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6. Show that the followings define metrics on R2: For r = (11, 12), y =...
(e) Let x = (T1,T2, . . . ,xn),y=(y1,y2, . . . ,Un) ER" (i) Show...
(e) Let x = (T1,T2, . . . ,xn),y=(y1,y2, . . . ,Un) ER" (i) Show that for any λ E R: 3 where llxll = 1/(x, x). x, y (ii) Use (7) for λ =- to show: 1a1 with equality, if and only if, there exists a λ E R such that y = 1x.
10. Define φ : R2 → R by φ(x,y-x + y for (x,y) E R2. Show...
10. Define φ : R2 → R by φ(x,y-x + y for (x,y) E R2. Show that φ is an onto homomorphism and find the kernel of φ (10 Points)
10. (10 points) A function f : R2 + R is called a probability density function...
10. (10 points) A function f : R2 + R is called a probability density function on D CR if (6) f(, y) 0 for all (x, y) E D and (i) SD. f(x,y)dA= 1. ſk(1 – 22 – y2) 22 + y2 <1 (a) For what constant k is the function f(z,y) a prob- 12 + y2 > 1 ability density function? Note that D= {(1, Y) ER? : x2 + y² <1}, the closed unit disk in R2...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
2. Show that B:R2 x R2 R be given by B ((21, 12), (81, y2)) =...
2. Show that B:R2 x R2 R be given by B ((21, 12), (81, y2)) = (x1 2a) ( ) (%) ax1y1 + bx1y2 + bx2yı + C2242 is negative definite iff a < 0 and ac- 62 > 0.
8.) (minimum along lines does not mean minimum) Define f: R2 and, if (a, y)0, R by f(0,0) (a) Pro...
8.) (minimum along lines does not mean minimum) Define f: R2 and, if (a, y)0, R by f(0,0) (a) Prove that f is continuous at (0,0). Hint: show that 4r4y2 < (z4 + y2)2. (b) Let & be an arbitrary line through the origin. Prove that the restriction of f [0, π) and t E R. (c) Show that f does not have a local minimum at (0,0). Hint: consider f(1,12). to ( has a strict local minimum at (0,0)....
Name: Hash Tables CS 2020 Hw 14 edefine MAX CAPACITY 11 #define R struct HashTable f...
Name: Hash Tables CS 2020 Hw 14 edefine MAX CAPACITY 11 #define R struct HashTable f int hashtable[MAX CAPACITY]: 1/ table to store values int countj // count of values currently stored int hash1(int value) ( return (value % MAX CAPACITY); int hash2(int value) ( return (R- (value % R)); 1) Use hash1 function and linear probing as a collision resolution strategy, insert the following elements into the correct index of the hash table: 1, 5, 4, 11, 21, 15,...
Let r= (11, 12) and y=(41,42) be vectors in the vector space Cover C, and define...
Let r= (11, 12) and y=(41,42) be vectors in the vector space Cover C, and define (): C2 x C2 C by (r,y) = r17 +iny2-irzyı + 2r272- 1 Apply the Gram-Schmidt orthogonalization process to {(1,0), (0, 1)} to conctruct an orthonor- mal basis for C2 with respect to (- -).
20. Show that the second derivative test is inconclusive when applied to f(r, y) 2 at (0,0). Desc...
20. Show that the second derivative test is inconclusive when applied to f(r, y) 2 at (0,0). Describe the behavior of the function at the critical point For the next few exercises things to know are: 1. In a closed and bounded region, a continuous function will assume a maximum value and it will assume ImIIm valuic. 2. These values have to be assumed either at a critical interior point or on the boundary. They canot be assumed anywhere else....
6. Let R be a ring, and let 11 and 12 be ideals of R. We...
6. Let R be a ring, and let 11 and 12 be ideals of R. We define the product of 11 and 12 to be 1112 = {TER:r => aibi, with k > 1, Q1, ..., ak € 11, b1,..., bk € 12 In other words, an element of the product 1.12 is a finite sum of products a;bi, where a, comes from I and bi comes from 12. (a) Prove that 11 12 is an ideal of R, contained...
(e) Let x = (T1,T2, . . . ,xn),y=(y1,y2, . . . ,Un) ER" (i) Show that for any λ E R: 3 where llxll = 1/(x, x). x, y (ii) Use (7) for λ =- to show: 1a1 with equality, if and only if, there exists a λ E R such that y = 1x.
10. Define φ : R2 → R by φ(x,y-x + y for (x,y) E R2. Show that φ is an onto homomorphism and find the kernel of φ (10 Points)
10. (10 points) A function f : R2 + R is called a probability density function on D CR if (6) f(, y) 0 for all (x, y) E D and (i) SD. f(x,y)dA= 1. ſk(1 – 22 – y2) 22 + y2 <1 (a) For what constant k is the function f(z,y) a prob- 12 + y2 > 1 ability density function? Note that D= {(1, Y) ER? : x2 + y² <1}, the closed unit disk in R2...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
2. Show that B:R2 x R2 R be given by B ((21, 12), (81, y2)) = (x1 2a) ( ) (%) ax1y1 + bx1y2 + bx2yı + C2242 is negative definite iff a < 0 and ac- 62 > 0.
8.) (minimum along lines does not mean minimum) Define f: R2 and, if (a, y)0, R by f(0,0) (a) Prove that f is continuous at (0,0). Hint: show that 4r4y2 < (z4 + y2)2. (b) Let & be an arbitrary line through the origin. Prove that the restriction of f [0, π) and t E R. (c) Show that f does not have a local minimum at (0,0). Hint: consider f(1,12). to ( has a strict local minimum at (0,0)....
Name: Hash Tables CS 2020 Hw 14 edefine MAX CAPACITY 11 #define R struct HashTable f int hashtable[MAX CAPACITY]: 1/ table to store values int countj // count of values currently stored int hash1(int value) ( return (value % MAX CAPACITY); int hash2(int value) ( return (R- (value % R)); 1) Use hash1 function and linear probing as a collision resolution strategy, insert the following elements into the correct index of the hash table: 1, 5, 4, 11, 21, 15,...
Let r= (11, 12) and y=(41,42) be vectors in the vector space Cover C, and define (): C2 x C2 C by (r,y) = r17 +iny2-irzyı + 2r272- 1 Apply the Gram-Schmidt orthogonalization process to {(1,0), (0, 1)} to conctruct an orthonor- mal basis for C2 with respect to (- -).
20. Show that the second derivative test is inconclusive when applied to f(r, y) 2 at (0,0). Describe the behavior of the function at the critical point For the next few exercises things to know are: 1. In a closed and bounded region, a continuous function will assume a maximum value and it will assume ImIIm valuic. 2. These values have to be assumed either at a critical interior point or on the boundary. They canot be assumed anywhere else....
6. Let R be a ring, and let 11 and 12 be ideals of R. We define the product of 11 and 12 to be 1112 = {TER:r => aibi, with k > 1, Q1, ..., ak € 11, b1,..., bk € 12 In other words, an element of the product 1.12 is a finite sum of products a;bi, where a, comes from I and bi comes from 12. (a) Prove that 11 12 is an ideal of R, contained...
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