Existence in the Large for Pressure-Gradient System

In this paper, the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data. To this end, some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions (see [Diperna, R. J., Existence in the large for quasilinear hyperbolic conservation laws, Arch. Ration. Mech. Anal., 52(3), 1973, 244–257]) are studied. Then they construct the approximate solution sequence through Glimm scheme. By establishing accurate local interaction estimates, they prove the boundedness of the approximate solution sequence and its total variation.     

Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

For the scalar conservation laws with discontinuous flux, an infinite family of (A, B)‐interface entropies are introduced and each one of them is shown to form an L¹‐contraction semigroup (see [2]). One of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [6]. In this paper, we discuss this particular issue in detail and produce a counterexample to show that the solution, in general, has unbounded total variation near the interface. In fact, this example illustrates that smallness of the BV norm of the initial data is immaterial. We hereby settle the question of determining for which of the aforementioned (A, B) pairs the solution will have bounded total variation in the case of strictly convex fluxes. © 2010 Wiley Periodicals, Inc.     

Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods

This article deals with development and analysis of a numerical method for a coupled system describing miscible displacement of one incompressible fluid by another through heterogeneous porous media. A mixed finite element (MFE) method is employed to discretize the Darcy flow equation combined with a conservative finite volume (FV) method on unstructured grids for the concentration equation. It is shown that the FV scheme satisfies a discrete maximum principle. We derive L^∞ and BV estimates under an appropriate CFL condition. Then we prove convergence of the approximate solutions to a weak solution of the coupled system. Numerical results are presented to see the performance of the method in two space dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On the Hughes' model for pedestrian flow: The one-dimensional case

In this paper we investigate the mathematical theory of Hughes' model for the flow of pedestrians (cf. Hughes (2002) [17]), consisting of a non-linear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task. For such an approximated system we prove existence and uniqueness of entropy solutions (in one space dimension) in the sense of Kru?kov (1970) [22], in which the boundary conditions are posed following the approach of Bardos et al. (1979) [7]. We use BV estimates on the density ρ and stability estimates on the potential ? in order to prove uniqueness. Furthermore, we analyze the evolution of characteristics for the original Hughes' model in one space dimension and study the behavior of simple solutions, in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves. The characteristic calculus is supported by numerical simulations.     

More consistency results in partition calculus

This paper has two principal aims. The first is to supply a proof of Theorem 6 of [ShSt1]:Theorem:If ZFC+ “there are c ⁺ measurable cardinals” is consistent, then so is ZFC+ “ ℵ _c +is not a strong limit cardinal and ”. This is done in sections 1 and 2. See the introduction for a discussion of the evolution of the proof and of some interesting questions which remain open, related to obstacles encountered in obtaining maximum freedom in arranging for any desired cardinal exponentiation in Theorems 4 and 6 of [ShSt1]. The method is quite generally applicable in partition calculus and variants of it have in fact been applied in recent work of the authors, see [ShSt2]. first, a preservation result is proved for the game-theoretic properties of the filters considered in [ShSt1]. Then, it is shown that the existence of a system of such filters yields a canonization-style result. Finally, it is shown that the canonization property gives the positive partition relation. The second aim makes the title of this paper slightly inaccurate (but we suspect this will be pardoned): we supply a (straightforward) proof of a result which shows that Theorem 2 of [ShSt1] in some sense is best possible. This is done in section 3. Research of both authors partially supported by NSF grants. Partially supported by the BSF. Paper number 293.     

On the system governed by parabolic partial delay-differential equations with first boundary conditions

Summary In this paper, we consider a system governed by second order parabolic partial delay-differential equations with first boundary conditions. All the coefficients of the system are assumed bounded measurable and contain delays in their arguments. The second order parabolic partial delay-differential equation is in ? divergence form ?. In Theorem4.1, we present results on the existence and uniqueness of weak solution in the sense of Aronson for this class of systems. In Theorem4.2, we prove that whenever the coefficients and forcing terms converge in the almost everywhere topology the corresponding solutions converge weakly in an appropriate Sobolev space. Entrata in Redazione il 9 novembre 1977.     

Convergence of the point vortex method for the 3-D euler equations

We prove consistency, stability, and convergence of a point vortex approximation to the 3-D incompressible Euler equations with smooth solutions. The 3-D algorithm we consider here is similar to the corresponding 3-D vortex blob algorithm introduced by Beale and Majda; see [3]. We first show that the discretization error is second-order accurate. Then we show that the method is stable in l^p norm for the particle trajectories and in w^?1.p norm for discrete vorticity. Consequently, the method converges up to any time for which the Euler equations have a smooth solution. One immediate application of our convergence result is that the vortex filament method without smoothing also converges.     

