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. ADVANCED MATHEMATICAL METHODS. METRIC SPACES Question 1....

. ADVANCED MATHEMATICAL METHODS. METRIC SPACES Question 1. Let d be some metric on X. Prove that for all x, y, w, and Z in X: d(x, y) + d(w,z) < d(x, w) + d(x,2 z) + d(y, w) + d(y, z). Question 2. Let X be a non-empty set and d : X X X R be the function that satisfies the following two p...

PROBLEM 1. Show that the Hamming distance is a metric on V&...

PROBLEM 1. Show that the Hamming distance is a metric on V" the set of all words of length n. PROBLEM 2. Compare the Hamming metric and the Levenshtein metric on the set V". Is either metric, in general, always less than or equal to the other? Prove your assertion. PROBLEM 3...

1. a) Suppose that j : M N is an injective immersion betwee...

1. a) Suppose that j : M N is an injective immersion between two smooth manifolds. (i) Is it true that j is an embedding? (ii) Assume. in addition, that J is a proper map. Is j is a embedding? b) Is a composition of two smooth embeddings an embedding? 2. Let \T : E M be a smooth rank k: vector ...

Problem 3. In this problem, we investigate several impossibi...

Problem 3. In this problem, we investigate several impossibility results. (a) Prove there is no continuous surjective function from the sphere S² to the real line R. (b) Prove that there is no continuous surjective function from the plane with two rays removed C \ R to the plane with thre...

Problem 8. Figure 2.42 below gives knot diagrams for three t...

Problem 8. Figure 2.42 below gives knot diagrams for three types of knots with crossing number 6. De- termine which of these are colorable with three colors. C A FIGURE 2.42 The three basic knot types of crossing number 6. Problem 9. This problem concerns the Alexander Polynomial. (a) Compute...

Problem 11. In this problem, you will prove a classification...

Problem 11. In this problem, you will prove a classification theorem for 1-dimensional manifolds. In particular, you will prove that every closed 1-dimensional manifold (without boundary) is homeomorphic to the circle S¹. Let M be a closed (i.e., compact and connected) 1-dimensional manifold....

Problem 4. Prove care about the lengths of distances, even ...

Problem 4. Prove care about the lengths of distances, even if points X in with the metric space become arbitrarily far away. More specifically, prove the that topology doesn't following. Given any metric space d: (a) the function d(x,y) = 1+d(x,y) d(x,y) is a metric on X. (b) the metrics &agra...

Problem 6. Show that the open ball of radius 1 in Rⁿ cente...

Problem 6. Show that the open ball of radius 1 in Rⁿ centered at the origin is homeomorphic to Rⁿ itself. Problem 7. Consider an oriented link of two components. (a) Show that changing an overcrossing to an undercrossing changes the sign of the number assigned to that crossing. (b) What effec...

Section 1.2 1. Let S and T be sets. Show that jSj jTj if ...

Section 1.2 1. Let S and T be sets. Show that jSj jTj if and only if there is a surjective map from T onto S. 2. Let S be a set. Show that jSj < jB(S). (Hint: Assume that there is a surjective map f : S ! B(S); and consider the set fx 2 S : x =2 f(x)g.) 3. Let (Sn)1 n=1 be a sequence o...

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