http://www.math.harvard.edu/~mazur/preprints/time.pdf Barry Mazur gave a talk about time and distance and the way we have "discovered" time by the virtues of cyclic states and uniformity of observations of state. Is it possible for me to divorce the concept of time from measure theory to be a t...

How do I attempt the following question, to find the volume of the cylinders: $$ z=\log x , z=\log y $$ and planes - $$z=0 , x+y=2 e , (x>=1)? $$ Is there a way out apart from double integration ?

(Here is a picture of the problem) I have doubt in that question (where he has used $e^{1/k}$) My doubt is as $k \to \infty$, $1\over k$ should tend to zero therefore $e^{1/k } \to 1$, so ${e^{1/k}\over k^2}$ should tend to zero because denominator is very huge so all series will end to zero but ...

Suppose that we have $G(\overline{x},\overline{y})=-\frac{1}{4 \pi} \frac{1}{||\overline{x}-\overline{y}||}$ for $\overline{x}, \overline{y} \in \mathbb{R}^3$. I want to calculate $\Delta{G(\overline{x}, \overline{y})}$. I have found that $\frac{\partial^2}{\partial{x_1^2}} \left( -\frac{1}{4 \...

I'm new to this concept of a $\sigma$-algebra generated by a collection of subsets. Let $\Omega = \{a, b, c, d\}$ and $$\begin{align} &\mathcal{F}_1 = \{\Omega, \emptyset, \{a\}\} \\ &\mathcal{F}_2 = \{\Omega, \emptyset, \{a\}, \{b, c, d\}\}\text{.} \end{align}$$ I wish to show that $$\sigma\la...

Determine the minimum polynomial of $v=\sqrt{3}+\sqrt[3]{2}$ Over Q[x]. Cant find the right calculations .I am trying to find another way.I know the minimum polynomial of $w=\sqrt{3}+\sqrt{2}$ but still that does not help.I know it must be of degree 4 since the field extension of $\sqrt{3}$ + $...

I know this is going to come across as a very strange question, but it's important that I know the answer. Say I have a degree $k$ polynomial (for my case, I need it to be a complex-valued polynomial with real coefficients, but I think the idea would work even if it were real-valued), $f(z) = a_...