# All Questions

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### How to show for every $p\in M$ there is a local chart $(U, x_1, \ldots, x_n)$ around $p$ with desired properties?

Let $M$ be an $n$-dimensional smooth manifold and let $\pi:TM\longrightarrow M$ be the tangent bundle of $M$. Furthermore, let $E$ be a vector subbundle of $TM$ such that $\textrm{dim}(E_p)=k$ for ...
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### $G$ be a finite simple group and $H,K$ be subgroups of prime index ; then is it true that $H,K$ are of same size?

Let $G$ be a finite simple group and $H,K$ be subgroups of prime index ; then is it true that $|H|=|K|$ ?
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### find the solution set of the following inequality

with so many radical I get lost $\sqrt[4]{\frac{\sqrt{x^{2}-3x-4}}{\sqrt{21}-\sqrt{x^{2}-4}}}\geqslant x-5$
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### If f(n) = Θ(g(n)) does that also mean 2^f(n) = Θ(2^f(n))?

I know that if f(n) = Θ(g(n)) does that also mean g(n) = Θ(f(n))? but is it true for this case please answer me. I didn't know how to prove it i tried using the definitions but it didn't work i'm i ...
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### Doubt in the proof of Banach Alaoglu Theorem

I'm reading a proof of Banach Alaoglu Theorem from Functional Analysis by S Keshavan on page 142: Theorem: Let $V$ be a Banach space. Then $B^{*}$, the closed unit ball in $V^*$ is weak$^*$-compact. ...
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### Matrix multiplication in inequalities

Let $x,y\in \mathbb{R^n}$ be such that $$x>y.$$ Is it possible to premultiply or postmultiply the above inequality by a positive definite diagonal matrix, say $A$, of ...
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### Derivative of the inverse of a matrix

I've seen in a scientific paper this equation: $\frac{\delta K^{-1}}{\delta p} = -K^{-1}\frac{\delta K}{\delta p}K^{-1}$ where K is a $n\times n$ matrix which depends on $p$. In my calculations I ...
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### Finding y-value in data point to determine coefficient on interpolating polynomial.

Let $P(x)$ be the interpolating polynomial for the data $(0, 0)$, $(0.5, y)$, $(1, 3)$ and $(2, 2)$. Find $y$ if the coefficient of $x^3$ in $P(x)$ is $6$. I tried finding the Lagrange interpolating ...
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### Recursive binary strings

Hi I have a problem that looks like this: The set S of binary strings is recursively defined in the following way: The string 00 is an element of the set S. The string 01 is an element of the set ...
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### Plotting points of Musical Note frequencies linearly on a Graph

I am trying to plot the frequencies of musical notes on a graph so that they are equally spaced apart. I have researched that the relationship between each note is: beginning_note_freq*(2^(X/12)) ...
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### Can you give me some example of this problem

How to this or how to subject contrains Here is the answers but doesn't have the Production Rate and natural gas usage am I not to solve it?Why is it that it turn to be equal?
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### Sections of the canonical bundle

This is maybe a stupid question. Let $M$ be a simply-connected complex (kahler?) manifold, is it true that the canonical bundle $K_M$ has always (global) sections? For example, we know that an ...
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### Is there a way to solve this differential equation with two functions?

Is there a method to simplify a differential equation with two variables? Specifically I am looking for a way to relate x(t) and y(t) in the following equation and I have a strong urge to say ...
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### how to evaluate this limit with n and p positive integers?

I'm wondering how it is possible to demonstrate the following : $$\lim_{x \rightarrow 1} \frac{1-x^{-\frac{1}{n}}}{x^{\frac{1}{p}}-1}=\frac{p}{n}$$ $$n, p > 1$$
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### The number of ways to fill a 3 by 3 grid

I am currently studying the problem of combination. And when I am doing an exercise, I saw the following question: There is a 3 x 3 grid, and for each cells in the grid, two players take turn to fill ...
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### What is a weight vectors weight really?

What does a weight vector with weight $(\lambda_1,\cdots,\lambda_n)$ actually mean? Let $V$ be a $gl(n)$-module and I have that $v\in V$ is a weight vector if it is an eigenvector for all elements of ...
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### Cartesian product of sets proof

Let A,B and C be some arbitrary sets.Prove the given equation : (A \ B) x C = (A x C) \ (B x C) , where x stands for Cartesian product of sets. I don't know how to start. Could you please help me ...
Someone asked me if the tautochronicity property of a cycloid would still hold if the cycloid were rotated, so that its lowest point (the equilibrium point) be no more the vertex. If $V$ is the ...