Integration of vector Sobolev-type PDE over octonions

Vector mixed Sobolev-type partial differential equations (PDE) are studied. The technique of their integration over the octonion algebra is developed. Theorems about integration of vector mixed Sobolev-type PDE over octonions are proved.     

Feynman integration over octonions with application to quantum mechanics

The article is devoted to Gaussian quasi‐measures and Feynman integrals on infinite‐dimensional spaces with values in the octonion algebra. Their characteristic functionals are studied. Products and convolutions of characteristic functionals and quasi‐measures are investigated. Theorems about properties of octonion‐valued Gaussian quasi‐measures and Feynman integrals are proved. Applications of the Feynman integration over octonions to quantum mechanics and partial differential equations are outlined. Copyright © 2009 John Wiley & Sons, Ltd.     

The BMV-conjecture over quaternions and octonions

This paper investigates generalizations of the BMV-conjecture for quaternionic and octonionic matrices. For quaternions the correctness of the formulation is shown as well as its equivalence to the original conjecture for complex matrices. General properties of octonions and Hermitian matrices over them are examined for the BMV-conjecture formulation over octonions.     

A note on the fundamental theorem of algebra for the octonions

This paper is a revision of a portion of the author's doctoral dissertation submitted to the University of Oregon. Using elementary concepts of K$K$-theory, the Brouwer degree of the power map in the octonions is computed. Later, a proof of a weaker version of the fundamental theorem of algebra for polynomials with coefficients in the octonions is given. As a partial complement, a lower bound to the number of solutions of a homogeneous monomial equation over the octonions is obtained.     

Affine and wrap quasi-algebras over octonions

The article is devoted to affine and wrap quasi-algebras over octonions. Residues of functions of octonion variables are studied. They are used for constructing such quasi-algebras. Their structure is investigated.     

Matrix representations of octonions and their applications

As is well-known, the real quaternion division algebra ℍ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because is a non-associative algebra over ℝ. However since is an extension of ℍ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.     

Associative Cones and Integrable Systems

We identify R7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S6. It is known that a cone over a surface M in S6 is an associative submanifold of R7 if and only if M is almost complex in S6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S6 are the equation for primitive maps associated to the 6-symmetric space G2/T2, and use this to explain some of the known results. Moreover, the equation for S1-symmetric almost complex curves in S6 is the periodic Toda lattice, and a discussion of periodic solutions is given.     

Characterization of orthomaps on the Cayley plane

We classify maps which preserve orthogonality on the Cayley projective plane over octonions. In addition, we also classify orthogonality preserving maps on finite dimensional projective spaces over reals, complexes, or quaternions. Unlike similar results which extend Uhlhorns’s theorem we assume neither injectivity/surjectivity nor that orthogonality is preserved in both directions.     

Split‐type octonion matrix

The split and hyperbolic (countercomplex) octonions are eight‐dimensional nonassociative algebras over the real numbers, which are in the form , where e_m's have different properties for them. The main purpose of this paper is to define the split‐type octonion and its matrix whose inputs are split‐type octonions and give some properties for them by using the real quaternions, split, and hyperbolic (countercomplex) octonions. On the other hand, to make some definitions, we present some operations on the split‐type octonions. Also, we show that every split‐type octonions can be represented by 2 × 2 real quaternion matrix and 4 × 4 complex number matrix. The information about the determinants of these matrix representations is also given. Besides, the main features of split‐type octonion matrix concept are given by using properties of  real quaternion matrices. Then, 8n × 8nreal matrix representations of split‐type octonion matrices are shown, and some algebraic structures are examined. Additionally, we introduce real quaternion adjoint matrices of split‐type octonion matrices. Moreover, necessary and sufficient conditions and definitions are given for split‐type octonion matrices to be special split‐type octonion matrices. We describe some special split‐type octonion matrices. Finally, oct‐determinant of split‐type octonion matrices is defined. Definitive and understandable examples of all definitions, theorems, and conclusions were given for a better understanding of all these concepts.     

Cayley–Dickson Procedure, Relativistic Wave Equations and Supersymmetric Oscillators

The relativistic first-order wave equations for massive particles with spin 0,1,1/2 are formulated in terms of a factorization of the Klein–Fock equation by means of the algebra of octonions. An analogous method applied to Hamiltonian of the quantum isotropic oscillator leads to the natural generalization of the model. The class of supersymmetric oscillators with dimension N7 associated with te algebras of the Cayley–Dickson series is introduced.     

