# All Questions

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2 views

### Specific cases of the tangle hypothesis in terms of “classical” n-categories

As is well known, the tangle hypothesis of Baez and Dolan proposes that, for suitable definitions, the $n$-category of framed $n$-tangles in $n+k$ dimensions is the free $k$-tuply monoidal ...

**-2**

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6 views

### Choosing Matrix Decomposition for Analysis

Consider a complex square matrix $A\in\mathcal{C}^{n\times n}$. Now let us discuss two kinds of the factorization of $A$, say, eigendecomposition and Schur decomposition because both of them are often ...

**1**

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14 views

### When is the image of a 2-dim l-adic representation associated to a modular form open

I know the following theorems by Serre:
1, The 2-dim l-adic representation associated to a non-CM elliptic curve is open.
2, The 2-dim l-adic representation associated the weight-12 cusp form ...

**5**

votes

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27 views

### What is the chromatic number of the “conic hypergraph” on a non-singular plane cubic?

Can we color the points of a complex non-singular plane cubic curve with finitely many colors so that no conic intersects the curve at 6 points of the same color?
If so, what is the smallest number ...

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8 views

### Construction of homogeneous Siegel domain from j-algebra

I am reading bounded homogeneous domain from Piatetski-Shapiro's
book ``Automorphic functions and the geometry of classical domains''
and have questions on how to construct homogeneous Siegel domain
...

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26 views

### General extension of scalars and tensor products

My goal is to in a sense, create an additional set of "scalars" from the field of complex numbers, and the ring of integers. That will also maintain the property of being bilinear under bilinear forms ...

**1**

vote

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28 views

### Can there be computable non-standard models of PA in a weaker sense?

By Tennenbaum's theorem, in the usual sense of computability for models,
neither addition nor multiplication can be computable in a non-standard model of PA.
Weak version:
Can addition or ...

**2**

votes

**1**answer

43 views

### optimal bound in diophantine representation question

Given that, with integers $t \geq 1$ and $q \geq 3,$ there are solutions to $$ x^2 - q x y + y^2 = - t q $$
with integers $x,y \geq 1,$ I was able to show that
$$ q \leq 1 + \frac{324}{25} t^2. $$
...

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29 views

### Invertibility of a Cauchy-type matrix

I am trying to show with general (structural) arguments, that the following matrix
\begin{align*}
\begin{pmatrix}
\frac{1}{3} & -\frac{1}{5} & -\frac{1}{7} & \frac{1}{9} \\
...

**0**

votes

**1**answer

28 views

### On the separability of operator range

Let $T$ be an operator from a Banach space $X$ into a Banach space $Y$ and $1\leq p<\infty$. If $ST$ is compact for any operator $S$ from $Y$ into $l_{p}$, Is $T(X)$ separable? Or under what ...

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19 views

### log convexity for the norm of vector-valued function

Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory.
Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...

**3**

votes

**1**answer

63 views

### Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations

Let $\pi$ be a generic irreducible admissible representation of $GL_n(L)$, where $L$ is a $p$-adic field, $R$ is its ring of integers, and $\mathfrak{p}$ is its prime ideal. The conductor of $\pi$ ...

**5**

votes

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162 views

### Lemma 1 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

Consider Lemma 1 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: ...

**2**

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16 views

### Is there a class of functions acting on a set of projected points that remain invariant under changes in projection parameters?

Suppose I have a set of $k$ points $\{x_1,x_2,\ldots,x_k\}$ in $\mathbb{R}^n$ that I can project into $\mathbb{R}^m$ with the linear operator $\mathcal{P}$, with $\alpha, \beta, \ldots$ parameters of ...

**3**

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**1**answer

52 views

### A question about simple closed plane polygons

For any positive integer $n$ greater than $3$, let $P(1),P(2),...,P(n)$ be a set of $n$ pairwise distinct points in the Euclidean plane, no three of which are collinear. Let $H(P(1),P(2),...,P(n))$ be ...

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24 views

### Is the heat kernel on complete Riemannian manifolds uniformly continuous in $t\in(0,\infty)$?

let $(M,g)$ be a complete Riemannian manifold (e.g. the hyperbolic space $\mathbb{H}^n$) and let $p(t,x,y)$ be the heat kernel for $M$.
I am wondering, if for any fixed $y\in M$ and $\epsilon >0$ ...

