Explicit hybrid six–step,sixth order,fully symmetric methods for solving y ″ = f (x,y)

A family of explicit, fully symmetric, sixth order, six‐step methods for the numerical solution of y′′ = f(x,y) is studied. This family wastes two function evaluations per step and can be derived through interpolation techniques. An interval of periodicity is possessed and the phase lag is of high order. Numerical instabilities usually present in such type of multistep methods were circumvented. We conclude with extended numerical tests over a set of problems justifying our effort of dealing with the new methods.     

New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients

Two‐derivative Runge‐Kutta methods are Runge‐Kutta methods for problems of the form y′ = f(y) that include the second derivative y′′ = g(y) = f ′(y)f(y) and were developed in the work of Chan and Tsai. In this work, we consider explicit methods and construct a family of fifth‐order methods with three stages of the general case that use several evaluations of f and g per step. For problems with oscillatory solution and in the case that a good estimate of the dominant frequency is known, methods with frequency‐dependent coefficients are used; there are several procedures for constructing such methods. We give the general framework for the construction of methods with variable coefficients following the approach of Simos. We modify the above family to derive methods with frequency‐dependent coefficients following this approach as well as the approach given by Vanden Berghe. We provide numerical results to demonstrate the efficiency of the new methods using three test problems.     

Optimal (r,N)‐policy for discrete‐time Geo ∕ G ∕ 1 queue with different input rate and setup time

This paper presents a queue‐length analysis of Geo ∕ G ∕ 1 queue with ( r , N )‐policy and different input rate. Using a different method, the recursive expressions of queue‐length distribution at different epochs are obtained. Furthermore, some performance measures are also investigated. Finally, the Tabu search algorithm is used to search the joint optimum value of ( r , N ), which minimizes the state‐dependent operating cost. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

The asymptotic behaviour and stability properties are studied for a real two‐dimensional system x^′(t) = A(t)x (t) + B(t)x (θ (t)) + h (t, x (t), x (θ (t))), with a nonconstant delay t ‐ θ (t) ≥ 0. It is supposed that A,B and h are matrix functions and a vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex‐valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov‐Krasovskii functional. The results generalize the great part of the results of J. Kalas and L. Baráková [J. Math. Anal. Appl. 269 , No. 1, 278–300 (2002)] for two‐dimensional systems with a constant delay (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Nonnegative Solutions of the Inequality Δ u + uσ ≤ 0 on Riemannian Manifolds

We study the uniqueness of a nonnegative solution of the differential inequality on a complete Riemannian manifold, where σ > 1 is a parameter. We prove that if, for some x₀ ? M and all large enough r where , and B(x,r) is a geodesic ball, then the only nonnegative solution of (*) is identically 0. We also show the sharpness of the above values of the exponents p,q. © 2014 Wiley Periodicals, Inc.     

Unification of Stieltjes‐Calogero type relations for the zeros of classical orthogonal polynomials

The classical orthogonal polynomials (COPs) satisfy a second‐order differential equation of the form σ(x)y^′′+τ(x)y^′+λy = 0, which is called the equation of hypergeometric type (EHT). It is shown that two numerical methods provide equivalent schemes for the discrete representation of the EHT. Thus, they lead to the same matrix eigenvalue problem. In both cases, explicit closed‐form expressions for the matrix elements have been derived in terms only of the zeros of the COPs. On using the equality of the entries of the resulting matrices in the two discretizations, unified identities related to the zeros of the COPs are then introduced. Hence, most of the formulas in the literature known for the roots of Hermite, Laguerre and Jacobi polynomials are recovered as the particular cases of our more general and unified relationships. Furthermore, we present some novel results that were not reported previously. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of a class of potential Korteweg‐de Vries‐like equations

We analyze a class of third‐order evolution equations, i.e. u_t = f(x, u_x, u_xx) u_xxx+g(x, u_x, u_xx) via the method of preliminary group classification. This method is a systematic means of analyzing the equation for symmetries. We find explicit forms of f and g, which allow for a larger dimensional Lie algebra of point symmetries. Copyright © 2009 John Wiley & Sons, Ltd.     

A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order

The paper deals with the following Kirchhoff‐type problem $$\left\{\begin{array}{lll}& M\left({?}_{{?}^{2N}}\frac{1}{p(x,y)}\frac{|v(x)?v(y){|}^{p(x,y)}}{|x?y{|}^{N+p(x,y)s(x,y)}}dxdy\right){(?\mathrm{\Delta })}_{p(·)}^{s(·)}v(x)=\mu g(x,v)+|v{|}^{r(x)?2}v& \text{in}\mathrm{\Omega },\\ & v=0& \text{in}{?}^N\backslash \mathrm{\Omega },\end{array}\right.$$ where M models a Kirchhoff coefficient, ${(?\mathrm{\Delta })}_{p(·)}^{s(·)}$ is a variable s(·) ‐order p(·) ‐fractional Laplace operator, with $s(·):{?}^{2N}\to (0,1)$ and $p(·):{?}^{2N}\to (1,\infty )$. Here, $\mathrm{\Omega }?{?}^N$ is a bounded smooth domain with N > p(x, y)s(x, y) for any $(x,y)\in \overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}$, μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent $p_s^{?}(x)=N\overline{p}(x)/(N?\overline{s}(x)\overline{p}(x))$, given with $\overline{p}(x)=p(x,x)$ and $\overline{s}(x)=s(x,x)$ for $x\in \overline{\mathrm{\Omega }}$. We prove the existence and asymptotic behavior of at least one non‐trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb‐type lemma for fractional Sobolev spaces with variable order and variable exponent.     

