Siberian Mathematical Journal最新文献

We prove that for a compact metric space ( X ) and for a nonnegative real ( b )not exceeding the lower box dimension of ( X ), there exists a maximal linkedsystem in ( lambda X ) with lower quantization dimension ( b ) and support ( X ).There also exists a maximal linked system in ( lambda X ) with support ( X ) whose lowerand upper quantization dimensions coincide respectivelywith the lower and upper box dimensions of ( X ).

Under study is the Tricomi–Neumann problem for a three-dimensional mixed-type equationwith three singular coefficients in a mixed domain consisting of a quarter of a cylinder and a triangular straight prism.We prove the unique solvability of the problem in the class of regular solutions by usingthe separation of variables inthe hyperbolic part of the mixed domain, which yields the eigenvalue problemsfor one-dimensional and two-dimensional equations.Finding the eigenfunctions of the problems, we usethe formula of the solution of the Cauchy–Goursat problem to construct a solution to the two-dimensional problem.In result, we find the solutions to eigenvalue problems for the three-dimensional equation in the hyperbolic part.Using the eigenfunctions and the gluing condition, we derive a nonlocal problemin the elliptic part of the mixed domain.To solve the problem in the elliptic part, we reformulate the problemin the cylindrical coordinate system and separating the variables leads tothe eigenvalue problems for two ordinary differential equations.We prove a uniqueness theorem by using the completeness propertyof the systems of eigenfunctions of these problems and constructthe solution to the problem as the sum of a double series.Justifying the uniform convergence of the series relies on someasymptotic estimates for the Bessel functions of the real and imaginary arguments.These estimates for each summand of the series made it possible to prove the convergence ofthe series and its derivatives up to the second order,as well as establish the existence theorem in the class of regular solutions.

We say that a graded ring (( * )-ring) ( R ) is a graded quasi-Baer ring (graded quasi-Baer ( * )-ring)if, for each graded ideal ( I ) of ( R ), the right annihilator of ( I ) is generated by a homogeneous idempotent (projection).We prove that a Leavitt pathalgebra is quasi-Baer (quasi-Baer ( * )) if and only if it is graded quasi-Baer (graded quasi-Baer ( * )).We show that a Leavitt path algebra is quasi-Baer (quasi-Baer ( * )) if its zero component is quasi-Baer (quasi-Baer ( * )).However, we give some example that showing that the converse implication fails.Finally, we characterize the Leavitt path algebras that are quasi-Baer ( * )-ringsin terms of the properties of the underlying graph.

This note addresses the problem of recovering a convex set of homogeneous polynomials from the subset of its extreme points,i.e., the justification of a polynomial version of the classical Krein–Milman theorem.Not much was done in this direction. The existing papers deal mostly with the description of the extreme points ofthe unit ball in the space of homogeneous polynomials in various special cases. Even in the case of linear operators,the classical Krein–Milman theorem does not work, since closed convex sets of operators turn out to be compactin some natural topology only in rather special cases. In the 1980s, a new approach to the study of theextremal structure of convex sets of linear operators was proposed on the basis of thetheory of Kantorovich spaces, which led to an operator form of the Krein–Milman theorem.Combining the approach with the linearization method for homogeneous polynomials, we obtain a version of theKrein–Milman theorem for homogeneous polynomials.Namely, a weakly order bounded, operator convex, and pointwise order closed set ( Omega ) ofhomogeneous polynomials from an arbitrary vector space to a Kantorovich space isthe pointwise order closure of the operator convex hull of the extremepoints of ( Omega ).We also establish Milman’s converse of the Krein–Milman theorem for homogeneous polynomials:The extreme points of the smallest operator convex pointwise order closed setincluding a given set ( Omega )of homogeneous polynomials are pointwise uniformlimits of appropriate nets of mixings in ( Omega ).A mixing of a family of polynomials with thevalues in a Kantorovich space is understood as the (infinite) sum of these polynomialsmultiplied by pairwise disjoint order projections with sum the identity operatorin the Kantorovich space.

