scholarly journals A New Approach on Proving Collatz Conjecture

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Wei Ren
Keyword(s):  
Natural Number ◽  
Natural Numbers ◽  
Induction Method ◽  
New Approach ◽  
Collatz Conjecture ◽  
The Future ◽  
Future Work ◽  
Left Portion

Collatz Conjecture (3x+1 problem) states any natural number x will return to 1 after 3⁎x+1 computation (when x is odd) and x/2 computation (when x is even). In this paper, we propose a new approach for possibly proving Collatz Conjecture (CC). We propose Reduced Collatz Conjecture (RCC)—any natural number x will return to an integer that is less than x. We prove that RCC is equivalent to CC. For proving RCC, we propose exploring laws of Reduced Collatz Dynamics (RCD), i.e., from a starting integer to the first integer less than the starting integer. RCC can also be stated as follows: RCD of any natural number exists. We prove that RCD is the components of original Collatz dynamics (from a starting integer to 1); i.e., RCD is more primitive and presents better properties. We prove that RCD presents unified structure in terms of (3⁎x+1)/2 and x/2, because 3⁎x+1 is always followed by x/2. The number of forthcoming (3⁎x+1)/2 computations can be determined directly by inputting x. We propose an induction method for proving RCC. We also discover that some starting integers present RCD with short lengths no more than 7. Hence, partial natural numbers are proved to guarantee RCC in this paper, e.g., 0 module 2; 1 module 4; 3 module 16; 11 or 23 module 32; 7, 15, or 59 module 128. The future work for proving CC can follow this direction, to prove that RCD of left portion of natural numbers exists.

scholarly journals New construction of algebras as quotients

2020 ◽  
Vol 5 (2) ◽  
pp. 53
Author(s):  
José Ramón Játem Lásser
Keyword(s):  
Vector Space ◽  
Natural Number ◽  
Clifford Algebras ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Natural Numbers ◽  
New Approach ◽  
Binary Operations ◽  
Direct Sum ◽  
New Construction

  In this article we have presented a new approach to define algebras using for a natural number k ≥ 2, the set of natural numbers in base k, none of their digits equal to zero. The study was developed in the context of vector R -spaces and the vector space definitions of the formal multiples of any element x of the field R, of the direct sum of vector spaces and binary operations on vector spaces were used. The results obtained were the construction of a vector space denoted by V, on the basis of the particular set of natural numbers in base k mentioned, which allowed novel ways of defining the well-known and very important algebras of complex numbers and that of quaternions on R as quotients of ideals of V, for suitably chosen ideals I. With this new approach and with the help of the vector spaces V, known algebras can be presented in a different way than those found up to now, by using certain ideals of those spaces in their quotient form. The spaces V can be over any field K and other algebras such as Clifford algebras can be constructed using this procedure.   Keywords: Algebras, Quotients in algebras, Complex numbers and quaternions as quotients of algebras.   Abstract En este artículo se ha presentado un nuevo enfoque para definir álgebras usando para un número natural k ≥ 2, el conjunto de números naturales en base k, ninguno de sus dígitos iguales a cero. El estudio se desarrolló en el contexto de los R-espacios vectoriales y se usaron las definiciones de espacio vectorial de los múltiplos formales de un elemento cualquiera x del cuerpo R, de la suma directa de espacios vectoriales y operaciones binarias sobre espacios vectoriales. Los resultados obtenidos fueron la construcción de un espacio vectorial denotado por V, sobre la base del particular conjunto de números naturales en base k mencionado, que permitió novedosas formas de definir las conocidas y muy importantes álgebras de los números complejos y la de los cuaterniones sobre R como cocientes de ideales de V, para ideales I convenientemente elegidos. Con este nuevo enfoque y con la ayuda de los espacios vectoriales V se pueden presentar álgebras conocidas de manera distinta a las encontradas hasta ahora, al usar en su forma de cociente ciertos ideales de esos espacios V. Los espacios V pueden ser sobre cualquier cuerpo K y otras álgebras como las álgebras de Clifford se pueden construir usando este procedimiento.   Palabras claves: Algebras, cocientes en álgebras, Números complejos y quaterniones como cocientes en álgebras.  

scholarly journals On the Probabilistic Proof of the Convergence of the Collatz Conjecture

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Kamal Barghout
Keyword(s):  
Probability Distribution ◽  
Random Numbers ◽  
Natural Numbers ◽  
New Approach ◽  
Collatz Conjecture ◽  
Probabilistic Proof ◽  
Number Of Iterations

A new approach towards probabilistic proof of the convergence of the Collatz conjecture is described via identifying a sequential correlation of even natural numbers by divisions by 2 that follows a recurrent pattern of the form x,1,x,1…, where x represents divisions by 2 more than once. The sequence presents a probability of 50:50 of division by 2 more than once as opposed to division by 2 once over the even natural numbers. The sequence also gives the same 50:50 probability of consecutive Collatz even elements when counted for division by 2 more than once as opposed to division by 2 once and a ratio of 3:1. Considering Collatz function producing random numbers and over sufficient number of iterations, this probability distribution produces numbers in descending order that lead to the convergence of the Collatz function to 1, assuming that the only cycle of the function is 1-4-2-1.

scholarly journals Supplement to the Collatz Conjecture

2018 ◽  
Vol 15 ◽  
pp. 8120-8132
Author(s):  
Anatoliy Nikolaychuk
Keyword(s):  
Natural Number ◽  
New Method ◽  
Natural Numbers ◽  
Original Sequence ◽  
Collatz Conjecture

For any natural number was created the supplement sequence, that is convergent together with the original sequence. The parameter - index was defined, that is the same tor both sequences. This new method provides the following results: All natural numbers were distributed into different classes according to the corresponding indexes; The analytic formulas ( not by computer performed routine calculations) were produced, the formulas for groups of consecutive natural numbers of different lengths, having the same index; The new algorithm to find index for any natural number was constructed and proved.

