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In combinatorics, the twentyfold way (and better known predecessor the twelvefold way) classifies counting problems via a single mathematical framework, functions with certain properties mapping domains with certain properties to codomains with certain properties. While not exhaustive of all...

I am brand new to studying machine learning. Why are the nodes in a neural network made to have values ranging between 0 and 1? It seems like they should range between 0 and \frac{1}{n}, where n is the number of nodes in that same layer. Is it wrong to think of a node's value as a probability...

Problem: Find the Sprague-Grundy values of the graphs, and find a winning move, if any. Solution: The SG-value of the three-leaf clover is 2. The SG-value of the girl is 3. The SG-value of the dog is 2. And the SG-value of the tree is 5. So there exists a winning move on the tree that...

The only difference between the definitions on Wikipedia for cross-correlation and cross-covariance (for deterministic signals) is that cross-correlation shows definite integration, while cross-covariance shows indefinite integration. Is this really the difference between the two, or is this an...

Can someone explain this Wikipedia example? I get the idea that W is a weighted average of the movie's average rating and the global average rating, but not how the formula relates to the broader article. Which piece in this example is the loss function and which piece is the Bayes Estimator?

I have worked out a few successive derivations of Bayes' Theorem... \Pr(W) = \Pr(W) \Pr(W\ |\ C_1)\ =\ \frac{\Pr(C_1\ |\ W)\Pr(W)}{\Pr(C_1)} \Pr(W\ |\ (C_1\ \cap\ C_2))\ =\ \frac{\Pr(C_2\ |\ (W\ \cap\ C_1))\Pr(W\ |\ C_1)}{\Pr(C_2\ |\ C_1)} \Pr(W\ |\ (C_1\ \cap\ C_2\ \cap\ C_3)) =...

According to Wikipedia... Suppose I want to relate \Pr(W\ |\ C), a degree of rational belief in W after evidence C becomes available, back to \Pr(W), a degree of rational belief in W before evidence C becomes available. Bayes' Theorem is just a double-application of the conditional...

According to this paper, "every submodular function can be represented as a maximum of additive valuations." It gives an algebraic description as well, but I'm having trouble seeing the high level idea. Is there a simple geometric intuition for this fact? Is there an obvious economic...

According to the Wikipedia entry on q-values... The "positive" qualifier here seems to be a distraction, as the article later states... So the q-value controls the False Discovery Rate. Similarly, the p-value appears to control the False Positive Rate, and the power appears to control the...

In this video, the profession gives what he calls a simulation argument. He goes through it pretty quickly, seemingly assuming that students are already familiar with that type of argument. All I can find when searching for this are philosophical considerations of simulation theory, the idea...

Does a mechanical set of rules or algorithm exist for doing forward induction on a game tree, or is it an "every problem is too unique and requires its own reasoning" type of situation? I have seen a few examples of forward induction explained, but with pretty basic game trees, and I fear I may...

Suppose I am told the outcomes (x_1 through x_n) of n rolls (X_1 through X_n) of a fair, \theta-sided die, and want to find a point estimate of \theta. Let's state the problem without using the lexeme "likely" in order to eliminate one layer of confusion, since its technical and colloquial...

In this video (37:27 - 42:21), it is proved that buying a Euro with a Dollar and buying a Dollar with a Euro can both have positive expected value without varying the probabilistic structure of the problem statement. My initial reaction was that this might be structurally equivalent to the...

In studying introductory level Partial Differential Equations, I have never come across methods for solving systems of linear PDEs (searching reveals a lot of research level papers). Is the basic tool for solving these matrix algebra, more or less identical to systems of linear ODEs? If not...

Suppose I want to solve u_{xy} = xy via Green's Function. This will correspond to the associated PDE G_{xy} = \delta(x - x_G,\ y - y_G), and I want my boundary conditions for this Green problem to be G(0,\ y) = 0,\ G_x(x,\ 0) = 0. The first step logically would actually be to find the adjoint...

I want to calculate Green's Function to solve \triangle u = f(x,\ y), using Laplace Transforms. My plan was to tailor boundary conditions to the problem which simplify the computation. Because the Laplacian is self-adjoint, my associated Green's Function equation can immediately be written as...

In this video, the professor solves a heat equation with two boundary conditions and one initial condition, u(x, 0) = 0, via the similarity solution method. The PDE is used to construct a second order ODE, and the two boundary conditions determine the two arbitrary constants that arise out of...

When solving a 2D Heat Equation, suppose I separate the solution into time and space, i.e., f_1(T(t)) = f_2(Z(x,\ y)) = \lambda, and then separate space into its dimensions, i.e., f_3(X(x)) = f_4(Y(y),\ \lambda) = r. The problem of this sort I worked seems to have two nontrivial paths, one in...

I am trying to understand Duhamel's Principle by applying it to a simple first order ODE and then generalizing. I am thinking of P(t) as expressing a bank account balance at time t, as my economics intuition is better than my physics intuition. The \pi forcing term is just chosen for algebraic...

Is there a nice geometric interpretation of reversing the components of a vector? So given some \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}, I want to visualize how it relates to \begin{bmatrix}x_3 \\ x_2 \\ x_1\end{bmatrix}. Is there a specific linear transformation that accomplishes this...