All Questions
3,561 questions
15
votes
3
answers
1k
views
Is this lower bound for a norm of some complex matrices true?
Let $A = [a_{ij}]_{n\times n}$ be a Hermitian matrix, such that $|a_{ij}| =1$ for $i \neq j$, and $a_{ii} = 0$ for each $i$.
I am interested in a tight lower bound of $\|A\|_*:=\sum_{i=1}^n |\lambda_i(...
- 1,991
9
votes
2
answers
667
views
Semi group of polynomials which all roots lie on the unit circle
Let $X=\{f\in \mathbb{C}[z]\mid |z| \neq 1 \implies f(z) \neq 0\} $.
The motivation for consideration of such an $X$ is the the concept of Lee-Yang polynomials.
With the standard multiplication, $X$...
- 366
8
votes
0
answers
339
views
The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures
Let $C\left(\mathbb{R}/\mathbb{Z}\right)$ denote the Banach space of continuous, $1$-periodic complex-valued functions on the unit interval, let $M\left(\mathbb{R}/\mathbb{Z}\right)$ denote its dual, ...
- 1,284
1
vote
1
answer
756
views
An integral involving the argument of the Gamma function and the Riemann Hypothesis
Evaluate $$I=\int_{0}^{\infty} \frac{t\arg
\Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$
where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$
Note that $I$ converges ...
- 105
1
vote
1
answer
489
views
Mandelbrot set and logistic map connection
I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, ...
16
votes
10
answers
5k
views
Good book on Riemann surfaces and Galois theory?
I'm supervising an undergraduate project on Galois theory and covering spaces. I want to have him read about the fact that from a branched cover of a Riemann surface you get an extension of its field ...
- 753
5
votes
1
answer
273
views
A domination property for the Hardy space $H^1$
In the theory of Hardy spaces of the unit disc, a fact that is implicitely used quite often is that if $f\in H^p, 1<p<\infty$, then there exists a function $F\in H^p$ such that $|f(z)| \leq |F(z)...
6
votes
2
answers
784
views
On some analytic property of the Riemann zeta function
Denote by $\zeta$ the Riemann zeta function. For $\Re(s)=\sigma>0$, it is well known that
$$\sum_{n\leq x} n^{-s} = \zeta(s) + \frac{x^{1-s}}{1-s}+ O(x^{-\sigma}).$$
But do there exist infinitely ...
user140392
2
votes
1
answer
163
views
Max-root inequality for convex combination of real-stable monic polynomials (Kadison-Singer Problem)
In the paper "The Kadison-Singer Problem" by Marcin Bownik (https://arxiv.org/pdf/1702.04578.pdf), the following Lemma (3.8) is proven:
Lemma:
Let $p, q\in \mathbb{R}[x]$ be stable monic ...
- 203
3
votes
2
answers
215
views
Radial limits of bounded outer functions
Let $f$ be a non-invertible bounded outer function on the unit disk. Does $f$ has radial limit $0$ somewhere? Note that such a property holds for singular inner functions.
- 687
1
vote
1
answer
261
views
Beltrami equation with harmonic coefficient
I need to find solutions to the Beltrami equation
$$
\frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z}
$$
for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, ...
- 134
9
votes
1
answer
350
views
Collinear Galois conjugates
This is inspired by this old question, which may provide a bit more background. But the two present questions seem somewhat more fundamental to me.
Let $p$ be an irreducible polynomial with integer ...
- 13.4k
0
votes
1
answer
335
views
On partial sum estimate on the series $S(p,q;s)=\sum_1^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$ and other Generalizations
Consider the following sum :
$$S(p,q;s)=\sum_{n=1}^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$$
Here , $p$ is a variable w.r.t which we are going to analyse the sum.
$s$ is another parameter with ...
- 375
7
votes
2
answers
358
views
Julia set containing smooth curve
I have two realted questions.
Let $R$ be a rational function on $\mathbb{C}$ with degree at least 2. We denote by $\mu$ the measure of maximal entropy for $R$ and recall that the Julia set coincides ...
- 589
4
votes
0
answers
176
views
Finding roots of equation with gamma functions
Encountered this function in one of my research problems
$$\frac{\Gamma \left(1-\dfrac{i c}{a}-\gamma \right) \Gamma \left(1+\dfrac{i c}{a}+\dfrac N 2-\gamma \right)}{\Gamma \left(1+\dfrac{i c}{a}-\...
