All Questions
152,901
questions
4
votes
0
answers
120
views
Reductive Operator Problem
In the 1972 paper ''An equivalent Formulation of the Invariant Subspace Conjecture'' Dyer, Pedersen, and Porcelli announce the following result:
The Invariant Subspace Problem has a positive ...
2
votes
1
answer
265
views
A generalization of Vaught's two cardinal theorem
I'm trying to determine whether or not the following generalization of Vaught's two cardinal theorem is true.
Let $T$ be a complete theory in a language with a unary predicate $U$ and a binary ...
4
votes
0
answers
223
views
Minimization over a convex function of equal vs unequal success probabilities of Bernoulli random variables
Let $U_1,U_2,\ldots,U_n$ be $n\geq 2$ mutually independent Bernoulli random variables. There are two cases of interest:
$1.$ The random variables $U_1,U_2,\ldots,U_n$ are identically distributed;
$...
19
votes
2
answers
1k
views
Obstruction to a general S^1-action
Question: Suppose you have a simply connected, closed, orientable, smooth manifold $M$. What are some restrictions to the existence of a smooth (non-trivial) $S^{1}$-action on $M$?
Note: 1.There are ...
4
votes
0
answers
174
views
What kind of module is this?
Recall that, if $R$ is a commutative ring, then a suitably finite $R$-module $M$ is projective if and only if the localization $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{...
4
votes
1
answer
316
views
Infinite-dimensional classical Lie algebras
The Bott periodicity theorem states that $\pi_k(O(\infty))=\pi_{k+8}(O(\infty))$ and similarly for other classical Lie groups.
But his groups are defined as an inductive limit. For e.g. $GL(n,F)$, for ...
1
vote
0
answers
67
views
When is $R/Soc(R)$ reduced?
Let $R$ be a ring with identity. It is readily checked that when the quotient $R/S_r$ is reduced, the nilpotent elements of $R$ fall into $S_r$, where $S_r$ is the right socle of $R$. Is the converse ...
1
vote
1
answer
293
views
Admissible k-tuples and primorials
Let $ (a_{1},\cdots,a_{k}) $ an admissible $ k $ -tuple and $ P_{k} $ the product of the first $ k $ primes. Do we have a conjectural expression for the number of positive integers $ n $ not ...
5
votes
2
answers
232
views
Determining the Largest Face of a Simplex
This question is in the vein of my former question Fast Comparing of the Volume of Simplices Defined by Sidelengths, but it has a different twist, that may allow for an easier answer:
Questions:
...
6
votes
1
answer
212
views
Nice S¹-action implies existence of unconditional basis?
Let $V$ be a Banach space equipped with a continuous linear action of $S^1$ (meaning, the map $S^1\times V\to V$ is continuous). Assume that all the eigenspaces of the $S^1$-action are finite ...
2
votes
1
answer
144
views
Residue of the following variant of Dirichlet function [closed]
I am working with the Piltz divisor problem where the number of ways in which a number $n$ can be written as a product of $k$ is of the form
$$D_{k}(x)=xP_{k}(\log x)+\Delta _{k}(x)$$
where $P_k$ is ...
4
votes
0
answers
88
views
Locus of Hodge classes
Let $\pi: X\to S$ be a proper smooth morphism of complex analytic spaces, with connected smooth $X$ and $S$ over $\mathbf{C}$, projective fibers, and $$\mathscr{H}_{X/S}^p := R^p\pi_*\Omega^{\bullet}_{...
1
vote
0
answers
73
views
Decompose elements in $SL_2$ as a pair of elements in $SL_2^*$.
I have a question about decomposing elements in $SL_2$ as a pair of elements in $SL_2^*$. Here $SL_2^*$ is the dual Poisson Lie group of $SL_2$ which is defined as follows.
Let $G$ be a Poisson-Lie ...
7
votes
1
answer
584
views
What is the distance between two points on the Berger metric of the squashed three-sphere?
The Berger metric on a "squashed" three-sphere is given (in Euler coordinates) by
4 $ds^2 = \lambda^2 (d \tau + \cos \theta d \phi)^2 + d \theta^2 + \sin^2 \theta d \phi^2$.
See for example Eq. 1....
1
vote
0
answers
67
views
Deriving the time evolution of the reflection coefficient for 1d cubic NLS
Update: I have found that the detailed answer to my questions is contained in the book "Solitons: an introduction" by P.G. Drazin and R.S. Johnson. Generally speaking, this seems to be a great book ...
2
votes
0
answers
92
views
Open problems concerning Araujo's biseparating maps
Araujo stated the following four open questions at the end of his paper, page $518$ and $519.$
Question $1:$ Assume that there exists a biseparating map $T:A^n(\Omega:E)\to A^m(\Omega',F)$ which is ...
4
votes
1
answer
1k
views
Existence of solutions to first-order PDE involving convolution
Let $f(x,\alpha)$ be a smooth function of compact support in $x$. Now, let its $\alpha$-dependence be determined by the following first-order equation,
\begin{align}
\frac{\partial}{\partial \alpha} ...
