Fractal analysis of polyferric chloride-humic acid(PFC-HA) flocs in different topological spaces

The fractal dimensions in different topological spaces of polyferric chloride-humic acid （PFC-HA） flocs, formed in flocculating different kinds of humic acids （HA） water at different initial pH （9.0, 7.0, 5.0） and PFC dosages, were calculated by effective densitymaximum diameter, image analysis, and N2 absorption-desorption methods, respectively. The mass fractal dimensions （De） of PFC-HA floes were calculated by bi-logarithm relation of effective density with maximum diameter and Logan empirical equation. The Df value was more than 2.0 at initial pH of 7,0, which was 11% and 13% higher than those at pH 9.0 and 5.0, respecively, indicating the most compact flocs formed in flocculated HA water at initial pH of 7.0. The image analysis for those flocs indicates that after flocculating the HA water at initial pH greater than 7.0 with PFC flocculant, the fractal dimensions of D2 （logA vs. logdL） and D3 （logVsphere vs. logdL） of PFC-HA floes decreased with the increase of PFC dosages, and PFC-HA floes showed a gradually looser structure. At the optimum dosage of PFC, the D2 （logA vs. logdL） values of the flocs show 14%-43% difference with their corresponding Dr, and they even had different tendency with the change of initial pH values. However, the D2 values of the floes formed at three different initial pH in HA solution had a same tendency with the corresponding Df. Based on fractal Frenkel-Halsey-HiU （FHH） adsorption and desorption equations, the pore surface fractal dimensions （Ds） for dried powders of PFC-HA flocs formed in HA water with initial pH 9.0 and 7.0 were all close to 2.9421, and the Ds values of flocs formed at initial pH 5.0 were less than 2.3746. It indicated that the pore surface fractal dimensions of PFC-HA floes dried powder mainly show the irregularity from the mesopore-size distribution and marcopore-size distribution.

γ and β Approximations via General Ordered Topological Spaces

In this paper, we introduce the concepts of g and b approximations via general ordered topological approximation spaces. Also, increasing (decreasing) g, b boundary, positive and negative regions are given in general ordered topological approximation spaces (GOTAS, for short). Some important properties of them were investigated. From this study, we can say that studying any properties of rough set concepts via GOTAS is a generalization of Pawlak approximation spaces and general approximation spaces.

EQUILIBRIUM PROBLEMS WITH LOWER AND UPPER BOUNDS IN TOPOLOGICAL SPACES

By employing a fixed point theorem due to Ding, Park and Jung, some existence theorems of solutions for equilibrium problems with lower and upper bounds are proved in noncompact topological spaces. These results further answer the open problem raised by Isac, Sehgal and Singh under much weaker assumptions.

μ-Separations in generalized topological spaces

This paper takes some investigations on generalized topological spaces with some μ- separations. Some characterizations of μTi-spaces for i = 0, 1, 2, 3, 4, μTn-spaces and μR0-spaces are obtained and some relations among these spaces are established.

T_2 Separation Subsets and T_2 Separateness in L-fuzzy Topological Spaces

The concept of α-CT2 separation L-fuzzy subsets in L-fuzzy topological spaces is presented by taking the stratiform structure of L-fuzzy subsets as the point of departure,and its basic characterizations and some topological properties are discussed,and the relation between it and other separateness is exposed,and the action is studied of α-CT2 separateness in N-compact spaces and N-paracompact spaces.

Almost Fuzzy Compactness in L-fuzzy Topological Spaces

In this paper, the notion of almost fuzzy compactness is dened in L-fuzzy topological spaces by means of inequality, where L is a completely distributiveDeMorgan algebra. Its properties are discussed and many characterizations of it arepresented.

An Introduction to Fuzzy Topological Spaces

Topology has enormous applications on fuzzy set. An attention can be brought to the mathematicians about these topological applications on fuzzy set by this article. In this research, first we have classified the fuzzy sets and topological spaces, and then we have made relation between elements of them. For expediency, with mathematical view few basic definitions about crisp set and fuzzy set have been recalled. Then we have discussed about topological spaces. Finally, in the last section, the fuzzy topological spaces which is our main object we have developed the relation between fuzzy sets and topological spaces. Moreover, this article has been concluded with the examination of some of its properties and certain relationships among the closure of these spaces.

A Ubi Sunt Motif of Life Philosophy:Construction of Three Topological Spaces in Shakespeare’s Merchant of Venice

Topology,as a geometrical branch,studies spaces,whose essence lies in its topological equivalence.As far as literary criticism is concerned,the author’s writing purpose can be regarded as an open set in topology,which contains a number of different topological spaces.No matter how these spaces transform,they remain fundamental properties as long as their essence is invariant.From the perspective of topology,Shakespeare constructed three equivalent metaphorical topological spaces:one-pound-flesh agreement,marriage casket“lottery”,and wedding ring trick.The present paper purports to explore how Shakespeare constructed the three equivalent topological spaces to express the theme of literature:the ubi sunt1 motif of life philosophy.

Inclusive Regular Separation in L-Fuzzy Topological Spaces

In this paper, we first introduce the concept of the inclusive regular separation in L-fuzzy topological spaces. Then we compare the inclusive regular separation with pointed regular separation and regular separation, and discuss the implicative and non-implicative relations among the above three separations. Finally, we illustrate that the inclusive regular separation is harmonic with the inclusive normal separation and inclusive completely regular separation.

