# Administrative Information - **Class Meeting**: Friday, October 9, 2026, 1:00–1:50pm — **Coolbaugh 212** - Previous meeting: [[MATH310F26(Day 18) - What the walk remembers]] - 📄 [[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 18 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121668)** · **[Statement](https://elearning.mines.edu/courses/81632/quizzes/121669)** — due **Friday 10/9, 11:59pm** > - 🎯 **[LR] [Day 19 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121759)** — 4 questions drawn from today's boards and the Media Reflection page, unlimited attempts, highest kept — due **Friday 10/16, 11:59pm** > - 🎯 **[LR] [Day 19 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121760)** — one free response, no wrong answers — due **Friday 10/16, 11:59pm** > - 🎯 **[MR] [Media Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121701)** · **[Statement](https://elearning.mines.edu/courses/81632/quizzes/121702)** · [[The Breakdown of Linear Stability Analysis and Critical Slowing Down|the page]] (applets, about an hour, quiz filled in as you go) · due **Friday 10/23, 11:59pm** > - 📝 **[MA1 — Model Selection](https://elearning.mines.edu/courses/81632/assignments/588667)** · [[MATH310F26-MA1-ModelSelection.pdf|guide]] — due **Friday 10/9, 11:59pm** > - 📝 Next on the calendar: **[MP3 — Initial Model and Test](https://elearning.mines.edu/courses/81632/assignments/590872)** and **[MA2 — Source and Background](https://elearning.mines.edu/courses/81632/assignments/590873)**, both due **Monday 10/26, 11:59pm** (MP3 moved off the fall-break week; both since moved from Friday 10/23; [[MATH310F26 - Project Work|Project Work]]) > - 🎯 **[WB] [Workbook 11](https://elearning.mines.edu/courses/81632/assignments/588368)** · [[MATH310F26_WB11.pdf|blank]] · [[MATH310F26(Workbook 11 Companion) - Three outcomes, two moments|companion]] — due **Friday 10/30, 11:59pm** > - ✍️ **Workbook check-in, in person** — done today, first 15 minutes, one workbook problem of your choice > - 🎬 [[MATH310F26 - Random numbers, two videos]] — the guided videos for this unit, if you haven't yet --- # Deliverables - 🎯 **[LR] [Day 19 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121759)** — 4 questions drawn from today's boards and the Media Reflection page, unlimited attempts, highest kept — due **Friday 10/16, 11:59pm** - 🎯 **[LR] [Day 19 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121760)** — one free response, no wrong answers — due **Friday 10/16, 11:59pm** - 🎯 **[MR] [Media Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121701)** · **[Statement](https://elearning.mines.edu/courses/81632/quizzes/121702)** — work through [[The Breakdown of Linear Stability Analysis and Critical Slowing Down|the page]] with its applets — due **Friday 10/23, 11:59pm** - 📝 **[MA1 — Model Selection](https://elearning.mines.edu/courses/81632/assignments/588667)** — due **Friday 10/9, 11:59pm** Questions and statement are due Friday 10/16 at 11:59 pm (Monday and Tuesday are fall break; the due date was moved to Friday). --- # Lecture Boards + Notes > [!warning] No audio for this meeting > The recorder was full on Friday, so there is no transcript. This section is reconstructed from the four board captures, the instructor's own notes on the hour, and the language of the [[The Breakdown of Linear Stability Analysis and Critical Slowing Down|Media Reflection page]] that was shown in class. Quotes are absent on purpose; the mathematics is as it appeared on the boards. **Nineteenth meeting, October 9 — the walk's memory, computed.