You can find anything on the web. Earlier today, I noticed that the Sterling approximation and the Gaussian distribution both had a $\sqrt{2 \pi}$ in them, so I started thinking that maybe you could apply Sterling’s approximation to the binomial distribution with $p=1/2$ and large $n$ and the central limit theorem to derive the Gaussian distribution. After making a few calculations it seemed to be working and I thought, “Hey, maybe someone else has done this.” Sure enough, Jake Hoffman did it and posted it here. The internet is pretty cool.
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I was reading MarkCC’s gripe about the misuse of the word dimension and it reminded me about the Hausdorff dimension. Generally speaking, the Hausdorff dimension matches our normal intuitive definition of dimension, i.e the H-dimension of a smooth curve or line is 1, the H-dimension of a smooth surface is 2, the H-dimension of the interior of a cube is 3. But for fractals, the H-dimension can be a non-integer. For the Cantor set, the H-dimension is 0.63. For more information, check out the Wikipedia article
http://en.wikipedia.org/wiki/Hausdorff_dimension
Check out Terence Tao‘s wonderful post “Matrix identities as derivatives of determinant identities“.
Thank you to Freakonometrics for pointing me toward the book “Proofs without words” by Rodger Nelson. Might be a nice Christmas present 
Matroids are an abstraction of Vector Spaces. Vectors spaces have addition and scalar multiplication and the axioms of vectors spaces lead to the concepts of subspaces, rank, and independent sets. All Vector Spaces are Matroids, but Matroids only retain the properties of independence, rank, and subspaces. The idea of the Matroid closure of a set is the same as the span of the set in the Vector Space.
So I have been thinking about which ideas in mathematics I use most often and I have listed them below. Please feel free to comment because I would like to use this list as a starting point for a future list of “The most useful ideas in Mathematics for Scientists and Engineers”.
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I showed this list to Carl and he said his list would be completely different. (He choked a bit when he saw primes at #71.) Hopefully we can post his list as well as others.
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- counting
- zero
- integer decimal positional notation 100, 1000, …
- the four arithmetic operations + – * /
- fractions
- decimal notation 0.1, 0.01, …
- basic propositional logic (Modus ponens, contrapositive, If-then, and, or, nand, …)
- negative numbers
- equivalence classes
- equality & substitution
- basic algebra – idea of variables, equations, …
- the idea of probability
- commutative and associative properties
- distributive property
- powers (squared, cubed,…), – compound interest (miracle of)
- scientific notation 1.3e6 = 1,300,000
- polynomials
- first order predicate logic
- infinity
- irrational numbers
- De Morgan’s laws
- statistical independence
- the notion of a function
- square root (cube root, …)
- inequalities (list of inequalities)
- power laws (i.e. $a^b a^c = a^{b+c}$ )
- Cartesian coordinate plane
- basic set theory
- random variable
- probability distribution
- histogram
- the mean, expected value & strong law of large numbers
- the graph of a function
- standard deviation
- Pythagorean theorem
- vectors and vector spaces
- limits
- real numbers as limits of fractions, the least upper bound
- continuity
- $R^n$, Euclidean Space, and Hilbert spaces (inner or dot product)
- derivative
- correlation
- central limit theorem, Gaussian Distribution, Properties of Guassains.
