All Questions
3,560 questions
0
votes
1
answer
2k
views
Dual of Zorn's Lemma? [closed]
It seems to me that the dual of Zorn's Lemma should be true: if $S$ is a non-empty partially ordered set and every chain of $S$ has a lower bound in $S$, then $S$ has at least one minimal element.
...
- 1
0
votes
3
answers
2k
views
Higher direct image of coherent sheaf
Hi.
Can any one me say if there is a simple proof of this claim which i can prove it by localization and no easy technique of nuclear spaces...
Let $f:X\rightarrow S$ be an open, surjective ...
- 435
3
votes
1
answer
2k
views
Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)
The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao....
- 1,795
3
votes
1
answer
675
views
Relation between entire function of exponential type and exponential polynomials
Is it true in general that the theory of entire function of exponential type and and that of exponential polynomials (with purely imaginary exponents) are analogous ?
Can one derive results about ...
- 1,795
2
votes
1
answer
286
views
Linear independence in the algebraic closure of $\mathbb{C}(z)$
Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.)
Define $w_i=(\...
- 454
2
votes
3
answers
632
views
How to find the almost period of an exponential polynomial
Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...
- 1,795
129
votes
2
answers
16k
views
What are the shapes of rational functions?
I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
- 25.1k
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 ...
38
votes
2
answers
13k
views
What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
- 1,990
3
votes
0
answers
649
views
Stability by flat base change of certain properties
Hi.
Let $f:X\rightarrow S$ be a surjective proper, open morphism of reduced or without embedded component complex spaces (or, in alg.geom, surjective proper, universally open morphism of excellent, ...
- 435
4
votes
1
answer
6k
views
Inverse of a function defined by an integral
Hi, I have a function defined by an integral as follows.
$$
z=f(w) = \int_0^w \frac{(\zeta-a_1)^{\alpha_1}(\zeta-a_2)^{\alpha_2}...}{(\zeta-b_1)^{\beta_1}(\zeta-b_2)^{\beta_2}...}\ d\zeta
$$
where $w$ ...
- 167
4
votes
0
answers
312
views
Transforming a multivariable integral to make it separable
In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine ...
- 37.7k
3
votes
0
answers
259
views
Quotient of manifolds by groups and embeddings
Let $f:X_1\to X_2$ be a closed submanifold. Let $\rho:G_1\to G_2$ be a closed Lie subgroup. Let $G_1$ acts on $X_1$ and $G_2$ on $X_2$ and suppose $f$ is $\rho$-equivariant. I would like to get a ...
- 411
5
votes
2
answers
904
views
Are there compact analogues of Cartan's theorems A and B?
Cartan's theorem A says that on for a coherent sheaf ${\mathcal{F}}$ on a Stein manifold X, the fibres ${\mathcal{F}}_x$ over each point x in X are generated by global sections.
I'm wondering if ...
user2529
2
votes
1
answer
2k
views
The normal derivative of the Green's function
I was wondering if anything was known about the following:
Let $\mathbb{D}^2=\lbrace x^2+y^2< 1 \rbrace \subset \mathbb{R}^2$ be the open unit disk.
Consider now the Green's functions $G(z; p)$ ...
- 2,299
4
votes
1
answer
822
views
Converse of Picard's Big Theorem?
The celebrated Big Theorem of Picard's is that, in every open set containing an essential singularity of a function $f(z)$, $f(z)$ takes on every value (except for at most one) of $\mathbb{C}$ ...
- 2,019
10
votes
8
answers
2k
views
Undergraduate Probability Topics
I am teaching undergraduate probability this semester, and I am looking for some suggestions about inspiring applications that could be reasonably covered over the course of two one-hour lectures or ...
0
votes
1
answer
261
views
Flat locus of $S_{1}$-morphism
Hi, everybody.
Consider an ${\rm S}_{1}$- morphism $f:X\rightarrow S$ of reduced complex spaces. Assume that $f$ is open (universally open in Alg.geom), equidimensional with $n$-pure dimensional ...
- 435
7
votes
2
answers
1k
views
Contour integration problem from probability
Can integrals of the form
$$
\int_{-\infty}^{\infty}{\exp\left(-\left[x - c\right]^{2}\right) \over 1 + x^{2}}\, {\rm d}x
$$
be computed in closed form using contour integration (or any other ...
- 5,227
9
votes
4
answers
2k
views
Applications of Math: Theory vs. Practice
I have a problem: I learned about a lot of the applications of mathematics from academics. Neither they nor I have had much contact with the "real world" to go and see for ourselves how mathematics ...
3
votes
2
answers
459
views
Infinite-dimensional complex polynomial or rational Lie algebras and their pseudogroups
In studying the transformation groups generated by holomorphic vector fields V(z) d/dz on ℂ, I've noticed the (surely well-known) fact that the complex quadratic vector fields:
&...
- 2,858
17
votes
2
answers
2k
views
Getting a differential equation for a function from a functional equation of its Mellin transform
If $f$ is a locally integrable function then its Mellin transform
$\mathcal{M}[f]$ is defined by
$$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$
This integral usually converges in a ...
- 1,412
5
votes
3
answers
942
views
Square of an elliptic curve and projective plane
Let's assume one takes $E = \mathbb{C}^* / \langle p \rangle$ an elliptic (Tate) curve over the complex field ($p = e^{2 \pi i \tau}$ where $1, \tau$ are the 2 periods in additive notation; $\Im \tau &...
