Consider a restricted 2d random walk the random on a square lattice with lattice constant a walker is not allowed to step back, but it ca go forward, turn right and turn left with can 1 equal probability. calculate the mean-square end-to-end distance for such a restricted n-step random walk starting with the general equation n n 2(cos (0)) r2) (3) = a i=1 j=1 which in this case can be evaluated analytically. note that the values for the cosines can be only 0, 1 or - on the square lattice. derive the exact result for the characteristic ratio cn.

Classify the following soils according to the uscs classification system. provide appropriate uscs designations. a) this sample of well-graded gravel with sand was obtained from a large earthen dam in vancouver, canada. the sample has 73% fine to coarse sub-angular gravel, 25% fine to coarse sub-angular sand and 2% fines. the maximum size of the particles is 75 mm. the coefficient of curvature is 2.7, while the uniformity coefficient is 12.4. b) this dark brown and wet soil with a “strong organic odor” has 100% passing the no. 200 sieve. the liquid limit of the material is 32% when not dried and is 21% when oven-dried. the plastic index is 21% when not dried. c) this sand has 61% predominantly fine sand, 23% silty fines, and 16% fine sub-rounded gravel size. the maximum size is 20 mm. the liquid limit is 33% and the plastic limit is 27%. d) this soil has 74% fine to coarse sub-angular reddish sand and 26% organic and silty dark brown fines. the liquid limit is 37% when not dried and is 26% when oven dried. the plastic index is 6% when not dried.