Related papers: Two-layered numbers
The aim of this paper is to study certain properties of the Kullback-Leibler distance between two positive integer numbers or between two ideals. We present some results related the entropy of a positive integer number and the divergence of…
We briefly describe some well-known means and their properties, focusing on the relationship with integer sequences. In particular, the harmonic numbers, deriving from the harmonic mean, motivate the definition of a new kind of mean that we…
2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed curves where several sheets meet. They arise in the study of categorical invariants of 3-manifolds and may have applications to topological data…
Let $\alpha_1, \cdots, \alpha_d$ be real numbers, and let $S$ be the set of integers $s$ so that $||\alpha_i s||_{\mathbb{R}/\mathbb{Z}}>\delta$ for some $i$ and some fixed $\delta>0$. We prove $S$ is not \enquote{$2$-large}, i.e. there is…
A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the…
Let $f(n)=\min_{p} |n-p|$, where $p$ is a prime. We show that there is a positive constant $\delta$ such that for any large integer $N$ there exist two positive integers $n_1$ and $n_2$ such that $N=n_1 + n_2$ and $f(n_i)\gg \ln N (\ln\ln…
The partial Stirling numbers T_n(k) used here are defined as the sum over odd values of i of (n choose i) i^k. Their 2-exponents nu(T_n(k)) are important in algebraic topology. We provide many specific results, applying to all values of n,…
We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different…
A positive integer $n$ is practical if every $m \leq n$ can be written as a sum of distinct divisors of $n$. One can generalize the concept of practical numbers by applying an arithmetic function $f$ to each of the divisors of $n$ and…
The Dyson rank of an integer partition is the difference between its largest part and the number of parts it contains. Using Fine-Dyson symmetry, we give formulas for the number of partitions of n with rank larger than n/2, and we prove…
The Z_2^n gradings of the classical Lie algebras are described. To elucidate the grading, the classical Lie algebras are expressed in terms of matrix algebras over one of eight fields or Clifford algebras which carry gradings ranging from…
The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…
All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classification of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
This paper concerns the \textbf{abstract geometry of numbers}: namely the pursuit of certain aspects of geometry of numbers over a suitable class of normed domains. (The standard geometry of numbers is then viewed as geometry of numbers…
Integer partitions express the different ways that a positive integer may be written as a sum of positive integers. Here we explore the analytic properties of a new polynomial $f_\lambda(x)$ that we call the partition polynomial for the…
A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3 \pmod{4}$, have even exponent in the prime…
In a recent survey paper we introduced one-sided multipliers between two different operator spaces. Here we give some basic theory for these maps.
Using factorization properties, we give several characterizations for an algebraic number ring to have class number 2.
Everybody has certainly heard about palindromes: words that stay the same when read backwards. For instance kayak, radar, or rotor. Mathematicians are interested in palindromic numbers: positive integers whose expansion in a certain integer…