convex linear combinations of compositions summary when a sampled target composition is suspected of being a mixture of different compositions from a number of independent sources the question of the nature of the mixing mechanism arises for the resolution of this question several models involving convex linear mixtures of compositions are considered and in particular the distributional problem of describing the pattern of variability of the target compositions given information about the source distributions is resolved in terms of approximations involving logistic normal and logistic skew normal distributions the quality of these approximations is shown to be satisfactory through a series of simulations briefly reported the modelling and subsequent statistical inference are motivated by an illustrative application to investigating the nature of pollution at three fishing locations in a scottish loch

a discontinuous galerkin finite element method for time dependent partial differential equations with higher order derivatives in this paper we develop a new discontinuous galerkin dg finite element method for solving time dependent partial differential equations pdes with higher order spatial derivatives unlike the traditional local discontinuous galerkin ldg method the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system stability is ensured by a careful choice of interface numerical fluxes the method can be designed for quite general nonlinear pdes and we prove stability and give error estimates for a few representative classes of pdes up to fifth order numerical examples show that our scheme attains the optimal k 1 th order of accuracy when using piecewise k th degree polynomials under the condition that k 1 is greater than or equal to the order of the equation

a finite volume scheme for nonlinear degenerate parabolic equations we propose a second order finite volume scheme for nonlinear degenerate parabolic equations which admit an entropy functional for some of these models porous media equation drift diffusion system for semiconductors it has been proved that the transient solution converges to a steady state when time goes to infinity the present scheme preserves steady states and provides a satisfying long time behavior moreover it remains valid and second order accurate in space even in the degenerate case after describing the numerical scheme we present several numerical results which confirm the high order accuracy in various regime degenerate and nondegenerate cases and underline the efficiency to preserve the large time asymptotic

linear unmixing of multivariate observations in many fields of science there are multivariate observations that may be assumed to be generated by a physical linear mixing process of contributions from different sources if the compositions of the sources are constant for different observations then these observations are up to a random error term nonnegative linear combinations of a fixed set of so called source profiles that characterize the sources the goal of linear unmixing is to recover both the source profiles and the source activities also called scores from a multivariate dataset we present a new parametric mixing model that assumes a multivariate lognormal distribution for the scores this model is proved to be identifiable moreover consistency and asymptotic normality of the maximum likelihood estimator mle are established in special cases to calculate the mle we propose the combination of two variants of the monte carlo em algorithm the proposed model is applied to simulated datasets and to a set of air pollution measure

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

perturbation of burkholder s martingale transform and monge ampere equation abstract given a sequence of martingale differences burkholder found the sharp constant for the l p norm of the corresponding martingale transform we are able to determine the sharp l p norm of a small quadratic perturbation of the martingale transform in l p by quadratic perturbation of the martingale transform we mean the l p norm of the square root of the squares of the martingale transform and the original martingale with a small constant the problem of perturbation of martingale transform appears naturally if one wants to estimate the linear combination of riesz transforms as for example in the case of ahlfors beurling operator let d k k 0 be a complex martingale difference in l p 0 1 where 1 p and e k k 0 a sequence in 1 we obtain the following generalization of burkholder s famous result if 1 2 1 2 and n z then k 0 n e k d k l p 0 1 c 2 p 1 2 2 1 2 k 0 n d k l p 0 1 c where p 1 2 2 1 2 is sharp and p 1 max p 1 1 p 1 for 2 p the result is also true with the sharp constant for r

third order maximum principle satisfying dg schemes for convection diffusion problems with anisotropic diffusivity abstract for a class of convection diffusion equations with variable diffusivity we construct third order accurate discontinuous galerkin dg schemes on both one and two dimensional rectangular meshes the dg method with an explicit time stepping can well be applied to nonlinear convection diffusion equations it is shown that under suitable time step restrictions the scaling limiter proposed in liu and yu 2014 23 when coupled with the present dg schemes preserves the solution bounds indicated by the initial data i e the maximum principle while maintaining uniform third order accuracy these schemes can be extended to rectangular meshes in three dimension the crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions this is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages numerical results are presented to validate the numerical methods and demonstrate their effectiveness

