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# Discrete Math

### Analytic Zeros Proof

Let f be analytic on a domain D. Prove that if f(z(0))=0 and if f is not identically zero, then z(0) is a zero of f of some finite order m.

### Isolated Zeros Proof

Let f be analytic on a domain D. Prove that if f is not identically zero, then the zeros of f in D are isolated. (That is, prove that if f is not identically zero and if z(0) is a point in D with f(z(0))=0, then there exists e>0 such that f(z)=/0 for all z in the region 0<|z-z(0)|<e.) e, epsilon. =/, does not equal.

### Propositional Logic

For the following expression, give one interpretation that makes it true and one interpretation that makes it false: {see attachment for expression}

### Propositional Logic: Blood Types

Let the propositions have the intuitive meanings given in the Figure attached. Write a clause or product of clauses that express the following ideas. (a) If test T is positive, then that person has blood type A or AB (b) If test S is positive, then that person has blood type B or AB (c) If a person has type A, the test

### Discrete Set of Values

If there is any reason why you do not want to answer the question (problem with attachment, bid price, ect.) please let me know.

### Proof that there must be 12 Pentagons on a Soccer Ball

A soccer ball is formed by stitching together pieces of material that are regular pentagons and regular hexagons. Each corner of a polygon is the meeting place for exactly three polygons. Prove that there must be exactly 12 pentagons. (Please see attachment for full question and background)

### Propositional Logic: Commutitivity of Sums and Products

Please see the attached file for the fully formatted problems. Prove that: p1+p2+...+pn is equivalent to the sum (logical OR) of the pi's in any order. and p1p2 ... pn is equivalent to the product (logical AND) of the pi's in any order.

### Propositional Logic : DeMorgan's Laws and Truth Tables

Please see the attached file for the fully formatted problems. Verify DeMorgan's laws (equation 1 and 2 below) using truth tables. Prove the generalized DeMorgan's laws: (1) (NOT(p1 p2 .... pk)) (2) (NOT(p1+p2+...+pk)) by induction on k, using the basic laws: NOT(pq) NOT(p+q) Then, justify the ge

### Newton's Method Proof

Please show that when n=1, Newtons method given by: x^k=x^(k-1)-(J(x^(k-1))^-1)(F(x^(k-1)) for k>=1 reduces to the familiar Newton's method given by: P_n=P_n-1 - f(p_n-1)/f'(P_n-1) for n>=1 Note: ^-1 is inverse J is the jacobian matrix The top equation is called newton's method for non linear systems. x is a vecto

### Math - Knots

Only solve 4 part A, and 5 using MATLAB Codes. See the attached file.

### Discrete 47.3

3. Let d1,d2...dn be .... prove that d1...dn are degrees of the vertices. (See attachment for full question).

### John Nash's Game Theory

Can anybody explain and summarize the detail of John Nash's paper please? It is in the attachment file.

### Chinese Remainder Theorem and Proofs

The Chinese Remainder Theorem (CRT) applies when the moduli ni in the system of equations x&#8801; a1 (mod n1) ... x&#8801; ar (mod nr) are pairwise relatively prime. When they are not, solutions x may or may not exist. However, the related homogeneous system (2'), in which all ai=0, always has a solution, namely the trivial

### Improvement for Solution Correctness

Respond/pick up the credit if you absolutely know the solution is correct. If you can make an improvement on the solution in correctness, clarity, presentation, or if a proof can be more elegant, than please rewrite the entire solution.

### Nonnegative Integers

If the solution to this nonnegative integer question is correct, then you may respond that it is. If the solution needs ANY kind of improvement, in presentation, in clarity, in correctness, if a proof can be more elegant, then please rewrite the entire solution. See the attached file.

### Analyticity Proof Derivative

Suppose that f: C->C and that f is analytic at a point z0 element of C. Prove that there exists a real number r>0 such that, the nth derivative of z0=[n!/(2 pi r^n)]x[int(e^(-niy)f(z0+re^(iy)) from 0 to 2pi with respect to y for all n element of Natural numbers.

