Quasi-linear Transformation Research Articles

A splitting method to solve a single nonlinear equation with derivative-free iterative schemes

In the paper, we convert a single nonlinear equation to a system consisting of two equations. While a quasi-linear term is added on the first equation, the nonlinear term in the second equation is decomposed at two sides through a weight parameter. After performing a linearization, an iterative scheme is derived, which is proven of third-order convergence for certain parameters. An affine quasi-linear transformation in the plane is established, and the condition for the spectral radius being smaller than one for the convergence of the iterative scheme is derived. By using the splitting method, we can further identify a sufficient condition for the convergence of the iterative scheme. Then, we develop a step-wisely quasi-linear transformation technique to solve nonlinear equations. Proper values of the parameters are qualified by the derived inequalities for both iterative schemes, which accelerate the convergence speed. The performances of the proposed iterative schemes are assessed by numerical tests, whose advantages are fast convergence, saving the function evaluation per iteration and without needing the differential of the given function.

Multi-scale Arithmetization of Linear Transformations

A constructive nonstandard interpretation of a multiscale affine transformation scheme is presented. It is based on the $$\Omega $$ -numbers of Laugwitz and Schmieden and on the discrete model of the real line of Reeb and Harthong. In this setting, the nonstandard version of the Euclidean affine transformation gives rise to a sequence of quasi-linear transformations over integer spaces, allowing integer-only computations.

Quasi-linear transformations and discrete tilings

Tilings of the discrete plane Z 2 generated by quasi-linear transformations (QLT) have been introduced by Nehlig [P. Nehlig, Applications quasi-affines: Pavages par images réciproques, Theoretical Computer Science 156 (1995) 1–38]. We studied these tilings and gave some results, such as periodicity and the number of neighbours of each of them [M.-A. Jacob-Da Col, Applications quasi-affines et pavages du plan discret, Theoretical Computer Science 259 (2001) 245–269. Also available in English: http://dpt-info.u-strasbg.fr/~jacob/articles/paving.pdf]. The aim of this paper is to go on with this study in the discrete n -dimensional space Z 2 ; we give a lower and an upper bound to the number of distinct tiles. We also give an algorithm to determine the points of a given tile, this algorithm will induce another algorithm to determine the number of distinct tiles associated to a QLT.

On the Transformation T+m Due Gray and Clark

We determine the values of the integer m for which the parametric transformation T+m due Gray and Clark is well conditionned. This process of acceleration being quasilinear transformation, we use an adequate definition of the condition numbers that apply to the real sequences of linear convergence. The results obtained for this set are enough meaning

On condition numbers of some quasi-linear transformations

Convergence acceleration processes are known to be well conditioned for alternating sequences and ill conditioned for monotonic ones. The aim of this paper is to adapt the definition of conditioning and to give a link between this notion and the property of convergence acceleration. The cases of the linear and logarithmic convergence are studied in details.

GL( E)-quasilinear transformations and acceleration

The aim of this paper is to show that Henrici transformations, and, more generally, GL( E)-quasilinear transformations, do not accelerate the convergence of linearly converging sequences.

Quasilinear sequence transformations

We give a new characterization of quasilinear sequence transformations; we prove that any quasilinear transformation can be represented by its kernel. This approach is new and allows one to give a general result of convergence acceleration and tools for the construction of new quasilinear transformations.

Three-valued quasi-linear transformation for logic synthesis

A nonlinear transformation, called the sign transform, is defined and studied. It maps all the ternary functions onto themselves, is uniquely invertible and is inherently related to Walsh and Walsh–Galois transforms. Decomposable Boolean functions and ternary product terms are mapped analytically into the sign domain. It is shown that sign coefficients describe the structure of a ternary output logic network implemented by a complete basis: EXOR of variables and sign function. Methods are discussed for direct transformation of switching functions without having to use the butterfly algorithm: Shannon's decomposition in the sign domain and, in general, merging sign spectra of subfunctions.As applications, spectra of arbitrary logic circuits is computed analytically by merging spectra of their gate components. Also, it is shown that testable realisations of incompletely specified Boolean functions can be designed by appropriate assignments of don't cares in the sign domain.

A Method of a Quasi-Linear Canonical Transformation for the Many-Body Problem. II

We apply the method of quasi-linear canonical transformation proposed in a previous paper to the system of N interacting bosons, and we calculate the excitation energy for the dilute hard-sphere Bose gas. In our formulation the total number of the particles are exactly conserved in contrast with the Bogoliubov theory. There appears no logarithmic divergent term in the physical quantities which correspond to the self-energy operators .Ell and .s02 appearing in the method of Green's function. In the prevIOUS paper referred to as I, the general theory of the: quasi-linear transformation has been investigated. In this paper we shall give its application to the system of N bosons at the zero temperature, and we evaluate the excitation energy and the chemical potential in the hard-sphere Bose gas. The system considered is enclosed in the cubic box of volume S2. The number of the par­ ticles, N, and the volume S2 are so large that we consider the 'problem. in the limit N ---,) 00, S2---') 00 as keeping N / S2 to be constant. In our formulation the total number of the particles are exactly conserved. It is assumed that the ground state of the system is given by the completely degenerate state of the quasi-particles denoted by (ao+)NIO) where ao+ is the creation operator for the quasi-particles with zero momentum, and the operators a p and a p + are introduced by the quasi-linear transformation. This assumption is very plausible as seen in the next section. Some theorems for the matrix' elements of the field operators are derived from the properties of the q~asi-linear transformation and from the assumption that' almost all the quasi-particles are condensed into the zero momentum state at the zero temperature. Usingthese theorems it can be shown that the excitation energy exhibits no gap, and the volocity of the quasi-particles with small momenta is equal to the hydrodynamical sound velocity. Moreover, it can be seen that there appears no divergent term in the physical quantities which corresponds to the self-energy operators appearing in the method of Green's function. For the case of the dilute hard-sphere Bose gas, we obtain the excitation energy for a small momentum and the chemical potential which

Thermodynamical Functions of a Dilute Hard-Sphere Bose Gas

We obtain thermodynamical functions for a dilute Bose gas of hard spheres by using the efffective Hamiltonian which is obtained by a quasi-linear transformation proposed previously and may be recognized as an improvement of the model Hamiltonian of Lee and Yang which is not correct about the terms of second order in the diameter of hard spheres when it is applied to the problem of the phase transition. The Bose condensation occurs at a temperature below which the degenerate phase of the quasi-particles appears. We evaluate the amount of the gap of the specific heat and that of the compressibility at the transition point.

Nonlinear effects and the transformation of wave energy in an inhomogeneous plasma

This paper discusses the problem: under what conditions does the quasilinear transformation of waves in plasma with an inhomogeneous beam remain a dominant effect if the nonlinear interaction of waves is respected. It is shown that in the region of densitiesω p 2 ω B 2 the nonlinear interaction between the high-frequency and the newly excited waves can already affect the initial stage of the process and can result in the establishment of the Rayleigh-Jeans distribution for the two groups of waves. If, on the other hand, the low-frequency waves grow first with a quasilinear increment, the nonlinear transformation of energy inside a group of low-frequency waves is again apt to affect substantially their final amplitude and thus also the total amount of energy that can be transformed into those waves. The conditions under which a particular nonlinear effect predominates are derived for various characteristic values of the density of a cold plasma.