In ferromagnetic spin models above the critical temperature ͑T Ͼ T cr ͒ the partition function zeros accumulate at complex values of the magnetic field ͑H E ͒ with a universal behavior for the density of zeros ͑H͒ ϳ͉H − H E ͉ . The critical exponent is believed to be universal at each space dimension and it is related to the magnetic scaling exponent y h via = ͑d − y h ͒ / y h . In two dimensions we have y h =12/ 5 ͑ =−1/ 6͒ while y h =2 ͑ =−1/ 2͒ in d = 1. For the one-dimensional Blume-Capel and Blume-Emery-Griffiths models we show here, for different temperatures, that a value y h =3 ͑ =−2/ 3͒ can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.

Here we study the partition function zeros of the one-dimensional Blume-Emery-Griffiths model close to their edge singularities. The model contains four couplings (H, J, , K) including the magnetic field H and the Ising coupling J. We assume that only one of the three couplings (J, , K) is complex and the magnetic field is real. The generalized zeros z i tend to form continuous curves on the complex z-plane in the thermodynamic limit. The linear density at the edges z E diverges usually with ρ(z) ∼ |z − z E | σ and σ = −1/2. However, as in the case of complex magnetic fields (Yang-Lee edge singularity), if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with σ = −2/3 can appear as we prove here explicitly for the cases where either or K is complex. Our proof applies for a general three-state spin model with short-range interactions. The Fisher zeros (complex J) are more involved; in practice, we have not been able to find an explicit example with σ = −2/3 as far as the other couplings (H, , K) are kept as real numbers. Our results are supported by numerical computations of zeros. We show that it is absolutely necessary to have a non-vanishing magnetic field for a new critical behavior. The appearance of σ = −2/3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher-dimensional spin models.

The aim of the present study was to investigate the cumulative effects of two bouts of maximal exercise on heart rate (HR) by spectral and detrended fluctuation analysis (DFA). Rowers (n=14, 11 males, 24±6 years old) performed two maximal (2k1 and 2k2) tests. HRV and DFA were calculated before (PRE1) and after 2k1 (POS1), four hours after POS1 (PRE2) and after 2k2 (POS2). The HF power was reduced from PRE1 (1527±1349 ms2) to POS1 (224±339 ms2) and from PRE2 (908±861 ms2) to POS2 (214±234 ms2, respectively, p<0.05) and in PRE2 was lower than PRE1 (p<0.05), with similar reductions in POS1 and POS2 (p>0.05). DFA in the time domain was used as a non‐linear method to quantify HR fluctuations detrended from its nonstationary background, giving a more precise measure of the fractality and long‐range correlations in the time series. DFA analysis is able to assess that HR fluctuations in POS2 were lower that POS1, moreover, the exponent α related to the fractal measure departs from a purely random one (α=0.5), evidencing long‐range correlations in HR variability, with slightly different exponents in PRE and POS groups. DFA showed that after 2k2 there was an additional reduction on HR fluctuations. These data suggest cumulative effects of exercise on HR dynamics and both methods give complementary information.Support: CNPq 481434/2008‐9 and FAPERJ E‐26/111.345/2011