We call a mapping f:X→Y an l -Darboux mapping if the image of any arcwise connected subset of X is connected. We prove that the class of l -Darboux F σ -measurable mappings of a topological space to a metric space is closed with respect to uniform limits.

In a free boundary domain there were established conditions of existence and uniqueness of the classical solution to the inverse problem for determination a time-dependent major coefficient in a parabolic equation with weak power degeneration near the derivative with respect to time. The Stefan condition and heat flux are given as overdetermination conditions.

Well-posedness of the initial-boundary-value problem for linear elliptic-parabolic-pseudoparabolic equations are proved. An estimate of the generalized solution of this problem are received.

The l -index boundedness of entire ridge functions of finite order with real zeros, entire characteristic functions of probability laws and finite Fourier-Stieltjes transforms was investigated.

We prove the Bari--Markus property for spectral projectors of non-self-adjoint Dirac operators on (0,1) with square-integrable matrix-valued potentials and some separated boundary conditions.

We find conditions under which the pretangent spaces to general metric spaces have the nonpositive Aleksandrov curvature or nonnegative one. The infinitesimal structure of general metric spaces with Busemann convex pretangent spaces is also described.

We provide bar and cobar constructions as functors between some categories of curved algebras and curved augmented coalgebras over a graded commutative ring. These functors are adjoint to each other.

It is proved that each multiplicatively periodic subharmonic function in R^m ∖{0} , m≥3 , is constant. Examples of non-constant multiplicatively periodic differences of subharmonic in R^m ∖{0} functions are constructed.

A description of solutions of some integral equations has been obtained. A two-radii theorem is obtained as well.

A version of Carleman's formula for functions holomorphic in a rectangle is proved. It is applied to the evaluation of the integral of ζ -function logarithm with the summing factor exp(−t) along the critical line. This allowed to obtain a new statement equivalent to the riemann hypothesis.