All Questions
3,561 questions
12
votes
9
answers
6k
views
Topics for an Undergraduate Expository Paper in Number Theory
I am teaching an undergraduate course in number theory and am looking for topics that students could take on to write an expository paper (~10 pages). No new results are expected of them. Many of the ...
3
votes
1
answer
194
views
Proper analytic embedding of $\overline{\Bbb C}$ minus a Cantor set into $\Bbb C^2$
I am a PhD student in several complex variables.
I am reading this paper by Orevkov proving that there exists a proper analytic embedding of $\overline{\Bbb C}$ minus a Cantor set into $\Bbb C^2$.
I ...
- 779
2
votes
1
answer
153
views
Regarding upper semicontinuity of a function
Let $E$ be a linear subspace of $\mathbb{C}^{n\times n}$.
Define the function $\mu_E:\mathbb{C}^{n\times n}\longrightarrow \mathbb{R}_+$ as
$$
\mu_E(A)=\frac{1}{\inf\{\|X\|:X\in E\text{ and }\det(I_n-...
- 31
8
votes
5
answers
6k
views
Advantages of the sequence definition of limits
I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of ...
4
votes
1
answer
211
views
Rate of convergence of Padé approximants
Let $f$ be an entire function of order $1$. Two questions:
1) Can one assert that the diagonal Padé approximants converge to $f$ (pointwise or uniformly over compacts of $\mathbb C$)?
2) if yes, can ...
- 3,998
4
votes
1
answer
271
views
On the roots of Bernoulli polynomials
Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It ...
- 43.2k
4
votes
1
answer
247
views
Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part?
Plotting $t\mapsto\zeta(1/2+it)$ on Wolfram alpha, it seems that the maxima of its real part are close to the zeros of its imaginary part, while the maxima of the latter seem close to the inflection ...
- 6,982
19
votes
6
answers
6k
views
an engineering Ph.D. teaching math in college
I have a friend who has been teaching college-level math (e.g., all levels of calculus)
for about 4 years, although all of his education, including his Ph.D., was in engineering.
Now he is ...
3
votes
0
answers
167
views
Does the maximum principle hold in this pluriharmonic setting?
Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and ...
- 5,215
0
votes
1
answer
75
views
Factorization of a polynomial involving cosine into $m$ second-order factors [closed]
For each $m\in\mathbb{N}$ and fixed $a>0,\theta\in\mathbb{R}$, I want to factorizate the polynomial $p_m(x) = x^{2m} - 2a^m\cos (m\theta)x^m + a^{2m}$ into $m$ polynomials of second order. Using ...
- 383
4
votes
9
answers
4k
views
Functions of one complex variable: geometric theory
Can someone recommend a good textbook on functions of one complex variable which have good chapters on geometric theory, in English?
When I studied complex analysis, I used two
textbooks:
An ...
- 371
1
vote
1
answer
162
views
Commuting matrices of complex functions
If $A(z) :=[A_{ij}(z)] $ and $B(z) :=[B_{ij}(z)] $ are two invertible $n\times n$ matrices of entire complex valued functions entries $A_{ij}(z)$, and $B_{ij}(z) $ with
(1). $AA^{\#}=A^{\#}A$ ...
- 39
4
votes
1
answer
1k
views
seeking proofs: infinite series inequalities
Question. Numerically, the following is convincing. However, is there a proof?
$$\left(\sum_{k\geq1}\frac1{\sqrt{2^k+3^k}}\right)^4
<\pi^2\left(\sum_{k\geq1}\frac1{2^k+3^k}\right)\left(\sum_{k\...
- 43.2k
7
votes
3
answers
3k
views
Roots of a polynomial inside the unit circle
Let $k$ be a even positive integer. Now, consider the polynomial
$$
p(x)=x^k-px^{k-1}-qx^{k-2}-x^{k-3}-\cdots -x-1,
$$
with $p$ and $q$ integers satisfying $q-1>p\geq 1$.
How to prove that this ...
- 515
22
votes
2
answers
3k
views
Reason for studying coherent sheaves on complex manifolds.
Hello everybody! I would be interested in knowing, what the reason is for investigating coherent sheaves on complex manifolds. By definition a sheaf $F$ on a complex manifold $X$ is coherent, when it ...
- 583
7
votes
2
answers
661
views
Locus of roots of all convex combinations of two monic polynomials
Let $p,q$ be monic polynomials in $\mathbb{C}[t]$ and for $\alpha \in [0,1]$, let $c_\alpha := \alpha p + (1-\alpha)q \in \mathbb{C}[t]$. Since the roots of a polynomial vary continuously with respect ...
