Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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0
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43 views

Decay estimate of an inverse Fourier transform in R^n [closed]

Is there decay estimate that $\int_{\mathbb R^n}\frac{1}{\xi^2+1} e^{ix\cdot\xi}$ decays like $log|x|$ when $n=2$ and $|x|^{1-n}$ when $n\geq 3$? I don't know how to estimate this non absolute ...
-3
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0answers
33 views

problem related to conformal map of doubly connected region [closed]

Is the explicit result known? Product[(1 - Cos[x]/Cosh[n*h]), {n, 1, Infinity}]
8
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0answers
125 views

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 ...
3
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1answer
84 views

maybe this conjecture also hold to this complex inequality

I have solve this following Question: Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{...
4
votes
2answers
109 views

Integral representation of a limit

Let $$ \tilde{\mathbb C}={\mathbb C}\smallsetminus (-\infty,0] $$ the complex plane without the negative real axis. Let $V$ denote the set of all holomorphic functions $f:\tilde{\mathbb C}\to{ \...
2
votes
1answer
163 views

Intersection of two curves is not Cohen Macaulay

Let be $R=\mathbb{C} \lbrace x,y,z \rbrace$ the formal series ring and let $f_{1},f_{2},f_{3} \in R$ be nonzero elements of $R$. (a) Consider the varieties $M:=V(f_{1},f_{2})$ and $N:=V(f_{2},f_{3})$ ...
1
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0answers
110 views

Is an bijective analytic map bi-analytic?

Suppose that $E$ and $F$ are complex Banach spaces and $U\subset E$ and $V\subset F$ are open subses. $f\colon U\to V$ is analytic $f\colon U\to V$ is bijective Is $f$ bi-analytic? (i.e. is its ...
0
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1answer
84 views

Polynomials satisfying $q(z, \bar{w}) + q(w, \bar{z}) \le q(z, \bar{z}) + q(w, \bar{w})$ for all complex z and w

Consider the polynomial $q(x, y) = a x y^3 + a x^3 y + b x^2 y^2$, where $a$ and $b$ are real constants. Suppose that $$ q(z, \bar{w}) + q(w, \bar{z}) \le q(z, \bar{z}) + q(w, \bar{w}) \quad \text{for ...
5
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2answers
242 views

A generalization of holomorphic functions

Assume that $U$ is an open set in the complex plane $\mathbb{C}$ and $A$ is a real $2\times 2$ matrix. We define $$\mathcal{S}_{A}=\{f:U\to \mathbb{C}\mid Df.A=A.Df \}$$ where $Df$ is the $2\...
0
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0answers
47 views

Who was first to use reproducing kernals in order to try to solve interpolation problems?

I understand that Sarason generalized the interpolation problem by taking it into the operator theoretic setting via reproducing kernels, but whose idea was it to use reproducing kernels such as the ...
0
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0answers
127 views

Localisation of the formal power series ring

Let $A \colon= K[[X_1,...,X_d]]$ be a formal power series ring of $d$-variables over a field $K$. Let ${\frak a}$ be a height $r$ prime of $A$ given by ${\frak a} \colon= (f_1,...,f_r)$, where $f_1 ...
5
votes
1answer
274 views

Confusion regarding statement of mirror symmetry for elliptic curves

I am a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
2
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0answers
132 views

Inverse limit of Noetherian rings

Let $R_i$ be a noetherian regular local ring of Krull dimension being finite. Suppose we are given a surjective homomorphism $\phi_{i,j} \colon R_{j} \twoheadrightarrow R_i$ for each $i,j$ with $j >...
3
votes
1answer
129 views

Xi Function on Critical Strip - Mellin Transform

Story I'm trying to prove following identity $$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$ where $$\psi(x)=\...
0
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0answers
68 views

On semi-complete ring K[X_1,X_2,…,X_∞]] and Popescu theorem

Let $P_n \colon= K[X_1,...,X_n]$ be a $n$-variables polynomial ring. We define 'semi-complete' polynomial ring $P_{\infty}$ by the following$\colon$ $P_{\infty} = K[X_1,...,X_\infty]] \colon = \...
1
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0answers
22 views

