Newest Questions
159,036 questions
3
votes
0
answers
121
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Which holomorphic curves can be leaves of a non-singular holomorphic foliation of $\mathbb C^2$?
It is easy to see that for any entire function $f : \mathbb C \to \mathbb C$, its graph $G(f) = \{(z,f(z)) \in \mathbb C^2 \mid z \in \mathbb C\}$ can be translated by $(0,c)$ for any $c \in \mathbb C$...
- 2,858
11
votes
1
answer
1k
views
Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions
Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:
Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
- 533
9
votes
1
answer
789
views
Sequence of real-rooted polynomials
I've been interested in proving a log-concavity result via proving that certain family of polynomials is real-rooted. By performing a sequence of transformations, I can reduce that problem to proving ...
- 1,889
2
votes
0
answers
169
views
$\mathrm{Ext}^i(\pi_1, \pi_2)\neq0$ implies same central character
If $\pi_1$ and $\pi_2$ are two smooth admissible representations of $\operatorname{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$ with central characters. I want to prove that if $\pi_1$ has ...
user515481
0
votes
0
answers
34
views
It is possible to limit a set of curves in the sense $f(x,y) \leq C f(x_0,y)$?
Suppose you have a continuous function $f:[a,b]\times (-\infty, \infty) \rightarrow \mathbb{R}$. I'm trying to understand if it's possible to conclude that due to the compactness of the interval $[a,b]...
- 677
0
votes
1
answer
107
views
An identity about Bessel potential operators
I'm reading this paper where I encounter below equality, i.e.,
$$
\begin{aligned} & \left|\int_{\mathbb{R}^d}\left\{\ell_s^2\left(a_s^{\gamma^2}-a_s^{\gamma^1}\right)_{i j} \partial_i \partial_j ...
- 825
1
vote
0
answers
46
views
Reference request: Hölder regularity of $(1-\Delta)^{\frac{\alpha}{2}}$ for $\alpha >0$
Let $j \in \mathbb N$ and $\alpha \in (0, 1)$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\mathbb R}^d)$ the usual Hölder space. For convenience, we denote $H^{\alpha} := H^{j + \alpha}$ for the ...
- 825
2
votes
1
answer
194
views
Continuity of Moore-Penrose generalized inversion
Any matrix $A\in\mathbb{C}^{m\times n}$ has a unique generalized inverse $A^{\dagger}\in\mathbb{C}^{n\times m}$ with the properties $$AA^{\dagger}A=A,\qquad A^{\dagger}AA^{\dagger}=A^{\dagger},\qquad (...
- 1,093
1
vote
0
answers
170
views
Asymptotic distribution of L infinity norm of Gaussian random vector
Let $\mathbf{X}_n = (X_{n,1}, \ldots, X_{n,n})$ be a $n$-dimensional random vector with $N_n( \mathbf{0}_n, \boldsymbol{\Sigma}_n )$ distribution. The asymptotic distribution of the $L_\infty$-norm of ...
- 119
1
vote
1
answer
194
views
Non-split extension of representations of $\mathrm{GL}_2$ and $\mathrm{Hom}$
Let $0\to V_1\to V\to V_1\to0$ be a sequence of representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$, where $V_1$ is irreducible, smooth and admissible. Assume that this ...
user515481
7
votes
0
answers
254
views
$C^0$-limit of volume-preserving maps on $\mathbb R^n$
Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\...
- 435
9
votes
1
answer
435
views
Natural set-theoretic principles implying the Ground Axiom
The Ground Axiom states that the set-theoretic universe is not a set-forcing extension of an inner model. By
Reitz, it is first-order expressible and easy to force over any given ZFC model with class-...
- 18.6k
9
votes
2
answers
383
views
Does the Affine Pappus Axiom imply the Affine Desargues Axiom in affine planes?
I am interested in the affine version of the well-known Hessenberg's Theorem (saying that Pappian projective planes are Desarguesian).
First I introduce all necessary definitions.
Definition L. A ...
- 42k
1
vote
0
answers
58
views
States on Bratteli diagrams
This a reference request. We are writing a paper on calculi on AF algebras and their relation to Dirac operators. This is quite simple for UHF algebras (and we have references), but AF algebras ...
- 1,143
12
votes
0
answers
764
views
Does this matrix norm inequality have interesting application in other areas of mathematics?
In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices:
Theorem 3. Let $A=[a_{ij}]$ be a real symmetric ...
- 4,484
2
votes
1
answer
158
views
Definition of average $\langle \langle \cdot \rangle \rangle$
I started reading the paper Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model and I would like to clarify the notation $\langle \langle \cdot \rangle\rangle$ the authors use in ...
- 3,535
5
votes
0
answers
306
views
Arithmetic derivatives and non-commutative generalizations
In the theory of arithmetic derivatives, in the simplest case an arithmetic derivative on $\mathbb{N}$ is defined via the rule $(a \times b)'= a \times b' + a' \times b$, mirroring the product rule ...
