All Questions
153,426
questions
28
votes
0
answers
2k
views
Supercompact and Reinhardt cardinals without choice
A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it:
Definition. A cardinal $\kappa$ is supercompact if for all ordinals $\...
- 1,103
18
votes
1
answer
718
views
Is the category Idem filtered?
I have recently been reading Lurie's "Higher Topos Theory", and come upon what I believe to be an erronous claim. However, the author goes to some pain as a result of that claim, and the error seems ...
- 65.1k
4
votes
1
answer
308
views
Shifted eigenvalues and Gershgorin theorem
Suppose we have a $n\times n$ symmetric positive semi-definite matrix $\mathbf{A}$.
Based on Gershgorin circles theorem all the eigenvalues of the, $\mathbf{A}=[a_{ij}]$, are located in the union of $...
2
votes
1
answer
236
views
Some question on Harish-Chandra height function
Let $G$ be a connected reductive group over a number field $F$ and fix a minimal parabolic subgroup $P_0$ of $G$. Let $K$ be a fixed good maximal compact subgroup of $G(\mathbb{A}_F)$ such that $G=PK$ ...
- 879
9
votes
1
answer
295
views
An extrapolation method
I've stumbled upon a method of extrapolation that I haven't seen before.
We are trying to approximate $f(0)$ for a certain function $f$, which we have only measured
at points $x_0, \ldots, x_N$ in ...
- 5,571
1
vote
1
answer
364
views
Sketching Frobenius norm of a tensor with a rank-1 random tensor
Let $A\in\mathbb{R}^{n^k}$ be a $k$-dimensional tensor with $n$ elements along each dimension. Moreover suppose $u_1,u_2,\dots,u_k\sim\text{Unif}(\pm1)^n$ are $n$ dimensional vectors with each of ...
- 118k
3
votes
0
answers
71
views
Ring epimorphisms and finiteness assumptions
Let $f : A\to B$ be a ring epimorphism. It is well-known that, under the extra assumption that $A$ and $B$ are commutative, then $f$ makes $B$ a finitely-generated $A$-module implies that $f$ must be ...
- 710
2
votes
0
answers
51
views
Integral of product of characters of GL(N, C)
We know that
$\int_{U(N)} \chi_{R}(A\Omega B\Omega^{\dagger}) [d\Omega ] = \chi_R(A)\chi_R(B)/\chi_R(1)$
where $[d\Omega ]$ is the Haar measure on the unitary group $U(N)$, $A, B$ are hermitian $N\...
- 21
19
votes
2
answers
753
views
Dual Borel conjecture in Laver's model
A set $X\subseteq 2^\omega$
of reals is of strong measure zero (smz) if $X+M\not=2^\omega$
for every meager set $M$. (This is a theorem of Galvin-Mycielski-Solovay,
but for the question I am going to ...
- 3,073
5
votes
0
answers
382
views
A subspace of the homogeneous Sobolev space $\dot{W}^{1,p}% (\mathbb{R}^{N})$
Given $1\leq p<\infty$, the homogeneous Sobolev space $\dot{W}^{1,p}
(\mathbb{R}^{N})$ is defined as the space of all functions $u\in
L_{\operatorname*{loc}}^{1}(\mathbb{R}^{N})$ such that the ...
- 411
1
vote
0
answers
68
views
Partitions corresponding to unipotent elements in simple classical algebraic groups
Let $G$ be a simple classical algebraic group with corresponding root system $\Phi$ and natural module $W$ of dimension $n$.
For a (closed) subsystem $\Psi$ of $\Phi$, let $G_\Psi = \{ U_\alpha \mid ...
- 121
2
votes
2
answers
168
views
enumerate line partitions of points in the plane
Let $S$ be a nonempty subset of $\mathbb{R}^2$ and $l$ a line in $\mathbb{R}^2$ disjoint from $S$. Then $l$ partitions $S$ into two disjoint sets $S = S_1 \cup S_2$ in the obvious way. Note that, at ...
