dmm

dmm, 4 months ago to math

One of the reasons I love Group Theory is its beautiful structure. For example, the figure on the left shows one consequence of Theorem 7.1, which states that if φ : G → G is a group homomorphism, then the kernel of φ is a normal subgroup of G. Pretty amazing really.

Some of my (older) notes outlining a tiny piece of that structure are here: https://davidmeyer.github.io/qc/groups.pdf. The LaTeX source is here: https://www.overleaf.com/read/xjmnmzvrgsfk.

As always, questions/comments/corrections/* greatly appreciated.

#math #maths #grouptheory
As always, questions/comments/corrections/* greatly appreciated.

#math #maths #grouptheory #beautifulstructure

Group theory

dmm, 4 months ago to math

Here's a curious and beautiful integral:

#math #maths #integralcalculus

dmm, 4 months ago to math

This is the Bailey-Borwein-Plouffe formula, which was discovered in 1995 [1]. It allows us to calculate any digit of π (in hex) without calculating the preceding digits. Pretty cool.

Here's a quick python implementation:

def pi(precision):
getcontext().prec = precision
return sum(1/Decimal(16)**k *
(Decimal(4)/(8k+1) -
Decimal(2)/(8k+4) -
Decimal(1)/(8k+5) -
Decimal(1)/(8k+6))
for k in xrange(precision))

References

[1] "Bailey–Borwein–Plouffe formula", https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula

#pi #math #maths #mathematics

dmm, 4 months ago to random

Gus, who was a great dog, enjoying some beautiful Oregon winter sunshine...

dmm, 4 months ago to math

Did you know that

2024 = 2³+3³+4³+5³+6³+7³+8³+9³

2024 is also an abundant number (that is, the sum of its divisors is greater than the number [1]) and a Harshad number (the number is an integer which is divisible by sum of its digits, here 2024 = 8 x 253 [2]).

Happy New Year 2024 everyone!

#happynewyear2024 #abundantnumber #harshadnumber #numbertheory #math #maths #mathematics

References

[1] "Abundant number", https://en.wikipedia.org/wiki/Abundant_number

[2] "Harshad number" https://mathworld.wolfram.com/HarshadNumber.html

[3] "2024", https://www.numbersaplenty.com/2024

dmm, 4 months ago to math

Born #onthisday in 1903, John von Neumann was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath.

von Neumann had perhaps the widest coverage of any mathematician of his time, integrating pure and applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics. He was a pioneer in building the mathematical framework of quantum physics, in the development of functional analysis, and in game theory, introducing or codifying concepts including cellular automata, the universal constructor and the digital computer. His analysis of the structure of self-replication preceded the discovery of the structure of DNA.

#johnvonneumann #math #maths #physics

[Image credit: https://en.wikipedia.org/wiki/John_von_Neumann]

dmm, 4 months ago to random

The Rayleigh–Jeans law, which is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature [1], is an interesting piece of scientific history. The Rayleigh-Jeans result lead to what became known as the "ultraviolet catastrophe", which in turn lead to the development of quantum mechanics [4]. As usual, I got distracted by the history and the fascinating result(s) and am now down a new rabbit hole...

Among the many interesting things that were going on in physics at the beginning of the 20th century was the attempt to explain how (and why) bodies radiate energy. It's an interesting question: why exactly do some objects glow red, then orange, yellow, and then bluish-white, and with greater intensity, as their temperature increases?

To answer this question much (if not all) of the foundational work in this area focused on so-called black bodies, objects that absorb all incident electromagnetic radiation at all wavelengths and also radiate at all wavelengths. The image below shows the empirical data for spectral radiance vs. wavelength compared to the classical theory (i.e., the Rayleigh–Jeans law). The problem that was discovered was that the classical physics of the day couldn't explain black body radiation; this is shown int the graph as the "classical theory" diverging as the wavelength goes to zero (alternatively, as the frequency goes to infinity). This is known as the "ultraviolet catastrophe" since the theory predicts that an ideal black body at thermal equilibrium would emit unbounded quantity of energy as wavelength decreases into the ultraviolet spectrum [2], clearly a problem.

(1/2)

dmm, 4 months ago to science

This paper describes the predatory bacterium Vampirococcus lugosii, which preys on members of the bacterial species of the genus Halochromatium [1].

This thing is incredible. For example: Vampirococcus lugosii has a severely reduced genome, something like 1.3 Mbp, and lacks the genes which code for many of the standard biosynthetic metabolic pathways (e.g. phospholipid synthesis, amino acid synthesis, and nucleotide synthesis). Yet it is somehow still alive.

