The Math Thread

[lpetrich]

I'll try to determine its general rule.

11, 2(12), 2(13)+(22), 2(14)+2(23), 2(15)+2(24)+(33), 2(16)+2(25)+2(34) over 1,2,3,4,5,6, ...

In other words, for positive-integer k, is there a set of a_k all
\( \frac1n \sum_{k=1}^{n} \, a_k a_{n-k+1} \)

are equal?

Let each one's value be S_n, and let us find the solution by folding the values together by multiplying by tⁿ and adding:
\( \sum_n n S_n t^n = A(t)^2 \)
where
\( A(t) = \sum_n a_n t^n = \sqrt{ \sum_n n S_n t^n } \)

If all the S_n's equal S₀, then
\( \sum_n n S_n t^n = S_0 \sum_n n t^n = \frac{S_0}{(1 - t)^2} \)

That gives A(t) = a0/(1 - t) or a_n = a0 -- all equal.

 

[rousseau]

So at some point I'd like to read this book because I figure it'd be a neat exercise, and provide a bit of novelty.

However, I wonder if there are any other mathematical topics out there that might be of interest to someone who's into history, sociology, biology, topics more directly related to human experience.

Basically I'm interested in better understanding people by any means, and math seems like one of those fields that I'm completely ignorant of, outside of some introductory calculus.

 

[beero1000]

So at some point I'd like to read this book because I figure it'd be a neat exercise, and provide a bit of novelty.

However, I wonder if there are any other mathematical topics out there that might be of interest to someone who's into history, sociology, biology, topics more directly related to human experience.

Basically I'm interested in better understanding people by any means, and math seems like one of those fields that I'm completely ignorant of, outside of some introductory calculus.

I've read that book (and know the authors) and while it's good, at only around 350 pages it's intended to be a (very, very) fast recap of main concepts for people going to grad school in math (i.e. those already familiar with the main concepts and who just need a refresher). If you've only had intro calculus, spending 20 pages on material that normally takes a full semester (or more) might not have much benefit other than as a light reference text, so it might not be the best fit for what you're looking for. A fuller reference would be 1000+ pages, like Kolmogorov's Content, Meaning, and Methods, or the Princeton Companions.

If you wanted to really learn the material, I think you'd probably be better off getting standalone introductory books on a few important topics and reading those. Start with linear algebra, calculus, and statistics. Throw in some graph theory and differential equations and you should be good to go for most applied math topics.

 

[steve_bank]

So at some point I'd like to read this book because I figure it'd be a neat exercise, and provide a bit of novelty.

However, I wonder if there are any other mathematical topics out there that might be of interest to someone who's into history, sociology, biology, topics more directly related to human experience.

Basically I'm interested in better understanding people by any means, and math seems like one of those fields that I'm completely ignorant of, outside of some introductory calculus.

Seriously, understanding why 1 + 1 = 2. It underlies all math.

Some here can talk to it, hopefully without being too pedantic.

 

[bilby]

Some here can talk to it,

*about it

hopefully without being too pedantic.

Oops. Too late.

 

[rousseau]

So at some point I'd like to read this book because I figure it'd be a neat exercise, and provide a bit of novelty.

However, I wonder if there are any other mathematical topics out there that might be of interest to someone who's into history, sociology, biology, topics more directly related to human experience.

Basically I'm interested in better understanding people by any means, and math seems like one of those fields that I'm completely ignorant of, outside of some introductory calculus.

I've read that book (and know the authors) and while it's good, at only around 350 pages it's intended to be a (very, very) fast recap of main concepts for people going to grad school in math (i.e. those already familiar with the main concepts and who just need a refresher). If you've only had intro calculus, spending 20 pages on material that normally takes a full semester (or more) might not have much benefit other than as a light reference text, so it might not be the best fit for what you're looking for. A fuller reference would be 1000+ pages, like Kolmogorov's Content, Meaning, and Methods, or the Princeton Companions.

