Showing papers on "Bounded function published in 1975"

An example concerning fixed points

Abstract: An example is given of a contractionT defined on a bounded closed convex subset of Hilbert space for which ((I+T)/2) n does not converge.

Spatial birth and death processes

Abstract: We examine birth and death processes where the birth/death rates depend on the particular configuration of the population. The easiest case is when the configurations form the subsets of some finite lattice (thus the state space is finite). In this case we look at the relationships between time-reversibility, nearest neighbour interactions, and the equilibrium state being a Markov random field. A more interesting case is when the entities which can be born or die can do so at any point in some bounded region of space (or the plane). This gives us a pure jump process, whose general properties are well-known, (see for example, Feller (1966) Vol. II Chapter X.3). We examine some particular examples and compute equilibrium distributions.

A Quadratic Measure of Deviation of Two-Dimensional Density Estimates and A Test of Independence

Abstract: This paper considers estimates of multidimensional density functions based on a bounded and bandlimited weight function. The asymptotic behavior of quadratic functions of density function estimates useful in setting up a test of goodness of fit of the density function is determined. A test of independent is also given. The methods use a Poissonization of sample size. The estimates considered are appropriate if interested in estimating density functions or determining local deviations from a given density function.

Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in C[n] with smooth boundary

Abstract: The Carathe'odory and Kobayashi distance functions on a bounded domain G in Cn have related infinitesimal forms. These are the Caratheodory and Kobayashi metrics. They are denoted by F(z, t) Oength of the tangent vector t at the point z). They are defined in terms of holomorphic mappings, from G to the unit disk for the Carathe'odory metric, and from the unit disk to G for the Kobayashi metric. We consider the boundary behavior of these metrics on strongly pseudoconvex domains in Cn with C2 boundary. t is fixed and z is allowed to approach a boundary point z0. The quantity F(z, t) d(z, 8G) is shown to have a finite limit. In addition, if t belongs to the complex tangent space to the boundary at z0, then this first limit is zero, and (F(z, t))2d(z, 8G) has a (nontangential) limit in which the Levi form appears. We prove an approximation theorem for bounded holomorphic functions which uses peak functions in a novel way. The proof was suggested by N. Kerzman. This theorem is used here in studying the boundary behavior of the Carathe'odory metric.

The synthesis of solids bounded by many faces

Abstract: A technique is presented which allows a class of solid objects to be synthesized and stored using a computer. Synthesis begins with primitive solids like a cube, wedge, or cylinder. Any solid can be moved, scaled, or rotated. Solids may also be added together or subtracted. Two algorithms to perform addition are described. For practical designers, the technique has the advantage that operations are concise, readily composed, and are given in terms of easily imagined solids. Quite short sequences of operations suffice to build up complex solids bounded by many faces.

States of classical statistical mechanical systems of infinitely many particles. I

Abstract: We study a general mathematical model of a classical system of infinitely many point particles. The space X of infinite particle configurations is equipped with a natural topology as well as a measurable structure related to it. It is also connected with a family {XA} of local spaces of finite configurations indexed by bounded open sets A in the one-particle space E. A theorem analogous to Kolmogoroff's fundamental theorem for stochastic processes is proved, according to which a consistent family {μA} of local probability measures μAdefined on the XAgives rise to a unique probability measure μ on X. We also study the problem of integral representation for positive linear forms defined over some linear space of real functions on X. We prove that a positive linear form F(f), defined for functions f in the class C+P, admits a uniquely determined integral representation F(f)=∝ f (ξ) dμ, where μ is a probability measure over X.

Non-commutative LP-spaces

Abstract: 1. Introduction. The spaces L1 and L2 of unbounded operators associated with a regular gauge space (von Neumann algebra equipped with a faithful normal semi-finite trace) are defined by Segal(5) definitions 3.3, 3.7. The spaces Lp (1 < p < ∞, p ± 2) are defined by Dixmier(2) as the abstract completions of their bounded parts. Dixmier makes use of the Riesz convexity theorem to prove the Holder inequality, and the uniform convexity, and hence reflexivity, of LLp (2 < p < ∞).