- 199
13
votes
1
answer
409
views
Does the $\overline{\partial}$ operator have closed image?
Let $X$ be a complex-analytic manifold, not necessarily compact.
Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
- 890
4
votes
1
answer
180
views
Why do the absolute values of functions in Hardy spaces tend to be non-oscillatory?
I'm trying to find a rigorous formulation of an impression/intuitive notion related to Hardy spaces. It seems to me that functions in Hardy spaces tend to have modulus functions which do not ...
11
votes
6
answers
3k
views
Explicit Spin Structures on the Torus
Basically, I am trying to build explicit examples of Dirac operators. To this end, I'm looking at the surface E = C/(Z + λZ) - for some λ in H \ SL(2,Z) - with the Euclidean metric and ...
- 22.8k
9
votes
5
answers
3k
views
Assessing effectiveness of (epsilon, delta) definitions [closed]
There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...
- 16.6k
6
votes
0
answers
144
views
What does it mean for the torsion to blow up?
Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian:
Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
- 651
3
votes
0
answers
71
views
Envelope of holomorphy for the union of two balls
Checking my file archive I found an unpublished note by Bros and Glaser in which they calculated the envelope of holomorphy of the union of two overlapping polydiscs $P_j\subset\mathbb C^n$, $j=1,2$. ...
- 246
2
votes
0
answers
90
views
Second order estimates for Dirichlet problem for complex Monge-Ampere equation
Let $\Omega\subset \mathbb{C}^n$ be a bounded pseudo-convex domain. Let $0<f\in C^{\infty}(\bar\Omega)$, $\phi\in C^\infty(\partial \Omega)$. Consider the Dirichlet problem for the complex Monge -...
- 21.8k
2
votes
1
answer
765
views
What about the other $f$ such that $f(f(x)) = \sin(x)$?
This question is inspired by the big MO question here; and also inspired by the big MO question here. The premise of this question requires a backdrop on fractional iteration; so I'll start slow.
...
user78249
5
votes
1
answer
199
views
An identity for the Lambert $W$ function
Expressing the integral in An integral identity in terms of residues, we come to the following supposed identity:
$$\sum_{k=-\infty}^\infty\frac1{1 + W_k(x)}=\frac12$$
for all $x\in(-1/e,0)$, where $...
- 128k
9
votes
1
answer
322
views
Cauchy path integral as a linear operator: kernel and image?
Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-...
- 5,438
3
votes
0
answers
151
views
When holomorphic convexity implies polynomial convexity
For what follows I refer to Forstneric's Book "Stein Manifolds and Holomorphic mappings", Theorem 4.14.6 p. 168 (second edition).
I need some clarifications.
It starts talking about a ...
- 779
3
votes
1
answer
214
views
Holomorphic vector fields with a non-degenerate isolated zero
Let $v$ be a holomorphic vector field defined in a neighbourhood of $0$ on $\mathbb C^n$ with an isolated zero at $0$. Let $\sum_{i,j}{a_{ij}}z_i\frac{\partial}{\partial z_j}$ be the linear term of $...
- 14.3k
12
votes
1
answer
890
views
The Koch snow flake, Holder exponents of conformal mappings
The Koch snow flake $K$ is a domain of $\mathbb{C}$, complex plane. Though, I do not state the precise definition, you can see the picture in wikipedia Koch snow flake.
The Koch snow flake $K$ is a ...
- 721
2
votes
4
answers
1k
views
Complex differential equations
I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs.
Mostly, I'm just ...
- 3,574
8
votes
1
answer
431
views
Holomorphic deformation of complex structure on the real plane
It is known that each complex structures on $\mathbb{R}^2$ is biholomorphic to either $\mathbb{C}$ or the open unit disk $\Delta$.
One can continuously deform one complex structure to the other as is ...
- 1,409
4
votes
2
answers
1k
views
What does analyticity imply in complex analysis? [closed]
In complex analysis, we're constantly faced with problems about the analyticity of a function, on which many theorems are developed. I of course know a bunch of formulas and theorems, but could not ...
- 77
10
votes
3
answers
1k
views
About the classification of commutative and of cocommutative, fin. dim. Hopf algebras
I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....
- 7,746
1
vote
1
answer
63
views
Existence of continuous family of uniformising parameters
I asked this question on MSE a while ago but didn't receive any useful answers.
