In the case of sample proportions, why do we not get a $t$-distribution when...
If $\bar{x}$ has a normal distribution (or approx normal via CLT), then:$z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$ (has a z-distribution)If we substitute the sample standard deviation $s$ for the...
View ArticleDoes it have to be $P = NP$ or $P \neq NP$?
(I know nothing about this topic beyond the popular level, so apologies if this question is not well-posed.)Every time I've seen this problem discussed, it's always implicit that P must either equal NP...
View Article"For some mathematicians there is algebra after Galois theory." line in the...
I was reading some topics from the Textbook :Associative algebras by Richard Price( Springer GTM) and I am a graduate student and I saw this line on the very 1st page of the textbook: "For many people...
View ArticleAdvice for Self Learning Higher Mathematics? [closed]
I'm currently in my last year of my CS undergraduate program and I have always been pretty interested in mathematics in general. Throughout my program I have always tried picking up higher level...
View ArticleIs there any known application for normal numbers?
Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving...
View ArticleThe Existence of "Simple" Prime Generating Functions
Obviously, we do not know an explicit and easily manipulable formula for finding every prime - that is, a function $f(n)$ which yields the $n^{th}$ prime. I've seen plenty of formulas that "cheat" in...
View ArticleAre there "simple" motivations for the profinite topology?
I'm taking a topology course right now, and a lot of people are (understandably) confused by the profinite topology (and pro-$\mathcal{C}$ topologies more generally) on a group. The definition is a bit...
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Why hasn't homologically $\ker f = \text{ im } g$ been generalized to $(\ker...
I came up with a generalization before of homology which calls the usual homology "forward" homology and the homology induced by $\ker f \subset \text{ im } g$ (notice opposite direction of inclusion)...
View ArticleExamples of when a "new way of thinking" led to a solution [closed]
I was reading William Thurston's On Proof and Progress in Mathematics in which he discusses the value of the different ways people think about the same mathematical structure. He claims that many...
View ArticleHow to prove that one normed vector space is the completion of another one?...
Suppose we are given two normed vector spaces $Y$ and $X$. $Y$ is said to be the completion of $X$ if $Y$ consists of all points in $X$ and all Cauchy sequences are given a limit belonging to $Y$.How...
View ArticleWhy is it so difficult to answer such a basic question in this field? ("Basic...
I am reading "Basic Manifold Theory" by Yukio Matsumoto:Even if a differential structure exists on a topological manifold, it is not necessarily unique. The first discovery of an example where the...
View ArticleDepth of mathematical abstraction
We often say that the types of abstract algebraic structures we learn in algebra are groups, rings, and fields. However, there are many types of algebraic structures such as magma and...
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An approximation to $\ln(1+e^x)$ and how to use it for splitting the...
Trying to split the logarithm of the sum of two exponential functions (this question), I found the following approximation for the Softplus function$f(x)=\ln(1+e^x)$:$$\ln(1+e^x) \approx \begin{cases}...
View ArticlePractical research using mathematicians' algebraic structures
I know that the algebraic structure represented by groups, rings, and fields is a great help in mathematical research because it abstracts the operation structure of number systems such as real numbers...
View ArticleWhat to know before solving sequence and series problems?
Talking about How to solve the sequence: $87, 89, 95, 107, ?, 157$ for example, I read the hint: The difference between each term goes like this: 2,6,12. Can you notice any pattern? Based on it, can...
View ArticleAppearance of Formal Derivative in Algebra
When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are...
View ArticleFamous Mathematical Problems Inspired by Chess
I'm currently preparing an abstract for a presentation, and I would like to mention classical or well-known mathematical problems that have been inspired by the game of chess.Question. In your opinion,...
View ArticleUnderstanding the construction of the natural, integer and real numbers and...
I am trying to semi self study Real Analysis from Terence Tao's textbook with a Discord study group of people. I have had some trouble with the material in 2.2 (pretty much the beginning of the text)...
View ArticleHow is a cube made?
Apologies in advance and I will try to be as precise as possible, I started to independently learn Category Theory from "Introducing Category Theory" by Peter Smith. I find this part very confusing:For...
View ArticleCollection of surprising identities and equations.
What are some surprising equations/identities that you have seen, which you would not have expected?This could be complex numbers, trigonometric identities, combinatorial results, algebraic results,...
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