Convergence of a large time step generalization of Godunov's method for conservation laws

A natural generalization of Godunov's method for Courant numbers larger than 1 is obtained by handling interactions between neighboring Riemann problems linearly, i.e., by allowing waves to pass through one another with no change in strength or speed. This method is well defined for arbitrarily large Courant numbers and can be written in conservation form. It follows that if a sequence of approximations converges to a limit u(x,t) as the mesh is refined, then u is a weak solution to the system of conservation laws. For scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence. It is conjectured that in fact every sequence converges to the (unique) entropy solution provided the correct entropy solution is used for each Riemann problem. If the true Riemann solutions are replaced by approximate Riemann solutions which are consistent with the conservation law, then the above convergence results for general systems continue to hold.     

Variation zum Konzept der Lusternik—Schnirelmann—Kategorie

For any non—empty class a of pointed spaces, we define the notions of a—cat and a—cat in analogy to T. Ganea's definitions of cat and Cat. Let e.g. s be the class of wedges of spheres Sⁿ, n ≥ 1, then the difference s—cat — cat can be arbitrarily large, s—Cat is well known as spherical cone—length, but what is s—Cat — s—cat? If σ is the class of suspensions we recover cat and Cat. We also investigate subclasses a ? σ such that a—cat(x) = σ–cat(X) for large classes of spaces X. As a byproduct we extend known mapping theorems for cat slightly. As an application we obtain a—cat as an upper bound for the solvability (or nilpotency) degree of certain groups of self—equivalences.     

The Goulden—Jackson cluster method: extensions,applications and implementations

Th powerful (and so far under-utilized) Goulden—Jackson Cluster method for finding the generating function for the number of words avoiding, as factors, the members of a prescribed set of ‘dirty words’, is tutorialized and extended in various directions. The authors' Maple implementations, contained in several Maple packages available from this paper's website www.math.temple.edu/zeilberg/gj.html, ar described and explained.     

Mathematics education research, its nature, and its purpose: A discussion of Lester's paper

This commentary discusses the framework for mathematics education researchers outlined in Lester's (2005) paper. The author reacts to (a) Lester's concern about the current political forces in the U.S. to define scientific research in education rigidly, and offers a possible reason—apart from political ideology—for the emergence of hese forces; (b) recapitulates lester's outline and model for theory-based research in mathematics education, and intereprets Lester's paper as a call to the MER community to respond to the current political forces that (inappropriately) shape our field and (c) addresses the role of mathematical context in MER, a topic absent from the paper's narrative.     

The evolution dam problem for a compressible fluid with nonlinear Darcy's law and Dirichlet boundary condition

In this paper, we consider the evolution dam problem (P) related to a compressible fluid flow governed by a generalized nonlinear Darcy's law with Dirichlet boundary conditions on some part of the boundary. We establish existence of a solution for this problem. We choose a convenient regularized problem (P_?) for which we prove the existence and uniqueness of solution using the comparison Lemma 2.1 and the Schauder fixed‐point theorem. Then, we pass to the limit, when ? goes to 0, to get a solution for our problem. Moreover, we will see another approach for the incompressible case where we pass to the limit in (P), when α goes to 0, to get a solution.     

Labeled sequent calculi for modal logics and implicit contractions

The paper settles an open question concerning Negri-style labeled sequent calculi for modal logics and also, indirectly, other proof systems which make (more or less) explicit use of semantic parameters in the syntax and are thus subsumed by labeled calculi, like Brünnler’s deep sequent calculi, Poggiolesi’s tree-hypersequent calculi and Fitting’s prefixed tableau systems. Specifically, the main result we prove (through a semantic argument) is that labeled calculi for the modal logics K and D remain complete w.r.t. valid sequents whose relational part encodes a tree-like structure, when the unique rule which contains an harmful implicit contraction—by which the condition that the premises be less complex than the conclusion is violated—is modified into a contraction-free one respecting the latter condition, thus making the proof-search space finite.     

