On ratio asymptotics for general polynomials

In this paper some characterizations of the ratio asymptotics for general polynomials are given. These results are extensions and improvements of the ratio asymptotics for orthogonal polynomials and are applicable to the ratio asymptotics for polynomials with disturbed nodes.     

On the number of bound states for Schrödinger operators with operator-valued potentials

Cwikel's bound is extended to an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for Schrödinger operators with operator-valued potentials. We recover Cwikel's bound for the Lieb-Thirring constantL _0,3 which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimensiond≥3) for the quotientL _0,d/L^cl _0,d is the so-called classical constant. This gives some improvement in large dimensions.     

Calculation of the asymptotics of the two-point correlation function for one-dimensional Bose gas

We consider a quantum field-theoretical model which describes spatially-nonhomogeneous, one-dimensional, repulsive Bose gas in an external harmonic potential. The two-point correlation function is calculated in the framework of functional integration. The corresponding functional integrals are estimated by means of stationary phase approximation. Asymptotic estimates are obtained in the limit as the temperature tends to zero while the volume occupied by the quasi-condensate increases. A power-law behavior is established for the correlation function in this limit. It is shown that the power-law behavior is governed by the critical exponent depending on spatial arguments. Bibliography 32 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 56–74.     

On the asymptotics of the Riesz-Bochner kernel

В работе рассматрива ется асимптотика в ме трике пространстваL _p (T ^N ),T ^N ={x∈R ^N , 0<x _i <2π} ядра Р исса-Бохнера $$\Theta ^s \left( {x, \lambda } \right) = \left( {2\pi } \right)^{ - N} \mathop \Sigma \limits_{\left| n \right|^2< \lambda } \left( {1 - \frac{{\left| n \right|^2 }}{\lambda }} \right)^s e^{inx} \left( {x \in T^N , s \geqq 0, \lambda \geqq 0} \right)$$ при λ→∞. Доказывается, что есл иN≧4,p≧2N/(N?1) иs>N((N?1)/2N?1/p), то для произвольной точкиx∈T ^N существует п остояннаяC=C _p (x, s) такая, что выполняется неравен ство $$\parallel \Theta ^s \left( {x - y, \lambda } \right) - \left( {2\pi } \right)^{ - {N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}} 2^s \Gamma \left( {s + 1} \right)\lambda ^{{N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}} J_{{N \mathord{\left/ {\vphantom {N {2 + s}}} \right. \kern-\nulldelimiterspace} {2 + s}}} {{\left( {\left| {x - y} \right|\sqrt \lambda } \right)} \mathord{\left/ {\vphantom {{\left( {\left| {x - y} \right|\sqrt \lambda } \right)} {\left( {\left| {x - y} \right|\sqrt \lambda } \right)^{{N \mathord{\left/ {\vphantom {N {2 + s}}} \right. \kern-\nulldelimiterspace} {2 + s}}} \parallel _{L_p \left( {T^N } \right)} \leqq }}} \right. \kern-\nulldelimiterspace} {\left( {\left| {x - y} \right|\sqrt \lambda } \right)^{{N \mathord{\left/ {\vphantom {N {2 + s}}} \right. \kern-\nulldelimiterspace} {2 + s}}} \parallel _{L_p \left( {T^N } \right)} \leqq }}$$ где нормаL _p (T ^N ) берется по пе ременнойy, а черезJ _v обозначена функция Б есселя первого рода порядкаv. СлучаиN=2 иN=3 рассматриваются отдельно.     

On asymptotics for Vaserstein coupling of Markov chains

We prove that strong ergodicity of a Markov process is linked with a spectral radius of a certain “associated” semigroup operator, although, not a “natural” one. We also give sufficient conditions for weak ergodicity and provide explicit estimates of the convergence rate. To establish these results we construct a modification of the Vaserstein coupling. Some applications including mixing properties are also discussed.     

On the asymptotics of supremum distribution for some iterated processes

In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes \(\{X(Y(t)) : t \in [0, \infty )\}\), where \(\{X(t) : t \in \mathbb {R} \}\) is a centered Gaussian process and \(\{Y(t): t \in [0, \infty )\}\) is an independent of {X(t)} stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of \(\mathbb {P}(\sup _{s\in [0,T]} X(Y (s)) > u)\) as \(u \to \infty \), where T>0, as well as \(\lim _{u\to \infty } \mathbb {P}(\sup _{s\in [0,h(u)]} X(Y (s)) > u)\), for some suitably chosen function h(u) are analyzed. As an illustration, we study the asymptotic behavior of the supremum distribution of iterated fractional Brownian motion process.     

On the asymptotics of some difference equations

We prove two inclusion theorems regarding a difference equation and apply them in getting the asymptotics of positive solutions of some concrete difference equations.     

On the asymptotics of the number of reduced integral quadratic forms with a condition of divisibility of the first coefficients

For the quadratic forms mentioned in the title, we use the discrete ergodic method to obtain asymptotic formulas with remainders depending on the L-Dirichlet function L(s, χ) and its behavior under some hypotheses.     

On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

In this paper we consider the Lane–Emden problem adapted for the p-Laplacian

On the asymptotics of one-sided large deviation probabilities

Institute Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 1, pp. 14–22, January–March, 1995.     

On the asymptotics of a Robin eigenvalue problem

The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to ?∞ as the perturbation goes to zero. We prove in this case that Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criterion to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter.     

Low-temperature asymptotics of correlators in a one-dimensional Bose gas

In completely integrable models the asymptotic form of correlators over great distances is universal in the case when the temperature is significantly less than the Fermi energy. The asymptotic forms of all the correlators are determined by a single critical exponent which depends only on the R matrix and the Fermi momentum.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 91–94, 1988.In conclusion, the author would like to express his thanks to N. M. Bogolyubov for useful discussions.     

On the asymptotics of some partial theta functions

Using the Euler–Maclaurin sum formula, we develop an asymptotic expansion for a fairly general sum of exponentials, which when specialized includes some common partial theta functions. Some conjectured asymptotic expansions for relevant integrals are given. We give a simple proof of a theorem by Bruce Berndt and Byungchan Kim generalizing a result found in Ramanujan’s second notebook.