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A mathematician is a device for turning coffee into theorems”. ―
Alfréd Rényi
"A mathematician will recognise Cauchy, Gauss, Jacobi or Helmholtz after reading a few pages, just as musicians recognise, from the first few bars, Mozart, Beethoven or Schubert."
Ludwig Boltzmann
Btw. Have you met the Michael Jordan from Statistics? (not Basketball).
Watch closely his PhD. Students.
Map of Mathematician
Map of Mathematician
Ihr seid Wanderer & Seefahrer.
Wie zeichnet man eine Karte? / ein Koordinatensystem?
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Mathe Einblick
Mathe Einblick
If you like to get a first impression of Math, I highly recommend to you the books from Matt Parker the Math Book.
Also from Clifford Pick to get an insight into the history of math.
(Also physics and medicine)
Andrew Wiles (field medal laureate) used to read a book about the history of math in the 19 century before he came up with his proof of Fermat's Last Theorem. There are a vast number of books, that displays these developments like the Andals of Math in Princeton.
These aren't however helpful, because they are written in the current language and understanding. Hence, I worked around that and used Wikipedia, Encyclopedias, and Books from friendly math enthusiasts to build the understanding layer by layer up.
Math was originally written in textform by the Greeks.
So everything can be interpreted as language itself.
From a picture to a diagram, map, or even video, that conveys some visible 'structure' to you. (when for example to lines cross each other).
This should be known.
Also 2^2^3 = 256, but (2^2)^3 is 64.
Many things are defined and you have to be careful if one symbol, function or term is meant ambivalent.
f(x) =
f'(x) = .. the f'(x) = 0 is 'maybe' the max/min in f(x). -> Relation
An sqrt of 2 is defined as irrational with weird consecutive numbers, you can easily draw it yourself when you follow Pythagoras. So it is contained in the symbol (sqrt2) as one container, like any other number.
One form of optimization:
Inside Mathematics can be found the highest optimization in a technical application.
From comparing the the sun and the rivers, we could develop simply by counting numbers our world.
Genesis: We used the day and night as * ** … values
Then we used to calculate from our hands :
*** *** *** *** .. 12 Finger Parts
by counting the thumb .. Nail
by counting the thumb .. Middle Part
by counting the thumb .. Finger End.
By using the left hand, we can use each Finger 5 * 12 Finger Parts = 60.
We could also draw from 6x circles, the same radius. (wiith same distance) (6x)
By connecting all points by the same radius, we get 6 same pieces of the one o circle. Hence we get 360°.
Sidenote:
Thus. Measuring the Earth's Radius r was by light and shadow of any wall possible (Babylonian, Greeks, etc.)
There was a sealed Röhre, where liquid went through. He drew 360 points from ancient greek. (Sacredo during Italy)
> The Philipines have a different counting technique, different from the Babylonians, which is surprising as well, like the Tibetans.
Single Functions, Derivatives and Integrals
Single Functions, Derivatives and Integrals
(University, 1st semester)
(University, 1st semester)
Ggf. Als Introduction:
Man geht in der Mathematik meist von Zahlenreihen aus.
Dadurch wird die y = m*x hergeleitet. f(x) = x
Man kann ähnlich zu den quadraten Gleichungen, kubischen oder weitere Gleichungen auch einfach das Pascalsche Dreieck verwenden.
Diese beschreiben die Faktoren (= "Koeffizienten" genannt) der einzelnen x in linearen, quadrat., kubischen oder weiteren Gleichungen.
Gleichzeitig kann jede Funktions mit einer einzigen Variable ersetzt werden.
Beispiel:
1 2 1 = a x² + b x + c = (a + b)² = a² + 2ab + b² **
für a=x ** x² + 2b * x + c² (c ist eine Constant)
etc.**
Hier wird 'f(x)' nach einer Variable abgeleiten.
Ab hier kann die Wurzel (-1), als i gesehen werden.
j für Elektrotechniker. (i schon für Strom besetzt gewesen)
Somit werden mithilfe von
1. Plus + Multiplikation
2. sinus+ cosinus dargestellt und
3. e-Funktionen dargestellt,
aber alle drei sind nur 3 unterschiedliche Schreibweisen.
"Sind das 'gleiche'." .. Ambivalent behaftet, wie f(x): und f'(x) := d f(x)/dt ... andere Notation.
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Fourier Reihen (like a simple ++sum and function.)
Fourier Reihen (like a simple ++sum and function.)
(University, 2nd semester)
(University, 2nd semester)
Graveyard Fourier. (in Paris)
P.S. I brought Fourier, Flowers.
Spektrum (Deutsch)
Spektrum (Deutsch)
Wortart: Substantiv, (sächlich)
Wortart: Substantiv, (sächlich)
Bedeutung/Definition
1) Vielfalt der Erscheinungen in einem bestimmten Bereich
2) Physik: die Farben, in die man mit einem Prisma das Licht zerlegen kann
3) Mathematik: die Menge der Eigenwerte eines Operators
The Ph.D. committee of Fourier consisted of:
Lagrange, Laplace and Monge.
What many don't know is, that he didn't receive his recognition for Fourier's law, but due to his work on the ballistic missiles under the command of Napoleon during the french revolution.
For his other work, Biot and Poisson also insisted on errors.
He had tough examiners.
