Narrow Results

Projective synchronization between two nonlinear systems with different dimension was investigated. The controllers were designed when the dimension of drive system greater than the one of response system. The opposite situation also was discussed. In addition, we found an approach to control the chaotic (hyperchaotic) system to exhibit the behaviors of hyperchaotic (chaotic) system. The numerical simulations were implemented on different chaotic (hyperchaotic) systems, and the results indicate that our methods are effective.

By a careful analysis on cyclotomic cosets, the maximal designed distance δ_new of narrow-sense imprimitive Euclidean dual containing q-ary BCH code of length is determined, where q is a prime power and l is odd. Our maximal designed distance δ_new of dual containing narrow-sense BCH codes of length n improves upon the lower bound δ_max for maximal designed distances of dual containing narrow-sense BCH codes given by Aly et al. [IEEE Trans. Inf. Theory53 (2007) 1183]. A series of non-narrow-sense dual containing BCH codes of length n, including the ones whose designed distances can achieve or exceed δ_new, are given, and their dimensions are computed. Then new quantum BCH codes are constructed from these non-narrow-sense imprimitive BCH codes via Steane construction, and these new quantum codes are better than previous results in the literature.

Currently, many new materials including nanomaterials exhibit exotic properties like, mention a few, unconventional superconductivity, reduced magnetism, reduced Pauli repulsion. These exotic properties could not be explained completely with the existing quantum mechanical theories which did not take into account the dimension of angle. Now, there arises a necessity to invoke the forgotten dimension of angle, for explaining these exotic interesting properties. So far angle is considered as a dimensionless quantity even though its all time partner $r$ is endowed with dimension. In materials, where surface plays a dominant role, this new dimension, of angle becomes an important tool for exploring their morphology and exotic properties. This new dimension of angle restores the orbital degrees of freedom to the s electrons whose orbital degrees of freedom are suppressed ($\ell =0$). It compensates for the missed signature of s electron orbital angular momentum.