All Questions
153,410
questions
15
votes
1
answer
1k
views
Uniform distribution of points on Riemannian manifolds
Recently, I came across a beautiful paper by Arnol'd and Krylov (Uniform distribution of points on a sphere...) that contains the following theorem:
Theorem: Let A and B be two rotations of the ...
- 1,054
5
votes
1
answer
897
views
Algorithms for projecting a point onto the convex hull spanned by a set of vectors
Given a set of vectors $V = \{ \mathbf{v}_1, \ldots, \mathbf{v}_n \} \subset \mathbb{R}^d$, I want to project a point $\mathbf{x}_0 \in \mathbb{R}^d$ onto the convex hull $\text{conv}(V)$ of the ...
- 271
3
votes
1
answer
2k
views
When is the matrix norm multiplicative
Let $|| = ||_{p,q}$ be an operator norm on $\mathbb R^{n \times m}$. In General, $\|AB\|\le \|A\|\|B\|$. Is there some criterion on $A, B$ (at least for some operator norms) so that $\|AB\| = \lVert ...
- 133
1
vote
1
answer
729
views
KL divergence and convolution of distributions
Let $P,Q,R$ be probability measures on the real line.
If these are discrete, we can show
\begin{equation}
D_{\mathrm{KL}}(P\ast R\,\Vert\,Q\ast R)\le D_{\mathrm{KL}}(P\,\Vert\,Q)
\end{equation}
by ...
- 13
3
votes
0
answers
80
views
The analogue of difference operator in Drinfeld module theory
The theory of Drinfeld modules is, in part, inspired by the analogy between Frobenius and the operator of differentiation. My question is: what should be the analogue of a difference operator $f(x)\...
- 365
4
votes
1
answer
334
views
ODE with a measurable vector field
Suppose we have a bounded Borel measurable vector field $F:\mathbb{R}^n\to\mathbb{R}^n$. To make the question non-trivial, assume that $F\neq 0$ eveywhere.
Question. Does there exist at least one ...
- 27.3k
5
votes
1
answer
288
views
harmonic coordinates on non-compact manifolds
Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...
- 103
12
votes
2
answers
441
views
Example of a finite group $G$ with low dimensional cohomology not generated by Stiefel-Whitney classes of flat vector bundles over $BG$
In Stiefel-Whitney classes of real representations of finite groups, J. Algebra 126 (1989), no. 2, 327–347, Gunawardena, Kahn and Thomas dealt with the question, whether the cohomology ring $H^...
- 2,590
3
votes
0
answers
66
views
Does this definition of the Fourier intensity measure make sense?
Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$.
For simplicity, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$ (although the following definition is ...
- 775
7
votes
3
answers
534
views
Endomorphism ring of $J_0(p)$ and Hecke operators
Let $J_0(p)$ be Jacobian of the modular curve $X_0(p)$ over $\mathbb Q$ where $p$ is a prime, consider the subring $\mathbb T$ inside $\newcommand{\End}{\operatorname{End}}\End_{\mathbb Q}(J_0(p))$ ...
- 6,158
7
votes
0
answers
160
views
Constructive proof of Swan theorem
Let $M$ be an $S_n$-lattice (so it is free as an abelian group), and assume that $M$ is projective (i.e. direct summand of some $\mathbb Z[S_n]^m$). A theorem of Swan implies that $M$ is stably ...
- 161
2
votes
0
answers
273
views
Compatibility of Kirillov-Kostant-Souriau form and Killing form
Let $\mathfrak{g}$ be a real semisimple Lie algebra. We know that a coadjoint orbit $\mathcal{O} \hookrightarrow \mathfrak{g}^*$ carries a natural symplectic form $\omega$, namely the Kirillov-Kostant-...
- 461
3
votes
0
answers
212
views
Is there a natural homotopy inverse to the map $∣Sing(X)∣\rightarrow X$
Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set).
Let $∣-∣: SSet \rightarrow C$ be the geometric realization functor.
For every $X$ we ...
2
votes
0
answers
161
views
Linear projection from a point preserves flatness
Let $\pi:\mathcal{X} \to S$ be a flat family of affine curves contained in $\mathbb{C}^n$ for $n \ge 3$ i.e., $\mathcal{X} \hookrightarrow \mathbb{C}^n_S$ and the inclusion commutes with the natural ...
- 1,573
4
votes
0
answers
260
views
Failure of Schur's lemma for topological group representations
Is there an example of $G$, $\rho$ as below?
$G$ is a locally compact group.
$\rho$ is an irreducible continuous representation of $G$ on a complex Hilbert space $V$. This means that we have a ...
2
votes
1
answer
300
views
optimal transport, measurable selection
Let $X = Y = \mathbb{R}^d$ and let $\nu$ be a probability measure on $\mathbb{R}^d$. Consider the collection of probability measure $\pi$ on $X\times Y$ such that $\pi$ has $y$-marginal $\nu$:
$$
\Pi(...
