Blog Profile / The Endeavour

Filed Under:Academics
Posts on Regator:1342
Posts / Week:4.8
Archived Since:April 26, 2011

Blog Post Archive

Area of a triangle and its projections

Let S be the area of triangle T in three-dimensional space. Let A, B, and C be area of the projections of T to the xy, yz, and xz planes respectively. Then S2 = A2 + B2 + C2. There’s an elegant proof of this theorem here using differential forms. Below I’ll sketch a less elegant but more elementary proof. You could prove the identity above by using the fact that the […]


Neal Stephenson coins a useful word Amistics in his novel Seveneves: … it was a question of Amistics, which was a term that had been coined ages ago by a Moiran anthropologist to talk about the choices that different cultures made as to which technologies they would, and would not, make part of their lives. […]

How many ways can you tile a chessboard with dominoes?

Suppose you have an n by m chessboard. How many ways can you cover the chessboard with dominoes? It turns out there’s a remarkable closed-form solution:     Here are some questions you may have. But what if n and m are both odd? You can’t tile such a board with dominoes. Yes, in that case the formula evaluates to […]

Acoustic roughness examples

Amplitude modulated signals sound rough to the human ear. The perceived roughness increases with modulation frequency, then decreases, and eventually disappears. The point where roughness reaches is maximum depends on the the carrier signal, but for a 1 kHz tone roughness reaches a maximum for modulation at 70 Hz. Roughness also increases as a function […]

Tensors 5: Scalars

There are two uses of the word scalar, one from linear algebra and another from tensor calculus. In linear algebra, vector spaces have a field of scalars. This is where the coefficients in linear combinations come from. For real vector spaces, the scalars are real numbers. For complex vector spaces, the scalars are complex numbers. […]

Tensors 4: Behavior under change of coordinates

In the first post in this series I mentioned several apparently unrelated things that are all called tensors, one of these being objects that behave a certain way under changes of coordinates. That’s what we’ll look at this time. In the second post we said that a tensor is a multilinear functional. A k-tensor takes k vectors and […]

Tensors 3: Tensor products

In the previous post, we defined the tensor product of two tensors, but you’ll often see tensor products of spaces. How are these tensor products defined? Tensor product splines For example, you may have seen tensor product splines. Suppose you have a function over a rectangle that you’d like to approximate by patching together polynomials so that […]

Tensors 2: Multilinear operators

The simplest definition of a tensor is that it is a multilinear functional, i.e. a function that takes several vectors, returns a number, and is linear in each argument. Tensors over real vector spaces return real numbers, tensors over complex vector spaces return complex numbers, and you could work over other fields if you’d like. A dot product is […]

Tensors 1: What is a tensor?

The word “tensor” is shrouded in mystery. The same term is applied to many different things that don’t appear to have much in common with each other. You might have heared that a tensor is a box of numbers. Just as a matrix is a rectangular collection of numbers, a tensor could be a cube of […]

From triangles to the heat equation

“Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.” — Joseph Fourier The above quote makes me think of a connection Fourier made between triangles and thermodynamics. Trigonometric functions were first studied because they relate angles in a right triangle to ratios of the lengths of the triangle’s sides. For […]

Contradictory news regarding ABC conjecture

“Research is what I’m doing when I don’t know what I’m doing.” — Wernher von Braun I find Shinichi Mochizuki’s proof of the abc conjecture fascinating. Not the content of the proof—which I do not understand in the least—but the circumstances of the proof. Most mathematics, no matter how arcane it appears to outsiders, is […]

Practical continuity

I had a discussion recently about whether things are really continuous in the real world. Strictly speaking, maybe not, but practically yes. The same is true of all mathematical properties. There are no circles in the real world, not in the Platonic sense of a mathematical circle. But a circle is a very useful abstraction, […]

Yet another way to define fractional derivatives

Fractional integrals are easier to define than fractional derivatives. And for sufficiently smooth functions, you can use the former to define the latter. The Riemann-Liouville fractional integral starts from the observation that for...Show More Summary

Quantifying how annoying a sound is

Eberhard Zwicker proposed a model for combining several psychoacoustic metrics into one metric to quantify annoyance. It is a function of three things: N5, the 95th percentile of loudness, measured in sone (which is confusingly called the 5th percentile) ?S, a function of sharpness in asper and of loudness ?FR, fluctuation strength (in vacil), roughness (in […]

Loudness units

I’ve posted an online calculator to convert between two commonly used units of loudness, phon and sone. The page describes the purpose of both units and explains how to convert between them.

Physical models

The most recent episode of 99% Invisible tells the story of the Corp of Engineers’ enormous physical model of the Mississippi basin, nearly half of the area of the continental US. Spanning over 200 acres, the model was built during WWII and was shut down in 1993. Here are some of my favorite lines from the show: […]

Another way to define fractional derivatives

There are many ways to define fractional derivatives, and in general they coincide on nice classes of functions. A long time ago I wrote about one way to define fractional derivatives using Fourier transforms. From that post: Here’s one way fractional derivatives could be defined. Suppose the Fourier transform of f(x) is g(?). Then for […]

Humble Lisp programmers

Maybe from the headline you were expecting a blank post? No, that’s not where I’m going. Yesterday I was on and noticed that nearly all the books they recommended for me were either about Lisp or mountain climbing. I thought this was odd, and mentioned it on Twitter. Carl Vogel had a witty reply: […]

Integral equation types

There are four basic types of integral equations. There are many other integral equations, but if you are familiar with these four, you have a good overview of the classical theory. All four involve the unknown function ?(x) in an integral with a kernel K(x, y) and all have an input function f(x). In all […]

What is a vacil?

Fluctuation strength is similar to roughness, though at much lower modulation frequencies. Fluctuation strength is measured in vacils (from vacilare in Latin or vacillate in English). Police sirens are a good example of sounds with high fluctuation strength. Show More Summary

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