All Questions
1,033 questions with no upvoted or accepted answers
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Padé Approximants of Power Series with Natural Boundaries
Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}...
- 1,284
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277
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Cohomology of complex manifold vs cohomology of its complex submanifold
Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that
$$H^i(Z, A|_Z)=0 \mbox{ for any } ...
- 21.8k
8
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0
answers
104
views
What is known about the following series?
For $k\in{\mathbb Z}^2$ write $|k|=\sqrt{k_1^2+k_2^2}$ for the euclidean norm. Then let $g(k)=gcd(k_1,k_2)$.
For $s\in\mathbb C$ let
$$
D(s)=\sum_{\substack{k\in{\mathbb Z}^2}\\ k\ne 0}\frac{|k|}{g(k)}...
user1688
8
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0
answers
175
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Smooth quotients of algebraic spaces that are varieties away from codimension $\ge 2$ subset
This is a question about when a smooth complex algebraic space that is very close to being an algebraic variety is actually an algebraic variety.
General question: Let $X$ be a smooth separated ...
- 81
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416
views
Pedagogical question on Lie groups vs. matrix Lie groups
There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...
- 28.1k
8
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403
views
rings of modular functions on the upper half plane
Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.
Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.
Let $M_0(\Gamma_i)$ denote the ring of modular ...
- 10.7k
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150
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Extremal length of graphs in surfaces
Given a surface $\Sigma$ with conformal structure $\omega$, the extremal length of a homotopy class $\gamma$ of curves in $\Sigma$ is defined to be
$$
\sup_{g \in \omega} \frac{\ell_g(\gamma)^2}{A_g(\...
- 10.1k
8
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139
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What does this number tell me about a convex lattice polygon?
EDIT: I realized I'd tricked myself by working with a too special case of $f$, the question is now updated (boundary lattice points replaced vertices).
Suppose I have a convex lattice polygon $P$, ...
- 1,488
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554
views
Lower semicontinuity of naive fiber size
I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
- 156k
8
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288
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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
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answers
964
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 ...
- 1,111
7
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answers
167
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Example of closed non-exact torsion differential form on variety
I asked this question some time ago on MSE and received close to no interest. I feel it is appropriate for this site:
I am interested in finding a particular example. I would like to find a variety (...
- 513
7
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answers
218
views
Analytic continuation of Dixon's identity
Many well-known combinatorial identities has an analytic version. For example, the following identities
$$
2^n = \sum_{k=0}^n \binom{n}{k}
$$
$$
\binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2
$$
can be ...
- 1,608
7
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answers
219
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Partitions, weights and polynomials with roots on the unit circle
Let us consider the set $[n]=\{1,\ldots,n\}$ and all of its partitions into exactly $m$ blocks, but let us allow each block to be internally ordered. For example, taking $n=6$ and $m=2$, we will ...
- 1,889
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306
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Gottfried Helms' tetra-eta series
Here Gottfried Helms introduces the following fascinating divergent series
$$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$
The terms don't go to zero, so technically the series does not converge ...
- 1,730
7
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answers
204
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Global generation of $S^n \Omega_X$ for a fake projective plane
Let $X$ be a fake projective plane, namely, a compact complex surface with
$$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample.
Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...
- 66.3k
7
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answers
168
views
The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space
Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
- 1,170
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130
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holomorphic functions on Stein manifolds reaching maxima in a given point on the boundary (Shilov boundary)
Let $M$ be a Stein manifold with smooth, strictly
pseudoconvex boundary, and $x$ a point on its
boundary. Is there a holomorphic function $f$ on
$M$, smooth on the boundary, with strict
maximum of $|f|...
- 9,237
7
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answers
203
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Polynomials having all zeros in the closed left half plane
Let $$P(z) =\sum_{k=0}^n(\alpha_k+e^{i\gamma}\beta_k)z^k=P_1(z)+e^{i\gamma}P_2(z)$$ be a polynomial of degree $n$ with $\alpha_k, \beta_k\geq 0$ for $0\leq k\leq n, $ where $$P_1(z) =\sum_{k=0}^n\...
- 559
7
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0
answers
348
views
Zero derivative on a connected set
I apologize in advance for this (not really research level) question whose answer should be well known. Complex differentiability of a function $f:A\to\mathbb C$ for $A\subseteq \mathbb C$ without ...
- 16.5k
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answers
169
views
Limiting behavior of a sequence of polynomials
Let $f(z)\in\mathbb{C}[z]$ have all its zeros on the line
$\Re(z)=\alpha$ for some $\alpha\in\mathbb{R}$. It is an elementary
fact (equivalent to Lemma 9.13 here) that
if $u\in\mathbb{C}$ and $|u|=1$, ...
- 50.8k
7
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answers
772
views
How much differs the category of real-analytic manifolds from $C^\infty$ ones?
I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
- 395
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354
views
If SO$(3,\mathbb C)$ is isomorphic to PGL$(2,\mathbb C)$, what objects do vectors in $\mathbb C^3$ represent in the context of Möbius geometry?
I hope this question isn't too basic or ambiguous for this site.
