Angel "Java" Lopez en Blog

Matemáticas


Publicado el 5 de Abril, 2020, 19:51

En estos días estuve leyendo temas de teoría algebraica de números. Algo publiqué por acá en Números Irreducibles y Primos. Hoy leo en el libro Algebraic Number Theory de Romyar Sharifi:

At its core, the ancient subject of number theory is concerned with the arithmetic of the integers. The Fundamental Theorem of Arithmetic, which states that every positive integer factors uniquely into a product of prime numbers, was contained in Euclid’s Elements, as was the infinitude of the set of prime numbers. Over the centuries, number theory grew immensely as a subject, and techniques were developed for approaching number-theoretic problems of a various natures. For instance, unique factorization may be viewed as a ring-theoretic property of Z, while Euler used analysis in his own proof that the set of primes is infinite, exhibiting the divergence of the infinite sum of the reciprocals of all primes.

Es importante conocer que la factorización única NO SIEMPRE está presente en otros anillos. Es parte de lo que la teoría algebraica de números tiene para ofrecernos.

Algebraic number theory distinguishes itself within number theory by its use of techniques
from abstract algebra to approach problems of a number-theoretic nature. It is also often considered, for this reason, as a subfield of algebra. The overriding concern of algebraic number theory is the study of the finite field extensions of Q, which are known as number fields, and their rings of integers, analogous to Z.

Curiosamente las extensions de campos comenzaron a aparecer en los trabajos para resolver los ecuaciones de grado mayor que 2. Pero si en esas extensions, definimos algo como "enteros", se nos abre la puerta a estudiar nuevos sistemas de números.

The ring of integers O of a number field F is the subring of F consisting of all roots of all
monic polynomials in Z[x]. Unlike Z, not all integer rings are UFDs, as one sees for instance
by considering the factorization of 6 in the ring Z[√−5]. However, they are what are known
as Dedekind domains, which have the particularly nice property that every nonzero ideal factors uniquely as a product of nonzero prime ideals, which are all in fact maximal. In essence, prime ideals play the role in O that prime numbers do in Z.

UFD es Unique Factorization Domain, dominio con factorización única. El caso del 6 mencionado se refiere a que 6 = 3 * 2 pero también en ese anillo Z[√−5]  el 6 es igual a (1 + √−5)(1 - √−5) y esos dos pares de factores no se dividen entre sí. Es un resultado un tanto inesperado, pero sumamente interesante.

Para recuperar la factorización única, hay que reemplazar los enteros por ideales, conjuntos de elementos de un anillo.

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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 4 de Abril, 2020, 17:42

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Se obtuvo una hermosa fórmula matemática usando el número de Napie
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Sphere Point Picking
https://mathworld.wolfram.com/SpherePointPicking.html

Unpredictability, Undecidability, and Uncomputability
https://www.youtube.com/watch?v=hDpEg881BnI

Mathematics as a Team Sport
https://www.quantamagazine.org/mathematics-as-a-team-sport-20200331/

Computing A Glimpse of Randomness
https://arxiv.org/abs/nlin/0112022

Mathematicians who revealed the power of random walks win Abel prize
https://www.newscientist.com/article/2237832-mathematicians-who-revealed-the-power-of-random-walks-win-abel-prize/

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https://www.quantamagazine.org/rainbows-are-a-mathematicians-best-friend-20200318/?mc_cid=559347b58d&mc_eid=c608d388a6

Euler"s number via products of primes
https://twitter.com/TamasGorbe/status/1240014581394702342

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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 31 de Marzo, 2020, 11:04

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Maths, Madness and the Manhattan Project: the Eccentric Lives of Steinhaus, Banach and Ulam
https://culture.pl/en/article/maths-madness-and-the-manhattan-project-the-eccentric-lives-of-steinhaus-banach-and-ulam

"A Singular Mathematical Promenade" is a delightful book by a gifted expositor (Ghys) that beautifully showcases both the unity of mathematics
https://twitter.com/AlexKontorovich/status/1244731670487609348

A Singular Mathematical Promenade
http://ghys.perso.math.cnrs.fr/bricabrac/promenade.pdf

The Grogono Generator: an Apologia
http://users.encs.concordia.ca/~grogono/RNG/grog-gen.html

Stefan Banach
http://mathshistory.st-andrews.ac.uk/Biographies/Banach.html

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https://time.com/5792615/emmy-noether-100-women-of-the-year/

How Rational Math Catches Slippery Irrational Numbers
https://www.quantamagazine.org/how-rational-math-catches-slippery-irrational-numbers-20200310/

Gömböc
https://es.wikipedia.org/wiki/G%C3%B6mb%C3%B6c

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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 29 de Marzo, 2020, 12:31

