Articles of reference request

Positive and negative logical connectives

By inspecting the rules of inference for (intuitionistic) predicate calculus (or, alternatively, thinking about double negation translation), one sees that there is a certain dichotomy between two groups of logical connectives: $\top$, $\land$, $\to$, $\forall$ $\bot$, $\lor$, $\exists$ For some reason, I have it in my mind that the first group is called positive and […]

Approximate largest (in quantity, not magnitude) eigenvalue and eigenvector of a matrix

I need to obtain an approximate expression for the eigenvector corresponding to the largest real eigenvalue of a matrix, as well as the largest eigenvalue. Note that I mean largest not in absolute value, but largest in real value. The matrix can be approximately diagonal in some cases, but not always. I have tried perturbation […]

Exercise book for Elementary/Introduction to Real Analysis?

I’m currently doing a course in Elementary Analysis (Intro to real analysis). My course focuses on the topics: sequences, limits of functions, continuity, uniform continuity and derivative/derivability of functions, all in R. I’ve studied on my own with the book “Elementary Analysis, the theory of calculus“, which has proved to be an extremely useful book […]

Real orthogonal Lie algebra isomorphic to Clifford bivectors

I’m studying Clifford algebras on this moment, and I frequently find the statement $$\left(\mathbb{R}_m^{(2)},[\cdot,\cdot]\right) \cong \mathfrak{so}_{\mathbb{R}}(m)$$ stating that the bivectors of a real Clifford algebra are isomorphic to the real special orthogonal Lie algebra. Unfortunately I’m not able to find the original theorem proving this. Is there anyone that can help me with the finding […]

Prove a categorical statement

In the answer to Direct products in subcategories it is said: If $\mathcal{D}$ is a full subcategory of $\mathcal{C}$ and $A \times_{\mathcal{C}} B$ is (isomorphic to) an object of $\mathcal{D}$, then it is isomorphic (in $\mathcal{D}$) to $A \times_{\mathcal{D}} B$. In other words, the embedding of a full subcategory reflects products. (In fact, it reflects […]

Must a certain continued fraction have “small” partial quotients?

I have reformulated the original question, which appears at the bottom, it a way that seems more likely to produce a reference. New version: Let $\Delta$ be a positive nonsquare integer congruent to $0$ or $1$ modulo $4$. If $\Delta$ is even, expand $\frac{\sqrt{\Delta}}{2}$ in a simple continued fraction. If $\Delta$ is odd, expand $\frac{\sqrt{\Delta}+1}{2}$ […]

A book to study about hyperbolic plane, hyperbolic translations, etc.

In this paper, page $6$, the authors state the following: The translations of the hyperbolic plane are defined as products of two central symmetries; the set of hyperbolic translations forms a sharply transitive set on the hyperbolic plane, the associated loop is the classical simple Bruck loop. I would like to have a refence to […]

Weitzenböck Identities

The Wikipedia page for Weitzenb√∂ck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In particular, I am interested in references which focus on these identities in complex geometry. I have already consulted Griffiths & Harris which is […]

Text on Group Theory and Graphs

A student and I are going to investigate the use of group theoretic techniques in graph theory. What are good texts in this area (introductory and otherwise)? We are particularly interested in studying automorphism groups of graphs, but a text with a broader view would also be welcome.

Are there results for relations between upward and downward closed partitions of some powerset?

I stumbled upon this, given some set $X$ and its powerset $\mathcal{P}(X)$ and some incomparable set $\mathbb{S}\subseteq\mathcal{P}(X)$, i.e. for any $S,S’\in\mathbb{S}$ we have $S\setminus S’\neq\emptyset$. Then upward closure of $S$ is defined as $up(\mathbb{S})=\{S’\supseteq S\mid S\in\mathbb{S}\}$. Given an upward closed set its minimal elements define an incomparable set. Similarly for incomparable $\mathbb{T}\subseteq\mathcal{P}(X)$ we can define […]