Collatz Research Articles

Collatz Conjecture Solution

The Collatz conjecture, also known as the 3n + 1 conjecture, stands as one of the most enduring unsolved problems in mathematics. Proposed by German mathematician Lothar Collatz in 1937, the conjecture poses a deceptively simple question: can iteratively applying two basic arithmetic operations—dividing even numbers by 2 and multiplying odd numbers by 3 and adding 1—lead any positive integer to converge to the value 1? Despite extensive computational verification for vast ranges of integers, a proof or disproof remains elusive. This abstract explores the historical background, various attempts at proving the conjecture, and its implications across mathematical domains.

Finding Saddle-Node Bifurcations via a Nonlinear Generalized Collatz–Wielandt Formula

The Collatz–Wielandt formula obtained by Lothar Collatz (1942) and Helmut Wielandt (1950) provides a simple variational characterization of the Perron–Frobenius eigenvalue of certain types of matrices. We introduce a nonlinear generalization of the Collatz–Wielandt formula and show that it enables finding the so-called maximal saddle-node bifurcations of systems of nonlinear equations. As a consequence, we obtain an easily verifiable criterion for the detection of such saddle-node bifurcations. We illustrate our approach by examples of finite-difference approximations of nonlinear partial differential equations and a system of power flow.

Walking on Plane and Matrix Square

We know Pascal’s triangle and planer graphs. They are mutually connected with each other. For any positive integer n, φ(n) is an even number. But it is not true for all even number, we could find some numbers which would not be the value of any φ(n). The Sum of two odd numbers is one even number. Gold Bach stated “Every even integer greater than 2 can be written as the sum of two primes”. Other than two, all prime numbers are odd numbers. So we can write, every even integer greater than 2 as the sum of two primes. German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio. We could find the series which is generated by one and inverse the golden ratio. Also we can note consecutive golden ratio numbers converge to the golden ratio. Lothar Collatz stated integers converge to one. It is also known as 3k + 1 problem. Tao redefined Collatz conjecture as 3k − 1 problem. We could not prove it directly but one parallel proof will prove this conjecture.

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.

Editorial

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 100, Issue 10 e202002041 EDITORIALFree Access Editorial First published: 19 October 2020 https://doi.org/10.1002/zamm.202002041AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Lothar Collatz passed his first mathematics university exam in Berlin in 1933 supervised by Richard von Mises, who founded Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) in 1921. Since his graduation L. Collatz published short contributions [1-3] in ZAMM that summarized his presentations from the annual GAMM conferences. His first regular article appeared in ZAMM in 1935, right after the completion of his PhD [4]. In his long career (from obtaining his habilitation in 1937 in Karlsruhe, being a researcher in Karlsruhe and Darmstadt until 1943, when he received a professorship in Hannover and then from 1952 in Hamburg until his retirement 1978) he continuously published in ZAMM contributing over 35 articles. Lothar Collatz (1910–1990, photography by Elfriede Liebenow) His work had a major impact on the development of numerical mathematics, particularly by his supervision of 42 PhD students and his involvement in 24 habilitation procedures. Before he moved to Hamburg, all his positions were at Technical Universities. His research is closely related to applications of numerical mathematics in mechanics. Many of his contributions to ZAMM are directly motivated by specific computational engineering challenges for membranes, plates, and bars [2, 5-8] at a time well before the development of modern computers. His engagement for the collaboration of applied mathematics and mechanics was continuously embedded in his activities in GAMM: he was involved in both the organization and management of the GAMM society, especially as President from 1964 to 1966 [9]. With this issue's special anniversary article “Error bounds and enclosures: The development of numerical analysis and the impact of the contributions by Lothar Collatz” we celebrate and remember his work. REFERENCES 1Collatz, L.: Bemerkungen zur Fehlerabschätzung für das Differenzenverfahren bei partiellen Differentialgleichungen. J. Appl. Math. Mech. 13(1), 56– 57 (1933) Wiley Online LibraryGoogle Scholar 2Collatz, L.: Die Berechnung von Eigenschwingungen einer gleichseitig dreieckigen Membran mit Hilfe von Sechsecksnetzen. J. Appl. Math. Mech. 14(5), 315– 317 (1934) Wiley Online LibraryGoogle Scholar 3Collatz, L.: Eine Verallgemeinerung des Differenzenverfahrens für Differentialgleichungen. J. Appl. Math. Mech. 14(5), 350– 351 (1934) Google Scholar 4Collatz, L.: Über das Differenzenverfahren bei Anfangswertproblemen partieller Differentialgleichungen. J. Appl. Math. Mech. 16(4), 239– 247 (1936) Wiley Online LibraryGoogle Scholar 5Vdi, T.P., Collatz, L.: Über die Berechnung und Darstellung der Eigenfrequenzen homogener Maschinen mit Zusatzdrehmassen. J. Appl. Math. Mech. 18(3), 186– 194 (1938) Wiley Online LibraryGoogle Scholar 6Collatz, L.: Über Stabilität von Regelungen mit Nachlaufzeit. J. Appl. Math. Mech. 25(2), 60– 63 (1947) Wiley Online LibraryGoogle Scholar 7Collatz, L.: Das Mehrstellenverfahren bei Plattenaufgaben. J. Appl. Math. Mech. 30(11–12), 385– 388 (1950) Wiley Online LibraryGoogle Scholar 8Collatz, L.: Zur Stabilität des Differenzenverfahrens bei der Stabschwingungsgleichung. J. Appl. Math. Mech. 31(11–12), 392– 393 (1951) Wiley Online LibraryGoogle Scholar 9Ansorge, R.: Zum Tode von Lothar Collatz. J. Appl. Math. Mech. 71(7–8), 306– 308 (1991) Wiley Online LibraryGoogle Scholar Volume100, Issue10October 2020e202002041 This article also appears in:100th Jubilee of ZAMM Journal: Historical Anniversary Articles ReferencesRelatedInformation

