Spaces of Densely Continuous Forms

The set C(X,Y) of continuous functions from a topological space X into a topological space Y is extended to the set D(X,Y) of densely continuous forms from X to Y, such form being a kind of multifunction from X to Y. The topologies of pointwise convergence, uniform convergence, and uniform convergence on compact sets are defined for D(X,Y), for locally compact spaces X and metric spaces Y having a metric satisfying the Heine–Borel property. Under these assumptions, D(X,Y) with the uniform topology is shown to be completely metrizable. In addition, if X is compact, D(X,Y) is completely metrizable under the topology of uniform convergence on compact sets. For this latter topology, an Ascoli theorem is established giving necessary and sufficient conditions for a subset of D(X,Y) to be compact.     

Sufficient Conditions for Error Bounds and Applications

Our aim in this paper is to present sufficient conditions for error bounds in terms of Fréchet and limiting Fréchet subdifferentials in general Banach spaces. This allows us to develop sufficient conditions in terms of the approximate subdifferential for systems of the form (x, y) C × D, g(x, y, u) = 0, where g takes values in an infinite-dimensional space and u plays the role of a parameter. This symmetric structure offers us the choice of imposing conditions either on C or D. We use these results to prove the nonemptiness and weak-star compactness of Fritz–John and Karush–Kuhn–Tucker multiplier sets, to establish the Lipschitz continuity of the value function and to compute its subdifferential and finally to obtain results on local controllability in control problems of nonconvex unbounded differential inclusions.     

Quasilinear operators and Hammerstein's equation

We describe the class of operators in a Hilbert space H, introduced by A. I. Perov, which can be represented in the form Ax = D(x)x, where D(x) is a self-conjugate operator satisfying the inequalities B_–D(x) B₊ (B_– and B₊ are fixed self-conjugate operators). As an application we obtain new theorems on the solvability of Hammerstein's equation.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 453–464, October, 1972.     

CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems

The goal of this work is to derive and justify a multilevel preconditioner of optimal arithmetic complexity for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. Our approach is based on the following simple idea given in [R.D. Lazarov, P.S. Vassilevski, L.T. Zikatanov, Multilevel preconditioning of second order elliptic discontinuous Galerkin problems, Preprint, 2005]. The finite element space of piece-wise polynomials, discontinuous on the partition , is projected onto the space of piece-wise constant functions on the same partition that constitutes the largest space in the multilevel method. The discontinuous Galerkin finite element system on this space is associated to the so-called “graph-Laplacian”. In 2-D this is a sparse M-matrix with -1 as off diagonal entries and nonnegative row sums. Under the assumption that the finest partition is a result of multilevel refinement of a given coarse mesh, we develop the concept of hierarchical splitting of the unknowns. Then using local analysis we derive estimates for the constants in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality, which are uniform with respect to the levels. This measure of the angle between the spaces of the splitting was used by Axelsson and Vassilevski in [Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569–1590] to construct an algebraic multilevel iteration (AMLI) for finite element systems. The main contribution in this paper is a construction of a splitting that produces new estimates for the CBS constant for graph-Laplacian. As a result we have a preconditioner for the system of the discontinuous Galerkin finite element method of optimal arithmetic complexity.     

Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a system of open neighbourhoods at infinity while an exterior map is a continuous map which is continuous at infinity. The category of spaces and proper maps is a subcategory of the category of exterior spaces.In this paper we show that the category of exterior spaces has a family of closed simplicial model structures, in the sense of Quillen, depending on a pair {T,T} of suitable exterior spaces. For this goal, for a given exterior space T, we construct the exterior T-homotopy groups of an exterior space under T. Using different spaces T we have as particular cases the main proper homotopy groups: the Brown–Grossman, erin–Steenrod, p-cylindrical, Baues–Quintero and Farrell–Taylor–Wagoner groups, as well as the standard (Hurewicz) homotopy groups.The existence of this model structure in the category of exterior spaces has interesting applications. For instance, using different pairs {T,T}, it is possible to study the standard homotopy type, the homotopy type at infinity and the global proper homotopy type.     

