the geometry of r adaptive meshes generated using optimal transport methods the principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods and a metric tensor and mesh metric are useful theoretical tools for understanding a method s level of mesh alignment or anisotropy we consider a mesh redistribution method based on the monge ampere equation which combines equidistribution of a given scalar density function with optimal transport it does not involve explicit use of a metric tensor although such a tensor must exist for the method and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh for model problems with a linear feature and with a radially symmetric feature we derive the exact form of the metric which involves expressions for its eigenvalues and eigenvectors the eigenvectors are shown to be orthogonal and tangential to the feature and the ratio of the eigenvalues corresponding to the level of anisotropy is shown to depend both locally and globally on the value of the density function and the amount of curvature we thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features including for a non trivial time dependant nonlinear pde which evolves narrow and curved reaction fronts

nonequilibrium fluctuations for a tagged particle in mean zero one dimensional zero range processes we prove a non equilibrium functional central limit theorem for the position of a tagged particle in mean zero one dimensional zero range process the asymptotic behavior of the tagged particle is described by a stochastic differential equation governed by the solution of the hydrodynamic equation

strong matching preclusion for n dimensional torus networks the strong matching preclusion number of a graph is the minimum number of edges and or vertices whose deletion results in the remaining graph that has neither perfect matchings nor almost perfect matchings in 14 wang et al proved that c k c k is super strongly matched where k 3 is odd in this paper we show that c k 1 c k 2 c k n is super strongly matched where n 3 is an integer and k i 3 is an odd integer for each i 1 n our studies generalize the results of wang et al 14

optimal mass transport for higher dimensional adaptive grid generation in this work we describe an approach for higher dimensional adaptive grid generation based on solving the l 2 monge kantorovich problem mkp which is a special case of the classical optimal mass transportation problem two methods are developed for computing the coordinate transformation used to define the grid adaptation for the first method the transformation is determined by solving a parabolic monge ampere equation for a steady state solution for the second method the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the l 2 mkp several numerical experiments are presented to demonstrate the performance of the mkp methods and to compare them with some related adaptive grid methods the experimental results demonstrate that the mkp methods show promise as effective and reliable methods for higher dimensional adaptive grid generation

a synchronous and iterative flux correction formalism for coupled transport equations many problems of fluid dynamics involve the coupled transport of several density like dependent variables for instance densities of mass and momenta in elastic flows in this paper a conservative and synchronous flux corrected transport fct formalism is developed which aims at a consistent transport of such variables the technique differs from traditional fct algorithms in two respects first the limiting of transportive fluxes of the primary variables e g mass and momentum does not derive from smooth estimates of the variables but it derives from analytic constraints implied by the lagrangian form of the governing continuity equations which are imposed on the specific mixing ratios of the variables e g velocity components second the traditional fct limiting based on sufficiency conditions is augmented by an iterative procedure which approaches the necessity requirements this procedure can also be used in the framework of traditional fct schemes and a demonstration is provided that it can significantly reduce some of the pathological behaviors of fct algorithms although the approach derived is applicable to the transport of arbitrary conserved quantities it is particularly useful for the synchronous transport of mass and momenta in elastic flows where it assures intrinsic stability of the algorithm regardless of the magnitude of the mass density variable this latter property becomes especially important in fluids with large density variations or in models with a material vertical coordinate e g geophysical hydrostatic stratified flows in isopycnic isentropic coordinates where material surfaces can collapse to zero mass layers admitting therefore arbitrarily large local courant numbers

estimates on translations and taylor expansions in fractional sobolev spaces abstract in this paper we study how the normalised gagliardo semi norms u w s p r n control translations in particular we prove that u y u l p r n c u w s p r n y s for n 1 s 0 1 and p 1 where c depends only on n we then obtain a corresponding higher order version of this result we get fractional rates of the error term in the taylor expansion we also present relevant implications of our two results first we obtain a direct proof of several compact embedding of w s p r n where the frechet kolmogorov theorem is applied with known rates we also derive fractional rates of convergence of the convolution of a function with suitable mollifiers thirdly we obtain fractional rates of convergence of finite difference discretisations for w s p r n

