Related papers: On polynomial functions on non-conmmutative groups
We denote by ${\rm Hess}^+$ the set of all points $p\in\mathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is positive definite. In this paper we provide a class of…
We obtain a class of examples of non-rational adjoint classical groups of type $^2A_n$ and a group of type $^2D_3$ over the function field $F$ of a smooth geometrically integral curve over a $p$-adic field with $p \neq 2$. We also show that…
In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear…
For any $n\geq 1$, let $T_n$ be the complete binary rooted tree of height $n$, and $f(x)=(x+a)^2-a-1$ such that $a\neq \pm b^2$ for any $b\in \mathbb{Z}$. In \cite{Settled}, Jones and Boston empirically observed that iteratively applying a…
We have established various criteria for the topological transitivity of families of continuous (holomorphic) functions. Furthermore, by leveraging the properties of expanding families of meromorphic functions, we offer an alternative proof…
We outline a general procedure that builds classifying spaces for generalized Thompson groups $\Gamma$. The construction depends on a small number of choices: (1) an inverse semigroup $S$ of partial transformations that ``locally determine"…
We show that any scalar differential operator with a family of polyno- mials as its common eigenfunctions leads canonically to a matrix differen- tial operator with the same property. The construction of the correspond- ing family of matrix…
Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…
Given a field $F$, an integer $n\geq 1$, and a matrix $A\in M_n(F)$, are there polynomials $f,g\in F[X]$, with $f$ monic of degree $n$, such that $A$ is similar to $g(C_f)$, where $C_f$ is the companion matrix of $f$? For infinite fields…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal…
We study the umbral "classical" orthogonal polynomials with respect to a generalized derivative operator $\cal D$ which acts on monomials as ${\cal D} x^n = \mu_n x^{n-1}$ with some coefficients $\mu_n$. Let $P_n(x)$ be a set of orthogonal…
Let $ G $ be a cyclic group, in this paper, we study the Herbrand quotient and $ 1-$th cohomology group on finitely generated $ G-$modules in some cases. When $ G $ is of order $ 2, $ the order of the cohomology group is explicitly related…
Let L be a bounded distributive lattice. We give several characterizations of those L^n --> L mappings that are polynomial functions, i.e., functions which can be obtained from projections and constant functions using binary joins and…
The article is devoted to investigation of classes of functions monotone as functions on general $C^*$-algebras that are not necessarily the $C^*$-algebras of all bounded linear operators on a Hilbert space as it is in classical case of…
Calogero-Sutherland models associated to the Weyl groups of type A and B with exchange terms included in the Hamiltonians systems have non-symmetric eigenfunctions, which are products of the ground state with members of a family of…
We consider four classes of polynomials over the fields $\mathbb{F}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+Ax^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+Ax^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+Ax^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+Ax^{q}-Bx$,…
We provide a uniform bound for the index of cohomology classes in $H^i(F, \mu_\ell^{\otimes i-1})$ when $F$ is a semiglobal field (i.e., a one-variable function field over a complete discretely valued field $K$). The bound is given in terms…
We give an answer to the abstract Capelli problem: Let $(G, V)$ be a multiplicity-free finite-dimensional representation of a connected reductive complex Lie group $G$ and $G'$ be its derived subgroup. Assume that the categorical quotient…
Many-to-one mappings and permutation polynomials over finite fields have important applications in cryptography and coding theory. In this paper, we study the many-to-one property of a large class of polynomials such as $f(x) = h(a x^q + b…