# All Questions

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### primitive recursive conditional

I am confronted with the assertion that the following expression describes the conditional: $\text{Cond}\left[ t, f, g \right] = \text{Pr} \left[pr^2_1, pr^4_2 \right] \circ(f,g,t)$. This is meant ...
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### finding the sum of this series $(2n-1)^2(1/2)^n$

$\sum\limits_{n=1}^{\infty}(2n-1)^2(\frac{1}{2})^n$ I know via Wolfram that the sum is 17, but I'm not sure I've ever found the sum of such a series before. Any help is appreciated.
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### Scale of Oscillations

I'm reading an article which claims the following result : (paragraph 2.2) for a function $f = \sin (N g(x)) h(x)$ where $g$ and $h$ are $C^{\infty}$ scalar functions non oscilattory and $N$ a large ...
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### Simplify and Denest $\sqrt[3]{2+\sqrt{3}+\sqrt[3]{4}}$

I'm not too sure how to begin in denesting $\sqrt[3]{2+\sqrt{3}+\sqrt[3]{4}}$. I thought about setting it equal to $2a+b\sqrt{3}+c\sqrt[3]{4}$, but that sounds a bit sketchy.
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### Parametric equations in rectangular form

$y=t^2+4, x=2t-5$ Graph and state any restrictions on the domain. I need to state any restrictions on the domain but don't know how. I added 5 to both sides of the x equation and then I divided by ...
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### Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...
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### How to prove that $\lim_{x \to 0^{+}} \frac{e^{-1/x}}{x} = 0$

Today someone asked me how to calculate $\lim_{x \to 0^{+}} \frac{e^{-1/x}}{x}$. At first sight that limit is $0$, because the exponential decreases faster than the lineal term in the denominator. ...
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### Notation: square brackets with a unique scalar?

my question is purely about notation. I am reading papers in computer science and I see that people use the following notation $[x]$ to denote $\{1,2,\ldots,x\}$. Is that correct? Or does it mean ...
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### Characteristic functions proof… $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$

I am trying to prove Lemma 4.1 (1) from Olav Kallenberg's Foundations of Modern Probability: $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$ From context, I believe that ...
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### Show that every polynomial of degree $1,2,$ or $4$ in $Z_2[x]$ has a root in $Z_2[x]/(x^4+x+1)$.

The problem: Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$. My attempt: I know that the polynomials $x$ and $x+1$ have ...
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### If $a$ and $b$ are nonzero integers such that each is a divisor of the other, show that $a = ± b$ .

I tried many approaches to this problem. I believe that if I did $b|a=m$ and $a|b=n$ and set $m=n$, then $a$ and $b$ would be equal. Is that how it should be done? If not, please help me out. Thanks.
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### Prove this series converges to a continuous function

My problem: Prove that the series $\sum\limits_{n=0}^\infty e^{n(\sin(nx)-2)}$ converges for all $x\in\mathbb{R}$ to a continuous function. By the root test it converges, but as far as the continuous ...
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### What is the significance of studying the steady state behaviors of a system? What information do steady state models provide us?

For example, http://www.jameslovelock.org/page31.html In this 1983 paper by Lovelock and Watson modeling Daisyworld, in equations (10) through (14), the paper considers the non zero steady state ...
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### Two decreasing, convex functions agreeing on a closed set

Fix a closed subset $B$ of $[0,\infty)$ and assume that $0\in B$. I am striving to construct two functions $f,g:[0,\infty)\to\mathbb R$ such that $f(0)=g(0)=1$; $f(x)\geq 0$ and $g(x)\geq0$ for each ...
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### solving the integral $\sqrt{\sin(y)}(4-y)$

So the question asked me to find the volume of the area between $\sqrt{\sin(y)}$ and $x=0$ rotated about $y=4$. I came up with the integral $$\int_0^{\pi}(4-y)\sqrt{\sin(y)}dy$$ I can't figure out ...
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### Poison's equation for charge distributed on a ring

Let's consider the problem of determining the electrostatic potential given a charge distribution $\rho : U\subset \mathbb{R}^3\to \mathbb{R}$. In that case, the potential satisfies the Poisson ...
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### Hypothesis Testing on Renewal Processes

We have time $[0,T]$ to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
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### Evaluate Distribution

Evaluate the distributions p(a), p(b|c), and p(c|a) corresponding to the joint distribution given in Table 8.2. Hence show by direct evaluation that p(a, b, c) = p(a)p(c|a)p(b|c). Table 8.2 ...
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### Prove that if $f$ and $g$ are integrable on $[a,b]$, then so are $\max{(f,g)}$ and $\min{(f,g)}$

Prove that if $f$ and $g$ are integrable on $[a,b]$, then so are $\max{(f,g)}$ and $\min{(f,g)}$. Since $f$ and $g$ are integrable, we know that $U(f,\mathcal{P})-L(f,\mathcal{P}) < \epsilon$ ...
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### Solving equations.

How would you solve these equations and show that they do not intersect each other? $$x^2+y^2=2x-2y$$ $$x^2+y^2=4(x^2+y^2)^{1/2} +y$$ It's isolating a term which I am struggling with. General ...
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### Integrate $\frac{\sin x^3}{x^3}$ as a power series

Today, I tried to do this by taking the MacLaurin of Sin to 4 terms, putting in $x^3$ in place of $x$, multiplying the terms by $x^{-3}$, and integrating. I came out with a sum that had $x^{6n+1}$ as ...
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### Convex subsets of a topological vector space

I'm trying to prove: Let $X$ be a topological vector space and $A \subseteq X$. $A$ is convex if and only if $$\forall s,t \in \mathbb{R}_{+}, (s+t)A = sA + tA$$ Let $sx + ty \in sA + tA$. How ...
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### What do you call the category of groups with surjections (injections)?

Take the category $Grp$ and throw out all morphisms that are not surjections. I think what's left is still a category. The composition of two surjections is surjective, and identities exist ...
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### Complex integration problem via Cauchy's integral formula

I want to integrate the following :$$\int_{|z|=2} \frac{dz}{z^{2}-1}$$ in the positive direction. So my idea is two split the integral into a sum of two integral , something like \int_{|z|=2} ...
### Problem in understanding the proof of lemma $7.2.5$ in Liu's book
Let's analyze the proof of the following lemma: Lemma: Let $X$ be an integral, Noetherian scheme and let $f\in K(X)^\ast$, then for all but finitely many points $x\in X$ of codimension $1$ we have ...
### $P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$
From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.