Uncategorized math stuff

frequentist probability: relative frequencies
Bayesian probability: degree of knowledge

Alternatively one can think of Riemann sheets whereby on crossing a branch cut
one moves onto a different Riemann sheet of the function; the number of branches equals
the number of Riemann sheets. This allows closed contours to be formed by going around
branch cuts as many times as required to get back to the original Riemann sheet. However
although this scheme is very elegant, for calculational purposes it is best to treat branch
cuts as barriers one cannot cross.

A harmonic function is a function whose value at a point is always equal to the average of its values on a sphere centered at that point (reference). This is why they show up as steady-state solutions to the heat equation: if this averaging property weren't true, then heat would be flowing either from or to a point.

In mathematics, the tensor product V ⊗ W of two vector spaces V and W (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by ⊗, from ordered pairs in the Cartesian product V × W into V ⊗ W, in a way that generalizes the outer product. The tensor product of V and W is the vector space generated by the symbols v ⊗ w, with v ∈ V and w ∈ W, in which the relations of bilinearity are imposed for the product operation ⊗, and no other relations are assumed to hold.

https://jeremykun.com/2014/01/17/how-to-conquer-tensorphobia/
basically, (a, b) (x) (c, d) is a completely different element unless a is a multiple of c or b is a multiple of d
that's the difference from adiition where (a, b) + (c, d) would be (a + c, b + d)

Consider the statement "a continuous function preserves closeness". That means if f ( x ) f(x) and f ( y ) f(y) are close to each other then x x and y y were close to each other originally.

In analysis, a function is continuous if you can make the image in the codomain as small as you like, by choosing a small enough part of the domain.

hmm interesting definition in terms of 'touches' (v)

• multiply both parts by e^{∫ f(t) dt}
  
• after that, we have
  (y exp(∫ f(t) dt)' = g(x) exp(∫ f(t) dt)
  Easy to solve.
  

Homogenous: Sum_n=0^N A_n y⁽ⁿ⁾ = 0

• Guess: y = e^{z x}, then everything reduces to the polynomial:
  
• Replace derivative y⁽ⁿ⁾ with zⁿ, solve for z, get the solution.
  
• If multiplicity of root z is m, then: x^k e^zx for k from 0 to m-1
  
• Linear independence can be shown using Vandermonde determinant
  

free monoids are lists
free semigroups are nonempty lists?
free magmas are nonempty binary trees (data is in lists)

https://www.scottaaronson.com/blog/?p=4684

As a simple example, I’m totally fine uttering statements like, “a family of Boolean functions with superquadratic gap between randomized and quantum query complexities exists“—even though the type of “existence” that we’re talking about clearly isn’t physical (some would call it Platonic).

https://xenaproject.wordpress.com/2019/12/17/lean-has-real-manifolds/#comments

https://xenaproject.wordpress.com/2019/12/07/prove-a-theorem-write-a-function/

• Definitional (intensional) : two = succ (succ zero)
  Might use reductions (computational/reductional equality) : (\x. x + x) two = succ (succ (succ (succ zero)))
  
• Propositional: ==
  
• Extensionally equal, with particular numbers
  ∀ x, y -> x + y == y + x
  

The point: infinite sum of positive numbers in not always unbounded.

The point: infinite sum of positive numbers in not always unbounded.

First, some numbers are squares, while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every square there is exactly one positive number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of the idea of one-to-one correspondence in the context of infinite sets.

Resolution: different defintions of equally sized sets.

• Explicit: forward time, central space
  
• Implicit: backward time, central space
  
• Crank–Nicolson: central time, central space
  

https://math.stackexchange.com/questions/908499/is-a-functional-derivative-a-generalized-function

Not really.. the derivatives are pretty different

todo
from ip Lean is better for proper maths than all the other theorem provers

[2017-11-01]

• .
  

