All Questions
3,561 questions
4
votes
0
answers
152
views
How to show the set of stable polynomials equals to the set of Lorentzian polynomials in degree 2
Give a homogenous polynomial $f\in \mathbb{R}[x_1,\dots,x_n]$ of degree $2$ in $n$ variables, we can consider $f$ as a quadratic form.
We call $L_n^2:=$ the set of quadratic forms with nonnegative ...
- 41
4
votes
1
answer
407
views
Solving equation of matrix valued functions
Given $n\times n$ matrices with entire functions entries (holomorphic on all of the complex plane $\mathbb{C}$)
$A(z)=[a_{ij}(z)],B(z)=[b_{ij}(z)]$,
i.e.,
$a_{ij}(z),b_{ij}(z)$ are entire functions ...
- 51
2
votes
1
answer
111
views
Quasiconformal map from a subset of $\mathbb{C}$ to a polytope
Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such
as a unit disc or rectangle) and a polytope?
Here, I take a polytope to be a two-dimensional surface that could be ...
- 547
22
votes
1
answer
3k
views
Why is Oka's coherence theorem a deep result?
This is a very naive question.
Let $X$ be a complex manifold. Let $\mathcal{O}_X$ be the structure sheaf of $X$, a sheaf of rings whose sections over opens $U\subset X$ are just the holomorphic ...
- 10.7k
2
votes
0
answers
92
views
Gysin homomorphism of an inclusion to Kähler tubular neighborhood
Let $Z\subset U$ be a Kähler tubular neighborhood of a compact manifold $Z$ of codimension $r$.
Consider de Rham complexes of smooth differential forms $\Lambda^{*,*}(Z),\Lambda^{*,*}(U)$, let $\...
- 789
5
votes
3
answers
363
views
Fully invariant measures for rational functions
Let $f(z)$ be a rational function of degree $d \geq 2$, with complex coefficients. I am interested in fully invariant measures for the dynamical system $(\mathbb C_\infty,f)$, where $\mathbb C_\infty$ ...
- 26.1k
21
votes
4
answers
3k
views
Is the Euler product formula always divergent for 0<Re(s)<1?
It is known that the Euler product formula converges for $\Re(s)>1$
(and there it represents the Riemann zeta function).
My question: Is the Euler product always divergent for
$0 < \Re(s) < ...
- 255
9
votes
0
answers
321
views
When exactly is the principal AGM equal to the optimal AGM?
Definitions
Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
- 83
1
vote
2
answers
784
views
Inverse of exponential integral function
The exponential integral function $x \mapsto E_1(x)$ is strictly decreasing on the positive real
axis and, so, is globally real analytically invertible there. Where can I find information concerning
...
- 19
6
votes
2
answers
201
views
holomorphy in infinite dimensions (holomorphic families of operators)
Let $X$ be a Banach space (over $\mathbb C$), and let $\mathcal L(X)$ be its algebra of bounded linear operators.
Let $U\subset \mathbb C^N$ be an open subset, and $f:U\to \mathcal L(X)$ a function ...
- 43.2k
8
votes
3
answers
1k
views
Behaviour at natural boundary
Suppose I have a holomorphic function $f$ in a domain $\Omega$ with natural boundary $\partial \Omega.$ Let $p \in \partial \Omega.$ Is it true that there is some analogue of Picard's little theorem - ...
- 96.4k
2
votes
0
answers
53
views
Approximating an infinite family of holomorphic functions by polynomials in relative error
I think I just proved a theorem I haven't found in the literature, and I think it must generalize. I therefore have two questions. First, if this is in the literature, what is it called? Second, what ...
- 981
5
votes
1
answer
1k
views
Mapping the doubly connected domain to an annulus
Is there a direct proof, using the Riemann mapping theorem for the Jordan domain, than every doubly connected domain in the complex plane can be mapped conformaly onto a round annulus.
- 83
5
votes
1
answer
280
views
First order PDE in complex variables?
Consider the equation
$$f'(x)+ g(x)f(x)=0$$
This equation is an ODE and has a solution $$ f(x)=C e^{ \int_1^x g(x) \ dx}.$$
Similarly, we can look at complex variables and consider the equation and ...
- 536
5
votes
0
answers
225
views
Belyi functions with prescribed image of a given point
$\newcommand{\bP}{\mathbb{P}}\newcommand{\bQ}{\mathbb{Q}}$Definition. A Belyi function is a non-constant rational function $f:\bP_{\bQ}^1\to \bP^1_{\bQ}$ such that the image of any of its critical ...
