The renormalization group and the implicit function theorem for amplitude equations

This article lays down the foundations of the renormalization group (RG) approach for differential equations characterized by multiple scales. The renormalization of constants through an elimination process and the subsequent derivation of the amplitude equation [Chen, Phys. Rev. E 54, 376 (1996)] are given a rigorous but not abstract mathematical form whose justification is based on the implicit function theorem. Developing the theoretical framework that underlies the RG approach leads to a systematization of the renormalization process and to the derivation of explicit closed-form expressions for the amplitude equations that can be carried out with symbolic computation for both linear and nonlinear scalar differential equations and first order systems but independently of their particular forms. Certain nonlinear singular perturbation problems are considered that illustrate the formalism and recover well-known results from the literature as special cases. © 2008 American Institute of Physics.

THE IMPLICIT FUNCTION THEOREM FOR CONTINUOUS FUNCTIONS

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence. some versions of these classic theorems are proved when we consider differenciable (not necessarily C-1) maps.

A homological version of the implicit function theorem

Our objective in this paper is to prove an Implicit Function Theorem for general topological spaces. As a consequence, we show that, under certain conditions, the set of the invertible elements of a topological monoid X is an open topological group in X and we use the classical topological group theory to conclude that this set is a Lie group.

A simple application of implicit function theorem

In this note we show that the roots of a polynomial are C∞ depend of the coefficients. The main tool to show this is the Implicit Function Theorem.

Function-based single and dual point haptic interaction in Cyberworlds

Polygon and point based models dominate virtual reality. These models also affect haptic rendering algorithms, which are often based on collision with polygons. With application to dual point haptic devices for operations like grasping, complex polygon and point based models will make the collision detection procedure slow. This results in the system not able to achieve interactivity for force rendering. To solve this issue, we use mathematical functions to define and implement geometry (curves, surfaces and solid objects), visual appearance (3D colours and geometric textures) and various tangible physical properties (elasticity, friction, viscosity, and force fields). The function definitions are given as analytical formulas (explicit, implicit and parametric), function scripts and procedures. We proposed an algorithm for haptic rendering of virtual scenes including mutually penetrating objects with different sizes and arbitrary location of the observer without a prior knowledge of the scene to be rendered. The algorithm is based on casting multiple haptic rendering rays from the Haptic Interaction Point (HIP), and it builds a stack to keep track on all colliding objects with the HIP. The algorithm uses collision detection based on implicit function representation of the object surfaces. The proposed approach allows us to be flexible when choosing the actual rendering platform, while it can also be easily adopted for dual point haptic collision detection as well as force and torque rendering. The function-defined objects and parts constituting them can be used together with other common definitions of virtual objects such as polygon meshes, point sets, voxel volumes, etc. We implemented an extension of X3D and VRML as well as several standalone application examples to validate the proposed methodology. Experiments show that our concern about fast, accurate rendering as well as compact representation could be fulfilled in various application scenarios and on both single and dual point haptic devices.

Differential geometry of intersection curves in R(4) of three implicit surfaces

We present algorithms for computing the differential geometry properties of intersection Curves of three implicit surfaces in R(4), using the implicit function theorem and generalizing the method of X. Ye and T. Maekawa for 4-dimension. We derive t, n, b(1), b(2) vectors and curvatures (k(1), k(2), k(3)) for transversal intersections of the intersection problem. (C) 2008 Elsevier B.V. All rights reserved.

Fitting 3D morphable models using implicit representations

We consider the problem of approximating the 3D scan of a real object through an affine combination of examples. Common approaches depend either on the explicit estimation of point-to-point correspondences or on 2-dimensional projections of the target mesh; both present drawbacks. We follow an approach similar to [IF03] by representing the target via an implicit function, whose values at the vertices of the approximation are used to define a robust cost function. The problem is approached in two steps, by approximating first a coarse implicit representation of the whole target, and then finer, local ones; the local approximations are then merged together with a Poisson-based method. We report the results of applying our method on a subset of 3D scans from the Face Recognition Grand Challenge v.1.0.