- 30.7k
5
votes
2
answers
1k
views
Who proved that the Mandelbrot set's Julia sets are locally connected?
I'd be greatly interested in a reference to the respective article.
Was it Douady? Julia? Hubbard? Fatou?
Bonus question: Who gave the proof that can be found in the Orsay notes?
EDIT: The question ...
- 88.9k
1
vote
0
answers
91
views
Equivalence of Cauchy Type Problems and Volterra Integral Equations
The following Theorem is from the book Theory and Applications of Fractional Differential Equations by Kilbas, Srivastava and Trujillo. It's Theorem 3.10 from page 163.
Theorem.
Let $ \alpha \in \...
- 11
0
votes
3
answers
3k
views
When is the union of embedded smooth manifolds a smooth manifold?
Suppose we have k embeddings of one single smooth manifold into one other, such that the intersections are manifolds,too. What are sufficient conditions, such that the union of those embeddings is a ...
- 27.4k
1
vote
0
answers
101
views
Size of a set defined by divisor function
After some computations, I guessed the following conjecture.
How can I prove or disprove it? thanks!
Let
$$
A(k)=\#\left\{\left(t,\frac{k+t+a}{4t-1}\right):1\leq t\leq k,\ 1\leq a\leq k+t,\ a\mid(k+...
- 4,618
0
votes
1
answer
700
views
Are all manifolds in $\mathbb R^n$ homeomorphic to a smooth manifold?
My question is whether all manifolds that can be embedded in $\mathbb R^n$ are homeomorphic to a smooth manifold?
I know that every smooth manifold can be triangulated which I think is a result of ...
- 27.4k
2
votes
0
answers
85
views
chromatic class of graphs of order $n$
Let $\mathcal{G}(n)$ be the isomorphism class of simple graphs of order $n$. We say two graphs in $\mathcal{G}(n)$ are chromatic equivalent if their chromatic polynomials have an equal linear ...
- 1,219
0
votes
0
answers
59
views
Concentration of Sample Mode
Is there a concentration bound for sample mode, when there exists a unique mode for a density $f$ that doesn't depend on $f$ (Essentially I am looking for a Chernoff type bound for mode)?.
- 53
7
votes
1
answer
2k
views
Z/48 and Moonshine Beyond the Monster
I am interested in pursuing an understanding of K-theory. Primarily, the
$K_3(\mathbb{Z})$ algebraic K-group over ring of integers of an algebraic number field and its relationship to the $\mathbb{Z}/...
- 21.5k
24
votes
0
answers
848
views
Have any of Maryam Mirzakhani's doodles been preserved?
I edit a magazine for High School students, and would very much like to get hold of a large image of one of the large sheets of paper with Maryam Mirzakhani's mathematical drawings on for the cover of ...
- 22.5k
2
votes
1
answer
118
views
Two cospectral (normal) digraphs which are not orthogonal similar
Preliminaries
A complex matrix $A$ is normal when $A$ and $A^*$ commute. A real matrix $A$ is normal when $A$ and $A^t$ commute.
Two complex matrices $A$ and $B$ are said to be unitary similar if ...
- 56
12
votes
0
answers
540
views
Elementary-ish geometric proof of Hirzebruch signature theorem for Riemannian 4-manifolds?
The Hirzebruch signature theorem tells us that for a smooth compact oriented 4-manifold, the signature $\sigma(M)$ is proportional to the first Pontryagin number of $M$:
$$
3\sigma(M)= p_1(M) = k \...
- 33.9k
1
vote
0
answers
36
views
Continuity for maximal destablizing sheaves of nef sequences
Let $X$ be an $n$-dimensional smooth projective variety and $\mathcal{E}$ a torsion-free sheaf.
Let $\overline{\rm{NA}}(X)^{n-1}:=\{\sum D_1\cdots D_{n-1}\,|\, D_i \text{ is nef $\mathbb{R}$-Cartier}\...
