All Questions
3,560 questions
4
votes
3
answers
507
views
Defining negation
I'm currently coauthoring a book intended to teach first-year students basic proof techniques. One of the chapters, written by my coauthor, is about basic logic. In that chapter the negation of a ...
- 42.8k
3
votes
1
answer
427
views
Possible condition for a many variable holomorphic map to be locally surjective
Suppose $a \in \mathbb C^n$, $U$ is a neighbourhood of $a$, and $f: U \to \mathbb C^n$ is analytic. Let $b = f(a)$ and suppose also that $f^{-1}(b) = \{a\}$. Must the image of $f$ contain a ...
- 6,377
4
votes
1
answer
771
views
Understanding Remmert-Stein extension theorem
I'm trying to study the Remmert-Stein theorem in analytic geometry. This is an important result which can be used to prove the Proper Mapping theorem.
A preliminary result is stated in various books (...
- 6,377
1
vote
1
answer
476
views
Gravitational instantons metric (change variables)
I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric:
$$
\gamma dz d\bar{z}+\gamma^{-...
CommunityBot
- 1
7
votes
2
answers
1k
views
Polynomials having all their zeros on the unit circle
Let $P(z)=\sum_{k=0}^na_kz^k$ be a polynomial of degree $n$ having all its zeros on the unit circle. Let $M=\max_{0\leq k\leq n}\lvert a_k\rvert$. The polynomial $P(z)=z^n+1$ has $\max_{\lvert z\...
- 12.9k
3
votes
1
answer
299
views
Example of a morphism of complex spaces or "nice schemes" that is not cohomologically flat in any point
Suppose that $f:X\rightarrow S$ is a proper, separated morphism of complex spaces (with $S$ reduced) and $\mathcal{F}$ a is $f$-flat coherent sheaf on $X$.
From (well-)known results it is known that ...
CommunityBot
- 1
2
votes
1
answer
106
views
A modified Paley–Wiener theorem with weaker condition
Let's consider the following argument: let $f$ be a function in $L^2(\mathbb R)$ such that $\hat f$ extends to an entire function on $\mathbb C.$ Assume that for each $t>0$ and $x \in \mathbb R$
$$
...
- 91.8k
2
votes
3
answers
264
views
Control of values of an entire function in a strip around the real line
Consider an entire function $f: \mathbb C \to \mathbb C$ such that $f|_{\mathbb R}(x)\to 0$ as $x \in \mathbb R \to \pm\infty.$ Does that imply that for each $T>0,$ we have $f(x+iy) \to 0$ as $x\to ...
- 2,145
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 1,446
0
votes
0
answers
50
views
pseudo inverse of a holomorphic multivariate injective map
Let $f:{\mathbb C}^n \rightarrow {\mathbb C}^N$, $N > n$, be holomorphic and injective on an open ball $B_n \subset {\mathbb C}^n$ such that the Jacobian matrices have full column rank at each ...
- 265
13
votes
1
answer
929
views
Sendov's conjecture
It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n ...
0
votes
1
answer
85
views
Criteria for Hardy space membership
Assume that $p>2$ and let $H^p$ be the Hardy space of holomorphic functions in the unit disk $D$. It seems that $f\in H^p$ if and only if $$P(f):=\int_0^{2\pi}\left(\int_0^{1}|f'(re^{it})|^2(1-r)dr\...
0
votes
2
answers
339
views
Error term in França-LeClair approximation of zeta zeros
The imaginary part of the $n$th critical zero of the Riemann zeta function with positive imaginary part (in increasing order) is asymptotically
$$
t_n \sim 2\pi\frac{n}{\log n}
$$
and has been ...
- 7,034
1
vote
1
answer
90
views
The number of roots of pseudo-exponential polynomials
Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
- 2,145
8
votes
1
answer
1k
views
Beautiful examples of arc-like continua
A continuum is a nonempty compact, connected metric space.
A continuum $X$ is called arc-like if, for every $\varepsilon>0$, there is a continuous and surjective function $f:X\to [0,1]$ such that $...
- 1,446
17
votes
2
answers
2k
views
Is this equivalent to RH - Riemann hypothesis?
$$\pi = 3\prod_{\zeta(1/2+it) = 0}\frac{9+4t^2}{1+4t^2}\iff\text{RH is true}.$$
- 105
0
votes
1
answer
128
views
Rational approximation for continuous function on curve $\Gamma$
Let $\Gamma \in C^{1,\lambda}$ be an oriented Jordan curve in complex plane $\mathbb{C} $, $\mathrm{R}(\Gamma)$ the set of all rational functions without poles on $\Gamma $. "$\mathrm{R}(\Gamma)$...
