**2**

votes

**0**answers

32 views

### prove that $\min\{|z_{j} - w_{1}|,|z_{j} - w_{2}|\}\leq 1$ holds

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $$ (z-z_{1})(z-z_{2})+(z-z_{2})(z-z_{3})+(z-z_{3})(z-z_{1})=0$$ ...

**9**

votes

**1**answer

359 views

### Is a one-dimensional compact complex analytic space necessarily projective?

Let $X$ be a compact complex analytic space with singular locus $X^{\mathrm{sing}}$. Suppose that $X\setminus X^{\mathrm{sing}}$ is a Riemann surface. If $X^{\mathrm{sing}} = \emptyset$, then $X$ is ...

**0**

votes

**0**answers

44 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 ...

**8**

votes

**0**answers

128 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**

votes

**1**answer

86 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

**2**answers

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

**1**answer

166 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**

vote

**0**answers

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**

votes

**1**answer

86 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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

275 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**

votes

**0**answers

134 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

**1**answer

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**

votes

**0**answers

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**

vote

**0**answers

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**

vote

**1**answer

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**

vote

**1**answer

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**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 $G$...

**1**

vote

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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

**2**answers

221 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**

votes

**0**answers

37 views

### Inverse Mellin transform of ratio of gamma functions

Any pointers on how to solve the inverse Mellin transform below:

**2**

votes

**1**answer

261 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 ...

**1**

vote

**0**answers

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{...

**1**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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

**1**answer

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

**3**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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 $s:A\to{\mathbb C}$, ...