**26**

votes

**0**answers

3k views

### A paper to the question, if the six dimensional sphere is a complex manifold

Hi,
for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf
Because I am not able to ...

**25**

votes

**0**answers

1k views

### Is S^2 x S^4 a complex manifold?

As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...

**19**

votes

**0**answers

338 views

### Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them.
Define the set $\mathcal{E}$ of ...

**19**

votes

**0**answers

912 views

### Is analytic capacity inner regular?

For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by
$$\gamma(K) := \sup |f'(\infty)|$$
where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ ...

**13**

votes

**0**answers

606 views

### What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula:
$$
f(\zeta) = ...

**12**

votes

**0**answers

204 views

### Analytic contraction of the Stone-Cech compactification of $\mathbb C$

Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$.
Do the meromorphic functions separate the points of $S$?
...

**12**

votes

**0**answers

245 views

### Are the zeros of Tutte polynomials dense in $\mathbb C^2$?

For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of ...

**11**

votes

**0**answers

725 views

### Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb ...

**9**

votes

**0**answers

575 views

### Convexity of Jacobi's theta function with zero argument

This question may be elementary, I have asked it on math.stackexchange.com but have not received any answer yet. Note that I am not an expert on theta/elliptic functions.
Define Jacobi's theta ...

**8**

votes

**0**answers

216 views

### Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number?

Let $X$ be an $n$-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i
\geq 0$ with $\sum ik_i = n$, bounded in terms ...

**8**

votes

**0**answers

363 views

### Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition ...

**7**

votes

**0**answers

382 views

### The natural generalization of Euler's derivation of the Basel sum

Euler proved that $$\sum_{n=0}^\infty \frac{1}{n^2} = \frac{{\pi}^2}{6}$$ by comparing the $z^3$ term in the power series expression of $\sin(z)$ given by
$$\sin(z) = z - \frac{z^3}{3!} + ...

**6**

votes

**0**answers

121 views

### Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...

**6**

votes

**0**answers

174 views

### A zeta function using half of the primes

It is well known that the zeta function satisfies the Euler product formula. See this wikipedia article.
Enumerate all primes by $p_1, p_2, \ldots $ in ascending order.
Set $S$ to be the set of all ...

**6**

votes

**0**answers

192 views

### Invariant curves of rational functions

Let $\gamma$ be a Jordan analytic curve on the Riemann sphere, and $f$ a rational function
of degree at least 2 which
maps $\gamma$ onto itself homeomorphically. The following examples of such ...

**6**

votes

**0**answers

153 views

### Multiplicity of zero (higher dimensional analog)

Consider a sistem of n holomorphic equations with n unknowns in a neighborhood of zero. Suppose that a solution in a neighborhood of 0 is a k-dimensional manifold.
I want to associate to it some ...

**5**

votes

**0**answers

65 views

### Density of rational functions in open Stein

I repost here, after I tried here.
Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb ...

**5**

votes

**0**answers

157 views

### proper mapping between Stein manifolds

My question is the following:
Let $b:X \rightarrow Y$ be a proper holomorphic mapping between two Stein manifolds $X$ and $Y$. For obvious reasons, $b^{-1}(y)$ are finite subsets of $X$. Is the set
...

**5**

votes

**0**answers

199 views

### Automorphisms of Compact Riemann Surfaces

I read a statement that for a compact Riemann surface $C$ with genus $g\geq 2$, one has
for the Jacobian $J(C)$ of the curve $C$:
$$ Aut (J(C))\sim Aut C$$
when $C$ is hyperelliptic and
...

**5**

votes

**0**answers

135 views

### Perturbations of zero-dimensional algebraic varieties

Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C$ its restriction to the real torus. Assume generic situation, so by ...

**5**

votes

**0**answers

154 views

### Complex manifold with non-finitely generated canonical ring

P.M.H. Wilson has an example of a compact non-Kahler manifold whose canonical ring is not finitely generated; see his article and this MO question. I'm trying to understand his construction and have ...

