Compactness Conditions for the Green Operator in Lp(ℝ) Corresponding to a General Sturm–Liouville Operator

We consider the equation ℝ, where , for ℝ, (ℝ), (ℝ), (ℝ), (ℝ) := C(ℝ)). We give necessary and sufficient conditions under which, regardless of , the following statements hold simultaneously: I) For any (ℝ) Equation (0.1) has a unique solution (ℝ) where $\int ^{\infty}_{-\infty}$ ℝ. II) The operator (ℝ) → (ℝ) is compact. Here is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator.     

Strengths and Weaknesses of LH Arithmetic

In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2^s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.     

On a free boundary problem in electrostatics

Let Ω_i ? ?^N, i = 0, 1, be two bounded separately star-shaped domains such that $ \Omega _0 \supset \bar \Omega _1 $. We consider the electrostatic potential u defined in $ \Omega : = \Omega _0 \backslash \bar \Omega _1 $: The geometry of the two boundary components Γ₀ and Γ₁ is not given, but instead the electrostatic potential u is supposed to satisfy the further boundary conditions Using a best possible maximum principle, we show that this free boundary problem has a unique solution which is radially symmetric.     

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem where H is a Hilbert space, A is a self‐adjoint linear non‐negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non‐local non‐linearities. Copyright © 1999 John Wiley & Sons, Ltd.     

Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor-Coefficients G and Applications to Elasticity

Let be bounded Lipschitz and relatively open. We show that the solution to the linear first order system 1 : (1) vanishes if and , (e.g. ). We prove to be a norm if with , for some p, q > 1 with 1/p + 1/q = 1 and . We give a new proof for the so called ‘in-finitesimal rigid displacement lemma’ in curvilinear coordinates: Let , satisfy for some with . Then there are and a constant skew-symmetric matrix , such that . (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Radially symmetric Poisson–Boltzmann equation in a domain expanding to infinity

The electric potential u in a solution of an electrolyte around a linear polyelectrolyte of the form of a cylinder satisfies We study the problem when R → ∞.     

Unions and Intersections of Families of Lp-Spaces

Let L^Ψ and E^Ψ be the ORLICZ space and the space of finite elements respectively, on a measure space (Ω, Σ, μ), and let T ? (0, ∞). It is proved that if inf {p: p ? T} ? T or sup {p: p ? T} ? T and μ is an infinite atomless measure, then there is no ORLICZ function Ψ such that: \documentclass{article}\pagestyle{empty}\begin{document}$ L^\varphi = Lin\mathop { \cup L^p }\limits_{p\varepsilon T} $\end{document} or \documentclass{article}\pagestyle{empty}\begin{document}$ E^\varphi = Lin\mathop { \cup L^p }\limits_{p\varepsilon T} $\end{document} and moreover, there is no ORLICZ function Ψ such that: \documentclass{article}\pagestyle{empty}\begin{document}$ L^\varphi = Lin\mathop { \cap L^p }\limits_{p\varepsilon T} $\end{document} or \documentclass{article}\pagestyle{empty}\begin{document}$ E^\varphi = Lin\mathop { \cap L^p }\limits_{p\varepsilon T} $\end{document}.     

Necessary and Sufficient Conditions for the Oscillation of a Second Order Linear Differential Equation

In this paper we give a necessary and sufficient condition for the oscillation of the second order linear differential equation where p is a locally integrable function and either or where We give some applications which show how these results unify and imply some classical results in oscillation theory.     

On the Points on the Unit Circle with Finite b–Adic Expansions

Let be an arbitrary integer base and let be the number of different prime factors of with , . Further let be the set of points on the unit circle with finite –adic expansions of their coordinates and let be the set of angles of the points . Then is an additive group which is the direct sum of infinite cyclic groups and of the finite cyclic group . If in case of the points of are arranged according to the number of digits of their coordinates, then the arising sequence is uniformly distributed on the unit circle. On the other hand, in case of the only points in are the exceptional points (1, 0), (0, 1), (–1, 0), (0, –1). The proofs are based on a canonical form for all integer solutions of .     

Comparing Packing Measures to Hausdorff Measures on the Line

For each 0 < s < 1, define where , denote respectively the s‐dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E ⊂ R with . In this paper we give a nontrivial estimation of c(s), namely, for each 0 < s < 1, where . As an application, we obtain a lower density theorem for Hausdorff measures.     

