Newest Questions
159,035 questions
1
vote
0
answers
103
views
Cyclic representation isomorphic to L2 space
This question is also posted on Math Stack Exchange.
I need some help understanding a proof of the following claim: every cyclic representation is isomorphic to some $L^2$ space.
First, formal ...
- 111
1
vote
0
answers
22
views
Parabolic equations and nonconvex domains
Is there a (system of) parabolic equation(s) where qualitative properties depend on whether or not the (say, smooth and bounded) domain $\Omega$ is convex? I am aware of a few cases where a proof ...
- 313
19
votes
0
answers
553
views
Talagrand's "Creating convexity" conjecture
We say a subset $A$ of $\mathbb{R}^N$ is balanced if
\begin{equation}
x \in A, \lambda \in [-1,1] \implies \lambda x \in A.
\end{equation}
Given a subset $A$ of $\mathbb{R}^N$, we write
\begin{...
- 484
3
votes
0
answers
74
views
Locally compact rings with reciprocals
A topological field is defined to be a topological ring $F$ with reciprocals such that the reciprocal function $F\setminus\{0\} \to F\setminus\{0\}$ is continuous. Locally compact topological fields ...
- 974
7
votes
1
answer
415
views
An example of radical ideal which is irreducible but not prime
$\DeclareMathOperator\rad{rad}$I am searching an example of ideal $I$ of a ring $R$ such that $\rad(I)$ is irreducible but not prime ideal.
In case $R$ is Noetherian, the radical of $I$ being ...
- 253
7
votes
1
answer
635
views
Is every smooth projective curve a modular curve?
I have seen a quote saying that
Every smooth projective curve over a number field is a modular curve, i.e. (compactification of) $\Gamma\backslash\mathcal{H}$ for some finite index subgroup $\Gamma&...
- 1,428
9
votes
2
answers
584
views
Does this integral condition characterize $L^\infty$?
Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$ with smooth boundary. For any $f \in L^1 (\Omega)$, is it true that $f \in L^\infty (\Omega)$ if and only if the following condition ...
- 6,321
1
vote
1
answer
212
views
The ideal in $\mathbb{Z}[x]$ of all vanishing polynomials of a curve automorphism
Let $C$ be a projective irreducible algebraic curve of genus $g$ over an algebraically closed field of characteristic $0$ (for simplicity). Given an automorphism $\alpha \in \mathrm{Aut}(C)$ of order $...
- 1,833
13
votes
1
answer
403
views
Egyptian fraction of a number in the interval (0.5,1)
An Egyptian fraction is a finite sum of distinct unit fractions, such as
$$\frac{43}{48} = \frac{1}{2}+\frac{1}{3}+\frac{1}{16}.$$
Does there exist a number in the range $(0.5, 1)$ that when written ...
- 243
11
votes
1
answer
681
views
Solve $\binom{n}{k}=m$ for $(n,k)$
For an integer $m>0$, put $X(m)=\{(n,k):4\leq 2k\leq n \text{ and } \binom{n}{k}=m\}$. Is there an efficient method to calculate $X(m)$? Is there a uniform upper bound for $|X(m)|$?
By ...
- 56.9k
1
vote
0
answers
132
views
Quotient of a polynomial ring with a prime ideal is Cohen$-$Macaulay
[Bruns-Herzog, Exercise 2.1.17] Let $k$ be a field and $R = k[x_1, . . . , x_n]$. Suppose $\mathfrak{p} \subset R$ is a prime ideal, $ht\mathfrak{p} \in \{0, 1, n − 1, n\}$. Show that $R/\mathfrak{p}$ ...
15
votes
3
answers
2k
views
May two Cohen reals collapse cardinals?
My question is the following:
Let $M$ be a c.t.m. of $\mathsf{ZFC}$. Are there two reals $r_0,r_1 \in \mathbb{R}$ such that $r_i$ is Cohen over $M$ for $i=0,1$ and such that $\omega_1^M$ is countable ...
- 2,286
0
votes
1
answer
117
views
Product decomposition for finite graphs
We say that a finite, simple, undirected graph $G = (V,E)$ is (product-)irreducible if it is connected and there are no graphs $A,B$ such that $G\cong A\times B$ (where $\times$ denotes the ...
- 48.9k
2
votes
1
answer
292
views
One unexpected observation related to algebraic curves and their Jacobians
Let $C$ be a projective irreducible algebraic curve over an algebraically closed field of characteristic $0$ (for simplicity). Assume that there is also a cover $\varphi\!: C \to E$ to an elliptic ...
- 1,833
2
votes
0
answers
112
views
Applications of the Riemann-Hilbert Correspondence
I am aware of the (first) proof of the Kazhdan–Lusztig conjectures using the Riemann-Hilbert Correspondence. Are there any other interesting applications of the RH correspondence?
