Various estimates of the Borsuk number are discussed. A method for constructing two-distance counterexamples to Borsuk’s conjecture by using $$(0,1)$$ -vectors in spaces with $$l_p$$ metrics is considered.

The inverse spectral problem method is applied to integrate a negative-order nonlinear Schrödinger equation in the class of periodic infinite-gap functions. The evolution of spectral data for a periodic Dirac operator whose coefficient is a solution to the negative-order nonlinear Schrödinger equation is introduced. An algorithm for deriving the Dubrovin differential equation system is proposed. The solvability of the Cauchy problem for the infinite Dubrovin differential equation system in the class of twice continuously differentiable periodic infinite-gap functions is proved. It is shown that, for sufficiently smooth initial data, there exists a global solution to the mixed problem for the negative-order nonlinear Schrödinger equation.

Nonstationary linear and nonlinear descriptor systems with discrete time are considered. In the linear case, criteria for state and output controllability on a finite horizon are obtained, and robust controllability conditions are found. For nonlinear systems, a theorem on local state controllability is proved using the properties of linear approximation.

It is shown that the so-called sparse clouds contract to a point in themselves, which gives a partial proof of Gromov’s remark that any cloud is contractible.

For a one-parameter family of linear operators, given an operator on a homogeneous Sobolev space with given value of the parameter, we consider the problem of optimal recovery of this operator from incomplete information about this operator with a different value of the parameter. A family of optimal recovery methods is constructed. As a corollary, we obtain optimal methods of recovery of solutions to some equations of mathematical physics.

We introduce and study weighted spaces of functions with mixed norm on the upper half-plane, defined in terms of the Fourier transform. We characterize analytic functions in these spaces and, in particular, give an analog of the Paley–Wiener theorem in this setting. As an application, we consider Toeplitz operators with vertical symbols in these new spaces.

In the paper, a new class of universal $$C^*$$ -algebras, namely, Blaschke $$C^*$$ -algebras, are constructed, for which the relations between generators are defined by a family of finite Blaschke products. Two approaches to constructing a universal object are proposed, one of which is related to the inductive limit of Toeplitz algebras, and the other to an isometric representation of the uniform Blaschke algebra. $$C^*$$ -algebras generated by isometric representations of a Blaschke algebra in the algebra of bounded linear operators on generalized Hardy spaces with a probability measure and, in particular, with a Haar measure are considered in detail.

We study boundedly weakly compact sets admitting, for each $$\varepsilon>0$$ , $$\mathrm{nw}$$ -continuous (norm-weak continuous) $$\varepsilon$$ -selections of the near-best metric projection operator. Such a set in a reflexive Kadec–Klee space is shown to be a sun. For an approximatively compact set it is verified that this set admits an $$\mathrm{nw}$$ -continuous $$\varepsilon$$ -selection for each $$\varepsilon>0$$ if and only if it admits an $$\mathrm{nn}$$ -continuous (norm-norm continuous) $$\varepsilon$$ -selection for each $$\varepsilon>0$$ .

In this paper our final objective is to develop a method for generation of fractals in fuzzy metric spaces. This is done by applying an iterated function system consisting of a fuzzy metric space and a family of continuous functions satisfying fuzzy Banach orbital condition. The Banach orbital condition is brought about by an application of concatenated words generated out of a set of finite alphabets. It is shown that the iterated function system considered here is weakly Picard and admits attractors. The process of fractal generation is illustrated.

Let $$D\subset \mathbb{R}^d$$ be a bounded Lipschitz domain, let $$\omega$$ be a modulus of continuity of a high order of smoothness, and let $$T$$ be a Calderón–Zygmund convolution operator with even kernel. Using the recent T(P) boundedness criterion found by the author with E. Doubtsov, we prove that the operator $$T$$ is bounded in the Zygmund space $$\mathcal{C}_{\omega}(D)$$ if the smoothness of the boundary of $$D$$ is by one higher than that of $$\mathcal{C}_{\omega}(D)$$ . The proof is based on estimates for the potentials with Calderón–Zygmund kernels of the characteristic function of a domain with a polynomial boundary.