Studying Mathematics Thread

[polymath257]

Is there any mathematician who chose Fractals as his specialy? Hard to imagine anyone who wants to dive into that madness

Other than Mandelbrot himself? Oh yes. At least one difficulty is that there is no agreed upon definition of what it means to be a fractal. yes, self-similarity is one key, but the notion of Hausdorff (fractal) dimension is also widely recognized. The two lead to different notions, however.

Here's an actual mathematics book:


https://www.amazon.com/Geometry-Fractal-...s+falconer

There are many unanswered mathematical questions even concerning the Mandlebrot set: for example, is it locally connected?

Here are some recent papers:
https://scholar.google.com/scholar?hl=en...nsion&btnG=

In any case, fractals seem to be less 'in vogue' than they were 20 years ago, but still being actively studied in various ways.

[Jehanne]

Is there any mathematician who chose Fractals as his specialy? Hard to imagine anyone who wants to dive into that madness

Other than Mandelbrot himself? Oh yes. At least one difficulty is that there is no agreed upon definition of what it means to be a fractal. yes, self-similarity is one key, but the notion of Hausdorff (fractal) dimension is also widely recognized. The two lead to different notions, however.

Here's an actual mathematics book:


https://www.amazon.com/Geometry-Fractal-...s+falconer

There are many unanswered mathematical questions even concerning the Mandlebrot set: for example, is it locally connected?

Here are some recent papers:
https://scholar.google.com/scholar?hl=en...nsion&btnG=

In any case, fractals seem to be less 'in vogue' than they were 20 years ago, but still being actively studied in various ways.

Problem is that this stuff is really, really hard, not only to learn but to actually improve upon the work of others.  My problem was that I had a love of physics and math; just was not particularly good at either subject.  Still, got through 3-semesters of calculus and taught myself ordinary and partial differential equations with a little tensor calculus over the years.  They are beautiful subjects, and I see no reason to believe that what modern science observes in the heavens and on earth is just the product of invisible angelic beings who are the true cause of particle motion, whether it be atomic, microscopic or macroscopic.

The Universe can be modeled just fine with higher math, although, just because something is correctly mathematically does not mean that such corresponds to physical reality.

[polymath257]

Other than Mandelbrot himself? Oh yes. At least one difficulty is that there is no agreed upon definition of what it means to be a fractal. yes, self-similarity is one key, but the notion of Hausdorff (fractal) dimension is also widely recognized. The two lead to different notions, however.

Here's an actual mathematics book:


https://www.amazon.com/Geometry-Fractal-...s+falconer

There are many unanswered mathematical questions even concerning the Mandlebrot set: for example, is it locally connected?

Here are some recent papers:
https://scholar.google.com/scholar?hl=en...nsion&btnG=

In any case, fractals seem to be less 'in vogue' than they were 20 years ago, but still being actively studied in various ways.

Problem is that this stuff is really, really hard, not only to learn but to actually improve upon the work of others.  My problem was that I had a love of physics and math; just was not particularly good at either subject.  Still, got through 3-semesters of calculus and taught myself ordinary and partial differential equations with a little tensor calculus over the years.  They are beautiful subjects, and I see no reason to believe that what modern science observes in the heavens and on earth is just the product of invisible angelic beings who are the true cause of particle motion, whether it be atomic, microscopic or macroscopic.

The Universe can be modeled just fine with higher math, although, just because something is correctly mathematically does not mean that such corresponds to physical reality.

Yes, research in math is hard.

The concept of a fractal dimension goes back to Hausdorff and is around a century old. Using various forms of iteration to produce rather complicated sets is even a bit older (the Cantor ternary set is, perhaps, the first example of a set now recognizable as a fractal).

All I can say is that a lot depends on your definitions. And there are several competing definitions for what constitutes a 'fractal'. But iterative methods involving self-similarity are widespread in math.

[Jehanne]

Problem is that this stuff is really, really hard, not only to learn but to actually improve upon the work of others.  My problem was that I had a love of physics and math; just was not particularly good at either subject.  Still, got through 3-semesters of calculus and taught myself ordinary and partial differential equations with a little tensor calculus over the years.  They are beautiful subjects, and I see no reason to believe that what modern science observes in the heavens and on earth is just the product of invisible angelic beings who are the true cause of particle motion, whether it be atomic, microscopic or macroscopic.

The Universe can be modeled just fine with higher math, although, just because something is correctly mathematically does not mean that such corresponds to physical reality.

Yes, research in math is hard.