On the theory of Schur-Rings

Zusammenfassung Diese Arbeit versucht, die von Issai Schur[1] entdcckte und von Wielandt ([14], [15], [16], [17]) betr?chtlich neiterentwickelte Methode zur Untersuchung von endlichen Permutationsgruppen zu einer Theorie der Schur-Ringe zu entfalten. Der Grundgedanke ist sehr einfach: Die Schur-Ringe werden nicht als eine spezielle Klasse von Ringen aufgefaβt, sondern als eine eigene mathematische Struktur. Nach unserer heutigen Ansicht f?llt der Begriff der mathematischen Struktur weitgehend mit dem Begriff der Kategorie zusammen. Daher wird für die Schur-Ringe (genauer: für die Schur-Algebren) ein eigener Homomorphiebegriff (Definition1.5) eingeführt, der eine Kategorie liefert (Theorem1.6). Ein weiterer Leitgedanke ist mit dem kategoriellen Grundgedanken sehr eng verknüpft. Die Theorie der Schur-Ringe wird als eine Verallgemeinerung der Theorie der endlichen Gruppen aufgefaβt und in diesem Sinne entwickelt. Dabei ist die Theorie der endlichen Gruppen vermittelst der Gruppenringe der endlichen Gruppen (die eine spezielle Teilkategorie der Kategorie aller Schur-Ringe sind) in die Theorie der Schur-Ringe eingefügt. Hierfür ist es wichtig, daβ die Morphismen der Gruppenringe in der Kategorie der Schur-Ringe genau die von den Gruppenhomomorphismen induzierten Gruppenringhomomorphismen sind. Die Einbettung der Theorie der endlichen Gruppen in die Theorie der Schur-Ringe vollzieht sich entlang dreier Entwicklungslinien. Die erste ist eine verallgemeinerte Charakterentheorie ([2], [3], [5], [6], [7] und[8]), die die Theorie der (gen?hnlichen) Charaktere von endlichen Gruppen als Spezialfall enth?lt. Die zweite ist die Verknüpfung der Struktur jedes Schur-Ringes T auf einer endlichen Gruppe G mit gewissen Klassen von Untergruppen von G. Es werden die Begriffe der T-Untergruppe (Abschnitt 3), des T-Normalteilers (Abschnitt 4), und der T-subnormalen Untergruppe (Abschnitt 8) eingeführt. Die T-Untergruppen bilden einen Teilverband des Verbandes aller Untergruppen von G (Theorem3.4). Die T-Normalteiler sind genau die Kerne (Definition6.1) der Homomorphismen der Schur-Algebren QT (Theoreme6.2 und6.3). Der dritte und wohl zugleich der wichtigste Aspekt ist die Gültigkeit des Homomorphiesatzes (Theorem6.2) und der Isomorphies?tze (Theoreme7.1 und7.2) für Schur-Algebren. Auf diese S?tze gründet sich der Vier-Untergruppen-Satz (Zassenhaus’ Lemma; Theorem9.1), der den Verfeinerungssatz für T-Subnormalketten (Theorem9.2) und den Jordan-H?lder Satz für T-Kompositionsketten (Theorem10.3) nach sich zieht. Als die Theorie der Schur-Ringe ungef?hr den soeben geschilderten Stand erreicht hatte, tauchte die Idee auf, diese Theorie auf beliebige Gruppen zu verallgemeinern ([9], [10], [11], [12], [13]). Das führte zum Begriff der Schur-Halbgruppe (Definition1.9). Der zugeh?rige Homomorphiebegriff (Definition1.11) liefert die Kategorie aller Schur-Halbgruppen (Theorem1.12), die die Kategorie aller Gruppen als echte Teilkategorie enth?lt. Jedem Schur-Ring T über einer endlichen Gruppe G wird eine Schur-HalbgruppeT über G zugeordnet (Theorem1.15). Jedem Homomorphismus ϕ einer Schur-Algebra ΘT über G wird ein Homomorphismus φ vonT zugeordnet (Theorem1.16). Das Paar der Zuordnungen ΘT →T, ϕ → Φ ist ein Funktor auf der Kategorie aller Schur-Algebren in die Kategorie aller Schur-Halbgruppen über endlichen Gruppen (Theorem1.17).      

On the Radial Behavior of Universal Taylor Series

In this work we deal with universal Taylor series in the open unit disk, in the sense of Nestoridis; see [12]. Such series are not (C,k) summable at every boundary point for every k; see [7], [11]. In the opposite direction, using approximation theorems of Arakeljan and Nersesjan we prove that universal Taylor series can be Abel summable at some points of the unit circle; these points can form any closed nowhere dense subset of the unit circle.     

Differentiable dependence upon the data in a one-phase Stefan problem

It is well known that the free boundary of the one-phase Stefan problem (1.1–5) depends continuously on the boundary data [1]. In this paper we prove that, in addition, the solution operator S which, to each g. assigns the corresponding free boundary, is continuously Frechet differentiable and we give the defining formulas of the Frechet derivative.     

