# All Questions

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### What is the difference between Representation and Fibre Bundle?

When a Group G have a homomorphism to General Liner Group GL(n, K), we call GL(n, K) Liner Representation. When a Space X have a map to another Space Y, We call the inverse image of y, or f~-1(y), ...
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### Did differential geometry undertgo a notation change?

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by ...
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### Sumset of parallel arithmetic progressions in Euclidean space

Let $A$ be a finite subset of $\mathbb R^n$ such that dim $A=n$. It is a known result from Freiman that the size of the sumset $A+A$ is lower bounded as $|A+A|\geq (n+1)|A|-\frac{n(n+1)}{2}$ where ...
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### Stalks of étale sheaves

I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F_{\bar{x}}\to G_{\bar{x}}\to H_{\bar{x}}\to 0$ is exact at ...
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### Asymptotic decay for the inhomogeneous Schrödinger equation

Let $H$ be an Schrödinger operator $(-\Delta+V(x))$ with a radially symmetric and smooth potential and $H$ is e.s.a. on $C^\infty_0(\mathbb{R}^n)$, furthermore let $\lambda$ be an element of the ...
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### Generalization of the alternating sign test for convergence of a series?

I'm struggling with a series of the form $$\sum_n |a_n|\, s_n$$ where $s_n$ is the sign of a simple function of $n$. The $|a_n|$ monotonically decrease and are relatively simple functions of ...
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### Hilbert triples

I apologize if this is a trivial matter for the analysts out there, but I looked this up in a number of books (e.g., H. Brezis, J. Wloka) and could not find a satisfying discussion, one that would ...
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### Is there a relationship between the standard conjectures and Langlands program?

I would like to know are there connections between Standard conjectures on algebraic cycles and Langlands program (in the light of Motives, I assume)? What implications would a development of the ...
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### A perturbation question for the intersection of C*-subalgebras

This feels like something I may have asked before (in which case, apologies) and it also might be some kind of "standard counterexample in a book on C* algebras". Let M be a unital C*-algebra and let ...
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### Old Math books, will research and sell most [on hold]

I saw an earlier thread on selling old math books. My dad was a professor at Manhattan College for many years, he passed away a couple of years ago and has tons of old math books. I promised him I ...
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### Fast computation of a Groebner basis - What is Possible

I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...
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### graphical estimate of convergence rates [looking for textbook reference] [on hold]

I asked this question already in mathematics, but got no sufficient answer. I am writing a paper for engineers. There, under other things, I compare convergence rates of sequences $x_n \to x^*$, ...
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### Lower central series in a free pro-p group

Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$. Is ...
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### Intersection of closed geodesics in hyperbolic surface

This question may be easy but I could not come up with a proof. Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed ...
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### The set of matrices with same spectral radius

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...
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### Trees with a maximal convex hull: are the only optimal solutions Steiner trees?

For given $n\geqslant 3$, I'm looking for a connected set composed of $n$ equal segments in the plane such that the convex hull of it has maximal area $A(n)$. To simplify notation, we'll take ...
I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$ and $|\epsilon_k|=1$ for all $k=1,2,\ldots,n$. We have \begin{align*} ...
In the bifurcation diagram, is it true that if the function $f_r(x)$ at $x^*$ (where $x^*$ is a fixed point where a bifurcation occurs) can be locally written as a smooth function then ...