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+...

A pplication-orien ted validation o f phonetic transcriptions: prelim inary results

There is an increasing need for automatic procedures to generate and validate phonetic transcriptions. As the production of manual phonetic transcriptions tends to be time-consuming, error-prone and costly, procedures have been developed to derive phonetic transcriptions automatically by means of automatic speech recogni­ tion technology. Such automatic phonetic transcrip­ tions are usually val...