nhaliday + math.nt   51

The Future of Mathematics? [video] | Hacker News
https://news.ycombinator.com/item?id=20909404
Kevin Buzzard (the Lean guy)

- general reflection on proof asssistants/theorem provers
- Kevin Hale's formal abstracts project, etc
- thinks of available theorem provers, Lean is "[the only one currently available that may be capable of formalizing all of mathematics eventually]" (goes into more detail right at the end, eg, quotient types)
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5 weeks ago by nhaliday
Factorization of polynomials over finite fields - Wikipedia
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them.

All factorization algorithms, including the case of multivariate polynomials over the rational numbers, reduce the problem to this case; see polynomial factorization. It is also used for various applications of finite fields, such as coding theory (cyclic redundancy codes and BCH codes), cryptography (public key cryptography by the means of elliptic curves), and computational number theory.

As the reduction of the factorization of multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with one variable are considered in this article.

...

In the algorithms that follow, the complexities are expressed in terms of number of arithmetic operations in Fq, using classical algorithms for the arithmetic of polynomials.

[ed.: Interesting choice...]

...

Factoring algorithms
Many algorithms for factoring polynomials over finite fields include the following three stages:

Square-free factorization
Distinct-degree factorization
Equal-degree factorization
An important exception is Berlekamp's algorithm, which combines stages 2 and 3.

Berlekamp's algorithm
Main article: Berlekamp's algorithm
The Berlekamp's algorithm is historically important as being the first factorization algorithm, which works well in practice. However, it contains a loop on the elements of the ground field, which implies that it is practicable only over small finite fields. For a fixed ground field, its time complexity is polynomial, but, for general ground fields, the complexity is exponential in the size of the ground field.

[ed.: This actually looks fairly implementable.]
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july 2019 by nhaliday
About - Project Euler
I've written my program but should it take days to get to the answer?
Absolutely not! Each problem has been designed according to a "one-minute rule", which means that although it may take several hours to design a successful algorithm with more difficult problems, an efficient implementation will allow a solution to be obtained on a modestly powered computer in less than one minute.
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june 2019 by nhaliday
Diophantine approximation - Wikipedia
- rationals perfectly approximated by themselves, badly approximated (eps>1/bq) by other rationals
- irrationals well-approximated (eps~1/q^2) by rationals:
https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem
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august 2017 by nhaliday
Main Page - Competitive Programming Algorithms: E-Maxx Algorithms in English
original russian version: http://e-maxx.ru/algo/

some notable stuff:
- O(N) factorization sieve
- discrete logarithm
- factorial N! (mod P) in O(P log N)
- flow algorithms
- enumerating submasks
- bridges, articulation points
- Ukkonen algorithm
- sqrt(N) trick, eg, for range mode query
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february 2017 by nhaliday
254A, Supplement 4: Probabilistic models and heuristics for the primes (optional) | What's new
among others, the Cramér model for the primes (basically kinda looks like primality is independently distributed w/ Pr[n is prime] = 1/log n)
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february 2017 by nhaliday
nt.number theory - Is $x^{2k+1} - 7x^2 + 1$ irreducible? - MathOverflow
Here is a proof, based on a trick that can be used to prove that x^n+x+1 is irreducible when n≠2 mod 3.
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january 2017 by nhaliday
The Mathematician Ken Ono’s Life Inspired By Ramanujan | Quanta Magazine
This intellectual crucible produced the desired results — Ono studied mathematics and launched a promising academic career — but at great emotional cost. As a teenager, Ono became so desperate to escape his parents’ expectations that he dropped out of high school. He later earned admission to the University of Chicago but had an apathetic attitude toward his studies, preferring to party with his fraternity brothers. He eventually discovered a genuine enthusiasm for mathematics, became a professor, and started a family, but fear of failure still weighed so heavily on Ono that he attempted suicide while attending an academic conference. Only after he joined the Institute for Advanced Study himself did Ono begin to make peace with his upbringing.
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may 2016 by nhaliday
Lean
https://lean-forward.github.io
The goal of the Lean Forward project is to collaborate with number theorists to formally prove theorems about research mathematics and to address the main usability issues hampering the adoption of proof assistants in mathematical circles. The theorems will be selected together with our collaborators to guide the development of formal libraries and verified tools.

mostly happening in the Netherlands

https://formalabstracts.github.io

A Review of the Lean Theorem Prover: https://jiggerwit.wordpress.com/2018/09/18/a-review-of-the-lean-theorem-prover/
- Thomas Hales
seems like a Coq might be a better starter if I ever try to get into proof assistants/theorem provers

edit: on second thought this actually seems like a wash for beginners

An Argument for Controlled Natural Languages in Mathematics: https://jiggerwit.wordpress.com/2019/06/20/an-argument-for-controlled-natural-languages-in-mathematics/
By controlled natural language for mathematics (CNL), we mean an artificial language for the communication of mathematics that is (1) designed in a deliberate and explicit way with precise computer-readable syntax and semantics, (2) based on a single natural language (such as Chinese, Spanish, or English), and (3) broadly understood at least in an intuitive way by mathematically literate speakers of the natural language.

The definition of controlled natural language is intended to exclude invented languages such as Esperanto and Logjam that are not based on a single natural language. Programming languages are meant to be excluded, but a case might be made for TeX as the first broadly adopted controlled natural language for mathematics.

Perhaps it is best to start with an example. Here is a beautifully crafted CNL text created by Peter Koepke and Steffen Frerix. It reproduces a theorem and proof in Rudin’s Principles of mathematical analysis almost word for word. Their automated proof system is able to read and verify the proof.

https://github.com/Naproche/Naproche-SAD
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january 2016 by nhaliday

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