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 […]

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 […]

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 […]

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 […]

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 […]

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 $\frac{\sqrt{\Delta}+1}{2}$ […]

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 […]

The Wikipedia page for Weitzenböck 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 […]

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.

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 […]

Intereting Posts

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Given an infinite poset of a certain cardinality, does it contains always a chain or antichain of the same cardinality?
One equation, three unknowns.
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How to solve ODEs by converting it to Clairaut's form through suitable substitutions.
A non-noetherian ring with $\text{Spec}(R)$ noetherian
Is there a division algorithm for any Euclidean Domain?
If $2^{\aleph_0}$ is weakly inaccessible, can every cardinal $\kappa$ in the interval $[\aleph_0,2^{\aleph_0})$ satisfy $2^\kappa = 2^{\aleph_0}$?
Hilbert polynomial Twisted cubic
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Is there a $C^1$ curve dense in the plane?
Asymptotics of $\prod_{x=1}^{\lceil\frac{n}{\log_2{n} }\rceil} \left(\frac{1}{\sqrt{n}} + x\left(\frac{1}{n}-\frac{2}{n^\frac{3}{2}} \right)\right) $
Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$
Showing that $x^n -2$ is irreducible in $\mathbb{Q}$
If A is a finite set and B is an infinite set, then B\A is an infinite set