All Questions
372 questions
21
votes
2
answers
2k
views
Real rootedness of a polynomial
Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $...
- 1,889
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
21
votes
5
answers
7k
views
References for complex analytic geometry?
I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc....
20
votes
1
answer
1k
views
Provable zero-free region for any entire function that analytically is similar to zeta(s)
Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$:
$f(z)$ is bounded when $\Re z>1+\delta$
$f(z)$ is unbounded when $\Re z=1$
$f(z)$ grows polynomially ...
- 1,243
18
votes
2
answers
984
views
A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?
In a 1986 paper, Harer and Zagier proved the recursion:
$$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$
where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
- 1,303
18
votes
1
answer
830
views
Cohomology of real analytic coherent sheaves
Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$
is both "...
- 9,237
18
votes
3
answers
2k
views
If a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence?
Let $ P(z) $ be a $\textit{formal}$ power series in $z$ that a priori may not have a non zero radius of convergence. Assume that $P(0) =0$.
Let $\Phi(w,z)$ be a polynomial in two variables, that ...
- 3,245
17
votes
2
answers
2k
views
Is this equivalent to RH - Riemann hypothesis?
$$\pi = 3\prod_{\zeta(1/2+it) = 0}\frac{9+4t^2}{1+4t^2}\iff\text{RH is true}.$$
- 1,305
17
votes
17
answers
3k
views
Readings for an honors liberal art math course
Our university has an Honors section of our "liberal arts mathematics" course. Typically 10-20 students enroll each Fall, with most of them extremely bright, but lacking the interest and/or ...
17
votes
4
answers
10k
views
Analytic implicit function theorem
I'm looking for a proof of the analytic implicit function theorem (IFT). The only related proof I could find was the holomorphic inverse function theorem (by Henri Cartan). On Wikipedia, the analytic ...
- 183
16
votes
2
answers
1k
views
Maximum of a function of one variable
Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs
of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie ...
- 91.8k
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 151k
16
votes
2
answers
1k
views
New series for $\pi$ from string theory
This is a direct followup to the post Possible new series for $\pi$ by Timothy Chow
and its numerous answers and comments.
Using another formula in the same string theory paper by Saha and Sinha one ...
- 13.1k
16
votes
5
answers
1k
views
Permission to use Online Notes
I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by ...
16
votes
0
answers
531
views
Aligned roots of irreducible polynomials
It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
- 13.4k
15
votes
7
answers
6k
views
Freshman's definition of sin(x)?
I would like to know how you would rigorously introduce the trigonometric functions ($\sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ ...
- 23.4k
14
votes
3
answers
778
views
Is analytic Quillen-Suslin simple?
This question is motivated by a sentence on the Wikipedia entry for Quillen-Suslin theorem. This theorem states that every algebraic vector bundle on affine space is trivial. The analogous result is ...
- 156k
14
votes
1
answer
3k
views
How is the "conformal prediction" conformal?
The question is clarified by Prof.V.Vovk. See his answer below for discussion.
Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
- 8,071
14
votes
1
answer
463
views
Obstruction to the existence of global resolution of coherent sheaf
It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-...
- 912
14
votes
2
answers
2k
views
Determining rational functions by their critical points
Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...
- 1,199
13
votes
2
answers
801
views
For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
Can we characterise the set of rationals $x$ for which the sum
$$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$
remains bounded ...
- 6,323
13
votes
1
answer
682
views
How can one "see" the Hopf fibration in the space of lattices in the plane?
This question is inspired from Etienne Ghys's talk on Knots and Dynamics from ICM 2006.
The map $L \mapsto (G_4(L), G_6(L))$ gives a bijection between all lattices $L\subset \mathbb{C}$ (including ...
- 1,821
13
votes
1
answer
453
views
$\pm1$-polynomials with a maximal non-real root
For given $n$, consider a polynomial $\sum_{k=0}^na_kz^k$ with all coefficients $a_k\in\{\pm1\}$. I am interested in the following:
How big can the modulus of a non-real root of such a polynomial ...
- 13.4k
13
votes
2
answers
1k
views
Is the exponential function the sole solution to these equations?
Let us take the exponential function $\lambda^z$ where $0 < \lambda < 1$. There are many great uniqueness conditions this holomorphic function satisfies. For example, it is the only function ...
user78249
13
votes
0
answers
385
views
Are the zeros of Tutte polynomials dense in $\mathbb C^2$?
For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\...
- 85.6k
13
votes
3
answers
1k
views
Do contact and CR structures have corresponding $G$-structures?
For an $n$-dimensional manifold $M$, almost complex and almost symplectic structures on $M$ correspond to reductions on the structure group of the tangent bundle, introducing a $\operatorname{GL}(n/2,\...
- 131
13
votes
1
answer
2k
views
Surgery in complex geometry
I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something ...
- 4,343
13
votes
1
answer
3k
views
When are complex polynomial maps almost surjective?
Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
- 133
13
votes
3
answers
1k
views
Is anything known about the series $\sum_{n=0}^{\infty} x^{\sqrt{n}} $?
It's well known that there are a shocking number of identities for the usual Jacobi theta function $$ \theta_3(x) = \sum_{n=-\infty}^{\infty} x^{n^2}. $$
So I wanted to turn my attention to slowly ...
- 2,689
13
votes
7
answers
35k
views
Real analysis has no applications?
I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications ...
