Divisions by Two in Collatz Sequences: A Data Science Approach

The Collatz conjecture is an unsolved number theory problem. We approach the question by examining the divisions by two that are performed within Collatz sequences. Aside from classical mathematical methods, we use techniques of data science. Based on the analysis of 10,000 sequences we show that the number of divisions by two lies within clear boundaries. Building on the results, we develop and prove an equation to calculate the maximum possible number of divisions by two for any given a Collatz sequence. Whenever this maximum is reached, a sequence leads to the result one, as conjectured by Lothar Collatz. Furthermore, we show how many divisions by two are required for a cycle of a specific length. The findings are valuable for further investigations and could form the basis for a comprehensive proof of the conjecture.

Remarks on Essential Maximal Numerical Range of Aluthge Transform

This paper focuses on the properties of the essential maximal numerical range of Aluthgetransform T. For instance, among other results, we show that the essential maximal numerical rangeof Aluthge transform is nonempty and convex. Further, we prove that the essential maximal numericalrange of Aluthge transform e T is contained in the essential maximal numerical range of T. This studyis therefore an extention of the research on Aluthge transform which was begun by Aluthge in hisstudy of p−hyponormal operators.

Certain Types of Continuity via Ig**α-Closed Sets

In this paper we introduce and investigate the notion of Ig**α-continuous functions, almost Ig**α-continuous functions and discussed the relationship withother continuous functions and obtained their characteristics. Finally we obtain the decomposition of *α-continuity.

Introducing a Finite State Machine for Processing Collatz Sequences

The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine’s word transformation is equivalent to the standard Collatz number transformation and then discuss the possibilities for utilizing this approach for solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation.

Locally gωα-Closed Sets in Topological Spaces

In the year 2014, the present authors introduced and studied the concept of gωα-closed sets in topological spaces. The purpose of this paper to introduce a new class of locally closed sets called gωα-locally closed sets (briefly gωαlc-sets) and study some of their properties. Also gωα-locally closed continuous (briefly gωαlc-continuous) functions and its irresolute functions are introduced and studied their properties in topological spaces.

On M-Projective Curvature Tensor of Lorentzian α-Sasakian Manifolds

In this paper, we study the nature of Lorentzianα-Sasakian manifolds admitting M-projective curvature tensor. We show that M-projectively flat and irrotational M-projective curvature tensor of Lorentzian α-Sasakian manifolds are locally isometric to unit sphere Sn(c) , wherec = α2. Next we study Lorentzianα-Sasakian manifold with conservative M-projective curvature tensor. Finally, we find certain geometrical results if the Lorentzianα-Sasakian manifold satisfying the relation M(X,Y)⋅R=0.

Some Properties of Soft β-Connected Spaces in Soft Topological Spaces

Soft set theory is a newly emerging tool to deal with uncertain problems and has been studied by researchers in theory and practice. In this paper, we investigated the properties and characterizations of softβ-connected spaces in soft topological spaces. We anticipate that the results in this paper can be stimulated to the further study on soft topology to accomplish genenral framework for the practical life applications.

Projectively Flat Finsler Space of Douglas Type with Weakly-Berwald (α,β)-Metric

The present article is organized as follows: In the first part, we characterize the important class of special Finsler (α,β)-metric in the form ofL=α+α2/β, whereαis Riemannian metric andβis differential 1-form to be projectively flat. In the second part, we describe condition for a Finsler spaceFnwith an (α,β)-metric is of Douglas type. Further we investigate the necessary and sufficient condition for a Finsler space with an (α,β)-metric to be weakly-Berwald space and Berwald space.

Topological Structure of Quasi-Partial b-Metric Spaces

In this paper we discuss the topological properties of quasi-partialb-metric spaces. The notion of quasi-partialb-metric space was introduced and fixed point theorem and coupled fixed point theorem on this space were studied. Here the concept of quasi-partialb-metric topology is discussed and notion of product of quasi-partialb-metric spaces is also introduced.

Traces in SL(3,C) and SU(2,1) Groups

In this paper we prove some trace identities in SL(3,C) and SU(2,1) groups. We also present the merits on how to parametrise pair of pants via traces and cross-ratio. Finally, we compute traces of matrices that are generated by complex reflections in complex triangle groups.