A blow-up criterion for 3D compressible viscoelastic flow with large initial data

In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).     

Numbers as functions

In this survey I discuss A. Buium’s theory of “differential equations in the padic direction” ([8]) and its interrelations with “geometry over field with one element”, on the background of various approaches to p-adic models in theoretical physics (cf. [1, 30]).     

Discretization of multiparameter eigenvalue problems

Summary Although multiparameter eigenvalue problems, as for example Mathieu's differential equation, have been known for a long time, so far no work has been done on the numerical treatment of these problems. So in this paper we extend the spectral theory for one parameter (cf. [7, II, VII]) to multiparameter eigenvalue problmes, formulate in the framework of discrete approximation a convergent numerical treatment, establish algebraic bifurcation equations for the intersection points of the eigenvalue curves and illustrate this with some numerical examples.     

Gorenstein gr-injective modules over graded isolated singularities

Graded isolated singularities appear very naturally in algebraic pro-jcctive geometry (cf. [18]) or invariant theory of the binary polyhedral groups (cf. [15], [17]). We define mock finitely generated modules in the category of graded modules and we use such graded modules which are also Gorenstein injective (cf. [1], [2]) in this category to characterize graded isolated Gorenstein singularities.     

Modes of transition from imperfection sensitivity to imperfection insensitivity

In order to a priori find the mode of conversion of a structure from imperfection sensitivity to insensitivity in the course of sensitivity analysis of the initial postbuckling path (cf. [2]), terms appearing in the consistently linearized eigenvalue problem (cf. [1]) are studied. Two main classes can be identified (cf. [4]). The first one is characterized by the restriction of the prebuckling deformations to axial deformations. For this class, the quality of the aforementioned conversion is better than the one for the other class, for which the above restriction does not hold. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A robin-neumann problem in bounded domains with splitting boundary

We consider a mixed boundary-value problem for the homogeneous Laplace equation in a bounded domain which boundary splits up into two disjoint smooth components. On the one boundary component we pose a homogeneous Robin condition and an inhomogeneous Neumann condition on the other. We give a weak formulation, interpret this problem as a generalized spectral (eigenvalue) problem in the sense of F.Stummel (cf.[12]) and investigate existence, uniqueness and regularity of weak solutions. This problem is a cut-off version of a basic problem in water-wave theory (cf.Ramm [8], pp.394-395, Simon/Ursell [10] Stoker [11])     

广义M?bius反演及其在积分方程与插值过程上的应用——纪念华罗庚教授

The object of this expository paper is to establish a kind of generalized M?bius-Rota inversion formula by the aid of non-standard analysis. Certain constructive applications to Volterra type integral equations and to equidistant smooth interpolation process are expounded in some detail. Moreover, some of our earlier results (cf. [1] ,[2] ,[4]) are refined and improved.     

Weak solutions for the fully hyperbolic phase-field system of conserved type

We consider a conserved phase-field system coupling two nonlinear hyperbolic integro-differential equations. The model results from the assumption that the material undergoing phase transition exhibits some thermal memory effects (cf. [15]) and that the response of the order parameter to the variation of the free-energy functional is delayed (cf. [10, 23]). We prove the existence of the solution to the corresponding initial-boundary value problem associated with the resulting PDE system and a (conditioned) continuous dependence estimate of the solution with respect to the data of the problem. This work is partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca, PRIN no. 2004011204, Project Analisi Matematica nei Problemi Inversi     

On characterizations of atomistic lattices

In the present paper we characterize atomistic lattices. These characterizations are given in terms of concepts related to pure elements (cf. [4], [6]) and neat elements (cf. [3]).     

On the Relation between Linear Difference and Differential Equations with Polynomial Coefficients

This paper represents the third part of a contribution to the “dictionary” of homogeneous linear differential equations with polynomial coefficients on one hand and corresponding difference equations on the other. In the first part (cf. [4]) we studied the case that the differential equation (D) has at most regular singularities at O and at ∞, and arbitrary singularities in the rest of the complex plane. We constructed fundamental systems of solutions of a corresponding difference equation (A), using integral transforms of microsolutions of (D) at its singular points in ?. In the second part ([5]) we considered differential equations having at most a regular singularity at O and an irregular one at O. We used integral transforms of asymptotically flat solutions of (D) to define it fundamental system of solutions of (Δ), holomorphic in a right half plane, and integral transforms of sections of the sheaf of solutions of (D) modulo solutions with moderate growth as t → 0 in some sector, to define a fundamental system of (Δ), holomorphic in a left half plane. In this final part we combine the techniques and results of the preceding papers to deal with the general case.     

