Video links for STAT students.

Dear class,
Not all YouTube math videos are created equal!! I will post links to the most useful ones here:

 

Confidence Intervals:

 

Central Limit Theorem:

 

If you want me to post more post a comment about the topic that you need and I will find the best (and mathematically correct) videos for the topic.

Summer courses almost over, "children's" books in the works.

This summer is one I will remember. I taught both differential equations and calculus II for the first time. Seeing these subjects from the instructors side has really opened my eyes to all kinds of details I never noticed before. One of the most striking new insights is how much these two courses have in common. They both rely deeply on sequences and series. Sequences are like a hallway in mathematics, one that connects many many many rooms.

I am working on two math book projects. The first is a Japanese-styled art book on the topic of sine and cosine. It's inspired by many of the lessons I taught this summer.

Japanese-style book about sine and cosine

I want to bring all of the different ways that sine and cosine are presented in elementary and undergraduate mathematics in to one (long) pictorial document. I start with the differential equation, y''+y=0 then solved it (using the series method from differential equations) producing \sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} and \cos x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n}

Next I wanted a pictorial way to relate these power series to the unit circle. I have found it in this spiral (the first image shows how it is constructed as an involution):

Constructing The Involute Pinwheel

a sequence of involutes: the vertical and horizontal components will form the power series for sine and cosine respectively.

The vertical and horizontal components will form the power series for sine and cosine respectively. Take the series of vertical line segments: \sin x = A_1A_2 - A_3A_4 + A_5A_6 - \cdots and so on, the segments repeatedly over and under-shoot the accutal value of sine. The full paper by Leo S. Gurin, "A problem", can be found here.

I'm going to incorporate Gurin's spiral in to my book. I want to show the power series literally flying out of it, like they have come to life. I wonder if I can make it like the famous drawing of the sine curve projecting out of the unit circle?

Naturally, I already have planned to put that diagram in my booklet.

Work in progress

The Japanese-style book is perfect for series and periodic functions It's one long continuous piece of paper:

Yet very compact:

I'm also working on a very silly book about hypercycloids (that's the "math" name for the shapes drawn by spirographs, did I mention I collect spirographs?):

I'm trying to make it like a children's book, fun, light, a little silly:

I can't wait to share the final product.

Of yarn and math.

A collection of hand-spun yarn and my spindle. I did not spin all of this yarn, some is from etsy.com

Yarn has been on my mind. It's on my mind because I love to knit, but also because of this post I read some weeks ago on the mathematics stack exchange. The author presents a paper about an "optimal yarn ball" that looks like this:

via. http://arxiv.org/abs/1005.4609

Very cool, but it would be impossible to do that with real yarn! (Though, I doubt that was their goal here.) When wrapping yarn around a ball you cannot make sharp turns or the yarn will fall off. If we think of the yarn as a curve on the surface of the sphere, we would say it must have curvature less than some small constant m. (meaning the best fit tangental circle must have radius greater than \frac 1m)

The yarn should be wrapped evenly: it should not overlap itself too often. (This would lead to the yarn ball taking on a non-spherical shape over time. No good!) The distribution of the self-intersections should be of nearly consistent density over the surface of the sphere. The yarn will divide the surface of the sphere in to regions. If A_i, A_j are the areas of any two of these regions then for some small constant F, |A_i- A_j| < F, \forall i, j.

As the curve is extended further and further (wrapped around more times) the density of intersections should increase and A_i \rightarrow 0 for all of the regions.

I know one answer is a randomly generated curve that deviates from a straight path. Overtime it produces a perfect yarn ball. That's how I make my yarn balls, randomness produces even inward tension which forms a sphere.

But, I want to know if there is a non-random answer to this question.

First Thoughts:
In trying to solving this myself I thought it might be a good idea to project the sphere on to the plane, as a Riemann surface. But, since so many of the requirements focus on uniformity, this makes the problem a little strage. Observe: The area of the regions will need to *increase* as we move away from the unit circle toward infinity. If the curve ever passes through the north pole, then the plane version would shoot off to infinity. What I found even more weird is what would happen if the curve simply passed very near, but not through, the north pole. Then, on the plane, the curve would go very far from the origin and then loop back. By making the curve near enough to the north pole the loops can be as large as we please.

