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(R6) Let P1, P2, ... Pn be positive numbers such that n 1. рі For j...
2. Let e1(x) = 1, ez(x) = x, p1(x) = 1 – x and p2(x) =...
2. Let e1(x) = 1, ez(x) = x, p1(x) = 1 – x and p2(x) = –2 + x. Let E = (e1,e2) and B = (P1, P2). 2 a) Show that B is a basis for P1(R). 4 b) Let ce : R → R3 be the change of coordinates from E to ß. Find the matrix representation of C. Leave your answer as a single simplified matrix. 6 c) Let (:,:) be an inner product on P1(R). Suppose...
Topology For all s ε 1-1 1], let P1 (s) = 0 and Pn+1(s) = Pn(s) +--Re) for all n > 1. (b) (i) For every n > 1 and Isl-1, show that 0 < pn(s) sl and Pn(s) Pn+1(s) Conclude that(PJnzi converge...
Topology
For all s ε 1-1 1], let P1 (s) = 0 and Pn+1(s) = Pn(s) +--Re) for all n > 1. (b) (i) For every n > 1 and Isl-1, show that 0 < pn(s) sl and Pn(s) Pn+1(s) Conclude that(PJnzi converges uniformly to ρ on [-1,1], where pls) = Isl. (ii)
For all s ε 1-1 1], let P1 (s) = 0 and Pn+1(s) = Pn(s) +--Re) for all n > 1. (b) (i) For every n >...
Many thanks!! (a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that...
Many thanks!!
(a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that {fn} is a bounded sequence in LP (R) for all p E [1, 00]. Show that fn >0 pointwise everywhere in R, i.e. fn(x) -> 0 for all x E R. Show that fn does not converge to 0 in LP (R) (b) Fix p E 1, o0). Let fn E LP(0, 1) be defined by fn(x) n1/? on [0,1/n), and fn(x)0 otherwise. Show...
Let po, P1, ...,Pn be boolean variables. Define ak = (Pk + (ak-1)), where ao =...
Let po, P1, ...,Pn be boolean variables. Define ak = (Pk + (ak-1)), where ao = po. Prove the following boolean-algebra identity using proof by induction and the rules of boolean algebra (given below). Poan = po, for all n > 1. Equivalently this can be written out as: po · (Pn + (Pn-1 +...+(p2 + (p1 + po)...)) = po, for all n > 1. (p')=P (a) Commutative p.q=qp p+q = 9+p (b) Associative (p. 9).r=p.(q.r) (p+q) +r=p+(q +...
For the truss shown below, if P1 = 78 N, P2 = 86 N, and P3...
For the truss shown below, if P1 = 78 N, P2 = 86 N, and P3 = 95 N, and if d = 8 m, calculate the force on member FC (positive for compression, negative for tension). For the truss shown below, if P1 = 66 N. P2 = 69 N and P3 = 76 N, and for d = 6 m, calculate the force in member FB. (Use positive numbers for compression forces, and negative numbers for tensile forces)...
TS2 TS1 EA2 EA1 Energy a) Product P1 is formed faster. b) Product P2 is formed...
TS2 TS1 EA2 EA1 Energy a) Product P1 is formed faster. b) Product P2 is formed faster. c) Product SM P1 is more P1 thermodynamically stable d) Product thermodynamically stable. e) Performing the reaction at low carefully controlled temperature will favor the formation of product P1. P2 is more P2 Reaction coordinate f) Performing the reaction at low, carefully controlled temperature will favor the formation of product P2. g) Running the reaction at sufficiently high temperature for a very short...
1.28. Let(P1,P2, . . . , pr} be a set of pri N pip.pr +1. Prove...
1.28. Let(P1,P2, . . . , pr} be a set of pri N pip.pr +1. Prove that N is divisible by some prime not in the original set. Use this fact to deduce that there must be infinitely many prime numbers. (This proof of the infini. tude of primes appears in Euclid's Elements. Prime numbers have been studied for thousands of years.)
1. Let {rn;n > 1} be a sequence of real numbers such that rn → x,...
1. Let {rn;n > 1} be a sequence of real numbers such that rn → x, where r is real. For each n let yn = (1/n) E*j. Show that yn + x. HINT: (xj – a) Let e >0 and use the definition of convergence. Split the summation into two parts and show that each is < e for all sufficiently large n.
(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set...
(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set of all polynomials. Let the polynomials li() defined by II;tilt - a;) i=0,1,...11 bi(T) = 11: a; - aj) where aj, j = 0,1,..., are distinct real numbers and aia . Show that (d) The change of basis transformation from the standard basis ', j = 0,1,...,n to l; () is given by the Vandermonde matrix (1 00 ... am 1 01 .01 1...