[–]skibilly 2 points 2 years ago*
When p-value is used it's because you are attempting to test a hypothesis of some sort. So let's first define a hypothesis. At its simplest, a hypothesis is an attempt to explain some observation (generally by linking it to other observations). To keep things simple, let's say that you have observed that every time a cat gives birth, you've also observed that the babies have been kittens. You therefore attempt to explain this with the following hypothesis: "When a cat gives birth, the resultant offspring will be kittens." We'll call this "H1" for hypothesis number 1.
Now, in statistics (and by extension, science in general) there is never just one hypothesis. There is your hypothesis AND there's something called the Null Hypothesis. The Null Hypothesis states that there is actually no relationship between the observations you're trying to link. In our example, the Null Hypothesis would state: "There is no relationship between pregnant cats and kittens." We'll call this "H0" for hypothesis number 0 or, null hypothesis.
But what's the point of having a null hypothesis?
Aside from math (as far as I know) nothing we know about the universe is absolute. All that we know is based on observation, and since we can't observe everything that there is, we can't say with complete certainty that any explanation of an observation is "true." However, we can say that any given hypothesis has a higher or lower probability of being true than other hypotheses. Through beating as many alternative explanations as possible, we can increase our confidence that our successful explanation is true....but we can't ever say that it is 100% true. This is called inductive reasoning and it is the basis for all that we think we know about the universe.
Back to our example of pregnant cats and kittens. Because we understand the limitations of inductive reasoning, we know that we can't ever prove 100% that a pregnant cat will always give birth to kittens (maybe on some far away planet cats give birth only to snakes). However, we can at least show that our hypothesis (H1) has a higher probability of being true than the null hypothesis (H0). In other words, we can say that there is a high probability that there is at least some relationship between pregnant cats and kittens.
So, we start tracking every cat birth that we possibly can. Each time a cat gives birth we mark down whether the offspring is a kitten or something else. Once we have a good size of observations, we run the result through a statistic called "p-value" or "calculated probability." The number that comes out at the end is the probability of getting the same results if the null hypothesis is true.
In our example, let's say we observed 100 pregnant cats. All 1,000 births resulted in kittens. No snakes, no turtles, no balls of anti-matter. Just kittens. Rather than doing all the math we'll just say that our resulting p-value is 0.01. This means that there is a 1% chance that all 100 pregnant cats gave birth to kittens and yet there's actually no relationship between the two. The p-value doesn't say that your hypothesis is correct, and it doesn't say the null hypothesis is incorrect. The p-value merely states the probability that you could get the same results as you observed due to random chance.
Typically in science we consider a p-value of 0.05 to be significant. Meaning that unless the probability of getting the same results due to random chance is 5% or less, we generally don't pursue the alternate hypothesis and instead either continue looking for something better or stick with the null hypothesis: the observations are unrelated.

• .
  

It's not significant with respect to your a priori cutoff value, so it is not significant. All these terms:
approaching
marginally
trending toward
are incorrect interpretations, as you have no idea what would happen if you were to collect more data (e.g. p could be 0.09 after collecting additional cases, so how can one justify saying that 0.055 is approaching significance?).

• .
  

