Determinants and Their Applications in Mathematical Physics
246 6. Applications of Determinants in Mathematical Physics Equation (6.2.25) was solved by Ohta, Kajiwara, Matsukidaira, and Sa ...
6.3 The Dale Equation 247 Note that the theorem cannot be applied toAdirectly sinceφmdoes not satisfy the Appell equation for an ...
248 6. Applications of Determinants in Mathematical Physics Applying double-sum identities (C) and (A), n ∑ r=1 n ∑ s=1 [x r+s− ...
6.4 The Kay–Moses Equation 249 =− 4 c(c−1)α 2 1 . (6.3.20) Referring to (6.3.16), ( y+4n 2 1+x )′ =4 [ (c−1)xA 11 +n 2 1+x ]′ =− ...
250 6. Applications of Determinants in Mathematical Physics (A ij ) ′ =− ∑ r e cru A rj ∑ s e csu A is , (6.4.4) 2 ∑ r brcrA rr ...
6.4 The Kay–Moses Equation 251 = ∑ i biciφi[(ci+cj)A ij −φiφj] = ∑ r brcrφr[(cr+cj)A rj −φrφj], ∑ r brcrA rj φ ′ r= ∑ r brcrφr[c ...
252 6. Applications of Determinants in Mathematical Physics =2 ∑ r brcrφr ∑ j e cju A rj +2 ∑ r brcr ∑ j φre cju A rj cj− 1 =2 ∑ ...
6.5 The Toda Equations 253 are proved in Theorem 4.30 in Section 4.8.5 on Turanians. Let the elements in bothAnandBnbe defined a ...
254 6. Applications of Determinants in Mathematical Physics =0, which proves the theorem whennis even. Theorem 6.2. The functi ...
6.5 The Toda Equations 255 The derivative ofAnwith respect tox, as obtained by differentiating the rows, consists of the sum ofn ...
256 6. Applications of Determinants in Mathematical Physics = ρ 2 Bn+1Bn− 1 e − 2 x B 2 n = Bn+1Bn− 1 B 2 n This equation is ide ...
6.5 The Toda Equations 257 whereAnandBnare Hankelians defined as An=|φm|n, 0 ≤m≤ 2 n− 2 , Bn=|φm|n, 1 ≤m≤ 2 n− 1 , φ ′ m=(m+1)φm ...
258 6. Applications of Determinants in Mathematical Physics Hence, referring to the first equation in (4.5.10), [ B (n+1) 1 ,n+1 ...
6.6 The Matsukidaira–Satsuma Equations 259 Hence applying the Jacobi identity (Section 3.6), ∣ ∣ ∣ ∣ τr+2 τr+1 τr+1 τr ∣ ∣ ∣ ∣ = ...
260 6. Applications of Determinants in Mathematical Physics Applying the Jacobi identity, FF 13 , 13 = ∣ ∣ ∣ ∣ F 11 F 13 F 31 F ...
6.6 The Matsukidaira–Satsuma Equations 261 6.6.2 A System With Two Continuous and Two Discrete Variables LetA (n) (r, s) denote ...
262 6. Applications of Determinants in Mathematical Physics Hence, ∣ ∣ ∣ ∣ (τr,s+1)y (τrs)y τr,s+1 τrs ∣ ∣ ∣ ∣ =A (n+1) (r, s)A ...
6.7 The Korteweg–de Vries Equation 263 urs= (τrs)y τrs , vrs= (τrs)x τrs , for all values ofnand all differentiable functionsfrs ...
264 6. Applications of Determinants in Mathematical Physics where A=|ars|n, ars=δrser+ 2 br+bs =asr, er= exp(−brx+b 3 rt+εr). Th ...
6.7 The Korteweg–de Vries Equation 265 Eliminating the sum common to (6.7.3) and (6.7.5) and the sum common to (6.7.4) and (6.7. ...
«
9
10
11
12
13
14
15
16
17
18
»
Free download pdf