Matemáticas

Matemáticas: Enlaces, Novedades y Recursos (228)

Publicado el 2 de Abril, 2021, 13:31

A glimpse of infinity. Radicals within radicals, forever.... 42
https://twitter.com/pickover/status/1373280568927002631

Infinitas Incógnitas
https://www.gaussianos.com/infinitas-incognitas/

Egyptian Fractions
https://brilliant.org/wiki/egyptian-fractions/

Mathematicians Find a New Class of Digitally Delicate Primes
https://www.quantamagazine.org/mathematicians-find-a-new-class-of-digitally-delicate-primes-20210330/

Traversal Optics and Polynomial Functors
https://github.com/BartoszMilewski/Publications/blob/master/Traversals.pdf

Srinivasa Ramanujan: The Mathematician & His Legacy
https://www.youtube.com/watch?v=uhNGCn_3hmc

Pioneers Linking Math and Computer Science Win the Abel Prize
https://www.quantamagazine.org/avi-wigderson-and-laszlo-lovasz-win-abel-prize-20210317/

I'm publishing traversal optics as a series of blog posts
https://twitter.com/BartoszMilewski/status/1377680473682366470

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

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Matemáticas: Enlaces, Novedades y Recursos (227)

Publicado el 27 de Marzo, 2021, 11:41

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Non-Periodic Tiling in Blender
http://cgfromspace.blogspot.com/2013/12/non-periodic-tiling-in-blender-chapter-2.html

2021 Global Online Celebration
https://www.idm314.org/launch-2021-program.html

¿Sabía que el número 11111…11111 (109297 unos seguidos) es un número primo casi seguro?
https://www.gaussianos.com/%C2%BFsabia-que-4/

What's Special About This Number?
https://www.archimedes-lab.org/numbers/Num24_69.html

The World of Pi: Ramanujan
http://www.pi314.net/eng/ramanujan.php

Four squares. What"s the total shaded area?
https://twitter.com/Cshearer41/status/1372545154607894528

Situación de las raíces de la derivada, o «el teorema más maravilloso de las matemáticas»
https://www.gaussianos.com/situacion-de-las-raices-de-la-derivada-o-el-teorema-mas-maravilloso-de-las-matematicas/

Wonderful 42. 42 is the largest number n such that there exist positive integers p, q, r with....
https://twitter.com/pickover/status/1373277523048550401

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

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Matemáticas: Enlaces, Novedades y Recursos (226)

Publicado el 24 de Marzo, 2021, 19:31

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Raíces Complejas y Espirales
https://www.gaussianos.com/raices-complejas-y-espirales/

It just occurred to me that the category of matrices is enriched over itself, which feels very weird
https://twitter.com/emilyriehl/status/1369063456507273216

Siegfried Heinrich Aronhold
https://mathshistory.st-andrews.ac.uk/Biographies/Aronhold/

El Teorema de Pick
https://www.gaussianos.com/el-teorema-de-pick/

Los centros del triángulo: el punto de Gergonne
https://www.gaussianos.com/los-centros-del-triangulo-el-punto-de-gergonne/

Matrix Multiplication Inches Closer to Mythic Goal
https://www.quantamagazine.org/mathematicians-inch-closer-to-matrix-multiplication-goal-20210323/

Emmy Noether and the Myth of Genius
https://www.ladyscience.com/essays/myth-of-genius-emmy-noether-gender

Hofstadter points
https://en.wikipedia.org/wiki/Hofstadter_points

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

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Matemáticas: Enlaces, Novedades y Recursos (225)

Publicado el 21 de Marzo, 2021, 14:18

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Nonlinearity in the hierarchy of large cardinal consistency strength
http://jdh.hamkins.org/nonlinearity-in-the-hierarchy-of-consistency-strength/

Olimpiada Matemática Española 2012 – Problema 6: Triángulo acutángulo y circunferencias
https://www.gaussianos.com/olimpiada-matematica-espanola-2012-problema-6-triangulo-acutangulo-y-circunferencias/

George Keith Batchelor
https://mathshistory.st-andrews.ac.uk/Biographies/Batchelor/