7
votes
2
answers
663
views
Is every true statement independent of $PA$ equivalent to some consistency statement?
Most true statements independent of PA that I know of is equivalent to some consistency statement. For example
Con(PA), Con(PA + Con(PA)), Con(PA + Con(PA) + Con (PA + Con(PA)), $\dots$
Goodstein's ...
4
votes
0
answers
161
views
Convergence of integral formula for Fourier inversion (and Hilbert transform) for integrable piecewise-smooth functions
I asked the question below on Math Stack Exchange, https://math.stackexchange.com/questions/2592555/convergence-of-integral-formula-for-fourier-inversion-and-hilbert-transform-fo, but [despite it ...
8
votes
1
answer
204
views
Integral complete 4-partite graphs
For given block sizes $a<b<c<d$, consider the complete 4-partite graph $K_{a,b,c,d }$.
Can such a graph be integral, i.e. have only integer eigenvalues?
It is easy to see that the ...
3
votes
1
answer
602
views
A particular embedding of a Lie group in Euclidean space
I apologize in advance if my question is elementary.
Before I present my question I mention my motivation:
Motivation:
A Lie group is a manifold. At the same time it is a Riemannian manifold ...
4
votes
0
answers
701
views
Pullback of the canonical bundle
Let $f : X\to Y$ be a morphism of smooth projective varieties over a field $k$.
Assume $f^*\omega_{Y/k} \simeq \omega_{X/k}$.
I'd like to collect a bestiary of the properties $f$ has, or even ...
8
votes
1
answer
953
views
Consistency strength of Berkeley cardinals
From Cantor's Attic:
A cardinal κ is a Berkeley cardinal, if for any transitive set $M$ with $κ∈M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(...
2
votes
0
answers
98
views
Why do "large" opens of abelian surfaces have "small" canonical bundle?
Let $A$ be an abelian surface, and let $S$ be a finite set of points. Let $U=A\setminus S$. Note that $U$ is a "large" open of $A$.
Let $B\to A$ be a proper birational surjective morphism with $B$ ...
3
votes
0
answers
144
views
Causal fermion systems fromm fractal geometry
Okay, first off- I apologise if this is a stupid question. I'm mainly a very young physics guy, but this has primarily math basis. I'm trying to build a theory that is, long story short, some ...
3
votes
1
answer
262
views
Are injective Hilbert Schmidt operators (measure theoretically) generic?
It's well known that when the elements of an $n \times n$ matrix $A$ are chosen independently from e.g. $U[0,1]$ distributions, then with probability $1$ the matrix $A$ will be injective (indeed, ...
9
votes
2
answers
3k
views
Localization and intersection
It is very well known that if $\mathfrak p_1, \ldots,\mathfrak p_n$ are prime ideal of an integral domain $A$, then we have the equality$$S^{-1}A=\bigcap_{i=1}^n A_{\mathfrak{p}_i},$$ where $S:=A\...
2
votes
0
answers
114
views
Do integral curves on simple abelian surfaces define big line bundles?
Let $A$ be a simple abelian surface over $\mathbb{C}$.
Let $C\subset A$ be an irreducible and reduced one-dimensional closed subscheme. Since $A$ is simple, the normalization of $C$ is of genus at ...
8
votes
1
answer
530
views
Vanishing of H-cohomology
This looks elementary, but somehow I am stuck, please bear with me:
Let $H$ be a differential 3-form, nowhere vanishing, but not necessarily closed. What is a sufficient condition that the sequence ...
11
votes
1
answer
1k
views
Why do we mainly integrate with respect to martingales?
Although my resarch focuses on PDEs (optimal transport, these days), I am currently trying to learn stochastic calculus and integration. I am just beginning in this topics, but I was wondering: why do ...
2
votes
1
answer
845
views
Partition set into equal-sum subsets
What is known about how to divide $\{1,4,9,16,...,81^2\}$ into $3$ equal-sum subsets with $27$ elements per subset?
4
votes
0
answers
217
views
Singularities of Viehweg's fiber product
Let $f:X\to Y$ be a fibration between smooth projective manifolds. Assume that $\Delta$ is a simple normal crossing divisor in $Y$ such that $f^*(\Delta)$ is normal crossing in $X$, and
$$
f_0:X_0=f^{...
3
votes
1
answer
426
views
Generator of Wiener process and its running maximum
This was originally posted on Math StackExchange a long time ago, but got no answer (even after a bounty).
Let $W$ be a standard linear Wiener process issued from zero and $M$ its running maximum
$$
...
3
votes
2
answers
229
views
Largest $A\subset \mathbb{F}_2^n$ such that no two $a\neq b$ in $A$ add to an element of $A.$
If such a set $A$ of size $m$ exists, all its admissible pairwise sums must lie in its complement, thus
$$
\binom{m}{2} \leq 2^n-m,
$$
which gives $$m\leq 2^{(n+1)/2}\qquad (1)$$.
