# All Questions

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### Is this language and its complement is Context free Language?

The language $\mathcal{L}=\{a^n b^n c^n\;|\;n={1,2,3,4}\}$ is not context free from my point of view. Because no. of a's pushed in to stack is totally compensated by no of b's. popped. Please help me. ...
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### First order differential equation

Im clueless on how to solve the following question... $xe^y\frac {dy}{dx} = e^y +1$ What i've done is... $\frac {dy}{dx} = \frac 1x + \frac {1}{xe^e}; \frac {dy}{dx} - \frac {1}{xe^e} = \frac 1x$ ...
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### If $\tau\in\mathscr{L}(V)$, why does $\tau^{-1}=p(\tau)$ for some polynomial $p$?

I'm working on an exercise from Steven Roman's Advanced Linear Algebra. He asks Let $\dim(V)<\infty$. If $\tau,\sigma\in\mathscr{L}(V)$, prove that $\sigma\tau=\mathrm{id}$ implies $\tau$ and ...
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### Show that: $\mu\left(\bigcup_N \bigcap_{n=N}^{\infty} A_n \right) \leq \lim \inf \mu(A_n)$

Let $(X,\mathcal{M},\mu)$ be a measure space. Let $A_1, A_2, \ldots \in \mathcal{M}$. Then, I want to show that: $$\mu\left(\bigcup_N \bigcap_{n=N}^{\infty} A_n \right) \leq \lim \inf \mu(A_n)$$ ...
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### Isomorphy of vector bundles over a variety $X$

As in my other question, let $X$ be a variety over a field $k$, and let $\pi:F\to X$, $\psi:G\to X$ be vector bundles of rank $r$ over $X$ defined on the same open cover $\{U_i\}$. That is, we have ...
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I am trying to solve a problem in the book Introduction to commutative algebra from Atiyah. Page 11, exercise 5 (iv). Let $A[\![x]\!]$ be the ring of all formal power series of the form ...
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### Question on a proof of a sequence

I have some questions 1) In the forward direction of the proof, it employs the inequality $|x_{k,i} - a_i| \leq (\sum_{j=1}^{n} |x_{k,j} - a_j|^2)^{\frac{1}{2}}$. What exactly is this inequality? ...
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### Need a reference for techniques for the evaluation of limits of functions

Is anyone familiar with a comprehensive source (a book or a web-site) for techniques used in evaluating the limits of functions ?
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### What are the morphisms $\mathcal{O}_X\to k_p$ on a variety $X$?

Let $X$ be a variety over a field $k$ (we always assume $k=\bar k$, but I think this doesn't matter here), and let $k_p$ denote the skyscraper sheaf on $X$ w.r.t. the point $p\in X$. I want to find ...
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### Finding all differentiable functions with a certain property

Find all differentiable functions $f \colon (0,\infty) \to \mathbb R$ for which there is a positive real number $k$ such that: $$f(x) \cdot f'(k/x) = x, \qquad\text{for all }x > 0.$$
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### Nonabelian order 28 group whose Sylow 2-subgroups are cyclic

Prove that if such a group exists it is unique (up to isomorphism). Also, determine the numbers of elements of each order, and the class equation of the group. I don't really know how to do the ...
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### Isomorphisms of linear representations of finite groups

Let $G$ be a finite group with representations $\rho_1, \rho_2:G\rightarrow GL(V)$. According to the definition of representation isomorphisms, $\rho_1$ and $\rho_2$ are isomorphic if there exists a ...
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### A question on groups actions of permutation groups

This is the final step into completing a problem and I am a bit stuck. I need to show that: Consider the action of $S_{n-1}$ on $S_n/S_{n-1}$ by left multiplication. Does this action have exactly two ...
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### Diagnostic Tests and Expected Values

I have this question from my textbook I'm not sure how to answer. I got the first part but the second part is a bit confusing. It goes something like this: "Two percent of the population has a ...
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### 7 card stud counting problem

Consider a standard deck of 52 cards. A game called 7 card stud is played where each player is dealt 7 cards. How many possible: 7 card hands are there? --my answer:133,784,560 hands from (1) ...
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### For any odd integer $x,y$, $(x^2+2) \nmid (y^2+4)$ [duplicate]

Possible Duplicate: Quadratics and divisibility Prove that for any odd integers $x$ and $y$, we have $(x^2+2) \nmid (y^2+4)$
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### Expectations Homework Questions Clarification

This is another one of the questions I didn't get a chance to ask the TA at my school today. I thought I had a pretty good grasp on Expectations but apparently I could still use some clarification. ...
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### Help with the proof that $A^*$ has the UMP of the free monoid

In the following proof $A^*$ here is a Kleele closure and $*$ is a product of $a$'s or "concatanation": Proposition 1.9. $A^*$ has the UMP of the free monoid on A. Proof. Given $f:A\to|N|$, ...
From $$1=\sum_{k\geq 1} a_k \sin((k\pi+\frac{\pi}{2})x),$$ I want to find $a_k.$ My unsuccessful approach is first multiplying both side by $\cos((k\pi+\frac{\pi}{2})x)$. That is, ...