On a family of singular measures related to Minkowski's?(x) function

In the present paper we are investigating a certain point measure of a distribution function arising in a paper by Grabner et al. [Combinatorica 22 (2002) 245-267]. This distribution function is defined by means of the subtractive Euclidean algorithm and bears a striking resemblance to the singular?(x)-function of H. Minkowski. Beyond it, we will also consider a whole family of distribution functions arising in a natural way from the above ones. Nevertheless we will prove that all of the corresponding measures of the mentioned functions are mutually singular by using dynamical systems and the ergodic theorem.     

Asymptotic behavior of the solutions of the p-Laplacian equation

The asymptotic behavior of the solutions for p-Laplacian equations as p→∞ is studied.     

Asymptotic behavior of the convex hull of a stationary Gaussian process?

Let X = {X(t), t ?? T} be a stationary centered Gaussian process with values in ? ^d , where the parameter set T equals ? or ?+. Let ?? _t = Cov(X ₀ ,X _t ) be the covariance function of X, and (??,?, P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W _t = conv{X _u , u ?? T ?? [0, t]} as t ?? +?? and show that under the condition ??t ?? 0, t????, the rescaled convex hull (2 ln t) ^?1/2 W _t converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ? associated to the covariance matrix ?? ₀. The asymptotic behavior of the mathematical expectations E f(W _t ), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171?C179, 2011].     

Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces

A boundedness result is established for multilinear singular integral operators on the homogeneous Morrey–Herz spaces. As applications, two corollaries about interesting cases of the boundedness of the considered operators on the homogeneous Morrey–Herz spaces are obtained.     

Asymptotic behavior of the Jost function of a two-dimensional Schrödinger operator

We find the asymptotic behavior of the Jost function(Z,) of a two-dimensional Schrödinger operator for arbitrary and Z/|Z|S¹ as |Z| We discuss consequences of the asymptotic formulas for the inverse scattering problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 96—103, 1988.     

Local behavior of solutions of the stationary Schrödinger equation with singular potentials and bounds on the density of states of Schrödinger operators

We study the local behavior of solutions of the stationary Schrödinger equation with singular potentials, establishing a local decomposition into a homogeneous harmonic polynomial and a lower order term. Combining a corollary to this result with a quantitative unique continuation principle for singular potentials, we obtain log-Hölder continuity for the density of states outer-measure in one, two, and three dimensions for Schrödinger operators with singular potentials, results that hold for the density of states measure when it exists.     

Asymptotic behavior of the Newton–Boussinesq equation in a two-dimensional channel

We prove the existence of a global attractor for the Newton–Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor.     

The spectral asymptotics of elliptic operators of Schrödinger type on a hyperbolic space

For self-adjoint second-order elliptic differential operators that satisfy the non-trapping condition on the n-dimensional hyperbolic space H ⁿ and coincide with the operator in a neighborhood of infinity, where is the Laplace-Beltrami operator on H ⁿ ,we obtain the complete asymptotic expansion of the spectral function as +.For self-adjoint operators of the form (–) +Q _m–r,where Q _m–r is a pseudodifferential operator of order m–r that is automorphic with respect to a discrete group of isometries of the spaceH ⁿ whose fundamental domain has finite volume, we introduce the spectral distribution function N(),which is the analog of the integrated state density, and we find its asymptotics up to order O(^(n–r)/m)as +.Bibliography: 49 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 4–32, 1991.     

Inverse spectral problem for singular AKNS and Schrödinger operators on [0,1]

We consider an inverse spectral problem for singular Sturm–Liouville equations on the unit interval with explicit singularity $a(a+1)/x^2$, $a\in \mathbb{N}$. This problem arises by splitting of the Schrödinger operator with radial potential acting on the unit ball of ${\mathbb{R}}^3$. Our goal is the global parametrization of potentials by spectral data noted by ${\lambda }^a$, and some norming constants noted by ${\kappa }^a$. For $a=0$ and 1, ${\lambda }^a\times {\kappa }^a$ was already known to be a global coordinate system on $L_{\mathbb{R}}^2(0,1)$. We extend this to any non-negative integer a. Similar result is obtained for singular AKNS operator. To cite this article: F. Serier, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Asymptotic behavior of the Landau—Lifshitz model of ferromagnetism

According to the classical theory of Weiss, Landau, and Lifshitz, on a microscopic scale a ferromagnetic body is magnetically saturated (i.e., |M| =: constant) and consists of regions in which the magnetization is uniform, separated by thin transition layers. Any stationary configuration corresponds to a minimum point of an energy functional in which a small parameter is present. The asymptotic behavior as 0 is studied here. It is easy to see that any sequence of minimizers contains a subsequenceM _j that converges to a fieldM. By means of a -limit procedure it is shown that this fieldM is a minimizer of a new functional containing a term proportional to the area of the surfaces separating different domains of uniform magnetization. TheC ^1, -regularity of these surfaces, for < 1/2, is also proved under suitable assumptions for the external magnetic field.     

