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Consider the set of integers Z with the metric da.y)-2supm e Nu (o): 2" divides (r-y)...
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in...
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in class is a valid norm on R2 (b) Prove that in R2, any open ball in 12 ("Euclidean metric") can be enclosed in an open ball in the loo norm ("sup" norm). (c). Say I have a collection of functions f:I R. Say I (1,2). Consider the convergence of a sequence of functions fn (z) → f(x) in 12-Show that the convergence amounts to...
Let R be the relation defined on Z (integers): a R b iff a + b...
Let R be the relation defined on Z (integers): a R b iff a + b is even. Then the distinct equivalence classes are: Group of answer choices [1] = multiples of 3 [2] = multiples of 4 [0] = even integers and [1] = the odd integers all the integers None of the above
Let R^2 be equipped by the metric ds^2 = (4/(1 + x^2 + y^2 )^2) (dx^2 + dy^2 ), i.e. its first fu...
Let R^2 be equipped by the metric ds^2 = (4/(1 + x^2 + y^2 )^2)
(dx^2 + dy^2 ), i.e. its first fundamental form is E = G =
4/(1+x^2+y^2)^2 , F = 0. Use the formula dω12 = −Kω1 ∧ ω2 to
calculate its Gauss curvature.
Let R2 be equipped by the metric i.e. its first fundamental form is E = G = TrtFF, F = 0. Use the formula dui,-- . КМ Л w2 to calculate its Gauss...
problem1&2 thx！ interval in R is a set IC R such that for all <y < z in R, if E I and z e I then Recall t...
problem1&2 thx！
interval in R is a set IC R such that for all <y < z in R, if E I and z e I then Recall that an points yE I. We call an interval non-degenerate if it contains at least two (1) Let I be a nondegenerate interval in R, and suppose f: IR is continuous (a) Show that f[] is an interval in R. (b) Show that if I is closed and bounded, then so is...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and Let R and S be arbitrary nonempty subsets of Z. Define an even indicator function F F: ZP by F(x) = (x + 1) mod 2 for x e Z That is, F(x) 1 if x is even, and F(x) = 0 if x is odd. or neither? Explain. a) Is F: Q P one-to-one, onto, both, or neither? Explain. b) Is F: (Pn...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
Th 5. (14pt) A function defined on D C R is said to be somewhat continuous if for any e >1, (51) ...
th 5. (14pt) A function defined on D C R is said to be somewhat continuous if for any e >1, (51) Prove or find a counter example for the statement "a somewhat continuous func- (52) Let EC Rm, f: ER, and a e E. Prove or disprove that f is continuous at a there is a 6 0 such that whenever z,y D and la -vl <6, then f()-)e es tion is continuous." if and only if given any...
1. (a) Let d be a metric on a non-empty set X. Prove that each of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 ...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in class is a valid norm on R2 (b) Prove that in R2, any open ball in 12 ("Euclidean metric") can be enclosed in an open ball in the loo norm ("sup" norm). (c). Say I have a collection of functions f:I R. Say I (1,2). Consider the convergence of a sequence of functions fn (z) → f(x) in 12-Show that the convergence amounts to...
Let R^2 be equipped by the metric ds^2 = (4/(1 + x^2 + y^2 )^2)
(dx^2 + dy^2 ), i.e. its first fundamental form is E = G =
4/(1+x^2+y^2)^2 , F = 0. Use the formula dω12 = −Kω1 ∧ ω2 to
calculate its Gauss curvature.
Let R2 be equipped by the metric i.e. its first fundamental form is E = G = TrtFF, F = 0. Use the formula dui,-- . КМ Л w2 to calculate its Gauss...
problem1&2 thx！
interval in R is a set IC R such that for all <y < z in R, if E I and z e I then Recall that an points yE I. We call an interval non-degenerate if it contains at least two (1) Let I be a nondegenerate interval in R, and suppose f: IR is continuous (a) Show that f[] is an interval in R. (b) Show that if I is closed and bounded, then so is...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and Let R and S be arbitrary nonempty subsets of Z. Define an even indicator function F F: ZP by F(x) = (x + 1) mod 2 for x e Z That is, F(x) 1 if x is even, and F(x) = 0 if x is odd. or neither? Explain. a) Is F: Q P one-to-one, onto, both, or neither? Explain. b) Is F: (Pn...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
th 5. (14pt) A function defined on D C R is said to be somewhat continuous if for any e >1, (51) Prove or find a counter example for the statement "a somewhat continuous func- (52) Let EC Rm, f: ER, and a e E. Prove or disprove that f is continuous at a there is a 6 0 such that whenever z,y D and la -vl <6, then f()-)e es tion is continuous." if and only if given any...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
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