Showing papers on "Bounded function published in 1973"

Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems

Abstract: This paper describes a technique for obtaining error bounds for certain characteristic subspaces associated with the algebraic eigenvalue problem, the generalized eigenvalue problem, and the singular value decomposition. The method also gives perturbation bounds for isolated eigenvalues and useful information about clusters of eigenvalues. The bounds are obtained from an iterative process for generating the subspaces in question, and one or more steps of the iteration can be used to construct perturbation estimates whose error can be bounded.

Convergence of Estimates Under Dimensionality Restrictions

Abstract: Consider independent identically distributed observations whose distribution depends on a parameter $\theta$. Measure the distance between two parameter points $\theta_1, \theta_2$ by the Hellinger distance $h(\theta_1, \theta_2)$. Suppose that for $n$ observations there is a good but not perfect test of $\theta_0$ against $\theta_n$. Then $n^{\frac{1}{2}}h(\theta_0, \theta_n)$ stays away from zero and infinity. The usual parametric examples, regular or irregular, also have the property that there are estimates $\hat{\theta}_n$ such that $n^{\frac{1}{2}}h(\hat{\theta}_n, \theta_0)$ stays bounded in probability, so that rates of separation for tests and estimates are essentially the same. The present paper shows that need not be true in general but is correct under certain metric dimensionality assumptions on the parameter set. It is then shown that these assumptions imply convergence at the required rate of the Bayes estimates or maximum probability estimates.

On search directions for minimization algorithms

Abstract: Some examples are given of differentiable functions of three variables, having the property that if they are treated by the minimization algorithm that searches along the coordinate directions in sequence, then the search path tends to a closed loop. On this loop the gradient of the objective function is bounded away from zero. We discuss the relevance of these examples to the problem of proving general convergence theorems for minimization algorithms that use search directions.

FK Spaces in Which the Sequence of Coordinate Vectors is Bounded

Abstract: The work presented in this paper was initially motivated by the following question of A. Wilansky: “Is there a smallest FK-space E in which is bounded?” Here FK-space means a complete linear metric space of real or complex sequences x = (x i ) upon which the coordinate functional x → xt are continuous for each i (see [10, p. 202]), and An FK-space need not be locally convex, and therein lies the difficulty of the problem since it is easy to see that l1 is the smallest locally convex FK-space.

The Rayleigh hypothesis and a related least‐squares solution to scattering problems for periodic surfaces and other scatterers

Abstract: The Rayleigh hypothesis is reviewed in relation to scattering by periodic surfaces, aperiodic surfaces, and bounded, two-dimensional bodies. Conditions for its validity are described, and explicit results are quoted for a sinusoidal grating. Some methods to solve scattering problems for periodic surfaces are outlined. One particular procedure for periodic surfaces and bounded scatterers is examined in detail. This involves an expansion for the scattered field in terms of the same sets of elementary wavefunctions that occur in connection with the Rayleigh hypothesis. The coefficients are determined by satisfying the boundary condition in the least-squares sense. It is shown that this solution converges uniformly to the scattered field at all points exterior to the boundary of the scatterer. Necessary completeness properties of the sets of wavefunctions are established in the appendices.

Bounded Production and Inventory Models with Piecewise Concave Costs

Abstract: An n period single-product single-facility model with known requirements and separable piecewise concave production and storage costs is considered. It is shown using network flow concepts that for arbitrary bounds on production and inventory in each period there is an optimal schedule such that if, for any two periods, production does not equal zero or its upper or lower bound, then the inventory level in some intermediate period equals zero or its upper or lower bound. An algorithm for searching such schedules for an optimal one is given where the bounds on production are -∞, 0 or ∞. A more efficient algorithm assumes further that inventory bounds satisfy “exact requirements.”

Set Merging Algorithms

Abstract: This paper considers the problem of merging sets formed from a total of n items in such a way that at any time, the name of a set containing a given item can be ascertained. Two algorithms using different data structures are discussed. The execution times of both algorithms are bounded by a constant times $nG(n)$, where $G(n)$ is a function whose asymptotic growth rate is less than that of any finite number of logarithms of n.

Positivity of the ϕ34 Hamiltonian

Abstract: The renormalized φ Hamiltonian is bounded from below by a constant proportional to the volume.

A-Stable Methods and Padé Approximations to the Exponential

Abstract: The set of Pade approximations to the exponential function is studied. It is shown that all entries on the first and second subdiagonal of the Pade table are analytic and bounded by 1 in the entire left half-plane. These results are then applied to the problem of producing A-stable numerical methods for solving initial value problems. It is shown that they easily permit one to generate several classes of methods of arbitrarily high order which are A-stable.

