Sizes and series of m35–atomic clusters in icosahedral quasicrystals and in m35–approximant crystals. Theory and examples

Abstract

A critical examination of the behaviour of three fragile glass forming liquids viz. Salol, α-phenyl-o-cresol and o-terphenyl, has been carried out using differential scanning calorimetry and dielectric relaxation technique (frequency: 10⁶-10^?3 Hz). Our study reveals two sub-T_g processes designated as β- and γ-processes, in addition to the primary (α-) process. The β-process has an activation energy of 36–50 kJ/mol. in these liquids, and is found to be intramolecular in nature. Our results along with the published viscosity data on these liquids indicate a decoupling of the viscous modes from the α-modes in the region of T_g, with the latter still retaining its non-Arrhenius character. The nature of this decoupling has been discussed in detail.     

3D Icosahedral aperiodic packings of spheres of different diameters from discrete icosahedral crystals

General results for transient pattern dynamics in the context of spinodal decomposition as well as domain-wall formation in the Fréedericksz transition in nematics are reviewed. The question of wavelength selection and the relevance of finite-size effects in this issue are considered in a general model of transient pattern dynamics. A finite-size scaling law for domain growth is shown to be satisfied by this model.     

On a Generalization of the Hermite Constant

We report some results on the theory of quasiperiodic sphere packings controlled by a finite group G in the n-dimensional Euclidean space, constructed by means of an higher dimensional Euclidean space which contains G-invariant lattices. A generalization of the Hermite constant is used to find upper bounds of the density.     

On the dichotomy of Perron numbers and beta-conjugates

Let β > 1 be an algebraic number. A general definition of a beta-conjugate of β is proposed with respect to the analytical function ${f_{\beta}(z) =-1 + \sum_{i \geq 1} t_i z^i}$ associated with the Rényi β-expansion d _β (1) = 0.t ₁ t ₂ . . . of unity. From Szeg?’s Theorem, we study the dichotomy problem for f _β (z), in particular for β a Perron number: whether it is a rational fraction or admits the unit circle as natural boundary. The first case of dichotomy meets Boyd’s works. We introduce the study of the geometry of the beta-conjugates with respect to that of the Galois conjugates by means of the Erd?s–Turán approach and take examples of Pisot, Salem and Perron numbers which are Parry numbers to illustrate it. We discuss the possible existence of an infinite number of beta-conjugates and conjecture that all real algebraic numbers > 1, in particular Perron numbers, are in ${{\rm C}_1 \cup \,{\rm C}_2 \cup \,{\rm C}_3}$ after the classification of Blanchard/Bertrand-Mathis.     

Covering a Ball with Smaller Equal Balls in ℝn

We give an explicit upper bound of the minimal number _T,n of balls of radius 1/2 which form a covering of a ball of radius T > 1/2 in ⁿ, n \geq 2. The asymptotic estimates of _T,n we deduce when n is large are improved further by recent results of Böröczky, Jr. and Wintsche on the asymptotic estimates of the minimal numberof equal balls of ⁿ covering the sphere S^n-1. The optimality of the asymptotic estimates is discussed.