Simpson’s paradox

Linear Algebra: (Determinants, Matrices, Spanning Sets, Basis & Dimensions, Linear Independence, Linear Dependence, Eigenvalues, Eigenvectors, Vector Spaces, Subspaces, Row Spaces, Column Spaces, Null Spaces, Orthogonality, Normalization, Diagonalization, Gram-Schmidt Orthogonalization, etc.)Tensor Analysis/Tensor Calculus: (Pseudotensors, Dual Tensors, Jacobians, Differential & Integral Forms)Ordinary Differential Equations: (First Order, Second Order, Constant Coefficients, Undetermined Coefficients, Linear ODE’s, Series Solutions: Frobenius Method & Legendre Equations, Dirac-Delta Functions, Homogeneous, Inhomogeneous, Nonlinear ODE’s)Sturm-Liouville Theory: (Hermitian Operators, ODE Eigenvalue Problems, Variation Method.)Partial Differential Equations: (First Order, Second Order, Canonical Forms, Separation of Variables, Method of Characteristics for: 1st Order Linear, Semi-Linear, and Quasi-Linear PDE’s, Eigenfunction Expansion Method, Applications of Fourier Integrals to PDE’s, Laplace & Poisson Equations, Wave Equation, Heat Equation.)Green’s Functions: (One-Dimensional, Two-Dimensional, and Three-Dimensional Problems)Complex Variable Theory: (a.k.a. Complex Analysis): (Complex Variables & Functions, Cauchy-Riemann Conditions, Cauchy’s Integral Theorem, Cauchy’s Integral Formula, Laurent Expansion, Singularities, Calculus of Residues, Evaluation of Sums & Definite Integrals.)Further Topics in Analysis: (a.k.a. Mathematical Analysis & Numerical Analysis): (Orthogonal Polynomials, Bernoulli Numbers, Euler-Maclaurin Integration Formula, Dirichlet Series, Asymptotic Series, Method of Steepest Descents, etc.)Gamma Functions: (Digamma & Polygamma Functions, The Beta Function, Stirling’s Series, Riemann Zeta Function, etc.)Bessel Functions: (Bessel Functions of the First & Second Kind, Neumann Functions, Hankel Functions, Asymptotic Expansions, Spherical Bessel Functions, etc.)Angular Momentum: (Angular Momentum Coupling & Operators, Spherical Tensors, Spherical Harmonics, Vector Spherical Harmonics, etc.)Group Theory: (Representation of Groups, Symmetry & Physics, Discrete Groups, Direct Products, Symmetric Groups, Continuous Groups, Lorentz Groups, Space Groups, Lorentz Covariance of Maxwell’s Equations.)Fourier Series: (General Properties, Applications of Fourier Series, Gibbs Phenomenon)Integral Transforms: (Fourier Transform, Laplace Transform, Fourier Convolution Theorem, Signal-Processing Applications, Discrete Fourier Transform, Fourier Sine & Cosine Transforms, Laplace Convolution Theorem, Properties & Inverses of Laplace Transforms)Integral Equations: (Neumann Series, Special Methods, Hilbert-Schmidt Theory)Topology: (Metric Spaces, Topological Spaces, Connectedness, Compactness, Injective, Surjective, Bijective and Continuous Maps between Topological spaces, Homeomorphism, Hausdorff Spaces, Subspace and Product Topologies.)Calculus of Variations: (Euler-Lagrange Equation, More General Variations, Constrained Minima/Maxima, Variation with Constraints.)Probability & Statistics: (Definitions & Simple Properties, Random Variables, Binomial Distribution, Poisson Distribution, Gauss’ Normal Distribution, Transformations of Random Variables, Statistics)

Herbert Gross

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Cell Size and Scale

Symmetry groups are pretty cool, so let's look at a simple example: the symmetry group of an equilateral triangle. What's the symmetry group of this triangle? To find out, we'll need to find some actions which realign the triangle with itself. One action which leaves the triangle looking unchanged is a rotation by 120 degrees. And another action which leaves the triangle looking unchanged is a horizontal flip. To help us keep track of what an action does to the triangle, I'll doodle some letters on it. Finally, on the right, I'll keep track of the actions you discover. We'll start with "do nothing", which I've labeled "e", and we'll draw more actions as we find them. Why "e"? The first group theory books were in German, they shortened "Einheit" to "e", and the name stuck. On the right, each circular shape represents an action in the group, and arrows will show you what happens if you rotate and flip the triangle starting from every action. Use these two buttons to explore the group! Find every action and arrow to proceed. Alright! Looks like you found everything! This way of drawing symmetry groups, where we draw one circle per action and arrows to represent certain actions, is called a Cayley graph. Doesn't the triangle's Cayley graph look somewhat triangular? That's no coincidence. Somehow, the symmetry group is capturing info about the original shape we started with. This group is called "the dihedral group of order 6". Since it's the symmetry group of a 3-sided triangle, the group's name is usually written "D3". Confusingly, some mathematicians call it D6 instead, because there are 6 actions in this group. Even more interesting, the resulting group doesn't depend on the actions we choose to build the graph with. For example, we built this Cayley graph using a rotation and flip. But what if we chose other actions? For example, what if I choose the action labeled "rf"? I'll draw it in purple. What happens if we try building this triangle's Cayley graph out of this purple flip and a horizontal flip? Try applying the actions in different orders. Well done. The arrows are a bit different, but we still reached all the same actions. It looks like the symmetry group of a shape is somehow independent of the actions we use to build it. Let's test it one more time. What if we choose these two actions as our starting actions? Let's try building the triangle's Cayley graph out of these two rotation actions. Looks like you filled out as much of the Cayley graph as you can. Huh. This isn't the full symmetry group of our triangle. Why can't we reach any of the flips? Isn't the symmetry group of a shape independent of the actions we started from?

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Web Apps

etiopathophysiology

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This video was filmed 5 years ago

Heather Staines, Hypothesis' Director of Partnerships, recreated this flash talk she gave at the STM Innovations Seminar 2017 in London.