Questions tagged [tag-removed]

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 is also used in the process of moderators deleting tags. Thus a question having this tag does most of the time not mean that somebody found it off-topic.

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2 answers
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How to specify a finite group up to inner automorphism?

I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...
John Pardon's user avatar
  • 18.3k
2 votes
1 answer
244 views

How many flavors should a notational system offer for rank-1 tensors?

The notation for tensors is like the plumbing in a very old Vermont farmhouse. It may once have been intentionally designed, but after that it just evolved. As an example, it seems that depending on ...
user avatar
4 votes
1 answer
446 views

Homotopy groups of K3

Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface. Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ ...
Mohammad Farajzadeh-Tehrani's user avatar
1 vote
0 answers
1k views

Properties of a rational function of multiple variables

Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/. Assume that all constants and exponents are integers within certain range. I ...
Alex_Waterloo's user avatar
13 votes
1 answer
502 views

Permanent of a matrix of odd integers

It is clear that the permanent of an $n\times n$ matrix which entries are odd integers, is an even number, as it is the sum of $n!$ odd numbers. I am interested in finding the highest power of $2$ ...
Tal H's user avatar
  • 273
14 votes
2 answers
1k views

Hopf Algebra for a physicist

Hello, for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I ...
understandhopf's user avatar
18 votes
1 answer
4k 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 ...
Wolfgang Loehr's user avatar
0 votes
1 answer
154 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 ...
user29733's user avatar
4 votes
1 answer
352 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 ...
innerproduct's user avatar
18 votes
2 answers
3k 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
215 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 ...
Rasmus's user avatar
  • 3,144
5 votes
2 answers
268 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 ...
sqrt2sqrt2's user avatar
1 vote
0 answers
121 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 ...
user13559's user avatar
  • 645
1 vote
0 answers
263 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 ...
Mike Battaglia's user avatar
6 votes
1 answer
416 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 ...
seub's user avatar
  • 1,337
1 vote
0 answers
172 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)-...
kwkwkw's user avatar
  • 11
3 votes
3 answers
297 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 $b$,...
B H's user avatar
  • 387
2 votes
3 answers
1k 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_{...
ljjpfx's user avatar
  • 19
0 votes
1 answer
203 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 $\bigcup_{\sigma} C_{\sigma} =X$, ...
QuantumLogarithm's user avatar
0 votes
1 answer
336 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 ...
QuantumLogarithm's user avatar
-1 votes
1 answer
942 views

Geometrically connected curve [closed]

What is the definition of a geometrically connected curve?
rukiye's user avatar
  • 21
0 votes
1 answer
1k 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 ...
Alex Johnson's user avatar
3 votes
1 answer
361 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 ...
sqrt2sqrt2's user avatar
5 votes
3 answers
989 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 ...
Marcus's user avatar
  • 328
0 votes
1 answer
228 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)...
user27381's user avatar
4 votes
1 answer
393 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^...
CuriousUser's user avatar
  • 1,420
0 votes
0 answers
243 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 ...
io0's user avatar
  • 1
5 votes
1 answer
388 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 ...
Pietro Majer's user avatar
  • 56.5k
0 votes
1 answer
453 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 that ...
kitefrog's user avatar
1 vote
1 answer
878 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 $a_n,...
user27381's user avatar
1 vote
0 answers
389 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 ...
Helmut Brandl's user avatar
3 votes
2 answers
286 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 ...
Col_Boogie's user avatar
7 votes
1 answer
445 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, ...
IMeasy's user avatar
  • 3,717
6 votes
1 answer
280 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 ...
GreginGre's user avatar
  • 183
1 vote
0 answers
115 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 ...
Adterram's user avatar
  • 1,361
26 votes
2 answers
2k 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 ...
CuriousUser's user avatar
  • 1,420
3 votes
1 answer
778 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 ...
Changyu Guo's user avatar
  • 1,861
16 votes
2 answers
1k 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 $...
user avatar
3 votes
0 answers
154 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 $E$ ...
Mohammad Farajzadeh-Tehrani's user avatar
6 votes
1 answer
557 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(...
Eleanor Von Hohlandsbourg's user avatar
4 votes
1 answer
291 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^...
Pooya's user avatar
  • 41
12 votes
1 answer
1k 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 $...
Francis Snapper's user avatar
5 votes
0 answers
396 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)$...
Wox's user avatar
  • 347
8 votes
4 answers
1k 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 ...
Joe Johnson's user avatar
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, ...
Gerard's user avatar
  • 195
4 votes
0 answers
386 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 ...
QcH's user avatar
  • 805
3 votes
2 answers
2k views

What's the meaning of pencils in birational geometry?

I see in some books the authors call a one dimensional linear system a pencil, but in other books one call a linear system $|D|$ is not compsited with a pencil if $\dim \phi_{|D|}(X)\geq 2$ and even ...
MZWang's user avatar
  • 843
2 votes
1 answer
549 views

Reference for a derivative formula for matrices

I found the identity $$ \frac{\partial( \det (X^T A X ))}{\partial X} = 2\det(X^TAX)AX(X^TAX)^{-1} $$ On the matrix cookbook (http://orion.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf). It is equation ...
user avatar
5 votes
2 answers
758 views

Can we calculate the inner product of a semicontinous function with the Dirac delta function?

Dear all, It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
Anand's user avatar
  • 1,619
4 votes
2 answers
251 views

Steinberg Group as a Lattice in a lie group

Given an integral domain $R$, the Steinberg group $St_n(R)$ is the group given by generators $e_{pq}(\lambda) := \mathbf{1} + a_{pq}(\lambda)$, $p\neq q$, $1\leq p,q \leq n$ Subject to ...
Nicolas's user avatar
  • 41

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