Error bounds and enclosures: The development of numerical analysis and the impact of the contributions by Lothar Collatz

AbstractIn 1933 Lothar Collatz published his very first article in Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) on error estimates for finite difference methods for partial differential equations. At that time, numerics meant calculations of approximations by hand and mechanical calculators. Then, in the next decades, parallel to the progress in computer technology, more and more methods were developed. Nevertheless, in all cases the accuracy of numerical approximations is limited, so that at least rough error bounds or, at best, tight enclosures are required for the reliability of the numerical scheme and the validation of the approximate results. Here, we recall the early development of numerical analysis of differential equations, of numerical iterations, and for the approximation of eigenvalues.

Richard von Mises’ work for ZAMM until his emigration in 1933 and glimpses of the later history of ZAMM

Richard von Mises’ work for ZAMM until his emigration in 1933 and glimpses of the later history of ZAMM

The 3n  p Conjecture: A Generalization of Collatz Conjecture

The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutanis problem (after Shizuo Kakutani) and so on. Several various generalization of the Collatz conjecture has been carried. In this paper a new generalization of the Collatz conjecture called as the 3n ± p conjecture; where p is a prime is proposed. It functions on 3n + p and 3n - p, and for any starting number n, its sequence eventually enters a finite cycle and there are finitely many such cycles. The 3n ± 1 conjecture, is a special case of the 3n ± p conjecture when p is 1.

Finding the Largest Eigenvalue of a Nonnegative Tensor

In this paper we propose an iterative method for calculating the largest eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating t...

An analytical approach for estimating CO 2 and heat fluxes over the Amazonian region

Accurate assessments of the CO 2 fluxes between the terrestrial ecosystems and the atmosphere are pressingly needed for the climate change and carbon cycle studies. The Collatz et al. parameterization of leaf photosynthesis-stomatal conductance has been widely applied in land surface parameterization schemes for simulating the land surface CO 2 fluxes. The study in this paper developed an analytical solution approach for the Collatz et al.’s parameterization for stable solution and computational efficiency. This analytical approach is then applied to the simplified biosphere model (SSiB), enhancing its capability of simulating land surface CO 2 fluxes. The enhanced SSiB model is tested with field observation data sets from two Amazonian field experiments (ABRACOS missions and Manaus Eddy Covariance Study). Simulations of the land surface fluxes of latent heat, sensible heat and soil heat by the enhanced SSiB agree very well with observations with correlation coefficients being larger than 0.80. However, the correlation coefficient for the daily means of CO 2 fluxes is only 0.42 for the Manaus data set. A day-time “square wave” in the simulated CO 2 flux diurnal curves is found. The discrepancies between simulation and observation were found to be the results of incorrect parameter setup or improper leaf to canopy scaling strategy. A modification to the scaling strategy improves significantly the accuracy of the photosynthesis-stomatal conductance model.

