7

This is a bit an opinion based question and answer. The RH is about $\log\zeta(s),\frac1{\zeta(s)},\frac{\zeta'(s)}{\zeta(s)}$, not $\zeta(s)$. On the $\zeta(s)$ side we can easily exploit that it is the Dirichlet series of the integers. Surprisingly (or not?) on the $\log\zeta(s),\frac1{\zeta(s)},\frac{\zeta'(s)}{\zeta(s)}$ side we can't, and complicated ...


5

Turning my comment above into an answer: As Don Thousand commented above, narrower classes of objects have more guaranteed properties a priori. You're focusing on tameness properties, but it's also worth noting that we can get guaranteed pathologies this way too. For example, here are a couple ways in which adding a natural requirement to a starting class of ...


2

The "paradox" of Achilles and the Tortoise has nothing to do with uncountable infinities. It can be easily be phrased within the (countable) rationals, since it is about $$\tag1 \sum_{n=1}^\infty\frac1{2^n}=1. $$ And in fact it doesn't require infinity to be phrased, as it could be written as $$\tag2 \forall \varepsilon>0,\ \exists N\in\mathbb ...


2

$$b^2 - 4ac = \begin{bmatrix} a\\ b \\ c\end{bmatrix}^\top \begin{bmatrix} 0 & t_1 & t_2 - 4 \\ -t_1 & 1 & t_3 \\ -t_2 & - t_3 & 0\end{bmatrix} \begin{bmatrix} a\\ b \\ c\end{bmatrix} $$ We would like to find $t_1, t_2, t_3 \in \Bbb R$ such that the rank of the matrix is minimized. The easy first step would be to force the matrix to ...


2

I'll try to answer to your questions following the order of their appearance. Are there interesting approaches that focus on the minimization of the energy directly? Yes, there are several approaches to the solution of this problem: they are all known under the collective name of "Variational Method". This "method" has its root in the ...


2

You should take "Let $x$ be an integer" as a synonym for "Assume that $x$ is an integer" or "Consider an integer $x$".


2

"Let" is usually used to introduce a new symbol along with an assumption about it ("Let $G$ be a group") or when defining notation ("Let $\mathcal P(S)$ denote the power set of $S$"). "Suppose" and "assume" are more conventional when introducing new assumptions using only existing symbols and notation ("...


2

This is another very good answer to your question. Roughly speaking, “let” typically declares variables, while “assume” & “suppose” typically specify antecedents/conditions/hypotheticals (say, in a proof-by-contradiction). “Let $x\in\mathbb{R},$ and suppose that $P(x)$” has the same meaning as “Let $x\in\mathbb{R}$ such that $P(x)$”.


1

This is unfortunately a nonsense question. There is a set of Cauchy sequences. This set has a relation on it ($(a_n) \sim (b_n)$ if $\lim_n (a_n - b_n) = 0$). There is therefore a quotient set or set of equivalence classes. None of this ever passes through anything with more structure than sets. (If you like, it passes through rings: the set of Cauchy ...


1

The real numbers can be defined by the process of completion with respect to the usual norm $\lvert\:\cdot\:\rvert:\Bbb{Q}\to \Bbb{Q}$. The norm $\lvert \:\cdot\:\rvert$ defines a metric on $\Bbb{Q}$ by $d(x,y)=\lvert x-y\rvert$. We note that $\Bbb{Q}$ is not complete with respect to this metric because there are Cauchy sequences which do not converge (this ...


1

First, regarding your first point, I agree with comments and answers pointing out that the Paradox of Achilles and the Tortoise has virtually nothing to do with uncountable infinities. Regarding your second point, I would like to add something which I think is not stressed enough in other answers: Namely that inclusion of structures $A \subset B$ says ...


1

Very informally, a function $f$ is convex if the value of $f$ at the average of two points lies below the average of the values at the two points. In symbols, $f\left(\frac{x + y}{2}\right) \leq \frac{1}{2} \left(f(x) + f(y)\right)$. (However, it's very good to experiment with drawing what a continuous function satisfying this inequality can possibly look ...


1

Here's a slight re-packaging of your mnemonic. We start with the figure I call the Fundamental Trigonograph (the inspiration for my avatar!), whose segment-lengths correspond to the trig values associated with (acute) angle $\theta$. Note that the "$1$" segment separates "ordinary" segments ($\sin$, $\tan$, $\sec$) from "...


1

I think it is really close to know the theorems and their proofs in the following sense. When doing mathematics (whether it is an exercise sheet or research) you often want to prove things by reducing it to things you know. So it is helpful if not necessary to know the theorems of the subject you are working in and have an idea how and why they work. By this ...


1

In your example, the columnspace is not a subspace of $\Bbb{R}^5$, but rather $\Bbb{R}^5 \times \{0\}$ (a subspace of $\Bbb{R}^6$). While this subspace of $\Bbb{R}^6$ is isomorphic to $\Bbb{R}^5$, it is not the same thing. The columnspace of an $n \times m$ real matrix is always a subset of $\Bbb{R}^n$. If $a \neq b$, then $\Bbb{R}^a \cap \Bbb{R}^b = \...


Only top voted, non community-wiki answers of a minimum length are eligible