Given an arbitrary function, f, we define a function f’ which returns 1 on input n if f halts on input n. Now, for some number x we define a function g which, on input n, returns 1 if n = x, and otherwise calls f'(n). If functional equivalence were decidable, then deciding whether g … Read more
TF uses automatic differentiation and more specifically reverse-mode auto differentiation. There are 3 popular methods to calculate the derivative: Numerical differentiation Symbolic differentiation Automatic differentiation Numerical differentiation relies on the definition of the derivative: , where you put a very small h and evaluate function in two places. This is the most basic formula and … Read more
The only reason to use the log-likelihood instead of the plain old likelihood is mathematical convenience, because it lets you turn multiplication into addition. The plain old likelihood is P(parameters | data), i.e. assuming your data is fixed and you vary the parameters of your model. Maximizing this is one way to do parameter estimation … Read more
Think about how to prove the result for a full binary tree, and you’ll see how to do it in general. For the full binary tree, say of height h, the number of nodes N is N = 2^{h+1} – 1 Why? Because the first level has 2^0 nodes, the second level has 2^1 nodes, … Read more
I use Roger’s Online Equation Editor. PNG, colors, transparent background and anti-aliasing are all included.
You could try using the parametrized equations for an ellipse rotated at an arbitrary angle: x = h + a*cos(t)*cos(phi) – b*sin(t)*sin(phi) [1] y = k + b*sin(t)*cos(phi) + a*cos(t)*sin(phi) [2] …where ellipse has centre (h,k) semimajor axis a and semiminor axis b, and is rotated through angle phi. You can then differentiate and solve … Read more
Wikipedia: Equivalence relation: In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being “equivalent” in some way. Let a, b, and c be arbitrary elements of some set X. Then “a ~ b” or “a ≡ b” denotes that a is equivalent to b. … Read more
Foreword: I released this utility in github.com/icza/gox, see mathx.Round(). Go 1.10 has been released, and it adds a math.Round() function. This function rounds to the nearest integer (which is basically a “round to nearest 1.0” operation), and using that we can very easily construct a function that rounds to the unit of our choice: func … Read more
“General Theory” of CAS is a pretty huge scope for a question. That being said, I’ll do my best to cover as much as I can in the hopes that something helps you find what you’re looking for 🙂 The proceedings of the ISSAC and SIGSAM groups would no doubt have some good stuff about … Read more
“yep base-10 is what I mean” In that case, yes, they can be extended to base-10 in several ways, though they aren’t nearly as useful as in binary. One idea is that &, |, etc. are the same as doing arithmetic mod-2 to the individual binary digits. If a and b are single binary-digits, then … Read more