The concept of a fractal dimension goes back to Hausdorff and is around a century old. Using various forms of iteration to produce rather complicated sets is even a bit older (the Cantor ternary set is, perhaps, the first example of a set now recognizable as a fractal).

All I can say is that a lot depends on your definitions. And there are several competing definitions for what constitutes a 'fractal'. But iterative methods involving self-similarity are widespread in math.

It is interesting because what is taught at the entire undergraduate level is mostly 19th-century math and science with some early 20th-century stuff tossed in for the junior and senior level students.  I have read that PhD students in math start learning the 1950s material around their 2nd or 3rd year of graduate studies.

[polymath257]

Yes, research in math is hard.

The concept of a fractal dimension goes back to Hausdorff and is around a century old. Using various forms of iteration to produce rather complicated sets is even a bit older (the Cantor ternary set is, perhaps, the first example of a set now recognizable as a fractal).

All I can say is that a lot depends on your definitions. And there are several competing definitions for what constitutes a 'fractal'. But iterative methods involving self-similarity are widespread in math.

It is interesting because what is taught at the entire undergraduate level is mostly 19th-century math and science with some early 20th-century stuff tossed in for the junior and senior level students.  I have read that PhD students in math start learning the 1950s material around their 2nd or 3rd year of graduate studies.

Actually, most of the undergraduate math is 18th century or before. The one exception at the 'lower' levels is linear algebra. If you get to the junior/senior classes, you get into the 19th century with groups and rings as well as some of the material on epsilon-delta proofs.

And yes, getting past 1950 or so is pretty much limited to PhD students, although some of the material relevant to theoretical computer science gets past that mark.

[Jehanne]

It is interesting because what is taught at the entire undergraduate level is mostly 19th-century math and science with some early 20th-century stuff tossed in for the junior and senior level students.  I have read that PhD students in math start learning the 1950s material around their 2nd or 3rd year of graduate studies.

Actually, most of the undergraduate math is 18th century or before. The one exception at the 'lower' levels is linear algebra. If you get to the junior/senior classes, you get into the 19th century with groups and rings as well as some of the material on epsilon-delta proofs.

And yes, getting past 1950 or so is pretty much limited to PhD students, although some of the material relevant to theoretical computer science gets past that mark.

I have read that the modern-day definition of the limit in calculus was near the last quarter of the 19th-century, with the last book (other than the University of Wisconsin's foray back in time) on infinitesimals being around 1915.

[polymath257]

Actually, most of the undergraduate math is 18th century or before. The one exception at the 'lower' levels is linear algebra. If you get to the junior/senior classes, you get into the 19th century with groups and rings as well as some of the material on epsilon-delta proofs.

And yes, getting past 1950 or so is pretty much limited to PhD students, although some of the material relevant to theoretical computer science gets past that mark.

I have read that the modern-day definition of the limit in calculus was near the last quarter of the 19th-century, with the last book (other than the University of Wisconsin's foray back in time) on infinitesimals being around 1915.

Cauchy did his stuff in the early part of the 19th. By mid-century, the epsilon-delta definition was standard among mathematicians (if not among others).

[Silver]

[Jehanne]

I have read that the modern-day definition of the limit in calculus was near the last quarter of the 19th-century, with the last book (other than the University of Wisconsin's foray back in time) on infinitesimals being around 1915.

Cauchy did his stuff in the early part of the 19th. By mid-century, the epsilon-delta definition was standard among mathematicians (if not among others).

Some final touches, apparently, a century ago:

Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime (Felscher 2000). Cauchy discussed variable quantities, infinitesimals, and limits and defined continuity of y = f ( x ) {\displaystyle y=f(x)} by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y in his 1821 book Cours d'analyse, while (Grabiner 1983) claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0 (Burton 1997).

The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908 (Miller 2004).

I lament the fact, though, that many modern calculus texts no longer have a separate, concluding chapter on ODEs. A step backward in my opinion.

[polymath257]

Cauchy did his stuff in the early part of the 19th. By mid-century, the epsilon-delta definition was standard among mathematicians (if not among others).

Some final touches, apparently, a century ago:

Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime (Felscher 2000). Cauchy discussed variable quantities, infinitesimals, and limits and defined continuity of y = f ( x ) {\displaystyle y=f(x)} by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y in his 1821 book Cours d'analyse, while (Grabiner 1983) claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0 (Burton 1997).

The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908 (Miller 2004).

I lament the fact, though, that many modern calculus texts no longer have a separate, concluding chapter on ODEs.  A step backward in my opinion.

And conic sections appear to have been eliminated from many curricula. Another sad fact.