On the automorphism group of a cone

Let V be a real finite dimensional vector space, and let C be a full cone in C. In Sec. 3 we show that the group of automorphisms of a compact convex subset of V is compact in the uniform topology, and relate the group of automorphisms of C to the group of automorphisms of a compact convex cross-section of C. This section concludes with an application which generalizes the result that a proper Lorentz transformation has an eigenvector in the light cone. In Sec. 4 we relate the automorphism group of C to that of its irreducible components. In Sec. 5 we show that every compact group of automorphisms of C leaves a compact convex cross-section invariant. This result is applied to show that if C is a full polyhedral cone, then the automorphism group of C is the semidirect product of the (finite) automorphism group of a polytopal cross-section and a vector group whose dimension is equal to the number of irreducible components of C. An example shows that no such result holds for more general cones.     

Continuity of operators in spaces of vector functions,with applications to the theory of bases

Banach spaces with unconditional martingale differences are investigated. In Sec. 1 a survey of their fundamental properties and connections with vector-valued harmonic analysis is given. In Sec. 2 new results are obtained regarding bases in the spaces E(X), where E is a symmetric space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 5–22, 1987.     

Infinite Faces of a Perfect Voronoi Polyhedron

Infinite faces of perfect Voronoi polyhedra are studied. The result substantially varies depending on whether the perfect Voronoi polyhedron is considered as the closed or nonclosed convex hull of the set of Voronoi points (see Theorem 1 in Sec. 5 and Theorem 2 in Sec. 7).     

Study of families of monotone continuous functions on Tychonoff spaces

The main object of study is the space of all monotone continuous functions CM(X) on a connected Tychonoff space X endowed with the topology of pointwise (CM _p (X)) or uniform (CM(X)) convergence. Technical questions concerning restriction and extension of monotone functions are considered in Sec. 2. Conditions for CM(X) to separate the points of X and for CM(X) to contain only constant functions are found in Sec. 3. In Sec. 4, the linear structure of CM(X) is studied and all linear subspaces of CM(X) for a certain class of spaces X are described. In Sec. 5, conditions under which CM(X) is closed and nowhere dense in C _p (X) and C(X) are determined. The metrizability of CM _p (X) is considered in Sec. 6; necessary and sufficient metrizability conditions for various classes of spaces X are obtained. In Sec. 7, criteria for σ-compactness and the Hurewicz property in the class of spaces CM _p (X) are given. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 34, General Topology, 2005.     

The Combinatorics of Polynomial Sequences

We present a combinatorial model for the several kinds of polynomial sequences of binomial type and develop many of the theorems about them from this model. In the first section, we present a prefab model for the binomial formula and the generating-function theorem. In Sec. 2, we introduce the notion of U-graph and give examples of binomial prefabs of U-graphs. The umbral composition of U-graphs provides an interpretation of umbral composition of polynomial sequences in Sees. 3 and 5. Rota's interpretation of the Stirling numbers of the first kind as sums of the Mobius function in the partition lattice inspired our model for inverse sequences of binomial type in Sec. 4. Section 6 contains combinatorial proofs of several operator-theoretic results. The actions of shift operators and delta operators are explained in set-theoretic terms. Finally, in Sec. 6 we give a model for cross sequences and Sheffer sequences which is consistent with their decomposition into sequences of binomial type. This provides an interpretation of shift-invariant operators. Of course, all of these interpretations require that the coefficients involved be integer and usually non-negative as well.     

Inertia and eigenvalue relations between symmetrized and symmetrizing matrices for the real and the general field case

Starting from a theorem of Frobenius that every n×n matrix is the product of two symmetric ones, we study relations between the similarity invariants of a square matrix and the congruence invariants of its symmetric factors. Section 1 treats the real case, Sec. 2 the arbitrary field case, and Sec. 3 the indefinite inner product case for Krein spaces. The proofs are obtained from the real canonical pair form in Secs. 1 and 3 and from the recently found rational canonical pair form in Sec. 2, each time via combinatorial type arguments on weighted partitions of n. The resulting theorems typically give bounds for the elementary divisor structure of A in terms of the index or signature of one or both of its symmetric factors (or vice versa). Our results greatly extend and generalize the classic results of Klein, Loewy, Taussky, et al., and in Sec. 2 put new light on Waterhouse's recent characterization of hereditarily euclidean fields. A short survey on the history of the subject from the early 1800s on completes the paper.     

Parabolic spaces

In the present paper we define parabolic spaces with simple fundamental Lie groups and we construct models of them, establish a relation between parabolic and reductive spaces, describe the topological structure of parabolic spaces with classical Lie fundamental groups.The introduction and Sec. 1 were written by B. A. Rozenfel'd, Sec. 2 by M. P. Zamakhovskii, Sec. 3 by T. A. Timoshenko.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 26, pp. 125–160, 1988.     