- 4,034
3
votes
0
answers
119
views
Basic obstruction to anything like holomophic symmetric functions of infinitely many variables?
The totality of all holomorphic functions on the unit disk forms some sort of infinite-dimensional complex manifold, where the coefficients of the Taylor expansion might serve as coordinates for the ...
- 17.6k
0
votes
1
answer
103
views
Probabilistic bounds of random polynomials
This is follow-up question to my previous question about the expected number of roots .
I am considering a random polynomial given by $$p(z) = \sum_{i=0}^{n} a_i z^i$$,
where each coefficient } $a_i$ ...
- 1,724
2
votes
1
answer
133
views
Expected fraction of roots in the unit disc of random polynomial with Gaussian coefficients
I am trying to find the expected fraction of roots located in the unit disc for a random polynomial with Gaussian coefficients. Given a random polynomial
$$P(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^...
- 91.8k
1
vote
3
answers
168
views
Can a disk that is limit of disks in a pseudoconvex domain be partially contained in the boundary?
A holomorphic disk is the image of an injective holomorphic map $f:\mathbb D \to \mathbb C^n$ from the unit disk $\mathbb D \subset \mathbb C$ to $\mathbb C^n$.
Let $\Omega$ be a pseudoconvex domain ...
- 246
23
votes
4
answers
2k
views
Lower bounding $|1+z+\cdots + z^{n-1}|$ when $z\approx 1$
I am trying to lower bound $|1+z+\cdots + z^{n-1}|$ when $z$ is a complex number close to $1$ (and $n$ is sufficiently large). My main concern occurs in the case $z = 1 + it$, where $t$ is small. In ...
- 188k
1
vote
0
answers
127
views
an eigenvalue problem for Jacobi Forms
Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$).
$\...
- 43.2k
2
votes
1
answer
236
views
Zero sets for entire functions $f$ with $|f(z)| \leq C_f e^{c|z|}$
Let $c>0$ and $X$ the collection of all entire functions $f$ for which there exists $C_f > 0$ s.t.
$$
|f(z)| \leq C_f e^{c|z|}.
$$
Given a sequence $(z_n) \subset (0,\infty)$ s.t.
$$
\frac {z_n}{...
3
votes
0
answers
115
views
Conformal welding and Jordan loop consequences?
In the similar context as Conformal welding of rectifiable curves
In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ ...
- 5,474
2
votes
0
answers
109
views
Two definitions for transverse $(p,p)$ form
Let $V$ be a complex vector space of dimension $n$ and let $V^*$ be its dual. Fix any integer $1\leq p\leq {n-1}.$ A $(p,p)$-form $\alpha\in\bigwedge^{p,p}V^*$ is said to be strictly weakly positive ...
- 295
3
votes
0
answers
146
views
A Hartogs analogue?
Let $0<r<1<R$, and $A:=\{z\in \ell^2: r<\|z\|_2<R\}$.
For $n\in \mathbb{N}$, let $e_n$ denote the sequence in $\ell^2$ all of whose terms are zero, except the $n^{\text{th}}$ term, ...
- 31
3
votes
0
answers
87
views
Bounding the degree of the Weierstrass polynomial of a product of a holomorphic function and a polynomial
In brief. For a fixed holomorphic function $v$, I want to bound the degree $q$ of the Weierstrass polynomial $Q$ in the Weierstrass decomposition $v^TP = uQ$, in terms of the degree $p$ of the ...
- 981
4
votes
1
answer
535
views
A converse of the maximum modulus Theorem
W. Rudin in Real and Complex Analysis (262) mentioned that
Theorem Suppose $M$ is a vector space of continuous complex functions
on the closed unit disc $\bar U$, with
the following properties:
(a) $...
- 12.9k
7
votes
1
answer
503
views
Combinatorial consequences of de Branges's Theorem?
I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining ...
- 191
1
vote
1
answer
100
views
Orbit closure of two elliptic Möbius transformations
Let $g_1$ and $g_2$ be two elliptic Möbius transformations of infinite order in $\mathrm{Aut}(\bar{\mathbb{C}})$. If $\mathrm{Fix} (g_1) \cap \mathrm{Fix} (g_2) = \emptyset$, then can we deduce that $...
- 28.2k
3
votes
0
answers
122
views
How to find the right path of integration to get the asymptotic partition formula
I am trying to understand how the asymptotic partition formula $p(n) \sim \frac{e^{\pi\sqrt{\frac{2n}{3}}}}{4n\sqrt3} $ was derived for a project and have been reading and following many papers.
I am ...