strong matching preclusion of n k star graphs the strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings this is an extension of the matching preclusion problem that was introduced by park and ihm the generalized n k star graph was introduced to address scaling issues of the star graph and it has many desirable properties in this paper the goal is to find the strong matching preclusion number of n k star graphs and to categorize all optimal strong matching preclusion sets of these graphs since bipartite graphs generally have low strong matching preclusion numbers we assume that k n 2

uniqueness in weighted lebesgue spaces for a class of fractional parabolic and elliptic equations abstract we investigate uniqueness in suitable weighted lebesgue spaces of solutions to a class of fractional parabolic and elliptic equations

fractional nonlinear degenerate diffusion equations on bounded domains part i existence uniqueness and upper bounds abstract we investigate quantitative properties of nonnegative solutions u t x 0 to the nonlinear fractional diffusion equation t u l f u 0 posed in a bounded domain x r n with appropriate homogeneous dirichlet boundary conditions as l we can use a quite general class of linear operators that includes the two most common versions of the fractional laplacian s 0 s 1 in a bounded domain with zero dirichlet boundary conditions but it also includes many other examples since our theory only needs some basic properties that are typical of linear heat semigroups the nonlinearity f is assumed to be increasing and is allowed to be degenerate the prototype is the power case f u u m 1 u with m 1 in this paper we propose a suitable class of solutions of the equation and cover the basic theory we prove existence uniqueness of such solutions and we establish upper bounds of two forms absolute bounds and smoothing effects as well as weighted l 1 estimates the class of solutions is very well suited for that work the standard laplacian case s 1 is included and the linear case m 1 can be recovered in the limit in a companion paper bonforte and vazquez in preparation we will complete the study with more advanced estimates like the upper and lower boundary behaviour and harnack inequalities for which the results of this paper are needed

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

symmetries of the self dual einstein equations i the infinite dimensional symmetry group and its low dimensional subgroups this is the first of two papers in which the authors give a complete classification of symmetry reduced solutions of plebanski s potential equation for self dual einstein spaces in this first part the infinite pseudogroup of symmetries of plebanski s equation is described and the conjugacy classes of all local subgroups of dimensions one two and three over both the real and complex numbers are classified then in the second paper this classification is used to obtain all symmetry reduced solutions

optimal mass transport for registration and warping image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times in this paper we present a method for computing elastic registration and warping maps based on the monge kantorovich theory of optimal mass transport this mass transport method has a number of important characteristics first it is parameter free moreover it utilizes all of the grayscale data in both images places the two images on equal footing and is symmetrical the optimal mapping from image a to image b being the inverse of the optimal mapping from b to a the method does not require that landmarks be specified and the minimizer of the distance functional involved is uniques there are no other local minimizers finally optimal transport naturally takes into account changes in density that result from changes in area or volume although the optimal transport method is certainly not appropriate for all registration and warping problems this mass preservation property makes the monge kantorovich approach quite useful for an interesting class of warping problems as we show in this paper our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the l2 kantorovich wasserstein or earth mover s distance under a mass preservation constraint we show how this approach leads to practical algorithms and demonstrate our method with a number of examples including those from the medical field we also extend this method to take into account changes in intensity and show that it is well suited for applications such as image morphing

parametric backlund transformations i phenomenology we begin an exploration of parametric backlund transformations for hyperbolic monge ampere systems the appearance of an arbitrary parameter in the transformation is a feature of several well known completely integrable pdes we compute invariants for such transformations and explore the behavior of four examples two of which are new in terms of their invariants symmetries and conservation laws we prove some preliminary results and indicate directions for further research

convergence analysis for a class of high order semi lagrangian advection schemes the convergence properties of a class of high order semi lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions we review the general convergence results for discrete time approximation schemes belonging to that class and we prove some a priori estimates in l infty and l2 for the rate of convergence of fully discrete schemes we prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions several examples of schemes and numerical tests are presented

strong matching preclusion for k composition networks abstract let g 1 v 1 e 1 g 2 v 2 e 2 g k v k e k be k graphs and let f 1 v 1 v 2 f 2 v 2 v 3 f k 1 v k 1 v k f k v k v 1 be k bijections the k composition networks g induced by g 1 g 2 g k is the graph with v g t 1 k v g t and e g t 1 k e g t a t f t a t a t v g t and 1 t k many interconnection networks such as n dimensional torus networks recursive circulant graphs and cayley graphs on abelian groups generated by minimal generating sets are special k composition networks the strong matching preclusion number of a graph is the minimum number of edges and or vertices whose deletion results in the remaining graph has neither perfect matchings nor almost perfect matchings in this paper we study the strong matching preclusion number and strong matching preclusion sets for k composition networks with odd order our results generalize the main conclusion in 12

