Proof of a Bonferroni inequality

Here is a very enjoyable theorem due to Bonferroni. Let \(n \geq 2 \) and consider the probability triple \( (\Omega, \mathcal{F}, P) \) and a collection of sets of \( \Omega \) in \( \mathcal{F} \) denoted \( ( A_i )_{i=1}^n \). Then the following holds:

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Brief and (somewhat) intuitive description of the Kolmogorov equations

I have never taken a course in probability, let alone a course in stochastic differential equations, so I am sort of winging it here. I think the following is one of the shorter “derivations” of the BKE and FKE that can be performed–but the disclaimer above is just in case there’s a well-known shorter derivation of which I’m unaware!

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Heat equation, apparently just for fun

It often happens in research that one writes a considerable amount of code only to find that it won’t turn out to be useful in whatever project one is working on. Such was the case with me today: I wrote a solver for the diffusion equation only to find that it won’t suit the purpose for which I originally wrote it. Oh well! Check out these cool numerical solutions anyway.

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Quantum particles on the torus

Here is a simple yet beautiful problem in quantum mechanics: find the explict form of the wavefunction in the position basis for a single free particle (or multiple, noninteracting, distinguishable free particles) confined to move on the surface of the two-dimensional torus \(\mathbb{T}^2\).

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An efficient mechanism for carbon trading

Here’s a simple mechanism for formulating a carbon-trading market. This won’t be the fanciest thing ever, but it is efficient (defined in a very precise way) and guarantees a cap on total emissions into the infinite future, provided cheating the mechanism isn’t possible.

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Health insurance is hard

Health insurance has been front-and-center in the news recently, following Congressional Budget Office (CBO) reports that the current House and Senate healthcare bills may cause millions of Americans to lose their healthcare coverage while also removing several taxes imposed by the Affordable Care Act, colloquially known as Obamacare. Insurers haven’t been silent; some have stated that the proposed restructuring of Affordable Care Act exchanges would do wonders for their business, while others have vocally advocated for retaining the individual mandate. Here, I’ll outline a simple model of a health insurance company to demonstrate why the problem of covering as many people as possible while also ensuring low premiums for those who can’t afford insurance is so difficult. I’ll also suggest guidelines for future policy.

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Random walks on networks

Given some network, weighted or unweighted, directed or undirected, what is the probability that a random walker starting at node \(i\) reaches node \(j\) after \(t\) timesteps?

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Making decisions heuristically

Many economic models of decision-making assume rational behavior; that is, agents use the information at their disposal as efficiently as possible and optimizing their behavior to maximize their utility. This view has recently (well, in the past sixty years) come under fire from the psychology, sociology, and behavioral economics communities as being overly optimistic regarding humans’ abilities to perform truly optimal activity. Proponents of this behavioral approach suggest instead a model of “bounded rationality” (term due to Herbert Simon) in which agents try to perform optimal decision-making, but are limited by their own cognitive abilities and resource constraints.

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Back to 2012...

In the wake of the 2016 presidential election, I thought I’d better rev up my natural language processing skills in preparation for the four years of adjective-laden data to come. I decided to get my feet wet with a much more tame data set: the 2012 American National Election Studies (ANES) time series free-response answers. I wondered if descriptions of the Democratic (Obama) and Republican (Romney) candidates for President were described differently by survey participants, and what words survey participants would use to describe them.

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Moral obligations in the 2016 presidential election

In the aftermath of Donald Trump’s recent statements that the presidential election is rigged against him and his supporters’ suggestions that his opponent be either assassinated or locked up for purely political reasons, I find it impossible to refrain from comment. Mr. Trump has done nothing but encourage his supporters’ intrinsically unpatriotic sentiments, tantamount to support for illegal and unquestionably immoral action against the legislative process inherent to American republican democracy. Furthering the crisis while sullying the name of their storied party, Republican leaders damn themselves with faint criticism of Mr. Trump’s actions. His racist rhetoric and boasts of sexual assault demanded withdrawal of any endorsement of his candidacy; his patently seditious assertions of late generate unconditional rejection as a requirement for maintenance of any semblance of decency. Mr. Trump’s supporters and Republican leaders have a moral obligation to disavow the entirety of his candidacy.

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Price supports? Really?

I dearly love NPR. Its coverage of global news, national and local elections, and its thought-provoking interviews with everyday folks are just some of the reasons I’m proud to support my local stations. But when it comes to issues of economic policy, many of the syndicated shows’ hosts, to say nothing of their guests, seem to lack any knowledge of the subject whatsoever. Take, for example, this story on New England’s dairy farms that I heard today. The theme of the piece appears to be the following: the local food movement, while an entertaining pastime for local yuppies, masks the dreadful reality that New England’s dairy farms are disappearing. Milk prices are too low for dairy farmers to support themselves. To compound the problem, suburban homeowners, while a fan of the concept of dairy farms, aren’t willing to make the sacrifices necessary to support their existence. (You can listen to the piece yourself if you want more details; I think that was a pretty fair summary.) In the piece, the filmmaker, Dr. Sarah Gardner, Associate Director for the Center for Environmental Studies at Williams College, suggests that perhaps greater milk price supports are one way to ensure that small dairy farms remain solvent.