Implicit linear time‐dependent differential‐difference equations and applications

The paper deals with the differential‐difference equation in a Banach space. The operator coefficient of the delay‐free derivative is allowed to be degenerate. Existence and uniqueness theorems are proved under the main assumption that for every the point is a polar singularity of the resolvent . The results are applied to evolution problems of microwave circuits. Copyright © 2000 John Wiley & Sons, Ltd.     

On the Eigenvalues of Half‐Linear Boundary Value Problems

We consider the half‐linear boundary value problem where and the weight function q is assumed to change sign. We prove the existence of two sequences , of eigenvalues and derive asymptotic estimates for as .     

The isoperimetric constant of the random graph process

The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of , taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, , is a sequence of graphs, where is the edgeless graph on n vertices, and is the result of adding an edge to , uniformly distributed over all the missing edges. The authors show that in almost every graph process equals the minimal degree of as long as the minimal degree is o(log n). Furthermore, it is shown that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to , its final value. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Stress-driven local solution approach to quasistatic brittle delamination

This contribution reports on a model describing the rate-independent evolution of brittle delamination between two elastic bodies and , bonded along a prescribed contact surface , over a fixed time interval (0, T). In contrast to [1], which is set in the framework of energetic solutions, here the existence and properties of so-called local solutions are studied; a detailed discussion and a rigorous mathematical analysis of this model can be found in [2]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Mixed Finite Element Formulation for Piezoelectric Materials

A geometrically nonlinear finite element formulation to analyze multi-field problems as they arise e.g. in piezoelectric or magnetostrictive materials is presented. Here we focus on piezoelectric problems. The formulation is based on a Hu-Washizu functional considering six independent fields. These are displacements u , electric potential ϕ, strains E , stresses S , electric field , and the electric flux density . The finite element approximation leads to an 8-node hexahedral element with u and ϕ as nodal degrees of freedom. The fields E , S , , and are interpolated on element level by employing some internal degrees of freedom. These fields do not require continuity across the element boundaries, thus the internal degree of freedoms are eliminated on element level by a static condensation. The geometrically non-linear theory allows large deformations and accounts for stability problems. To fulfill the charge conservation law in bending dominated situations exactly a quadratic approximation of the electric potential is necessary. This leads in general to additional nodal degrees of freedom, which is circumvented by the presented formulation by employing appropriate interpolations of and . Numerical examples show that the locking effect which arise in low order elements are significantly reduced and that the element provides good accuracy with respect to experimental data. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of Inertial Manifolds and Application to Modulated Systems

We consider two dissipative systems having inertial manifolds and give estimates which allow us to compare the flows on the two inertial manifolds. As an example of a modulated system we treat the Swift–Hohenberg equation , ∈ ℝ, with periodic boundary conditions on the interval . Recent results in the theory of modulation equation show that the solutions of this equation can be described over long time scales by those of the associated Ginzburg–Landau equation ∈ ℂ, with suitably generalized periodic boundary conditions on . We prove that both systems have an inertial manifold of the same dimension and that the flows on these finite dimensional manifolds converge against each other for .     

On the Range of Two Convolution Operators

Let _(ω)(ℝ) denote the non–quasianalytic class of Beurling type on ℝ. For μ, ν ∈ ′_(ω)(ℝ) we give necessary conditions for the inclusion T_ν( _(ω)(ℝ)) ⊂ T_μ( _(ω)(ℝ)), thus extending previous work of Malgrange and Ehrenpreis .     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

Data Based Regularization Matrices for the Tikhonov-Phillips Regularization

In Tikhonov-Phillips regularization of general form the given ill-posed linear system is replaced by a Least Squares problem including a minimization of the solution vector x, relative to a seminorm with some regularization matrix L. Based on the finite difference matrix L_k, given by a discretization of the first or second derivative, we introduce the seminorm where the diagonal matrix and is the best available approximate solution to x. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hamilton decompositions of complete multipartite graphs with any 2-factor leave

For m ≥ 1 and p ≥ 2, given a set of integers s₁,…,s_q with for and , necessary and sufficient conditions are found for the existence of a hamilton decomposition of the complete p-partite graph , where U is a 2-factor of consisting of q cycles, the jth cycle having length s_j. This result is then used to completely solve the problem when p = 3, removing the condition that . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 208–214, 2003