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

semi lagrangian methods for level set equations a new numerical method for solving geometric moving interface problems is presented the method combines a level set approach and a semi lagrangian time stepping scheme which is explicit yet unconditionally stable the combination decouples each mesh point from the others and the time step from the cfl stability condition permitting the construction of methods which are efficient adaptive and modular analysis of a linear one dimensional model problem suggests a surprising convergence criterion which is supported by heuristic arguments and confirmed by an extensive collection of two dimensional numerical results the new method computes correct viscosity solutions to problems involving geometry anisotropy curvature and complex topological events

eulag a computational model for multiscale flows an mhd extension eulag is an established high performance computational model for simulating fluid flows across a wide range of scales and physical scenarios prusa et al comput fluids 37 2008 1193 historically driven by interests in simulating weather and climate processes the numerics of eulag are unique owing to a synergistic blend of non oscillatory forward in time mpdata methods robust elliptic solver and generalized coordinate formulation enabling grid adaptivity in this paper the numerical apparatus of an ideal magnetohydrodynamic mhd extension of the eulag model is discussed the robust workings of which have been recently revealed in global large eddy simulations of solar magneto convection producing solar like magnetic cycles and dynamo action ghizaru et al apjl 715 2010 l133 racine et al apj 735 2011 46 here a specialized nonoscillatory forward in time scheme for integrating ideal anelastic mhd equations is presented in detail and illustrated with an abstract example of magnetized three dimensional flow in time dependent geometry for a weak moderate and strong magnetic field an analysis of the model performance reveals that multiple solutions of elliptic problems do not have to imply proportionally larger computational expense

strong matching preclusion for augmented cubes the strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings the concept was introduced by park and son in this paper we study the strong matching preclusion problem for the augmented cube graphs as a result we find smp aqn and classify all optimal solutions

generalised lagrangian solutions for atmospheric and oceanic flows atmospheric or oceanic flows strongly constrained by rotation and stratification can be described by a set of lagrangian partial differential equations called the semigeostrophic equations in these equations the trajectories must be determined implicitly generalised solutions of these equations are defined as a sequence of rearrangements of the fluid which need not be smooth these solutions are closely related to generalised solutions of the monge ampere equation existence and uniqueness of such solutions is proved the evolution is shown to be a sequence of minimum energy states of the fluid giving strong physical plausibility to the solutions

strong matching preclusion for non bipartite torus networks the strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in the remaining graph that has neither perfect matchings nor almost perfect matchings the torus network is one of the most popular interconnection network topologies for massively parallel computing systems because of its desirable properties it is known that bipartite torus networks have low strong matching preclusion numbers hu et al 13 proved that non bipartite torus networks with an odd number of vertices have good strong matching preclusion properties to complete the study of strong matching preclusion problem for non bipartite torus networks in this paper we establish the strong matching preclusion number and classify all optimal strong matching preclusion sets for the n dimensional non bipartite torus network with an even number of vertices where n 3

an all scale anelastic model for geophysical flows dynamic grid deformation we have developed an adaptive grid refinement approach for simulating geophysical flows on scales from micro to planetary our model is nonoscillatory forward in time nft nonhydrostatic and anelastic the major focus in this effort to date has been the design of a generalized mathematical framework for the implementation of deformable coordinates and its efficient numerical coding in a generic eulerian semi lagrangian nft format the key prerequisite of the adaptive grid is a time dependent coordinate transformation implemented rigorously throughout the governing equations of the model the transformation enables mesh refinement indirectly via dynamic change of the metric coefficients while retaining advantages of cartesian mesh calculations speed low memory requirements and accuracy conducted fully in the computational domain diverse test results presented in this paper simulations of a traveling stratospheric inertio gravity wave packet with numerically advected dense mesh region and an idealized climate of the earth with analytically prescribed adaptive transformations demonstrate the potential and the efficacy of the new deformable grid model for tracing targeted flow features and dynamically adjusting to prescribed undulations of model boundaries

