Type-Changing Transformations of Hurwitz Pairs, Quasiregular Functions, and Hyper-Kählerian Holomorphic Chains Ii

Ławrynowicz, Julian; Nowak-Kȩpczyk, Małgorzata; Sánchez, Luis Manuel Tovar

doi:10.1142/9789812701763_0015

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“…In contrast, those corresponding to complex numbers (p = 2) and quaternions (p = 4) have an infinite number of members. As far as our graded fractal bundles are concerned, Now let us come back to the relationship between the gradation and inoculation of fractals [20,21]. Recall [22,23] that each fractal Σ of the flower type has its dual Ξ of the branch type and vice versa.…”

Section: From Alloys To Fractals Related To Clifford Structuresmentioning

confidence: 99%

“…They will be called gemmae of Ξ. Extending our Atomization Theorem (on isometric imbeddings) [24,25], we are going to state the following Atomization Theorem for Fractals [21]:…”

Section: Atomization Theorem For Fractals and Hurwitz Twistor-like Stmentioning

confidence: 99%

“…By analogy with the fact that 0 ∈ C lies in the real axis and in the imaginary axis as well, we complete both sequences (6) and (7) by 0 at the beginning, so that we get the objects 0 and i 0 . This is caused by the fact that we have no pairing for i 3 [21], so we need this extension. Table 1 is illustrated by Fig.…”

Section: (15)mentioning

confidence: 99%

“…The other cases appearing in (15) were not discussed in [21] because of the (8, 8)-periodicity of the Clifford structure. Table 2.…”

Section: (15)mentioning

confidence: 99%

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

2008

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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).

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Section: From Alloys To Fractals Related To Clifford Structuresmentioning

confidence: 99%

“…They will be called gemmae of Ξ. Extending our Atomization Theorem (on isometric imbeddings) [24,25], we are going to state the following Atomization Theorem for Fractals [21]:…”

Section: Atomization Theorem For Fractals and Hurwitz Twistor-like Stmentioning

confidence: 99%

Section: (15)mentioning

confidence: 99%

“…The other cases appearing in (15) were not discussed in [21] because of the (8, 8)-periodicity of the Clifford structure. Table 2.…”

Section: (15)mentioning

confidence: 99%

See 2 more Smart Citations

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

2008

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Type-Changing Transformations of Hurwitz Pairs, Quasiregular Functions, and Hyper-Kählerian Holomorphic Chains Ii

Cited by 1 publication

References 0 publications

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

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

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