Linear Difference Equations with Constant Coefficients Combinatorics Week 10 Difference Equations Discrete Math April 30, 2020 Marie Demlova: Discrete Math
Homogeneous Linear Difference Equations with Constant Coefficients Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co Combinatorics A Procedure for Solving Linear Difference Equations with Constant Linear Difference Equations with Constant Coefficients The difference equation a n + k + c k − 1 a n + k − 1 + . . . + c 1 a n + 1 + c 0 a n = b n , n ≥ n 0 , c i ∈ R , i.e. coefficients c i ( n ) are constant functions, is difference equation with constant coefficients. Characteristic equation of the equation above is λ k + c k − 1 λ k − 1 + . . . + c 1 λ + c 0 = 0 . Any λ satisfying characteristic equation leads to one solution a n = { λ n } ∞ n = n 0 . Marie Demlova: Discrete Math
Homogeneous Linear Difference Equations with Constant Coefficients Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co Combinatorics 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 n = 0 , { n 2 λ n } ∞ n = 0 , . . . , { n t − 1 λ n } ∞ { λ n } ∞ n = 0 , { n λ n } ∞ n = 0 . Marie Demlova: Discrete Math
Homogeneous Linear Difference Equations with Constant Coefficients Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co Combinatorics 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 { r n cos n ϕ } ∞ n = 0 and { r n sin n ϕ } ∞ n = 0 Marie Demlova: Discrete Math
Homogeneous Linear Difference Equations with Constant Coefficients Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co Combinatorics 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 ) β n is a quasi-polynomial. Marie Demlova: Discrete Math
Homogeneous Linear Difference Equations with Constant Coefficients Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co Combinatorics 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 a n + k + c k − 1 a n + k − 1 + . . . + c 1 a n + 1 + c 0 a n = b n , where b n is a quasi-polynomial, b n = P ( n ) λ n . We seek one of its solutions of the form a n = Q ( n ) n t β n , ˆ 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
Homogeneous Linear Difference Equations with Constant Coefficients Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co Combinatorics 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 a n } 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 of the polynomial Q ( n ) , ◮ if the estimate is correct, the system has a unique solution. Marie Demlova: Discrete Math
Homogeneous Linear Difference Equations with Constant Coefficients Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co Combinatorics 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 a n + 2 + 4 a n + 1 − 5 a n = 36 n with the initial conditions a 0 = − 1, a 1 = 10. Marie Demlova: Discrete Math
Homogeneous Linear Difference Equations with Constant Coefficients Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co Combinatorics 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 b n = P ( n ) β n we made an estimate ˆ a n = Q ( n ) n t β n , 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 a n 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
Homogeneous Linear Difference Equations with Constant Coefficients Linear Difference Equations with Constant Coefficients Non-homogeneous Linear Difference Equations with Constant Co Combinatorics 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 of a general solution of the associated homogeneous equation and the solution ˆ a n from 3). 5) If initial conditions a 0 , a 1 , . . . , a k − 1 are given, we substitute into the general solution n = 0, n = 1, . . . , n = k − 1 and obtain the unknown coefficients from the general solution of the associated homogeneous equation. Marie Demlova: Discrete Math
Binomial coefficients Linear Difference Equations with Constant Coefficients Binomial Theorem Combinatorics Pigeon Hole Principle Binomial coefficients Let k ≤ n be two natural numbers. Then the number � n � n ! = k k ! ( n − k )! is a binomial coefficient (or a combinatorial number ). Marie Demlova: Discrete Math
Binomial coefficients Linear Difference Equations with Constant Coefficients Binomial Theorem Combinatorics Pigeon Hole Principle Binomial coefficients Proposition. � n � 1) For all n ∈ N we have = 1 0 � n � 2) For all n ∈ N we have = n . 1 3) For all k ≤ n , k , n ∈ N , we have � n � � n � = . k n − k 4) For all k ≤ n , k , n ∈ N , it holds that � n � � n � � n + 1 � + = . k − 1 k k Marie Demlova: Discrete Math
Binomial coefficients Linear Difference Equations with Constant Coefficients Binomial Theorem Combinatorics 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 of binomial coefficients). Marie Demlova: Discrete Math
Binomial coefficients Linear Difference Equations with Constant Coefficients Binomial Theorem Combinatorics Pigeon Hole Principle Binomial Theorem Theorem. Let n be a natural number. Then for every real numbers x , y it holds that n � n � ( x + y ) n = x n − k y k . � k k = 0 Proposition. The number of different subsets of an n element set is 2 n . Proof. The number is n � n � = ( 1 + 1 ) n = 2 n . � i i = 0 Marie Demlova: Discrete Math
Binomial coefficients Linear Difference Equations with Constant Coefficients Binomial Theorem Combinatorics 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
Binomial coefficients Linear Difference Equations with Constant Coefficients Binomial Theorem Combinatorics Pigeon Hole Principle Pigeon Hole Principle Proposition. Let A and B be two sets, | A | = n , | B | = k . Then there are k n 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
Binomial coefficients Linear Difference Equations with Constant Coefficients Binomial Theorem Combinatorics Pigeon Hole Principle Pigeon Hole Principle Example 1. Let P = { p 1 , p 2 , p 3 , p 4 , p 5 } 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
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