Showing papers on "Bounded function published in 1977"

Stochastic Inequalities on Partially Ordered Spaces

Abstract: In this paper we discuss characterizations, basic properties and applications of a partial ordering, in the set of probabilities on a partially ordered Polish space $E$, defined by $P_1 \prec P_2 \operatorname{iff} \int f dP_1\leqq \int f dP_2$ for all real bounded increasing $f$. A result of Strassen implies that $P_1 \prec P_2$ is equivalent to the existence of $E$-valued random variables $X_1 \leqq X_2$ with distributions $P_1$ and $P_2$. After treating similar characterizations we relate the convergence properties of $P_1 \prec P_2 \prec \cdots$ to those of the associated $X_1 \leqq X_2 \leqq \cdots$. The principal purpose of the paper is to apply the basic characterization to the problem of comparison of stochastic processes and to the question of the computation of the $\bar{d}-$distance (defined by Ornstein) of stationary processes. In particular we get a generalization of the comparison theorem of O'Brien to vector-valued processes. The method also allows us to treat processes with continuous time parameter and with paths in $D\lbrack 0, 1\rbrack$.

On the diffusion of biological populations

Abstract: This paper develops a model for the spatial diffusion of biological populations. Arguments are given in support of a degenerate, non-linear partial differential equation for the population density. Because of this degeneracy, a population which is initially confined to a bounded region spreads out at a finite speed, and may even remain confined for all time. A transformation is given which reduces our equation to an equation which arises in the theory of porous media. Using this transformation we are able to carry over to our theory theorems of existence and uniqueness for the one-dimensional initial-value problem as well as the solution for an initial point source.

Quelques propriétés des opérateurs angle-bornés et n -cycliquement monotones

Abstract: The paper deals with properties of σ-angle bounded linear and nonlinear operators. In particular it is proved that ifA is a σ-angle bounded linear maximal monotone operator in a real Hilbert space then −A is the infinitesimal generator of an analytic semi-group.

The kakeya maximal function and the spherical summation multipliers.

Abstract: obvious cases: n = 1 or p =2; and also Fefferman [4] proved that TX is bounded on L P (Rn) provided that p (X) (n - 1)/4. This result has been sharpened by Tomas [15] to X > (n - 1)/2(n + 1). Finally Carleson and Sjolin [3], Fefferman [6] and Hormander [10] proved that, in R2, TA is bounded on LP whenever X >0 and p(X) 2 we have the natural question: is TX (X >0) bounded on L P(R'), p(X) < p < p'(X)? Our approach to the problem is inspired by the work of Fefferman and it is as follows: The multiplier theorem for TX can be easily reduced to this problem:

A Note on the Lack of Exact Controllability for Mild Solutions in Banach Spaces

Abstract: It is shown that exact controllability in finite time for linear control systems given on an infinite dimensional Banach space in integral form (mild solution) can never arise using locally $L_1 $-controls, if the associated $C_0 $ semigroup is compact for all $t > 0$. This includes, in particular, the class of parabolic partial differential equations defined on bounded spatial domains.

On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space

Abstract: In some problems related to the spectral theory in Hubert space it is more natural and at the same time often less restrictive to use symmetric linear relations (in the terminology of [1], subspaces in the terminology of [2-4]) instead of symmetric operators. Hence the question arises if the theory of generalized resolvents of symmetric operators can be extended to symmetric linear relations. In [4] a description of all generalized resolvents of a symmetric linear relation was given, following the lines of A. V. Straus [5] in the operator case. It is the aim of this paper to generalize M. G. Kreϊn's formula for the generalized resolvents of a symmetric operator (see [6, 7]) to the symmetric linear relation case. This can be done rather easily by means of the Gayley transformation, using the results of [8]. However, in this connection there arise natural problems and questions: To introduce and to study the Q-function of a linear relation, to prove criteria for the selfadjoint extension of the given symmetric linear relation being an operator, to study the special case of a bounded nondensely defined operator etc. After the necessary definitions and their simple consequences in §1, the §2 is devoted to a study of the Q-f unction. From arguments similar to those in [9, 10] it follows that every function Q, whose values are bounded operators in a Hubert space and which is holomorphic in the upper half plane and has the property

Plant location, Set Covering and Economic Lot Size: An 0 (mn)-Algorithm for Structured Problems

Abstract: For the simple (uncapacitated) plant location problem for which the coefficient matrix possesses certain properties, we devise a polynomially bounded algorithm by means of which an optimal solution can be derived after a single pass.

A Note on Weak Stabilizability of Contraction Semigroups

Abstract: A recent result on weak stabilizability is that the system $\dot x = Ax + Bu$, where A is the infinitesimal generator of a contraction semigroup over a Hilbert space H, and B is linear bounded, is ...

Inverses of Band Matrices and Local Convergence of Spline Projections

Abstract: It is shown that the size of the entries in the inverse of a band matrix can be bounded in terms of the norm of the matrix, the norm of its inverse and the bandwidth In many cases this implies that the entries of the inverse decay to zero exponentially as they move away from the diagonal These results are used to obtain local convergence theorems for some spline projections

On Optimal Control Problems with Bounded State Variables and Control Appearing Linearly

Abstract: Necessary conditions for the switching function, holding at junction points of optimal interior and boundary arcs or at contact points with the boundary, are given. These conditions are used to derive necessary conditions for the optimality of junctions between interior and boundary arcs. The junction theorems obtained are similar to those developed for singular control problems in [1] and establish a duality between singular control problems and control problems with bounded state variables and control appearing linearly. The transition from unconstrained to constrained extremals is discussed with respect to the order p of the state constraint. A numerical example is given where the adjoins variables are not unique but form a convex set which is determined numerically.

