## Ice Storm of 2007

So last time we were working on finding a good way to say “the darts I throw went more or less HERE”.

Points Distance $(x_i-\bar{x})^2$ $(y_i-\bar{y})^2$
(-2.2,4.1) 4.6 7.8 13.7
(-1.1,2.1) 2.4 2.9 2.9
(-0.5,0.7) 1.1 1.2 0.1
(-2.5,-3.1) 4.7 9.6 12.3
(-0.3,-3.5) 4 0.8 15.2
(2.2,3.8) 3.8 2.6 11.6
(2.4,1.8) 2.3 3.2 2
(3.8,-0.4) 3.3 10.2 0.6
(3.2,-1.8) 3.4 6.8 4.8
Avg=(0.6,0.4) Max=4.7 Avg=5 Avg=7
$\sqrt{Avg}=2.2$ $\sqrt{Avg}=2.6$

$(\sigma_x,\sigma_y)=\sqrt{\frac{1}{n}\sum_{i=1}^n((x_i-\bar{x})^2,(y_i-\bar{y})^2)}$

And in the end we came up with a formula for “standard deviation”:$\sigma=\sqrt{\frac{1}{n}\sum(x_i-\bar{x})}$
It looks like the box is at least moderately representative of where the darts went, but not many of the darts actually landed in the box. Why not? We came up with an oddball average distance away from center in the x- and y-directions, and based our rectangle on that, so why does it contain so few of the points? The picture above only contains two of the nine darts, so how can we justify calling it an ‘average’ center of any sort?

Consider for a moment what this would look like if all of the darts had a zero y-value (orange) or a zero x-value (green):

Looking at the orange dots, it looks like there are a bunch to the left and a bunch to the right and not many in the middle, so if our box is in the middle it’s probably OK if it only has a few darts in it. For the greens, five of the nine are in the box so I can’t complain too much about that. So where is the problem? The box we have drawn contains only darts that fit into BOTH categories!

Revisiting basic probability for a moment, recall how we would calculate the probability of rolling a ‘3’ on a normal die while also flipping a coin and getting ‘heads’: the probability of rolling a ‘3’ is $\frac{1}{6}$ and the probability of flipping for ‘heads’ is $\frac{1}{2}$ so the probability of both at the same time is $\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$. We have to multiply the fractions to find the probability of two independent events happening at the same time.

If we assume that our left-right error has nothing to do with the up-down error in our dart-throwing skills, then we can treat those two axes as independent and do a little multiplication to find out what is likely to be in the box. Just to see what happens, let’s redo the experiment with 100 points and keep an eye on how many points appear in the box when reduced to each axis, and also how many un-moved points stay in the box. We’ll call this our probability of being ‘in the box’. I’ll spare you the listing of 100 points, and finding the averages and standard deviations: we did that in the previous article about darts.

Wow. That graph is a bit of a mess. Even making the blue and red points semi-transparent didn’t help much. If you could count the individual dots, you would find that there are 72 blue dots in the box, 68 red dots in the box, and 50 orange ones. Coincidentally, $\frac{72}{100} \times \frac{68}{100} = 0.4896 \approx \frac{50}{100}$, which is about what we would expect since measured probability is not guaranteed to be exactly the same as a prediction.

Arguably, a dart at the far corner of the box is not as ‘good’ a shot as one in the middle of an edge, if you consider the distance from center. Perhaps something useful would be to consider a circle, but if we put a circle inside a rectangle, we lose more darts (we’re already down to about half, not very good if we want to refer to ‘most’ of them) and also circles don’t fit very well inside rectangles. It seems to me that the obvious solution is then to use an ellipse and put it outside, killing both birds with one equation. The math is going to get a little rough here, since finding out whether or not a point is inside an ellipse is a little harder than for a rectangle, and finding the equation of the right ellipse is a little tougher as well. Let’s start with a simple example and work our way up to the darts.

In this graph, there is a rectangle extending 3 units left and right from center, and 2 units up and down. The ellipse fits into the same space, and clearly is smaller than the rectangle. How are we going to figure out how big the ellipse ought to be in order to be outside the rectangle instead of inside? Let’s see if there is a value $k$ by which we can scale both the width $w$ and height $h$ of our ellipse in order to include the point $(w,h)$.

\begin{align*} y&=\frac{h}{w}\sqrt{w^2-x^2} \\ y&=\frac{h\cdot k}{w\cdot k}\sqrt{(w\cdot k)^2-x^2} \\ y&=\frac{h\cdot k}{w\cdot k}\sqrt{(w\cdot k)^2-x^2} \\ y&=\frac{h}{w}\sqrt{w^2\cdot k^2-x^2} \\ \end{align*}

