scholarly journals Reduced Collatz Dynamics Data Reveals Properties for the Future Proof of Collatz Conjecture

Data ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 89 ◽  
Author(s):  
Ren
Keyword(s):  
Upper Bound ◽  
Residue Class ◽  
Data Set ◽  
Source Codes ◽  
Collatz Conjecture ◽  
The Future ◽  
One To One ◽  
Positive Integers ◽  
Minimal Integer

Collatz conjecture is also known as 3X + 1 conjecture. For verifying the conjecture, we designed an algorithm that can output reduced dynamics (occurred 3 × x+1 or x/2 computations from a starting integer to the first integer smaller than the starting integer) and original dynamics of integers (from a starting integer to 1). Especially, the starting integer has no upper bound. That is, extremely large integers with length of about 100,000 bits, e.g., 2100000 − 1, can be verified for Collatz conjecture, which is much larger than current upper bound (about 260). We analyze the properties of those data (e.g., reduced dynamics) and discover the following laws; reduced dynamics is periodic and the period is the length of its reduced dynamics; the count of x/2 equals to minimal integer that is not less than the count of (3 × x + 1)/2 times ln(1.5)/ln(2). Besides, we observe that all integers are partitioned regularly in half and half iteratively along with the prolonging of reduced dynamics, thus given a reduced dynamics we can compute a residue class that presents this reduced dynamics by a proposed algorithm. It creates one-to-one mapping between a reduced dynamics and a residue class. These observations from data can reveal the properties of reduced dynamics, which are proved mathematically in our other papers (see references). If it can be proved that every integer has reduced dynamics, then every integer will have original dynamics (i.e., Collatz conjecture will be true). The data set includes reduced dynamics of all odd positive integers in [3, 99999999] whose remainder is 3 when dividing 4, original dynamics of some extremely large integers, and all computer source codes in C that implement our proposed algorithms for generating data (i.e., reduced or original dynamics).

scholarly journals Collatz Dynamics is Partitioned by Residue Class Regularly

2020 ◽  
Author(s):  
Wei Ren
Keyword(s):  
Positive Integer ◽  
Residue Class ◽  
Collatz Conjecture ◽  
If Current ◽  
Positive Integers

<div>We propose Reduced Collatz Conjecture that is equivalent to Collatz</div><div>Conjecture, which states that every positive integer can return to</div><div>an integer less than it, instead of 1. Reduced Collatz Conjecture</div><div>should be easier because some properties are presented in reduced</div><div>dynamics, rather than in original dynamics (e.g., ratio and period).</div><div>Reduced dynamics is a computation sequence from starting integer to</div><div>the first integer less than it, and original dynamics is a</div><div>computation sequence from starting integer to 1. Reduced dynamics is</div><div>a component of original dynamics. We denote dynamics of x as a</div><div>sequence of either computations in terms of ``I'' that represents</div><div>(3*x+1)/2 and ``O'' that represents x/2. Here 3*x+1 and x/2 are</div><div>combined together, because 3*x+1 is always even and followed by x/2.</div><div>We formally prove that all positive integers are partitioned into</div><div>two halves and either presents ``I'' or ``O'' in next ongoing</div><div>computation. More specifically, (1) if any positive integer x that</div><div>is i module $2^t$ (i is an odd integer) is given, then the first t</div><div>computations (each one is either ``I'' or ``O'' corresponding to</div><div>whether current integer is odd or even) will be identical with that</div><div>of i. (2) If current integer after t computations (in terms of ``I''</div><div>or ``O'') is less than x, then reduced dynamics of x is available.</div><div>Otherwise, the residue class of x (namely, i module $2^t$) can be</div><div>partitioned into two halves (namely, i module $2^{t+1}$ and $i+2^t$</div><div>module $2^{t+1}$), and either half presents ``I'' or ``O'' in</div><div>intermediately forthcoming (t+1)-th computation.</div>

scholarly journals Collatz Dynamics is Partitioned by Residue Class Regularly

2020 ◽  
Author(s):  
Wei Ren
Keyword(s):  
Positive Integer ◽  
Residue Class ◽  
Collatz Conjecture ◽  
If Current ◽  
Positive Integers

<div>We propose Reduced Collatz Conjecture that is equivalent to Collatz</div><div>Conjecture, which states that every positive integer can return to</div><div>an integer less than it, instead of 1. Reduced Collatz Conjecture</div><div>should be easier because some properties are presented in reduced</div><div>dynamics, rather than in original dynamics (e.g., ratio and period).</div><div>Reduced dynamics is a computation sequence from starting integer to</div><div>the first integer less than it, and original dynamics is a</div><div>computation sequence from starting integer to 1. Reduced dynamics is</div><div>a component of original dynamics. We denote dynamics of x as a</div><div>sequence of either computations in terms of ``I'' that represents</div><div>(3*x+1)/2 and ``O'' that represents x/2. Here 3*x+1 and x/2 are</div><div>combined together, because 3*x+1 is always even and followed by x/2.</div><div>We formally prove that all positive integers are partitioned into</div><div>two halves and either presents ``I'' or ``O'' in next ongoing</div><div>computation. More specifically, (1) if any positive integer x that</div><div>is i module $2^t$ (i is an odd integer) is given, then the first t</div><div>computations (each one is either ``I'' or ``O'' corresponding to</div><div>whether current integer is odd or even) will be identical with that</div><div>of i. (2) If current integer after t computations (in terms of ``I''</div><div>or ``O'') is less than x, then reduced dynamics of x is available.</div><div>Otherwise, the residue class of x (namely, i module $2^t$) can be</div><div>partitioned into two halves (namely, i module $2^{t+1}$ and $i+2^t$</div><div>module $2^{t+1}$), and either half presents ``I'' or ``O'' in</div><div>intermediately forthcoming (t+1)-th computation.</div>

