By Richard H Enns; George C McGuire

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**Scissors Congruences, Group Homology & C **

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**Sample text**

In the ¯rst chemical reaction, the rate of producing X is +k1 A Y . In the second reaction, there is a decrease in X, the rate contribution being ¡k2 X Y . The third reaction provides a positive contribution +k3 A X. Finally, noting that 2 X in the fourth reaction is treated as X + X, this reaction provides a rate decrease given by ¡k4 X 2 . The other rate equations may be similarly understood. The factor of 2 in the Z_ equation appears because in the third reaction, two of Z appear for each net (2 X ¡ X) one of X.

2. THREE-DIMENSIONAL AUTONOMOUS SYSTEMS 39 (c) How long does it take for the system to be within 1% of this point? (d) If the spring system is initially at rest in the right potential well, what is the minimum value that the total energy must have so that it asymptotically approaches the stationary point in the left potential well? Problem 1-10: Period 8 In the text example, determine an approximate F for which period 8 occurs. Problem 1-11: A di®erent response In the text example, what e®ect does changing the driving frequency to !

Right: Current-voltage curve for diode. org). 1. PHASE-PLANE PORTRAITS 29 The battery voltage VB is adjusted to coincide with the in°ection point VS of the iD vs. , VB = VS . Near this operating point, one may write i = ¡a v + b v 3 , where i = iD ¡ iS and v = VD ¡ VS , and a and b are positive. 8) x Ä ¡ ² (1 ¡ x2 ) x_ + x = 0; ² > 0; with x proportional to v. 8) is just the simple harmonic oscillator equation for unit frequency and mass with an amplitude-dependent damping term. For x < 1, the damping contribution is negative, so that oscillations tend to grow, while for x > 1 the damping is positive, tending to reduce the oscillations.