Xah Math Blog

been slacking in the math department. Going to pick up again. Each day, i spend 1 hour reading math, and post whatever. Today, let's learn about “smooth map”.

2019-05-06

xah talk show 2019-05-06 geometric inversion, stereographic projection, their relations

2019-04-29

this is logic, part of proof theory. this is what intent to learn in next 5 years, as opposed to category theory. (i got asked about the latter often, from programer idiots) https://twitter.com/johncarlosbaez/status/1122976661132021760

Peano axioms

Tennenbaum's theorem

Tennenbaum's theorem is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff).

Compactness theorem

In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.

Finite model theory

Finite model theory (FMT) is a subarea of model theory (MT). MT is the branch of mathematical logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). FMT is a restriction of MT to interpretations on finite structures, which have a finite universe.

Interpretation (logic)

An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.

The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate T (for “tall”) and assign it the extension {a} (for “Abraham Lincoln”). Note that all our interpretation does is assign the extension {a} to the non-logical constant T, and does not make a claim about whether T is to stand for tall and ‘a’ for Abraham Lincoln. Nor does logical interpretation have anything to say about logical connectives like ‘and’, ‘or’ and ‘not'. Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.

An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.

Lindström's theorem

In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logic[1] (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.[2]

Modal logic

2019-04-08

these shapes form hex stars. #geometry https://twitter.com/CGTNOfficial/status/1115445476537450496

[Too good to be Truchet 2018-10-18 By Colin Beveridge. At http://chalkdustmagazine.com/features/too-good-to-be-truchet/ ]

[Truchet 2007-04- By Cameron Browne. At http://cambolbro.com/games/truchet/ ]

[Truchet, Braille and Euler 2010-02-11 By Peter Rowlett. At https://aperiodical.com/2010/02/truchet-braille-and-euler/ ]

Zernike polynomials

2019-03-22

todo

Math and Illustrations

2019-03-17

xah talk show 2019-03-17, python, iterator, iterators, #math intro to group theory for programers

1:53 emacs xah html mode as org babel
6:46 python 3 iterator, OOP language vs functional language
15:50 starts on group theory intro

2019-03-16

xah talk show 2019-03-16. history of python perl ruby JavaScript golang, neural network, complex numbers.

2019-03-15

xah talk show, 2019-03-14, calculus, analysis, free text books, regex

2019-02-28

Module (Algebra)

2019-02-26

Linear Map, Bilinear Map, Multilinear Map

2019-02-22

Geometric algebra

todo read

[Let's remove Quaternions from every 3D Engine (An Interactive Introduction to Rotors from Geometric Algebra) 2019-02-22 By Marc Ten Bosch. At http://marctenbosch.com/quaternions/ ]

Geometric algebra

i no unstand.

help:

[Linear and Geometric Algebra 2011-01-19 By Alan Macdonald. At Buy at amazon ]

https://enkimute.github.io/ganja.js/examples/coffeeshop.html

[Geometric Algebra for Computer Science 2007-05-22 By Leo Dorst , Daniel Fontijne , Stephen Mann. At Buy at amazon ]

2019-02-21

some old articles.

2019-02-14

2019-02-04

Hyperboloid of Two Sheet http://VirtualMathMuseum.org/Surface/hyperboloid2/hyperboloid2.html

Dirac Belt Trick http://VirtualMathMuseum.org/Surface/dirac-belt/DiracBelt.html

2019-02-04

Mathematician Gaston Julia

this guy, is the first to study julia set. #math #geometry

2019-02-02

Mathematician Pierre Wantzel

Pierre Wantzel

Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.[1]

