We present a necessary and sufficient condition for a continuous function on a plane to be topologically equivalent to a projection onto one of the coordinates.

We investigate elliptic boundary-value problems with additional unknown functions on the boundary of a Euclidean domain. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on two-sided refined scales built on the base of inner product isotropic H\"ormander spaces. The regularity of the distributions forming these spaces are characterized by a real number and an arbitrary function that varies slowly at infinity in the sense of Karamata. For the generalized solutions to the problem, we prove theorems on a priori estimates and local regularity in these scales. As applications, we find new sufficient conditions under which the solutions have continuous classical derivatives of a prescribed order.

Let $f:M\to \mathbb{R}$ be a Morse function on a connected compact surface $M$, and $\mathcal{S}(f)$ and $\mathcal{O}(f)$ be respectively the stabilizer and the orbit of $f$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(M)$. In a series of papers the first author described the homotopy types of connected components of $\mathcal{S}(f)$ and $\mathcal{O}(f)$ for the cases when $M$ is either a $2$-disk or a cylinder or $\chi(M)<0$. Moreover, in two recent papers the authors considered special classes of smooth functions on $2$-torus $T^2$ and shown that the computations of $\pi_1\mathcal{O}(f)$ for those functions reduces to the cases of $2$-disk and cylinder.

In the present paper we consider another class of Morse functions $f:T^2\to\mathbb{R}$ whose KR-graphs have exactly one cycle and prove that for every such function there exists a subsurface $Q\subset T^2$, diffeomorphic with a cylinder, such that $\pi_1\mathcal{O}(f)$ is expressed via the fundamental group $\pi_1\mathcal{O}(f|_{Q})$ of the restriction of $f$ to $Q$.

This result holds for a larger class of smooth functions $f:T^2\to \mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}$ without multiple factors.

We continue our study of a $q$-difference version of a second-order differential operator which depends on a real parameter. This version was introduced in our previous three articles on the subject. First we study general symmetric and scale-invariant operators on a Hilbert space. We show that if the index of defect of the operator under consideration is $(1,1)$, then the operator either does not admit any scale-invariant self-adjoint extension, or it admits exactly one scale-invariant self-adjoint extension, or it admits exactly two scale-invariant self-adjoint extensions, or all self-adjoint extensions are scale invariant. We then apply these results to the differential operator and the corresponding difference operator under consideration. For the continuous case, we show that the interval of the parameter, for which the differential operator is not semi-bounded, contains an infinite sequence of values for which all self-adjoint extensions are scale-invariant, while for the remaining values of the parameter from that interval, there are no scale-invariant self-adjoint extensions. For the corresponding difference operator, we show that if it is not semi-bounded, then it does not admit any scale-invariant self-adjoint extension. We also show that both differential and difference operators, at value(s) of the parameter that cor espond to the endpoint(s) of the interval(s) of semi-boundedness, have exactly one scale-invariant self-adjoint extension.

The paper specifies necessary and sufficient conditions for regularity of an infinite matrix of real numbers, which determines some summation method for a class of functions that are analytic on the unit disk and continuous on the closed circle.

A positive Borel measure $\mu$ on $\mathbb R$, which possesses all power moments, is N-extremal if the space of all polynomials is dense in $L^2(\mu)$. If, in addition, $\mu$ generates an indeterminate Hamburger moment problem, then it is discrete. It is known that the class of N-extremal measures that generate an indeterminate moment problem is preserved when a finite number of mass points are moved (not ``removed''!). We show that this class is preserved even under change of infinitely many mass points if the perturbations are asymptotically small. Thereby ``asymptotically small'' is understood relative to the distribution of ${\rm supp}\mu$; for example, if ${\rm supp}\mu=\{n^\sigma\log n:\,n\in\mathbb N\}$ with some $\sigma>2$, then shifts of mass points behaving asymptotically like, e.g. $n^{\sigma-2}[\log\log n]^{-2}$ are permitted.

Let $H$ be a finite dimensional Hilbert space and let $[H]$ be the set of all li ear operators in $H$. We consider first-order symmetric system $J y'-B(t)y=\Lambda(t) f(t)$ with $[H]$-valued coefficients defined on an interval $[a,b) $ with the regular endpoint $a$. It is assumed that the corresponding minimal relation $T_{\rm min}$ has maximally possible deficiency index $n_+(T_{\rm min})=\dim H$. The main result is a parametrization of all characteristic matrices and pseudospectral (spectral) functions of a given system by means of a Nevanlinna type boundary parameter $\tau$. Similar parametrization for regular systems has earlier been obtained by Langer and Textorius. We also show that the coefficients of the parametrization form the matrix $W(\lambda)$ with the properties similar to those of the resolvent matrix in the extension theory of symmetric operators.