Classification of Critical Stationary Points in Unconstrained Optimization

1992 ◽  
Vol 2 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Stefan Schäffler
Keyword(s):  
Unconstrained Optimization ◽  
Stationary Points

scholarly journals Isotropic Points on Glaciers

1986 ◽  
Vol 32 (112) ◽  
pp. 363-365 ◽  
Author(s):  
J. F. Nye
Keyword(s):  
Strain Rate ◽  
Pattern Classification ◽  
Transverse Direction ◽  
Transverse Velocity ◽  
Longitudinal Direction ◽  
Stationary Points ◽  
Star Pattern ◽  
Isotropic Point ◽  
Made In

AbstractTwo isotropic points measured by Meier and others (1985) on Columbia Glacier, Alaska, are examined. The pattern classification of the upper one is on the borderline between monstar and lemon, and this is traced to the fact that the variation of strain-rate in the longitudinal direction is approximately equal to that in the transverse direction, contrary to the assumption made in Nye (1983). The conditions for the lower isotropic point to have the star pattern, as observed, are believed to be typical for a glacier that ends in an ice cliff, like this one, which calves icebergs. Where, as in this case, there is only a small transverse velocity, the isotropic points on a glacier must nearly coincide with stationary points for the speed, and these are almost always either maxima or saddles, alternating. The maxima correspond to lemon or monstar patterns, and the saddles to star patterns.

scholarly journals Isotropic Points on Glaciers

1986 ◽  
Vol 32 (112) ◽  
pp. 363-365 ◽  
Author(s):  
J. F. Nye
Keyword(s):  
Strain Rate ◽  
Pattern Classification ◽  
Transverse Direction ◽  
Transverse Velocity ◽  
Longitudinal Direction ◽  
Stationary Points ◽  
Star Pattern ◽  
Isotropic Point ◽  
Made In

AbstractTwo isotropic points measured by Meier and others (1985) on Columbia Glacier, Alaska, are examined. The pattern classification of the upper one is on the borderline between monstar and lemon, and this is traced to the fact that the variation of strain-rate in the longitudinal direction is approximately equal to that in the transverse direction, contrary to the assumption made in Nye (1983). The conditions for the lower isotropic point to have the star pattern, as observed, are believed to be typical for a glacier that ends in an ice cliff, like this one, which calves icebergs. Where, as in this case, there is only a small transverse velocity, the isotropic points on a glacier must nearly coincide with stationary points for the speed, and these are almost always either maxima or saddles, alternating. The maxima correspond to lemon or monstar patterns, and the saddles to star patterns.

scholarly journals BIFURCATION ANALYSIS OF THE DYNAMICAL SYSTEM FOR A THREELAYERED VALVE WITH PERPENDICULAR ANISOTROPY

2018 ◽  
Vol 185 ◽  
pp. 01008
Author(s):  
Natalia Ostrovskaya ◽  
Vladimir Skidanov ◽  
Iulia Iusipova ◽  
Maxims Skvortsov
Keyword(s):  
Magnetization Vector ◽  
Stationary Points ◽  
Magnetization Distribution ◽  
Perpendicular Anisotropy ◽  
Magnetization Dynamics ◽  
Switching Dynamics ◽  
Uniform Magnetization ◽  
Plane Anisotropy ◽  
The Stability

The features of switching dynamics in a model of a three-layered valve have been investigated theoretically and numerically. For this purpose, the system of ordinary differential equations in the approximation of the uniform magnetization distribution for the magnetization dynamics in the valve with perpendicular anisotropy was derived. It was shown that in such a system, in contrast with the system for the in-plane anisotropy, there are only two equilibrium positions of the magnetization vector. The stability analysis of the stationary points of the system has been carried out. With its help, the classification of types of dynamics versus field and current values was performed. The regions of limit cycles existence and the regions of optimal magnetization switching were revealed.