How does this work?

One mechanism that Vampirococcus uses is to get these raw materials from its prey. An example of this are the nucleotides that Vampirococcus lugosii gets by chopping up the DNA that it sucks out of its prey. And amazingly, Vampirococcus lugosii uses a CRISPR-Cas system and various restriction enzymes to accomplish this. See the image for a cartoon of this system.

Predatory microbes.

Crazy.

---

Description of “Candidatus Vampirococcus lugosii”]

Lugosii after Bela Lugosi (1882–1956), who played the role of the vampire in the iconic 1931’s film “Dracula”. Epibiotic bacterium that preys on anoxygenic photosynthetic gammaproteobacterial species of the genus Halochromatium. Non-flagellated, small flat rounded cells (500–600 nm diameter and 200–250 nm height) that form piles of up to 10 cells attached to the surface of the host. Gram-positive cell wall structure. Complete genome sequence, GenBank/EMBL/DDBJ accession number PRJNA678638.

#biology #vampirococcuslugosii

References

[1] "Reductive evolution and unique infection and feeding mode in the CPR predatory bacterium Vampirococcus lugosii", https://www.nature.com/articles/s41467-021-22762-4

dmm, 4 months ago to quantumcomputing

Interesting commentary if you happen to be interested in these topics...

https://spectrum.ieee.org/quantum-computing-skeptics

#quantumcomputing

dmm, 4 months ago to random

Have a safe and happy Christmas everyone!

dmm, 5 months ago to math

One of my favorite mathematicians, Srinivasa Ramanujan, was born #onthisday in 1887 [1]. Ramanujan had many important and fascinating results in number theory including his famous formula for π shown below. There is also the Hardy–Ramanujan number, 1729, and the wonderful story that goes along with it [2].

Ramanujan has so many important results that would be hard to list even a small fraction of them here. That said, one of my favorite Ramanujan results concerns what are called "continued fractions" and is known as the Rogers–Ramanujan continued fraction [3]. A few of my notes on Ramanujan and nested radicals are here: https://davidmeyer.github.io/qc/nested_radicals.pdf. The LaTeX source is here: https://www.overleaf.com/read/qwhvvhrzrgct. As always, questions/comments/corrections/* greatly appreciated.

There is also a nice movie about Ramanujan's life called "The Man Who Knew Infinity" [4]. Burkard Polster (aka Mathologer) has many interesting videos about Ramanujan and his results, e.g. [5].

Sadly Ramanujan died of tuberculosis in 1920 at the young age of 32.

Read more about Ramanujan's life and times here: https://royalsociety.org/blog/2018/10/revisiting-ramanujan.

#ramanujan #nestedradicals #math #maths #mathematics

[Right image credit: https://in.pinterest.com/SHUBHAMRAJ54/srinivasa-ramanujan]

References

[1] "Srinivasa Ramanujan", https://en.wikipedia.org/wiki/Srinivasa_Ramanujan

[2] "1729_(number)", https://en.wikipedia.org/wiki/1729_(number)

[3] "Rogers–Ramanujan continued fraction", https://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fraction

[4] "The Man Who Knew Infinity", https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity

[5] "Ramanujan: Making sense of 1+2+3+... = -1/12 and Co.", https://www.youtube.com/watch?v=jcKRGpMiVTw

Ramanujan

dmm, 5 months ago to math

A factorion is an interesting kind of natural number that equals the sum of the factorials of its decimal digits [1].

40585 is the largest known factorion and was discovered in 1964 by Leigh Janes [2].

#factorion #math #maths

References

[1] "Factorion", https://en.wikipedia.org/wiki/Factorion

[2] "Mathematician:Leigh Janes", https://proofwiki.org/wiki/Mathematician:Leigh_Janes

dmm, 5 months ago to random

Born #onthisday 208 years ago, Ada Lovelace wrote the world’s first computer program (which calculated Bernoulli numbers) and worked with Charles Babbage on the Analytical engine [1,2,3].

She was also the only legitimate child of poet Lord Byron and Lady Byron [4].

#adalovelace #analyticalengine

References

[1] "Ada Lovelace’s skills with language, music and needlepoint contributed to her pioneering work in computing", https://theconversation.com/ada-lovelaces-skills-with...