Generalizing an expected increase in autocorrelation near a bifurcation point to a system of ODE

Near a bifurcation point, a stochastically forced dynamical system should show an increase in autocorrelation and variance. This is due to critical slowing (a loss in resilience to perturbations). ...
1
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1answer
251 views

A Geometric proof of the Gauss Lucas theorem

Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask: Is there a geometric proof for the Gauss-Lucas theorem ?Since we are working on a half plane, can one imagine a possible ...
1
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1answer
49 views

On the limit set of eigenvalues of banded Toeplitz Hessenberg matrices

Let $T_{n}(b)$ be the $n\times n$ Toeplitz matrix determined by the symbol $$ b(z)=\frac{1}{z}+\sum_{j=0}^{k}a_{j}z^{j} $$ where $k\in\mathbb{N}$ and $a_{0},\dots,a_{k}\in\mathbb{R}$, $a_{k}\neq0$. ...
2
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0answers
122 views

Parametric normalized Fermat curves (Fermat functions)

The normalized Fermat curve is $X^n+Y^n=1$. We have of course infinite possibilities for parametrisation, but the periodicity is a special characteristic here. E.g. $\cos^2x+\sin^2x=1$ has ...
1
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1answer
145 views

Complete subring of F_p[[X]]

Pointed out on famous disbelief, I know now that there is an embedding $\iota_n \colon {\Bbb F}_p[[T_1,...,T_n]] \hookrightarrow {\Bbb F}_p[[X,Y]]$ for any $n \leq \infty$. Then I would like to ask ...
2
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1answer
291 views

Half spaces free of roots of a given polynomial

I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version This question is motivated by the following fact in complex variable:(I learned this fact from the book of ...
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0answers
110 views

Gravitational instantons metric (change variables)

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
1
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0answers
56 views

Coherence of subrings of K[[X,Y]]

Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$. Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...
2
votes
1answer
60 views

Uniform Mahler Measure Lower Bound

I'm working on a problem in multiplicative ergodic theory, and Mahler measure has just made another appearance. I am looking for a uniform lower bound on Mahler measure over all polynomials of fixed ...
3
votes
1answer
117 views

Analytic Combinatorics: upper bound for sum of absolute values of two complex functions: $|z f'(z)| + |2 f(z) - zf'(z)| \leq 2f(|z|)$

Let $f \colon \mathbb C \to \mathbb C$ be a complex-valued analytic function with non-negative coefficients of Taylor series at 0 (suppose that radius of convergence is $+\infty$ for simplicity): $$ ...
4
votes
2answers
111 views

Bieberbach-type bound for bounded univalent functions

Suppose $f: \mathbb{D}\to \mathbb{C}$ is a univalent function with $$f(z)=z+a_2z^2+a_3z^3+\cdots.$$ The Bieberbach conjecture/de Branges' theorem asserts that $|a_n|\leq n$ with equality for the Koebe ...
6
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1answer
490 views

Is this a semi algebraic set?

Put $$A=\{(a_{0},a_{1},\ldots,a_{n}) \in \mathbb{C}^{n+1}\mid p(z)=a_{0}+a_{1}z+\ldots a_{n}z^{n} \;\;\text{is a one-to one function on the unit disc} \{z\in \mathbb{C} \mid |z|\leq 1\}$$ Is $\{(...
2
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1answer
89 views

Neighborhoods with proper multiplication

The following question was originally asked here, by C. Dubussy: http://math.stackexchange.com/questions/1802111/neighbourhoods-with-proper-multiplication Assume we have two closed subsets $F$ and $G$...
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0answers
46 views

Does cutting off the taylor expansion of e^x always give an irreducible polynomial? [duplicate]

I am talking of the polynomials: $P_n(x)$ = $1+x..+x^n/n!$ I've tested this for the first 10 values and it seems so. I know this might be random but I've got a hunch that there's something deeper ...
1
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0answers
113 views

Can an algebraic function be zero both at $z=0$ and at its leading singularity?

