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Huan Huang

Publications and source records attributed to Huan Huang.

47 records · Page 3Linked to original sources
arXiv2019

Some properties of Skorokhod metric on fuzzy sets

In this paper, we have our discussions on normal and upper semi-continuous fuzzy sets on metric spaces. The Skorokhod-type metric is stronger than the Skorokhod metric. It is found that the Skorokhod metric and the Skorokhod-type metric are equivalent on compact fuzzy sets. However, the Skorokhod metric and the Skorokhod-type metric need not be equivalent on $L_p$-integrable fuzzy sets. Based on this, we investigate relations between these two metrics and the $L_p$-type $d_p$ metric. It is found that the relations can be divided into three cases. On compact fuzzy sets, the Skorokhod metric is stronger than the $d_p$ metric. On $L_p$-integrable fuzzy sets, which take compact fuzzy sets as special cases, the Skorokhod metric is not necessarily stronger than the $d_p$ metric, but the Skorokhod-type metric is still stronger than the $d_p$ metric. On general fuzzy sets, even the Skorokhod-type metric is not necessarily stronger than the $d_p$ metric. We also show that the Skorokhod metric is stronger than the sendograph metric.

math.GM↗
arXiv2018

Characterizations of endograph metric and $Γ$-convergence on fuzzy sets

This paper is devoted to the relationships and properties of the endograph metric and the $Γ$-convergence. The main contents can be divided into three closely related parts. Firstly, on the class of upper semi-continuous fuzzy sets with bounded $α$-cuts, we find that an endograph metric convergent sequence is exactly a $Γ$-convergent sequence satisfying the condition that the union of $α$-cuts of all its elements is a bounded set in $\mathbb{R}^m$ for each $α> 0$. Secondly, based on investigations of level characterizations of fuzzy sets themselves, we present level characterizations (level decomposition properties) of the endograph metric and the $Γ$-convergence. It is worth mentioning that, using the condition and the level characterizations given above, we discover the fact: the endograph metric and the $Γ$-convergence are compatible on a large class of general fuzzy sets which do not have any assumptions of normality, convexity or star-shapedness. Its subsets include common particular fuzzy sets such as fuzzy numbers (compact and noncompact), fuzzy star-shaped numbers (compact and noncompact), and general fuzzy star-shaped numbers (compact and noncompact). Thirdly, on the basis of the conclusions presented above, we study various subspaces of the space of upper semi-continuous fuzzy sets with bounded $α$-cuts equipped with the endograph metric. We present characterizations of total boundedness, relative compactness and compactness in these fuzzy set spaces and clarify relationships among these fuzzy set spaces. It is pointed out that the fuzzy set spaces of noncompact type are exactly the completions of their compact counterparts under the endograph metric.

math.GM↗
arXiv2016

Characterizations of compact sets in fuzzy sets spaces with $L_p$ metric

In this paper, we present characterizations of totally bounded sets, relatively compact sets and compact sets in the fuzzy sets spaces $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ equipped with $L_p$ metric, where $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ are two kinds of general fuzzy sets on $\mathbb{R}^m$ which do not have any assumptions of convexity or star-shapedness. Subsets of $F_B(\mathbb{R}^m)^p$ include common fuzzy sets such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and the general fuzzy star-shaped numbers introduced by Qiu et al. The existed compactness criteria are stated for three kinds of fuzzy sets spaces endowed with $L_p$ metric whose universe sets are the former three kinds of common fuzzy sets respectively. Constructing completions of fuzzy sets spaces with respect to $L_p$ metric is a problem which is closely dependent on characterizing totally bounded sets. Based on preceding characterizations of totally boundedness and relatively compactness and some discussions on convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy sets spaces mentioned in this paper can be obtained by using the $L_p$-extension. We also clarify relation among all the ten fuzzy sets spaces discussed in this paper, which consist of five pairs of original spaces and the corresponding completions. Then, we show that the subspaces of $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ mentioned in this paper have parallel characterizations of totally bounded sets, relatively compact sets and compact sets. At last, as applications of our results, we discuss properties of $L_p$ metric on fuzzy sets space and relook compactness criteria proposed in previous work.

math.GM↗
arXiv2016

Approximation of fuzzy numbers by convolution method

In this paper we consider how to use the convolution method to construct approximations, which consist of fuzzy numbers sequences with good properties, for a general fuzzy number. It shows that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers which have finite non-differentiable points. In the previous work, this convolution method only can be used to construct differentiable approximations for continuous fuzzy numbers whose possible non-differentiable points are the two endpoints of 1-cut. The constructing of smoothers is a key step in the construction process of approximations. It further points out that, if appropriately choose the smoothers, then one can use the convolution method to provide approximations which are differentiable, Lipschitz and preserve the core at the same time.

math.GM↗
arXiv2015

A countable set derived by fuzzy set

In this paper, it shows that for each fuzzy set $u$ on $\mathbb{R}^m$, the set $D(u)$ is at most countable. Based on this, it modifies the proof of assertion (I) in step 2 of the sufficiency part of Theorem 4.1 in paper: Characterizations of compact sets in fuzzy sets spaces with $L_p$ metric, http://arxiv.org/abs/1509.00447.

math.GM↗
arXiv2012

Some notes on characterizations of compact sets in fuzzy number spaces

In this paper, we presents a characterization of compact subsets of the fuzzy number space equipped with the level convergence topology. Based on this, it is shown that compactness is equivalent to sequential compactness on the fuzzy number space equipped with the level convergence topology. Diamond and Kloeden gave a characterization of compact sets in fuzzy number spaces equipped with the supremum metric, Fang and Xue also gave a characterization of compact sets in one-dimensional fuzzy number spaces equipped with supremum metric. The latter characterization is just the one-dimensional case of the former characterization. There exists conflict between the characterization given by us and the characterizations given by the above mentioned authors. We point out the characterizations gave by them is incorrect by a counterexample.

math.GM↗
arXiv2011

Femtosecond Laser-induced Crystallization of Amorphous Sb2Te3 film and Coherent Phonon Spectroscopy Characterization and Optical Injection of Electron Spins

A femtosecond laser-irradiated crystallizing technique is tried to convert amorphous Sb2Te3 film into crystalline film. Sensitive coherent phonon spectroscopy (CPS) is used to monitor the crystallization of amorphous Sb2Te3 film at the original irradiation site. The CPS reveals that the vibration strength of two phonon modes that correspond to the characteristic phonon modes of crystalline Sb2Te3, enhances with increasing laser irradiation fluence (LIF), showing the rise of the degree of crystallization with LIF and that femtosecond laser irradiation is a good post-treatment technique. Time-resolved circularly polarized pump-probe spectroscopy is used to investigate electron spin relaxation dynamics of the laser-induced crystallized Sb2Te3 film. Spin relaxation process indeed is observed, confirming the theoretical predictions on the validity of spin-dependent optical transition selection rule and the feasibility of transient spin-grating-based optical detection scheme of spin-plasmon collective modes in Sb2Te3-like topological insulators.

cond-mat.str-el↗