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Lord Brouncker"s continued fraction for π
https://web.maths.unsw.edu.au/~mikeh/webpapers/paper168.pdf

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http://www.aimspress.com/article/10.3934/mbe.2020148

Wolfe Conditions
https://en.wikipedia.org/wiki/Wolfe_conditions

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https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space

Korovkin-type Theorems and Approximation by Positive Linear Operators
https://arxiv.org/abs/1009.2601

Estructuras Algebraicas
http://www.ehu.eus/juancarlos.gorostizaga/apoyo/estruct_alg.htm

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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 28 de Marzo, 2020, 18:22

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Irreducible and prime elements
https://math.stackexchange.com/questions/1076517/irreducible-and-prime-elements

In algebraic topology, cohomology classes are cocycle classes. In algebraic geometry, they are Poincare duals of subvarieties. In differential geometry, they are differential forms
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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 21 de Marzo, 2020, 13:02

Hoy leo en la introducción del excelente "Algebraic number theory and Fermat's last theorem", de Ian Stewart y David Tall

For organizational reasons rather than mathematical necessity, the book is divided into four parts. Part I develops the basic theory from an algebraic standpoint, introducing the ring of integers of a number field and exploring factorization within it. Quadratic and cyclotomic fields are investigated in more detail, and the Euclidean imaginary fields are classified. We then consider the notion of factorization and see how the notion of a 'prime' p can be pulled apart into two distinct ideas. The first is the concept of being 'irreducible' in the sense that p has no factors other than 1 and p. The second is what we now call 'prime': that if p is a factor of the product ab (possibly multiplied by units—invertible elements) then it must be a factor of either a or b. In this sense, a prime must be irreducible, but an irreducible need not be prime. It turns out that factorization into irreducibles is not always unique in a number field, but useful sufficient conditions for uniqueness may be found. The factorization theory of ideals in a ring of algebraic integers is more satisfactory, in that every ideal is a unique product of prime ideals. The extent to which factorization is not unique can be 'measured' by the group of ideal classes (fractional ideals modulo principal ones).

Es un tema más que interesante: uno, basado en el manejo de enteros y naturales, tiende a poner como equivalentes los conceptos de número primo y número irreducible. Pero se vió (justamente en el siglo XIX, tratando de demostrar el ultimo teorema de Fermat) que no es el caso: hay sistemas de números (anillos) donde no se cumple la equivalencia.

Ver

Irreducible and prime elements
https://math.stackexchange.com/questions/1076517/irreducible-and-prime-elements

Luego, si quieren algo más en profundidad, y cómo afecta esto a varias estructuras algebraicas:

Irreducible Elements
https://en.wikipedia.org/wiki/Irreducible_element

Any Prime is Irreducible
https://math.stackexchange.com/questions/69504/any-prime-is-irreducible

Prime implies Irreducible
https://math.stackexchange.com/questions/1149078/prime-implies-irreducible

Irreducible Elements in a Principal Ideal Domain are Prime
https://math.stackexchange.com/questions/770731/irreducible-elements-in-a-pid-are-prime

Irreducible Elements in an Unique Factorization Domain are Prime
https://math.stackexchange.com/questions/257955/irreducibles-are-prime-in-a-ufd

A principal ideal ring that is not a euclidean ring
http://www.math.buffalo.edu/~dhemmer/619F11/WilsonPaper.pdf

Ring of integers is a Principal Ideal Domain but not a Euclidean domain
https://math.stackexchange.com/questions/857971/ring-of-integers-is-a-pid-but-not-a-euclidean-domain

An example of a principal ideal domain which is not a Euclidean domain
http://www.maths.qmul.ac.uk/~raw/MTH5100/PIDnotED.pdf

En este blog, algo traté del tema cuando comenté

Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (4)
http://ajlopez.zoomblog.com/archivo/2016/06/28/libro-Abstract-Algebra-de-Carstensen-F.html

Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (3)
http://ajlopez.zoomblog.com/archivo/2016/06/27/libro-Abstract-Algebra-de-Carstensen-F.html

Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (2)
http://ajlopez.zoomblog.com/archivo/2016/06/26/libro-Abstract-Algebra-de-Carstensen-F.html

Libro: Abstract Algebra Structure and Application, de Finston y Morandi
http://ajlopez.zoomblog.com/archivo/2016/06/14/libro-Abstract-Algebra-Structure-and-A.html

En esos libros aparece más detallado la evolución del concepto, en especial, la aparición de ideales primos, que de nuevo, tuvo su origen en los intentos de demostración del ultimo teorema de Fermat, por parte de Kummer y sus números ideales, una extension para conseguir la factorización única, luego levantada por Dedekind para formar los ideales primos. Esa extension del concepto de número resultó fructífera, como se ve en los capítulos de los libros mencionados arriba.

