Group Divisible Designs with Block Size Four and Group Type for

Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of ‐GDDs of type for . First, we determine completely the spectrum of ‐GDDs of types and . Furthermore, for general cases, we show that for each and , a ‐GDD of type exists if and only if , and , except possibly for , and .     

Constructions of Group Divisible Designs

In this paper we present constructions for group divisible designs from generalized partial difference matrices. We describe some classes of examples.     

Group Divisible Designs with Block Size Four and Group Type gum1 with Minimum m

In this paper, we continue to investigate the spectrum for {4}-GDDs of type g^u m¹ with m as small as possible. We determine, for each admissible pair (g,u), the minimum values of m for which a {4}-GDD of type g^um¹ exists with four possible exceptions.Gennian Ge-Researcher supported by NSFC Grant 10471127.Alan C. H. Ling-Researcher supported by an ARO grant 19-01-1-0406 and a DOE grant.classification Primary 05B05     

Group Divisible Covering Designs with Block Size 4: A Type of Covering Array with Row Limit

A k‐GDCD, group divisible covering design, of type is a triple , where V is a set of gu elements, is a partition of V into u sets of size g, called groups, and is a collection of k‐subsets of V, called blocks, such that every pair of elements in V is either contained in a unique group or there is at least one block containing it, but not both. This family of combinatorial objects is equivalent to a special case of the graph covering problem and a generalization of covering arrays, which we call CARLs. In this paper, we show that there exists an integer such that for any positive integers g and , there exists a 4‐GDCD of type which in the worst case exceeds the Schönheim lower bound by δ blocks, except maybe when (1) and , or (2) , , and or . To show this, we develop constructions of 4‐GDCDs, which depend on two types of ingredients: essential, which are used multiple times, and auxiliary, which are used only once in the construction. If the essential ingredients meet the lower bound, the products of the construction differ from the lower bound by as many blocks as the optimal size of the auxiliary ingredient differs from the lower bound.     

Semiframes with Block Size Three and Odd Group Size

A ‐semiframe of type is a ‐GDD of type , , in which the collection of blocks can be written as a disjoint union where is partitioned into parallel classes of and is partitioned into holey parallel classes, each holey parallel class being a partition of for some . A ‐SF is a ‐semiframe of type in which there are p parallel classes in and d holey parallel classes with respect to . In this paper, we shall show that there exists a (3, 1)‐SF for any if and only if , , , and .     

The Existence of Mixed Orthogonal Arrays with Four and Five Factors of Strength Two

Symmetric orthogonal arrays and mixed orthogonal arrays are useful in the design of various experiments. They are also a fundamental tool in the construction of various combinatorial configurations. In this paper, we investigated the mixed orthogonal arrays with four and five factors of strength two, and proved that the necessary conditions of such mixed orthogonal arrays are also sufficient with several exceptions and one possible exception.     

A construction of resolvable nested 3‐designs

A method of constructing resolvable nested 3‐designs from an affine resolvable 3‐design is proposed with one example. © 2004 Wiley Periodicals, Inc.     

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.     

Convergence analysis of the Navier–Stokes alpha model

We consider the Navier–Stokes‐alpha (NS‐α) model as an approximation of turbulent flows under nonperiodic boundary conditions. We prove global existence and uniqueness of weak solutions of the particular model. Further, we give full discretization of the model using the finite element approximations. Finally, we prove convergence of the method to the continuous NS‐α solution as h → 0 for a constant α. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

The spectrum of BSA(v, 3, λ; α) with α = 2,3

In this article, a kind of auxiliary design BSA^* for constructing BSAs is introduced and studied. Two powerful recursive constructions on BSAs from 3‐IGDDs and BSA^*s are exploited. Finally, the necessary and sufficient conditions for the existence of a BSA(v, 3, λ; α) with α = 2, 3 are established. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 61–76, 2007     

2‐(v,k,1) Designs Admitting a Primitive Rank 3 Automorphism Group of Affine Type: The Extraspecial and the Exceptional Classes

2‐(v,k,1) designs admitting a primitive rank 3 automorphism group , where G₀ belongs to the Extraspecial Class, or to the Exceptional Class of Liebeck's Theorem in [23], are classified.     

Numerical calculation of the discrete spectra of one‐dimensional Schrödinger operators with point interactions

In this paper, we consider one‐dimensional Schrödinger operators S_q on with a bounded potential q supported on the segment and a singular potential supported at the ends h₀, h₁. We consider an extension of the operator S_q in defined by the Schrödinger operator and matrix point conditions at the ends h₀, h₁. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator . Moreover, we provide closed‐form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self‐adjoint and nonself‐adjoint problems involving general point interactions described in terms of δ‐ and δ′‐distributions.     