Now that we have an equation with the proper scaling factor in it, let’s replace $(x,y)$ with the specific point $(h,w)$ which, coincidentally, has the same values as the height and width (measured from center, not all the way across) of our ellipse.
\begin{align*} y&=\frac{h}{w}\sqrt{w^2\cdot k^2-x^2} \\ h&=\frac{h}{w}\sqrt{w^2\cdot k^2-w^2} \\ h&=\frac{h}{w}\sqrt{w^2(k^2-1)} \\ h&=\frac{h}{w}\cdot w\cdot \sqrt{k^2-1} \\ h&=h\cdot \sqrt{k^2-1} \\ 1&=\sqrt{k^2-1} \\ 1&=k^2-1 \\ 2&=k^2 \\ \sqrt{2}&=k \\ \end{align*}

Well, perhaps that wasn’t too bad. Starting with the equation of an ellipse of width $w$ and height $h$, substitute $w$ in for $x$ and set the result to $h$, and also make sure to use $hk$ and $wk$ for the height and width of the ellipse we want to find. After than, cancel common factors, square everything to get rid of the square root, and solve for $k^2$. Finally, take the square root to get our answer in terms of $k$. Whoever would have guessed that all we have to do is multiply $w$ and $h$ by a simple constant? Then the final equation, which circumscribes the rectangle extending $w$ units left and right and $h$ units up and down, looks like this:

\begin{align*} y&=\frac{h'}{w'}\sqrt{w'^2-x^2} \\ y&=\frac{h\sqrt{2}}{w\sqrt{2}}\sqrt{((w\sqrt{2})^2-x^2} \\ y&=\frac{h}{w}\sqrt{2w^2-x^2} \\ \end{align*}

And the resulting graph looks like this:

Beautiful! Now how many of the points is this ellipse going to contain?

There are 32 orange dots outside the blue ellipse, so there are 68 inside. I just eyeballed that from the graph, so if that’s off by one don’t be too surprised. $100-32 = 68$, so 68% of the darts are within the ellipse. That’s in line with the 72 blue and 68 red dots above! So perhaps we are at the point where we can say, “Most of my darts are inside that ellipse!”

But…

What happens if our high darts tend to be to the right, and low ones to the left?

That is a story for next time. Until then, keep throwing those darts!

## A Collection of LaTeX symbols

More symbols to come!

Note 1: There are too many to do all at once

Note 2: Any symbols in red (which seem to be limited to the uppercase Greek letters) are ones that for some reason my installation of mathjax is not configured properly to handle. With luck, this will also convince me it is time to do a little tweaking of the mathjax configuration to sort that out.

Greek Symbols
Symbol $\LaTeX$ Symbol $\LaTeX$
$\Alpha$ and $\alpha$ \Alpha and \alpha $\Nu$ and $\nu$ \Nu and \nu
$\Beta$ and $\beta$ \Beta and \beta $\Xi$ and $\xi$ \Xi and \xi
$\Gamma$ and $\gamma$ \Gamma and \gamma $\Omicron$ and $\omicron$ \Omicron and \omicron
$\Delta$ and $\delta$ \Delta and \delta $\Pi$, $pi$ and $\varpi$ \Pi, \pi and \varpi
$\Epsilon$, $epsilon$ and $\varepsilon$ \Epsilon, \epsilon and \varepsilon $\Rho$, $rho$ and $\varrho$ \Rho, \rho and \varrho
$\Zeta$ and $\zeta$ \Zeta and \zeta $\Sigma$, $sigma$ and $\varsigma$ \Sigma, \sigma and \varsigma
$\Eta$ and $\eta$ \Eta and \eta $\Tau$ and $\tau$ \Tau and \tau
$\Theta$, $theta$ and $\vartheta$ \Theta, \theta and \vartheta $\Upsilon$ and $\upsilon$ \Upsilon and \upsilon
$\Iota$ and $\iota$ \Iota and \iota $\Phi$, $phi$ and $\varphi$ \Phi, \phi and \varphi
$\Kappa$, $kappa$ and $\varkappa$ \Kappa, \kappa and \varkappa $\Chi$ and $\chi$ \Chi and \chi
$\Lambda$ and $\lambda$ \Lambda and \lambda $\Psi$ and $\psi$ \Psi and \psi
$\Mu$ and $\mu$ \Mu and \mu $\Omega$ and $\omega$ \Omega and \omega
Relational Operators
Symbol $\LaTeX$ Symbol $\LaTeX$
$<$ and $>$ < and > $\nless$ and $\ngtr$ \nless and \ngtr
$\leq$ and $\geq$ \leq and \geq $\leqslant$ and $\geqslant$ \leqslant and \geqslant
$\leqslant$ and $\geqslant$ \leqslant and \geqslant $\leqslant$ and $\geqslant$ \leqslant and \geqslant
$\prec$ and $\succ$ \prec and \succ $\nprec$ and $\nsucc$ \nprec and \nsucc
$\ll$ and $\gg$ \ll and \gg $\lll$ and $\ggg$ \lll and \ggg
$\subset$ and $\supset$ \subset and \supset $\not\subset$ and $\not\supset$ \not\subset and \not\supset
$\subseteq$ and $\supseteq$ \subseteq and \supseteq $\nsubseteq$ and $\nsupseteq$ \nsubseteq and \nsupseteq
$\sqsubset$ and $\sqsupset$ \sqsubset and \sqsupset $\sqsubseteq$ and $\sqsupseteq$ \sqsubseteq and \sqsupseteq
Equality
Symbol $\LaTeX$ Symbol $\LaTeX$
$=$, $\ne$ and $\neq$ =, \ne and \neq $\equiv$, $\approx$ and $\doteq$ \equiv, \approx and \doteq
$\cong$ and $\simeq$ \cong and \simeq $\sim$ and $\propto$ \sim and \propto
Binary Operators
Symbol $\LaTeX$ Symbol $\LaTeX$
$+$ and $-$ + and - $\times$, $\div$ and $\neg$ \times, \div and \neg
$\pm$ and $\mp$ \pm and \mp $\vee$ and $\wedge$ \vee and \wedge
$\ast$ and $\star$ \ast and \star $\dagger$ and $\ddagger$ \dagger and \ddagger
$\cap$, $\cup$ and $\uplus$ \cap, \cup and \uplus $\sqcap$ and $\sqcup$ \sqcap and \sqcup
$\triangleleft$ and $\triangleright$ \triangleleft and \triangleright $\bigtriangleup$ and $\bigtriangledown$ \bigtriangleup and \bigtriangledown
$\bullet$ and $\wr$ \bullet and \wr $\diamond$ and $\bigcirc$ \diamond and \bigcirc
$\oplus$, $\ominus$ and $\otimes$ \oplus, \ominus and \otimes $\oslash$, $\odot$ and $\circ$ \oslash, \odot and \circ

