All Questions
3,561 questions
1
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927
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canonical divisor on singular curves with nodal point
What's the definition of canonical divisor(or whatever related concept) on singular curve with nodal point. More generally, what the definition of canonical divisor on a singular variety X, which is ...
- 623
2
votes
1
answer
186
views
Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set
Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...
- 739
5
votes
0
answers
161
views
On the asymptotics of some sum involving the Mertens function
Let $a_n$ be a sequence of nonnegative real numbers such that $\sum_{n\leq x} a_n \gg \frac{\sqrt x}{\log x}$ for large enough $x$. Denote by $\mu$ the Mobius function, and let $M(N)=\sum_{n\leq N} \...
- 1,019
7
votes
3
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603
views
Closed, sum-free form for the $n$-th derivative of $\operatorname{arcsinh}(\frac1x)$ in $x=1$
During research involving the Born–Jordan quantization I came across the expression
$$
\frac{d^k}{dx^k}\operatorname{arcsinh}\Big(\frac1x\Big)\Big|_{x=1}\tag1
$$
for $k\in\mathbb N_0$. It is not too ...
4
votes
1
answer
103
views
Deriving integral in Gaiotto-Tommasiello theory
I was looking at a paper by Takao Suyama on GT theory, and I couldn't figure out how he derived his formula (3.59):
$$\frac{1}{\pi}\int_a^bdx\frac{1}{z-x}\frac{\sqrt{(z-a)(z-b)}}{\sqrt{|(x-a)(x-b)|}}\...
- 161
4
votes
1
answer
466
views
multivalued holomorphic function on Riemann surfaces
Let $M$ be an open Riemann surface and $f$ a multivalued holomorphic function from $M$ to $\mathbb{H}$, where $\mathbb{H}$ is the upper half plane. Suppose that the monodromy of $f$ lies in the two-...
- 401
1
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0
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105
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question about the book "Holomorphic Morse Inequalities" by Marinescu-Ma
Could somebody please explain to me the proof of proposition 1.6.4 (page 52) in the book "Holomorphic Morse Inequalities" by Marinescu-Ma? I am completely lost. One point that is really ...
1
vote
3
answers
207
views
Existence of solution to linear fractional equation
We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...
user121558
3
votes
0
answers
260
views
Stokes's Theorem with singularities on projective line
Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
- 789
5
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0
answers
189
views
Extension of holomorphic maps to smooth family of holomorphic maps
Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a ...
- 1,409
4
votes
0
answers
251
views
Two questions on asymptotic expansion of confluent hypergeometric functions for real variable $x, |x| \to \infty$
I'm looking into the asymptotic expansion for confluent hypergeometric function $_1F_1(a;b;z) \equiv M(a;b;z)$ and I've two quick questions regarding its asymptotic behavior for real values $x,$ i.e. ...
- 223
27
votes
2
answers
3k
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Teaching the fundamental group via everyday examples
This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...
5
votes
1
answer
139
views
Interference between entire functions with separated indicator diagrams
Let $F,H:\mathbb{C}\to\mathbb{C}$ be entire functions of mean exponential type and of completely regular growth. Assume further that the indicator diagrams $I_F$ and $I_H$ are on the imaginary axis ...
- 1,959
1
vote
0
answers
140
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Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?
Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form:
$$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$
where the ...
- 435
8
votes
2
answers
1k
views
How small (in modulus) can a polynomial get?
Question. If $f(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$, is it true that
$$\max\{\,\vert f(x)\vert: \, -1\leq x\leq 1\}\geq 2^{1-n} \,\, ?$$
Context. This came up while working on ...
- 43.2k
4
votes
2
answers
302
views
Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability
We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...
- 366
2
votes
1
answer
348
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Harmonic functions with boundary condition
I have a question on a harmonic function and the boundary behavior.
Let $\mathbb{U} \subset \mathbb{C}$ be a unit disk. We denote by $\overline{\mathbb{U}}$ the closure of $\mathbb{U}$ in $\mathbb{C}$...