- 1,347
0
votes
0
answers
115
views
Conjecture that there are finitely many integer powers $a^x$ and $b^y$ such that $b^y - a^x = n$: who first came up with it?
I came up with an interesting mathematical conjecture: for every natural number $n$ there is only a finite number of integer powers $a^x$ and $b^y$ such that $b^y - a^x = n$.
I would like to find out ...
- 161
1
vote
0
answers
125
views
Transforming nilpotency into diagonalizability [closed]
We designate the $k$-th standard vector as $e_k$ in $\mathbb{C}^n$.
We consider the backward shift operator, denoted as $T: \mathbb{C}^n \to \mathbb{C}^n$, which is defined as follows:
$Te_1=0$ and $...
- 4,058
1
vote
0
answers
41
views
Potential function in the smoothing of toric degenerations when not collapsing all $-2$-Spheres
In the paper "An-type singularity and nondisplaceable Lagrangian tori", https://arxiv.org/pdf/1710.11221.pdf, by Sun the author proves that when taking the toric degeneration of a semi-Fano ...
- 791
4
votes
1
answer
361
views
Spaces that are contractible mod diagonal
I bumped into a seemingly natural strengthening of contractibility, which I refer to as "contractible mod diagonal". I'd like to know if this is something standard and whether it appears ...
- 6,652
11
votes
2
answers
1k
views
Twice continuously differentiable implied by existence of limit
I have the following question. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions (vanishing at infinity) and assume that
$$
\frac{f(x+t)+f(x-t)-2f(x)}{t^2}\to g(x)
$$
for all $x\in X$ when ...
- 129
2
votes
0
answers
135
views
Need help understanding the geometry of a particular building structure
$\DeclareMathOperator\SL{SL}$I’m not primarily a geometer, so apologies if this question is worded poorly. I’ve been looking at asymptotic cones of connected semisimple Lie groups with at least one ...
- 21
7
votes
1
answer
581
views
The tangent map of the exponential map
Let $(M,g)$ be a Riemannian manifold and let $\exp^M:TM\to M$ denote the exponential map. Its tangent map $T\exp^M$ is a map $TTM\to TM$. The connection on $TM$ induces a canonical metric on $TM$, the ...
- 797
3
votes
1
answer
123
views
Vanishing of self-hom in Spanier–Whitehead stabilization category
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...
- 413
2
votes
1
answer
179
views
On segments of equal area cut from planar convex regions by chords
Consider a planar convex region $C$ of unit area and all chords of it that cut off a segment of area $\alpha$ from $C$. Obviously, if $C$ is a circular disk of unit area, all segments of area $\alpha$ ...
- 5,979
16
votes
0
answers
434
views
University library dropping independent journal subscriptions. What to do?
We received an email from our university library stating that they plan to drop subscriptions to the following journals. These are some of the best journals published by independent sources. Of course,...
- 2,751
6
votes
0
answers
149
views
Tweak the numerators in the alternating harmonic series so that the partial sums alternate across $0$. What's the pattern in the numerators?
I was thinking about the alternating harmonic series: $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$.
I wondered what would happen if we tweak the numerators so that the partial sums alternate between ...
- 3,577
2
votes
1
answer
180
views
liftability of isomorphism of curves in $P^3$
It is well known that the isomorphism between smooth curves $C$ and $C'$ in $\mathbb{P}^2$ can be lifted to an automorphism of $\mathbb{P}^2$ if degree of $C$ and $C'\geq 4$. Now I am considering an ...
- 1,982
4
votes
0
answers
192
views
Simple continued fraction of Freiman's constant
The quadratic irrational $\frac{2221564096+283748\sqrt{462}}{491993569}$ is known as Freiman's constant and arises in the theory of continued fractions. I'm curious as to its simple continued ...
- 4,985
6
votes
2
answers
494
views
Can every finitely generated field extension of $\mathbb{Q}$ be embedded into a local field?
Let $K$ be a finitely generated field extension of $\mathbb{Q}$, and let $p$ be a prime number. Can $K$ must be embedded into a p-adic local field (i.e. a finite field extension of $\mathbb{Q}_p$) ?
- 95
2
votes
1
answer
173
views
Estimates on perturbation of drift of SDEs
Let $\mu_1,\mu_2:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $\sigma:\mathbb{R}^n\rightarrow \mathbb{R}^{n\times n}$ be Lipschitz functions, of at-most linear growth; i.e. $\|\sigma(x)\|\lesssim \|x\|,\|...
- 364
1
vote
1
answer
146
views
Cohen–Macaulayness of $k[[x^2, x^3, xy, y]]$ over $k[[x^2, y]]$
Let $k$ be a field and $R = k[[x^2, y]]$ and $S = k[[x^2, x^3, xy, y]]$. Since $R \subset S$, is $S$ Cohen–Macaulay as $R$-module?
To check this, what I have observed is that in $S$, the maximal ...
