Definition 4. A symmetry generator of (4) is denoted by U and is given by + SQ- +...+ (9) dun+k-1 une z =n [+une. Q is the characteristic of the group of transformations. 27 In the above definition, SİQ(n,un) = Q(n+j,u,n+i). The operator S is known as the shift operator. Consider the system of ordinary difference equations of the form 28 29 Un+k+2 = w1(un+k, Vn, Vn+2), Vn+k+2 = w2 2(un, Un+2, Vn+k), (10) where the independent variable here is denoted by n and the dependent variables are denoted by un, Vn and their shifts. Consider the group of transformations (n, un, Vn) → (n, ũn = Un + ɛQ1(n, un) +0(?), 0n (11) = Un + EQ2(n,vn)+O(ɛ²)). In (11), the characteristic of the group of transformations is Q = (Q1, Q2). The infinitesimal generator corresponds to U = Q1du, + Q2do, (12) where d, = . In this work, we will need the kth extension dx uk = Qidu, + Q2do, + s²Q1ðun+2+ S?Q2dv+2+ S*Qiðun+* +s*Q2dv+* (13) Un+k Un+k of (12). For set solutions of (10) to be mapped to itself, the following linearised symmetry conditions s(k+2)Q1 - ukwi = 0 and s*+2) Q2 – uklw2 = = 0, (14) whenever (10) is true, must be satisfied. If the conditions given in(14), that is, Q;(n+k+ 2,0;) – uk (n;) = 0, j = 1,2, are satisfied, then the group of transformations (11) is a 32 symmetry group. 30 31 3. Symmetries and Solutions of the System of Difference Equations (1) 33 Equivalently, equation (3) can be written as Un+kVn Un+k+2 =W1 = Vn+2(an + bnUn+kOn)' UnVn+k (15) Un+k+2 =W2 = Un+2(Cn+ dnUnVn+k)' where an and bn are real sequences. Applying (14) onto (15) yields,
Definition 4. A symmetry generator of (4) is denoted by U and is given by + SQ- +...+ (9) dun+k-1 une z =n [+une. Q is the characteristic of the group of transformations. 27 In the above definition, SİQ(n,un) = Q(n+j,u,n+i). The operator S is known as the shift operator. Consider the system of ordinary difference equations of the form 28 29 Un+k+2 = w1(un+k, Vn, Vn+2), Vn+k+2 = w2 2(un, Un+2, Vn+k), (10) where the independent variable here is denoted by n and the dependent variables are denoted by un, Vn and their shifts. Consider the group of transformations (n, un, Vn) → (n, ũn = Un + ɛQ1(n, un) +0(?), 0n (11) = Un + EQ2(n,vn)+O(ɛ²)). In (11), the characteristic of the group of transformations is Q = (Q1, Q2). The infinitesimal generator corresponds to U = Q1du, + Q2do, (12) where d, = . In this work, we will need the kth extension dx uk = Qidu, + Q2do, + s²Q1ðun+2+ S?Q2dv+2+ S*Qiðun+* +s*Q2dv+* (13) Un+k Un+k of (12). For set solutions of (10) to be mapped to itself, the following linearised symmetry conditions s(k+2)Q1 - ukwi = 0 and s*+2) Q2 – uklw2 = = 0, (14) whenever (10) is true, must be satisfied. If the conditions given in(14), that is, Q;(n+k+ 2,0;) – uk (n;) = 0, j = 1,2, are satisfied, then the group of transformations (11) is a 32 symmetry group. 30 31 3. Symmetries and Solutions of the System of Difference Equations (1) 33 Equivalently, equation (3) can be written as Un+kVn Un+k+2 =W1 = Vn+2(an + bnUn+kOn)' UnVn+k (15) Un+k+2 =W2 = Un+2(Cn+ dnUnVn+k)' where an and bn are real sequences. Applying (14) onto (15) yields,
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 9AEXP
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