scholarly journals The Collatz Problem in the Light of an Infinite Free Semigroup

2014 ◽  
Vol 2014 ◽  
pp. 1-21
Author(s):  
Manfred Trümper
Keyword(s):  
Free Semigroup ◽  
Rational Functions ◽  
Intimate Relationship ◽  
Parameter Family ◽  
Group Operation ◽  
Important Notion ◽  
Collatz Problem

The Collatz (or 3m+1) problem is examined in terms of a free semigroup on which suitable diophantine and rational functions are defined. The elements of the semigroup, called T-words, comprise the information about the Collatz operations which relate an odd start number to an odd end number, the group operation being the concatenation of T-words. This view puts the concept of encoding vectors, first introduced in 1976 by Terras, in the proper mathematical context. A method is described which allows to determine a one-parameter family of start numbers compatible with any given T-word. The result brings to light an intimate relationship between the Collatz 3m+1 problem and the 3m-1 problem. Also, criteria for the rise or fall of a Collatz sequence are derived and the important notion of anomalous T-words is established. Furthermore, the concept of T-words is used to elucidate the question what kind of cycles—trivial, nontrivial, rational—can be found in the Collatz 3m+1 problem and also in the 3m-1 problem. Furthermore, the notion of the length of a Collatz sequence is discussed and applied to average sequences. Finally, a number of conjectures are proposed.

scholarly journals Perfect Matchings for the Three-Term Gale-Robinson Sequences

10.37236/214 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
James Propp ◽  
Julian West
Keyword(s):  
Recurrence Relations ◽  
Initial Conditions ◽  
A Priori ◽  
Rational Functions ◽  
Rational Numbers ◽  
Perfect Matchings ◽  
Parameter Family ◽  
Integer Coefficients ◽  
Special Cases

In 1991, David Gale and Raphael Robinson, building on explorations carried out by Michael Somos in the 1980s, introduced a three-parameter family of rational recurrence relations, each of which (with suitable initial conditions) appeared to give rise to a sequence of integers, even though a priori the recurrence might produce non-integral rational numbers. Throughout the '90s, proofs of integrality were known only for individual special cases. In the early '00s, Sergey Fomin and Andrei Zelevinsky proved Gale and Robinson's integrality conjecture. They actually proved much more, and in particular, that certain bivariate rational functions that generalize Gale-Robinson numbers are actually polynomials with integer coefficients. However, their proof did not offer any enumerative interpretation of the Gale-Robinson numbers/polynomials. Here we provide such an interpretation in the setting of perfect matchings of graphs, which makes integrality/polynomiality obvious. Moreover, this interpretation implies that the coefficients of the Gale-Robinson polynomials are positive, as Fomin and Zelevinsky conjectured.

HOW IS THE DYNAMICS OF KÖNIG ITERATION FUNCTIONS AFFECTED BY THEIR ADDITIONAL FIXED POINTS?

Fractals ◽  
1999 ◽  
Vol 07 (03) ◽  
pp. 327-334 ◽  
Author(s):  
V. DRAKOPOULOS
Keyword(s):  
Fixed Points ◽  
Rational Functions ◽  
Parameter Family ◽  
Quadratic Polynomials ◽  
Iteration Functions ◽  
Extraneous Fixed Points ◽  
Roots Of Equations ◽  
The One ◽  
Cubic Polynomials ◽  
Raphson Method

König iteration functions are a generalization of Newton–Raphson method to determine roots of equations. These higher-degree rational functions possess additional fixed points, which are generally different from the desired roots. We first prove two new results: firstly, about these extraneous fixed points and, secondly, about the Julia sets of the König functions associated with the one-parameter family of quadratic polynomials. Then, after finding all the critical points of the König functions as applied to a one-parameter family of cubic polynomials, we examine the orbits of the ones available for convergence to an attracting periodic cycle, should such a cycle exist.

Approaching Mull? ?adr? as Scriptural Exegete: A Survey of Scholarship on His Quranic Works

2010 ◽  
Vol 4 (1-2) ◽  
pp. 75-96
Author(s):  
Mohammed Rustom
Keyword(s):  
Intimate Relationship ◽  
Diverse Range ◽  
Scholarly Literature ◽  
Religious Worldview ◽  
Comprehensive Survey ◽  
Methodological Problems

This article offers the first comprehensive survey of scholarly literature devoted to the Qur??nic works of the famous Muslim philosopher, Mull? ?adr? (d. 1050/1640). While taking account of the merits and shortcomings of studies on ?adr?’s Qur??nic writings, we will also be concerned with highlighting some of the methodological problems raised by the diverse range of approaches adopted in these studies. Chief amongst them is the tendency to pit ?adr? the philosopher against ?adr? the scriptural exegete. Such a dichotomy is not entirely helpful, both with respect to painting a clearer picture of ?adr?’s religious worldview, and to addressing broader questions pertaining to the intimate relationship shared between the “act” of philosophy and the “act” of reading scripture.

Intimate Partner Violence and Its Under-Reporting in Pakistan

2018 ◽  
Vol 12 (1) ◽  
pp. 254
Author(s):  
Abdul Hadi
Keyword(s):  
Intimate Partner Violence ◽  
Partner Violence ◽  
Intimate Relationship ◽  
Intimate Partner ◽  
Power Relationship ◽  
Gender Based Violence ◽  
Patriarchal System ◽  
Attempted Murder ◽  
Gender Based ◽  
Private Matter

Intimate Partner violence is not a culturally limited practice, but prevails in every country, irrespective of culture, class, and ethnicity. Violence is one of the mechanisms used by men to control and subjugate women; and is a manifestation of unequal power relationship sustained by patriarchy. Patriarchy makes violence necessary for the sake of its existence. Intimate partner violence, the most common forms of gender-based violence entails the exertion of power over a partner in an intimate relationship through a behavior that is intimidating, threatening, harassing or harmful. The spouse can be harmed physically, as well sexually, emotionally, and psychologically, the violence can occur multiple times. Intimate partner violence in Pakistan persists almost in every family because women have subjugated and vulnerable status and are generally treated as second class citizens. Generally, the occurrence of violence at home is effectively condoned and regarded it as ‘private matter’ which does not require any intervention. it is seldom recognized as a crime socially unless it takes an extreme form of murder or attempted murder which could range from driving a woman to suicide or engineering an accident (frequently the bursting of a kitchen stove). This study aims to find out the factors which precipitate Intimate partner violence in Pakistan and what are the factors which preclude the reporting of Intimate partner violence and seeking legal redress. This study has found that strict cultural and patriarchal system and values precipitate intimate partner violence and also preclude victims to report the incidences by not giving them appropriate moral, cultural and legal support.

VBTree Tutorial: Automatic Completion of Group Operation on Structural Dataset

2020 ◽  
Author(s):  
Chen Zhang ◽  
Huakang Bian ◽  
Kenta Yamanaka ◽  
Akihiko Chiba
Keyword(s):  
Group Operation
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