**25**

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 ...

**23**

votes

**0**answers

923 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 ...

**18**

votes

**0**answers

312 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 ...

**18**

votes

**0**answers

894 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 $1$ ...

**16**

votes

**0**answers

537 views

### Computing Self-Intersections with Complex Analysis

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...

**12**

votes

**0**answers

232 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 ...

**12**

votes

**0**answers

583 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) = ...

**11**

votes

**0**answers

691 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 ...

**10**

votes

**0**answers

193 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 $S$?
...

**9**

votes

**0**answers

549 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

211 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

355 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

379 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

111 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

166 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

177 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

191 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

131 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

148 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

330 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

702 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

493 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

714 views

### Brownian Motion Winding Number

Take a simple random walk $\gamma$ in the complex plane conditioned to start at point $a$ and end at point $b$. For this random walk, we can define the winding number $W_\gamma(a,b)$ around $b$ in the ...

**5**

votes

**0**answers

361 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

399 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

75 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

123 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

147 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
...

**4**

votes

**0**answers

246 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

272 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

308 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 CarathÃ©odory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction ...

**4**

votes

**0**answers

261 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

196 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

65 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.

**3**

votes

**0**answers

131 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

95 views

### Lower bounds of laplace transform of characteristic functions

Cross-posted on maths.stackexchange
I have the following integral:
\begin{equation}
f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt
\end{equation}
where $\varphi_X(t)$ is the characteristic function ...

**3**

votes

**0**answers

186 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 ...

**3**

votes

**0**answers

229 views

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

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

**3**

votes

**0**answers

189 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

150 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

196 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

277 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

400 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

165 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

150 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

303 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

235 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 ...

**3**

votes

**0**answers

249 views

### Quotient of manifolds by groups and embeddings

Let $f:X_1\to X_2$ be a closed submanifold. Let $\rho:G_1\to G_2$ be a closed Lie subgroup. Let $G_1$ acts on $X_1$ and $G_2$ on $X_2$ and suppose $f$ is $\rho$-equivariant. I would like to get a ...

**3**

votes

**0**answers

120 views

### When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?

If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when is ...