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Recently, Acharjee, ¨Ozko¸c and Issaka introduced a new structure in topology named primal. Primal is the dual structure of grill. The main purpose of this paper is to introduce an operator using primal and obtain some of its fundamental properties. Also, we define the notion of topology suitable for a primal. We not only obtain some characterizations of this new notion, but also investigate many properties. [ABSTRACT FROM AUTHOR]

We introduce a topology on the set of vertices of a directed graph and we call the topological space as pathless directed topological space. We study relation between the relative topologies and pathless directed topological spaces of E-generated subdirected graphs. Then, we study connectedness, isomorphic and homeomorphic properties in digraphs and pathless directed topological spaces. Moreover, we apply our results to blood circulation process in human heart and disprove Shokry and Aly [M. Shokry and R. E. Aly, Topological properties on graph vs medical application in human heart, Int. J. Appl. Math., 15 (2013), 1103-1109], Nada et al. [S. Nada, A. E. F. El Atik and M. Atef, New types of topological structures via graphs, Math. Method. Appl. Sci., 41 (2018), 5801-5810] and Nawar et al. [A. S. Nawar and A. E. F. A. El-Atik, A model of a human heart via graph nano topological spaces, Int. J. Biomath., 12 (2019), p.1950006]. We show that pathless directed topology is accurately describing the circulation of blood in the heart of human body. [ABSTRACT FROM AUTHOR]

Topological dynamical system is an area of dynamical system to investigate dynamical properties in terms of a topological space. Nada and Zohny [Nada, S.I. and Zohny, H., An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202-204] applied topological dynamical system to explore the development process of an embryo from the zygote until birth and made three conjectures. In this paper, we disprove conjecture 3 of Nada and Zohny [Nada, S.I. and Zohny, H., An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202-204] by applying some of our mathematical results of bitopological dynamical system. Also, we introduce forward iterated Hausdorff space, backward iterated Hausdorff space, pairwise iterated Hausdorff space and establish relations between them in bitopological dynamical system. We formulate the function that represents cell division (specially, mitosis) and using this function we show that in the development process of a human baby from the zygote until its birth, there is a stage where the developing stage is forward iterated Hausdorff. [ABSTRACT FROM AUTHOR]

The forecasting graphs of World Bank, Reserve Bank of India, etc. are mostly line graphs or time series graphs. Any forecasting contains "standard error" as an error with complicated statistical formulae. A keen observation shows that mathematical patterns are available in nature, but in most of the cases, it is difficult for us to recognize these patterns. Similarly, it is most important for us to know the least upper bounds of these line graphs or time series graphs so that peaks of the prices with respect to time will not exceed these least upper bounds. It is hard to find any statistical or mathematical tool to determine these least upper bounds. Thus we give methodology to obtain these least upper bounds. We show existence of an equilibrium between the expected price and the original price of a commodity with the help of local functions and expansion operators of a bitopological space. These methods are based on choice of a consumer. Examples are provided to show that price of a commodity cannot exceed the interval of expected price. Moreover, we try to provide possible answers to the problem of "Control of Economic Variable" of Morgenstern [O. Morgenstern, Thirteen critical points in contemporary economic theory: An interpretation, Journal of Economic Literature10(4) 1972 1163–1189] by determining least upper bounds. [ABSTRACT FROM AUTHOR]

Sarma introduced the concept of (i; j)-bI-compact spaces with respect to an ideal in a bitopological space. In this paper, our aim is to present several characterizations of these spaces. Also, the class of (i; j)-bI-countably compact spaces is introduced with some of its properties. [ABSTRACT FROM AUTHOR]

We introduce the idea of using aggregation operators to replace the concepts of minimum and maximum in the basic definitions and results involving fuzzy relations and fuzzy graphs. This development has begun with the replacement of minimum and maximum with t -norms and t -conorms, respectively. With this new theory, it will be possible to open the door to new theoretical results and at times more useful application methods. We apply our results in the areas of human trafficking and slavery. [ABSTRACT FROM AUTHOR]

In this paper, we first correct a recent misconception about a topology that was suggested by Zeeman as a possible alternative to his fine topology. This misconception appeared while trying to establish the causality in the ambient boundary-ambient space cosmological model. We then show that this topology is actually the intersection topology (in the sense of Reed [The intersection topology w.r.t. the real line and the countable ordinals, Trans. Am. Math. Soc. 297(2) (1986) 509-520]) between the Euclidean topology on and the order topology whose order, namely horismos, is defined on the light cone and we show that the order topology from horismos belongs to the class of Zeeman topologies. These results accelerate the need for a deeper and more systematic study of the global topological properties of spacetime manifolds. [ABSTRACT FROM AUTHOR]

