# Administrative Information - **Class Meeting**: Friday, September 4, 2026, 1:00–1:50pm — **Coolbaugh 212**. *(Monday 9/7 is Labor Day — no class.)* - Previous meeting: [[MATH310F26(Day 5) - The Dirac delta, and the idea of a distribution]] - 📄 [[Teaching/MATH310/MATH310F26/Administration/MATH310_F26_Syllabus_1.0.pdf|MATH 310 F26 Syllabus, v1.0]] > [!info]- Admin. Notes > Expect an LR and WB for each lecture. WBs target 15-minute engagements (longer early on); LRs are short, ~5 minutes. > > - 🎯 **[LR] [Day 6 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121144)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Wednesday 9/9, 11:59pm** (Labor Day pushes the usual Monday) > - 🎯 **[LR] [Day 6 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121145)** — one free response, no wrong answers — due **Wednesday 9/9, 11:59pm** > - 🎯 **[LR] [Day 5 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121118)** — 4 questions drawn from Wednesday's material, unlimited attempts, highest kept > - 🎯 **[LR] [Day 5 Reflection](https://elearning.mines.edu/courses/81632/quizzes/121119)** — one free response, no wrong answers > - 🎯 **[WB] [Workbook 5](https://elearning.mines.edu/courses/81632/assignments/585806)** — prove $e^{i\theta}=\cos\theta+i\sin\theta$; guardrails on the sheet; upload one PDF > - 📄 [[MATH310F26_WB5.pdf|Workbook 5 (blank, 1 page)]] > - Start with the companion: [[MATH310F26(Workbook 5 Companion) - Splitting a series by parity]] > - 🎯 **[CA] [Dot-in-a-square uploads](https://elearning.mines.edu/courses/81632/assignments/585632)** — Thursday 11:59pm target, Labor Day outer bound; unprimed participants only > - 🎯 **[CA] Three Questions, Revisited** — restate your three questions, then push one toward a first result; type your responses in the [Canvas quiz](https://elearning.mines.edu/courses/81632/quizzes/121139), **or** print the two-sided form and attach it there — due **Wednesday 9/9, 10:00am** > - 📄 [[MATH310F26-ThreeQuestionsRevisited.pdf|Response form (2-sided, print at home)]] > - 🎯 **[CA] [Yahtzee Dice Data](https://elearning.mines.edu/courses/81632/assignments/586037)** — at least 3 games under the **one-roll house rules** (announcement below); upload a scan or clear photo of your play sheet — due **Friday 9/11, 10:00am** > - 📄 [[Yahtzee Score Sheet - Large Print.pdf|Score card (front-side scoring rules)]] > - A Canvas check-in on **project leanings** lands after today, over the weekend > [!info]- 📣 Announcement — Yahtzee data: one roll per turn (no re-rolling) > A couple of you asked after class how Yahtzee is played, and specifically about re-rolling. Good question, and it surfaced something worth fixing: **for our purposes we are NOT playing standard Yahtzee** (I had totally forgotten this aspect of the game). > > **New house rules — one roll per turn:** roll all five dice once, choose an open category, write the five dice values in that category's row on the **play sheet** (the back side of the handout), and enter the score using the front card's rules. Each category gets used once, so thirteen turns completes a game. **No re-rolling at any point.** > > **The reason:** with a single roll, every turn is a fresh sample of five fair dice, and that keeps the data clean for what we will do with it. The interesting decision becomes *where you put the hand you rolled*. > > Please play and record **at least 3 games**; more are welcome. If you already played some turns in class *with* re-rolling, keep that data but write **"RR"** at the top of those games so I can separate them. > > Upload a scan or clear photo of your play sheet to the Canvas assignment (*Class Activity — Yahtzee Dice Data*) by **Friday 9/11 at 10:00am**. > [!example]- 🗃️ COMAP Corner — three from the archive > A standing feature: each day, three problems out of the [MCM/ICM archive](https://www.contest.comap.com/undergraduate/contests/mcm/previous-contests.php) — one statistics-flavored, one computational, one weird. Any of them can seed a Project 1 or a Project 2; the full 208-row catalog lives on the [[MATH310F26(Day 4) - Differencing, the regression run, and a name for the pipeline|Day 4 page]]. > > - **Computational — Space Junk (2016 MCM B).** With over 500,000 tracked pieces of orbital debris threatening spacecraft — dramatized by the 2009 Kosmos–Iridium collision — teams build a time-dependent model evaluating removal methods (lasers, water jets, sweeper satellites) and work out whether a private debris-removal company could actually turn a profit, or whether avoidance is the only realistic strategy. Classic optimization-under-simulation, with a genuinely current-feeling premise. [comap.com](https://www.contest.comap.com/undergraduate/contests/mcm/contests/2016/problems/2016_MCM_Problem_B.pdf) > - **Data/stats — Predicting Wordle Results (2023 MCM C).