Articles of reference request

Is there a Lucas-Lehmer equivalent test for primes of the form ${3^p-1 \over 2}$?

I’m reviewing the cyclotomic form $f_b(n)= {b^n-1 \over b-1}$ for various properties to extend an older treatize of mine on that form. With respect to primality there is the Lucas-Lehmer-test for primeness of $f_2(p)$ where of course $p$ itself must be a prime. I was now looking, whether I can say some things for primes […]

Explicit Riemann mappings

Typical proofs of the Riemann mapping theorem are not terribly explicit (one maximizes a functional, or something equivalent, such as using Dirichlet’s principle). The theorem states that if $U$ is a simply connected open subset of the plane, then there is a biholomorphism between $U$ and the unit disk. I imagine, due to the wild […]

What is the connection between linear algebra and geometry?

I am currently studying linear algebra. Yet, I found discussions about linear algebra usually explain things in a geometric fashion. I am quite confused on how to link up these two topics. Can anyone kindly recommend some books/readings for me as an introduction to the concept of combining these two topics?

What's the dual of a binary operation?

I have a binary operation: $ \diamond : M\times M \to M $ . I want to dualize the binary operation by flipping the arrow, giving me: $$ f : M \to M\times M $$ Now, I can define a coassociativity law as: $$ ((f \circ fst \circ f)(m), (snd \circ f)(m)) = ((fst \circ […]

On the possible values of $\sum\varepsilon_na_n$, where $\varepsilon_n=\pm1$ (i.e., changing signs of the original series)

I have used the following result in an answer on this site. Suppose that $a_n>0$ are positive real numbers such that $\sum\limits_{n=1}^\infty a_n=+\infty$ and $\lim\limits_{n\to\infty} a_n=0$. Ten for any choice of $A,B\in\mathbb R\cup\{\pm\infty\}$ such that $A\le B$ there exists a sequence $\varepsilon_n$ such that each $\varepsilon_n\in\{\pm1\}$, $$\liminf\limits_{n\to\infty} \sum_{k=1}^n \varepsilon_k a_k=A \qquad\text{ and }\qquad \limsup\limits_{n\to\infty} \sum_{k=1}^n […]

mathematical books

I am a Japanese so it is difficult for me to read English and I may make some grammatical mistakes. I have little experience with mathematics but plan to self-study mathematics by reading mathematics books in English. I would like to know what the best text is in each of the following subjects, set theory […]

Proof of the extreme value theorem without using subsequences

I am preparing a lecture on the Weierstrass theorem (probably best known as the Extreme Value Theorem in english-speaking countries), and I would propose a proof that does not use the extraction of converging subsequences. I did not explain subsequences in my calculus course, and I must choose between skipping the proof of the theorem […]

Advice about taking mathematical analysis class

I appologize if this isn’t the place to ask, if it’s not could you let me know and I will take it to meta? Anyway, so I am planning on taking a mathematical analysis course next spring, and I’m really excited about it because it seems so interesting and fun. However, I know this will […]

Are hyperoperators primitive recursive?

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related to these functions, but I am not sure if they still keep the property of being NOT primitive recursive. My intuition is […]

Understanding measures on the space of measures (via examples)

Let $X$ be a Polish space. If it allows for an interesting answer, you may assume $X$ is compact or even $X=[0,1]$. The space $\mathcal{P}(X)$ of Borel probability measures on $X$ is also Polish (via the Prokhorov metric). Measures on $\mathcal{P}(X)$ (i.e. elements of $\mathcal{P}(\mathcal{P}(X))$) arise, for example in the ergodic decomposition. I’m looking to […]