# 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 $\text{MaxSpec}(R) = \mathbb{C}^n$.
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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 ...
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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 ...
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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 ...
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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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### 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. $$f^*(y)= sup_{x \in dom f } (y^Tx-f(x))$$
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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 $a\in T$...
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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 ...
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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 ...
I know what is discrete form of $$\int_\Omega\nabla\phi dV$$ in which $\phi$ is vector in 2D and $\Omega$ is volume of cell in CFD field. The result is: \frac{1}{\text{Volume}}\sum_{\text{face}} n\... 0answers 7 views ### Computationally Efficient Way to Partition N-Dimensional Space Around Distinct Values Sorry if the title isn't super helpful, I'm really just looking for someone to point me in the right direction or let me know if there is a standard way of doing this. What I am wondering is, if I ... 0answers 9 views ### Ranking: How to adjust for underlying variability (risk) 63 stores. Each day a store sells x number of products. An observation is defined as the percentile rank of a store for a given day. In terms of volume of products sold, the store that sells the least ... 2answers 26 views ### Simplifying this fraction in a different base Note: I would appreciate a solution that DOES NOT convert back to base 10. How would one simplify \frac{43}{70}_8? I assume, like in decimal, I must recognize a common factor and divide by that ... 1answer 10 views ### Cokernel of a module homomorphism Let A a K-algebra. Let M, N A-modules and f:M\rightarrow N a module homomorphism. The cokernel of f is Cokerf=N/Imf I define a homomorphism \rho:N\rightarrow N/Imf by \rho(n)=n+Imf.... 2answers 51 views ### Does not exist cover of \mathbb{R}^n by disjoint closed balls Does not exist cover of \mathbb{R}^n by disjoint closed balls with positive radius. My attempt: Suppose that exists, we can write: \mathbb{R}^n=\displaystyle\bigcup_{i=1}^{\infty} B_{i}. Let C... 0answers 6 views ### Does the derivative by extension of Holder continuity coincides original derivative? Let B be an open bounded convex domain in \mathbb{R}^{n} and F:\mathbb{R}^{n}-> \mathbb{R} be a function that is differentiable at any x_{0}\in\overline{B}. Assume further that F\in C^{1,1}... 0answers 3 views ### Marginal stability with non-simple poles on imaginary axis It is known that a system marginally stable if and only if the real part of every pole in the system's transfer-function is non-positive, one or more poles have zero real part, and all poles with zero ... 1answer 24 views ### Set of points near zero The subset A of the positive segment of real line has 0 as a limit point (that is has points of distance less than \epsilon to zero for every positive \epsilon). Let I(x) be an interval ... 1answer 32 views ### Math Explanation Induction [on hold] How would you explain to this person why it is that induction actually does work. In my follow up responses to you, I may ask you questions that your friend might as in response to your explanation. ... 1answer 14 views ### Universal Cover of wedges S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2} and \mathbb{R}P^{2} \vee \mathbb{R}P^{2}. We are asked to find the universal cover of the wedges S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2} and \mathbb{R}P^{2} \vee \mathbb{R}P^{2}. I am second guessing myself on this problem because I ... 4answers 125 views ### Solve \cos^n x + \sin^n x =1  the solutions of this equation as a function of the value of n?? \begin{align} \cos^n x + \sin^n x =1 \end{align} I already found the solution if n is odd, 1answer 24 views ### Walking to infinity stepping on randomly selected lattice points Suppose you randomly fill the infinite non-negative quadrant of \mathbb{Z}^2 with 1's and 0's, with 1 occurring with probability p (and 0 with probability 1-p). The lowerleft corner of ... 2answers 22 views ### Elementary Set Theory proof regarding infinite and finite sets Suppose X is an infinite set and Y is a finite set. Show that exists a surjective function f:X\rightarrow Y and an injective function g:Y\rightarrow X. 1answer 35 views ### Integer solutions of a^{(b^c)}=b^{(a^c)}=c^{(b^a)} Let a,b,c be positive integers and leta^{(b^c)}=b^{(a^c)}=c^{(b^a)}. Are there any nontrivial positive integer solutions to this set of equations? This is a question of my own musings. I know ...
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 \$\...