** Given a year of daily NYT Wordle data — the word, number of reporters, hard-mode participation, the guess distribution — teams have to explain why the reported-results count swings so much day to day, build a prediction interval for a future date, and forecast the guess distribution for a word that hasn't been played yet. Straightforward regression and interval estimation on a dataset everyone already has intuition for. [comap.org](https://www.comap.org/membership/member-resources/item/predicting-wordle-results) > - **Just weird — The Velociraptor Problem (1997 MCM A).** Model how a velociraptor — fast but low-stamina — should pursue an enduring-but-agile prey animal like *Thescelosaurus*, accounting for detection range, turning radius, top speed, and fatigue; then extend it to a coordinated two-raptor hunt, and check the results against real lion/tiger predator–prey data. Genuinely a dinosaur pursuit-and-evasion differential equations problem. [comap.com](https://www.contest.comap.com/undergraduate/contests/matrix/PDF/1997mcmProblems.htm) > [!abstract]- The random-dots experiment has a paper — and the data in it is ours > The dot-in-a-square experiment we are re-running was published as: **Amanda S. Hering, Luke Durell & Grant Morgan (2021), "Illustrating Randomness in Statistics Courses With Spatial Experiments,"** *The American Statistician*, 75:3, 343–353. [doi:10.1080/00031305.2020.1871070](https://doi.org/10.1080/00031305.2020.1871070). The first author taught spatial statistics here before moving to Baylor, and the original data was collected at Mines — "the survey was distributed to the students in two large sections of ordinary differential equations, collecting nearly 200 responses." > > **Their abstract, in full:** > > > Understanding the concept of randomness is fundamental for students in introductory statistics courses, but the notion of randomness is deceivingly complex, so it is often emphasized less than the mechanics of probability and inference. The most commonly used classroom tools to assess students' production or perception of randomness are binary choices, such as coin tosses, or number sequences, such as dice rolls. The field of psychology has a long history of research on random choice, and we have replicated some experiments that support results seen there regarding the collective distribution of individual choices in spatial geometries. The data from these experiments can easily be incorporated into the undergraduate classroom to visually illustrate the concepts of random choice, complete spatial randomness (CSR), and Poisson processes. Furthermore, spatial statistics classes can use this point pattern data in exploring hypothesis tests for CSR along with simulation. To foster student engagement, it is simple to collect additional data from students to assess agreement with existing data or to develop related, unique experiments. All R code and data to duplicate results are provided. > > **What was done.** Three surveys — the same slip we used, plus two variants with a printed "landmark" dot (Figure 1): 314 replies for the empty square, 340 with a dot in the center, 267 with a dot in the upper right. > > ![[hering2021_fig1.png]] > > The pattern they expected comes from **Blum's grassfire theory**: "imagine that the edges of a shape are all simultaneously lit on fire, and the fire consumes the shape as it moves toward the center" — the fire fronts meet along "lines of annihilation" and end at the "quench point," and "Blum hypothesized that participants will choose points close to the lines of annihilation and the quench point" (Figure 2). For a square, the lines of annihilation are the diagonals — the **X** we talked about Wednesday. > > ![[hering2021_fig2.png]] > > **What was found.** Exactly that (Figure 3): "The results were surprising to us because many selections were made in the very center of the square, as if students were trying to identify the 'bullseye.'" And the punchline for our data-gathering: "while each point is chosen independently of the others, a collective preference appears to exist for making choices in some locations over others." The landmark dot did *not* do what they hypothesized: "we hypothesized that the presence of the landmark dot would repel choices nearby, but that does not seem to be the case. In fact, the landmark dot appears to attract *more* attention" — with the finer verdict that "there is evidence of attraction to the landmark dot at small distances, but evidence of repulsion at medium distances." > > ![[hering2021_fig3.png]] > > **Our data so far.