- 7,377
0
votes
1
answer
379
views
On some property of the zeros of $\zeta(s)$ in the complex plane
This property is rather elementary, and not at all specific to $\zeta$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well ...
- 3,098
3
votes
0
answers
193
views
Fermat's Last Theorem (FLT) in standard model space corresponding to an infinite Blaschke product
Let $u$ be an inner function and denote by $H^2$ the Hardy space on the open unit disc D. A model space $K_u$ associated to $u$ is a Hilbert space of the form $K_u=(uH^2)^⊥$ where ⊥ denotes the ...
3
votes
2
answers
289
views
Regularity of a conformal map
Let $D$ be a domain in $\mathbb{C}$ with $n$ boundary components. From the work of Koebe, we know that $D$ can be conformally mapped to a parallel slit domain of a specified angle of inclination (...
- 1,379
1
vote
1
answer
377
views
Holomorphic function bounded in a sector with angle $>\pi$ [closed]
I know that according to Liouville’s theorem, if a holomorphic function is bounded on all of C, it is constant. This got me thinking if I could find holomorphic non-constant functions that are bounded ...
- 11
1
vote
2
answers
106
views
'Partial boundedness' of continuously parametrised power series
Let $a_1, a_2, \ldots : D\rightarrow\mathbb{R}$ be a sequence of continuous functions, with $D$ a compact metric space.
Suppose that the function $f : D\times[0,\infty)\rightarrow\mathbb{R}$ given by
$...
- 463
14
votes
9
answers
2k
views
math circles video lectures for school children?
Hello,
I am from India. I find the mathoverflow amazing.
I have a question: Are there any good quality video lectures on school math topics?
There are a lot of high quality lectures available on ...
1
vote
0
answers
78
views
Complex manifold associated to analytic function germ
When learning about Riemann surfaces we encounter the fact that an analytic function germ on a small complex disk yields a Riemann surface via analytic continuation. It seems to me that the same ...
- 2,788
4
votes
0
answers
104
views
Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$
Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
- 41
27
votes
3
answers
948
views
A point set of power series with coefficients in {-1, 1}. Connected or not?
Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ ...
- 373
2
votes
2
answers
491
views
On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I_s$ be defined as above.
Is $I_s$ known to be convergent for any real number $s<1$ ?
user140392
19
votes
2
answers
11k
views
Meaning of $\Subset$ notation
The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...
- 1,601
4
votes
0
answers
109
views
Quasi-crystaline generalization of elliptic functions
I came across some meromorphic function, call it $f(z)$, which is "quasicrystalline" in the following sense: one can write $f$ as:
$$
f(z)=\frac{\sum_i a_i e^{i(q_{i,x}x+q_{i,y}y)}}{\sum_i ...
- 219
1
vote
0
answers
112
views
Semipositive curvature on holomorphic line bundle
Let $(X,\omega)$ be a (possibly non-Kähler) compact hermitian manifold and let $L\rightarrow X$ be a holomorphic line bundle. Is there an algebraic characterization of (Griffiths) semi-positivity of $...
4
votes
0
answers
549
views
Understanding vector-valued analytic functions vs holomorphic functional calculus
Let $A$ be a unital Banach algebra over complex numbers and call elements of $A$ "vectors". Let $\Omega$ be an open set in $\mathbb{C}$ and $H(\Omega)$ the space of analytic functions on $\...
- 83
3
votes
0
answers
875
views
Hard problems solving tricks
This question is motivated by this one that I posted on math.stackexchange.
When I fail to solve a hard math problem (like the ones I presented in the linked post), I read a solution and I noticed ...
- 161
3
votes
1
answer
294
views
Equivalent definitions of normality for complex algebraic varieties
In Kollár's article The structure of algebraic threefolds: an introduction to Mori's program he gives the following definition of a normal variety:
Definition 5.4. Let $V \subset \mathbb{C}^n$ be an ...
11
votes
1
answer
490
views
turn $\pi/n$, move $1/n$ forward
start at the origin, first step number is 1.
turn $\pi/n$
move $1/n$ units forward
Angles are cumulative, so this procedure is equivalent (finitely)
to
$$
u(k):=\sum_{n=1}^{k} \frac{\exp(\pi i H_{n}...
- 975
16
votes
2
answers
1k
views
Teaching Steenrod Operations
I am teaching a class on topology and want to introduce Steenrod Operations. I have talked about simplicial sets and classifying spaces of groups but have not talked about Eilenberg–MacLane spaces. ...
