A (T,ψ)‐ψe decoupled scheme for a time‐dependent multiply‐connected eddy current problem

The aim of this paper is to develop a fully discrete ( T ,ψ)‐ψ_e finite element decoupled scheme to solve time‐dependent eddy current problems with multiply‐connected conductors. By making ‘cuts’ and setting jumps of ψ_e across the cuts in nonconductive domain, the uniqueness of ψ_e is guaranteed. Distinguished from the traditional T ‐ ψ method, our decoupled scheme solves the potentials T and ψ‐ ψ_e separately in two different simple equation systems, which avoids solving a saddle‐point equation system and leads to a remarkable reduction in computational efforts. The energy‐norm error estimate of the fully discrete decoupled scheme is provided. Finally, the scheme is applied to solve two benchmark problems—TEAM Workshop Problems 7 and IEEJ model. Copyright © 2013 John Wiley & Sons, Ltd.     

Perfect coloring and linearly χ‐bound P6‐free graphs

We derive decomposition theorems for P₆, K₁ + P₄‐free graphs, P₅, K₁ + P₄‐free graphs and P₅, K₁ + C₄‐free graphs, and deduce linear χ‐binding functions for these classes of graphs (here, P_n (C_n) denotes the path (cycle) on n vertices and K₁ + G denotes the graph obtained from G by adding a new vertex and joining it with every vertex of G). Using the same techniques, we also obtain an optimal χ‐binding function for P₅, C₄‐free graphs which is an improvement over that given in [J. L. Fouquet, V. Giakoumakis, F. Maire, and H. Thuillier, 11 , Discrete Math, 146, 33–44.]. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 293–306, 2007     

Fixed points of (α,ψ)‐contractions in Hausdorff partial metric spaces

The concept of (α,ψ)‐contractions was introduced by Samet et al. in this paper, we introduce (α,ψ)‐generalized contractions in a Hausdorff partial metric space. We discuss its significance and obtain some common fixed point theorems for a pair of self‐mappings. Some examples are given to support the theory.     

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Fully discrete T‐ψ finite element method to solve a nonlinear induction hardening problem

We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.     

The ramsey number r(k1 + c4, k5 − e)

The graph K₅ - e is obtained from the complete graph K₅ by deleting one edge, while K₁ + C₄ is obtained from K₅ by deleting two independent edges. With the help of a computer it is shown that r(K₁ + C₄, K₅ - e) = 17. © 1995 John Wiley & Sons, Inc.     

Global minimum and orthogonality in Cp ‐classes

In this paper we use recent results [14] to establish various characterizations of the global minimum of the map F_ψ : U → ?⁺ defined by F_ψ (X) = ‖ψ (X)‖_p (1 < p < ∞) where ψ: U → C_p is a map defined by ψ (X) = S +? (X), with ?: B (H) → B (H) a linear map and S ∈ C_p , and U = {X ∈ B (H): ? (X) ∈ C_p }. Further, we apply these results to characterize the operators which are orthogonal to the range of elementary operators. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Note on multidimensional empirical processes for φ-mixing random vectors

In 1974, Sen proved weak convergence of the empirical processes (in the J₁-topology on D^p[0, 1]) for a stationary φ-mixing sequence of stochastic p( 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants {φ_n}, i.e., φ_n = O(n^−1−δ) for some δ > 0.     

ON THE ITERATED ω-RULE

Let Γ_n(φ) be a formula of L_PA (PA = Peano Arithmetic) meaning “there is a proof of φ from PA-axioms, in which ω-rule is iterated no more than n times”. We examine relations over pairs of natural numbers of the kind. (n, k) ≦_H (n', k') iff PA + RFN_n' (H_k') ? RFN_n (H_k). Where H denotes one of the hierarchies ∑ or Π and RFN_n(C) is the scheme of the reflection principle for Γ_n restricted to formulas from the class C(Γ_n(φ) implies “φ is true”, for every φ ∈ C). Our main result is that. (n, k) ≦H (n', k') if n ≦ n' and k ≦ max (k', 2n' + 1).     

A small generating set for the 3‐BD closed set of block sizes ≥ 6

Let v, k be positive integers and k ≥ 3, then K_k = : {v: v ≥ k} is a 3‐BD closed set. Two finite generating sets of 3‐BD closed sets K₄ and K₅ are obtained by H. Hanani [5] and Qiurong Wu [12] respectively. In this article we show that if v ≥ 6, then v ∈ B₃(K,1), where K = {6,7,…,41,45,46,47,51,52,53,83,84}\{22,26}; that is, we show that K is a generating set for K₆. Finally we show that v ∈ B₃(6,20) for all v ∈ K\{35,39,40,45}. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 128–136, 2008     

Maximal σ‐polynomials of connected 3‐chromatic graphs

In the set of graphs of order n and chromatic number k the following partial order relation is defined. One says that a graph G is less than a graph H if c_i(G) ≤ c_i(H) holds for every i, k ≤ i ≤ n and at least one inequality is strict, where c_i(G) denotes the number of i‐color partitions of G. In this paper the first ? n/2 ? levels of the diagram of the partially ordered set of connected 3‐chromatic graphs of order n are described. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 210–222, 2003     

Stability of the fractional Volterra integro‐differential equation by means of ψ‐Hilfer operator

In this paper, using the Riemann‐Liouville fractional integral with respect to another function and the ψ?Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro‐differential equation. In this sense, for this new fractional Volterra integro‐differential equation, we study the Ulam‐Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed‐point theorem. As an application, we present the Ulam‐Hyers stability using the α‐resolvent operator in the Sobolev space .     