If you wanted to really learn the material, I think you'd probably be better off getting standalone introductory books on a few important topics and reading those. Start with linear algebra, calculus, and statistics. Throw in some graph theory and differential equations and you should be good to go for most applied math topics.

Yea, that sounds about right. That specific text was picked for no real rhyme or reason except that I wanted to process a bit of math in my head, really in any form. A challenge for my brain, if nothing else.

Sounds like I'll have to commit some time, research a bit more, and find a few more books to throw on my e-reader.

 

[beero1000]

So at some point I'd like to read this book because I figure it'd be a neat exercise, and provide a bit of novelty.

However, I wonder if there are any other mathematical topics out there that might be of interest to someone who's into history, sociology, biology, topics more directly related to human experience.

Basically I'm interested in better understanding people by any means, and math seems like one of those fields that I'm completely ignorant of, outside of some introductory calculus.

I've read that book (and know the authors) and while it's good, at only around 350 pages it's intended to be a (very, very) fast recap of main concepts for people going to grad school in math (i.e. those already familiar with the main concepts and who just need a refresher). If you've only had intro calculus, spending 20 pages on material that normally takes a full semester (or more) might not have much benefit other than as a light reference text, so it might not be the best fit for what you're looking for. A fuller reference would be 1000+ pages, like Kolmogorov's Content, Meaning, and Methods, or the Princeton Companions.

If you wanted to really learn the material, I think you'd probably be better off getting standalone introductory books on a few important topics and reading those. Start with linear algebra, calculus, and statistics. Throw in some graph theory and differential equations and you should be good to go for most applied math topics.

Yea, that sounds about right. That specific text was picked for no real rhyme or reason except that I wanted to process a bit of math in my head, really in any form. A challenge for my brain, if nothing else.

Sounds like I'll have to commit some time, research a bit more, and find a few more books to throw on my e-reader.

I'm all for it. Let me know if I can help.

 

[beero1000]

So at some point I'd like to read this book because I figure it'd be a neat exercise, and provide a bit of novelty.

However, I wonder if there are any other mathematical topics out there that might be of interest to someone who's into history, sociology, biology, topics more directly related to human experience.

Basically I'm interested in better understanding people by any means, and math seems like one of those fields that I'm completely ignorant of, outside of some introductory calculus.

Seriously, understanding why 1 + 1 = 2. It underlies all math.

Some here can talk to it, hopefully without being too pedantic.

Russell and Whitehead, in attempting to axiomatize a consistent and complete mathematical system, finally got around to proving that 1 + 1 = 2 on page 86.... of volume 2. They comment that "The above proposition is occasionally useful. It is used at least three times..."

 

[Kharakov]

Has anyone ever attempted a "smooth" proof of why 1+1=2?

I'm imagining using the number of gradient direction reversals that one crosses while traveling along a path, but can't really think of a reason "why" crossing 2 gradients would mean you crossed 2 gradients, without inputting information into a mathematical system.

 

[beero1000]

Has anyone ever attempted a "smooth" proof of why 1+1=2?

I'm imagining using the number of gradient direction reversals that one crosses while traveling along a path, but can't really think of a reason "why" crossing 2 gradients would mean you crossed 2 gradients, without inputting information into a mathematical system.

I would say that 1 + 1 = 2 is more foundational than ideas of 'smoothness' or 'gradient' or even 'direction'. Any attempt at a proof using those would need to take real care to not be circular.

 

[steve_bank]

So at some point I'd like to read this book because I figure it'd be a neat exercise, and provide a bit of novelty.

However, I wonder if there are any other mathematical topics out there that might be of interest to someone who's into history, sociology, biology, topics more directly related to human experience.

Basically I'm interested in better understanding people by any means, and math seems like one of those fields that I'm completely ignorant of, outside of some introductory calculus.

Seriously, understanding why 1 + 1 = 2. It underlies all math.

Some here can talk to it, hopefully without being too pedantic.

Russell and Whitehead, in attempting to axiomatize a consistent and complete mathematical system, finally got around to proving that 1 + 1 = 2 on page 86.... of volume 2. They comment that "The above proposition is occasionally useful. It is used at least three times..."

