Volume entropy based on integral Ricci curvature and volume ratio

For a compact Riemannian manifold M, we obtain an explicit upper bound of the volume entropy with an integral of Ricci curvature on M and a volume ratio between two balls in the universal covering space.     

Development and Analysis of Higher Order Finite Volume Methods over Rectangles for Elliptic Equations

Currently used finite volume methods are essentially low order methods. In this paper, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. In particular, we restrict ourselves to finite volume methods posed over rectangular partitions and begin by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of index two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we derive finite volume methods of all orders from the Raviart–Thomas two-dimensional rectangular elements; we also find finite volume methods to associate with BDFM ₂ three-dimensional rectangles. In each case, we obtain optimal error estimates for both the scalar variable and the recovered flux.     

涉及两个质点组的一个恒等式及应用(英文)

In this paper, we first give and prove a geometric identity for distance between any point and mass-points of two mass-points systems in n-dimensional Euclidean space Rn.As its application, we obtain an equality for two mass-points systems and its radius of the circumscribed sphere and an inequality for distance between mass-points of two mass-points systems and its k-dimensional volume.     

Adaptive cell-centered finite volume method for diffusion equations on a consistent quadtree grid

Models applied in image processing are often described by nonlinear PDEs in which a good approximation of gradient plays an important role especially in such cases where irregular finite volume grids are used. In image processing, such a situation can occur during a coarsening based on quadtree grids. We present a construction of a deformed quadtree grid in which the connection of representative points of two adjacent finite volumes is perpendicular to their common boundary enabling us to apply the classical finite volume methods. On the other hand, for such an adjusted grid, the intersection of representative points connection with a finite volume boundary is not a middle point of their common edge and standard methods cannot achieve a good accuracy. In this paper we present a new cell-centered finite volume method to evaluate solution gradients, which results into a solution of a simple linear algebraic system and we prove its unique solvability. Finally we present numerical experiments for the regularized Perona-Malik model in which we applied this new method.     

Relaxation theorems in nonlinear elasticity

Relaxation theorems which apply to one, two and three-dimensional nonlinear elasticity are proved. We take into account the fact an infinite amount of energy is required to compress a finite line, surface or volume into zero line, surface or volume. However, we do not prevent orientation reversal.     

Efficient Scheduling Rules in a Combined Make-to-Stock and Make-to-Order Manufacturing System

In this paper, we consider a mixed MTS/MTO policy to manage a single manufacturing facility producing two classes of end-products. A few end-products have high volume demands, whereas a fairly large number of end-products have low volume demands. In this situation, it is appealing to try to produce the high volume products according to an MTS policy and the low volume products according to an MTO policy. The purpose of this paper is to analyze and compare the impact of the choice of the scheduling policy on the overall performance of the system. We consider two policies: the classical FIFO policy and a priority policy (PR). The PR policy gives priority to production orders corresponding to low volume products over production orders corresponding to high volume products. Under some simple stochastic modeling assumptions, we develop analytical/numerical solutions to optimise each system. We then provide insights regarding this issue with the help of numerical examples. It appears that for some range of parameters, the PR rule can outperform the FIFO rule in the sense that, to achieve the same service level constraint, the corresponding cost under the PR rule is much lower. This situation is encountered when the low volume products can be managed with an MTO policy under the PR scheduling rule, while they have to be managed according to an MTS policy under the FIFO scheduling rule. We also derive some theoretical properties that support our empirical findings.     

The Mixed Volume of Two Finite Vector Sets

We introduce the concept of the mixed volume of two finite vector sets in Rⁿ. By employing the exterior differential, we prove a new and powerful inequality and establish a series of quantity relations associated with the mixed volume of two finite vector sets. As applications, we discuss some well-known results of simplices and the Hadamard inequality.     

Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling

We consider a model system consisting of two reaction–diffusion equations, where one species diffuses in a volume while the other species diffuses on the surface which surrounds the volume. The two equations are coupled via a nonlinear reversible Robin-type boundary condition for the volume species and a matching reversible source term for the boundary species. As a consequence of the coupling, the total mass of the two species is conserved. The considered system is motivated for instance by models for asymmetric stem cell division.Firstly we prove the existence of a unique weak solution via an iterative method of converging upper and lower solutions to overcome the difficulties of the nonlinear boundary terms. Secondly, our main result shows explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate for the considered nonlinear volume-surface reaction–diffusion system.     

