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Compact-Like Operators in Vector Lattices Normed by Locally Solid Lattices
Published in Hemen Dutta, Topics in Contemporary Mathematical Analysis and Applications, 2020
An ordered vector E is said to be vector lattice (or, Riesz space) if, for each pair of vectors x,y ∈ E, the supremum x ∨ y = sup{x,y} and the infimum x ∧ y = inf{x,y} both exist in E. Then, x+ := x ∨ 0, x− :=(−x) ∨ 0, and |x| := x ∨ (−x) are called the positive part, the negative part, and the absolute value of x ∈ E, respectively. Also, two vectors x and y in a vector lattice are said to be disjoint whenever |x|∧|y| = 0. A vector lattice is called order complete if every nonempty bounded above subset has a supremum (or, equivalently, whenever every nonempty bounded below subset has an infimum).
On some consequences of Mazur–Orlicz theorem to Hahn–Banach–Lagrange theorem
Published in Optimization, 2018
Jerzy Grzybowski, Diethard Pallaschke, Hubert Przybycień, Ryszard Urbański
Throughout the paper by X we will denote a nontrivial vector space over the field of real numbers. By we denote a Dedekind complete real vector lattice and for we put Here vector lattice means that the partial order in F satisfies the following property: for all and and that for every there exists the infimum or the greatest lower bound . By Dedekind complete we mean that every nonempty bounded from below subset of F has its infimum. For the sake of convenience we write that the infimum of a nonempty unbounded from below subset of F is equal to .
Laser cooling of rubidium atoms in a 2D optical lattice
Published in Journal of Modern Optics, 2018
Chunhua Wei, Carlos C. N. Kuhn
In order to compare the scalar and vector lattice configurations, the polarization of the horizontal lattice beam changes from circular to linear. The trap frequencies of both lattices were measured by modulating the lattice potential, resulting in resonant heating of the atoms when the modulation frequency is twice the oscillation frequency. For strong heating, a readily detectable loss of atoms from the trap will result. Figure 6 shows the remain trapped atoms as a function of the applied modulation frequency. The maximum atom loss (valley in the plot) is observed for both configurations at , implying a similar lattice depth for both schemes. These data assert that the observed cooling is not due to different potential depths between the lattices, and is, therefore, the cooling observed in this work is not only due to the minimization of the re-absorption of scattered light during the PGC as demonstrated in [17,35].
Separation, convexity and polarity in the space of normlinear functions
Published in Optimization, 2022
To conclude the paper, we would like to explore the differences and the similarities between downward X-convex sets, and sets of lower bounds in the ordered Banach space . We noticed at the end of Section 5 that the latter are a particular case of the former, since sets of lower bounds in Y are characterized (see [12]) as those sets which are downward, bounded above, and sup-containing, where L is bounded above in Y if there exists such that for all . The family of sets of lower bounds for Y is known to form the Dedekind vector lattice completion of Y, that is, the least Dedekind complete vector lattice in which Y is embedded. Dedekind completeness refers to the special property of a vector lattice Z that every subset W in Z which is bounded above admits a supremum in Z. This matter has been studied under a different point of view in [12,23,24]. The main result in [24], rephrased to meet the present setting, says that there is a vector lattice isomorphism between and the space of bounded normal lower semicontinuous functions defined on , the set of extreme points of the base of the polar cone .