All Questions
367 questions
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Periodicities of a Complex Dynamical System
Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number.
It is easy to ...
- 149
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164
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Generalization of the Hermite-Biehler-Kakeya Theorem (2)
This is a follow up to the questions posed in Generalization of the Hermite-Bielher-Kakeya Theorem. Here is an interesting follow up to those comments.
Firstly we remark that: $f(x)+g(x)\cdot w$ is ...
- 402
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400
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Cohomology of lattice with coefficients in field of rational functions
In my research, I came across a 1-cocycle in the following group cohomology complex:
Let $\Lambda_\mathbb{Z}$ be a lattice (i.e. isomorphic to $\mathbb{Z}^n)$; let $\Lambda_\mathbb{C} = \Lambda_\...
- 4,991
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93
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Lucas Theorem on a sector
Let $P(z)$ be a polynomial with non-negative real coefficients. Suppose $P(z)$ has all its zeros in a sector $S $ with sector angle greater than $\pi$ contains all the zeros of $P(z).$ My ...
- 559
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715
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The dual space of the Dirac measures on an Abelian group
Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group.
Question. What would be natural vector space $\mathcal{R}$ of ...
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2k
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When are conformal maps holomorphic?
It is a standard fact from elementary complex analysis that a holomorphic function $f:\mathbb{C}\to \mathbb{C}$ is a conformal mapping. Now, suppose I have a map $f':\mathbb{R}^2\to \mathbb{R}^2$ ...
- 640
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102
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How to construct non-abelian functions?
I have found some functions $t_g, g \in G$ for cyclic groups $G=C_n$ which seem to satisfy the following convolution identity:
$$t_g(x+y) = \sum_{h \in G} t_{gh^{-1}}(x) t_h(y)$$
Example of such ...
- 3,074
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137
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Zeros of entire functions with parameter
Let $f_w:\mathbb C \to \mathbb C$ be an entire function with $f_w(0)=1$ and at least one root for any choice of $w \in (0,1)$. Assume further that for a dense set of $w$ the function $f_w$ has ...
- 192
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703
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flow of holomorphic vector field [closed]
Let $(M,J)$ be a complex manifold, where $J$ is the integrable complex structure. Let $X$ be a holomorphic vector field on $M$ and let $\varphi_{t} : M\rightarrow M $ be its flow. Question: Is $\...
- 21
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2
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285
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When I know self convolution of the complex function can I recover function itself or its modulus?
I have a function $A : \mathbb{R} \to \mathbb{C}$.
I know there exists unknown function $u: \mathbb{R} \to \mathbb{C}$, such that $A$ is convolution of $u$ and its complex conjugate $A = u * u^*$.
I ...
- 151
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1
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860
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Sierpinski Triangle and the Chaos Game
The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...
- 87
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1
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376
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Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
- 213
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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
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290
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Analytic continuation of Euler product $\prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1}$
Is anything useful known about the function defined by
\[
f(s, \alpha) = \prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1} \quad ?
\]
Here, $\alpha$ is real. When $\alpha = 1$, this is certainly the ...
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1
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406
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Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
- 362
-2
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1
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203
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Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)
Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question
We consider the following two classes of smooth maps on $...
- 366
-6
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1
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614
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Proof of formula for $\pi$ [closed]
The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...
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