# All Questions

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### Clarification on Limit Comparison Test

For one of my classes we are using Manfred Stoll's, $\textit{Introduction to Real Analysis}$, and I had a question regarding the Limit Comparison Test. Here is the definition that they have: ...
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### Determining all functions $f(x+c)=-1/(f(x)+1)$

I've noticed in my free time when the functional mapping $f(x+c)=-1/(f(x)+1)$ is iterated twice, it yields the original function $f(x)$ (i.e. $f(x+3c)=f(x)$). So I thought to study it as a periodic ...
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### Elements of $\text{Spec}(\mathbb{C}[x_1, …, x_n])$

I'm just curious as to what the elements of $\text{Spec}(R)$ are when $R = \mathbb{C}[x_1,..., x_n]$. I'm aware that $\text{MaxSpec}(R) = \mathbb{C}^n$.
1answer
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### Proving that $\lim_{(x,y) \to (0,0)} (x^2 +y^2 -x^3 y^3)/(x^2 +y^2) =1$

How can I go about proving that $$\lim_{(x,y) \to (0,0)} \frac{x^2 +y^2 -x^3 y^3}{x^2 +y^2} = 1 ?$$ I checked some lines along $x, y$ and $x=y$ and it all gave $1$
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### Can any tell me how to simplify this statistical thing by using maple code [duplicate]

I have a problem similar to the problem given in the link Addition of two Binomial Distribution Mr. Robert found some kind of solution using maple but I do not know how to use maple. Can any one ...
1answer
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### Commutativity and associativity of isomorphic binary structures

Assume that $(S,*)$ and $(T,\circ)$ are isomorphic binary structures. (a) Show that $(S,*)$ is commutative if and only if $(T,\circ)$ is commutative. (b) Show that $(S,*)$ is associative if ...
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### good book for algebra elementary

For us,the prescribed text book was algebra vol 1 by manicavasagom pillay. But I found lot of error in text and the book has only formula and problem ,there is no theory in it. So I think that it is ...
1answer
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### Deriving equation of ellipse from expanded form?

The equation of an ellipse centered around the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ The expanded form is $Ax^2 + By^2 + Cx + Dy + E = 0$ How do I derive the second from the first? I have ...
1answer
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### If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$.. [on hold]

If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$ dimensional subspace of $\mathbb{R}^n$ that's contained in $H$, show that $G = H$ I know that a subspace of $\mathbb{R}^n$ is ...
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### Class Number of $\mathbb{Q}(\sqrt[3]{19})$ and Hilbert class field

Finding the class number of $\mathbb{Q}(\sqrt[3]{19})$ is an exercise from Marcus 'Number Field'. This question was uploaded by some other user, but it was removed by now. I have worked on details and ...
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1answer
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### F(x) = 0 for all points except c. Show F is integrable.

Suppose $c$ is a point in the closed $[a,b]$ and that $F(x) = 0$ for all $x$ in $[a,b]$ except for $c$ and that $F(c) = 1$. Show that $F$ is integrable on $[a,b]$ and that $\int_a^bF(x)dx = 0$. By ...
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### What is the Inverse Fourier Transform of this function $\frac{e^ \frac{1}{2}t(-w^2 +2 \sqrt{2 \pi} u \delta''(w)) }{ \sqrt{2 \pi}}$?

As the title says, what is the Inverse Fourier Transform of this function: $$\frac{e^ \frac{1}{2}t(-w^2 +2 \sqrt{2 \pi} u \delta''(w)) }{ \sqrt{2 \pi}}$$ The inverse should be taken with ...
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### Conjugate function

I am looking for a geometric and intuitive explanation of the conjugate function and how it maps to the below analytical formula. $$f^*(y)= sup_{x \in dom f } (y^Tx-f(x))$$
1answer
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### Proof about binary structures and binary operations

Suppose that $*$ is an associative and commutative binary operation on a set S. Show that the subset $$T=\{a \in S \mid a*a=a \}$$ of $S$ is closed under $*$. Proof: We need to show that if $a\in T$...
1answer
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### Surjective continuous function

I am trying to learn how can construct onto continuous function from rational(irrational) numbers to integers. I believe, I have an example helpfully it is true example let $Q$ denoted the rational ...
1answer
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### solve $\cos^n x - \sin^nx=1$

I already post a question on the solution of \begin{align} \cos^n x + \sin^nx=1 \end{align} but it's just a mistake My real question is \begin{align} \cos^n x - \sin^nx=1 \end{align}
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### Infinitesimal generator of Brownian motion with additional jumps

A compound Poisson process is a jump process with two parameters, the rate of the jumps $\lambda$ and the distribution of the jumps $\mu$ ($\mu$ is a probability measure on $\mathbb{R}$). The ...
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### In what sense is metric space completion universal?

The completion of a metric space is unique up to metric monomorphism (usually called isometry). It is also the "obvious" way to make all Cauchy sequences convergent. Structures which are unique up ...
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1answer
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### Restriction of sheaf of modules

Let $\mathcal{F}$ be an $\mathcal{O}_x$-module. Let $V \subset U$ open subsets of the scheme $X$. Then the restriction map $\mathcal{F}(U) \to \mathcal{F}(V)$ is compatible with the restriction map \$\...

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