Math and CS Research Departments

Princeton

Some notable contemporary names at Princeton include Brian Kernighan (of awk fame - Aho, Weinberger and Kernighan) and Andrew Appel. BK is also involved in a very interesting project called AMPL a modelling language for mathematical programming.

Princeton is also famous as the place Alonzo Church, the American mathematician, did his PhD in mathematics and published a paper in 1936 on an "undecidable problem" which introduced the lambda calculus that inspired the creation of the LISP programming language.

http://www.cs.princeton.edu/

Inria

Inria have published an interesting series of monographs for 2008, including the use of temporal logic as a verification tool.
http://www.inria.fr/publications/editeurs/edit_monographies.fr.html

UNSW

Project on compiling Haskell to Java.
http://www.cse.unsw.edu.au/~pls/thesis-topics/ghcjava.html

Project Martingale (Java API for Derivatives Pricing)

http://martingale.berlios.de/Martingale.html#code

Sample signatures: public class BrownianMotion extends StochasticProcess; public class DigitalRandomSequence extends LowDiscrepancySequence (methods to compute L2-discrepancy). Utilises JFreeChart created by Object Refinery based ici.

Basic Calculus and Vector Calculus Concepts for Computer Programmers

The MIT website outlines the basic rules of differentiation, which should be second-nature to any mathematical programmer.

It revises rules like product rule (to differentiate xln(x), breaking the product xln(x) into the sum of two products), chain rule (to differentiate sin(ln(x)) and quotient rule (to differentiate (ln(x)/cos(x)).

Programmers should also have a basic knowledge of limits. Like...what is the limit of sinx as x tends to zero? (you can guess this one from the graph). More tricky, what is the limit of sinx/x as x tends to zero? Don't know? Find out here.

Concepts of vector calculus (a.k.a. vector analysis, which has its roots in the theory of quaternions) like scalar and vector fields and grad (or "nabla"), div ("divergence") and curl should also be clearly understood and examples done. The link above also covers basic 2D and 3D co-ordinate systems (i.e. cylindrical and spherical coordinates versus cartesian and polar co-ordinates) essential in graphics programming.

An example of a scalar field would be temperature in a room, where a scalar temperature (in Kelvins say) can be associated to every 3D vector in the room. An example of a vector field would be an electrical force field where each vector has a magnitude and direction in which the force is acting.

Puzzle: is light a vector or a scalar field? Why?

K-Maps

I was reading about black-box testing when I came across a technique involving Karnaugh maps
(K-maps, or Veitch diagrams). These optimisation diagrams were invented by American mathematician Edward Veitch in 1952 and further refined by Maurice Karnaugh (BSc, Msc, PhD in Physics) at Bell Labs (now part of Alcatel-Lucent). In K-maps, boolean variables are taken (e.g. from a truth table) and ordered by Gray Code (aka reflected binary code) principles (where successive members of the code differ by only one digit).

Here is a software that creates K-Maps: http://www.sharewareplaza.com/images/screenshot/44948.gif

Functional Analysis, Operational Calculus and Operator Theory

Banach spaces feature prominently in courses on functional analysis, including theoretical numerical analysis. Stefan Banach initially referred to these as "spaces of type B".

Functional analysis is basically the study of vector spaces and the operators that act upon them. The word "functional" originates from the calculus of variations, which deals in higher-order functions, or functions whose arguments are functions.

Here is a history on the related domain of Operator Theory.

The article makes mention of Oliver Heaviside's operational calculus of which a short summary can be found here. Oliver Heaviside was born in Camden Town, London in 1850 (near what is now Mornington Crescent) and worked in Newcastle as a telegraph operator. Interested in telegraphy, mathematics and electrical engineering, he quit his job and studied full-time from home. He invented transmission line theory and developed techniques equivalent to Laplace transforms for solving differential equations.

Heaviside read Maxwell's famous treatise on electricity and magnetism as a young man. He writes: "I saw that it was great, greater and greatest with prodigious possibilities in its power. I was determined to master the book and set to work...it took me several years to understand as much as I possibly could. Then I set Maxwell aside and followed my own course..I progressed much more quickly".

Oliver Heaviside was the first recipient of the Faraday Medal in 1922 awarded by the Institute of Engineering and Technology.

Bonjour, Monsieur Martingale!

Here is a conversation with Joseph ("Joe") Leo Doob, the American mathematician from Ohio famous for his martingale convergence theorems. http://www.dartmouth.edu/~chance/Doob/conversation.html 

What motivated Joe to develop deep results in applied probability? In the interview, Joe reveals it was his desire to convert common probabilistic intuition into mathematics (statements and proofs) that drove him to discover his theorems. 

 The wikipedia entry on martingales is also interesting. Doob's soiree into martingale theory coincided with the Great Depression of 1929. 

Other notable statistical activity at that time included Kolmogorov's axiomatisation of probability in 1933. Read Doob's bio here.

Math Jargon for November

November's math jargon:

supremum, sup - simply a jargonista way of saying "least upper bound"
sufficient statistic - concept introduced by Fisher in 1920, refers to a function of a data set that summarises all the sample information in that set. want more precision, and a chat about Rao-Blackwellisation? Go here.
Goldbach conjecture - if n>4 is even, then n is the sum of two odd primes (can be rephrased: even numbers >4 are the sum of two odd primes). The conjecture is named after Prussian mathematician Christian Goldback.

Math-programmers are also well-advised to know ALL the greek letters used in mathematics (commonly forgotten or under-used ones are eta, kappa, nu, upsilon (looks like nu), psi).