#realanalysis — Public Fediverse posts

https://www.twitch.tv/zapwai
Streaming some Problems and Theorems in Analysis by Pólya & Szegő
#Mathematics #polya #realanalysis

https://www.twitch.tv/zapwai
Streaming some Problems and Theorems in Analysis by Pólya & Szegő
#Mathematics #polya #realanalysis

Keeping busy with Sudoku, Wordle, Crosswords, and mining textbooks for statements to practice proving in #Metamath ...

Inspired by this video playlist on #RealAnalysis #Math

https://www.youtube.com/playlist?list=PLYPU-RArkLZNwMZ55-y8FZZEeY7zCJX59

Which was created from https://www.jirka.org/ra/ _Basic Analysis:
Introduction to Real Analysis_ by Jiří Lebl. #JiříLebl a #CreativeCommons (4.0) free #math #textbook

In Metamath's compressed format, my proofs of strong induction over the natural numbers took 316 bytes, well-ordering of the natural numbers: 376 bytes, 2ⁿ⁻¹ ≤ n! : 1204 bytes, formula for finite sum of geometric series: 2566 bytes. The last has 147 steps, some of which are reused and some of which depend on up to 8 prior steps and on a truncated library of 30477 statements of syntax, axioms, definitions, and theorems from the wider world of Metamath.

Keeping busy with Sudoku, Wordle, Crosswords, and mining textbooks for statements to practice proving in #Metamath ...

Inspired by this video playlist on #RealAnalysis #Math

https://www.youtube.com/playlist?list=PLYPU-RArkLZNwMZ55-y8FZZEeY7zCJX59

Which was created from https://www.jirka.org/ra/ _Basic Analysis:
Introduction to Real Analysis_ by Jiří Lebl. #JiříLebl a #CreativeCommons (4.0) free #math #textbook

In Metamath's compressed format, my proofs of strong induction over the natural numbers took 316 bytes, well-ordering of the natural numbers: 376 bytes, 2ⁿ⁻¹ ≤ n! : 1204 bytes, formula for finite sum of geometric series: 2566 bytes. The last has 147 steps, some of which are reused and some of which depend on up to 8 prior steps and on a truncated library of 30477 statements of syntax, axioms, definitions, and theorems from the wider world of Metamath.

A FORGOTTEN EPISODE in French-occupied Naples in the years around 1800—just after the French Revolution—illustrates why it makes sense to see mathematics and politics as entangled. The protagonists of this story were gravely concerned about how mainstream mathematical methods were transforming their world—somewhat akin to our current-day concerns about how digital algorithms are transforming ours. But a key difference was their straightforward moral and political reading of those mathematical methods. By contrast, in our own era we seem to think that mathematics offers entirely neutral tools for ordering and reordering the world—we have, in other words, forgotten something that was obvious to them.

In this essay, I’ll use the case of revolutionary Naples to argue that the rise of a new and allegedly neutral mathematics—characterized by rigor and voluntary restriction—was a mathematical response to pressing political problems. Specifically, it was a response to the question of how to stabilize social order after the turbulence of the French Revolution. Mathematics, I argue, provided the logical infrastructure for the return to order. This episode, then, shows how and why mathematical concepts and methods are anything but timeless or neutral; they define what “reason” is, and what it is not, and thus the concrete possibilities of political action. The technical and political are two sides of the same coin—and changes in notions like mathematical rigor, provability, and necessity simultaneously constitute changes in our political imagination.

#Mathematics #Math #Analysis #MassimoMazzotti #LAReviewOfBooks #Epistemology #Revolution #RealAnalysis #HistoryOfMath #HistoryOfMathematics

https://lareviewofbooks.org/article/foundational-anxieties-modern-mathematics-and-the-political-imagination/

A FORGOTTEN EPISODE in French-occupied Naples in the years around 1800—just after the French Revolution—illustrates why it makes sense to see mathematics and politics as entangled. The protagonists of this story were gravely concerned about how mainstream mathematical methods were transforming their world—somewhat akin to our current-day concerns about how digital algorithms are transforming ours. But a key difference was their straightforward moral and political reading of those mathematical methods. By contrast, in our own era we seem to think that mathematics offers entirely neutral tools for ordering and reordering the world—we have, in other words, forgotten something that was obvious to them.

