A generalization of Kakutani’s splitting procedure

In this paper, we will study uniformly distributed sequences of partitions of [0, 1], a concept which has been introduced by Kakutani (Measure Theory 541:369–375, 1976). We will construct new families of u.d. sequences of partitions and study relations with the classical concept of uniformly distributed sequences of points.     

Uniform convergence of Green’s functions

Given a sequence of regular planar domains converging in the sense of kernel, we prove that the corresponding Green’s functions converge uniformly on the complex sphere, provided the limit domain is also regular, and the connectivity is uniformly bounded.     

Composition analogues of Beck’s conjectures on partitions

In this work, we first present new bijections for the composition analogues of the partition theorems of Euler, Glaisher, and Franklin respectively. Then we generalize one composition analogue of Beck’s conjectures found by Huang recently, and give a purely combinatorial proof applying our bijection. Finally, a new and general composition analogue of Beck’s conjectures is established.     

Benford’s law and distribution functions of sequences in (0, 1)

Applying the theory of distribution functions of sequences x _n ∈ [0, 1], n = 1, 2, ..., we find a functional equation for distribution functions of a sequence x _n and show that the satisfaction of this functional equation for a sequence x _n is equivalent to the fact that the sequence x _n to satisfies the strong Benford law. Examples of distribution functions of sequences satisfying the functional equation are given with an application to the strong Benford law in different bases. Several direct consequences from uniform distribution theory are shown for the strong Benford law.     

On a Uniform Treatment of Darboux’s Method

Let F(z) be an analytic function in |z| < 1. If F(z) has only a finite number of algebraic singularities on the unit circle |z| = 1, then Darbouxs method can be used to give an asymptotic expansion for the coefficient of zⁿ in the Maclaurin expansion of F(z). However, the validity of this expansion ceases to hold, when the singularities are allowed to approach each other. A special case of this confluence was studied by Fields in 1968. His results have been considered by others to be too complicated, and desires have been expressed to investigate whether any simplification is feasible. In this paper, we shall show that simplification is indeed possible. In the case of two coalescing algebraic singularities, our expansion involves only two Bessel functions of the first kind.     

Pseudo-randomness of van der Corput’s sequences

The aim of this paper is to find main terms of the star D _N ^* and extremal D _N discrepancies of the two dimensional sequence (x _n , x _n+1), n = 0, 1, 2, ..., N − 1, where x _n , n = 0, 1, 2, ..., is the van der Corput sequence. This give a quantitative form of a well-known result that van der Corput sequence is not pseudorandom. This research was supported by the Slovak Academy of Sciences Vega Grant No. 2/7138/27.     

Pillai’s generalized function

The distribution of values of Pillai’s function over the ring of Gauss integers $ \mathbb{Z}\left[ i \right] $ is studied. The asymptotic formulas for the summators $ {\varSigma_{{N\left( \alpha \right)\leq x}}}\frac{{g\left( \alpha \right)}}{{{N^{\alpha }}\left( \alpha \right)}} $ , where $ a\in \mathbb{R} $ , are found.     

The analyticity of a generalized Ruelle’s operator

In this work we propose a generalization of the concept of Ruelle’s operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle’s operator. Our operator generalizes both the Ruelle operator proposed in [2] and the Perron Frobenius operator defined in [7]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle’s operator and present some examples.     

The ring of Colombeau’s full generalized quaternions

In this paper, we introduce the ring of Colombeau full generalized quaternion, and we study its algebraic and topological properties. We prove that the ring of Colombeau full generalized quaternion is Gelfand, normal, duo and some related properties. Moreover, we study the essential ideals of this ring.     

Regev’s and Amitsur’s conjectures for codimensions of generalized polynomial identities

Let A be a finite-dimensional associative algebra over a field of characteristic 0. Then there exist C ∈ ℚ₊ and t ∈ ℤ₊ such that gc _n (A) ∼ Cn ^t d ⁿ as n → ∞, where d = PIexp(A). In particular, Amitsur’s and Regev’s conjectures hold for the codimensions gc _n (A) of generalized polynomial identities.     

The stability of set of generalized Ky Fan’s points

This paper is concerned with a generalized Ky Fan’s inequality. We first give an existence result of generalized Ky Fan’s (weak) efficient points, and then establish a complete metric space. Based on these results, we obtain the sufficient and necessary conditions of upper semicontinuity of efficient solution mapping to a generalized Ky Fan’s inequality. We also obtain the sufficient conditions of lower semicontinuity and continuity of efficient solution mapping to a generalized Ky Fan’s inequality. Our results are new and different from the corresponding ones in the literature.     

On generalized Rubel’s equation

We solve generalized the generalized Rubel equation on the space of analytic functions in domains.     

λ-Equidistributed sequences of partitions and a theorem of the De Bruijn-Post type

Summary The notion of uniform distribution of a sequence is generalized to sequences of partitions in a separable metric space X. Results concern Riemann integrability with respect to a probability on X, and Riemann approximations of Lebesgue integrals.Lavoro presentato al terzo Convegno nazionale «Analisi reale e teoria della misura» (Capri, 12–16 settembre 1988).     

A survey of Skolem-type sequences and Rosa’s use of them

Let D be a set of positive integers. A Skolem-type sequence is a sequence of i ∈ D such that every i ∈ D appears exactly twice in the sequence at positions a _i and b _i , and |b _i − a _i | = i. These sequences might contain empty positions, which are filled with null elements. Thoralf A. Skolem defined and studied Skolem sequences in order to generate solutions to Heffter’s difference problems. Later, Skolem sequences were generalized in many ways to suit constructions of different combinatorial designs. Alexander Rosa made the use of these generalizations into a fine art. Here we give a survey of Skolem-type sequences and their applications. Supported by an NSERC Graduate fellowship. This work is in partial fulfillment of an M.Sc.     

Presentations for a class of generalized Thompson’s groups

For a given group G and a monomorphism φ:G→G×G there is a group ?_φ(G), introduced by the author, which blends Thompson’s group F with G. Given a presentation of G we determine a presentation of ?_φ(G). In particular, we prove that ?_φ(G) is finitely generated (resp. finitely presented) if G is finitely generated (resp. finitely presented).     

Uniform Zariski’s theorem on fundamental groups

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundamental groups of the complements to the hypersurface in the plane and in the space. If a family of hypersurfaces depends algebraically on parameters then it is not true in general that there exists a plane such that the natural embedding generates an isomorphism of the fundamental groups of the complements to each hypersurface from this family in the plane and in the space. But we show that in the affine case such a plane exists after a polynomial coordinate substitution. The research was partially supported by an NSA grant.