** Three movements. The first fifteen minutes were the in-person workbook check-in. Then a tour of the course's first **Media Reflection** — a page of readings and watchings built around applets, so that a two-week discussion from an ODE course (bifurcations of scalar systems, the breakdown of linear stability, [critical slowing down](https://en.wikipedia.org/wiki/Critical_transition)) is compressed to its basic principles and *seen* rather than derived — ending with a truck, a flatbed, and a sway. Then back to chalk: the rest of the [covariance](https://en.wikipedia.org/wiki/Covariance) properties, the easy payoff that **the [variance](https://en.wikipedia.org/wiki/Variance) of a sum of [independent](https://en.wikipedia.org/wiki/Independence_(probability_theory)) variables is the sum of the variances**, and the start of the computation this unit has been aiming at — the covariance of the walk with itself $k$ steps later — stopped just short of defining their *correlation*, which is Monday's. ## The workbook check-in Fifteen minutes, one workbook problem of your choice, on your own paper or the printed sheet you asked for on Wednesday. As advertised: not a quiz, a verification that a person was present and did the thing. ## The Media Reflection, and what it is for The page shown was [[The Breakdown of Linear Stability Analysis and Critical Slowing Down]], the first of the course's Media Reflections: a reading with things to *do* — three applets and a video — and a Canvas quiz you fill in as you go (questions in the **Quiz** boxes on the page; multiple choice into the Questions quiz, free response into the Statement quiz, due Friday 10/23). Budget about an hour. The applets run only on **scottastrong.org**; an empty box means you've landed on the publish.obsidian.md copy. What the page is for, in the language it uses. A **[bifurcation](https://en.wikipedia.org/wiki/Bifurcation_theory)** is a change in the number of [equilibria](https://en.wikipedia.org/wiki/Equilibrium_point_(mathematics)) of an [autonomous system](https://en.wikipedia.org/wiki/Autonomous_system_(mathematics)) *or* in their stability, caused by turning a parameter. The page produces two, in the pond from the first week, $P'=rP(1-P/N)-H$ with $r=N=2$: with a **constant quota** $H$ (Model A), the two equilibria $P^{*}_{\mp}=1\mp\sqrt{1-H}$ slide toward each other and **collide and vanish** at $H^{*}=1$ — a **[saddle-node](https://en.wikipedia.org/wiki/Saddle-node_bifurcation)** (fold, or "blue-sky") bifurcation; with a **constant effort** $HP$ (Model B), the equilibria $P=0$ and $P=2-H$ **pass through each other and swap stability** at $H=r$ — a **[transcritical](https://en.wikipedia.org/wiki/Transcritical_bifurcation)** bifurcation. In both, the bifurcating equilibrium is a root of $f$ *and* a local extremum, $f(y^{*})=0$ and $f'(y^{*})=0$, which is exactly where **[linear stability analysis](https://en.wikipedia.org/wiki/Linear_stability) has nothing to say**: the linearization $\eta'\approx f'(y^{*})\eta$ gives $\eta(t)\approx\eta_{0}e^{f'(y^{*})t}$ with **[relaxation time](https://en.wikipedia.org/wiki/Relaxation_(physics))** $\tau=1/|f'(y^{*})|$, and $\tau$ grows without bound as the bifurcation approaches — for Model A, $\tau=1/(2\sqrt{1-H})$, which is $0.71$ at $H=0.5$, $1.58$ at $H=0.9$, $5.0$ at $H=0.99$. At the bifurcation the quadratic term takes over, $\eta'\approx c\,\eta^{2}$, and the return is **algebraic**, $\eta\sim 1/(|c|t)$, instead of exponential; the equilibrium is **half-stable** (the half-filled circle on the [phase line](https://en.wikipedia.org/wiki/Phase_line_(mathematics))). The slowing that is visible *before* the bifurcation — a healthy-looking equilibrium whose recovery from a small disturbance gets slower and slower — is **critical slowing down**. The race applet makes it a number: same push, same 1% target, the return