In this essay, I’ll use the case of revolutionary Naples to argue that the rise of a new and allegedly neutral mathematics—characterized by rigor and voluntary restriction—was a mathematical response to pressing political problems. Specifically, it was a response to the question of how to stabilize social order after the turbulence of the French Revolution. Mathematics, I argue, provided the logical infrastructure for the return to order. This episode, then, shows how and why mathematical concepts and methods are anything but timeless or neutral; they define what “reason” is, and what it is not, and thus the concrete possibilities of political action. The technical and political are two sides of the same coin—and changes in notions like mathematical rigor, provability, and necessity simultaneously constitute changes in our political imagination.

#Mathematics #Math #Analysis #MassimoMazzotti #LAReviewOfBooks #Epistemology #Revolution #RealAnalysis #HistoryOfMath #HistoryOfMathematics

https://lareviewofbooks.org/article/foundational-anxieties-modern-mathematics-and-the-political-imagination/

I recently dumped my notes on modulus of convergence for hypergeometric functions on my website. I also had some thoughts on numerical accuracy. Not very valuable thoughts, but thoughts nonetheless.

For those who are wondering why anyone would care, many math libraries such as GSL, boost, and so on, suck. If you are trying to do intense calculations with any sort of decent accuracy, these libraries have dusty corners that fail. And they don't document where those corners are. Or they don't have routines for complex arguments. I write my own routines for these cases.

#Math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/numerical-accuracy-of-generalized-hypergeometric-series

I recently dumped my notes on modulus of convergence for hypergeometric functions on my website. I also had some thoughts on numerical accuracy. Not very valuable thoughts, but thoughts nonetheless.

For those who are wondering why anyone would care, many math libraries such as GSL, boost, and so on, suck. If you are trying to do intense calculations with any sort of decent accuracy, these libraries have dusty corners that fail. And they don't document where those corners are. Or they don't have routines for complex arguments. I write my own routines for these cases.

#Math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/numerical-accuracy-of-generalized-hypergeometric-series

A few years ago (mid 2023), I wrote up some research notes regarding the modulus of convergence of the generalized hypergeometric functions. This month I wrote up and posted a series of articles that are those notes, cleaned up a bit, and this article is the wrap-up of the series.

#Math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/modulus-of-convergence-for-generalized-hypergeometric-functions

A few years ago (mid 2023), I wrote up some research notes regarding the modulus of convergence of the generalized hypergeometric functions. This month I wrote up and posted a series of articles that are those notes, cleaned up a bit, and this article is the wrap-up of the series.

#Math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/modulus-of-convergence-for-generalized-hypergeometric-functions

We all get the feeling that, day by day, the world is converging towards disaster. But what tells us how fast? The Modulus of Convergence does!

#Math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/modulus-of-convergence

We all get the feeling that, day by day, the world is converging towards disaster. But what tells us how fast? The Modulus of Convergence does!

#Math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/modulus-of-convergence

In mathematics, we say that a function is bounded if can restrict its image. Oddly, we never seem to 'bind' a function, though. I can find bounds on the the remainder of generalized hypergeometric functions, and I never used the word 'bind' either. Maybe 'bounding' refers to bunnies and deer?

#math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/remainder-of-the-conway-maxwell-poisson-distribution-normalization-factor

In mathematics, we say that a function is bounded if can restrict its image. Oddly, we never seem to 'bind' a function, though. I can find bounds on the the remainder of generalized hypergeometric functions, and I never used the word 'bind' either. Maybe 'bounding' refers to bunnies and deer?

#math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/remainder-of-the-conway-maxwell-poisson-distribution-normalization-factor

The generalized hypergeometric series are ubiquitous in the world of computing special functions, for certain amounts of ubiquity. Turns out the speed of convergence is related to the obscure Conway-Maxwell-Poisson distribution, which no one has ever heard of - pretty much the opposite of ubiquity.

#math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/bounding-the-remainder-of-generalized-hypergeometric-series

The generalized hypergeometric series are ubiquitous in the world of computing special functions, for certain amounts of ubiquity. Turns out the speed of convergence is related to the obscure Conway-Maxwell-Poisson distribution, which no one has ever heard of - pretty much the opposite of ubiquity.

