Articles of reference request

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 . 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 […]

moment generating function

can anyone help me to inform the name of any book where I can get the following theorem, or give some detailed hint to solve this one: Let $X$ and $Y$ be two random variables, if the moment generating functions of $X$ and $Y$ are equal then the probability distributions of $X$ and $Y$ will […]

Is there anything special about a transforming a random variable according to its density/mass function?

Lets say that $X\sim p$, where $p:x\mapsto p(x)$ is either a pmf or a pdf. Does the following random variable possess any unique properties: $$Y:=p(X)$$ It seems like $E[Y]=\int f^2(x)dx$ is similar to the Entropy of $Y$, which is: $$ H(Y):=E[-\log(Y)]$$ It seems like we can always make this transformation due to the way random […]

How do we know that certain concrete nonstandard models of the natural numbers satisfy the Peano axioms?

It is easy to come up with objects that do not satisfy the Peano axioms. For example, let $\Bbb{S} = \Bbb N \cup \{Z\}$, and $SZ = S0$. Then this clearly violates the axiom that says that $Sa=Sb\to a = b$, unless we agree that $Z=0$, in which case what we have is exactly the […]

Exercise books in functional analysis

I’m studying functional analysis and I was wondering if there are some exercise books (that is, books with solved problems and exercises) The books I’m searching for should be: full of hard, non-obvious, non-common, and thought-provoking problems; rich of complete, step by step, rigorous, and enlightening solutions;

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i’m not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, and let $$ BL(\mathcal{X})=\{f:\mathcal{X}\to \mathbb{R}\, \, | \, \, f \text{ is Lipschitz and bounded}\} $$ denote the bounded real-valued Lipschitz […]

Global convergence for Newton's method in one dimension

I’m looking for a Theorem that I can cite which proves that Newton’s method for finding a zero of a function converges globally and quadratically if the function $f : [a, b] \rightarrow \mathbb{R}$ is increasing and convex and has a zero $r \in [a,b]$ with $f(r) = 0$ and $f'(r) \neq 0$, with starting […]

Analytic Capacity

For a compact set $K\subset\mathbb{C}$ the analytic capacity is defined as $$\gamma(K)=\sup\{|f^\prime(\infty)|:f\in M_K\}$$ where $M_K$ is the set of bounded holomorphic functions on $\mathbb{C}\backslash K$ with $\|f\|_\infty\le 1$ and $f(\infty)=0$. I have two questions. What is the intuition behind this definition? What does $f^\prime(\infty)$ mean? It doesn’t seem to be $\lim_{z\rightarrow\infty}f^\prime(z)$ so I’m confused. It […]

Are there books introducing to Complex Analysis for people with algebraic background?

I’m a third year student who is mostly interested in commutative algebra. In Algebraic Geometry a lot of example come from Complex Analysis. So to deepen my understanding/intuition, I’ll finally attend an Introduction to Complex Analysis lecture. (Which is actually made for second year students, but let’s say I’ve not been a big fan of […]