Articles of reference request

Good introductory probability book for graduate level?

Would you please suggest a good, readable introductory probability book for graduate level ? I have Shiryaev ‘s Probability with me, however i want to find another one. Preferably with solution manual for self study.

$\mathcal{C}^1$ implies locally Lipschitz in $\mathbb{R}^n$

This question already has an answer here: A continuously differentiable map is locally Lipschitz 3 answers

What textbook should I get to self-learn Calculus?

This question already has an answer here: What are the recommended textbooks for introductory calculus? 8 answers Which calculus text should I use for self-study? 6 answers

Understanding differentials

What is a good reference to learn about differentials and related topics. Some of my questions are: Why is it possible to split $dy/dx$ into individual terms $dx$ and $dy$? In a separated differential equation such as $F(x)dx + G(y)dy = 0$, what is the physical intuition behind “$F(x)dx$”? When integrating the latter equation, what […]

Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $.

Gödel’s Second Incompleteness Theorem says that if $ \mathsf{ZFC} $ is consistent, then $ \mathsf{ZFC} \nvdash \text{Con}(\mathsf{ZFC}) $, i.e., $ \text{Con}(\mathsf{ZFC}) $ is not provable in $ \mathsf{ZFC} $. Does anyone know of an authoritative reference that contains the claim that $ \neg \text{Con}(\mathsf{ZFC}) $ is also not provable in $ \mathsf{ZFC} $ if $ […]

An easy reference for genetic algorithm

My field is Coding Theory and my background is Algebraic, there are many applications of Genetic Algorithm in Coding Theory, I would to know the easiest and the most elementary and introductory note about “Genetic Algorithm in Coding Theory”, also is this algorithm using in Crypto too?

Equivalence between norms in $H_0^1(\Omega)\cap H^2(\Omega)$.

Based in many questions and answers like [1, 2, 3 ] and a comment a good comment here [4]. I would like to know that the space $H=H_0^1(\Omega)\cap H^2(\Omega)$ can be equiped with this norm $$\tag{1}\|\cdot\|_H=||\Delta\cdot||_{L^2}+||\nabla\cdot||_{L^2}+||\cdot||_{L^2},$$ this is the $H^2$-norm. Or, is it equipped with this one $$\tag{2}\|\cdot\|_H=||\Delta\cdot||_{L^2}+||\cdot||_{L^2}.$$ Can I say that if $H$ is […]

The derivation of the Weierstrass elliptic function

I am wondering if any of you could point me to any books and/or lecture notes that explain the Weierstrass $\wp$ function for a self-studying student of elliptic curves and functions. I am interested in any resources that may give the history of the Weierstrass function and its derivation. I do understand the basics of […]

Proving the Kronecker Weber Theorem for Quadratic Extensions

I am starting to read about the Kronecker-Weber Theorem. It says that any abelian extension of $\mathbb{Q}$ is contained in a cyclotomic extension. I read somewhere that for quadratic extensions the proof is not very difficult. Can anyone tell me any reference material for the proof of the kronecker weber theorem for quadratic extensions ?

Lie algebra associated to an arbitrary discrete group

I read somewhere that there is a classical (due to Philip Hall?) construction of a Lie algebra associated to any discrete group $\pi$ which is obtained from filtration quotients of the descending central series of $\pi$. Can anyone recommend some introductory material on this construction?