## Where did my darts go?

Suppose for a moment that you throw darts at a dartboard occasionally, and want to know ahead of time the most likely place for your darts to go when you are aiming at a particular spot. Assuming you are not a professional, this could be a fairly large patch. With that in mind, let us now take a journey through one of my other self-refocussing exercises that turned out to be useful.

Where is the center of the three darts thrown? That’s pretty easy: take the average of the points, which means take the average of the x-values, and of the y-values, and plot that as another point:

$(\bar{x},\bar{y}) = \frac{1}{n} \sum_{i=0}^{n}(x_i,y_i)$

More darts just means more points on the graph, but doesn’t really change any of the math, so we’ll go on. If we considered the average of the points as the center of a circle with the radius reaching to the farthest point, what would that circle look like?

Great! Now we know that of the darts thrown, all of them landed in that circle. True, yes, useful…not so much. What if had kept track of more darts and wanted to know where “most” of them went? That question is just a little bit more interesting.

Let’s start with a bunch of (made up, but work with me for a bit here) thrown darts in orange, and the average of them marked in blue:

Points
(-2.2,4.1)
(-1.1,2.1)
(-0.5,0.7)
(-2.5,-3.1)
(-0.3,-3.5)
(2.2,3.8)
(2.4,1.8)
(3.8,-0.4)
(3.2,-1.8)
Average=(0.6,0.4)

The question now is how far away is the farthest point? Sadly, the easiest way to find it is to calculate the distance for each point from center.

Points Distance
(-2.2,4.1) 4.7
(-1.1,2.1) 2.4
(-0.5,0.7) 1.1
(-2.5,-3.1) 4.7
(-0.3,-3.5) 4
(2.2,3.8) 3.8
(2.4,1.8) 2.3
(3.8,-0.4) 3.3
(3.2,-1.8) 3.4
Average=(0.6,0.4) Maximum=4.7

Since I’m not really sure what to do next, let’s repeat ourselves for a moment and consider taking another average. Let’s add up the distances for each of the $x$ and $y$ values from our center point and find the averages of those, and that will at least tell us whether my aim is better up-and-down or left-to-right. Since I’m not really sure what to call this average distance, I’ll just pick a random letter $\sigma$ and use that. Since we have two distances, one left-right and the other up-down, let’s subscript the $\sigma$ as $\sigma_x, \sigma_y$ so we know which one is which.

$(\sigma_x,\sigma_y)=\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x},y_i-\bar{y})$
Points Distance $x_i-\bar{x}$ $y_i-\bar{y}$
(-2.2,4.1) 4.6 -2.8 3.7
(-1.1,2.1) 2.4 -1.7 1.7
(-0.5,0.7) 1.1 -1.1 0.3
(-2.5,-3.1) 4.7 -3.1 -3.5
(-0.3,-3.5) 4 -0.9 -3.9
(2.2,3.8) 3.8 1.6 3.4
(2.4,1.8) 2.3 1.8 1.4
(3.8,-0.4) 3.3 3.2 -0.8
(3.2,-1.8) 3.4 2.6 -2.2
Avg=(0.6,0.4) Max=4.7 Avg=0 Avg=0

Oops. What happened? Some of our $x_i-\bar{x}$ were positive, and some negative, so in the end they cancelled out. So how do we ensure that all of the differences are positive? The easiest way I can think of is to ignore the sign, but I don’t really know how to do sums when I have to treat some of the terms differently. The next best choice may be to square everything, which we know makes numbers positive.

