Siberian Mathematical Journal最新文献

Let ( w_{k} ) be the maximum of the minimum degree-sum (weight) of vertices in ( k )-vertex paths (( k )-paths) in 3-polytopes.Trivially, each 3-polytope has a vertex of degree at most 5, and so ( w_{1}leq 5 ).Back in 1955, Kotzig proved that ( w_{2}leq 13 ) (so there is an edge of weight at most 13), which is sharp.In 1993, Ando, Iwasaki, and Kaneko proved that ( w_{3}leq 21 ), which is also sharpdue to a construction by Jendrol’ of 1997.In 1997, Borodin refined this by proving that ( w_{3}leq 18 ) for 3-polytopes with ( w_{2}geq 7 ),while ( w_{3}leq 17 ) holds for 3-polytopeswith ( w_{2}geq 8 ), where the sharpness of 18 was confirmed by Borodin et al. in 2013,and that of 17 was known long ago.Over the last three decades, much research has been devoted to structural and coloring problemson the plane graphs that are sparse in this or that sense.In this paper we deal with 3-polytopes without adjacent 3-cycles that is without chordal 4-cycle(in other words, without ( K_{4}-e )).It is known that such 3-polytopes satisfy ( w_{1}leq 4 ); and, moreover, ( w_{2}leq 9 ) holds, whereboth bounds are sharp (Borodin, 1992).We prove now that each 3-polytope without chordal 4-cycleshas a 3-path of weight at most 15; and so ( w_{3}leq 15 ), which is sharp.

We establish two new criteria for the closedness of Archimedean cones in countable-dimensional locally convex spacesin terms of projective parallelotopes and projective automorphisms.We also answer some open questions about quasidenseness and quasi-interior.

Abstract

We locate Musial and Sagher’s concept of ( operatorname{HK}_{r} ) -integration within the approximate Henstock–Kurzweil integral theory. If we restrict the ( operatorname{HK}_{r} ) -integral by the requirement that the indefinite ( operatorname{HK}_{r} ) -integral is continuous, then it becomes included in the classical Denjoy–Khintchine integral. We provide a direct argument demonstrating that this inclusion is proper.

Abstract

We consider bounded selfadjoint linear integral operators  ( T_{1} ) and  ( T_{2} ) in the Hilbert space ( L_{2}([a,b]times[c,d]) ) which are usually called partial integral operators. We assume that  ( T_{1} ) acts on a function  ( f(x,y) ) in the first argument and performs integration in  ( x ) , while  ( T_{2} ) acts on  ( f(x,y) ) in the second argument and performs integration in  ( y ) . We assume further that  ( T_{1} ) and  ( T_{2} ) are bounded but not compact, whereas  ( T_{1}T_{2} ) is compact and ( T_{1}T_{2}=T_{2}T_{1} ) . Partial integral operators arise in various areas of mechanics, the theory of integro-differential equations, and the theory of Schrödinger operators. We study the spectral properties of  ( T_{1} ) , ( T_{2} ) , and ( T_{1}+T_{2} ) with nondegenerate kernels and established some formula for the essential spectra of  ( T_{1} ) and  ( T_{2} ) . Furthermore, we demonstrate that the discrete spectra of  ( T_{1} ) and  ( T_{2} ) are empty, and prove a theorem on the structure of the essential spectrum of  ( T_{1}+T_{2} ) . Also, under study is the problem of existence of countably many eigenvalues in the discrete spectrum of  ( T_{1}+T_{2} ) .

We obtain two-sidedestimates for Alexandrov’s ( n )-width ofthe compact set of infinitely smooth functionsboundedly embedded into the space of continuous functions on a finite segment.

We found the geodesics, shortest arcs, cut loci, and injectivity radiusof any oblate ellipsoid of revolution in three-dimensional Euclidean space.

Consider the fundamental group ( {mathfrak{G}} )of an arbitrary graph of groupsand some root class ( {mathcal{C}} )of groups,i.e., a class containing a nontrivial groupand closed under subgroups,extensions,and unrestricted direct products of the form( prod_{yin Y}X_{y} ),where( X,Yin{mathcal{C}} )and ( X_{y} )is an isomorphic copy of ( X )for each( yin Y ).We provide some criterion for the separability by ( {mathcal{C}} )of a finitely generated abelian subgroup of ( {mathfrak{G}} )valid whenthe group satisfies an analog of the Baumslag filtration condition.This enables us to describethe ( {mathcal{C}} )-separable finitely generated abelian subgroupsfor the fundamental groups of some graphs of groupswith central edge subgroups.

We prove thatevery mapping with finite distortion on a Carnot groupis open and discrete provided that it is quasilight and the distortion coefficient is integrable.Also, we estimate the Hausdorff dimension of the preimages of pointsfor mappings on a Carnot groupwith a bounded multiplicity functionand summable distortion coefficient.Furthermore, we give some example showing thatthe obtained estimates cannot be improved.

We construct two new infinite series of irreducible components ofthe moduli space of semistable nonlocally free reflexive rank 2 sheaveson the three-dimensional complex projective space.In the first seriesthe sheaves have an even first Chern class,and in the second seriesthey have an odd one,while the second and third Chern classescan be expressed as polynomials of a special formin three integer variables.We prove the uniqueness of components in these seriesfor the Chern classesgiven by those polynomials.

We show that the Levi class of the quasivariety of right-orderable groups strictlyincludes this quasivariety.