We consider the problem of finding lower bounds for growth of solutions of a fractional differential equation in the complex plane. We estimate a fractional integral of the logarithmic derivative of a meromorphic function.

Let sn i z be algebraically independent Jacobi elliptic functions, (4K i ,2iK ′ i ) be main periods and ϰ 1 ,ϰ 2 be their moduli sn i z (i∈{1,2} ). We estimate from below the simultaneous approximation ϰ 1 ,ϰ 2 ,sn 1 K 2 ,sn 2 iK ′ 1 .

In the paper an analytical solution of the analytic continuation problem in the unit N-dimensional ball by its values on the inner sphere is found. A generalization of the classical Poisson formula is obtained. A solution of the similar analytic continuation problem for certain known operator expressions of a given potential on the inner sphere is obtained. A new result of the moment problem making a connection between the moment problem on a segment and on the half-axis is found.

A new criterion of boundedness of L-index in direction is obtained.

We consider the relationships between the local Artin mapp θ:K ∗ →Gal(K ab /K) and the Hilbert symbol (⋅,⋅):K ∗ /K ∗m ×K ∗ /K ∗m ⟶μ m for a general local field, as well as between the Tate pairing and the Weil pairing for elliptic curves with good reductions over pseudolocal fields (complete discretely valued fields with pseudofinite residue fields). It is known that the Weil pairing {⋅,⋅}:E(K ¯ ¯ ¯ ) m ×E(K ¯ ¯ ¯ ) m ⟶μ m and the Tate pairing ⟨⋅,⋅⟩:E(K)/mE(K)×H 1 (G K ,E(K ¯ ¯ ¯ )) m ⟶Z/mZ satisfy ζ ⟨c 1 ,c 2 ⟩ ={e 1 ,e 2 } , where E is an elliptic curve with good reduction over local field and ζ an appropriate m th root of 1. This is Kolyvagin's formula. It is proved that the same holds true for elliptic curves with good reductions over pseudolocal fields.

We consider the question of precise conditions ensuring that a function having zero integrals over all balls of fixed radius is equal to zero. We completely investigate the case where together with zero integrals over congruent balls a function has zero first moments over these balls.

We consider pairs of linear mappings (A,B) of the form \pudVWAB in which V and W are finite dimensional unitary or Euclidean spaces over C or R , respectively. Let (A,B) be transformed to \pudV ′ W ′ A ′ B ′ } by bijections φ 1 :V→V ′ and φ 2 :W→W ′ . We say that (A,B) and (A ′ ,B ′ ) are linearly equivalent if φ 1 and φ 2 are linear bijections and topologically equivalent if φ 1 and φ 2 are homeomorphisms. We prove that (A,B) and (A ′ ,B ′ ) are topologically equivalent if and only if their regular parts are topologically equivalent and their singular parts are linearly equivalent.

We consider a convolution type equation in the Smirnov spaces in a semi-strip. We obtain a description of solutions for the case when the characteristic function of the equation has a "deep zero" at infinity.

Let φ be the characteristic function of a probability law F that is analytic in D R. A connection between the growth of M(r,φ) and the decrease of W F (x) is investigated in terms of estimates from below. For entire characteristic functions is proved, for example, that if lnx k ≥λln(1 x k ln1 W F (x k ) ) for some increasing sequence (x k ) such that x k+1 =O(x k ),k→∞, then lnlnM(r,φ) r ≥(1+o(1))λlnr as r→+∞.

The conditions of solvability in the generalized sense of a problem with unknown breaking line of initial data for a hyperbolic quasilinear system with two independent variables in the sector were established. Applying the method of characteristics a solution of the problem is reduced to finding the fixed point of the operator whose existence and uniqueness are proved using the Banach theorem.

We consider hypercyclic operators on free Banach spaces and little Lipschitz spaces which are some kind of generalizations of shift operators and composition operators respectively.

We solve the problem on a construction of a separately continuous mapping with the given diagonal, which is the pointwise limit of a sequence of continuous mappings with values in an equiconnected space. We construct an example of a closed-valued separately continuous mapping f:[0,1] 2 ⊸R with an everywhere discontinuous diagonal. The example shows that the results on points of joint continuity for compact-valued separately continuous mappings can not be generalized to the case of closed-valued mappings.

One of the Ph.D. students in mathematics that belonged to Lviv Mathematical School of 1920–1939 years is Jozef Schreier. In the article, the authors provide an overview of main results of the unpublished thesis “On finite base in topological groups” by Jozef Schreier as well as the connection of the ideas of this work with modern investigations in the theory of topological groups and semigroups.

The properties of the prime and the congruence-prime acts over monoid with zero are investigated.

Various theorems on the convergence of general spatial homeomorphisms are proved and, on this basis, convergence theorems for classes of the so-called ring Q --homeomorphisms are obtained. These results will have wide applications to Sobolev's mappings.