- 507
6
votes
1
answer
431
views
A strange product forcing
Given two forcing notions $\mathbb{P}_0,\mathbb{P}_1$, we know that the product $\mathbb{P_0}\times\mathbb{P_1}$ will produce the following diagram of models ordered by inclusion:
where $M$ is the ...
- 2,286
5
votes
0
answers
214
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
- 1,429
2
votes
0
answers
120
views
Is there an explicit description of a gauge transformation $F$ such that $U_{\hbar}(\mathfrak{g})$ and $(U(\mathfrak{g})[[\hbar]])_F$ are isomorphic?
Let $\mathfrak{g}$ be a semisimple Lie algebra, let $t$ be its canonical 2-tensor, and let $\Phi_{KZ}$ be a Drinfeld associator.When $R_{KZ}=e^{\hbar t/2}$, $(U(\mathfrak{g})[[\hbar]],\Phi_{KZ},R_{KZ})...
- 255
1
vote
0
answers
112
views
Perron-Frobenius theorem to positive delay differential equations
The Perron-Frobenius theorem is that the largest eigenvalue (in modulus) of a non-negative matrix is real (and simple) and corresponds to a non-negative eigenvector. It is applicable to the positive ...
- 73
1
vote
0
answers
112
views
Diophantine approximation away from $0$
Let $\alpha$ be a real irrational algebraic number. The now-classic Thue-Siegel-Roth theorem asserts that for any $\varepsilon > 0$ there exists a positive number $c = c(\alpha, \varepsilon)$ such ...
- 26.9k
2
votes
1
answer
250
views
A concentration inequality related to suprema of sub-Gaussian processes
Let $x_1,\dots,x_n$ be deterministic points in some space $X$ and consider a class of real-valued functions $\mathcal G$ on $X$. We further assume that for any $g \in \mathcal G$,
$$
\Bigl(\frac1n \...
- 1,731
1
vote
0
answers
50
views
Bounds on the spectral radius of a perturbed directed graph
Suppose $(G_n)$ is a sequence of strongly connected directed graphs (without multiple edges) with $G_n$ having $n$ edges such that the adjacency matrix $A_n$ of $G_n$ is primitive, and let $(G_n’)$ be ...
- 11
1
vote
0
answers
99
views
Number of points in a ball in positive characteristic
Let $w_1,\cdots,w_n$ be of elements of $\Omega$, that is the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by the $-\deg$ valuation.
Assume that $w_1,...
- 3,998
0
votes
0
answers
393
views
Spiralling cycles surrounding zeros
The following came up, as a vague idea, in dialogue with a bright, female, 20 year old student of mine. It is a bit vague, but it seems that conjecture 1 is not present in the literature, which seems ...
- 232
1
vote
0
answers
145
views
Multiplicity and the perfect projective line
Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$.
Let $\Gamma$ be the ...
- 85
1
vote
0
answers
98
views
Periodicity in one Fourier variable
Let $f:[0,1]\times [0,1] \to \mathbb C$ be a double periodic function (periodic in both variables) that depends real-analytically on its argument.
We can thus write $f$ as $$ f(x) = \sum_{n \in \...
2
votes
0
answers
87
views
Is there a set of ℵ₁ sequences that can dominate any sequence? [duplicate]
Is there a set $S$ of $\mathbb \aleph_1$ sequences of natural numbers such that for any sequence not in $S$, there is a sequence in $S$ that grows faster than it?
Assuming the continuum hypothesis ...
- 6,353
16
votes
2
answers
943
views
Higher-rank Archimedean valuations of $\mathbb{Q}$, does it exist?
I was reading the proof of Ostrowski's theorem, with an eye toward the Zariski-Riemann space (as well as adic space, Berkovich space, etc.) In the proof, the value group is always assumed to be in $\...
- 343
0
votes
2
answers
280
views
Bounds tighter than the additive Chernoff
Additive Chernoff
Suppose $X_1, \ldots, X_n$ are i.i.d. random variables, taking values in $\{0,1\}$. Let $p=\mathrm{E}\left[X_i\right]$ and $\varepsilon>0$.
\begin{gather*}
\operatorname{Pr}\left(\...
- 105
3
votes
0
answers
106
views
Multiplication map by a ring element on an object vs. all its suspensions in singularity category
Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
- 412
3
votes
2
answers
219
views
Heat equation with nonlocal boundary condition
$\newcommand{\R}{\mathbb R}$I am looking for any possible references (modelling, mathematical and numerical analysis, well-posedness, regularity, anything really!) for heat equation-like problems with ...
- 5,380
7
votes
1
answer
421
views
Consistency strength of some problems about singular cardinals
What is the consistency strength of singularizing a regular cardinal with forcing? Is it exactly a measurable cardinal? Of course the consistency strength of "$V\subseteq W$, $\kappa$ regular in $...