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 23 de Febrero, 2020, 8:42

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The science checklist applied: Mathematics
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Rainbow Proof Shows Graphs Have Uniform Parts
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Equations that changed the world
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Introduction to Clifford Algebra
https://www.av8n.com/physics/clifford-intro.htm

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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 16 de Febrero, 2020, 15:06

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Great Woman of Mathematics: Katherine Johnson
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Color-Changing Material Unites the Math and Physics of Knots
https://www.quantamagazine.org/color-changing-material-unites-the-math-and-physics-of-knots-20200210/

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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 15 de Febrero, 2020, 14:33

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Karmarkar's algorithm
https://en.wikipedia.org/wiki/Karmarkar%27s_algorithm

Auction Theory
https://en.wikipedia.org/wiki/Auction_theory

It may surprise you to learn how much insight into auctions mathematics has been able to provide in recent years...
http://www.ams.org/publicoutreach/feature-column/fc-2011-09

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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 9 de Febrero, 2020, 11:32

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https://twitter.com/gabrielpeyre/status/1222398971043205120
https://en.wikipedia.org/wiki/Harris_Corner_Detector

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https://en.wikipedia.org/wiki/Ellipsoid_method

Probability Distribution Fitting
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El Oráculo de Aaronson
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Continuous Fraction for Square Root
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Mathematics is the only infinite human activity
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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 8 de Febrero, 2020, 12:34

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https://en.wikipedia.org/wiki/Metaballs

Jim Blinn
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The Quasi-Stationary Distribution of the Subcritical Contact Process
https://twitter.com/pgroisma/status/1225749509747150848
https://arxiv.org/pdf/1908.04175.pdf

A Chemist Shines Light on a Surprising Prime Number Pattern
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Reverse Mode Differentiation is Kind of Like a Lens II
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Ring of Periods
https://twitter.com/fermatslibrary/status/1220710857170210819nn
https://en.wikipedia.org/wiki/Ring_of_periods

Projective Geometric Algebra Done Right
http://terathon.com/blog/projective-geometric-algebra-done-right/

A survey of the Schrödinger problem and some of its connections with optimal transport
https://arxiv.org/abs/1308.0215

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 25 de Enero, 2020, 11:29

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Grothendieck reformulated algebraic geometry using "schemes"
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How Pi Connects Colliding Blocks to a Quantum Search Algorithm
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Cómo dibujar un pentágono regular
https://twitter.com/HdAnchiano/status/1216078131720867843

Lyapunov function
https://en.wikipedia.org/wiki/Lyapunov_function

God the Geometer
https://twitter.com/pickover/status/1219007599858921472

Some abstract theorems can be used as tools to simplify matrices to practical ends
https://twitter.com/SamuelGWalters/status/1218720944220332032

Jules Hedges - compositional game theory - part I
https://www.youtube.com/watch?v=5Qny8YmLUzk&feature=emb_logo

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 19 de Enero, 2020, 11:26

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The MacTutor History of Mathematics Archive
http://mathshistory.st-andrews.ac.uk/

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https://www.bloomberg.com/news/features/2018-05-03/the-gambler-who-cracked-the-horse-racing-code

Möbius strip hidden inside Klein Bottle
https://twitter.com/InertialObservr/status/1216111529516355584

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https://www.youtube.com/watch?v=sOIjL-wlfk4

Rhodonea
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Searching for Positive Returns at the Track
https://www.researchgate.net/publication/292145708_Searching_for_Positive_Returns_at_the_Track

Equation inspired by Ramanujan notebooks
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Drawing an elephant with four complex parameters
https://twitter.com/fermatslibrary/status/1214545733048774663
https://fermatslibrary.com/s/drawing-an-elephant-with-four-complex-parameters#email-newsletter

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 18 de Enero, 2020, 15:42

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https://perlitasmatematicas.wordpress.com/2020/01/14/cuanto-da-0-elevado-a-la-0/

This is the largest known Fibonacci number that's also prime
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About Noether's Theorem
https://twitter.com/InertialObservr/status/1217545021810888706

Programming with Categories - Lecture 6
https://www.youtube.com/watch?v=xStrvUgN51A&feature=emb_logo

Tai-Danae Bradley: Modeling Language with Tensor Networks
https://www.youtube.com/watch?v=12j8OV-ptC4

Berstein Polynomial
https://en.wikipedia.org/wiki/Bernstein_polynomial

Choose 2 numbers at random - here's the probability they will have no common factor
https://twitter.com/fermatslibrary/status/1216350626055184385

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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 11 de Enero, 2020, 15:22

