Related papers: Functional identities of one variable

Let $f:\mathbb{Q}\to \mathbb{Q}$ be a function definable in an o-minimal expansion of $(\mathbb{Q},<,+,0)$. We show that $f$ is eventually linear. In addition, we show that this holds in every elementary equivalent structure.

We study families of self-adjoint operators with given spectra whose sum is a scalar operator. Such families are $*$-representations of certain algebras which can be described in terms of graphs and positive functions on them. The main…

This article describes local normal forms of functions in noncommuting variables, up to equivalence generated by isomorphism of noncommutative Jacobi algebras, extending singularity theory in the style of Arnold's commutative local normal…

In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $\rho$, independent…

The aim of this paper is to present a general algebraic identity. Applying this identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic numbers. We also recover some known identities including an…

The article continues the work on the description of integrable nonlinear chains with three independent variables of the following form $u^j_{n+1,x}=u^j_{n,x}+f(u^{j+1}_{n}, u^{j}_n,u^j_{n+1 },u^{j-1}_{n+1})$ by the presence of a hierarchy…

By means of principal isotopes lH(a,b) of the algebra lH [Ra 99] we give an exhaustive and not repetitive description of all 4-dimensional absolute-valued algebras satisfying (x^p, x^q, x^r) = 0 for fixed integers p, q, r \in\{1,2\}. For…

We prove that if $A \subset {\Bbb F}_q$ is such that $$|A|>q^{{1/2}+\frac{1}{2d}},$$ then $${\Bbb F}_q^{*} \subset dA^2=A^2+...+A^2 d \text{times},$$ where $$A^2=\{a \cdot a': a,a' \in A\},$$ and where ${\Bbb F}_q^{*}$ denotes the…

Let $M$ be a random matrix chosen according to Haar measure from the unitary group $\mathrm{U}(n,\mathbb{C})$. Diaconis and Shahshahani proved that the traces of $M,M^2,\ldots,M^k$ converge in distribution to independent normal variables as…

Given a base $b$, a "digit map" is a map $f: \mathbb{Z}^{\ge 0} \to \mathbb{Z}^{\ge 0}$ of the form $f(\sum_{i=0}^n a_ib^i) = \sum_{i=0}^n f_*(a_i)$, $0 \le a_i \le b-1$ for each $i$, where $f_* : \{0,1,\dots, b-1\} \to \mathbb{Z}^{\ge 0}$…

Let $\mathbb{K}$ be an algebraically closed field, and $A \subset \mathbb{K}[x_{1}, \ldots, x_n]$ be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of $\mathbb{K}$-algebras \[…

In this note, we will prove that a finite dimensional Lie algebra $L$ of characteristic zero, admitting an abelian algebra of derivations $D\leq Der(L)$ with the property $$ L^n\subseteq \sum_{d\in D}d(L) $$ for some $n\geq 1$, is…

We study identities of finite dimensional algebras over a field of characteristic zero, graded by an arbitrary groupoid $\Gamma$. First we prove that its graded colength has a polynomially bounded growth. For any graded simple algebra $A$…

In this paper, we prove the convexity of trace functionals $$(A,B,C)\mapsto \text{Tr}|B^{p}AC^{q}|^{s},$$ for parameters $(p,q,s)$ that are best possible, where $B$ and $C$ are any $n$-by-$n$ positive definite matrices, and $A$ is any…

We investigate the algebraic genericity of various families of continuous functions exhibiting extreme irregularity, focusing on fractal dimensions, H\"older regularity, and fractional differentiability. Our first main result shows that for…

UNIFORM algebras have been extensively investigated because of their importance in the theory of uniform approximation and as examples of complex Banach algebras. An interesting question is whether analogous algebras exist when a complete…

Let $q$ be an integer. A $D(q)$-$m$-tuple is a set of $m$ distinct positive integers ${a_1, a_2, . . . , a_m}$ such that $a_ia_j + q$ is a perfect square for all $1 \leq i < j \leq m$. By counting integer solutions $x \in [1, b]$ of…

Let k be an algebraically closed field of characteristic zero. An element F from k(x_1,...,x_n) is called a closed rational function if the subfield k(F) is algebraically closed in the field k(x_1,...,x_n). We prove that a rational function…

Here, we establish a polynomial identity in three variables $a, b, c$, and with the degree of the polynomial given in terms of two integers $L, M$. By letting $L$ and $M$ tend to infinity, we get the 1993 Alladi-Gordon $q$-hypergeometric…

Let $q\ge 2$ and $N\ge 1$ be integers. W. Zhang (2008) has shown that for any fixed $\epsilon> 0$, and $q^{\epsilon} \le N \le q^{1/2 -\epsilon}$, $$ \sum_{\chi \ne \chi_0} |\sum_{n=1}^N \chi(n)|^2 |L(1, \chi)|^2 = (1 + o(1)) \alpha_q q N…