Theorems linking topology and analysis concepts
I'd like to find examples of theorems that express a link between the topology of, say, a manifold and some differential concept. So far I've thought about the de Rham theorem (isomorphism between de...
View ArticleIs the power of $2$ in the Euclidean norm related to the fact that the...
Consider any local field$K$, endowed with its topological field structure. We define the function $| \cdot | : K \to \mathbb{R}_{\ge 0}$ as$$|x| = \frac{\mu(xS)}{\mu(S)},$$where $\mu$ is any Haar...
View ArticleWhat does "maximally" breaks the symmetry in Galois theory mean as expressed...
I have read the paper A view of mathematics by Alain Connes. If you read page 28 of this paper, you will read the following paragraph:Abel and Galois analyzed the symmetries of functions of roots...
View ArticleVisualising Spectra?
I have recently started to learn about Spectra. To state the definition (that I have learned), a spectrum $X = \{X_{n} \}_{n \geq0}$ is a sequence of based spaces $X_{n}$, with basepoint preserving...
View ArticleCan combining elements of a binary property result in the opposite property?
For non-binary properties, like size, we can have states like "large" and "small", and elements with the state "small", like atoms, can sum to an element with the state "large", like an elephant. In...
View ArticleIntuition behind nets
How can we think of nets as a generalization of sequences?Basically, we can see sequences as a way of enumerating elements of a set: in fact a sequence is a function $x_n:\mathbb{N}\rightarrow X$.Now,...
View ArticleCan I represent a transcendental function as an infinite product?
Let's say $f(x)$ is a real-valued transcendental function which has complex roots. If I know the formula for all roots, can I then immediately represent it as an infinite product of the form$$f(z) =...
View ArticleWhat is the connection between game theory and (modal) logic?
I'm interested in dynamic epistemic logic lately (reasoning about information and change in multi-agent systems). I also like game theory. I'm looking for some good resources about the connection...
View ArticleNoncommutative analogues of integration
In a paper I’m writing, I take the partial sums of sequences of elements from a monoid. Much of the paper has analogous theorems for continuous cases using integration. In particular, one of the proofs...
View ArticleUnexpected examples of natural logarithm
Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common...
View ArticleHow can I find an invertible function to flatten a probability distribution...
Let us assume that I have a Probability Density Function $t \to f(t)$, s.t. $$\int_{-\infty}^{\infty} f(\tau)d\tau = 1$$ and it is very "long-tailed".What I seek is given an estimate of $f$ - for...
View ArticleLagrange's theorem in symmetric polynomial
I read the book Unknown Quantity: A Real and Imaginary History of Algebra - John derbysheire If you read page 124 of this book, you will find the following paragraph.Suppose you switch around the...
View ArticleIs “reduct” just another word for “image through a forgetful functor”?
I have just started learning Mathematical Logic in greater depth and I was assigned some lecture notes to read.I noticed how the lecturer often used the word reduct, to mean… a structure which has gone...
View ArticleRamanujan&Types of mathematical intelligence [closed]
I have long been wondering about working mechanism of Ramanujan's brain. It seems to me that he'd been given some sort of mysterious intelligence that no one else ever had been. Even if one regards it...
View ArticleDoes excluding or including zero from the definitions of "positive" and...
I was absolutely certain that zero was both positive and negative. And zero was neither strictly positive nor strictly negative.But today I made a few Google searches, and they all say the same thing:...
View ArticleMoment generating function for Hypergeometric Distribution
For reference,$$P_{X}(k)=\frac{\binom{K}{k}\binom{N-n}{K-k}}{\binom{N}{n}}$$${\displaystyle N}$ is the population size${\displaystyle K}$ is the number of success states in the...
View ArticleA $\sigma$-algebra $\mathcal{S}$ on $\mathbb{R}$ that is larger than...
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.The author wrote as follows on p.52 in this book:We have accomplished the major goal of this section, which was to show that...
View ArticleWhat are some general guidelines or tips to reading a mathematics textbook?
I am an undergraduate who started down the math educator path but found pure math to be very intriguing. As I enrolled in higher math courses, I found reading the textbook to be ever more important. My...
View ArticleI'm just starting grad school and I'm already finding baby rudin's exercises...
I'm an economics bsc just starting grad school in quantitative finance, my first course is calculus, on week 1 we're studying the first two chapters of principles of mathematical analysis by Walter...
View Article$12$ Balls, $3$ bins. Find the Probability.
I'm not sure if a question like this might be discouraged, but I'm genuinely confused now.I've been going over a question for some time, which is:If $12$ distinct balls are to be placed in 3 identical...
View Article