In a paper from 1837,[2] Wantzel proved that the problems of

doubling the cube, and

trisecting the angle

are impossible to solve if one uses only compass and straightedge. In the same paper he also solved the problem of determining which regular polygons are constructible:

a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e. that the sufficient conditions given by Carl Friedrich Gauss are also necessary) The solution to these problems had been sought for thousands of years, particularly by the ancient Greeks. However, Wantzel's work was neglected by his contemporaries and essentially forgotten. Indeed, it was only 50 years after its publication that Wantzel's article was mentioned either in a journal article[3] or in a textbook.[4] Before that, it seems to have been mentioned only once, by Julius Petersen, in his doctoral thesis of 1871. It was probably due to an article published about Wantzel by Florian Cajori more than 80 years after the publication of Wantzel's article[1] that his name started to be well-known among mathematicians.[5]

Wantzel was also the first person who proved, in 1843,[6] that when a cubic polynomial with rational coefficients has three real roots but it is irreducible in Q[x] (the so-called casus irreducibilis), then the roots cannot be expressed from the coefficients using real radicals alone, that is, complex non-real numbers must be involved if one expresses the roots from the coefficients using radicals. This theorem would be rediscovered decades later by (and sometimes attributed to) Vincenzo Mollame and Otto Hölder.

“Ordinarily he worked evenings, not lying down until late; then he read, and took only a few hours of troubled sleep, making alternately wrong use of coffee and opium, and taking his meals at irregular hours until he was married. He put unlimited trust in his constitution, very strong by nature, which he taunted at pleasure by all sorts of abuse. He brought sadness to those who mourn his premature death.” — Adhémar Jean Claude Barré de Saint-Venant on the occasion of Wantzel's death.[1]

2019-02-02

Mathematician Pierre Fatou and Julia Set

this guy, is the first to study Julia set.

the math work of Pierre Fatou. Analysis, analytical functions, dynamical systems, chaos theory.

2019-02-01

Mathematician Adrien Douadly, and Mandelbrot Set

this guy started research on Mandelbrot set. one major result is that the Mandelbrot set is connected. #math #geometry

the Mandelbrot set, is truly one of the most incredible thing in math. Incredible is the word, and amazing, cosmetic, and in a visual way too. It shows the deep mystery of math.

here's the deepest zoom. Zoom to: 3.4 * 10^1091, and the video plays for 70 minutes.

here's the deepest zoom of Mandelbrot set, Zoomed to 3.4 * 10^1091. If you zoom this much on a atom, it'd be 10 followed by 109 zeros times larger than the screen. it's 70 minutes video.

Eye of the Universe - Mandelbrot Fractal Zoom (e1091) (4k 60fps)

and i always thought, if you zoom to a particular point deep enough, the face of god would suddenly appear, and the universe would blow up. It's, like, playing lottery, you wouldn't know there's no jackpot untill you played all the possible numbers. #math

2019-02-01

annulus, math

2019-01-23

differential geometry site

been helping mat professors build differential geometry site.

latest are Soliton Surface and others, see

visit the whole gallery at http://virtualmathmuseum.org/index.html

we've been working on it in past year.

graduate level differential geometry

math Three-Soliton Surface http://VirtualMathMuseum.org/Surface/three-soliton/three-soliton.html

Breather Surface http://VirtualMathMuseum.org/Surface/breather/breather.html

2019-01-06

#math Chaitin's constant Chaitin's constant this is when, computer science enters the twilight zone. if you understand it, let me know. i want to kiss your ass.

2019-01-06

Calculus, Gradient

Gradient

2019-01-04

Derivative and Jacobian Matrix

Jacobian matrix and determinant

#math derivative. if you are rusty with calculus, or programer who never learned it, it's good time to revisit. join my journey. read my daily snippet, and lookup Wikipedia.

Wikipedia is usually chaotic, rambling on and on, touching highschool stuff to research stuff. if calculus is new to you, read textbooks. Here's Free Math Textbooks i verified quality. Free Math Textbooks

2019-01-02

Proof theory

2019-01-01

#math programers, if u haven't seen Conway's Game of Life yet, look into. it's eye opening. In 1990s, i spent years “playing” it. 1st deep theory you learn: deterministic system can be unpredictable. https://twitter.com/icm7216/status/1080209582041907200

Xah Math Blog Archive

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