- 73
32
votes
20
answers
6k
views
What are your favorite puzzles/toys for introducing new mathematical concepts to students?
We all know that the Rubik's Cube provides a nice concrete introduction to group theory. I'm wondering what other similar gadgets are out there that you've found useful for introducing new math to ...
2
votes
2
answers
356
views
Coefficients of holomorphic functions defined by Borel probability measures on the unit disc
Let be $\mathcal M(\partial\mathbb D)$ denote the set of all Borel complex probability measures on $\partial\mathbb D$ (unit circle in the complex plane). Define a mapping $\Phi:\mathcal M(\partial\...
- 2,044
3
votes
2
answers
957
views
Simple definition of the Hausdorff measure using squared paper
I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.
For simplicity, I was hoping to give a more intuitive ...
- 20.2k
5
votes
4
answers
3k
views
Minimizing the modulus of a polynomial around a circle
I'm probably missing something elementary here, but I guess the only way to be sure is to ask here.
Now, I have encountered a situation where given an nth-degree polynomial $p_n(z)$ with complex ...
35
votes
19
answers
9k
views
Interesting applications (in pure mathematics) of first-year calculus
What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...
4
votes
3
answers
2k
views
Coprimality and squarefree numbers
As observed on Mathworld, "Amazingly, the probabilities for random pairs of integers and Gaussian integers being relatively prime are the same as the asymptotic densities of squarefree integers of ...
- 3,612
28
votes
5
answers
3k
views
Continuous + holomorphic on a dense open => holomorphic?
Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halfs $D_1$ and $D_2$.
Let ...
- 43.2k
24
votes
7
answers
8k
views
How do professional mathematicians learn new things? [closed]
How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues?
52
votes
9
answers
26k
views
Is Galois theory necessary (in a basic graduate algebra course)?
By definition, a basic graduate algebra course in a U.S. (or similar) university with
a Ph.D. program in mathematics lasts part or all of an academic year and is taken
by first (sometimes second) ...
1
vote
2
answers
541
views
Weierstrass Theorem [closed]
Hi--
Where can I find a proof of this theorem: For each $r \in \mathbb{Z}_{+}$,
there exists a complex entire function $f(z)$ such that $f(r) \neq 0$ but
$f(r+1)=f(r+2)=\cdots =0$,
i.e. $f(z) \in ...
- 4,795
18
votes
5
answers
2k
views
What is the spectrum of the ring of entire functions?
Let $\mathcal{O}(\mathbb{C})$ be the ring of entire functions, that is, those functions $f : \mathbb{C} \to \mathbb{C}$ which are holomorphic for all $z \in \mathbb{C}.$ For each $z_0 \in \mathbb{C}$.
...
- 3,268
51
votes
22
answers
19k
views
Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...
11
votes
0
answers
528
views
Adeles of Holomorphic Functions
In number theory, an adele is a restricted product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...
- 15.4k
0
votes
1
answer
652
views
Fiberwise torsion free and generically null sheaf for flat morphism
Hi.
Has some one an example of sheaf $A$ on flat morphism $f:X\rightarrow S$ of reduced complex spaces with fibers of constant positive dimension (or locally noetherian excellent schemes without ...
- 435
17
votes
3
answers
2k
views
What is a reasonable finitary analogue of the statement that harmonic functions are smooth?
In my answer to this question on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way ...
- 118k
3
votes
1
answer
575
views
Kahler forms on Cohen Macaulay spaces
Hi.
Can anyone answer the two following questions:
For $n$-dimensional $X$ Cohen-Macaulay complex space, is it true that the sheaf of top degree homolorphic forms $\Omega^{n}_{X}$ has no ...
- 435
6
votes
1
answer
978
views
Simply-connected domain around a curve
In a current project with a colleague, we have come across the following reasonably classical-sounding geometric question. While not vital to our work, it would be interesting if anyone has seen this ...
- 6,548
24
votes
7
answers
4k
views
Why are two notions of Gaussian curvature are the same - what is the simplest & most didactic proof?
This question is still wide open - all of the answers so far rely on magical calculations. I've only accepted an answer because, by bounty rules, otherwise one would be accepted automatically. I can't ...
- 3,203
7
votes
0
answers
189
views
When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?
If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma:=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when ...
- 103
17
votes
1
answer
847
views
Irrational Numbers and the Riemann Surface of a Multi-Valued Function
Suppose a meromorphic function $f(z)$ has two poles, with residues $1$ and $\gamma$, respectively. Then the topology of the Riemann surface of the anti-derivative of $f(z)$ depends on whether or not $\...
- 15.4k
13
votes
4
answers
1k
views
"Simple" Kahler manifolds
I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$...
- 4,343
7
votes
2
answers
2k
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Vinogradov's Elements of Number Theory
I can't be the only person here who has fond memories of the problems in Vinogradov's Elements of Number Theory. (For people who have not read it - the text itself is just a concise basic number ...
5
votes
3
answers
799
views
Euclidean function of Euclidean domain defined at 0
In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things as ...
39
votes
3
answers
6k
views
On linear independence of exponentials
Problem.
Let $\{\lambda_n\}_{n\in\mathbb N}$ be a sequence of complex numbers . Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is ...
- 22.3k
24
votes
11
answers
8k
views
The role of the mean value theorem (MVT) in first-year calculus
Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...
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 ...
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990