"The set of topologically equivalent maps from the circle into some space is known as the fundamental group of X."
"Now since the infinite expressions are equal in a way, by expanding the infinite product we expect to recover the infinite sum."
"Bayes theorem allows you to turn that around and ask about it in terms of some other parameters."
"Think of it as a conceptual shorthand for 'the best constant approximation for the rate of change'."
"Power laws are scale-free, there's no special size, no favored size, and mathematically there's no mean."
"So whenever we see the derivative of one of these sigmoids with respect to its input, we can just write the output times one minus alpha, and we've got it."
"Something that grows logarithmically with the size of the input. It's a slow growth."
"Infinity is not a number. It's more a process. Infinity never ends."
"Eigenvalues tell you about powers of a matrix in a way that we had no way to approach previously."
"The identity Combinator is a function that takes in an input a and it returns an output a."
"As the number of rays approaches infinity, the shape approaches the smooth spiral."
"This is the vector gives you each direction independently and that's kind of nice in some circumstances we like this."
"Infinity is weird and 0.999 repeating out to infinity is actually literally equal to one."
"Don't forget triangular numbers, very important."
"Diagonalization is the essence of many theorems."
"Ok, so the Galois group from Q to the biggest field is solvable."
"If the Galois group is not solvable, there is no way that the polynomial is solvable by radicals and we are done."
"Remember that throughout the video, we established that if the equation is solvable by radicals, then the corresponding Galois group is solvable."
"The breakthrough was finding a way to define limits while banishing all talk of infinitely small numbers."
"The row vector bra psi is the conjugate transpose of the vector ket psi."
"It's 10 seconds to the power of 4 then 10 seconds to the power of 3 then 10 seconds to the power of 2."
"If you have such a thing as Hilbert's hotel, that you're not looking at a feature, you're probably looking at a bug."
"Composition takes a function that goes from B to C, a function that goes from A to B, and input A, and then returns a C."
"As we approach an x value of zero from the left side, notice that the y value approaches zero as well."
"Conformal Maps, my favorite part."
"Things which are orthogonal stay orthogonal."
"So, what the Markov structure allows us to do, with the fact that the present that the future is independent of the past given the present, is it allows us to decompose the value function."
"For any prime, there will always be a bigger prime."
"The perimeter simply refers to the distance around the figure."
"The Hilbert transform is actually very straightforward if you understand the Fourier transform and the purpose of negative frequencies."
"Let's remind ourselves what P actually is... log base 3x is equal to -2."
"The number infinity is not a number, it's a concept; it's the largest possible thing that you never ever get to."
"Asymptotes are lines that the hyperbola will never touch."
"The idea of a conjugate is not going to go away, it's just a special way of multiplying by a very similar term that gets rid of the radicals."
"The supremum of a set is the least upper bound of that set."
"It's just an equation that has a derivative."
"This is the idea behind the exact equation."
"What is probability? It's a measure that tells you roughly on average how likely an event is to occur."
"It's the exact reverse of the vectorial summation of all terrestrial gravitational forces."
"Probability distribution is just the probabilities of occurrence of different possible outcomes."
"An infinite series cannot be resolved in a finite time."
"Here's the kicker: this point right here, the theoretical previous sharp corner, will always cut that angle right down the middle."
"This type of structure that repeats infinitely on smaller and smaller scales is called a fractal."
"These special points are called Sigma points."
"Mutually exclusive is when there's no overlap; those are the easy ones, we just add up the probabilities."
"What exactly is a proportion? A proportion is a ratio that's equal to another ratio."
"If observing what y is doesn't tell you anything about what x is, that's really your intuitive definition of independence."
"For a given involution on a set, each element must either map to itself or be paired with a different element."
"Maximization is equivalent to minimizing only the negative of the function."
"The convolution of two time functions is the product of their Fourier transforms."
"If the distance between x and a is less than delta, then the distance between f(x) and our limit is less than epsilon."
"The absolute value of \( x \) is just \( x \) if \( x \) is a positive number."
"The range is \( y \) is greater than or equal to 0."
"Taking the uniqueness and combining it to come up with something larger that is made up of those things, that incorporates those things within this new object that you created, and that would be the least common multiple."
"Maximum likelihood estimation... you actually choose the parameter value X that maximizes this quantity."
"Although these models start off looking like an exponential, they don't forever live as an exponential."
"Probability is a number between zero and one that a stated event will occur."
"Partial derivatives with respect to X or Y provide the rate of change in the direction of either the x-axis or the y-axis."
"Lyapunov functions are another way of defining stability of a function."
"The gradient points to the steepest ascent."
"If we think of X of Z as a product of zeros divided by a product of poles, one effect is that there's a constant that perhaps collects out in front."
"Cosine similarity is also very nice because it works well in high dimensional spaces."
"A set E is open if every point of E is an interior point of E."
"A set is closed if it contains all its limit points."
"A set is dense in X if its closure is all of X."
"Let's consider 'N' to infinity limit."
"A set is countable if its elements can be put in one-to-one correspondence with the set of natural numbers."
"Supremum means something super, so something related to the maximum."
"Any sequence has a limit point, which means it has a convergent subsequence."
"C delta t over delta x is a non-dimensional number, it is actually called the Courant number or CFL number."