a general fractional porous medium equation we develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion u t 2 u m 1 u 0 x r n t 0 u x 0 f x x r n we consider data f l 1 r n and all exponents 0 0 existence and uniqueness of a weak solution is established for m m n n giving rise to an l 1 contraction semigroup in addition we obtain the main qualitative properties of these solutions in the lower range 0 m m existence and uniqueness of solutions with good properties happen under some restrictions and the properties are different from the case above m we also study the dependence of solutions on f m and moreover we consider the above questions for the problem posed in a bounded domain

the fractional laplacian in power weighted l p spaces integration by parts formulas and self adjointness abstract we consider the fractional laplacian operator s let s 0 1 on euclidean space and investigate the validity of the classical integration by parts formula that connects the l 2 r d scalar product between a function and its fractional laplacian to the nonlocal norm of the fractional sobolev space h s r d more precisely we focus on functions belonging to some weighted l 2 space whose fractional laplacian belongs to another weighted l 2 space we prove and disprove the validity of the integration by parts formula depending on the behaviour of the weight x at infinity the latter is assumed to be like a power both near the origin and at infinity the two powers being possibly different our results have direct consequences for the self adjointness of the linear operator formally given by 1 s the generality of the techniques developed allows us to deal with weighted l p spaces as well

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

notes on limits of sobolev spaces and the continuity of interpolation scales we extend lemmas by bourgain brezis mironescu 2001 and maz ya shaposhnikova 2002 on limits of sobolev spaces to the setting of interpolation scales this is achieved by means of establishing the continuity of real and complex interpolation scales at the end points a connection to extrapolation theory is developed and a new application to limits of sobolev scales is obtained we also give a new approach to the problem of how to recognize constant functions via sobolev conditions

mesh adaptation on the sphere using optimal transport and the numerical solution of a monge ampere type equation abstract an equation of monge ampere type has for the first time been solved numerically on the surface of the sphere in order to generate optimally transported ot meshes equidistributed with respect to a monitor function optimal transport generates meshes that keep the same connectivity as the original mesh making them suitable for r adaptive simulations in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers the semi implicit solution of the monge ampere type equation involves a new linearisation of the hessian term and exponential maps are used to map from old to new meshes on the sphere the determinant of the hessian is evaluated as the change in volume between old and new mesh cells rather than using numerical approximations to the gradients ot meshes are generated to compare with centroidal voronoi tessellations on the sphere and are found to have advantages and disadvantages ot equidistribution is more accurate the number of iterations to convergence is independent of the mesh size face skewness is reduced and the connectivity does not change however anisotropy is higher and the ot meshes are non orthogonal it is shown that optimal transport on the sphere leads to meshes that do not tangle however tangling can be introduced by numerical errors in calculating the gradient of the mesh potential methods for alleviating this problem are explored finally ot meshes are generated using observed precipitation as a monitor function in order to demonstrate the potential power of the technique

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

a new characterization of sobolev spaces on mathbb r n in this paper we present a new characterization of sobolev spaces on mathbb r n our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of euclidean space an interesting feature of our condition is that depends only on the metric of mathbb r n and the lebesgue measure so that one can define sobolev spaces of any order of smoothness on any metric measure space

uniqueness and properties of distributional solutions of nonlocal equations of porous medium type abstract we study the uniqueness existence and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation t u l u 0 here l can be any nonlocal symmetric degenerate elliptic operator including the fractional laplacian and numerical discretizations of this operator the function r r is only assumed to be continuous and nondecreasing the class of equations include nonlocal generalized porous medium equations fast diffusion equations and stefan problems in addition to very general uniqueness and existence results we obtain stability l 1 contraction and a priori estimates we also study local limits continuous dependence and properties and convergence of a numerical approximation of our equations

optimization based mesh correction with volume and convexity constraints we consider the problem of finding a mesh such that 1 it is the closest with respect to a suitable metric to a given source mesh having the same connectivity and 2 the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh this volume correction problem arises in important simulation contexts such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi lagrangian transport schemes in this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective while the prescribed cell volumes mesh validity and or cell convexity specify the constraints we solve this problem numerically using a sequential quadratic programming sqp method whose performance scales with the mesh size to achieve scalable performance we develop a specialized multigrid based preconditioner for optimality systems that arise in the application of the sqp method to the volume correction problem numerical examples illustrate the importance of volume correction and showcase the accuracy robustness and scalability of our approach

