# Tagged Questions

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 ...

2k views

### reference for “X compact <=> C_b(X) separable” (X metric space)

I know (and am able to prove via Stone-Čech compactification) that the following is correct: Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ...
152 views

### Boundary Problem with an Area Constraint

Consider a boundary given by vertices (0,a), (0,0) and (1,0) (an 'L' shaped boundary). The problem is to find the equation that passes between the endpoints (0,a) (1,0) of minimum length that ...
242 views

### Convex polyhedral decomposition of spheres

Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$? Remarks: A polyhedron is defined as an area enclosed by a ...
2k views

### The non-traveling mathematician problem

This is a career question. I have just begun a research postdoc position in Southern California. It has been hard, but I've enjoyed teaching my first graduate courses and working on research and ...
136 views

### Inductive vs projective limit of sequence of split surjections

Let $$A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots$$ be an inductive sequence of countable abelian groups, the connecting homomorphisms of which are ...
198 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 ...
105 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 ...
411 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 ...
823 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 ...
3k 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 ...
7k 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 ...
94 views

282 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 ...
191 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 $\bigcup_{\sigma} C_{\sigma} =X$, ...
299 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 ...
414 views

### Geometrically connected curve [closed]

What is the definition of a geometrically connected curve?
192 views

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 ...
644 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 ...
261 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 ...
540 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 ...
199 views

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 ...
315 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 ...
386 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 ...
511 views

140 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 $E$ ...
337 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))= Hom(A,Gamma(...
281 views

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 $H^... 1answer 423 views ### What are the applications of Dowker's theorem? Let$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$consists of finitely many elements ... 0answers 282 views ### Azimuthal and polar integration of a 3D Gaussian Numerical evaluation of the following integral of a 3D gaussian$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= g(R)$...
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 ...
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, ...