225 research outputs found
El problema de los mínimos cuadrados con restricciones de igualdad mediante la factorización QR generalizada
The generalized QR factorization, also known as GQR factorization, is a method that simultaneously transforms two matrices A and B in a triangular form. In this paper, we show the application of GQR factorization in solving linear equality-constrained least square problems; in addition, we explain how to use GQR factorization for solving quaternion least-square problems through the matrix representation of quaternions.La factorización QR generalizada, también conocida como factorización GQR, permite descomponer dos matrices A y B simultáneamente a una forma triangular. En este artículo, se muestra cómo aplicar la factorización GQR para resolver problemas de mínimos cuadrados con restricciones de igualdad; además, se emplea esta factorización para resolver problemas de mínimos cuadrados sobre cuaterniones
Algebraic technique for mixed least squares and total least squares problem in the reduced biquaternion algebra
This paper presents the reduced biquaternion mixed least squares and total
least squares (RBMTLS) method for solving an overdetermined system in the reduced biquaternion algebra. The RBMTLS method is suitable when
matrix and a few columns of matrix contain errors. By examining real
representations of reduced biquaternion matrices, we investigate the conditions
for the existence and uniqueness of the real RBMTLS solution and derive an
explicit expression for the real RBMTLS solution. The proposed technique covers
two special cases: the reduced biquaternion total least squares (RBTLS) method
and the reduced biquaternion least squares (RBLS) method. Furthermore, the
developed method is also used to find the best approximate solution to over a complex field. Lastly, a numerical example is presented to
support our findings.Comment: 19 pages, 3 figure
A noncommutative framework for topological insulators
We study topological insulators, regarded as physical systems giving rise to
topological invariants determined by symmetries both linear and anti-linear.
Our perspective is that of noncommutative index theory of operator algebras. In
particular we formulate the index problems using Kasparov theory, both complex
and real. We show that the periodic table of topological insulators and
superconductors can be realised as a real or complex index pairing of a
Kasparov module capturing internal symmetries of the Hamiltonian with a
spectral triple encoding the geometry of the sample's (possibly noncommutative)
Brillouin zone.Comment: 32 pages, final versio
The Octonions
The octonions are the largest of the four normed division algebras. While
somewhat neglected due to their nonassociativity, they stand at the crossroads
of many interesting fields of mathematics. Here we describe them and their
relation to Clifford algebras and spinors, Bott periodicity, projective and
Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also
touch upon their applications in quantum logic, special relativity and
supersymmetry.Comment: 56 pages LaTeX, 11 Postscript Figures, some small correction
Quaternionic R transform and non-hermitian random matrices
Using the Cayley-Dickson construction we rephrase and review the
non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I.
Zahed, Nucl.Phys. B , 603 (1997)], that generalizes the free
probability calculus to asymptotically large non-hermitian random matrices. The
main object in this generalization is a quaternionic extension of the R
transform which is a generating function for planar (non-crossing) cumulants.
We demonstrate that the quaternionic R transform generates all connected
averages of all distinct powers of and its hermitian conjugate :
\langle\langle \frac{1}{N} \mbox{Tr} X^{a} X^{\dagger b} X^c \ldots
\rangle\rangle for . We show that the R transform for
gaussian elliptic laws is given by a simple linear quaternionic map
where
is the Cayley-Dickson pair of complex numbers forming a quaternion
. This map has five real parameters , ,
, and . We use the R transform to calculate the limiting
eigenvalue densities of several products of gaussian random matrices.Comment: 27 pages, 16 figure
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