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THE OCTONIONIC EIGENVALUE PROBLEM
, 1998
"... Abstract. We discuss the eigenvalue problem for 2×2 and 3×3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with nonreal eigenvalues. 1. ..."
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Abstract. We discuss the eigenvalue problem for 2×2 and 3×3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with nonreal eigenvalues. 1.
Finding Octonionic Eigenvectors Using Mathematica
, 1998
"... The eigenvalue problem for 3 × 3 octonionic Hermitian matrices contains some surprises, which we have reported elsewhere [1]. In particular, the eigenvalues need not be real, there are 6 rather than 3 real eigenvalues, and the corresponding eigenvectors are not orthogonal in the usual sense. The non ..."
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The eigenvalue problem for 3 × 3 octonionic Hermitian matrices contains some surprises, which we have reported elsewhere [1]. In particular, the eigenvalues need not be real, there are 6 rather than 3 real eigenvalues, and the corresponding eigenvectors are not orthogonal in the usual sense. The nonassociativity of the octonions makes computations tricky, and all of these results were first obtained via brute force (but exact) Mathematica computations. Some of them, such as the computation of real eigenvalues, have subsequently been implemented more elegantly; others have not. We describe here the use of Mathematica in analyzing this problem, and in particular its use in proving a generalized orthogonality property for which no other proof is known.
Locality, Weak or Strong Anticipation and Quantum Computing. I. Nonlocality in Quantum Theory
"... Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Categ ..."
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Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Category theory provides the necessary coordinatefree mathematical language which is both constructive and nonlocal to subsume the various interpretations of quantum theory in one pullback/pushout Dolittle diagram. This diagram can be used to test and classify physical devices and proposed algorithms for weak or strong anticipation. Quantum Information Science is more than a merger of ChurchTuring and quantum theories. It has constructively to bridge the nonlocal chasm between the weak anticipation of mathematics and the strong anticipation of physics.
A DUALITY METHOD IN PREDICTION THEORY OF MULTIVARIATE STATIONARY SEQUENCES
, 2001
"... Hermitian matrixvalued weight function, trigonometric approximation. Abstract. Let W be an integrable positive Hermitian q × qmatrix valued function on the dual group of a discrete abelian group G such that W −1 is integrable. Generalizing results of T. Nakazi and of A. G. Miamee [N] and M. Pourah ..."
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Hermitian matrixvalued weight function, trigonometric approximation. Abstract. Let W be an integrable positive Hermitian q × qmatrix valued function on the dual group of a discrete abelian group G such that W −1 is integrable. Generalizing results of T. Nakazi and of A. G. Miamee [N] and M. Pourahmadi [MiP] for q = 1 we establish a correspondencebetween trigonometric approximation problems in L 2 (W) and certain approximation problems in L 2 (W −1). The result is applied to prediction problems for qvariate stationary processes over G, in particular, to the case G = Z. 1 2 M. FRANK AND L. KLOTZ 1.