scholarly journals Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes

2021 ◽  
Vol 7 (6) ◽  
Author(s):  
Emil M. Prodanov
Keyword(s):  
Stationary Points ◽  
Model Parameters ◽  
The Real ◽  
Quartic Equation ◽  
Real Roots ◽  
Quadratic Equations ◽  
Parallel Lines ◽  
Intersection Points

AbstractPresented is a very detailed two-tier analysis of the location of the real roots of the general quartic equation $$x^4 + a x^3 + b x^2 + c x + d = 0$$ x 4 + a x 3 + b x 2 + c x + d = 0 with real coefficients and the classification of the roots in terms of a, b, c, and d, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and solving these, together with some of the resolvent quadratic equations, allows the full classification of the roots of the general quartic and also the determination of the isolation interval of each root. These isolation intervals involve the stationary points of the quartic (among others) and, by solving some of the resolvent quadratic equations, the isolation intervals of the stationary points of the quartic are also determined. The presented classification of the roots of the quartic equation is particularly useful in situations in which the equation stems from a model the coefficients of which are (functions of) the model parameters and solving cubic equations, let alone using the explicit quartic formulæ , is a daunting task. The only benefit in such cases would be to gain insight into the location of the roots and the proposed method provides this. Each possible case has been carefully studied and illustrated with a detailed figure containing a description of its specific characteristics, analysis based on solving cubic equations and analysis based on solving quadratic equations only. As the analysis of the roots of the quartic equation is done by studying the intersection points of the “sub-quartic” $$x^4 + ax^3 + bx^2$$ x 4 + a x 3 + b x 2 with a set of suitable parallel lines, a beautiful Pythagorean analogy can be found between these intersection points and the set of parallel lines on one hand and the musical notes and the staves representing different musical pitches on the other: each particular case of the quartic equation has its own short tune.

Levenberg–Marquardt method for unconstrained optimization

2019 ◽  
pp. 60-74
Author(s):  
Alexey Izmailov ◽  
Alexey Kurennoy ◽  
Petr Stetsyuk
Keyword(s):  
Objective Function ◽  
Unconstrained Optimization ◽  
Optimization Problems ◽  
Stationary Points ◽  
Systems Of Nonlinear Equations ◽  
Fermat Principle ◽  
Marquardt Method ◽  
Unconstrained Optimization Problems ◽  
High Convergence Rate ◽  
Levenberg Marquardt

We propose and study the Levenberg–Marquardt method globalized by means of linesearch for unconstrained optimization problems with possibly nonisolated solutions. It is well-recognized that this method is an efficient tool for solving systems of nonlinear equations, especially in the presence of singular and even nonisolated solutions. Customary globalization strategies for the Levenberg–Marquardt method rely on linesearch for the squared Euclidean residual of the equation being solved. In case of unconstrained optimization problem, this equation is formed by putting the gradient of the objective function equal to zero, according to the Fermat principle. However, these globalization strategies are not very adequate in the context of optimization problems, as the corresponding algorithms do not have “preferences” for convergence to minimizers, maximizers, or any other stationary points. To that end, in this work we considers a different technique for globalizing convergence of the Levenberg–Marquardt method, employing linesearch for the objective function of the original problem. We demonstrate that the proposed algorithm possesses reasonable global convergence properties, and preserves high convergence rate of the Levenberg–Marquardt method under weak assumptions.

scholarly journals Cálculo variacional e aplicações

2020 ◽  
Vol 42 ◽  
pp. 50
Author(s):  
Jardel Carpes Meurer ◽  
Lucas Tavares Cardoso ◽  
Glauber Rodrigues de Quadros
Keyword(s):  
Sufficient Conditions ◽  
Variational Calculus ◽  
Necessary And Sufficient Conditions ◽  
Stationary Points ◽  
Second Variation ◽  
Rigorous Treatment ◽  
Necessary And Sufficient ◽  
First And Second Variation ◽  
Lagrange Mechanics