Since wrapping increases the density uniformly, if we observe the wrapping pattern projected on to the plane it would be like loopy knot that, over time, keeps casting bigger and bigger outlier loops, it would always grow and be un-bounded. (even without any intersection with the north pole!)
The requirement about curvature, would mean that the knot could be more curvy inside of the unit circle, and then grow less curvy as we move away from the origin.

Next to find such a curve... Maybe if I projected the following polar plot back on to the Riemann sphere?

The graph of the form r= \theta^2 \sin (k \theta) produced good-looking results for 0 < k <1... but all of the graphs are far too dense near the origin. Also, wrapping would start in a lop-sided manner, filling in the bottom of the sphere first then moving up... and never reaching the north pole! But, maybe there is some way to tweak this?

(This graph shows the kind of non-random pattern I have in mind for a solution. In a practical sense, a solution to this problem could be used to wind spherical balls of ropes, yarns, or cords in a factory setting. Now, I will think about how I would instruct a robot to wind a yarn ball... if the robot is unable to do things randomly. What would you tell the robot to do?)

This post grew out of a question I asked at the Stack Exchange.

Almost Convergence.

The title of my last post "Almost Cauchy Sequence" made me wonder if "Almost Cauchy" was already defined in some formal sense. I could not find much on "Almost Cauchy" ... but I did find out about "Almost Convergence." (I think I have ran in to this before... but it is very neat... observe:)

A sequence S = S_n is almost convergent to L if for any \epsilon > 0 we can find an integer n such that the average of n or more consecutive terms in the sequence is within \epsilon of L. Formally, \forall \epsilon > 0 \quad \exists N \ni \quad \displaystyle \left| \frac{1}{n} \sum_{i=0}^{n-1} S_{k-i} -L    \right| < \epsilon \quad \forall n > N, \; k \in \mathbb{N}. (I took this definition from here, and I wonder why they didn't mention that we need k \geq n or else we are looking at negative terms in the sequence... But, maybe there is an interpretation of something like S_{-3}? Or maybe the author thinks that would be obvious? I'm going to ask at the mathematics stack exchange!)

Almost Cauchy Sequence, unfinished business.

I just remembered a question from beginning analysis that my professor mentioned in class:

"Find a sequence A that is not Cauchy, yet any consecutive terms A_n, A_{n+1} will become arbitrarily close."

It was one of those questions that he mentioned in class for us to do on our own, but I never quite got to it. I was thinking about convergence and complete metric spaces today (and about unfinished business, as I've made a lot of my own since I started teaching.) And the question came back to me.

We seek a sequence, A, where \forall \; \epsilon > 0 \quad \exists N \in \mathbb{N} \;  \ni \; \forall n > N \quad |A_n - A_{n+1}| < \epsilon and \exists \epsilon > 0 \; \ni  \quad \forall  N \in \mathbb{N} \quad \exists m,n > N \quad |A_m - A_n| > \epsilon. (Writing the negation of what it means for a sequence to be Cauchy was a little more tricky than I thought...)

The natural log function is the most obvious answer. For large vales of n \ln (n) - \ln (n+1) is small. That is, \displaystyle \lim_{n \rightarrow \infty} \ln \frac{n}{n+1} = 0. This is true becuase after taking EXP of both sides we have: \displaystyle \lim_{n \rightarrow \infty}  \frac{n}{n+1} = 1

Next, it seems obvious that the natural log can't be Cauchy, but just to be certian if we let \epsilon = \frac{1}{2} then, \forall N \in \mathbb{N} if we choose some n > N and let m= ne (so, n,m > N) then

| \ln m - \ln n | = |\ln \frac{m}{n}| = |\ln \frac{ne}{n}| = 1 > \frac{1}{2} = \epsilon.

NOT Cauchy at all!

Ok, I imagine my professor was thinking of the natural log... but, what other functions also have this property? \sqrt[n]{x}? So, I guess this is really asking when \displaystyle \lim_{x \rightarrow \infty} f(x) - f(x+1) = 0 while \displaystyle \lim_{x \rightarrow \infty} f(x) = \pm \infty. It is necesairy but not sufficent for f'(x) > 0 \; \forall x and f''(x) < 0 \; \forall x , both will have the x-axis as a horizontal asymptote.

