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Exercise 8.7. Recall that the modified Dirichlet function is defined to be 9(2) = Sa ila...
2. Let U be a set, and let A CU. Recall the indicator function XA: U → Z2 defined by XA() : S 1, XEA 10, x¢ A. Now, let A, B CU and consider the symmetric difference of A and B defined by A A B = (A – B) U (B – A). (a) Show that A AB CU, and compute Ø A A. (b) Prove that Vx € U, XAAB(x) = x1(x) + XB(x), where addition is...
(a) Let u: R2R be a harmonic function. Show that the function v: R2R defined by is also harmonic. (b) Show that the tranformation maps the positive quadrant Q+-[(x,y): x > 0&y to the upper half plane c)Find the Dirichlet Green function for the positive quadrant + (a) Let u: R2R be a harmonic function. Show that the function v: R2R defined by is also harmonic. (b) Show that the tranformation maps the positive quadrant Q+-[(x,y): x > 0&y to...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
1. Using the Epsilon Criterion on a function with one discontinuity Consider the function g : [0, 2] + R where g(1) = 5 and g(x) = 1 otherwise. a) Find a partition P of (0, 2) so that U(9, P) - L(9, P) < 1/10. b) Is there a partition of (0, 2) so that U(9,Q) - L(9,Q) < 1/600? If so, find one! c) Suppose e > 0. Construct a partition Pof (0, 2) so that U(g, P.)...
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
Let U be a set, and let A CU. Recall the indicator function XA: U + Z, defined by XA(x) = ſi, rEA 0, A. Now, let A, B CU and consider the symmetric difference of A and B defined by A AB= (A - B)U(B - A). (a) Show that AAB CU, and compute Ø A A. (b) Prove that Ve EU, XAAB(C) = XA(2) + XB(2), where addition is taken modulo 2 (so that 1+1 = 0).
Consider a potential problem in the half-space defined by 2 20, with Dirichlet boundary conditions on the plane z = 0 and at infinity). (a) Write down the appropriate Green function G(x, x'). (b) If the potential on the plane z = 0 is specified to be 0 Vinside a circle of radius a centered at the origin, and Ø = 0 outside that circle, find an integral expression for the potential at the point P specified in terms of...
Let f be the function defined below on the given region R, and let P be the partition P = P x P. Find L(P). f (x, y) = 2x – 2y R:03 51, 0 Sy 31 1 P = - [#. - ( a) OL(P) = b) OL(P) = 1 -0) OL(P) = 1 1 1 1) O L;(P) = 12 ) OL(P) 7 12 None of these.
Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all partitions P of [0, 1]. b) Let m E N. Show that the cardinality of the set A :-{х є [0, 1] : f(x) > 1/m} is bounded by m(m...
In the exercise that chains are defined as in problem 4.6 All function are to be implemented by employing Chainletor to traverse a chain. Do it blow for the case of circularly linked lists : Let X11,x2,...Xbox be the elements of a chain. Each circuit is a integer. Write a C++ function to compute the expression 4. L nunction copy together with an as xi an integer. w Let x 1,x 2 be the elements of a chain. Each is...