Angel "Java" Lopez en Blog

Publicado el 31 de Mayo, 2020, 18:00

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Coronavirus and Flu
https://www.nytimes.com/2020/02/29/health/coronavirus-flu.html

Spreading the Word on a Possible Alzheimer’s Treatment
https://www.quantamagazine.org/stimulated-brain-waves-offer-a-possible-treatment-for-alzheimers-20200527/

Black Hole Paradoxes Reveal a Fundamental Link Between Energy and Order
https://www.quantamagazine.org/black-hole-paradoxes-reveal-a-fundamental-link-between-energy-and-order-20200528/

Proxima Centauri Exoplanet
https://twitter.com/johncarlosbaez/status/1266428654721961984

Integrated Information in Process Theories
https://arxiv.org/abs/2002.07654

Particle Quiz
https://scoollab.web.cern.ch/sites/scoollab.web.cern.ch/files/ParticleGame/

How to Desing a Perpetual Energy Machine
https://www.quantamagazine.org/how-to-design-a-perpetual-energy-machine-20200401/

Epidemic Modeling 101: Or why your CoVID-19 exponential fits are wrong
https://medium.com/data-for-science/epidemic-modeling-101-or-why-your-covid19-exponential-fits-are-wrong-97aa50c55f8

Nos leemos!

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

Por ajlopez, en: Ciencia

Publicado el 30 de Mayo, 2020, 13:18

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Triángulo y Circunferencia Circunscrita
https://www.gaussianos.com/triangulo-y-circunferencia-circunscrita/

Mathematician Measures the Repulsive Force Within Polynomials
https://www.quantamagazine.org/new-math-measures-the-repulsive-force-within-polynomials-20200514/

Isbell duality
https://twitter.com/johncarlosbaez/status/1266029338190680064

Put coplay / lens update into a state monad where the state variable is a payoff vector
https://twitter.com/_julesh_/status/1251507999606276104

Kurt Gödel Wanted to Revise Our Concept of Time
https://www.thegreatcoursesdaily.com/kurt-godel-wanted-to-revise-our-concept-of-time/

Highly composite numbers
http://wwwhomes.uni-bielefeld.de/achim/highly.html

Symbolic Mathematics Finally Yields to Neural Networks
https://www.quantamagazine.org/symbolic-mathematics-finally-yields-to-neural-networks-20200520/

Graduate Student Solves Decades-Old Conway Knot Problem
https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/

Nos leemos!

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

Publicado el 25 de Mayo, 2020, 15:07

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Los conceptos de campo, partícula, partícula virtual y vacío
https://francis.naukas.com/2012/08/15/los-conceptos-de-campo-particula-particula-virtual-y-vacio/

Así inactivan el coronavirus los productos de limpieza
https://www.lasexta.com/el-muro/deborah-garcia/asi-inactivan-coronavirus-productos-limpieza_202003185e71f764cf7ab300010f99cc.html

What Other Coronaviruses Tell Us About SARS-CoV-2
https://www.quantamagazine.org/what-can-other-coronaviruses-tell-us-about-sars-cov-2-20200429/

Study: Could dark matter be hiding in existing data?
https://phys.org/news/2020-05-dark.html

Physicists have just seen "Pauli crystals"
https://twitter.com/johncarlosbaez/status/1263128844850081792

The Great Debate of Shapley and Curtis — 100 years later
https://astronomy.com/news/2020/04/the-great-debate-of-shapley-and-curtis--100-years-later

Physicists identify unique signature to confirm quark-gluon plasma in Universe
https://arstechnica.com/science/2020/05/physicists-identify-unique-signature-to-confirm-quark-gluon-plasma-in-universe/

Sugary Camouflage on Coronavirus Offers Vaccine Clues
https://www.quantamagazine.org/sugars-on-coronavirus-spike-protein-offer-vaccine-clues-20200505/

Nos leemos!

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

Por ajlopez, en: Ciencia

Publicado el 24 de Mayo, 2020, 11:44

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The role of voting intention in public opinion polarization
https://arxiv.org/abs/1909.07092

Mirth programming language
https://twitter.com/typeswitch/status/1249368983729471488
https://github.com/mirth-lang/mirth

Rule 110
https://en.wikipedia.org/wiki/Rule_110

The Computer Scientist Who Can"t Stop Telling Stories
https://www.quantamagazine.org/computer-scientist-donald-knuth-cant-stop-telling-stories-20200416/

Bourbaki's final perfected definition of the number 1, printed out on paper, would be 200,000 as massive as the Milky Way!
https://twitter.com/johncarlosbaez/status/1250868989414137857

Math3ma
https://www.math3ma.com/

The Mathematics of Privacy
https://becominghuman.ai/the-mathematics-of-privacy-361742827f08

Reflections on monadic lenses
http://www.cs.ox.ac.uk/jeremy.gibbons/publications/mlenses.pdf

Nos leemos!