13
votes
4
answers
2k
views
Complex evaluation of a classical (real) integral
There are several ways to compute the classical integral
$$
\int_{\mathbb R}e^{-x^2}dx=\sqrt{\pi}.
$$
Probably, best known are
(1) squaring the integral with subsequent change
of (now two) variables ...
- 13.5k
12
votes
1
answer
2k
views
When does continuity imply holomorphy?
I was studying the construction of the modular lambda function and I started thinking about the following question. Suppose that $\Omega\subset \mathbb{C}$ is an open connected set and $f:\Omega\to \...
- 255
12
votes
1
answer
1k
views
How to best distribute points on two concentric circles?
An N-subset $\{x_1,\dots,x_N\}$ of a compact set $X\subset \mathbb R^d$ is called a set of Fekete points (named after Michael Fekete) if it maximizes the product $$\prod_{1\le k<j\le N}|x_k-x_j|\...
12
votes
1
answer
528
views
Are the fibers of a surjective holomorphic submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?
Are the fibers of a surjective holomorphic submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?
For $n=1$ this means that a surjective entire function $\mathbb{C}\to\mathbb{C}$ without critical ...
- 173
12
votes
1
answer
735
views
Parametrisations for null temperature functions: nonuniqueness of solutions to the heat equation
Disclaimer. I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition....
- 1,990
12
votes
2
answers
489
views
Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$?
Let $f(z)$ be an entire holomorphic function in $\mathbb{C}$, and consider the real-valued function
$$g_f(z)=\frac{|f'(z)|}{1+|f(z)|^2}.$$
If $f(z)$ is a polynomial, then it is easy to prove that $\...
- 1,350
12
votes
2
answers
2k
views
Universal covering of a 2-sphere without $n$ points
Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic ...
- 21.8k
12
votes
4
answers
614
views
Why does the parameterization (F:F':1) happen?
1) To parameterize the conic $x^2+y^2=1$, we can use $(x,y)=(\sin t,\sin't)$ ($\sin'$ meaning the derivative of $\sin$, namely $\cos$).
2) To parameterize an elliptic curve $y^2=4x^3-g_2x-g_3$, we ...
- 18.7k
12
votes
2
answers
851
views
Visualizing holomorphic differentials on a compact Riemann surface?
It is a classical result that the vector space of holomorphic differentials on a compact Riemann surface of genus $g$ has dimension $g$. I am wondering if there is a way of visualizing this wonderful ...
- 82.7k
12
votes
1
answer
858
views
Is this function concave?
Let
$$h(u):=u^3 \left|\int_u^\infty \frac{e^{-i t}}{t^3} \, dt\right|$$
for $u>0$. Is the function $h$ concave on $(0,\infty)$?
(For context, see Proposition 4.4.4 and formula (4.4.21) in this ...
- 128k
11
votes
6
answers
2k
views
Hard problems with an easy-to-understand answer
I am very interested by problem in mathematics which are difficult (go at least 10 years without a resolution, say) but which have a solution that is short and elementary.
In this video Launay gave an ...
11
votes
4
answers
2k
views
Routh-Hurwitz for eigenvalues
The Routh-Hurwitz criterion provides a convenient test, even for hand calculation, of whether a polynomial with real coefficients has all its roots in the left half plane. I'm wondering about a ...
- 111
11
votes
3
answers
748
views
Explicit triples of isomorphic Riemann surfaces
Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows.
A compact Riemann surface can be presented in many different ways....
11
votes
5
answers
4k
views
Applications of Liouville's theorem
I'm looking for "nice" applications of Liouville's theorem (every bounded entire map is constant) outside the area of complex analysis.
An example of what I'm not looking for : a non-constant entire ...
11
votes
1
answer
3k
views
When are entire functions surjective?
Is there some useful criterion to determine whether or not an entire function is surjective?
- 211
11
votes
0
answers
201
views
Holomorphically convex manifolds and Bergman complete manifolds
Suppose $X$ is a complex manifold which admits the Bergman metric (for definitions, see for instance Kobayashi's book "Hyperbolic Complex Spaces"). Suppose moreover that the Bergman metric of $X$ is ...
- 7,902
11
votes
6
answers
6k
views
Why $\partial$ and $\bar{\partial}$ defined in that way (the Wirtinger derivatives)?
For $\mathbb{C}$-valued functions, why are $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$ defined as
$$
\frac{\partial}{\partial z}=
\frac{1}{2}\left(
\frac{\partial}{\...
- 1,111
11
votes
4
answers
3k
views
Riemann mapping theorem and smoothness on the boundary
Let $U\subset \mathbb C$ be open, bounded, simply connected, with $C^\infty$ boundary. Apply the Riemann mapping theorem to get a bilolomorphic isomorphism
$$
f:U\to \mathbb D
$$
between $U$ and the ...
- 43.2k
11
votes
1
answer
860
views
Is every surjective holomorphic self-map on a compact complex manifold finite-to-one?
I have already asked this question on stack exchange, but I didn’t get any answer.
Let $X$ be a compact connected complex manifold.
Let $f:X \to X$ be a surjective holomorphic map. Is it true that $f$...
- 337
11
votes
2
answers
1k
views
Jacobi's elliptic functions and plane sections of a torus
In $\mathbb R^3$ with Cartesian coordinates $(x,y,z),$ revolve the circle $(x-\sqrt 2)^2+z^2 =1,\ y=0$ about the $z$-axis.
This yields a torus embedded in $3$-space that is conformally equivalent to ...