- 1,719
5
votes
0
answers
188
views
Proof of Tian's constant
Basically Tian's invariant relies fundamentally on a lemma of Homander which says that $e^{- \phi}$ is integrable for $\phi$ plurisubharmonic on a ball of radius $1$ under some extra assumption. I am ...
- 51
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
24
votes
3
answers
9k
views
Functions of several complex variables: book recommendations?
Can anyone recommend a good comprehensive introduction to functions of several complex variables that a) is fairly up to date, b) isn't a geometry or an algebra book only, but takes multiple ...
13
votes
2
answers
539
views
$f$ real-rooted forbid truncated $\frac1f$ to be so?
Let $f(x)$ be a polynomial in the ring $\mathbb{R}[x]$, the roots are all real and $f(0)=1$. Write the Taylor series of $1/f(x)$ around the origin as
$$\frac1{f(x)}=\sum_{k=0}^{\infty}a_kx^k,$$
and ...
- 43.2k
6
votes
3
answers
618
views
Does the proof of Picard's theorem become simpler by increasing the number of points that are not attained?
Let $f$ be an entire analytic function which attains all but $k$ complex numbers $z_1,\ldots,z_k$. Is there any elementary proof, for some $k$, that $f$ is constant?
- 761
11
votes
1
answer
602
views
How do analysts think about functions with poles at all roots of unity?
In branches of algebra impinging on the enumeration of partitions, one often encounters formulas like
$$\prod_i \left( \frac{1}{1-q^i} \right)^{n_i}$$
for some integers $n_i$. E.g., with $n_i = 1$, ...
- 8,733
2
votes
1
answer
272
views
Subharmonic in any holomorphic coordinates = Plurisubharmonic?
An upper semi-continuous function $u : \Omega \to \mathbb{R}$, $\Omega \subseteq \mathbb{C}^n$ is said to be subharmonic if it satisfies the submean inequality $u(a) \leq \mu_S(u;a,r)$, where $\mu_S(...
- 1,457
7
votes
1
answer
193
views
Greatest lower bound for subordination
Consider the set $X$ of all analytic functions $f$ in the unit disk $U$ satisfying
$f(0)=0, f'(0)\neq 0$. We say that $f\prec g$ if there exists
$\phi\in X$ which maps $U$ into itself, and $f=g\circ\...
- 91.8k
11
votes
1
answer
1k
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 ...
- 1,141
1
vote
0
answers
55
views
What are we to deduce from a structure theorem of this type concerning totally geodesic maps?
I apologise in advance for the vague nature of the question, but some insight would be greatly appreciated.
I'm reading a paper of Lei Ni concerning structure theorems for Kähler manifolds. Here is an ...
- 651
22
votes
2
answers
2k
views
Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?
QUESTION
I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex logarithm is ...
- 7,500
9
votes
1
answer
413
views
Existence of Laurent series with zeroes at $e^{2n}$ ($n \in \Bbb{N}_0$) and extremely fast coefficient decay
I am working on a problem in harmonic analysis, which I converted into the following existence problem concerning Laurent series. I am a bit at a loss concerning this problem, since my knowledge of ...
- 734
18
votes
2
answers
2k
views
Homotopy types of schemes
Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...
- 15.5k
4
votes
2
answers
439
views
Simple Closed Hyperbolic Geodesics on Punctured Spheres
Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...
- 1,761
5
votes
1
answer
780
views
Do real analytic functions on $\mathbb{C}\mathbb{P}^n$ form a Noetherian ring?
Question: Is the ring of real-analytic functions on $\mathbb{C}\mathbb{P}^n$ (real valued)
a Noetherian ring?
References or counterexamples are welcome.
I know that the ring of germs of holomorphic ...
- 589
5
votes
1
answer
236
views
Which plane curves can be harmonically parametrized?
In this question, a “(closed oriented plane) curve” $\Gamma$ will mean a continuous map $f \colon \mathbb{U} \to \mathbb{C}$ where $\mathbb{U} := \{z\in\mathbb{C} : |z|=1\}$ is the unit circle, modulo ...
