Articles of reference request

Proof that if an algebraic integer is rational, it is integer?

There is a well-known fact that the intersection of 𝔸 and ℚ is ℤ. It is mentioned in many places, including Wikipedia, without proof. Does this theorem have a well-known name, and where can i find the proof?

Convolution Error Estimate Reference Request

I am looking for a reference for the $L_p$ error of the difference of a Sobolev function and its convolution with a band-limited mollifier. The type of estimate that is quoted in a paper without a source is as follows: Consider $f\in W_p^k(\mathbb{R})$, and $\phi$ is a band-limited function ($\widehat{\phi}$ is supported on the interval […]

Looking for a book: $B(H)$ not reflexive

I’m looking for a book with a proof that for an infinite dimensional Hilbert space, $B(H)$ is not reflexive. Thank you.

How to show existence and uniqueness of a SL problem with von Neumann BCs

Let $f\in C[0,1]$ be a continuous function and consider for $x\in(0,1)$ the Sturm-Liouvile problem $$ -u”(x)+x\cdot u(x)=f(x) \tag1$$ where $u'(0)=u'(1)=0.$ I need to show that for any $f\in C[0,1]$ there is a unique $u\in C^2[0,1]$ that satisfies (1). Is there someone who knows a good book where I can find this result?

Representation theory of associative algebras applied to Lie Algebras

I’m given to understand that a lot of result on representation theory of Lie Algebras can be obtained by applying known result of representation theory of associative algebras to the enveloping algebras of lie algebras. I understand why it works and that’s not the problem. I’d just like to have a reference of some book […]

On terms “Orientation” & “Oriented” in different mathematical areas?

The goal of this question is to help to deal with different meanings of the words such as “orientation“ and “oriented” in different mathematical areas. Are different oriented concepts somehow related to each other in different mathematical areas? Oriented matroids (Matroid Theory) Oriented graphs (Graph Theory) Orientation (Topology, Global Analysis) Other areas?

Definition of Upper & Lower Riemann Sum

I recently came across the terms: ‘upper Riemann sum’ and ‘lower riemann sum’. Are they represent the same things as of ‘upper sum’ and ‘lower sum’ defined as follows.

Are there online English proofs of preservation of properties of morphisms by fpqc descent?

Accroding to Wikipedia’s article, the following propositions are proved in EGA. Are there online proofs written in English? Suppose that $f\colon X \rightarrow Y$ is a morphism of $S$-schemes. Let $g\colon S’ \rightarrow S$ be faithfully flat and quasi-compact, and let $X’, Y’$, and $f’$ denote the base changes by $g$. Then for each of […]

Reference on Infinite Dimensional Manifold

I am studying manifold. For comprehension, I read the site, and there is some information about infinite dimensional manifold. Now I have two questions or requests: (1) When was infinite dimensional manifold introduced? I guess this may be related to Functional Analysis. But I want more details about its history. (2) I am still […]

Key reference book on toric ideals: normal or not? Which definition to follow?

I want to understand sum of binomials better in terms of ideals such as binomial ideals, normal ideals and so by toric ideals. Examples about toric ideals contain $$\sum x^\alpha+\sum x^\beta\in\mathbb R$$ where $\alpha,\beta$ are multidegrees. I have a gut feeling that toric varieties and this Sparse elimination theory is crucial to understand features of […]