«Assoluta continuità» e «Quasi sub-additività rispetto a una coppia» in spazi di misura

A well-known theorem due to J. C. Burkill and some existence theorems for Cesari's integral obtained in [1] are generalized here in measure spaces. Moreover, the last result, where is used the concept of quasi sub-additivity with respect to two families \((\mathfrak{D},\mathfrak{D}'),\mathfrak{D}' \subset \mathfrak{D}\) , and to the mesh for a set function ψ:I?ψ(I),I ∈ {I}, introduced in [1], is an extension of a theorem obtained by L. Cesari in [5].     

Existence of solutions for discrete p-Laplacian with potential boundary conditions

We are concerned with the existence of solutions for some discrete p-Laplacian equations subjected to a potential type boundary condition. Our approach is a variational one and relies on Szulkin's critical point theory. We obtain the existence of solutions in a coercive case as well as the existence of non-trivial solutions when the corresponding energy functional has a ‘mountain pass’ geometry.     

Convergence analysis of some first order and second order time accurate gradient schemes for semilinear second order hyperbolic equations

This work is devoted to the convergence analysis of finite volume schemes for a model of semilinear second order hyperbolic equations. The model includes for instance the so‐called Sine‐Gordon equation which appears for instance in Solid Physics (cf. Fang and Li, Adv Math (China) 42 (2013), 441–457; Liu et al., Numer Methods Partial Differ Equ 31 (2015), 670–690). We are motivated by two works. The first one is Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043) where a recent class of nonconforming finite volume meshes is introduced. The second one is Eymard et al. (Numer Math 82 (1999), 91–116) where a convergence of a finite volume scheme for semilinear elliptic equations is provided. The mesh considered in Eymard et al. (Numer Math 82 (1999), 91–116) is admissible in the sense of Eymard et al. (Elsevier, Amsterdam, 2000, 723–1020) and a convergence of a family of approximate solutions toward an exact solution when the mesh size tends to zero is proved. This article is also a continuation of our previous two works (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321; Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39) which dealt with the convergence analysis of implicit finite volume schemes for the wave equation. We use as discretization in space the generic spatial mesh introduced in Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043), whereas the discretization in time is performed using a uniform mesh. Two finite volume schemes are derived using the discrete gradient of Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043). The unknowns of these two schemes are the values at the center of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The first scheme is inspired from the previous work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39), whereas the second one (in which the discretization in time is performed using a Newmark method) is inspired from the work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321). Under the assumption that the mesh size of the time discretization is small, we prove the existence and uniqueness of the discrete solutions. If we assume in addition to this that the exact solution is smooth, we derive and prove three error estimates for each scheme. The first error estimate is concerning an estimate for the error between a discrete gradient of the approximate solution and the gradient of the exact solution whereas the second and the third ones are concerning the estimate for the error between the exact solution and the discrete solution in the discrete seminorm of and in the norm of . The convergence rate is proved to be for the first scheme and for the second scheme, where (resp. k) is the mesh size of the spatial (resp. time) discretization. The existence, uniqueness, and convergence results stated above do not require any relation between k and . The analysis presented in this work is also applicable in the gradient schemes framework. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 5–33, 2017     

Mathematical study of a tissue-implant connection in a ventral hernia repair in a context of the system's parameters

Ventral hernia repair is nowadays a well recognized matter in surgery and advanced techniques of operating are developed. However, mechanical properties of the system after an operation are not known so recurrences of the illness happen. The paper describes a simple mathematical model of a human fascia-synthetic implant system. The synthetic mesh is modeled by a cable structure and its elastic supports simulate the human fascia. The junction force in the fascia-implant system is sought. A relation of the force change to variations of some parameters of the system is also studied by applying a sensitivity analysis. The authors conclude that a crucial role for the junction force and a patient's comfort plays the implant's elasticity modulus and initial tension of the mesh is the least important factor. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of solutions to a phase transition model with microscopic movements

We prove the existence of weak solutions for a 3D phase change model introduced by Michel Frémond in (Non‐smooth Thermomechanics. Springer: Berlin, 2002) showing, via a priori estimates, the weak sequential stability property in the sense already used by the first author in (Comput. Math. Appl. 2007; 53 :461–490). The result follows by passing to the limit in an approximate problem obtained adding a superlinear part (in terms of the gradient of the temperature) in the heat flux law. We first prove well posedness for this last problem and then—using proper a priori estimates—we pass to the limit showing that the total energy is conserved during the evolution process and proving the non‐negativity of the entropy production rate in a suitable sense. Finally, these weak solutions turn out to be the classical solution to the original Frémond's model provided all quantities in question are smooth enough. Copyright © 2008 John Wiley & Sons, Ltd.     

