New Estimates for the Rate of Convergence of the Method of Subspace Corrections

We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections. We provide upper bounds and in a special case, a lower bound for preconditioners defined via the method of successive subspace corrections.     

A Criterion for the Uniqueness of the Solution of the Tricomi Problem for the Multidimensional Lavrent’ev–Bitsadze Equation

Differential Equations - An example of a mixed domain in which the solution of the Tricomi problem is trivial is given. A criterion for the uniqueness of the classical solution of this problem is...     

Estimation for the Dilatation of the Generalized Beurling-Ahlfors' Extension

51.IntroductionLethbeahomeomorphismontherealaxisR,andsatisfyh(lco)=loo.Wecallaquasisymmetricfunction.lfthereexistsa'constantMsuchthatM-1Sp(x,t)SM,x,teR,(l.2)then(1.2)iscalledtheM-condition-Byvirtueoftheclassicresultofpaper[lj,ifp(x,t)sat-isfiestheM-condition,thenthereexistsaK(M')-quasiconformaIextensiononthel1alfplaneH,whichisthewell-knownBeurling-Ahlfors'extension:ytz)=u(x,y) iv(x,y),whereAfterpaper[lj,manyworkshavebeendoneforestimatingK-Particularly,itshouldbere-markedheretl1atM.Leht…     

On the failure of concentration for the ℓ∞-ball

Let (X, d) be a compact metric space and µ a Borel probability on X. For each N ≥ 1 let d_∞^N be the ?_∞-product on X^N of copies of d, and consider 1-Lipschitz functions X^N → ? for d_∞^N.     

Estimate for the amplitude of the limit cycle of the Liénard equation

We consider the nonlinear Liénard equation \(\ddot x\left( t \right) + f\left( x \right)\dot x\left( t \right) + g\left( x \right) = 0\). Liénard obtained sufficient conditions on the functions f(x) and g(x) under which this equation has a unique stable limit cycle. Under additional conditions, we prove a theorem that permits one to estimate the amplitude (the maximum value of x) of this limit cycle from above. The theorem is used to estimate the amplitude of the limit cycle of the van der Pol equation \(\ddot x\left( t \right) + \mu \left[ {{x^2}\left( t \right) - 1} \right]\dot x\left( t \right) + x\left( t \right) = 0\).     

Asymptotic of the Solutions of the Initial Boundary Value Problem for the Diffusion Equations for Semiconductors（Ⅱ）

The paper deal with the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial value, we prove that the smooth solutions to these evolutionary problems tend to the unique stationary solution exponentially as time tends to infinity.     

Decrease rate of the probabilities ofε-deviations for the means of stationary processes

The asymptotic behavior asn → ∞ of the normed sumsσn =n ^?1 Σ _k ^=0n?1 Xk for a stationary processX = (X _n ,n ∈ ?) is studied. For a fixedε > 0, upper estimates for P(sup _k≥n ¦σ _k ¦ ≥ε) asn → ∞ are obtained.     

On the Distribution of the Cardinality of Row Space for Boolean Matrices

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Note on the coefficients of Ramanujan’s expansion for the logarithm of the factorial

Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique.     

Asymptotics at t→∞ of the solution of the Cauchy problem for the Landau-Lifshitz equation

Conclusions The formulas obtained in the present paper for the leading term in the asymptotic behavior of the solution of the Cauchy problem for the LL equation subject to the boundary conditions L₃1, x± describe the solitonless sector. The transition to the general case, which takes into account the presence in the solution of soliton formations, can be made on the basis solely of algebraic considerations that use the procedure of soliton dressing developed in [17, 18] for the LL equation. In particular, applying to the obtained asymptotic formulas the procedure for a dressing of domain wall type (see [17]), we arrive at formulas that describe the asymptotic solution of the Cauchy problem for the LL equation with boundary conditions of the form L₃±1, x±.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 76, No. 1, pp. 3–17, July, 1988.     

Well-posedness of the cauchy problem for the coupled system of the Schrödinger-KdV equations

This paper is devoted to the study of the Cauchy problem for the coupled system of the Schrödinger-KdV equations which describes the nonlinear dynamics of the one-dimensional Langmuir and ion-acoustic waves. Global well-posedness of the problem is established in the spaceH ^κ ×H ^κ (κ εZ ⁺), the first and second components of which correspond to the electric field of the Langmuir oscillations and the low-frequency density perturbation respectively.     

