If I consider an algebraic group as a functor $\mathbf{G}$ from the category of $\mathbb{Q}$-algebras to the category of groups, can I consider $\mathbf{G}(\mathbb{Q})$ as a subgroup of $\mathbf{G}(\mathbb{R})$ as a subgroup of $\mathbf{G}(\mathbb{C})$?
But functors need not send monomorphisms to monomorphism, so even through $\mathbb{Q} \hookrightarrow \mathbb{R}$ is injective, I don't see why the induced map $\mathbf{G}(\mathbb{Q}) \to \mathbf{G}(\mathbb{R})$ should be injective.
Cannot think of anything exotic. Easy ways to change the value of the limit is to multiply it with some other limit expression that equals to some desired real number r
I understand that people wants to criticize capitalism but putting imperialism in a higher pedestal to build an argument against capitalism is naive and wrong-headed
Very conservative. But I don’t want to argue whether him being a Justice is good/bad, though I do have my opinions. My point in bringing him up is just to emphasize that Trump’s actions will outlive his presidency
I'm looking for a counterexample, specifically a normed vector space $X$ and two convex subsets $A$ and $B$ one of which is open that are separated but not strictly by an hyperplane, do you happen to know one or where to find one?
@Balarka my anti-imperialist feelings are definitely guided by emotional concerns is the thing, I'm a native person from a place where imperialism has devastated native peoples, so IDT I can give an argument without getting really really heated
@EricSilva I understand. I have strong opinions against colonialisms too, but very obviously it would be nowhere close to as strong as the arguments you would have.
It's an umbrella term for a lot of different structures and institutions that manifest differently over time and space but there's probably a few characteristics that are present in all its forms or the term would be useless
At least in Latin America there's some degree of continuity between explicit political domination in the form of colonialism and economic imperialism in the form of multinationals turning Nations into enclave economies @Semi
@Semiclassical The multinational corporations that emerged in New Imperialism is of a distinctly different flavor than the ones appearing now, of course (one is an attempt at colonial exploitation, the other is a characteristic of capitalism). But it's a good point of similarity
Great line from the talk I’m at just now, after the speaker picked up two pads of paper to illustrate: “Even as an Italian I can’t handwave with more than two hands”
I have the following limit $\lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0}$. I know it's the definition of the derivative but I want to treat it has a formal limit to apply L'Hospital to it. But to be able to use L'Hospital I need to justify that the limit is one of the two cases "$\frac{0}{0}$" or "$\frac{\infty}{\infty}$". I believe it's the "$\frac{0}{0}$" case but how do I justify it?
An exercise asks to find the values of $a\in\Bbb R$ such that, given a linear system with $A_a=\begin{bmatrix}2 & a & 0 \\ a&8&a \\ 0&a&2 \end{bmatrix}$, the Jacobi method converges.
I've found $\rho(A_a)=5-\sqrt{9+2a^2}$ and setting it to be smaller than $1$ I get $\lvert a\rvert <2\sqrt2$. How...
Is it OK to call $s = \sum \left\{x \in \mathbb{Z}\,\middle|\, 0 \leq x < i \land (3 \!\mid\! x \,\lor\, 5 \!\mid\! x) \right\}$ Mathematical logic?
I'm trying to find a name for a first-order logic in which mathematics is used (mathematical symbols like +, -, <, >, summation, set theory symbols, etc.)
@anakhronizein I'm mostly interested in mathematical symbols. I'm translating English sentences into first-order logic, and the "domain" that the sentences use is mathematical. For example, the sum of all the multiples of 3 or 5 below 1000.
I don't quite follow what you are asking for though. Symbols aren't inherently mathematical or not mathematical. + has no universal meaning. I can define + to mean very well whatever I want it to be, as long as I make it clear.
For other sentences I've used quantifiers, that could make the use of first-order logic more evident
@anakhronizein So, say that you are given the task of translating an English sentence that involves mathematical concepts into a formal language, like first-order logic. Would you say that the task is "Translate the following English sentences into first-order logic" or something else? Like "Translate the following English sentences into mathematical logic"?
It's cool how the symplectic and complex geometry is so intimately connected... $d\omega = 0$ is the condition for integrability of almost symplectic structures to symplectic ones, and Nijenhuis tensor = 0 is the analogue for that.
I guess I'm doing research, but not on mathematics. I'm just asking because I want to make a new question but not quite sure where to post it. Basically I'm looking for algorithms to solve a system of linear Diophantine equations. The problem itself is of course mathematical but since I would like to know if there are computer algorithms for it then it's sort of computer science too..
If you're looking for algorithmic solutions to mathematical problems, I don't think math.stackexchange.com is the most appropriate site to ask your question.
Furthermore, I doubt people are going to be happy if your question is: "is there an algorithm to solve a system of equations". At least, provide context, what you have searched so far, etc.