I remember in the last century there was a systematic review of the foundations to check for any ambiguities and inconsistencies.

I was thinkig proof by induction and Peano's arithmetic. I know about it, but little details. Most use arithmetic without questioning if it can ever be wrong.

 

[beero1000]

Russell and Whitehead, in attempting to axiomatize a consistent and complete mathematical system, finally got around to proving that 1 + 1 = 2 on page 86.... of volume 2. They comment that "The above proposition is occasionally useful. It is used at least three times..."

I remember in the last century there was a systematic review of the foundations to check for any ambiguities and inconsistencies.

I was thinkig proof by induction and Peano's arithmetic. I know about it, but little details. Most use arithmetic without questioning if it can ever be wrong.

That they do.

Showing that 1 + 1 = 2 is a little anticlimactic though, but here goes.

1. The number 0 exists axiomatically, as does the equality relation =, and a successor function S. (See  Peano's axioms)
2. The number 1 is defined to be the successor of 0, i.e. S(0) = 1
3. The number 2 is defined to be the successor of 1, i.e. S(1) = 2
4. For any number n, n + 0 is defined to be n
5. For any numbers n and m, n + S(m) is defined to be S(n + m)

Then, 1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2

 

[Kharakov]

Has anyone ever attempted a "smooth" proof of why 1+1=2?

I'm imagining using the number of gradient direction reversals that one crosses while traveling along a path, but can't really think of a reason "why" crossing 2 gradients would mean you crossed 2 gradients, without inputting information into a mathematical system.

I would say that 1 + 1 = 2 is more foundational than ideas of 'smoothness' or 'gradient' or even 'direction'. Any attempt at a proof using those would need to take real care to not be circular.

Ok, you can't count gradient changes without 1+1=2... although wouldn't there have to be some distinct boundary perception to have knowledge of more than 1?

I'd think a boundary condition knowledge/belief must exist before there can be a 1 and another 1.


Is 1+1=2 is more foundational than 1+1 > 1?


Also, is there a name for division of infinite series? I use a method that uses diagonals, but I can't find literature on it to see if there are other methods. I'll write mine out when I figure out the latex code for tables (again).

 

[beero1000]

Ok, you can't count gradient changes without 1+1=2... although wouldn't there have to be some distinct boundary perception to have knowledge of more than 1?

I'd think a boundary condition knowledge/belief must exist before there can be a 1 and another 1.


Is 1+1=2 is more foundational than 1+1 > 1?


Also, is there a name for division of infinite series? I use a method that uses diagonals, but I can't find literature on it to see if there are other methods. I'll write mine out when I figure out the latex code for tables (again).

I'd say that 1 + 1 > 1 is comparable to 1 + 1 = 2.

Long division of series? I'm not sure what you mean by diagonals, but I think the standard approach is to reverse the Cauchy product, i.e. if F(x) has coefficients a_i and G(x) has coefficients b_i, the H(x) = F(x)/G(x) with coefficients c_i would be given recursively by \(c_i = \frac{1}{b_0}(a_i - \sum_{j=1}^i b_jc_{i-j})\) as long as everything is well-defined.

 

[Kharakov]

That's it. I was writing 2 series over one another (long division style), then apparently taking the reverse Cauchy product as I wrote the answer in a vertical column besides the Cauchy product's a_i and b_i (which was arranged in a grid, the diagonals being the Cauchy product's c_i).

 

[rousseau]

Ok, you can't count gradient changes without 1+1=2... although wouldn't there have to be some distinct boundary perception to have knowledge of more than 1?

I'd think a boundary condition knowledge/belief must exist before there can be a 1 and another 1.


Is 1+1=2 is more foundational than 1+1 > 1?


Also, is there a name for division of infinite series? I use a method that uses diagonals, but I can't find literature on it to see if there are other methods. I'll write mine out when I figure out the latex code for tables (again).

I'd say that 1 + 1 > 1 is comparable to 1 + 1 = 2.

