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Open AccessArticle

Regular Partial Residuated Lattices and Their Filters

by Nan Sheng and Xiaohong Zhang

Mathematics 2022, 10(14), 2429; https://doi.org/10.3390/math10142429 - 12 Jul 2022

Cited by 11 | Viewed by 1586

To express wider uncertainty, Běhounek and Daňková studied fuzzy partial logic and partial function. At the same time, Borzooei generalized t-norms and put forward the concept of partial t-norms when studying lattice valued quantum effect algebras. Based on partial t-norms, Zhang et al. studied partial residuated implications (PRIs) and proposed the concept of partial residuated lattices (PRLs). In this paper, we mainly study the related algebraic structure of fuzzy partial logic. First, we provide the definitions of regular partial t-norms and regular partial residuated implication (rPRI) through the general forms of partial t-norms and partial residuated implication. Second, we define regular partial residuated lattices (rPRLs) and study their corresponding properties. Third, we study the relations among commutative quasi-residuated lattices, commutative Q-residuated lattices, partial residuated lattices, and regular partial residuated lattices. Last, in order to obtain the filter theory of regular partial residuated lattices, we restrict certain conditions and then propose special regular partial residuated lattices (srPRLs) in order to finally construct the quotient structure of regular partial residuated lattices. Full article

(This article belongs to the Special Issue Fuzzy Logic and Its Applications)

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13 pages, 1362 KiB

Open AccessFeature PaperArticle

Puzzling Low-Temperature Behavior of the Van Der Waals Friction Force between Metallic Plates in Relative Motion

by George Dedkov

Universe 2021, 7(11), 427; https://doi.org/10.3390/universe7110427 - 9 Nov 2021

Cited by 5 | Viewed by 1755

Abstract

This paper presents the results of calculating the van der Waals friction force (dissipative fluctuation-electromagnetic force) between metallic (Au) plates in relative motion at temperatures close to 1 K. The stopping tangential force arises between moving plates along with the usual Casimir force of attraction, which has been routinely measured with high precision over the past two decades. At room temperatures, the former force is 10 orders of magnitude less than the latter, but at temperatures

T < 50 K,

friction increases sharply. The calculations have been carried out in the framework of the Levin-Polevoi-Rytov fluctuation electromagnetic theory. For metallic plates with perfect crystal lattices and without defects, van der Waals friction force is shown to increase with decreasing temperature as

T^{- 4}

. In the presence of residual resistance

ρ_{0}

of the metal, a plateau is formed on the temperature dependence of the friction force at

T \to 0

with a height proportional to

ρ_{0}^{- 0.8}

. Another important finding is the weak force-distance dependence

~ a^{- q}

(with

q < 1

). The absolute values of the friction forces are achievable for measurements in AFM-based experiments. Full article

(This article belongs to the Special Issue The Casimir Effect: From a Laboratory Table to the Universe)

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