- 85
1
vote
0
answers
71
views
Polytopal domains in non-archimedean torus
Given a non-archimedean field $\mathbb K$, there is a natural map
$$
\mathrm{val}: (\mathbb K^*)^n\to\mathbb R^n$$
(See Section 4 of Gubler's paper).
Gubler mentions there $\mathrm{val}$ is a ...
- 2,719
1
vote
1
answer
270
views
Bounding Coefficients of Dirichlet Series
Consider the exponentiated Riemann-Zeta function $\zeta(s)^p$. If it is represented as
$$\zeta(s)^p = \sum_{n=1}^\infty\frac{a_n}{n^s}$$
Is there any upper bound we can put on $|a_n|$ in terms of ...
- 99.2k
2
votes
2
answers
489
views
Hilbert Scale Inclusions
I'm looking at properties of the scale of Hilbert spaces $(X_s)_{s\in \mathbb{R}}$, which are constructed as follows. Starting with $A:D(A)\subset H\to H$, $A$ a densely defined, strictly positive ($...
- 22.6k
0
votes
2
answers
477
views
Derivatives of delta function as a basis for distributions [closed]
Is there some sense in which one could write any distribution as a sum of this sort?
$$A(x,y)=\sum_{n=0}^{\infty}a_n(x)i^n\frac{\partial^n}{\partial x^n}\delta (x-y)$$
Provided that the rhs acting ...
- 311
3
votes
1
answer
560
views
Eigenfunctions and eigenvalues of an operator defined by a certain integral
Let $k :[0,1]^2 \rightarrow \bf{R} $ be a kernel function definded by
$ k(x,y)= (1- max(x,y))^2 .$ Now, let $ L $ be a linear operator defined on $ L^2 [0,1] $ by $$ Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...
- 245
3
votes
2
answers
595
views
Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad
If $X$ is a based connected topological space, it is well-known what the homology of $\Omega\Sigma X$ is: according to the Bott-Samelson theorem, it is a tensor algebra over reduced homology of $X$. (...
- 10.6k
3
votes
0
answers
79
views
Image of Obstruction Map for Relative Quot-scheme
Let $f: X \to S$ be a projective morphisms between finite-type projective schemes, and $\mathcal{O}_{X}(1)$ an $f$-ample line bundle. Given an $S$-flat coherent $\mathcal{O}_{X}$-module $\mathcal{H}$ ...
- 1,701
4
votes
1
answer
256
views
On the convergence of the ratio of order statistics of gaps induced by $n$ uniform points on $[0,1].$
In an MO question here @IosifPinelis shows that the ratio of expectations $\mathbb{E}(A)/\mathbb{E}(B)$ of the largest (say $A$) and smallest (say $B$) gap resulting from $n$ uniform random variables ...
- 118k
1
vote
1
answer
166
views
$S^1$ normal bundle on divisor and Serre spectral sequence
Let $D\subset X$ be a smooth divisor in a smooth complex variety. On $D$ we have the normal bundle $N$. Removing the zero section and retracting we get an $S^1$ bundle. Call this bundle $N'$. Now I'd ...
- 35k
4
votes
1
answer
705
views
On the largest and smallest spacings for the uniform distribution
Let $Z_1,\dots,Z_n$ be iid random variables (r.v.'s) each uniformly distributed on $[0,1]$. Let $Z_{n:1}\le\cdots\le Z_{n:n}$ be the corresponding order statistics. For $i=1,\dots,n-1$, let $G_i:=Z_{n:...
- 118k
0
votes
1
answer
333
views
Inclusion of closed submanifolds of a manifold
Consider a smooth compact manifold $M$ of dimension $n$, with or without boundary. Choose a submanifold $N$ of $M$ of dimension $k$, where $1 \leq k \leq n - 1$, such that $N$ is either without ...
- 43.1k
29
votes
4
answers
4k
views
Model structure on Simplicial Sets without using topological spaces
The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
- 29.8k
2
votes
2
answers
470
views
Probability space with exactly one Brownian motion
Very recently, the following question was asked:
Often, we encounder the assumption that $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ is a stochastic base on which a Brownian motion is defined. ...
