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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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9 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.]
wiki  reference  concept  algorithms  calculation  nibble  numerics  math  algebra  math.CA  fields  polynomials  levers  multiplicative  math.NT 
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
nibble  wiki  reference  math  math.NT  approximation  accuracy  levers  pigeonhole-markov  multi  tidbits  discrete  rounding  estimate  tightness  algebra 
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

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