On the choice of parameters in MAOR type splitting methods for the linear complementarity problem

In the present work we consider the iterative solution of the Linear Complementarity Problem (LCP), with a nonsingular H ₊ coefficient matrix A, by using all modulus-based matrix splitting iterative methods that have been around for the last couple of years. A deeper analysis shows that the iterative solution of the LCP by the modified Accelerated Overrelaxation (MAOR) iterative method is the “best”, in a sense made precise in the text, among all those that have been proposed so far regarding the following three issues: i) The positive diagonal matrix-parameter Ω ≥ diag(A) involved in the method is Ω = diag(A), ii) The known convergence intervals for the two AOR parameters, α and β, are the widest possible, and iii) The “best” possible MAOR iterative method is the modified Gauss-Seidel one.     

Branches of the essential spectrum of the lattice spin-boson model with at most two photons

We consider a lattice analogue of the A_m model of light radiation with a fixed atom and at most m photons (m = 1, 2). We describe the essential spectrum of the operator A₂ in terms of the spectrum of the operator A₁, i.e., we find the “two-particle” and “three-particle” branches of the essential spectrum of A₂. We prove that the essential spectrum is a union of at most six intervals, and we study their positions. We derive an estimate for the lower bound of the “two-particle” and “three-particle” branches.     

Convergence analysis of the direct extension of ADMM for multiple-block separable convex minimization

Recently, the alternating direction method of multipliers (ADMM) has found many efficient applications in various areas; and it has been shown that the convergence is not guaranteed when it is directly extended to the multiple-block case of separable convex minimization problems where there are m ≥ 3 functions without coupled variables in the objective. This fact has given great impetus to investigate various conditions on both the model and the algorithm’s parameter that can ensure the convergence of the direct extension of ADMM (abbreviated as “e-ADMM”). Despite some results under very strong conditions (e.g., at least (m ? 1) functions should be strongly convex) that are applicable to the generic case with a general m, some others concentrate on the special case of m = 3 under the relatively milder condition that only one function is assumed to be strongly convex. We focus on extending the convergence analysis from the case of m = 3 to the more general case of m ≥ 3. That is, we show the convergence of e-ADMM for the case of m ≥ 3 with the assumption of only (m ? 2) functions being strongly convex; and establish its convergence rates in different scenarios such as the worst-case convergence rates measured by iteration complexity and the globally linear convergence rate under stronger assumptions. Thus the convergence of e-ADMM for the general case of m ≥ 4 is proved; this result seems to be still unknown even though it is intuitive given the known result of the case of m = 3. Even for the special case of m = 3, our convergence results turn out to be more general than the existing results that are derived specifically for the case of m = 3.     

Convergence of some finite element iterative methods related to different Reynolds numbers for the 2D/3D stationary incompressible magnetohydrodynamics

Based on finite element method (FEM), some iterative methods related to different Reynolds numbers are designed and analyzed for solving the 2D/3D stationary incompressible magnetohydrodynamics (MHD) numerically. Two-level finite element iterative methods, consisting of the classical m-iteration methods on a coarse grid and corrections on a fine grid, are designed to solve the system at low Reynolds numbers under the strong uniqueness condition. One-level Oseen-type iterative method is investigated on a fine mesh at high Reynolds numbers under the weak uniqueness condition. Furthermore, the uniform stability and convergence of these methods with respect to equation parameters R_e,R_m, S_c, mesh sizes h,H and iterative step m are provided. Finally, the efficiency of the proposed methods is confirmed by numerical investigations.     

On the divergence of trigonometric Fourier series in classes <Emphasis Type="Italic">ϕ</Emphasis>(<Emphasis Type="Italic">L</Emphasis>) close to <Emphasis Type="Italic">L</Emphasis>

We show the unimprovability of a theorem on sufficient convergence conditions for the trigonometric Fourier series of a function in classes ?(L) in the case when the class ?(L) is “close” to L.     

On the Conservativeness of Some Markov Processes

We give a unified method to obtain the conservativeness of a class of Markov processes associated with lower bounded semi-Dirichlet forms on L ²(X;m), including symmetric diffusion processes, some non-symmetric diffusion processes and jump type Markov processes on X, where X is a locally compact separable metric space and m is a positive Radon measure on X with full topological support. Using the method, we give an example in each section, providing the conservativeness of the processes, that are given by the “increasingness of the volume of some sets(balls)” and “that of the coefficients on the sets” of the Markov processes.     