the monge ampere trajectory correction for semi lagrangian schemes requiring that numerical estimates of the flow trajectories comply with the fundamental euler expansion formula that governs the evolution of a volume of fluid leads to a second order nonlinear monge ampere partial differential equation mae in cossette and smolarkiewicz 2011 15 a numerical algorithm based on solving the mae with an inexact newton krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi lagrangian scheme in the context of an incompressible fluid here we extend the theoretical analysis of the elemental rotational and deformational motions presented in cossette and smolarkiewicz 2011 15 in particular closed form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness scalar advection shows that the ma correction improves mass conservation substantially by suppressing anomalous fluid contraction the impact of the ma correction on complex flows is studied in the framework of an ideal magneto fluid in which the formation of current sheets leads to topological changes and reconnection of field lines the use of the ma correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy 2d and 3d examples are presented and the computational performance of the solver is documented

strong matching preclusion for torus networks the torus network is one of the most popular interconnection network topologies for massively parallel computing systems strong matching preclusion that additionally permits more destructive vertex faults in a graph is a more extensive form of the original matching preclusion that assumes only edge faults in this paper we establish the strong matching preclusion number and all minimum strong matching preclusion sets for bipartite torus networks and 2 dimensional nonbipartite torus networks

generalized gagliardo nirenberg inequalities using lorentz spaces bmo holder spaces and fractional sobolev spaces abstract the main purpose of this paper is to prove some generalized gagliardo nirenberg interpolation inequalities involving the lorentz spaces l p bmo and the fractional sobolev spaces w s p including also c holder spaces although some of the results can be alternatively obtained by using interpolation spaces specifically the reiteration theorem the precise form of the inequalities stated here appears to be novel and moreover the proofs given in the present paper are self contained save for the use of the john nirenberg inequality for the bmo result in contrast to the other mentioned approach the use of c holder spaces in such gagliardo nirenberg inequalities seems to be new in the literature

nonlocal discrete diffusion equations and the fractional discrete laplacian regularity and applications abstract the analysis of nonlocal discrete equations driven by fractional powers of the discrete laplacian on a mesh of size h 0 h s u f for u f z h r 0 s 1 is performed the pointwise nonlocal formula for h s u and the nonlocal discrete mean value property for discrete s harmonic functions are obtained we observe that a characterization of h s as the dirichlet to neumann operator for a semidiscrete degenerate elliptic local extension problem is valid regularity properties and schauder estimates in discrete holder spaces as well as existence and uniqueness of solutions to the nonlocal dirichlet problem are shown for the latter the fractional discrete sobolev embedding and the fractional discrete poincare inequality are proved which are of independent interest we introduce the negative power fundamental solution u h s f which can be seen as the neumann to dirichlet map for the semidiscrete extension problem we then prove the discrete hardy littlewood sobolev inequality for h s as applications the convergence of our fractional discrete laplacian to the continuous fractional laplacian as h 0 in holder spaces is analyzed indeed uniform estimates for the error of the approximation in terms of h under minimal regularity assumptions are obtained we finally prove that solutions to the poisson problem for the fractional laplacian s u f in r can be approximated by solutions to the dirichlet problem for our fractional discrete laplacian with explicit uniform error estimates in terms of h

the monge ampere equation various forms and numerical solution we present three novel forms of the monge ampere equation which is used e g in image processing and in reconstruction of mass transportation in the primordial universe the central role in this paper is played by our fourier integral form for which we establish positivity and sharp bound properties of the kernels this is the basis for the development of a new method for solving numerically the space periodic monge ampere problem in an odd dimensional space convergence is illustrated for a test problem of cosmological type in which a gaussian distribution of matter is assumed in each localised object and the right hand side of the monge ampere equation is a sum of such distributions