Weak and Semi Compatible Maps in Probabilistic Metric Space Using Implicit Relation

2000 Mathematics Subject Classification: 54H25, 47H10.

Wavelike patterns in one-dimensional coupled map lattices

We investigate the existence of wavelike solution for the logistic coupled map lattices for which the spatiotemporal periodic patterns can be predicted by a simple two-dimensional mapping. The existence of such wavelike solutions is proved by the implicit function theorem with constraints. We also examine the stabilities of these wave solutions under perturbations of uniform small deformation type. We show that in some specific cases these perturbations are completely general. The technique used in this paper is also applicable to investigate other space-time regular patterns.

When do export subsidies have a redistributional role?

If an export subsidy is efficient, that is, has a surplus-transfer role, then there exists an implicit function relating the optimal level of the subsidy to the income target in the agricultural sector. If an export subsidy is inefficient no such function exists. We show that dependence exists in large-export equilibrium, not in small-export equilibrium and show that these results remain robust to concerns about domestic tax distortions. The failure of previous work to produce this result stems from its neglect of the income constraint on producer surplus in the programming problem transferring surplusfrom consumersand taxpayers to farmers.

Discourses of anxiety in nursing practice : a psychoanalytic case study of the change-of-shift handover ritual

This paper reports on the findings of a study that considered how anxiety might function to organise nurses' practice. With reference to psychoanalytic theory this paper analyses field notes taken during a series of nursing change-of-shift handovers. The handover practices analysed met all the criteria for a ritual, as understood in psychoanalytic theory, and functioned to alleviate anxiety in the short term while symbolically expressing a forbidden and unknown knowledge. We argue that the handover ritual contained certain prohibitions, yet allowed some expression of the prohibited knowledge in a disguised way. The prohibition concerned how the patient affected the nurse, that is, moved the nurse to love and hate the patient. We argue that this prohibition is expressed, in disguise, via the displacement of affection for the patient onto other nurses and through negative stereotyping of some patients. We also argue that these prohibitions of the handover mirror broader prohibitions within nursing, and thus the rituals of the handover become an expression of how professional prohibitions are enacted in practice. We conclude that the important implicit function of the handover ritual is to keep anxiety at bay, thereby enabling the nurse to commence practice rather than being immobilised by the effect of potentially overwhelming anxiety.

Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144: 13-29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144: 13-29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green's theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.

SUFFICIENT CONDITIONS ON DIFFUSIVITY FOR THE EXISTENCE AND NONEXISTENCE OF STABLE EQUILIBRIA WITH NONLINEAR FLUX ON THE BOUNDARY

A reaction-diffusion equation with variable diffusivity and non-linear flux boundary condition is considered. The goal is to give sufficient conditions on the diffusivity function for nonexistence and also for existence of nonconstant stable stationary solutions. Applications are given for the main result of nonexistence.

Singularities and undefinitions in the calibration functions of sonic anemometers

A mathematical model of the process employed by a sonic anemometer to build up the measured wind vector in a steady flow is presented to illustrate the way the geometry of these sensors as well as the characteristics of aerodynamic disturbance on the acoustic path can lead to singularities in the transformation function that relates the measured (disturbed) wind vector with the real (corrected) wind vector, impeding the application of correction/calibration functions for some wind conditions. An implicit function theorem allows for the identification of those combinations of real wind conditions and design parameters that lead to undefined correction/ calibration functions. In general, orthogonal path sensors do not show problematic combination of parameters. However, some geometric sonic sensor designs, available in the market, with paths forming smaller angles could lead to undefined correction functions for some levels of aerodynamic disturbances and for certain wind directions. The parameters studied have a strong influence on the existence and number of singularities in the correction/ calibration function as well as on the number of singularities for some combination of parameters. Some conclusions concerning good design practices are included.

Robust infraestructure design in rapid transit rail systems

Incidents and rolling stock breakdowns are commonplace in rapid transit rail systems and may disrupt the system performance imposing deviations from planned operations. A network design model is proposed for reducing the effect of disruptions less likely to occur. Failure probabilities are considered functions of the amount of services and the rolling stock’s routing on the designed network so that they cannot be calculated a priori but result from the design process itself. A two recourse stochastic programming model is formulated where the failure probabilities are an implicit function of the number of services and routing of the transit lines.