Iterative roots of multidimensional operators and applications to dynamical systems

Solutions φ(x) of the functional equation φ(φ(x)) = f (x) are called iterative roots of the given function f (x). They are of interest in dynamical systems, chaos and complexity theory and also in the modeling of certain industrial and financial processes. The problem of computing this “square root” of a function or operator remains a hard task. While the theory of functional equations provides some insight for real and complex valued functions, iterative roots of nonlinear mappings from \({\mathbb{R}^n}\) to \({\mathbb{R}^n}\) are less studied from a theoretical and computational point of view. Here we prove existence of iterative roots of a certain class of monotone mappings in \({\mathbb{R}^n}\) spaces and demonstrate how a method based on neural networks can find solutions to some examples that arise from simple physical dynamical systems.     

On local equivalence of the majorant of partial sums and Paley function of Franklin series

In this paper we prove that the majorant of partial sums and the Paley function of Franklin series have equivalent norms in the space L _p (I), p > 0, provided that the “peak” intervals of Franklin functions with non-vanishing coefficients lie in I. Examples of series emphasizing that this condition is essential are also given.     

Dynamic study of Schröder’s families of first and second kind

The Schröder iterative families of the first and second kind are of great importance in the theory and practice of iterative processes for solving nonlinear equations f(x) = 0. In both cases, the methods E _r (first kind) and S _r (second kind) converge locally to a zero α of f as O(|x _k ? α| ^r ). Although characteristics of these families have been studied in many papers, their dynamic and chaotic behavior has not been completely investigated. In this paper, we compare convergence properties of both iterative schemes using the two methodologies: (i) comparison by numerical examples and (ii) comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. Apart from the visualization of iterative processes, basins of attraction reveal very useful features on iterations such as consumed CPU time and average number of iterations, both as functions of starting points. We demonstrate by several examples that the Schröder family of the second kind S _r possesses better convergence characteristics than the Schröder family of the first kind E _r .     

Algebraic Theory of Causal Double Products

Corresponding to each “rectangular” double product in the form of a formal power series R[h] with coefficients in the tensor product ?(?)⊙ ? (?) with itself of the Itô Hopf algebra, we construct “triangular” elements T[h] of ?(?) satisfying ΔT[h] = T[h]⁽¹⁾ R[h]T{h]⁽²⁾. In Fock space representations of ?(?) by iterated quantum stochastic integrals when ? is the algebra of Itô differentials of the calculus, these correspond to “causal” double product integrals in a single Fock space.     

The Fourier-Walsh series of the functions absolutely continuous in the generalized restricted sense

We consider the questions of convergence in Lorentz spaces for the Fourier-Walsh series of the functions with Denjoy integrable derivative. We prove that a condition on a function f sufficient for its Fourier-Walsh series to converge in the Lorentz spaces “near” L _∞ cannot be expressed in terms of the growth of the derivative f′.     

Globally convergent Jacobi methods for positive definite matrix pairs

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem A x = λ B x, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begovi? Kova? (Trans. Numer. Anal. (ETNA) 47, 107–147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of Δ _A AΔ _A and Δ _B BΔ _B are small, for some nonsingular diagonal matrices Δ _A and Δ _B .     

On iterative methods for solving equations with covering mappings

In this paper we propose an iterative method for solving the equation Υ(x, x) = y, where the mapping Υ acts in metric spaces and is covering in the first argument and Lipschitzian in the second one. Each subsequent element x _i+1 of the sequence of iterations is defined by the previous one as a solution to the equation Υ(x, x _i) = y _i, where y _i can be an arbitrary point sufficiently close to y. Conditions for convergence and error estimates are obtained. The method proposed is an iterative development of the Arutyunov method for finding coincidence points of mappings. In order to determine x _i+1 in practical implementation of the method in linear normed spaces, it is proposed to perform one step by using the Newton–Kantorovich method. The thus-obtained method of solving the equation of the form Υ(x, u) = ψ(x) ? φ(u) coincides with the iterative method proposed by A.I. Zinchenko,M.A. Krasnosel’skii, and I.A. Kusakin.     

