(a) for a function f() that can be expanded in a taylor series, show that where p is an arbitrary angle. for this reason, lz/h is called the generator of rotations about the z axis. hint: use eq. (4.129) and refer to problem 3.39 more generally, l n/h is the generator of rotations about the direction n, in the sense that explil-mp/h] produces a rotation of the angle in the right- hand sense about the n axis. for spins, the generator of rotations is s·n/h in particular, for spin 1/2, tells us how spinors rotate (b) construct the 2 × 2 matrix representing a rotation by 180° about the x-axis and show that it converts "spin up" (x+ into "spin down" ( as one would expect classically (c). construct the matrix representing a rotation by 90° about the y-axis, and check what it does to x+. (d). construct the matrix representing a rotation by 360° about the z-axis. if the answer is not exactly what you would have expected, discuss its implications (e). show that e'(a-n)v/2 cosp/2 ) + i(σ . n) sin(p/2) - cos

Two resistors r1 and r2 may be connected either in series or parallel across an ideal battery with emf ε. we desire the rate of energy dissipation of the parallel combination to be 8.75 times that of the series combination. if r1 = 105 ω, what are the (a) smaller and (b) larger of the two values of r2 that result in that dissipation rate?

1. the materials manager for a billiard ball maker must periodically place orders for resin, one of the raw materials used in producing billiard balls. she knows that manufacturing uses resin at a rate of 50 kilograms each day, and that it costs $.04 per day to carry a kilogram of resin in inventory. she also knows that the order costs for resin are $100 per order, and that the lead time for delivery is four (4) days.2. at what point should resin be reordered? a. 0 kilograms remainingb. 50 kilograms remainingc. 200 kilograms remainingd. 400 kilograms remaininge. 500 kilograms remaining3. if order size was 1,000 kilograms of resin, what would be the length of an order cycle? a. 0.05 daysb. 4 daysc. 16 daysd. 20 dayse. 50 days4. if the order size was 1,000 kilograms of resin, what would be the average inventory level? a. 50 kilogramsb. 200 kilogramsc. 500 kilogramsd. 800 kilogramse. 1,000 kilograms5. if the order size was 1,000 kilograms of resin, what would be the daily total inventory costs, excluding the cost of the resin? a. $5b. $10c. $20d. $25e. $40

Give me a example of a decomposer. explain what would happen if decomposers were absent from a forest ecosystem