# Toronto Math Forum

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

Title: HAY--as preparation for TT2
Post 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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Title: Re: HAY--as preparation for TT2
Post by: Vitaly Shemet on November 11, 2012, 09:06:22 PM
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?

Title: Re: HAY--as preparation for TT2
Post by: Zarak Mahmud on November 11, 2012, 09:51:28 PM
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 $\beta$.

Title: Re: HAY--as preparation for TT2
Post by: Vitaly Shemet on November 11, 2012, 10:28:14 PM
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)
Title: Re: HAY--as preparation for TT2
Post by: Zarak Mahmud on November 11, 2012, 10:38:16 PM
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
Title: Re: HAY--as preparation for TT2
Post by: Victor Ivrii on November 12, 2012, 04:06:52 AM
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).
Title: Re: HAY--as preparation for TT2
Post by: Zarak Mahmud on November 12, 2012, 11:44:04 AM
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.  :)
Title: Re: HAY--as preparation for TT2
Post by: Victor Ivrii on November 12, 2012, 01:49:49 PM
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.
Title: Re: HAY--as preparation for TT2
Post by: Zarak Mahmud on November 12, 2012, 03:49:15 PM
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.
Title: Re: HAY--as preparation for TT2
Post by: Vitaly Shemet on November 12, 2012, 09:34:32 PM
Thank You!
Title: Re: HAY--as preparation for TT2
Post by: Vitaly Shemet on November 13, 2012, 07:05:16 PM
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?
Title: Re: HAY--as preparation for TT2
Post by: Victor Ivrii on November 13, 2012, 07:34:20 PM
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 $\beta=\alpha$).