We review recent investigations of the critical behavior of ferromagnetic q-state Potts models on a class of hierarchical lattices, with exchange interactions according to some deterministic but aperiodic substitution rules. The problem is formulated in terms of exact recursion relations on a suitable parameter space. The analysis of the fixed points of these equations leads to a criterion to gauge the relevance of the aperiodic geometric fluctuations. For irrelevant fluctuations, the critical behavior remains unchanged with respect to the underlying uniform models. In the presence of relevant fluctuations, a non-trivial symmetric fixed point, associated with the critical behavior of the uniform model, becomes fully unstable, and there appears a two-cycle of the recursion relations. A scaling analysis, supported by direct numerical thermodynamical calculations, shows the existence of a novel critical universality class associated with relevant geometric fluctuations.

Desde o início do tratamento cirúrgico da doença coronária obstrutiva com Vineberg (1) até os dias atuais com o emprego de vários tipos de enxertos, os cirurgiões procuram aumentar a sobrevida bem como melhorar a qualidade de vida dos pacientes. Paciente de 60 anos de idade, com angina e síncope, cujo estudo hemodinâmico mostrou lesão de 90% no ramo interventricular anterior (RIA) e 99% na coronária direita (CD) (Figura 1). Foi revascularizado com artéria torácica interna esquerda (ATIE) para o RIA e ressecção da lesão da CD, seguida de anastomose dos cotos (Figura 2). Paciente de 40 anos, com lesão de 80% da CD e de 99% do RIA foi igualmente revascularizado com enxerto livre de ATIE para o RIA e um segmento de 4 cm de ATIE em ponte coronária-coronária sobre o segmento lesado (Figura 3). Ambos os pacientes evoluíram bem na fase hospitalar. Após um ano, permanecem assintomáticos, com teste ergométrico negativo.

From the early days of surgical treatment of coronary artery disease by Vineberg until the present time using several types of grafts, we are hoping to improve outcome. Case 1 - A 60 year old man with angina and syncope. An angiographic study has shown a 90% obstruction in A. D., 99% in the right coronary (Fig. 1). An anastomosis between the left Internal thoracic artery and the right coronary artery. A resection of the narrow segment with an end to end anastomosis (Fig. 2). Case 2 - A 40 year old man, with 80% obstruction in the right coronary artery (Fig. 4) and 99% in the descending aorta. A 4 cm segment of ATIE was replaced with a bypass from the right coronary artery (Fig. 3). And a free graft of ATIE from descending anterior artery. The one year follow up of both patients has been good with normal stress test and symptoms.

We write exact renormalization-group recursion relations for a ferromagnetic Ising model on the diamond hierarchical lattice with an aperiodic distribution of exchange interactions according to a class of generalized two-letter Fibonacci sequences. For small geometric fluctuations, the critical behavior is unchanged with respect to the uniform case. For large fluctuations, the uniform fixed point in the parameter space becomes fully unstable. We analyze some limiting cases, and propose a heuristic criterion to check the relevance of the fluctuations.