New Algorithm Breaks Speed Limit for Solving Linear Equations
https://www.quantamagazine.org/new-algorithm-breaks-speed-limit-for-solving-linear-equations-20210308/

In the movie "A Beautiful Mind," Russell Crowe scribbles these formulas. Here's why
https://twitter.com/pickover/status/1369037610404941828

The history of the concept of function and some educational implications
https://www.researchgate.net/publication/251211596_The_history_of_the_concept_of_function_and_some_educational_implications

Great geophysicists #10: Joseph Fourier
https://agilescientific.com/blog/2013/11/12/great-geophysicists-10-joseph-fourier.html

Sunday workout: prove this curious identity
https://twitter.com/nntaleb/status/1373609971036921856

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

Por ajlopez, en: Matemáticas

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Matemáticas: Enlaces, Novedades y Recursos (224)

Publicado el 20 de Marzo, 2021, 15:18

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Prime Number: 19191...19191 (33-digits)
https://primes.utm.edu/curios/page.php?curio_id=20136

Great Mathematicians Playing Cards
https://www.reddit.com/r/math/comments/m3g3lf/great_mathematicians_playing_cards_inclusion/

Mathematicians Of The Day: 16th March
https://mathshistory.st-andrews.ac.uk/OfTheDay/oftheday-03-16/

James Joseph Sylvester
https://mathshistory.st-andrews.ac.uk/Biographies/Sylvester/

Number Theory, Topology, and Fractals with Wacław Sierpiński
http://scihi.org/waclaw-sierpinski/

Identifiability Analysis of Linear Ordinary Differential Equation Systems with a Single Trajectory
https://arxiv.org/abs/2103.05660

Math Online Celebration
https://www.idm314.org/launch-2021-program.html

The Responsibility of the Scientist Today
by Alexandre Grothendieck, 1970
http://matematicas.unex.es/~navarro/res/scientist.pdf

Angel "Java" Lopez
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Matemáticas: Enlaces, Novedades y Recursos (223)

Publicado el 13 de Marzo, 2021, 18:20

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¡Viene la derivada!
https://www.gaussianos.com/%C2%A1viene-la-derivada/

A Minimal Proof of the Fundamental Theorem of Algebra
https://twitter.com/theoremoftheday/status/1370773386658471938
https://www.daniellitt.com/blog/2016/10/6/a-minimal-proof-of-the-fundamental-theorem-of-algebra

A beautiful result, hope we can prove it
https://twitter.com/nntaleb/status/1369768934753308678

Louis Bachelier
https://mathshistory.st-andrews.ac.uk/Biographies/Bachelier/

Maryam Mirzakhani, the Only Woman to Have Won Math's Highest Honor
https://interestingengineering.com/maryam-mirzakhani-the-only-woman-to-have-won-maths-highest-honor

The mystifying Göbel's Sequence
https://twitter.com/pickover/status/1305318978236100610

Three Month Plan to Learn Mathematics Behind Machine Learning
https://towardsdatascience.com/three-month-plan-to-learn-mathematics-behind-machine-learning-74335a578740

Statistics Postdoc Tames Decades-Old Geometry Problem
https://www.quantamagazine.org/statistics-postdoc-tames-decades-old-geometry-problem-20210301/

Angel "Java" Lopez
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Matemáticas: Enlaces, Novedades y Recursos (222)

Publicado el 7 de Marzo, 2021, 20:13

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The Napkin Ring Problem
https://www.youtube.com/watch?v=J51ncHP_BrY

MATHCOUNTS Practice Competition #4 Live Solve by Po-Shen Loh SAT
https://www.youtube.com/watch?v=vgWV-6l3ouM&feature=emb_logo

Martin"s doileys
https://www.johndcook.com/blog/2021/02/17/martins-doileys/

In 1918, Albert Einstein wrote to Felix Klein decrying the great injustice that brilliant mathematician Emmy Noether, because of being a woman, was denied the right to teach in Germany as a university professor:
https://twitter.com/phalpern/status/1361745091526864903
https://einsteinpapers.press.princeton.edu/vol8-trans/742