Collatz cycles with few descents

Collatz cycles with few descents

Ludwig Elsner and his contributions to core, applied and numerical linear algebra

Ludwig Elsner was bona on 17 January 1939 in Grog-Strehlitz, Upper Silesia in Germany. He studied mathematics, physics and astronomy at the universities of Hamburg and Freiburg and received his diploma in mathematics in 1963 at the University of Hamburg. As a member of the lively and inspiring group of Lothar Collatz in Hamburg he worked on his Ph.D., while holding a position at the computing center of the University of Hamburg. He received his Ph.D. in 1965 and became an Assistant Professor at Collatz' Institute of Applied Mathematics at the University of Hamburg for six years. During this period he spent one year, 1968-1969, as senior officer at the CSIR in Pretoria, South Africa. In 1971 Ludwig Eisner became Associate Professor at the University of Erlangen, where he worked at the Institute for Applied Mathematics until 1976. Since 1976 he has been a Full Professor at the University of Bielefeld where he has built up the Numerical Analysis group. Among many other activities in Bielefeld, he has twice been Chairman of the Department of Mathematics and has planned and formed the numerical analysis group of the Sonderforschungsbereich Diskrete Strukturen in der Mathematik, which since 1989 has attracted many specialists in linear algebra and numerical linear algebra to work with Eisner in Bielefeld. Numerous publications prove the success of these collaborative efforts.

On the Error Estimate of Approximate Solutions of the Discrete Volterra Equations of K‐th Order

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 71, Issue 7-8 p. 293-295 Short Communication On the Error Estimate of Approximate Solutions of the Discrete Volterra Equations of K-th Order† Prof. Dr. M Kwapisz, Prof. Dr. M Kwapisz University of Gdańsk, Department of Mathematics, ul. Wita Stwosza 57, PL-80-952 Godańsk, PolandSearch for more papers by this author Prof. Dr. M Kwapisz, Prof. Dr. M Kwapisz University of Gdańsk, Department of Mathematics, ul. Wita Stwosza 57, PL-80-952 Godańsk, PolandSearch for more papers by this author First published: 1991 https://doi.org/10.1002/zamm.19910710725 † Dedicated to the memory of LOTHAR COLLATZ AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume71, Issue7-81991Pages 293-295 RelatedInformation

Lothar Collatz—On the occasion of his 75th birthday

Lothar Collatz—On the occasion of his 75th birthday

Numerical Solution of Two-Point Boundary Value Problems on Total Differential Equations

