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.     

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.     

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.     

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.     

Convergence Laws for Very Sparse Random Structures with Generalized Quantifiers

We prove convergence laws for logics of the form , where is a properly chosen collection of generalized quantifiers, on very sparse finite random structures. We also study probabilistic collapsing of the logics , where is a collection of generalized quantifiers and k ∈ ℕ₊, under arbitrary probability measures of finite structures.     

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)     

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)     

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 .     

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     

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.     

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 .     

Spectral Estimations for Can nical Systems

Two–dimensional canonical systems are boundary value problems of the form with y₁(0) = 0 and Weyl's limit point case at L. The 2 × 2 matrix valued function H is real, symmetric and nonnegative, . The correspondence between canonical systems and their Titchmarsh–Weyl coefficients Q is a bijection between the class of all matrix functions H with tr H(x) = 1 a.e. on [0, L) and the class of the Nevanlinna functions ℕ augmented by the function Q ≡ 8. Each Titchmarsh–Weyl coefficient Q ∈ ℕ can be represented by means of a measure σ, the so–called spectral measure of the canonical system. In this note matrix functions H are specified whose corresponding spectral measures σ satisfy conditions of the form or . Herewith we generalize corresponding results of M.G. Krein and I. S. Kac for so–called vibrating strings.     

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)     

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     

An Lq-theory for weak solutions of the stokes system in exterior domains

We introduce a new concept for weak solutions in L^q-spaces, 1 < q < ∞, of the Stokes system in an exterior domain Ω ? ?ⁿ, n ? 2. Defining the variational formulation in the homogeneous Sobolev space $ \mathop H\limits^.{_{0}}^{1,q} (\Omega )^n = \{ u \in L_{1{\rm oc}}^q (\overline \Omega )^n;\nabla u \in L^q (\Omega )^{n^2 },u\left| {_{\partial \Omega } = 0} \right.\},$ we prove existence and uniqueness of weak solutions for an arbitrary external force and a prescribed divergence g = div u. On the other hand, solutions in the sense of distributions which are defined by taking test functions only in C(Ω)ⁿ are not unique if q > n/(n?1). In this case, a hidden boundary condition related to the force exerted on the body may be imposed to single out a unique solution.     

The PENROSE Transform for Bundles Non-Trivial on the General Line

Let L₀ be a fixed projective line in CP ³ and let M ? C ⁴ be the complexified MINKOWSKI space interpreted as the manifold of all projective lines L ? CP ³ with L ∩ L ₀ ?? Ø. Let D ? M , D ′ ? CP ³/ L ₀ be open sets such that \documentclass{article}\pagestyle{empty}\begin{document}$ D' = \mathop \cup \limits_{L \in D} $\end{document}. Under certain topological conditions on D, R. S. WARD'S PENROSE transform sets up an 1–1 correspondence between holomorphic vector bundles over D ′ trivial over each L ? D and holomorphic connections with anti-self-dual curvature over D (anti-self-dual YANG-MILLIS fields). In the present paper WARD'S construction is generalized to holomorphic vector bundles E over D′ satisfying the condition that \documentclass{article}\pagestyle{empty}\begin{document}$ E|_L \cong E|_{\tilde L} $\end{document} for all \documentclass{article}\pagestyle{empty}\begin{document}$ L,\tilde L \in D $\end{document}.     

Derived Functors of Hom Relative to Flat Covers

We study the derived functors of Hom that are computed by using flat resolutions of Hom. These are denoted ⁿ. We compare these with the usual Extⁿ's and show that ¹⊂ Ext¹and indicate (using MacLane's terminology) why the class of associated short exact sequences is a proper class. When the ring is a Dedekind domain we classify the N such that ⁿ(–, N) = 0 and show that unlike the situation for other classically defined right derived functors of Hom, Hom is not balanced relative to the two classes of modules that make ¹ vanish.     

On the Harnack principle for strongly elliptic systems with nonsmooth coefficients

In this paper we prove the following theorem (for notation and definitions, see the paragraphs below): “Let Ω ⊆ ℝⁿ be a domain, m ∈ ℕ, and λ, q > 0. Then, there exists r (= r(λ, q)) > 1 such that for every 0 < p < q, whenever are weak solutions of a strongly elliptic system with m equations of ellipticity λ satisfying ∈ 𝒫_r a.e. and Ω′ ⊆ Ω subdomain, the following inequalities hold: where C (= C(n,m,λ,q,p,Ω,Ω′)) is a positive constant.” © 1999 John Wiley & Sons, Inc.     

A list analogue of equitable coloring

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when unless G contains or k is odd and . For forests, the threshold improves to . If G is a 2-degenerate graph (given k ≥ 5) or a connected interval graph (other than ), then G is equitably k-choosable when . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 166–177, 2003     

Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen-Suslin) pour le Calcul Formel

Let k be an arbitrary field, X₁,….,X_n indeterminates over k and F₁…, F₃ ε ∈ k[X₁…,X_n] polynomials of maximal degree $ d: = \mathop {\max }\limits_{1 \le i \le a} \deg $ (F_i). We give an elementary proof of the following effective Nullstellensatz: Assume that F₁,…,F have no common zero in the algebraic closure of k. Then there exist polynomials P₁…, P₃ ε ∈ k[X₁…,X_n] such that $ 1: = \mathop \Sigma \limits_{1 \le i \le a} $ P_iF_i and This result has many applications in Computer Algebra. To exemplify this, we give an effective quantitative and algorithmic version of the Quillen-Suslin Theorem baaed on our effective Nullstellensatz.