Nina Holden just won the 2021 Maryam Mirzakhani New Frontiers Prize for her work on random surfaces and the mathematics of quantum gravity
https://twitter.com/johncarlosbaez/status/1304450533051097090

El día que el arte me ayudó a escribir un artículo de matemáticas
https://culturacientifica.com/2021/03/07/el-dia-que-el-arte-me-ayudo-a-escribir-un-articulo-de-matematicas/

Groups in the form of objects and morphisms
https://twitter.com/SchmiegSophie/status/1367308613153222656

When Math Gets Impossibly Hard
https://www.quantamagazine.org/when-math-gets-impossibly-hard-20200914/

Angel "Java" Lopez
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Matemáticas: Enlaces, Novedades y Recursos (221)

Publicado el 6 de Marzo, 2021, 15:12

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Imaginary Numbers May Be Essential for Describing Reality
https://www.quantamagazine.org/imaginary-numbers-may-be-essential-for-describing-reality-20210303/

Good math problems for high school students
https://twitter.com/AbellsTheorem/status/1362268697482596352

Grover"s Algorithm
https://en.wikipedia.org/wiki/Grover's_algorithm

World's hardest easy geometry problem
https://twitter.com/pickover/status/1361453165187633153
https://mathsmed.co.uk/hardest/taskD.html

The Coach Who Led the U.S. Math Team Back to the Top
https://www.quantamagazine.org/po-shen-loh-led-the-u-s-math-team-back-to-first-place-20210216/

Fast Factoring Integers by SVP Algorithms
https://eprint.iacr.org/2021/232

Lattice problem
https://en.wikipedia.org/wiki/Lattice_problem

Isadore Singer Transcended Mathematical Boundaries
https://www.quantamagazine.org/isadore-singer-bridged-math-and-physics-20210304/

Angel "Java" Lopez
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Matemáticas: Enlaces, Novedades y Recursos (220)

Publicado el 28 de Febrero, 2021, 16:50

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A quantum algorithm for the direct estimation of the steady state of open quantum systems
https://quantum-journal.org/papers/q-2021-02-22-399/

On Computable Numbers, with an Application to the Entscheidungsproblem
https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

The Halting Problem - An Impossible Problem to Solve
https://www.youtube.com/watch?v=t37GQgUPa6k&t=1s

Turing Machines - The Accidental Birth of Computer Science
https://www.youtube.com/watch?v=PLVCscCY4xI&t=1s

Una integral muy «dorada»
https://www.gaussianos.com/una-integral-muy-dorada/

Tübinger Dreiecke
https://commons.wikimedia.org/wiki/File:T%C3%BCbinger_Dreiecke.gif

Which great mathematicians had great political commitments?
https://mathoverflow.net/questions/384230/which-great-mathematicians-had-great-political-commitments

Is there any integral for the Golden Ratio?
https://twitter.com/pickover/status/1362458565198237696
https://math.stackexchange.com/questions/1653979/is-there-any-integral-for-the-golden-ratio

Angel "Java" Lopez
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Matemáticas: Enlaces, Novedades y Recursos (219)

Publicado el 27 de Febrero, 2021, 6:55

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The larger square from the seventh simple nonperiodic tiling using pattern blocks
https://ar.pinterest.com/pin/121949102394724563/

Hay una transición de fase en el modelo de Erdös-Rényi
https://twitter.com/pgroisma/status/1365271264542261248

Hausdorff and Box dimensions of continuous functions and lineability
https://www.tandfonline.com/doi/abs/10.1080/03081087.2019.1612832?journalCode=glma20#.YDeBCccVH0E.twitter

How (1-z)/(1+z) transforms the unit disk
https://twitter.com/AnalysisFact/status/1364984115376902147

Mathematicians Set Numbers in Motion to Unlock Their Secrets
https://www.quantamagazine.org/with-arithmetic-dynamics-mathematicians-unlock-new-insights-20210222/

42 métodos de demostración en Matemáticas
https://www.gaussianos.com/42-metodos-de-demostracion-en-matematicas/