CommunityBot
- 1
2
votes
1
answer
193
views
Interpretation of an equivalence to the Riemann hypothesis due to de Reyna and Toulisse in the spirit of a formula from an article
In [1] the authors present an equivalence to the Riemann hypothesis that is the Theorem 6.2.
On the other hand I know a statement from [2], in English this is the article Andrew Granville and Greg ...
- 12.7k
34
votes
7
answers
13k
views
Textbook for Etale Cohomology
What is the best textbook (or book) for studying Etale cohomology?
- 10.7k
3
votes
0
answers
97
views
Minimal localization need it to "diagonalize" a matrix
Let $A$ be an $n\times n$-matrix over $\mathbb Z[t^\pm]$. In general doesn't exist $P,Q\in GL(n,\mathbb Z[t^\pm])$ such that $PAQ$ is a diagonal matrix (this happens cause $\mathbb Z[t^\pm]$ is not a ...
7
votes
0
answers
96
views
Rough classification of Peano Curves
By Peano curve I mean a continuous map from the unit interval that fills the unit square in $\mathbb R^2$.
In the paper:
Shchepin, E. V.; Bauman, K. E., Minimal Peano curve, Proc. Steklov Inst. Math....
- 4,792
4
votes
1
answer
2k
views
Is a free and discrete group action on the plane a covering space action?
Let $\mathbb{R}^2$ be the plane, and let a group $G$ act on it with orientation preserving homeomorphisms, and assume that
every orbit of $G$ is a discrete subset in $\mathbb{R}^2$
$G$ acts freely: ...
- 31k
10
votes
0
answers
371
views
Hyperkähler structure on the moduli space of tetrahedra?
Consider a moduli space of geodesic tetrahedra in the hyperbolic space $\mathbb{H}^3.$ In the Klein's model the hyperbolic space can be presented as the interior of a unit ball
$$
\mathbb{H}^3=\{(x_1,...
- 4,296
3
votes
1
answer
274
views
Bounds on the size of isogeny classes (over number fields)
I am reading B. Mazur's seminal paper "Rational isogenies of prime degree" (Invent. Math. 44 (1978), 129-162), and Theorem 5 of this paper caught my attention; it states that there exists an absolute ...
- 2,481
5
votes
1
answer
480
views
Isomorphism between the hyperbolic space and the manifold of SPD matrices with constant determinant
I'm studying properties of the manifold of symmetric positive-definite (SPD) matrices and I've learnt about the following connection to the hyperbolic space [1, Section 2.2], $$\mathcal{P}(n) = \...
- 271
6
votes
1
answer
636
views
How wide is the Birkhoff Polytope?
This question is migrated from MSE where it turned out to be much harder than I thought. I still cannot figure this out. Does anyone have any ideas?
Define the width of a polytope $P \subset \mathbb ...
- 151k
3
votes
0
answers
200
views
Maximize an $L^p$-functional subject to a set of constraints
Let
$(E,\mathcal E,\lambda)$ and $(E',\mathcal E',\lambda')$ be measure spaces
$f\in L^2(\lambda)$
$I$ be a finite nonempty set
$\varphi_i:E'\to E$ be bijective $(\mathcal E',\mathcal E)$-measurable ...
- 161
1
vote
1
answer
432
views
Quotient graph of a tree
We know that every graph is isomorphic to a subgraph of a complete graph. Similarly, can we say that every graph is isomorphic to a quotient graph of a tree?
- 12.6k
4
votes
4
answers
279
views
How many residues mod p do you need to take to ensure that you can always find some multiple that contains 3 elements within ϵ of each other
For $\epsilon<p$, let $N(\epsilon,p)$ be the smallest value of $n$ such that for any set $S \subset \mathbb Z_p$ of size $n$, there exists $\lambda\in \mathbb Z_p^{*}$, $\mu \in \mathbb Z_p$ s.t $\...
- 4,792
3
votes
0
answers
254
views
Homotopicity of $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ as morphisms from $A$ to $A\otimes A$
let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.
Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\...
- 312