2
votes
0answers
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
1answer
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
0answers
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
1answer
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
0answers
24 views

What is the conditional probability of the following expression? [on hold]

Conditional probability of the following expression
0
votes
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
41 views

Embedded submanifold [closed]

Show that an embedded submanifold is closed if and only if the inclusion map is proper.
5
votes
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
3answers
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{...

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