La infinitud de los números primos y Fermat
https://www.gaussianos.com/la-infinitud-de-los-numeros-primos-y-fermat/

Descenso infinito: un método de demostración poco conocido
https://www.gaussianos.com/descenso-infinito-un-metodo-de-demostracion-poco-conocido/

Angel "Java" Lopez
http://www.ajlopez.com
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Matemáticas: Enlaces, Novedades y Recursos (218)

Publicado el 21 de Febrero, 2021, 20:17

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Ludwig Boltzmann
https://mathshistory.st-andrews.ac.uk/Biographies/Boltzmann/

Classical Mechanics and Category Theory
https://twitter.com/johncarlosbaez/status/1291048030515339264

0019: Article 9 (More Pi Formulas)
https://sites.google.com/site/tpiezas/0019

Mathematicians Open a New Front on an Ancient Number Problem
https://www.quantamagazine.org/mathematicians-open-a-new-front-on-an-ancient-number-problem-20200910/

Seven Sketches in Compositionality: An Invitation to Applied Category Theory
https://arxiv.org/abs/1803.05316

At the Math Olympiad, Computers Prepare to Go for the Gold
https://www.quantamagazine.org/at-the-international-mathematical-olympiad-artificial-intelligence-prepares-to-go-for-the-gold-20200921/

Legendre"s Constant
https://en.wikipedia.org/wiki/Legendre%27s_constant

The Simple Math Problem We Still Can"t Solve
https://www.quantamagazine.org/why-mathematicians-still-cant-solve-the-collatz-conjecture-20200922/

Angel "Java" Lopez
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Matemáticas: Enlaces, Novedades y Recursos (217)

Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time
https://twitter.com/OxUniMaths/status/1356724980894072833
https://www.maths.ox.ac.uk/node/38304

Dynamic Rectangles are rectangles generated from the unit square
https://twitter.com/fermatslibrary/status/1129747875523645441

Renaissance Science – II
https://thonyc.wordpress.com/2021/01/27/renaissance-science-ii/

The Future of Mathematics
https://www.microsoft.com/en-us/research/video/the-future-of-mathematics/

Lean Prover Tutorial
https://leanprover.github.io/tutorial/index.html

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
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Matemáticas: Enlaces, Novedades y Recursos (213)

Publicado el 6 de Febrero, 2021, 11:17

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Look-and-say y la constante de Conway
https://www.gaussianos.com/look-and-say-y-la-constante-de-conway/

Look-and-say sequence
https://en.wikipedia.org/wiki/Look-and-say_sequence

Elliptic Curve PKC and Key Sizes
https://medium.com/iovlabs-innovation-stories/elliptic-curve-pkc-and-key-sizes-941db80040b6

Jost Bürgi
https://mathshistory.st-andrews.ac.uk/Biographies/Burgi/

Carl Friedrich Gauss: el príncipe de las matemáticas
https://www.gaussianos.com/carl-friedrich-gauss-el-principe-de-las-matematicas/

246B, Notes 3: Elliptic functions and modular forms
https://terrytao.wordpress.com/2021/02/02/246b-notes-3-elliptic-functions-and-modular-forms/

Mark Kac y su nacionalidad compleja
https://www.gaussianos.com/mark-kac-y-su-nacionalidad-compleja/

The Cistercian monks invented a numbering system in the 13th century
https://twitter.com/MathematicsUCL/status/1356558846093914114

Nos leemos!

