Articles of reference request

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 […]

Essay about the art and applications of differential equations?

I teach a high school calculus class. We’ve worked through the standard derivatives material, and I incorporated a discussion of antiderivatives throughout. I’ve introduced solving “area under a curve” problems as solving differential equations. Since it’s easy to see the rate at which area is accumulating (the height of the function), we can write down […]

Laguerre polynomials and inclusion-exclusion

Does anyone know a reference for the solution of the generalized derangement problem via Laguerre polynomials? The Wikipedia article here says that this is an application of inclusion-exclusion, but I don’t see how. This formula was used by joriki in a nice MSE answer here. The article below solves the generalized derangement problem with inclusion-exclusion, […]

Proof of Hasse-Minkowski over Number Field

Can anyone point me towards a good reference which contains the proof of the Hasse-Minkowski theorem of quadratic forms over a number field? Serre’s “A Course in Arithmetic” has a self contained proof for the theorem when the base field is $\mathbb{Q}$, but I am looking for the more general setting.

Where can I find good, free resources on differential equations?

I’d like to know if there are any good online books, lecture notes, videos, tutorials, or similar that are free to the public (on differential equations). Suggestions are welcome!