mesh free semi lagrangian methods for transport on a sphere using radial basis functions abstract we present three new semi lagrangian methods based on radial basis function rbf interpolation for numerically simulating transport on a sphere the methods are mesh free and are formulated entirely in cartesian coordinates thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface based coordinate systems for problems involving tracer transport in a given velocity field the semi lagrangian framework allows these new methods to avoid the use of any stabilization terms such as hyperviscosity during time integration thus reducing the number of parameters that have to be tuned the three new methods are based on interpolation using 1 global rbfs 2 local rbf stencils and 3 rbf partition of unity for the latter two of these methods we find that it is crucial to include some low degree spherical harmonics in the interpolants standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy efficiency conservation properties and dissipation dispersion errors for global rbfs spectral spatial convergence is observed for smooth solutions on quasi uniform nodes while high order accuracy is observed for the local rbf stencil and partition of unity approaches

uniqueness of very weak solutions for a fractional filtration equation abstract we prove existence and uniqueness of distributional bounded solutions to a fractional filtration equation in r d with regards to uniqueness it was shown even for more general equations in 22 that if two bounded solutions u w of 1 1 satisfy u w l 1 r d 0 t then u w we obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions for nonnegative initial data we first show that a minimal solution exists and then that any other solution must coincide with it a similar procedure is carried out for sign changing solutions as a consequence distributional solutions have locally finite energy

matching preclusion for balanced hypercubes matching preclusion is a measure of robustness in the event of edge failure in interconnection networks the matching preclusion number of a graph g is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching and the conditional matching preclusion number of g is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or an almost perfect matching in this paper we consider balanced hypercubes we obtain that an n dimension balanced hypercube bh n has the matching preclusion number 2n and mainly prove that for the balanced hypercube bh n each matching preclusion set of cardinality 2n is trivial and the conditional matching preclusion number of balanced hypercube is 4n 2 whenever n 2

a widder s type theorem for the heat equation with nonlocal diffusion the main goal of this work is to prove that every non negative strong solutionu x t to the problem u t delta alpha 2 u 0 rm for x t in mathbb r n times 0 t 0 alpha 2 can be written as u x t int mathbb r n p t x y u y 0 dy where p t x frac 1 t n alpha p left frac x t 1 alpha right and p x int mathbb r n e i x cdot xi xi alpha d xi this result shows uniqueness in the setting of non negative solutions and extends some classical results for the heat equation by widder in 15 to the nonlocal diffusion framework

a fractional porous medium equation abstract we develop a theory of existence uniqueness and regularity for the following porous medium equation with fractional diffusion u t 1 2 u m 1 u 0 x r n t 0 u x 0 f x x r n with m m n 1 n n 1 and f l 1 r n an l 1 contraction semigroup is constructed and the continuous dependence on data and exponent is established nonnegative solutions are proved to be continuous and strictly positive for all x r n t 0

continuity of the temperature in boundary heat control problems abstract motivated by the boundary heat control problems formulated in the book of duvaut and lions we study a boundary stefan problem and a boundary porous media problem we prove continuity of the solution with the appropriate modulus we also extend the results to the fractional order case and to the anomalous diffusion problems

on the bourgain brezis and mironescu theorem concerning limiting embeddings of fractional sobolev spaces the article is concerned with the bourgain brezis and mironescu theorem on the asymptotic behaviour of the norm of the sobolev type embedding operator ws p lpn n sp as s 1 and s n p thei

uniqueness of entropy solutions to fractional conservation laws with fully infinite speed of propagation our goal is to study the uniqueness of bounded entropy solutions for a multidimensional conservation law including a non lipschitz convection term and a diffusion term of nonlocal porous medium type the nonlocality is given by a fractional power of the laplace operator for a wide class of nonlinearities the l 1 contraction principle is established despite the fact that the finite infinite speed of propagation alibaud jee 2007 cannot be exploited in our framework existence is deduced with perturbation arguments the method of proof adapted from andreianov maliki nodea 2010 requires a careful analysis of the action of the fractional laplacian on truncations of radial powers

short note a simple and conservative operator splitting semi lagrangian volume of fluid advection scheme a semi lagrangian operator splitting high performance volume of fluid advection scheme slosh vof is developed in the backward time integration a directional splitting strategy is adopted which greatly simplifies the definition and locating of the departure volume and reduces it to a grid cell expanded or compressed in one grid direction the vof value in the departure cell is estimated using a geometrical interpolation algorithm with a piecewise linear interface calculation plic scheme for the interfacial cells the proposed slosh vof method is unconditionally stable and very large time steps can be used to significantly speed up the overall computations it is conceptually simple and can be very easily implemented due to the direction split advection its performance is evaluated through several benchmark advection tests the slosh vof results are comparable to those from an eulerian vof method in terms of interface position errors and improved with regard to mass conservation even with cfl numbers increased up to one order of magnitude

numerical solution of the optimal transportation problem using the monge ampere equation a numerical method for the solution of the elliptic monge ampere partial differential equation with boundary conditions corresponding to the optimal transportation ot problem is presented a local representation of the ot boundary conditions is combined with a finite difference scheme for the monge ampere equation email protected s method is implemented leading to a fast solver comparable to solving the laplace equation on the same grid several times theoretical justification for the method is given by a convergence proof in the companion paper 4 solutions are computed with densities supported on non convex and disconnected domains computational examples demonstrate robust performance on singular solutions and fast computational times

fast finite difference solvers for singular solutions of the elliptic monge ampere equation the elliptic monge ampere equation is a fully nonlinear partial differential equation which originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail in this article we build a finite difference solver for the monge ampere equation which converges even for singular solutions regularity results are used to select a priori between a stable provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain this allows singular solutions to be computed using a stable method and regular solutions to be computed more accurately the resulting nonlinear equations are then solved by newton s method computational results in two and three dimensions validate the claims of accuracy and solution speed a computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities

robust numerical methods for nonlocal and local equations of porous medium type part ii schemes and experiments we develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations partial t u mathfrak l varphi u f x t in math

quantitative local and global a priori estimates for fractional nonlinear diffusion equations abstract we establish quantitative estimates for solutions u t x to the fractional nonlinear diffusion equation t u s u m 0 in the whole range of exponents m 0 0 s 1 the equation is posed in the whole space x r d we first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data in the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions depending only on the norm of the initial data in a certain ball the estimates take a different form in three exponent ranges slow diffusion good range of fast diffusion and very fast diffusion finally we show existence and uniqueness of initial traces

monge ampere based moving mesh methods for numerical weather prediction with applications to the eady problem we derive a moving mesh method based upon ideas from optimal transport theory which is suited to solving pde problems in meteorology in particular we show how the parabolic monge ampere method for constructing a moving mesh in two dimensions can be coupled successfully to a pressure correction method for the solution of incompressible flows with significant convection and subject to coriolis forces this method can be used to resolve evolving small scale features in the flow in this paper the method is then applied to the computation of the solution to the eady problem which is observed to develop large gradients in a finite time the moving mesh method is shown to work and be stable and to give significantly better resolution of the evolving singularity than a fixed uniform mesh

aspects of the dynamical core of a nonhydrostatic deep atmosphere unified weather and climate prediction model the dynamical core which governs the evolution of resolved fluid dynamical processes is a critical element of any atmospheric model its governing equations must include all relevant dynamical terms and the numerical formulae used to approximate them must be accurate stable and efficient this is particularly so in a unified modeling environment in which the same dynamical core is used for both operational weather prediction and long term climate simulations recent research at the met office on unified dynamical core issues is reviewed aspects covered include properties of various equation sets vertical coordinates semi lagrangian advection and conservation trajectory computation and dynamical equivalence horizontal and vertical discretization and coupling of physical parameterizations to a dynamical core

optimal existence and uniqueness theory for the fractional heat equation abstract we construct a theory of existence uniqueness and regularity of solutions for the fractional heat equation t u s u 0 0 s 1 posed in the whole space r n with data in a class of locally bounded radon measures that are allowed to grow at infinity with an optimal growth rate we consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions which is given in one direction by the representation formula in the other one by the initial trace we review many of the typical properties of the solutions in particular we prove optimal pointwise estimates and new harnack inequalities

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

strong matching preclusion the matching preclusion problem introduced by brigham et al r c brigham f harary e c violin and j yellen perfect matching preclusion congressus numerantium 174 2005 185 192 studies how to effectively make a graph have neither perfect matchings nor almost perfect matchings by deleting as small a number of edges as possible extending this concept we consider a more general matching preclusion problem called the strong matching preclusion in which deletion of vertices is additionally permitted we establish the strong matching preclusion number and all possible minimum strong matching preclusion sets for various classes of graphs