All Questions
3,561 questions
-1
votes
2
answers
129
views
Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]
Under which condition can it form a Hilbert space? Or what space can it form?
You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
- 11
3
votes
4
answers
1k
views
$Aut(\mathbb{CP}^n)$ [..especially $n=1$ and $n=2$..]
I am confused and curious about the meaning of the $Aut(\mathbb{CP}^n)$.
Is what is called the "linear automorphism group" of $\mathbb{CP}^n$ the same as $Aut(\mathbb{CP}^n)$? It somehow seems to me ...
- 3,541
7
votes
0
answers
354
views
If SO$(3,\mathbb C)$ is isomorphic to PGL$(2,\mathbb C)$, what objects do vectors in $\mathbb C^3$ represent in the context of Möbius geometry?
I hope this question isn't too basic or ambiguous for this site.
The following is an explicit isomorphism from $\mathrm{PGL}(2,\mathbb C)$ to $\mathrm{SO}(3,\mathbb C)$:
$$\left[\begin{matrix}p & ...
- 4,943
3
votes
1
answer
300
views
Example of a morphism of complex spaces or "nice schemes" that is not cohomologically flat in any point
Suppose that $f:X\rightarrow S$ is a proper, separated morphism of complex spaces (with $S$ reduced) and $\mathcal{F}$ a is $f$-flat coherent sheaf on $X$.
From (well-)known results it is known that ...
- 373
13
votes
2
answers
1k
views
Is the exponential function the sole solution to these equations?
Let us take the exponential function $\lambda^z$ where $0 < \lambda < 1$. There are many great uniqueness conditions this holomorphic function satisfies. For example, it is the only function ...
user78249
6
votes
1
answer
642
views
Practically calculating the domain of a power series for function of several complex variables
For simplicity, let us consider a function $f$ holomorphic on a domain $D \subseteq \mathbb{C}^2$. We may therefore write $f$ as a sum of power series $$f(z) = \sum_{\nu_1 \nu_2 =0}^{\infty} c_{\nu_1 \...
- 1,379
0
votes
1
answer
105
views
Explicit entire functions with infinitely many prescribed values
A set $S$ of entire functions $\mathbb{C}\to \mathbb{C}$ is called interpolating if for any sequence $a_1<a_2<\dots$ tending to infinity and any sequence $b_1, b_2, \dots$ there exists $f\in S$ ...
2
votes
1
answer
128
views
Infinitely many deformation equivalent Hodge diamonds
Let $S$ be a connected open complex manifold. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
An ...
user164740
1
vote
0
answers
113
views
Quasi-plurisubharmonic function with polynomial decay
Let $(M, \omega)$ be a compact Kähler manifold. An $\omega$-quasi-plurisubharmonic function on $M$ is an upper semi-continuous function $\varphi : M \to \mathbb{R} \cup \{ - \infty \}$ such that $\...
- 651
4
votes
2
answers
986
views
Reference request: Oldest complex analysis books with (unsolved) exercises?
Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ...
12
votes
2
answers
555
views
$\zeta^{(k)}(s) < 0$ for $s\in (0,1)$
A bit of plotting suggests that $\zeta^{(k)}(s) < 0$ for all $s\in (0,1)$ and all integers $k\geq 0$. (Or, what is the same: $\zeta^{(k)}(s)$ has no zeroes on $(0,1)$.) Is there a brief, clean ...
- 20.2k
18
votes
1
answer
833
views
Cohomology of real analytic coherent sheaves
Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$
is both "...
- 9,237
3
votes
0
answers
233
views
Sequence unifomly bounded
Let $f(\lambda,z)$ be a continuous function on $\Bbb R^2$ such that
I) For $n\in\Bbb N$ and $x\in\Bbb R^*_+$ we have : $f(n,x)=\cos(nx)+ x O\big(\frac{1}{n}\big)$ as $n\to\infty$ and $x\in[n^{-1}\...
- 81
2
votes
1
answer
1k
views
Essential singularity [closed]
In shaum's outline complex analysis,definition of essential point is:
An isolated singularity that is not pole or removable singularity is called essential singularity
Now in the same book there is an ...
8
votes
2
answers
2k
views
Applications of the Small and Great Theorems of Picard
I just presented the two famous theorems of Picard (Small and Great) in a graduate course, but I have not managed to discover a good number of interesting applications.
List of applications (rather ...
- 2,933
11
votes
0
answers
307
views
More identities for the Lambert $W$ function
In view of Iosif Pinelis's question about $\sum_{k\in\Bbb Z}1/(1+W_k(x))$, I played a little with
$S(x,j)=\sum_{k\in\Bbb Z}1/W_k(x)^j$. It seems that if $x$ is a rational number with small denominator,...
