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    Discrete Math

    Algorithm

    Let T[1..n] be a sorted array of distinct integers, some of which may be negative. Give an algorithm that can find an index i such that 1 <= i <= n and T[i] = i, provided such an index exists. Your algorithm should take a time in Big "O" (log n) in worst case.

    Proof that a sequence is monotone increasing

    (See attached file for full problem description and equations) --- Prove that the sequence is monotone increasing. Use the following hints: 1) If ln f(x) is increasing, then so is f(x). 2) If , then f is increasing. 3) ln x is defined to be . ---

    Correspondence of Borel sets

    If f is one-to-one, f, f^-1 are continuous, then f is called a homeomorphism. Now I want you to prove the following: Let f : X -> Y, ( X and Y are topological spaces)be homeomorphism, prove that it establishes one-to-one correspondence between Borel sets in X and Y.

    Proof Limit Solved

    I received the following proof, can someone show all steps of how the solution was formed? Proof: Let n=2^2^k, then we have T(n)=T(2^2^k)=2T(n^(1/2))+log n ***How do you get n^(1/2) equals 2^2^(k-1) =2T(2^2^(k-1))+2^k ***How do you get 2T(2^2^(k-1)) equals 2(2T(2^2^(k-2))+2^(k-1)) =2(2T(2^2^(k-2))+2^(k-1))+

    Solving a recurrence

    Solve the following recurrence exactly for n of the form 2^2^k. T(2) = 1 T(n) = 2T(n^(1/2)) + log n Express your answer as simply as possible using theta notation. note added ** theta notation is based on big O notation Show all work!

    Irreducible representation proofs

    1. Assume that the field is algebraically closed and has zero characteristic, G is finite and representations are finite-dimensional. Show that this statement is true under the above assumptions: "Let p be an irreducible representation of G, and q be an irreducible representation of H. Is it always true that the exterior t

    Calculating Time from the Number of Loops

    How much time does the following algorithm require as a function of n? Express your answer in theta notation in the simplest form. Consider each individual instruction (including loop control) as elementary. l = 0 for i = 1 to n for j = 1 to n^2 for k = 1 to n^3 l = l + 1

    Proof of Differentiability

    Let f(x) = { x^2 if x is rational { 0 if x is irrational Show that f is differentiable at x=0 but not at any other point. --- Please see the attached file for the fully formatted problem.

    Sets of Six, Seven and Eight-Letter Words

    (a) Give an example of a set S such that the language S* has more six-letter words than seven-letter words. (b) Give an example of a set S such that the language S* has more six-letter words than eight-letter words. (c) Does there exist a set S such that S* contains more six-letter words than twelve-letter words. Give reas

    5 Finite Math Problems

    1) A man walking in the woods encounters a stream. Because he is unsure of the stream depth, he measures how deep the water is in many random spots along the entire width of the stream. After 1000 measurements each with a depth of 6 inches, he concludes the probability of the water being 6 inches deep the entire way across to

    Finite Math

    1.) Dan borrows $500 at 9% per annum simple interest for 1 year and 2 months. What is the interest charged, and what is the amount due? ________________________________________ 2.) A mutual fund pays 9% per annum compounded monthly. How much should I invest now so that 2 years from now I will have $100 in the account?

    Transitive Closures

    I've attached the problem I'm having trouble with. Please provide assistance. (See attached file for full problem description).

    Removable singularity

    (See attached file for full problem description) --- Let have an isolated singularity at and suppose that is bounded in some punctured neighborhood of . Prove directly from the integral formula for the Laurent coefficients that for all j = 1,2,3,..., i.e. must have a removable singularity at . The integ

    Finite Math

    (See attached file for full problem description) --- 1. What is marginal cost? Fixed cost? Find the slope for each line that has a slope. 3. Through (-2, 5) and (4, 7) 5. Through the origin (11, -2) 7. 2x + 3y = 15 9. y + 4 = 9 11. y = -3x Find an equation in the form y = mx + b (where possible) for each line.

    Numerical Methods Proofs

    a) Show that the l_inf vector norm satisfies the three properties i. ||x|| > 0 if x =/ 0 and x belongs to R^n ii. ||lambda(x)|| |lambda| ||x|| if lambda belongs to R and x belongs to R^n iii. ||x + y|| <= ||x|| + ||y|| for x, y belonging to R^n. b) consider the matrix 4 2 1 A = 2 5 -2 1 -

    Finite or Countable Collection

    a) If {I } is a finite or countable collection of disjoint open intervals with I (a, b), Prove that (m: Measure). b) If U (empty set) is an open set of R, Prove That (U)> 0. c) Let P denote the cantor set in [0, 1]. Prove that m(P) = 0. d) Suppose that U, V are open subsets of R, a,b R with U [a,b] = V [a,

    Algorithm to convert to binary

    The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. 3. The following algorithm... Please see attached.

    Finding a lighter or heavier ball out of 12 balls

    You have 12 balls out of which one is odd ie lighter or heavier than others. You can't identify the odd ball just by looking at it. So, how you would find out the ball, using the weighing scale and maximum weighings allowed are three only?

    Finite Mathematics - 20 Review Questions

    Please see the attachment. Problems 1-3 (also attached): Acme Inc. has just purchased new computers for $60,000. Suppose that in 8 years the computers will have a salvage value of $8,000. The college plans to depreciate the value of the computers linearly over the 8 years. 1. Based on the above information, value

    Bijections, Denumerability, Tautologies, Sets, Logic and Proofs

    Consider the compound statement (P ^ Q) V (~ P ^ R) a) Find the truth table for the statement. b) IS the statement a tautology? c) In the following program code, what has to be the output for the answr to be "yes"? ... 2. Given the following sets A = {1.2,3.4}, B = {2,3,4,5}, and C= {2,4,6} a) Find .... 4. Prove that

    Venn Diagram and Intersection Proof

    Draw Venn diagrams for each side of the equation below, shading the appropriate region. Then give a careful proof using the element method (A-C)&#8745;(B-C) = (A&#8745;B) - C Please see the attached file for the fully formatted problem.

    Logic and Logistics : Crossing a Desert and Supply Chain

    This is a workbook example called crossing a desert. Suppose you and your truck are all set to leave the town of Ucnestahn on the edge of Ucnahara desert. you want to travel across the desert as quickly as possible. You have calculated that all the provisions you need for one day's travel (food, fuel, lager, chewing gum, 'w

    Partially ordered set Proof

    A set A is call a partially-ordered set if there is a relation R on A such that R is reflexive and transitive. If A is any set, prove that the power set P(A) is a partially ordered set under the relation "is a subset of".

    Abelian groups

    Let G ---> H be a group homomorphism and... Show that if... then &#934;[G] is albelian. Please see attached for full question.