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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 …
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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 …
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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 …
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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 …
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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}} … 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. … 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: … 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 … 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 … 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 … 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 … 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 … 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 … 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 … 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 … 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 … 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 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 … 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 … 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 … 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 $$By applying one of the identities in Exton's … 1answer 131 views a question of local field Let K be a local field with mix char, k residue field. We have an exact sequence 0 \longrightarrow I \longrightarrow G_{K} \longrightarrow G_{k} \longrightarrow 0 Then we o … 2answers 98 views Quotients in Sums of Rings Suppose we are given a commutative ring R with unit-element. Now we have a composition of R as the direct product of two rings R\cong R_1\times R_2. It is now straight forward, … 0answers 49 views How fast is discrete-time diffusion on a continuous set? This question is inspired by Joseph O'Rourke's beautiful answer to my previous question. Let \mathbb{S}^{d\times n} denote the set of real d\times n matrices whose columns hav … 1answer 69 views Upper bound of a series Given N and a positive integers, with a\ge 2 is it possible to prove the inequality:$$\sum_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}\le\frac{N}{2}$$1answer 100 views Embedded associated prime and non zero divisor M is a finitely generated A-module of dimension d such that G(M) is eqidimensional and M does not have any embedded prime. Given x\in I where I is an ideal of A an … 2answers 4k views Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville: “The big experts in the field had already tried to make this approach w … 0answers 50 views Antiderivative of an absolute function [closed] sgn(x) is the Sign-Function, F is an antiderivative of f and S(x) := F(x) \cdot sgn(f(x))$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) …
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\ \}\$ …