**5**

votes

**0**answers

339 views

### Jet differentials and hyperbolicity: possible mistake in the literature?

I was reading this note by Jingzhou Sun http://arxiv.org/abs/1109.1329
about Demailly's approach to hyperbolicity using jet differentials. The author seems to claim that there is a mistake in one of ...

**5**

votes

**0**answers

709 views

### Is this Fourier integral well-known?

The following integral is a special case of one that arises in an economics problem:
$I(u_{1}, u_{2}) := \displaystyle \int_{z_{1}=-\infty}^{\infty} \int_{z_{2}=-\infty}^{\infty} \frac{ \displaystyle ...

**5**

votes

**0**answers

533 views

### Computing the Chern class for a flat line bundle using the holonomy group?

Let $X$ be a Riemann surface of genus $g \geq 2$. I would like to consider flat line bundles on $X$. Flat line bundles can be identified with representations $\pi_1(X) \rightarrow U(1)$. From ...

**5**

votes

**0**answers

364 views

### Lacunar series with an interesting (in-formula) symmetry.

So, I wrote out a table of functions like so:
$\sum_{n=1}^{\infty} (-1)^{n+1}q^{n}=$ $+q^{1}$ $-q^{2}$ $+q^{3}$ $-q^{4}$ $+q^{5}$ + $\ldots$
$\sum_{n=1}^{\infty} (-1)^{n}q^{n^{2}}=$ $-q^{1}$ ...

**5**

votes

**0**answers

407 views

### Two meromorphic functions with overlapping sets of poles

Assume that we have two meromorphic functions $f(z,w)$ and $g(z,w)$, where $z$ and $w$ are complex (we are interested only in behavior on compact sets). Fix $z$ and assume that the sets of poles of ...

**4**

votes

**0**answers

85 views

### Can Voronin's universality theorem be used to show that $\sigma\circ\zeta=\zeta\circ\sigma$ implies $\sigma$ continuous?

Let $\sigma$ be a field automorphism of $\mathbb{C}$ that commutes with the Riemann Zeta function. Can we use Voronin's universality theorem to prove that $\sigma$ is necessarily continuous?
Thanks in ...

**4**

votes

**0**answers

75 views

### status of Invariant subspace problem on Krein Space

What is the status of Invariant subspace problem on Krein Space? What sort of developments have taken place in this area.

**4**

votes

**0**answers

132 views

### semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.
\[ \int ...

**4**

votes

**0**answers

204 views

### What is the spectrum of a ring of holomorphic rational power series?

Let $R_\infty$ be the ring of power series in a single variable with rational coefficients that converge in the whole complex plane. Let $R_\rho$ be the subring of $R_\infty$ that defines holomorphic ...

**4**

votes

**0**answers

258 views

### Etymology of the O-notation for algebras of holomorphic functions

The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...

**4**

votes

**0**answers

279 views

### Weight-2 modular forms under $\Gamma(N)$

I'm looking for explicit bases of weight 2 modular forms under $\Gamma(N)$, for small N (<16 would be enough). (Ideally in terms of Theta- or Eta-Functions)
It seems to me that this should ...

**4**

votes

**0**answers

311 views

### Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction ...

**4**

votes

**0**answers

273 views

### Studying primes via the gamma function alone: $(x+1)\prod_n \Gamma(\frac{x}{n}+1)^{\mu(n)}$

Various questions on MO concerning the "surprise" occurrence of the gamma function in the functional equation of the Riemann zeta function got me wondering whether the Gamma function alone suffice for ...

**3**

votes

**0**answers

74 views

### Homogenous polynomially convex hull of $[0,1]^n$

I would like to calculate the set of $z\in \mathbb{C}^d$ such that there exists a constant $C >0$ such that for every homogeneous polynomial $p$ in $d$ variables $$|p(z)|\leq C\sup_{x\in [0,1]^d} ...

**3**

votes

**0**answers

90 views

### Is there a coordinate free proof of the Morrey--Kohn--Hormander identity?

The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the ...

