Foundations of Analysis Let X = {A, B,C,D}. ·Describe a function δ : X × X-> R that is a metric on X. . Describe a function δ : X x X-R that satisfies δ(z,x)-0 and δ(x,y)-δ(y,x) for all z, y E X b...
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) = 2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
Consider the set of integers Z with the metric da.y)-2supm e Nu (o): 2" divides (r-y) (a) Describe the open balls of radius 1 around the centres 0 and 1 (b) Let f : Z -R be defined as f()0 if is even and (x)1 f r is odd. Is f a continuous function from (Z, d) to R equipped with the standard metric? Himt: Use the criterion of continuity in terms of open sets
A function f : R^2 ↦R is called a metric if f(x, y) >= 0 for all x, y that are an element of R, f(x,y) = 0 if and only if x = y, and for any x, y, z we have f(x, z) <= f(x,y) + f(y, z). Is the function f(x, y) = the square root of |x - y| a metric or not?
Consider the function Let where f(t) is differentiable for all t ∈ R. Show that z satisfies the partial differential equation (x2 − y2 ) ∂z/∂x + xy ∂z/∂y = xyz for all (x, y) ∈ R2 \ { (t, 0)|t ∈ R }.
A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f"-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C...
Does it satisfies the completeness axiom? R3 = {(x, y, z) x,y,z E R}, with the usual metric de obtained from Pythagoras's theorem. di ((x1, 71,21). (X2, Yz, 22)) = ((x1 - x2)2 + (y1 - y2)2 + (21 - 22)
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
Let x, y e R" for n e N, writing x-(n, ,%), similarly for y. The Euclidean Metric of $2.2 is often called the 12 metric and written |x - y l2 for x,y e R. Show that the following three similar relations are also metrics: (a) the tancab, or 11 metric: lx-wi : = Σ Iri-Vil (b) the marirnum, or lo-metric. Ilx-ylloo:=max(zi-yil c) the comparison, or 10-metric. Ix_ylo rn (ri-Vi) where δ(t)- if t = 0
Let 0< a<b<e<d for a, b, c, d E R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation x=au + bu, y=cu + dv. (a) Sketch D in the r-y plane for the case ad -bc > 0. (a) Sketch D in the r-y plane for the case ad bc < 0. (c) Calculate the area of D. Show all working. Let 0
6 6. Let (X, d) be a metric space and T the topology induced on X by d. Let Y be a subset of X and di the metric on Y obtained by restricting d; that is, di(a, b) d(a, b) for all a and b in Y. If T1 is the topology induced orn Y by di and T2 is the subspace topology on Y (induced by T on X), prove that Ti -T2. [This shows that every subspace...