- 721
24
votes
1
answer
4k
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analytic vs. algebraic Gauss-Manin connection
There are the following two notions of "Gauss-Manin connection":
The complex-analytic one: let $f:X\to S$ be a smooth family of complex manifolds. Then we obtain a local system $R^nf_{\ast}\...
- 1,298
1
vote
2
answers
243
views
Thinness and polarity
Let $D$ be a bounded open set in $\mathbb{R}^{n}$ with $n\geq2$ and $E$ a subset of the boundary $\partial D$ of $D$. $D$ is said to be thin at a point $y\in D$ if there is a superharmonic function $u$...
- 411
11
votes
1
answer
1k
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Extending an assignment property from Q to R (or C)
Property of any odd number of nonnegative integers:
Given $x_1 \leq \cdots \leq x_{2n + 1}$ with each $x_i \in \mathbb{Z}_{\geq 0}$, suppose that for any $x_i$ we remove, the remaining numbers can be ...
- 7,831
2
votes
0
answers
366
views
The conformal map from interior of ellipse to interior of the unit disk (property check)
Based on example 5 (page 546) and example 7 (page 550) of the book "Applied and computational complex analysis. Volume 3, Wiley, 1986" written by Peter Henrici, if $a,b>0$ satisfies $a^2-...
- 730
3
votes
0
answers
49
views
Convexity of integral trajectories of rational vector field
Suppose we have a vector field determined by a rational function,
of the form
$$
R(z) = \alpha i z + \sum_{j=1}^k \frac{c_j}{z-r_j}
$$
where $\alpha \in \mathbb{R}$, and the other constants are in $\...
- 15.8k
5
votes
1
answer
616
views
On limits of manifolds
This question should be fairly elementary. I’d just like to check I’m not missing anything.
Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$,...
user131191
2
votes
1
answer
184
views
Conformal maps between simply connected domains with piecewise real algebraic boundary
Between polygons in $\mathbb C\cup\{\infty\}$ (including the "single side polygons", hemispheres, disks) the Schwartz-Christoffel mappings give arguably explicit conformal maps. For polygons with few ...
- 982
3
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0
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187
views
No common roots of complex polynomial and of its derivative
Our specific context
Here is our specific contour integral
$$\int_{\Gamma_{0}}F\big(\sum_{w:p_{z}(w)=0}\frac{1}{w^{a}}\frac{1}{n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}} \big)\frac{dz}{z},$$
...
- 5,474
1
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0
answers
76
views
Second question on a real sequence
I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
- 903
3
votes
0
answers
116
views
A problem on polynomial operators
Let $p(z)=\sum_{k=1}^na_kz^k$ be a polynomial of degree $n.$ Then we have these two results;
If $p(z)\neq 0 $ in $\{|z|<1\}$ then
$$
np(z)+(\alpha-z)p'(z)\neq 0\label{1}\tag{1}
$$ for all $z,\...
- 559
32
votes
2
answers
2k
views
Example of a compact Kähler manifold with non-finitely generated canonical ring?
A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely ...
- 6,871
1
vote
1
answer
655
views
Series involving factorials
Playing around with this series for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+ k+1)! }=\frac{a!\ b!\ (c-1)!}{...
- 13.4k
22
votes
6
answers
2k
views
Elementary solutions to f(z+1)-f(z)=g(z) in entire functions
Let g(z) be an entire function of a complex variable z. Does there exist an entire function f(z) such that f(z+1)-f(z)=g(z)? As I learned several years back, the answer to this is apparently 'yes', ...
- 7,836
33
votes
2
answers
6k
views
Which almost complex manifolds admit a complex structure?
I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau'...
- 4,343
0
votes
0
answers
211
views
Understanding Krantz's proof of Hefer's lemma in $\mathbb{C}^2$
Note: I initially phrased the question in a different way, and it did not receive much attention. In the hope to make it more interesting, I have included a (long) introduction to contextualize and ...
- 11
4
votes
2
answers
2k
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Non-bijective conformal maps between annuli
I need to answer the following question, hopefully in the negative.
Question: Does there exist a conformal map $f$ of degree $1$ from the annulus $\{1<|z|<R\}$ to the punctured disk $\{0<|...
