# All Questions

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### A set, which appropriately scaled is expressible as sums of elements of a compact set is pre-compact

Assume $X$ is a Banach space and $K\subseteq X$ is compact. Let $C\subseteq X$ be such that $(\forall x\in C)(\exists x_1,x_2\in K)(2x=x_1+x_2)$ Does it follow that $C$ is pre-compact? In particular ...
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### The $\cos(\sin 60^\circ)$

I stumbled across this question and I cannot figure out how to use the value of $\cos(\sin 60^\circ)$ which would be $\sin 0.5$ and $\cos 0.5$ seems to be a value that you can only calculate using a ...
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### Does every countable subset of the set of all countable limit ordinals have the least upper bound in it?

I'm sorry if the question is that kind of trivial, I just feel uncertain about these ordinals all the time. Is the answer to the following question "yes": Denote by A the set of all countable limit ...
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### Closed set in $l^1$ space

Let $$X := \left \{ (a_n) : \sum_{n=0}^\infty |a_n| < \infty \right\}$$ with the metric $d(a_n,b_n) := \sum_n |a_n-b_n|$. Let $\delta_j^{(n)} := 1$ if $n = j$ and $0$ otherwise. Denote ...
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### Extension of a metric defined on a closed subset

If $X$ is any metrizable space, $A$ is a closed subset of $X$. Let $d$ be a compatible metric on $A$ then $d$ can be extended to a compatible metric on $X$.
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### Integral $\int_{-\infty}^{\infty} e^{−k^2/(k-k_0)^2}dk$

So I have here the integral $\int_{-\infty}^{\infty} e^{−k^2/(\Delta k)^2}dk$, but then $\Delta k$ is equal to $k-k_0$ so this equation becomes $\int_{-\infty}^{\infty} e^{−k^2/(k-k_0)^2}dk.$ The ...
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### Concept Of Double Integration

Can someone explain how double integration is equivalent to calculating volume as single integration is calculating area.
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### Non-trivial Topology

I can't understand the differences between a non-trivial topology and a trivial one. Whuat's the meaning of "non-trivial" topology? Is there a link with connection's properties? For example, could we ...
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### Fundamental theorem of Morse theory for $\Omega(S^n )$

Using the Fundamental theorem of Morse Theory we can prove that $\Omega(S^n)$ is homotopically equivalent to a CW complex with one cell each in dimensions $o,n-1,2(n-1), \cdots$ and so on. But how can ...
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### Calculating $1819^{13} \pmod{2537}$ using Fermat's little theorem

Can anyone make me understand how to calculate $1819^{13} \pmod{2537}$ using Fermat's little theorem? Here $p=2537$ and $p-1=2537-1=2536$. I am unable to understand how to express $1819^{13}$ in ...
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### Find k such that f is density function

I have the following function: $f_X(x, \theta) = \left\{ \begin{array}{lr} k/x^3 & : x \leq \theta \\ 0 & : x > \theta \end{array} \right.$ and $\theta >0$. I ...
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### Which sentences are preserved by equivalence of quasi-ordered sets?

Given a quasi-ordered set $(X,\lesssim)$, we can define that for $a,b \in X$ we have $$a \sim b \;\Leftrightarrow\; a \lesssim b \;\wedge\; b \lesssim a.$$ and thereby obtain a partially ordered set ...
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### What does it mean by piecewise smooth boundary?

I will be highly obliged if anyone can give me any reference where i can get the definition of domain (in $\mathbb{R^n}$) with piecewise smooth boundary. My question is when a domain in ...
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### Specific Modular Arithmetic Question with Exponentiation

Are there any theorems that can be used to reduce $1213^{797} \pmod {2591}$ without using a computer?
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### Factorial related problems

How many zeros are there in $25!$ ? My answer was $6$. But i solved it by finding how many numbers are divisible by $5$ and $2$.here i was told to find out the zeros at the last end. But what is the ...
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### Express $[\cos(x) + \sqrt3 \sin(x)]$ in the form $[r\cos(x-a)]$

Express $[\cos(x) + \sqrt3\sin(x)]$ in the form $[r\cos(x-a)]$, where $r>0$ and $0\leq360$, hence solve the equation $[\cos(x) + \sqrt3\sin(x)= \sqrt2]$ This is as far as i have completed. I ...
### Calculating radius for $\sum\frac{\sin{n^3x}}{n^2}$
Let $\displaystyle \sum\limits_{n=1}^{\infty}\frac{\sin{n^3x}}{n^2}$ be a series: a. Find where the series converge pointwise and where uniformally. b. Does its derivative is continuous? About A: ...
### A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$
This is a problem from "A Course of Pure Mathematics" by G H Hardy. Find the limit $$\lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$$ I had solved it long back (solution presented in my blog ...