On Gauss-Kuz’min statistics for finite continued fractions

The article is devoted to finite continued fractions for numbers a/b when integer points (a, b) are taken from a dilative region. Properties similar to the Gauss-Kuz’min statistics are proved for these continued fractions. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 195–208, 2005.     

L-Functions of Jacobi forms with Shimura type

For every Jacobi form of Shimura type over H × ℂ, a system of L-functions associated to it is given. These L-functions can be analytically continued to the whole complex plane and satisfy a kind of functional equation. As a consequence, Hecke’s inverse theorem on modular forms is extended to the context of Jacobi forms with Shimura type.     

Propriétés diophantiennes du développement en cotangente continue de Lehmer

Diophantine Properties of Lehmer’s Continued Cotangent Developments. This article deals with an algorithm devised by Lehmer which enables us to write any real positive number as the sum of an alternating series of cotangents of integers n _ν, ν ≥ 0, in a unique way. We continue the work begun by Lehmer and continued by Shallit: amongst other things, we give explicitly the link between the rational approximations of a given real number coming from this algorithm and the usual convergents of the same real number and we produce a quasi-optimal bound for the growth of the sequence (n _ν)_{ν ≥ 0} associated to an algebraic number. We also determine the regular continued fractions of an exceptional class of continued cotangent developments, which enables us to produce optimal irrationality measures of these expansions.     

Geometric proof of R?dseth’s formula for Frobenius numbers

Using a geometric interpretation of continued fractions, we give a new proof of R?dseth’s formula for Frobenius numbers.     

Continued fraction of e2 with confluent hypergeometric functions

The tanh-type, tan-type, and e-type Hurwitz continued fractions have been generalized by the author. In this paper, we study a generalized form of e²-type Hurwitz continued fractions by using confluent hypergeometric functions. We also obtain a similar type of Tasoev continued fractions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 513–531, October–December, 2006.     

On the convergence of continued fractions at Runckel’s points

We consider the limit periodic continued fractions of Stieltjes type

On a random variable associated with excursions in an M/M/∞ system

We show in this paper how the theory of continued fractions can be used to invert the Laplace transform of a transient characteristic associated with excursions in an M/M/∞ system with unit service rate and input intensity u. The characteristic under consideration is the area V swept under the occupation process of an M/M/∞ queue during an excursion period above a given threshold C. The Laplace transform V ^⋆ of this random variable has been established in earlier studies and can be expressed as a ratio of Tricomi functions. In this paper, we first establish the continued fraction representation of V ^⋆, which allows us to obtain an alternative expression of the Laplace transform in terms of Kummer functions. It then turns out that the continued fraction considered is the even part of a Stieltjes (S) fraction, which provides information on the location of the poles of V ^⋆. It appears that the Laplace transform has simple poles on the real negative axis. Taking benefit of the fact that the spectrum is compact and that the numerical values of the Laplace transform can easily be computed by means of the continued fraction, we finally use a classical Laplace transform inversion technique to numerically compute the survivor probability distribution function x➙ℙ{V > fx} of the random variable V, which exhibits an exponential decay only for very large values of the argument x when the ratio u/C is sufficiently smaller than one. This revised version was published online in June 2006 with corrections to the Cover Date.     

Problem of interpolation of functions by two-dimensional continued fractions

We investigate the problem of interpolation of functions of two real variables by two-dimensional continued fractions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 842–851, June, 2006.     

Quasi-Elliptic Integrals and Periodic Continued Fractions

 In this report we detail the following story. Several centuries ago, Abel noticed that the well-known elementary integral

The extension of the first main theorem for π-blocks

Let π be a set of primes and G a π-separable group. Isaacs defines the B _π characters, which can be viewed as the “π-modular” characters in G, such that the B _p′ characters form a set of canonical lifts for the p-modular characters. By using Isaacs’ work, Slattery has developed some Brauer’s ideals of p-blocks to the π-blocks of a finite π-separable group, generalizing Brauer’s three main theorems to the π-blocks. In this paper, depending on Isaacs’ and Slattery’s work, we will extend the first main theorem for π-blocks.     

