# All Questions

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### Must a complex power series *fail* to be convergent somewhere on its circle of convergence?

My textbook (Saff & Snider, pg. 256) asserts so, but I can't seem to find its proof of the claim. On the other hand, a uniMelb lecture slide I'm cross-referencing claims that a power series is ...
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### Any way to simplify this gcd totient function

I have the following expression $$\frac{gcd(a,b)}{\varphi(gcd(a,b))}$$ $a,b$ are known positive integers. Is there any way to rephrase this or simplify it?
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### Coset of W containing $v$ is a subspace of $V$ iff $v \in W$

I would like if someone could look over my proof. It feels odd to me. Let $W$ be a subspace of a vector space $V$ over a field $F$. Prove that $v + W = \{v + w \mid w \in W\}$ is a subspace of $V$ ...
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### The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $p-(\frac{-ab}{p})$

What I need to show is that For $\gcd(ab,p)=1$ and p is a prime, the number of solutions of the equation $ax^2+by^2\equiv 1\pmod{p}$ is exactly $p-(\frac{-ab}{p})$. I got a hint that I have to use ...
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### Any arbitrary closed smooth curve bounds a orientable surface?

I've got a question that, given an arbitrary closed smooth curve $C:[0,1]\rightarrow\mathbb{R}^3$, can you always find a orientable surface $\Omega$ which satisfy $\partial\Omega=C[0,1]$ ? I have no ...
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### Difference of two powers of $3$ divisible by $2011$

How to prove that there exists two powers of $3$ that differ by a number that is divisible by $2011$?
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### Differentiation - Limits Equal (Possibly MVT, Rolle's or L'Hopital)

Got a quick question from a past exam paper. If $f:R \rightarrow R$ is differentiable, and $f$ is such that $\lim_{ x\rightarrow \infty } f(x)=\lim_{ x \rightarrow -\infty} f(x)=0$ and there is a ...
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### The spectrum of an invertible element $x$ is $\sigma(x^{-1})=\{\lambda^{-1}: \lambda\in \sigma(x)\}$

Suppose $x$ is invertible in the unital Banach algebra $A$. How can I prove that $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$
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### algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
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### Help solving this Diophantine equation

For a problem that I'm working on, I need to solve this Diophantine equation:- $-2a^3 + b^3 + c^3 = 36650$, where $a, b, c > 0$ are all DISTINCT positive integers, and $a, b, c \notin$ { 2, 9, ...
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### Action of $S_7$ on the set of $3$-subsets of $\Omega$

Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem: Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
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### Spectrum of T in $B(\ell^2)$

Let $T:\ell^2 \to \ell^2$ be an operator on $\ell^2$ is defined as follows: $$T\{a_1,a_2,\dots\}=\{0,a_1,a_2,\dots\}$$ What is spectrum of T in $B(\ell^2)$?
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### How to detect ellipse [closed]

I tried to detect the rim of the this cup I've tried this but the detection result is not quite good. The reason might be that ChanVeseBinarize function can't separate the rim and the body. Therefore ...
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### Multiplication in $\mathcal D'(R)$.

I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ ...
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### Set of maximal ideals such that localization not integrally closed

Let $A$ be a Noetherian domain that is also finitely generated as a $k$ - algebra over a field $k$. Form the quotient field of $A$ (which I denote $K$) and consider the integral closure $\overline{A}$ ...
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### Do the adjoint functor theorems usefully dualise?

The special and general adjoint functor theorems exist to construct left adjoints to particular functors given certain conditions on them. However, I've not been able to find much mention – at least, ...
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### which of the followings are true for bijective functions

which of the followings are true:- 1. There is a continuous bijection from $\mathbb{R}^2\to \mathbb{R}$. 2. There is a bijection between $\Bbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}$. Can somebody ...
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### Tutte's 1947 proof and paper on a family of cubical graphs

It is known that if a graph is connected, cubic, simple and $t$-transitive, then $t \le 5$. A proof is given in [Biggs, Algebraic Graph Theory, Chapter 18], and this result is due to [Tutte, A ...
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### Career in Number Theory?

I am about to get my B.S. in Mathematics, and I will be applying for PhD in pure mathematics next year, with future plans of teaching and doing research. Over the past year, I have developed a great ...
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### Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0$

$$\int_{-\infty}^{\infty} \sin x \, dx$$ When I am doing the proof for this, why do i have to split it into $\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx$? where a is a constant
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### Antiderivative of an absolute function

$sgn(x)$ 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) \lim_{x \to ...
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### Rolling dices and simple problem

I'm facing the following problem. Let's say I have N dices in a hand. I need to calculate how much time I should roll my dices to make all of them equal selected (pre-defined) number. Each time when ...
I'm looking for a series for $W_0(x)$ for x $\in [\frac{-1}{e},\infty [$ but every time i found only for $x\in [\frac{-1}{e}, \frac{1}{e}]$ and what about a series for $W_-1(x)$ if it is no series ...