- integrals
- chain rule
- modular arithmetic
- sine cosine tangent
- $\pi$, circumference, area, and volume formulas for circles, rectangles, parallelograms, triangles, spheres, cones,…
- linear regression
- Taylor’s theorem
- the number e and the exponential function
- Rolle’s theorem, Karush–Kuhn–Tucker conditions, derivative is zero at the maximum
- the notion of linearity
- Big O notation
- injective (one-to-one) / surjective (onto) functions
- imaginary numbers
- symmetry
- Euler’s Formula $e^{i \pi} + 1 = 0$
- Fourier transform, convolution in time domain is the product in the frequency domain (& vice versa), the FFT
- fundamental theorem of calculus
- logarithms
- matrices
- conic sections
- Boolean algebra
- Cauchy–Schwarz inequality
- binomial theorem – Pascal’s triangle
- the determinant
- ordinary differential equation (ODE)
- mode (maximum likelihood estimator)
- cosine law
- prime numbers
- linear independence
- Jacobian
- fundamental theorem of arithmetic
- duality – (polyhedron faces & points, geometry lines and points, Dual Linear Program, dual space, …)
- intermediate value theorem
- eigenvalues
- median
- entropy
- KL distance
- binomial distribution
- Bayes’ theorem
- $2^{10} \approx 1000$
- compactness, Heine – Borel theorem
- metric space, Triangle Inequality
- Projections, Best Approximation
- $1/(1-X) = 1 + X + X^2 + \ldots$
- partial differential equations
- quadratic formula
- Reisz representation theorem
- Fubini’s theorem
- the ideas of groups, semigroups, monoids, rings, …
- Singular Value Decomposition
- numeric integration – trapezoidal rule, Simpson’s rule, …
- mutual information
- Plancherel’s theorem
- matrix condition number
- integration by parts
- Euler’s method for numerical integration of ODEs (and improved Euler & Runge–Kutta)
- pigeon hole principle
There is a long list of mathematical ideas that I use less often. Here’s a sampling: Baire category theorem, Banach Spaces, Brouwer Fixed Point Theorem, Carathéodory’s Theorem, Category Theory, Cauchy integral formula, calculus of variations, closed graph theorem, Chinese remainder theorem, Clifford algebra (quaternions), Context Free Grammars, countable vs uncountable infinity, Cramer’s Rule, cohomology, Euclidean algorithm, fundamental group, Gauss’ Law, Grassmannian algebra , Graph Theory, Hahn-Banach Theorem, homology, Hairy Ball Theorem, Hölder’s inequality, inclusion-exclusion, Jordan Decomposition, Kalman Filters, Markov Chains (Hidden Markov Models), modules, non-associative algebras, Picard’s Great Theorem, Platonic/Euclidean solids, Principle of Induction, Probabilistic Graphical Models (Bayesian Networks, Markov Random Fields), Pontryagain duality, Quaternions, Spectral Theorem, Sylow p subgroup, repeating decimals equal a fraction, ring ideals, sine law, tensors, tessellation, transcendental numbers, Uniform Boundedness Theorem, Weierstrass approximation theorem, …
Gauss did a lot of math, but sometimes I am surprised when I find something new to me done by Gauss
$\tan(x) = { 1 \over{1/x \ – {1\over{3/x \ – {1\over{5/x\ – \ldots}}}}}}$
Given a directed a-cyclic graph (DAG) find an ordering of the nodes which is consistent with the directed edges. This is called Topological Sorting and several algorithms exist.
Gaussian distributions are the most “natural” distributions. They show up everywhere. Here is a list of the properties that make me think that Gaussians are the most natural distributions:
- The sum of several random variables (like dice) tends to be Gaussian. (Central Limit Theorem).
- There are two natural ideas that appear in Statistics, the standard deviation and the maximum entropy principle. If you ask the question, “Among all distributions with standard deviation 1 and mean 0, what is the distribution with maximum entropy?” The answer is the Gaussian.
- Randomly select a point inside a high dimensional hypersphere. The distribution of any particular coordinate is approximately Gaussian. The same is true for a random point on the surface of the hypersphere.
- Take several samples from a Gaussian Distribution. Compute the Discrete Fourier Transform of the samples. The results have a Gaussian Distribution. I am pretty sure that the Gaussian is the only distribution with this property.
- The eigenfunctions of the Fourier Transforms are products of polynomials and Gaussians.
- The solution to the differential equation y’ = -x y is a Gaussian. This fact makes computations with Gaussians easier. (Higher derivatives involve Hermite polynomials.)
- I think Gaussians are the only distributions closed under multiplication, convolution, and linear transformations.
- Maximum likelihood estimators to problems involving Gaussians tend to also be the least squares solutions.
- I think all solutions to stochastic differential equations involve Gaussians. (This is mainly a consequence of the Central Limit Theorem.
- “The normal distribution is the only absolutely continuous distribution all of whose cumulants beyond the first two (i.e. other than the mean and variance) are zero.” – Wikipedia.
- For even n, the nth moment of the Gaussian is simply an integer multiplied by the standard deviation to the nth power.
- Many of the other standard distributions are strongly related to the Gaussian (i.e. binomial, Poisson, chi-squared, Student t, Rayleigh, Logistic, Log-Normal, Hypergeometric …)
- “If X1 and X2 are independent and their sum X1 + X2 is distributed normally, then both X1 and X2 must also be normal.” — From the Wikipedia.
- “The conjugate prior of the mean of a normal distribution is another normal distribution.” — From the Wikipedia.
- When using Gaussians, the math is easier.
- The Erdős–Kac theorem implies that the distribution of the prime factors of a “random” integer is Gaussian.
- The velocities of random molecules in a gas are distributed as a Gaussian. (With standard deviation = $z*\sqrt{ k\, T / m} $ where $z$ is a “nice” constant, $m$ is the mass of the particle, and $k$ is Boltzmann’s constant.)
- “A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.” — From Wikipedia
- Kalman Filters.
- The Gauss–Markov theorem.