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roblems 2.1. Show that the decimation transformation of a one-dimensional Ising model, with l = 2, can be written in terms of the transfer matrix P as P'{K'} = p (K), (1) where K and K' are the coupling constants of the original and the decimated lattice, respectively. Next show that, with P given by (eki+K2 e-ki (P(K) = Ky - K2) see equation (13.2.4), relation (1) leads to the same transformation equations among K and K' as (14.2.8a, b, and c). 4.2. Verify that expression (15) of Section 14.2 indeed satisfies the functional equation (14) for the field-free Ising model in one dimension. Next show (or at least verify) that, with the field present, the functional equation (11), with K given by (8), is satisfied by the more general expression (2) eki K2 - In[e cosh Kz + {e-26+ 2* sinh? ks)"] f(K,Kx) = - In ek 4.3. Verify that expression (32) of Section 14.2 indeed satisfies the functional equation (31) for the field-free spherical model in one dimension. Next show (or at least verify) that , with the field present the functional equation (27), with K given by (25), is satisfied by the more general expression A+/A² - K} K} f(Kı,K2,1)= 4(A - Ki)' 1 In 20 where A is determined by the constraint equation 1 =1 + af ал K} 4(A - KU)2 2/12 -K}