Bounded compactness of sets in linear metric spaces

An analog of the well known concept of bounded compactness from the theory of normed spaces is investigated in a linear metric space X. From the results obtained follows, for example, a sufficient condition for approximative compactness of a finite-dimensional subspace Lx.Translated from Matematicheskie Zametki, Vol. 11, No. 6, pp. 651–660, June, 1972.     

Hyperspaces of compact sets in metric linear spaces

We write 2^x for the hyperspace of all non-empty compact sets in a complete metric linear space X topologized by the Hausdorff metric. Using the notation $\text{F}$(X) = {A ϵ 2^X: A is finite}, l^f₂ = {x} = (x_i) ϵ l₂: x_i = 0 for almost all i}, and l^σ₂ = {x = (x _i) ϵ l₂:σ^∞_i=1 (ix_i)² < ∞}, we have the following theorem:A family $\text{G}$ ⊂$\text{F}$(X) is homeomorphic to l^f₂ if $\text{G}$ is σ-fd-compact, the closure $\text{G}$ of $\text{G}$ in 2^x is not locally compact and if whenever A, B ∈ $\text{G}$, λ ∈ [0, 1] and C ⊂ λA + (1 - λ)B with card C⩽ max{card A, card B} then C ϵ $\text{G}$.Moreover, for any G_δ-AR-set $\text{G}$$\text{G}$ of $\text{G}$ with $\text{G}$$\text{G}$ ⊃ $\text{G}$ we have ($\text{G}$$\text{G}$, $\text{G}$)≅(l₂, l^ƒ₂).Similar conditions for hyperspaces to be homeomorphic to l^σ₂ are also established.     

Connectedness and solar properties of sets in normed linear spaces

The connectedness of certain classes of suns is studied. In particular, it is proved that a compact sun is connected in a normed space. Every sun is connected in a uniformly non-square Banach space.Translated from Matematicheskii Zametki, Vol. 19, No. 2, pp. 267–278, February, 1976.     

The connectivity and approximative properties of sets in linear normed spaces

The connectivity of suns is investigated. It is shown that a sun is connected in finite-dimensional space. A set M in uniformly convex space X is shown to be approximatively compact if and only if M is P-compact and the metric projection of X onto M is upper semicontinuous.     

Hybrid triple systems and cubic feedback sets

Ac-hybrid triple system of orderv is a decomposition of the completev-vertex digraph intoc cyclic tournaments of order 3 and transitive tournaments of order 3. Hybrid triple systems generalize directed triple systems (c = 0) and Mendelsohn triple systems (c = v(v – 1)/3); omitting directions yields an underlying twofold triple system. The spectrum ofv andc for which ac-hybrid triple system of orderv exists is completely determined in this paper. Using (cubic) block intersection graphs, we then show that every twofold triple system of order underlies ac-hybrid triple system with . Examples are constructed for all sufficiently largev, for which this maximum is at most . The lower bound here is proved by establishing bounds onF _i (n, r), the size of minimum cardinality vertex feedback sets inn-vertexi-connected cubic multigraphs havingr repeated edges. We establish that , 8$$ " align="middle" border="0"> . These bounds are all tight, and the latter is used to derive the lower bound in the design theoretic problem.     

On the reachable sets in linear systems

We consider the finite-dimensional linear control system, with minimal assumptions concerning the control restraint set . Specifically, we consider the set of reachable states and conclude that as far as controllability is concerned, can be replaced by convhull().     

Bounded linear regularity of convex sets in Banach spaces and its applications

This paper deals with bounded linear regularity, linear regularity and the strong conical hull intersection property (CHIP) of a collection of finitely many closed convex intersecting sets in Banach spaces. It is shown that, as in finite dimensional space setting (see [6]), the standard constraint qualification implies bounded linear regularity, which in turn yields the strong conical hull intersection property, and that the collection of closed convex sets {C ₁, . . . ,C _n } is bounded linearly regular if and only if the tangent cones of {C ₁, . . . ,C _n } has the CHIP and the normal cones of {C ₁, . . . ,C _n } has the property (G)(uniformly on a neighborhood in the intersection C). As applications, we study the global error bounds for systems of linear and convex inequalities. The work of this author was partially supported by the National Natural Sciences Grant (No. 10471032) and the Excellent Young Teachers Program of MOE, P.R.C The authors thank professor K.F.Ng for his helpful discussion and the referee for their helpful suggestions on improving the first version of this paper     

Approximate convexity of average sums of sets in normed linear spaces

This note gives a sufficient condition under which the average sum of a large but finite number of sets in an arbitrary normed linear space is approximately convex. The result can be deduced as a simple corollary of an analogous result for a nonstandard universe. The latter has independent interest.     

Test closures of sets in Banach spaces generated by linear functionals

Some criteria are proved for an element of a Banach space to belong to the test or Korovkin closure of a given set. The operations of closure in question are determined by linear functionals. The cases of arbitrary linear functionals and of positive functionals are considered. Some sufficient conditions are given for the test closure to coincide with the Korovkin closure. Bibliography: 33 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 114–145. Translated by S. Yu. Pilyugin.     

Attainable sets for linear stochastic control systems

Let x_t^u(w) be the solution process of the n-dimensional stochastic differential equation dx_t^u = [A(t)x_t^u + B(t) u(t)] dt + C(t) dW_t, where A(t), B(t), C(t) are matrix functions, W_t is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rⁿ is (?, δ) attainable if there exists an admissable control u such that P{x_t^u?S_?(x)} ? δ, where S_?(x) is the closed ?-ball in Rⁿ centered at x. The set of all (?, δ) attainable points is denoted by $\text{A}$(t). In this paper, we derive various properties of $\text{A}$(t) in terms of K(t), the attainable set of the deterministic control system $\text{x}\text{?}\text{= A(t)x + B(t)u}$. As well a stochastic bang-bang principle is established and three examples presented.     

Hybrid procedures for solving linear systems

Summary. In this paper, we introduce the notion of hybrid procedures for solving a system of linear equations. A hybrid procedure consists in a combination of two arbitrary approximate solutions with coefficients summing up to one. Thus the combination only depends on one parameter whose value is chosen in order to minimize the Euclidean norm of the residual vector obtained by the hybrid procedure. Properties of such procedures are studied in detail. The two approximate solutions which are combined in a hybrid procedure are usually obtained by two iterative methods. Several strategies for combining these two methods together or with the previous iterate of the hybrid procedure itself are discussed and their properties are analyzed. Numerical experiments illustrate the various procedures. Received October 21, 1992/Revised version received May 28, 1993     

Zero sets and interpolating sets in Fock spaces

An example is constructed to show that interpolating sets for Fock spaces are not necessarily zero sets.