- 667
2
votes
1
answer
227
views
Component group of stabilizer group of a nilpotent element
Let $G$ be a semisimple complex algebraic group, and $\mathfrak{g}$ be its Lie algebra. $G$ acts on $\mathfrak{g}$ by adjoint action. Let $x$ be an nilpotent element in $\mathfrak{g}$, and $G(x)\...
- 653
1
vote
1
answer
181
views
Upper-half space model of $\text{H}_3$
Does $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$ have a natural (unitary) left-action on $\mathcal{L}^2(\text{H}^{+}_{3})$?
If $G$ is a unimodular Lie group and $K$ is a compact subgroup, $G\times G$ ...
user515818
1
vote
0
answers
108
views
Iterated quotients in GIT
Suppose that $G$ is a reductive group that acts algebraically on an affine variety $X$ over an algebraically closed field $k$.
Suppose also that $G$ is equipped with a normal abelian subgroup $N$ such ...
- 18.9k
14
votes
1
answer
413
views
Product analogue of Egyptian fractions
Background
An Egyptian fraction is a finite sum of distinct unit fractions, in which each denominator is not bigger than the next one. In other words, it is a representation of $a/b$ such that $$\frac{...
- 4,829
2
votes
0
answers
178
views
Regularity of linear Bellman equation
Let $f(x,t):B_1 \times [0,1] \to \mathbb{R}$ be Lipschitz function on both $x$ and $t$, $\varphi$ be Lipschitz function on both $x$ and $t$ on the parabolic boundary of $B_1 \times [0,1]$. Let $A$ be ...
- 54
4
votes
0
answers
116
views
Lipschitz extension of a flow can still be a flow?
Consider a map $\Phi: [0,T] \times \mathbb{R}^d \to \mathbb{R}^d$, and assume that there exists a set $U \subset \mathbb{R}^d$ such that $\Phi\rvert_{[0,T] \times U}$ is $L$-Lipschitz. It is well ...
- 729
2
votes
1
answer
169
views
Selberg sieve for counting monic irreducible $P \in \mathbb{F}_q[t]$ such that $P + K$ is also irreducible
In a 1983 paper by William Webb (link below), the author gives a version of the Selberg sieve for function fields and uses it to prove that for a fixed $K \in \mathbb{F}_q[t]$, the number $\mathcal{N}(...
- 23
2
votes
1
answer
190
views
What's the benefit of adding a well-ordering over all classes to $\textsf{MK}$?
Working in $\textsf{ZF} + \text {there is a strongly inaccessible cardinal}$.
Let $\kappa$ be the first strongly inaccessible cardinal, and let $\lvert V_\kappa\rvert= \kappa$, then $(V_{\kappa+1}, \...
- 11.3k
7
votes
0
answers
150
views
The space of analytic associative operations
This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
- 20.5k
0
votes
1
answer
173
views
Is the integral closure of a henselian local domain of dimension $1$ again local?
Let $(R,\mathfrak m)$ be a local domain of dimension $1$. Let $\overline R$ be the integral closure of $R$ in the field of fractions $Q(R)$.
If $R$ is henselian, then is $\overline R$ also a local ...
- 412
1
vote
1
answer
253
views
A vanishing sum and related $p$-adic congruences
Recently I had a curious discovery. Namely, I have made the following conjectures.
Conjecture 1. We have the identity
$$\sum_{k=0}^\infty\frac{(10k-1)\binom{3k}k\binom{6k}{3k}}{(2k+1)512^k}=0.\label{1}...
- 15.6k
0
votes
0
answers
181
views
How does the extra rope length of this link/tangle scale with the inner triangle size?
The symmetric chiral link made of three long intertwined/linked/tangled flexible ropes of radius 1 shown in the figure, whose 6 ends all lie in a plane at spatial infinity and which are pulled ...
- 9
1
vote
2
answers
242
views
Link invariants from Hecke relations of higher order
Alexander theorem says oriented links in $\mathbb{R}^3$ can be
represented by closures of braids. Markov theorem says that
braids related by Markov moves produce isotopic braid closures,
and vice ...
- 5,230
1
vote
0
answers
107
views
Planar sections of convex sets in Cartan-Hadamard manifolds
Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...
- 7,179
2
votes
1
answer
277
views
Understanding an example of abelian-type Shimura varieties
I'd like some help understanding the idea of abelian-type Shimura varieties. In paricular, I understand an abelian-type Shimura datum $(G,X)$ generally parameterizes non-rational Hodge structures ...
- 2,054
3
votes
2
answers
978
views
The adjoint representation of a Lie group
Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
- 131
1
vote
1
answer
370
views
Self-intersection of the diagonal on a surface
Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
- 85
1
vote
1
answer
161
views
Generating function of the stopped simple random walk
Let $\varepsilon_i$ be independent random variables such that $\mathbb{P}(\varepsilon_i = \pm 1)= 1/2$ and denote $W_n = \sum_{i=1}^{n}\varepsilon_i$. That is, $W_n$ is the simple random walk on $\...
- 13