There is space for new ideas of the essence and the entity of time. The article refers to our time concept as a special wave type and presents results of our investigations on this subject. Thus, time defined as waves and an energy carrier could give explanation to multiple unclear phenomena. It could explicate gravity, organization in the planetary systems and light speed limit. A hypothesis t...

Introduction. The problem of integration in finite terms asks for an algorithm for deciding whether an elementary function has an elementary indefinite integral and for finding the integral if it does. "Elementary" is used here to denote those functions built up from the rational functions using only exponentiation, logarithms, trigonometric, inverse trigonometric and algebraic operations. This...

We consider methods for finding high-precision approximations to simple zeros of .smooth functions. As an application, we give fast methods for evaluating the elementary functions log(x), exp(x), sin(x) etc. to high precision. For example, if x is a positive floating-point number with an n-bit fraction, then (under rather weak assumptions) an n-bit approximation to log(x) or exp(x) may be compu...

We investigate collapse of a spherical cloud of counter-rotating particles. An explicit solution is given using an elliptic integral. If the speci c angular momentum L(r) = O(r) at r! 0, no central singularity occurs. With L(r) like that, there is a nite region around the center that bounces. On the other hand, if the order of L(r) is higher than that, a central singularity occurs. In marginall...

It is well known that the conformal group of a non-flat vacuum pp-wave is at most 7-dimensional. Here we explicitly determine all solutions with a 7-dimensional group in three particular families of gravitational plane waves. All of them are exact solutions in terms of elementary functions. Furthermore, it turns out that a gravitational plane wave with a 6-dimensional homothety group (larger th...

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by “explaining” their zeros using an appropriate combinatorial extension of the objects under consideration to negative integer parameters. We apply this method to p...

This paper is part of the collection initiated in [12], aiming to evaluate the entries in [8] and to provide some context. This table contains a large variety of entries involving the Bessel functions. The goal of the current work is to evaluate some entries in [8] where the integrand is an elementary function and the result involves the so-called modified Bessel function of the second kind, de...