$(\sigma_x,\sigma_y)=\frac{1}{n}\sum_{i=1}^n((x_i-\bar{x})^2,(y_i-\bar{y})^2)$
Points Distance $(x_i-\bar{x})^2$ $(y_i-\bar{y})^2$
(-2.2,4.1) 4.6 7.8 13.7
(-1.1,2.1) 2.4 2.9 2.9
(-0.5,0.7) 1.1 1.2 0.1
(-2.5,-3.1) 4.7 9.6 12.3
(-0.3,-3.5) 4 0.8 15.2
(2.2,3.8) 3.8 2.6 11.6
(2.4,1.8) 2.3 3.2 2
(3.8,-0.4) 3.3 10.2 0.6
(3.2,-1.8) 3.4 6.8 4.8
Avg=(0.6,0.4) Max=4.7 Avg=5 Avg=7

I’m not sure I want to make a box that extends 5 units left and right, and 7 units up and down (the blue box in the graph below). It seems way too big, well outside the darts farthest from center, and doesn’t look at all like an average of the darts. Perhaps I should follow the advice of my science teacher: “When something looks really wrong at the end, follow the units.” Let’s call everything inches even though the graph doesn’t have anything on it (my dartboard doesn’t either). In $(x_i-\bar{x})^2$ inches minus inches is still inches, and inches times inches gives inches squared. Wait! the $n$ in $\frac{1}{n}$ doesn’t have any units – it is just the number of darts thrown, so dividing by $n$ doesn’t change the square inches. So really, what I want to do is take the square roots of 5 and 7 to get the actual size (in inches from center, not square inches) of the box to draw, colored green in the graph below.

Points Distance $(x_i-\bar{x})^2$ $(y_i-\bar{y})^2$
(-2.2,4.1) 4.6 7.8 13.7
(-1.1,2.1) 2.4 2.9 2.9
(-0.5,0.7) 1.1 1.2 0.1
(-2.5,-3.1) 4.7 9.6 12.3
(-0.3,-3.5) 4 0.8 15.2
(2.2,3.8) 3.8 2.6 11.6
(2.4,1.8) 2.3 3.2 2
(3.8,-0.4) 3.3 10.2 0.6
(3.2,-1.8) 3.4 6.8 4.8
Avg=(0.6,0.4) Max=4.7 Avg=5 Avg=7
$\sqrt{Avg}=2.2$ $\sqrt{Avg}=2.6$
$(\sigma_x,\sigma_y)=\sqrt{\frac{1}{n}\sum_{i=1}^n((x_i-\bar{x})^2,(y_i-\bar{y})^2)}$

I’m still not really happy about those boxes, though. The green one is too small, and the blue one too big. On the bright side, we did ‘accidentally’ come up with the formula for standard deviation:$\sigma=\sqrt{\frac{1}{n}\sum(x_i-\bar{x})^2}$

The biggest problem that I see is that the green box clearly does not contain enough of the darts to be useful. We’ll figure out why next time, and see if we can work our way towards a solution.

## Fun With Series

In hindsight, one of the good things about teaching where I did was that I had time to think about other stuff during school. This was especially handy when I was studying for the actuarial exams (I passed the first two on first attempt) and needed to practice some of the little things I haven’t done in a while and that never comes up in the course of what little I was able to teach. So just for grins, here’s a little derivation practice.

Yes, $i$ is interest rate. It looks funny not being part of a complex number, but that’s how it goes sometimes
For instance, deriving the formula for annuities: $\annuai{n}{i}$

Since an annuity is just the sum of the present values of a series of payments, it’s pretty easy to derive. The key is to remember the equation for the present value $PV$ of some amount of money at a later date $FV$, given the periodic interest rate (per period) $i$ expected between now and later, and the number of periods $t$ between now and later. Valuation equations all derive from this most basic premise.

\begin{align} PV=\frac{1}{(1+i)^t} \cdot FV\label{a1} \end{align}

A useful shortcut. Why? Rearrangements.
\begin{align} v &=\frac{1}{1+i} \label{v1}\\ (1+i)\cdot v &=1 \notag\\ v + vi &= 1 \notag\\ vi &=1-v \label{v2} \end{align}
Consider the worst way to describe the present value of an annuity: “Figure a payment of, say, $50 at the end of each year for the next 10 years. Take the first payment and find its present value. Then the second, and add it to your first answer…” Ouch. Well, that’s what math is for. How about we write it in math instead? $PV_{annuity}=PV(first payment)+PV(second payment)+ \dotsb + PV(last payment) \notag$ Slightly closer, but not clean yet. We keep adding up the same sort of thing over and over again. Perhaps there’s a way to write that a little more concisely. $PV_{annuity}=\sum_{t=1}^{10}PV_{t} \notag$ The jump is a little big, but now it is starting to look like math!1 Substitute our definition of$PV$from$\eqref{a1}$, then use our definition of$v$at$\eqref{v1}, factor out the constant, and we get something useful. And as a side benefit, we also see that the amount of the payment really is irrelevant. \begin{align*} PV_{annuity}&=\sum_{t=1}^{10}50\cdot \frac{1}{(1+i)^t} \\ PV_{annuity}&=\sum_{t=1}^{10} 50 \cdot v^t \\ PV_{annuity}&=50 \cdot \sum_{t=1}^{10} v^t \\ \end{align*} The purpose of this isn’t the deriving of a formula, it’s about how we think about things and make them easier for ourselves. Thinking about it in English is difficult because the language is less than conducive to compacting a complex thought into an easily manipulated form. Crossing the bridge towards mathematical thinking is the single biggest hurdle my students face, and one they typically stumble on. Why? <rant>To paraphrase (and make suitable for a family publication): “Because math is a class to blow off, not a tool to be used to make life easier”2</rant> I take every opportunity I can find to point out that formalized math is there to let us both be lazier and also accomplish more at the same time. Back to the point… \begin{align} \annuai{n}{i}&=\sum_{t=1}^n v^t \label{q1} \\ \annuai{n}{i}&=v^1+v^2+v^3+\dotsb+v^{n-1}+v^n \label{q2} \\ v\cdot \annuai{n}{i}&=v\cdot v^1+v\cdot v^2+v\cdot v^3+\dotsb+v\cdot v^{n-1}+v\cdot v^n \label{q3} \\ v\cdot \annuai{n}{i}&=v^2+v^3+v^4+\dotsb+v^{n}+v^{n+1} \label{q4} \\ \annuai{n}{i} - v\cdot \annuai{n}{i}&=v^1 - v^{n+1} \label{q5} \\ \annuai{n}{i}(1-v)&=v^1 - v^{n+1} \label{q6} \\ \annuai{n}{i}\cdot i \cdot v&=v\cdot (1-v^n) \label{q7} \\ \annuai{n}{i}&=\frac{1-v^n}{i} \label{q8} \\ \end{align} In general, there are often steps that are either necessary to show or not. One such example here is\eqref{q3}$, which could easily be omitted. But if what you mean to do is multiply every term by$v$, then perhaps writing it out will help reduce mistakes. Another place is at$\eqref{q5}$, which could easily have another equation in front of it explicitly showing the subtraction. Getting MathJax up and running, and putting together the other pieces for this blog, have made this post take WAAAY longer than it should have. Doing a little math on paper is a good mind-clearing exercise sometimes, but this was nearly the opposite. Given our basic definition$\act{a}{n}{I}=\frac{1-v^n}{i}$, there are a few other useful varieties of basic annuities worth having fun with. Consider an annuity that pays 1 at the beginning of each month, instead of at the end. This is referred to as an “annuity due” and uses the symbol$\actd{a}{n}{i}. Each payment has one fewer compounding period, so \begin{align*} \actd{a}{n}{i} &= v^0 + v^1 + \dotsb + v^{n-2} + v^{n-1} \\ \actd{a}{n}{i} &= \frac{v^1}{v} + \frac{v^2}{v} + \dotsb + \frac{v^{n-1}}{v} + \frac{v^{n}}{v} \\ \actd{a}{n}{i} &= \frac{\act{a}{n}{i}}{v} \\ \actd{a}{n}{i} &= \frac{1-v^n}{iv} \\ \end{align*} Consider an annuity that pays 1 at the end of the first month, 2 the second, three the third, and so on fornmonths. The usual name for this is an increasing annuity. \begin{align*} \act{(Ia)}{n}{i} &= PV(1) + PV(2) + \dotsb + PV(n-1) + PV(n) \\ \act{(Ia)}{n}{i} &= \sum_{t=1}^n PV(t) \\ \act{(Ia)}{n}{i} &= \sum_{t=1}^n tv^t \\ \act{(Ia)}{n}{i} &= 1v^1 + 2v^2 + \dotsb + (n-1)v^{n-1} + nv^n \\ v\left(\act{(Ia)}{n}{i}\right) &= 1v^2 + 2v^3 + \dotsb + (n-1)v^n + nv^{n+1} \\ (1-v)\left(\act{(Ia)}{n}{i}\right) &= v^1 + v^2 + \dotsb + v^{n-1} + v^n - v^{n+1} \\ (iv)\left(\act{(Ia)}{n}{i}\right) &= \act{a}{n}{i} - v^{n+1} \\ \act{(Ia)}{n}{i} &= \frac{\act{a}{n}{i} - v^{n+1}}{iv} \\ \act{(Ia)}{n}{i} &= \frac{\actd{a}{n}{i} - v^n}{i} \\ \end{align*} And then there is the decreasing annuity, which starts by payingn$at the end of the first month,$(n-1)$the second, and so on down to 1 the$nth month. \begin{align*} \act{(Da)}{n}{i} &= PV(n) + PV(n-1) + \dotsb + PV(2) + PV(1) \\ \act{(Da)}{n}{i} &= \sum_{t=1}{n} PV(n-t+1) \\ \act{(Da)}{n}{i} &= \sum_{t=1}{n} (n-t+1)v^t \\ \act{(Da)}{n}{i} &= (n)v^1 + (n-1)v^2 + \dotsb + 2v^{n-1} + 1v^n \\ v\left(\act{(Da)}{n}{i}\right) &= (n)v^2 + (n-1)v^3 + \dotsb + 2v^{n} + 1v^{n+1} \\ (1-v)\left(\act{(Da)}{n}{i}\right) &= nv^1 - 1v^2 - v^3 - \dotsb - v^n - v^{n+1} \\ (iv)\left(\act{(Da)}{n}{i}\right) &= nv - v\left(\act{a}{n}{i}\right) \\ \act{(Da)}{n}{i} &= \frac{nv - v\left(\act{a}{n}{i}\right)}{vi} \\ \act{(Da)}{n}{i} &= \frac{n - \left(\act{a}{n}{i}\right)}{i} \\ \end{align*} Yes, this is what I did for relaxation when I was stressed at school. There’s another one that is entertaining (but I’m not sure I see the point) in which the annuity pays 1 the first month, 2 the second, up ton$in the$n^{\text{th}}$month, and then decreases back to 1 in the$(2n-1)^{\text{th}}$month. I have heard it called a “rainbow annuity” (even though it is a triangle, not an arc) and I don’t know the symbol for it so I’ll just call it$\act{RA}{n}{i}$. (Mind you,$nrefers to the peak payment, not the number of payments) \begin{align*} \act{RA}{n}{i} &= v^1 + 2v^2 + \dotsb + (n-1)v^{n-1} + nv^n + (n-1)v^{n+1} + \dotsb + 2v^{2n-2} + v^{2n-1} \\ v\left(\act{RA}{n}{i}\right) &= v^2 + 2v^3 + \dotsb + (n-1)v^{n} + nv^{n+1} + (n-1)v^{n+2} + \dotsb + 2v^{2n-1} + v^{2n} \\ (1-v)\left(\act{RA}{n}{i}\right) &= v + v^2 + \dotsb + v^n + (-1)v^{n+1} + (-1)v^{n+2} + \dotsb + (-1)v^{2n-1} + (-1) v^{2n} \\ (iv)\left(\act{RA}{n}{i}\right) &= \act{a}{n}{i} + (-1)v^{n+1} + (-1)v^{n+2} + \dotsb + (-1)v^{2n-1} + (-1)v^{2n} \\ (iv)\left(\act{RA}{n}{i}\right) &= \act{a}{n}{i} - v^n\left(v^1 + v^2 + \dotsb + v^{n-1} + v^{n}\right) \\ (iv)\left(\act{RA}{n}{i}\right) &= \act{a}{n}{i} - v^n\left(\act{a}{n}{i}\right) \\ (iv)\left(\act{RA}{n}{i}\right) &= (1-v^n)\left(\act{a}{n}{i}\right) \\ \act{RA}{n}{i} &= \frac{1-v^n}{iv} \act{a}{n}{i} \\ \act{RA}{n}{i} &= \frac{1}{v} \frac{1-v^n}{iv} \act{a}{n}{i} \\ \act{RA}{n}{i} &= \frac{1}{v} \act{a}{n}{i} \act{a}{n}{i} \\ \act{RA}{n}{i} &= \frac{\left(\act{a}{n}{i}\right)^2}{v} \\ \end{align*} One of the neat bits that shows up in each and every one of these is the ongoing pattern of “take the first line, multiply byv$, and subtract it off” which collapses the sequence into something more manageable, followed by representing the left side with a factor of$(1-v)$which turns into$iv\$, conveniently canceling out something on the right. One tool, many variations. Isn’t that what math is all about?

## Moonlight By Mirror (2018) “Moonlight by Mirror” is the first music I have written completely without a hardware synthesizer. After a long hiatus, I decided it was past time I started writing music again, and actually succeeded where my previous attempts over the past few years had failed. While I was teaching, I explored the notion of doing another work using some sort of algorithmic composition, and sketched out some ideas, but none bore fruit. With the change to a new career, I was able to push past that idea and start writing again. I kept the same working title because I had wanted to use the title for quite some time. The cover art comes from a photo I took of the moon: for this photo I attached my camera to a catadioptric telescope, so it truly is “Moonlight by Mirror.”
“Mons Hadley Delta” is a mountain the lower slopes of which were visited on the Apollo 15 mission. In form, there is a harmonic pattern which first gets longer with each repetition, then after peaking decreases in length until disappearing.
“Rima Cauchy” is a relaxed, slightly wandering ABA formed composition inspired by the 200+ kilometer long depression in the surface of the moon of the same name, which wanders slightly as it passes the small impact crater also called Cauchy.
“Landing” is an octatonic (8-tone) composition, the scale chosen to represent the binary nature of the computers that ran the space program, and ultimately the Apollo Lunar Module.
“Mare Desiderii” (Sea of Dreams) is a far-side feature no longer recognized, having been determined to be composed of a smaller mare with a cluster of dark craters.
“Alien Suits” is a whole-tone (six note) composition in an odd time signature representing the alien clothing used by the astronauts as they explored the Lunar surface. Given that the astronauts were, relatively speaking, the aliens, this is my interpretation of what it may have felt like wandering the landscape wearing one of those space suits.
“Vallis Schrödinger” is named for a nearly straight valley of the same name, possibly formed by the impact that created Schrödinger basin on the far side of the moon.
“The Crater’s Rim” takes us on a journey around the rim of a crater: one of the fundamental lunar features. Craters are round, and one idea that describes roundness is the mathematical constant pi, so the harmonic structure of this piece is determined by the first 100 digits of pi. Listen for the diminished arpeggios to locate all the 9’s.
“Meteor” is an expression of the eventfulness of the lunar surface: a relaxed stroll around the crater park transitions into an octatonic fugue when a meteor comes streaking in and raises a cloud of dust, which quickly settles due to a lack of atmosphere. If you really want to, calculate the maximum height of the debris using 1.62 m/s^2 for g and the time from the start of the fugue until the return to tonic for t.

