# All Questions

**12**

votes

**1**answer

457 views

### Is the wonderful compactification of a spherical homogeneous variety always projective?

Let $G/H$ be a spherical homogeneous variety, where $G$ is a complex semisimple group. Assume that the subgroup $H$ is self-normalizing, i.e., $\mathcal{N}_G(H)=H$. Then by results of Brion and Pauer
...

**-4**

votes

**0**answers

28 views

### Convergence/Divergence of Integral, can P-test be used here? [closed]

I have an integral like this:
integral
How do I check it's convergence? As far as I know, P-test can be used for integrals from 0 to 1, or A to infinity, what would I do in this case?

**3**

votes

**0**answers

45 views

### Matroid Representation of the Antichains of a Poset

Introduction
I am studying a problem in which the antichains of a poset are of key importance. They are naturally geometrically embedded as vectors in the space $\mathbb{R}^P$, where $P$ is the poset,...

**1**

vote

**1**answer

100 views

### Gaussian bounds on Dirichlet heat kernel

Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0
$ such that $$\frac{c}{t^{n/2}}
e^{-\frac{1}{4t}d(x, y)^2} \leq ...

**0**

votes

**1**answer

197 views

### The spherical harmonics are the EIGENVECTORS of Beltrami operator [on hold]

In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...

**5**

votes

**4**answers

436 views

### List of invariants that distinguish homotopy equivalent non-homeomorphic spaces

It is written on wikipedia article (https://en.wikipedia.org/wiki/Analytic_torsion) that the Reidemeister torsion is the first invariant that could distinguish between spaces which are homotopy ...

**2**

votes

**0**answers

68 views

### Self-adjoint, strictly singular operators on Hilbert spaces

Let $X$ and $Y$ be infinite-dimensional Banach spaces. Recall that an operator $T: X\rightarrow Y$ is strictly singular if it is not an isomorphic embedding when restricted to any infinite-dimensional ...

**8**

votes

**1**answer

232 views

### Do you know this Burnside ring module?

Let $G$ be a finite group and $\Omega(G)$ its Burnside ring. There is a certain $\Omega(G)$-module, let's call it $M(G)$, that appears in something that I am thinking about. As an abelian group $M(G)$ ...

**0**

votes

**0**answers

65 views

### Is the Lebesgue measure zero for the discontinuous set of a semicontinuous function? [migrated]

[Q.]
Is there a semicontinuous function, which has its discontinuous set with non-zero measure?
Remark:
Given a semicontinuous function, the set of all discontinuous points may be uncountable, for ...

**2**

votes

**0**answers

107 views

### Inverting a function

I posted this question on crypto.SE but got no answer:
Let $w = a_0 \cdot a_1 \cdots a_{n-1} $ be a word from $ \{0,1\}^n $, $|w| = n$
Let $m = \sum_{i=0}^{n-1}{ a_i \cdot 2 ^ {n-1-i} } $ be the ...

**6**

votes

**0**answers

112 views

### Symmetry of function defined by integral

(Originally posed in Math.SE in Jan 2013. Received no complete answers as of yet.)
Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as
$$ f(\alpha, \beta) = \int_0^{\...

**4**

votes

**2**answers

242 views

### Critical points in $ZF$ without Choice

Recall the definition of critical point for set theory:
A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to ...

**6**

votes

**0**answers

363 views

+100

### Counting limit cycles via curvature in Riemannian geometry

In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem
First we give a short introduction:
A quadratic system is a polynomial vector field on ...

**2**

votes

**0**answers

60 views

### Characterizing the optimimum over the space of probability measures

Consider the following optimization problem:
\begin{equation}
\max_{\mu \in \mathcal{M}} \int \log\left( \int e^{\alpha U(x,y)} d\mu(y) \right) d\nu(x)
\end{equation}
where $\mathcal{M}$ is the space ...

**1**

vote

**1**answer

175 views

### An answer to this system of PDE's

Planning of the question:
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle
The isotropic almost complex structures $J_{\delta , \sigma}$ were introduced by Aguilar on the ...

