The generating degree $\text{g}\deg \left( A \right)$ of a topological commutative ring $A$ with char $A\,=\,0$ is the cardinality of the smallest subset $M$ of $A$ for which the subring $\mathbb{Z}\left[ M \right]$ is dense in $A$ . For a prime number $p$ , ${{\mathbb{C}}_{p}}$ denotes the topological completion of an algebraic closure of the field ${{\mathbb{Q}}_{p}}$ of $p$ -adic numbers. We prove that $\text{g}\deg \left( {{\mathbb{C}}_{p}} \right)\,=\,1$ , i.e., there exists $t$ in ${{\mathbb{C}}_{p}}$ such that $\mathbb{Z}\left[ t \right]$ is dense in ${{\mathbb{C}}_{p}}$ . We also compute $\text{gdeg}\left( A\left( U \right) \right)$ where $A\left( U \right)$ is the ring of rigid analytic functions defined on a ball $U$ in ${{\mathbb{C}}_{p}}$ . If $U$ is a closed ball then $\text{gdeg}\left( A\left( U \right) \right)\,=\,2$ while if $U$ is an open ball then $\text{gdeg}\left( A\left( U \right) \right)$ is infinite. We show more generally that $\text{gdeg}\left( A\left( U \right) \right)$ is finite for any affinoid $U$ in ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$ and $\text{gdeg}\left( A\left( U \right) \right)$ is infinite for any wide open subset $U$ of ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$ .

This paper is concernedwith the structure of the arithmetic sum of a finite number of central Cantor sets. The technique used to study this consists of a duality between central Cantor sets and sets of subsums of certain infinite series. One consequence is that the sum of a finite number of central Cantor sets is one of the following: a finite union of closed intervals, homeomorphic to the Cantor ternary set or an $M$ -Cantorval.

In this note the multiplicities of binary recurrences over algebraic number fields are investigated under some natural assumptions.

Precise, elegant evaluations are given for Gauss sums of orders six and twelve.

We show that an $\text{RA}$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\text{RA}$ loop can be normal in its unit loop. Over fields, this can never happen.

The Hamiltonian potentials of the bending deformations of $n$ -gons in ${{\mathbb{E}}^{3}}$ studied in $\left[ \text{KM} \right]$ and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra ${{P}_{n}}$ of the pure braid group ${{P}_{n}}$ on the moduli space ${{M}_{r}}$ of $n$ -gon linkages with the side-lengths $r\,=\,\left( {{r}_{1}},\ldots ,{{r}_{n}} \right)$ in ${{\mathbb{E}}^{3}}$ . If $e\,\in \,{{M}_{r}}$ is a singular point we may linearize the vector fields in ${{P}_{n}}$ at $e$ . This linearization yields a flat connection $\nabla$ on the space $\mathbb{C}_{*}^{n}$ of $n$ distinct points on $\mathbb{C}$ . We show that the monodromy of $\nabla$ is the dual of a quotient of a specialized reduced Gassner representation.

The paper is dealing with determination of the integral ${{\int }_{\gamma }}\,f$ along the fractal arc $\gamma $ on the complex plane by terms of polynomial approximations of the function $f$ . We obtain inequalities for polynomials and conditions of integrability for functions from the Hölder, Besov and Slobodetskii spaces.

Motivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complexmanifold with an almost complex structure. We describe various properties of this structure.

Applying the Sullivan conjecture we construct compacta of certain cohomological and extensional dimensions.

Let $p$ be prime and let $\vartheta \,\in \,\mathbb{Z}_{p}^{*}$ be of multiplicative order $t$ modulo $p$ . We consider exponential sums of the form

$$S\left( a \right)\,=\,\sum\limits_{x=1}^{t}{\exp \left( 2\pi ia{{\vartheta }^{{{x}^{2}}}}\,/\,p \right)}$$

and prove that for any $\varepsilon \,>\,0$

$$\underset{\gcd (a,\,p)\,=\,1}{\mathop{\max }}\,\,\left| S\left( a \right) \right|\,=\,O\left( {{t}^{5/6+\varepsilon }}\,{{p}^{1/8}} \right)$$

A challenge by R. Padmanabhan to prove by group theory the commutativity of cancellative semigroups satisfying a particular law has led to the proof of more general semigroup laws being equivalent to quite simple ones.

An asymptotic formula is obtained for the number of cyclic quartic fields over $Q$ with discriminant $\le \,x$ .

The singular spectrum of $u$ in a convolution equation $\mu *u\,=\,f$ , where $\mu$ and $f$ are tempered ultra distributions of Beurling or Roumieau type is estimated by

$$SSu\,\subset \,\left( {{\mathbf{R}}^{n}}\,\times \,\text{Char}\,\mu\right)\,\cup \,SSf$$

The same is done for $S{{S}_{*}}u$ .

The purpose of this paper is to show the limitations of the conjectures of algebraic approximation. For this, we construct points of ${{\mathbf{C}}^{m}}$ which do not admit good algebraic approximations of bounded degree and height, when the bounds on the degree and the height are taken from specific sequences. The coordinates of these points are Liouville numbers.

Simple necessary conditions for weak type (1, 1) of invariant operators on $L\left( {{\mathbb{R}}^{d}} \right)$ and their applications to rational Fourier multiplier are given.

Around 1995, Professors Lupacciolu, Chirka and Stout showed that a closed subset of ${{\mathbb{C}}^{N}}\left( N\ge 2 \right)$ is removable for holomorphic functions, if its topological dimension is less than or equal to $N\,-\,2$ . Besides, they asked whether closed subsets of ${{\mathbb{C}}^{2}}$ homeomorphic to the real line (the simplest 1-dimensional sets) are removable for holomorphic functions. In this paper we propose a positive answer to that question.