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Q1:Determine whether each series is convergent or divergent. For each alternating series that is convergent, determine whether it is conditionally or absolutely convergent.See Answer
Q2:(1 + 3 Points) A 2 (a) Calculate the expression 1 @ 2@...@(n-2) @ (n-1) in the group (Zn,) for (i) n € {3,4,5,6) (ii) n N, n > 2. (b) Terence Tao's birthday is on July 17th, which is a Monday in 2023. To what days of the week is his/her birthday in 2050 and 2010? Justify the answer in a group-theoretical way. (2 + 2 Points)See Answer
Q3:(2 + 2 Points) A 3 Let Pk: Z10 -> Z10 be an illustration with the rule Pk (n) = (k n) mod 10, k N. The figure yr is an encryption of Z10 if yr is injective. (a) Alice would like to send her friend Bob her phone number 0152-347896. For security reasons, she would like to encrypt the number and add a Find encryption of the form pr. First, she tries to do this with the- educations y2 and p3. What is your phone number in each case, after you click on every digit that has applied figures y2 and p3 respectively? (b) Show that v3 is an encryption. For which k € Z10 N N is the decoding (i.e. reverse graphing) of y3? (c) Why is 22 not an encryption? (d) For which k € N is yr an encryption and for which k € N is yk no encryption? Justify your answer. 4k (1+2+1 +2 Points) For the summer semester beginners: A figure f: X-> Y is called injective if there are no two x values that are mapped to the same y value: for all x1 # x2 € x, f(x1) # f(x2) apply. An equivalent formulation is: if f(x1) = f(x2) for two values x1, x2 € X, then x1 must be = x2 apply.See Answer
Q4:A 4 (a) Determine all the elements of Z35. (b) Show that U = {1, 8, 22, 29} a subgroup of Z35 is, so (U, O) < (Z85, 0). (c) Find a subgroup of Z35 that contains 11 as an element. (1+2+1 Points) Please justify all answers. Solutions without a solution path will not be evaluated.See Answer
Q5:Task 45: Show the following addition theorem of the tangent' for x, y, x + ye (-1, 1): tan(x) + tan(y) tan(x +)1 - tan(x) tan(y)See Answer
Q6:Task* 46: Solve one of the following subtasks: (a) Show, lim (1 + 2)" = Σo (b) If (an)nen is a zero sequence with 0 # an eRso for all n e N, then the following applies lim (1 + an) = e. (c) For x € R20 show lim0 (1 + )" = e. AR Notes: a. if sn denotes the nth partial sum of the series and tn = (1+), then let tn Ssn and limn-00 tn 2 sm for all M. Note that the convergence of the two sequences (sn)nen and (tn)n21 is already known, see task 29. For B., divide the sequence into partial sequences of (tn)n >1.See Answer
Q7:Task 47: Decide at which points the following functions differ- are ornamental, and prove the statement: (a) f: R -> R: x x Ixl. (b) g:R-->R:x2x2, for x $1, |x. + 1, for x > 1. 1See Answer
Q8:Task 48: Edit one of the two subtasks: (a) By defining the derivative as the limit value of the difference between- that the root function f : (0, 00) -> R : x - x is differentiable with f': (0, 00) -> R: x- 1 2. √x (b) Determine the derivative for one of the functions with the following functions- with the help of the derivation rules at the points where they are differentiable: (1) f(x) -x22x+1 X-+4 (2) g(x) = (x4 + 2x3 + 1) Vx4 + 1.See Answer
Q9:Presence task 1: Determine the derivatives of the functions with the following Functional regulations with the help of the derivation rules at the points where they are differentiable: (a) f(x) = ₂. (b) g(x) = 4x3 + 9 - sin(x) + tan(x) - 5 in(x). (c) h(x) = e**-3x+1 (d) k(x) = sin (x5 + !).See Answer
Q10:Exercise 1 du Ət u(x, 0) = 0, ди Consider the problem J²u D = əx² Solution Әх u(l, t) = 0, Where Q and a are positive constants, with a <D()². Determine the solution and study its trend for 0 (0, t) = Qe-at u(x, t) = Qe-at (x - 1) + with A₁ = [(n+1)4]². 2Q 1 ²0+ [e-² n=0 -Xn Dt + in (0,1) × (0, ∞) per x = [0,1] per t€ [0, ∞) t→∞. Dλn -α -Xn Dt 2-at) ] COS √VANT - eSee Answer
Q11:Homogeneous heat equation with homogenous starting condition, one Dirichlet boundary condition and another condition concerning the spatial derivative. Requires Fourier analysis. The correct solution is written below the problem in the picture. Student just need the procedure for solving this partial differential equation, so provide detailed step wise solution Student tried separation of variables but he is still struggling/nExercise 1 ди Consider the problem J²u Ət əx² u(x, 0) = 0, ди Solution u(l, t) = 0, Where Q and a are positive constants, with a <D() ². Determine the solution and study its trend for əx = 0 (0, t) = Qe-at, with n= [(n+1) 7]². u(x, t) = Qe-at (x − 1) + 2Q 20 Lt le n=0 -Xn Dt + t→∞. a in (0,7) x (0, ∞) per x = [0, 1] per t = [0, ∞) Dλn - α -An Dt - e-at)] cos cos AnxSee Answer
Q12:Problem 1 Let LE Rnxm and A = LLT. a Prove that A is positive semidefinite. b Prove that A is positive definite iff L has full row rank.See Answer
Q13:Problem 2 Show that: a If Qis positive definite, then the diagonal elements are positive. b Let Q be a symmetric matrix. If there exist positive and negative elements in the diagonal, then Q is indefinite.See Answer
Q14:Problem 3 i) Write a program in MATLAB/Octave/Scilab or Python (any other language please contact me) that randomly generates a positive definite matrix. ii) In order to check your previous code, write a function that implements Sylvester's criterion and use this function to check every matrix created in i). ProblemSee Answer
Q15:Problem 4 Consider the matrix = (ad). A = = When a + b = c+d, show that the vector (1, 1)T is an eigenvector of A and calculate both eigenvalues of A in terms of a, b, c, d.See Answer
Q16:Problem 5 Let 2 05 3 -4 3 1 0 3-4 3 3 -14 9 Find the following subspaces associated with A: A = (a) The column space of A (b) The row space of A (c) The nullspace of A (d) The nullspace of AT 1 -2 -1 -2 2See Answer
Q17:Problem 7 Let u = (1,0,-2), v = (1,2,A), and w = (2,1,-1)T. Find all values of A which make {u, v, w} a linearly dependent set of vectors in R³.See Answer
Q18:Problem 8 Compute the gradient and the Hessian of: a f(x) = ell, with x € R¹ and |||| = Σ₁x². b g(x) = II-12j.See Answer
Q19:Problem 9 Compute using Taylor's Theorem. lim (z,y) →(0,0) ry - sin(x) sin(y) x² + y²See Answer
Q20:Problem 10 Develop a quadratic approximation for the following functions: (a) g: R³ → R defined as g(x, y, z) = €²e³ + e²e³+ y²(1 + x + z) around the point (0, 1,0). (b) h: R¹ → R defined as h = x¹ Ar + ex around the point (1,..., 1); here A is an n x n symmetric matrix and c E R is a given vector.See Answer

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