It took far too long to put this together. Hopefully, the next project will be a little quicker.

## The Open Book (2009) Once I started teaching again, I stepped out of my comfort zone and did something a little different for this collection: it is based mainly on improvisational themes supplemented and cleaned up via MIDI post-processing. The title came from that which was in the way when I went to work on it. In an interesting twist, I had recently had an inspiration for the artwork: a page from Archimedes’ Palimpsest. When I wrote a paper for one of my math classes on Archimedes, I found that the Palimpsest changed hands in the 90’s — and that the new (anonymous) owner handed it over to researchers for imaging with the intent that the document be made public on the web. I realized then at this was as close as I would ever come to being able to use a multi-millennia old primary source for a research paper. Thus, this set of pieces took on the name “The Open Book” as inspired by the opening of the Palimpsest to the public. The cover art was my own attempt at hiding one set of text underneath another. The three instruments used are three synthesizers each of which uses a completely different approach to creating sound: The Alesis A6 Andromeda may have digital controls, but the signal path is all analog (subtractive synthesis); the Yamaha SY77 is a revolutionary digital synthesizer using FM synthesis (frequency modulation) combined with small-scale sample playback; and the Kawai K5000s builds sounds through additive synthesis.

## Taking Flight (2009) In the mid-2000s, I married and found I had a new resource for inspiration. When I returned to school in the latter half of the decade to learn a little more math, I discovered I could write a little music in-between classes so I set about writing a new piece, one that would be more complete than anything I had written before. Since writing “Thought Foundry,” I had accumulated a pair of new synthesizers and wanted to write something that would push one of them to the edge of its capacity. “Taking Flight” was recorded using mostly (but not quite solely) an Alesis A6 Andromeda, a truly astounding synthesizer. In the end, I pushed it hard enough that it couldn’t keep up (not enough voices — I was wondering I didn’t hear all the notes I knew were there) with everything at once, so I ended up tracking each instrument individually.
When I went to name the tracks, I asked my wife for some naming advice and ended up with some names that I think fit better than what I was using as working titles. My goal during composition was to tell the story of a character traveling from the country to the city and from there to an orbital colony.

“En[chant|trap]ment” is the beginning of the story, the journey to the Enchanted City, where people get sucked into a life of sustenance rather than beauty and plenty.

“Intrigue” reflects upon the decision to move onward: while there is always something trying to grab the attention of our wanderer, it is not truly in character.

“Core,” “Taking Time,” “Off-Track,” and “High Wire Act” collectively represent the stages of the journey from the city to the spaceport, and getting distracted along the way until realizing that the alternatives are stay and be sucked into the city permanently or go through security and make the commitment to leaving the surface.

“Golf Cart Blues”: A spaceport must be a large place, and like any modern airport, have places to stop for a festive beverage, a place to instruct the automated personal transportation (a self-driving golf cart) to stop for a few minutes to relax while not in constant motion.

“Setting Sail,” “Suborbital (Taking Flight),” and “S/F Cowboy Suite” detail the first stage of the outward journey: the lightness of leaving the world behind, the launch sequence, and passing through the low-orbit transfer station.

“Slow Boat to L5” and “Time to Reflect” are the longer journey spiraling outward towards a point in space trailing the Moon in orbit around the Earth, when our wanderer has time to consider all of what has transpired.

“Archipelago” is the end of the journey, arriving at a cluster of islands in space.

## Thought Foundry (1997) In the late nineties, just before I went back to school to get my teaching certificate, I decided it was time to write a piece of music that was a little more serious than the pieces I had been recording over the last few years. The result was “Thought Foundry.” The cover art is a photo of an iceberg, with a little Photoshop tomfoolery to keep it in line with the synthesized nature of the music.

