Showing papers on "Maxima and minima published in 1975"
TL;DR: In this paper, necessary and sufficient conditions for a local minimum to be global are derived, and the main result is that every local minimum is global if, and only if, its level sets are lower-semicontinuous point-to-set mappings.
Abstract: In this paper, necessary and sufficient conditions for a local minimum to be global are derived. The main result is that a real function, defined on a subset ofRn, has the property that every local minimum is global if, and only if, its level sets are lower-semicontinuous point-to-set mappings.
45 citations
01 Jan 1975
TL;DR: The theory of minima of a function J on an abstract space V was introduced in this paper. But in many problems of interest the domain of the function to be miminized is not a portion of some finite-dimensional space V, rather, the function is defined on some portion of an infinite dimensional space if.
Abstract: In calculus one studies the problem of minimizing f(x), where x is real or more generally x=(x1…,xn) denotes an n-tuple of real numbers. However, in many problems of interest the domain of the function to be miminized is not a portion of some finite-dimensional space V. Rather, the function is defined on some portion of an infinite-dimensional space if. We shall begin by outlining the elementary theory of minima of a function J on an abstract space V (§2).
22 citations
TL;DR: Weak convergence of the multivariate point process of upcrossings of several high levels by a stationary Gaussian process is established in this article, which is then used to determine the joint asymptotic distribution of heights and locations of highest local maxima over an increasing interval.
Abstract: Weak convergence of the multivariate point process of upcrossings of several high levels by a stationary Gaussian process is established. The limit is a certain multivariate Poisson process. This result is then used to determine the joint asymptotic distribution of heights and locations of the highest local maxima over an increasing interval. The results are generalized to upcrossings and local maxima of two dependent Gaussian processes. To prevent nuisance jitter from hiding the overall structure of crossings and maxima the above results are phrased in terms of varepsilon-crossings and varepsilon-maxima, but it is shown that under suitable regularity conditions the results also hold for ordinary upcrossings and maxima.
14 citations
Patent•
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TL;DR: A pulse train analyzer utilizing digital clock frequencies corresponding to the minima and maxima of the acceptable performance ranges provides a pass-fail visual indication depending upon whether the pulses are uniform, and the speed and percent break figures are within the specified ranges as discussed by the authors.
Abstract: A pulse train analyzer utilizing digital clock frequencies corresponding to the minima and maxima of the acceptable performance ranges provides a pass-fail visual indication depending upon whether the pulses are uniform, and the speed and percent break figures are within the specified ranges.
10 citations
TL;DR: In this paper, a class of apodized pupil functions is derived which have the property of producing a point-spread function which decreases monotonically with increasing radius, which in turn produces coherent line and edge images, free of the usual subsidiary diffraction maxima and minima.
8 citations
TL;DR: In this paper, some Wiener-Hopf type results are collected, related and given more direct proofs, where transform equations are interpreted as decompositions of time-changed processes.
Abstract: Some Wiener-Hopf type results are collected, related and given more direct proofs. Spitzer's random-walk method for the Wiener-Hopf integral equation also produces his factorisation relating functionals of maxima and minima. Transform equations are interpreted as decompositions of time-changed processes. Discrete- and continuous-time versions are related. Prabhu's factorisation
8 citations
TL;DR: For a sequence of independent and identically distributed positive random variables, the almost sure convergence of sums of maxima to appropriate constants is proved for both bounded and unbounded random variables as discussed by the authors.
Abstract: For a sequence of independent and identically distributed positive random variables, the almost sure convergence of sums of maxima (when suitably normalized) to appropriate constants is proved for both bounded and unbounded random variables. A similar result is also proved for sums of minima of such variables.
4 citations
Journal Article•
1 citations
TL;DR: In this article, it was shown that there is a relation between the best-possible error function? n and a polynomialR N, which is an estimate for the number of maxima and minima of? n.
Abstract: Using the ?-method of Lanczos, it is shown, that there is a relation between the best-possible errorfunction? n and a polynomialR N . The result is an estimate for the number of maxima and minima of? n .
1 citations
Patent•
30 Jan 1975
TL;DR: In this article, the photo-electronic components are distributed over a test field in an optical system image plane, and the display devices produce control signals for detection of brightness maxima and minima which consist of charge quantities in stores whose charge or discharge time, or charge intensity is disjunctively controlled by the output signals of the above photoelectronic component.
Abstract: The photo-electronic components are distributed over a test field in an optical system image plane. Evaluating and/or display devices produce control signals for detection of brightness maxima and minima which consist of charge quantities in stores whose charge or discharge time, or charge intensity is disjunctively controlled by the output signals of the above photoelectronic components. Connections between these components and the evaluation and/or display devices contain logarithmic elements in the form of diodes directly connected in series with the photoelectronic components.
01 Aug 1975
TL;DR: In this paper, a necessary condition for a real valued Frechet differentiable function of a vector variable having an extremum at a vector x sub 0 is that the Frechet derivative vanishes at X sub 0.
Abstract: A necessary condition for a real valued Frechet differentiable function of a vector variable have an extremum at a vector x sub 0 is that the Frechet derivative vanishes at x sub 0. A relationship between Frechet differentials and matrix derivatives was established that obtains a necessary condition on the matrix derivative at an extrema. These results are applied to various scalar functions of matrix variables which occur in statistical pattern recognition.