"Read Euler, read Euler, he is the master of us all"
- Pierre Simon Laplace
A student of Fourier was Dirichlet.
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Multivariable Calculus, Matrix and Graphs
Multivariable Calculus, Matrix and Graphs
(University, 3rd semester)
(University, 3rd semester)
Waveequation:
Source: https://en.wikipedia.org/wiki/Wave_equation
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Probability
Probability
(University, 4th semester)
(University, 4th semester)
To wrap up, here is how Maxwell — the famous physicist — thinks about probability.
"The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on."
Hint: Obviously, He was wrong. (Regard Bells Theorem for current maximum reasoning)
"Who ... is not familiar with Maxwell's memoirs on his dynamical theory of gases? ... from one side enter the equations of state; from the other side, the equations of motion in a central field. Ever higher soars the chaos of formulae. Suddenly we hear, as from kettle drums, the four beats 'put n = 5.' The evil spirit v vanishes; and ... that which had seemed insuperable has been overcome as if by a stroke of magic ... One result after another follows in quick succession till at last ... we arrive at the conditions for thermal equilibrium together with expressions for the transport coefficients." -- Ludwig Boltzmann
Another good note is, that many Russian mathematicians like Chebychev (Prof of the following Student), Lyapunov, and Markov advanced in these fields.
Dyson considered studying instead of the US in the former Sowjetunion. (It became to me obvious why.)
Nevertheless, he later studied under Hans Bethe (Lead Theoretical Physician of Los Alamos) and later under Feynmann.
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from Brown University (Basics of Probability)
http://www.math.wm.edu/~leemis/chart/UDR/UDR.html
Extended Math (Not study relevant)
Extended Math (Not study relevant)
(Institute)
(Institute)
"I write as an engineer and do not pretend to be a mathematician. I lean for support, and expect always to lean, upon the mathematician, just as I must lean upon the chemist, the physician, or the lawyer.
Norbert Wiener has patiently guided me around many a mathematical pitfall ... he has written an appendix to this text on certain mathematical points. I did not know an engineer and a mathematician could have such good times together. I only wish that I could get the real vital grasp of mathematics that he has of the basic principles of physics."
Vannevar Bush,
Context:
When Harold Jeffreys in Cambridge, England, offered his mathematical treatment in Operational Methods in Mathematical Physics (1927), Bush responded with his seminal textbook Operational Circuit Analysis (1929) for instructing electrical engineering students. In the preface he wrote: (...)
Detailed overview:
https://twitter.com/dzackgarza/status/1534375102934044673
Tensor
https://www.youtube.com/playlist?list=PLlXfTHzgMRULkodlIEqfgTS-H1AY_bNtq
Friendly Note from Peyman Milanfar publicly in Twitter: ->
Galois Theory: https://t.co/UY5MJqUYpc
Lie Algebra: https://francisrlb.wordpress.com/2023/02/25/how-does-a-lie-algebra-encode-a-space-part-1/
Map of Math: https://gogeometry.com/education/mathematics_fields_mind_map.html
Unknown Author (Please reference the original author)
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Unknown Author (Please reference the original author)
" W. Havens: Where does one find out about a Wishart distribution?
E. Wigner: A Wishart distribution is given in S. S. Wilks book about statistics and I found it just by accident."
"Any number either is prime or is measured by some prime number.
or: If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it."
- Euclid, Elements Book VII,
Current space "distance" interpretation mathematically
Sources: https://www.lix.polytechnique.fr/~nielsen/FrankNielsen-distances-figs.pdf
https://twitter.com/FrnkNlsn/status/1588356787031285761
Source: https://www.youtube.com/watch?v=c81ilGlIH1I&ab_channel=OxfordMathematics
Prime Numbers are the most important concepts.
Current Research: P-Adic Numbers: (Bridge between Number Theory and Geometric), Peter Schulze (2021)
https://www.quantamagazine.org/with-a-new-shape-mathematicians-link-geometry-and-numbers-20210719/
Current Research: Random Matrix Theorys:
Hurwitz (1897) ,Wishart (1928), Wigner (1955), Manchenko+Pastur(1967) Montgomery+Dyson + Odlyzko(1972), Alain Connes (1980ties), Andrew Wiles (1986), 1993 on - going, >2021 ICML Presentation
(today: Epsilon Subsampling, which indicates Phase transition of spectral clustering)
by Romain Couillet (2022)
Partial results for general RBM (d = 1):
(Random band matrices in )
• Schenker (2009): ( < W° localization techniques; improved to W7;
• Erdös, Yau, Yin (2011): l ≥ W - RM methods;
• Erdös, Knowles (2011): ( » W7/6 (in a weak sense);
• Erdös, Knowles, Yau, Yin (2012): & » W5/4 (in a weak sense, not
uniform in N);
• Bourgade, Erdos, Yau, Yin (2016): gap universality for W ~ N;
• Bourgade, Yang. Yau, Yin (2018): W » N3/4 (quantum unique
ergodicity);
by Tatyana Shcherbina
Research and Edge of Mathematics
Research and Edge of Mathematics
(Institute)
(Institute)
"When Hilbert's collected works were published, the editors realized they couldn't publish the originals as they were full of errors, some serious. They hired a young & unemployed Olga Taussky-Todd to go over and correct them. It took 3 yrs"
Peyman Milanfar, Google
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