De Morgan Heyting algebras satisfying the identity xn(′*) ≈ x(n+1)(′*)

In this paper we investigate the sequence of subvarieties $ {\mathcal {SDH}_n} $of De Morgan Heyting algebras characterized by the identity x^n(′*) ≈ x^(n+1)(′*). We obtain necessary and sufficient conditions for a De Morgan Heyting algebra to be in $ {\mathcal {SDH}_1} $ by means of its space of prime filters, and we characterize subdirectly irreducible and simple algebras in $ {\mathcal {SDH}_1} $. We extend these results for finite algebras in the general case $ {\mathcal {SDH}_n} $. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

The unconditional stability and mass‐preserving S‐DDM scheme for solving parabolic equations in three dimensions

In this paper, the unconditional stability and mass‐preserving splitting domain decomposition method (S‐DDM) for solving three‐dimensional parabolic equations is analyzed. At each time step level, three steps (x‐direction, y‐direction, and z‐direction) are proposed to compute the solutions on each sub‐domains. The interface fluxes are first predicted by the semi‐implicit flux schemes. Second, the interior solutions and fluxes are computed by the splitting implicit solution and flux coupled schemes. Last, we recompute the interface fluxes by the explicit schemes. Due to the introduced z‐directional splitting and domain decomposition, the analysis of stability and convergence is scarcely evident and quite difficult. By some mathematical technique and auxiliary lemmas, we prove strictly our scheme meet unconditional stability and give the error estimates in L²‐norm. Numerical experiments are presented to illustrate the theoretical analysis.     

For most large underdetermined systems of equations,the minimal 𝓁1‐norm near‐solution approximates the sparsest near‐solution

We consider inexact linear equations y ≈ Φx where y is a given vector in ?ⁿ, Φ is a given n × m matrix, and we wish to find x_0,? as sparse as possible while obeying ‖y ? Φx_0,?‖₂ ≤ ?. In general, this requires combinatorial optimization and so is considered intractable. On the other hand, the ??₁‐minimization problem is convex and is considered tractable. We show that for most Φ, if the optimally sparse approximation x_0,? is sufficiently sparse, then the solution x_1,? of the ??₁‐minimization problem is a good approximation to x_0,?. We suppose that the columns of Φ are normalized to the unit ??₂‐norm, and we place uniform measure on such Φ. We study the underdetermined case where m ～ τn and τ > 1, and prove the existence of ρ = ρ(τ) > 0 and C = C(ρ, τ) so that for large n and for all Φ's except a negligible fraction, the following approximate sparse solution property of Φ holds: for every y having an approximation ‖y ? Φx₀‖₂ ≤ ? by a coefficient vector x₀ ∈ ?^m with fewer than ρ · n nonzeros, This has two implications. First, for most Φ, whenever the combinatorial optimization result x_0,? would be very sparse, x_1,? is a good approximation to x_0,?. Second, suppose we are given noisy data obeying y = Φx₀ + z where the unknown x₀ is known to be sparse and the noise ‖z‖₂ ≤ ?. For most Φ, noise‐tolerant ??₁‐minimization will stably recover x₀ from y in the presence of noise z. We also study the barely determined case m = n and reach parallel conclusions by slightly different arguments. Proof techniques include the use of almost‐spherical sections in Banach space theory and concentration of measure for eigenvalues of random matrices. © 2006 Wiley Periodicals, Inc.     

{2, 3}‐perfect m‐cycle systems are equationally defined for m = 5, 7, 8, 9, and 11 only

An m‐cycle system (S,C) of order n is said to be {2,3}‐perfect provided each pair of vertices is connected by a path of length 2 in an m‐cycle of C and a path of length 3 in an m‐cycle of C. The class of {2,3}‐perfect m‐cycle systems is said to be equationally defined provided, there exists a variety of quasigroups V with the property that a finite quasigroup (Q, , \, /) belongs to V if and only if its multiplicative (Q, ) part can be constructed from a {2,3}‐perfect m‐cycle system using the 2‐construction (a a = a for all a ∈ Q and if a ≠ b, a b = c and b a = d if and only if the m‐cycle (…, d, x, a, b, y, c, …) ∈ C). The object of this paper is to show that the class of {2,3}‐perfect m‐cycle systems cannot be equationally defined for all m ≥ 10, m ≠ 11. This combined with previous results shows that {2, 3}‐perfect m‐cycle systems are equationally defined for m = 5, 7, 8, 9, and 11 only. © 2004 Wiley Periodicals, Inc.     

