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The single exponential (SE) and double exponential (DE) formulas are widely recognized as efficient quadrature formulas for evaluating integrals with endpoint singularity. For integrals exhibiting algebraic singularity, explicit error…
We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point.…
The criterion of differentiability of functions of quaternion variable is used as the basis of some algebraic field theory. Its necessary consequences are free Maxwell and Yang-Mills equations. The differentiability equations may be…
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized…
G\"odel's Dialectica has been introduced and developed in the tradition of the so-called functional interpretations. Only recently has it been related with the a priori unrelated notion of differentiation, by taking a program-theoretic…
Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch's algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They…
The exponential trapezoidal rule is proposed and analyzed for the numerical integration of semilinear integro-differential equations. Although the method is implicit, the numerical solution is easily obtained by standard fixed-point…
This paper shows that the plane wave expansion can be a useful tool in obtaining analytical solutions to infinite integrals over spherical Bessel functions and the derivation of identites for these functions. The integrals are often used in…
An algebraic technique adapted to the problems of the fundamental theoretical physics is presented. The exposition is an elaboration and an extension of the methods proposed in previous works by the aut
Using tools from the Siegel-Shidlovskii theory of transcendental numbers, we prove that a nontrivial solution of the Airy equation, its derivative, and an antiderivative are algebraically independent over the field of rational functions.…
Employing a limiting case of a conjecture for constructing piecewise separable-variables functions, the elements of the Pseudoanalytic Function Theory are used for numerically approaching solutions of the forward Dirichlet boundary value…
Representable implication algebras are known to be axiomatised by a finite number of equations (making the representation and finite representation problems decidable here). We show that this also holds in the context of unary (and binary)…
We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals…
This paper analyzes the structure of the set of nodal solutions of a class of one-dimensional superlinear indefinite boundary values problems with an indefinite weight functions in front of the spectral parameter. Quite astonishingly, the…
We introduce a symbolic method for the evaluation of definite integrals containing combinations of various functions, including exponentials, logarithm and products of Bessel functions of different types. The method we develop is naturally…
There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only…
This paper presents the basic ideas and properties of elliptic functions and elliptic integrals as an expository essay. It explores some of their numerous consequences and includes applications to some problems such as the simple pendulum,…
This text is addressed to mathematicians who are interested in generalized functions and unbounded operators on a Hilbert space. We expose in detail (in a "formal way" - as done by Heisenberg and Pauli - i.e. without mathematical…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of classical correspondence…