** The first 54 selections from your returned slips, extracted to coordinates (upload spot stays open through Labor Day): > > ![[day6_dots_scatter_n54.png]] > > Squint: the diagonal tendency and an off-center cluster are already visible at $n=54$. And because our slips use the same unit square as the paper's survey, we can draw our selections directly onto the paper's empty-square data, Figure-9 style — **F26 in blue over the original 314 in gray:** > > ![[day6_dots_on_fig3.png]] > > **What we are about to do is literally Figure 9.** They overlaid a later class's 34 fresh selections on the original 314 and asked whether the new points are consistent with the old pattern — closing on the lesson our own scans will test: "even though each selected point is independent of the others, a pattern among the collective points does form, illustrating that subconscious similarities of thought or choice may exist among people, so guarding against biases in random samples is extremely important." Keep their §4.2 warning in mind as our count grows and the heat map comes back: "'Random' does not mean 'uniform' or 'regular.' Sometimes a random sample may not look representative, but that does not make it any less random." > > ![[hering2021_fig9.png]] > > *Figures 1, 2, 3, and 9 reproduced from Hering, Durell & Morgan (2021) for classroom discussion; captions in the original. The acknowledgments thank the Mines students and faculty who collected the data — including a familiar name.* # Deliverables - 🎯 **[LR] [Day 6 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121144)** — 4 questions drawn from today's material, unlimited attempts, highest kept - 🎯 **[LR] [Day 6 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121145)** — one free response, no wrong answers - 🎯 **[CA] [Three Questions, Revisited](https://elearning.mines.edu/courses/81632/quizzes/121139)** — type in the quiz or attach the printed form; due Wednesday 9/9 at 10:00 am - 🎯 **[CA] [Yahtzee Dice Data](https://elearning.mines.edu/courses/81632/assignments/586037)** — at least 3 games, one-roll house rules; due Friday 9/11 at 10:00 am Questions and statement are due Wednesday 9/9 at 11:59 pm (Labor Day Monday, no class). --- # Lecture Boards + Transcript + GenAI **Friday before Labor Day** — no class Monday, and old man Scott will have you know that once, long ago, we used to go to school on Labor Day. Sixth lecture, and the agenda is a send-off: close Wednesday's cliffhanger (what the derivative of the step *is*), answer the why-care with a die roll, hand out actual dice, and load the long weekend with things to wonder about. In the words of a certain cardiganed philosopher: *do you ever just wonder?* Classic Mr. Rogers. ## The website, mirrored Class opened with a tour of this very page — the stated policy being to stop *extending* the site silently and start *mirroring* it in the room. Two features got their introductions: **COMAP Corner.** The contest archive lives in a spreadsheet of some 208 rows, each row a real open-ended problem with links back to [COMAP](https://www.comap.com/) for details (the half-pipe-for-maximal-trick-points problem got quoted again). That spreadsheet went into an LLM, and the standing request is three problems a day — one computational, one data/stats, one weird — "until it really starts getting mind-numbing, and then maybe I'll stop. But when it ends with a velociraptor, that's just asking me to look again." Today's three are in the callout above: space junk (a pseudo-economic cost-benefit of clearing orbit), Wordle prediction, and the velociraptor pursuit problem — turning radius, top speed, fatigue. **The random dots, and where our data comes from.