Controllability and Observability in Banach Space with Bounded Operators

Abstract: The classical theory of (state and output) controllability and observability in finite-dimensional spaces is extended to linear abstract systems defined on infinite-dimensional Banach spaces, under the basic assumption that the operator acting on the state be bounded. Tests for approximate controllability as well as observability, expressed only in terms of the coefficients of the system, are proved via a consequence of the Hahn–Banach theorem, and new phenomena arising in infinite dimensions are studied : for instance, by using Baire category arguments, it is shown that state exact controllability, under large conditions met in cases of physical interest, never arises in infinite-dimensional Banach spaces, even with free final instant. Several examples are presented throughout ; in particular, for dynamical systems modeled by integro-differential equations of Volterra type, the present theory leads in turn to explicit, easy-to-check criteria for approximate controllability and observability. An example s...

Univalent Functions - Selected Topics

Abstract: Functions with positive real part.- Special classes: convex, starlike, real, typically real, close-to-convex, bounded boundary rotation.- The Polya-Schoenberg conjecture.- Representation of continuous linear functionals.- Faber polynomials.- Extremal length and equicontinuity.- Compact families ?(D,?1,?2,P,Q) of univalent functions normalized by two linear functionals.- Properties of extreme points for some compact families ?(D,?1,?2,P,Q).- Elementary variational methods.- Application of Schiffer's boundary variation to linear problems.- Application to some nonlinear problems.- Some properties of quasiconformal mappings.- A variational method for q.c. mappings.- Application to families of conformal and q.c. mappings.- Sufficient conditions for q.c. extensions.

Second-Order Linear Differential Equations with Two Turning Points

Abstract: Differential equations of the form d 2 w / d x 2 = { u 2 f ( u , a , x ) + g ( u , a , x ) } w are considered for large values of the real parameter u . Here x is a real variable ranging over an open, possibly infinite, interval ( x 1 , x 2 ), and a is a bounded real parameter. It is assumed that f {u, a, x) and g{u,a, x) are free from singularity within ( x 1 , x 2 ), and f (u, a, x) has exactly two zeros, which depend continuously on a and coincide for a certain value of a . Except in the neighbourhoods of the zeros, g(u,a,x) is small in absolute value compared with u 2 f(u, a, x ). By application of the Liouville transformation, the differential equation is converted into one of four standard forms, with continuous coefficients. Asymptotic approximations for the solutions are then constructed in terms of parabolic cylinder functions. These approximations are valid for large u , uniformly with respect to x e ( x 1 , x 2 ) and also uniformly with respect to a . Each approximation is accompanied by a strict and realistic error bound. The paper also includes some new properties of parabolic cylinder functions.

Recent advances in gradient algorithms for optimal control problems

Abstract: This paper summarizes recent advances in the area of gradient algorithms for optimal control problems, with particular emphasis on the work performed by the staff of the Aero-Astronautics Group of Rice University. The following basic problem is considered: minimize a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter π are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. First, the sequential gradient-restoration algorithm and the combined gradient-restoration algorithm are presented. The descent properties of these algorithms are studied, and schemes to determine the optimum stepsize are discussed. Both of the above algorithms require the solution of a linear, two-point boundary-value problem at each iteration. Hence, a discussion of integration techniques is given. Next, a family of gradient-restoration algorithms is introduced. Not only does this family include the previous two algorithms as particular cases, but it allows one to generate several additional algorithms, namely, those with alternate restoration and optional restoration. Then, two modifications of the sequential gradient-restoration algorithm are presented in an effort to accelerate terminal convergence. In the first modification, the quadratic constraint imposed on the variations of the control is modified by the inclusion of a positive-definite weighting matrix (the matrix of the second derivatives of the Hamiltonian with respect to the control). The second modification is a conjugate-gradient extension of the sequential gradient-restoration algorithm. Next, the addition of a nondifferential constraint, to be satisfied everywhere along the interval of integration, is considered. In theory, this seems to be only a minor modification of the basic problem. In practice, the change is considerable in that it enlarges dramatically the number and variety of problems of optimal control which can be treated by gradient-restoration algorithms. Indeed, by suitable transformations, almost every known problem of optimal control theory can be brought into this scheme. This statement applies, for instance, to the following situations: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with equality constraints on the time rate of change of the state, (iv) problems with control inequality constraints, (v) problems with state inequality constraints, and (vi) problems with inequality constraints on the time rate of change of the state. Finally, the simultaneous presence of nondifferential constraints and multiple subarcs is considered. The possibility that the analytical form of the functions under consideration might change from one subarc to another is taken into account. The resulting formulation is particularly relevant to those problems of optimal control involving bounds on the control or the state or the time derivative of the state. For these problems, one might be unwilling to accept the simplistic view of a continuous extremal arc. Indeed, one might want to take the more realistic view of an extremal arc composed of several subarcs, some internal to the boundary being considered and some lying on the boundary. The paper ends with a section dealing with transformation techniques. This section illustrates several analytical devices by means of which a great number of problems of optimal control can be reduced to one of the formulations presented here. In particular, the following topics are treated: (i) time normalization, (ii) free initial state, (iii) bounded control, and (iv) bounded state.

