By Paulo Ney de Souza, Jorge-Nuno Silva (auth.)

In 1977 the math division on the collage of California, Berkeley, instituted a written exam as one of many first significant specifications towards the Ph.D. measure in arithmetic. Its function was once to figure out no matter if first-year scholars within the Ph.D. software had effectively mastered easy arithmetic as a way to proceed within the application with the chance of luck. on the grounds that its inception, the examination has develop into an immense hurdle to beat within the pursuit of the measure. the aim of this booklet is to publicize the cloth and reduction within the education for the exam in the course of the undergraduate years seeing that a) scholars are already deeply concerned with the cloth and b) they are going to be ready to take the examination in the first month of the graduate application instead of within the center or finish of the 1st 12 months. The e-book is a compilation of roughly 9 hundred difficulties that have seemed at the initial tests in Berkeley during the last two decades. it really is a useful resource of difficulties and options for each arithmetic scholar who plans to go into a Ph.D. application. scholars who paintings via this e-book will increase challenge fixing abilities in components comparable to actual research, multivariable calculus, differential equations, metric areas, complicated research, algebra, and linear algebra. the issues are geared up by way of topic and ordered in an expanding point of trouble. Tags with the precise examination yr give you the chance to rehearse whole examinations. The appendix comprises directions on getting access to digital types of the assessments in addition to a syllabus, information of passing rankings, and a Bibliography used in the course of the strategies. This re-creation comprises nearly one hundred twenty new difficulties and 2 hundred new suggestions. it truly is a fantastic capacity for college kids to reinforce their origin in simple arithmetic and to organize for graduate reports.

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**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.