All Questions
3,560 questions
0
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Is there an Isomorphism between R and C under addition? [duplicate]
Possible Duplicate:
AC in group isomorphism between R and R^2
Somewhere, I recall being told that there is an isomorphism between $\mathbb{R}$ and $\mathbb{C}$ under addition. However, despite a ...
- 5,759
15
votes
1
answer
2k
views
Dirichlet series expansion of an analytic function
Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$
$$\lim_{T\to\infty}\frac{1}...
- 7,127
0
votes
0
answers
520
views
Motivation of proof of Riemann-Roch for elliptic curve and generalizations
Given a lattice $L \subseteq \mathbb{C}$, Alain Robert defines a theta function as a meromorphic function such that $\theta(z+\omega)=a(\omega) e^{\pi h(\omega)(z+\frac{\omega}{2})} \theta(z)$ for all ...
- 15.4k
0
votes
1
answer
2k
views
pure dimensional and embedded components
Hi.
Let $X$ be a pure $n$-dimensional complex subspace of manifold $Z$. It is true that $X$ has no embedded components? (perhaps that is obvious with Weierstrass preparation theorem or Noether ...
- 435
1
vote
1
answer
535
views
Points at twice the distance from (-1, 0) that they are from (1, 0) in hyperbolic geometry [closed]
In answer to the question Demystifying complex numbers, Charles Matthews suggests "finding the points at twice the distance from (-1, 0) that they are from (1, 0)." as a motivation for complex numbers....
- 1,190
1
vote
1
answer
201
views
real-valued functions on the modular surface
How does one write down $\mathbb{R}$-valued functions on the modular surface? I am considering taking an arbitrary function on the upper half plane $f:\mathbb{H} \to \mathbb{R}$ and averaging over ...
- 22.8k
86
votes
44
answers
21k
views
Demystifying complex numbers
At the end of this month I start teaching complex analysis to
2nd year undergraduates, mostly from engineering but some from
science and maths. The main applications for them in future
studies are ...
1
vote
1
answer
1k
views
Duality and isomorphism of functor
Hi.
First of all thanks to Zsolt for the answer to the question on "Cohen Macaulay morphism".
I want to show for which proper and flat morphisms $f:X\rightarrow S$ of complex spaces with $n$ pure ...
- 435
1
vote
1
answer
448
views
Cohen macaulay morphism
Hi.
I have a doubt about this fact:
Let f:XS be a flat, proper and surjective morphism of complex spaces (or locally noetherian, excellent schemes) with n-pure dimensional fibers. Then f is Cohen-...
- 435
41
votes
2
answers
4k
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Must the set of lines through the origin on which a nonconstant entire function is bounded be finite?
If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \...
- 4,014
4
votes
1
answer
2k
views
torsion freeness of tensor product continued
Hi.
Question 1: If $f:A\rightarrow B$ be a morphism of local noetherian rings with $B$ is $A$-flat. Let $M$ (resp. $N$) be a $B$ (resp. $A$-)-module of finite type (fin. generated). We assume that $...
- 435
3
votes
0
answers
479
views
torsion freeness of tensor product
Hi.
Let $f:A\rightarrow B$ be a morphism of local noetherian rings, $M$ (resp. $N$) a $B$ (resp. $A$-)-module of finite type. We assume that $prof_{A}(M)\geq 2$ and $N$ is torsion free.
Then it is ...
- 435
1
vote
0
answers
991
views
Trigonometric identities and (several?) complex variables
I don't know anything about several complex variables nor whether that topic will answer my questions below, but in one complex variable one learns that since $\sin x$ and $\cos x$ are entire ...
1
vote
2
answers
701
views
Extension of harmonic function at infinity
Can a harmonic function defined on the upper half-plain (or any domain which is unbounded) be extended to the point at infinity. If so, under what condition. What happens to the mean value property ...
- 1,795
4
votes
0
answers
715
views
some questions about properties of harmonic measure
The original post
The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle ...
- 1,795
6
votes
1
answer
2k
views
Approximation by analytic functions
Dear all.
Let
$$
f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx)
$$
be a function given by usual fourier series.
Since my original question hasn't got any answer yet, and I ...
- 3,343
4
votes
1
answer
513
views
Is the following a sufficient condition for flatness?
Hi.
Let $f\rightarrow S$ be an open morphism of reduced finite dimensional complex spaces (or a universally open morphism of locally noetherian excellents without embedded components or reduced ...
- 435
33
votes
20
answers
5k
views
Do names given to math concepts have a role in common mistakes by students?
Perhaps this question overlaps with similar ones, ... but I want to focus on a particular possible cause of confusion. I notice that students are often confused by the concepts of "infinite" and "...
27
votes
5
answers
7k
views
References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
2
votes
1
answer
959
views
finite tor dimension
Hi. Can, every one, give me an example of finite surjective morphism of finite tor dimension (but not flat!) between reduced schemes or complex analytic spaces... Thank you.
- 435
19
votes
9
answers
5k
views
Mathematics and autodidactism
Mathematics is not typically considered (by mathematicians) to be a solo sport; on the contrary, some amount of mathematical interaction with others is often deemed crucial. Courses are the student's ...
15
votes
2
answers
2k
views
Picard-Fuchs equations for modular functions
Hello, MathOverflow community!
Suppose we have a modular curve of genus $0$, whose rational function field is generated by the modular function $f$. We can view $f$ as the parameter for some pencil ...
