We prove that, for a normed space X of dimension dim(X)≥2 the space PConv H (X) of non-empty polyhedral convex subsets of X endowed with the Hausdorff metric is homeomorphic to the topological sum {0}⊕|X ∗ |×(R⊕(R×R ¯ + )⊕l f 2 ) , where the cardinal |X ∗ | is endowed with the discrete topology.

A critical note on the book “Fermat Last Theorem. Three elementary proofs” by V. T. Ryabukha.

In this paper we introduce and study a new topologo-algebraic structure called a (di)topological unosemigroup. This is a topological semigroup endowed with continuous unary operations of left and right units (which have certain continuous division property called the dicontinuity). We show that the class of ditopological unosemigroups contains all topological groups, all topological semilattices, all uniformizable topological unosemigroups, all compact topological unosemigroups, and is closed under the operations of taking subunosemigroups, Tychonoff product, reduced product, semidirect product, and the Hartman-Mycielski extension.

Let H be an open upper half-space in R n , n≥2 , and assume that A is a non-empty, open, bounded subset of R n such that A ¯ ¯ ¯ ⊂H and the exterior of A is connected. Let p∈[2,+∞). It is proved that there is a nonzero function with zero integrals over all sets in R n congruent to A if and only if the indicator function of A is the limit in L p (H) of a sequence of linear combinations of indicator functions of balls in H with radii proportional to positive zeros of the Bessel function J n/2 . The proportionality coefficient here is the same for all balls and depends only on A .

We obtain the integral and differential conditions for convergence of operator series.

We introduce a new generalization of matrix (1+1)-dimensional k -constrained KP hierarchy. The new hierarchy contains matrix generalizations of stationary DS systems, (2+1)-dimensional modified Korteweg-de Vries equation and the Nizhnik equation. A binary Darboux transformation method is proposed for integration of systems from this hierarchy.

Exact-order estimates for the approximations of functions of classes S Ω p,θ B(R d ) by entire functions with the spectrum of a special form in the space L q (R d ) for some relations between the parameters p and q , are established.

We present a unified approach to define {dimension functions like} trind , trind p , trt and p . We show how some similar facts on these functions can be proved similarly. Moreover, several new classes of infinite-dimensional spaces close to the classes of countable-dimensional and σ -hereditarily disconnected ones are introduced. We prove a compactification theorem for these classes.

In this paper we prove some analogue of Wiman's inequality for analytic functions f(z 1 ,z 2 ) in the domain T={z∈C 2 :|z 1 |<1, |z 2 |<+∞} . The obtained inequality is sharp.

We construct explicitly in any finite field of the form F p p elements with multiplicative order larger than 4 p .

Let Y be an infinite discrete set of points in R , satisfying the condition inf{|y−y ′ |,y,y ′ ∈Y,y ′ ≠y}>0. In the paper we prove that the systems {δ(x−y)} y∈Y ,{δ ′ (x−y)} y∈Y ,{δ(x−y),δ ′ (x−y)} y∈Y form Riesz bases in the corresponding closed linear spans in the Sobolev spaces W −1 2 (R) and W −2 2 (R) . As an application, we prove the transversalness of the Friedrichs and Kre\u\i n nonnegative selfadjoint extensions of the nonnegative symmetric operators A 0 , A ′ , and H 0 defined as restrictions of the operator A=−d 2 dx 2 , dom(A)=W 2 2 (R) to the linear manifolds dom(A 0 )={f∈W 2 2 (R):f(y)=0,y∈Y} , dom(A ′ )={g∈W 2 2 (R):g ′ (y)=0,y∈Y}, and dom(H 0 )={f∈W 2 2 (R):f(y)=0,f ′ (y)=0,y∈Y} , respectively. Using the divergence forms, the basic nonnegative boundary triplets for A ∗ 0 , A ′∗ , and H ∗ 0 are constructed.