Search results for "Mathematica"

Smooth Feshbach map and operator-theoretic renormalization group methods

2003

Abstract A new variant of the isospectral Feshbach map defined on operators in Hilbert space is presented. It is constructed with the help of a smooth partition of unity, instead of projections, and is therefore called smooth Feshbach map . It is an effective tool in spectral and singular perturbation theory. As an illustration of its power, a novel operator-theoretic renormalization group method is described and applied to analyze a general class of Hamiltonians on Fock space. The main advantage of the new renormalization group method over its predecessors is its technical simplicity, which it owes to the use of the smooth Feshbach map.

Simplification of Models

2016

In practical applications the “complete” model, i.e., a model that contains all features that the experts in the application domain consider important, is often quite complicated and difficult to analyse mathematically. A straightforward numerical realization is often costly and may give very little qualitative understanding of the situation. It is therefore important to study if the model can be systematically simplified in order to enhance a qualitative analysis/understanding.

Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity

2001

Geometric Singular Perturbation theory has traditionally dealt only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.

Multiple Canard Cycles in Generalized Liénard Equations

2001

AbstractThe paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields. The results deal with any number of parameters. Proofs are based on the techniques introduced in “Canard Cycles and Center Manifolds” (F. Dumortier and R. Roussarie, 1996, Mem. Amer. Math. Soc., 121). The presentation is limited to generalized Liénard equations εx+α(x, c)x+β(x, c)=0.

New solvability conditions for the Neumann problem for ordinary singular differential equations

2000

Collision Orbits in the Isosceles Rectilinear Restricted Problem

1995

In the study of the Collinear Three-Body Problem, McGehee (1974) introduced a new set of coordinates which had the effect of blowing up the triple collision singularity. Subsequently, his method has been used to analyze some other collision or singularities. Recently, Wang (1986) introduced another transformation which differs from the McGehee’s coordinates in the fact that the blowing-up factor is now the potential function, U, instead of the moment of inertia, I. Meyer and Wang (1993) have applied this method to the Restricted Isosceles Three-body Problem with positive energy and Cors and Llibre (1994) to the hyperbolic restricted three-body problem. In this paper we study the singulariti…

A mathematical approach to predicting the percutaneous absorption enhancing effect of sodium lauryl sulphate.

2003

A study has been made of the effect of sodium lauryl sulphate (SLS) at several concentrations from 0.24 to 5% (w/w) on skin permeability. Seven model drugs were selected for this study on the basis of their lipophilicity as represented by their logP(oct) values (from -0.95 to 4.2). Skin pre-treatment with aqueous solutions of SLS does not increase the permeability coefficient of the most lipophilic compounds (logP(oct)> or =3). For the other compounds assayed the increase in the permeability coefficients depends on the concentration of SLS used in the skin pre-treatment, and on the lipophilicity of the compounds tested.The correlation between the inverse of SLS efficacy as an enhancer (1/ER…

Advances in designing drip irrigation laterals

2018

It is known that using paired laterals, in which two distribution pipes extend in opposite directions from a common manifold, contribute to increasing water use efficiency (WUE). Recently, an analytical procedure to optimally design paired drip laterals on uniform slopes was proposed. More recently, this design procedure was simplified by deriving simple explicit relationships, as a function of 16 calibration constants, with relative errors that were less than 2%. In this paper, further simple design relationships are derived that require only 3 calibration constants, thus more readily obtainable results are produced and the influence of the flow rate and diameter exponents of resistance eq…

Cyclicity of common slow–fast cycles

2011

Abstract We study the limit cycles of planar slow–fast vector fields, appearing near a given slow–fast cycle, formed by an arbitrary sequence of slow parts and fast parts, and where the slow parts can meet the fast parts in a nilpotent contact point of arbitrary order. Using the notion slow divergence integral, we delimit a large subclass of these slow–fast cycles out of which at most one limit cycle can perturb, and a smaller subclass out of which exactly one limit cycle will perturb. Though the focus lies on common slow–fast cycles, i.e. cycles with only attracting or only repelling slow parts, we present results that are valid for more general slow–fast cycles. We also provide examples o…

Test module filtrations for unit $F$-modules

2015

We extend the notion of test module filtration introduced by Blickle for Cartier modules. We then show that this naturally defines a filtration on unit $F$-modules and prove that this filtration coincides with the notion of $V$-filtration introduced by Stadnik in the cases where he proved existence of his filtration. We also show that these filtrations do not coincide in general. Moreover, we show that for a smooth morphism $f: X \to Y$ test modules are preserved under $f^!$. We also give examples to show that this is not the case if $f$ is finite flat and tamely ramified along a smooth divisor.