- 13.1k
16
votes
3
answers
3k
views
When is a holomorphic submersion with isomorphic fibers locally trivial?
A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a ...
- 47.5k
3
votes
1
answer
124
views
Use of Invariant metric/distances to classify domains in $\mathbb{C}^n$
I am a graduate student in mathematics, who works usually in operator theory. Lately I had to read about about the Lempert’s theorem(a theorem regarding when some pseudometric/distances coincide) and ...
- 195
0
votes
1
answer
133
views
Minimal modulus over a set in $\mathbb C^d$
Suppose that $\mathbf a = (a_1,...,a_d)$ is a constant in $\mathbb R_{+}^d \setminus \{\mathbf 0\}$ (some $a_i$ can be $0$ but not all, all are $\ge 0$).
Define the set $$S(\mathbf a) = \left\{\mathbf ...
- 686
5
votes
1
answer
302
views
An entire function all whose forward orbits are bounded
Edit: I revise the question according to the comment of Gabe Conant.
What is an example of a non constant entire function $f:\mathbb{C}\to \mathbb{C}$ which satisfy the following?:
For every $...
- 366
2
votes
0
answers
210
views
Binomial transform of Dirchlet series (2)
Referring to this MO question, i managed to do the following :
We denote by $J(k+1,z)$ the sum :
$$J(n+1,z)=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{\theta(k+1)}{(k+1)^{z}}$$
and by $S(k+1,z)$ the sum :...
- 181
2
votes
1
answer
304
views
3D similarities and quaternions?
As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form
$$\forall z \...
- 2,306
3
votes
0
answers
168
views
On a certain inverse Mellin transform involving $\zeta$
Let $c \in (1/2,3/2)$. In my research I have encountered the inverse Mellin transform of $\zeta(s-1)/s$:
$$f(x) = \frac{1}{2\pi i} \int_{(c)} \frac{\zeta(s-1)}{s}x^{-s} ds.$$
Does it have a closed ...
- 14.7k
3
votes
1
answer
591
views
Reference on boundary behavior of conformal maps
I am looking for some results on the boundary behavior of conformal maps between simply connected domains. In particular I am interested in conformal maps between $\mathbb{C}-\Delta$, where $\Delta$ ...
1
vote
1
answer
287
views
Regular singular point of non-linear ODE: $\dot{x}(t) + t^{-1}Ax(t) = Q(x(t))$
Consider a system of ordinary differential equations of the form
$$
\dot{x}(t) + \frac{1}{t}Ax(t) = Q(x(t))
$$
where $x(t) \in \mathbb{C}^n$, $A \in \mathrm{Mat}_{n\times n}(\mathbb{C})$ is a constant ...
- 1,383
3
votes
1
answer
530
views
Does the Riemann Xi function possess the universality property?
Here is the question.
Does the Riemann Xi function possess the universality property, or something similar to Voronin's universality property?
Here is why the answer to this question is important. ...
11
votes
2
answers
1k
views
Does this product have analytic continuation?
The product
$$
F(s)=\prod_{p}\frac1{(1-p^{-s})^p},
$$
converges for $\mathrm{Re}(s)>2$, when $p$ runs over all primes. Does it admit analytic continuation beyond the line $\mathrm{Re}(s)=2$? Any ...
user1688
0
votes
0
answers
59
views
Arranging the $k$ solutions of $r(z)=te^{i\theta}$ into $k$ continuous functions of $(t,\theta)$
I have originally opened this question on MSE, but I migrated here, since I realized this environment is more suitable.
Let $r$ be a rational function, that is, quotient of two coprime polynomials $p,...
- 779
1
vote
1
answer
1k
views
What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in C^n ( n>1)? [closed]
What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.
- 11
3
votes
1
answer
278
views
Mittag-Leffler for non-compact Riemann surfaces
Quote from Grauert & Remmert's Theory of Stein spaces: 'Behnke and Stein showed in 1948 that the Mittag-Leffier Partial Fraction Theorem and the Weierstrass
Product Theorem (i.e. the Cousin ...
- 637
3
votes
1
answer
157
views
Inequality for generalized Laguerre polynomials
Please. Does anybody know a proof of this inequality
$$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and
$L^{\alpha}_n$ ...
- 81
7
votes
1
answer
244
views
Volume of solution sets for polynomials in $\mathbb{C}[x]$
Denote $\pmb{a}=(a_1,\dots,a_d)\in\mathbb{R}^d$ and consider the set
$$\mathcal{E}_d=\{\pmb{a}\in\mathbb{R}^d: \text{each root $\xi$ of $x^d+a_dx^{d-1}+\cdots+a_2x+a_1=0$ lies in $\vert\xi\vert<1$}\...
