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4 Feb 2015: Find an optimal binarycode. Determine whether there are optimal binary codes with either (a) all but one codeword ofthe same length, or (b) each codeword a different length. ... Show that in the case where all of the odds are equal this maximum andthe

3 Nov 2017: 3. Show that the function f : Rn R given by f(v) = ‖v‖2 is differentiable at all nonzerov V. ... Show that f is continuousat (0, 0) and that it has directional derivatives in all directions there.

14 May 2003: Everything takes place inside the vector space of all real-valued functions on B. ... ij. =a. i. b. j. Notice that matrices of the form v@w are only a small subset of all m-by-n matrices.

19 Feb 2009: Deduce that traces of all words in A and Band their inverses (i.e. ... of all elements of the group 〈A, B〉 generated by Aand B) are determined by the three numbers {tr(A), tr(B), tr(AB)}.

1 Nov 2010: Is it Lipschitz equivalent. to the uniform norm? (b) Let R[0, 1] denote the vector space of all integrable functions on [0, 1]. ... Usingthe fact that all norms on a finite-dimensional space are Lipschitz equivalent, deduce that α iscontinuous.

5 May 2016: Show that f is continuous ifand only if f(A) f(A) for all A X. ... Show thatthis is a topology, that all points of R are closed with respect to it, but thatthe topology is not Hausdorff.

31 Oct 2011: Is it Lipschitz equivalent. to the uniform norm? (b) Let R[0, 1] denote the vector space of all integrable functions on [0, 1]. ... Usingthe fact that all norms on a finite-dimensional space are Lipschitz equivalent, deduce that α iscontinuous.

8 May 2007: 2. (a) Define a subset of the integers Z to be open either if it is empty or if for some k Zthe set S contains all integers k. ... b) For a function f : X Y , show f is continuous if and only if for all A X,f (cl(A)) cl(f (A)).

9 Oct 2017: 0, 1] |. 10 f(x) dx = 0}? 5. Let 0 be the set of real sequences (xn) such that all but finitely many xn are 0. ... a) Show that |ϕ(s) ϕ(t)| |s t| for all s,t R.(b) Define f(x) =. n=0. (34. )nφ(4nx). Prove that f is well-defined and continuous.

2 Nov 2015: 1. Quickies: (a) Describe all continuous functions f : [0, 1] Rn satisfying ‖ 10f‖ =. 10‖f‖. (b) Show that two norms ‖ ‖, ‖ ‖′ on a vector space V are Lipschitz equivalent if and only ... denote the vector space of all bounded Riemann

21 May 2005: Suppose that for. x [a,b], |N. 0fn(x)| K, where K is constant, for all N; and suppose that gn(x) is monotonic. ... Suppose that. 0fn(x) is. uniformly convergent on [a,b], that each gn(x) is bounded on [a,b], and that gn(x) gn1(x) 0for all x [a,b].

8 Oct 2006: b) Symmetry: d(x, y) = d(y, x) for all x, y X;. ... z for all z C {}. This occurs if and only if a = d and b = c = 0, so(.

3 Sep 2003: S. That sounds pretty convincing. So let's include all algebraic numbers then. ... All right. It says that if f is a continuous function, a< b are real numbers and f(a)=u, f(b)=v, then for any w between u and v

29 Jan 2002: Yes it is: all our numbers are integer combinations of things like 2. ... But what is the dimension of Q(b) over Q(a)? It means the dimension of the vector space of all linear combinations of powers of b, with coefficients in

19 Jan 2018: 8. Suppose X and Y are metric spaces. A mapf : X Y is an isometric embedding ifdX(x1,x2) = dY (f(x1),f(x2)) for all x1,x2 X. ... 10. Let f : Rn Rn be a C1 map. Suppose that there is some constant µ < 1 such that‖Df|x I‖op < µ for all x Rn.

17 Feb 2014: Deduce that traces of all words in A and B and theirinverses (i.e. ... of all elements of the group 〈A,B〉 generated by A and B)are determined by the three numbers {tr(A), tr(B), tr(AB)}.(ii) Suppose A and B are

28 May 2007: T (x) = B(x) v for all x R2. When T is an isometry with det B = 1, show that T is either a reflection or a glide reflection.2. ... Define the Schottky group corresponding to K pairs of discs all of which are disjoint.

19 Nov 2010: Deducethat the set GLn(R) of all invertible nn real matrices is an open subset of Mn(R). ... some r > 0 and a continuous functiong : B(I; r) Mn such that g(X)2 = X for all X B(I; r).Is it possible to define a square-root

16 May 2016: for every x X, there is an open neighborhood U of x such thatf(y) = f(x) for all y U. ... Hint:first find a subsequence (fni ) such that fni (x) converges for all x Q [0, 1].)Deduce that the closure of B in (C[0, 1], d) is compact.

22 Mar 2018: Show that the collection of all such subsets of X form a base for a topology on X. ... Let. Y := {f : A {0, 1} : f(α) = 0 for all but countably many α A} X.