**0**

votes

**0**answers

35 views

### Inner Products vs Discretizations of Functions

Let $f:(0,1)\rightarrow [0,1]$ be a $\mathcal{C}^{\infty}$ function.
We say that a function $D_r^f:\{1,...,r\}\rightarrow [0,1]$ is an $r$-discretization of $f$ if $$D_r^f(j) = \frac{1}{|a(j)-b(j)|}\...

**4**

votes

**1**answer

207 views

### Hodge decomposition for Bott-Chern cohomology

$\DeclareMathOperator{im}{im}$
I want to prove that Bott-Chern cohomology group $H^{p,q}_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge ...

**-4**

votes

**0**answers

24 views

### What is the conditional probability of the following expression? [closed]

Conditional probability of the following expression

**0**

votes

**1**answer

63 views

### Reference request: Existence of plurisubharmonic potential for positive (1,1)-current on Stein (affine) manifold

I am looking for a reference for the following fact, which I believe should be true.
Let $X$ be a Stein manifold (or smooth affine variety over $\mathbb{C}$).
If $\omega$ is a positive closed $(1,...

**3**

votes

**1**answer

298 views

### What is the mathematics behind the random experiment which produces the data with this strange property?

I have a following scenario. there is a huge collection of data resulting from a random experiment $E$ (I do not say random variable yet, for reasons that you will need to explain in your answer). Let ...

**1**

vote

**1**answer

44 views

### Reference request: compatibility conditions of four versions of Yetter-Drinfeld modules

There are four versions of compatibility conditions of Yetter-Drinfeld modules (left-left, left-right, right-left, right-right) in the article. Are there some references which derive these ...

**0**

votes

**0**answers

97 views

### Blow-up of projective space is a projective bundle [on hold]

Suppose that $k$ is an algebraically closed field and let $X= \mathrm{Bl}_p(\mathbb{P}^n_k)$ be the blow-up of $\mathbb{P}^n_k$ at a point, and let $Y = \mathbb{P}^{n-1}_k$.
I read something that ...

**-4**

votes

**0**answers

22 views

### Condensing two conditional constraints (with different signs) into one single constraint [closed]

How can I condense these 2 constraints in the attached image into one linear constraint?
Constraints

**-7**

votes

**0**answers

60 views

### W^{∞,p}(IRⁿ) are separable space for 1<p<∞ [closed]

How can prove that the space W^{∞,p}(IRⁿ) are separable space

**7**

votes

**1**answer

227 views

### Are There Mutually Exclusive Large Cardinal Axioms in ZFC?

The various large cardinal axioms are usually described in terms of some roughly linear hierarchy of varying consistency strengths. However, some cardinal axioms potentially contradict one another. As ...

**1**

vote

**1**answer

81 views

### How to show that Yetter-Drinfeld condition is equialent to $\Psi$ is a braiding

Let $H$ be a Hopf algebra and $V$ a right $H$-module and right $H$-comodule. The module $V$ is a Yetter-Drinfeld module over $H$ if and only if
\begin{align}
( v \triangleleft h_{(2)} )_{(0)} \otimes ...

**4**

votes

**2**answers

189 views

### Invariant theory for parabolics

Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \...

**1**

vote

**1**answer

334 views

### Doing graph theory after a thesis in pure mathematics [closed]

I've just went through the 1st year of my PhD in France, it is related to Floer Homology. I didn't know what it was really about at that time, I chosed this subject because I thought it would combine ...

**3**

votes

**1**answer

129 views

### Characterizing the image of $D(A_f) \rightarrow D(A)$

Let $A$ be a ring and $f \in A$ an element. If $M$ is an $A$-module on which multiplication by $f$ is an isomorphism, then $M$ is in fact an $A_f$-module.
Now suppose that $C \in D(A)$ is a complex ...

**3**

votes

**0**answers

146 views

### Construction of model of arithmetic from an arbitrary model

Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:
$M'\models PA^-$ (or ...

**-1**

votes

**0**answers

24 views

### Complex fixed points on the bifurcation diagrams [closed]

I'm working with bifurcation diagrams, an extesion that is being made of them is the determination of complex fixed points in addition to the real fixed points.
Given a dynamic system (e.g. an ...

