Vol. 54 No. 2 (252) (2020)

In this paper we prove that the set of non-isomorphic 2-generated $C^*$-simple relatively free groups has the cardinality of the continuum. A non-trivial identity is satisfied in any (not absolutely free) relatively free group. Hence, they cannot contain a non-abelian absolutely free subgroups. The question of the existence of $C^*$-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.

In this paper a special class of infinite nonlinear system of algebraic equations with Teoplitz matrix is studied. The mentioned system arises in the mathematical theory of the spatial temporal spread of the epidemic. The existence and the uniqueness of the solution in the space of bounded sequences are proved. It is studied also the asymptotic behavior of the constructed solution at infinity. At the end of the work specific examples are given.

Let $A_1, \dots, A_n$ be fixed positive semi-definite matrices, i.e. $A_i \in \mathbb{S}_p^{+}(\mathbf{R})$ $\forall i \in \{1, \dots, n\}$ and $u_1, \dots, u_n$ are i.i.d. with $u_i \sim \mathcal{N}(1, 1)$. Then, the object of our interest is the following probability

$$\mathbb{P}\bigg(\sum_{i=1}^n u_i A_i \in \mathbb{S}_p^{+}(\mathbf{R})\bigg).$$

In this paper we examine this quantity for pairwise commutative matrices. Under some generic assumption about the matrices we prove that the weighted sum is also positive semi-definite with an overwhelming probability. This probability tends to $1$ exponentially fast by the growth of number of matrices $n$ and is a linear function with respect to the matrix dimension $p.$

In this paper we obtain balance formulas for the logarithmic means of Blaschke type functions and investigate their boundary values.

In this paper we consider the concept of the multiplicity of intersection points of plane algebraic curves $p,q=0,$ based on partial differential operators. We evaluate the exact number of maximal linearly independent differential conditions of degree $k$ for all $k\ge 0.$ On the other hand, this gives the exact number of maximal linearly independent polynomial and polynomial-exponential solutions, of a given degree $k,$ for homogeneous PDE system $p(D)f=0,$ $q(D)f=0.$

In the present study, the methods of circular dichroism and UV/Vis spectrophotometry were used to study the influence of urea on the structural transitions i-motif $\leftrightarrows$ unfolded single strand in cytosine-rich ${\text{d[3}^{\prime}\text{-(CCCAAT)}_{3}\text{CCC-5)}^{\prime}]}$ region of telomeric DNA (Tel22C) and G-quadruplex $\leftrightarrows$ unfolded single strand in complementary guanine-rich strand ${\text{d[5}^{\prime}\text{-A(GGGTTA)}_{3}\text{GGG-3}^{\prime}]}$ (Tel22G) at pH 5.5 and 400 mM Na⁺. Under these conditions, Tel22C and Tel22G were found to form stable i-motif and G-quadruplex structures. It has been shown that urea (0-8 M) destabilizes the i-motif and G-quadruplex structures, but unlike thermal denaturation, it does not destroy the structures completely. The melting processes of G-quadruplex and i-motif are separated in the temperature scale (at any concentration of urea, the melting of the G-quadruplex starts at temperatures where the melting of the i-motifs has already been completed).