- 325
2
votes
0
answers
154
views
Classification of mod p Galois Representations for l not equal to p
Let $l\neq p$ be primes and let $\text{G}_l:=\text{G}_{\mathbb{Q}_l}$. Let $k$ be a finite field of characteristic $p$ and $\bar{\rho}:\text{G}_l\rightarrow \text{GL}_2(k)$ a local Galois ...
user130124
2
votes
1
answer
169
views
How can a correlation structure for a stochastic process not correspond to a Toeplitz matrix?
I was reading the following question: A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary
The following matrix is given representing the correlation ...
- 121
148
votes
26
answers
27k
views
Good "casual" advanced math books
I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks ...
3
votes
1
answer
60
views
Looking for an Alternative Characterization of the Upper Orthant of a Convex Hull
I have a question on convex analysis, which I require in one step of my research work. Before stating it, let me give a small background.
Let $Y, X_1,\ldots,X_n$ be $n+1$ points in $\mathbb{R}^d$. ...
- 31
3
votes
0
answers
68
views
A notion of largeness somewhere between $\mathrm{IP}_+$ and $\mathrm{IP}_+^*$
It is a well known fact that a set $A \subset \mathbb{N}$ is $\mathrm{IP}$ if and only if there exists an idempotent $p \in \beta \mathbb{N}$ (i.e., $p+p=p$) such that $A \in p$. Similarly, $B \subset ...
- 1,762
1
vote
1
answer
161
views
Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive
Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty.
Let $S : X → X$ and ...
- 895
9
votes
1
answer
339
views
Is restricting Replacement and Separation enough to make $Q+I\Sigma_n$ bi-interpretable with Set Theory?
We have the result that $\mathsf{ZFCfin}$, the usual $\mathsf{ZFC}$ axioms with the axiom of infinity replaced by its negation, is bi-interpretable with $\mathsf{PA}$, first order Peano Arithmetic. We ...
- 482
2
votes
1
answer
238
views
Holonomy of a Warped Product Metric
A warped product metric on the "cone" $\tilde{M} = \mathbb{R}^{+} \times M$ is $\tilde{g} =dr^2 + r^2g_M$ where $g_M$ is the metric on $M$.
If we know the holonomy group of the manifold $(M,g_M)$, ...
user117832
13
votes
1
answer
426
views
What are some good lower bounds on the consistency of the failure of the PCF conjecture?
Shelah's celebrated theorem states that $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{\omega_4}$.
But the conjecture is that $\omega_4$ can be provably replaced by $\...
- 38.2k
1
vote
0
answers
41
views
Subspace of smooth Foliations inside smooth Distributions - Is it a Frechet submanifold or atleast a Retraction?
Assume M is a closed manifold. Does the set of smooth foliations on M form a Frechet submanifold inside the Frechet manifold consisting of smooth distributions?
If not, can the set of smooth ...
- 183
3
votes
1
answer
328
views
Bounds on the size of rough numbers
A positive integer $n$ can be described as $B$-rough if all of the prime factors of $n$ strictly exceed $B$.
The first five 2-rough numbers are 1, 3, 5, 7, 9. We always include 1 by convention.
It ...
- 3,195
0
votes
0
answers
137
views
Reference for convergence to a Poisson Point Process
Edited after comment by Ofer Zeitouni
I have a sequence of discrete time stochastic processes $\big((S_n(i))_{i \geq 1}\big)_{n\geq 1}$ such that for every $n$, $i$,
\begin{equation}S_n(i)=\sum_{j=...
- 66
0
votes
1
answer
89
views
Convergence of Stochastic Flow but not Flow
Suppose that $\phi(t,x):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d$ is a flow. Is it possible to extend $\phi$ to a family of stochastic flows $\{\Phi(t,x,\sigma)\}_{\sigma \in [0,1]}$ ...
- 4,969
5
votes
0
answers
335
views
Equivalence of categories of $D$-modules on a singular $X$
Is $D^b(Mod_{qc}(D_X)) \to D^b_{qc}(D_X)$ an equivalence of categories for singular $X$? Where $Mod_{qc}(D_X)$ is the category of quasi-coherent modules on $X$ and $D^b_{qc}(D_X)$ is the full ...
- 367
2
votes
2
answers
172
views
Prove that given root of a polynomial is zero by approximation
Let $\alpha$ be a root of a polynomial $a_nx^n + \ldots + a_1x$ with integral coefficients.
I would like to determine $\varepsilon > 0$ depending on $a_1, \ldots, a_n$ so that $|\alpha| < \...
- 1,262
1
vote
1
answer
210
views
Graph Laplacian Operator
Consider the linear operator $\mathbb{L} : L^2([0,1])\to L^2([0,1])$ defined by
$$
(\mathbb{L}f)(x) = \int_0^1 xy(f(x)-f(y)) \mathrm{d}y
$$
for all $f\in L^2([0,1])$ and $x \in [0,1]$. Is $\mathbb{L}$...
- 31
34
votes
2
answers
1k
views
Representations of $\zeta(3)$ as continued fractions involving cubic polynomials
$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$
where $k\in\mathbb Q$ and $p$ is a ...
- 13.2k
2
votes
1
answer
267
views
Does the Hopf construction work for $S^0$?
Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I ...