Homological Invariance For Asymptotic Invariants and Systolic Inequalities

We show that the systolic constant, the minimal volume entropy, and the spherical volume of a manifold depend only on the image of the fundamental class under the classifying map of the universal covering. Moreover, we compute the systolic constant of manifolds with fundamental group of order two (modulo the value for the real projective space) and derive an inequality between the minimal volume entropy and the systolic constant. Received: February 2007, Revised: November 2007, Accepted: December 2007     

网络服务中非线性多重两阶段定价策略

本文在基于流量的定价策略基础上,结合非线性定价方法给出了一种多重两阶段定价模型,并结合原单一定价策略构造混合定价,通过分析生产者剩余和消费者剩余,推出该模型存在帕累托改进并给出了具体的改进方法。     

Integral foliated simplicial volume of aspherical manifolds

Simplicial volumes measure the complexity of fundamental cycles of manifolds. In this article, we consider the relation between the simplicial volume and two of its variants — the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space.     

A comparison of routing sets for robust network design

Designing a network able to route a set of non-simultaneous demand vectors is an important problem arising in telecommunications. In this paper, we compare the optimal capacity allocation costs for six routing sets: affine routing, volume routing and its two simplifications, the routing based on an unrestricted 2-cover of the uncertainty set, and the routing based on a cover delimited by a hyperplane.     

Superconvergence and Asymptotic Expansions for Bilinear Finite Volume Element Approximations

Aiming at the isoparametric bilinear finite volume element scheme, we initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise by employing the energy-embedded method on uniform grids. Furthermore, we prove that the approximate derivatives are convergent of order two. Finally, numerical examples verify the theoretical results.     

On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

We study a family of geometric variational functionals introduced by Hamilton, and considered later by Daskalopulos, Sesum, Del Pino and Hsu, in order to understand the behavior of maximal solutions of the Ricci flow both in compact and noncompact complete Riemannian manifolds of finite volume. The case of dimension two has some peculiarities, which force us to use different ideas from the corresponding higher-dimensional case. Under some natural restrictions, we investigate sufficient and necessary conditions which allow us to show the existence of connected regions with a connected complementary set (the so-called “separating regions”). In dimension higher than two, the associated problem of minimization is reduced to an auxiliary problem for the isoperimetric profile (with the corresponding investigation of the minimizers). This is possible via an argument of compactness in geometric measure theory valid for the case of complete finite volume manifolds. Moreover, we show that the minimum of the separating variational problem is achieved by an isoperimetric region. The dimension two requires different techniques of proof. The present results develop a definitive theory, which allows us to circumvent the shortening curve flow approach of the above mentioned authors at the cost of some applications of the geometric measure theory and of the Ascoli-Arzela's Theorem.     

Mathematical and numerical study of a corrosion model

In this paper, we consider a PDE system arising in corrosion modelling. This system consists in two convection-diffusion equations on the densities of charge carriers and a Poisson equation on the electric potential. Boundary conditions are Robin boundary conditions. We discretize each equation by a finite volume scheme and we prove the convergence of the scheme towards a weak solution to the initial system. Finally, we provide numerical results describing the behaviour of the solutions with respect to an applied voltage.     

On the volume of the convex hull of two convex bodies

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$ -space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.     

Meshfree Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations

In this paper, we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients. There are two contributions of this paper. Firstly, we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space, which is competitive for high-dimensional random inputs. Secondly, the a priori error estimates are derived for the state,the co-state and the control variables. Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.     

Numerical Solution of Hamilton-Jacobi-Bellman Equations by an Upwind Finite Volume Method

In this paper we present a finite volume method for solving Hamilton-Jacobi-Bellman(HJB) equations governing a class of optimal feedback control problems. This method is based on a finite volume discretization in state space coupled with an upwind finite difference technique, and on an implicit backward Euler finite differencing in time, which is absolutely stable. It is shown that the system matrix of the resulting discrete equation is an M-matrix. To show the effectiveness of this approach, numerical experiments on test problems with up to three states and two control variables were performed. The numerical results show that the method yields accurate approximate solutions to both the control and the state variables.     

Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions

We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretization and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains.     

有限体积KFVS方法在二维溃坝中的应用

本文采用了基于KFVS格式的有限体积方法 (FVM)求解了控制水流运动的二维浅水方程 ,建立了二维水坝瞬间溃坝的洪水演进模型 .并应用此模型模拟了二维非对称溃坝和对称溃坝情形下坝左下角有障碍物时的洪水波演进过程 .模拟结果表明该数学模型对二维浅水运动的模拟很有效 .