Asymptotics of the basis functions of generalized Taylor series for the class H ρ,2

We study the basis functions φ _n,k and ψ _n,p of generalized Taylor series for the classH _ρ,2 and obtain asymptotic expansions of the functions φ _n,0 ^(l) and $\psi _{n,2 \cdot 4^n - 1}^{(l)} $ . We prove the existence of an asymptotics for the functions φ _n,k and ψ _n,p for k ≠ 0 and p ≠ 2·4 ⁿ ? 1. The first term of the asymptotic expansions of these functions is obtained.     

Distribution of zeros of the Hermite-Padé polynomials for a system of three functions,and the Nuttall condenser

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite-Padé polynomials for a set of m multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) m-sheeted Riemann surface possessing certain properties. In this paper, for m = 3, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface $\Re _3$ that has a canonical decomposition. We consider a system of three functions $\mathfrak{f}_1 ,\mathfrak{f}_2 ,\mathfrak{f}_3$ that are rational on the constructed Riemann surface and satisfy the independence condition det . In the case of m = 3, we refine the main theorem from Nuttall’s paper of 1981. In particular, we show that in this case the complement ?? \ B of the open (possibly, disconnected) set B ? ?? introduced in Nuttall’s paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.     

Solvability in the large on (0, ∞) of the fundamental initial-boundary value problem for the two-dimensional equations of the motion of Oldroyd fluids

The unqiue global solvability for t (0, ) is proved for the system describing the two-dimensional motion of an Oldroyd fluid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 69–72, 1986.     

Weak type inequality for the maximal operator of the (C, α) means of two-dimensionalWalsh-Fourier series

The main aim of this paper is to prove that the maximal operator σ _* ^α of the (C, α) means of the cubical partial sums of the two-dimensional Walsh-Fourier series is bounded from the Hardy space H _2/(2+α) to the space weak-L _2/(2+α).     

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…     

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.     

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...     

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 \(\ddot x\left( t \right) + \mu \left[ {{x^2}\left( t \right) - 1} \right]\dot x\left( t \right) + x\left( t \right) = 0\).     

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.     

Accuracy-optimal evaluation of the fourier transform for CL,N,ɛ functions

We consider minimum-error evaluation of integrals of rapidly oscillating functions of the form $$I(\omega ) = \int\limits_a^b {f(x)\left\{ {\begin{array}{*{20}c} {\sin \omega x} \\ {\cos \omega x} \\ \end{array} } \right\}dx}$$ where f(x) is in the class of interpolation Lipschitzian functions and the information given about f(x) is approximate. The boundary function method is applied to derive lower bounds on the numerical integration error of l(ω) in this class and quadrature formulas attaining these bounds are constructed.     

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

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Necessity of the Cramer, Linnik, and Statulevičius Conditions for Large-Deviation Probabilities

In this paper, we study the necessity of the Cramer, Linnik, and Statuleviius conditions in theorems on large-deviation probabilities. We also show that the Statuleviius condition is a generalization of the Cramer and Linnik conditions. Bibliography: 7 titles.     

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.