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Q1:(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
Q2: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
Q3: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
Q4: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
Q5:p. 53, #37 We will do this in three problems: Note here I want you to assume f: R→ R is Lipschitz 3) Prove that if S CR has measure zero then so does f(S). Hint: For 0 pick a cover of A by finite open intervals whose sum of lengths is smaller than e. Now naturally form a cover of f(A) by closed intervals whose sum of lengths is also small. Now use countable subadditivity! 4) Prove that if A C R is F。 then so is f(A). Hint: This just requires f to be con- tinuous! Note that A' closed doesn't necessarily mean that f(A') is closed, BUT A' compact implies f(A') is compact. 5) Using Theorem 11 on p. 40 (i.e. E measurable iff there exists Fo set F C E where m*(E\F) = 0, prove that E measurable implies f(E) measurable. NOTE: this is FALSE for f RR continuous! See p. 52.See Answer
Q6:Q2: Consider the following infinite series. f(n) ΣΑΣ [ h(n) = Σ 1/1 n=0 n=0 g(n) (a) If the series converges, what can you say about the relationship between f(n) and g(n)? (b) If the series diverges, what can you say about the relationship between f(n) and g(n)? Hint: You may specify the functions: f(n) = x" and g(k) = xk. What is the relationship between n and k?See Answer
Q7: Notes: Assignment #1 Please refer to Homework & How to turn-in your assignments sections of "Course_Outline_Details" file for details regarding groupwork, how to turn in your assignments. Q1: Determine whether the following statements are true or false. Provide reasoning as well. (a) If X₁ ≤ X2 and X2 X3, then X₁ ≤ X3. Tru (b) Continue from part (a). What is the necessary condition that you need to be able to state that X₁ = X2 = X3? (c) Let X3 = X₁ UX₂ and X₁ = X₁ NX2. If X₁ = X2, then X4 C X3. Q2: Consider the following infinite series. 8 ∞ Σh(n) = f(n) Σ n=0 n=0 (a) If the series converges, what can you say about the relationship between f(n) and g(n)? (b) If the series diverges, what can you say about the relationship between ƒ (n) and g(n)? Hint: You may specify the functions: f(n) = x^and g(k) = xk. What is the relationship between n and k? Q3: In general, fogg of. Provide an example of such f(x) and g(x). Bonus Q: ex = 00 .n x n=0 x x² x3 2! 3! = 1+ + + + n! 1! Using the fact that ex and ln(x) are inverses of each other, explain why ln(0) is undefined. Hint: First, start off by looking at first few terms of ex. That is, see if ex > 0Vx ≥ 0. Then, do the same thing for x < 0. You should be able to determine the range for ex Vx = R at this point. Now, what are the domain and range of In(x)?See Answer
Q8:Q3: In general, fog #gof. Provide an example of such f(x) and g(x). Bonus Q: x x3 ex = 2! 3! n=0 Using the fact that ex and In(x) are inverses of each other, explain why In(0) is undefined. Hint: First, start off by looking at first few terms of ex. That is, see if ex > 0Vx ≥0. Then, do the same thing for x < 0. You should be able to determine the range for ex Vx E R at this point. Now, what are the domain and range of In(x)?See Answer
Q9:Assume ECR is measurable throughout this problem set. #1) Let f : E [0, ∞) be measurable and assume that f is bounded, so that f(E) C [0, d) for some d>0. Here we see how to directly approximate f by an increasing sequence of sim- ple functions: For nЄ N and j = {0,1,...,2"} set y;= jd/2" and set Ij,n = [j-1, j) so that {n} partitions [0, d). Also, set Ej,n = f¹(In) so that {E}}=1 partitions E. If on is defined by 2" On(x)=-1XE₁n (x) j=1 where XE,,, is the characteristic function of Ej,n then prove that n→f uniformly on E and that for all x € E we have 41(x) ≤ 42(x) ≤ 43(x) ≤ ··· ≤ f(x). (In other words, {n(x)}=1 is an increasing sequence of nonnegative numbers.) Hint: Fix n = N and consider r = Ej,n. Look at two cases: f(x) is in the left half of Ij.n and f(x) is in the right half of Ij.n. Notice that these two halves are intervals in {I},n+1}}=-11 which allows you to explicitly compute n+1(x) depending on these two cases! #2) Here we prove a version of Egoroff's theorem for sets of infinite measure. Note that this is a slight generalization of August 2023 problem #7. Let E be measurable with m(E) = ∞. Let {f} be a sequence of measurable real valued functions on E where fn → f pointwise a.e. on E. =1 a) Suppose E' CE is measurable with m(E') < ∞. Show that we can write E' == UEk where each Ek is measurable, fnf uniformly on each Ek for k > 1 and m(E1) 0 Hint: By Egoroff's theorem, there exists measurable Ek for k > 2 where m(E\Ek) < 1/k and fnf uniformly on Ek. Now define E₁ in the obvious way... == b) Using a) show that we can write E=UEk where each Ek is measurable, fn → f uniformly on each Ek for k > 1 and m(E₁) = 0. Hint: First write E as the countable union of bounded, measurable sets like we have many times before!See Answer
Q10:2. Prove that a metric space X is discrete if and only if every function on X to an arbitrary second metric space is continuous.See Answer
Q11:3. Let X be a dense subset of a metric space Y. Suppose that every Cauchy sequence in X converges to a point in Y. Prove that Y is complete.See Answer
Q12:1. Explain why the Comparison Test can or cannot be used to decide if the series converges or diverges.See Answer
Q13:2. Use the Comparison Test or Limit Comparison Test to determine whether each series is convergent or divergent. Carefully explain your justification for using each test.See Answer
Q14:3. Explain why the Integral Test can or cannot be used to determine whether the series is convergent.See Answer
Q15:Let z= {x ε r: f(x)=0} f is a continuous function. Show thatSee Answer
Q16:2. For the functions f and g defined by, f(x) = cosh (3x), g(x) = log(x) find the composite functions fog and go f stating clearly the intended domain and range.See Answer
Q17:4. Determine whether the following series converge or diverge. In the case of convergence, find the limit.See Answer
Q18: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
Q19:(a) Draw and shade in the rectangles needed to represent a left sum approximation of the distance traveled on the interval [0,30] with n = 6 rectangles. (b) Use the graph to estimate the distance traveled on the interval [0,30] with a left Riemann sum with n = 6 rectangles. Note: 1 second is equal to 1/3600 hours. (c) Does this give an overestimate or an underestimate of the distance traveled? (d) Interpret the Riemann Sum in the context of the situation. Include units in your answer.See Answer
Q20:Sketch the rectangles for the specified Riemann Sums to estimate the area under the curve of f on the interval [2, 8].See Answer

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