Example 2.1.4 A counterexample. Let with probability 1-Pn 7L n with probability Pn Then Yn--1, provided...
Example 2.1.4 A counterexample. Let with probability 1-Pn 7L n with probability Pn Then Yn--1, provided Pn → 0 (Problem 1.2(i). On the other hand, E(%) = (1-m) + npn which tends to a if Pn = a/n and to oo if, for example, Pn = 1/vn. This shows that (i) need not hold, and thati) need not hold is seen analogously (Problem 1.2( 1.2 In Example 2.1.4, show that p (i) Y, 1 if Pn → 0; (ii) EK,-,...
2. Let e1(x) = 1, ez(x) = x, p1(x) = 1 – x and p2(x) = –2 + x. Let E = (e1,e2) and B = (P1, P2). 2 a) Show that B is a basis for P1(R). 4 b) Let ce : R → R3 be the change of coordinates from E to ß. Find the matrix representation of C. Leave your answer as a single simplified matrix. 6 c) Let (:,:) be an inner product on P1(R). Suppose...
Topology
For all s ε 1-1 1], let P1 (s) = 0 and Pn+1(s) = Pn(s) +--Re) for all n > 1. (b) (i) For every n > 1 and Isl-1, show that 0 < pn(s) sl and Pn(s) Pn+1(s) Conclude that(PJnzi converges uniformly to ρ on [-1,1], where pls) = Isl. (ii)
For all s ε 1-1 1], let P1 (s) = 0 and Pn+1(s) = Pn(s) +--Re) for all n > 1. (b) (i) For every n >...
Many thanks!!
(a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that {fn} is a bounded sequence in LP (R) for all p E [1, 00]. Show that fn >0 pointwise everywhere in R, i.e. fn(x) -> 0 for all x E R. Show that fn does not converge to 0 in LP (R) (b) Fix p E 1, o0). Let fn E LP(0, 1) be defined by fn(x) n1/? on [0,1/n), and fn(x)0 otherwise. Show...
Let po, P1, ...,Pn be boolean variables. Define ak = (Pk + (ak-1)), where ao = po. Prove the following boolean-algebra identity using proof by induction and the rules of boolean algebra (given below). Poan = po, for all n > 1. Equivalently this can be written out as: po · (Pn + (Pn-1 +...+(p2 + (p1 + po)...)) = po, for all n > 1. (p')=P (a) Commutative p.q=qp p+q = 9+p (b) Associative (p. 9).r=p.(q.r) (p+q) +r=p+(q +...
For the truss shown below, if P1 = 78 N, P2 = 86 N, and P3 = 95 N, and if d = 8 m, calculate the force on member FC (positive for compression, negative for tension). For the truss shown below, if P1 = 66 N. P2 = 69 N and P3 = 76 N, and for d = 6 m, calculate the force in member FB. (Use positive numbers for compression forces, and negative numbers for tensile forces)...
TS2 TS1 EA2 EA1 Energy a) Product P1 is formed faster. b) Product P2 is formed faster. c) Product SM P1 is more P1 thermodynamically stable d) Product thermodynamically stable. e) Performing the reaction at low carefully controlled temperature will favor the formation of product P1. P2 is more P2 Reaction coordinate f) Performing the reaction at low, carefully controlled temperature will favor the formation of product P2. g) Running the reaction at sufficiently high temperature for a very short...
1.28. Let(P1,P2, . . . , pr} be a set of pri N pip.pr +1. Prove that N is divisible by some prime not in the original set. Use this fact to deduce that there must be infinitely many prime numbers. (This proof of the infini. tude of primes appears in Euclid's Elements. Prime numbers have been studied for thousands of years.)
1. Let {rn;n > 1} be a sequence of real numbers such that rn → x, where r is real. For each n let yn = (1/n) E*j. Show that yn + x. HINT: (xj – a) Let e >0 and use the definition of convergence. Split the summation into two parts and show that each is < e for all sufficiently large n.
(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set of all polynomials. Let the polynomials li() defined by II;tilt - a;) i=0,1,...11 bi(T) = 11: a; - aj) where aj, j = 0,1,..., are distinct real numbers and aia . Show that (d) The change of basis transformation from the standard basis ', j = 0,1,...,n to l; () is given by the Vandermonde matrix (1 00 ... am 1 01 .01 1...
Example 2.1.4 A counterexample. Let with probability 1-Pn 7L n with probability Pn Then Yn--1, provided Pn → 0 (Problem 1.2(i). On the other hand, E(%) = (1-m) + npn which tends to a if Pn = a/n and to oo if, for example, Pn = 1/vn. This shows that (i) need not hold, and thati) need not hold is seen analogously (Problem 1.2( 1.2 In Example 2.1.4, show that p (i) Y, 1 if Pn → 0; (ii) EK,-,...
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