Edit: The upper ...
2
votes
1
answer
494
views
A complex limit cycle not intersecting the real plane
Edit: This is a real coefficient version of the current post.
Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow?
There is a ...
3
votes
0
answers
61
views
Fast Comparing of the Volume of Simplices Defined by Sidelengths
I have a problem, that requires sorting a set of simplices, that are defined via their sidelengths, according to volume; the value of the individual volumes isn't relevant in my problem.
Question:
...
3
votes
0
answers
155
views
Decomposition of injective modules over Noetherian rings
Let $A=\mathbb{C}[x_1,\ldots,x_n]$ be a polynomial algebra over the complex numbers.
I am interested in injective modules over $A$.
Since $A$ is projective over itself, the $\mathbb{C}$-dual module $...
3
votes
2
answers
303
views
Log-concavity of the maximum of gaussians
Let $Z_1,\ldots, Z_n$ be independent standard gaussian random variables. Is it true that $X=\max\{Z_1,\ldots,Z_n\}$ has a log-concave distribution function?
6
votes
1
answer
326
views
Does exist a Kahler-Einstein metric on the blow-up of $\mathbb{P}^3$ along a smooth plane cubic?
This might be well known for the experts but I am not able to find a reference. I was wondering if there exists a Kahler-Einstein metric on the Fano threefold given by blow-up of $\mathbb{P}^3$ along ...
5
votes
3
answers
372
views
Graphs of groups with homomorphisms not necessarily injective
I'm wondering if there is any literature on graphs of groups where the maps $G\to H$ from an edge group $G$ to its endpoint group $H$ are not necessarily $\pi_1$-injective. Or is this just too general ...
7
votes
1
answer
990
views
Specialization map étale cohomology
Let $R$ be a henselian dvr, $s,\eta\in\text{Spec}(R)$ the closed and generic points, and $f : X\to \text{Spec}(R)$ a proper smooth scheme.
For a prime $\ell$ invertible on $R$, is there a ...
5
votes
1
answer
272
views
Moy-Prasad and lattice stablizers
Consider $\mathrm{SO}(5)$, or maybe $\mathrm{SO}(n)$ over your favorite locally compact non-Archimedian field of characteristic $0$. There are two interesting families of compact open subgroups. The ...
4
votes
2
answers
332
views
algorithm for finding radical expressions of all conjugates of an arbitrary algebraic number expressed in radicals
By an algebraic number expressed in radicals, I mean one that is an element of a set $S$ characterized as follows:
$\mathbb{Z}\subset S$.
For any $a,b\in S$, $a+b,a·b\in S$.
For $a,b\in S$ with $b\...
7
votes
1
answer
1k
views
Matrix elements of exponential of tridiagonal matrices
Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements?
Motivation: I'm trying to find the first passage time ...
9
votes
1
answer
494
views
Regular $p$-norm of a matrix
Let $n \in \mathbb{N}$ and $p \in [1,\infty]$ be fixed and endow $\mathbb{C}^n$ with the $p$-norm $\|\cdot\|_p$. For every matrix $A \in \mathbb{C}^{n \times n}$ we denote the operator norm of $A$ as ...
0
votes
0
answers
74
views
Looser condition for regularity for Neumann problems
If $u(x) = g(|x|)$ is a rotationally symmetric function in $\mathbb{R}^{n+1}$ then
$$\Delta u = g''(|x|) + n |x|^{-1} g'(|x|).$$
Let's say we are studying rotationally symmetric solutions to ...
6
votes
1
answer
143
views
Spinor bundle tensored with certain line bundle gives the dual spinor bundle
Let $E$ be a $spin^c$ bundle and $L_E$ be a (complex) line bundle defined using transition functions $\nu \circ g_{U,V}$ where $\nu:spin^c(n) \to \mathbb{T}$ is map such that $\ker \nu=spin(n)$ and $...
7
votes
1
answer
536
views
First Chern class of a specific line bundle
Let $E$ be a spin$^c$ bundle and $spin^c(E)$ the corresponding $spin^c(n)$-principial bundle. Let $g_{U,V}: U \cap V \to spin^c(n)$ denote transition functions for this principial bundle and consider ...
2
votes
0
answers
535
views
Eigenvalues of a specific Hankel matrix
I have an $\frac{N}{2} \times \frac{N}{2}$ matrix $G$ with entries given by
\begin{equation}
G_{ij} = \frac{1}{\sin(\frac{\pi}{N}(i+j-\frac{3}{2}))}, \;\;\;\;\;\;\;\; 1 \le i,j \le \frac{N}{2},
\end{...
5
votes
3
answers
1k
views
PDE-oriented textbook on probability and random processes?
I was trained in reaction-diffusion (parabolic/elliptic) PDEs, and my research now focuses on applied optimal tranport. I'd like to learn probability and stochastic processes, mostly their connection ...