New approach to the Lax‐Wendroff modified differential equation for linear and nonlinear advection

The paper presents an enhanced analysis of the Lax‐Wendroff difference scheme—up to the eighth‐order with respect to time and space derivatives—of the modified‐partial differential equation (MDE) of the constant‐wind‐speed advection equation. The modified equation has been so far derived mainly as a fourth‐order equation. The Π ‐form of the first differential approximation (differential approximation or equivalent equation) derived by expressing the time derivatives in terms of the space derivatives is used for presenting the MDE. The obtained coefficients at higher order derivatives are analyzed for indications of the character of the dissipative and dispersive errors. The authors included a part of the stencil applied for determining the modified differential equation up to the eighth‐order of the analyzed modified differential equation for the second‐order Lax‐Wendroff scheme. Neither the derived coefficients at the space derivatives of order p ∈ (7 – 8) in the modified differential equation for the Lax‐Wendroff difference scheme nor the results of analyses on the basis of these coefficients of the group velocity, phase shift errors, or dispersive and dissipative features of the scheme have been published. The MDEs for 2 two‐step variants of the Lax‐Wendroff type difference schemes and the MacCormack predictor–corrector scheme (see MacCormack's study) constructed for the scalar hyperbolic conservation laws are also presented in this paper. The analysis of the inviscid Burgers equation solution with the initial condition in a form of a shock wave has been discussed on their basis. The inviscid Burgers equation with the source is also presented. The theory of MDE started to develop after the paper of C. W. Hirt was published in 1968.     

A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

This paper is concerned with computing $\mathrm{??}$‐eigenpairs of symmetric tensors. We first show that computing $\mathrm{??}$‐eigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a modified normalized Newton method (MNNM) for it. Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions, which greatly improves the Newton correction method and the orthogonal Newton correction method recently provided by Jaffe, Weiss and Nadler since these two methods only enjoy a quadratic rate of convergence. As an application, the unitary symmetric eigenpairs of a complex‐valued symmetric tensor arising from the computation of quantum entanglement in quantum physics are calculated by the MNNM method. Some numerical results are presented to illustrate the efficiency and effectiveness of our method.     

Ergodic stationary distribution and practical application of a hybrid stochastic cholera transmission model with waning vaccine-induced immunity under nonlinear regime switching

Considering the effect of stochasticity including white noise and colored noise, this paper aims to study a hybrid stochastic cholera epidemic model with waning vaccine-induced immunity and nonlinear telegraph perturbations. First, we derive a critical value ${?}_0^C$ related to the basic reproduction number ${?}_0$ of the deterministic model. The key aim of this paper is to generalize the θ-stochastic criterion method proposed by the recent work (Han et al. in Chaos Solit Fract 140:110238, 2020) to eliminate nonlinear telegraph perturbations. Next, via constructing several θ-stochastic Lyapunov functions and using the generalized method, we further prove that the stochastic model have a unique ergodic stationary distribution under ${?}_0^C>1$. Results show that the prevention and control of cholera epidemic depend on low transmission rate and small telegraph perturbations. Finally, the corresponding numerical simulations are performed to illustrate our analytical results and a practical application on the Somalia cholera outbreak is shown at the end of this paper.     

A computational study of the one‐dimensional parabolic equation subject to nonclassical boundary specifications

Certain problems arising in engineering are modeled by nonstandard parabolic initial‐boundary value problems in one space variable, which involve an integral term over the spatial domain of a function of the desired solution. Hence, in the past few years interest has substantially increased in the solutions of these problems. As a result numerous research papers have also been devoted to the subject. Although considerable amount of work has been done in the past, there is still a lack of a completely satisfactory computational scheme. Also, there are some cases that have not been studied numerically yet. In the current article several approaches for the numerical solution of the one‐dimensional parabolic equation subject to the specification of mass, which have been considered in the literature, are reported. Finite difference methods have been proposed for the numerical solution of the new nonclassic boundary value problem. To investigate the performance of the proposed algorithm, we consider solving a test problem. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

The topological structure of fuzzy sets with endograph metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rⁿ, let be the family of all fuzzy sets ofRⁿ, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space with the topology of endograph metric is homeomorphic to the Hilbert cube Q=[-1,1]^ω iff Y is compact; and the space is homeomorphic to {(x_n)Q:sup|x_n|<1} iff Y is non-compact and locally compact.     

Third order accurate variable mesh discretization and application of TAGE iterative method for the non-linear two-point boundary value problems with homogeneous functions in integral form

In this paper, we discuss the application of two-parameter alternating group explicit (TAGE) iterative method to an efficient third order variable mesh method for the solution of non-linear differential equations with integral homogeneous functions subject to natural boundary conditions. The proposed method is applicable only when the internal grid points of the solution space are odd in number. The proposed iterative method is also applicable to the integro-differential equations with singular coefficients. Comparative numerical results are given to demonstrate the usefulness of the proposed method.     

The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility

In this paper, firstly we show that the determining equations of the (1+1) dimension nonlinear differential equation with arbitrary order for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the (m+1) dimension differential equations. The nonlinear Klein–Gordon equation, the (2+1)-dimensional Boussinesq equation and the generalized Nizhnik–Novikov–Veselov equation serve as examples illustrating this method.