2
votes
0
answers
70
views
Zero sets of integral power series that converge on disks
Fix a radius $r \leq 1$. I'm interested in any necessary conditions, or any sufficient conditions, for a subset $S$ of $B(0,r)$, the origin-centered open disk of radius $r$, for $S$ to be the set of ...
- 17.6k
10
votes
0
answers
654
views
Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$
My question seems elementary, yet I could not find the solution after working on and searching for several days...
I'd like to find the eigenfunctions of a simple integral kernel:
\begin{equation}
\...
- 628
2
votes
1
answer
194
views
Minimal degree of a polynomial such that $|p(z_1)| > |p(z_2)|, |p(z_3)|, ..., |p(z_n)|$
I was investigating the behavior of $p(x)^n \mod {q(x)}$, for some polynomials $p, q \in \mathbb{C}[x]$. We'll assume $q$ is squarefree. If $q(x) = (x - z_1) (x - z_2) (x - z_3) ... (x - z_n)$ for ...
- 109k
3
votes
1
answer
237
views
Quadrature domains for arc length
Is ellipse a quadrature domain for arc-length measure?
More precisely does there exist points $z_1,\cdots,z_n$ inside an ellipse $E$ and non zero constants $c_1,\cdots,c_n$ such that $$\int _Ep(z)\,...
- 191
3
votes
0
answers
245
views
Norm on the space of real analytic functions
The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
- 203
5
votes
1
answer
855
views
$L\log L$ and Hardy space on the upper half plane
Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane.
It is well-known that the Cauchy ...
CommunityBot
- 1
0
votes
1
answer
177
views
Green's function in terms of logarithmic potential and energy of a measure
Let $\mu$ be a finite (Borel) measure on $\mathbb{C}$ with compact support $K := \mbox{supp } \mu$.
The logarithmic potential associated to the measure $\mu$ is
\begin{equation}
\Phi_{\mu}(z) = - \...
- 63.9k
1
vote
0
answers
126
views
Canonical basis of cycles of Riemann surfaces
Let $\Gamma$ be the compact Riemann surface defiend by the algebraic curve
$$
f(x, y) = y^n + a_1(x)y^{n-1} + a_2(x) y^{n-2} + \dots + a_{n-1}(x)y + a_n(x) = 0,
$$
where $a_1(x), \dots, a_n(x)$ are ...
- 89
24
votes
12
answers
4k
views
2D problems which are easier to solve in 3D
It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex ...
- 2,821
1
vote
1
answer
88
views
Mean values of polynomial and holomorphic matrices
Lemma. Assume $H: \mathbb{R} \to \mathbb{R}^{d \times d}$ is a polynomial of degree $m$, such that for all $x \in \mathbb{R}$, $H(x)$ is a symmetric semidefinite matrix. For all $n \geq 0$ and real ...
- 981
3
votes
1
answer
309
views
Zeros of the derivative of $\xi$
In his paper on Zeros of derivatives of Riemann $\xi$-function on critical line Brian Conrey mention that
It can be shown that the Riemann hypothesis implies that all zeros of $\xi (s)$, the ...
- 61
0
votes
0
answers
85
views
Meromorphic extension of a limit function
Suppose $f_j(z)$, $j=1,2,..$ is a sequence of meromorphic functions on the complex plane $\mathbb{C}$. With a common set of all poles given by $S = \{-i,-2i, -3i,..\}$.
Assume that each of them is ...
1
vote
0
answers
113
views
Computing a limit for the Weierstrass function
Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
- 4,115
1
vote
1
answer
95
views
Common holomorphic forms for two distinct complex structures
Let $S$ be a closed real surface having two complex structures $c_1$ and $c_2$ which are not biholomorphic (so $S$ is a Riemann surface with genus at least 1). Consider $\omega$ a 1-form on $S$ which ...
- 26.3k
12
votes
2
answers
1k
views
Landau's constant
(Hi. This is my first question here.)
A well known result in complex analysis says that there is an $\varepsilon\gt 0$ such that if $f$ is holomorphic in (a neighborhood of) the closed disk ${\mathbb ...
5
votes
4
answers
954
views
Limit of an integral vs limit of the integrand
I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral
$$
I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
4
votes
1
answer
519
views
Are there any necessary conditions of lacunary functions known?
On the internet, most theorems about lacunary function only give the sufficient conditions. For example, Ostrowski-Hadamard Gap Theorem concerns the asymptotic length of null Taylor coefficients, ...
- 6,377
7
votes
0
answers
218
views
Analytic continuation of Dixon's identity
Many well-known combinatorial identities has an analytic version. For example, the following identities
$$
2^n = \sum_{k=0}^n \binom{n}{k}
$$
$$
\binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2
$$
can be ...
- 1,608