A more conclusive and more inclusive second derivative test

2020 ◽  
Vol 104 (560) ◽  
pp. 247-254
Author(s):  
Ronald Skurnick ◽  
Christopher Roethel
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Local Minimum ◽  
Local Maximum ◽  
Number Line ◽  
Second Derivative ◽  
First Derivative ◽  
Line Test ◽  
First Derivative Test ◽  
Derivative Test

Given a differentiable function f with argument x, its critical points are those values of x, if any, in its domain for which either f′ (x) = 0 or f′ (x) is undefined. The first derivative test is a number line test that tells us, definitively, whether a given critical point, x = c, of f(x) is a local maximum, a local minimum, or neither. The second derivative test is not a number line test, but can also be applied to classify the critical points of f(x). Unfortunately, the second derivative test is, under certain conditions, inconclusive.

Lagrangean conditions and quasiduality

1977 ◽  
Vol 16 (3) ◽  
pp. 325-339 ◽  
Author(s):  
B.D. Craven
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Local Minimum ◽  
Constraint Qualification ◽  
Necessary Conditions ◽  
Weak Duality ◽  
Partially Ordered Vector Space ◽  
Counter Example ◽  
Cone Constraints ◽  
Constrained Minimization Problem

For a constrained minimization problem with cone constraints, lagrangean necessary conditions for a minimum are well known, but are subject to certain hypotheses concerning cones. These hypotheses are now substantially weakened, but a counter example shows that they cannot be omitted altogether. The theorem extends to minimization in a partially ordered vector space, and to a weaker kind of critical point (a quasimin) than a local minimum. Such critical points are related to Kuhn-Tucker conditions, assuming a constraint qualification; in certain circumstances, relevant to optimal control, such a critical point must be a minimum. Using these generalized critical points, a theorem analogous to duality is proved, but neither assuming convexity, nor implying weak duality.

A second derivative formula of the Liouville entropy at spaces of constant negative curvature

1997 ◽  
Vol 17 (5) ◽  
pp. 1131-1135 ◽  
Author(s):  
GERHARD KNIEPER
Keyword(s):  
Critical Points ◽  
Compact Manifold ◽  
Negative Curvature ◽  
Symmetric Spaces ◽  
Local Maximum ◽  
Second Derivative ◽  
Constant Negative Curvature ◽  
Locally Symmetric Spaces ◽  
Derivative Formula ◽  
Total Scalar Curvature

In this paper we study a new functional on the space of metrics with negative curvature on a compact manifold. It is a linear combination of Liouville entropy and total scalar curvature. Locally symmetric spaces are critical points of this functional. We provide an explicit formula for its second derivative at metrics of constant negative curvature. In particular, this shows that a metric of constant curvature is a local maximum.

The Stationary Phase Method for Certain Degenerate Critical Points I

1989 ◽  
Vol 41 (5) ◽  
pp. 907-931 ◽  
Author(s):  
Milos Dostal ◽  
Bernard Gaveau
Keyword(s):  
Stationary Phase ◽  
Asymptotic Behaviour ◽  
Critical Point ◽  
Critical Points ◽  
Compact Support ◽  
Fourier Integral ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Second Derivative ◽  
Image Position

We consider, in this work, the asymptotic behaviour for large λ, of a Fourier integralwhere 𝜑(x) is in general a C∞ function and a(x) a C∞ function with compact support. It is well known that the asymptotic behaviour of this integral is controlled by the behaviour of 𝜑 at its critical points (i.e., points where 𝜕𝜑/𝜕xj(x) = 0) and is given by local contributions at these points ([1], [3], [7], [9]).In general, one assumes the hypothesis of non degenerate isolated critical point, namely that the determinant of the second derivative at the critical point is non zero.

Phase space correspondence between Jordan and Einstein frames

2021 ◽  
pp. 2150087
Author(s):  
Amin Salehi
Keyword(s):  
Phase Space ◽  
Critical Point ◽  
Critical Points ◽  
Dynamical Behavior ◽  
Cosmological Parameters ◽  
Jordan Frame ◽  
Field Equations ◽  
Theories Of Gravity ◽  
The Stability ◽  
Jordan And Einstein Frames

Scalar–tensor theories of gravity can be formulated in the Einstein frame or in the Jordan frame (JF) which are related with each other by conformal transformations. Although the two frames describe the same physics and are equivalent, the stability of the field equations in the two frames is not the same. Here, we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans–Dicke theory in a Friedmann–Lemaitre–Robertson–Walker universe, but the results can easily be generalized. Our analysis shows that while there is a one-to-one correspondence between critical points in two frames and each critical point in one frame is mapped to its corresponds in another frame, however, stability of a critical point in one frame does not guarantee the stability in another frame. Hence, an unstable point in one frame may be mapped to a stable point in another frame. All trajectories between two critical points in phase space in one frame are different from their corresponding in other ones. This indicates that the dynamical behavior of variables and cosmological parameters is different in two frames. Hence, for those features of the study, which focus on observational measurements, we must use the JF where experimental data have their usual interpretation.

scholarly journals Local and global extrema for functions of several variables

1980 ◽  
Vol 29 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Bruce Calvert ◽  
M. K. Vamanamurthy
Keyword(s):  
Critical Point ◽  
Local Minimum ◽  
Global Minimum ◽  
Sufficient Conditions ◽  
Subject Classification ◽  
General Function ◽  
Several Variables ◽  
Open Question

AbstractLet p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

Carrier Concentrations and Drift Currents in the Depletion Region of a p–n Junction

2003 ◽  
Vol 10 (04) ◽  
pp. 649-660
Author(s):  
D. K. Mak
Keyword(s):  
Solid State ◽  
Applied Voltage ◽  
Local Minimum ◽  
Local Maximum ◽  
Depletion Region ◽  
Electron Drift ◽  
Electron Carrier ◽  
Quantitative Explanation ◽  
State Device ◽  
Solid State Device

It has always been stated in electronics, semiconductor and solid state device textbooks that the hole drift and electron drift currents in the depletion region of a p–n junction are constant and independent of applied voltage (biasing). However, the explanations given are qualitative and unclear. We extrapolate the existing analytic theory of a p–n junction to give a quantitative explanation of why the currents are constant. We have also shown that the carrier concentrations in the depletion region, as depicted in some of the textbooks, are incorrect, and need to be revised. Our calculations further demonstrate that in reverse biasing, both hole and electron carrier concentrations each experience a local maximum and a local minimum, indicating that their diffusion currents change directions twice within the depletion region.

scholarly journals Global Phase Portrait and Bifurcation Diagram for a Multi-Parametric Linear System

2020 ◽  
Vol 55 (4) ◽  
Author(s):  
Jorge Rodríguez Contreras ◽  
Alberto Reyes Linero ◽  
Juliana Vargas Sánchez
Keyword(s):  
Linear System ◽  
Critical Point ◽  
Phase Portrait ◽  
Bifurcation Diagram ◽  
Critical Points ◽  
Global Dynamics ◽  
Parametric Surfaces ◽  
Critical Points At Infinity ◽  
Global Phase Portrait ◽  
The Stability

The goal of this article is to conduct a global dynamics study of a linear multiparameter system (real parameters (a,b,c) in R^3); for this, we take the different changes that these parameters present. First, we find the different parametric surfaces in which the space is divided, where the stability of the critical point is defined; we then create a bifurcation diagram to classify the different bifurcations that appear in the system. Finally, we determine and classify the critical points at infinity, considering the canonical shape of the Poincaré sphere, and thus, obtain a global phase portrait of the multiparametric linear system.

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