We consider the problem of optimal quantization with norm exponent r > 0 for Borel probability measures on ℝd under constrained Rényi-α-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization
error are well-known in the special cases α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization
error under large entropy bound with entropy parameter α ∈ [1+r/d,∞]. For α ∈ [0,1 + r/d] we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization
error is decreasing exponentially fast with the entropy bound and the exact rate is determined for all α ∈ [0, ∞].
Authors:S. N. Kreitmeier, G. L. Liang, D. W. Noid, and B. G. Sumpter
Thermal analysis by classical molecular dynamics simulations is discussed on hand of heat capacity of crystals of 9600 atoms. The differences between quantum mechanical and classical mechanical calculations are shown. Anharmonicity is proven to be an important factor. Finally, it is found that defects contribute to an increase in heat capacity before melting. The energy of conformational gauche defects within the crystal is only about 10% due to internal rotation. The other energy must be generated by cooperative strain. The conclusion is that the next generation of faster computers may permit wider use of molecular dynamics simulations in support of the interpretation of thermal analysis.