This paper consists of a brief review and introduction to the main concepts of Classic Variational Calculus. Starting from thedefinitions of the concepts of first and second variation of a functional, we present a mathematically rigorous treatment for theVariational Calculus, establishing necessary and sufficient conditions for obtaining extrema. In this context, the notion of conjugatepoints is introduced, which is fundamental for the classification of weak extrema. Some simple and enlightening examples are dealtwith throughout the paper. Strong extrema are characterized and sufficient conditions for their occurrence are given. The paperconcludes with a brief application to Lagrange mechanics, showing the existence of actions whose stationary points are saddlepoints instead of minima.

Note on the spectral classification of carbon stars in the photographic infrared region

1966 ◽  
Vol 24 ◽  
pp. 21-23
Author(s):  
Y. Fujita
Keyword(s):  
Infrared Region ◽  
Spectral Classification ◽  
Carbon Stars

We have investigated the spectrograms (dispersion: 8Å/mm) in the photographic infrared region fromλ7500 toλ9000 of some carbon stars obtained by the coudé spectrograph of the 74-inch reflector attached to the Okayama Astrophysical Observatory. The names of the stars investigated are listed in Table 1.

Diagnostic Value of Electron Microscopy in Soft Tissue Tumors

1970 ◽  
Vol 28 ◽  
pp. 220-221
Author(s):  
Gerald Fine ◽  
Azorides R. Morales
Keyword(s):  
Cardiac Myxoma ◽  
Osteogenic Sarcoma ◽  
Diagnostic Value ◽  
Soft Tissue Tumors ◽  
Amelanotic Melanoma ◽  
Growth Patterns ◽  
Light Microscopic Level ◽  
Mesenchymal Tumors ◽  
Good Agreement

For years the separation of carcinoma and sarcoma and the subclassification of sarcomas has been based on the appearance of the tumor cells and their microscopic growth pattern and information derived from certain histochemical and special stains. Although this method of study has produced good agreement among pathologists in the separation of carcinoma from sarcoma, it has given less uniform results in the subclassification of sarcomas. There remain examples of neoplasms of different histogenesis, the classification of which is questionable because of similar cytologic and growth patterns at the light microscopic level; i.e. amelanotic melanoma versus carcinoma and occasionally sarcoma, sarcomas with an epithelial pattern of growth simulating carcinoma, histologically similar mesenchymal tumors of different histogenesis (histiocytoma versus rhabdomyosarcoma, lytic osteogenic sarcoma versus rhabdomyosarcoma), and myxomatous mesenchymal tumors of diverse histogenesis (myxoid rhabdo and liposarcomas, cardiac myxoma, myxoid neurofibroma, etc.)

Ultrastructure of mesotheliomas

1984 ◽  
Vol 42 ◽  
pp. 98-101
Author(s):  
Irving Dardick
Keyword(s):  
Electron Microscopy ◽  
Latent Period ◽  
Spindle Cell ◽  
Spindle Cell Sarcoma ◽  
Metastatic Adenocarcinoma ◽  
Cell Sarcoma ◽  
Cytological Features ◽  
Industrial Use ◽  
Degree Of Differentiation

With the extensive industrial use of asbestos in this century and the long latent period (20-50 years) between exposure and tumor presentation, the incidence of malignant mesothelioma is now increasing. Thus, surgical pathologists are more frequently faced with the dilemma of differentiating mesothelioma from metastatic adenocarcinoma and spindle-cell sarcoma involving serosal surfaces. Electron microscopy is amodality useful in clarifying this problem.In utilizing ultrastructural features in the diagnosis of mesothelioma, it is essential to appreciate that the classification of this tumor reflects a variety of morphologic forms of differing biologic behavior (Table 1). Furthermore, with the variable histology and degree of differentiation in mesotheliomas it might be expected that the ultrastructure of such tumors also reflects a range of cytological features. Such is the case.

Sign in / Sign up

Export Citation Format

Share Document