Previous article Next article Numerical Solution of Two-Point Boundary Value Problems on Total Differential EquationsDavid GlasserDavid Glasserhttps://doi.org/10.1137/0706054PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Henry J. Kelley, G. Leitman, Method of gradientsOptimization techniques, Academic Press, New York, 1962, 205–254, Chap. 6. MR0162671 CrossrefGoogle Scholar[2] L. Fox, Numerical solution of ordinary and partial differential equations., Based on a Summer School held in Oxford, August-September 1961, Pergamon Press, Oxford, 1962ix+509 MR0146969 0101.09904 Google Scholar[3] Lothar Collatz, Functional analysis and numerical mathematics, Translated from the German by Hansjörg Oser, Academic Press, New York, 1966xx+473 MR0205126 0148.39002 Google Scholar[4] E. S. Lee, Quasi-linearization, non-linear boundary value problems and optimization, Chem. Eng. Sci., 21 (1966), 183–194 10.1016/0009-2509(66)85007-8 CrossrefISIGoogle Scholar[5] F. H. Deist and , L. Sefor, Solutions of systems of non-linear equations by parameter variation, Comput. J., 10 (1967), 78–82 10.1093/comjnl/10.1.78 0171.35603 CrossrefISIGoogle Scholar[6] C. B. Haselgrove, The solution of non-linear equations and of differential equations with two-point boundary conditions, Comput. J., 4 (1961/1962), 255–259 10.1093/comjnl/4.3.255 MR0130114 0121.11301 CrossrefISIGoogle Scholar[7] O. Vejvoda, I. Babushka, Nonlinear boundary-value problems for differential equations, Differential Equations and Their Applications (Proc. Conf., Prague, 1962), Publ. House Czechoslovak Acad. Sci., Prague, 1963, 199–215 MR0173072 0196.40102 Google Scholar[8] Thomas L. Saaty and , Joseph Bram, Nonlinear mathematics, McGraw-Hill Book Co., New York, 1964xv+381 MR0164812 0198.00102 Google Scholar[9] L. Lapidus, Digital Computations for Chemical Engineers, McGraw-Hill, New York, 1962 Google Scholar[10] E. S. Lee, Optimization by Pontryagin's maximum principle on the analog computer, American Inst. Chem. Eng. J., 10 (1964), 309–315 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A Continuation Method for Nonlinear RegressionNoël de Villiers and David GlasserSIAM Journal on Numerical Analysis, Vol. 18, No. 6 | 17 July 2006AbstractPDF (1889 KB)Numerical solution of nonlinear equations by one-parameter imbedding methodsNumerical Functional Analysis and Optimization, Vol. 3, No. 2 | 26 June 2007 Cross Ref Suboptimal control of systems with multiple delaysJournal of Optimization Theory and Applications, Vol. 30, No. 4 | 1 Apr 1980 Cross Ref Solution of nonlinear boundary value problems—XIChemical Engineering Science, Vol. 34, No. 5 | 1 Jan 1979 Cross Ref One-parameter imbedding techniques for the solution of nonlinear boundary-value problemsApplied Mathematics and Computation, Vol. 4, No. 4 | 1 Oct 1978 Cross Ref Optimal control of nonlinear power systems by an imbedding methodAutomatica, Vol. 11, No. 6 | 1 Nov 1975 Cross Ref NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS: SURVEY AND SOME RECENT RESULTS ON DIFFERENCE METHODSNumerical Solutions of Boundary Value Problems for Ordinary Differential Equations | 1 Jan 1975 Cross Ref On the optimal control solution of nonlinear power systemsIFAC Proceedings Volumes, Vol. 7, No. 2 | 1 Oct 1974 Cross Ref Generalized Quasi-Einearization MethodAIAA Journal, Vol. 12, No. 9 | 1 Sep 1974 Cross Ref Parameter variation for the solution of two-point boundary-value problems and applications in the calculus of variationsJournal of Optimization Theory and Applications, Vol. 13, No. 2 | 1 Feb 1974 Cross Ref Extension of a perturbation technique for nonlinear two-point boundary-value problemsJournal of Optimization Theory and Applications, Vol. 12, No. 5 | 3 August 2013 Cross Ref An indirect trajectory optimization algorithm based on the continuation method for solution of nonlinear equationsGuidance and Control Conference | 16 August 1973 Cross Ref The epsilon variation method in two-point boundary-value problemsJournal of Optimization Theory and Applications, Vol. 12, No. 2 | 3 August 2013 Cross Ref Numerical Solutions by the Continuation MethodE. WasserstromSIAM Review, Vol. 15, No. 1 | 18 July 2006AbstractPDF (2435 KB)A manifold imbedding algorithm for optimization problemsAutomatica, Vol. 8, No. 5 | 1 Sep 1972 Cross Ref On the imbedding solution of a class of optimal control problemsAutomatica, Vol. 8, No. 5 | 1 Sep 1972 Cross Ref A Manifold Imbedding Algorithm for Optimization ProblemsIFAC Proceedings Volumes, Vol. 5, No. 1 | 1 Jun 1972 Cross Ref Solving Boundary-Value Problems by ImbeddingJournal of the ACM, Vol. 18, No. 4 | 1 Oct 1971 Cross Ref Volume 6, Issue 4| 1969SIAM Journal on Numerical Analysis523-616 History Submitted:02 July 1968Published online:14 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0706054Article page range:pp. 591-597ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

Generalized Matrix Functions and the Graph Isomorphism Problem

Generalized Matrix Functions and the Graph Isomorphism Problem

Computational Aspects of Three-Term Recurrence Relations

Computational Aspects of Three-Term Recurrence Relations