# Berkeley Problems in Mathematics by Paulo Ney de Souza, Jorge-Nuno Silva (auth.) By Paulo Ney de Souza, Jorge-Nuno Silva (auth.)

Similar linear books

Banach Algebras (Modern Analytic and Computational Methods in Science and Mathematics)

Banach algebras are Banach areas built with a continuing binary operation of multiplication. various areas thought of in practical research also are algebras, e. g. the distance C(0, 1) with pointwise multiplication of capabilities, or the gap l1 with convolution multiplication of sequences. Theorems of the final idea of Banach algebras, utilized to these areas, yield a number of classical result of research, e.

The Linear Algebra a Beginning Graduate Student Ought to Know, Second Edition

This e-book carefully offers with the summary conception and, whilst, devotes massive house to the numerical and computational features of linear algebra. It encompasses a huge variety of thumbnail graphics of researchers who've contributed to the advance of linear algebra as we all know it at the present time and in addition comprises over 1,000 routines, lots of that are very tough.

Descriptive Topology and Functional Analysis: In Honour of Jerzy Kakol's 60th Birthday

Descriptive topology and useful research, with wide fabric demonstrating new connections among them, are the topic of the 1st element of this paintings. functions to areas of continuing services, topological Abelian teams, linear topological equivalence and to the separable quotient challenge are incorporated and are offered as open difficulties.

Extra resources for Berkeley Problems in Mathematics

Example text

Denote it by Sh(nh), 3. Let x be fixed and h = x / n. Show that lim Sx/n(nx/n) n-+oo = sinx . 1 (Su78) Let E be the set offunctions f : ~ tions to the differential equation f'" + f" - 2f = O. ~ ~ which are solu- 1. Prove that E is a vector space and find its dimension. 2. Let Eo C E be the subspace of solutions g such that lim g(t) = O. Find 1--+00 g E Eo such that g(O) = 0 and g' (0) = 2. 2 (Fa98) Find a function y(x) such that y(4) y(O) = 0, y'(O) = 1 and lim y(x) = lim y'(x) = O. 3 (Sp87) Let V be a finite-dimensional linear subspace of COO (~) (the space of complex valued, infinitely differentiable functions).

7 (Sp93) Prove that every solution x(t) (t 2: 0) of the differential equation dx 2 6 - = x -X dt with x(O) > 0 satisfies lim x(t) t--+oo = 1. S (SpSO) Consider the differential equation , x 3 -x x=--· 1 + eX I. Find all its constant solutions. 2. 10 (Fa82) Let f : JR --* JR be a continuous nowhere vanishing function, and consider the differential equation ~~ = f(y)· 1. For each real number c, show that this equation has a unique, continuously differentiable solution y = y(x) on a neighborhood of 0 which satisfies the initial condition y(O) = c.

Define the map J : l'2 -+ ~ by J(f) = = 10 1 f(x)2 dx. = Let Q (f E l'2 I f(l) I}. Show that J attains a minimum value on Q and determine where the minimum occurs. 16 (Su79) Let X be the space of orthogonal real n x n matrices. Let Vo E ~n. Locate and describe the elements of X, where the map f: X f(A) -+~, = (vo, Avo) takes its maximum and minimum values. 17 (Fa78) Let W C ~n be an open connected set and f a real valued function on W such that all partial derivatives of f are O. Prove that f is constant.