Bound Of Function Research Articles

Visits, crosses, and reversals for nondeterministic off-line machines

The different concepts involved in “reversal complexity”counting reversals (sweeps), visits to a square, or crossing sequences—are discussed for non-deterministic off-line Turing machines with one working tape and for preset Turing machines, a generalization of two-way checking automata. Restriction to finite reversals or visits or crosses yields the same family, NSPACE(log2n), for off-line one working tape Turing machines or for two-way checking automata. For each k, a k-reversal bounded machine has the power of a nondeterministic k-head finite automaton. Finite visit preset Turing machines with working tapes selected from context-free languages yield %plane1D;4AB;. For an arbitrary bounding function T(n), a T(n) reversal or visit bound on a nondeterministic off-line Turing machine corresponds to a T(n) log2 n space bound within a linear factor. However, there is no general linear speedup theorem for reversal bounds on a nondeterministic off-line Turing machine.

Global minimization of a cooled reactor by using a posynomial lower bounding function

AbstractThe cost of a reactor‐heater system with an auxiliary cooler can have at least two local minima and a local maximum, with respect to the design variables: temperature and extent of reaction. This cost can be bounded below by a unimodal function which, being a posynomial (a sum of power functions), can be minimized efficiently by geometric programming. Construction of this lower bounding function is not obvious, since the cost involves both a definite intergral and economies of scale.

An Inverse Problem for Second Order Differential Systems and Waveform Design

This paper concerns the problem of synthesizing a second order self-oscillating system which generates a prescribed periodic waveform. The class of realizable waveforms is characterized completely for two types of differential systems and the procedures for synthesis are given. The concept of a bounding function, which plays the same role as a Lyapunov function, provides an effective means for synthesis.

Bounds for Initial Value Problems

A new method has been developed for finding rigorous upper and lower bounds to the solution of a wide class of initial value problems. The method is applicable to initial value problems of the following type: x(¨t)+f(t,x,x)˙=0,x(0)=X0,x(˙0)=V0, where f is continuous with continuous first derivatives, Lipschitzian, and ∂f/∂x ≥ 0. An original bounding theorem has been formulated and proven and a numerical technique has been developed for finding the bounding functions in analytic form as linear combinations of Tchebyshev polynomials. The method has been applied to several problems of engineering interest.

Branch-and-Bound Methods: General Formulation and Properties

The branch-and-bound procedure is formulated in rather general terms and necessary conditions for the branching and bounding functions are precisely specified. Results include the standard properties for finite procedures, plus several convergence conditions for infinite procedures. Discrete programming which includes integer programming and combinatorial optimization problems, is discussed and Fibonacci search is presented as an example of a nonfinite branch-and-bound procedure employing an optimal convergence rule.

Uniform Bounding Functions for a Certain One-Dimensional Form of Poisson's Equation

A classical one-dimensional specialization of the more general Poisson–Boltzman equation for potential space distributions associated with the existence of a double layer at an electrode surface leads to a certain nonlinear two-point boundary value problem. The classical approach to solving this problem involves a formal procedure which assumes a certain preknowledge of the actual unknown solution, and it involves finding an approximate solution which is obtained by deleting “higher-order” terms in a certain expansion. This paper presents a different approach in that it makes use of a known existence and uniqueness theorem to establish bounds on the actual solution to the problem. In certain examples, as shown in the paper, the actual effect of ignoring the so called higher-order terms in the classical approach can be clearly established.

ELECTROMAGNETIC RADIATION-PATTERN EFFECTS OF DIELECTRIC LAYERS

A formulation for the radiation pattern of a grounded magnetic line source in the presence of dielectric layers is obtained and the effects of differing permittivity variations are investigated. It is shown that relatively simple bounding functions may be established, and that layers may be employed to obtain both directive and smoothed patterns.

Heat Transfer of Thin Fins With Temperature-Dependent Thermal Properties and Internal Heat Generation

An analytical study of the temperature distribution along thin fins with temperature-dependent thermal properties and internal heat generation is presented. The analysis utilizes a recently published bounding procedure which yields analytical and continuous bounding functions for the temperature distribution. Several numerical examples are considered. Tabular and graphical results are given. The effects of variable thermal properties and internal heat generation are also shown.

Bounds for moment generating functions and for extinction probabilities

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

Bounds for moment generating functions and for extinction probabilities

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

Correlation Functions for Eigenvalues of Real Quaternian Matrices

The eigenvalue density, the two-, and the three-point correlation functions for the ensemble of real quaternion matrices are calculated. The forms suggest a generalization for the n-point correlation function. The probability that no eigenvalues lie inside a circle of radius r around the origin is also calculated for the ensemble of real quaternion matrices as well as for that of complex matrices. Upper and lower bounding functions for the last probability density are given.

Analytical mechanics and the analysis of linear, time-varying networks

The formalism of classical mechanics is extended to yield an energy function analysis of single degree of freedom, linear time-varying networks. With the aid of the newly found modifications of the classical Lagrangian and Hamiltonian, the behavior of the single loop R(t) - L(t) - C(t) network is discussed through analogy with familiar classical quantities. The Hamiltonian formulation, through its associated canonical equations, leads to a qualitative phase plane specification of single degree of freedom network behavior. Moreover, the Hamiltonian formulation leads directly to stability criteria which are expressed in terms of the network element values only. Upper and lower bounding functions for the network stored energy are obtained which lead to separate necessary and sufficient conditions for network stability.