A p value is not a statistical analysis or a test, it is the outcome of a statistical test. One of the issues in relying on p values is that there are a lot of different ways to get to a p value, and not all of these may be appropriate (this is sometimes called torturing, submitting data to various statistical tests until something comes up significant).
The other point to make here is that we don't prove or disprove hypotheses. We can reject the null hypothesis and provide support for the alternate hypotheses, or we can fail to reject the null hypothesis and find, well, not much.
So, what does a p value mean?
Let's use an example. Say we are interested in testing a new drug to treat depression. One good way to test this is to get a big group of depressed individuals and to give them either this new drug or a placebo. Before doing this, we measure their level of depression on the Fictitious Depression Inventory (FDI). Before giving them a course of our new drug and placebo, our two groups have rather similar FDI scores:
Drug: 44.2 Placebo: 46.7
We then give them the appropriate pills (double blind and randomised, of course). After, say, eight weeks of this, our groups don't look as similar anymore:
Drug: 36.7 Placebo: 42.4
Both have improved in their depression scores (as frequently happens, depression improves over time). We run some statistics on our two groups, and we find that the two groups are different, with an obtained p value of 0.038 (this is usually written as p = .038). This is smaller than our critical value of 0.05 and we can conclude that there is indeed a difference between our groups as a result of the drug being better at lowering FDI scores than a placebo control.
So, what does this all mean? Well, there's two important things here. One is the meaning of the null hypothesis, and the other is the meaning of p = .038.
The null hypothesis states that the difference between the two groups we have is not real. Any difference that we see in those numbers is just because of chance or sampling or something else, because that difference does not exist in the population. What we are working with is a sample, we have a small group of depressed individuals relative to all the people in the world who are depressed. We want to know if what we see in our sample applies to everyone else (the population). If the null hypothesis is true, it says that, yeah, okay we have a difference in our sample, because our two numbers are different, but this difference is just noise or something else, it's not about the pills. When we reject the null, we say that the difference we see in our numbers is not due to chance, and we infer that this effect exists in the population.
So what is the p value? The p value is a way for us to formally acknowledge that there is some risk that we are wrong when we say the null hypothesis is false. Specifically, the p value is how likely we are to see the two numbers we're seeing (or another two numbers that different from one another) if the null is true. The cutoff that we have all accepted is at most 5% chance of accepting a result when the null is in fact true (p < .05). Why 5%? Well, when Fisher developed the idea of null hypothesis testing, he threw number out as an example and people kind of ran with it.
What are the issues with null hypothesis significance testing? Well, one big one is that it dichotomises thinking. Everything ends up being significant or not, when there is a little more finesse in there. The other issues, closely related to this, is that people misinterpret the p value so that a smaller p (say 0.0001 instead of 0.001 or 0.049) means that your experiment worked better. This really isn't true. You can get a "really significant" finding that really isn't important. Going back to our fake example from above, say your groups looked like this:
Drug: 36.8 Placebo: 38.4
If this effect comes out to be "highly significant" (p = .0013), it doesn't mean that the results are all that important because the difference between the two groups is so tiny. The drug did have an effect, but it didn't really help people much with their depression. And given the side effects that drugs can have, and the cost, you're probably better off in the long run in just taking that sugar pill.
This dichotomous thinking has other downsides. It forces people to push for significance when maybe that's not the best thing given the data. You can slice a data pie in many different ways, and eventually you may be able to get a significant result, but perhaps not one that is meaningful or that will replicate.
Whether or not you get a significant result is also dependent on the sample size you're working with. Running experiments is very laborious, and it's in everyone's benefit to test the least number of participants to give robust findings. The issue is that testing too few people, however, can make it more likely to find a significant effect (and testing way too many does the same by inflating the degrees of freedom).
Edit: Apologies if any of this is incoherent, please feel free to ask followup questions.

A new random variable Y can be defined by applying a real Borel measurable function {\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} } g\colon \mathbb {R} \rightarrow \mathbb {R}  to the outcomes of a real-valued random variable {\displaystyle X} X. The cumulative distribution function of {\displaystyle Y\,\!} Y\,\! is

{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y).} F_{Y}(y)=\operatorname {P} (g(X)\leq y).
If function {\displaystyle g} g is invertible, i.e., {\displaystyle g^{-1}} g^{-1} exists, and is either increasing or decreasing, then the previous relation can be extended to obtain

{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq g^{-1}(y))=F_{X}(g^{-1}(y)),&{\text{if }}g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq g^{-1}(y))=1-F_{X}(g^{-1}(y)),&{\text{if }}g^{-1}{\text{ decreasing}}.\end{cases}}} F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq g^{-1}(y))=F_{X}(g^{-1}(y)),&{\text{if }}g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq g^{-1}(y))=1-F_{X}(g^{-1}(y)),&{\text{if }}g^{-1}{\text{ decreasing}}.\end{cases}}

Algebraic multiplicity: multiplicity as a root of the characteristic polynomial.
In case of the field of complex numbers, the sum of algebraic mult. is exactly n.