The following is an explicit isomorphism from $\mathrm{PGL}(2,\mathbb C)$ to $\mathrm{SO}(3,\mathbb C)$:
$$\left[\begin{matrix}p & ...
- 4,943
7
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answers
461
views
On a paper of Alain Connes entitled 'Around Wilson's Theorem '
A relatively recent paper Alain Connes - Around Wilson's theorem
introduced the function
$$
S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr).
$$
In the same paper, he proved that the ...
user145059
7
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0
answers
198
views
Does this bound on an average over character sums have a more direct proof?
A special case of a well known result of Ingham is that
$$\sum_{n\leq x} d(n)d(n+1)=\frac{6}{\pi^2}x(\log x)^2+O(x\log x)$$
where $d(n)$ is the number of divisors of $n$.
Ingham's results, which ...
- 2,480
7
votes
0
answers
452
views
Sufficient condition on coefficients for a complex power series to be bounded
Let $f(z)$ be an entire function (on $\mathbb{C}$). Assume it has a power series of the form
$$\displaystyle \sum_{n=0}^\infty (-1)^nc_{2n}z^{2n},$$
where $c_{2n}\geq 0$ for all $n$.
Is there a ...
- 783
7
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answers
202
views
Biholomorphic neighborhoods of the boundary of Stein domains
Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
- 773
7
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160
views
Finite covers in complex analytic geometry
Given a complex manifold or complex analytic space, one has the standard notion of open set. There are two different Grothendieck topologies that one can define using this notion, one where covers ...
- 1,180
7
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745
views
What function space does holomorphic functional calculus give us?
Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function $f:U\rightarrow\...
- 175
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461
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Convergence at the radius of convergence
Suppose I have (roughly speaking) a multivalued meromorphic function $f(z)$ on all of $\mathbb{C}$ that is single-valued and holomorphic on the open unit disc and has some branch points of finite ...
- 56.9k
7
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answers
227
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 $...
- 2,374
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answers
896
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 ...
- 430
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votes
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answers
496
views
planar mappings 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 $f:...
- 551
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0
answers
189
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 ...
- 103
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0
answers
160
views
Is the map $G^g/G \to \operatorname{Bun}_G X$ locally an isomorphism in good cases?
$\DeclareMathOperator\Bun{Bun}$Suppose you have a closed Riemann surface $X$ constructed by cutting out $2g$ holes into a sphere and sewing pairs of holes together. Given elements $g_1, \dotsc g_{g}$ ...
- 563
6
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208
views
Partial fraction expansions of meromorphic functions
Sorry if this question (inspired by the recent flurry of activity around a "new" formula for $\pi$) is too naive.
Imitating what one does with Hadamard products, one can try to do the same ...
- 13.1k
6
votes
0
answers
342
views
Does the Poincaré lemma (Dolbeault–Grothendieck lemma) still hold on singular complex space?
Let $X$ be a complex manifold, then we have the Poincaré lemma (or say, Dolbeault-Grothendieck lemma) (locally) on $X$, whose formulation is as follows:
( $\bar{\partial}$-Poincaré lemma) If $\...
- 327
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votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
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0
answers
200
views
Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
- 1,286
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answers
220
views
Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
- 370
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0
answers
165
views
Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...
user516086
6
votes
0
answers
131
views
Complex beta function $\int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\beta-1} \,dx\,dy$
I am interested in showing that the integral
\begin{align}
& \int_{\mathbb{C}} |z|^{2\alpha-2}|1-z|^{2\beta - 2} \,dA(z) \\[8pt]
= {} & \int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\...
- 177
6
votes
0
answers
171
views
Computing residues at $\infty$
As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
- 1,730
6
votes
0
answers
252
views
Picard-Lefschetz formula for the quotient of a degenerating family of curves by a cyclic group
$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$(This is a slight rephrasing and modification of the original question)
Let $D\subset\mathbb{C}$ be the complex unit disk. Let $X$ be a ...
- 10.7k
6
votes
0
answers
395
views
Is there a residue sum formula in quaternionic analysis?
In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series.
If the function $f: \mathbb{C} \to \mathbb{C} $ ...
- 4,829
6
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0
answers
327
views
Are the two-valued homogeneous harmonic functions classified?
Question. Is there a classification of homogeneous two-valued harmonic functions on $\mathbf{R}^n$, valid in dimensions $n \geq 3$?
For reference, multi-valued functions are familiar objects in ...
- 5,048
6
votes
0
answers
78
views
Implications of combinatorial results towards discrete function theory on circle packings
Spurred primarily by a conjecture of Thurston in 1985, there was a series of developments in creating a "discrete analytic function" theory for maps between circle packings of complex ...
- 161
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
6
votes
0
answers
755
views
Discriminant of $\alpha P(u) + (z-u) P'(u)$
I'm trying to find a “closed form” of $\textrm{Discriminant}_u(f(u))$, where $f(u) := \alpha P(u) + (z-u) P'(u)$.
Here $P(u)$ is a monic polynomial of degree $d > 1$ with $u\in\mathbb{C}$, $\alpha$ ...
- 41
6
votes
0
answers
228
views
All complex surfaces embed into a common complex manifold
Is there a closed complex manifold into which every closed complex surface embeds?
user164740