We describe the algebras of binary formulas for countably categorical weakly circularly minimal theories with 1-transitive nonprimitive automorphism group and trivial definable closure having convexity rank 1. We find some criterion for commutativity of the algebras.
We establish two new criteria for the closedness of Archimedean cones in countable-dimensional locally convex spaces in terms of projective parallelotopes and projective automorphisms. We also answer some open questions about quasidenseness and quasi-interior.
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.
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-sided estimates for Alexandrov’s ( n )-width of the compact set of infinitely smooth functions boundedly embedded into the space of continuous functions on a finite segment.
We found the geodesics, shortest arcs, cut loci, and injectivity radius of any oblate ellipsoid of revolution in three-dimensional Euclidean space.
Consider the fundamental group ( {mathfrak{G}} ) of an arbitrary graph of groups and some root class ( {mathcal{C}} ) of groups, i.e., a class containing a nontrivial group and 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 when the group satisfies an analog of the Baumslag filtration condition. This enables us to describe the ( {mathcal{C}} )-separable finitely generated abelian subgroups for the fundamental groups of some graphs of groups with central edge subgroups.
We prove that every mapping with finite distortion on a Carnot group is open and discrete provided that it is quasilight and the distortion coefficient is integrable. Also, we estimate the Hausdorff dimension of the preimages of points for mappings on a Carnot group with a bounded multiplicity function and summable distortion coefficient. Furthermore, we give some example showing that the obtained estimates cannot be improved.
We show that the Levi class of the quasivariety of right-orderable groups strictly includes this quasivariety.
We construct two new infinite series of irreducible components of the moduli space of semistable nonlocally free reflexive rank 2 sheaves on the three-dimensional complex projective space. In the first series the sheaves have an even first Chern class, and in the second series they have an odd one, while the second and third Chern classes can be expressed as polynomials of a special form in three integer variables. We prove the uniqueness of components in these series for the Chern classes given by those polynomials.
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