The bilateral Laguerre transform

In the earlier paper [13], the crucial role of the complex plane in the formulation of the algorithms was evident, even though the algorithms were entirely in the real domain. For the bilateral transform, the complex plane is again very much present, with Laurent expansions, bilateral Laplace transformation, and conformal mapping entering as crucial tools. The first section extends the earlier formalism to the full continuum. That this extension is natural, and not just an artificial piecing together of the formalism for each half line, will be clear from (1.9), (1.12), and (1.13). The harmony of the basis will also emerge vividly in Sec. 3, which deals with the extent of the transform coefficients and associated uncertainty relations. The topic of extent is crucial to the utility of the Laguerre-transform method as a numerical tool. Numerical examples are presented in Sec. 5.     

On the semiring of cone preserving maps

If K is a proper cone in Rⁿ, then the cone of all linear operators that preserve K, denoted by π(K), forms a semiring under usual operator addition and multiplication. Recently J.G. Horne examined the ideals of this semiring. He proved that if K₁, K₂ are polyhedral cones such that π(K₁) and π(K₂) are isomorphic as semirings, then K₁ and K₂ are linearly isomorphic. The study of this semiring is continued in this paper. In Sec. 3 ideals of π(K) which are also faces are characterized. In Sec. 4 it is shown that π(K) has a unique minimal two-sided ideal, namely, the dual cone of π(K¹), where K¹ is the dual cone of K. Extending Horne's result, it is also proved that the cone K is characterized by this unique minimal two-sided ideal of π(K). The set of all faces of π(K) inherits a quotient semiring structure from π(K). Properties of this face-semiring are given in Sec. 5. In particular, it is proved that this face-semiring admits no nontrivial congruence relation iff the duality operator of π(K) is injective. In Sec. 6 the maximal one-sided and two-sided ideals of π(K) are identified. In Sec. 8 it is shown that π(K) never satisfies the ascending-chain condition on principal one-sided ideals. Some partial results on the question of topological closedness of principal one-sided ideals of π(K) are also given.     

New Series of Rational Approximations and Some of Their Applications

We consider the well-known discrete logarithm problem in a finite simple field GF( $p$ ), where $p$ is a prime number, which has several application in problems of information protection. In Sec. 1, we introduce and study some number sequences arising in the continued fraction expansion of a real number. The results obtained are used in Sec. 2, where we introduce a new algorithm based on rational approximations for solving the problem of representing the discrete logarithm of a given number as the sum of logarithms of small primes; this problem is an important part of the discrete logarithm problem. We obtain several results necessary to construct and justify the representation algorithm. This algorithm is stated exactly in Sec. 3. We present several experimental results illustrating the work of the algorithm for prime numbers of the order of 10¹⁶10³¹.     

Maximum and minimum weighted diagonal sums of certain non-negative matrices

This paper extends the notion of diagonal sums of a square matrix to “weighted diagonal sums”. Using simple probabilistic arguments, most of the results of Wang [5] concerning the maximum and minimum diagonal sums of doubly stochastic matrices are extended to maximum and minimum weighted diagonal sums of stochastic matrices (Sec. 3). Two stronger versions of one of Wang's conjectures are also proven (Theorems 4.1 and 5.1), of which the latter easily generalizes to the case of non-negative matrices (Theorem 5.2). The paper ends with a few open questions and counter-examples.     

A Theorem on Multiplicators in Nonhomogeneous Hölder Spaces and Some of Its Applications

In Sec. 2, sufficient conditions of the boundedness of convolution operators with kernel m in a nonhomogeneous Hölder space are given in terms of the Fourier-image of m. After that, the results of Sec. 2 are used to prove the solvability in a Hölder space of the Cauchy problem for linear systems of the hydrodynamic type. Bibliography: 6 titles.     

Einstein metrics on five-dimensional Seifert bundles

The aim of this article is to study Seifert bundle structures on simply connected 5-manifolds. We classify all such 5-manifolds which admit a positive Seifert bundle structure, and in a few cases all Seifert bundle structures are also classified. These results are then used to construct positive Ricci curvature Einstein metrics on these manifolds. The proof has 4 main steps. First, the study of the Leray spectral sequence of the Seifert bundle, based on work of Orlik-Wagreich. Second, the study of log Del Pezzo surfaces. Third, the construction of Kähler-Einstein metrics on Del Pezzo orbifolds using the algebraic existence criterion of Demailly-Kollár. Fourth, the lifting of the Kähler-Einstein metric on the base of a Seifert bundle to an Einstein metric on the total space using the Kobayashi-Boyer-Galicki method.     

