Toward an Octonionic World

In order to obtain a consistent formulation ofoctonionic quantum mechanics (OQM), we introduceleft/right-barred operators. Such operators enable us tofind the translation rules between octonionic numbers and 8 × 8 real matrices (a translation isalso given for 4 × 4 complex matrices). The use ofa complex geometry allows us to overcome the hermiticityproblem and define an appropriate momentum operator within OQM. As an application of our results,we develop an octonionic relativistic free waveequation, linear in the derivatives. Even if the wavefunctions are only one-component, we show that fourindependent solutions, corresponding to those of the Diracequation, exist.     

Octonionic representations of Clifford algebras and triality

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the nonassociativity and noncommutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octoninic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest ₃ ×SO(8) structure in this framework.     

A class of Lorenz-like systems

The transformation of a three-dimensional dynamical system to its differential model can be used to identify different nonlinear dynamical systems that share the same time series of one of its variables. This transformation then can be used to find classes of nonlinear dynamical systems with similar dynamical behavior.     

Complex dynamical behavior and chaos control in fractional-order Lorenz-like systems

In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilities of the equilibrium points are analyzed as one of the system parameters changes. The pitchfork bifurcation is discussed for the first time, and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived. The largest Lyapunov exponents and phase portraits are given to check the existence of chaos. Finally, the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable. Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.     

Octonionic Yang–Mills instanton on quaternionic line bundle of Spin(7) holonomy

The total space of the spinor bundle on the four-dimensional sphere S⁴ is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang–Mills instanton on this eight-dimensional gravitational instanton. This is a higher dimensional generalization of (anti-) self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang–Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7). The existence of more general Spin(7) valued solutions can be seen by making an asymptotic expansion.     

Insights into the algebraic structure of Lorenz-like systems using feedback circuit analysis and piecewise affine models

The characterization of chaotic attractors has been a widely addressed problem and there are now many different techniques to define their nature in a rather accurate way, at least in the case of a three-dimensional system. Nevertheless, the link between the structure of the ordinary differential equations and the topology of their solutions is still missing and only a few necessary conditions on the algebraic structure are known today. By using a feedback circuit analysis, it is shown that it is possible to identify the relevant terms of the equations, that is, the terms that really contribute to the structure of the phase portrait. Such analysis also provides some guidelines for constructing piecewise affine models. Moreover, equivalence classes can be defined on the basis of the active feedback circuits involved.     

An algebraic spectrum condition

A condition, necessary and sufficient for the existence of a vacuum representation with positive energy of the quasilocal algebra, is formulated.     

Insufficiency of Karmarkar's condition

It is shown that Karmarkar's condition which has been claimed to be a necessary and sufficient condition for the spherically symmetric (s.s.) space-time to be of class 1 ceases to be sufficient in caseR ₂₃₂₃=0. An s.s. line element which satisfies Karmarkar's condition but ceases to be of class 1 is given. This line element describes a conformally flat perfect fluid distribution with zero density.     

An algebraic spectrum condition

It is shown that a translationally invariant algebra of local observables (see [1]) admits a representation in a Hilbert space having a vacuum state. Furthermore an algebraic criterion is given which is necessary and sufficient for the existence of at least one representation of in which the usual spectral condition for the energy-momentum operators holds.     

On Dirac's quantization condition

The consequences of Dirac's quantization condition are discussed. Observing that this condition generally leads to contradiction and that the normal explanation to resolve the contradiction seems unsatisfactory, we propose a modified quantization condition. It is shown that operator ordering ambiguity could be represented as the arbitrariness of the distribution functions in the modified condition.     

Glauber model and unitary condition

The Glauber model is discussed from the point of view of unitary condition. It is shown that the model amplitudes slightly violate the unitary condition unless the longitudinal momentum transfers of all contributing processes are much smaller (in Lab) than a nucleon mass. Physical origin of this fact and its consequence are briefly indicated.The author thanks Dr. R. J. N.Phillips for his hospitality in the Theory Group of Rutherford Laboratory.     

