mathematics

Nathan Fine

Nathan Fine is a mathematics 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 Nathan Fine rather than just read about it. In short: Nathan Jacob Fine (22 October 1916 in Philadelphia – 18 November 1994 in Deerfield Beach, Florida) was an American mathematician who worked on basic hypergeometric series. Сareer He is best known for his lecture notes on the subject which for four decades served as an inspiration to experts in the field until they were finally published as a book. He solved the Jeep problem in 1946.

Key takeaways

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

Reference excerpt

Nathan Jacob Fine (22 October 1916 in Philadelphia – 18 November 1994 in Deerfield Beach, Florida) was an American mathematician who worked on basic hypergeometric series.

Сareer He is best known for his lecture notes on the subject which for four decades served as an inspiration to experts in the field until they were finally published as a book. He solved the Jeep problem in 1946. Nathan Fine retired in 1978, as a professor at Pennsylvania State University. Prior to that he had been on the faculties of the University of Pennsylvania and Cornell University. For a brief period (1946–1947) he also worked at the Operations Evaluation Group, affiliated with the Massachusetts Institute of Technology. Beside the book he published about 40 papers in several fields of mathematics. He is known for the Rogers-Fine identity. Nathan Fine received his Ph.D. in 1946 from University of Pennsylvania, where he was a student of Antoni Zygmund. Fine was at the Institute for Advanced Study for the three academic years 1953–1954, 1958–1959, and 1959–1960. Fine's doctoral students include J. J. Price. He wrote the book Basic Hypergeometric Series and Applications ISBN 0-8218-1524-5.

Selected publications ——; Niven, Ivan (1944). "The probability that a determinant be congruent to a (mod m)". Bulletin of the American Mathematical Society. 50 (2): 89–94. doi:10.1090/S0002-9904-1944-08085-3. —— (1948). "Some New Results on Partitions". Proceedings of the National Academy of Sciences of the United States of America. 34 (12): 616–618. Bibcode:1948PNAS...34..616F. doi:10.1073/pnas.34.12.616. JSTOR 88084. PMC 1079179. PMID 16588844. —— (1949). "On the Walsh functions". Transactions of the American Mathematical Society. 65 (3): 372–414. doi:10.1090/S0002-9947-1949-0032833-2. —— (1950). "Proof of a theorem of Jacobi". Proceedings of the American Mathematical Society. 1 (5): 666–667. doi:10.1090/S0002-9939-1950-0038397-5. —— (1950). "The generalized Walsh functions". Transactions of the American Mathematical Society. 69: 66–77. doi:10.1090/S0002-9947-1950-0042535-2. —— (1951). "Note on the Hurwitz zeta-function". Proceedings of the American Mathematical Society. 2 (3): 361–364. doi:10.1090/S0002-9939-1951-0043194-1. —— (1954). "On the asymptotic distribution of certain sums". Proceedings of the American Mathematical Society. 5 (2): 243–252. doi:10.1090/S0002-9939-1954-0064337-2. —— (1955). "Cesàro Summability of Walsh-Fourier Series". Proceedings of the National Academy of Sciences. 41 (8): 588–591. Bibcode:1955PNAS...41..588F. doi:10.1073/pnas.41.8.588. PMC 528139. PMID 16578449. —— (1957). "Fourier-Stieltjes series of Walsh functions". Transactions of the American Mathematical Society. 86: 246–255. doi:10.1090/S0002-9947-1957-0091371-6. ——; Gillman, L. (1960). "Extension of continuous functions in β N {\displaystyle \beta N} ". Bulletin of the American Mathematical Society. 66 (5): 376–382. doi:10.1090/S0002-9904-1960-10460-0. ——; Gillman, L. (1962). "Remote points in β R {\displaystyle \beta R} ". Proceedings of the American Mathematical Society. 13: 29. doi:10.1090/S0002-9939-1962-0143172-5. ——; Wilf, H. S. (1965). "Uniqueness theorems for periodic functions". Proceedings of the American Mathematical Society. 16: 109–114. doi:10.1090/S0002-9939-1965-0174934-9. —— (1965). "The distribution of the sum of digits (mod p)". Bulletin of the American Mathematical Society. 71 (4): 651–653. doi:10.1090/S0002-9904-1965-11381-7.

References

External links "Nathan Fine 1916–1994" – biography article by George Andrews Fine's Equation – in MathWorld "On Fine's Partition Theorems, Dyson, Andrews and Missed Opportunities" Archived 2006-08-31 at the Wayback Machine – popular article by Igor Pak Nathan Fine at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Nathan Fine", MacTutor History of Mathematics Archive, University of St Andrews

Worked examples

Example 1 — a first encounter with Nathan Fine

Start with the simplest possible case. Write down what Nathan Fine claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Nathan Fine 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 Nathan Fine 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 Nathan Fine

In research: Nathan Fine appears in mathematics 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 Nathan Fine 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: Nathan Fine is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1916 births, 1994 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Nathan Fine 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 Nathan Fine in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Nathan Fine 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 Nathan Fine out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Nathan Fine in simple terms?

Nathan Jacob Fine (22 October 1916 in Philadelphia – 18 November 1994 in Deerfield Beach, Florida) was an American mathematician who worked on basic hypergeometric series. Сareer He is best known for his lecture notes on the subject which for four decades served as an inspiration to experts in the…

Why does Nathan Fine matter?

Because it connects several mathematics 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 Nathan Fine?

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 Nathan Fine.