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.
346
questions
5
votes
2
answers
494
views
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 ...
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 ...
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 $H_2(X)$ ...
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 ...
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 $2$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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)-...
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 $b$,...
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_{...
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 $\bigcup_{\sigma} C_{\sigma} =X$, ...
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 ...
-1
votes
1
answer
942
views
Geometrically connected curve [closed]
What is the definition of a geometrically connected curve?
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 ...
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 ...
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 ...
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)...
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^...
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 ...
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 ...
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 ...
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,...
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 ...
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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 $...
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 $E$ ...
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(...
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^...
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 $...
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 $g$:
$$\int_{0}^{2\pi}\int_{0}^{\pi}G(R,\phi,\theta)\sin\theta\ \text{d}\theta \ \text{d}\phi= g(R)$...
8
votes
4
answers
1k
views
Multivariable Calculus Lecture Ideas
I am teaching a course in multivariable calculus this semester. We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces. After that ...
7
votes
11
answers
2k
views
A function that is defined everywhere but has unknown values [closed]
For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, ...
4
votes
0
answers
386
views
A question on an intuitive way to look at stacks
I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack ...
3
votes
2
answers
2k
views
What's the meaning of pencils in birational geometry?
I see in some books the authors call a one dimensional linear system a pencil, but in other books one call a linear system $|D|$ is not compsited with a pencil if $\dim \phi_{|D|}(X)\geq 2$ and even ...
2
votes
1
answer
549
views
Reference for a derivative formula for matrices
I found the identity
$$ \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 ...
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 ...
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 ...