Related papers: Transcendental numbers as solutions to arithmetic …
After H\"older proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental i.e. they cannot be solution to an algebraic…
We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic…
An original approach to solving rather difficult probabilistic problems arising in studying the readout of random discrete fields and having no exact analytical solutions at the moment is proposed. Several algorithms for direct, iterative,…
We construct some explicit formulas of rational maps and transcendental meromorphic functions having Herman rings of period strictly larger than one. This gives an answer to a question raised by Shishikura in the 1980s. Moreover, the…
The validation and parallel implementation of a numerical method for the solution of the time-dependent Dirac equation is presented. This numerical method is based on a split operator scheme where the space-time dependence is computed in…
We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this…
We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides; the solvable character of these dynamical systems…
Transcendental numbers play an important role in many areas of science. This paper contains a short survey on transcendental numbers and some relations among them. New inequalities for transcendental numbers are stated in Section 2 and…
In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with non integer exponents. We apply the method to equations resembling generalizations…
The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there…
In this note, a numerical method based on finite differences to solve a class of nonlinear advection-diffusion fractional differential equation is proposed. The fractional operator considered here is the fractional Riemann-Liouville…
Carmichael quotients for an integer $m\ge 2$ are introduced analogous to Fermat quotients, by using Carmichael function $\lambda(m)$. Various properties of these new quotients are investigated, such as basic arithmetic properties, sequences…
Difference schemes are considered for dynamical systems $ \dot x = f (x) $ with a quadratic right-hand side, which have $t$-symmetry and are reversible. Reversibility is interpreted in the sense that the Cremona transformation is performed…
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
Fractional analysis is applied to describe classical dynamical systems. Fractional derivative can be defined as a fractional power of derivative. The infinitesimal generators {H, .} and L=G(q,p) \partial_q+F(q,p) \partial_p, which are used…
Differential categories provide the categorical foundations for the algebraic approaches to differentiation. They have been successful in formalizing various important concepts related to differentiation, such as, in particular,…
Periods are numbers represented as integrals of rational functions over algebraic domains. A survey of their elementary properties is provided. Examples of periods includes Feynman Integrals from Quantum Physics and Multiple Zeta Values…
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…
I consider differential of mapping $f$ of continuous division ring as linear mapping the most close to mapping $f$. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux…