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Recurrence Relation

A recurrence relation is an equation that recursively defines a sequence, once one more initial terms are given. Each further term of the sequence is defined as a function of the preceding terms. The term difference equation is sometimes referred to as a specific type of recurrence relation. However, difference equation is frequently used to refer to any recurrence relation.

An example of a recurrence relation is the logistic map:

Xn+1 = rxn(1-xn)

With a given constant r, given the initial term x0 each subsequent term is determined by this relation. Some simply defined recurrence relations can have very complex behaviours and they are a part of the field of mathematics known as nonlinear analysis. Solving the recurrence relation means obtaining a closed-form solution: a non-linear recursive function of n.

Recurrence relation has many different applications. These can include biological applications, digital signal processing and economics. In biology, some of the best-known difference equations have their origins in the attempt to model population dynamics. The logistical map is used either directly to model population growth, or as a starting point for more detailed models. In digital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in infinite impulse response digital filters. Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics. 

Relations and Graph Theory

1. List all the functions from the three element set {1, 2, 3} to the set {a, b}. Which functions, if any, are one-to-one? Which functions, if any, are onto? 2. How many base 10 numbers have 3 digits? How many three-digits numbers have no two consecutive digits equal? How many have at least one pair of consecutive digits equal?

Predicate logic

PREDICATE LOGIC Note: For your convenience the truth tables are included. You may not need to use all of the columns in the skeleton tables. 1. Use modus ponens or modus tollens to fill in the blanks in the arguments so as to produce a valid inference. If this polygon is a triangle, then the sum of its interior an

Truth Tables Equivalent Statements

LOGIC 1. Indicate and justify which of the following sentences are statements. a) She is mathematics major. b) 128 = 26 c) x = 26 2. Write the statement bellow in symbolic form using the symbols ~, , and  and the indicated letters to represent component statements. (m = "Juan is a math major", c = "J

Combinations and Permutations

COUNTING PERMUTATION COMBINATIONS 1. In a monthly test, the teacher decides that there will be three questions, one from each chapter I, II and III of the book. If there are 12 questions in chapter I, 10 in chapter II and 6 in chapter III. In how many ways can three questions be selected? Justify your answer. 2. Find the tot

One-To-One Functions

FUNCTIONS 1. Find all functions from X = {a, b, c} to Y = {u, v}. 2. Let F and G be functions from the set of all real numbers to itself. Define new functions F - G: R R and G - F: R R as follows: (F - G)(x) = F(x) - G(x) for all x  R, (G - F)(x) = G(x) - F(x) for all x  R. Does F - G = G - F? Explain. 3. Le

Set Relations Binary Relations

Relations 1. Let C = {2, 3, 4, 5} and D = {3, 4} and define a binary relation S from C to D as follows: for all (x, y)  C  D, (x, y)  S  x  y. a) Write S as a set of ordered pairs. b) Is 2 S 4? Is 4 S 3? Is (4, 4)  S? Is (3, 2)  S? 2. Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the "divides" rel

Set Theory questions

1. Let T = {m   m = 1 + (-1)i, for some integer i}. Describe T. 2. Let A = {m   m = 2i - 1, for some integer i}, B = {n   n = 3j + 2, for some integer j}, C = {p   p = 2r + 1, for some integer r}, and D = {q   q = 3s - 1, for some integer s}. a) Describe the 4 sets (by enumerat

Rate of Return Probability Question

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Capital Asset Pricing Model; Risk Return Analysis

Tony is wondering how much risk he must accept in order to generate a reasonable return on his portfolio. The risk-free return currently is 5 percent. The return on the market portfolio is 16 percent. Use the Capital Asset Pricing Model to calculate the beta coefficient associated with each of the following portfolio returns.

Recursion and Iteration

Can someone please help with my practice questions? I would like details and explanations using words to show how you got the answer. The questions cover the concepts of linear recurrence, iterations, and power functions.

Discrete Math-Induction

A flagpole is n feet tall. On this flag pole we display flags of the following types: red flags that are 1 foot tall, blue flags that are 2 feet tall and green flags that are 2 feet tall. The sum of the heights is exactly n feet. Prove that there are exactly [(2/3)*(2^n)] + [(1/3)*(-1)^n] ways to display the flags.

The Newton-Raphson Method: Consider the Function

See the attached file. The Newton-Raphson Method 1. Consider the function . i) Show, graphically or otherwise, that the equation has a root in the interval (1,2). Show that . ii) Use the secant method, with the function f(x) and starting values, 1 and 2, to find an estimate of correct to three decimal places.

Recurrence relations - initial conditions

A) Find a recurrence relation for the number of bit strings of length n that contain three consecutive 0s. b) What are the initial conditions? c) How many bit strings of length seven contain three consecutive 0s?

Recurrence relations, compound interest, polynomials

Please help with the following discrete math involving recurrence relations, compound interest, polynomials, number of combinations and iteration. 14. Individual membership fees at the evergreen tennis club were $50 in 1970 and have increased by $2 per year since then. Write a recurrence relation and initial conditions for

Matrix Relation

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Combinations, Directed Graphs and Recurrence Relations

1. A baseball manager has decided who his 5 starting hitters are to be, but not the order in which they will bat. How many possibilities are there? 2. Susan has a fondness for chocolate desserts, in particular, pudding, pie, ice cream, éclairs, and cookies. Her preference is for ice cream over pie and cookies, éclairs over

Sets, Relations, Prim's Algorithm, State Tables and Recurrence Relations

1. Let A = {1, 2, 3, 4}, B = {3, 4, 5}, C = {1}, and D = {x: 3 < x < 10}. Are each of the following true or false? b. B &#8838; D c. &#8709; &#8838; D 2. Calculate the following: a. P(8, 4) 3. Let A = {1, 2, 3, 4}, B = {1, 4, 5}, C = {3, 5, 6}, and the universal set U = {1, 2, 3, 4, 5, 6}. a. Determi

Recursions, Recurrence Relations, Difference Equations

1. Solve xn=axn-1+b when a=1. 2. An isotope of carbon called carbon-14 (14C) is used to establish the age of artifacts and fossils. It decays so that every 5000 years an amount of 14C is reduced to 54.44256% of its initial valued A archaeologist finds a fossil that contains 16% of the amount of 14C it contained when it was al

Recurrence Relation : Compound Interest

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Probability : Mean, Standard Deviation and Recurrence Relation

In order to test a vaccine, we have to find patients with a certain blood type, that is found in 20% of the population. Model W, the number of people sampled until we have found one with thhis blood type; X, the number sampled to find four with the blood type; and Y, the number with this blood type among 20 people. Find the mean

Domain and Range of a Relation

1. Find the domain and range of the relation {(x,y)&#9474;5x < -5} 2. Find the domain of the relation A={(x,y)&#9474;x^2+y^2=4}

Discrete Structures

1. Consider the sequence of triangles Ti, i >= 2: T2 is simply a triangle sitting upright, on its base. T3 is T2, except that an additional straight line is drawn from the upper vertex, down to somewhere on the base. For each Ti+1, one more line is added to triangle Ti (such that each line meets the base at a different point).