On dentability and the Bishop-Phelps property

Abstract: It is shown that for Banach spaces the Radon-Nikodym property and the Bishop-Phelps property are equivalent. Using similar techniques, we prove that ifC is a bounded, closed and convex subset of a Banach space such that every nonempty subset ofC is dentable, then the strongly exposing functionals ofC form a denseGδ-subset of the dual.

A regularity condition at the boundary for solutions of quasilinear elliptic equations

Abstract: In this paper we are concerned with the behavior at the boundary of weak solutions of the Dirichlet problem for quasilinear elliptic equations of second order in an open set O ~ IR". The main result is that a bounded weak solution is continuous at a boundary point of f2 provided that the complement of f2 in a neighborhood of this point is sufficiently " thick" when measured by an appropriate capacity. In the case of Laplace's equation this condition reduces to that considered by WIENER [W1], [W2"]. The equations considered are of the general divergence structure type

Supergravity Has Positive Energy

Abstract: We show that the total energy in supergravity theory is nonnegative. The derivation is based on the fact that for bounded systems the Hamiltonian operator is the square of the spinor supercharge just as for globally supersymmetric systems. The result holds for any of the pure, matter-coupled, or extended versions of supergravity, except those with cosmological term for which the concept of total energy does not exist.

The integrated squared error estimation of parameters

Abstract: SUMMARY The paper develops in a more general setting the methods used' by Paulson, Holcomb & Leitch (1975) to estimate the parameters of a stable law The statistic considered minimizes a distance function determined, by the empirical characteristic function Consistency is established under the condition of differentiability of the characteristio function and the existence of bounded second derivatives is required to obtain a central limit theorem for the estimators of one or more parameters Questions concerning efficiency and robustness are discussed

Point-Wise Compact Subsets of the First Baire Class

Abstract: Let X be a complete separable metric space. Various characterizations of point-wise compact subsets of the first Baire class of real-valued functions on X are obtained. For example, it is proved that a compact subset is sequentially compact in the topology of point-wise convergence, and moreover (in the case where it is additionally uniformly bounded) that it is compact with respect to the topology induced by the set of Borel probability measures on X. The results are applied to show that 11 imbeds in a separable Banach space B provided there exists a bounded sequence in B** which has no weak*-convergent subsequence.

A Characterization of Stability in Linear Programming

Abstract: We prove that a necessary and sufficient condition for the primal and dual solution sets of a solvable, finite-dimensional linear programming problem to be stable under small but arbitrary perturbations in the data of the problem is that both of these sets be bounded. The distance from any pair of solutions of the perturbed problem to the solution sets of the original problem is then bounded by a constant multiple of the norm of the perturbations. These results extend earlier work of Williams.

Differential Stability in Nonlinear Programming

Abstract: This paper consists of a study of stability and differential stability in nonconvex programming. For a program with equality and inequality constraints, upper and lower bounds are estimated for the potential directional derivatives of the perturbation function (or the extremal-value function). These results are obtained' with the help of a constraint qualification which is shown to be necessary and sufficient to have bounded multipliers. New results on the continuity of the perturbation function are also obtained.

Remarks on Weak Compactness in L1(μ,X)

Abstract: Let (Ω,Σ,μ) be a finite measure space and X a Banach space. Denote by L1 (μ,X) the Banach space of (equivalence classes of) μ-strongly measurable X-valued Bochner integrable functions f:Ω→X normed byThe problem of characterizing the relatively weakly compact subsets of L1(Ω, X) remains open. It is known that for a bounded subset of L1(μ, X) to be relatively weakly compact it is necessary that the set be uniformly integrable; recall that K ⊆ L1, (μ, X) is uniformly integrable whenever given e >0 there exists δ > 0 such that if μ (E) ≦ δ then ∫E∥f∥ dμ ≦ δ, for all f ∈ K. S. Chatterji has noted that in case X is reflexive this condition is also sufficient [4]. At present unless one assumes that both X and X* have the Radon-Nikodym Property (see [1]), a rather severe restriction which, for purposes of potential applicability, is tantamount to assuming reflexivity, no good sufficient conditions for weak compactness in L1(μ, X) exist. This note puts forth such sufficient conditions; the basic tool is the recent factorization method of W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski [3].

A bounded operator approach to tomita-takesaki theory

Abstract: of the paper we show that the operators Δu and J can, in fact, be associated with any fairly general real subspace of a complex Hubert space, and that many of their properties, for example the characterization of Au in terms of the K.M.S. condition, can be derived in this less complicated setting. In the second half of the paper we show, by using some of the ideas from the first half, that a simplified proof of the Tomita-Takesaki theory given recently by the second author can be reformulated entirely in terms of bounded operators, thus further simplifying it by, among other things, eliminating all considerations involving domains of unbounded operators.

On Banach spaces whose dual balls are not weak∗ sequentially compact

Abstract: Theorem 1. LetX be a Banach space. (a) IfX ∗ has a closed subspace in which no normalized sequence converges weak∗ to zero, thenl 1 is isomorphic to a subspace ofX. (b) IfX ∗ contains a bounded sequence which has no weak∗ convergent subsequence, thenX contains a separable subspace whose dual is not separable.

Smoothing and extending cosmic time functions

Abstract: Any continuous time function on aC k space-timeV (ie, a scalar onV that increases along any causal curve) can be approximated by smoothC k time functions A time function defined on a (bounded) subset of a stably causalV can be extended to a time function on the whole ofV