the multivariate skew normal distribution summary the paper extends earlier work on the so called skew normal distribution a family of distributions including the normal but with an extra parameter to regulate skewness the present work introduces a multivariate parametric family such that the marginal densities are scalar skew normal and studies its properties with special emphasis on the bivariate case

a sharp weak type p p inequality p 2 for martingale transforms and other subordinate martingales if d n n o is a martingale difference sequence e n n 0 a sequence of numbers in 1 1 and n a positive integer then p max 0 m n m k 0 e k d k 1 p n k 0 d k p p here p denotes the best constant if 1 2 and that p p 1 2 is also the best constant in the analogous inequality for two martingales m and n indexed by 0 right continuous with limits from the left adapted to the same filtration and such that m m t n n t is nonnegative and nondecreasing in t in section 7 we prove a similar inequality for harmonic functions

on positivity preserving high order discontinuous galerkin schemes for compressible navier stokes equations we construct a local lax friedrichs type positivity preserving flux for compressible navier stokes equations which can be easily extended to multiple dimensions for generic forms of equations of state shear stress tensor and heat flux with this positivity preserving flux any finite volume type schemes including discontinuous galerkin dg schemes with strong stability preserving runge kutta time discretizations satisfy a weak positivity property with a simple and efficient positivity preserving limiter high order explicit runge kutta dg schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy numerical tests suggest that the positivity preserving flux and the positivity preserving limiter do not induce excessive artificial viscosity and the high order positivity preserving dg schemes without other limiters can produce satisfying non oscillatory solutions when the nonlinear diffusion in compressible navier stokes equations is accurately resolved

l p bounds for the beurling ahlfors transform let b denote the beurling ahlfors transform defined on l p c 1 p 00 the celebrated conjecture of t iwaniec states that its l p norm b p p 1 1 where p max p p p 1 in this paper the new upper estimate b p 1 575 p 1 1 p is found

a free energy satisfying discontinuous galerkin method for one dimensional poisson nernst planck systems we design an arbitrary order free energy satisfying discontinuous galerkin dg method for solving time dependent poisson nernst planck systems both the semi discrete and fully discrete dg methods are shown to satisfy the corresponding discrete free energy dissipation law for positive numerical solutions positivity of numerical solutions is enforced by an accuracy preserving limiter in reference to positive cell averages numerical examples are presented to demonstrate the high resolution of the numerical algorithm and to illustrate the proven properties of mass conservation free energy dissipation as well as the preservation of steady states

entropy stable high order discontinuous galerkin methods with suitable quadrature rules for hyperbolic conservation laws abstract it is well known that semi discrete high order discontinuous galerkin dg methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws jiang and shu 1994 39 and symmetric hyperbolic systems hou and liu 2007 36 in any space dimension and for any triangulations however this property holds only for the square entropy and the integrations in the dg methods must be exact it is significantly more difficult to design dg methods to satisfy entropy inequalities for a non square convex entropy and or when the integration is approximated by a numerical quadrature in this paper we develop a unified framework for designing high order dg methods which will satisfy entropy inequalities for any given single convex entropy through suitable numerical quadrature which is specific to this given entropy our framework applies from one dimensional scalar cases all the way to multi dimensional systems of conservation laws for the one dimensional case our numerical quadrature is based on the methodology established in carpenter et al 2014 5 and gassner 2013 19 the main ingredients are summation by parts sbp operators derived from legendre gauss lobatto quadrature the entropy conservative flux within elements and the entropy stable flux at element interfaces we then generalize the scheme to two dimensional triangular meshes by constructing sbp operators on triangles based on a special quadrature rule a local discontinuous galerkin ldg type treatment is also incorporated to achieve the generalization to convection diffusion equations extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable dg methods