The Separation and N-Compactness of Induced R（L）-Fuzzy Topological Spaces

In this paper, we prove that （L^X,5） is T0,T1, T2, regular （T3）, normal （T4） and completely regular spaces if and only if （R（L）^X, ω（δ）） is T0, T1, T2, regular （T3）, normal （T4） and completely regular spaces, respectively, and （L^X,δ） is N-compact if and only if （R（L）^X, ω（δ）） is N-compact.

Mathematical Morphology View of Topological Rough Sets and Its Applications

This article focuses on the relationship between mathematical morphology operations and rough sets,mainly based on the context of image retrieval and the basic image correspondence problem.Mathematical morphological procedures and set approximations in rough set theory have some clear parallels.Numerous initiatives have been made to connect rough sets with mathematical morphology.Numerous significant publications have been written in this field.Others attempt to show a direct connection between mathematical morphology and rough sets through relations,a pair of dual operations,and neighborhood systems.Rough sets are used to suggest a strategy to approximatemathematicalmorphology within the general paradigm of soft computing.A single framework is defined using a different technique that incorporates the key ideas of both rough sets and mathematical morphology.This paper examines rough set theory from the viewpoint of mathematical morphology to derive rough forms of themorphological structures of dilation,erosion,opening,and closing.These newly defined structures are applied to develop algorithm for the differential analysis of chest X-ray images from a COVID-19 patient with acute pneumonia and a health subject.The algorithm and rough morphological operations show promise for the delineation of lung occlusion in COVID-19 patients from chest X-rays.The foundations of mathematical morphology are covered in this article.After that,rough set theory ideas are taken into account,and their connections are examined.Finally,a suggested image retrieval application of the concepts from these two fields is provided.

On the Reflectivity and Coreflectivity of L-Fuzzifying Topological Spaces in L-Topological Spaces

In this paper it is proved that for all completely distributive lattices L. the category of L-fuzzifying topological spaces can be wmbedded in the category of L-topological spaces (stratified Chang-Goguen spaces) as a simultaneously bireflective and bicoreflective full subcategory.

Separation axioms in topological ordered spaces

The Ti-axiom,the Ti-ordered axiom and Ti-pairwise axiom(i = 0,1,2,3,4) of topological ordered space are discussed and proved that they are equivalence under the certain conditions.

Generalized Bi quasi variational Inequalities in Locally Convex Topological Vector Spaces

In this paper we investigate generalized bi quasi variational inequalities in locally convex topological vector spaces. Motivated and inspired by the recent research work in this field,we establish several existence theorems of solutions for generalized bi quasi variational inequalities,which are the extension and improvements of the earlier and recent results obtained previously by many authors including Sun and Ding [18],Chang and Zhang [23] and Zhang [24].

On Irresolute Topological Vector Spaces

In this paper, our focus is to investigate the notion of irresolute topological vector spaces. Irresolute topological vector spaces are defined by using semi open sets and irresolute mappings. The notion of irresolute topological vector spaces is analog to the notion of topological vector spaces, but mathematically it behaves differently. An example is given to show that an irresolute topological vector space is not a topological vector space. It is proved that: 1) Irresolute topological vector spaces possess open hereditary property;2) A homomorphism of irresolute topological vector spaces is irresolute if and only if it is irresolute at identity element;3) In irresolute topological vector spaces, the scalar multiple of semi compact set is semi compact;4) In irresolute topological vector spaces, every semi open set is translationally invariant.

Locomotion Paradigms of Topological Space in Shakespeare’s Double Falsehood

In the 16th and 17th century,Britain was constantly strained by sporadic plagues,famines,fires and political conflicts.As a response in literature,Shakespeare has constructed an annular locomotion paradigm in Fletcher and his collaboration work Cardenio.Be more specific,the locomotion routes can be exemplified in Möbius strip,a rhizome system,a reciprocating juxtaposition between the foreground and the background,either of which finally runs to an annular schema.The annular in narration corresponds with The Globe theater and the round world.Shakespeare may express his expectation and prospect for the uprising bourgeoise and the newly found world by the resurrection theme in literature.

Prime MP-filter Spaces of Fuzzy Implication Algebras

In this paper,the topological space(PF_(MP)(X),T) based on prime MP-filters of a lattice FI-algebra X is constructed firstly and we proved that it is a compact T_0-space if X with condition(P).Secondly,we restricted T to the set of all maximal MP-filters MF_(MP)(X) of X and concluded that(PF_(MP)(X),T |_(PF_(MP)(X)) )is a compact T_2 space if X with conditions(P) and(S).

LI-ideal Spaces of Lattice Implication Algebra

In this paper, a topological space based on LI-ideals of a lattice implication algebra is constructed, and its topological properties, such as separability, compactness and connectedness are discussed.

Some New Fixed Point Theorems in Probabilistic Metric Spaces

In this paper, we introduce the concept of the Z-M-PN space and obtain somenew fixed point theorems in probabilistic metric spaces Meanwhile,some famous fixedpoint theorems are generalized in probabilistic metric spaces, such a.s fixed point theorem of Schauder, Guo's theorem and fixed point theorem of Petryshyn are generalized in Menger PN-space. And fixed point theorem of Altman is also generalized in the Z-M-PN space.

About Compact and Sequentially Compact Coincide in Topological Groups with the Axiom (A_1)

This paper is concerned with the relation between the compactness and sequential compactness in a topological space or a topological group, and show that the compactness and sequential compactness coincide in a topolpgical group with the axiom (A1).