On principal minors of Bezout matrix

Let $x_1,...,x_{n}$ be real numbers, $P(x)=p_n(x-x_1)...(x-x_n)$, and $Q(x)$ be a polynomial of degree less than or equal to $n$. Denote by $\Delta(Q)$ the matrix of generalized divided differences of $Q(x)$ with nodes $x_1,...,x_n$ and by $B(P,Q)$ the Bezout matrix (Bezoutiant) of $P$ and $Q$. A relationship between the corresponding principal minors, counted from the right-hand lower corner, of the matrices $B(P,Q)$ and $\Delta(Q)$ is established. It implies that if the principal minors of the matrix of divided differences of a function $g(x)$ are positive or have alternating signs then the roots of the Newton's interpolation polynomial of $g$ are real and separated by the nodes of interpolation.Comment: 15 page

Dynamical systems method for solving operator equations

Consider an operator equation $F(u)=0$ in a real Hilbert space. The problem of solving this equation is ill-posed if the operator $F'(u)$ is not boundedly invertible, and well-posed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear ill-posed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or non-linear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of well-posed problems as well.Comment: 21p

Example of two different potentials which have practically the same fixed-energy phase shifts

It is shown that the Newton-Sabatier procedure for inverting the fixed-energy phase shifts for a potential is not an inversion method but a parameter-fitting procedure. Theoretically there is no guarantee that this procedure is applicable to the given set of the phase shifts, if it is applicable, there is no guaran- tee that the potential it produces generates the phase shifts from which it was reconstructed. Moreover, no generic potential, specifically, no potential which is not analytic in a neighborhood of the positive real semiaxis can be reconstructed by the Newton-Sabatier procedure. A numerical method is given for finding spherically symmetric compactly supported potentials which produce practically the same set of fixed-energy phase shifts for all values of angular momentum. Concrete example of such potentials is given

Generalization of a theorem of Gonchar

Let $X, Y$ be two complex manifolds, let $D\subset X,$ $G\subset Y$ be two nonempty open sets, let $A$ (resp. $B$) be an open subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Under a geometric condition on the boundary sets $A$ and $B,$ we show that every function locally bounded, separately continuous on $W,$ continuous on $A\times B,$ and separately holomorphic on $(A\times G) \cup (D\times B)$ "extends" to a function continuous on a "domain of holomorphy" $\hat{W}$ and holomorphic on the interior of $\hat{W}.$Comment: 14 pages, to appear in Arkiv for Matemati

Mathematical Modeling of Boson-Fermion Stars in the Generalized Scalar-Tensor Theories of Gravity

A model of static boson-fermion star with spherical symmetry based on the scalar-tensor theory of gravity with massive dilaton field is investigated numerically. Since the radius of star is \textit{a priori} an unknown quantity, the corresponding boundary value problem (BVP) is treated as a nonlinear spectral problem with a free internal boundary. The Continuous Analogue of Newton Method (CANM) for solving this problem is applied. Information about basic geometric functions and the functions describing the matter fields, which build the star is obtained. In a physical point of view the main result is that the structure and properties of the star in presence of massive dilaton field depend essentially both of its fermionic and bosonic components.Comment: 16 pages, amstex, 5 figures, changed conten

Structure of W3(OH) from Very High Spectral Resolution Observations of 5 Centimeter OH Masers

Recent studies of methanol and ground-state OH masers at very high spectral resolution have shed new light on small-scale maser processes. The nearby source W3(OH), which contains numerous bright masers in several different transitions, provides an excellent laboratory for high spectral resolution techniques. We present a model of W3(OH) based on EVN observations of the rotationally-excited 6030 and 6035 MHz OH masers taken at 0.024 km/s spectral resolution. The 6.0 GHz masers are becoming brighter with time and show evidence for tangential proper motions. We confirm the existence of a region of magnetic field oriented toward the observer to the southeast and find another such region to the northeast in W3(OH), near the champagne flow. The 6.0 GHz masers trace the inner edge of a counterclockwise rotating torus feature. Masers at 6030 MHz are usually a factor of a few weaker than at 6035 MHz but trace the same material. Velocity gradients of nearby Zeeman components are much more closely correlated than in the ground state, likely due to the smaller spatial separation between Zeeman components. Hydroxyl maser peaks at very long baseline interferometric resolution appear to have structure on scales both smaller than that resolvable as well as on larger scales.Comment: 21 pages using emulateapj.cls including 16 figures and 2 tables, accepted to Ap

Semantic and Syntaxic Features of the Opposition Design and Ways of Its Expression in a Legal Article

The work is devoted to the analysis of the semantic and syntactic features of the construction of opposition, as one of the most common figures of contrast. This technique is considered in a legal article to identify the functions performed by opposition in terms of semantics and syntax. Specific examples of the use of oppositions are given and analyzed, and it is concluded that the antithesis in a legal article to a lesser extent performs an aesthetic, expressive function, but provides subject-logical content, acting as a request, suggestion or persuasion.Работа посвящена анализу семантических и синтаксических особенностей конструкции противопоставления, как одной из самых распространенных фигур контраста. Данный приём рассматривается в статье юридической направленности для выявления функций, выполняемых противопоставлением с точки зрения семантики и синтаксиса. Приводятся и анализируются конкретные примеры использования противопоставлений и делается вывод о том, что антитеза в статье юридической тематики в меньшей степени выполняет эстетическую, экспрессивную функции, а обеспечивает предметно-логическое содержание, выступая в функции просьбы, предложения или убеждения