Related papers: Insolvability of $x^x = a$ in elementary functions

Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…

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…

We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential…

Necessary and sufficient conditions for the solvability of boundary value problems for a family of functional differential equations with a non-integrable singularity are obtained.

We present an application of the basic mathematical concept of complex functions as topological solitons, a most interesting area of research in physics. Such application of complex theory is virtually unknown outside the community of…

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…

Several older and more recent results on the boundaries of solvability and unsolvability in tag systems are surveyed. Emphasis will be put on the significance of computer experiments in research on very small tag systems.

Based on earlier work on regular quantum graphs we show that a large class of scaling quantum graphs with arbitrary topology are explicitly analytically solvable. This is surprising since quantum graphs are excellent models of quantum chaos…

We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the $a$-points of the zeta-function, i.e., the roots of the equation $\zeta(s)=a$, where $a$ is an arbitrary fixed complex…

We investigate the issues of inductive problem-solving and learning by doxastic agents. We provide topological characterizations of solvability and learnability, and we use them to prove that AGM-style belief revision is "universal", i.e.,…

We prove that the following problem is decidable: given a finite set of relations, decide whether this set admits a near-unanimity function.

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of…

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…

Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…

The function $ \tan(\pi x / 2) / (\pi x / 2) $ is expanded into a Laurent series of $ 1 - x^2 $, where the coefficients are given explicitly as combinations of zeta function of even integers. This is used to achieve a sequence of upper and…

We prove a general solvable subgroup theorem in terms of length functions. As applications, we obtain a solvable subgroup theorem in dynamical systems: any solvable group of finite Hirsch length acting on a smooth manifold with uniformly…

We prove that the pattern matching problem is undecidable in polymorphic lambda-calculi (as Girard's system F) and calculi supporting inductive types (as G{\"o}del's system T) by reducing Hilbert's tenth problem to it. More generally…

We introduce a fairly general concept of functional equation for $k$-tuples of functions $f_1,\dots,f_k\colon X \to Y$ between arbitrary sets. The homomorphy equations for mappings between groups and other algebraic systems, as well as…

We give a new simple proof of Dehn's theorem by generalizing the notion of area. The method proposed in the present article is actually the "translation" of the method of additive functions into the elementary math language.

Imposing some conditions on derivatives of the known functions, using the Fiber Contraction Theorem we prove the existence of $C^1$ solutions of a class of iterative functional equations which involves iterates of the unknown functions and…