# 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

Model Categories and Their Localizations

###############################################################################################################################################################################################################################################################

Uniqueness of the Injective III1 Factor

In keeping with lectures brought to the Seminar on Operator Algebras at Oakland college through the iciness semesters of 1985 and 1986, those notes are an in depth exposition of contemporary paintings of A. Connes and U. Haagerup which jointly represent an evidence that every one injective elements of kind III1 which act on a separable Hilbert area are isomorphic.

Linear Triatomic Molecules - CCH

With the appearance of contemporary tools and theories, a large amount of spectroscopic info has been amassed on molecules in this final decade. The infrared, specifically, has noticeable remarkable task. utilizing Fourier rework interferometers and infrared lasers, exact information were measured, frequently with severe sensitivity.

Extra resources for Berkeley Problems in Mathematics

Sample text

Show that if Xo i= 0 and YO i= 0, the solution referred to in Part 1 approaches the circle x 2 + y2 = 1 as t -+ 00. 5 (SpS4) Show that the system of differential equations ~(~)=(~ ~ ~)(~) 3 dt zOO has a solution which tends to 00 as t -+ -00 z and tends to the origin as t -+ +00. 6 (Sp91) Let x(t) be a nontrivial solution to the system dx = Ax, dt where 1 A= ( -4 -3 6 4 -9 Prove that IIx(t) II is an increasing function oft. 7 (Su84) Consider the solution curve (x(t), y(tÂ» to the equations dx 1 2 .

Prove that F is continuous and periodic with period 1. 2. Prove that if G is continuous and periodic with period 1, then t F(x)G(x)dx = 1 f(x)G(x)dx. 31 (Sp79) Show that for any continuous function f : [0, 1] and e > 0, there is afunction of the form -7 JR. 7 E :l;, where Co, ... , en E Q and Ig(x) - f(x)1 < eforall x in [0,1]. 1 (SpSO) Let f : JR. -7 JR. be the unique function such that f(x) = x if -T( :s x < T( and f(x + 2nT() = f(x)forall n E IZ. 1. Prove that the Fourier series of f is t n=l (_l)n+12sinnx.

Show that L is 'an open subset of Mmxn. 2. Show that there is a continuous function T : L -+ Mm xn such that T(A)A = 1m for all A, where 1m is the identity on IRm.