Week 10 Difference Equations Discrete Math April 30, 2020 Marie - - PowerPoint PPT Presentation

Linear Difference Equations with Constant Coefficients Combinatorics

Week 10 Difference Equations

Discrete Math April 30, 2020

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Homogeneous Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co A Procedure for Solving Linear Difference Equations with Constant

Linear Difference Equations with Constant Coefficients

The difference equation an+k + ck−1 an+k−1 + . . . + c1 an+1 + c0 an = bn, n ≥ n0, ci ∈ R, i.e. coefficients ci(n) are constant functions, is difference equation with constant coefficients. Characteristic equation of the equation above is λk + ck−1λk−1 + . . . + c1λ + c0 = 0. Any λ satisfying characteristic equation leads to one solution an = {λn}∞

n=n0.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Homogeneous Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co A Procedure for Solving Linear Difference Equations with Constant

Homogeneous Linear Difference Equations with Constant Coefficients

Real roots of characteristic equation. If λ is a root of the characteristic equation of multiplicity t then the following are linearly independent solutions of its homogeneous equation {λn}∞

n=0, {n λn}∞ n=0, {n2 λn}∞ n=0, . . . , {nt−1 λn}∞ n=0.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Homogeneous Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co A Procedure for Solving Linear Difference Equations with Constant

Homogeneous Linear Difference Equations with Constant Coefficients

Complex roots of characteristic equation. If λ = a + ı b is a complex root of the characteristic equation of multiplicity t then the following are linearly independent complex solutions of its homogeneous equation {(a + ı b)n}∞

n=0 and {(a − ı b)n}∞ n=0

and the following real solutions {rn cos nϕ}∞

n=0 and {rn sin nϕ}∞ n=0

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Homogeneous Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co A Procedure for Solving Linear Difference Equations with Constant

Linear Difference Equations with Constant Coefficients

A Quasi-polynomial. A function of the form f (n) = P(n) βn is a quasi-polynomial.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Homogeneous Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co A Procedure for Solving Linear Difference Equations with Constant

Non-homogeneous Linear Difference Equations with Constant Coefficients

An Estimate of One Solution of a Non-homogeneous Equation. Given a linear equation with constant coefficients an+k + ck−1 an+k−1 + . . . + c1 an+1 + c0 an = bn, where bn is a quasi-polynomial, bn = P(n)λn. We seek one of its solutions of the form ˆ an = Q(n) nt βn, where Q(n) is a suitable polynomial of the same degree as P(n), and t is the multiplicity of β as a root of the characteristic equation.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Homogeneous Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co A Procedure for Solving Linear Difference Equations with Constant

Non-homogeneous Linear Difference Equations with Constant Coefficients

How to Use the Estimate. Once we have got an estimate {ˆ an} of one solution of the non-homogeneous equation, then ◮ we substitute it into the non-homogeneous equation, ◮ we get a system of linear equations for unknown coefficients

• f the polynomial Q(n),

◮ if the estimate is correct, the system has a unique solution.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Homogeneous Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co A Procedure for Solving Linear Difference Equations with Constant

Linear Difference Equations with Constant Coefficients

Example. Find one solution of the following non-homogeneous equation an+2 + 4 an+1 − 5 an = 36 n with the initial conditions a0 = −1, a1 = 10.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Homogeneous Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co A Procedure for Solving Linear Difference Equations with Constant

A Procedure for Solving Linear Difference Equations with Constant Coefficients.

The procedure 1) We calculate the general solution of the associated homogeneous equation. 2) For bn = P(n) βn we made an estimate ˆ an = Q(n) nt βn, where Q(n) is a general polynomial of the same degree as P(n), t is the multiplicity of β as a root of the characteristic equation. 3) ˆ an is substituted into the non-homogeneous equation; comparing coefficients of the two (equal) polynomials we get the coefficients of Q(n).

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Homogeneous Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co A Procedure for Solving Linear Difference Equations with Constant

A Procedure for Solving Linear Difference Equations with Constant Coefficients.

4) General solution of the non-homogeneous equation is the sum

• f a general solution of the associated homogeneous equation

and the solution ˆ an from 3). 5) If initial conditions a0, a1, . . . , ak−1 are given, we substitute into the general solution n = 0, n = 1, . . . , n = k − 1 and

• btain the unknown coefficients from the general solution of

the associated homogeneous equation.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Binomial coefficients Binomial Theorem Pigeon Hole Principle

Binomial coefficients

Let k ≤ n be two natural numbers. Then the number n k

• =

n! k! (n − k)! is a binomial coefficient (or a combinatorial number).

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Binomial coefficients Binomial Theorem Pigeon Hole Principle

Binomial coefficients

Proposition. 1) For all n ∈ N we have n

• = 1

2) For all n ∈ N we have n

1

• = n.

3) For all k ≤ n, k, n ∈ N, we have n k

• =
• n

n − k

• .

4) For all k ≤ n, k, n ∈ N, it holds that

• n

k − 1

• +

n k

• =

n + 1 k

• .

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Binomial coefficients Binomial Theorem Pigeon Hole Principle

Binomial coefficients

Example. Consider the following problem: From a set of n people a committee of r people should be chosen, and from the committee k members have to be chosen to form a steering committee. Give two possible ways how it can be calculated (leading to an equality

• f binomial coefficients).

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Binomial coefficients Binomial Theorem Pigeon Hole Principle

Binomial Theorem

Theorem. Let n be a natural number. Then for every real numbers x, y it holds that (x + y)n =

n

• k=0

n k

• xn−k yk.
• Proposition. The number of different subsets of an n element set

is 2n.

• Proof. The number is

n

• i=0

n i

• = (1 + 1)n = 2n.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Binomial coefficients Binomial Theorem Pigeon Hole Principle

Pigeon Hole Principle

Principle of inclusion and exclusion. For any sets A, B, C we have |A ∪ B| = |A| + |B| − |A ∩ B|. |A∪B ∪C| = |A|+|B|+|C|−|A∩B|−|A∩C|−|B ∩C|+|A∩B ∩C|.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Binomial coefficients Binomial Theorem Pigeon Hole Principle

Pigeon Hole Principle

Proposition. Let A and B be two sets, |A| = n, |B| = k. Then there are kn distinct mappings from A to B. Theorem (Pigeon hole principle). Let A and B be two sets, |A| = n, |B| = k. If n > k then there does not exist a one-to-one mapping from A to B.

Marie Demlova: Discrete Math

Linear Difference Equations with Constant Coefficients Combinatorics Binomial coefficients Binomial Theorem Pigeon Hole Principle

Pigeon Hole Principle

Example 1. Let P = {p1, p2, p3, p4, p5} be five distinct point in the Euclidean plane, where each of them has integer coordinates. Show that there must be a pair of points which has midpoint also with integer coordinates. Example 2. A 3 × 7 rectangle is divided into 21 squares which are coloured by two colours: red and green. Show that there is a non trivial rectangle (not 1 × k or k × 1) such that it has all its four corners coloured by the same colour.

Marie Demlova: Discrete Math