Consider the master equation for a random walker on a one dimensional lattice of sites with m = 0, plus or minus 1, plus or minus 2, and discrete time t = 0, 1, 2, as discussed in class, p(m)_t + 1 = (1 - p - q)p(m)_t + pp(m - 1)_t + qp(m + 1)_t the particle jumps with probability p(q) to the nearest neighbor site to the right (left), and does nor move with probability 1 - p - q. the probability distribution of the this random walk process, when p + q = 1 has the binomial form p(m, t) =p^n_r q^n_l t! /n_r! n_l! with n_r - n_l! = m and n_r + n_b = t. explain in your own words what this has to do with the binomial expansion of the mathematical form (a + b)^t. also comment on what this has to do with the pascal triangle. the moments of the probability distribution are defined as w_n(t) = ([m - (m)]^n) in class we derived how the first and second moments (n = 1, 2) behave as function of time n starting from a delta function distribution at time tau = 0 or the special binomial case where p + q=1. m_av(t) = (m) = (p - q)t and sigma^2 = ((m - m_av)^2) = 4pqt repeat these derivations here. derive that the third moment of the binomial case p + q = 1 is equal to m_3 = ((m - m_av)^3) = 8[(n^3_r) - 3(n^2_r)(n_r) + 2(n_r)^3] = -8pq(p - q)t with n_r the number of right jumps, such that m = n_r - n_l is the displacement at time t = n_r + n_l.

You have been consulted by the mobile phone company hookup about a new augmented reality product called “brows”. they want to develop an alternative to google glasses, which, you recall, is a proposed new product which plans to superimpose a whole range of wireless google services over the user’s ordinary vision of the world. these appear in little circles and ovals, and let you phone people, take photographs, navigate across cities and inside buildings, find shops and update your social media. brows would do this too, but not be limited to special google services. the user would wear a small cylinder-shaped device designed to fit on top of their existing eyewear (glasses or sunglasses). it would project a reversed image onto the inside of the lenses, the reflection of which would be in focus for the user. instead of voice control, the user would control the device by eye and neck movements, which the device would track. the device would connect via bluetooth to the user’s phone, which would supply most of the computing power and wireless connectivity. hookup also wants to know what existing services would be most in demand, and which would not work well with brows. using your knowledge of augmented reality, wearable technology and interface design, is the project feasible? if not, why not? if feasible, how could the product need to be modified for a better experience?

Aheat engine running backward is called a refrigerator if its purpose is to extract heat from a cold reservoir. the same engine running backward is called a heat pump if its purpose is to exhaust warm air into the hot reservoir. heat pumps are widely used for home heating. you can think of a heat pump as a refrigerator that is cooling the already cold outdoors and, with its exhaust heat qh, warming the indoors. perhaps this seems a little silly, but consider the following. electricity can be directly used to heat a home by passing an electric current through a heating coil. this is a direct, 100% conversion of work to heat. that is, 19.0 \rm kw of electric power (generated by doing work at the rate 19.0 kj/s at the power plant) produces heat energy inside the home at a rate of 19.0 kj/s. suppose that the neighbor's home has a heat pump with a coefficient of performance of 4.00, a realistic value. note: with a refrigerator, "what you get" is heat removed. but with a heat pump, "what you get" is heat delivered. so the coefficient of performance of a heat pump is k=qh/win. an average price for electricity is about 40 mj per dollar. a furnace or heat pump will run typically 200 hours per month during the winter. what does one month's heating cost in the home with a 16.0 kw electric heater? what does one month's heating cost in the home of a neighbor who uses a heat pump to provide the same amount of heating?

Atank has a shape of a cone with a radius at the top of 2 m and a height of 5 m. the tank also has a 1 m spout at the top of the tank. the tank is filled with water up to a height of 2 m. find the work needed to pump all the water out the top of the spout. (use 9.8 m/s2 for g and the fact that the density of water is 1000 kg/m3.)