Long division of series? I'm not sure what you mean by diagonals, but I think the standard approach is to reverse the Cauchy product, i.e. if F(x) has coefficients a_i and G(x) has coefficients b_i, the H(x) = F(x)/G(x) with coefficients c_i would be given recursively by \(c_i = \frac{1}{b_0}(a_i - \sum_{j=1}^i b_jc_{i-j})\) as long as everything is well-defined.

Presumably what steve_bank was getting at, is that math usually involves proving how two different things are equivalent?

 

[beero1000]

Ok, you can't count gradient changes without 1+1=2... although wouldn't there have to be some distinct boundary perception to have knowledge of more than 1?

I'd think a boundary condition knowledge/belief must exist before there can be a 1 and another 1.


Is 1+1=2 is more foundational than 1+1 > 1?


Also, is there a name for division of infinite series? I use a method that uses diagonals, but I can't find literature on it to see if there are other methods. I'll write mine out when I figure out the latex code for tables (again).

I'd say that 1 + 1 > 1 is comparable to 1 + 1 = 2.

Long division of series? I'm not sure what you mean by diagonals, but I think the standard approach is to reverse the Cauchy product, i.e. if F(x) has coefficients a_i and G(x) has coefficients b_i, the H(x) = F(x)/G(x) with coefficients c_i would be given recursively by \(c_i = \frac{1}{b_0}(a_i - \sum_{j=1}^i b_jc_{i-j})\) as long as everything is well-defined.

Presumably what steve_bank was getting at, is that math usually involves proving how two different things are equivalent?

Did you mean to quote that post? It doesn't seem to match what you were talking about.

I wouldn't say usually, but a lot of the time, sure. There's the Poincare quote that "Mathematics is the art of giving the same name to different things."

 

[Kharakov]

Is there a closed form equation that does a smooth transition between the following series? I'm looking for something so n(n+1) on the bottom smoothly increases to n(n+1)(n+2) and the top does the same (for the summations). I have an idea off the top of my head, but don't know if it's Kosher, I'll explain it later if it works out.

Straight up harmonic/geometric results in:

\(\frac {1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...}{x^0 + x^1 +x^2 +x^3 +x^4 +x^5 +...} = 1 - \sum_{n=2}^\infty \frac {n(x-1) +2x -1}{n(n+1)}=0\)
You've got 2*triangulars on the bottom...

However when you multiply an increasing coefficient to the geometric series...
\(\frac {1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...}{ 1 x^0 + 2x^1 +3x^2 +4x^3 +5x^4 +6x^5 +...}= 1 - \frac{2x-1}{2} + \sum_{n=3}^\infty\frac{(x-1)n^2+(3-x)n+2}{n(n+1)(n+2)}\)

If you increase the coefficient of the geometric series to triangulars n(n+1)/2, you end up with 4*pentatopes on the bottom (of the summation) instead of 3* tetrahedrals. Haven't figure out the top term for x>1 yet.

Show

You end up with sums that work another way entirely, something like:
harm/modified geo(x=1)
\(1 - \frac {3}{2} +\frac {1}{3} +\frac {1}{12} +\frac {1}{30} +\frac {1}{60} +... \)
which is -1/2 + 1/3 * sum of the reciprocals of the tetrahedrals =0

harm/mod geo(x=2) = harmonic series / geometric series of 2 with modified coefficients
\(1 - \frac {7}{2} +\frac {7}{3} +\frac {11}{12} +\frac {16}{30} +\frac {22}{60} +... \)

harm/ mod geo(3)
\(1 - \frac {11}{2} +\frac {19}{3} +\frac {33}{12} +\frac {51}{30} +\frac {73}{60} +... \)

 

[beero1000]

Is there a closed form equation that does a smooth transition between the following series? I'm looking for something so n(n+1) on the bottom smoothly increases to n(n+1)(n+2) and the top does the same (for the summations). I have an idea off the top of my head, but don't know if it's Kosher, I'll explain it later if it works out.