- 161
31
votes
3
answers
2k
views
Polynomials with the same values set on the unit circle
Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...
- 109k
2
votes
0
answers
359
views
Triangulating Riemann surfaces by using non-constant meromorphic functions
Let $X$ be a connected Riemann surface, i.e. $X$ is a one dimensional connected complex manifold (Hausdorff and second-countable as a topological space). The following is a classical result:
Theorem (...
- 395
4
votes
1
answer
746
views
Fun examples relating to Hopf surfaces
A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following ...
- 1,379
16
votes
1
answer
2k
views
A conjecture in which both "if" and "only if" are near misses
[Migrated from Math Stack Exchange]
More than a year ago, I posted the following on the Math Stack Exchange.
Consider $2^n-1$. Based on checking a few small numbers for $n$ (in
fact, the first ...
- 2,437
18
votes
14
answers
3k
views
Teaching a pedagogy course
At my institution incoming graduate students must take a semester long course on pedagogy taught by current grad students. I may soon be in the position of having to teach this course and I'm looking ...
4
votes
2
answers
332
views
Exponential iterates of a complex number
Let $f:\mathbb C\to \mathbb C$ be defined by $f(z)=e^z-1$. Let $f^n$ denote the $n$-fold composition of $f$.
In my new paper Erdős space in Julia sets I show that $$Z:=\{z\in \mathbb C:\lvert\...
- 3,317
3
votes
3
answers
257
views
Computing the maximum modulus
For each $a\in \mathbb C$ define $f_a:\mathbb C\to \mathbb C$ by $f_a(z)=\exp(z)+a$. I am primarily interested in real values $a\in (-\infty,-1)$.
For each $r\in [0,\infty)$ define $M_a(r)=\max\{|f_a(...
- 3,317
4
votes
3
answers
600
views
Meaning of divergent integrals
In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. ...
5
votes
2
answers
850
views
Local phase statistics of the nontrivial Riemann zeros
(The question is inspired by Owen Maresh's post)
The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$.
Numerical results on the first 10000 zeros suggest ...
- 9,501
6
votes
3
answers
933
views
Any closed form for series like $F(x)=\sum\limits_{p=2}^{\infty}x^p,$ where $p$ is prime?
Any closed form for series like $$F(x)=\sum_{p=2}^{\infty}x^p,\quad p\text{ is prime}$$ or $$F(x)=\sum_{i=0}^{\infty}x^{i!}\quad ?$$
More generally, we can obtain a power series from decimal expansion ...
- 3,104
3
votes
0
answers
320
views
is $\sum_{i=1}^{n}\sum_{j=1}^{n}|a_{i}+a_{j}|\cdot p_{i}p_{j}\ge \sum_{i=1}^{n}p_{i}|a_{i}|$?
see:old post
and Make the integral discrete, we have
conjecture :for any complex numbers $a_{i},i=1,2,\cdots,n$,and $p_{i}\ge 0$ such $p_{1}+p_{2}+\cdots+p_{n}=1$,then maybe be have
$$\sum_{i=1}^{n}\...
- 4,280
20
votes
2
answers
2k
views
Bitcoin Research
I have recently been assigned to advise a student on a senior thesis. She has taken linear algebra, introductory real analysis, and abstract algebra. Her interest is in cryptography. And she has a ...
- 822
1
vote
1
answer
133
views
A subharmonic function with a growth property
Let $B=\left\{ \left(x,y\right)\in\mathbb{R}^{2}:x^{2}+y^{2}<1\right\} $
be the unit ball in $\mathbb{R}^{2}.$
Can we construct a subharmonic
function $f:B\rightarrow\left[-\infty,0\right]$ such ...
5
votes
0
answers
186
views
Examples of partial adjoints
Recall that a functor $$R: D \to C$$ is said to have a partial left adjoint $L$ defined at an object $X \in C$ if the functor
$$D \to Sets, Y \mapsto Hom_C(X, R(Y))$$
is corepresentable by some object ...
- 2,040
6
votes
1
answer
289
views
Bounded non-symmetric domains covering a compact manifold
This question is somewhat related to this other question of mine.
I was wondering which are the known examples of bounded domains $\Omega$ in $\mathbb C^n$ admitting a compact free quotient.
By a ...
- 7,902
1
vote
0
answers
381
views
Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$
I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...
- 3,098
1
vote
0
answers
144
views
Zeroes of Mellin transform
There exist a "standard" or canonical way to construct a real valued function whose Mellin transform has a prescribed set of zeroes? Clearly for some set of zeroes this could be impossible ...
- 131