On a theorem of Wald and Wolfowitz on randomization in statistics

In [9] it is proved that to each randomized decision function, there exists an equivalent randomized decision rule (cf. [3]), provided that the space of decisions is Polish. The object of the present paper is to show that the above theorem is a special case of risk-equivalence (cf. [5]) even if the space of decisions is analytic.     

Filtered and stable boolean powers are relativized full boolean powers

The question whether the extended theorem of Feferman and Vaught (cf. Comer [7], Volger [13]) can be applied to filtered boolean powers with arbitrary base was raised by J. H. Schmerl [11]. It can be answered with the help of a relativized version of the theorem (cf. Comer [7], Volger [12], [14]) which seems to have escaped notice. Therefore we shall present in this note a precise formulation of the relativized version and apply it to filtered boolean powers with arbitrary base and to stable boolean powers with finite base and finite group. The former application was mentioned in Comer [7]. The latter application yields a decidability result which seems to be new. The question whether the second application can be extended to arbitrary bases remains open.Presented by E. Nelson.     

Identification of two independent coefficients with one observation for the Schrödinger operator in an unbounded strip

This article is devoted to prove a stability result for two independent coefficients (each one depending on only one variable) for a Schrödinger operator in an unbounded strip with only one observation on an unbounded subset of the boundary. For that, we first use the global Carleman estimate proved in Cardoulis et al. (2008) [3]. Then, with a Carleman-type estimate for a first order differential operator (cf. Immanuvilov and Yamamoto (2005) [4]) and an energy estimate, we prove the simultaneous identification of the diffusion coefficient and the potential with only one observation.     

New convergence results for continued fractions generated by four-term recurrence relations

In the study of simultaneous rational approximation of functions using rational functions with a common denominator (which can be viewed as the ‘German polynomial’ problem in simultaneous Pade´approximation, cf. [1]) the quest for convergence results lead to the study of generalized continued fractions, a type of Jacobi—Perron algorithm [2,3]. It then becomes important to exploit the connection between the convergence of the generalized continued fraction and the solutions of the associated difference equation (cf. [4,5]).     

Remarks on Steinhaus' property and ratio sets of sets of positive integers

This paper is closely related to an earlier paper of the author and W. Narkiewicz (cf. [7]) and to some papers concerning ratio sets of positive integers (cf. [4], [5], [12], [13], [14]). The paper contains some new results completing results of the mentioned papers. Among other things a characterization of the Steinhaus property of sets of positive integers is given here by using the concept of ratio sets of positive integers.     

On the approximation of the solutions of the Riemann problem for a discontinuous conservation law

For a class of discontinuous flux functions introduced in [3] (cf. also [4]), we prove, for the Riemann problem, an extension of the existence result proved in [2] for a Lipschitz continuous flux function. In the last section, and based in the previous results, we apply the Lax-Friedrichs approximation method and the limiters technique (cf.[6]) to compute the quoted solution in a numerical example. For related results see [5].     

On the Resolution of the Heat Equation with Discontinuous Coefficients

We give in this work some results about the existence and uniqueness with optimal regularity for solutions of a parabolic equation in nondivergence form in L^q(0,T;L^p(Omega)) where 1 < p,q < infinity in two cases. We use Lamberton's results (cf. [9]) in the first case and Dore-Venni's results (cf. [6]) in the second case.     

Generalized Cauchy equations on groups

It is well known ([1], [3]) that any measurable solution of the Cauchy functional equationf(x+y)=f(x)+f(y) must actually be continuous. The same is true of some other functional equations likef(x+y)=f(x)f(y),f(x+y)f(x–y)=f(x) ² –f(y) ², etc. (cf. [1]). In this note we prove a general result of this type for functional equations on groups.     

A note on Krylov subspace methods for singular systems

Recently, Calvetti et al. have published an interesting paper [Linear Algebra Appl. 316 (2000) 157–169] concerning the least-squares solution of a singular system by using the so-called range restricted GMRES (RRGMRES) method. However, one of the main results (cf. [loc. cit., Theorem 3.3]) seems to be incomplete. As a complement of paper [loc. cit.], in this note we first make an example to show the incompleteness of that theorem, then we give a modified result.     

Determination of the Transverse Shear Stress in an Euler-Bernoulli Beam Using Non-Admissible Virtual Displacements

The point of view that a beam can be considered as a three-dimensional continuum with a constrained position field (cf. [1]) together with the virtual work principle and the concept of perfect constraint stresses, leads to a systematic way to reduce the equilibrium equations of the continuous body to an ordinary differential equation describing the constrained displacement field of the beam. Using virtual displacements, being non-admissible with respect to the constrained beam kinematics, together with the solution of the boundary value problem, allows us to analytically determine the constraint stresses and consequently the total stresses of a beam up to a certain indeterminacy. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)