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Calculus I

I will be teaching MATH 019 B at UVM this fall semester. This is a calculus I course for business, economics, biology, etc. majors. The course webpage is here (note: it’s a github page). I’ll be tweeting information about the class regularly, and posting about it here too. You’ll be able to find information about it by looking for the teaching, calculus, or UVM tags.

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Rationality and textualism

In conversation with mathematicians regarding the interpretation of constitutional law, I have noticed that many seem to be in favor of a “living Constitution” whose meaning may be interpreted while also taking into account the spirit of the times. (A perfectly valid survey sample, I’m aware.) I find that a mathematician would support this concept to be entirely strange.

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An elucidated proof of Tauber's theorem

Titchmarsh’s (in)famous The Theory of Functions proves Tauber’s theorem in five lines. It’s a wonderful proof, but if you’re like me and prefer physical explanations of proofs—or just take awhile to wrap your head around analytical proofs in general—this concision can be a little overwhelming. Here, I’ll proove Tauber’s theorem in essentially the same manner that Titchmarsh does, but I’ll explain the steps in much greater detail.

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Collisions

Brief update to my ongoing physics notes series: a page on collisions. Coming next: rotational kinematics, in which we’ll introduce the vector product for the first time! Excitement.

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Momentum

Contained within these notes: General statement of the conservation of energy, center of mass and introduction to the concept of a density function, and momentum, including the case of variable mass.

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Conservation of energy

We’re really moving now—on to the conservation of energy already! Here, I derive the principle of the conservation of energy in the presence of conservative forces, introducing the concept of path dependence along the way. We then cover simple harmonic motion by finding the potential energy of a mass attached to a perfect spring and solving its equation of motion. In doing so, we figure out the mystery of why \(i\) is showing up in a mechanics problem by introducing phase space.

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Symbolic--a wrapper for sympy

Sympy is a fantastic computer algebra system written in python. I feel that it could use a few added functions and a convenient wrapper, so I’m creating symbolic, available on my github. Feel free to fork and work on it! Email me or dm me on twitter with any questions.

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Work-energy theorem

New notes are up! In this set, I introduce the fundamental concepts of work and energy. I also briefly cover line integrals and the two most important ways in which they’re defined, and Einstein summation notation.

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Particle dynamics, continued

New notes are up, and quite late. Apologies, and I’ll try to get caught up over the weekend. In this set, I introduce friction, the dynamics of uniform circular motion, and the cylindrical coordinate system.

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Derivation of the diffusion-transport equation

Think of a flowing stream, and imagine the motion of a certain amount of substance in a portion of the stream. You’d notice two things about it: it would diffuse—that is, the concentration of it would lessen in the area in which you originally spotted it—and it would move down the stream at the stream’s velocity, call it \(v\). The equation that describes the amount of substance in any given place \(x\) in the stream at \(t\) is called the concentration equation (or the diffusion-transport equation). Let’s derive it!

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Particle dynamics

New notes are up, albeit a day late. My apologies. Here, we’re talking about force, with special appearances from blocks, ropes, ramps, and even a plumb bob.

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Conditional probability with trees

Here’s an interesting question: consider a tree structure, where each branch (edge) has a leaf set \(L\) containing two nodes. Choosing an edge at random, what is the probability that \(a\) is in the leaf set given that \(b\) is in the leaf set? Assume that there are a countable number of leaf set states.

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Learning for free!

I want more people to have access to quality materials on mathematics and the sciences. It’s time for me to put my money where my mouth is, so to speak, so I’ll be uploading course notes for a variety of physics, math, and economics courses on my github.

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Choose a firm...

Consider the following (entirely unrealistic) scenario: suppose a firm invents a new product; they are, by definition, the only producer of the product, and if the product is new enough, there are no consumers. When there are \(N\) firms in the market making identical products, a consumer that enters at discrete time step \(t_i\) must choose between starting a new firm and producing the product herself with probability \(\rho\), and choosing to purchase the product with probability \(1-\rho\) from one of the existing firms with likelihood of choosing any particular firm proportional to the number of consumers \(x\) already purchasing the product from the firm.

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Website? Website.

Here you’ll eventually find posts about economics, politics, and applied math. Courses I teach will also have course updates, notes, homework, etc. posted here. All posts will be tagged and can be found at the tags page.

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