New approximate solutions of the problem of the thermomechanics of a hollow cylinder with a material of variable properties

We use the method of equivalent sources to obtain approximate solutions of an unrestricted quasistatic axisymmetric problem of the thermomechanics of a long hollow cylinder with ends fastened and materials whose properties depend on the temperature. We perform a comparative analysis of the results with the solutions obtained by the method of perturbations and the method of finite differences.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 71–74.     

Numerical study of the two-dimensional problem of the theory of elasticity using the boundary-element method

We propose a method of solving the two-dimensional problem of the theory of elasticity by a direct boundary-element method. We apply Galerkin's method with linear and quadratic approximations of the forces and displacements. We give the numerical results of solving a model problem that shows the effectiveness of the proposed approach.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 60–63.     

Jacobi Spectral Methods for Volterra-Urysohn Integral Equations of Second Kind with Weakly Singular Kernels

Abstract

In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L²-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L²-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.     

Upper and Lower Bounds on Maximum Nonlinearity of n-input m-output Boolean Function

The paper presents lower and upper bounds on the maximumnonlinearity for an n-input m-output Booleanfunction. We show a systematic construction method for a highlynonlinear Boolean function based on binary linear codes whichcontain the first order Reed-Muller code as a subcode. We alsopresent a method to prove the nonexistence of some nonlinearBoolean functions by using nonexistence results on binary linearcodes. Such construction and nonexistence results can be regardedas lower and upper bounds on the maximum nonlinearity. For somen and m, these bounds are tighter than theconventional bounds. The techniques employed here indicate astrong connection between binary linear codes and nonlinear n-input m-output Boolean functions.     

A Method of Deducing L-Polyhedra for n-Lattices

We suggest a method for selecting an L-simplex in an L-polyhedron of an n-lattice in Euclidean space. By taking into account the specific form of the condition that a simplex in the lattice is an L-simplex and by considering a simplex selected from an L-polyhedron, we present a new method for describing all types of L-polyhedra in lattices of given dimension n. We apply the method to deduce all types of L-polyhedra in n-dimensional lattices for n=2,3,4, which are already known from previous results.     

Transportation problems which can be solved by the use of hirsch-paths for the dual problems

Balinski uses his signature method for the proof of the Hirsch-conjecture for dual transportation polyhedra to obtain an efficient algorithm for the assignment problem. We will show how to extend this method to other primal transportation problems, including transportation problems with unit demands. We then prove that Balinski's assignment algorithm is equivalent, cycle by cycle, to that of Hung and Rom. We demonstrate that, under some assumptions for our probability model, a modification of the latter algorithm has an average complexity of O(n ²logn) and present some computational results confirming this. We also present results that indicate that this modification compares favorably with Balinski's algorithm and other codes. Research of both authors supported, in part, by grants of the Alexander von Humboldt-Stiftung. Supported, in part, by NSF grant DMS-8504050.     

Global and Finite Convergence of a Generalized Newton Method for Absolute Value Equations

We investigate an efficient method for solving the absolute value equation Ax−|x|=b when the interval matrix [A−I,A+I] is regular. A generalized Newton method which combines the semismooth and the smoothing Newton steps is proposed. We establish global and finite convergence of the method. Preliminary numerical results indicate that the generalized Newton method is promising.     

Outer approximation methods for solving variational inequalities in Hilbert space

In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima’s method has so far been considered only in the Euclidean setting with different conditions on F. We provide several examples for the case where C is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.     

Numerical solution of two-dimensional time fractional cable equation with Mittag-Leffler kernel

The main motive of this article is to study the recently developed Atangana-Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag-Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two-dimensional cable equation in which time-fractional derivative is of Mittag-Leffler kernel type. First, we derive an approximation formula of the fractional-order ABC derivative of a function t^k using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two-dimensional model of the time-fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two-dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag-Leffler fractional derivative.     

Empirical Likelihood for a Class of Functionals of Survival Distribution with Censored Data

The empirical likelihood was introduced by Owen, although its idea originated from survival analysis in the context of estimating the survival probabilities given by Thomas and Grunkemeier. In this paper, we investigate how to apply the empirical likelihood method to a class of functionals of survival function in the presence of censoring. We define an adjusted empirical likelihood and show that it follows a chi-square distribution. Some simulation studies are presented to compare the empirical likelihood method with the Studentized-t method. These results indicate that the empirical likelihood method works better than or equally to the Studentized-t method, depending on the situations.     