On the Existence of Nontrivial Solutions of Quasi-asymptotically Linear Problem for the P-Laplacian

In this paper, we study the existence of nontrivial solutions for the following Dirichlet problem for the p-Laplacian (p > 1):where Ω is a bounded domain in Rn (A≥1) and f(x,u) is quasi-asymptotically linear with respect to |u|p-2 u at infinity. Recently it was proved that the above problem has a positive solution under the condition that f(x, s)/sp-1 is nondecrcasing with respect to s for all x ∈Ω and some others. In this paper. by improving the methods in the literature, we prove that the functional corresponding to the above problem still satisfies a weakened version of (P.S.) condition even if f(x, s)/sp-1 isn't a nondecreasing function with respect to s, and then the above problem has a nontrivial weak solution by Mountain Pass Theorem.     

Algorithm for solving the Navier–Stokes equations for the modeling of creeping flows

We study a coupled algorithm for solving the two-dimensional Navier–Stokes equations in the stream function–vorticity variables. The algorithm is based on a finite-difference scheme in which the inertial terms in the vortex transport equation are taken from the lower time layer and the dissipative terms, from the upper time layer. In the linear approximation, we study the stability of this algorithm and use test computations to show its advantages when modeling flows at moderate Reynolds numbers.     

A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations

In this work the solution of the Volterra–Fredholm integral equations of the second kind is presented. The proposed method is based on the homotopy perturbation method, which consists in constructing the series whose sum is the solution of the problem considered. The problem of the convergence of the series constructed is formulated and a proof of the formulation is given in the work. Additionally, the estimation of the approximate solution errors obtained by taking the partial sums of the series is elaborated on.     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Conditions for the uniqueness of statistic's distribution in the class of spherical distridutions

This paper gives the necessary and sufficient conditions, which are very simple for checking the uniqueness of the distribution of a statistic in those classes of the matrix spherical distributions and improve the results of Kariya [1].     

Application of the differential transformation method for the solution of the hyperchaotic Rössler system

The aim of this paper is to investigate the accuracy of the differential transformation method (DTM) for solving the hyperchaotic Rössler system, which is a four-dimensional system of ODEs with quadratic nonlinearities. Comparisons between the DTM solutions and the fourth-order Runge–Kutta (RK4) solutions are made. The DTM scheme obtained from the DTM yields an analytical solution in the form of a rapidly convergent series. The direct symbolic-numeric scheme is shown to be efficient and accurate.     

Some results of the implementation of the “podmodeli” program for the gas dynamics equations

The results of investigations using the PODMODELI (SUBMODELS) program, which is aimed at the exhaustive use of the symmetry of the gas dynamics equations to construct classes of exact solutions (submodels) of these equations, are summarized. The starting point is the fact that any Lie group of transformations of the basis space (of all independent and dependent variables) which is admitted by the gas dynamics equations can generate certain submodels. In order to describe the infinite set of submodels which arises from this in a compact manner, a number of techniques for ordering them is proposed: with respect to the equation of state of a gas, using a similarity criterion, according to their types (rank, defect), with respect to the property of evolutionarity and according to the criterion of regularity. Examples of new submodels are presented. The most significant work carried out using the PODMODELI program up to the present time is indicated in a list of references.     

On the law of the iterated logarithm for the discrepancy of 〈n k x〉

By a well known result of Philipp (1975), the discrepancy D _N (ω) of the sequence (n _k ω) _k≥1 mod 1 satisfies the law of the iterated logarithm under the Hadamard gap condition n _{k + 1}/n _k ≥ q > 1 (k = 1, 2, …). Recently Berkes, Philipp and Tichy (2006) showed that this result remains valid, under Diophantine conditions on (n _k ), for subexpenentially growing (n _k ), but in general the behavior of (n _k ω) becomes very complicated in the subexponential case. Using a different norming factor depending on the density properties of the sequence (n _k ), in this paper we prove a law of the iterated logarithm for the discrepancy D _N (ω) for subexponentially growing (n _k ) without number theoretic assumptions. C. Aistleitner, Research supported by FWF grant S9603-N13. I. Berkes, Research supported by FWF grant S9603-N13 and OTKA grants K 61052 and K 67961. Authors’ addresses: C. Aistleitner, Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria; I. Berkes, Institute of Statistics, Graz University of Technology, Steyrergasse 17/IV, 8010 Graz, Austria