On Some Metaplectic Eisenstein Series

We study cubic metaplectic Eisenstein series connected with the Jacobi maximal parabolic subgroup of a symplectic group. We use the so-called ``sl(2)-triples' technique in order to evaluate the Fourier coefficients of these series. In Secs. 1 and 2, we introduce the necessary notation and study the group and its subgroups in detail. In Sec. 3, we prove the main result of the present paper (Theorem 1). Section 4 is devoted to the study of the Dirichlet series appearing in Theorem 1. Bibliography: 5 titles.     

On Isomorphisms of Relations Embedded into Each Other

In this paper, we consider analogs of the Cantor--Bernstein theorem for sets with binary relations. In Sec. 1, we prove an analog of this theorem for arbitrary binary relations; in Sec. 2, we consider an application; in Sec. 3, we study a class of relations with the Cantor-- Bernstein property and a class of exact relations, and prove that these classes are closed under certain operations.     

Parseval's equality in the abstract interpolation problem and the coupling of open systems. II

The constructions described in Sec. 1 are applied to the investigation of the abstract interpolation problem. The general solution of the problem is the characteristic function of an operator colligation, obtained by the closure of fixed colligation by means of an arbitrary colligation with definite exterior spaces. The complete integral representation of a nonnegative quadratic form is obtained by applying Parseval's equality, considered in Sec. 1.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 98–103, 1988.     

Geometric Dynamics

A kinematic differential system on a Riemann (or semi-Riemann) manifold induces a Lorentz-Udrite world-force law, i.e., any local group with one parameter (any local flow) on a Riemann (or semi-Riemann) manifold induces the dynamics of the given vector field or of an associated particle, which will be called geometric dynamics.The cases of Riemann-Jacobi or Riemann-Jacobi-Lagrange structures are imposed by the behavior of an external tensor field of type (1,1). The case of the Finsler-Jacobi structure appears if the initial metric is chosen such that the energy of the given vector field is constant (Sec. 1). At the end of Sec. 1 are formulated open problems regarding some extensions of geometric dynamics.Adequate structures on the tangent bundle describe the geometric dynamics in the Hamilton language (Sec. 2).Section 3 proves the existence of a Finsler-Jacobi structure induced by an almost contact metric structure.The theory is applied to electromagnetic dynamical systems (the starting point of our theory), offering new principles of unification of the gravitation and the electromagnetism. Also, here, one enounces open problems regarding the geometric dynamics induced by the electric intensity and magnetizing force (Sec. 4).From the geometrical point of view, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds ensuring that all trajectories of a given vector field are geodesics. Having T₁M²ⁿ⁺¹ in mind, the problem of creating a wider class of Riemannian manifolds, in which there exists a vector field such that (1) all trajectories of the vector field are geodesics; (2) the flow defined by is incompressible; (3) the condition which corresponds to the property that is the associate vector field of the contact structure is satisfied;was studied intensively by S. Sasaki. The results were not satisfactory, but Sasaki discovered (, , )-structures [10].AMS Subject Classification (1991): 70H35, 53C22, 58F25, 83C22     

Structures of topological type

In this paper we continue the study of structures of various types initiated by the author in the earlier paper Structures of extensions (Ref. Zh. Mat., 1974, 4A361). The present paper is devoted to the so-called structure of topological type. By a structure of topological type on the set X is meant a topological structure, defined on some set obtained from X, and possibly additional sets, by a totally ordered sequence of operations of unions of sets, products of sets, and passage to the set of subsets. We study certain structures of topological type: bitopological (Sec. 2) and settopological (Sec. 3). A bitopological structure on the set X is any topological structure on the set X×X, and a bitopological space is a pair (X,). This concept is a natural extension of the concept of a bitopological space as a set X on which there are given two topological structures ₁ and ₂-these structures define a structure =₁×₂ on the set X×X. A settopological structure on the set X is any topological structure on the set={A¦A. There are given representations of piecewise-linear structures (Sec. 4) and smooth structures (Sec. 5) as settopological structures.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 83, pp. 5–62, 1979.     

Exponential Function of Pseudo-Differential Operators in Gevrey Spaces

The article is devoted to the construction of the exponential function of the matrix pseudo-differential operator, which does not satisfy conditions of any known theorem (see, e.g. Sec. 8 Ch. VIII and Sec. 2 Ch. XI of Treves in Introduction to the theory of pseudodifferential and Fourier integral operator, vols. 1 & 2, Plenum Press, New York, 1980). An application of the exponential function to the fundamental solution of the Cauchy problem for the hyperbolic operators with the characteristics of variable multiplicity is given.