Angel "Java" Lopez
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Matemáticas: Enlaces, Novedades y Recursos (212)

Publicado el 17 de Enero, 2021, 15:36

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Some Fixed Point Theorems for Partially Ordered Sets
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F7BE2EFF0FF71A8CBFC34FF973BED5AA/S0008414X00030704a.pdf/some_fixed_point_theorems_for_partially_ordered_sets.pdf

Theoretical Physics
http://www.tug.org/texshowcase/cheat.pdf

New Geometric Perspective Cracks Old Problem About Rectangles
https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/

Enlaces útiles para la clase de Matemáticas
https://docs.google.com/document/d/174DY7xK8rLVvVwqYCweQDZ0tO4Zluf8RiG2k-cTWPDw/edit
https://twitter.com/lolamenting/status/1285510978445225989

Applied Category Theory 2020
https://twitter.com/johncarlosbaez/status/1277643847363772416

The Mathematical Structure of Particle Collisions Comes Into View
https://www.quantamagazine.org/new-particle-collision-math-may-offer-quantum-clues-20200820/

The Crooked Geometry of Round Trips
https://www.quantamagazine.org/the-crooked-geometry-of-round-trips-20210113/

Mathematicians Probe Unsolved Hilbert Polynomial Problem
https://www.quantamagazine.org/mathematicians-probe-unsolved-hilbert-polynomial-problem-20210114/

Nos leemos!

Angel "Java" Lopez
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Emily Riehl y la Teoría de Categorías

Publicado el 10 de Enero, 2021, 19:08

El año pasado (2020) leí un artículo de Quanta Magazine que me hizo conocer el trabajo de Emily Riehl en teoría de categorías (el artículo abarca más que esa actividad), Quiero comentar hoy brevemente los primeros párrafos de su excelente libro Category Theory in Context. Serán comentarios livianos, el tema da para mucho más, sirva esto como una leve introducción.

Teoría de categorías es un tema que goza de la fama de abstracto, y en gran parte es así. Tambien va surgiendo en este siglo una rama aplicada de la teoría, pero en general, aún muchos matemáticos profesionales ven a la teoría como abstracta. Algo que no colabora para su diffusion, es que, como muchas otras ramas, si uno comienza a estudiarla se encuentra con muchas definciones y conceptos, que si bien son interesantes, NO PARECEN tener motivación que justifique su desarrollo. Es por eso que es bienvenido este libro de Riehl (la pueden segurr en su twitter @emilyriehl donde se describe como working mathematician, supongo que un guiño a uno de los más conocidos libros de teoría de categorás) que pone en contexto, como dice su título, las ideas que se desarrollan en el pensamiento de categorías. Leo al comienzo del libro:

Atiyah described mathematics as the "science of analogy." In this vein, the purview of category theory is mathematical analogy. Category theory provides a cross-disciplinary language for mathematics designed to delineate general phenomena, which enables the transfer of ideas from one area of study to another. The category-theoretic perspective can function as a simplifying1 abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline.

Suenore recuerdo el comentario de Bourbaki sobre Gauss, que en su Disquisitione Mathematica probaba una y otra vez una propiedad que quedaba evidente dentro de la teoría de grupos, pero debía hacerlo así porque esa teoría no estaba desarrollada. Las matemáticas como que extraen lo común a varias situaciones y trabajan sobre ello. Por ejemplo, las propiedades de un anillo, luego de haber establecido los axiomas a cumplir por todo anillo. O los espacios topológicos, dados las propiedades de sus conjuntos abiertos. La teoría de categorías va un paso (yo diría más de un paso), en esa dirección.

A subtle shift in perspective enables mathematical content to be described in language that is relatively indifferent to the variety of objects being considered. Rather than characterize the objects directly, the categorical approach emphasizes the transformations between objects of the same general type. A fundamental lemma in category theory implies that any athematical object can be characterized by its universal property—loosely by a representation of the morphisms to or from other objects of a similar form. For example, tensor products, "free" constructions, and localizations are characterized by universal properties in appropriate categories, or mathematical contexts. A universal property typically expresses one of the mathematical roles played by the object in question. For instance, one universal property associated to the unit interval identifies self-homeomorphisms of this space with re-parameterizations of paths. Another highlights the operation of gluing two intervals end to end to obtain a new interval, the construction used to define composition of paths.

El punto importante, que exige un esfuerzo intellectual, es justamente ver el objeto a través de sus relaciones con otros objetos parecidos. El tema de propiedad universal es fundamental en teoría de categorías, pero también cuesta aprehenderlo porque en general lo manejos en dominios particulares, como anillos o campos.

