# All Questions

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### Let $P(x)$ be any polynomial and suppose that $a_n \rightarrow a$. Prove $lim_{n\rightarrow\infty} P(a_n) = P(a)$

I know the limit rules but that's not helping me out much here this seems so simple but I don't even know where to start. I read the chapter on this and they don't do any examples like this. I ...
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### Notation Question regarding Ring-mod-Number and Ring-mod-Some Kernel

I'm having trouble linking the notation of something like $\mathbb{Z}/n$ and $R/H$ where $n \in \mathbb{Z}$, $R$ is a ring, and $H$ is the kernel of some homomorphism from that ring to another. In ...
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### Multiples of frequencies with high Fourier coefficients

As I understand it, any signal that satisfies certain properties can be represented as an infinite sum of sine waves. And the amplitude of each sine wave is reflected in the Fourier coefficient, so ...
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### Finding the $100$th derivative of $f(x) = 1 / (x^2 - 1)$ [on hold]

Let $f(x) = 1 / (x^2 - 1)$. Find the $100$th derivative of $f(x)$.
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### A basic question on expectation of distribution composed random variables

Suppose that $X$ and $Y$ are random variables with distribution functions $F$ and $G$. If $F$ and $G$ have no common jumps then I need to show that $E[F(Y)] + E[G(X)] = 1$. How to proceed here ? ...
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### Solving $\frac{x}{1-x}$ using definition of derivative

I was trying to find the equation of the tangent line for this function. I solved this using the quotient rule and got $\frac{1}{(x-1)^2}$ but I can't produce the same result using definition of ...
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### Definition of a splitting field of a finite group

This is a basic question from the journal 'Mathematische Zeitschrift' 208 (1991) page 243. Let $K/F$ be a finite Galois extension of number fields and $G={\rm Gal}(K/F)$. Also let $L$ be any number ...
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### Is $a \sim b$ such that $\gcd(a,b) > 1$ an equivalence relation?

Is $a \sim b$ such that $\gcd(a,b) > 1$ an equivalence relation? I know that it's reflexive, since $\gcd(a,a) > 1$. It's also symmetric since $\gcd(a,b) > 1$ iff $\gcd(b,a) > 1$. ...
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### Lagrange Identity Proof

Was reading through Lagrange Identity Proof. However, one thing the proof assumes is $$\sum_{i=1}^p\sum_{j=1}^q a_i b_j=\sum_{i=1}^pa_i\sum_{j=1}^qb_j$$ which seems intuitive - but I wonder if ...
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### integral involving hypergeometric function

I've obtained that the eigenfunctions of a certain Sturm-Liouville problem are: $$\phi(x,\lambda) = C\cdot(1/x)^{-1/2\pm i\lambda}\Psi(-1/2\pm i\lambda, 1\pm2i\lambda,1/x),$$ where $C$ is a ...
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### Product of topological vector spaces endowed with the product topology is a topological vector space

Any help is appreciated. Any proof from internet is also highly appreciated.
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### Irreducible polynomial over a field

I'm working with the idea of irreducible and prime and am struggling to figure out how to show this. Anybody who could figure this example out, it would be much appreciated. Let p(X) be a polynomial ...
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### Showing that a function is not computable.

the following function was shown not to be computable: $h(x) = \begin{cases} \mu n.\Phi_x(n) \downarrow & \mbox{if } \exists n \Phi_x(n) \downarrow \\ \uparrow & \mbox{otherwise} \end{cases}$ ...
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### Subtract matrices of different sizes

I have 2 matrices of different sizes, say A - 8*3 and B - 2*3. I need to subtract from each row of B, a particular row of A and sum these values. This I need to do for all the rows for A and store the ...
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### What's the intuition on integration of forms?

In vector calculus, given a vector field $F : \mathbb{R}^3\to T\mathbb{R}^3$ we can consider integrating it over a surface. This integral represents then the flux of the vector field accross the ...
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### Scale a Point onto Plane

I'm trying to find the scale factor that scale a point onto plane in 3D Space. I have the following information: Point on a plane: $a = (x_1,y_1,z_1)$ Plane equation: $P\colon Ax + By + Cz +D =0$; ...
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### Proving $\prod_{i=1}^n (\frac{1}{i} + 1) = n+1$

Prove using a direct proof that $$\prod_{i=1}^n \left(\frac{1}{i} + 1\right) = n+1$$ Okay, so I think I have done it correctly using an inductive proof: Base case: $(1+\frac11)=2$, ...
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### Sylvester-Gallai Theorem

How is this theorem used in applications? I've been searching for it on the web but can't seem to find. Only to "correct codes". Can someone please give a few simple examples? /lost student
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### Equation of tangent line for $y' = \frac{x}{(1-x)^2}$ at point $(0,0)$

I tried to solve this by plugging zero into x the $x$ values and I end up getting $\frac{0}{1}$, which obviously is $0$. From there I multiply out and get all zeros. What am I doing wrong? More ...
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### Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
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### An open cover $\{U_\alpha\}$ of $X$ is locally finite iff each $U\alpha$ intersects $U_\beta$ for only finitely many $\beta$

I am trying to prove: Lee, Smooth Manifolds, Exercise 2.9. Show that an open cover $\{U_\alpha\}$ of $X$ is locally finite if and only if each $U\alpha$ intersects $U_\beta$ for only finitely many ...
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### Convexity w.r.t a vector, convexity w.r.t elements [on hold]

In general, if f(a, b) = f(x) is convex w.r.t a vector x, which is constructed using some combination of elements of a and b and zero (i.e. x = ($a_1$, 0, $a_3$, $b_4$, 0,...)) is it also convex w.r.t ...
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### Proving the zeros of the chebyshev points

I am trying to prove that the zeros of $T_n(x)$, also called the chebyshev points are, $x_i = \cos ((2i + 1)\frac{π}{2n}) \in (−1, 1), i = 0, 1, . . . , n − 1.$ I believe I have to use the fact that ...
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### Prove that $\mathcal F_1\otimes\cdot\cdot\cdot\otimes\mathcal F_n=2^\Omega.$

How can I prove that if $n\in\mathbb{N}, i\in\{1,...,n\},$ $\Omega_i=\{0,1\},$ $\mathcal F_i=2^{\Omega_i}=\{\emptyset, \{0\}, \{1\}, \{0,1\}\},$ ...
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### how to prove this limit, convolution?

I am wondering how to prove that $\vert (f*g)(x) \vert \rightarrow 0$ when $\vert x \vert \rightarrow \infty$ if we assume $f \in L^{p}(\mathbb R)$ and $g \in L^{q}(\mathbb R)$ where $1/p+1/q=1$ ...