Bounds on the Non-real Spectrum of a Singular Indefinite Sturm-Liouville Operator on ℝ

A simple explicit bound on the absolute values of the non-real eigenvalues of a singular indefinite Sturm-Liouville operator on the real line with the weight function sgn(·) and an integrable, continuous potential q is obtained. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the spectral properties of discrete Schrödinger operators

We use the method of the conjugate operator to prove a limiting absorption principle and the absence of the singular continuous spectrum for discrete Schrödinger operators. We also obtain local decay estimates. Our results apply to a large class of perturbating potentials V decaying arbitrarily slowly to zero at infinity.     

Asymptotic behavior of a Lotka–Volterra food chain stochastic model in the Chemostat

This article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Itô's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results.     

Asymptotic behavior on the Hénon equation with supercritical exponent

The main purpose of this paper is to analyze the asymptotic behavior of the radial solution of Hénon equation −Δu = |x| ^α u ^p−1, u > 0, x ∈ B _R (0) ⊂ ℝ ⁿ (n ⩾ 3), u = 0, x ∈ ∂B _R (0), where $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} from left side, α > 0.     

Are unbounded linear operators computable on the average for Gaussian measures?

Traub and Werschulz [Complexity and Information, Cambridge University Press, New York, 1999] ask whether every linear operator S:⊆X→Y$S:\subseteq X\to Y$ is “computable on the average” w.r.t. a Gaussian measure on X. The question is inspired by an analogous result in information-based complexity on the average-case solvability of linear approximation problems. We give several interpretations of Traub and Werschulz’ question within the framework of type-2 theory of effectivity. We have negative answers to all of these interpretations but the one with minimal requirements on the algorithm's uniformness. On our way to these results, we give an effective version of the Mourier–Prokhorov characterization of Gaussian measures on separable Hilbert spaces.     

Asymptotic behavior of eigenvalues of the two-particle discrete Schrödinger operator

We consider two-particle Schrödinger operator H(k) on a three-dimensional lattice ? ³ (here k is the total quasimomentum of a two-particle system, $k \in \mathbb{T}^3 : = \left( { - \pi ,\pi ]^3 } \right)$ . We show that for any $k \in S = \mathbb{T}^3 \backslash ( - \pi ,\pi )^3$ , there is a potential $\hat v$ such that the two-particle operator H(k) has infinitely many eigenvalues z_n(k) accumulating near the left boundary m(k) of the continuous spectrum. We describe classes of potentials W(j) and W(ij) and manifolds S(j) ? S, i, j ∈ {1, 2, 3}, such that if k ∈ S(3), (k ₂ , k ₃ ) ∈ (?π,π) ² , and $\hat v \in W(3)$ , then the operator H(k) has infinitely many eigenvalues z_n(k) with an asymptotic exponential form as n → ∞ and if k ∈ S(i) ∩ S(j) and $\hat v \in W(ij)$ , then the eigenvalues z_nm(k) of H(k) can be calculated exactly. In both cases, we present the explicit form of the eigenfunctions.     

Asymptotic behavior of integer programming and the stability of the Castelnuovo–Mumford regularity

Mathematical Programming - The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer...     

Asymptotic behavior of the loss rate for Markov-modulated fluid queue with a finite buffer

We consider a Markov-modulated fluid queue with a finite buffer. It is assumed that the fluid flow is modulated by a background Markov chain which may have different transitions when the buffer content is empty or full. In Sakuma and Miyazawa (Asymptotic Behavior of Loss Rate for Feedback Finite Fluid Queue with Downward Jumps. Advances in Queueing Theory and Network Applications, pp. 195–211, Springer, Cambridge, 2009), we have studied asymptotic loss rate for this type of fluid queue when the mean drift of the fluid flow is negative. However, the null drift case is not studied. Our major interest is in asymptotic loss rate of the fluid queue with a finite buffer including the null drift case. We consider the density of the stationary buffer content distribution and derive it in matrix exponential forms from an occupation measure. This result is not only useful to get the asymptotic loss rate especially for the null drift case, but also it is interesting in its own light.     

Asymptotic behavior of a generalized Cahn–Hilliard equation with a mass source

We consider in this article a generalized Cahn–Hilliard equation with mass source (nonlinear reaction term) which has applications in biology. We are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Dirichlet boundary conditions and then Neumann boundary conditions. The latter require additional assumptions on the mass source term to obtain the dissipativity. Indeed, otherwise, the order parameter u can blow up in finite time. We also give numerical simulations which confirm the theoretical results.