2
votes
1
answer
752
views
why is counit called the trace map
Let $f: X \to Y$ be a morphism of schemes, then
$f_*$ and $f^*$ form an adjoint pair inducing natural
correspondence
$\text{Hom}_{\mathcal{O}_X}(f^*\mathcal{G},\mathcal{F})=
\text{Hom}_{\mathcal{O}_Y}(...
- 619
9
votes
1
answer
456
views
Growth of powers of symmetric subsets in a finite group
(This question was originally asked on Math.SE, where it was answered in the abelian case)
Let $G$ be a finite group, and let $A$ be a symmetric subset of $G$ containing the identity (i.e., $A^{-1}=A$ ...
- 2,889
0
votes
1
answer
153
views
Unitary representation of a group of automorphism on an abelian algebra
Given an abelian C*-algebra $\mathcal{A}$, a state $\omega$, a strongly continuous group of *-automorphism $\{\tau_t : t \in \mathcal{R}\}$, and given a representation $ (\pi(\mathcal{A}), \mid \...
- 33
3
votes
0
answers
215
views
Global class field theory and closure of unit groups
I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
- 85
1
vote
0
answers
64
views
distance between consecutive eigenvalues for the laplacian on cubes
The asymptotic expansion of the eigenvalues of the Dirichlet Laplacian on a cube $[0,\pi]^d$ is given by Weyl's asymptotic, namely it starts with
$$
\lambda_n = C(d) n^{2/d}+o(n^{2/d}).
$$
This fact ...
- 2,494
4
votes
1
answer
160
views
DG algebra structure on minimal free resolution of modules over regular local ring
Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,...
- 412
5
votes
1
answer
306
views
Explicit description for action of Weyl element in Whittaker model for GL2
Let $F$ be a non-archimedean local field and let $\pi =\mathscr{B}(\chi, \chi^{-1})$ be a principal series representation of $\mathrm{PGL}_2(F)$ induced from a character $\chi$ of $F^\times$. Let $w = ...
- 53
1
vote
1
answer
210
views
Connected vertex-transitive graphs of fixed chromatic number and arbitrary size
A simple, undirected graph $G=(V,E)$ is said to be vertex-transitive if for all $v,w\in V$ there is a graph isomorphism $\varphi:V\to V$ such that $\varphi(v) = w$.
The cyclic graphs $C_{2n+1}$ are ...
- 48.9k
2
votes
0
answers
121
views
Can this theory of classes of ordinals interpret ZFC?
The following theory is a theory of classes of ordinals.
Language: Bi-sorted FOL with identity. First sort in lower case ranging over ordinals. Second sort in upper case ranging over classes of ...
- 11.3k
4
votes
1
answer
193
views
Identity related to Ramanujan's congruences
A very simple question: how do you prove the following identity:
$$\sum_{k=0}^\infty p_{5k+4}x^k=5\frac{\phi(x^5)^5}{\phi(x)^6},$$
where
$$\phi(x)=\prod_{n=1}^\infty 1-x^n,$$
and $p_n$ is the ...
- 75
-2
votes
1
answer
211
views
Giving meaning to and solving a second-order stochastic differential equation with white noise
I have encountered a second-order stochastic differential equation (SDE) of the form:
$$
\frac{d^2 T}{dr^2} = (1 + W(r)) (r - A)(r - B)$$
where $r \in (A, B)$ and $W(r)$ is, for example, white noise. ...
- 31
3
votes
0
answers
241
views
A few questions on Feller processes
Update. Most of my questions have been answered in the comments. I am adding these answers to the post.
There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \...
- 620
5
votes
0
answers
128
views
Fusion categories with $\mathrm{PSU}(2)_k$ fusion rules
Let $R_k$ be a fusion ring with $\mathrm{SU}(2)_k$ fusion rules (or equivalently $A_{k+1}$ fusion rules).
All fusion categories with such fusion rules have been classified by Frohlich and Kerler in ...
- 303
2
votes
1
answer
223
views
Finitely generated $\mathbb{Z}$-algebra embeds into unramified $p$-adic ring
Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded ...
- 423
1
vote
1
answer
214
views
Proof in Higher Algebra that $\mathcal{C}at(\mathcal{K})$ is presentable
In Higher Algebra Lemma 4.8.4.2, Lurie shows that for $\mathcal{K}$ a small set of simplicial sets, the $\infty$-category $\mathcal{C}at(\mathcal{K})$ of small $\infty$-categories with $K$-shaped ...
- 898
6
votes
5
answers
2k
views
Standard models of N and R: An Alice/Bob approach
This is a question about a comment in a recent publication by Roman
Kossak. Kossak wrote:
"Nonstandardness in set theory has a different nature. In
arithmetic, there is one intended object of ...
- 16.6k
5
votes
2
answers
868
views
If every ascending chain of ideals leading up to an ideal stabilises, is the ideal finitely generated?
I'm a fourth-year undergraduate currently studying a master's degree. In the last couple of weeks, I have been wondering about the interaction of the Noetherian condition with the prime ideals of a ...
- 53