Geometric multiplicity: dimenstion of the eigenspace, associated with the eigenvalue.

Sum of geometric multiplicities for A - λ I is equal to dim ket (A - λ I)

Jordan cell corresponds to each subspace

L^∞ space
A sequence of bumps of height 1 functions:

f(x) = 1 if x ∈ [n; n + 1], 0 otherwise

Does not have a convergent subsequence!

Fischer-Rietz theorem: L^p is a Banach space.
Dominated convergence theorem.

https://news.ycombinator.com/item?id=21628449

https://intuitiveexplanations.com/math/differential-equation-solution-strategies

http://immersivemath.com/ila/index.html

https://news.ycombinator.com/item?id=16373386
like physics travel guide, but for math

https://en.wikipedia.org/wiki/Bourbaki_dangerous_bend_symbol

The dangerous bend or caution symbol ☡ (U+2621 ☡ CAUTION SIGN) was created by the Nicolas Bourbaki group of mathematicians and appears in the margins of mathematics books written by the group. It resembles a road sign that indicates a "dangerous bend" in the road ahead, and is used to mark passages tricky on a first reading or with an especially difficult argument.[2]

interesting, there are not that many different computer algebra systems. Also look that sagemath is updated more often than sympy?

https://www.youtube.com/watch?v=UVuKyZ4pBzg&list=WL&index=71&t=0s
18:42 if you're a mathematician, you're not gonna escape, the universe will find applications

https://www.youtube.com/watch?v=ffUnNaQTfZE
good video on hypercomputation and infinite series

https://www.youtube.com/watch?v=Dp-mQ3HxgDE&list=WL&index=48&t=1s
great talk (advocates Lean prover)

https://math.stackexchange.com/questions/320220/intuitively-what-is-the-difference-between-eigendecomposition-and-singular-valu/320232#320232
difference between svd and eigendecomposition

https://stats.stackexchange.com/questions/22/bayesian-and-frequentist-reasoning-in-plain-english#comment3052_56

some practical example with beta distribution for bayes vs freq

good talk, nice explanation of bayes vs frequentist
takeaway: bayes os more natural for communicating scientific results to public: model parameter is 95% likely to be within the specific confidence interval

https://www.lesswrong.com/posts/mQfNymou9q5riEKrf/frequentist-vs-bayesian-breakdown-interpretation-vs

https://twitter.com/Meaningness/status/1193683556364513280

Excellent explanation of the emotional fallout from the crisis of the foundations of mathematics

Parabolic PDEs (e.g. heat) smooth out singularities. Hyperbolic PDEs (e.g. wave) displace singularities. 

excellent videos

if f(y, x) and df/dy(y, x) are continuous around y₀, x₀, there exists a local solution
Note that the theorem only guarantees the existence of solution nearby the initial values,and one cannot expect the solution to be defined for allx

https://www.amazon.co.uk/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260

some good explanations for calculus of variations

Reading great mathematicians like Terence Tao discuss the heuristics they use on unsolved problems25, they bear some resemblances to computer science techniques.

Lean: https://leanprover.github.io/
Repo: https://github.com/leanprover/lean/
Chat: https://leanprover.zulipchat.com/
The maths course (in French) that can be seen during the presentation: https://www.math.u-psud.fr/~pmassot/enseignement/math114/

Lean deserves wide recognition. First, it's fast enough for highly interactive theorem development. Second it can also be used as a programming language. And lastly its syntax is pleasant to work with which is important to the experience.
If you have only heard about interactive theorem provers and don't yet have any opinions I'd give Lean a try first. The interactive tutorials are nice and the aforementioned features make it pleasant to work with.

