# FREE Materials Science Lecture Notes

1. Jul 1, 2007

### neutrino

2. Jul 14, 2007

### bham10246

Consider the following space $X$, consisting of two $2$-spheres and two arcs glued together. Compute its fundamental group.

Since I can't draw a picture online, call the first sphere $S_1$ and call the second sphere $S_2$. Then one arc connects $x_1 \in S_1$ to $x_2 \in S_2$ and another arc connects $y_1\in S_1$ to $y_2 \in S_2$, where all the points are distinct.

I thought about this problem and contracted the arcs (so it looks like two 2-spheres identified in two points), and I want to use van Kampen. But I'm having a hard time figuring out two open sets $A$ and $B$ so that $A \cup B = X$ and $A\cap B$ is path connected.

Thank you!

Actually, can I contract one of the arcs so that the two 2-spheres touch at one point, then move the points $x_1, x_2, y_1, y_2$ to the point of intersection? Then I have $S^2 \vee S^1\vee S^2$. Then $\Pi_1(X) = \mathbb{Z}[\itex]. Is this a correct analysis? Last edited: Jul 14, 2007 3. Aug 9, 2007 ### Astronuc ### Staff: Mentor This subject seems to be coming up quite often, especially for those studying wave mechanics. Here is a brief intro. I'll be looking for better articles and I hope others will contribute references or insight from personal experience. http://en.wikipedia.org/wiki/Dispersion_relation and related topics http://en.wikipedia.org/wiki/Group_velocity http://en.wikipedia.org/wiki/Phase_velocity http://tosio.math.toronto.edu/wiki/index.php/Dispersion_relation - note that this is a wiki page from Department of Mathematics at U. Toronto I also hope to address anharmonic and non-linear systems. I think it important for students to understand the significance of the higher order spatial (and temporal) derivates with respect to the dependent variable, uxxx or [itex]\partial^3_x\phi$ in the KdV equation.

Last edited: Aug 9, 2007
4. Aug 14, 2007

### rewebster

I was looking for a reference for something relating to something mentioned, and ran across a nice site for a lot of references. If there is a thread that it can be re-posted--let me know.

This site organizes topics from various sources including arXiv alphabetically for you.

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oh--and I still couldn't find what I was looking for though

Last edited: Aug 14, 2007
5. Sep 24, 2007

### EnumaElish

6. Sep 24, 2007

### MathematicalPhysicist

well i dont know how much you can trust a site where it says there something like:
"physics hypothesis that there is no speed barrier in the universe"

i would say quite unpopular approach is it not?

7. Sep 28, 2007

### fizixx

Agreed 'loop"...and if you nav to the home page it is filled with java errors that make looking frustrating and counterproductive.

8. Jan 27, 2008