The first part of the paper presents a unique reduction of some general Riccati-type differential equation to the general Bernoulli-type equation. Furthermore four special theorems on reduction of some Riccati-type differential equations are considered in the second part of the paper. A number of examples are also presented to illustrate the considered issues.

Undoubtedly, one of the most powerful applications that allow symbolic computations is Wolfram Mathematica. However, it turns out that sometimes Mathematica does not give the desired result despite its continuous improvement. Moreover, these gaps are not filled by many authors of books and tutorials. For example, our attempts to obtain a compact symbolic description of the roots of polynomials or coefficients of a polynomial with known roots using Mathematica have often failed and they still fail. Years of our work with theory, computations, and different kinds of applications in the area of polynomials indicate that an application ‘offering’ the user alternative methods of solving a given problem would be extremely useful. Such an application would be valuable not only for people who look for solutions to very specific problems but also for people who need different descriptions of solutions to known problems than those given by classical methods. Therefore, we propose the development of an application that would be not only a program doing calculations but also containing an interactive database about polynomials. In this paper, we present examples of methods and information which could be included in the described project.

Ramsey’s theorem states that for any natural numbers n, m there exists a natural number N such that any red–blue coloring of the graph $K_N$ contains either a red $K_n$ or blue $K_m$ as a subgraph. The smallest such N is called the Ramsey number, denoted as $R(n,m)$. In this paper, we reformulate this theorem and present a proof of Ramsey’s theorem that is novel as far as we are aware.

Our goal is to prove the Euler–Maclaurin summation formula using only the Taylor formula. Furthermore, the Euler–Maclaurin summation formula will be considered for the case of functions of the class $C^{2k}\left([a,b]\right)$. A stronger version of the estimation of the rest for functions of class $C^n\left([a,b]\right)$ is given. We will also present the application of the obtained Euler–Maclaurin summation formulas to determine the asymptotics of Riemann sums for uniform partitions of intervals of integration. This paper also introduces the concepts of the asymptotic smoothing of the graph of a given function, as well as the asymptotic uniform distribution of the positive and negative values of the given function (generalizing the concept of the symmetric distribution of these values with respect to the x-axis, as in the case of the Peano–Jordan measure).

The goal of this work is to present a new theoretical and simultaneously practical approach to the conception of improper integrals of type II and, consequently, to improper integrals of type I. Seven distinct definitions of improper integrals of type II are presented, each illustrated by suitable examples. Among them, particularly important are those defined on nets composed of sets symmetric with respect to points belonging to a given finite family of points. Connections between improper integrals of type II defined here are described in two main theorems. It should be emphasized that all definitions of improper integrals presented here are special cases of a very general definition of improper integrals on non-compact sets, which is presented at the end of this work. This definition is inspired by a concept introduced by Maurin.