-1
votes
0answers
43 views
Are there any precise results about the intuition behind Morse functions?
A Morse funnction on a smooth manifold is usually intuitively interpreted as follows: Imagine the manifold to be a mountainous landscape and the Morse function as the elevation of …
0
votes
2answers
35 views
Eigenvalues of an amplification matrix
Let $A$ and $B$ square real matrices.
I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1.
Can we say something a …
0
votes
0answers
65 views
About the curvature of a connection?
In "Lectures on gauge theory and integrable systems" of M.Audin, she identifies the space of conections $\mathcal{A}$ on the trivial bundle $G\times S$ ($G$ Lie group, $S$ surface …
14
votes
1answer
279 views
How much of character theory can be done without Schur’s lemma or the Artin-Wedderburn theorem?
This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question wi …
0
votes
1answer
49 views
enumerative Gromov-Witten invariants
Let $X$ be a sympletic manifold and $A\in H_2(X;\mathbb{Q})$. Let $g$ and $k$ be nonnegative integers.
Assume that $$\mathcal{M}_{g,k}(X;A)$$
is dense in
$$\overline{\mathcal{M}} …
0
votes
0answers
16 views
Equivariant versus retractive spaces: a reference request
Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G. …
0
votes
0answers
38 views
Is a certain group related to a primitive L function isomorphic to $Gal(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ for some $\ell$?
I define the notion of "Galois class of L functions" in the following way:
$A$ is a Galois class of L functions if and only if the follwing three conditions hold simultaneously: …
0
votes
2answers
103 views
What does a singular simplex with real coefficient mean
For an $n$-dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$-norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which repre …
2
votes
0answers
23 views
Duality between K-theory and K-homology in the non-compact, spin$^c$ case
Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms …
4
votes
0answers
45 views
Smith Normal Form of powers of a matrix
What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix?
The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R …
1
vote
3answers
207 views
Help with this Diophantine equation
Note: This question was posted in error, and should be closed as no longer relevant. The correct question is posted at http://mathoverflow.net/questions/131353/help-with-this-sys …
2
votes
2answers
107 views
Help with this system of Diophantine equations
A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it.
This is the actual question which I'm tr …
16
votes
14answers
762 views
objects which can’t be defined without making choices but which end up independent of the choice
It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data one first chooses some additional structure …
0
votes
2answers
69 views
Why every complex of injectives is homotopically injective (provided that, the injective dimension is finite)?
Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain c …
0
votes
0answers
39 views
Circle segment of exact length [closed]
I need to find:
a spherical line
that passes through the points 0,0 and 8,8
and the distance of the line between those two points must be exactly 12
I imagine the answer will b …
1
vote
0answers
23 views
Non-crystallographic cluster algebras
Background
Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Her …
-1
votes
0answers
49 views
What is the perimeter of the figure shown on the coordinate plane? [closed]
What is the perimeter of the figure shown on the coordinate plane?
Picture
http://imgur.com/r4CNd4y
1
vote
1answer
87 views
fundamental class is the sum of simplices of triangulation of the manifold?
M is an n-dimensional closed orientable manifold. I find in a book "Intuitively,the fundamental class can be thought of as the sum of the (top-dimension) simplices of a suitable tr …
2
votes
1answer
67 views
Field of definition of canonical morphism between (congruence) modular curves
Let $\Gamma\subseteq \Gamma'\subset SL_2(\mathbb Z)$ be congruence subgroups, and
$X(\Gamma)$, $X(\Gamma')$ be the associated smooth projective modular curves over $\mathbb C$. Th …
1
vote
1answer
61 views
Non-(stable)-triviality of the tautological bundles
This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/396217/7110/
The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann m …
0
votes
0answers
19 views
How to simplify this Kampé de Fériet function?
I was dealing with a convolution type integral
$$
\int^z_0 t^m {}_0F_1(;1;-t) \: {}_2F_3\Big( 1,1;2,m,m+1 ; -a t\Big) \:\mathrm{d}t
$$
By applying one of the identities in Exton's …
1
vote
1answer
131 views
a question of local field
Let $K$ be a local field with mix char, $k$ residue field. We have an exact sequence
$0 \longrightarrow I \longrightarrow G_{K} \longrightarrow G_{k} \longrightarrow 0$
Then we o …
3
votes
2answers
98 views
Quotients in Sums of Rings
Suppose we are given a commutative ring R with unit-element. Now we have a composition of R as the direct product of two rings $R\cong R_1\times R_2$. It is now straight forward, …
2
votes
0answers
49 views
How fast is discrete-time diffusion on a continuous set?
This question is inspired by Joseph O'Rourke's beautiful answer to my previous question.
Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns hav …
0
votes
1answer
69 views
Upper bound of a series
Given $N$ and $a$ positive integers, with $a\ge 2$ is it possible to prove the inequality:
$$\sum_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}\le\frac{N}{2}$$
-2
votes
1answer
100 views
Embedded associated prime and non zero divisor
$M$ is a finitely generated $A$-module of dimension $d$ such that $G(M)$ is eqidimensional and $M$ does not have any embedded prime.
Given $x\in I$ where $I$ is an ideal of $A$ an …
38
votes
2answers
4k views
Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture
Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville:
“The big experts in the field had
already tried to make this approach
w …
-1
votes
0answers
50 views
Antiderivative of an absolute function [closed]
$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$
$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) …
4
votes
2answers
184 views
Are sums of the inverses of prime siblings finite?
PART I (Initial version)
Let $P$ be the set of all primes $2\ 3\ \ldots$. Let
$$P_d\ \ :=\ \ \{\ p\in P\ :\ \exists_{q\in P}\ \ 0 < |p-q|\le d\ \}$ …
1
vote
0answers
38 views
Chern Character Isomorphism for non-finite CW complexes, resp. for non-CW complexes
This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/397164/7110/
Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous …
