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Showing papers on "Maxima and minima published in 1972"


Journal ArticleDOI
TL;DR: A simple algorithm for computing channel capacity is suggested that consists of a mapping from the set of channel input probability vectors into itself such that the sequence of probability vectors generated by successive applications of the mapping converges to the vector that achieves the capacity of the given channel.
Abstract: By defining mutual information as a maximum over an appropriate space, channel capacities can be defined as double maxima and rate-distortion functions as double minima. This approach yields valuable new insights regarding the computation of channel capacities and rate-distortion functions. In particular, it suggests a simple algorithm for computing channel capacity that consists of a mapping from the set of channel input probability vectors into itself such that the sequence of probability vectors generated by successive applications of the mapping converges to the vector that achieves the capacity of the given channel. Analogous algorithms then are provided for computing rate-distortion functions and constrained channel capacities. The algorithms apply both to discrete and to continuous alphabet channels or sources. In addition, a formalization of the theory of channel capacity in the presence of constraints is included. Among the examples is the calculation of close upper and lower bounds to the rate-distortion function of a binary symmetric Markov source.

1,472 citations


Journal ArticleDOI
TL;DR: In this paper, the authors introduce the basic techniques for locating extrema (minima or maxima) of a function of several variables, and present a standard set of techniques for unconstrained function extremization.
Abstract: : The purpose of the book is to introduce the basic techniques for locating extrema (minima or maxima) of a function of several variables. Such a need arises naturally in various design optimization and planning problems. A standard set of techniques for unconstrained function extremization are presented. Small-step and large-step gradient methods: methods involving second partial derivatives of the function, such as the Newton-Raphson method and the Davidon-Fletcher-Powell method; and several other direct search methods are discussed. There are also discussions on elementary aspects of function extremization subject to linear or nonlinear constraints-such as the concept of constraint qualification, Fritz-John and Kuhn-Tucker theorems, penalty function method, etc. assuming differentiability and convexity of objective functions and constraint equations. In addition to presenting various standard algorithms for function extremization, the book also contains some simplified accounts of optimization problems drawn from various branches of engineering and operations research. (Author)

48 citations



Journal ArticleDOI
TL;DR: A study is made of various statistical properties of a particular non-Gaussian process, the low-pass filtered random telegraph signal, and general expressions for the average numbers of crossings of random processes with arbitrary curves are derived.
Abstract: A study is made of various statistical properties of a particular non-Gaussian process, the low-pass filtered random telegraph signal. Results are obtained for the distribution of slope, the average numbers of zero and level crossings, the average number of crossings with a straight line, distributions of maxima and minima and the average number of specular points. As a prelude to these specific investigations, general expressions for the average numbers of crossings of random processes with arbitrary curves are derived. In particular, for the average number of zero-crossings of a stationary random process, the result does not explicitly involve the joint distribution of the process and its derivative.

11 citations


Journal ArticleDOI
TL;DR: In this article, a discrete generalized coordinate energy method is applied to a general, symmetric, multidegree-of-freedom system experiencing branching of equilibrium into more than one mode of buckling at a single value of loading parameter.
Abstract: A discrete generalized-coordinate energy method is applied to a general, symmetric, multidegree-of-freedom system experiencing branching of equilibrium into more than one mode of buckling at a single value of loading parameter. Maxima and minima are established for the potentially large number of post-buckling paths which exist. The maximum number of distinct post-buckling path curvatures among these paths is established, as well as the maximum number of load-corresponding deflection relationships. The case where the branching points on the basic state are not coincident, but nearly so, is examined. The possibilities for secondary bifurcations on post-buckling paths are demonstrated in this latter case.

6 citations


Patent
18 Sep 1972
TL;DR: In this article, the envelope of the minima is defined by a similar apparatus, consisting of a maximum follower, which follows the input signal till a peak is reached and which holds this peak value, and a minimum detector, which gives a logic signal when the inputs have passed a minimum; the said logic signal of the minimum detector introducing the following three actions: firstly the peak value held by the maximum follower is given to a memory element, for example a sample-hold module, the value stored in this memory element being the readout of the apparatus, secondly the said maximum follower
Abstract: An apparatus for defining continually the envelope of the maxima of a varying input signal, consisting of: a maximum follower, which follows the input signal till a peak is reached and which holds this peak value, and a minimum detector, which gives a logic signal when the input signal has passed a minimum; the said logic signal of the minimum detector introducing the following three actions: firstly the peak value held by the maximum follower is given to a memory element, for example a sample-hold module, the value stored in this memory element being the readout of the apparatus, secondly the said maximum follower is set back to a zero value and thirdly the maximum follower is released for following the varying input signal in order to measure the next maximum which will replace the previous maximum stored in the memory element when the next minimum is detected by the minimum detector. The first and second action can be introduced by a maximum detector. Instead of the memory element one can use a second maximum follower in combination with a maximum selector. The envelope of the minima is defined by a similar apparatus. When an apparatus for defining the maxima is combined with an apparatus for defining the minima one gets an apparatus, which can define continually the modulation depth of a varying input signal.

5 citations


Patent
17 May 1972
TL;DR: In this paper, a hybrid computer is used to solve nonlinear programming problems, in which and analog computing unit comprises: a channel of initial conditions producing, on the basis of signals proportional to current values of variables, and on a basis of preset signals proportionally to required or desired values of the initial conditions, a signal representing a time derivative of an axuiliary function of the original conditions; and an auxiliary function being solved which produces, according to conditions of the problem introduced in an analog problem model, the latter channels being connected to the input of an output amplifier via a
Abstract: A hybrid computer to solve nonlinear programming problems, in which and analog computing unit comprises: a channel of initial conditions producing, on the basis of signals proportional to current values of variables, and on the basis of preset signals proportional to required or desired values of the initial conditions, a signal representing a time derivative of an axuiliary function of the initial conditions; and a channel of the problem being solved which produces, on the basis of signals proportional to current values of variables and according to conditions of the problem introduced in an analog problem model, a signal representing a time derivative of an auxiliary function, the latter channels being connected to the input of an output amplifier via a switch. The computer is provided with analog-todigital and digital-to-analog converters and with a digital computing unit. While finding the minima of the auxiliary functions to solve non-linear programming problems, the computer uses the descent method.

4 citations


Journal ArticleDOI
TL;DR: A computational method is presented to analyze the worst case sensitivity of electrical networks over a band of frequencies using nonlinear programming techniques to find the maxima and minima of a network function at discrete frequencies within the frequency interval.
Abstract: A computational method is presented to analyze the worst case sensitivity of electrical networks over a band of frequencies. The method uses nonlinear programming techniques to find the maxima and minima of a network function at discrete frequencies within the frequency interval over the entire element constraint space in order to calculate the sensitivity.

4 citations


Journal ArticleDOI
TL;DR: Convenient second-order sufficiency conditions are given first for constrained relative minima and then implemented as computational methods for solving constrained parameter optimization problems arising in control.
Abstract: Convenient second-order sufficiency conditions are given first for constrained relative minima. These sufficiency conditions then are implemented as computational methods for solving constrained parameter optimization problems arising in control. Second-order partial derivatives need not be computed directly when a min-max modification of these computational methods is used in conjunction with the Fletchor-Powell method for unconstrained problems. Comparisons with previously published methods are included in the form of an example.

3 citations


Journal ArticleDOI
TL;DR: In this article, a method with quadratic convergence is described for minimizing a maximum function on a set specified by a system of inequalities, which is effective in the neighbourhood of a stationary point and can therefore be recommended for the final stage of computations.
Abstract: A METHOD possessing quadratic convergence is described, for minimizing a maximum function on a set specified by a system of inequalities. The method is effective in the neighbourhood of a stationary point, and can therefore be recommended for the final stage of computations. Its “sphere of efficiency” may be reached by using any method which converges to the stationary point. The various methods now available for finding the minima of a maximum function are all essentially of the first order (they utilize the first directional derivatives in order to discover the direction of descent). Like all first-order methods, they converge slowly in the vicinity of the minimum. When minimizing in E n a continuously differentiable function, some rapidly convergent method, possessing quadratic convergence (e.g. Newton's method), is used at the final stage. The present paper aims at developing a method with quadratic convergence, representing an extension of the method of maxima equalization [1], for minimizing a maximum function on a set specified by a system of inequalities. The method is effective in the neighbourhood of a stationary point and may there-fore be recommended for the final stage of computations. Its “sphere of effectiveness” may be reached by using any method converging to the stationary point.

2 citations


Journal ArticleDOI
TL;DR: In this paper, a SIMPLE necessary and sufficient condition is found, such that a vertex of the transport polyhedron is a local minimum of a Schur-concave target function.
Abstract: A SIMPLE necessary and sufficient condition is found, such that a vertex of the transport polyhedron is a local minimum of a Schur-concave target function. The local minima are shown in this case to be independent of the behaviour of the target function.

Dissertation
01 Jan 1972
TL;DR: A map, drawing or chart was part of the material being p h o r ing to g rap h ed the photographer followed a definite m ethod in "sectioning" the material.
Abstract: 3. When a map, drawing or chart, etc., was part of the material being p h o to g rap h ed the photographer followed a definite m ethod in "sectioning" the material. It is customary to begin photoing a t the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again beginning below the first row and continuing on until complete.

Journal ArticleDOI
TL;DR: A modification of the method to reduce its sensitivity to sampling fluctuations of maxima and minima, or to accidentally grossly different values (blunders), is included in the program, as well as a method for avoiding the bias in estimating error probabilities that is inherent in any classification method based on samples.

Journal ArticleDOI
TL;DR: A method for finding the non-stationary extrema of non-linear scalar-valued functions of a vector argument in the presence of restrictions determining a set of feasible values of the argument is presented.
Abstract: A method for finding the non-stationary extrema of non-linear scalar-valued functions of a vector argument in the presence of restrictions determining a set of feasible values of the argument is presented in the paper. It is based upon the geometric interpretation of the problem. Further, it is shown that the same procedure can be applied for finding either stationary or non-stationary optima, the objective function having been suitably adapted.