Toronto Math Forum
APM3462012 => APM346 Math => Home Assignment Y => Topic started by: Victor Ivrii on November 09, 2012, 09:07:51 AM

http://www.math.toronto.edu/courses/apm346h1/20129/HAY.html (http://www.math.toronto.edu/courses/apm346h1/20129/HAY.html)
http://www.math.toronto.edu/courses/apm346h1/20129/HAY.pdf (http://www.math.toronto.edu/courses/apm346h1/20129/HAY.pdf)
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In last year TT2 #2 Solution. I can't understand the following reasons. Why do we need to say that tanh(beta l) intersects 1/
alpha?

The only way we can have a negative eigenvalue is if the line $y=\frac{1}{\alpha}$ intersects $\tanh \beta l$. This can't happen if $\alpha$ is positive. Have you tried drawing the graph?
By the way, for negative eigenvalues the convention is to use $\gamma$ instead of $\beta$.

Where this reasoning came from? (I mean what is connection between sign of eigenvalues and intersection of these two graphs, and why do we take specifically 1/alpha or 1/alpha)

It is discussed in lecture 13 (http://www.math.toronto.edu/courses/apm346h1/20129/L13.html). If you have the Strauss textbook, it is discussed in quite a bit of detail in chapter 4.3

It is discussed in lecture 13 (http://www.math.toronto.edu/courses/apm346h1/20129/L13.html). If you have the Strauss textbook, it is discussed in quite a bit of detail in chapter 4.3
See Appendix C (http://www.math.toronto.edu/courses/apm346h1/20129/LC.html) and Appendix B (http://www.math.toronto.edu/courses/apm346h1/20129/LB.html).

It is discussed in lecture 13 (http://www.math.toronto.edu/courses/apm346h1/20129/L13.html). If you have the Strauss textbook, it is discussed in quite a bit of detail in chapter 4.3
See Appendix C (http://www.math.toronto.edu/courses/apm346h1/20129/LC.html) and Appendix B (http://www.math.toronto.edu/courses/apm346h1/20129/LC.html).
You linked to appendix C twice. :)

It is discussed in lecture 13 (http://www.math.toronto.edu/courses/apm346h1/20129/L13.html). If you have the Strauss textbook, it is discussed in quite a bit of detail in chapter 4.3
See Appendix C (http://www.math.toronto.edu/courses/apm346h1/20129/LC.html) and Appendix B (http://www.math.toronto.edu/courses/apm346h1/20129/LC.html).
You linked to appendix C twice. :)
Yes, right. Fixed.

By the way, I just wanted to remark  the hyperbola dividing the $(\alpha , \beta)$ plane is a great way to keep everything straight. I was getting mixed up with the signs until I start thinking of it in this way.

Thank You!

about condition of a+b+a b l=o and b=a in Appendix C. It does not cross BOTH branches. They intersect at origin. Maybe b=a?

about condition of a+b+a b l=o and b=a in Appendix C. It does not cross BOTH branches. They intersect at origin. Maybe b=a?
Yes, fixed (one can see from (3) that it was exactly $\beta=\alpha$).