Recycling. The Future Work for the Recycling of Steel.

1997 ◽  
Vol 113 (12) ◽  
pp. 986-988
Author(s):  
Shozo MIZOGUCHI
Keyword(s):  
The Future ◽  
Future Work

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

Welcome to the marriage of human capability and artificial intelligence

2019 ◽  
Vol 19 (1) ◽  
pp. 10-14
Author(s):  
Ryan Scott ◽  
Malcolm Le Lievre
Keyword(s):  
Artificial Intelligence ◽  
Team Building ◽  
Workplace Change ◽  
Content Type ◽  
The Future ◽  
Human Capability ◽  
Mind Set ◽  
Future Work ◽  
Crucial Part ◽  
Practical Implications

Purpose The purpose of this paper is to explore insights methodology and technology by using behavioral to create a mind-set change in the way people work, especially in the age of artificial intelligence (AI). Design/methodology/approach The approach is to examine how AI is driving workplace change, introduce the idea that most organizations have untapped analytics, add the idea of what we know future work will look like and look at how greater, data-driven human behavioral insights will help prepare future human-to-human work and inform people’s work with and alongside AI. Findings Human (behavioral) intelligence will be an increasingly crucial part of behaviorally smart organizations, from hiring to placement to adaptation to team building, compliance and more. These human capability insights will, among other things, better prepare people and organizations for changing work roles, including working with and alongside AI and similar tech innovation. Research limitations/implications No doubt researchers across the private, public and nonprofit sectors will want to further study the nexus of human capability, behavioral insights technology and AI, but it is clear that such work is already underway and can prove even more valuable if adopted on a broader, deeper level. Practical implications Much “people data” inside organizations is currently not being harvested. Validated, scalable processes exist to mine that data and leverage it to help organizations of all types and sizes be ready for the future, particularly in regard to the marriage of human capability and AI. Social implications In terms of human capability and AI, individuals, teams, organizations, customers and other stakeholders will all benefit. The investment of time and other resources is minimal, but must include C-suite buy in. Originality/value Much exists on the softer aspects of the marriage of human capability and AI and other workplace advancements. What has been lacking – until now – is a 1) practical, 2) validated and 3) scalable behavioral insights tech form that quantifiably informs how people and AI will work in the future, especially side by side.

scholarly journals Industrial policy-making after COVID-19: Manufacturing, innovation and sustainability

2021 ◽  
pp. 103530462110147
Author(s):  
Mark Dean ◽  
Al Rainnie ◽  
Jim Stanford ◽  
Dan Nahum
Keyword(s):  
Recent Work ◽  
Industrial Policy ◽  
Industrial Revolution ◽  
Manufacturing Sector ◽  
The Future ◽  
Australian Economy ◽  
Industry Policy ◽  
Future Work ◽  
Policy Proposals ◽  
Recent Policy

This article critically analyses the opportunities for Australia to revitalise its strategically important manufacturing sector in the wake of the COVID-19 pandemic. It considers Australia’s industry policy options on the basis of both advances in the theory of industrial policy and recent policy proposals in the Australian context. It draws on recent work from The Australia Institute’s Centre for Future Work examining the prospects for Australian manufacturing renewal in a post-COVID-19 economy, together with other recent work in political economy, economic geography and labour process theory critically evaluating the Fourth Industrial Revolution (i4.0) and its implications for the Australian economy. The aim of the article is to contribute to and further develop the debate about the future of government intervention in manufacturing and industry policy in Australia. Crucially, the argument links the future development of Australian manufacturing with a focus on renewable energy. JEL Codes: L50; L52; L78; O10; O13: O25; O44; P18; Q42

Improved estimation is a prerequisite for successful terminal guidance

2005 ◽  
Vol 219 (2) ◽  
pp. 145-156 ◽  
Author(s):  
J Shinar ◽  
V Turetsky
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Estimation Errors ◽  
New Approach ◽  
Target Acceleration ◽  
The Future ◽  
New Ideas ◽  
Estimation System ◽  
Terminal Guidance

Successful interception of manoeuvring anti-surface missiles that are expected in the future can be achieved only if the estimation errors against manoeuvring targets can be minimized. The paper raises new ideas for an improved estimation concept by separating the tasks of the estimation system and by explicit use of the time-to-go in the process. The outcome of the new approach is illustrated by results of Monte Carlo simulations in generic interception scenarios. The results indicate that if an eventual ‘jump’ in the commanded target acceleration is detected sufficiently rapidly, small estimation errors and consequently precise guidance can be obtained.

Analysis on Characters of Technology Innovations in Shenzhen Enterprises

2014 ◽  
Vol 638-640 ◽  
pp. 2383-2387
Author(s):  
Tong Hua Mou ◽  
Fang Du
Keyword(s):  
Technological Innovation ◽  
Evaluation Method ◽  
Service System ◽  
Research Result ◽  
Research Centers ◽  
Government Regulations ◽  
External Finance ◽  
Technology Innovations ◽  
The Future ◽  
Future Work

According to current research result on the meaning, foundation, and evaluation method of technological innovation of enterprise in Shenzhen, the character of the innovation can be summarized as: utilizing the demand of the current market as a guideline, cooperating with research centers and universities, applying most of the innovation in practice quickly. For the future work, the research could be focused on the factors which might affect the ability of technological innovation. Those factors could be either external such as external finance and government regulations, or it could be internal such as the size of the enterprise. At the same time it is necessary to establish the innovation and evaluation service system for small and medium sized enterprises (SMES).

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