- 1,804
4
votes
0
answers
151
views
Entire function which modulus grows only along the real axis
My research in theoretical physics led to the necessity of constructing an entire function with modulus decaying in the significant part of the complex plane. I wonder whether this is possible because ...
- 41
11
votes
1
answer
752
views
Gluing Riemann surfaces
Let $X$ be a compact Riemann surface with boundary $\partial X$. Assume (for simplicity only) that $\partial X$ has a single connected component. Let us fix an orientation preserving diffeomorphism $\...
- 21.8k
0
votes
2
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392
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What is the relationship between the abscissa of holomorphy and abscissa of convergence of a Dirichlet series
Given a Dirichlet series $$\phi(s)=\sum_{n\ge1}\frac{a_n}{n^s}$$
let $\sigma_{\text{conv}}\in\bar{\mathbb{R}}$ its abscissa of convergence, then we know that $\phi(s)$ is holomorphic on the half-plan $...
- 93
4
votes
3
answers
644
views
Traceless GUE : Four Centered Fermions
The proof of the Wigner Semicircle Law comes from studying the GUE Kernel
$$ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...
- 22.8k
0
votes
1
answer
343
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Integrate Faddeeva function
I came across this integration in my studies.
$\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$
It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I ...
2
votes
0
answers
108
views
The dual of the Lefschetz operator under a perturbation
Let $(X, \omega)$ be a compact Kähler, or more generally, Hermitian manifold. Let $L_{\omega} : \Omega^k(X) \to \Omega^{k+2}(X)$ denote the Lefschetz operator given by $$L_{\omega}(\alpha) : = \omega \...
- 137
8
votes
4
answers
1k
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Monge Ampere equations
I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook ...
- 1,211
16
votes
3
answers
1k
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Analog of Newlander–Nirenberg theorem for real analytic manifolds
It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost ...
- 21.6k
20
votes
2
answers
1k
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Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles
Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that $...
- 6,548
1
vote
1
answer
125
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Applying analytic coordinate changes to singular function germs [closed]
Suppose we are given a function germ
\begin{align}
f = \sum a_{ijk}x^iy^jz^k
\end{align}
such that $f\in \mathfrak{m}^2$, where $\mathfrak{m}$ is the ideal in $\mathbb{C}\{x,y,z\}$ of holomorphic ...
- 19
4
votes
2
answers
450
views
Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
- 395
4
votes
1
answer
234
views
The Dirichlet space of martingales
The Hardy space of martingales can be defined in terms of martingale differences. I'll stick to the simplest case of dyadic martingales.
Notation The underlying probability space can be taken to be $[...
- 325
5
votes
2
answers
274
views
An equality relation for complex numbers off the nonnegative real axis [closed]
For every complex number $z$ off the nonnegative real axis there exist positive numbers $p_0,... ,p_n$ such that $\sum_{i=0}^n p_iz^i = 0$.
Finding difficulty in proceeding with the problem. Need ...
- 199
3
votes
1
answer
181
views
Is a domain biholomorphic to the unit ball a Runge domain?
Let $\Omega \subset \mathbb C^n$ be a bounded domain which is biholomorphic to the unit ball $B^n=\{|z|<1 \mid z\in \mathbb C^n\}$. Can we show $\Omega$ must be a Runge domain? By definition, $\...
- 2,535
4
votes
1
answer
199
views
Distribution boundary value of analytic function and wave front sets
Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
- 335
0
votes
0
answers
105
views
About the definition of lineal convexity
I have been trying to understand the definition of lineal convexity. I am reading the article Duality of functions defined in lineally convex sets by Christer O. Kiselman. For a set $A\subset \mathbb{...
- 31
2
votes
0
answers
158
views
Pull back of a Bounded form
Let $(X, \omega) $ be a complex manifold and let $\alpha $ be a $p$-form $\omega$-bounded on $X$.
Let $f:Y\to X$ be a holomorphic function.
Is $f^*\alpha$ $f^*(\omega)$-bounded on $Y$ ?
- 21