Angel "Java" Lopez en Blog


Publicado el 24 de Mayo, 2020, 11:44

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The role of voting intention in public opinion polarization

Mirth programming language

Rule 110

The Computer Scientist Who Can’t Stop Telling Stories

Bourbaki's final perfected definition of the number 1, printed out on paper, would be 200,000 as massive as the Milky Way!


The Mathematics of Privacy

Reflections on monadic lenses

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Angel "Java" Lopez

Publicado el 17 de Mayo, 2020, 10:02

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A simulation of sunflower growth

To Win This Numbers Game, Learn to Avoid Math Patterns

In Star Trek Discovery we heard about the "Logic Extremists"

What does "birational equivalence" mean in a cryptographic context?

If you view any ellipse from just the right distance, it will *always* occupy 90 degrees

Here"s the axioms for a group in strings

Modeling Opinion Dynamics: Theoretical analysis and continuous approximation

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Angel "Java" Lopez

Publicado el 3 de Mayo, 2020, 14:36

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Interface Scaling in the Contact Process

Radical Solutions: Evariste Galois

Explicación de teoría de colas y ejercicio resuelto

Mathematical proof that rocked number theory will be published

Conmutative Semigroups

Quantum Natural Language Processing

Graced With Knowledge, Mathematicians Seek to Understand

Landmark Computer Science Proof Cascades Through Physics and Math

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Angel "Java" Lopez

Publicado el 2 de Mayo, 2020, 17:55

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Conway"s Impact on the Theory of Random Tilings

Ha muerto John Horton Conway (1937-2020) - Gaussianos

Math After COVID-19

Method of estimating the number of infected people

Tikzcd Editor Category Diagrams

Uncomputable Numbers

The Quasi-Stationary Distribution of the Subcritical Contact Process

Simulation of quasi-stationary distributions on countable spaces

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Angel "Java" Lopez

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 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

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

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

"A Singular Mathematical Promenade" is a delightful book by a gifted expositor (Ghys) that beautifully showcases both the unity of mathematics

A Singular Mathematical Promenade

The Grogono Generator: an Apologia

Stefan Banach

Woman of the Year 1921: Emmy Noether

How Rational Math Catches Slippery Irrational Numbers


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Angel "Java" Lopez

Publicado el 29 de Marzo, 2020, 12:31

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Lord Brouncker"s continued fraction for π

A mathematical model for the novel coronavirus epidemic in Wuhan, China

Wolfe Conditions

Christoph and the Calendar

Linear Regression and Kernel Methods

Reproducing kernel Hilbert space

Korovkin-type Theorems and Approximation by Positive Linear Operators

Estructuras Algebraicas

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Angel "Java" Lopez

Publicado el 28 de Marzo, 2020, 18:22

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Matemáticas online mientras dure la cuarentena

Catriona Shearer is great at inventing geometry problems that seem impossible to solve

Aryabhatta I (ca. 5th CE) was an astronomer and a mathematician and hypothesized the rotation of the earth

Mathgen; Randomly generated mathematics research papers!

This past Monday marked what would have been mathematician Emmy Noether"s 138th birthday

Catriona Shearer in Twitter

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

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.


Irreducible and prime elements

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

Irreducible Elements

Any Prime is Irreducible

Prime implies Irreducible

Irreducible Elements in a Principal Ideal Domain are Prime

Irreducible Elements in an Unique Factorization Domain are Prime

A principal ideal ring that is not a euclidean ring

Ring of integers is a Principal Ideal Domain but not a Euclidean domain

An example of a principal ideal domain which is not a Euclidean domain

En este blog, algo traté del tema cuando comenté

Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (4)

Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (3)

Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (2)

Libro: Abstract Algebra Structure and Application, de Finston y Morandi

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.

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Angel "Java" Lopez

Publicado el 23 de Febrero, 2020, 8:42

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The Category Theory Behind UMAP

The science checklist applied: Mathematics

Rainbow Proof Shows Graphs Have Uniform Parts

Equations that changed the world

Every number greater than 55 is the sum of distinct primes of the form 4n+3

The Map of Mathematics

The Freyd-Mitchell Embedding Theorem

Introduction to Clifford Algebra

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Angel "Java" Lopez

Publicado el 16 de Febrero, 2020, 15:06

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Mathematical model reveals best auction strategy

Why Is M-Theory the Leading Candidate for Theory of Everything?

But what is the Fourier Transform? A visual introduction.

There"s something special about four-sided figures packed into circles

Construcciones con regla y compás (III): Los polígonos regulares - Gaussianos

Números primos gemelos y demás familia - Gaussianos

Great Woman of Mathematics: Katherine Johnson

Color-Changing Material Unites the Math and Physics of Knots

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Angel "Java" Lopez

Publicado el 15 de Febrero, 2020, 14:33

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Escher ¿Qué hay en el agujero del medio?

Quantum Questions Inspire New Math

Proof that √2 is irrational

A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory☆

A Fight to Fix Geometry"s Foundations

Karmarkar's algorithm

Auction Theory

It may surprise you to learn how much insight into auctions mathematics has been able to provide in recent years...

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Angel "Java" Lopez

Publicado el 9 de Febrero, 2020, 11:32

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A Visual Proof

Los secretos matemáticos del embaldosado

Harris Corner Detector

Ellipsoid Method

Probability Distribution Fitting

El Oráculo de Aaronson

Continuous Fraction for Square Root

Mathematics is the only infinite human activity

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Angel "Java" Lopez

Publicado el 8 de Febrero, 2020, 12:34

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Jim Blinn

The Quasi-Stationary Distribution of the Subcritical Contact Process

A Chemist Shines Light on a Surprising Prime Number Pattern

Reverse Mode Differentiation is Kind of Like a Lens II

Ring of Periods

Projective Geometric Algebra Done Right

A survey of the Schrödinger problem and some of its connections with optimal transport

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Angel "Java" Lopez

Publicado el 25 de Enero, 2020, 11:29

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Poetic Equation

Grothendieck reformulated algebraic geometry using "schemes"

How Pi Connects Colliding Blocks to a Quantum Search Algorithm

Cómo dibujar un pentágono regular

Lyapunov function

God the Geometer

Some abstract theorems can be used as tools to simplify matrices to practical ends

Jules Hedges - compositional game theory - part I

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Angel "Java" Lopez

Publicado el 19 de Enero, 2020, 11:26

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The MacTutor History of Mathematics Archive

The Gambler Who Cracked the Horse-Racing Code

Möbius strip hidden inside Klein Bottle

Acylindrically hyperbolic structures on groups - Balasubramanya


Searching for Positive Returns at the Track

Equation inspired by Ramanujan notebooks

Drawing an elephant with four complex parameters

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Angel "Java" Lopez

Artículos anteriores en Matemáticas