#math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/bounding-the-remainder-of-generalized-hypergeometric-series

Accurately computing generalized hypergeometric functions is hard. How many terms do we need? Guess we need a general expression for the size of the terms in the series. Oh, wait, I've got one right here!

#math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/estimating-the-terms-of-generalized-hypergeometric-series

Accurately computing generalized hypergeometric functions is hard. How many terms do we need? Guess we need a general expression for the size of the terms in the series. Oh, wait, I've got one right here!

#math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/estimating-the-terms-of-generalized-hypergeometric-series

Making Magic with MCP: From Data Retrieval to Real Analysis and Insights

https://jellyfish.co/blog/how-ably-makes-magic-with-jellyfishs-mcp/

#HackerNews #Making #Magic #with #MCP #DataRetrieval #RealAnalysis #Insights #Innovation

Making Magic with MCP: From Data Retrieval to Real Analysis and Insights

https://jellyfish.co/blog/how-ably-makes-magic-with-jellyfishs-mcp/

#HackerNews #Making #Magic #with #MCP #DataRetrieval #RealAnalysis #Insights #Innovation

#realanalysis #complexanalysis #complexnumbers

On the first use of decimal points: https://www.npr.org/2024/02/24/1233702474/the-decimal-point-was-in-use-150-years-before-previously-thought-research-shows

My discomfort with the decimal / base 10 system is...it's too popular. For too many people, it's our only conceptual way of understanding real numbers.

I remember taking real analysis as an undergrad, and not really getting it. It took me years until I could pry my brain away from conceiving of real numbers as something different from their representation as decimal expansions.

For example, people say π "is" 3.14159...and that it "goes on forever". Well, no: the sequence of symbols that starts with 3.14159 is one way of *representing* π, and it's only that representation that goes on forever; as a number, π is, well: π.

If I ever get the chance to teach undergrad real analysis, I want to focus the course on showing students just what the real numbers (and functions thereof) *are*, and how unspeakably strange they are.

#math #realanalysis #decimals

On the first use of decimal points: https://www.npr.org/2024/02/24/1233702474/the-decimal-point-was-in-use-150-years-before-previously-thought-research-shows

My discomfort with the decimal / base 10 system is...it's too popular. For too many people, it's our only conceptual way of understanding real numbers.

I remember taking real analysis as an undergrad, and not really getting it. It took me years until I could pry my brain away from conceiving of real numbers as something different from their representation as decimal expansions.

For example, people say π "is" 3.14159...and that it "goes on forever". Well, no: the sequence of symbols that starts with 3.14159 is one way of *representing* π, and it's only that representation that goes on forever; as a number, π is, well: π.

If I ever get the chance to teach undergrad real analysis, I want to focus the course on showing students just what the real numbers (and functions thereof) *are*, and how unspeakably strange they are.

#math #realanalysis #decimals

So is the only difference between a Darboux sum and a Riemann sum how you choose to calculate the height of a rectangle? I’m thinking there must be more to the story. #ITeachMath #Mathematics #RealAnalysis

So is the only difference between a Darboux sum and a Riemann sum how you choose to calculate the height of a rectangle? I’m thinking there must be more to the story. #ITeachMath #Mathematics #RealAnalysis

I always struggle with epsilon-delta style proofs 💀

what are some things people do to make these more palatable? #calculus #realanalysis

Video lectures for a #RealAnalysis class that I've taught at #UCLA:

Feel free to e-mail me if you are interested in the problem sets and other study materials.

https://youtube.com/playlist?list=PL54Pt_mZzBqjcodJ_GqXxqG1WCCmq433k

Video lectures for a #RealAnalysis class that I've taught at #UCLA:

Feel free to e-mail me if you are interested in the problem sets and other study materials.

https://youtube.com/playlist?list=PL54Pt_mZzBqjcodJ_GqXxqG1WCCmq433k

For some reason there's a small, almost trivial early result in #RealAnalysis that I really like:

Every bounded infinite collection of rational numbers has an accumulation point.

Is there a similar, simple, elegant result that you like?