**0**

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47 views

### notation for vector product in the space

The notation for vector (a.k.a. cross) product in $\mathbb{R}^3$ I usually see is $\times$.
However, some places use $\wedge$ instead, which IMHO creates a lot of confusion, as $\wedge$ usually is ...

**10**

votes

**2**answers

205 views

### Woodin on Posner-Robinson for the hyperjump and sharp

The Posner-Robinson theorem states that, if $X$ is noncomputable, there is some $G$ such that $X\oplus G=G'$; that is, even though genuine jump inversion only works above $0'$, every (nontrivial) $X$ ...

**2**

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**0**answers

49 views

### On a theorem of Dwork and totally ramified extensions

Suppose that $K \subset L$ is a totally abelian ramified extension of local fields. Let $\pi_L$ be a prime element of $L^*.$ $F \in Gal(\tilde{L}/L)$ is the Frobenius, where $\tilde{L}$ is the maximal ...

**4**

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**1**answer

55 views

### Characteristic Variety of the Principal Symbol solves PDE system?

In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see, ...

**8**

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**1**answer

193 views

### 'Updated' book in the same spirit as Dieudonné's Panorama des mathématiques pures

Today a colleague of mine asked me if I knew of any "more modern version" of J. Dieudonné's Panorama des mathématiques pures. Le choix bourbachique.
The very first thing that instantly came to my ...

**2**

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**0**answers

39 views

### Invertible combinations of linear maps on infinite-dimensional vector spaces

Let $V$ be a real infinite-dimensional vector space of cardinality $\kappa$. Does there exist a set $\Omega$ of cardinality $\kappa$ of linear maps from $V$ to $V$ such that for every $n\geq 1$, every ...

**-2**

votes

**0**answers

33 views

### Calculate elliptic arc length from total ellipse length [on hold]

Assume you have the total length $S_{total}$ of the circumference of an ellipse in $\mathbb{R}^2$.
Is there an exact analytic relationship between $S_{total}$ and the length of a segment of the ...

**0**

votes

**1**answer

35 views

### A question on Ahlfors covering surface

Given a transcendental entire function $f$, and three disjoint Jordan domains $D_1$, $D_2$, and $D_3$. Then from Ahlfors covering surface (See in Walter Hayman's book), we can choose $i\in {1,2,3}$ ...

**2**

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58 views

### Intuitive (?) inequality extremal inequality

Consider $N$ pairs of random variables $(X_i, Y_i)$. $X_i$ are iid, with $EX_i=0$ and $EX_i^2=1$. The same conditions hold for $Y_i$. Moreover all $X_i$ are independent of all $Y_j$. It seems very ...

**0**

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47 views

### Question about the generalized Jacobian

Assume that we are given the function
$$f(s_1,s_2)=|s_1|P_1^{\textrm{sgn}(s_1)}+|s_2|P_1^{\textrm{sgn}(s_2)},$$ where $P_1^\pm$, $P_2^\pm$ are positive or negative constants and of the same sign as ...

**3**

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**1**answer

47 views

### Exponential of approximate quadratic variation of Brownian motion

Let $X_t$ be a Brownian motion or a Brownian Bridge on a (\edit: compact) Riemannian manifold. Let $T>0$ be given.
The question is: Does there exists a constant $C>0$ such that for all ...

**5**

votes

**1**answer

195 views

### Motivating Lubin-Tate theory

The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated ...

**5**

votes

**1**answer

147 views

### What are some examples of total derived functors that can't be computed from a functorial replacement?

(Or more generally, what are some examples of Kan extensions which are not pointwise?)
By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along ...

**6**

votes

**3**answers

496 views

### What are the usual deadlines in paper submission procedure?

I've submitted a paper to a journal 10 days ago, and I did not yet get any news from the handling editor.
Of course, 10 days is quite short, but I hope I will not wait one year without any news for ...

**5**

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**0**answers

136 views

### Can you have many independent reals?

Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that:
$\Bbb P_\alpha$ is c.c.c.
$\Bbb P_\alpha$ adds a real ...

**2**

votes

**2**answers

76 views

### Density of Gaussian measures on Banach spaces

I am trying to get my head around this question and was reading (1) which states the same a little bit more general:
Let $X$ be a separable Banach space and $X^*$ the dual space. The mean
value ...

**-1**

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**0**answers

71 views

### How to denote RHS with the help of LHS Mathematically [on hold]

$A=\left[
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right]<==>\left[\begin{array}{cccc}
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 \\
1 & 1 & 0 & 0 \\
...

