Related papers: Various Arithmetic Functions and their Application…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
Selection of 25 examples from extensive nontrivial families for different types of nonlinear PDEs and their formal general solutions are given. The main goal here is to show on examples the types of solvable PDEs and what their general…
Thirty research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone.
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…
This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises…
The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…
Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important…
We investigate the special class of formulas made up of arbitrary but finite com- binations of addition, multiplication, and exponentiation gates. The inputs to these formulas are restricted to the integral unit 1. In connection with such…
We briefly describe each of the four topics: Schubert Calculus, Schubert Cell, Schubert Cycle, and Schubert Polynomials.
We give some remarks on some manifolds K3 surfaces, Complex projective spaces, real projective space and Torus and the classification of two dimensional Riemannian surfaces, Green functions and the Stokes formula. We also, talk about traces…
Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…
An extensive table of pairs of functions linked by the Legendre transformation is presented. Many special functions and formulas that are used in the sciences are included in the pairs. Formulations are provided for finding the Legendre…
In this paper, we give one possible definition for functions of several variables applied to endomorphisms of finite dimensional C-vector spaces. This definition is consistent with the usual notion of a function of a square matrix. Some…
Existence results for Hilbert's problem 13th mean that any equation constructed by continue functions can be given solution represented as a superposition of continue functions of one variable or of continue functions of two variables.…
Rationals are known to form interesting and computationally rich structures, such as Farey sequences and infinite trees. Little attention is being paid to more general, systematic exposition of the basic properties of fractions as a set.…
Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen…
Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals.…
The following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second…
The goal of this paper is to count the number of distinct functions of n variables, up to permutation of the variables, that can be constructed using each variable exactly once, without constants, using only the operations of addition,…