**12**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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 ...

**3**

votes

**1**answer

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 ...

**14**

votes

**2**answers

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 ...

**0**

votes

**1**answer

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 ...

**5**

votes

**2**answers

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 ...

**1**

vote

**0**answers

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 ...

**2**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**29**

votes

**9**answers

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 ...

**13**

votes

**5**answers

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 ...

**0**

votes

**1**answer

94 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^+}\frac{\...

**0**

votes

**0**answers

87 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

**1**answer

106 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 ...

**4**

votes

**3**answers

818 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

**0**answers

254 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 ...

**6**

votes

**1**answer

267 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 ...

**31**

votes

**5**answers

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

**0**answers

166 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 (n-1)-...

**3**

votes

**3**answers

266 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 $b$,...

**2**

votes

**3**answers

540 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 $G_{...

**7**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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 $\bigcup_{\sigma} C_{\sigma} =X$, ...

**0**

votes

**1**answer

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 ...

**-1**

votes

**1**answer

414 views

**0**

votes

**1**answer

192 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

**1**answer

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 ...

**3**

votes

**1**answer

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 ...

**5**

votes

**2**answers

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 ...

**0**

votes

**1**answer

199 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 $\lim\limits_{x\to\infty}f_1(x)...

**4**

votes

**1**answer

329 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 $S^...

**0**

votes

**0**answers

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

**1**answer

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 ...

**0**

votes

**1**answer

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 ...

**1**

vote

**1**answer

511 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

**0**answers

364 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

**2**answers

254 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

**1**answer

346 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

**1**answer

240 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

**0**answers

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

**2**answers

1k 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

**1**answer

492 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 ...

**12**

votes

**2**answers

631 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

**0**answers

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 $E$ ...

**6**

votes

**1**answer

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(...

**4**

votes

**1**answer

281 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 $H^...

**7**

votes

**1**answer

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 ...

**5**

votes

**0**answers

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)$...

**5**

votes

**4**answers

832 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 ...

**7**

votes

**11**answers

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, ...