# All Questions

132 views

### Conservative Measures under a group action (reference request)

I was reading a paper and the author define the concept of conservative measure: Let $(X,\mathcal{B})$ a measurable space and $G$ a group that acts on $X$ by $$G\times X:(g,x)\mapsto T_g(x)$$ where ...
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### Inverse with respect to a given circle

Determine the inverse with respect to a given circle $g:\mathbb{R}^{2} \to \mathbb{R}^{+}, g(x,y)=x^{2}+y^{2}$. I have looked around for non geometric derivations without finding any of value. ...
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### How to find union and intersection of events?

I have sample space of experiment $S=\left\{x|-\infty<x<\infty\right\}$. I consider events $$A_i=\left\{x \;\middle|\;\frac{1}{2^{i-1}}\le x<\frac{3}{2^i}\right\};i=1,2...$$ And I want to ...
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### When $a\to \infty$, $\sqrt{a^2+4}$ behaves as $a+\frac{2}{a}$?

What does it mean that $\,\,f(a)=\sqrt{a^2+4}\,\,$ behaves as $\,a+\dfrac{2}{a},\,$ as $a\to \infty$? How can this be justified? Thanks.
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### Normed space and convex hull of closed subset

Let $(V, ||\cdot||)$ be a normed space. If $C\subseteq V$ is a closed set we do not know if $ch(C)$ is closed or not. The professor provided this example that as of now I'm not getting: Consider the ...
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### Bounded quantifier and it's meaning

It's explained in Velleman's how to prove book that $\exists x \in AP(x)$ means that there is at least one value of x in the set A such that P(x) is true. Then ...
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### Finding pooled variance

Find the variance of $S^2_p$ under the conditions; $\bar{x_1}, \bar{x_2}, s_1, s_2$ are the means and standard deviations of independent random samples of sizes $n_1$ and $n_2$ from normal populations ...
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### Isomorphism of ring localized twice - Atiyah Macdonald Exercise 3.3

I studied AM before studying universal properties. When I solved the following exercise, I had a tedious solution that involved dealing with elements. Let $A$ be a ring with multiplicatively ...
202 views

### Hartshorne II prop 6.6

I'm having a really hard time understanding the proof of this proposition. $X$ is a noetherian integral separated scheme that is regular in codimension 1. We consider $X\times \mathbb{A}^1$ and the ...
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### Calculating the limit of the “$\dfrac{volume}{area}$” ratio for a 2D function

Let's assume that we have a well behaving, continuous function $f(x,y)$ defined on $\mathbb{R^2}$. The double integral $\int_{x_0}^{x_1}\int_{y_0}^{y_1}f(x,y)dxdy$ gives the volume of the space ...
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### Non-trivial centers, abelian towers, Lang

I am currently reading a proof in Lang's algebra that says that: because $G$ is a finite $p$-group, with non-trivial center "we have an abelian tower for $G/Z$ by induction, we can lift this abelian ...
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### For small $x$, one has $\ln(1+x)=x$?

What does it mean that for small $x$, one has $\ln(1+x)=x$? How can you explain this thing ? Thanks in advance for your reply.
147 views

### Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.
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### Evaluate and find the principal value of $(-1+i)^ {2-i}$

Can anyone please help me evaluate and find the principal value of $(-1+i)^{2-i}$ I got up to $=e^{2-i}(ln(-1+i))$ $=e^{(2-i)(1/2 ln(2)+i(3pi/4))}$
consider the heat equation $u_t=a(t)u_{xx}+f(x,t)$, $0<x<L$, $0<t<T$ subject to the initial condition $u(x,0)=g(x)$ and boundary conditions $u(1,t)=0,$ $u_x(0,t)+hu(0,t)=0$ where ...