Continuous dependence of fixed point sets

Abstract: The stability of the fixed point sets of a uniformly convergent sequence of set valued contractions is proved under the assumption that the maps are defined on a closed bounded subset B of Hubert space and take values in the family of nonempty closed convex subsets of B. In (1) the convergence of a sequence of fixed points of a convergent sequence of set valued contractions was investigated in a metric space setting. By restricting the underlying space to be a Hubert space we prove the convergence of the sequence of fixed point sets of a convergent sequence of set valued contractions. This also extends a similar result for point valued maps (2, Theorem (10.1.1)) to the set valued case. Let A be a closed bounded subset of a Hubert space H, d the norm of H, and D the Hausdorff metric on the closed subsets of A generated by d. We assume that the family of set valued maps Fk, k=0, I, ■ ■ ■ , satisfy

Semi-Markov Decision Processes with Unbounded Rewards

Abstract: We consider a semi-Markov decision process with arbitrary action space; the state space is the nonnegative integers. As in queueing systems, we assume that {0, 1, 2,..., n + N} is the set of states accessible from state n in one transition, where N is finite and independent of n. The novel feature of this model is that the one-period reward is not required to be uniformly bounded; instead, we merely assume it to be bounded by a polynomial in n. Our main concern is with the average cost problem. A set of conditions sufficient for there to be an optimal stationary policy which can be obtained from the usual functional equation is developed. These conditions are quite weak and, as illustrated in several queueing examples, are easily verified.

On normal derivations

Abstract: Let A, be the derivation on S3(.X) defined by AT(X)= TX-XT (T, X E 8(,)). We prove that if T is an isometry or a normal operator, then the range of AT is orthogonal to the null space of AT. Also, we prove that if T is normal with an infinite number of points in its spectrum then the closed linear span of the range and the null space of AT is not all of 98(,). Introduction. If *' is a Hilbert space and 3(X) is the algebra of all bounded linear operators on X, then for each fixed T E 23(X)f) the operator equation AT(X) = TX XT defines a bounded linear operator on 23(X)AT is called a derivation because, for all X, Yin 23(X), AT(XY) = AT(X) Y + XAT(Y). When N is a normal operator in Q3(5X6) we will say that AN is a normal derivation. If T E 23(X) has a particular property it is often the case that AT has a similar property. For example if Tis selfadjoint then it is easy to show that the numerical range of AT is real; i.e., that AT is Hermitian in the sense of Lumer and Vidav (see [4]). Also, if Nis normal then it is shown in [1] that AN is a generalized scalar operator. When N is a normal operator in 23(9X) with null space X(N) and range J(N) it is elementary that (i) 91(N) I X(N), (ii) 91(N)-EDJ(N) = X. In this note we study the extent to which AN shares these properties. We find that the range 9 (AN) and the null space X(IAN) are "orthogonal" in a certain sense so that (i) holds, but that (ii) holds if and only if the spectrum of N contains only a finite number of points. In the last section we mention some open questions. Received by the editors September 21, 1970 and, in revised form, July 14, 1972. AMS (MOS) subject classifications (1969). Primary 4710, 4755; Secondary 4610.

Uniform estimates for the\(\bar \partial \)-equation on domains with piecewise smooth strictly pseudoconvex boundaries

Abstract: Henkin [3] and Grauert-Lieb I7] pioneered in the investigation of uniform estimates for the ~-equation. They independently proved the following: if D is a strictly pseudoconvex domain in fEn with smooth boundary and i f f is a uniformly bounded R-closed C °~ (0, 1)-form on D, then there exists a uniformly bounded C ° function u on D with du = f. Their proofs depend on the explicit construction of a solution by means of holomorphic kernels introduced by Henkin [2] and Ramirez [19]. Later, by using a local version of the Grauert-Lieb method, Kerzman 1'9] generalized the result to a strictly pseudoconvex domain with smooth boundary in a Stein manifold. Moreover, he showed that the type of solution constructed by Grauert-Lieb for such a domain yields also LP-estimates and H61der estimates with exponents 1/2. By modifying Henkin's solution of the ~-equation, Henkin and Romanov 1'5] obtained the H61der estimate with exponent t/2. Making use of Koppelman's results 1'1 t], Lieb [13, 14] proved that there exist uniform estimates and H/ilder estimates with exponent < 1/2 for the ~-equation for (0, q)-forms on strictly pseudoconvex domains with smooth boundary. Independently ~vrelid [16] also obtained uniform estimates and L p estimates for the ~-equation for (0, q)-forms on the same class of domains. Recently Henkin 1'4] announced the solution of the R-equation with uniform bounds for (0, 1)-forms for certain analytic polyhedra and indicated that his proof makes use of the type of Cauchy-Fantappie kernel introduced by Norguet 1'15]. In this paper we investigate uniform estimates for solutions of the R-equation on a domain with piecewise smooth strictly pseudoconvex boundary. More precisely, suppose D is a bounded domain in C n with aD covered by finitely many open subsets Uj(l _~j_~ k) ofC n and suppose 01 is a ¢2 strictly plurisubharmonic function on UI(1 < j < k) such that