Certain classes of universal properties define blueprints which specify how a new object may be built out of a collection of existing ones. A great variety of mathematical constructions fit into this paradigm: products, kernels, completions, free products, "gluing" constructions, and quotients are all special cases of the general category-theoretic notion of limits or colimits, a characterization that makes it easy to define transformations to or from the objects so- defined. The input data for these constructions are commutative diagrams, which are themselves a vehicle for mathematical definitions, e.g., of rings or algebras, representations of a group, or chain complexes.

Es por todo esto, que este libro es para matemáticos: RIehl pone en contexto las ideas de teoría de categorías, pero apelando a varios conocimientos, desde el algebra conmutativa y no conmutativa hasta topología general.

Important technical differences between particular varieties of mathematical objects can be described by the distinctive properties of their categories: that rings have all limits and colimits while fields have few, that a continuous bijection defines an isomorphism of compact Hausdorff spaces but not of generic topological spaces. Constructions that convert mathematical objects of one type into objects of another type often define transformations between categories, called functors. Many of the basic objects of study in modern algebraic topology and algebraic geometry involve functors and would be impossible to define without
category-theoretic language.

De alguna forma, teoría de categorías expone lo que muchos matemáticos hacen: encontrar analogías entre distintos objetos y sacar provecho de ello. Sí, Atiyah tenia razón: es la ciencia de la analogia.

Nos leemos!

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

Por ajlopez, en: Matemáticas

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Matemáticas: Enlaces, Novedades y Recursos (211)

Publicado el 3 de Enero, 2021, 15:04

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Transit Across the Purple Sun
https://twitter.com/pickover/status/1296101195447140353

Computer Search Settles 90-Year-Old Math Problem
https://www.quantamagazine.org/computer-search-settles-90-year-old-math-problem-20200819/

Pi in Pascal's Triangle
https://www.cut-the-knot.org/arithmetic/algebra/PiInPascal.shtml

This formula was first discovered by Laplace and later proved by Jacobi.
https://twitter.com/johncarlosbaez/status/1299442096621760512

The Anatomy of Integers and Permutations
https://twitter.com/Marco_Piani/status/1296088988642615296
https://dms.umontreal.ca/~andrew/MSI/AnatomyForTheBook.pdf

How Close Are Computers to Automating Mathematical Reasoning?
https://www.quantamagazine.org/how-close-are-computers-to-automating-mathematical-reasoning-20200827/

Julio Rey Pastor
https://twitter.com/borcherix/status/1294347487319400454

La hipótesis del continuo: del susto de Cantor a la prueba de Cohen
https://www.gaussianos.com/la-hipotesis-del-continuo-del-susto-de-cantor-a-la-prueba-de-cohen/

Nos leemos!

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

Por ajlopez, en: Matemáticas

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Matemáticas: Enlaces, Novedades y Recursos (210)

Publicado el 2 de Enero, 2021, 12:05

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Computer Scientists Attempt to Corner the Collatz Conjecture
https://www.quantamagazine.org/can-computers-solve-the-collatz-conjecture-20200826

Does 2 + 2 equal 4.00?
https://twitter.com/wtgowers/status/1296463616590393346

This quilt is a multiplication table (also called a Cayley table) for the symmetric group on 4 letters.
https://twitter.com/DaveJensenMath/status/1344740179966357505

Binomial coefficients mod primes
https://www.johndcook.com/blog/2020/06/24/binomial-coefficients-mod-primes/

Chasing the Tail of the Gaussian (Part 1)
https://golem.ph.utexas.edu/category/2020/08/chasing_the_tail_of_the_gaussi.html
https://twitter.com/johncarlosbaez/status/1300811563180240897

Conducting the Mathematical Orchestra From the Middle
https://www.quantamagazine.org/emily-riehl-conducts-the-mathematical-orchestra-from-the-middle-20200902/

Algebraic Number Starscapes
https://arxiv.org/abs/2008.07655

The envelope of Simson lines
https://twitter.com/74WTungsteno/status/1300765678522245120

Nos leemos!

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

Por ajlopez, en: Matemáticas

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