Apologies for asking possibly strange questions, but I am just a poor computer scientist trying to understand a mathematical paper on singularity analysis of algebraic functions that is apparently not ...
6
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0answers
187 views

Is the space of holomorphic maps a manifold

To be more specific: Let $Q\subset\mathbb{C}$ be a Lipschitz bounded domain, and $V$ is a compact complex manifold without boundary. Consider the set of holomorphic maps $f:Q\rightarrow V$, and $f\in ...
4
votes
1answer
67 views

Integral Expression in Complex Dynamics

Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\...
6
votes
1answer
293 views

Is polynomial convexity a topological invariant?

Is the property of being polynomially convex a topological invariant? In other words, let $M$ and $N$ be two homeomorphic, compact subsets of $n$-dimensional complex Euclidean space, and assume in ...
3
votes
1answer
110 views

Moving from $\Re(s) = 1+\epsilon$ to $\Re(s) = \frac{1}{2}$ in the proof of the Weil-Guinand explicit formula

In all proofs of the Weil-Guinand explicit formula, there's this step (this is from Paul Garrett's notes): Now consider this: (1) $\frac{\zeta^\prime(s)}{\zeta(s)}$ has poles at $s=1$ and $s=\...
1
vote
1answer
69 views

Absolutely continuous and rectifiable boundary

Assume that $\gamma$ is a Rectifiable curve in $\mathbf{C}$ and ssume that $f$ is a bounded holomorphic function on the unit disk $U$ such that if $z_n$ converges to a boundary point of $\mathbf{U}$, ...
1
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0answers
56 views

How quickly can we mutliply Cayley-Dickson hypercomplexes?

Assuming that all of the coordinates of two Cayley-Dickson Hypercomplex numbers are non-negative integers less than a prime $p$, how quickly can we multiply these numbers? I'm also interested in what ...
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0answers
82 views

Continuation to holomorphic function

Let G be a k-fold connected riemann surface with boundary given by $k >2$ non-intersecting Jordan-curves and $\alpha : \partial G \longrightarrow S^1$ a continuous map. Now i constructed a ...
0
votes
2answers
220 views

Absolute value inequality with complex numbers

Following a problem I found on mathstack, with no solution, and no comment, so I think this inequality is not easy, so I post it here (because I think there are more some good math job, maybe someone ...
0
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0answers
37 views

Inverse Mellin transform of ratio of gamma functions

Any pointers on how to solve the inverse Mellin transform below:
2
votes
1answer
260 views

upper bound on derivatives of a function defined on an arc

This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here. You can skip examples below and read from "General setting" at ...
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0answers
98 views

Applications of Iss'sa's theorem on homomorphisms between algebras of meromorphc functions

In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved: Let $U,V \subseteq \mathbb{C}$ be open subsets. Every $\mathbb{...
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0answers
81 views

extension for a complex operator

Let be $\lambda>0$. Put $$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial \overline{z}}-z\frac{\...
0
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1answer
83 views

Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...
2
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2answers
164 views

Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...
7
votes
1answer
288 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
6
votes
3answers
214 views

Logarithms of matrices in the disk-algebra

It is easy to see that within the disk algebra $A(D)$ $$\Delta(z):= \begin{pmatrix} 1&0\\z&1 \end{pmatrix}\; \begin{pmatrix} 1&1\\0&1 \end{pmatrix}= \begin{pmatrix} 1&1\\z&1+...
2
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0answers
59 views

Functions of form $f(z)/f(z^*)$

I am doing my research in mathematical physics, and in the process I am getting functions of complex variable $z$ of form $F(z) = \frac{f(z)}{f(z^*)}$ In my case $f$ doesn't have any interesting ...
5
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0answers
152 views

Complex Stone-Weierstrass Type problem

I have come across this problem which resembles complex Stone-Weierstrass theorem except for a problem that the conjugate of the functions are not necessarily in the sub algebra. Suppose $\Omega$ is ...
2
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1answer
162 views

The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
9
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3answers
521 views

Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?

Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am ...