带奇异系数的有限维与无穷维随机微分方程

本文对具非Lipschitz系数的随机微分方程给出解的存在唯一性与非爆炸性的新判别条件,少许改进了文\cite{4}的有关结果. 通过控制交互作用, 该结果还被推广到无穷维情形.     

The transverse curvature effect on the incompressible laminar boundary layer for longitudinal flow over a cylinder

A solution to the partial differential equations governing incompressible laminar flow in the axial direction over a circular cylinder is presented. A method is employed which reduces the partial differential equations to a set of ordinary differential equations which are then solved consecutively. The solution is initiated at a leading edge and proceeds far downstream. Velocity profiles and values for the coefficient of skin friction, displacement thickness and momentum thickness are obtained. The results are compared to previously obtained solutions valid near the leading edge and asymptotic solutions valid far downstream.     

Quantization as geometry in phase space

The geometrical interpretation of quantum mechanics in relativistic phase space proposed by this writer leads, under the sole assumption of a connection from which commutation rules between covariant derivatives ensue that reproduce Heisenberg’s, to an 8-dimensional metric space which contains and generalizesall of first quantization. Some of the new results (to be tested) are: Existence of a maximal proper acceleration; The reality condition on geodesic lines is identical with Bohr-Sommerfeld quantization; The Klein-Gordon and Dirac equations thus generalized give positive mass spectra lying on Regge trajectories; There exists (only in 8 dimensions) a natural supersymmetry (Cartan’s triality); Pure spinors and octonions appear as natural tools for describing particles, and for deeper analyses.     

Noncommutative,quasi-conformal integral transforms over quaternions and octonions

This paper is devoted to the study of holomorphic and meromorphic functions of quaternion and octonion variables. New classes of quasi-conformal and quasi-meromorphic mappings are defined and studied. Different properties of these functions such as residues and the argument principle are studied. We prove that the family of all quasi-conformal diffeomorphisms of a domain is a topological group relative to composition of mappings. In particular, cases where it is a finite-dimensional Lie group over R are studied. Relations between quasi-conformal functions and integral transformations of functions over quaternions and octonions are investigated. Noncommutative analogs of the Mellin transformations are studied and used. In addition, examples of such functions are given. Finally, applications to problems of complex analysis are discussed. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 52, Functional Analysis, 2008.     

The all-associativity of octonions and its applications

Using an elementary method, we give a new proof of the all-associativity of octonions. As some applications, the known Taylor theorem is improved, and a new definition and new properties of octonionic determinant are also obtained.     

New explicit approximate solution of MHD viscoelastic boundary layer flow over stretching sheet

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non‐Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non‐uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical methods for structured population models: The Escalator Boxcar Train

A numerical integration method is introduced for the class of hyperbolic partial differential equations that occur in physiologically structured population models. These equations describe the time evolution of the population density-function over the individual state-space Ω. Exploiting the biological interpretation of the equation, this density-function is represented by a set of moments over a collection of subdomains in Ω, moving along the characteristics of the partial differential equation (PDE). These moments are readily interpreted as numbers of individuals in a cohort, mean individual state in a cohort, etc. The numerical method consists of approximating the differential equations. The method appears to be very efficient for this special type of PDE. It combines such desirable properties as as lack of dispersion and dissipation and a preservation of monotonicity of the density function along the characteristics with a strong relation to the biological background of the equations for these moments to arrive at a closed system of ordinary differential equations. The method also allows one to incorporate the usual type of nonlinearities that occur in physiologically structured population models. A numerical example is presented as an illustration of the application of the method.     

Group solution for unsteady free-convection flow from a vertical moving plate subjected to constant heat flux

The problem of heat and mass transfer in an unsteady free-convection flow over a continuous moving vertical sheet in an ambient fluid is investigated for constant heat flux using the group theoretical method. The nonlinear coupled partial differential equation governing the flow and the boundary conditions are transformed to a system of ordinary differential equations with appropriate boundary conditions. The obtained ordinary differential equations are solved numerically using the shooting method. The effect of Prandlt number on the velocity and temperature of the boundary-layer is plotted in curves. A comparison with previous work is presented.     

Differential equations over subalgebras of the full matrix algebra

We consider a general matrix differential equation whose parameters and solution are defined over subalgebras of the full matrix algebra. Some subalgebras are presented over which the equation splits into a set of equations over matrices of smaller orders. Multiplicative matrix equations of higher order in normal form are introduced and studied over some subalgebras of 2 × 2 matrices.