math

I’ve tutored it from middle school level up to helping with dissertation work. The way it is taught varies so dramatically from class to class and field to field. There’s so many subtle and weird factors in interpretation that it’s honestly the most stressful thing to tutor.

With a newly discovered mathematical tool, researchers are hoping to gain unprecedented insight into the structure of complex knots.

Saw another few of these today and thought I'd collect related links somewhere. Comment with any that have been useful to you!

• T. Tao on career and writing
• S. Billey's advice collection
• I. Pak's blog
• C. Aten's 'one pager' for all stages in math
• University of michigan grad advice (note: use the dropdown menu at the top; links in main content are broken atm)
• ICERM professional development materials
• AMS profiles of women mathematicians
• I. Musson's LGBT resources links
• Academic Stack exchange on postdocs

not math specific, but grad/research related:

• thesis whisperer
• care and maintenance of your advisor
• (comp-sci inclined) the huge advice collection maintained by Tao Xie and Yuan Xie (I surely duplicate some of their items)

I am so fascinated by the regions where the number of lines to cover the primes levels out, what Brady calls "golden lines". Part 2 is here https://www.youtube.com/watch?v=u-_8wX4cECo

I remember back in 2007 feeling that my TI-89 was almost unfair. Pretty sure I used it for the ACT, and you can solve something like 60% of problems with no effort if you just learn how to graph. And now you can use Desmos on the ACT.

Nowadays, most students in higher level math are equipped with calculators that can just solve things. You don’t need to learn how to convert fractions to decimals, or work with a percentage, or even do algebra (80% of working with calc 1 students can often be “yeah here’s how to put it into solver”).

I’m very torn on this. On one hand, I think that doing it by hand is the only way to develop on understanding of what it all means. There’s patterns to what a base system mean that you start to “get” once you’ve done enough borrows and carries. Small and consistent practice in the small skills adds resonance to the major skills you are building to.

On the other, there are things like dysgraphia that just there’s no reason to not work around. Some people can’t hold onto times tables. There are amazing ways to do multiplication that are slow but work for people (draw a rectangle - 3x4 best for demo purposes - have person count squares, bam, you have now outdone their 2nd grade math teacher.) Why bar someone who can’t memorize things but can understand why things work from further study of math?

I do sorta wish that the SAT kept its no calculator section though. It would be interesting to make a bunch of adults take the modern tests and compare their scores to they got 20+ years ago…

cross-posted from: https://lemmy.sdf.org/post/37414239

I've read the old papers proving that fact, but honestly it seems like some of the terminology and notation has changed since the 70's, and I roundly can't make heads or tails of it. The other sources I can find are in textbooks that I don't own.

Ideally, what I'm hoping for is a segment of pseudocode or some modern language that generates an n-character string from some kind of seed, which then cannot be recognised in linear time.

It's of interest to me just because, coming from other areas of math where inverting a bijective function is routine, it's highly unintuitive that you provably can't sometimes in complexity theory.

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Extremely similar to tikz syntax for simple things, but it does have much more 'normal' variables, loops, etc. See the lovely and extensive tutorial by Staats.

I’m studying concrete mathematics by Donald Knuth and I find myself consulting ChatGPT quite a lot. It’s an effective study partner in my opinion if used sincerely.

However I feel this approach is missing a core element of mathematics which is collaboration with other people.

I’ve tried to look on the main maths discord channel but the society is too big and my queries get drowned out. Is there a book specific forum out there that I can find targeted musings on the questions I’m trying to solve?

Perhaps this is an opportunity to start my own server?

Just letting my mind wonder a bit but requirements for such a server would be:

• specific books
• question numbers
• maybe those two can be achieved with tagging
• latex integration

You’d also imagine that to get a server like this to actually take off we must all decide we’re studying the same book

This is pretty exciting! Even my understanding of the blog is not fully yet, but overall seems intriguing. Here's a direct link to the article https://arxiv.org/pdf/2210.09581

Joseph Fourier was a French mathematician, born on 21 March 1768 in Auxerre, France. His name is closely associated with the Fourier series and Fourier transform, mathematical tools utilised in a wide range of applications. He is also one of the 72 scientists and mathematicians whose names are engraved on the Eiffel Tower.

Fourier is celebrated for his extraordinary analytical skills and his pioneering theory of heat, which opened up an entirely new branch of physics and mathematics.

According to Cédric Villani, a mathematician and Fields Medal laureate, Fourier's work placed him at the forefront of scientific innovation.

