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In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

With the untimely passing of Patrick Dehornoy in September 2019, the world of mathematics lost a brilliant scholar who made profound contributions to set theory, algebra, topology, and even computer science and cryptography. And I lost a dear friend and a strong influence in the direction of my own research in mathematics. In this paper, I will concentrate on his remarkable discovery that the braid groups are left-orderable, and its consequences, and its strong influence on my own research. I'll begin by describing how I learned of his work. [ABSTRACT FROM AUTHOR]

A many-to-many k -disjoint path cover (k -DPC) of a graph G is a set of k vertex-disjoint paths joining k distinct pairs of source and sink in which each vertex of G is contained exactly once in a path. The balanced hypercube B H n , a variant of the hypercube, was introduced as a desired interconnection network topology. Let S = { s 1 , s 2 , ... , s 2 n − 2 } and T = { t 1 , t 2 , ... , t 2 n − 2 } be any two sets of vertices in different partite sets of B H n (n ≥ 2). Cheng et al. in [Appl. Math. Comput. 242 (2014) 127–142] proved that there exists paired many-to-many 2-disjoint path cover of B H n when | S | = | T | = 2. In this paper, we prove that there exists unpaired many-to-many (2 n − 2) -disjoint path cover of B H n (n ≥ 2) from S to T , which has improved some known results. The upper bound 2 n − 2 is best possible in terms of the number of disjoint paths in unpaired many-to-many k -DPC of B H n . [ABSTRACT FROM AUTHOR]

Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl.264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except (2 , p) -torus links has no gap. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the research results of DAD, Daily Artificial Dispatcher, published in 2010 [S. Z. Stefanov, New Mathematics and Natural Computation6 (2010) 275–283], and give it new interpretation bring out the best of its meaning, as well as some more quantitative novel formula. In metaphorical sense, DAD is the analogue to DNA and hence the Smart Grid is analogue to the Creature by invoking the postmodern theory [J.-F. Lyotard, Moralités Postmodernes (Éditions Galilée, 1993) (in French)]. Specifically, the DAD’s binary expressions are generated by an innocent dialogue between DAD and Smart Grid in the form of world-strings. Namely, the living space-time of the DAD’s binary symbols is generated via a discussion between DAD and Smart Grid. Metaphorically speaking, DAD’s world is digitally described as a dramatic game and the Smart Grid as a creating cartoon loaded with an integral holographic complexity. Overall, the so called creatures capable of innocent discussions under near-zero temperature are generated by the epidemic growth of the Smart Grid cartoon. It is further concluded that the number of the Smart Grid creatures is inversely proportional to the half-time of the DAD’s binary “DNA” life cycle. Each Smart Grid can be coded in one of eight colors pool, as an aging in one of the four ways and as being in one of three possible development phases, dynamically. Finally, we take the liberty of calling the DAD’s world, a string prescribed as topology and landscape, to be “Aria”. [ABSTRACT FROM AUTHOR]

An -string tangle is a pair such that is a disjoint union of properly embedded arcs in a topological -ball . And an -string tangle is said to be trivial (or rational), if it is homeomorphic to as a pair, where is a 2-disk, is the unit interval and each is a point in the interior of . A stick tangle is a tangle each of whose arcs consists of finitely many line segments, called sticks. For an -string stick tangle its stick-order is defined to be a nonincreasing sequence of natural numbers such that, under an ordering of the arcs of the tangle, each denotes the number of sticks constituting the th arc of the tangle. And a stick-order is said to be trivial, if every stick tangle of the order is trivial. In this paper, restricting the -ball to be the standard 3-ball, we give the complete list of trivial stick-orders. [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]

The object of the present paper is to study mixed quasi-Einstein manifolds. Some geometric properties of mixed quasi-Einstein manifolds have been studied. We also discuss spacetime with space-matter tensor and some properties related to it. Finally, we construct an example of a mixed quasi-Einstein spacetime. [ABSTRACT FROM AUTHOR]