Nice to have finally gotten around to doing this problem! I think it will be good practice to find more unfinished business in my notes.

Introducing the multi-zeta function!

Contemplating:

\frac{\pi ^{2n}}{(2n+1)!} = \displaystyle \sum_{j_1,...j_n=1 \atop j_i \neq j_k}^{\infty} (j_1j_2...j_n)^{-2}

My friend, and fellow math blogger, Owen, tried to tell me I was dealing with the multi-zeta function the other day, but the very general definition on WolframMathworld left me feeling a little mystified. I could see how the identities I was playing with fit in to it, but what was all of that other stuff (like \sigma_1,...,\sigma_k? I guess in my equation they are all 1? Why isn't it dealing with powers of the active variable in the product in the denominator? Why the extra level of subscripts?) So, I asked about it on stackexchange to gain more insight, and, thanks to Marni, I was promptly pointed to this lovely paper:

Now, I can see easily that the function is:

A(i_1, i_2, ... , i_k) = \sum_{n_1 > n_2 > \cdots > n_k > 1} \frac{1}{n_1^{i_1}n_2^{i_2} \cdots n_k^{i_k}}

With i_1, i_2, ... , i_k = 2.

More on this later.

Meaning one may repeat values,

Several series for powers of pi.

The Weierstrass factorization theorem gives us this identity:

\displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = \frac{\sin(\pi q)}{\pi q}

Take the left side:

\left( 1 - \frac{1}{1}q^2 \right) \left( 1 - \frac{1}{4}q^2 \right) \left( 1 - \frac{1}{9}q^2 \right) \cdots

Thinking of this in a combinatorial way we can expand it. Each even power of q has a sum of combinations of the inverse squares as its coefficient. That is:

1 + \displaystyle \left(  q^2 \sum_{j_1=1}^{\infty} -j_1^{-2} \right) + \left( q^4 \sum_{j_1,j_2=1 \atop j_1 \neq j_2}^{\infty} (j_1j_2)^{-2} \right) + \left( q^6  \sum_{j_1,j_2,j_3=1 \atop j_i \neq j_k} (j_1j_2j_3)^{-2} \right) + \cdots + \displaystyle q^{2n} \sum_{j_1,...j_n=1 \atop j_i \neq j_k}^{\infty}( -1)^n (j_1j_2...j_n)^{-2} + \cdots

Now on the right side of
\displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = \frac{\sin(\pi q)}{\pi q}

Expand using the Taylor series for the sine.

\frac{\sin(\pi q)}{\pi q} = 1 - \frac{(\pi q)^2}{3!} + \frac{(\pi q)^4}{5!} - \frac{(\pi q)^6}{7!} + \cdots

Equate the coefficients of the terms with the same exponent.:

  • 1  = 1
  • - \frac{\pi ^2}{3!} = \displaystyle \sum_{j_1=1}^{\infty} -j_1^{-2}
  • \frac{\pi ^4}{5!} = \displaystyle  \sum_{j_1,j_2=1 \atop j_1 \neq j_2}^{\infty} (j_1j_2)^{-2}
  • - \frac{\pi ^6}{7!} = \displaystyle \sum_{j_1,j_2,j_3=1 \atop j_i \neq j_k} - (j_1j_2j_3)^{-2}
  • \vdots
  • \frac{\pi ^{2n}}{(2n+1)!} = \displaystyle \sum_{j_1,...j_n=1 \atop j_i \neq j_k}^{\infty} (j_1j_2...j_n)^{-2}
  • \vdots

The second one is famous, more often written:

\frac{\pi ^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{25} \cdots .
And the rest of these series I do not know so well. Do you know them?

Next step with the interesting difference from last night...

From last night's post:

\displaystyle  \prod_{j=1}^{\infty} (1 +x^j) - \displaystyle  \prod_{j=1}^{n} (1 +x^j)

This difference is the generating function for partitions into distinct parts from the set \{ n+1, n+2, ...  \}.

I think my next task is to graph the upper bound I found for these coefficients, then see how it differs from the actual number of partitions for this set. The difference should grow pretty quickly. I wonder how mathematica deals with generating functions. I know some of the popular functs are pre-loaded... But, can I give it an expression and have it list exponents for me?

So much to learn!