This tag is to be used only when re-tagging highly(!) off-topic questions where none of the actual tags would make sense; all actual tags the questioner has used are removed and something is needed to have some tag, which is enforced by the software, so this tag is used. However note that this tag ...

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4
votes
2answers
179 views

Branch locus of a 6:1 cover of the grassmannian G(1,3)

Given a general quartic surface $S$ in $\mathbf{P}^3$, there is a natural 6:1 surjective map $\phi: Hilb^2(S) \to G(1,3)$ sending $\{P,Q\}$ to the line through them in $\mathbf{P}^3$. Can you ...
1
vote
0answers
102 views

Existence of open dense subset in a Lie group

Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group and $\Gamma$ a discrete subgroup of $G$ such that the subgroups $\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...
1
vote
1answer
304 views

The PDE $u_t=u_{xx}-u_{yy}$: The simplest linear second-order PDE that isn't elliptic, parabolic, or hyperbolic.

I know that there have been several questions on here and stackexchange about linear PDE's which don't fall into the standard classification, but I had a more focused question which I haven't seen ...
1
vote
0answers
373 views

Area Under Generalized Parabolas and Hyperbolas without Calculus.

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...
27
votes
9answers
2k views

Covering maps in real life that can be demonstrated to students

Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...
10
votes
5answers
4k views

If d/dx is an operator, on what does it operate?

If $\frac{d}{dx}$ is a differential operator, what are its inputs? If the answer is "(differentiable) functions" (i.e., variable-agnostic sets of ordered pairs), we have difficulty distinguishing ...
0
votes
1answer
88 views

example of a concave function with $\lim_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ which fullfills some additional condition

I'm looking for the example of a concave function $g\colon [0,1]\mapsto \mathbb{R}$, with $g(0)=0$, for which $\lim\limits_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ and $\lim\limits_{x\to ...
0
votes
0answers
83 views

name for a class of functions

I would like to ask whether is used some name for functions $g:A\to\mathbb{R}$, $A\subset \mathbb{R}$, for which $$\exists \lambda>1:\;\; \lim_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}>1.$$
1
vote
1answer
102 views

concave functions of different behaviour in the neighbourhood of 0 from Shannon function

I'm looking for an example of a concave function $g\colon [0,1]\to\mathbb{R}$, $g(0)=0$ such that: $$\liminf_{x\to 0^+}\frac{g(x)}{-x\ln x}\neq \limsup_{x\to 0^+}\frac{g(x)}{-x\ln x}.$$ Moreover is ...
9
votes
4answers
842 views

Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...
3
votes
3answers
522 views

Connection between properties of Dynamical and Ergodic Systems

Hi All While studying Topological and Ergodic Dynamics, I've got quite preplexed by the different Properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, ...
1
vote
0answers
199 views

Average weighted value of a linear functional over increasing bounded subsets of Z^n

Say you're working within the finite-dimensional free Z-module $\mathbb{Z}^n$, and you want to impose a "norm" on this module. By a "norm" I mean a function $\|·\|: \mathbb{Z}^n \to \mathbb{R}$ which ...
2
votes
1answer
505 views

Are there such numbers?

Maybe better to ask for help on this question here: Are there eight numbers $0< x_{1},\ldots,x_{4},y_{1},\ldots,y_{4}<1$, such that $$ ...
6
votes
1answer
231 views

Poincaré lemma in infinite dimensions

Hi everyone, Is the Poincaré lemma true in infinite dimensions? Here's a precise statement: Let $X$ be a Banach (or maybe Hilbert) vector space, $U$ a simply connected open set in $X$. Is it true ...
0
votes
0answers
129 views

Foliation over characteristic positive

Ekedahl wrote about foliation in characteristic positive, over the field $/frac{Z}{pZ}$ as a subsheaft of the tangent sheaft, that is closed over involution and $p$-power, My question is if there ...
28
votes
5answers
2k views

Can pure mathematics harness citizen science?