Hilbert Spaces Contractively Included in the Hardy Space of the Bidisk

We study the reproducing kernel Hilbert spaces with kernels of the form

Regular-Norm Balls can be Closed in the Strong Operator Topology

The main results obtained are:– A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T Lr(X,Y): ||T||r 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y).– A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y).– A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm.– A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm.Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance:– If X = Lp() and Y = Lq(), where 1 < p,q < , then Lr(X,Y) is a first category subset of L(X,Y).     

Invariant subspaces of the shift operator. Axiomatic approach

There is axiomatically described the class of spaces (resp. ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) z f (z) (resp. the inverse shift f(z)z^–1(f(z)–f (0))) are constructed just like the Hardy space H². It is proved that as one can take, for example, the space H¹, the disk-algebra C_A, the space U_A of all uniformly convergent power series; and as the space of integrals of Cauchy type L¹/H _– ¹ , the space VMO_A. There is also obtained an analog for the space U_A of W. Rudin's theorem on z-invariant subspaces of the space C_A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 113, pp. 7–26, 1981.     

On the rate of convergence of certain random series in terms of norms of Orlicz spaces

There are obtained conditions for the convergence and estimates of the rate of convergence in terms of norms of Orlicz spaces of random series of the form where f_k(¯t) is an orthonormal system of eigenfunctions of a certain integral equation, k are random variables in an Orlicz space, ¯t Rⁿ.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 841–848, June, 1991.     

A canonical form for self-adjoint pencils in Hilbert space

We characterize those pencils P=A–B of operators on a separable Hilbert space H for which a linear homeomorphism D of H exists satisfying the following: (i) H decomposes into a direct sum F+G, orthogonal in a (perhaps indefinite) inner product induced by A, where F is finite dimensional (ii) D^*PD=P_F(I_G–C) where P_F is a (congruence) canonical form for the general self-adjoint pencil on F, and C is a bounded self-adjoint operator on G. For a given P, explicit constructions are given for C, D, F, G and P_F.     

Some spectral properties of a bundle of linear operators in Banach space

The spectral properties of a bunchA–B, D(A)D(B), of linear closed densely defined operators in Banach space are considered. Our main result is a theorem to the effect that the spectrum of the bunch can be expanded with respect to a pair of direct sums; the theorem generalizes the celebrated theorem of Riesz concerning the expansion of the spectrum of an operator.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 847–857, December, 1977.     

The grothendieck property in the spacel ∞ (E) and the weak Banach-Saks property inc 0 (E)

We prove the following: 1) if E is a B-convex Banach lattice, the space l(E) is a Grothendieck space; 2) if the space E has the p-Banach-Saks property for some p>1, the space c₀(E) also has this property. It is shown by examples that these conditions are close to being necessary. These results are applied to study the geometric structure of limits of symmetric spaces.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 39–44.     

Dirac equation and spin 1 representations,a connection with symmetries of the Maxwell equations

A unitary operator on the space of spinors that makes it possible to associate each transformation in this space with a transformation in the space of electromagnetic field strengths is found. A connection is established by means of this operator between representations in the space of spinors and the space of field strengths for the Lorentsz, Poincaré, and conformal groups. Unusual symmetries of the Dirac equation are found on this basis. It is noted that the Pauli—Gürsey symmetry operators (without the ⁵ operator) of the Dirac equation withm=0 form the same representation D(1/2, 0)D(0, 1/2) of the O(1, 3) algebra of the Lorentz group as the spin matrices of the standard spinor representation. It is shown that besides the standard (spinor) representation of the Poincaré group, the massless Dirac equation is invariant with respect to two other representations of this group, namely, the vector and tensor representations specified by the generators of the representations D(1/2, 1/2) and D(1, 0) D(0, 0) of the Lorentz group, respectively. Unusual families of representations of the conformal algebra associated with these representations of the group O(1, 3) are investigated. Analogous O(1, 2) and P(1, 2) invariance algebras are established for the Dirac equation withm>0.Institute of Nuclear Research, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 388–406, March, 1992.     