Straight up harmonic/geometric results in:

\(\frac {1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...}{x^0 + x^1 +x^2 +x^3 +x^4 +x^5 +...} = 1 - \sum_{n=2}^\infty \frac {n(x-1) +2x -1}{n(n+1)}=0\)
You've got 2*triangulars on the bottom...

However when you multiply an increasing coefficient to the geometric series...
\(\frac {1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...}{ 1 x^0 + 2x^1 +3x^2 +4x^3 +5x^4 +6x^5 +...}= 1 - \frac{2x-1}{2} + \sum_{n=3}^\infty\frac{(x-1)n^2+(3-x)n+2}{n(n+1)(n+2)}\)

If you increase the coefficient of the geometric series to triangulars n(n+1)/2, you end up with 4*pentatopes on the bottom (of the summation) instead of 3* tetrahedrals. Haven't figure out the top term for x>1 yet.

Show

You end up with sums that work another way entirely, something like:
harm/modified geo(x=1)
\(1 - \frac {3}{2} +\frac {1}{3} +\frac {1}{12} +\frac {1}{30} +\frac {1}{60} +... \)
which is -1/2 + 1/3 * sum of the reciprocals of the tetrahedrals =0

harm/mod geo(x=2) = harmonic series / geometric series of 2 with modified coefficients
\(1 - \frac {7}{2} +\frac {7}{3} +\frac {11}{12} +\frac {16}{30} +\frac {22}{60} +... \)

harm/ mod geo(3)
\(1 - \frac {11}{2} +\frac {19}{3} +\frac {33}{12} +\frac {51}{30} +\frac {73}{60} +... \)

Those denominators are simple enough that you don't need to use the Cauchy product at all. Just multiply by the reciprocal of the closed form of the sum.

So:

\(\frac {1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...}{x^0 + x^1 +x^2 +x^3 +x^4 +x^5 +...} = \frac {1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...}{\frac{1}{1 - x}} = (1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...)(1-x)\)

and

\(\frac {1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...}{ 1 x^0 + 2x^1 +3x^2 +4x^3 +5x^4 +6x^5 +...} = \frac {1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...}{\frac{1}{(1-x)^2}} = (1+ \frac {1}{2} +\frac {1}{3} +\frac {1}{4} +\frac {1}{5} +\frac {1}{6} +...)(1-x)^2\)

... where I'm ignoring issues of convergence and what the numerator sum formally means.

 

[lpetrich]

Kharakov's posted expressions seem almost too easy to evaluate. Did he have in mind continued fractions? One way of writing them can easily be confused with the usual fraction notation.

I'll try to ASCIIfy a continued fraction as much as possible: a0 + b1/(a1 + b2/(a2 + b3/(a3 + ...)))

A "simple" continued fraction has all the b's equal to 1.

One might naively think that the only way to calculate a continued fraction is in the backward direction, but there is a recurrence that allows one to calculate a continued fraction in the forward direction.

Calculate the numerators and denominators together: X_n = {num_n, den_n} for n = 1, 2, 3, ....

This sequence has initial values X₁ = {1, 0} and X₂ = {a(0), 1}. The next member is calculated with
X_n+2 = a*X_n+1 + b*X_n

for n = 1, 2, 3, ...

-

If all the a's are equal, and likewise for all the b's, then the X's have form X<sub>n[/sup] = X1*r1ⁿ + X2*r2ⁿ

where r1 and r2 are (a/2) +- sqrt((a/2)² + b)

The sequence's initial conditions give X1 and X2 as {r1/(r1-r2),1/(r1-r2)} and {-r2/(r1-r2),-1/(r1-r2)}

If |r1| > |r2|, then the limit of the continued fraction is r1.

 Continued fraction has some more on them, with an ingenious way of finding continued-fraction representations of certain functions in  Gauss's continued fraction. It uses interrelationships of "hypergeometric functions", a superset of many common sorts of functions, like trigonometric functions, Bessel functions, and orthogonal polynomials.</sub>