Some twin regions of convergence for branched continued fractions

We study twin regions of convergence for branched continued fractions and establish an estimate of the rate of convergence; we construct a counterexample showing that the natural formulation of Thron's convergence criterion for continued fractions does not extend to branched continued fractions. Translated fromMatematichni Metodi ta Fiziko-Makhanichni Polya, Vol. 39, No. 2, 1996, pp. 62–64.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

On relations for the multiple <Emphasis Type="Italic">q</Emphasis>-zeta values

We prove a new relation for the multiple q-zeta values (MqZV’s). It is a q-analogue of the Ohno-Zagier relation for the multiple zeta values (MZV’s). We discuss the problem of determining the dimension of the space spanned by MqZV’s over ℚ, and present an application to MZV. The first author is supported by Grant-in-Aid for Young Scientists (B) No. 17740026 and the second author is supported by Grant-in-Aid for Young Scientists (B) No. 17740089.     

An analog of the Vorpits'kii convergence criterion for branched continued fractions of special form

We study branched continued fractions of a special form with inequivalent variables. We establish a multidimensional analog of the Vorpits'kii convergence criterion for continued fractions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 35–38.     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

Applications of Klee’s Dehn–Sommerville Relations

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges. I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748. E. Swartz research partially supported by NSF grant DMS-0600502.     

Vincent’s theorem of 1836: overview and future research

In this paper, we present two different versions of Vincent’s theorem of 1836 and discuss various real root isolation methods derived from them: one using continued fractions and two using bisections, the former being the fastest real root isolation method. Regarding the continued fractions method, we first show how, using a recently developed quadratic complexity bound on the values of the positive roots of polynomials, its performance has been improved by an average of 40% over its initial implementation, and then we indicate directions for future research. Bibliography: 45 titles.     

The Borel—Bernstein Theorem for multidimensional continued fractions

A central result in the metric theory of continued fractions, the Borel—Bernstein Theorem gives statistical information on the rate of increase of the partial quotients. We introduce a geometrical interpretation of the continued fraction algorithm; then, using this set-up, we generalize it to higher dimensions. In this manner, we can define known multidimensional algorithms such as Jacobi—Perron, Poincaré, Brun, Rauzy induction process for interval exchange transformations, etc. For the standard continued fractions, partial quotients become return times in the geometrical approach. The same definition holds for the multidimensional case. We prove that the Borel—Bernstein Theorem holds for recurrent multidimensional continued fraction algorithms. Supported by a grant from the CNP _q -Brazil, 301456/80, and FINEP/CNP _q /MCT 41.96.0923.00 (PRONEX).     

S-convexity revisited (fuzzy): long version

In this revisional article, we criticize (strongly) the use made by Medar et al., and those whose work they base themselves on, of the name ‘convexity’ in definitions which intend to relate to convex functions, or cones, or sets, but actually seem to be incompatible with the most basic consequences of having the name ‘convexity’ associated to them. We then believe to have fixed the ‘denominations’ associated with Medar’s (et al.) work, up to a point of having it all matching the existing literature in the field [which precedes their work (by long)]. We also expand his work scope by introducing s ₁-convexity concepts to his group of definitions, which encompasses only convex and its proper extension, s ₂-convex, so far. This article is a long version of our previous review of Medar’s work, published by FJMS (Pinheiro, M.R.: S-convexity revisited. FJMS, 26/3, 2007).     

On L 1 Bounds for Asymptotic Normality of Some Weakly Dependent Random Variables

In the present paper, we consider L ₁ bounds for asymptotic normality for the sequence of r.v.’s X ₁,X ₂,… (not necessarily stationary) satisfying the ψ-mixing condition. The L ₁ bounds have been obtained in terms of Lyapunov fractions which, in a particular case, under finiteness of the third moments of summands and the finiteness of ∑ _r≥1 r ² ψ(r), are of order O(n ^−1/2), where the function ψ participates in the definition of the ψ-mixing condition.