Clifford algebras and vector-valued rational forms. I

1990 ◽  
Vol 431 (1882) ◽  
pp. 285-300 ◽  
Keyword(s):  
Continued Fractions ◽  
Clifford Algebras ◽  
Recurrence Relations ◽  
Generalized Inverse ◽  
Multiplicative Group ◽  
Group Inverse ◽  
Scalar Theory ◽  
Real Analytic ◽  
Usual Scalar ◽  
Vector Valued

We demonstrate how Clifford algebras offer a framework for the construction of vector-valued rational forms possessing features of the usual scalar theory, including three-term recurrence relations for continued fractions. The price for this advantage is that the Moore–Penrose generalized inverse is replaced by the multiplicative group inverse of a Clifford algebra. However, the connection between the new vector-valued rational forms and generalized inverse rational forms is a close one; in fact, the two forms are identical for real analytic data.

scholarly journals On real-analytic recurrence relations for cardinal exponential B-splines

2007 ◽  
Vol 145 (2) ◽  
pp. 253-265 ◽  
Author(s):  
J.M. Aldaz ◽  
O. Kounchev ◽  
H. Render
Keyword(s):  
Recurrence Relations ◽  
B Splines ◽  
Real Analytic

Extended continued fractions, recurrence relations and two-dimensional Markov processes

1989 ◽  
Vol 21 (2) ◽  
pp. 357-375 ◽  
Author(s):  
C. E. M. Pearce
Keyword(s):  
Markov Processes ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Equilibrium Distribution ◽  
Linear Recurrence ◽  
Natural State ◽  
Two Dimensional ◽  
State Spaces ◽  
General Order ◽  
Dimension 2

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

FEWNOMIALS AND INTERSECTIONS OF LINES WITH REAL ANALYTIC SUBGROUPS IN Gnm

2002 ◽  
Vol 34 (1) ◽  
pp. 21-32 ◽  
Author(s):  
PAULA B. COHEN ◽  
UMBERTO ZANNIER
Keyword(s):  
Special Class ◽  
Multiplicative Group ◽  
Uniform Bounds ◽  
Complex Dimension ◽  
Real Analytic

In this paper, the authors study intersections of a special class of curves with algebraic subgroups of the multiplicative group of complex dimension at least 2. They show how results of Khovanskii on fewnomials can be used to derive finiteness results and bounds for the degrees of algebraic points for such intersections from more general results on intersections of curves with non-algebraic subgroups. They thereby generalise their earlier results, and recover in some cases, using different methods, more uniform bounds than those given in related work of Bombieri, Masser and Zannier.

Extended continued fractions, recurrence relations and two-dimensional Markov processes

1989 ◽  
Vol 21 (02) ◽  
pp. 357-375 ◽  
Author(s):  
C. E. M. Pearce
Keyword(s):  
Markov Processes ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Equilibrium Distribution ◽  
Linear Recurrence ◽  
Natural State ◽  
Two Dimensional ◽  
State Spaces ◽  
General Order ◽  
Dimension 2

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

Interpolation of Vector-Valued Real Analytic Functions

2002 ◽  
Vol 66 (2) ◽  
pp. 407-420 ◽  
Author(s):  
José Bonet ◽  
Paweł Domański ◽  
Dietmar Vogt
Keyword(s):  
Analytic Functions ◽  
Real Analytic Functions ◽  
Real Analytic ◽  
Vector Valued

Electrical a.c. conductivity for hot isotropic collisional plasmas

1986 ◽  
Vol 35 (1) ◽  
pp. 151-164 ◽  
Author(s):  
Helmut Hebenstreit ◽  
Kurt Suchy
Keyword(s):  
Boltzmann Equation ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Infinite System ◽  
Landau Damping ◽  
Balance Equations ◽  
The Boltzmann Equation ◽  
Collision Models

With an infinite system of balance equations, derived from the Boltzmann equation, conductivity expressions are obtained in the form of three-term recurrence relations leading to continued fractions. Without collisions, only Landau damping causes attenuation. Its modification by collisions is illustrated for some simple collision models.

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