# All Questions

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### How do I prove that an open ball on $S^1$ divide it into two pieces?

Define $\gamma:[0,2\pi]\rightarrow S^1 : t\mapsto e^{it}$. Let $f:S^1\rightarrow \mathbb{R}$ be a continuous function and $z_0\in S^1$. Then it must be true that: There exists $\delta>0$ such ...
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### Lagrange multipliers problem with system of equations

$f(x,y,z)=xz+yz$ with two constraint $g(x,y,z)=x^2+z^2-2$ and $h(x,y,z)=yz-2$ Here my procedure: Is the seocnd time I find this negative delta and I can't figure out why. Appreciate any help.
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### Special Fourier series concerning Gibbs constant and the divisor function.

It is quite a remarkable function I found. It seems, though, that I may be staring at something trivial, which is hopefully not the case. I would like some opinions. The function is ...
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### Prove that these two equations are equal

Q1: Prove that this is equal: 1 + 2^p + + (2^p)^2 + (2^p)^3 + ... + (2^p)^(q-1) = ((2^p)^q - 1) / 2^p - 1 = ? Q2: And can this be divided? (2^pq - 1) / (2^p - 1) Thank you!
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### local ring of a smooth point

Local ring of a point is a regular local ring iff the point is smooth? What confuse me is what is the dimension of a local ring at a point x. By definition this ring is just the localization at x ...
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### Structure theorems for infinitely generated Abelian groups

The classification theorem for finitely generated Abelian groups is well known and plays big role in mathematics. Are there any structure theorems about infinitely generated Abelian groups known?
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### Existence of a function in differential forms

The book is Spivak, Calculus on manifolds, page 100. I don't understand why the existence of a unique function "$f$". I hope you can help me on this. Thank you all.
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### Show $D_t = S_{\tau_t} - t$ is a homogenous Markov Chain and that $u(i) = \mathbb{E}(X_1)^{-1}\sum\limits_{k\geq i}q_k$ is an invariant measure for it

Let $X_i$ be independent with $\mathbb{P}(Xi=k)=q_k$, $\sum\limits_{k=1}^\infty q_k = 1$ and $\tau_t = \inf(n \,\vert\, S_n = X_1+...+X_n \geq t)$. Prove that $D_t = S_{\tau_t} - t$ is a homogenous ...
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### How can I solve for this variable?

I'm only studying for the final (during thanksgiving break no less :) ) I'm in PreCalc, and I've been struggling this entire semester. -.- So any help with studying would be really helpful I ...
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### sequence where n-th member expresses $n-1$ chords of a circle

I am totally lost with this problem Let $\lbrace a_{n} \rbrace$ is sequence, where $n^{th}$ member expreses the number of parts, that are cut on $n-1$ chords in general pose(no three chords intersect ...
You are given a bag with eleven tiles, one each marked with a different digit $0$ to $9$ or a decimal point. The tiles will be pulled from the bag one at a time and placed in sequence on the table to ...