TY - JOUR

T1 - Possible sets of autocorrelations and the Simplex algorithm

AU - Keren, Shahar

AU - Kfir, Haggai

AU - Kanter, Ido

PY - 2006/4/21

Y1 - 2006/4/21

N2 - The problem of imposing a set of correlations, , of any order, , on binary sequences is addressed. The entropy of infinitely long sequences obeying such a given set was calculated in previous works using the saddle-point method, and it was observed that a finite fraction of sets are characterized by a non-extensive entropy. In this paper, the region of finite entropy, the allowed region of sets of correlations, is found to be a convex hyper-polygon in the space of correlation-sets, using the Simplex algorithm. Outside of this region the Simplex solution indicates that sequences obeying the correlations cannot be found; therefore, the entropy is -∞. In particular, the boundaries of the allowed region for {C1, Cm} are presented. At the boundaries, the entropy drops in a first-order phase transition fashion, and this drop can be explained from a combinatorial point of view. Finally, we observe that the fraction of the volume occupied by allowed correlation-sets drops exponentially with the number of correlations imposed, and a qualitative explanation of this scaling phenomenon is provided.

AB - The problem of imposing a set of correlations, , of any order, , on binary sequences is addressed. The entropy of infinitely long sequences obeying such a given set was calculated in previous works using the saddle-point method, and it was observed that a finite fraction of sets are characterized by a non-extensive entropy. In this paper, the region of finite entropy, the allowed region of sets of correlations, is found to be a convex hyper-polygon in the space of correlation-sets, using the Simplex algorithm. Outside of this region the Simplex solution indicates that sequences obeying the correlations cannot be found; therefore, the entropy is -∞. In particular, the boundaries of the allowed region for {C1, Cm} are presented. At the boundaries, the entropy drops in a first-order phase transition fashion, and this drop can be explained from a combinatorial point of view. Finally, we observe that the fraction of the volume occupied by allowed correlation-sets drops exponentially with the number of correlations imposed, and a qualitative explanation of this scaling phenomenon is provided.

UR - http://www.scopus.com/inward/record.url?scp=33645656529&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/39/16/004

DO - 10.1088/0305-4470/39/16/004

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AN - SCOPUS:33645656529

SN - 0305-4470

VL - 39

SP - 4161

EP - 4171

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

IS - 16

ER -