Suppose I have a $1$-parameter family continuous maps $f_t: \mathbb{S}^2\rightarrow \mathbb{C}P^1$ from a topological $2$...
- 173
4
votes
0
answers
210
views
A more general version of the Fejér-Riesz theorem
A classical result, known as the Fejér-Riesz theorem, states that any Laurent polynomial $p(z)=\sum_{|k|\leq N} c_kz^k$
(the coefficients $c_k$ are complex numbers) which is nonnegative on the torus $\...
2
votes
1
answer
305
views
Reconstructing the metric on $CP^2$ with special one forms
I know that $(z_1,z_2)$ are the affine\inhomogeneous coordinates on the complex projective space $CP^2$. Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$. I want to rewrite the Fubini Study metric on $...
- 69
3
votes
2
answers
498
views
if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?
This question is related to solving $f(f(x))=g(x)$.
Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...
user147204
7
votes
0
answers
169
views
Limiting behavior of a sequence of polynomials
Let $f(z)\in\mathbb{C}[z]$ have all its zeros on the line
$\Re(z)=\alpha$ for some $\alpha\in\mathbb{R}$. It is an elementary
fact (equivalent to Lemma 9.13 here) that
if $u\in\mathbb{C}$ and $|u|=1$, ...
- 50.8k
5
votes
0
answers
206
views
Gluing together holomorphic functions without Mergelyan theorem
Consider the unit ball $\Delta=\{|z|<1\}$ and a Lipschitz (meaning that it is the graph of some Lip. real function) segment, say $\Gamma=[1,7]$.
Consider $f$ holomorphic in a neighborhood of the ...
- 779
4
votes
1
answer
150
views
Characterization of a domain of holomorphy
I need to show that the property of being a domain of holomorphy is the same as being a holomorphically convex domain (this result is known as Cartan-Thullen theorem). However, the proofs I found in ...
- 165
10
votes
1
answer
813
views
"unexpected" residue formula for $\Gamma^3(s)/(\Gamma(3s)(e^{2\pi is}-1)) $
There is a related problem in my current work: to find the residue of the following function at any negative integer $s=-n$:
$$f(s)=\frac{\Gamma^3(s)}{\Gamma(3s)(e^{2\pi is}-1)}$$
It seems to be a ...
- 101
1
vote
1
answer
149
views
Threefolds with Kodaira dimension 2 and non-isotrivial Iitaka map
Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve.
Q1. How many such threefolds exist, and ...
- 137
18
votes
3
answers
1k
views
Is the Hodge Conjecture an $\mathbb{A}^1$-homotopy invariant?
Let $X$ and $Y$ be two nonsingular projective varieties defined over the complex numbers.
If $X$ and $Y$ are $\mathbb{A}^1$-homotopy equivalent, then does $X$ satisfying the Hodge conjecture imply ...
user94803
22
votes
4
answers
2k
views
Technical issue in the approach to Lie groups taken in a book
I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer), which I've enjoyed using. I'm confused about ...
- 28.1k
4
votes
1
answer
172
views
Stein manifolds with "wrong" minimal dimension of embedding
Let $\Sigma^k$ be a $k$-dimensional Stein manifold with embedding as a real manifold (let's assume that that embedding is analytic if it makes things easier) $\Sigma^k \hookrightarrow \Bbb R^{2k}$.
...
- 4,599
18
votes
6
answers
3k
views
What's the use of Malgrange preparation theorem?
The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
- 1,644
14
votes
1
answer
3k
views
An elementary proof that the degree of a map of spheres determines its homotopy type
I'm helping to teach an undergraduate algebraic topology course (out of Hatcher's textbook). We have recently defined the degree of a map of spheres using homology, and the professor and I thought it ...
- 7,338
16
votes
5
answers
6k
views
Complex Analysis applications toward Number Theory
I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory.
So I would like to ask some guidelines about which theorems/concepts should I focus on in order ...
- 321
29
votes
1
answer
4k
views
Zeros of polynomials with real positive coefficients
The following problem arose in collaborative work with Subhro Ghosh:
Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros
$L_n=n^{...
- 7,569
18
votes
2
answers
5k
views
How did Riemann calculate the first few non-trivial zeros of the zeta-function?
Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z)...
- 3,699
3
votes
1
answer
300
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 ...
- 373