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

Publicado el 23 de Mayo, 2020, 12:28

En estos días vuelvo a leer a Max Jammer, a quien mencioné en:

Uhlenbeck, Goudsmit y el spin del electron
Max Jammer y el Formalismo de la Mecánica Cuántica

Estoy leyendo su excelente libro "The Conceptual Development of Quantum Mechanics", edición 1966. Leo:

It is the purpose of this study to trace the conceptual development of  quantum mechanics from its inception to its formulation as a full-fledged theory of atomic physics, from its status as a rather doubtful ad hoc hypothesis to that of an imposing intellectual structure of great beauty.

Trata de la mecánica cuántica no relativista de una cantidad finite de grados de libertad, que abarca los primeros desarrollos exitoso, digamos, hasta parte de la tercera década del siglo XX. Las primeras preguntas que nos aparecen son: ¿qué es el desarrollo conceptual? ¿es necesario en este tema?.

All the great theories in the history of physics—from Aristotelian mechanics and its medieval elaborations, through Newtonian dynamics with its Lagrangian or Hamiltonian modifications, to Maxwellian  electrodynamics and* Einsteinian relativity—have been subjected repeatedly to historico-critical investigations, and their conceptual foundations have been thoroughly analyzed. But no comprehensive scholarly study of the conceptual development of quantum mechanics has heretofore appeared. The popular or semiscientific publications available hardly skim the surface of the subject. And the few, though important, essays on the topic written by the originators of the theory themselves are mostly confined to a  particular aspect or to the defense of a specific philosophical position. The publication of a comprehensive and coherent analysis of the conceptual development of quantum mechanics, the only consistent theory of atomic processes, and hence a foundation of modern science, seems therefore to fill an important lacuna in the literature on the history and philosophy of physics.

Primero, es un libro sobre el desarrollo de las ideas. No es un libro de texto para APRENDER mecánica cuántica, sino para entender cómo se fue desarrollando. Por otro lado, no es un libro solamente con el desarrollo histórico: habrá ocasiones donde Jammer se concentra en el desarrollo de una idea, sin importar que ese desarrollo se superponga en años históricos con otro desarrollo de otra idea.

Such a study, however, should not be regarded as an end in itself. Never before has a theory been treated in so many excellent texts within so short a period of time. Since every additional text, as is natural in a developing branch of science, attempts to construct the theory on the basis of an ever more concise logical structure, the traditional texts presenting the material in the order of its historical development are gradually losing ground. Students, it is rightfully claimed, may be saved much trouble "if they are not led through all the historical pitfalls, and instead acquainted from the very beginning with concepts, such as the spin, that cannot be grasped except by quantum mechanical means." [cita 1] Admittedly, a good working knowledge, even a high standard of efficiency and competence in applying the theory to physical problems may be obtained by this method of instruction. But as long as it is true that "however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms," [cita 2] a profound comprehension of quantum mechanics requires more than a study of the subject in its shortest possible logical formulation. "It is impossible to understand the methods of modern quantum mechanics without a knowledge of the way in which the theory has been developing." [cita 3] Indeed, some knowledge of the dramatic struggle of ideas that preceded the formation of quantum  conceptions and of the intricate ways of reasoning that led to the generally accepted formulation of the theory is indispensable for a profound comprehension of its physical significance, for an intelligent appreciation of its philosophical implications, and even for an enlightened understanding of its logical structure. Duhem's statement "faire l'histoire d'un principe physique, c'est en meme temps en faire l'analyse logique" is particularly meaningful for the science of quantum mechanics.

Hoy aprendemos física clásica, digamos la mecánica newtoniana, sin estudiar ni su desarrollo histórico ni conceptual (¿cómo surgió el concepto de energía? ¿y el de momento angular?). Pero en cuántica, y en especial en mecánica cuántica, pasa que si bien se puede aprenderla sin necesitar de la historia o el desarrollo, sus conceptos, operaciones y formalismos son tan extraños a la intuición que para realmente comprenderla, atraparla en su esencia, puede que no haya otro camino que el emprendido por Jammer: el desarrollo de las ideas, para explicar fenómenos que no cuajan en el marco "clásico". Es cierto que hay muy buenos libros para comenzar a entender la mecánica cuántica (y luego la física cuántica) desde algunos principios, sin tener que seguir todo el desarrollo ni histórico ni conceptual. Pero es como que se pierde algo. Un excelente libro que recuerdo ahora, que sigue un desarrollo histórico, más que conceptual, es el Física Cuántica, de Eisberg y Resnick, que era un clásico en mis días de estudiante universitario.

