# All Questions

**2**

votes

**1**answer

162 views

### Scheme of Higgs reductions

I'm reading the Bruzzo and Graña Otero's paper Semistable and Numerically Effective Principal (Higgs) Bundles; here: $X$ is a smooth, complex, projective variety; $G$ is a connected, complex, ...

**4**

votes

**0**answers

160 views

### Homotopical interpretation of flatness?

I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like:
A map of commutative algebras is flat if pushing it out ...

**-1**

votes

**0**answers

26 views

### Can using Principal Components Analysis allow you identify redundant data and remove it from a dataset? [migrated]

I hope this is suitable for this forum: I am new to PCA and what I ultimately want to do is perform cluster analysis on my dataset.
I have 20 physical descriptor variables for organisms, each with ...

**1**

vote

**1**answer

144 views

### Well-ordering of power set of omega

Assuming initially the background set theory ZF, what is the exact status of the existence of a well-ordering on the power set of $\omega$? How much needs to be added to guarantee this?

**1**

vote

**2**answers

89 views

### Topological properties via properties continuous maps

A topological space $(X,\tau)$ is connected if and only if the only continuous maps $f:X\to\{0,1\}$ (where $\{0,1\}$ carries the discrete topology) are the constant maps.
Are there other examples of ...

**-4**

votes

**0**answers

78 views

### Everywhere holomorphic functions over C [on hold]

Let $f, g$ be two everywhere holomorphic functions over ${\Bbb C}$. We consider the local representation of $f, g$ at the origin of ${\Bbb C}$, i.e. $z = 0$. That is, we can consider $f, g \in {\Bbb C}...

**0**

votes

**0**answers

54 views

### quadratic forms in number theory and prime numbers [duplicate]

what are the prime numbers represented by $x^2-2y^2$ ?
I have seen the claim that any prime congruent to 1 or -1 modulo 8
is of this form. True ? Reference ?

**0**

votes

**0**answers

25 views

### Relaxed path decomposition of a graph

Definition
Given a directed connected graph $G$ without multiple edges or self loops. We call a final path of $G$ a path ending with a vertex with no successor (the path can not be extended anymore) ...

**11**

votes

**1**answer

253 views

### Does there exist a ``continuous measure'' on a metric space?

Let $X$ be a separable complete metrizable space. Does there exist a complete metric $d$ and a Borel measure $\mu$ such that
(a)
$\mu(B_r(x))<\infty$ for every open ball $B_r(x)$ of radius $r>...

**2**

votes

**0**answers

131 views

### Card Game Feasibility 2

We play the following card game:
We are given a deck of $M$ unique cards, each with a color $c\in\left\{ 1,\dots,C\right\}$ , and a number $d\in\left\{ 1,\dots,D\right\}$, so $M=CD$.
We wish to ...

**5**

votes

**1**answer

67 views

### Decay estimates for wave and Klein-Gordon equation in “generic” curved backgrounds

Consider a $d$-dimensional smooth Lorentzian manifold $(M,g)$ (we assume $d\geq 3$, $M$ to be Hausdorff, paracompact and connected, hence second-countable, and that the signature convention of $g$ is $...

**0**

votes

**0**answers

84 views

### Weyl sums with polynomial coefficients

Let
$$ f(x,N) = \sum_{0 \leq n\leq N} e(x n^2).$$
Weyl's inequality gives an estimate for $f(x,N)$ when $x$ is near a rational with small denominator. My question is:
What estimates are ...

**1**

vote

**0**answers

68 views

### Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

I also put this question on MSE here
Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable).
Let $\...

**1**

vote

**0**answers

61 views

### Hausdorff Dimension of Exceptional Set for Carleson's Theorem

In Mattila's book Fourier Analysis and Hausdorff Dimension, Mattila presents a result of Barcelo, Bennett, Carbery, and Rogers about convergence of solutions of the Schrodinger equation to the initial ...

**6**

votes

**0**answers

115 views

### What is known about the $q$-analogue of the simplex?

I am interested in the field with one element. I am thus interested in combinatorial interpretations of the Gaussian binomial coefficients. Richard Stanley's "Enumerative combinatorics" mentions ...

**1**

vote

**0**answers

17 views

### Generalizing an expected increase in autocorrelation near a bifurcation point to a system of ODE

Near a bifurcation point, a stochastically forced dynamical system should show an increase in autocorrelation and variance. This is due to critical slowing (a loss in resilience to perturbations). ...

**5**

votes

**1**answer

134 views

### Showing that $\alpha$ isn't a cardinal in $J_{\alpha+1}^{\vec E}$ for a fine extender sequence $\vec E$

In [FSIT] and [OIMT] it is claimed that there is a surjection from $P(\kappa)\cap J^{\vec E}_{\nu(E_\alpha)}\times[\nu(E_\alpha)]^{<\omega}$ onto $\alpha$, and that this surjection lies in $J_{\...

**2**

votes

**1**answer

112 views

### Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings
$$
\begin{array}{}
R & \xrightarrow{f_2} & R_2 \\
\downarrow{f_1} & &...

**12**

votes

**1**answer

455 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? [on hold]

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

99 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

196 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

430 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

67 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

231 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

104 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

110 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

240 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

304 views

+100

### Counting limit cycles via curvature in Riemannian geometry

In this post we would like to give a possible new approach in 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

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

295 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 [on hold]

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<∞ [on hold]

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

**7**

votes

**1**answer

223 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

75 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

329 views

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

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

143 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 ...