A relation between polynomials

This is a brief foray into algebra from a 2006 REU project at Texas A&M.

Given two polynomials P,Q \in \mathbb C[z_1,\dots,z_n], we write Q\preccurlyeq P if there is a differential operator T\in \mathbb C[\frac{\partial}{\partial z_1},\dots, \frac{\partial}{\partial z_n}] such that Q=T P.

The relation \preccurlyeq  is reflexive and transitive, but is not antisymmetric. If both Q\preccurlyeq P and Q\preccurlyeq P hold, we say that P and Q are \partial-equivalent, denoted P\thicksim Q.

A polynomial is \partial -homogeneous if it is \partial -equivalent to a homogeneous polynomial. Obviously, any polynomial in one variable has this property. Polynomials in more than one variable usually do not have it.

The interesting thing about \partial -homogeneous polynomials is that they are refinable, meaning that one has a nontrivial identity of the form P(z)=\sum_{j\in\mathbb Z^n} c_{j} P(\lambda z-j) where c_{j}\in \mathbb C, j\in \mathbb Z^n, and only finitely many of the coefficients c_j are nonzero. The value of \lambda does not matter as long as |\lambda|\ne 0,1. Conversely, every \lambda -refinable polynomial is \partial -homogeneous.

Controlled bilipschitz extension

A map f\colon X\to Y is L-bilipschitz if L^{-1} |a-b| \le |f(a)-f(b)| \le L |a-b| for all a,b\in X. This definition makes sense if X and Y are general metric spaces, but let’s suppose they are subsets on the plane \mathbb R^2.

Definition 1. A set A\subset \mathbb R^2 has the BL extension property if any bilipschitz map f\colon A\to\mathbb R^2 can be extended to a bilipschitz map F\colon \mathbb R^2\to\mathbb R^2. (Extension means that F is required to agree with f on A.)

Lines and circles have the BL extension property. This was proved in early 1980s independently by Tukia, Jerison and Kenig, and Latfullin.

Definition 2. A set A\subset \mathbb R^2 has the controlled BL extension property if there exists a constant C such that any L-bilipschitz map f\colon A\to\mathbb R^2 can be extended to a C L-bilipschitz map F\colon \mathbb R^2\to\mathbb R^2.

Clearly, Definition 2 asks for more than Definition 1. I can prove that a line has the controlled BL extension property, even with a modest constant such as C=2000. (Incidentally, one cannot take C=1.) I still can’t prove the controlled BL extension property for a circle.

Update: extension from line is done in this paper.

WeBWork class roster import

One way to import classroster into WeBWork (at SU):

  1. Download the roster from Blackboard Grade Center and import it into a spreadsheet
  2. The first four columns A,B,C,D will be Last Name, First Name, UserName, Student ID.
  3. Append the column with the function
    =CONCATENATE(D2,",";A2,",",B2,",C,,,,",C2,"@syr.edu,",C2)

    (second row shown). Or

    =CONCATENATE(D2;",";A2;",";B2;",C,,,,";C2;"@syr.edu,";C2)

    if using OpenOffice.

  4. Using WeBWork file manager, create a file roster.lst and paste this new column into it.
  5. Use the Import Users command under Classlist editor.

Almost norming functionals, Part 1

Let E be a real Banach space with the dual E^*. By the Hahn-Banach theorem, for every unit vector e\in E there exists a functional e^*\in E^* of unit norm such that e^*(e)=1. One says that e^* is a norming functional for e. In general, one cannot choose e^* so that it depends continuously on e. For example, the 2-dimensional space with \ell_1 norm does not allow such a continuous selection.

Fix \delta\in (0,1) and call a linear functional e^* almost norming for e if |e|=|e^*|=1 and e^*(e)\ge \delta. In any Banach space there exists a continuous selection of almost norming functionals.

Continue reading “Almost norming functionals, Part 1”

Expanding distances on a sphere

This came up in a discussion at AIM.

Let \mathbb S^{n-1}\subset \mathbb R^n be the unit sphere. Suppose that f\colon \mathbb S^{n-1}\to\mathbb R^n is a continuous map which does not decrease distances: that is, |f(x)-f(y)|\ge |x-y| for all x,y\in \mathbb S^{n-1}. By the generalized Jordan theorem the complement of f(\mathbb S^{n-1}) has two components, precisely one of which, denoted \Omega, is bounded.

Prove that \Omega contains an open ball of radius 1.

Continue reading “Expanding distances on a sphere”

Generalized Birthday Problem

Facebook currently allows up to 5000 friends. Assume that birthdays are uniformly distributed among 365 days. It is well-known that for a user with 23 friends there is a 50% chance of having to write ”Happy Birthday” on at least two walls in the same day.

How many facebook friends should one acquire to have a 50% chance of at least 10 birthdays falling on the same day? With 3286 friends this is guaranteed to happen, but one should expect that 50% probability will be reached with a substantially smaller number.

[Update] In 1989 Diaconis and Mosteller found an approximate solution that gives an approximate number of people required to have at least 50% probability of of at least k birthdays on the same day. The formula can be found on MathWorld and its output is the OEIS sequence A050255. For comparison, Diaconis and Mosteller quote exact values found by Bruce Levin. The exact solution is the sequence A014088, which begins thus:

1, 23, 88, 187, 313, 460, 623, 798, 985, 1181, 1385, 1596, 1813, 2035, 2263

So, with 1181 Facebook friends there is a 50% chance of having to write “Happy Birthday” at least ten times in the same day.