- 1,185
3
votes
0
answers
120
views
Degeneration of cycle class map
Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \...
- 1,573
3
votes
4
answers
822
views
Compute the two-fold partial integral, where the three-fold full integral is known
I have the following trivariate ($\rho_{11}, \rho_{22}, \mu$) function
\begin{equation}
4 \mu ^{3 \beta +1} \rho_{11}^{3 \beta +1} \left(-\rho_{11}-\rho_{22}+1\right){}^{3 \beta +1}
\rho_{22}^{3 \...
- 723
24
votes
0
answers
2k
views
How did Gauss find the units of the cubic field $\mathbb Q[n^{1/3}]$?
Recently I read the National Mathematics Magazine article "Bell - Gauss and the Early Development of Algebraic Numbers", which gives a good description of the genesis of Gauss's ideas ...
- 1,869
3
votes
2
answers
1k
views
Completely positive matrix with positive eigenvalue
A matrix $A \in \mathbb{R}^{n \times n}$ is called completely positive if there exists an entrywise nonnegative matrix $B \in \mathbb{R}^{n \times r}$ such that $A = BB^{T}$.
All eigenvalues of $A$ ...
- 133
6
votes
1
answer
138
views
$p$-groups with isomorphic subgroup lattices
Given two non-abelian finite p-groups $P_1$ and $P_2$ of the same order that are not isomorphic.
Can $P_1$ and $P_2$ have isomorphic subgroup lattices?
(I'm not experienced with group theory, ...
- 26.1k
3
votes
0
answers
62
views
Metrically homogeneous spaces as inverse limits
Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following:
Is $X$ ...
- 541
2
votes
0
answers
160
views
Coend of full subcategory
$\require{AMScd}$Let $F:\mathcal{C}^{op}\times \mathcal{C} \to \mathcal{D}$ be a functor and $\mathcal{C}' \subseteq \mathcal{C}$ a full subcategory. Assume that the coends $C$ over $F$ and $C'$ over $...
- 1,661
7
votes
1
answer
403
views
Kontsevich Formality sign convention
Since my question is related to sign convention, I want to define everything from the very beginning. $T_{poly}^k(M)=\Gamma(\wedge^{k+1} TM)$ are the multi vector fields with shifted degree and with ...
12
votes
1
answer
588
views
An inverse problem for Grothendieck rings of varieties
Suppose $A$ is a given commutative ring, and suppose that one knows that $A$ is isomorphic to the Grothendieck ring of $k$-varieties for some unknown field $k$.
Can $k$ be recovered from $A$ ? If ...
- 4,353
4
votes
1
answer
508
views
Trace-class properties of integral operator
Let $k$ be a smooth, compactly supported function defined on $\mathbb{R}^{2}$ and let $Op(k)$ denote the integral operator on $L^{2}(\mathbb{R})$ defined as $$Op(k)f(\cdot)=\int_{\mathbb{R}}k(y,\cdot)...
- 43
4
votes
2
answers
4k
views
Expected distance from the origin for a recurrent 1D random walk in a random environment
It is well known that for a discrete random walk on the integers with a fair coin, the expected distance of the walker from the origin after $N$ time steps is $\sqrt{\frac{2N}{\pi}}$ if $N$ is large. ...
- 775
3
votes
4
answers
430
views
Solution of a 2D Recurrence sequence
Can we solve the following recurrence relation:
$$a_{m,n} = 1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}$$
with $a_{0,n}=a_{m,0}=0$? If not, can we get an estimate of the growth of $a_{m,n}?$
I encountered this ...
- 1,475
7
votes
1
answer
470
views
Deligne Pairing v.s. Weil Pairing on a Family of curves
We have the Deligne Pairing on a family of curve $\pi:X\to S$ by using
$$\langle L,M\rangle_{\mathrm{Pic}^0(X/S)}=\det R\pi_*(L\otimes M) \otimes (\det R\pi_*L)^{-1}\otimes (\det R\pi_*M)^{-1} \otimes ...
- 347
5
votes
0
answers
289
views
Symmetry between V and HOD
Is it consistent that the set of ordinal definable real numbers is countable, but for every $y∈\mathrm{OD}∩ℝ$, every true $Σ_2^{V,y}$ statement holds in $\mathrm{HOD}$?
Note that $Σ_2^V$ is the best ...
- 6,112
0
votes
0
answers
65
views
Normality of certain subrings of polynomial rings in characteristic p
Let $k$ be an algebraically closed field of characteristic p. Let
$Z\subset k[x_1,\cdots,x_n]$ be a graded $k$-subalgebra of a
polynomial ring, such that for any $f\in Z,$ any divisor of $f$ (in
$k[...
0
votes
1
answer
43
views
For an one-order Linear Recurrence of a vector sequence, does the corresponding item follow a Linear Recurrence? [closed]
Consider an one-order Linear Recurrence of a vector sequence, such as
$${\bf x}_{n+1}={\bf A}{\bf x}_n$$
where ${\bf x}_n \in \mathbb{R}^m (\forall n)$, and ${\bf A} \in \mathbb{R}^{m\times m}$ and ${\...
- 601