bandali on May 28, 2018 [-]
An Introduction to Lean [0] is another nice (albeit incomplete) tutorial.
There’s a fairly active community over on Zulip [1] if you like to drop by for a chat or get some help.
[0]: https://leanprover.github.io/introduction_to_lean/
[1]: https://leanprover.zulipchat.com

An observation with an internally studentized residual that is larger than 3 (in absolute value) is generally deemed an outlier.

https://stats.stackexchange.com/questions/31867/bayesian-vs-frequentist-interpretations-of-probability

The snag is that we have to introduce the prior distribution into our analysis - this reflects our belief about the value of p before seeing the actual values of the Xi. The role of the prior is often criticised in the frequentist approach, as it is argued that it introduces subjectivity into the otherwise austere and object world of probability.
It might also be good to mention that the gap between the frequentist and Bayesian approaches is not nearly as great on a practical level: any frequentist method that produces useful and self-consistent results can generally be given a Bayesian interpretation, and vice versa. In particular, recasting a frequentist calculation in Bayesian terms typically yields a rule for calculating the posterior given some specific prior. One can then ask "Well, is that prior actually a reasonable one to assume?"
You're right about your interpretation of Frequentist probability: randomness in this setup is merely due to incomplete sampling. From the Bayesian viewpoint probabilities are "subjective", in that they reflect an agent's uncertainty about the world. It's not quite right to say that the parameters of the distributions "change". Since we don't have complete information about the parameters, our uncertainty about them changes as we gather more information.
A Bayesian may say that the probability that there was life on Mars a billion years ago is 1/2.
A frequentist will refuse to assign a probability to that proposition. It is not something that could be said to be true in half of all cases, so one cannot assign probability 1/2.
Frequentists posit that the probability of an event is its relative frequency over time,[1] (3.4) i.e., its relative frequency of occurrence after repeating a process a large number of times under similar conditions. This is also known as aleatory probability. The events are assumed to be governed by some random physical phenomena, which are either phenomena that are predictable, in principle, with sufficient information (see determinism); or phenomena which are essentially unpredictable. Examples of the first kind include tossing dice or spinning a roulette wheel; an example of the second kind is radioactive decay. In the case of tossing a fair coin, frequentists say that the probability of getting a heads is 1/2, not because there are two equally likely outcomes but because repeated series of large numbers of trials demonstrate that the empirical frequency converges to the limit 1/2 as the number of trials goes to infinity.

https://www.reddit.com/r/statistics/comments/ywrba/eli5_bayesian_statistics/

Bayesian statistics rely more on computational simulations and have become more common because computers have become much faster. A lot of people are put off by the fact that you incorporate prior believes into your estimate of the "truth", but you can use non-informative (vague) priors like in the example above (each possible outcome gets equal probability) and in any case with a lot of data your priors will become less important.

Copy and paste the code at the other end of that link into VS Code if you have Lean installed locally if you want a much speedier experience. The game is to prove symm and trans from refl and ind. I would work out the maths proofs first.

@XenaProject: Isabelle/HOL has schemes! https://t.co/p8SbYF90aw . The formalisation is due to Bordg, Paulson and Li. In this thread I will try to explain why this is a big deal. 1/n

Studying or teaching Complex Analysis? This online book might be useful.
⭐️Complex Analysis: A Visual and Interactive Introduction⭐️
Thanks everyone for your suggestions and reporting issues to improve this project. https://t.co/ElaWs4XjIV

https://twitter.com/PonceCampuzano/status/1214850688506716160

Referring to the same notion as Frunobulax did in their answer, you can define a notion of a point x∈X touching a subset A⊆X.
For example in the reals 0 is intuitively touching (0,1], and that stays true no matter how far you zoom in!
In usual topological terms this relation is expressed as "x is in the closure A¯¯¯¯ of A".
And this does indeed describe continuity in a forward fashion: If x is touching A, then f(x) should be touching f(A). In other words