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For 2008 Fall, I took Analysis I, measure theory. This is my cheat sheet that I made for the final, just for studying purposes. It's written on a letter paper with a HITEC-C 0.3 black pen.
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Let there be two identical, symmetrically coupled oscillators, denote the phase as If the coupling is pulse coupled with some sort of transfer function The phase dynamics can be described by, Now, in case of phase synchrony, Therefore, is the key function to consider. If this function is 0 at 0, it means there exist a stable synchronizing solution. And 0 at other phase differences implies phase synchrony. Note that this function has the same sign and solution as the odd portion of As says the second derivative of [1] Dong-Uk Hwang, Sang-Gui Lee, Seung K Han, Hyungtae Kook, "Phase-model analysis of coupled neuronal oscillators with multiple connections", Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 74, No. 3. (2006) [doi] [2] Hyungtae Kook , Sang-Gui Lee , Dong-Uk Hwang and Seung Kee Han Synchronization of a Neuronal Oscillator Network with multiple connections of time delays Journal of the Korean Physical Society, 2007 50:341-345. [3] Hideyuki Cateau, Katsunori Kitano, Tomoki Fukai, "Interplay between a phase response curve and spike-timing-dependent plasticity leading to wireless clustering", Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 77, No. 5. (2008) [doi] |
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When I was a teenager, I used to use these kinds of tricks to do well on tests. Especially useful for multiple choices but works very well otherwise. The course material is inspiring and well organized. I highly recommend to take a look. http://ocw.mit.edu/OcwWeb/Mathematics/18-098January--IAP--2008/CourseHome/index.htm Sanjoy Mahajan, course materials for 18.098 / 6.099 Street-Fighting Mathematics, IAP 2008. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [5 Aug 2008]. |
Let
For every i Proof (adapted from Karlin and Taylor 1981) First we define For each t, we decompose as Finally, Reference
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This is the algorithm I have been working on for a few days. Guess what it is before I publish it. BTW, Merry Christmas, everyone.  update 2008 Feb: This algorithm is published as Il Park, António R. C. Paiva, José Príncipe, Thomas B. DeMarse. An Efficient Algorithm for Continuous-time Cross Correlogram of Spike Trains, Journal of Neuroscience Methods, Volume 168, Issue 2, 15 March 2008, 514-523 |
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where This summation converges for You can solve this my differentiating the following easy sum however, it is always convenient to know the short cut answer. BTW, I wonder why nobody makes equation search engine for the web. Searching equations with reordering and change of variables would be very helpful. I know mathworld supports a little bit of search but it is not enough. If I only had some time, I would start implementing for Wikipedia formulas with Prolog. Reference: http://www.jstor.org/view/07468342/di020761/02p0031j/0 |
