Related papers: Correction to a theorem of Schoenberg

The Baer theorem states that for a group $G$ finiteness of $G/Z_i(G)$ implies finiteness of $\gamma_{i+1}(G)$. In this paper we show that if $G/Z(G)$ is finitely generated then the converse is true.

An $f$-subgroup is a linear recurring sequence subgroup, a multiplicative subgroup of a field whose elements can be generated (without repetition) by a linear recurrence relation, with characteristic polynomial $f$. It is called…

A classical theorem of S. Bochner states that a function $f:R^n \to C$ is the Fourier transform of a finite Borel measure if and only if $f$ is positive definite. In 1938, I. Schoenberg found a beautiful complement to Bochner's theorem. We…

Let $F$ be a field and let $E$ be an \'etale algebra over $F$, that is, a finite product of finite separable field extensions $E = F_1 \times \dots \times F_r$. The classical primitive element theorem asserts that if $r = 1$, then $E$ is…

As a serious attempt for constructing a new foundation for describing micro-entities from a causal standpoint, it was explained before in [1, 2, 3] that by unifying the concepts of information, matter and energy, each micro-entity is…

We establish a central limit theorem for counting large continued fraction digits $(a_n)$, i.e. we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp which…

It is well known that in Zermelo-Fraenkel (ZF) set theory any finite set is decidable. In this paper we discuss an extension of ZF where this result is no longer valid. Such an extension is quasi-set theory and it has its origin on problems…

A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, where $F_1 = 1$, $F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$. For general recurrences…

A finite subset of the natural numbers is weak-Schreier if $\min S \ge |S|$, strong-Schreier if $\min S>|S|$, and maximal if $\min S = |S|$. Let $M_n$ be the number of weak-Schreier sets with $n$ being the largest element and $(F_n)_{n\geq…

Let $f(z)=e^{-bz^2}f_1(z)$ where $b \geq 0$ and $f_1(z)$ is a real entire function of genus 0 or 1. We give a necessary and sufficient condition in terms of a sequence of inequalities for all of the zeros of $f(z)$ to be real. These…

We study the classes of filters F on N such that the weak and strong F-convergence of sequences in l1 coincide. We study also an analogue of l1 weak sequential completeness theorem for filter convergence.

Let $r_1,\ldots,r_s:\mathbb{Z}_{n\geqslant 0}\to\mathbb{C}$ be linearly recurrent sequences whose associated eigenvalues have arguments in $\pi\mathbb{Q}$ and let $F(z):=\sum_{n\geqslant 0}f(n)z^n$, where $f(n)\in\{r_1(n),\ldots,$…

This paper is a contribution to the study of extensions of arbitrary models of ZF (Zermelo-Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. We present some new constructions of certain types…

A nonempty set $F$ is Schreier if $\min F\ge |F|$. Bird observed that counting Schreier sets in a certain way produces the Fibonacci sequence. Since then, various connections between variants of Schreier sets and well-known sequences have…

We prove that Menger's theorem is valid for infinite graphs, in the following strong form: let $A$ and $B$ be two sets of vertices in a possibly infinite digraph. Then there exist a set $\cp$ of disjoint $A$-$B$ paths, and a set $S$ of…

Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langle f_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that the limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2, \dots$…

We give a new proof of Tietze Theorem on the convergence of infinite semi-regular continued fractions.

For $p, q\in \mathbb{N}$, a finite nonempty set $F$ is said to be $(p,q)$-Schreier (or maximal $(p,q)$-Schreier, respectively) if $q\min F\ge p|F|$ (or $q\min F = p|F|$, respectively). For $n\in \mathbb{N}$, let $$\mathcal{S}^{p/q}_{n}\ :=\…

In the geometry of polynomials, Schoenberg's conjecture, now a theorem, is a quadratic inequality between the zeros and critical points of a polynomial whose zeros have their centroid at the origin. We call its generalizations to other…

Let f be a smooth diffeomorphism of the half-line fixing only the origin and Z^r its centralizer in the group of C^r diffeomorphisms. According to well-known results of Szekeres and Kopell, Z^1 is a one-parameter group. On the other hand,…