Consider a family of polynomials $\{P_0, P_1, P_2, \cdots\}$ where each $P_n$ has degree $n$. The polynomials are orthogonal if there is an inner product such that $(P_i, P_j) = 0$ if and only if $i\neq j$. Now, any given $P_{n+1}$ can be written in terms of all the lower-degree polynomials: $$\displaystyle P_{n+1} = r_{n+1,n+1}xP_n … Read More “Recurrence Relations for Some Orthogonal Polynomials” »
A sultan has granted a commoner a chance to marry one of his $N$ daughters. The commoner will be presented with the daughters one at a time and, when each daughter is presented, the commoner will be told the daughter’s dowry (which is fixed in advance). Upon being presented with a daughter, the commoner must … Read More “The Sultan’s Daughters Problem” »
$\pi$ is the most well-known special number, but there is also $e$. Whereas $\pi$ has the easy interpretation of being the ratio of a circle’s circumference to its diameter, and everyone knows what a circle is, $e$ arises in calculus with which many people have no familiarity. Here’s something interesting… Without calculus, the closest one … Read More “What is e?” »
A previous post used the typical series-expansion-plus-linear-algebra approach for finding finite-difference formulas to derive approximations to the first derivative of any desired order of accuracy. If you’ve ever used that method yourself, you probably know how tedious it is, and that post doesn’t make it look much less so. On top of that, for all … Read More “All Finite-Difference Formulas for All Derivatives” »
Several competitive decks in Vintage Magic: The Gathering are powered by the card Bazaar of Baghdad. Deck construction rules mandate at least 60 total cards in a deck with at most four copies of any given card. The game starts by each player drawing seven cards then performing, if he wishes, a series of “mulligans” … Read More “Mulligans in the Bazaar” »
In celebration of the unofficial end of summer (Labor Day in the U.S.), consider a snowpile on a trash bin: Observe how the shape of the lid (straight edges, rounded corners) seems to propagate upward, and how the sides of each pile are nearly planar with nearly the same slope on each side. Is this … Read More “Snowpile Math” »
Ridders’ method is a single-variable root-finding method that is more efficient than the basic regula falsi method. The formula is easy enough to find online (e.g. the first link), but its derivation is not. This post fills in the blanks. Suppose the function $f$ has a zero between $x_L$ and $x_R$, i.e. $f(x_L)f(x_R) < 0$. … Read More “Notes on the Derivation of Ridders’ Method” »
Imagine you have a fluid inside a cylinder that suddenly starts rotating at angular velocity $\omega$: The fluid is initially stationary but eventually settles to a flow field that does not change further with time. What is the velocity of the fluid in the meantime? This question can be answered using Bessel functions. Simplifying the … Read More “Transient Flow in a Rotating Cylinder” »
While a student I often encountered sources claiming something to the effect of “no other special functions have received such detailed treatment […] as the Bessel functions”, which struck me as odd because in the undergraduate differential equations courses I took and taught no mention whatsoever was ever made of them, though the textbook might … Read More “Bessel Functions” »
In multivariable calculus courses, one usually first encounters the divergence as: $$\displaystyle \nabla\cdot\vec{u} = \frac{\partial{u_x}}{\partial x} + \frac{\partial{u_y}}{\partial y} + \frac{\partial{u_z}}{\partial z} $$ where $\vec{u} = [u_x,\,u_y,\,u_z]$ in cartesian coordinates. Then one learns the divergence theorem: $$\displaystyle \int_\Omega \nabla\cdot\vec{u}\, dx = \int_{\partial\Omega} \vec{u}\cdot\hat{n}\,dS $$ and then, since cylindrical coordinates have usually already been introduced, one … Read More “Divergence in Cylindrical Coordinates – The Right Way” »