# All Questions

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### Vector space judgment

Define $(a_1,a_2)+(b_1,b_2)=(a_1+b_1,0)$ and $c(a_1,a_2)=(ca_1,0)$ With these operations, the following conditions (1)There exists an element in $V$ denoted by $0$ such that $x+0=x$ for each $x$ in ...
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### For homework questions what is the difference between being asked to verify something and being asked to prove something?

I've always been curious if there is a definite difference between the terms or if they just depend on the context of a problem.
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### Solutions to $(z+1)^n = z^n$ using conformal maps.

I'm doing a homework problem where I have to find all roots of $(z+1)^7 - (z)^7 = 0$ using the roots of unity for $z^7$ I noticed that if $a$ is a root of unity for $z^7$, then $1/(a-1)$ maps the ...
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### If $T\subset\mathbb{R}$ is bounded and $S \subset T$, then $\sup S \leq \sup T$ and $\inf T \leq\inf S$

Let $S$ and $T$ be nonempty sets of $\mathbb{R}$, with $T$ a bounded set and $S \subset T$. Prove that $\sup S \leq \sup T$ and $\inf T \leq\inf S$.
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### Orthogonal intersection in a Riemannian manifold

Following is a question which I also asked in stats.stackexchange where I haven't got a response yet. Here is a link. Let $S$ be the set of all probability distributions on $\mathbb{R}$ and ...
878 views

### The equivalence between Cauchy integral and Riemann integral for bounded functions

Definitions Suppose $P\colon a=x_0<x_1<\dotsb<x_n=b$ is a partition of $[a,b]$. Let $\Delta x_k=x_k-x_{k-1}$ and $\lVert P\rVert$ denotes $\max_{0<k\le n}\Delta x_k$. The Cauchy integral ...