takes about $1.8$ time units at $a=1$, about $13$ at $a=0.01$, and $98$ at the bifurcation $a=0$. **The modeling point**, stated in class and on the page: complex, nonlinear, interacting dynamical systems can have *fundamental changes of state* — bifurcations — and while we probably cannot write down the system of initial value problems behind a given high-dimensional dynamic, we can **infer that a bifurcation may be occurring if we are witness to slowing down**. In many variables, the [Jacobian](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant) $J$ replaces $f'(y^{*})$, a disturbance breaks into pieces that go like $e^{\lambda t}$ for each [eigenvalue](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors), and a bifurcation is an eigenvalue (or a pair) being pushed to zero real part; the piece whose $\lambda$ is closest to zero is the last to die out, and that is what slows. **The flatbed-loading problem.** The page's Part 3, discussed in class: the Weigh Safe tongue-weight demonstration — a truck towing a flatbed, where the only thing changed between runs is *where the load sits*, which sets how much of its weight presses down on the hitch. Two steady behaviors: the flatbed **tracks straight** behind the truck and shrugs off a bump, or it settles into a **long-lived oscillation**, swaying and dragging the truck with it — not an equilibrium point but playing the same role (a [limit cycle](https://en.wikipedia.org/wiki/Limit_cycle), in more than one dimension). Somewhere between a load position that tracks and one that sways there is a value where one gives way to the other: that value plays the role of $H^{*}$, and it is a bifurcation. Set the load on the stable side but near that value, and a bump should die out — slowly, and more slowly the closer you are. That is the abstraction the page wants carried: a parameter, two kinds of steady behavior, a value in between, and slowing before the switch. > [!example]- The Media Reflection page, shown in class > [[The Breakdown of Linear Stability Analysis and Critical Slowing Down]] — Model A and Model B harvesting applets, the exponential-vs-algebraic race applet, and the flatbed video. Related: [[MATH310F26(Day 1) - Instructor information and introduction to the course|Day 1]] (the pond), [[MATH310F26(Workbook 1 Companion) - The phase line and what f-prime tells you|the Workbook 1 companion]] ($f'$ at an equilibrium is a rate). ## Covariance, the rest of the properties Recall, for random variables $X$, $Y$: $\mathrm{Cov}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\,\mathbb{E}[Y]$, and Wednesday's properties restated on the board — 1. $\mathrm{Cov}(X,X)=\mathbb{E}[X^{2}]-(\mathbb{E}[X])^{2}=\mathrm{Var}[X]$. 2. $X,Y$ independent $\Rightarrow\ \mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]\ \Rightarrow\ \mathrm{Cov}(X,Y)=0$. 3. $\mathrm{Cov}(X,Y)=\mathrm{Cov}(Y,X)$. 4. The linear-combination property, this time **derived** rather than announced: $\mathrm{Cov}(c_{1}X_{1}+c_{2}X_{2},\,Y)=\mathbb{E}\big[(c_{1}X_{1}+c_{2}X_{2})Y\big]-\mathbb{E}[c_{1}X_{1}+c_{2}X_{2}]\,\mathbb{E}[Y]$ $=c_{1}\mathbb{E}[X_{1}Y]+c_{2}\mathbb{E}[X_{2}Y]-\big(c_{1}\mathbb{E}[X_{1}]\mathbb{E}[Y]+c_{2}\mathbb{E}[X_{2}]\mathbb{E}[Y]\big)=c_{1}\,\mathrm{Cov}(X_{1},Y)+c_{2}\,\mathrm{Cov}(X_{2},Y),$ the white and pink underlines on the board pairing the $c_{1}$ terms and the $c_{2}$ terms. Linearity of the expectation does all the work; with property 3, the same holds in the second slot. 5. **New, and the easy payoff.** If $X$ and $Y$ are independent, $\mathrm{Var}[X+Y]=\mathrm{Cov}(X+Y,\,X+Y)=\mathrm{Cov}(X,X+Y)+\mathrm{Cov}(Y,X+Y)=\underbrace{\mathrm{Cov}(X,X)}_{\mathrm{Var}[X]}+\underbrace{\mathrm{Cov}(X,Y)}_{0}+\underbrace{\mathrm{Cov}(Y,X)}_{0}+\underbrace{\mathrm{Cov}(Y,Y)}_{\mathrm{Var}[Y]}=\mathrm{Var}[X]+\mathrm{Var}[Y].