Towards the development of the quantum mechanics of a subspace

Abstract: The development of the quantum mechanics for a subspace of a total system is pursued via the derivation or introduction of the following: (1) the variational principle for a subspace; (2) the relationship between the time dependent subspace average of an operator and its associated commutator; (3) the unique set of quantum properties found for a subspace bounded by a surface of zero flux as defined by ∇ρ (r) ⋅n (r) =0, ‐r‐S (r), and denoted as a ’’virial fragment’’; (4) the special variational properties of the zero‐flux boundaries in one‐electron systems; (5) operators which cause the energy integral to be stationary over a subspace; (6) the fragment virial theorem; (7) the importance of the scaling of the coordinates of a single electron in the determination of subspace properties; (8) a reformulation of the total Hamiltonian for a many‐particle system as a sum of single‐particle Hamiltonians through the use of a complete set of virial sharing operators; (9) new interpretation of the energy of the equil...

Describing functions revisited

Abstract: The well-known graphical describing function procedure can be simply modified to provide a completely reliable method for predicting whether or not certain kinds of nonlinear feedback system can oscillate. The modified method is easy to use and quantifies, in a natural way, the usual intuitive ideas about describing function reliability. In addition to the usual graphs, the user has to draw a band which measures the amount of uncertainty introduced by the approximations inherent in the method; in return for this extra work, the method gives error bounds for oscillation predictions, as well as ranges of frequency and amplitude over which oscillation is impossible. The main restriction is that the nonlinear element must be single valued and have bounded slope.

Solution of Generalized Geometric Programs

Abstract: A cutting plane algorithm for the solution of generalized geometric programs with bounded variables is described and then illustrated by the detailed solution of a small numerical example. Convergence of this algorithm to a Kuhn-Tucker point of the program is assured if an initial feasible solution is available to initiate the algorithm. An algorithm for determining a feasible solution to a set of generalized posynomial inequalities which may be used to find a global minimum to the program as well as test for consistency of the constraint set, is also presented. Finally an application in optimal engineering design with seven variables and fourteen nonlinear inequality constraints is formulated and solved.

Analyse Numerique d’un Probleme de Stefan a Deux Phases Par une Methode d’Elements Finis

Abstract: Our purpose is to solve numerically a two-phase Stefan problem in a bounded open set $\Omega $ of the two-dimensional space First, we describe briefly the equations and the theoretical method of resolutionThe numerical approach, which is expected to follow as well as possible the variations of the free boundary, is based on finite elements of first order; then we describe the different schemes used and the nature of their convergenceLastly, we discuss two examples for which numerical results have been computed

Inequalities for Ising models and field theories which obey the Lee-Yang Theorem

Abstract: A series of inequalities for partition, correlation, and Ursell functions are derived as consequences of the Lee-Yang Theorem. In particular, then-point Schwinger functions ofeven φ4 models are bounded in terms of the 2-point function as strongly as is the case for Gaussian fields; this strengthens recent results of Glimm and Jaffe and shows that renormalizability of the 2-point function by fourth degree counter-terms implies existence of a φ4 field theory with a moment generating function which is entire of exponential order at most two. It is also noted that ifany (even) truncated Schwinger function vanishes identically, the resulting field theory is a generalized free field.

