The Fixing Block Method in Combinatorics on Words

We give an overview of the method of fixing blocks introduced by Shelton. We apply the method to words which are nonrepetitive up to mod k.     

Application of the cross-entropy method to clustering and vector quantization

We apply the cross-entropy (CE) method to problems in clustering and vector quantization. The CE algorithm for clustering involves the following iterative steps: (a) generate random clusters according to a specified parametric probability distribution, (b) update the parameters of this distribution according to the Kullback–Leibler cross-entropy. Through various numerical experiments, we demonstrate the high accuracy of the CE algorithm and show that it can generate near-optimal clusters for fairly large data sets. We compare the CE method with well-known clustering and vector quantization methods such as K-means, fuzzy K-means and linear vector quantization, and apply each method to benchmark and image analysis data.     

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.     

A nondestructive numerical-experimental method of determining the residual stresses in welded shells of revolution

We propose a numerical-experimental method of determining the residual stresses in welded shells of revolution. We solve the inverse conditionally correct problem of recovering the complete picture of the residual stress state from part of the experimental values obtained by the method of photoelasticity. We apply the numerical spline-collocation method. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 149–151.     

Walsh Series Analysis of the L 2-Discrepancyof Symmetrisized Point Sets

 We present a method to estimate the L ₂-discrepancy of symmetrisized point sets from above and from below with the help of Walsh series analysis. We apply the method to a class of two-dimensional net-type point sets, thereby generalizing results of Halton and Zaremba and of Proinov.     

Numerical computation of the stress-strain state at the end of a crack under dynamic load

We obtain the values of the stress intensity factor at the end of a stationary crack under dynamic load using a numerical method. We compare the results with the existing experimental data. We establish that it is possible to apply the proposed method to compute stationary cracks under dynamic load. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 61–65.     

Preservation of local dynamics when applying central difference methods: application to SIR model

We apply the central difference method (u _t+1 ? u _t ? 1)/(2Δt) = f(u _t ) to an epidemic SIR model and show how the local stability of the equilibria is changed after applying the numerical method. The above central difference scheme can be used as a numerical method to produce a discrete-time model that possesses interesting local dynamics which appears inconsistent with the continuous model. Any fixed point of a differential equation will become an unstable saddle node after applying this method. Two other implicitly defined central difference methods are also discussed here. These two methods are more efficient for preserving the local stability of the fixed points for the continuous models. We apply conformal mapping theory in complex analysis to verify the local stability results.     

Walsh Series Analysis of the L 2-Discrepancyof Symmetrisized Point Sets

 We present a method to estimate the L ₂-discrepancy of symmetrisized point sets from above and from below with the help of Walsh series analysis. We apply the method to a class of two-dimensional net-type point sets, thereby generalizing results of Halton and Zaremba and of Proinov. (Received 14 September 2000)     

The Additivity Theorem in K-Theory

We present a method for converting Theorem B style proofs in algebraic K-theory to Theorem A style proofs and apply it to the additivity theorem.     

Long-time asymptotics of perturbed finite-gap Korteweg-de Vries solutions

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of solutions of the Korteweg-de Vries equation which are decaying perturbations of a quasi-periodic finite-gap background solution. We compute a nonlinear dispersion relation and show that the x/t plane splits into g+1 soliton regions which are interlaced by g + 1 oscillatory regions, where g + 1 is the number of spectral gaps.     

A geometric view of Krylov subspace methods on singular systems

We give a geometric framework for analysing iterative methods on singular linear systems A x = b and apply them to Krylov subspace methods. The idea is to decompose the method into the ?(A) component and its orthogonal complement ?(A)^?, where ?(A) is the range of A. We apply the framework to GMRES, GMRES(k) and GCR(k), and derive conditions for convergence without breakdown for inconsistent and consistent singular systems. The approach also gives a geometric interpretation and different proofs of the conditions obtained by Brown and Walker for GMRES. We also give examples arising in the finite difference discretization of two‐point boundary value problems of an ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Method of separation of the variables for basic analogs of equations of mathematical physics

We apply method of separation of the variables to q-analogs of several equations of mathematical physics. Dedicated to Dick Askey on his 70th birthday. 2000 Mathematics Subject Classification Primary—33D45, 42C10; Secondary—33D15.     

Practical problems arising for finding roots of nonlinear equations

Consider the numerical solutions of a nonlinear equation f(x)=0. We apply Newton's and Newton's-like methods to solve the equation.First, we discuss the attainable correct digits for the approximated root. Second, we give a practical termination criterion and the method to estimate the accuracy of the obtained root. Some numerical examples are presented.     

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.     

Renormalization group second‐order approximation for singularly perturbed nonlinear ordinary differential equations

We consider a 2 time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ϵ of the time scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ϵ²). A classical approach to study these problems is Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in 2 steps. In the first step, we show that the RG method allows for approximation of the fast time variables by their RG expansion taken at the slow time unknowns. Next, we study the slow time equations, where the fast time unknowns are replaced by their RG expansion. This allows to rigorously show the second order uniform error estimate. Our result is a higher order extension of Hoppensteadt's work on the Tikhonov singular perturbation theorem for infinite times. The proposed procedure is suitable for problems from applications, and it is computationally less demanding than the classical Vasil'eva‐O'Malley expansion. We apply the developed method to a mathematical model of stem cell dynamics.     

On the spread of a hermitian matrix and a conjecture of thompson

We prove some inequalities involving the eigenvalues of an nxn Hermitian matrix and the eigenvalues of the (n?1)x(n?1) principal submatrices. We apply this inequality to generalize a known result on the numerical range to the lth numerical range. The method used yields an elegant proof of the converse to the interlacing theorem, which we include. A counterexample to the quardratic spread inequality conjectured by R. C. Thompson is also given.     

Regular Path Systems and (Bi)automatic Groups

We develop a new method of proving that groups acting on various classes of spaces, with nontrivial stabilizers admitted, are automatic or biautomatic. We apply this method to buildings, CAT(0) cubical complexes, 6-systolic simplicial complexes and Cayley graphs of groups acted upon by other groups.Partially supported by the Polish State Committee for Scientific Research (KBN) grant 2 P03A 017 25.     

Oscillation of delay differential equations on time scales

Consider the following equation: , where t is in a measure chain. We apply the theory of measure chains to investigate the oscillation and nonoscillation of the above equation on the basis of some well-known results. And in some sense, we show a method to unify the delay differential equation and delay difference equation.     

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.     

Multifractional, Multistable, and Other Processes with Prescribed Local Form

We present a general method for constructing stochastic processes with prescribed local form, encompassing examples such as variable amplitude multifractional Brownian and multifractional α-stable processes. We apply the method to Poisson sums to construct multistable processes, that is, processes that are locally α(t)-stable but where the stability index α(t) varies with t. In particular we construct multifractional multistable processes, where both the local self-similarity and stability indices vary.