Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction

Abstract

The idea of fractal modeling of crystals dates back to Bethe [1], who observed its convenience when coming to first, second, and third nearest neighbors of an atom. Taking into account that it is a neighbor of two or more other atoms, even in the case of one layer with a lattice formed by squares, one naturally comes to the notion of cluster [2,3]. It is then natural to cut the plane of the lattice correspondingly to the cluster involved and construct a Riemann surface or a Bethe lattice-a fractal set of the branch type [4,5]. The construction is parallel to that related to the holomorphic function f (z) = exp z 2 in C (Fig. 1). The example already shows the importance of the corresponding group Γ of cover symmetry transformations (Decktransformationengruppe), inoculation (of the branch corresponding to no. 1 on the branch corresponding to no. 4), and gradation related to the points , •, , and •.The next important step was done by Kikuchi [6], who-within his theory of cooperative phenomenadeveloped a method of approximation for order-disorder phenomena.In this context, Sukiennicki, Wojtczak, Zasada, and Castillo Alvarado [7] investigated an infinite thin film of an AB 3 alloy. As examples we may take Ni 3 Fe or Cu 3 Au. They assumed the (111) orientation of the alloy. Let z(j) denote the concentration of A-atoms in the layer j = 0, 1, . . . , where j = 0 corresponds to the surface. Then the concentration of B-atoms in that layer is 1 − z(j). If U is the energy of interaction of the system, T is the absolute temperature, and g is the number of possible configurations, then the entropy S and the free energy F of the system are given byrespectively, where k is the Boltzmann constant, and the conditions for thermodynamic equilibrium at a given temperature T are ∂ ∂τ F τ =τ (j) = 0, ∂ ∂z F z=z(j) = 0 for each j, λ = const. Thus, 1 T ∂ ∂τ U τ =τ (j) = ∂ ∂z S τ =τ (j).

Fractals and Chaos Related to Ising-Onsager-Zhang Lattices. Quaternary Approach vs. Ternary Approach

“…We can prove the above assertions with the help of the two enclosed product tables. Now, owing to hints [13] and motivation given in references [7,8,13,15], following some earlier demands appearing in references [11,12,[16][17][18], we can prove the following basic theorems on the Galois extensions.…”

“…is the quaternionic algebra, so making its extension by T 2 as in (8) with e k , e k , k = 1, 2, 3, as in (17) and, in analogy to (21) and 22: can be introduced in an analogous way as shown in Fig. 9.…”

(Para) quaternionic geometry, harmonic forms, and stochastical relaxation

“…The results obtained have a clear physical significance [2], [5], [9], [11], [16], [17], [25], [26].…”

“…(b) Then the general solution of the system (33) is a linear combination (27) with complex coefficients of 42 fundamental solutions which, in the case of (16), are explicitly given by the formulae (29)-(31).…”

“…(ii) Then a solution s of (33) can be expressed as in (25) with coefficients as in (26) and (24), and the matrices (18) can be specified according to (20) in terms of para-quaternions. Moreover, the general solution of the system (33) is a linear combination (27) with complex coefficients of 42 fundamental solutions which, in the case of (16), are given explicitly by the formulae (29)-(31), where s 0 is determined by (16).…”