### Convex sets

A Theorem states: Convex Sets are closed under convex combinations. That is, if C is a convex set, and if x1,x2,...,xm belongs to C then for all non-negative real numbers %1,%2,...,%m are non-negative real numbers such that \$1+\$2+....+\$m=1, we have \$1x1 + \$2x2 + ... + \$mxm belonging to C. Furthermore, I want to prove that i

### Divisors and relative primes

Let a be an integer. Prove that 2a + 1 and a^2+ 1 are relatively prime. ( relative primes are numbers that their largest common divisor is 1). ONLY RESPOND IF THERE IS A a. mistake b. something is unclear c. proof is correct, but solved incorrectly (does not follow instructions) Exm. Instruction says proof by inducti

### Incorrect Proof Explanation

Take an in-depth look into this proof. Obviously it is wrong. Where is it wrong and why? This is obviously wrong. Where and why? Detailed explanation is needed of where and why it is wrong with all examples. Thanks Let a = b. Multiply both sides by a (OK because we don't violate the equal sign). We get a² = ab. Subtrac

### Total Maximization of Interest

At the start of the year, a company wants to invest excess cash in one-month, three-month and six-month Certificates of Deposit (CD's). (Purchase price and yields for the different CD's appear in the table below). The company is somewhat conservative, however, and wants to make sure that it has a safety margin of cash-on-hand e

### Finite Math - Probability

I need assistance with following problem along with steps to arrive at the solution/answer. Three envelopes are addressed for 3 secret letters written in invisible ink. A secretary randomly places each of the letters in an envelope and mails them. What is the probability that at least 1 person receives the correct letter?

### Finite Math - Sets Counting Techniques

The problem I am working is the following; please provide step-by-step to obtain solution's. I am unable to figure (c) out the ANSWER IS ... 7805 There are 5 rotten plums in a crate of 25 plums, How many samples of 4 of the 25 plums contain at least one rotten plum?

### Combinatorial Proof Based on the Binomial Formula

Please see the attached file for the fully formatted problem. Provide a combinatorial proof: For postive integers N....

### Discrete Math: Proof of the Product of Odd Numbers

Let Pn be the product of the first n odd numbers. For example P_3 = 1 x 3 x 5 = 15. Prove that for positive integers n, P_n = (2n)!/(2^n)(n!) Your proof will be graded for style, clarity and completion.

### Discrete Math: Subsets and Elements

Let S = {1,2,3,4,5,6,7,8}. Determine: (a) The number of subsets of S (b) The number of subsets of S with at most four elements (c) The number of ordered lists with elements chosen form S (with possible repetitions) (d) The number of ordered lists with nine elements chosen form S with no repetitions (e) The number

### Recursive definition

We can define sorted lists of integers as follows: Basis - A list consisting of a single integer is sorted. Induction - If L is a sorted list in which the last element is a and if b >= a, then L followed by b is a sorted list. Prove that this recursive definition of "sorted list" is equivalent to our original, nonrecurs

### Finding a Code with the Given String Characteristics

If no two strings in a code differ in fewer than three positions, the we can actually correct a single error, by finding the unique string in the code that differs from the received string in only one position. It turns out that there is a code of 7-bit strings that corrects single errors and contains 16 strings. Find such a cod

### Discrete Math : N multichoose K Proof

Please see the attached file for the full problem description. The double bracket notation is pronounced " n multichoose k". The doubled parentheses remind us that we may include elements more than once.

### Matlab : Topography of a Tank

I would like to use Matlab to make a graph of the topography of the tank. I would like some one to show me how to do it because I don't understand how to plot all the points. Can someone give me a 3D graph of the topography? In the attached file row one is my X-axis in cm and my y-axis is 1-41.

### Binary tree

A. Write 3n &#8722; (k + 5) in prefix notation: ????. b. If T is a binary tree with 100 vertices, its minimum height is ????. c. Every full binary tree with 50 leaves has ???? vertices.