**5**

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**0**answers

65 views

### A completeness criterion for $\infty$-categories

We all know that for ordinary categories $\mathscr{C}, \mathscr{D}$ (with $\mathscr{C}$ small) the limit of a functor $F:\mathscr{C} \to \mathscr{D}$, if it exists, can be constructed by using ...

**2**

votes

**1**answer

79 views

### Equivalence between Diffie Hellman and Discrete Log

For which non-trivial groups, do we know that the Diffie Hellman problem and the Discrete Log are equivalent?
Is there any group for which we suspect them to be different?
Could there be a finite ...

**0**

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**0**answers

29 views

### Maximize mutual information

Assume $P \in \mathbb{R}^{n \times n}$ describe the joint distribution of the random variable $J$ over the finite set $\mathcal{X}\times \mathcal{X} $.
I am interested in finding a right stochastic ...

**6**

votes

**0**answers

47 views

### Sitefel-Whitney class of bundles induced by a covering map

Let $S^m$ be the unit $m$-sphere. Let $P$ be a $m$-dimensional polyhedron with $k$-vertices, such that all the vertices of $P$ lie in $S^m$. The isometry group of $S^m$ is $Iso(S^m)=O(m+1)$. Let ...

**3**

votes

**2**answers

54 views

### Questions concerning convergence rate of Iterated Projections

Assume for simplicity $C_1,C_2,...,C_n\subset \mathbb{R}^m$ to be closed and convex subsets with $\underset{i=1}{\overset{n}{\bigcap{}}}C_i\neq\emptyset$.
Let $x^0\in\mathbb{R}^n$ and define the ...

**9**

votes

**1**answer

239 views

### “Pythagoras number” for integral matrices

It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric ...

**4**

votes

**1**answer

76 views

### Critical with respect to chromatic, but not Hadwiger number

For any simple, undirected graph $G=(V,E)$ where $V$ is finite, we define the Hadwiger number $\eta(G)$ to be the maximum $n$ such that $K_n$ is a minor of $G$.
Is there a graph $G$ on such that ...

**2**

votes

**0**answers

44 views

### Linearisation of Bianchi-gauged Einstein operator on Einstein manifolds

Let $(M,g)$ be a compact $(m+1)$-dimensional Einstein manifold with boundary $\partial M\neq\emptyset$ and Ricci curvature $\mathrm{Ricc}_g=-mg$. Consider the Bianchi-gauged Einstein operator
$$
...

**0**

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110 views

### question from Terry Tao's blog regarding taking the inner product of fourier transformed Navier Stokes equation

Hi I have a question on a calculation that is on Terence Tao's blog, https://terrytao.wordpress.com/tag/navier-stokes-equations/. In terms of his blog, I am asking how does he go from line (1) to ...

**9**

votes

**2**answers

266 views

### Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$

I am looking at a certain sequence, and consequently I am wondering if anyone knows if there happens to exist a closed form solution for this sum:
$$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$
...

**1**

vote

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124 views

### Characteristic subgroups and automorphisms [migrated]

$\DeclareMathOperator{\Char}{char}$$\DeclareMathOperator{\Aut}{Aut}$
Let $G$ be a group, $H$ a subgroup, such that $H \Char G$. If $\phi \in \Aut(H)$ is there an automorphism $\widehat{\phi} \in ...

**2**

votes

**1**answer

163 views

### Where does the name $NE(X)$ come from?

Why do we call the cone of curves(effective one cycles) on a variety $X$ as $NE(X)$, what does $NE$ stand for?

**1**

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153 views

### Generalization for Leray Hirsch theorem for Principal $G$-bundle

This is a general question:
Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...

**4**

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**0**answers

64 views

### Outline of Generic Seperable Banach Spaces don't have a Schauder Basis

So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces ...

**1**

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68 views

### Representation theory of groupoids

Is there for a groupoid an representation theorem as Representation theorem of Wagner-Preston for inverse semigroups?

**2**

votes

**1**answer

43 views

### When are simple foliations strictly simple?

Any submersion $f: M → N$ defines a foliation of M whose
leaves are the connected components of the fibres of $f$. Foliations
associated to the submersions are called simple foliations. The foliations
...

**3**

votes

**0**answers

50 views

### On Holonomy in (regular) Riemannian Foliations

Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid:
Let $\mathcal{F}\subset M$ be a ...