Related papers: Solvability of equations in elementary functions
We provide an analytical closed-form solution of the exponential equation $a^x+a^{-x}=x$ for a specific value $a$, discuss the number of roots in general case, and provide bounds on the roots.
The aim of this paper is to exhibit a method for proving that certain analytic functions are not solutions of algebraic differential equations. The method is based on model-theoretic properties of differential fields and properties of…
We provide an elementary proof of the asymptotic behavior of solutions of second order differential equations.
Let $N$ be a positive integer. We say a non-constant rational function $U(x)\in{\mathbb C}(x)$ is $N$-\emph{unital} if all the zeros and poles of both $U(x)$ and $1-U(x)$ are either 0 or $N$-th roots of unity. These functions are called…
In an attempt to regularize a previously known exactly solvable model [Yang and Zhang, Eur. J. Phys. \textbf{40}, 035401 (2019)], we find yet another exactly solvable toy model. The interesting point is that while the Hamiltonian of the…
In the paper it is demonstrated that Bells theorem is an unprovable theorem.
The functional equation $\varphi(Fx) - \varphi(x) = \gamma(x)$ is considered in topological, measurable and related categories from the point of view of functional analysis and general theory of dynamical systems. The material is presented…
The class of differential-equation eigenvalue problems $-y''(x)+x^{2N+2}y(x)=x^N Ey(x)$ ($N=-1,0,1,2,3,...$) on the interval $-\infty<x<\infty$ can be solved in closed form for all the eigenvalues $E$ and the corresponding eigenfunctions…
The general solution to the Complex Bateman equation is constructed. It is given in implicit form in terms of a functional relationship for the unknown function. The known solution of the usual Bateman equation is recovered as a special…
We prove in this article the well posedness of non - linear Ordinary Differential Equations (ODE) of first and second order in Orlicz spaces with unbounded domain of definition.
We prove that the equational theory of Kleene algebra with commutativity conditions on primitives (or atomic terms) is undecidable, thereby settling a longstanding open question in the theory of Kleene algebra. While this question has also…
The problem of root mean square approximation of a square integrable function by finite linear combinations of exponential functions is considered. It is subdivided into linear and nonlinear parts. The linear approximation problem is…
Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family…
This paper shows, in the framework of the logical method,the unsolvability of $k$-set agreement task by devising a suitable formula of epistemic logic. The unsolvability of $k$-set agreement task is a well-known fact, which is a direct…
We will study the solutions to the equation $f(n) - g(n) = c$, where $f$ and $g$ are multiplicative functions and $c$ is a constant. More precisely, we prove that the number of solutions does not exceed $c^{1-\epsilon}$ when $f, g$ and…
We prove that the equational complexity function for the variety of representable relation algebras is bounded below by a log-log function.
The Takagi function is a classical example of a continuous nowhere differentiable function. In this paper we prove that it is nowhere approximately derivable.
It is proved that the Halphen system of ordinary differential equations has no non-trivial rational first integrals.
We show that Artin's conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r>1.
We consider regularity for solutions of a class of de Rham's functional equations. Under some smoothness conditions of functions consisting the equation, we improve some results in Hata (Japan J. Appl. Math. 1985). Our results are…