astronomy

Lothar Collatz

Lothar Collatz is a astronomy topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lothar Collatz rather than just read about it. In short: Lothar Collatz (German: [ˈkɔlaʦ]; July 6, 1910 – September 26, 1990) was a German mathematician. The "3x + 1" problem is also known as the Collatz conjecture and remains unsolved.

Key takeaways

Lothar Collatz belongs to astronomy; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Lothar Collatz to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lothar Collatz from memory before moving on to harder problems.

Reference excerpt

Lothar Collatz (German: [ˈkɔlaʦ]; July 6, 1910 – September 26, 1990) was a German mathematician. The "3x + 1" problem is also known as the Collatz conjecture and remains unsolved. The Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a positive square matrix was also named after him. Collatz's 1957 paper with Ulrich Sinogowitz, who had been killed in the bombing of Darmstadt in World War II, founded the field of spectral graph theory.

Biography Collatz studied at universities in Germany including the University of Greifswald and the University of Berlin, where he was supervised by Alfred Klose, receiving his doctorate in 1935 for a dissertation entitled Das Differenzenverfahren mit höherer Approximation für lineare Differentialgleichungen (The finite difference method with higher approximation for linear differential equations). He then worked as an assistant at the University of Berlin, before moving to the Technische Hochschule Karlsruhe (now Karlsruhe Institute of Technology) in 1935 where he remained through 1937. From 1938 to 1943, he worked as a Privatdozent in Karlsruhe. In the war years he worked with Alwin Walther at the Institute for Practical Mathematics of the Technische Hochschule Darmstadt. From 1943 to 1952, Collatz held a chair at the Technische Hochschule Hannover (now Leibniz University Hannover). From 1952 until his retirement in 1978, Collatz worked at the University of Hamburg, where he founded the Institute of Applied Mathematics in 1953. After retirement as professor emeritus, he continued to be very active at mathematical conferences. For his many contributions to the field, Collatz had many honors bestowed upon him in his lifetime, including:

election to the Academy of Sciences Leopoldina, the Academy of Sciences of the Institute of Bologna, and the Academy at Modena in Italy honorary member of the Hamburg Mathematical Society honorary degrees from the University of São Paulo, the Vienna University of Technology, the University of Dundee in Scotland, Brunel University in England, the University of Hannover in 1981, and the Technische Universität Dresden. He died unexpectedly from a heart attack in Varna, Bulgaria, while attending a mathematics conference.

Selected works

Das Differenzenverfahren mit höherer Approximation für lineare Differentialgleichungen (= Schriften des mathematischen Seminars und des Instituts für angewandte Mathematik der Universität Berlin – Band 3/Heft 1), Leipzig 1935 Eigenwertprobleme und ihre numerische Behandlung. Leipzig 1945 Eigenwertaufgaben mit technischen Anwendungen. Leipzig 1949, 1963 Numerische Behandlung von Differentialgleichungen. Berlin 1951, 1955 (Eng. trans. 1966) Differentialgleichungen für Ingenieure. Stuttgart 1960 with Wolfgang Wetterling: Optimierungsaufgaben Berlin 1966, 1971 (Eng. trans. 1975) Funktionalanalysis und Numerische Mathematik. Berlin 1964 Differentialgleichungen. Eine Einführung unter besonderer Berücksichtigung der Anwendungen. Stuttgart, Teubner Verlag, 1966, 7th edn. 1990 with Julius Albrecht: Aufgaben aus der angewandten Mathematik I. Gleichungen in einer und mehreren Variablen. Approximationen. Berlin 1972 Numerische Methoden der Approximationstheorie. vol. 2. Vortragsauszüge der Tagung über Numerische Methoden der Approximationstheorie vom 3.-9. Juni 1973 im Mathematischen Forschungsinstitut Oberwolfach, Stuttgart 1975 Approximationstheorie: Tschebyscheffsche Approximation und Anwendungen. Teubner 1973

References

Sources Lothar Collatz (July 6, 1910 – September 26, 1990), Journal of Approximation Theory, vol. 65, issue 1, April 1991, page II by Günter Meinardus and Günther Nürnberger Collatz, Lothar (1942), "Einschließungssatz für die charakteristischen Zahlen von Matrizen", Mathematische Zeitschrift, 48 (1): 221–226, doi:10.1007/BF01180013, S2CID 120958677

Further reading J Albrecht, P Hagedorn and W Velte, Lothar Collatz (German), Numerical treatment of eigenvalue problems, vol. 5, Oberwolfach, 1990 (Birkhäuser, Basel, 1991), viii–ix. I Althoefer, Lothar Collatz zwischen 1933 und 1950 - Eine Teilbiographie (German), 3-Hirn-Verlag, Lage (Lippe), 2019. R Ansorge, Lothar Collatz (6 July 1910 – 26 September 1990) (German), Mitt. Ges. Angew. Math. Mech. No. 1 (1991), 4–9. U Eckhardt, Der Einfluss von Lothar Collatz auf die angewandte Mathematik, Numerical mathematics, Sympos., Inst. Appl. Math., Univ. Hamburg, Hamburg, 1979 (Birkhäuser, Basel-Boston, Mass., 1979), 9–23. L Elsner and K P Hadeler, Lothar Collatz – on the occasion of his 75th birthday, Linear Algebra Appl. 68 (1985), vi; 1–8. R B Guenther, Obituary : Lothar Collatz, 1910–1990, Aequationes Mathematicae 43 (2–3) (1992), 117–119. H Heinrich, Zum siebzigsten Geburtstag von Lothar Collatz, Z. Angew. Math. Mech. 60 (5) (1980), 274–275. G Meinardus, G Nürnberger, Th Riessinger and G Walz, In memoriam : the work of Lothar Collatz in approximation theory, J. Approx. Theory 67 (2) (1991), 119–128. G Meinardus and G Nürnberger, In memoriam : Lothar Collatz (July 6, 1910 – September 26, 1990), J. Approx. Theory 65 (1) (1991), i; 1–2. J R Whiteman, In memoriam : Lothar Collatz, Internat. J. Numer. Methods Engrg. 31 (8) (1991), 1475–1476.

External links O'Connor, John J.; Robertson, Edmund F., "Lothar Collatz", MacTutor History of Mathematics Archive, University of St Andrews Lothar Collatz at the Mathematics Genealogy Project

Illustrations

Worked examples

Example 1 — a first encounter with Lothar Collatz

Start with the simplest possible case. Write down what Lothar Collatz claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lothar Collatz before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lothar Collatz ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lothar Collatz

In research: Lothar Collatz appears in astronomy research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Lothar Collatz in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Lothar Collatz is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1910 births, 1990 deaths, 20th-century German mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Lothar Collatz outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Lothar Collatz in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lothar Collatz means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Lothar Collatz out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lothar Collatz in simple terms?

Lothar Collatz (German: [ˈkɔlaʦ]; July 6, 1910 – September 26, 1990) was a German mathematician. The "3x + 1" problem is also known as the Collatz conjecture and remains unsolved.

Why does Lothar Collatz matter?

Because it connects several astronomy ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lothar Collatz?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lothar Collatz.