La cita 1 es F.A.Kaempffer, Concepts in Quantum Mechanics. La cita 2 es de Niels Bohr, "Discussion with Einstein on epistemological problems in atomic physics". La cita 3 es A.March, Quantum Mechanics of Particles and Wave Fields.

Seguiré citando y comentando algunos temas de este libro. Es interesante, por ejemplo, cómo plantea que los desarrollos de Jordan y de Dirac, a partir de 1925, son como un contrapunto: mientras el primero se concentra en teorías de campo, el segundo prefirió una vision desde el modelo de partículas.

Nos leemos!

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

Por ajlopez, en: Ciencia

Publicado el 17 de Mayo, 2020, 10:02

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MIP*=RE
https://arxiv.org/abs/2001.04383

A simulation of sunflower growth
https://twitter.com/ZenoRogue/status/1247900522905886723

To Win This Numbers Game, Learn to Avoid Math Patterns
https://www.quantamagazine.org/to-win-this-numbers-game-learn-to-avoid-math-patterns-20200507/

In Star Trek Discovery we heard about the "Logic Extremists"
https://twitter.com/johncarlosbaez/status/1258475087704895488

What does "birational equivalence" mean in a cryptographic context?
https://crypto.stackexchange.com/questions/43013/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
https://twitter.com/gregeganSF/status/1243318369983418368

Here"s the axioms for a group in strings
https://twitter.com/CreeepyJoe/status/1248736658612031488

Modeling Opinion Dynamics: Theoretical analysis and continuous approximation
https://arxiv.org/abs/1606.00662

Nos leemos!

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

Publicado el 10 de Mayo, 2020, 13:21

Por estos lares seguimos en cuarentena. Mucho trabajo y estudio, pero pudo escribir poco. Repaso de las resoluciones del mes pasado:

- Escribir sobre Matemáticas [pendiente]
- Escribir sobre Física [pendiente]
- Escribir sobre Historia de las Matemáticas [pendiente]
- Escribir sobre Historia de la Ciencia [pendiente]
- Estudiar blues en guitarra [completo]

Tengo que poner voluntad para pasar por escrito lo que estuve aprendiendo de matemáticas y física. Así que sigo insistiendo con estas resoluciones para mayo:

- 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
http://www.ajlopez.com
http://twitter.com/ajlopez

Publicado el 3 de Mayo, 2020, 14:36

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Interface Scaling in the Contact Process
https://arxiv.org/pdf/cond-mat/0003480.pdf

Radical Solutions: Evariste Galois
https://www.damninteresting.com/radical-solutions/

Explicación de teoría de colas y ejercicio resuelto
https://www.youtube.com/watch?v=t3x2KinUqAA&feature=emb_logo

Mathematical proof that rocked number theory will be published
https://www.nature.com/articles/d41586-020-00998-2

Conmutative Semigroups
https://twitter.com/johncarlosbaez/status/1246470112321953792

Quantum Natural Language Processing
https://medium.com/cambridge-quantum-computing/quantum-natural-language-processing-748d6f27b31d
https://github.com/oxford-quantum-group/discopy/blob/ab2b356bd3cad1dfb55ca6606d6c4b4181fe590c/notebooks/qnlp-experiment.ipynb

Graced With Knowledge, Mathematicians Seek to Understand
https://www.quantamagazine.org/mathematicians-grapple-with-sudden-answer-to-connes-embedding-conjecture-20200408/

Landmark Computer Science Proof Cascades Through Physics and Math
https://www.quantamagazine.org/landmark-computer-science-proof-cascades-through-physics-and-math-20200304/

Nos leemos!

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

Publicado el 2 de Mayo, 2020, 17:55

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Conway"s Impact on the Theory of Random Tilings
https://twitter.com/JimPropp/status/1249189041343475712

Ha muerto John Horton Conway (1937-2020) - Gaussianos
https://www.gaussianos.com/ha-muerto-john-horton-conway-1937-2020/

Math After COVID-19
https://www.quantamagazine.org/how-has-coronavirus-affected-mathematics-20200428/

Method of estimating the number of infected people
https://twitter.com/wtgowers/status/1242823142088851456

Tikzcd Editor Category Diagrams
https://tikzcd.yichuanshen.de/

Uncomputable Numbers
https://medium.com/cantors-paradise/uncomputable-numbers-ee528830d295

The Quasi-Stationary Distribution of the Subcritical Contact Process
https://twitter.com/pgroisma/status/1242941761057632257
https://arxiv.org/pdf/1908.04175.pdf

Simulation of quasi-stationary distributions on countable spaces
https://arxiv.org/abs/1206.6712

Nos leemos!