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals Consistent satellite XCO<sub>2</sub> retrievals from SCIAMACHY and GOSAT using the BESD algorithm

2015 ◽  
Vol 8 (2) ◽  
pp. 1787-1832 ◽  
Author(s):  
J. Heymann ◽  
M. Reuter ◽  
M. Hilker ◽  
M. Buchwitz ◽  
O. Schneising ◽  
...  
Keyword(s):  
Carbon Cycle ◽  
Detailed Comparison ◽  
Mean Difference ◽  
Total Carbon ◽  
Retrieval Algorithm ◽  
Data Sets ◽  
Data Set ◽  
Time Period ◽  
Comparison Results ◽  
The Future

Abstract. Consistent and accurate long-term data sets of global atmospheric concentrations of carbon dioxide (CO2) are required for carbon cycle and climate related research. However, global data sets based on satellite observations may suffer from inconsistencies originating from the use of products derived from different satellites as needed to cover a long enough time period. One reason for inconsistencies can be the use of different retrieval algorithms. We address this potential issue by applying the same algorithm, the Bremen Optimal Estimation DOAS (BESD) algorithm, to different satellite instruments, SCIAMACHY onboard ENVISAT (March 2002–April 2012) and TANSO-FTS onboard GOSAT (launched in January 2009), to retrieve XCO2, the column-averaged dry-air mole fraction of CO2. BESD has been initially developed for SCIAMACHY XCO2 retrievals. Here, we present the first detailed assessment of the new GOSAT BESD XCO2 product. GOSAT BESD XCO2 is a product generated and delivered to the MACC project for assimilation into ECMWF's Integrated Forecasting System (IFS). We describe the modifications of the BESD algorithm needed in order to retrieve XCO2 from GOSAT and present detailed comparisons with ground-based observations of XCO2 from the Total Carbon Column Observing Network (TCCON). We discuss detailed comparison results between all three XCO2 data sets (SCIAMACHY, GOSAT and TCCON). The comparison results demonstrate the good consistency between the SCIAMACHY and the GOSAT XCO2. For example, we found a mean difference for daily averages of −0.60 ± 1.56 ppm (mean difference ± standard deviation) for GOSAT-SCIAMACHY (linear correlation coefficient r = 0.82), −0.34 ± 1.37 ppm (r = 0.86) for GOSAT-TCCON and 0.10 ± 1.79 ppm (r = 0.75) for SCIAMACHY-TCCON. The remaining differences between GOSAT and SCIAMACHY are likely due to non-perfect collocation (±2 h, 10° × 10° around TCCON sites), i.e., the observed air masses are not exactly identical, but likely also due to a still non-perfect BESD retrieval algorithm, which will be continuously improved in the future. Our overarching goal is to generate a satellite-derived XCO2 data set appropriate for climate and carbon cycle research covering the longest possible time period. We therefore also plan to extend the existing SCIAMACHY and GOSAT data set discussed here by using also data from other missions (e.g., OCO-2, GOSAT-2, CarbonSat) in the future.

Intersubjectivity and Materiality: A Multimodal Perspective

2016 ◽  
Vol 5 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jesse Pirini
Keyword(s):  
Data Set ◽  
Methodological Tool ◽  
Action Analysis ◽  
One To One

AbstractResearchers seeking to analyse how intersubjectivity is established and maintained face significant challenges. The purpose of this article is to provide theoretical/methodological tools that begin to address these challenges. I develop these tools by applying several concepts from multimodal (inter)action analysis to an excerpt taken from the beginning of a tutoring session, drawn from a wider data set of nine one-to-one tutoring sessions. Focusing on co-produced higher-level actions as an analytic site of intersubjectivity, I show that lower-level actions that co-constitute a higher-level action can be delineated into tiers of materiality. I identify three tiers of materiality: durable, adjustable and fleeting. I introduce the theoretical/methodological tool

Forecast of the efficiency confidence interval of decision-making units in data envelopment analysis