A continuous transformation useful for districting

Abstract: Suppose that one could stretch a geographical map so that areas containing many people would appear large, and areas containing few people would appear small. On a rubber map, for example, every person might be represented by an inked dot. We now imagine the rubber sheet to be stretched so that all the dots are at an equal distance from each other (see Ruston'). If such a map could be constructed, then all perfect political districts should be the same size, for they should contain equal numbers of people. Alternately, one might wish to construct district boundaries by drawing them as hexagons on such a map. These general notions are made more precise and given mathematical definition in the paragraphs that follow. The political problem of distiicting is related to a classical theoretical problem in the field of geography. The location-theoretic problem of positioning service facilities (schools, hospitals, stores, cities, and so forth) in a geographic area of varying population density has a structure similar to the districting problem (see Bunge2). This theoretical problem in geography provided the main impetus for my research, and it is reflected in the results to be demonstrated. Assume first that the relevant population density is described as a continuous nonnegative function of position, h(x, y). A small rectangle bounded by x, x + Ax, y, y + Ay then contains h(x, y)AxAy people. This number is to be the same as the area of a small rectangle bounded by the lines u, u + Au, v, v + Av on the final diagram. The condition equation thus becomes

On two conjectures in the fixed membrane eigenvalue problem

Abstract: In this paper we give conditions sufficient to insure that the following two results hold for eigenfunctions of the fixed membrane problem defined over a bounded region of the plane:

Global Controllability of Linear Systems with Positive Controls

Abstract: A linear autonomous differential control system is considered, where the zero control is an extreme point of the restraint set. Necessary and sufficient conditions are given for global controllability in the case of bounded or unbounded scalar control.

The Pseudo-Inverse of a Product

Abstract: Let A and B be bounded linear operators on a complex Hilbert space H, such that the range of each is a closed subspace of H. The following three conditions are necessary and sufficient for the pseudo-inverse of $AB$to be the pseudo-inverse of A followed by the pseudo-inverse of B : (i) the range of $AB$ must be closed; (ii) the range of $A^ * $ must be invariant under $BB^ * $; (iii) the intersection of the range of $A^ * $ and the kernel of $B^ * $ must be invariant under $A^ * A$. We use this basic result to obtain a simple technique for computing the pseudo-inverse of a given operator, particularly a given matrix.

State Reduction in Incompletely Specified Finite-State Machines

Abstract: The problem of reducing the number of states in an arbitrary incompletely specified deterministic finite-state machine to k states (for a given k) has proven intractible to solution within "reasonable" time; most techniques seem to require exponential time. Two reduction techniques–state assignment to the DON'T CARE entries, and so-called "state splitting"–are investigated. For both of the techniques, the question, "Can I achieve an equivalent k state machine?" is shown to be polynomial complete, with the resulting conjecture that neither is solvable in time bounded by a polynomial function of the size of the machine.

Controllability of discrete bilinear systems with bounded control

Abstract: Controllability of time-invariant discrete-time bilinear systems is discussed. Bilinear systems are classified into two categories: homogeneous and inhomogeneous. Sufficient conditions which ensure the global controllability of discrete-time bilinear systems are obtained by localized analysis in control variables.

Galerkin Methods for First Order Hyperbolics: An Example

Abstract: The continuous-time Galerkin method is studied for the equation $u_t + u_x = 0$ with periodic solution. If the space of possible approximate solutions is taken to be $C^1 $ piecewise cubic polynomials on mesh of size h, then the $L^2 $-norm of the error is in general no better than $ch^3 $; if the class of possible approximate solutions is taken to be $C^2 $ piecewise cubic polynomials on this mesh, the error is bounded by $ch^4 $ for sufficiently smooth solutions.

Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces

Abstract: Let X be a real Banach space, D a bounded open subset of X, and D the closure of D. In ?1 of this paper we establish a general fixed point theorem (see Theorem 1 below) for I-set-contractions and 1-ball-contractions T: D? X under very mild conditions on T. In addition to classical fixed point theorems of Schauder, Leray and Schauder, Rothe, Kransnoselsky, Altman, and others for T compact, Theorem 1 includes as special cases the earlier theorem of Darbo as well as the more recent theorems of Sadovsky, Nussbaum, Petryshyn, and others (see ?1 for further contributions and details) for T k-set-contractive with k < 1, condensing, and l-set-contractive. In ??2, 3, 4, and 5 of this paper Theorem 1 is used to deduce a number of known, as well as some new, fixed point theorems for various special classes of mappings (e.g. mappings of contractive type with compact or completely continuous perturbations, mappings of semicontractive type introduced by Browder, mappings of pseudo-contractive type, etc.) which have been recently extensively studied by a number of authors and, in particular, by Browder, Krasnoselsky, Kirk, and others (see ?1 for details), Introduction. Let X be a real Banach space, D a bounded open subset of X9 D and OD the closure and the boundary of D9 respectively. The object of this paper is two-fold. First, in ?1 we extend our main fixed point result (see Theorem 7' in Petryshyn [34]) by proving (see Theorem 1 below) that if T: D X is either a I-set or 1-ball contraction which satisfies the Leray-Schauder condition on OD, then T has a fixed point in D if and only if T satisfies condition (c). As will be seen9 Theorem 1 unifies and extends in some cases to nonconvex domains and/or to more general boundary conditions a number of known9 as well as some new, fixed Received by the editors October 4, 1971 and, in revised form, November 2, 1972. AMS (MOS) subject classifications (1970). Primary 47H10; Secondary 47H99.

Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non polynomial interactions

Abstract: We consider quantum field theoretical models inn dimensional space-time given by interaction densities which are bounded functions of an ultraviolet cut-off boson field. Using methods of euclidean Markov field theory and of classical statistical mechanics, we construct the infinite volume imaginary and real time Wightman functions as limits of the corresponding quantities for the space cut-off models. In the physical Hilbert space, the space-time translations are represented by strongly continuous unitary groups and the generator of time translationsH is positive and has a unique, simple lowest eigenvalue zero, with eigenvector Ω, which is the unique state invariant under space-time translations. The imaginary time Wightman functions and the infinite volume vacuum energy density are given as analytic functions of the coupling constant. The Wightman functions have cluster properties also with respect to space translations.

The approximation property does not imply the bounded approximation property

Abstract: There is a Banach space which has the approxi- mation property but fails the bounded approximation property. The space can be chosen to have separable conjugate, hence there is a nonnuclear operator on the space which has nuclear adjoint. This latter result solves a problem of Grothendieck (2).

Asymptotic solutions of the coupled equations of electron-atom collision theory for the case of some channels closed

Abstract: Solutions of coupled equations for electron-atom collision theory, having known asymptotic forms, can be calculated using asymptotic expansions at some large values of r, say r=rp and rp+1. At some smaller values of r, say r=r1 and r2, it is required to match the asymptotic solutions to solutions which are bounded at the origin. When some channels are closed, inwards integrations from (rp,rp+1) to (r1,r2) can give solutions which have poor linear independence at (r1,r2). It is shown that good linear independence can be obtained using a Fox-Goodwin technique for the solution of two-point boundary value problems.

Another Note on the Borel-Cantelli Lemma and the Strong Law, with the Poisson Approximation as a By-product

Abstract: Here is another way to prove Levy's conditional form of the Borel-Cantelli lemmas, and his strong law. Consider a sequence of dependent variables, each bounded between 0 and 1. Then the sum $S$ of the variables tends to be close to the sum $T$ of the conditional expectations. Indeed, the chance that $S$ is above one level and $T$ is below another is exponentially small. So is the chance that $S$ is below one level and $T$ is above another. The inequalities also show that for a sequence of dependent events, such that each has uniformly small conditional probability given the past, and the sum of the conditional probabilities is nearly constant at $a$, the number of events which occur is nearly Poisson with parameter $a$.

Multipliers for (C,α)-bounded Fourier expansions in Banach spaces and approximation theory

Abstract: General theory.- Multiplier criteria for (C,?)-bounded expansions.- Particular summation methods.- Applications to particular expansions.

On the Regularity of the Solution of the Biharmonic Variational Inequality.

Abstract: It is shown that the solution of the biharmonic variational inequality has bounded second derivatives provided that the obstacle and the data are smooth.