Translation group and spectrum condition

Let {A, ^d ,} be aC*-dynamical system, where ^d is thed-dimensional vector group. LetV be a convex cone in ^d and its dual cone. We will characterize those representations ofA with the properties (i) _a ,a ^d is weakly inner, (ii) the corresponding unitary representationU(a) is continuous, and (iii) the spectrum ofU(a) is contained in .     

Representations obeying the spectrum condition

We show that every properly infinite, injective von Neumann algebra acting on a separable Hilbert space is isomorphic to the weak closure of some translation covariant representation, obeying the spectrum condition for the generators of the translation group, of theC*-algebra of quasilocal observables of a free massless spinor field. We construct explicitly such representations in the case ofII andIII factors, 0<<1.     

The generic condition is generic

We consider the generic condition for vectors—both null and non-null—at a fixed pointp of a spacetime, and ask just how generic this condition is. In a general spacetime, if the curvature is not zero at the pointp, then the generic condition is found to be generic in the mathematical sense that it holds on an open dense set of vectors atp; more specifically, if there are as many as five non-null vectors in general position atp which fail to satisfy the generic condition, then the curvature vanishes atp. If the Riemann tensor is restricted to special forms, then stronger statements hold: An Einstein spacetime with three linearly independent nongeneric timelike vectors atp is flat atp. A Petrov type D spacetime may not have any nongeneric timelike vectors except possibly those lying in the plane of the two principal null directions; if any of the non-null vectors in such a plane are nongeneric, then so are all the vectors of that plane, as well as the plane orthogonal to it.     

Adiabatic condition for nonlinear systems

The adiabatic approximation is an important concept in quantum mechanics. In linear systems, the adiabatic condition is derived with the help of the instantaneous eigenvalues and eigenstates of the Hamiltonian, a procedure that breaks down in the presence of nonlinearity. Using an explicit example relevant to photoassociation of atoms into diatomic molecules, we demonstrate that the proper way to derive the adiabatic condition for nonlinear mean-field (or classical) systems is through a linearization procedure, using which an analytic adiabatic condition is obtained for the nonlinear model under study.     

Prebreakdown condition in selenium rectifiers

This paper reports data from an experimental investigation of the prebreakdown and breakdown conditions of factory production selenium rectifiers in the temperature range from 100 ° to –196 ° C when a unit voltage pulse with steep leading edge (rise time about 10^–8 sec) is applied to the rectifier in the non-conducting direction. Change in voltage across the rectifier and in current through it were recorded by a two-channel high-speed electronic oscilloscope. On the basis of this test data, the variation of discharge rise-time in the selenium p-n junction as a function of overvoltage and temperature was determined, the formation of current pulses was detected in the junction, both before and during breakdown in the temperature range from 100 ° to –196 ° C, and it was also established that breakdowns of selenium rectifier p-n junctions are due to joint action of the Zener effect and impact ionization.We are indebted to Prof. Kh. I. Amirkhanov for his constant interest in this work.     

Wetting failure condition on rough surfaces

Wetting states and processes attract plenty of interest of scientific and industrial societies. Air entrainment, i.e.,wetting failure, on smooth plate in wetting process has been investigated carefully before. Liquid bath entries of "rough"silicon wafers are studied experimentally in the present work, and the air entrainment condition is analyzed specially with the lubrication theory. The roughness effects on the moving contact lines are therefore explored. The contact line pinning is found to be the main reason for the dynamically enhanced hydrophobicity of rough surface, which implies an effective microscopic contact angle of θ_e = θ_Y + 90° where θY is the Young's contact angle of the material. Our results suggest that the solid surfaces can be considered as hydrophobic ones for a wide range of dynamic process, since they are normally rough. The work can also be considered as a starting point for investigating the high-speed advancing of moving contact line on rough surfaces.     

Instabilities and the null energy condition

We show that violation of the null energy condition implies instability in a broad class of models, including gauge theories with scalar and fermionic matter as well as any perfect fluid. When applied to the dark energy, our results imply that w=p/ρ$w=p/\rho$ is unlikely to be less than −1.