- 43.2k
17
votes
10
answers
109k
views
What are the qualities of a good (math) teacher? [closed]
In forming your answer you may treat the qualifier math or maths as optional, since part of the question is whether there is anything peculiar to the subject of mathematics that demands anything ...
3
votes
0
answers
192
views
Holomorphic sectional curvature and Kobayashi hyperbolicity
Let $(M,g)$ be a compact Hermitian manifold. Let $\text{HSC}(g)$ denote the holomorphic sectional curvature of $g$. The implication $$\text{HSC}(g) < 0 \implies M \ \text{is Kobayashi hyperbolic}$$ ...
- 651
21
votes
4
answers
2k
views
Holomorphic vector fields acting on Dolbeault cohomology
The question.
Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L_u$ on forms preserves the (p,q) ...
- 6,247
7
votes
2
answers
3k
views
An extension of Morera's Theorem
Morera's Theorem states that
If $f$ is continuous in a region $D$ and satisfies $\oint_{\gamma} f = 0$ for
any closed curve $\gamma$ in $D$, then $f$ is analytic in $D$.
I have two questions:
...
- 398
1
vote
0
answers
31
views
Helgason's support theorem type result in 2 dimensions
I had posted this question in math stackexchange here. Let $\Omega \subset \mathbb{R}^2$ be an open domain with smooth boundary. Identifying $\mathbb{R}^2$ with $\mathbb{C},$ consider the following ...
4
votes
1
answer
773
views
A question about openness theorem
The openness theorem says that:
If $\varphi$ be a negative plurisubharmonic function
in the unit ball $B(0,1)$ in $\mathbb{C}^{n}$ satisfying
$$
\intop_{B(0,1)}e^{-\varphi}<\infty,
$$
then there ...
- 41
4
votes
3
answers
715
views
Jordan curve theorem for cylinders
Hello,
I would like to know if the following result is true:
Let $A,B$ be two embedded circles in $S^2$ which do not intersect and let $C$ be the $\textit{closed}$ region bounded by $A$ and $B$ (...
- 41
27
votes
5
answers
7k
views
References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
4
votes
0
answers
229
views
Real part of a holomorphic section of a vector bundle
Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...
- 163
14
votes
1
answer
465
views
Obstruction to the existence of global resolution of coherent sheaf
It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-...
- 912
14
votes
5
answers
2k
views
What is $\sum (x+\mathbb{Z})^{-2}$?
This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges ...
- 13k
3
votes
1
answer
1k
views
When may function (meromorphic) be expanded as power series with coefficients of integers?
Let $F$ be meromorphic function. With what properties may it be expanded as power series with coefficients of integers in such a form
$$
F=\sum_0^{\infty}a_i x^i,a_i\in \mathbb{N} \cup \{0\},\exists M ...
- 3,104
6
votes
1
answer
220
views
Restriction of holomorphic functions on $G$-invariant subspace
Let $X$ be a complex manifold with a holomorphic action of a complex reductive group $G$. Let $Y \subset X$ be a $G$-invariant reduced complex analytic subspace. Is the restriction
$$
\mathcal{O}_X^G \...
- 1,383
4
votes
1
answer
395
views
A possible generalization of Gauss Lucas theorem to higher dimension
A real half space in $\mathbb{C}^2$ is $$\{(z,w)\in \mathbb{C}^2\mid \phi(z,w) > \lambda\}$$ where $\lambda$ is a real number and $\phi$ is a $\mathbb{R}$- linear functional from $\mathbb{C}^2$ to $...
- 366
3
votes
1
answer
396
views
Conformal map onto a circle, from an identification space composed of five squares
I am looking to derive a conformal map for the problem illustrated in this image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at ...
- 167
4
votes
2
answers
171
views
Why do we study biholomorphically invariant pseudodistances/metrics
It is said that pseudodistances/metrics which are invariant under biholomorphic maps are used to determine whether domains in $\mathbb{C}^n$ are biholomorphically equivalent or not.
Suppose $\Omega_1$ ...
- 41
4
votes
2
answers
426
views
On the search for an explicit form of a particular integral
Let $f$ be integrable over the interval $(0, 1)$, and
$$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$
Suppose $f(x) = f(1-x)$; we can then show that
$$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, I_{k}...
- 1,549
4
votes
2
answers
248
views
Holomorphic union of sets
Let $\mathbb{D}$ be the unit disk on the plane and let $U,V\subset \mathbb{D}$ be open and such that $U\cup V=\mathbb{D}$.
Is there a holomorphic map $\varphi:\mathbb{D}\times \mathbb{D}\to \mathbb{...
- 5,529