a discontinuous galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials abstract we consider a class of time dependent second order partial differential equations governed by a decaying entropy the solution usually corresponds to a density distribution hence positivity non negativity is expected this class of problems covers important cases such as fokker planck type equations and aggregation models which have been studied intensively in the past decades in this paper we design a high order discontinuous galerkin method for such problems if the interaction potential is not involved or the interaction is defined by a smooth kernel our semi discrete scheme admits an entropy inequality on the discrete level furthermore by applying the positivity preserving limiter our fully discretized scheme produces non negative solutions for all cases under a time step constraint our method also applies to two dimensional problems on cartesian meshes numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics

a variational finite volume scheme for wasserstein gradient flows we propose a variational finite volume scheme to approximate the solutions to wasserstein gradient flows the time discretization is based on an implicit linearization of the wasserstein distance expressed thanks to benamou brenier formula whereas space discretization relies on upstream mobility two point flux approximation finite volumes our scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level our scheme can be applied to a wide range of energies guarantees non negativity of the discrete solutions as well as decay of the energy we show that our scheme admits a unique solution whatever the convex energy involved in the continuous problem and we prove its convergence in the case of the linear fokker planck equation with positive initial density numerical illustrations show that it is first order accurate in both time and space and robust with respect to both the energy and the initial profile

a martingale study of the beurling ahlfors transform inrn abstract we apply the theory of martingale transforms to study the beurling ahlfors transform s in dimensions n 2 this operator reveals a rich structure through its representation as a martingale and we obtain new results concerning the operator norm of s acting on the class of differential forms having l p coefficients in particular we show that its norm is independent of the dimension when restricted to k forms and we present new essen type norm inequalities related to this martingale structure finally we suggest a purely analytic method to further investigate these norms which up to now has been lacking

boundary conditions for the microscopic fene models we consider the microscopic equation of finite extensible nonlinear elasticity fene models for polymeric fluids under a steady flow field it is shown that for the underlying fokker planck type of equations any preassigned distribution on the boundary will become redundant once the nondimensional number text it li frac hb k bt geq 2 where h is the elasticity constant sqrt b is the maximum dumbbell extension t is the temperature and k b is the usual boltzmann constant moreover if the probability density function is regular enough for its trace to be defined on the sphere m sqrt b then the trace is necessarily zero when text it li 2 these results are consistent with our numerical simulations as well as some recent well posedness results by preassuming a zero boundary distribution

on the approximation of the fokker planck equation of the finitely extensible nonlinear elastic dumbbell model i a new weighted formulation and an optimal spectral galerkin algorithm in two dimensions we propose a new weighted weak formulation for the fokker planck equation of the finitely extensible nonlinear elastic dumbbell model and prove its well posedness in weighted sobolev spaces we also propose simple and efficient semi implicit time discretization schemes which are unconditionally stable i e the step size of time marching does not depend on the number of the bases used in configurational space we then restrict ourselves to the two dimensional case and construct two fourier jacobi spectral galerkin algorithms which enjoy the following properties i they are unconditionally stable spectrally accurate and of optimal computational complexity ii they conserve the volume and provide accurate approximation to higher order moments of the distribution function and iii they can be easily extended to coupled nonhomogeneous systems numerical results are presented to show how to choose a proper weight to get the best numerical results of the distribution function and the polymer stress

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

sharp inequalities for the distribution of a stochastic integral in which the integrator is a bounded submartingale we obtain a sharp probability bound on the maximal function of a strong subordinate of a bounded submartingale an analogous inequality also holds for stochastic integrals in which the integrator is a bounded submartingale and the integrand is a bounded predictable process

maximum principle satisfying third order discontinuous galerkin schemes for fokker planck equations we design and analyze up to third order accurate discontinuous galerkin dg methods satisfying a strict maximum principle for fokker planck equations a procedure is established to identify an effective test set in each computational cell to ensure the desired bounds of numerical averages during time evolution this is achievable by taking advantage of the two parameters in the numerical flux and a novel decomposition of weighted cell averages based on this result a scaling limiter for the dg method with first order euler forward time discretization is proposed to solve the one dimensional fokker planck equations strong stability preserving high order time discretizations will keep the maximum principle it is straightforward to extend the method to two and higher dimensions on rectangular meshes we also show that a modified limiter can preserve the strict maximum principle for dg schemes solving fokker planck equations as a consequence the present schemes preserve steady states numerical tests

a structure preserving discontinuous galerkin scheme for the fisher kpp equation an implicit euler discontinuous galerkin scheme for the fisher kolmogorov petrovsky piscounov fisher kpp equation for population densities with no flux boundary conditions is suggested and analyzed using an exponential variable transformation the numerical scheme automatically preserves the positivity of the discrete solution a discrete entropy inequality is derived and the exponential time decay of the discrete density to the stable steady state in the l 1 norm is proved if the initial entropy is smaller than the measure of the domain the discrete solution is proved to converge in the l 2 norm to the unique strong solution to the time discrete fisher kpp equation as the mesh size tends to zero numerical experiments in one space dimension illustrate the theoretical results

on burkholder function for orthogonal martingales and zeros of legendre polynomials burkholder obtained a sharp estimate of bf e w p via bf e z p for martingales w differentially subordinated to martingales z his result is that bf e w p le p 1 p bf e z p where p max big p p over p 1 big what happens if the martingales have an extra property of being orthogonal martingales this property is an analog for martingales of the cauchy riemann equation for functions and it naturally appears in a problem on singular integrals see the references at the end of section 1 we establish here that in this case the constant is quite different actually bf e w p le big 1 z p over 1 z p big p bf e z p p ge 2 where z p is a specific zero of a certain solution of the legendre ode we also prove the sharpness of this estimate asymptotically 1 z p 1 z p 4j 2 0 o 1 p p to infty where j 0 is the first positive zero of the bessel function of zero order this connection with zeros of special functions and orthogonal polynomials for p n n 1 is rather unexpected

a positivity preserving high order discontinuous galerkin scheme for convection diffusion equations abstract for constructing high order accurate positivity preserving schemes for convection diffusion equations we construct a simple positivity preserving diffusion flux discontinuous galerkin dg schemes with such a positivity preserving diffusion flux are nonlinear schemes which can be regarded as a reduction of the high order positivity preserving dg schemes for compressible navier stokes equations in 1 to scalar diffusion operators in this paper we focus on the local dg method to discuss how to apply such a flux a limiter on the auxiliary variable for approximating the gradient of the solution must be used so that the diffusion flux is positivity preserving in the sense that dg schemes with this flux satisfies a weak positivity property together with a positivity preserving limiter high order dg schemes with strong stability preserving time discretizations can be rendered positivity preserving without losing conservation or high order accuracy for convection diffusion problems with periodic boundary conditions or a special class of dirichlet or neumann boundary conditions numerical tests on a few parabolic equations and an application to modeling electrical discharges are shown to demonstrate the performance of this scheme

differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities let x and y be two continuous time martingales if quadratic variation of x minus that of y is a nondecreasing and nonnegative function of time we say that y is differentially subordinate to x and prove that y p p 1 x p for 1 p where p p v q and q is the conjugate of p this inequality contains burkholder s l p inequality for stochastic integrals which implies that the above inequality is sharp we also extend his concept of strong differential subordination and several other of his inequalities and sharpen an inequality of banuelos

a skew symmetric discontinuous galerkin spectral element discretization and its relation to sbp sat finite difference methods this paper shows that the discontinuous galerkin collocation spectral element method with gauss lobatto points dgsem gl satisfies the discrete summation by parts sbp property and can thus be classified as an sbp sat simultaneous approximation term scheme with a diagonal norm operator in the same way sbp sat finite difference schemes can be interpreted as discontinuous galerkin type methods with a corresponding weak formulation based on an inner product formulation common in the finite element community this relation allows the use of matrix vector notation common in the sbp sat finite difference community to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for dgsem gl and diagonal norm sbp sat based on this result a skew symmetric energy stable discretely conservative dgsem gl formulation applicable to general diagonal norm sbp sat schemes for the nonlinear burgers equation is constructed