Download e-book for kindle: An Introduction to Semigroup Theory (L.M.S. Monographs ; 7) by John M. Howie

By John M. Howie

ISBN-10: 0127546332

ISBN-13: 9780127546339

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The Hilbert space of the IR Dε (κ, j, τ ) of the P (1, 4) algebra, corresponding to P 2 = κ 2 > 0, is expanded into the direct integral of the subspaces, which correspond to the IR of the G(3) algebra with the following values of the invariant operators: C1 = κ 2 , C2 = m2 s(s + 1), C3 = εm, |κ| ≤ m < ∞, |j − τ | ≤ s ≤ j + τ . 17). To conclude this section we consider the IR of the P (1, 4) algebra, corresponding to P 2 = 0. The realisations of such an IR have been obtained in the form (Fushchych and Krivsky [9, 10]): P0 = εE0 ≡ ε p2 + p24 J0a = −iεE0 J4a = i pa 1/2 , Pa = p a , ∂ Sab pb −ε , ∂pa E0 + p 4 ∂ ∂ − p4 ∂p4 ∂pa +ε P4 = p 4 , J04 = −iεE0 ∂ , ∂p4 Sab pb , E0 + p 4 where Sab are the generators of the IR D(s) of the SO(3) group.

So we reach the following result: Theorem. The Hilbert space of the IR Dε (κ, j, τ ) of the P (1, 4) algebra, corresponding to P 2 = κ 2 > 0, is expanded into the direct integral of the subspaces, which correspond to the IR of the G(3) algebra with the following values of the invariant operators: C1 = κ 2 , C2 = m2 s(s + 1), C3 = εm, |κ| ≤ m < ∞, |j − τ | ≤ s ≤ j + τ . 17). To conclude this section we consider the IR of the P (1, 4) algebra, corresponding to P 2 = 0. The realisations of such an IR have been obtained in the form (Fushchych and Krivsky [9, 10]): P0 = εE0 ≡ ε p2 + p24 J0a = −iεE0 J4a = i pa 1/2 , Pa = p a , ∂ Sab pb −ε , ∂pa E0 + p 4 ∂ ∂ − p4 ∂p4 ∂pa +ε P4 = p 4 , J04 = −iεE0 ∂ , ∂p4 Sab pb , E0 + p 4 where Sab are the generators of the IR D(s) of the SO(3) group.

2 ) было инвариантным относительно масштабного преобразования координат, необходимо и достаточно, чтобы правые части уравнений удовлетворяли условиям (9), где F1 и F2 — произвольные дифференцируемые функции. 3. Пусть задано масштабное преобразование времени t = bt, x1 = x1 , (3IV ) x2 = x2 , b — вещественный параметр. ˜ имеем выражение Согласно (4) для оператора X ∂ ∂ ˜ = t ∂ − x˙ 1 ∂ − x˙ 2 ∂ − 2¨ X x1 − 2¨ x2 − ∂t ∂ x˙ 1 ∂ x˙ 2 ∂x ¨1 ∂x ¨2 ∂ ∂ ··· ··· (4) ∂ (4) ∂ − 4x2 , −3 x 1 ··· − 3 x 2 ··· − 4x1 (4) (4) ∂x1 ∂x2 ∂ x1 ∂ x2 а в качестве условия инвариантности (5) получим систему уравнений в частных производных первого порядка: ∂fα ∂fα ˜ = t ∂fα − x˙ 1 ∂fα − x˙ 2 ∂fα − 2¨ X x1 − 2¨ x2 − ∂t ∂ x˙ 1 ∂ x˙ 2 ∂x ¨1 ∂x ¨2 ··· ∂fα ··· ∂fα α = 1, 2, −3 x 1 ··· − 3 x 2 ··· = −4fα , ∂ x1 ∂ x2 (10) общее решение которой имеет вид ··· ··· Φα x1 , x2 , x˙ 1 t, x˙ 2 t, x ¨ 1 t2 , x ¨2 t2 , x 1 t3 , x 2 t3 , f1 t4 , f2 t4 = 0, (11) где Φα — произвольные дифференцируемые функции.

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An Introduction to Semigroup Theory (L.M.S. Monographs ; 7) by John M. Howie


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