“Foundation” was the starting point: I had a variety of themes I wanted to use, so put them all into the mix. Fundamentally, it is in sonata form but with three primary themes instead of the usual two. Listen for the intermixing of themes as they cross the boundaries between instruments.

“Twilight” is a rondo following up on a few of the themes that didn’t make it into “Foundation”

## Stringy (1997) Like “Sun Return,” “Stringy” is a collection of pieces that didn’t originally fit together. These, however have one element in common: all use some subset of a string quartet plus an organ. The cover art is water cascading down the side of a fjord in Alaska, which in my mind looked like white threads being braided together (or unraveling, depending on point of view) from mountaintop to ocean.

“Prelude” serves simply as an introduction to the set of pieces.

“Stringy” began life in one of my music theory classes as the centerpiece of a string quartet, but I never got around to writing the other movements, so it got swept in with the other string-oriented pieces. It is the only one of the pieces here that lacks the organ, and is also the only one in a traditional form: sonata form, by the book.

“Threesome” and “Whip Tail” are two short pieces each of which I wrote solely to get out of my head and onto a piece of paper.

## Random Thoughts (1997) Once I had recorded everything I had written that I thought I should record, I set about a new project: compose a piece of music with some very basic rules to be interpreted using a source of random numbers. As a result, “Random Thoughts” was written mostly with pennies and a few dice, along with a set of rules on how to proceed. Each piece had the same set of rules: start with a set of notes generated by 12-sided dice, play a rhythmic section determined with coin flips for whether notes were ‘on’ or ‘off,’ and follow it with a melodic section where the flips stand for up and down within the set of pitches.

Each piece follows the same rules, but had different coin-flips and die-rolls. I admit to having ‘influenced’ the dice and coins from time to time, but in general I attempted to keep my fingers out of the pie.
Being a set of randomly generated pieces, there isn’t really a theme separating one from another or even any way to reliably distinguish one from the next aside from memorizing the pitch class sets. Even the names were generated randomly: “four-letter word ending in ‘-ats’” “multisyllable preposition” the “noun describing a location or noun expected to be in some location.”

The cover art is a heavily processed photo of the hand of an articulated suit of plate armor in a museum in Louisville, Kentucky.

## Electrons II (1996) The other half of “Electrons I,” this part has the more classically-oriented pieces. The cover art is another of my attempts at capturing lightning, this time at the beach on Siesta Key, Florida.

“Invention” follows the standard Baroque model for an invention: two part counterpoint with an exposition, development, and short recapitulation.

“Fugue” is a four-part fugue.

“Synfonia” is a three-part invention, which historically was referred to as a “sinfonia” and I renamed slightly to correspond with the instrumentation.

## Winter Weather (1996) In the mid-nineties, I set about recording all of the music I had that was finished so I could tackle new projects with a clear mind. “Winter Weather” is a collection of a few short thoughts that I had sketched out, but didn’t want to leave lying around for years. The cover art is a photo of a Maple tree after an ice storm. My wife once described these as evidence of my ADD — lots of short ideas all smashed together.

“Freezing Rain” is characterized by a series of rising themes that never quite make it above freezing far enough to melt, with short stabs of sleet throughout and a little ordinary rain in a few places in the middle.

“Thin Ice” has one theme punctuated by a reminder in the horns that the ice may be to thin to walk on.

“Lake Flurries” just sits lightly, occasionally getting thicker, sometimes lighter. Time signatures change with the direction of the wind.

“Storm Warning” comes from a need for dissonance in time and space, akin to the disruption that winter storms can cause in those places that get maybe one a year — enough to be familiar with them, but not enough so to deal with them. Listen for the dual themes of swirling wind and heavy blobs of snow pounding (as much as they can) on the windows.

## Sun Return (1996) During the writing and recording process, there were several works that didn’t fit into anything else. These three pieces have very little in common aside from being the only ones that didn’t fit in with anything else, so when I cleaned house they all went into the same bucket. The name comes from the last piece, “Fanfare to the Sun,” which was written for one of my Mother’s “Solstice Parties,” which she held for several years instead of a Christmas party: the darkest time of year turning around and heading for spring. The cover art is a photo of one of the spring Crocus in my yard.

“Aori Harp” started life in the dorm my senior year at college, but languished for several years. Parts of it are antiphonal, parts interwoven.

After I graduated from college, I acquired a small folk harp which played a central part in the creation of “Firelight Dance.” Again, it languished for some time before begin expanded into an ABA form and recorded.

“Fanfare to the Sun” was originally written for one of my mother’s Solstice parties, performed via tape (before CD burners became affordable) for the attendees at the height of the party, where it was well-received. Being wholly done via synthesizer my “performance” of it was limited to sticking a tape in the tape player and pressing play, but that’s close enough for me. Listen for the marimba to come alive near the end to usher in the lengthening days.