Angel "Java" Lopez en Blog

Abril del 2020

Publicado el 25 de Abril, 2020, 16:42

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James Propp "Conway's Impact on the Theory of Random Tilings" 03 23 14

Haskell implementation of open games

Gambit: Software Tools for Game Theory

The category-theoretic formulation of 1

Linear Algebra Course

Important Formulas(Part 7) - Permutation and Combination

Dr. Mary Cartwright, 1900-1998

Tiling with polyominoes and combinatorial group theory

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Angel "Java" Lopez

Publicado el 21 de Abril, 2020, 20:39

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John Horton Conway has died

John H. Conway, 1937–2020

Predicting the number of reported and unreported cases for the COVID-19 epidemics in China, South Korea, Italy, France, Germany and United Kingdom

My PhD thesis "At the Interface of Algebra and Statistics" is now on the arXiv

John Conway Solved Mathematical Problems With His Bare Hands

Compositional game theory

Recomendando películas matemáticas

Determining the length of paper on a toilet paper roll

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Angel "Java" Lopez

Publicado el 18 de Abril, 2020, 11:05

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El teorema de Mills: mucho ruido y pocas nueces - Gaussianos

"Amazing" Math Bridge Extended Beyond Fermat"s Last Theorem

Langlands Program

The E8 lattice

John Horton Conway: the world"s most charismatic mathematician

Mathematician and genius John Conway, inventor of The Game of Life has succumbed to COVID-19 today

John Conway Reminiscences about Dr. Matrix and Bourbaki

Monster Group (John Conway) - Numberphile

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Angel "Java" Lopez

Publicado el 10 de Abril, 2020, 13:11

En medio de la cuarentena, estuve bastante concentrado estudiando y trabajando. Primero, revision de mis resoluciones del mes pasado:

- Escribir sobre Matemáticas [completo] ver Números Irreducibles y Primos y Notas sobre Teoría Algebraica de Números (1)
- Escribir sobre Física [pendiente]
- Escribir sobre Historia de las Matemáticas [pendiente]
- Escribir sobre Historia de la Ciencia [pendiente]
- Estudiar blues en guitarra [parcial]

Mis resoluciones para el nuevo mes:

- Escribir sobre Matemáticas
- Escribir sobre Física
- Escribir sobre Historia de las Matemáticas
- Escribir sobre Historia de la Ciencia
- Estudiar blues en guitarra

Angel "Java" Lopez

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.

Nos leemos!

Angel "Java" Lopez

Publicado el 4 de Abril, 2020, 17:42

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Se obtuvo una hermosa fórmula matemática usando el número de Napie

Sphere Point Picking

Unpredictability, Undecidability, and Uncomputability

Mathematics as a Team Sport

Computing A Glimpse of Randomness

Mathematicians who revealed the power of random walks win Abel prize

"Rainbows" Are a Mathematician"s Best Friend

Euler"s number via products of primes

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Angel "Java" Lopez