Related papers: A note on internality of certain differential syst…
The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations.…
This article establishes a complete approximate axiomatization for the real-closed field $\mathbb{R}$ expanded with all differentially-defined functions, including special functions such as $\sin(x), \cos(x), e^x, \dots$. Every true…
In this paper, we prove some uniqueness results which improve and generalize several earlier works. Also, we prove a value distribution result concerning $f^{(k)}$ which provides a partial answer to a question of Fang and Wang [A note on…
A new ansatz is presented for a Lax pair describing systems of particles on the line interacting via (possibly nonsymmetric) pairwise forces. Particular cases of this yield the known Lax pairs for the Calogero-Moser and Toda systems, as…
We demonstrate how to separate the rotational degrees of freedom in a quantum N-body problem completely from the internal ones. It is shown that any common eigenfunction of the total orbital angular momentum ($\ell$) and the parity in the…
We prove that the partial zeta function introduced in [9] is a rational function, generalizing Dwork's rationality theorem.
A topological dynamical system $(X,f)$ induces two natural systems, one is on the probability measure spaces and other one is on the hyperspace. We introduce a concept for these two spaces, which is called entropy order, and prove that it…
Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the image \begin{equation*} f_{U}(C,C)=\{f(x,y):(x,y)\in (C\times C)\cap U\}. \end{equation*} If $\partial…
Let $f$ be a meromorphic function. We suggest a generalization of $f$ and its derivative $f'$ sharing a nonzero value $a$ IM that does not impose any a priori restrictions on the ramification of $f$. Then we discuss some results around the…
The real Jacobian conjecture claims that if $F=\left(f^1,\ldots,f^n\right):\mathbb{R}^n\rightarrow \mathbb{R}^n$ is a polynomial map such that $\det DF$ is nowhere zero, then $F$ is a global injective. The first part is to study the…
It is shown that every Feynman integral can be interpreted as Green function of some linear differential operator with constant coefficients. This definition is equivalent to usual one but needs no regularization and application of…
One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f(g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule…
This paper is devoted to the study of meromorphic solutions of nonlinear differential equations, specifically the equation \[ (f^n)^{(k)}(g^n)^{(k)} = \alpha^2, \] where $k$ and $n$ are positive integers with $n>2k$, and $\alpha$ is a…
It is shown that a chain of closed systems of first order ordinary differential equations describing the evolution of moments can be constructed using the Jacobi equation. It is shown that Wronsky determinants for fundamental matrices of…
In this paper we provide sufficient conditions that ensure the monotonicity, respectively the global injectivity of an operator. Further, some new analytical conditions that assure the injectivity/univalence of a complex function of one…
Multivariable generalizations of the continuous Hahn and Wilson polynomials are introduced as eigenfunctions of rational Ruijsenaars type difference systems with an external field.
Considering differential equation f''+A(z)f'+B(z)f=0, where A(z) and B(z) are entire complex functions, our results revolve around proving all non-trivial solutions are of infinite order taking various restrictions on coefficients A(z) and…
Fair termination is the property of programs that may diverge "in principle" but that terminate "in practice", i.e. under suitable fairness assumptions concerning the resolution of non-deterministic choices. We study a conservative…
General formula for causal Green's function of linear differential operator of given degree in one variable is given according to coefficient functions of differential operator as a series of integrals. The solution also provides analytic…
The problem to decide whether a given rational function in several variables is positive, in the sense that all its Taylor coefficients are positive, goes back to Szeg\H{o} as well as Askey and Gasper, who inspired more recent work. It is…