value problem of the fractional diffusion equation has the required regularity for each right hand side in this paper we show that the true solution to the dirichlet boundary value problem of a conservative fractional diffusion equation of order 2 with 0 1 as well as a constant diffusivity coefficient and a constant source term is not in the fractional sobolev space h 3 2 in general but is still in the besov space b 3 2 l 2 hence the provable convergence rate of a spectral galerkin method in the l 2 norm is at most of the order o n 3 2 where n is the degree of the polynomial space in the numerical method numerical experiments show that the spectral galerkin method exhibits a subquadratic convergence in the l 2 norm for any 0 1 we develop a high accuracy preserving spectral galerkin method for the dirichlet boundary value problem of one sided variable coefficient conservative fractional diffusion equations the method has a proved high order convergence rate of arbitrary order i without requiring the smoothness of the true solution u to the given boundary value problem but only assuming that the diffusivity coefficient and the right hand source term have the desired regularity ii for a variable diffusivity coefficient and iii for an inhomogeneous dirichlet boundary condition numerical experiments substantiate the theoretical analysis and show that the method exhibits exponential convergence provided the diffusivity coefficient and the right hand source term have the desired regularity

discontinuous galerkin method for fractional convection diffusion equations we propose a discontinuous galerkin method for fractional convection diffusion equations with a superdiffusion operator of order alpha 1 alpha 2 defined through the fractional laplacian the fractional operator of order alpha is expressed as a composite of first order derivatives and a fractional integral of order 2 alpha the fractional convection diffusion problem is expressed as a system of low order differential integral equations and a local discontinuous galerkin method scheme is proposed for the equations we prove stability and optimal order of convergence cal o h k 1 for the fractional diffusion problem and an order of convergence of cal o h k frac 1 2 is established for the general fractional convection diffusion problem the analysis is confirmed by numerical examples

a modified lindstedt poincare method for a strongly non linear two degree of freedom system abstract a modified lindstedt poincare l p method for extending the validity of perturbation expansions to strongly non linear oscillations of two degree of freedom dof systems is presented a parameter transformation ii i is adopted such that a strongly non linear system with a large parameter is transformed into a small parameter system with respect to show its essential features three examples are presented to show the efficiency and the advantages of this method

the lp minkowski problem for polytopes for 0 p 1 abstract necessary and sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for the critical case where 0 p 1