Non‐Abelian homomorphism testing,and distributions close to their self‐convolutions

In this paper, we study two questions related to the problem of testing whether a function is close to a homomorphism. For two finite groups G,H (not necessarily Abelian), an arbitrary map f : G,H, and a parameter 0 < ε < 1, say that f is ε‐close to a homomorphism if there is some homomorphism g such that g and f differ on at most ε | G | elements of G, and say that f is ε‐far otherwise. For a given f and ε, a homomorphism tester should distinguish whether f is a homomorphism, or if f is ε‐far from a homomorphism. When G is Abelian, it was known that the test which picks O(1/ε) random pairs x,y and tests that f(x) + f(y) = f(x + y) gives a homomorphism tester. Our first result shows that such a test works for all groups G. Next, we consider functions that are close to their self‐convolutions. Let A = {a_g | g ε G} be a distribution on G. The self‐convolution of A, A^′ = {a | g ε G}, is defined by It is known that A= A^′ exactly when A is the uniform distribution over a subgroup of G. We show that there is a sense in which this characterization is robust—that is, if A is close in statistical distance to A^′, then A must be close to uniform over some subgroup of G. Finally, we show a relationship between the question of testing whether a function is close to a homomorphism via the above test and the question of characterizing functions that are close to their self‐convolutions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Generalized Dirichlet‐to‐Neumann Map in Time‐Dependent Domains

We study the heat, linear Schrödinger (LS), and linear KdV equations in the domain l(t) < x < ∞ , 0 < t < T , with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.     

Ω‐estimates related to irreducible algebraic integers

We study large values of the remainder term E_K (x) in the asymptotic formula for the number of irreducible integers in an algebraic number field K. We show that E_K (x) = Ω_± (√(x)(log x)) for certain positive constant B_K, improving in that way the previously best known estimate E_K (x) = Ω_± (x^(1/2)‐ε) for every ε > 0, due to A. Perelli and the present author. Assuming that no entire L‐function from the Selberg class vanishes on the vertical line σ = 1, we show that E_K (x) = Ω_± (√(x)(log log x)^{D (K)‐1}(log x)^‐1), supporting a conjecture raised recently by the author. In particular, it follows that the last omega estimate is a consequence of the Selberg Orthonormality Conjecture (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Single‐cell discretization of O(kh2 + h4) for ∂u/∂n for three‐space dimensional mildly quasi‐linear parabolic equation

In this article, using a single computational cell, we report some stable two‐level explicit finite difference approximations of O(kh² + h⁴) for ?u/?n for three‐space dimensional quasi‐linear parabolic equation, where h > 0 and k > 0 are mesh sizes in space and time directions, respectively. When grid lines are parallel to x‐, y‐, and z‐coordinate axes, then ?u/?n at an internal grid point becomes ?u/?x, ?u/?y, and ?u/?z, respectively. The proposed methods are also applicable to the polar coordinates problems. The proposed methods have the simplicity in nature and use the same marching type of technique of solution. Stability analysis of a linear difference equation and computational efficiency of the methods are discussed. The results of numerical experiments are compared with exact solutions. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 327–342, 2003.     

On the asymptotic solution of the Graetz‐Nusselt problem for short x → 0 and large x → ∞ with partial usage of finite differences

The principal goal of this article is to present two asymptotic solutions for the classical Graetz‐Nusselt problem. The method of lines (MOL) has been adopted for solving the governing partial differential energy equation in two independent variables in an asymptotic manner. Two temperature subfields are determined semianalytically: one for small x (x → 0) and the other for large x (x → ∞). Later, the two asymptotic mean Nusselt number subdistributions, Nu _X→0(x) and Nu _X→∞(x), blend themselves into a generalized correlation equation for the mean Nusselt number distribution Nu (x) covering the entire x‐domain. The simplicity of the MOL procedure, combined with the high quality asymptotic mean Nusselt number subdistributions, provides an alternative methodology for solving the Graetz‐Nusselt problem without using higher level mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

The wi‐club filter on 𝒫κ λ

We develop the theory of C_{κ, λ}ⁱ, a strongly normal filter over ??_κ λ for Mahlo κ. We prove a minimality result, showing that any strongly normal filter containing {x ∈ ??_κ λ: |x | = |x ∩ κ | and |x | is inaccessible} also contains C_{κ, λ}ⁱ. We also show that functions can be used to obtain a basis for C_{κ, λ}ⁱ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotational k‐cycle systems of order v < 3k; another proof of the existence of odd cycle systems

We give an explicit solution to the existence problem for 1‐rotational k‐cycle systems of order v < 3k with k odd and v ≠ 2k + 1. We also exhibit a 2‐rotational k‐cycle system of order 2k + 1 for any odd k. Thus, for k odd and any admissible v < 3k there exists a 2‐rotational k‐cycle system of order v. This may also be viewed as an alternative proof that the obvious necessary conditions for the existence of odd cycle systems are also sufficient. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 433–441, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10061