Quantitative Research Guidelines IH Research Dept Version 1.1 08/01/2007 1 of 3 CRITICAL REVIEW OF QUANTITATIVE RESEARCH - GUIDELINES The following guidelines were adapted from many sources that outline how to critically evaluate the research literature (refer to reference section at the end of ...

Section4.3: Relatively Prime Integers Letaandbbe integers, not both zero (sogcd(a; b) exists). Letd=gcd (a; b) and let S=fc 2Zjthereexist integersmandn such **thatc**=ma+nbg: We have seen, in Theorem 5 of Section 4.2, **thatc** 2 Sifandonlyifd dividesc; that is, S consists of all integer multiples ofd.

It took 20 years to show **thatc** (n) =O (n/logn) proved by Chiniforooshan. Today we know that the cop number is at most n2 (1+o (1)) p log 2 n (whichisstilln 1 o (1)) for any connected graph onnvertices (the result obtained independently by Luand Peng, Scott and Sudakov, and Frieze, Krivelevichand Loh). 1

By a theorem of Doob (1953), this implies a finite random variabled such that v + (a t,z t) converges tod with probability 1.If >1, v + (a t,z t) E[v + (a t+1,z t+1) |z t]implies v + (a t,z t) 0 from which it is easy to show **thatc** t must converge to infinity.

It is possible that the quadratic equation is degenerate in the sense **thatc** 2 =0. In this case the equation is linear, but even that might be degenerate in the sense **thatc** 1 =0.

(a) Assume **thatc** t is chosen after R t is observed, formulate the Bellman equation and derive the Euler equation. (b) Now assume **thatc** t must be chosen before R t is observed.

If the linear loop program P1, defined by an (N◊N) -matrix A and anonzeroN◊1 -vectorc, is nonterminating then there exists areal eigenvector vofA, corresponding to positive eigenvalue, such **thatc** T v 0.

Linear dependence and independence Definitions: •AfinitesetS={x 1, x 2,..., x m}ofvectorsinR n is said to be linearly dependent if there exist scalars (real numbers) c 1,c 2,...,c m, not all of which are 0, such **thatc** 1 x 1 +c 2 x 2 +... +c m x m =0.

Substituting THAT 2180-and 2181-Series VCAs forT HAT 2150-Series VCAsinExisting Designs THAT Corporation Design Note 137

Assuming the Birch and Swinnerton-Dyer conjecture (or even the weaker statement **thatC** n (Q) is infinite, L (C n, 1) =0) one can show that any n 5,6,7 mod8 is a congruent number, and, moreover, Tunnellhasshown, again assuming the conjecture, that for noddandsquare-free nisacongruent number () #{x,y,z 2Z: 2 x 2 ...