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By Chaiho Rim, Ryu Sasaki, Changrim Ahn

This quantity comprises numerous lecture notes at the basics and basic options of integrable box theories and on their functions to low-dimensional physics platforms contributed via prime scientists within the respective fields. the most subject matters lined are a variety of elements of the thermodynamic Bethe ansatz, shape components, Calogero (and comparable) versions, sigma types, conformal boundary stipulations, and so on. the quantity offers either pedagogical fabric and a present learn pattern within the box.

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Obviously a2 is invariant under r. One needs to prove that I{a,t) is invariant. Let us rearrange I(a,t) in Eq. (113) as I(a,t) = -[J{a,t)-J(ai=0,t)] (118) where J(a, t) = ^2 cosh i2b(a -a-Q-a)t + h(l + b2)t)\,. (119) a>0 The problem reduces to prove J(a, t) invariant under r. Under r, if a positive root goes to another positive root, this only reshuffles the terms in J(a,t). However, there are cases where a postive root goes to a negative root, —/3 = TOL. Then apart from the reshuffling, J(a,t) contains the term, cosh (2ba • (-/3)f - 26Q • at + h{\ + b2)t) (120) This is the case when r changes a simple root aT to zeroth root eo : raT = eo.

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