[2] "Charles Babbage", https://en.wikipedia.org/wiki/Charles_Babbage

[3] "ADA BYRON, COUNTESS OF LOVELACE", https://www.sdsc.edu/ScienceWomen/lovelace.html

[4] "Ada Lovelace", https://www.biography.com/scholar/ada-lovelace

dmm, 5 months ago to math

Analytic geometry is a fantastic area of mathematics which is populated by all kinds of crazy objects. Being that it is Christmastime (and therefore Christmas math time) please check out the paradoxical object shown on the left. This object is sometimes known as the Infinite Gift. A related object is known as Gabriel's Wedding Cake [7].

In the Infinite Gift the length of the side of the nth box is 1/√n, so the area of one side of the nth box equals (1/√n)² = 1/n. Since a box has 6 sides the surface area of the nth box is 6·(1/n). Then what you find is that in the limit as n → ∞ that the Infinite Gift has infinite surface area but finite volume!

Here's an interesting aside: In the limit the area of the Infinite Gift equals 6 times the harmonic series (which we know diverges).

The continuous version of this object is known as Gabriel’s Horn (aka Torricelli’s Trumpet) and is shown in the figure on the right [1,2]. Gabriel's Horn is the surface of revolution of the function y = 1/x about the x-axis for x ≥ 1 [1,2]. As we can see in image, in the limit Gabriel’s Horn has volume = π and area = ∞.

These properties lead to an interesting situation known as the Painter’s Paradox [3,4].

This is the Painter's Paradox: Somehow even though you can fill Gabriel’s Horn with paint (its volume is finite), you still won’t have enough paint to cover its inside surface (its area is infinite)!

Merry Christmas everyone!

#christmastimeischristmasmathtime #infinitegift #gabrielshorn
#torricellistrumpet #math #maths #analyticgeometry

(1/2)

Gabriel's Horn

dmm, 5 months ago to math

In 1683 the Swiss mathematician Jacob Bernoulli discovered this beautiful expression for the constant e while studying continuous compound interest.

See https://en.wikipedia.org/wiki/Jacob_Bernoulli for more on Bernoulli's work and life.

#math #maths #eulersnumber #compoundinterest #jacobbernoulli

dmm, 5 months ago to machinelearning

This is a nice tutorial if you are interested in how and why overfitting (etc) happens, how/why regularization helps to alleviate overfitting, ...

"The Theory Behind Overfitting, Cross Validation, Regularization, Bagging, and Boosting: Tutorial", Benyamin Ghojogh, Mark Crowley

#machinelearning #overfitting

https://arxiv.org/abs/1905.12787

dmm, 5 months ago to physics

#onthisday in 1915: Albert Einstein submitted a paper to the journal "Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin" that would fundamentally alter our understanding of the universe [1]. The four page paper contained what became known as the Einstein field equations, which relate the geometry of spacetime to the distribution of matter within it [2].

Einstein's field equations were presented in the form of a tensor equation which related the local spacetime curvature (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor) [3].

#einstein #generalrelativity #einsteinfieldequations #physics #maths #maths

[Image credit: https://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/echo/einstein/sitzungsberichte/6E3MAXK4/index.meta]

References

[1] "Die Feldgleichungen der Gravitation", https://einsteinpapers.press.princeton.edu/vol6-doc/273

[2] "Einstein field equations", https://en.wikipedia.org/wiki/Einstein_field_equations

3] "Einstein tensor", https://en.wikipedia.org/wiki/Einstein_tensor

dmm, 5 months ago to math

Did you know that the roots of a cubic polynomial can be visualized using an equilateral triangle?

In this incredibly cool animation from Freya Holmér (@acegikmo):

🔵 the vertices of the triangle map to the roots
🔴 the incenter is the inflection point
🟢 the incircle boundaries are the local minima/maxima

#math #maths #mathematics #cuberooots #cubicpolynomial

video/mp4

dmm, 6 months ago to math

Functional Analysis is a very cool topic which sits between infinite dimensional linear algebra and real and complex analysis (a couple of mind-blowing topics in their own right...).

A few of my (very nascent) notes are here: https://davidmeyer.github.io/qc/functional_analysis.pdf. The LaTeX source is here: https://www.overleaf.com/read/fgrrxmkycvry. As always, questions/comments/corrections/* greatly appreciated.

Happy Thanksgiving everyone!

#math #maths #mathematics #functionalanalysis

dmm, 6 months ago (edited 6 months ago) to pnw

More fall in the #PNW. Looking east down 21st from University in Eugene, OR.