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

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
https://www.youtube.com/watch?v=e_729Ehb4vQ&feature=emb_logo

Haskell implementation of open games
https://github.com/jules-hedges/open-games-hs

Gambit: Software Tools for Game Theory
http://gambit-project.org/

The category-theoretic formulation of 1
https://twitter.com/andrejbauer/status/1253357470250123265

Linear Algebra Course
https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/

Important Formulas(Part 7) - Permutation and Combination
https://www.careerbless.com/aptitude/qa/permutations_combinations_imp7.php

Dr. Mary Cartwright, 1900-1998
https://twitter.com/GWOMaths/status/1196367452394938374

Tiling with polyominoes and combinatorial group theory
https://www.sciencedirect.com/science/article/pii/0097316590900574

Nos leemos!

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

Publicado el 21 de Abril, 2020, 20:39

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John Horton Conway has died
https://twitter.com/CardColm/status/1249038195880341505

John H. Conway, 1937–2020
https://www.math.princeton.edu/news/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
https://www.medrxiv.org/content/10.1101/2020.04.09.20058974v1

My PhD thesis "At the Interface of Algebra and Statistics" is now on the arXiv
https://twitter.com/math3ma/status/1249862830670729217
https://arxiv.org/abs/2004.05631
https://www.youtube.com/watch?v=wiadG3ywJIs&feature=youtu.be

John Conway Solved Mathematical Problems With His Bare Hands
https://www.quantamagazine.org/john-conway-solved-mathematical-problems-with-his-bare-hands-20200420/

Compositional game theory
https://arxiv.org/abs/1603.04641v3

Recomendando películas matemáticas
https://twitter.com/edusadeci/status/1242397128120578049

Determining the length of paper on a toilet paper roll
https://twitter.com/fermatslibrary/status/1242811233637666818

Nos leemos!

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

Publicado el 18 de Abril, 2020, 11:05

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El teorema de Mills: mucho ruido y pocas nueces - Gaussianos
https://www.gaussianos.com/el-teorema-de-mills-mucho-ruido-y-pocas-nueces/

"Amazing" Math Bridge Extended Beyond Fermat"s Last Theorem
https://www.quantamagazine.org/amazing-math-bridge-extended-beyond-fermats-last-theorem-20200406/

Langlands Program
https://www.quantamagazine.org/tag/langlands-program/

The E8 lattice
https://twitter.com/johncarlosbaez/status/1241764718999506944
https://johncarlosbaez.wordpress.com/2020/03/20/from-the-octahedron-to-e8/

John Horton Conway: the world"s most charismatic mathematician
https://www.theguardian.com/science/2015/jul/23/john-horton-conway-the-most-charismatic-mathematician-in-the-world?CMP=share_btn_tw

Mathematician and genius John Conway, inventor of The Game of Life has succumbed to COVID-19 today
https://twitter.com/AdiShavit/status/1249055920111452163

John Conway Reminiscences about Dr. Matrix and Bourbaki
https://blogs.scientificamerican.com/guest-blog/john-conway-reminiscences-about-dr-matrix-and-bourbaki/

Monster Group (John Conway) - Numberphile
https://www.youtube.com/watch?v=jsSeoGpiWsw&feature=emb_logo

Nos leemos!

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

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
http://www.ajlopez.com
http://twitter.com/ajlopez

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
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
https://twitter.com/thaiselenags/status/1237618582026059776

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/

"Rainbows" Are a Mathematician"s Best Friend
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

Nos leemos!

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

Woman of the Year 1921: Emmy Noether
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

Nos leemos!

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

A mathematical model for the novel coronavirus epidemic in Wuhan, China
http://www.aimspress.com/article/10.3934/mbe.2020148

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

Christoph and the Calendar
https://thonyc.wordpress.com/2016/02/24/christoph-and-the-calendar/

Linear Regression and Kernel Methods
http://www.numerical-tours.com/matlab/ml_2_regression/

Reproducing kernel Hilbert space
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

Nos leemos!

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

Publicado el 28 de Marzo, 2020, 18:22

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Matemáticas online mientras dure la cuarentena
http://www.aulamagna.com.es/matematicas-online-mientras-dura-la-cuarentena/

Catriona Shearer is great at inventing geometry problems that seem impossible to solve
https://twitter.com/johncarlosbaez/status/1230520859795677184

Aryabhatta I (ca. 5th CE) was an astronomer and a mathematician and hypothesized the rotation of the earth
https://twitter.com/VatsalTrivedi18/status/1230906845839020033

Mathgen; Randomly generated mathematics research papers!
https://thatsmathematics.com/mathgen/

This past Monday marked what would have been mathematician Emmy Noether"s 138th birthday
https://twitter.com/QuantaMagazine/status/1243659920916000772

Catriona Shearer in Twitter
https://twitter.com/Cshearer41/status/1231489403761176576

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
https://twitter.com/YuhangChen13/status/1238260647227543557

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Angel "Java" Lopez
http://www.ajlopez.com
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Publicado el 22 de Marzo, 2020, 13:00

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

Por ajlopez, en: Ciencia

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