Roots and Extremal Points

2013 ◽  
pp. 83-111
Author(s):  
Philipp O. J. Scherer
Keyword(s):  
Extremal Points

scholarly journals B∗algebra unit ball extremal points

1964 ◽  
Vol 14 (2) ◽  
pp. 627-637 ◽  
Author(s):  
Philip Miles
Keyword(s):  
Unit Ball ◽  
Extremal Points

scholarly journals Quantum limit of genuine tripartite correlations by bipartite extremality

2020 ◽  
Vol 18 (04) ◽  
pp. 2050014
Author(s):  
Hiroyuki Ozeki ◽  
Satoshi Ishizaka
Keyword(s):  
Probability Space ◽  
Quantum Correlations ◽  
Quantum Limit ◽  
Sufficient Criterion ◽  
Extremal Points ◽  
Chsh Inequality ◽  
Nonlocal Correlations ◽  
Necessary And Sufficient

The characterization of the extremal points of the set of quantum correlations has attracted wide interest. In the simplest bipartite Bell scenario, a necessary and sufficient criterion for identifying extremal correlations has recently been conjectured, but extremality of tripartite correlations is not well known. In this study, we analyze tripartite extremal correlations in terms of the conjectured bipartite extremal criterion, and we demonstrate that the bipartite part of some extremal correlations satisfies the bipartite criterion, even though they violate Svetlichny’s inequality, and therefore are considered (stronger) genuine tripartite nonlocal correlations. This phenomenon arises from the fact that the conjectured extremal criterion is automatically satisfied when the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality exceeds a certain threshold, the value of which is given by the maximum CHSH violation at the edges of the probability space. This also suggests the possibility that the extremality of bipartite correlations can be certified by verifying whether the CHSH violation exceeds the threshold.

scholarly journals Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 2. Mid-wave regimes

2019 ◽  
Vol 863 ◽  
pp. 185-214 ◽  
Author(s):  
Alexander L. Frenkel ◽  
David Halpern ◽  
Adam J. Schweiger
Keyword(s):  
Normal Modes ◽  
Bond Number ◽  
Base Shear ◽  
System Parameters ◽  
Critical Curves ◽  
Extremal Points ◽  
Insoluble Surfactant ◽  
Fluid Layers ◽  
Instability Regions ◽  
Unstable Branch

The joint effects of an insoluble surfactant and gravity on the linear stability of a two-layer Couette flow in a horizontal channel are investigated. The inertialess instability regimes are studied for arbitrary wavelengths and with no simplifying requirements on the system parameters: the ratio of thicknesses of the two fluid layers; the viscosity ratio; the base shear rate; the Marangoni number $Ma$; and the Bond number $Bo$. As was established in the first part of this investigation (Frenkel, Halpern & Schweiger, J. Fluid Mech., vol. 863, 2019, pp. 150–184), a quadratic dispersion equation for the complex growth rate yields two, largely continuous, branches of the normal modes, which are responsible for the flow stability properties. This is consistent with the surfactant instability case of zero gravity studied in Halpern & Frenkel (J. Fluid Mech., vol. 485, 2003, pp. 191–220). The present paper focuses on the mid-wave regimes of instability, defined as those having a finite interval of unstable wavenumbers bounded away from zero. In particular, the location of the mid-wave instability regions in the ($Ma$, $Bo$)-plane, bounded by their critical curves, depending on the other system parameters, is considered. The changes of the extremal points of these critical curves with the variation of external parameters are investigated, including the bifurcation points at which new extrema emerge. Also, it is found that for the less unstable branch of normal modes, a mid-wave interval of unstable wavenumbers may sometimes coexist with a long-wave one, defined as an interval having a zero-wavenumber endpoint.

SALIENT POINTS OF Q-CONVEX SETS

2001 ◽  
Vol 15 (07) ◽  
pp. 1023-1030 ◽  
Author(s):  
ALAIN DAURAT
Keyword(s):  
Finite Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Salient Point ◽  
Extremal Points ◽  
Salient Points

The Q-convexity is a kind of convexity in the discrete plane. This notion has practically the same properties as the usual convexity: an intersection of two Q-convex sets is Q-convex, and the salient points can be defined like the extremal points. Moreover a Q-convex set is characterized by its salient point. The salient points can be generalized to any finite subset of ℤ2.

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