A dual theorem on the summation of return differences

1985 ◽  
Vol 73 (1) ◽  
pp. 157-159 ◽  
Author(s):  
Wai-Kai Chen
Keyword(s):  
Dual Theorem

Ball characterizations in spaces of constant curvature

2018 ◽  
Vol 55 (4) ◽  
pp. 421-478
Author(s):  
Jesus Jerónimo-Castro ◽  
Endre Makai, Jr.
Keyword(s):  
Convex Sets ◽  
Central Symmetry ◽  
Connected Components ◽  
Plane Convex ◽  
Converse Implication ◽  
Plane Convex Body ◽  
Dual Theorem ◽  
Centrally Symmetric ◽  
Whole Space ◽  
Straight Lines

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class C+2 with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds). Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either d ≥ 3, or d = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds). We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points Sd for Sd of radius less than π/2- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.

scholarly journals A Note on a Lower Bound on the Minimum Rank of a Positive Semidefinite Hankel Matrix Rank Minimization Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yi Xu ◽  
Xiaorong Ren ◽  
Xihong Yan
Keyword(s):  
Lower Bound ◽  
Minimization Problem ◽  
Optimization Problem ◽  
Global Minimum ◽  
Hankel Matrix ◽  
Positive Semidefinite ◽  
Linear Constraints ◽  
Matrix Rank ◽  
Dual Theorem ◽  
Infinite Programming

This paper investigates the problem of approximating the global minimum of a positive semidefinite Hankel matrix minimization problem with linear constraints. We provide a lower bound on the objective of minimizing the rank of the Hankel matrix in the problem based on conclusions from nonnegative polynomials, semi-infinite programming, and the dual theorem. We prove that the lower bound is almost half of the number of linear constraints of the optimization problem.

A Practical Guide to the Dual Theorem

1958 ◽  
Vol 6 (3) ◽  
pp. 364-384 ◽  
Author(s):  
Harvey M. Wagner
Keyword(s):  
Practical Guide ◽  
Dual Theorem

Finite nonabelian p-groups all of whose subgroups are q-self dual

2014 ◽  
Vol 13 (06) ◽  
pp. 1450008
Author(s):  
Zvonimir Janko
Keyword(s):  
Dual Theorem ◽  
Abelian Theorem ◽  
Additional Assumptions

A finite p-group G is q-self dual if every quotient of G is isomorphic to a subgroup of G. Here, we determine finite 2-groups G all of whose subgroups are q-self dual (Theorem 3) and in case p > 2 we get a classification of such groups only under the additional assumptions that Ω1(G) is abelian (Theorem 4).

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