Berkeley Problems in Mathematics (Problem Books in by Paulo Ney de Souza, Jorge-Nuno Silva PDF

By Paulo Ney de Souza, Jorge-Nuno Silva

ISBN-10: 0387008926

ISBN-13: 9780387008929

This booklet collects nearly 9 hundred difficulties that experience seemed at the initial assessments in Berkeley over the past two decades. it really is a useful resource of difficulties and strategies. Readers who paintings via this publication will enhance challenge fixing talents in such components as actual research, multivariable calculus, differential equations, metric areas, advanced research, algebra, and linear algebra.

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20 (Fa93) Let K be a continuous real valued function o n [0,1] x [0,1]. Let F be the family of functions f o n [0,1] of the f o r m with g a real valued continuous function o n [0,1] satisfying )g)5 1 everywhere. Prove that the family F is equicontinuous. 21 (Fa78) Let {gn} be a sequence of Riemann integrable functions from [O, 11 into R such that lglL(x)l 5 1for all n, x. Define Prove that a subsequence of {G,} converges uniformly. 22 (Su79) Let { f n } be a sequence of continuous maps [0,1] 4 R such that I’ for all 71.

Prove that the map f : R --+ R, attains a maximum value. 5 (Su84) Let z ( t ) be the solution of the differential equation x”(t) + 8x’(t) + 2 5 x ( t ) = 2 cost with initial conditions x ( 0 ) constants a and 6 , = 0 and x’(0) = 0. Show that for suitable lim ( x ( t )- a c o s ( t - 6 ) ) = 0. 6 (Fa79, Su81, Fa92) Let y = y ( x ) be a solution of the diflerential equation y” = -1yI with --oo < x < 00, y ( 0 ) = 1 and y’(0) = 0. 1. Show that y is an even function. 2. Show that y has exactly one zero on the positive real axis.

Zn), dt where F = (Fl,.. , F,) : R” 1. Let ---f R” is a C1 vector field. U and V be two solutions o n a < t < b. Assuming that ( D F ( z ) z ,2 ) I 0 for all z, z in R”, show that lU(t) - V(t)I2is a decreasing function oft. 2. Let W(t) be a solution defined f o r t > 0. Assuming that ( D F ( z ) z 2, ) I -142, show that there exists C E R” such that l i m W ( t )= C. dV ax, a - Let z ( t )= (zl(t),. . , x,(t)) be a solution of this system o n a finite interval a

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Berkeley Problems in Mathematics (Problem Books in Mathematics) by Paulo Ney de Souza, Jorge-Nuno Silva

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