It’s my birthday today, and it happens to be Chicago’s first snow of 2025 winter. Couldn’t it be happier! And let me share the happiness vicariously with all of you reading. It’s been a lot for the past year for me. Here’s a lovely, interesting song to go with the day — Thank Goodness performed live by Ariana Grande — “getting your dream seems a little strange, well, complicated.” But after all...

P. D. Q. Bach is a fictional composer created by Peter Schickelem (1935-2024). Schickele studied composition at Juilliard and was an accomplished bassoonist. His wrote an awful lot of (brilliant) parodies of classical music tributed to the name P. D. Q. Bach eg the 1712 Overture, The Civilian Barber, etc. Read more about him: NYTimes on Peter Schickele The Boston Musical Intelligencer Article about P. Schikele

Another convenient way to view, compare, and solve Cheap Talk vs. Bayesian Persuasion. Linear-Programming Formulation: Cheap Talk vs. Bayesian Persuasion Consider an information environment $\langle \Omega, A, M, \mu_0, u, v \rangle$ with state space $\Omega$, action space $A$, signal space $M$, prior $\mu_0$, receiver utility $u(\omega,a)$, and sender utility $v(\omega,a)$. (Let every set be finite). Consider direct mechanisms. In general, the sender’s strategy $\sigma:\Omega \to \Delta(A)$ maps each state to a distribution over recommended actions, and the receiver’s strategy $\rho:A \to \Delta(A)$ specifies how she responds to each recommendation....

A general continuous-time dynamic programming problem is given by reward function $h(x, u)$ control function $g(x, u)$ and initial state $x_0$: $$ \begin{align*} V^\star (x_0) :=& \max_{\lbrace u_t\rbrace_{t\ge 0}} \int_0^\infty e^{-\rho t} h(x_t, u_t),\text{d}t\cr & \text{s.t. }\dot x_t = g(x_t, u_t). \end{align*} $$ The Hamiltonian functino $H(x, u, \lambda)$, and Maximized Hamiltonian $H^\star(x, \lambda)$ is defined as $$ \begin{align*} & H(x, u, \lambda) = h(x, u) + \lambda^Tg(x, u),\cr & H^\star(x, \lambda) = \max_u H(x, u, \lambda)....

When I learnt the continuous time and discrete time neoclassical model, it’s feels so tempting to try to put them together in a uniform view. It’s not trivially easy though. And here’s one way to do it. Continuous-Time Neoclassical Growth Model Given $k_0$, $f(\cdot), U(\cdot)$: $$ \begin{align*} & \max_{\lbrace c_t\rbrace_{t\ge 0}} \int_0^\infty e^{-\rho t} U(c_t),\text{d}t\cr & \text{s.t. }\dot k_t = f(k_t) - \delta k_t - c_t \end{align*}\tag{1} $$ Taking (appropriate) derivative of the Hamiltonian function solves this problem — in general, for any continuous-time problem in the following form: $$ \begin{align*} V^\star (x_0) :=& \max_{\lbrace u_t\rbrace_{t\ge 0}} \int_0^\infty e^{-\rho t} h(x_t, u_t),\text{d}t\cr & \text{s....

I thought it was another song Johannes Brahms wrote for Clara Schumann. In fact, Brahms had it dedicated to one of his former lover: The cradle song was dedicated to Brahms’s friend, Bertha Faber, on the occasion of the birth of her second son. Brahms had been in love with her in her youth and constructed the melody of the “Wiegenlied” to suggest, as a hidden counter-melody, a song she used to sing to him....

You can but don’t necessarily have to use fancy harmonic to write a masterpiece (Like math for research). Chopin proves this point — you need only two chords to write a masterpiece. Here’s his wonderful Berceuse (Op. 57). It’s a very sophisticated, rich work full of room for imagination. Note: I really likes the following Sumino’s recording where, it sounds like either he’s using the sostenuto pedal or it’s a really soft piano....

The musical scale is arguably humanity’s oldest algorithm for beauty. Music theory can seem intimidating (for instance, I probably should have learned tuning theory back in 16 when I was sitting in orchestra). But it’s actually quite straightforward. Imagine you’re a piano designer: you want your instrument to sound nice, but you also don’t want it to have an infinite number of keys. Axiom 1 (Sounding nice) Two notes sound consonant when their frequencies form a ratio of small integers....

AI can now make decent, impressive music. Check out Tunee and their example projects. Tunee’s interface features a conversational input, a project diagram, and options to preview AI-generated music with lyrics. However, there’s no way to export stems or sheet music yet. Out of curiosity, I impatiently splashed some random prompts on my keyboards into Tunee: the results were decent Bach, whimsical bouncy cool woodwinds, and a few suspiciously catchy ballads—of course, since AI is now quite good at generating lyrics:...

After surviving a macro midterm where the True/False section felt like coin flipping, I rushed downtown for Dvořák’s Symphony No. 9 with the Chicago Symphony Orchestra under Sir Riccardo Muti (Concert Note digital version) Can you believe that Muti’s 84?! And he probably definitely counts bars better than I do. I sat in the center front row of the Terrace—right behind the stage. That seat feels almost illicit, as if you’ve joined the orchestra....