# All Questions

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### Control theory: when does $G(s) = \frac{1}{P_\lambda(A)}$

In other words, under what condition is the system transfer function G(s) = Y(s)/U(s) equivalent to the reciprocal of the characteristic equation of the $A$ matrix in state space realization?
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### Random word problem for unknown groups.

Suppose that I pick a group $G$ from a distribution $X$, unknown to you, and I give you a generating set $|S|$ for $G$, and a word $x$ in $S$ and their formal inverses, and a set $T$ of sentences of ...
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### A closed form for the sum of $(e-(1+1/n)^n)$ over $n$

I have been having some trouble trying to find a closed form for this sum. It seems to converge really slowly
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### Matrix diagonalization - eigenvalues on diagonal

Diagonalization of a square matrix $A$ consists in finding matrices $P$ and $\Delta$ such that $A=PD P^{-1}$ where $D$ is a diagonal matrix. What theorem tells us that $P$ is a matrix composed of the ...
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### How to prove Existence and unicity

Prove or give a counterexample: Let $g$ be a continuous extension of a continuous function $f: \mathbb{Q} \rightarrow \mathbb{R}$ to $\mathbb{R}$. $g$ exists $g$ is unique I am interested in how ...
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### base of open neighborhood for dual group in k-topology

I wanted to ask the following: Suppose I have an abelian topological $G$, and $G^*$ is its dual group (all the continuous homomorphisms from $G$ to the circle group $T$). How can I show that the ...
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### Integrating $\int_{0}^{2} (1-x)^2 dx$

I solved this integral $$\int_{0}^{2} (1-x)^2 dx$$ by operating the squared binomial, first. But, I found in some book, that it arrives at the same solution and I don't understand why it appears ...
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### Let $S(n)$ be the sum of the digits of a number $n$. Solve $n+S(n)+S(S(n))=1993$

My first step is to just to try to understand the problem by considering specific values of $n$. In the simplest case, $n=1$, and then $S(n)=1$ and $S(S(n))=1$, and the sum of these values is 3, which ...
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### Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$

Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$. Now, I know that this involves the Sylow Theorems, of course. This is very new ...
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### Estimating the error of $\sin x \approx x-\frac {x^3}{6}$ for $|x|\le \frac 12$

Estimate the error of $\sin x \approx x-\frac {x^3}{6}$ for $|x|\le \frac 12$ It seems too easy so I just want to make sure: Since $f(x)-p(x)\le R(x)$ and $R_5(x)=\cos (c) \frac {x^5} {5!}$ So we ...
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### Continued Fraction Algorithm for 113/50

The numbers $a_k$ can be found for $\frac{113}{50}$ by using a continued fraction algorithm. Note that $\frac{113}{50}$ is rational, and as a result it will have to terminate. Can anyone help me ...
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### Differentiability of Function of Two Variables

Define $g(x, y) := (|x| + |y|)^{1/2}$. Find those points in $\Bbb R^{2}$ at which $g$ is differentiable. My Idea of a Solution: In the $1^{st}$ Quadrant, $g(x,y)= (x + y)^{1/2}$, which is a ...
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### Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
Given a presheaf $\mathcal{F}$ on a space $X$ and a map $f: X \rightarrow Y$, when does $f_* A(\mathcal{F}) = A(f_* \mathcal{F})$, where $A$ is the associated sheaf/sheafification functor? Since ...
Find the number of trees of $n$ vertices in which a given vertex is a leaf. I am having difficulty understanding what this question wants me to find. We know the total amount of trees with $n$ ...