**12**

votes

**2**answers

394 views

### Is SO(2n+1)/U(n) a symmetric space?

I am a physics student with only a rudimentary knowledge of differential geometry, so please feel free to point out if I miss something elementary / trivial.
According to https://arxiv.org/abs/1408....

**-5**

votes

**0**answers

53 views

### How many non-isomorphic Fano planes exist? [closed]

How many non-isomorphic Fano planes exist?
1
down vote
favorite
1
The Fano plane is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 ...

**4**

votes

**1**answer

138 views

### Differential structures on compact Lie groups

Given a compact Lie group can there be a differential structure on it with respect to which one cannot define a smooth group operation?

**2**

votes

**1**answer

171 views

### Spin structure for varieties, especially finite field

I wonder about the notion of a spin structure for varieties over any field and results in this direction. For example, I wonder if there is something like a spin-bundle for the sphere $x^2+y^2+z^2=R^2$...

**0**

votes

**0**answers

68 views

### Looking for an exposition of a certain theorem of Talagrand

The following is a theorem by Talagrand (as stated here, http://arxiv.org/pdf/1511.08609v1.pdf),
Let $(X, \mu)$ be a probability space. Let $F : X \rightarrow \{0,1\}$
be a family of functions ...

**14**

votes

**2**answers

341 views

### Can you use Chevalley‒Warning to prove existence of a solution?

Recall the Chevalley‒Warning theorem:
Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If
$$d_1 + \ldots + d_r < n,$$
then the ...

**8**

votes

**1**answer

308 views

### Trinity College, Cambridge, circa 1896 maths scholarship papers [on hold]

I've been searching around looking for the (maths component) of the scholarship papers to Trinity College (Cambridge) from around 1890. Can anyone provide a link to a pdf scan of these papers?
Was ...

**7**

votes

**1**answer

201 views

+50

### Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure?

Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure? Every paper or article I looked at, talk only about singular laminations on $\mathbb{CP}^2$. I was wondering why. If you know ...

**-5**

votes

**0**answers

25 views

### periodic solutions of this second order differential equation y“(t)+f”(t)y(t)=0 [closed]

I need to find a smooth non trivial periodic solution that is strictly positive or negative of the second order ODE y"(t)+f"(t)y(t)=0 such that f(t) be periodic.

**3**

votes

**0**answers

131 views

### Under what condition is a fiber bundle cobordant to the trivial bundle?

Let $E$ be the total space of a fiber bundle with base $B$ and fiber $F$, where $B$ and $F$ are smooth manifolds.
Under what condition is $E$ unoriented cobordant to $B\times F$?
And what happens ...

**-8**

votes

**0**answers

40 views

### Jonah bicycled 12 miles in 4 hours. What is the unit rate? [closed]

Jonah bicycled 12 miles in 4 hours. What is the unit rate?
3 miles per hour
8 miles per hour
16 miles per hour
48 miles per hour

**4**

votes

**0**answers

115 views

### Question on Weil-Petersson metric on Teichmuller space

I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space".
The tangent space at ...

**5**

votes

**1**answer

157 views

### Hyperplane sections with chi non-zero

Let $X$ be a smooth, projective variety over $\mathbb{C}$ for which $\chi(X) = 0$. Here by $\chi$, I mean the topological Euler characteristic of $X(\mathbb{C})$; this number can also be computed as ...

**5**

votes

**1**answer

208 views

### Constructing groups of Type E7 with certain Tits Index

In a new survey on $E_8$, namely
Skip Garibaldi - E8 the most exceptional group
, the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-...

**2**

votes

**0**answers

78 views

### Irreducible representations of Sp(2)

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$.
I can ...

**5**

votes

**1**answer

273 views

### Does the topology induced by the Hausdorff-metric and the quotient topology coincide?

Assume that $X$ is a metric space, and $\sim$ is an equivalence relation on $X$.
Furthermore we assume that the number of elements in each equivalence class
is bounded by a positive constant.
Does ...