Related papers: Solvability of equations in elementary functions

$K$ is an algebraically closed field with characteristic $0$ and $f$ is a polynomial or a holomorphic function. We study all solutions of the equation $XA-AX=f(X)$, in the unknown $X\in M_n(K)$, when $A\in M_n(K)$ is diagonalizable.

We prove that an element $g$ of prime order $>3$ belongs to the solvable radical $R(G)$ of a finite (or, more generally, a linear) group if and only if for every $x\in G$ the subgroup generated by $g, xgx^{-1}$ is solvable. This theorem…

Let f be an analytic function defined on a complex domain Omega and A be a (n,n) complex matrix. We assume that there exists a unique alpha satisfying f(alpha)=0. When f'(alpha)=0 and A is non derogatory, we solve completely the equation…

In this paper we deal with a non-linear Diophantine equation which arises from the determinant computation of an integer matrix. We show how to find a solution, when it exists. We define an equivalence relation and show how the set of all…

We show that the entailment problem, for a given entailment problem for DL-Lite$_{core}$ ontology, and given conjunctive query with inequalities, is undecidable. We also show that this problem remains undecidable if conjunctive queries with…

Let $[\, \cdot\,]$ be the floor function. In the present paper we prove that when $1<c<\frac{12}{11}$ and $\theta>1$ is a fixed, then there exist infinitely many prime numbers of the form $[n^c \tan^\theta(\log n)]$.

We prove in ZFC the existence of a definable, countably saturated elementary extension of the reals. It seems that it has been taken for granted that there is no distinguished, definable nonstandard model of the reals. (This means a…

We will show certain functional inequalities between some products of $x^p - 1$.

Let $\xi$ be a value, at an algebraic point, of a Siegel $E$-function. As a special case of a very general interpolation result, we prove that there exists an $E$-function $f$ such that $f(1)=\xi$, and such that 1 is not a singularity of…

In the previous work [2] (i.e., arXiv:2105.03385), we considered continuous solutions of an iterative equation involving the multiplication of iterates. In this paper, we continue to investigate this equation for differentiable solutions.…

We prove the existence of fractal solutions to a class of linear ordinary differential equations.This reveals the possibility of chaos in the very short time limit of the evolution even of a linear one dimensional dynamical system.

We suggest a closed solution for the roots of polynomial trinomial algebraic equation $$z^n+xz^{n-1}-1=0$$ with an appropriate $x$. This solution is a minor modification to the work of Mikhalkin (Mikhalkin E N, 2006. On solving general…

We show that there is always a uniformly antisymmetric f:A-> {0,1} if A subset R is countable. We prove that the continuum hypothesis is equivalent to the statement that there is an f:R-> omega with |S_x| <= 1 for every x in R. If the…

The arithmetic of ternary diagonal equation is considered for degree d >1, with the outcome that the set of coefficients for which the equation admits a non-zero integer solution is shown to have density zero.

We prove the $C^{\alpha}$ regularity for weak solutions to a class of ultraparabolic equation, with measurable coefficients. The results generalized our recent $C^{\alpha}$ regularity results of Prandtl's system to high dimensional cases.

The purpose of the present paper is to solve (under some assumption on the domain) the equation $$ g(x+y)-g(x)-g(y)=xf(y)+yf(x). $$ After determining the general solutions, we will investigate the so--called alien solutions. %More…

We present an elementary Functional Analytic proof of the roughness of Exponential Dichotomy of Ordinary Differential Equations (with exponential growth) on an arbitrary Banach Space.

We study the solvability of a quadratic integral equation of fractional order with linear modification of the argument. This equation is considered in the Banach space of real functions defined, bounded and continuous on an unbounded…

In this paper we present another proof of the analytic version of the Hahn-Banach theorem in terms of convex functionals.

We give a new elementary proof of existence and uniqueness of a solution to the Sylvester equation $AX-XB=Y$