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https://en.wikipedia.org/wiki/Strassen_algorithm

Chaos Machine
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Science history: Al-Khwarizmi, master of maths
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Facebook's AI mathematician can solve university calculus problems
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https://cosmosmagazine.com/physics/mathematicians-provide-explanation-for-an-uncertain-law-of-physics

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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 8 de Enero, 2020, 17:49

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Here's a proof that e^π is transcendental
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La conjetura ABC y el último teorema de Fermat
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The equation XAX=B is surprisingly simple to solve on the set of positive matrices
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The Suble Art of the Mathematical Conjecture
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Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 5 de Enero, 2020, 11:41

En su libro The Principles of Mathematics, Bertrand Russel describe lo que considera matemáticas puras:

Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.

Es una definición seca, llevada a la base de toda la matemática pura, pero que deja afuera (o al menos lejos) a cantidad de ramas activas y fructíferas de las matemáticas, puras o no. Pero en aquellos años (1903 fue publicado por primera vez ese libro), Russel buscaba los fundamentos de la matemática, que habían entrado en cierta crisis (recordemos la crítica de Russel al libro sobre conjuntos y lógica de Frege). En su infancia, había encontrado en las matemáticas una roca firme, donde la verdad era clara, en medio quizás de un mundo humano cambiante. Pero para fines del siglo XIX, los fundamentos de las matemáticas no estaban claros ni firmes.

Desde entonces, se ha avanzado, y hoy, gran parte de los matemáticos no se preocupan de los fundamentos, sino del gran juego que es el pensamiento matemático. No lo hacen en general por ser arriesgados, mas bien los fundamentos de la matemática se han ido desarrollando en el siglo XX en varias ramas, desde una teoría de conjuntos más elaborada, hasta teoría de categorías, pasando por el formalismo de Hilbert,  los teoremas de Gödel y la evolución de la lógica matemáticas más allá de verdadero o falso. No es un tema que ha quedado abandonado, sino que sigue siendo bien atendido. Pero hay tantas ramas interesantes y activas, que muchos matemáticos simplemente descansan sobre los hombros de los colegas que se ocupan de los fundamentos.

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Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Publicado el 4 de Enero, 2020, 15:27

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Rank of an elliptic curve
https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve

Mathematics
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From sequences to nets
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Beal Conjecture
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https://en.wikipedia.org/wiki/Subnet_(mathematics)

An Essay towards solving a Problem in the Doctrine of Chance
https://en.wikipedia.org/wiki/An_Essay_towards_solving_a_Problem_in_the_Doctrine_of_Chances

Lo que considero lo mejor de 2019 en gaussianos
https://www.gaussianos.com/lo-que-yo-considero-lo-mejor-de-2019-en-gaussianos/

Decoding the ancient greek astronomical calculator known as the antikythera mechanism
https://fermatslibrary.com/s/decoding-the-ancient-greek-astronomical-calculator-known-as-the-antikythera-mechanism

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 1 de Enero, 2020, 12:34

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Apolo, Pitágoras y la Música
https://twitter.com/pgroisma/status/1197826329346863104

Thirty-Six Unsolved Problems in Number Theory
https://arxiv.org/abs/math/0010143

Circular Prime
https://twitter.com/fermatslibrary/status/1212382480717754369

Solution Randomness from Deteterminism
https://www.quantamagazine.org/solution-randomness-from-determinism-20191122/

George Dantziq
https://en.wikipedia.org/wiki/George_Dantzig

Simplex Algorithm
https://en.wikipedia.org/wiki/Simplex_algorithm

In 1947 W. H. Mills proved that there is a number A, such that the following number is prime for all integer values of n
https://twitter.com/fermatslibrary/status/1197877654285889541

On Computing the Rank of Elliptic Curves
https://www.math.colostate.edu/~achter/math/brown.pdf

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 31 de Diciembre, 2019, 14:20

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Erdős–Rényi_model
https://en.wikipedia.org/wiki/Erdős–Rényi_model

Matthias Goerner made an in-space viewer for hyperbolic 3-manifolds
https://twitter.com/henryseg/status/1211226130167910400

The Central Limit Theorem and its misuse
https://lambdaclass.com/data_etudes/central_limit_theorem_misuse/

Transportation theory (mathematics)
https://en.wikipedia.org/wiki/Transportation_theory_(mathematics)

Gaspard Monge
https://en.wikipedia.org/wiki/Gaspard_Monge

Compositionality First Issue
https://johncarlosbaez.wordpress.com/2019/12/30/compositionality-first-issue/

Compositionality Journal
https://compositionality-journal.org/

Start with 82 and go backwards to 1
https://twitter.com/fermatslibrary/status/1197153129646694406

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

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