This paper focuses on two very important questions: "what is the future of a hybrid mathematical structure of soft set in science and social science?" and "why should we take care to use hybrid structures of soft set?". At present, these are the most fundamental questions; which encircle a few prominent areas of mathematics of uncertainties viz. fuzzy set theory, rough set theory, vague set theory, hesitant fuzzy set theory, IVFS theory, IT2FS theory, etc. In this paper, we review connections of soft set theory and hybrid structures in a non-technical manner; so that it may be helpful for a non-mathematician to think carefully to apply hybrid structures in his research areas. Moreover, we must express that we do not have any intention to nullify contributions of fuzzy set theory or rough set theory, etc. to mankind; but our main intention is to show that we must be careful to develop any new hybrid structure with soft set. Here, we have a short discussion on needs of artificial psychology and artificial philosophy to enrich artificial intelligence. [ABSTRACT FROM AUTHOR]

In E. Lutz and R. Lotspich, sex markets and human trafficking: Cause-effect and policy interventions, The Protection Project Journal of Human Rights and Civil Society 2 (2009) 199-124, causal linkages between commercial sex and human trafficking were examined. A three-link chain of necessary conditions exists. All three are required for trafficking into commercial sex. Consequently, trafficking can be addressed by policy intervention at any link. Prospects for policy success at the three points of intervention were compared. It was shown that a strategy of suppressing sex markets is least likely to be successful in reducing the associated human trafficking. In this paper, we create a mathematical model of the work developed above. We use five methods to derive five different linear equations to measure the success of policy intervention. The methods are the analytical hierarchy process, the Guiasu method, the Dempster rule of combination method, the Yen method, and the set-valued statistical method. The equations are based on an expert's opinion of the relative importance of the factors involved in sex trafficking. These equations are used to determine a measure of how well a particular state in the USA is doing in policy intervention. [ABSTRACT FROM AUTHOR]

It is well known that statistics deals with quantitative analysis. Thus, there is a lack of approach to do quantitative analysis in the presence of qualitative attributes. Soft set theory has the freedom to deal with attributes, [0, 1], etc. along with quantity. Thus, we introduce some fundamental ideas of soft statistics. Here, soft mean, soft standard deviation, soft coefficient of variation, soft correlation coefficient are introduced and some theorems are proved with respect to utility. Utility theory provides an analysis of choice behavior. As an application of our notions and results, we find soft correlation coefficients between vulnerability and government responses of various regions across the world. The data from 'The Global Slavery Index 2016' are considered for application purposes. [ABSTRACT FROM AUTHOR]

Bigeneralized topology was introduced in 2010. In this paper, we aim to contribute in the development of this area by proving results with respect to the hereditary class that defines two types of Lindelöf spaces in a bigeneralized topological space (BGTS). Some results on strong BGTS (briefly SBGTS) are proved by defining a -perfect mapping. We also prove some equivalent conditions with respect to the hereditary class. [ABSTRACT FROM AUTHOR]

In this paper, we prove that Lemma 5.7, Proposition 6.8 and Proposition 6.9 of Kharal [A. Kharal, Distance and similarity measures for soft sets, New Mathematics and Natural Computation 6(3) (2010) 321-334] are not correct w.r.t his Definition 5.5. We give suitable examples to show that Lemma 5.7 is not correct. Moreover, we rectify Lemma 5.7, Proposition 6.8 and Proposition 6.9 by providing suitable proofs. [ABSTRACT FROM AUTHOR]

The area of investigation at and around Mashak Pahar, Bankura district, West Bengal, India comprises a number of rock types namely: granite gneiss, migmatized quartz tourmaline gneiss, quartz pebble conglomerate, ferruginous quartzite, quartz tourmaline veins (as veins) and graphite schists. Interestingly, the study area lies in the region extending South Purulia Shear Zone (∼Tamar-Porapahar Shear Zone) which marks the boundary between two contrasting tectonic blocks of eastern India, namely, the Chhotanagpur Gneissic Terrane (CGC) to the north and Singhbhum Group of rocks to the south. The rocks of the study area are poly-phasedly deformed by three phases of folding, namely, F, F and F. All the tourmalines are classified to be of 'Alkali Group'. Chemistry of tourmalines from migmatized quartz tourmaline gneiss and those from quartz tourmaline veins are in conformity with their relation to (earthquake induced) shear system evolution in this terrain. In general, the compositional evolution of tourmaline during prograde metamorphism (∼400°-730°C) has been supported by both petrographic and chemical evidences. Assessment of mineral-chemical data of constituent tourmaline grains clearly suggests compositional variations across zonal boundaries within tourmaline that was controlled by changing metamorphic milieu in this terrane. Field and petrographic evidences clearly indicate activation of earlier and later shears in this region accompanied by infiltration of boron and formation of zoned tourmaline crystals. [ABSTRACT FROM AUTHOR]

Topological indices describe mathematical invariants of molecules in mathematical chemistry. M-polynomials of chemical graph theory have freedom about the nature of molecular graphs and they play a role as another topological invariant. Social networks can be both cyclic and acyclic in nature. We develop a novel application of M-polynomials, the (m , n , r) -agent recruitment graph where n > 1 , to study the relationship between the Dunbar graphs of social networks and the small-world phenomenon. We show that the small-world effects are only possible if everyone uses the full range of their network when selecting steps in the small-world chain. Topological indices may provide valuable insights into the structure and dynamics of social network graphs because they incorporate an important element of the dynamical transitivity of such graphs. [ABSTRACT FROM AUTHOR]