A Stone–Weierstrass Type Theorem for an Unstructured Set – with Applications

A general version of the Stone–Weierstrass theorem is presented – one which involves no structure on the domain set of the real valued functions. This theorem is similar to the Stone–Weierstrass theorem which appears in the book by Gillman and Jerison, but instead of involving the concept of stationary sets the one presented here involves stationary filters. As a corollary to our results we obtain Nel's theorem of Stone–Weierstrass type for an arbitrary topological space. Finally, an application is made to the setting of Cauchy spaces.     

Axiomatic theory of convexity

The axiomatic construction of the theory of convexity proceeds from an arbitrary set M and a mapping l: M² 2^M, i.e., from a pair (M, l). It is shown that such a space of a certain type is domain finite. A condition is given which, for such spaces, implies join-hull commutativity. A connection is established between the Carathéodory number and join-hull commutativity. Conditions are given which imply a separation property of the space (M, l). Convexity spaces which are domain finite are characterized.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 761–770, November, 1976.     

On the modulus of -convexity

In this paper, we prove that the moduli of W^*-convexity, introduced by Ji Gao [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386], of a Banach space X and of the ultrapower of X itself coincide whenever X is super-reflexive. Moreover, we improve a sufficient condition for uniform normal structure of the space and its dual. This generalizes and strengthens the main results of [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386].     

Low-type submanifolds of real space forms via the immersions by projectors

We undertake a comprehensive study of submanifolds of low Chen-type (1, 2, or 3) in non-flat real space forms, immersed into a suitable (pseudo) Euclidean space of symmetric matrices by projection operators. Some previous results for submanifolds of the unit sphere (obtained in [A. Ros, Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z. 187 (1984) 393–404; M. Barros, B.Y. Chen, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, Nagoya Math. J. 108 (1987) 77–91; I. Dimitrić, Spherical hypersurfaces with low type quadric representation, Tokyo J. Math. 13 (1990) 469–492; J.T. Lu, Hypersurfaces of a sphere with 3-type quadric representation, Kodai Math. J. 17 (1994) 290–298]) are generalized and extended to real projective and hyperbolic spaces as well as to the sphere. In particular, we give a characterization of 2-type submanifolds of these space forms with parallel mean curvature vector. We classify 2-type hypersurfaces in these spaces and give two sets of necessary conditions for a minimal hypersurface to be of 3-type and for a hypersurface with constant mean curvature to be mass-symmetric and of 3-type. These conditions are then used to classify such hypersurfaces of dimension n5. For example, the complete minimal hypersurfaces of the unit sphere Sⁿ⁺¹ which are of 3-type via the immersion by projectors are exactly the 3-dimensional Cartan minimal hypersurface and the Clifford minimal hypersurfaces M_k,n−k for n≠2k. An interesting characterization of horospheres in is also obtained.     

Certain classes of continuous linear operations

Certain classes of continuous linear operators in Banach and locally convex spaces are studied. A characterization of operators T: X Y, transforming bounded sets of the Banach space X into conditionally weakly compact sets of the Banach space Y, is given, and also a particular case where X = C(K) is considered. It is proved that if E is a Fréchet space and F is a complete ()-space, then the classes of absolutely summing and Nikodýmizing operators from E into F coincide.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 285–296, February, 1978.     

Sur des notions de n—categorie et n—groupoide non strictes via des ensembles multi-simpliciaux (On the notions of a nonstrict n–category and n–groupoid via multisimplicial sets)

In this paper we first give a simplicial approach to the definition of a nonstrict n–category that we call a n–nerve following the idea that a category could be interpreted as a simplicial set (its nerve). Then we prove that for n=2 our construction is equivalent to the usual nonstrict 2–category (bicategory). Next,we give a simplicial definition of a nonstrict n–groupoïd, and we associate to any topological space a n–groupoïd n (X) which generalises the famous Poincaré groupoïd 1 (X) and embodies the n–truncated homotopy type of . Conversely, we construct for each n–groupoïd a geometric realisation and we show that the functors geometric realisation and Poincaré n–groupoïd induce an equivalence between the category of n–groupoids and the category of n–truncated topological spaces, when we localise both categories by weak equivalence.