Articles of reference request

Is every irreducible operator unitary equivalent to a banded operator?

This issue continues this question. Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis. $T \in B(H)$ is banded if $\exists r \in \mathbb{N}$ such that $ (Te_{n}, e_{m})\ne 0 \Rightarrow \vert n-m \vert \leq r$. Definition : […]

Introductory Treatment of Differential Geometry

I’m going to be taking a graduate course in differential geometry, this coming fall, but I am not prepared for it. Can anyone recommend a good introductory treatment of the background materials? The list of topics in the course is: Manifolds, Local Study of Manifolds, Vector bundles, Submanifolds, Vector Fields, Lie Groups (brief treatment), Differential […]

Turing's 1939 paper on ordinal logic

I am reading Turing’s 1939 paper on ordinal logic (“Systems of Logic Based on Ordinals”, A. M. Turing, Proc. London Math. Soc. ser. 2, 45 (1939), #1, 161-228, DOI: 10.1112/plms/s2-45.1.161.) Immediately after talking about successively extending axiomatic systems using transfinite iteration of the reflection principle, he says, on p. 197: “Another ordinal logic of this […]

A question regarding Frobenius method in ODE

Suppose $b(x),c(x)$ are real functions analytic at $0$. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well known that the differential equation $$x^2y”+xb(x)y’+c(x)y=0$$ has a solution of the form $$y_1=x^r(1+\sum_{i=1}^\infty a_ix^i),$$ where the series $\sum_{i=1}^\infty a_ix^i$ has radius of convergence $\ge R$ (e.g., see Tyn Myint-U, […]

The integral $\int \frac{J_{d/2}^{2}(x)}{x} \ \mathrm{d}x$

I’d like to know what the explicit solution to the following integral is: $$\displaystyle \int_{t}^{\infty} \frac{J_{d/2}^{2}(x)}{x} \ \mathrm{d}x,$$ where $t > 0$, $d \in \mathbb{N}$, and $J_{\nu}$ denotes the Bessel function of the first kind. Using Mathematica, I’ve been able to get some results when $d$ is even. For instance, if we take $d = […]

Matrix-product-integrals?

Whereas the conventional “sum integral” is $$ \lim_{\Delta x\to 0} \sum_i f(x_i)\,\Delta x, $$ a “product integral” is $$ \lim_{\Delta x\to 0} \prod_i f(x_i)^{\Delta x}. $$ Now you’re thinking: just take logarithms and it’s a “sum integral”, so why bother having this additional concept? Somewhere I heard this answer proposed: Because one does this with […]

What is Ramsey Theory ? what is its own importance in maths?

3 days ago , i had a discussion with a close friend who studies physics – still a student – . and i was telling her about the biggest known numbers in maths , so i told her about numbers such googol and googolplex, then about Graham number and she asked me , what is […]

Reference request for a Meijer G identity

I found the following identity while I was working on a recent question of mine: For $z\in\mathbb{C}$, $$G_{2,3}^{3,0}\left(z\left|\begin{array}{c}1,1\\0,0,0\\\end{array}\right.\right)=\gamma\ln{z}+\frac12\ln^2(z)-z\,_3F_3(1,1,1;2,2,2;-z)+\frac{\gamma^2}2+\frac{\pi^2}{12}.\tag{1}$$ WolframAlpha confirms it, but I couldn’t find the formula stated anywhere online from my searches. Has this result already been discovered? If so, could someone point me in the right direction? I’ve made a proof for the […]

Frattini subgroup of a finite group

I have been looking for information about Frattini subgroup of a finite group. Almost all the books dealing with this topic discuss this subgroup for p-groups. I am actually willing to discuss the following questions : Let G be a finite group, is the Frattini subgroup of G abelian ? Why is the order of […]

Find area bounded by two unequal chords and an arc in a disc

Math people: This question is a generalization of the one I posed at https://math.stackexchange.com/questions/348296/find-area-bounded-by-two-chords-and-an-arc-in-a-disc . Below is an image of a unit circle with center $O$. $\theta_1, \theta_2 \in (0, \pi)$ and $\gamma \in (0, \min(\theta_1,\theta_2))$. $\theta_1 = \angle ROS$, $\theta_2 = \angle POQ$, and $\gamma = \angle ROQ$. I want to find the area […]