**3**

votes

**0**answers

207 views

### The Poisson-kernel in the plane and polynomials

Let
\begin{align*}
p(z) & = \ \prod_{j = 1}^{n} (z - z_j) \, ; \ |z_j| \ = \ 1 \\
& = \ \prod_{j = 1}^{l+1} (z - z_j)^{M_j}
\end{align*}
be a non-constant complex polynomial with ...

**3**

votes

**0**answers

138 views

### elementary proof for existence of point with minimal period 2 for entire function

Fatou proved a very interesting result:
for a transcendental entire function $f$, the second itarate
$f^{2}$ has at least has one fixed point.
(Using the technique of Picard theorem)
This result ...

**3**

votes

**0**answers

131 views

### Bounding an integral transform ouside a circle (or inside a strip)

Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane.
We call
$$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$
the dispersion function of $g$.
Now, one can ...

**3**

votes

**0**answers

101 views

### How did Nochka find weights in his proof of Cartan's conjecture?

Good evening,
I have just read Nochka's proof of Cartan's conjecture (Second Main Theorem of Nevanlinna Theory for linearly degenerate meromorphic curves in $\mathbb{CP}^n$). To prove the conjecture, ...

**3**

votes

**0**answers

243 views

### A Dedekind Eta trajectory / horocyclic flow (Reference request)

I've been exploring the composition of essentially the Dedekind $\eta$-function with
parabolic Möbius transformations,
...

**3**

votes

**0**answers

201 views

### generalizations of the interplay between Cauchy Riemann equations and complex-linearity

A function $f: \mathbb{C} \rightarrow \mathbb{C}$ is naturally viewed as mapping $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose $f = (u,v)$ is continuously differentiable on $D \subseteq ...

**3**

votes

**0**answers

156 views

### What is the status of the subadditivity problem for analytic capacity?

Hi,
Here is another question that concerns analytic capacity. For a compact set $K$ in the plane, define the analytic capacity of $K$ by
$$\gamma(K):=\sup|f'(\infty)|,$$
where the supremum is taken ...

**3**

votes

**0**answers

197 views

### Is it possible to “approximate” a compact set of the plane by compact sets with smooth boundary?

Hi,
Let $K \subseteq \mathbb{C}$ compact. Suppose that $K$ is connected, and that the boundary of $K$ is a simple, closed, piecewise $C^1$ curve. Denote by $A(K)$ the set of all functions holomorphic ...

**3**

votes

**0**answers

290 views

### Why is Mellin-inverse of Gamma periodic?

Specific Case
The periodicity is obvious from computation:
$$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$
However, is there a way to see directly from the integral ...

**3**

votes

**0**answers

405 views

### When checking if a harmonic function is continuous on its boundary, is a dense subset enough?

Let $U$ be an open connected subset of $\mathbb{C}$ and let $u:U\rightarrow \mathbb{R}$ be harmonic and bounded on $U$.
Let $f:\partial_\infty U \rightarrow \mathbb{R}$ be a continuous function, ...

**3**

votes

**0**answers

166 views

### One-parameter groups acting on dual Banach spaces

Let $E$ be a Banach space, and $M=E^*$ (my application has $M$ a von Neumann algebra, but this is unimportant). Let $(\sigma_t)$ be a SOT cts one-parameter group on $E$: so for $t\in\mathbb R$, we ...

**3**

votes

**0**answers

154 views

### plurisubharmonic sublevel sets

Let $X$ be a complex manifold, let $\Omega \subseteq {\bf C} \times X$ be defined by
$\Omega = \{ (z,p) \in {\bf C} \times X : a(p) < Im z < - b(p) \} $ where $a$ and $b$ are plurisubharmonic ...

**3**

votes

**0**answers

311 views

### Stability by flat base change of certain properties

Hi.
Let $f:X\rightarrow S$ be a surjective proper, open morphism of reduced or without embedded component complex spaces (or, in alg.geom, surjective proper, universally open morphism of excellent, ...

**3**

votes

**0**answers

241 views

### Transforming a multivariable integral to make it separable

In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine ...