# All Questions

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### Boolean rings, subring of the ring its elements come from?

If $B=\{x \in C:x=x^2\}$ then is the boolean ring $B$ a subring of the ring $C$?
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### finding the number of sub fields such that $(K : Q) = 2$

Consider the polynomial $f(x) = x^5 - 4x + 2$. Let $L$ be the complex splitting field of $f(x)$ over $\mathbb{Q}$. I want to find the number of subfields $K$ of $L$ such that $(K : \mathbb{Q}) = 2$. ...
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How did this: $2(1-\sin^2x)=1+\sin x$ Become this: $2\sin^2x+\sin x-1=0$ Wouldn't it be: $2 -2\sin^2x-1+\sin x=0 ?$
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### Taylor series for $\log(1+x)$ and its convergence

I know the taylor series of $\log(1+x)$. However I don't understand how to find the convergence for $x>1$ and divergence if $x<1$.
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### Alternating Series

A professor of mine gave me this problem and asked me to figure it out. I can not seem to figure it out. Express ln($2/3$) as an alternating series and use alternating series estimates to find ...
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### Limiting values for logistic function

Given the logistic function (map) $x_{n+1} = r\cdot{xn}\cdot (1 - {xn})$and an initial value $x_{0} = 0.4$ When r = 0.5, i worked out $x_{1}=0.12$ and $x_{2}=0.0528$ How do i work out the ...
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### Computing the Fourier transform of a function not in $L^1$?

The Fourier transform is defined on $L^2$ by a density argument. It doesn't seem like it's constructive. So how would one go about computing the Fourier transform of a function in $L^2$ but not $L^1$? ...
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### poisson's equation with robin's boundary, boundary value problem

Consider the Poisson’s equation with Robin’s boundary conditions as follows \begin{array}{ll} −\Delta u = f, &\text{in $U$,}\\ \frac{\partial u}{\partial \nu}+u=g, &\text{on $\partial U$,} ...
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### Unknown number of colours Bernoulli Urn

Okay, so, in the traditional Bernoulli Urn problem, we have an urn with a number N, possibly infinite, of coloured balls, and there are k possible colours. That one I grok. However, what if I don't ...
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### Easy probability ..

I want to make sure these answers are correct A class has 10 freshmen, 8 sophomores, and 12 seniors. On a recent test, 3 freshmen, 5 sophomores and 3 seniors got an A. a) What is the probability ...
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### Find $\vec{QB}$ in terms of $\mathbf c$

I've managed to work out “$\vec{AM}$ in terms of $\mathbf a$ and $\mathbf b$” to be $3\mathbf a+\mathbf b$. But how can I work out “$\vec{QB}$ in terms of $\mathbf c$”?
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### Transfinite derivatives

I don't know if this is exactly research level, as I am only starting college. But I feel like this is the best place to ask the question. We all know of 1st, 2nd, 3rd, nth derivatives. Is there a way ...
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### $p$-adic completion of $\mathbb{Z}[X]$ and $\mathbb{Z}[[X]]$.

Let $p$ be prime. The $p$-adic completion of $\mathbb{Z}$ is the ring $\mathbb{Z}_p$ of $p$-adic integers, and its elements can be thought of as power series in $p$. Is there a nice description of the ...
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### Number of subgroups and elements

I have a question that I feel I am going about in a roundabout way, and would like some help on. I am preparing for an exam. Problem: Let $G$ be a group with $|G|=150.$ Let $H$ be a non-normal ...
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### Differential Equation: Determine $x''$ when $x'=xt^2+x^3+e^xt$

Determine $x''$ when $x'=xt^2+x^3+e^xt$ I haven't worked with DE's in a while...would the best approach be to use separation of variables?
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### problem with quadratic equation two variable

I have following equation $a^2+4.8ab-b^2=0$ and I have problem with solving it, I don't know why $a=-5$ or $a=0.2$
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### (Obvious?) Half-Space Poisson Kernel Estimate

In Stein's Singular Integrals and Differentiability Properties of Functions, Theorem 1 (a) of Chapter 8 (pg. 197) he makes the claim without proof that the Poisson Kernel for the half-space ...
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### The diophantine equation $a^n - 1 = (a-1)^m$.

Let $a,n,m$ be odd integers larger than one. The diophantine equation $a^n - 1 = (a-1)^m$ fascinates me. I know that Catalan's conjecture has been proven and that Pillai's conjecture has not been ...
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### Curve is a submanifold

In our class today we said today that an 1-times continuously differentiable immersion $\phi:[0,1] \rightarrow \mathbb{R}^n$ is a submanifold of dimension 1 if it is closed such that $\phi(0)=\phi(1)$ ...
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Problem: Let $(x_i)$ be a sequence in (0,1) such that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n x_i^k$ exists for every $k=0,1,2,...$ and $f\in C[0,1]$. Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n ... 1answer 55 views ### irreducibility of$m$th cyclotomic polynomial in$\mathbb Z[x]$implies irreducibility in$\mathbb Q[x]$? (Ireland and Rosen) In Chapter 13 of Ireland and Rosen's An Introduction to Classical Modern Algebra, they prove that the$m$th cyclotomic polynomial is irreducible in$\mathbb Z[x]$. Immediately afterwards they state a ... 0answers 67 views ### Geometrically, what is the difference between a “flat face” and a “non-flat” face? I was curious when I was checking sites like MathisFun, and I came across a pretty unclear system that defines a "flat face" and as a "non-curving" face of a shape; a polyhedron. However, I have to ... 3answers 68 views ### Prove, by induction, that$5^n + 5 < 5^{n+1}$for all$n\in\Bbb N$. Prove, by induction, that$5^n + 5 < 5^{n+1}$for all$n\in\Bbb N$. Attempt: If$n = 1$, then$5^1 + 5 < 5^{2}$=>$10 < 25$which is a true statement so the base case holds. Assume$5^k + ...
I have a very big problem with the following question: Is the operator $T$ defined by $(Tx)t=tx(t)$, $(0<t<1)$ compact in $L_2(0,1)$? My guess is no and I've tried 3 different approaches to ...