0 - Prelim

Prelim Problem Solutions

1. Complex Analysis Practice Problems 2.0 Complex 2.0 #9.2 Let D be a domain which contains in its interior the closed unit disk |z| ≤ 1. Let f(z) be analytic in D except at a finite number of points z1, . . . , zk on the unit circle |z| = 1 where f(z) has first order poles with residues s1, . . . , sk. Let the Taylor series of f(z) at the origin be f(z) = ∑∞ n=0 anz n. Prove that there exists ...

Algebra Prelim Fall 2013

First we will prove a small lemma. Lemma 1. Let P be a p-Sylow subgroup of G. Let N be a normal subgroup of G such that P ⊂ N . Then all p-Sylow subgroups are in N . Proof. Let P̃ be a p-Sylow subgroup that is not P . Then by the Sylow Theorems, we know that gPg−1 = P̃ for some g ∈ G. Since P ⊂ N , and N is normal, we know that gPg−1 ⊂ N and hence P̃ ⊂ N . Thus all p-Sylow subgroups are in N . Now...

Algebra Prelim Fall 2012

Macro Prelim Q1 -answers (by K. Fukushima)

1. A competitive equilibrium is a sequence (p * t , c * t , s * t+1) T t=0 such that (i) (c * t , s * t+1) T t=0 solves max (ct,s t+1) T t=0 T t=0 β t u(c t), subject to (1) c t + p * t s t+1 ≤ (p * t + y t)s t , ∀t c t , s t ≥ 0, ∀t s 0 = 1 and (ii) markets clear: c * t = y t , ∀t s * t = 1, ∀t 2. The equation is: p * t = β u (y t+1) u (y t) (p * t+1 + y t+1) Proof: Let (p * t , c * t , s * t+...

بررسی‌های ساختاری و مغناطیسی نانوذرات‌ پروسکایت LaMn1-xCoxO3 (0/1، 75/0، 5/0، 25/0، 0/0 = x)

In this work, the structural and magnetic properties of LaMn1-xCoxO3 (x = 0.00, 0.25, 0.50, 0.75, 1.00) are investigated. The structural characterization of compounds by X-ray powder diffraction is evidence for a rhombohedral structure (R -3c space group). Much less increase of the unit cell volume suggests that Co enters in lattice as Co+2 for x ≤ 0.5 while much more decrease of the unit cell ...