A combined splitting—cross entropy method for rare-event probability estimation of queueing networks

We present a fast algorithm for the efficient estimation of rare-event (buffer overflow) probabilities in queueing networks. Our algorithm presents a combined version of two well known methods: the splitting and the cross-entropy (CE) method. We call the new method SPLITCE. In this method, the optimal change of measure (importance sampling) is determined adaptively by using the CE method. Simulation results for a single queue and queueing networks of the ATM-type are presented. Our numerical results demonstrate higher efficiency of the proposed method as compared to the original splitting and CE methods. In particular, for a single server queue example we demonstrate numerically that both the splitting and the SPLITCE methods can handle our buffer overflow example problems with both light and heavy tails efficiently. Further research must show the full potential of the proposed method.     

IPO estimation of heaviness of the distribution beyond regularly varying tails

Abstract

We introduce a completely novel method for estimation of the parameter which governs the tail behavior of the cumulative distribution function of the observed random variable. We call it Inverse Probabilities for p-Outside values (IPO) estimation method. We show that this approach is applicable for wider class of distributions than the one with regularly varying tails. We demonstrate that IPO method is a valuable competitor to regularly varying tails based estimation methods. Some of the properties of the estimators are derived. The results are illustrated by a convenient simulation study.     

A note on the girth of digraphs

Behzad, Chartrand and Wall conjectured that the girth of a diregular graph of ordern and outdegreer is not greater than [n /r]. This conjecture has been proved forr=2 by Behzad and forr=3 by Bermond. We prove that a digraph of ordern and halfdegree ≧4 has girth not exceeding [n / 4]. We also obtain short proofs of the above results. Our method is an application of the theory of connectivity of digraphs.     

Asymptotic profile of solutions for some wave equations with very strong structural damping

We consider the Cauchy problem in R ⁿ for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L^1,1( R ⁿ) initial data by using a simple method introduced in by the first author. The obtained results will include regularity loss type estimates, which are essentially new in this kind of equation.     

An approach for the Steiner problem in directed graphs

We present a scheme to solve the Steiner problem in directed graphs using a heuristic method to obtain upper bounds and thek shortest arborescences algorithm to compute lower bounds. We propose to combine these ideas in an enumerative algorithm. Computational results are presented for both thek shortest arborescences algorithm and the heuristic method, including reduction tests for the problem.This work was partially supported by CNP_q, FINEP, CAPES and IBM do Brasil.     

R-functions in problems of bending of three-layer plates

We study the application of the method of R-functions to the solution of problems of bending of elastic three-layer plates of arbitrary shape. We obtain new solution structures for plates that are freely supported over the whole boundary. We solve test problems and compare the experimental and theoretical results. All numerical results were obtained using the POLE programming system.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 77–79.     

On the complexity of some subgraph problems

We study the complexity of the problem of deciding the existence of a spanning subgraph of a given graph, and of that of finding a maximum (weight) such subgraph. We establish some general relations between these problems, and we use these relations to obtain new NP-completeness results for maximum (weight) spanning subgraph problems from analogous results for existence problems and from results in extremal graph theory. On the positive side, we provide a decomposition method for the maximum (weight) spanning chordal subgraph problem that can be used, e.g., to obtain a linear (or O(nlogn)) time algorithm for such problems in graphs with vertex degree bounded by 3.     

Smoothed penalty algorithms for optimization of nonlinear models

We introduce an algorithm for solving nonlinear optimization problems with general equality and box constraints. The proposed algorithm is based on smoothing of the exact l ₁-penalty function and solving the resulting problem by any box-constraint optimization method. We introduce a general algorithm and present theoretical results for updating the penalty and smoothing parameter. We apply the algorithm to optimization problems for nonlinear traffic network models and report on numerical results for a variety of network problems and different solvers for the subproblems.     

A new logarithmic-quadratic proximal method for nonlinear complementarity problems

In this paper, we propose a new modified logarithmic-quadratic proximal (LQP) method for solving nonlinear complementarity problems (NCP). We suggest using a prediction-correction method to solve NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed by using a new step size α_k. Under suitable conditions, we prove that the new method is globally convergent. We report preliminary computational results to illustrate the efficiency of the proposed method. This new method can be considered as a significant refinement of the previously known methods for solving nonlinear complementarity problems.     

On quasiperiodic solutions of semilinear systems of differential equations

We prove existence theorems for analytic quasi-periodic solutions for analytic systems of differential equations in a Banach space by the method of accelerated convergence. The results obtained are new even in the finite-dimensional case. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.