2020 ◽  
Vol 33 (02) ◽  
pp. 431-445
Author(s):  
Azarnoosh Kafi ◽  
Behrouz Daneshian ◽  
Mohsen Rostamy-Malkhalifeh ◽  
Mohsen Rostamy-Malkhalifeh
Keyword(s):  
Decision Making ◽  
Data Envelopment Analysis ◽  
Confidence Interval ◽  
Data Envelopment ◽  
Data Set ◽  
Interval Model ◽  
Time Period ◽  
Proposed Model ◽  
The Future ◽  
Decision Making Units

Data Envelopment Analysis (DEA) is a well-known method for calculating the efficiency of Decision-Making Units (DMUs) based on their inputs and outputs. When the data is known and in the form of an interval in a given time period, this method can calculate the efficiency interval. Unfortunately, DEA is not capable of forecasting and estimating the efficiency confidence interval of the units in the future. This article, proposes a efficiency forecasting algorithm along with 95% confidence interval to generate interval data set for the next time period. What’s more, the manager’s opinion inserts and plays its role in the proposed forecasting model. Equipped with forecasted data set and with respect to data set from previous periods, the efficiency for the future period can be forecasted. This is done by proposing a proposed model and solving it by the confidence interval method. The proposed method is then implemented on the data of an automotive industry and, it is compared with the Monte Carlo simulation methods and the interval model. Using the results, it is shown that the proposed method works better to forecast the efficiency confidence interval. Finally, the efficiency and confidence interval of 95% is calculated for the upcoming period using the proposed model.

scholarly journals The Minimum Size of Signed Sumsets

10.37236/4881 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Béla Bajnok ◽  
Ryan Matzke
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Finite Abelian Group ◽  
Minimum Size ◽  
Positive Integers

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho(G, m, h) = \min \{ |hA| \; : \; A \subseteq G, |A|=m\}$$ and$$\rho_{\pm} (G, m, h) = \min \{ |h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $\rho(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $\rho_{\pm}(G, m, h)$ equals $\rho (G, m, h)$ when $G$ is cyclic, and establish an upper bound for $\rho_{\pm} (G, m, h)$ that we believe gives the exact value for all $G$, $m$, and $h$.

scholarly journals Nutrigenomic Information in the openEHR Data Set

2018 ◽  
Vol 09 (01) ◽  
pp. 221-231 ◽  
Author(s):  
Priscila Maranhão ◽  
Gustavo Bacelar-Silva ◽  
Duarte Ferreira ◽  
Conceição Calhau ◽  
Pedro Vieira-Marques ◽  
...  
Keyword(s):  
Human Genome ◽  
Clinical Data ◽  
Literature Search ◽  
Systemic Review ◽  
Personalized Nutrition ◽  
Clinical Practices ◽  
Modeling Tools ◽  
Data Set ◽  
Traditional Concept ◽  
The Future

Background The traditional concept of personalized nutrition is based on adapting diets according to individual needs and preferences. Discussions about personalized nutrition have been on since the Human Genome Project, which has sequenced the human genome. Thenceforth, topics such as nutrigenomics have been assessed to help in better understanding the genetic variation influence on the dietary response and association between nutrients and gene expression. Hence, some challenges impaired the understanding about the nowadays important clinical data and about clinical data assumed to be important in the future. Objective Finding the main clinical statements in the personalized nutrition field (nutrigenomics) to create the future-proof health information system to the openEHR server based on archetypes, as well as a specific nutrigenomic template. Methods A systematic literature search was conducted in electronic databases such as PubMed. The aim of this systemic review was to list the chief clinical statements and create archetype and templates for openEHR modeling tools, namely, Ocean Archetype Editor and Ocean Template Design. Results The literature search led to 51 articles; however, just 26 articles were analyzed after all the herein adopted inclusion criteria were assessed. Of these total, 117 clinical statements were identified, as well as 27 archetype-friendly concepts. Our group modeled four new archetypes (waist-to-height ratio, genetic test results, genetic summary, and diet plan) and finally created the specific nutrigenomic template for nutrition care. Conclusion The archetypes and the specific openEHR template developed in this study gave dieticians and other health professionals an important tool to their nutrigenomic clinical practices, besides a set of nutrigenomic data to clinical research.

Blockchain Technology in International Trade in Goods

2019 ◽  
pp. 184-202
Author(s):  
Dagmar Gesmann-Nuissl
Keyword(s):  
International Trade ◽  
Data Set ◽  
Blockchain Technology ◽  
Legal Challenges ◽  
The Future ◽  
Transport Industry ◽  
Technical Revolution

No other technology has been mentioned as frequently as blockchain technology. No less than a technical revolution should go along with it. In addition to a brief introduction to the functionality of the blockchain technology, this chapter will also highlight various fields of application and the related legal challenges in international trade. The transport industry may be mentioned as an example. Especially in the transportation of goods, a large number of documentation obligations must be adhered to and permits must be obtained. Packing lists, export licenses, and product certificates are examples of this. All mentioned documents are based on the same data set, namely the details of the type, origin, and quantity of the goods. Large parts of these documents must be carried along with the goods and presented on instructions, which causes considerable bureaucracy and makes transportation more difficult and expensive. These documents could be displayed on the blockchain in the future.

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