The Eight Cayley–Dickson Doubling Products

The purpose of this paper is to identify all eight of the basic Cayley–Dickson doubling products. A Cayley–Dickson algebra \({\mathbb{A}_{N+1}}\) of dimension \({2^{N+1}}\) consists of all ordered pairs of elements of a Cayley–Dickson algebra \({\mathbb{A}_{N}}\) of dimension \({2^N}\) where the product \({(a, b)(c, d)}\) of elements of \({\mathbb{A}_{N+1}}\) is defined in terms of a pair of second degree binomials \({(f(a, b, c, d), g(a, b, c,d))}\) satisfying certain properties. The polynomial pair\({(f, g)}\) is called a ‘doubling product.’ While \({\mathbb{A}_{0}}\) may denote any ring, here it is taken to be the set \({\mathbb{R}}\) of real numbers. The binomials \({f}\) and \({g}\) should be devised such that \({\mathbb{A}_{1} = \mathbb{C}}\) the complex numbers, \({\mathbb{A}_{2} = \mathbb{H}}\) the quaternions, and \({\mathbb{A}_{3} = \mathbb{O}}\) the octonions. Historically, various researchers have used different yet equivalent doubling products.     

Levels and sublevels of algebras obtained by the Cayley–Dickson process

In this paper, we generalize the concepts of level and sublevel of a composition algebra to algebras obtained by the Cayley–Dickson process and we will show that, in the case of level for algebras obtained by the Cayley–Dickson process, the situation is the same as for the integral domains, proving that for any positive integer n, there is an algebra A obtained by the Cayley–Dickson process with the norm form anisotropic over a suitable field, which has the level ${n \in \mathbb{N}-\{0\}}$ .     

Ternary Derivations of Generalized Cayley–Dickson Algebras

Abstract

Ternary derivations, ternary Cayley derivations and ternary automorphisms are computed over fields of characteristic ≠ 2, 3 for the algebras A _t obtained by the Cayley–Dickson duplication process. While the derivation algebra of A _t stops growing after t = 3, the ternary derivation algebra significantly decreases in the step from the octonions A ₃ to the sedenions A ₄, revealing the symmetry lost on that stage.     

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.     

Codes Over Subsets of Algebras Obtained by the Cayley–Dickson Process

In this paper, we define binary block codes over subsets of real algebras obtained by the Cayley–Dickson process and we provide an algorithm to obtain codes with a better rate. This algorithm offers more flexibility than other methods known until now.     

The Ehrlich–Aberth method for palindromic matrix polynomials represented in the Dickson basis

An algorithm based on the Ehrlich–Aberth root-finding method is presented for the computation of the eigenvalues of a T-palindromic matrix polynomial. A structured linearization of the polynomial represented in the Dickson basis is introduced in order to exploit the symmetry of the roots by halving the total number of the required approximations. The rank structure properties of the linearization allow the design of a fast and numerically robust implementation of the root-finding iteration. Numerical experiments that confirm the effectiveness and the robustness of the approach are provided.     

The application of Cayley–Bacharach Theorem to bivariate Lagrange interpolation

In this paper, we give a new proof of the famous Cayley–Bacharach theorem by means of interpolation, and deduce a general method of constructing properly posed set of nodes for bivariate Lagrange interpolation. As a result, we generalize the main results in Liang (On the interpolations and approximations in several variables, Jilin University, 1965), Liang and Lü (Approximation Theory IX, Vanderbilt University Press, 1988) and Liang et al. (Analysis, Combinatorics and Computing, Nova Science Publishers, Inc., New York, 2002) to the more extensive situations.     

Krylov–Safonov estimates for a degenerate diffusion process

This paper proves a Krylov–Safonov estimate for a multidimensional diffusion process whose diffusion coefficients are degenerate on the boundary. As applications the existence and uniqueness of invariant probability measures for the process and Hölder estimates for the associated partial differential equation are obtained.     

Parameter identification for the Hermite Ornstein–Uhlenbeck process

Statistical Inference for Stochastic Processes - By using the analysis on Wiener chaos, we study the behavior of the quadratic variations of the Hermite Ornstein–Uhlenbeck process, which is...     

The density evolution of the killed McKean–Vlasov process

ABSTRACT

The density evolution of McKean–Vlasov stochastic differential equations in the presence of an absorbing boundary is analysed where the solution to such equations corresponds to the dynamics of partially killed large populations. By using a fixed point theorem, we show that the density evolution is characterized as the solution of an integro-differential Fokker–Planck equation with Cauchy–Dirichlet data. This problem arises naturally within mean field game theory.     

Integration by parts for two–parameter single jump process

We consider a process with a single jump at a random two parameter ‘time’ (T_l,T₂). We consider the case when (T_l,T₂) follows an exponential distribution.By employing a time change technique we derive an integration by parts formula for the process     

Sharp large deviations for the non-stationary Ornstein–Uhlenbeck process

For the Ornstein–Uhlenbeck process, the asymptotic behavior of the maximum likelihood estimator of the drift parameter is totally different in the stable, unstable, and explosive cases. Notwithstanding this trichotomy, we investigate sharp large deviation principles for this estimator in the three situations. In the explosive case, we exhibit a very unusual rate function with a shaped flat valley and an abrupt discontinuity point at its minimum.     

The Kelvin–Nevanlinna–Royden criterion for p-parabolicity

We generalize the so called Kelvin–Nevanlinna–Royden criterion for the parabolicity of manifolds to the case of p-parabolicity for all . Received July 29, 1998; in final form September 25, 1998