An instability of the Godunov scheme

We construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme [13] can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing‐viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or L¹‐stability estimates can in general be valid for finite difference schemes. © 2006 Wiley Periodicals, Inc.     

Almost sure exponential stability for stochastic neutral partial functional differential equations

In this paper, we initiate a study on stochastic neutral partial functional differential equations in a real separable Hilbert space. Our goal here is to study the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths. The results obtained here generalize the main results from [Taniguchi, Stochastics and Stochastics Reports, 53, (1995) 41–52], [Taniguchi, Stochastic Analysis and Applications, 16, (1998) 965–975] and [Liu and Truman, Statistics Probability Letters, 50, (2000) 273–278]. An example is given to illustrate the theory.     

Anisotropic error estimates for elliptic problems

Summary.  Moving from the anisotropic interpolation error estimates derived in [12], we provide here both a-priori and a-posteriori estimates for a generic elliptic problem. The a-priori result is deduced by following the standard finite element theory. For the a-posteriori estimate, the analysis extends to anisotropic meshes the theory presented in [3–5]. Numerical test-cases validate the derived results. Received July 22, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002 Mathematics Subject Classification (1991): 65N15, 65N50     

First Kind Cordial Volterra Integral Equations 1

For (pure) cordial Volterra integral equations of the first kind, we establish stability estimates of the solution in the scales of Banach spaces of C ^m -smooth functions on (0, T] and on [0, T]. This enables to estimate the error of polynomial or generalized polynomial approximate solutions which are easy to be determined.     

New constructions of hypohamiltonian and hypotraceable graphs

A graph G is hypohamiltonian/hypotraceable if it is not hamiltonian/traceable, but all vertex‐deleted subgraphs of G are hamiltonian/traceable. All known hypotraceable graphs are constructed using hypohamiltonian graphs; here we present a construction that uses so‐called almost hypohamiltonian graphs (nonhamiltonian graphs, whose vertex‐deleted subgraphs are hamiltonian with exactly one exception, see [15]). This construction is an extension of a method of Thomassen [11]. As an application, we construct a planar hypotraceable graph of order 138, improving the best‐known bound of 154 [8]. We also prove a structural type theorem showing that hypotraceable graphs possessing some connectivity properties are all built using either Thomassen's or our method. We also prove that if G is a Grinbergian graph without a triangular region, then G is not maximal nonhamiltonian and using the proof method we construct a hypohamiltonian graph of order 36 with crossing number 1, improving the best‐known bound of 46 [14].     

On the Convergence of Finite Element Method for Second Order Elliptic Interface Problems

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L ² and H ¹ norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established.     

A geometry for groups of <Emphasis Type="Italic">J</Emphasis><Subscript>3</Subscript>-type

The proof of the existence and of the uniqueness of groups of J₃-type by G. Higman and J. McKay is based on the fact that a group of J₃-type is a faithful completion of an amalgam of J₃-type, see [11]. In this paper here, we provide a direct reference for that fact. The proofs in this paper are elementary and we do not use any character theory. Received: 1 December 2005     

On sequences of maps into <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> with equibounded <Emphasis Type="Italic">W</Emphasis><Superscript>1/2</Superscript> energies

In the last years there has been some interest in studying mappings in the fractional Sobolev space W^1/2(, S¹), see e.g., [4] [3] [15] [12] and the paper [5]. Motivated by these papers, we characterize here in the framework of Cartesian currents, see [9], the class of weak limits of sequences of smooth mappings with values into S¹ with equibounded W^1/2 energies.     

A Priori Estimate for the Discontinuous Oblique Derivative Problem for Elliptic Systems

Recently [6] an existence as well as a uniqueness theorem for the discontinuous oblique derivative problem for nonlinear elliptic system of first order in the plane, see [12, 19, 23] was proved, based on some a priori estimate from [20]. This estimate, however, is deduced by reductio ad absurdum. Therefore the constants in this estimate are unknown so that the estimate cannot be used for numerical procedures, e.g. for approximating the solution of a nonlinear problem by solutions of related linear problems, see [24, 3, 4]. In this paper a direct proof of an a priori estimate is given using some variations of results from [14], see also [11], where the constants can explicitely be estimated. For related a priori estimates see [1 – 5, 8, 16, 17, 20, 21, 24 – 26]. A basic reference for the oblique derivative problem is [9].     

Amalgamation bases for the class of lattice-ordered groups

We prove Theorem A. The cardinal product of two copies of the integers is an amalgamation base for the class of all lattice-ordered groups but their lexicographic product is not. This answers Problem 27 of [Black Swamp Problem Book (W. Charles Holland, ed.), Bowling Green State University, 1982]. We also prove Theorem B. he cardinal product of n copies of the integers is not an amalgamation base for the class of all lattice-ordered groups if n ≥ 3.