- 3,910
14
votes
2
answers
780
views
Highly connected, compact complex manifolds
Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$:
If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a ...
- 56.6k
2
votes
2
answers
679
views
L^2 space of holomorphic functions with given weight
Hi folks, what is known about the $L^2$ space of holomorphic functions of 1 complex variable with the scalar product
$\langle f, g \rangle = \int dzd{\bar z} \frac{ {\bar f(z)} g(z) }{(1 + z{\bar z})^...
- 362
7
votes
2
answers
3k
views
Relative canonical sheaf
Hi.
I want to know if for $f:X\to S$ a proper flat holomorphic map with n-dimensionnal fibers over reduced complex space S, the relative canonical sheaf $w_{X/S}:=H^{-n}(f^{!}O_{S})$ is a dualizing ...
- 435
32
votes
7
answers
8k
views
Interpreting the Famous Five equation [closed]
$$e^{\pi i} + 1 = 0$$
I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us?
Best that I can figure out is that it just ...
- 425
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 ...
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
20
votes
0
answers
666
views
Polynomials with roots in convex position
Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...
- 17.9k
24
votes
15
answers
5k
views
Applications of connectedness
In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line).
What are nice examples of applications of the idea of ...
3
votes
3
answers
865
views
Analytic ODE with complex time
Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r.
I would like to understand:
1) if there exists an analytic flow $\phi_t(x)...
- 303
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....
4
votes
2
answers
627
views
The link of a singular quintic hypersurface in CP^4
Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by
$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$
we get a singular variety for $\epsilon=0$ with 125 singular points.
I know ...
- 93
154
votes
7
answers
85k
views
Where to buy premium white chalk in the U.S., like they have at RIMS? [closed]
While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek.
At RIMS (in Kyoto) in 2005, they had the best white ...
15
votes
2
answers
1k
views
Asymptotic approximation of $x^\alpha$ by entire functions
Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$
for $x\rightarrow+\infty$ (with $...
- 17.9k
8
votes
3
answers
2k
views
The harmonic (series) beetle: live illustrations of mathematical theorems
In my analysis class I use the following problem to illustrate the divergence
of the harmonic series (consider this as a hint for solving it).
Exercise.
A beetle creeps along a 1-meter infinitely ...
14
votes
3
answers
3k
views
Analytic continuation of holomorphic functions
Analytic/meromorphic continuation is a difficult problem in general. For "motivic L-functions", the idea of proving their analytic continuations by first proving their modularity goes back, I guess, ...
- 4,265
-4
votes
1
answer
514
views
Meaning of the Mobius transformations video [closed]
What is this video trying to tell us?
http://www.youtube.com/watch?v=JX3VmDgiFnY
The statement that fractional linear transformations correspond to rotations of the sphere under the stereographic ...
- 1,783
12
votes
5
answers
2k
views
Introducing Cryptology to Undergraduates
This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures on,...
- 4,842
14
votes
1
answer
3k
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An elementary proof that the degree of a map of spheres determines its homotopy type
I'm helping to teach an undergraduate algebraic topology course (out of Hatcher's textbook). We have recently defined the degree of a map of spheres using homology, and the professor and I thought it ...
- 7,338
27
votes
4
answers
3k
views
Genealogy of the Lagrange inversion theorem
A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, ...
- 60.6k
30
votes
3
answers
4k
views
What is special about polylogarithms that leads to so many interesting identities and applications?
I have heard that Polylogarithms are very interesting things. The wikipedia page shows a lot of interesting identities. These functions are indeed supposed to have caught the attention of Ramanujan. ...
- 3,699
6
votes
0
answers
161
views
Multiplicity of zero (higher dimensional analog)
Consider a sistem of n holomorphic equations with n unknowns in a neighborhood of zero. Suppose that a solution in a neighborhood of 0 is a k-dimensional manifold.
I want to associate to it some ...
- 61
2
votes
2
answers
801
views
Domains of holomorphy in the complex plane
There is a proof of Mittag-Leffler's theorem with an explicit construction of a holomorphic function with the prescribed poles with prescribed order and residues, for a countable discrete set of ...
- 3,699
1
vote
2
answers
534
views
Local representation of an analytic sets
Let V be a analytic set of $C^n$, $I(V)$ is the sheaf of ideals of V (the sheaf whose stalks are ideals defining germs of V at its points). Since $I(V)$ is a coherent analytic sheaf, we see that in a ...
- 750
7
votes
1
answer
2k
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Current status of Bloch constant and Landau constant bounds
The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance ...
- 948
4
votes
2
answers
442
views
Elementary functions with zeros only at the positive integers
Does there exist a (meromorphic) elementary function $f(z)$ that is zero at all the positive integers $z = 1, 2, 3, \ldots$ and only at those points?
Edit: an elementary function can be written as a ...
- 2,235
3
votes
2
answers
619
views
Schwarz Lemma in terms of conformal surfaces or holomorphic curves?
Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...
user2529
11
votes
2
answers
2k
views
Complex analytic vs algebraic families of manifolds
I'm studying the deformation theory of compact complex manifolds as developed by Kodaira and Spencer. On the side I'm reading as much about deformation theory in general as I can get my hands on (and ...
- 4,343
15
votes
3
answers
1k
views
Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?
The following describes an extension of the Descartes Rule of Signs to polynomials with complex coefficients.
First, I need to define the notion of a "sweep"... Given a complex polynomial p(z) := c0 ...
- 1,230