- 32.5k
4
votes
1
answer
535
views
Real and imaginary parts of $\ln \Gamma(i b)$
The imaginary part of the digamma function when its argument is pure imaginary is known as
$$\Im\psi(\mathrm{i}b)=\frac{1}{2}b^{-1}+\frac{1}{2}\pi\coth{\pi b},$$ and its real part is much more ...
user147095
6
votes
1
answer
261
views
The state of art of the singular Levi problem -- and hyperkähler quotients
One of the versions of the classical Levi problem asks the following:
Let $X$ be a complex manifold. Is it true that $X$ is Stein iff
$X$ admits a smooth exhaustion strictly plurisubharmonic ...
- 2,305
16
votes
1
answer
1k
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 ...
user94803
27
votes
5
answers
5k
views
Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties
This really is two questions, but they are kind of related so I would like to ask them at the same time.
Question 1:
In a question asked by Amitesh Datta, BCnrd commented that it is important to ...
4
votes
2
answers
397
views
Sums of entire surjective functions
Suppose $(f_n)_n$ is a countable family of entire, surjective functions, each $f_n:\mathbb{C}\to\mathbb{C}$. Can one always find complex scalars $(a_n)_n$, not all zero, such that $\sum_{n=1}^{\...
- 51
3
votes
0
answers
266
views
Does the minimal surface system in the plane have the weak unique continuation property?
Let $\Omega \subset \mathbf{R}^2$ be a domain in the plane and suppose that $u : \Omega \to \mathbf{R}^k$ is a smooth function for which the graph of $u$ is a smooth minimal surface in $\Omega \times \...
- 1,179
0
votes
1
answer
346
views
Variance of spectral density is related to the gradient of signal?
Define the frequency variance as:
$$ \sigma^2 = \int^\infty_{-\infty}\omega^2 P(\omega) d\omega$$
Where $P(\omega)$ is the spectral density function, which is the same as normalized power. Therefore,
$...
- 433
1
vote
0
answers
74
views
Is $A^{-\infty}$ a $Q$-algebra?
Let $\mathbb{D}=\{z\in \mathbb{C}: |z|<1\}$ and
$$
A^{-\infty}=\Big\{f:\mathbb{D}\rightarrow \mathbb{C}\;|\; \exists n\in \mathbb{N} \textrm{ such that }\|f\|_{-n}= \sup\limits_{z\in \mathbb{D}}\;\!...
- 21
-1
votes
1
answer
87
views
Inferring polynomial rate of convergence from polynomial bound
Let $x_n$ be a non-negative valued sequence and suppose that the following hold:
$\lim\limits_{n\to\infty} x_n =0$
There exists some polynomial function $p$ of degree at-least $1$ such that:
$$
\|x_n\...
- 5,405
3
votes
1
answer
107
views
Best bound of complex Hilbert transform
It is well-known (see Grafakos' Classical Fourier Analysis, Exercise 5.1.12) that if $f$ is a real valued $L^p(\mathbb R)$ function and $1<p<2$ , then we have the following inequality:
$$
\|Hf\|...
- 751
2
votes
0
answers
327
views
Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide
There are two definitions of intersection multiplicity of two complex algebraic plane curves. One of them is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t) )$ be ...
- 1,168
1
vote
0
answers
47
views
Uniformization of triangulation on a sphere up to Moebius transformations
This is not the most precise question but rather a hope that someone has seen something like this.
I am given a triangulation of the 2-sphere $S^2$ which I only know up to Moebius transformations. I ...
- 41
2
votes
1
answer
202
views
Can an "annular" subset of an annulus be conformally equivalent to the whole annulus?
Assume we are given an annulus
$$A = \{ z \in \mathbb{C}: 1< |z| < R\}.$$
Let $\phi\colon A \to A$ be a univalent map such that the image of $\phi$ contains a curve around the unit disk. Does ...
14
votes
4
answers
3k
views
A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst
I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability.
A quick google search gave a lot of references on SLE ...
- 2,154
6
votes
0
answers
283
views
Interesting things you learned while grading/marking? [closed]
What are some interesting mathematical things you have learned while grading (or marking, if you prefer) student work? For example, clever proofs that students came up with; nice counterexamples or ...
1
vote
0
answers
927
views
canonical divisor on singular curves with nodal point
What's the definition of canonical divisor(or whatever related concept) on singular curve with nodal point. More generally, what the definition of canonical divisor on a singular variety X, which is ...
- 623
1
vote
2
answers
353
views
Non-self-intersecting paths on $\mathbb{C}\setminus\{0,1\}$ [closed]
Let us make two small holes around points $0$ and $1$ on the complex plane and consider non-self-intersecting paths that start on the boundary of one hole and finish at the boundary of the another. It ...
- 29
9
votes
0
answers
891
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.9k