** The dot-in-a-square adventure has a Mines origin story: the data in the 2021 paper was collected here in the early teens — a statistician ran the study, your instructor helped hand out the slips, and it took a computer-science faculty member and a student to deal with digitizing the returned squares. Our re-run uses the same idea, now bot-programmed: pixelize each scan, find the square by its dark edges, and if a square comes back skewed, use linear algebra to transform it back; then difference against a blank original, and whatever changed is your dot. Current census: 73 captures examined, 54 verified cleanly (those are the dots in the overlay above), and about 19 more promoted after inspecting what the code identified — if you drew an X and the code called the crossing your point, that seems reasonable; keep it. Eyeball verdict from the room: yes, it's leaning the way the teens data leaned — center-heavy, with the X forming. > [!tip]- Project bait, hiding in the dots > It's not entirely clear what "the model" is here — and that's the point. "Sometimes in the modeling world, it's just as valuable to ask an interesting question and start probing the question, and then think about the model afterwards." The paper's own variants (move the landmark dot around) suggest others: change the shape entirely. "I don't know what I would do with a triangle. I barely even knew what I was doing with a square. I just put a dot in there. It's the data accumulated over people that tells us some structure." ## Closing the cliffhanger: what $H'$ is **Recall.** A function $f$ *induces* a distribution: $T_{f}(\varphi)=\langle f,\varphi\rangle=\int_{-\infty}^{\infty}f(t)\,\varphi(t)\,dt$, a machine that eats a test function $\varphi$ and reports a scalar. The test function is *very nice*: smooth, with $|\varphi|\to 0$ as $|t|\to\infty$ — and not just $\varphi$ but every derivative $\varphi^{(n)}$, all decaying faster than any polynomial. (Wednesday's Schwartz class.) For a physical picture, the analyst Robert Strichartz of Cornell would have you think of $\varphi$ as your literal *probe* stuck into a cooked object whose temperature field is $f$: the integral accumulates the field over the probe's profile and hands you the reading. A distribution is what a thermometer knows. **The target** is $T_{H'}(\varphi)$, where $H$ is the Heaviside step — $0,0,0,\dots$ then a jump at $t_{0}$, then $1,1,1,\dots$ — $T_{H'}(\varphi)=\langle H',\varphi\rangle=\int_{-\infty}^{\infty}H'(t-t_{0})\,\varphi(t)\,dt.$ The symbols write down fine; the rub is $H'$. A jump discontinuity has no instantaneous rate of change at the jump, so the integrand contains a thing we cannot classically name. Meta-problem noted — now forget the meta and treat it as a single-variable integral, because integrals we have skills for. Can't do it directly (it's no power, exponential, or sinusoid), so pull the next trick from the calculus toolbox: $u$-substitution — which fails spectacularly, as $u$-subs do. Choose $u=H'$ and you're differentiating the thing you were worried about in the first place; choose $u=\varphi$ and you need a $\varphi'$ in the integrand that isn't there. After $u$-sub fails, who do you try next? **Integration by parts** — the wild one, the one that sometimes solves your problem and sometimes just keeps cycling. Choose it to do the one thing we need: *get the derivative off* $H$. Take $dv=H'(t-t_{0})\,dt$ so $v=H(t-t_{0})$, which forces $u=\varphi$ and $du=\varphi'\,dt$ — and $\varphi$ is great, $\varphi$ has all the derivatives you could ever want to hand it: $T_{H'}(\varphi)=\varphi(t)\,H(t-t_{0})\Big|_{t=-\infty}^{t=\infty}-\int_{-\infty}^{\infty}H(t-t_{0})\,\varphi'(t)\,dt.$ The boundary term dies the way these arguments always hope it will: at the top, $H\to 1$ but $\varphi\to 0$, and $0\cdot 1$ cannot oppose anything; at the bottom both are $0$. So $0\cdot 1-0\cdot 0=0$. In the remaining integral the step does its only trick — it is $0$ for $t<t_{0}$ and $1$ for $t\ge t_{0}$ — so the integration domain truncates: $T_{H'}(\varphi)=-\int_{t_{0}}^{\infty}\varphi'(t)\,dt=-\Big(\varphi(t)\Big|_{t_{0}}^{\infty}\Big)=-\big(0-\varphi(t_{0})\big)=\varphi(t_{0}).$ Ta-da — big finish. Look at what came out: of all the heights $\varphi$ takes, the integral *sifted out exactly one*, the value at $t_{0}$. That is precisely delta's defining behavior. So the consequence stands: $H'(t-t_{0})=\delta(t-t_{0})\quad\text{in the sense of distributions.}$ Delta isn't a function. Delta is a distribution, made sense of by pairing with a test function; the abuse of notation from differential equations is accepted because *this* is how the thing works underneath. Any attempt to read $H'$ as a proper function is just sweeping Schwartz's 1950s theory under the rug — and better this derivation, which is honest integration by parts, than the three axioms alone, "because we can handle the integration." > [!warning] The red-chalk moment > Mid-derivation a negative sign went missing on the board — "what got lost in the mix?" — and the room chased it down; the recovery is in red on the second board below, along with the little green $0$ and red $1$ marking which factor kills the boundary term at each end. A minus times a minus made the big finish positive, as it must: $T_{H'}(\varphi)$ had better report $+\varphi(t_{0})$ if $H$ rises. > [!tip]- The transferable trick > Integration by parts is *another way of moving a derivative around inside an integral*: it switches who takes the derivative. Here $H$ couldn't accept one, $\varphi$ accepts them all, so the derivative moved. That single trick — pass derivatives from the rough function to the smooth one — is the engine of weak derivatives, of distribution theory, and of half the numerical methods you will ever meet. It comes up a lot. ## Why care? A die roll, distributionally Suppose $X$ is a random variable chosen from $U(1,6)$, the continuous uniform distribution: any number between $1$ and $6$, all equally likely, density $\tfrac{1}{5}$ on the interval (normalize your rectangle) and $0$ outside. Numbers came out of actual rolls: a 3, another 3, a 1 — definitely in there. But what will a die *never* produce? A $1.5$, or any of the decimals between the integers. The uniform model is silly for a die. So go Feynman-style space alien: you don't know how dice work, you just see outcomes clustering on six values. A first guess that respects your ignorance: narrow Gaussians localized at each integer. Then idealize — shrink them. Set the variance to $\tfrac{1}{n}$ and let $n$ grow: the width collapses while the normalization pins one unit of area under each curve throughout the squeeze. What delta *is*, operationally, is the distributional pairing that results in that limit. ![[day6_delta_comb.png]] > [!note]- The kooky fine print > The Gaussian is not special: infinitely many sequences of wild functions — bumpy ones, oscillatory ones — limit to the same spike. Delta doesn't remember which sequence built it. In that way it's a little kooky, and entirely a creature of the pairing $\langle\delta,\varphi\rangle=\varphi(t_{0})$ rather than of any pointwise graph. Now write the die's probability density in delta language: $f_{X}(x)=\delta(x-1)+\delta(x-2)+\cdots+\delta(x-6)$. What's the probability of finding $X$ at $1.5$? No delta spikes there — zero, exactly as a die demands. But run the sanity check that every density must pass — integrate over all events: $\int_{-\infty}^{\infty}f_{X}(x)\,dx=\sum_{i=1}^{6}\int_{-\infty}^{\infty}\delta(x-i)\,dx=\sum_{i=1}^{6}1=6.$ That's not right — and no, the 600% chance joke cannot be said. Nothing was normalized: six equally weighted outcomes need weight $\tfrac{1}{6}$ each, and the green ink on the board supplied it: $f_{X}(x)=\frac{1}{6}\sum_{i=1}^{6}\delta(x-i),\qquad\int_{-\infty}^{\infty}f_{X}(x)\,dx=\sum_{i=1}^{6}\frac{1}{6}=1.$ This is the codex between the two probability worlds: continuous probability *densities* and discrete probability *mass functions* are one theory once deltas are legal, everything living inside integrals. Physicists run the same codex in the other direction — delta as a *point mass*. A thrown eraser spins wildly, but the parabola you see traced is the center of mass, the average point where the mass accumulates over the mass density — no different from your statistical averages. > [!success]- Numbers checked in one sandbox run > With $\varphi(t)=e^{-t^{2}/8}\cos t$ and $t_{0}=1.7$: the nascent-delta pairing $\int\delta_{n}(t-t_{0})\varphi\,dt$ ran $-0.0579,\,-0.0864,\,-0.0897,\,-0.089779$ at $n=10,10^{2},10^{4},10^{6}$, against $\varphi(t_{0})=-0.089779$; the by-parts side $-\int_{t_{0}}^{\infty}\varphi'\,dt$ returned $-0.089779$ exactly. The unweighted comb integrated to $6.000000$, the $\tfrac{1}{6}$-weighted comb to $1.000000$, and the weighted density at $x=1.5$ evaluated to $5\times 10^{-6}$ at moderate width, $0$ in the limit. ## Zero-dimensional tweezers One last piece of funk, freely admitted. Picture $\varphi$ as a graph — a curve. A curve lives in two dimensions but *is* one-dimensional: one independent variable goes in, and even bent all around it's a bent-up line. After delta does its magical work, what comes out is a *point* on that curve — dimension zero. So the delta is a pair of very special zero-dimensional tweezers reaching into a one-dimensional object and plucking out a single point. "That is what we made rigorous in the '50s. I know of no tools that do this, but there are tools that come close." ## Dice out, data in Why roll dice at all? Because the course will eventually define **stochastic processes in time**, and that requires random variables — more rolls than any one person can produce. So: everyone got a bag of five dice (office dice, kid dice, store dice — provenance varies, fairness assumed), and the front of the handout is a Yahtzee card while the back is a grid of play sheets. The ask, over the long weekend: play at least three games and **record the five dice values every turn** — the values are the data; the game is the trade for collecting it. Fold your record sheet when done, and the dice come back to the office. A sharp question from the room: doesn't *choice* corrupt the distributions — three 6's and two 5's could be scored as sixes, full house, or three-of-a-kind depending on what you think pays? Answer, live: yes, the category outcomes are choice-dependent (there's even a temporal strategy to it — bank now or hope), and in post we could compare everyone's achieved scores against the hindsight-optimal score their rolls allowed. But the *dice values themselves* are untouched by strategy — and that's the data we're after. > [!warning] Rules amended after class > The in-class handout conversation assumed re-rolling was fine; it isn't, for our purposes. See the announcement above: **one roll per turn, no re-rolling** — every turn a fresh sample of five fair dice. Games already played with re-rolls stay in, marked "RR." Upload details and the due date (Friday 9/11, 10:00am) are in the Deliverables list. ## Wild ideas to carry into the weekend The send-off had two exhibits. First, the **El Niño shelf** went up on Canvas — the ENSO chapter and a stack of videos; that material now lives in the next section, for the many who signaled interest. Second, the room watched the class cut of the *Earworm* episode — the self-similarity matrices of the **Modeling Perspectives** section below, live. The cut kept the good cognition bits: the speech-to-song illusion (repeat a spoken phrase enough times and it starts to *sing* — discovered by the psychologist Diana Deutsch in 1995 while editing a CD), and the study in which excerpts of Elliott Carter's deliberately non-repetitive music, digitally adulterated to insert literal repetition, were rated not only more enjoyable but *more likely to have been composed by a human* than the originals. Repetition isn't beneath music; it's load-bearing. And then the modeling punchline, delivered over the credits: they took song lyrics as rows and columns and put a $1$ wherever words matched. "Now you've got a matrix. What's the first thing you're going to want to find out about a matrix? Its **eigenstructure**, right? Maybe that tells us something. If nothing else, there are more analyses to do on a matrix." The problems can come from anywhere. Closing instruction for the weekend, verbatim spirit: *do not pigeonhole yourself too strongly into one idea, at least for the time being — go forth over Labor Day and imagine and wonder.* See you Wednesday. > [!example]- Board scans — September 4, 2026 > ![[MATH310F26-Day6-Board-1.jpg]] > ![[MATH310F26-Day6-Board-2.jpg]] > ![[MATH310F26-Day6-Board-3.jpg]] > ![[MATH310F26-Day6-Board-4.jpg]] > ![[MATH310F26-Day6-Board-5.jpg]] > > 📄 [[MATH310F26-Day6-Boards-2026-09-04.pdf|Full board capture (5 pages, PDF)]] --- # Project steering — the El Niño shelf Brainstorming land stays open — nothing needs to be locked in for a while yet, and the Canvas check-in on your leanings arrives over the weekend. For the sizable contingent leaning toward climate: The first reading is already up: the El Niño (ENSO) chapter of Hans Kaper & Hans Engler, *Mathematics and Climate* (SIAM, 2013) — [on Canvas, under Modeling Materials → Weather and climate](https://elearning.mines.edu/courses/81632/files/folder/Modeling%20Materials/Weather%20and%20climate?preview=9614583). - The chapter is short on climatology and long on modeling. After a few pages on what ENSO *is* — trade winds piling warm water in the west, a tilted thermocline, the three-to-seven-year swing between El Niño and La Niña — it builds two toy oscillators for the eastern-Pacific temperature anomaly: a **recharge oscillator** (a planar ODE system, analyzed by trace/determinant and a Hopf bifurcation, Scott has more information about Hopf bifurcations for those who want to work with these materials) and a **delayed oscillator**, in which Kelvin and Rossby waves crossing the basin (roughly two and six months, respectively) become time delays in a scalar equation for $T(t)$. Both wave types get a derivation from first principles — dispersion relations, phase vs. group velocity — and the chapter closes with a primer on delay differential equations and some numerical experiments. - The videos below supply the physical picture the chapter assumes. The first two cover ENSO itself — normal vs. El Niño conditions in the tropical Pacific, and the eastward Kelvin wave as a subsurface rebound of the thermocline (a tank demo, not a breaking surface wave). The last two cover Rossby (planetary) waves, mostly in their atmospheric guise as meanders of the jet stream, whose stalling and resonance are now linked to persistent weather extremes.     --- # Modeling Perspectives The dot experiment above and everything below share a pretense worth naming: **a model does not have to look like mathematics to be one.** On Day 1 the claim was $\text{Context}+\text{Symbols}=\text{Math Model}$, and nothing in that equation says the symbols must be of a particular form. In the following, I offer to you a [Vox Earworm documentary](https://www.imdb.com/title/tt9563798/) (many of which were quite good BTW), but first we define the mathematical object/symbols and their visualization, which they will use. The point here is to encourage you to **think broadly about context** before trying to pin down the mathematics. Surely, *we* have something to say about the matter. ## The self-similarity matrix and its recurrence plot Take any sequence — the words of a song/book, samples of a signal, states of a system — $s_{1}, s_{2}, \dots, s_{n}$. Build the $n\times n$ grid whose cells record exactly one question, asked of every pair: $M_{ij} = \begin{cases} 1 & \text{if } s_{i} = s_{j},\\ 0 & \text{otherwise,}\end{cases}$ and darken the cells where the answer is yes. This is a **[self-similarity matrix](https://en.wikipedia.org/wiki/Self-similarity_matrix)** — the name SongSim and the video below use — with equality as the bluntest possible notion of "similar." The same grid answers to two other names. For continuous states one asks $\lVert s_{i}-s_{j}\rVert < \varepsilon$ instead and calls the picture a **[recurrence plot](https://en.wikipedia.org/wiki/Recurrence_plot)** — a standard tool in dynamical systems, which is not a coincidence given where this course lives. And a graph theorist reads $M$ as an **[adjacency matrix](https://en.wikipedia.org/wiki/Adjacency_matrix)**: make each position in the sequence a node, join two positions whenever their entries match, and $M$ is exactly the 0–1 bookkeeping of that graph's edges. Same object, three vocabularies. Three facts follow immediately from the definition, and they are the reading guide: - The **main diagonal** is always dark — everything equals itself. - The picture is **symmetric** across that diagonal — if $s_i=s_j$ then $s_j=s_i$. - A **dark line parallel to the diagonal** means a whole *passage* repeated later, entry for entry: that is a chorus. Solid **blocks** are one token repeated back-to-back; **checkerboards** are two tokens alternating; a dark full **row/column stripe** is a token that shows up everywhere. Here is a real one — each cell compares two words of a song's lyrics: ![[day6_recurrence_matrix.png]] Read it like terrain: the off-diagonal streaks are the chorus returning, the dense blocky neighborhoods are repetition-heavy sections, and the empty regions are verse text that never comes back. Notice what we modeled *away*: every actual word. The matrix cannot tell you what the song says — only how the song is *built* — and for some questions that is precisely the information you want. **Make your own.** These pictures come from [SongSim](https://colinmorris.github.io/SongSim/#/gallery), Colin Morris's browser tool — one word per cell, any lyrics you paste, and a gallery from Bohemian Rhapsody to Bad Romance. Ten minutes in the gallery will teach your eye the vocabulary above. ## The video 🎬 The class cut, edited down for our time budget:  > [!quote] Full attribution > The video above is my edit of **"Why we really really really like repetition in music"** from **Vox's *Earworm* series** (Estelle Caswell, featuring Colin Morris and his SongSim matrices) — the original, worth watching in full: [youtube.com/watch?v=HzzmqUoQobc](https://www.youtube.com/watch?v=HzzmqUoQobc). All credit for the reporting, animation, and analysis belongs to Vox. The *Earworm* episode takes the matrix further than we do here: what repetition has done to pop lyrics across six decades, and why the brain seems to *want* the diagonal streaks. As you watch, keep the modeling frame on: someone chose to represent a song as a yes/no matrix, and that strange choice made a sixty-year trend in music **visible, countable, and arguable** — which is what a model is for. ---