A TWO-GRID METHOD FOR THE C0 INTERIOR PENALTY DISCRETIZATION OF THE MONGE-AMP(E)RE EQUATION

The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.     

Infrared absorption spectra of some lanthanide acetylacetonate complexes

The infrared absorption spectra of 12 lanthanide acetylacetonate complexes were measured in the region 400–2000 cm^?1 and discussed. Assignments of the bands especially those due to metal-oxygen (M—O), C=O and C=C stretching vibration is given. It is found that the band at 530 ± 5 cm^?1 is due mainly to Ln—O stretching vibration. In the carbonyl region, it is confirmed that the band at lower frequency is a C=C stretching vibration while that at higher frequency is due to C=O stretching vibration. Integral intensities for the M—O, C=O and C=C stretching vibrations were calculated in KB₁ and CHCl₃ solutions. The variation of the intensities of the M—O bands witn complexed cation were discussed in terms of crystal field stabilization energy (CFSE).     

A Comparative Numerical Study of Meshing Functionals for Variational Mesh Adaptation

We present a comparative numerical study for three functionals used for variational mesh adaptation. One of them is a generalization of Winslow's variable diffusion functional while the others are based on equidistribution and alignment. These functionals are known to have nice theoretical properties and work well for most mesh adaptation problems either as a stand-alone variational method or combined within the moving mesh framework. Their performance is investigated numerically in terms of equidistribution and alignment mesh quality measures. Numerical results in 2D and 3D are presented.     

Density of Space-Time Distribution of Brownian First Hitting of a Disc and a Ball

We compute the joint distribution of the site and the time at which a d-dimensional standard Brownian motion ((B˙t)) hits the surface of the ball ((U(a) ={—x—<a})) for the first time. The asymptotic form of its density is obtained when either the hitting time or the starting site ((B˙0)) becomes large. Our results entail that if Brownian motion is started at ((x)) and conditioned to hit ((U(a))), at time t, the distribution of the hitting site approaches the uniform distribution or the point mass at ((ax/—x—)) according as ((—x—/t)) tends to zero or infinity; in each case we provide a precise asymptotic estimate of the density. In the case when ((—x—/t)) tends to a positive constant we show the convergence of the density and derive an analytic expression of the limit density.     

Layered elastic solid method for the generation of unstructured dynamic mesh

The layered elastic solid method (LESM), a modified elastic solid method (ESM) was put forward in the present study. In LESM, the computational zone is divided into several layers and material properties of these layers, which stay a certain value in ESM, are changed with mesh deformation. The deformation capability of mesh in LESM is better than ESM and the quality of the mesh generated by LESM is superior to that generated by ESM when they undergo the same deformation. In ESM, the main influence factors on mesh quality and deformation capability are Poisson’s ratio and single-step rotation angle. In LESM, mesh quality and deformation capability reach a largest value with an increase in Young’s modulus. Meanwhile, the mesh can achieve a larger deformation capability when single-step rotation angle is 0.25°. Finally, numerical simulation on a two-dimensional aerofoil using LESM was carried out. It is found that the results of LESM show a better agreement with experiment results.     

The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H C Yang

In this paper we present a unified and simplified approach to the universal eigenvalue inequalities of Payne—Pólya—Weinberger, Hile—Protter, and Yang. We then generalize these results to inhomogeneous membranes and Schrödinger’s equation with a nonnegative potential. We also show that Yang’s inequality is always better than HileProtter’s (and hence also better than Payne—Pólya—Weinberger’s). In fact, Yang’s weaker inequality (which deserves to be better known), $\lambda _{k + 1}< \left( {1 + \frac{4}{n}} \right)\frac{1}{k}\sum\limits_{i = 1}^k {\lambda _i } $ , is also strictly better than Hile—Protter’s. Finally, we treat Yang’s (and related) inequalities for minimal submanifolds of a sphere and domains contained in a sphere by our methods.