Quadrature formulae connected to σ-orthogonal polynomials

Let dλ(t) be a given nonnegative measure on the real line , with compact or infinite support, for which all moments exist and are finite, and μ₀>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes

where σ=σ_n=(s₁,s₂,…,s_n) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness d_max=2∑_ν=1ⁿs_ν+2n−1 if and only if

The proof of the uniqueness of the extremal nodes τ₁,τ₂,…,τ_n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ_ν, ν=1,2,…,n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.     

Principal Eigenvalues and Anti – Maximum Principle for Some Quasilinear Elliptic Equations on ℝN

We improve some previous existence and nonexistence results for positive principal eigenvalues of the problem —Δ_pu = λg(x)ψ^p(u), x ∈ ℝ^N, lim_{‖x‖⇒+∞}u(x) = 0. Also we discuss existence, nonexistence and antimaximum principle questions concerning the perturbed problem —Δ_pu = λg(x)ψ^p(u) + f(x), x∈ ℝ^N.     

On some sets of dictionaries whose ω ‐powers have a given

A dictionary is a set of finite words over some finite alphabet X. The ω ‐power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [10] the complexity of the set of dictionaries whose associated ω ‐powers have a given complexity. In particular, he considered the sets ??( Σ ⁰k) (respectively, ??( Π ⁰_k), ??( Δ ¹₁)) of dictionaries V ? 2* whose ω ‐powers are Σ ⁰_k‐sets (respectively, Π ⁰_k‐sets, Borel sets). In this paper we first establish a new relation between the sets ??( Σ ⁰₂) and ??( Δ ¹₁), showing that the set ??( Δ ¹₁) is “more complex” than the set ??( Σ ⁰₂). As an application we improve the lower bound on the complexity of ??( Δ ¹₁) given by Lecomte, showing that ??( Δ ¹₁) is in Σ ¹ ₂(2^2*)\ Π ⁰₂. Then we prove that, for every integer k ≥ 2 (respectively, k ≥ 3), the set of dictionaries ??( Π ⁰_k+1) (respectively, ??( Σ ⁰_{k +1})) is “more complex” than the set of dictionaries ??( Π ⁰_k) (respectively, ??( Σ ⁰_k)) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time-periodic solutions of wave equations on ℝ1 and ℝ3

The classical Paiey-Wiener theorem and Hilbert space methods are used to show the existence of time-periodic solutions of the wave equation w_tt?w_rr+λw=h, 0 < r < + ∞, which are radially symmetric and have exponential decay as r → + ∞. This problem is obtained when considering a one-dimensional or three-dimensional problem, and should be thought of as a linearization of a semilinear problem in which the associated linear operator has point spectrum (? ∞, λ). When λ ? 0 there is uniqueness, otherwise there is a non-trivial finite-dimensional null space. Estimates on w, w_t, w_r are obtained, which show that in either case there is a continuous correspondence h → w, where w is a uniquely characterized solution.     

ψ‐Hyperholomorphic functions and a Kolosov–Muskhelishvili formula

Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In , similar formulae were already developed in recent papers, using quaternionic monogenic functions as a generalization of holomorphic functions. However, the existing representations use functions from to , embedded in . It is not completely appropriate for applications in . In particular, one has to remove many redundancies while constructing basis solutions. To overcome that problem, we propose an alternative Kolosov–Muskhelishvili formula for the displacement field by means of a (paravector‐valued) monogenic, an anti‐monogenic and a ψ‐hyperholomorphic function. Copyright © 2015 John Wiley & Sons, Ltd.     

Upper bounds for ramsey numbers R(3, 3, ⋖, 3) and Schur numbers

In this paper we show that for n ≥ 4, R(3, 3, ⋖, 3) < + 1. Consequently, a new bound for Schur numbers is also given. Also, for even n ≥ 6, the Schur number S_n is bounded by S_n < - n + 2. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 119–122, 1997     

Characterization of g∞,σ‐integral operators

The application of the general tensor norms theory of Defant and Floret to the ideal of (p, σ)‐absolutely continuous operators of Matter, 0 < σ < 1, 1 ≤ p < ∞ leads to the study of g_p′,σ‐nuclear and g_p′,σ‐integral operators. Characterizations of such operators has been obtained previously in the case p > 1. In this paper we characterize the g_∞,σ‐nuclear and g_∞,σ‐integral operators by the existence of factorizations of some special kinds. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.