# All Questions

**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

174 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

206 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? [on hold]

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

297 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

96 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

226 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

77 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

188 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

332 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 [on hold]

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

390 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

137 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

340 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

307 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

193 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

24 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

207 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

272 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 ...

**7**

votes

**0**answers

110 views

### Representability of Weil Cohomology Theories in Stable Motivic Homotopy Theory

My understanding is that one purpose of stable motivic homotopy theory is to emulate classical stable homotopy theory. In particular, we would like Weil cohomology theories to be representable by ...

**2**

votes

**1**answer

91 views

### $H^1$-continuity of Laplace's equation with respect to boundary data

Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...

**1**

vote

**0**answers

44 views

### Second Countability hypothesis for a Banach manifold

Is the second countability hypothesis necessary to rigoruosly define a Banach Manifold (say in infinite dimension)?
In the finite-dimensional theory of manifolds, that request is included in the ...

**-2**

votes

**0**answers

41 views

### Embedded submanifold [closed]

Show that an embedded submanifold is closed if and only if the inclusion map is proper.

**5**

votes

**1**answer

60 views

### Optimal constant for a Sobolev-type inequality

Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that
$$
\lvert u\rvert_s = \...

**4**

votes

**1**answer

256 views

### Can K[[T_1,…,T_∞]] be embedded into K[[X,Y]]?

In the MathOverflow question about common false beliefs, the following answer teaches us that there is an embedding $\iota_n \colon K[[T_1,...,T_n]] \hookrightarrow K[[X,Y]]$. Now let us define the ...

**0**

votes

**1**answer

79 views

### Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators

Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the ...

**2**

votes

**1**answer

69 views

### Is there a characterization of r(M) by local cohomology instead of Ext

For a Noetherian local ring $(R,m)$, and a finite $R$-module $M$ with $\operatorname{depth} M=t,$ type of $M$ is defined to be $r(M):=dim_{R/m}Ext^t \ (R/m, M).$
Is there a characterization of $...

**2**

votes

**1**answer

75 views

### Representation of support of Gaussian measure by kernels of no-variance functionals

Let $\mu$ be a Gaussian measure on a separable Banach space $X$ and $q$ is the covariance operator of $\mu$. I am reading a proof for
$$\operatorname {supp} \mu = \bigcap_{q(f, f) = 0} \ker f =: E$$
...

**2**

votes

**1**answer

91 views

### Spaces $Y$ such that $C(-, Y)$ is always acceptable

Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to ...

**2**

votes

**1**answer

60 views

### When are maximal compacts same as maximal parahorics?

Let $G$ be a reductive algebraic group over a complete non-archimedean field $k$. We know that maximal compacts are exactly the same as maximal parahorics when the Iwahori is open compact subgroup of $...

**2**

votes

**0**answers

119 views

### Is there a model of ZF+ACC where transfer fails for the definable hyperreals?

A decade ago Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go ...

**12**

votes

**1**answer

588 views

### Non-isomorphic rings that are localizations of each other

Do there exist commutative rings $A$ and $B$ and multiplicative subsets $S\subseteq A$, $T\subseteq B$ such that $A\not\simeq B$ but $S^{-1}A \simeq B$ and $T^{-1} B\simeq A$?
This question comes ...

**-4**

votes

**0**answers

17 views

### Is it possible to say the worst case of distribution if the cluster has centrality [closed]

Thank you for reading my question.
There is a cluster or ball which consist of many n-dimensional points and the cluster has centrality(more probability mass closer to the cluster's center).
Then, ...

**2**

votes

**3**answers

268 views

### Geometry of numbers argument: counting integers with some linear condition

I am interested in the proof of the following result:
Suppose that $A > 1$, $\lambda \in \mathbb{R}$, and for $0 < Z \leq 1$, let $U(Z)$ be the number of integer solutions $v$ of
\begin{...