Having just finished Michael Nielsen's book "Reinventing Discovery", I find myself wondering if there are ways that pure mathematics research can engage the public in the way that GalaxyZoo or Foldit ...
1
vote
0answers
165 views

Is the canonical morphism $\mathbb A^n \to\mathbb A^{n-1}$ a projective morphism? [closed]

Let $\mathbb A^n$ be the n-dimensional affine space over a field K (algebraically closed if that makes it easier), so $\mathbb A^n= \text{Spec }K[x_1,...,x_n]$, and $\mathbb A^{n-1}$ the ...
2
votes
3answers
242 views

Space filling curve to simplify vector addition? [closed]

Since points on a euclidean plane can be represented by one coordinate on a space-filling curve, is there any curve such that if two vectors $(x_0,y_0)$ and $(x_1,y_1)$ were represented by $a$ and ...
2
votes
3answers
381 views

On the image of a G_\delta set under a continuous bijection

Let $X, Y$ be two metric spaces and $f$ be a continuous bijection (i.e. one-to-one map) from $X$ to $Y$. Let $E$ be a $G_{\delta}$ subset of $X$. I want to know weather the image $f(E)$ is also a ...
5
votes
1answer
235 views

stackification commutes with finite limits?

Suppose we work on the Grothendieck site $\mathcal{C}$ of all schemes in the fpqc topology. If it helps it is also fine with me to work only over affine schemes. Let us denote the category of stacks ...
0
votes
1answer
189 views

Relation between measure of sets

Is it true that, given a space $X$ and a probability measure $\mu$ on it, given some sets $A, B \subset X$ and a finite number of disjoint sets $C_{\sigma}$ such that $\bigcup_{\sigma} C_{\sigma} =X$, ...
0
votes
1answer
292 views

Integral inequality

Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...
0
votes
1answer
278 views

Geometrically connected curve

What is the definition of a geometrically connected curve?
0
votes
1answer
189 views

tensorproduct, p-adic groupring

Suppose there is a cyclic group $G$ and a prime $p$. Why can one write $$ \mathbb{Z}_p[G] \cong \mathbb{Z}_p \otimes _\mathbb{Z} \mathbb{Z}[G]$$ Is this some theorem which has a name? Thanks for ...
0
votes
1answer
366 views

Function (/matrix) to generate linearly independent vectors.

Hi, I want to whether there is a vector generating function (/matrix) such that it can generate a m-dimensional vector which will always be linearly independent of the set of m-dimensional vectors ...
3
votes
1answer
240 views

Hilbert scheme of 2 points on an elliptic curve

The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$. What can we say about the ...
5
votes
2answers
368 views

Is the hyperspace of the Hilbert cube homeomorphic to the Hilbert cube

Question: Is the hyperspace of the Hilbert cube $H=[0,1]^\mathbb {N}$ homeomorphic to $H$? Remarks and definitions: 1) The Hilbert cube $H$ is a compact metric space, where the metric is given by ...
0
votes
1answer
192 views

Hardy theorem on elementary functions

Say we have two elementary functions (see http://mathworld.wolfram.com/ElementaryFunction.html for the definition) $f_1,f_2\colon [0,\infty)\mapsto \mathbb{R}$ such that ...
4
votes
1answer
302 views

(Non)-exoticness of a diffeomorphism of a sphere

Suppose you have a standard sphere $S^n$ and a "standard" $S^{n-2}\subset S^n$. I am really thinking about $S^{n}\subset \mathbb{R}^{n+1}$ the usual sphere, and $S^{n-2}=S^n\cap \{x_0=x_1=0\}$. Let ...
0
votes
0answers
239 views

Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of ...
5
votes
1answer
287 views

Computing the limit of a certain recursively defined sequence

The following is not exactly a research question (it was originated from manufacturing of exercises for calculus), and has no other motivation than explaining a phenomenon. I apologize if it is ...
0
votes
1answer
247 views

How many 3-flip Mobius strip knots are there?

Take a clock-wise 3-flip mobius strip, Cut it down the middle and then let the ribbon cross itself 6 times. This forms a framed knot of which there are many. Get the knot diagram. I've found ...
1
vote
1answer
314 views

concave function with sublinear growth

Does there exist a concave, increasing function $h\colon[0,\infty)\to\mathbb{R}$ such that $\lim_{x\to\infty} h(x)=\infty$ $\lim_{x\to\infty} h(x)/x=0$ There exist sequences of positive numbers ...
1
vote
0answers
304 views

The used symbols for equality and equivalence

Background: I am currently developing a general purpose programming language which allows formal verification (i.e. correctness proofs) of programs. During the development it came out that a lot of ...
3
votes
2answers
239 views

Jacobian and determinants

Start with variables $(a_1, a_2, a_3, … a_n)$ and transform it to the system $(x_1, x_2, x_3, … x_n)$ where the xi’s are the solutions to $x^n + a_1x^{n-1} + a_2x^{n-2} + a_3x^{n-3} +…+ a_n$. The ...
5
votes
1answer
279 views

kapranov's realization of $\overline{M}_{0,n}$ over other fields

Kapranov gave a very nice desciption, over $\mathbb{C}$ of the moduli space of stable pointed rational curves $\overline{M}_{0,n}$ as a series of blow-ups of $P^{n-3}$. Does this, or a similar result, ...
6
votes
1answer
222 views

Algebraic integers in skew fields

Hi everyone, let $D$ be a skew field, which is finite dimensional over its center $k$. Assume that $k$ is a number field, and let $\mathcal{O}_D$ be the set of elements $z\in D$ which are roots of a ...
1
vote
0answers
112 views

A question about smoothness

$f$ is a smooth function on $[0,+\infty)$ and $f(x)>0$ for all $x>0$. Then does the following equivalence hold : $\phi(x,y)=f(\sqrt{x^2+y^2})$ is smooth if and only if $f^{(k)}(0)=0$ for all ...
24
votes
2answers
926 views

Euler characteristic and universal cover

Let $M$ be a compact manifold, let $\tilde{M}$ be its universal cover, and suppose that the Euler characteristic $\chi(\tilde{M})=0$. My question is: does this imply that $\chi(M)=0$? This is clear if ...
3
votes
1answer
367 views

A closed connected component in a topological space does not contain any path-connected subset?

Does there exist such a non-trivial closed connected component U of some connected topological space X or a non-trivial connected topological space X that do not contain any non-trivial path-connected ...
11
votes
2answers
540 views

Independence of Leibniz rule and locality from other properties of the derivative?

The following is meant to be an axiomatization of differential calculus of a single variable. To avoid complications, let's say that $f$, $g$, $f'$, and $g'$ are smooth functions from $\mathbb{R}$ to ...
3
votes
0answers
136 views

Topology of K3 as a sum of two abelian fibrations.

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$). K3 surfaces is obtained by removing a fiber from two copies of $E$ ...
6
votes
1answer
304 views

question about higher geometric stacks

I have a naive question I am asking. Given a higher geometric stack X in the sense of Simpson, Toen etc is it true that there is an affinization Spec Gamma(O_X) such that Hom(X, Spec(A))= ...
4
votes
1answer
278 views

Invariant lattice of algebraic surface.

Given an algebraic surface $S$ with action of a finite group $G$. Is it true that the invariant lattice $H^2(X,\mathbb{Z})^G$ is generated by elements pulled back from the $H^2(X/G,\mathbb{Z})$ (or ...
7
votes
1answer
327 views

What are the applications of Dowker's theorem?

Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$: a simplex in $K$ consists of finitely many elements ...
5
votes
0answers
225 views

Azimuthal and polar integration of a 3D Gaussian

Numerical evaluation of the following integral of a 3D gaussian $G$ seems to result in a 1D Gaussian $g$: $$\int_{0}^{2\pi}\int_{0}^{\pi}G(R,\phi,\theta)\sin\theta\ \text{d}\theta \ \text{d}\phi= ...
5
votes
4answers
518 views

Multivariable Calculus Lecture Ideas

I am teaching a course in multivariable calculus this semester. We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces. After that ...
6
votes
11answers
2k views

A function that is defined everywhere but has unknown values [closed]

For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, ...
3
votes
1answer
393 views

What grows faster? [closed]

What grows faster: Busy Beaver function, TREE function or BEAF largest resursive functions (legions, lugions, lagions...)?
4
votes
0answers
314 views

A question on an intuitive way to look at stacks

I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack ...