The present paper develops the approach to the famous problem presented by L. Onsager (1944) and its further investigation proposed in a recent work by Z.-D. Zhang (2007). The above works give quaternion-based two- and three-dimensional (quantum) models of order-disorder transition and simple orthorhombic Ising lattices (1925). The general methods applied by Zhang refer to opening knots by a rotation in a higher dimensional space, introduction of weight factor (his Conjecture 1 and 2) and important commutators. The main objective of the present paper is to reformulate the algebraic part of the theory in terms of the quaternionic sequence of Jordan algebras and to look at some of the geometrical aspects of simple orthorhombic Ising-Onsager-Zhang lattices. The present authors discuss the relationship with Bethe-type fractals, Kikuchi-type fractals, and fractals of the algebraic structure and, moreover, the duality for fractal sets and lattice models on fractal sets. A simple description in terms of fractals corresponding to algebraic structure involving the quaternionic sequence (H(q)(4)) of P. Jordan's algebras appears to be possible. Physically we obtain models of (H(q)(4)) for q = 5 . 2(2) = 20 for the melting, q = 9 . 2(6) = 576 for binary alloys, and q = 13 . 2(10) = 13 312 for ternary alloys
The paper is inspired by a spectral decomposition and fractal eigenvectors for a class of piecewise linear maps due to Tasaki et al. [1994] and by an ad hoc explicit derivation of the Heisenberg uncertainty relation based on a Peano–Hilbert planar curve, due to El Nashie [1994]. It is also inspired by an elegant generalization by Zhang [2008] of the exact solution by Onsager [1944] to the problem of description of the Ising lattices [Ising, 1925]. This generalization involves, in particular, opening the knots by a rotation in a higher dimensional space and studying important commutators in the corresponding algebra. The investigations of Onsager and Zhang, involving quaternion matrices of order being a power of two, can be reformulated with the use of the "quaternionic" sequence of Jordan algebras implied by the fundamental paper of Jordan et al. [1934]. It is closely related to Heisenberg's approach to quantum theories, as summarized by him in his essay dedicated to Bohr on the occasion of Bohr's seventieth birthday (1955). We show that the Jordan structures are closely related to some types of fractals, in particular, fractals of the algebraic structure. Our study includes fractal renormalization and the renormalized Dirac operator, meromorphic Schauder basis and hyperfunctions on fractal boundaries, and a final discussion.
Abstract. We continue to modify and simplify the Ising-Onsager-Zhang procedure for analyzing simple orthorhombic Ising lattices by considering some fractal structures in connection with Jordan and Clifford algebras and by following Jordan-von Neumann-Wigner (JNW) approach. We concentrate on duality of complete and perfect JNW-systems, in particular ternary systems, analyze algebras of complete JNW-systems, and prove that in the case of a composition algebra we have a self-dual perfect JNW-system related to quaternion or octonion algebras. In this context, we are interested in the product table of the sedenion algebra.Mathematics Subject Classification (2010). Primary 82C44; Secondary 82D25, 81R05, 15A66.
It is well known that starting with real structure, the Cayley–Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteraton process p → p + 2 → p + 4 → ⋯, they have constructed 24-dimensional "bipetals" for p = 9 and 27-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens." The present paper aims at an effective, explicit determination of the periods and expressing them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. The proof of the Periodicity Theorem is given in the case where the index of the generator of the algebra in question exceeds the order of the initial algebra. In contrast to earlier results, the fractal bundle flower structure, in particular sepals, bisepals, perianths, and calyces are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. The same concerns canonical two-layer pairing of prianth sepals and the relationship of fractal bundles of algebraic structure with the Hurwitz problem.
An almost complex manifold description of elementary particles is proposed which links the approaches given independently by J. Ławrynowicz and L. Wojtczak, and by C. von Westenholz. This description leads to relations between the curvature form of an almost complex manifold, which accounts for the symmetry classification schemes within the frame of principal fibre bundles, and a curved Minkowski space-time via induced smooth mappings characterizing nuclear reactions of type N+π⇄N, where N is some nucleon and π the virtual π-meson of this reaction. Both approaches follow the same main idea of D. A. Wheeler developed in a different way by A.D. Sakharov.
A system of coupled anharmonic oscillators with fluctuatious of t h e crystallographic lattice is considered. The anharmonicity is included as an effective field in t.he form of pseudoharmonic interactions. The description of the motion far from a Brownian pa.rtiele embedded in tho heat bath shows that. thc flnctuat,ions lead to an explanation of t.hc origin of stochast.ic forces appearing in the system and having the charrzctrr of rz black noise. The Brownian particle is, in general, nonMarkoffian if tho fluctuations are taken ir1t.o account, bnt it ca.n be expressed as a slim of Markoffian processes at least. Thc appearance of fluc!tuations is relatrd to the damping of motion in t.he heal, bath formed by harmonic oscillat,ors.Ein System von gekoppelten anhnrmonischen Oseillatoren mit Pluktuationen rlcs kristallographischen Gitters wird unt.ersuclit. Die Anharmonizit,itt mird a,ls effelrt.ives Feld in Form von pscudohsrmonischen Wcchselwirkiingcn eingefiihrt. Die Beschreibung der Bewegung, die weit entfernt ist von cinem Rrownschen Partikel in einmn Warmebad, zeigt, da8 die Fluktuationen zu eincr Erklarung dcs Ursprmngs drr stoohastischcn Brafte fiihrt, die in dein System auftretcn und den Chara,lrter einrs schwnrzrn Rauschens haben. Das Brownschc Partikcl ist im allgemeincn kein Markoff-'l'eilchen, wenn Fluktuationen beriicksichtigt uwden, la8t sich jcdoch wenigstens als Summe von Nlnrkoff-Pr ausdriickcn. I)as Auftrcten von Plukt~uat~ionen ist mit der llampfung der Bewcgung im bad, das durch die harmonkchen Oszillat,oren gebildet wird, verkniipfl,. I. IntrodlietionI n the present paper we are dealing with the non-equilibriixm proccsses in systems (considered as solid state) which are inhomogeneous. Tho inhornogeneities are treated as being of dynamic character varying in time. Their description may he given within the model of local fluctuations which appear in every system slid play an important role in the vicinity of phase transition points. In the case of solids the inodel has been considered in terms of pseudoharmonic approximation [l 1 and then applied in the context of local fluctuations for systems treated as being in equilibrium conditions [Z].
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