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The exact finite-size partition function for the nonhomogeneous one-dimensional (1D) Ising model is found through an approach using algebra operators. Specifically, in this paper we show that the partition function can be computed through a trace from a linear second-order recurrence relation with nonconstant coefficients in matrix form. A relation between the finite-size partition function and the generalized Lucas polynomials is found for the simple homogeneous model, thus establishing a recursive formula for the partition function. This is an important property and it might indicate the possible existence of recurrence relations in higher-dimensional Ising models. Moreover, assuming quenched disorder for the interactions within the model, the quenched averaged magnetic susceptibility displays a nontrivial behavior due to changes in the ferromagnetic concentration probability.

Background: Speckle tracking echocardiography (STE) allows for the study of myocardial strain (ε), a marker of early and subclinical ventricular systolic dysfunction. Cardiac disease may be present in patients with acromegaly; however, STE has never been used to evaluate these patients., Objective: To evaluate left ventricular (LV) global longitudinal strain in patients with active acromegaly with normal LV systolic function., Design: Cross-sectional clinical study., Methods: Patients with active acromegaly with no detectable heart disease and a control group were matched for age, gender, arterial hypertension and diabetes mellitus underwent STE. Global LV longitudinal ε (GLS), left ventricular mass index (LVMi), left ventricular ejection fraction (LVEF) and relative wall thickness (RWT) were obtained via two-dimensional (2D) echocardiography using STE., Results: Thirty-seven patients with active acromegaly (mean age 45.6 ± 13.8; 48.6% were males) and 48 controls were included. The mean GLS was not significantly different between the acromegaly group and the control group (in %, -20.1 ± 3.1 vs. -19.4 ± 2.2, p = 0.256). Mean LVMi was increased in the acromegaly group (in g/m 2 , 101.6 ± 27.1 vs. 73.2 ± 18.6, p < 0.01). There was a negative correlation between LVMi and GLS (r = -0.39, p = 0.01)., Conclusions: Acromegaly patients, despite presenting with a higher LVMi when analyzed by 2D echocardiography, did not present with impairment in the strain when compared to a control group; this finding indicates a low chance of evolution to systolic dysfunction and agrees with recent studies that show a lower frequency of cardiac disease in these patients.