$ Property 1 turns the variance into a covariance, property 4 breaks it over both slots, property 2 zeroes the two cross terms (pink arrows to $0$ on the board), and property 1 turns the survivors back into variances. This is the fact [[MATH310F26(Day 15) - The second moment, and the bands|Day 15]] used to get $\mathrm{Var}[S_{n}]=n\,\mathrm{Var}[X]$ — "the variance of a sum of independent things is the sum of the variances" — now with a two-line reason instead of an appeal. ## The walk against itself, $k$ steps later Now suppose we check in with the walk process at time $n$ and at time $n+k$: **how are these random variables [correlated](https://en.wikipedia.org/wiki/Correlation)?** Write $S_{n+k}=S_{0}+\sum_{i=0}^{n+k-1}X_{i}$ and use property 4 to break the first slot apart: $\mathrm{Cov}(S_{n+k},S_{n})=\mathrm{Cov}\Big(S_{0}+\sum_{i=0}^{n+k-1}X_{i},\ S_{n}\Big)=\mathrm{Cov}(S_{0},S_{n})+\sum_{i=0}^{n+k-1}\mathrm{Cov}(X_{i},S_{n}).$ The board then opened the first piece, writing $S_{n}=S_{0}+\sum_{j=0}^{n-1}X_{j}$ in the second slot: $\mathrm{Cov}(S_{0},S_{n})=\mathrm{Cov}\Big(S_{0},\ \sum_{j=0}^{n-1}X_{j}+S_{0}\Big)=\mathrm{Cov}(S_{0},S_{0})+\sum_{j=0}^{n-1}\mathrm{Cov}(S_{0},X_{j}),$ with each $\mathrm{Cov}(S_{0},X_{j})=\mathbb{E}[S_{0}X_{j}]-\mathbb{E}[S_{0}]\mathbb{E}[X_{j}]$ begun in the corner when time ran out. That is where the capture ends; the finish is two lines, in the same spirit. The starting location is deterministic (or at least independent of every step), so every $\mathrm{Cov}(S_{0},X_{j})=0$ and $\mathrm{Cov}(S_{0},S_{0})=\mathrm{Var}[S_{0}]=0$ for a fixed start. The second piece opens the same way: $\mathrm{Cov}(X_{i},S_{n})=\mathrm{Cov}(X_{i},S_{0})+\sum_{j=0}^{n-1}\mathrm{Cov}(X_{i},X_{j})$, and by independence of distinct steps the only surviving terms are $i=j$ with $j\le n-1$ — one $\mathrm{Var}[X]$ for each step the two walks **share**: $\mathrm{Cov}(S_{n+k},S_{n})=\mathrm{Var}[S_{0}]+\sum_{i=0}^{n+k-1}\sum_{j=0}^{n-1}\mathrm{Cov}(X_{i},X_{j})=n\,\mathrm{Var}[X]\qquad(S_{0}\ \text{fixed}).$ The walk remembers exactly its shared history and nothing of the $k$ steps that only one of them took. What that number means as a *correlation* — divided by the spreads $\sqrt{\mathrm{Var}[S_{n}]\,\mathrm{Var}[S_{n+k}]}$ — and what happens in the limits of small and large $k$, is next. > [!success]- Numbers checked in one sandbox run > Property 5 on 300,000 draws of an exponential $X$ and an independent uniform $Y$: $\mathrm{Var}[X+Y]=5.314$ against $\mathrm{Var}[X]+\mathrm{Var}[Y]=5.311$. The walk, with a *random* start $S_{0}\sim\mathcal{N}(0,3^{2})$ to test the $\mathrm{Var}[S_{0}]$ term: $\mathrm{Cov}(S_{37},S_{25})=33.91$ against $\mathrm{Var}[S_{0}]+n\,\mathrm{Var}[X]=9+25=34$; with the start removed, the correlation came out $0.821$ against Monday's $\sqrt{n/(n+k)}=\sqrt{25/37}=0.822$. The double sum, done symbolically for $n=3$, $k=2$ with $\mathrm{Cov}(X_{i},X_{j})=v\,\delta_{ij}$, is $3v=nv$. The Media Reflection page's numbers, re-run: $\tau=0.71,\ 1.58,\ 5.0$ at $H=0.5,\ 0.9,\ 0.99$; time to 1% at $a=1$ is $\tfrac12\ln\tfrac{101}{3}=1.758$ (linearized $1.956$); integrating $y'=a-(y-1)^{2}$ from $1.5\,y^{*}$ gives $13.2$ at $a=0.01$, $29.2$ at $a=0.001$, $52.7$ at $a=0.0001$, and $98$ exactly at $a=0$ — the race applet's "about 13" and the page's $98$. > [!example]- Board scans — October 9, 2026 > ![[MATH310F26-Day19-Board-1.jpg]] > ![[MATH310F26-Day19-Board-2.jpg]] > ![[MATH310F26-Day19-Board-3.jpg]] > ![[MATH310F26-Day19-Board-4.jpg]] > > 📄 [[MATH310F26-Day19-Boards-2026-10-09.pdf|Full board capture (4 pages, PDF)]]