Extended Lorentz Transformations in Clifford Space Relativity Theory

Some novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) were explored recently. In particular, generalized photon dispersion relations allowed for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, but breaking the extended Lorentz symmetry in C-spaces. In this work we analyze in further detail the extended Lorentz transformations in Clifford Space and their physical implications. Based on the notion of “extended events” one finds a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the Doubly Special Relativity (DSR) framework. A generalized Weyl-Heisenberg algebra, involving polyvector-valued coordinates and momenta operators, furnishes a realization of an extended Poincare algebra in C-spaces. In addition to the Planck constant ?, one finds that the commutator of the Clifford scalar components of the Weyl-Heisenberg algebra requires the introduction of a dimensionless parameter which is expressed in terms of the ratio of two length scales : the Planck and Hubble scales. We finalize by discussing the concept of “photons”, null intervals, effective temporal variables and the addition/subtraction laws of generalized velocities in C-space.     

On properties of infimal topology of a map space

We study properties of the infimal topology τ_inf which is the infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. One of the main results of the research consists in obtaining necessary and sufficient condition for the topology τ_inf to have the Fréchet–Urysohn property. We also establish necessary and sufficient conditions for coincidence of the topology τinf and a topology of uniform convergence τ_μ (“attaining” the infimum). We prove that for this coincidence it is sufficient for the topology τ_inf to satisfy the first axiom of countability.     

Noetherian Rings whose Modules are Prime Serial

A theorem due to Nakayama and Skornyakov states that “a ring R is an Artinian serial ring if and only if all left R-modules are serial” and a theorem due to Warfield state that “a Noetherian ring R is serial if and only if every finitely generated left R-module is serial”. We say that an R-module M is prime uniserial (?-uniserial, for short) if for every pair P, Q of prime submodules of M either \(P\subseteq Q\) or \(Q\subseteq P\), and we say that M is prime serial (?-serial, for short) if it is a direct sum of ?-uniserial modules. Therefore, two interesting natural questions of this sort are: “Which rings have the property that every module is ?-serial?” and “Which rings have the property that every finitely generated module is ?-serial?” Most recently, in our paper, Prime uniserial modules and rings (submitted), we considered these questions in the context of commutative rings. The goal of this paper is to answer these questions in the case R is a Noetherian ring in which all idempotents are central or R is a left Artinian ring.     

Analogues of some Tauberian theorems for bounded variation

In this paper definitions for “bounded variation”, “subsequences”, “Pringsheim limit points”, and “stretchings” of a double sequence are presented. Using these definitions and the notion of regularity for four dimensional matrices, the following two questions will be answered. First, if there exists a four dimensional regular matrix A such that Ay = Σ _k,l=1,1 ^∞∞ a _m,n,k,l y _k,l is of bounded variation (BV) for every subsequence y of x, does it necessarily follow that x ∈ BV? Second, if there exists a four dimensional regular matrix A such that Ay ∈ BV for all stretchings y of x, does it necessarily follow that x ∈ BV? Also some natural implications and variations of the two Tauberian questions above will be presented.     

On the strong Brillinger-mixing property of α-determinantal point processes and some applications

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function C(x, y) defining an α-determinantal point process (DPP). Assuming absolute integrability of the function C₀(x) = C(o, x), we show that a stationary α-DPP with kernel function C₀(x) is “strongly” Brillinger-mixing, implying, among others, that its tail-σ-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications to statistical second-order analysis of α-DPPs.     

Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem

In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume.The related questions that will also be studied are the following: given a contractible k-dimensional sphere in M ⁿ , how “fast” can this sphere be contracted to a point, if π _i (M ⁿ )={0} for 1≤i<k. That is, what is the maximal length of the trajectory described by a point of a sphere under an “optimal” homotopy? Also, what is the “size” of the smallest non-contractible k-dimensional sphere in a (k-1)-connected manifold M ⁿ providing that M ⁿ is not k-connected?     

On needed reals

Given a binary relationR, we call a subsetA of the range ofR R-adequate if for everyx in the domain there is someyεA such that (x, y)εR. Following Blass [4], we call a realη ”needed” forR if in everyR-adequate set we find an element from whichη is Turing computable. We show that every real needed for inclusion on the Lebesgue null sets,Cof(\(\mathcal{N}\)), is hyperarithmetic. Replacing “R-adequate” by “R-adequate with minimal cardinality” we get the related notion of being “weakly needed”. We show that it is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetic reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999 and in [4].