Reidemeister and Nielsen zeta-functions

In this paper we formulate a rationality theorem for the Reidemeister and Nielsen zeta-functions modulo a normal subgroup of the fundamental group. We give conditions under which these zeta-functions coincide. We formulate a conjecture aboutentropy for the Reidemeister numbers. We show that the radius of convergence of the Nielsen zeta-function for an orientation-preserving homeomorphism f of a compact surface is an invariant of a three-dimensional manifold, the torus of the map f, and a special flow on it. In special cases we derive a functional equation for the Nielsen zeta-function. We give an example of a transcendental Nielsen zeta function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 164–168, 1988.In conclusion the author expresses thanks to V. B. Piloginaya, V. G. Turaev, Boju Jiang, N. V. Ivanov for stimulating discussions and to D. Fried for sending preprints.     

Bounds for triple zeta-functions

In the present paper we consider the problem of the order of magnitude for the triple zeta-functions of Euler-Zagier type in the region 0≤ ℜsj < 1 (j = 1,2,3). We apply the Euler-Maclaurin summation formula and van der Corput's method of multiple exponential sums to the triple zeta-functions.     

Asymptotic properties of isometries

In this paper, we prove two theorems on the local stability of isometries in connection with (ε,p)-isometries. These theorems reveal that a large class of (ε,p)-isometries, defined on various restricted domains, are stable.     

Multifractal spectra and multifractal zeta-functions

We introduce multifractal zetafunctions providing precise information of a very general class of multifractal spectra, including, for example, the multifra     

On the secondary zeta-functions

Using summability it is shown that $\text{∑}{}_{\mathrm{n}}\text{?2 (Λ(n) ? 1) n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(log n)}{}^{\mathrm{?8}}$ defines an entire function in the s-plane. Its asymptotic nature is found and a functional equation relating it to the series $\text{∑{i(}\text{1}\text{2}\text{? p)}}{}^{\mathrm{1?8}}$, Im p = γ > 0,is obtained where p = β + iγ are the nontrivial zeros of Riemann's zeta-function.     

Deformations of polynomials and their zeta-functions

For an analytic in σ ∈ (ℂ 0) family P _σ of polynomials in n variables a monodromy transformation h of the zero level set V _σ ={P _σ =0} for sufficiently small σ ≠ 0 is defined. The zeta-function of this monodromy transformation is written as an integral with respect to the Euler characteristic of the corresponding local data. This leads to a study of deformations of holomorphic germs and their zeta-functions. We give some examples of computations with the use of this technique. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Joint discrete universality for periodic zeta-functions

Abstract

In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint universality theorems on the approximation of a collection of analytic functions by discrete shifts of the above zeta-functions are obtained. For this, certain linear independence hypotheses are applied.     

Asymptotic properties in dynamic programming

In the framework of dynamic programming we provide two results:

Asymptotic properties of near Pfeifer records

Asymptotic properties of the number of near records are known in the literature. We generalize these results to the Pfeifer model which has a wider application. In particular we establish convergence in probability, in the almost sure sense and in distribution for the number of near records under the Pfeifer model.     

Asymptotic properties of locally extensible designs

An (n+1, 1)-design D is locally extensible at a block B if D can be embedded in an (n+1, 1)-design having a block B ^* of cardinality n+1 and such that BB ^*. If D is embeddable in a finite projective plane of order n, then D is called globally extensible. In this paper, we investigate the asymptotic behaviour of locally extensible designs and Euclidean designs. We study the relationship between locally extensible and extensible designs and the uniqueness of such embeddings. It is shown that, for n, l and t sufficiently large, any (n+1, 1)-design which has minimum block length l and which is locally extensible at t of its blocks is globally extensible.     

Asymptotic properties of matrix differential operators

For given matrices A(s) and B(s) whose entries are polynomials in s, the validity of the following implication is investigated: ?_ylim_{t → ∞}A(D) y(t) = 0 ? lim_{t → ∞}B(D) y(t) = 0. Here D denotes the differentiation operator and y stands for a sufficiently smooth vector valued function. Necessary and sufficient conditions on A(s) and B(s) for this implication to be true are given. A similar result is obtained in connection with an implication of the form ?_yA(D) y(t) = 0, lim_{t → ∞}B(D) y(t) = 0, C(D) y(t) is bounded ? lim_{t → ∞}E(D) y(t) = 0.     

On joint universality of periodic Hurwitz zeta-functions

We prove a joint universality theorem for a collection of periodic Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers.     

Asymptotic properties of von Neumann algebras

One gives a survey of the results regarding asymptotic commutativity in von Neumann algebras.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 27, pp. 129–166, 1985.