According to Ptolemy’s theorem, the product of the lengths of the diagonalsof a quadrilateral inscribed in a circle on the Euclidean plane equals the sum of the products of the lengths of oppositesides. This theorem has various generalizations. In one of thegeneralizations on the plane, a quadrilateral is replaced with an inscribed hexagon.In this event the lengths of the sides and long diagonals of aninscribed hexagon is called Ptolemy’s theorem for a hexagon or Fuhrmann’s theorem. Casey’s theoremis another generalization of Ptolemy’s theorem.Four circles tangent to this circle appear instead of four points lying on some fixed circlewhilst the lengths of the sides and diagonals are replaced by the lengths of the segmentstangent to the circles.If the curvature of the Lobachevsky plane is ( -1 ), then in the analogs of the theorems of Ptolemy, Fuhrmann and Casey forthe polygons inscribed in a circle or circles tangent to one circle, the lengths of thecorresponding segments, divided by 2, will be under the signs of hyperbolic sines.In this paper, we prove some theorems that generalize Casey’s theorem and Fuhrmann’s theorem on theLobachevsky plane. The theorems involve six circlestangent to some line of constant curvature.We prove the assertions that generalize these theorems forthe lengths of tangent segments. If, in addition to the lengths of the segments ofthe geodesic tangents, we consider the lengths of the arcs of the tangent horocycles,then there is a correspondence between the Euclidean and hyperbolic relations, whichcan be most clearly demonstrated if we take a set of horocycles tangent to one line of constantcurvature on the Lobachevsky plane. In this case, if the length of the segment of the geodesic tangent tothe horocycles is ( t ), then the length of the “horocyclic” tangent to them is equal to ( sinhfrac{t}{2} ). Hence, if the geodesic tangents are connected by a “hyperbolic” relation, then the“horocyclic” tangents will be connected by the corresponding “Euclidean” relation.

We prove that the category of the complete bi-algebraic (0, 1)-latticesbelongingto the quasivariety generated by a certain finite lattice with completelattice homomorphisms, considered as a concrete category, is duallyequivalent tothe category of certain spaces with an additional structure.

We describe the ideals for Mizuhara extensions and findsome necessary and sufficient conditions for the simplicity ofthe direct Mizuhara extension.Also, we study the Mizuhara constructionfor the matrix algebra and Burde algebras.We construct some various generalizationsof the Mizuhara construction and exhibit some examplesof the simple pre-Lie algebras that are obtained by thisconstruction; in particular, we construct the simple Witt doubles( {mathcal{A}}_{d} ) and ( {mathcal{W}}_{d}({mathcal{A}}) ) for a unital associative commutativealgebra ( {mathcal{A}} ) with derivation ( d ).

Let ( N ) be a nonassociative finite-dimensional simple Novikovalgebra over an algebraically closed field ( F ) of characteristic ( p>0 ). Thenthe right multiplication algebra ( R )is a differential simple algebrawith respect to some derivation ( d ). The algebra ( N ) is isomorphicto a Novikov algebra ( (R,d,R_{x}) )for some operator of right multiplication by ( x ) and multiplicationis given by ( ucirc w=ud(w)+R_{x}uw ).Moreover, the algebra ( R ) is a truncated polynomial algebra.

We consider the Cauchy problem for a one-dimensional hyperbolic equation whose lower-order coefficient and right-hand sideoscillate in time with a high frequency and the amplitude of the lower-order coefficient is small.Under study is the reconstruction of the cofactors of these rapidly oscillating functions independentof the space variable from a partial asymptotics of a solution at some point of the space.The classical theory of inverse problems examines the numerous problems of determining unknown sources, and coefficients withoutrapid oscillations for various evolutionary equations, where the exact solutionto the direct problem appears in the additional overdetermination condition.Equations with rapidly oscillating data are often encountered in modeling the physical, chemical, andother processes that occur in media subjected to high-frequency electromagnetic, acoustic, vibrational, and others fields,which demonstrates the topicality of perturbation theory problems on the reconstruction of unknown functionsin high-frequency equations.We give some nonclassical algorithm for solving such problems that lies at the junction ofasymptotic methods and inverse problems. In this case the overdetermination condition involvesa partial asymptotics of solution of a certain lengthrather than the exact solution.

We consider a hyperbolic equation with variable leading part and nonlinearity in the lower-order term.The coefficients of the equation are smooth functionsconstant beyond some compact domain in the three-dimensional space.A plane wave with direction ( ell ) falls to the heterogeneity from the exterior of this domain.A solution to the corresponding Cauchy problem for the original equation is measured at boundary points of the domain fora time interval including the moment of arrival of the wave at these points.The unit vector ( ell ) is assumed to be a parameter of the problem andcan run through all possible values sequentially.We study the inverse problem of determining the coefficient of the nonlinearity on using thisinformation about solutions. We describe the structure of a solution to the direct problem anddemonstrate that the inverse problem reduces to an integral geometry problem.The latter problem consists of constructing the desired function on using given integralsof the product of this function and a weight function.The integrals are taken along the geodesic lines of the Riemannian metricassociated with the leading part of the differential equation. We analyze this new problemand find some stability estimate for its solution, which yieldsa stability estimate for solutions to the inverse problem.