“For the generations after him, Fourier remained the emblematic hero of mathematical physics. Some people even say he's the creator of mathematical physics,” Villani said.

Fourier was among the first mathematicians to express heat phenomena using mathematical equations. His groundbreaking work, presented in his 1822 publication The Analytical Theory of Heat, demonstrated how heat flows through materials, using what are now known as Fourier series and Fourier transforms.

These tools have since become indispensable in physics, engineering, and applied mathematics.

Fourier famously stated in Latin, "Fire obeys the laws of numbers," encapsulating his belief in the universality of mathematical principles.

His methods and equations extended far beyond the study of heat, finding applications in diverse areas such as signal processing, climate science, and quantum mechanics.

Villani noted that Lord Kelvin, another prominent mathematician and physicist, described Fourier’s contributions as "a great mathematical poem," emphasising their elegance and profound impact. Fourier’s theory of heat conduction not only revolutionised physics but also laid the foundation for mathematical physics as a discipline.

Fourier’s influence was not confined to academia. In 1802, he was appointed prefect of the Isère regional district under Napoleon's administration. During his tenure, he contributed significantly to the region's infrastructure, including supporting the construction of roads and canals.

Despite his remarkable achievements, Fourier’s life was tragically cut short. He died in 1830 at the age of 62, leaving behind a legacy that continues to shape modern science and mathematics.

I’m gathering data for a hobby project and notating where the triangles in the data correspond to Pythagorean triples, but sometimes it doesn’t seem clear to me with certain data.

Can there exist Pythagorean triples in which the leg lengths are not coprime with each other but both are coprime with the hypotenuse? i.e., a right triangle in which (leg~1~, leg~2~, hypotenuse) = (a * n, b * n, c), in which a, b, c, and n are whole numbers and n is not a factor of c?

How can I determine if a right triangle with given lengths can scale to be a Pythagorean triple? If any of the values in (leg~1~: leg~2~: hypotenuse) are irrational, that does indeed mean the values cannot scale to be whole numbers?

Once it is determined that the triangle can scale to a Pythagorean triple, what is the best method of scaling the values to three whole numbers?

Thanks for any help

Edit: I’ve found an effective way to determine primitive Pythagorean triples from given leg lengths. Using a calculator that can output in fractional form, such as wolfram alpha, input leg~1~ / leg~2~ and the output will be a fraction with the numerator and denominator denoting the leg lengths of a primitive Pythagorean triple. Determining the hypotenuse is then simply using the Pythagorean Theorem.

• Breaking Math

Breaking Math Podcast is the #1 Ranked Podcast in Math in the US & UK since 2016. We’ve worked with world famous mathematicians, cartoonists, and authors to bring you the rabbit hole, niche topics you love.

Hosted by Gabriel Hesch and Autumn Phaneuf, who have advanced degrees in electrical engineering and industrial engineering/operations research respectively, come together to discuss mathematics as a pure field all in its own as well as how it describes the language of science, engineering, and even creativity.  

https://www.breakingmath.io/

https://feeds.zencastr.com/f/FisVYUQH.rss

• New Books in Mathematics

By Marshall Poe

Interviews with Mathematicians about their New Books

https://feeds.megaphone.fm/LIT8989774512

• Math Ed Podcast

Interviews with mathematics education researchers about recent studies. Hosted by Samuel Otten, University of Missouri.

www.mathedpodcast.com

https://mathed.podomatic.com/rss2.xml

For those who don't know this book of Topology is rated very positively and Free ! Some professors teaching this branch of Mathematic may be able to confirm this.

Topology is an important and interesting area of mathematics, the study of which will not only introduce you to new concepts and theorems but also put into context old ones like continuous functions. However, to say just this is to understate the significance of topology. It is so fundamental that its influence is evident in almost every other branch of mathematics. This makes the study of topology relevant to all who aspire to be mathematicians whether their first love is (or will be) algebra, analysis, category theory, chaos, continuum mechanics, dynamics, geometry, industrial mathematics, mathematical biology, mathematical economics, mathematical finance, mathematical modelling, mathematical physics, mathematics of communication, number theory, numerical mathematics, operations research or statistics. Topological notions like compactness, connectedness and denseness are as basic to mathematicians of today as sets and functions were to those of last century.

For the reader who has not previously studied an axiomatic branch of mathematics such as abstract algebra, learning to write proofs will be a hurdle. To assist you to learn how to write proofs, in the early chapters I often include an aside which does not form part of the proof but outlines the thought process which led to the proof.

Since 1985 with the last update dating from this year... Amazing commitment !