Generalized Solutions of the Hamilton-Jacobi Equations of Eikonal Type. I. Formulation of the Problems; Existence, Uniqueness and Stability Theorems; Some Properties of the Solutions

Abstract: A nonlocal theory of boundary value problems in bounded and unbounded domains is constructed for nonlinear equations of first order of the type of the classical eikonal equation in geometric optics with Cauchy-Dirichlet boundary conditions and with additional conditions at infinity in the case of unbounded domains. Existence, uniqueness and stability theorems are proved; some representations of the solutions are indicated; and properties of the solutions generalizing the principles of Fermat and Huygens in geometric optics are established.Bibliography: 14 titles.

A Note on Tape-Bounded Complexity Classes and Linear Context-Free languages

Abstract: Let LINEAR denote the family of hnear context-free languages and DSPACE(L(n)) [NSPACE(L(n))] denote the family of languages recognized by determmmtm [nondetermimstie] off-line L(n)-tape bounded Turmg machines. The equivalence of the following statements, for 0 g ~ < 1, m shown by describing a log(n)-complete hnear language. (1) LINEAR C DSPACE(logl+'(n)). (2) The hnear context-free language L(B1) ~s in DSPACE(logl+'(n)). (3) NSPACE(L(n)) C DSPACE([L(n)]~+'), for all L(n) > log(n). All the above statements are known to be true when e = 1

Selecting sequence numbers

Abstract: This paper discusses techniques for selecting and synchronizing sequence numbers such that no errors will occur if certain network characteristics can be bounded and if adequate data error detection measures are taken. The discussion specifically focuses on the protocol described by Cerf and Kahn, (1) but the ideas are applicable to other similar protocols.

$H^2 $-Functions and Infinite-Dimensional Realization Theory

Abstract: In this paper the realization question for infinite-dimensional linear systems is examined for both bounded and unbounded operators. In addition to obtaining realizability criteria covering the basic cases, we discuss the relationship between canonical realizations of the same system. What one finds is that the set of transfer functions which are realizable by triples $(A,b,c)$ with A bounded is related in a close way to the space of complex functions analytic and square integrable on the disk $| {s < 1} |$, and that the set of transfer functions which are realizable by triples $(A,b,c)$ with A unbounded but generating a strongly continuous semigroup is related in a close way to functions analytic and square integrable on a half-plane. This relation makes possible a deeper study between the transfer function and the models which realize it. Some examples illustrate the results and their applications.

Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables

Abstract: Algorithms for the numerical solution of optimal control problems with bounded state variables are developed. Two main cases are considered: either the control variable appears nonlinearly or the control variable appears linearly. In the first case, an extremal are touching the boundary or containing a boundary arc, is shown to satisfy a suitable two-point boundary value problem. In the second case, a numerical idea for solving the problem in statespace is presented which dispenses with the Lagrange-multipliers. Three numerical examples are discussed illustrating the efficiency of the different algorithms. The encountered two-point boundary value problems are solved with the method of multiple shooting.

Stabilizing Control for Linear Systems with Bounded Parameter and Input Uncertainty

Abstract: We consider dynamical systems with norm-bounded uncertainty in (i) the system parameters (model uncertainty) or in (ii) the input (disturbance).

Splitting an -recursively enumerable set

Abstract: We extend the priority method in a-recursion theory to certain arguments with no a priori bound on the required preservations by proving the splitting theorem for all admissible ca. THEOREM: Let C be a regular car.e. set and D be a nonrecursive cs-r.e. set. Then there are regular or-r.e. sets A and B such that A U B = C, A n B=0, A, B

Wigner's Theorem on Symmetries in Indefinite Metric Spaces*

Abstract: The description of symmetries in indefinite metric spaces is investigated. It is shown that ray transformations preserving the modulus of an indefinite scalar product can be implemented by linear or antilinear vector transformations which are generalized unitary or antiunitary operators with respect to the indefinite scalar product. A numbero f interesting features arise since such operators need not to be bounded.