Fixed point equation

Fixed Point Solutions We begin by asking whether there are any values of a for which the system has fixed points. Solving the fixed-point equation... 

Kyf — 0. This is the fixed-point equation for every positive normalized r there exists unique positive normalized steady state c (r) KrC (r) — 0, c >0 and c (r) = 1. We have to solve the equation r = c (r). The solution exists because the Brauer fixed point theorem. 

This relationship applies to a wide range of straight-run and heat-treated pitches but not to fluxed pitches. If the K-and-S (Kramer and Samow) softening point is taken as a fixed point, equation 2 can be written as... 

The conditions for the existence of a pT-periodic solution to the forced system can be written in terms of a fixed point equation for the pth iterate of the stroboscopic map... 

Critical fixed points that are undergoing bifurcation can be found by augmenting the set of fixed-point equations (5) with one of these Floquet multiplier conditions ... 

As an immediate consequence we can determine the scale factor A(A. i%) at the fixed point. Equation (8,7) yields... 

Formula (1.3) is a fixed point equation since it has the general form... 

Should the solid undergo a two-particle or a classical crossover before the single-particle one, nesting at Q0 is not relevant. Below the two-particle crossover, the RPA pole defines the attractive fixed point. Equation (32) can be simplified to the bare essential elements... 

X2 is attracting whereas is repelling. This accounts for the sharp corner in the graph of Xoo r) at r — tq. For r > ri = 3, neither of the two fixed points of fr is stable. In order to understand the period doubling at r = ri, it is necessary to consider the fixed points (and their stability properties) of the second iterate of fr. The second iterate f x) is a quartic polynomial in x. Therefore, the fixed point equation X xf ) has four solutions given by... 

Of high interest for machining processes are fixed points. A fixed point Xf fulfills the fixed point equation ... 

Because of the fixed point equation, Xf vanishes from the equation, and we get a new dynamic system describing the progression of the distortion ... 

A T-periodic solution is a solution of the fixed point equation x = F(x). Methods to solve this equation are called shooting methods. The four methods for obtaining a periodic state of a cyclic chemical system differ in the way the fixed point equation is solved. 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Theorem 1.21. If S is a contraction mapping in a Hilbert space V then there exists a fixed point u such that Su = u and solutions u of the equation... 

In this section we derive a nonpenetration condition between crack faces for inclined cracks in plates and discuss the equilibrium problem. As it turns out, the nonpenetration condition for inclined cracks is of nonlocal character. This means that by writing the condition at a fixed point we have to take into account the displacement values both at the point and at the other point chosen at the opposite crack face. As a corollary of this fact, the equilibrium equations hold only in a domain located outside the crack surface projection on the mid-surface of the plate. This section follows the papers (Khludnev, 1997b Kovtunenko et ah, 1998). 

The invariant distribution, p x), is a fixed point of this equation ... 

Theorem 5 [goles87a] If the synaptic-weight matrix A is symmetric, and the number of sites in the lattice is finite, then the orbits of the generalized threshold rule (equation 5.121) are either fixed points or cycles of period two. 

Equation 5.121 is then reproduced in full by setting bi =5-1/2 for all i. Since aij is symmetric, the condition of theorem 5 is met. We conclude that (f)2d majority can only yield either fixed points or cycles of period two. 

The possible asymptotic, or equilibrium, values of the density Poo = hmt >oo p(t) are obtained by solving for the stable fixed points of the equation p = f p). Recall that a given solution p is stable if f p) < 1. 

Let us begin by considering the stability of homogeneous solutions and/or initial-conditions i.e. by considering the stability of a simple-diffusive CML when cri(O) = a for all sites i , where a is a fixed point of the local logistic map F(cr) = acr(l—cr). Following Waller and Kapral [kapral84], we first recast equations 8.23 and 8.24... 

These fixed points are used to calibrate a different kind of thermometer that is easier to use than a gas thermometer. Over the temperature range from 13.8033 to 1234.93 °A (or K), which is the temperature interval most commonly encountered, the thermometer used for ITS-90 is a platinum resistance thermometer. In this thermometer, the resistance of a specially wound coil of platinum wire is measured and related to temperature. More specifically, temperatures are expressed in terms of W(T9o), the ratio of the resistance R(Ttriple point of water R (273.16 K), as given in equation (1.11)... 

The international temperature scale is based upon the assignment of temperatures to a relatively small number of fixed points , conditions where three phases, or two phases at a specified pressure, are in equilibrium, and thus are required by the Gibbs phase rule to be at constant temperature. Different types of thermometers (for example, He vapor pressure thermometers, platinum resistance thermometers, platinum/rhodium thermocouples, blackbody radiators) and interpolation equations have been developed to reproduce temperatures between the fixed points and to generate temperature scales that are continuous through the intersections at the fixed points. 

Approximately every twenty years, the international temperature scale is updated to incorporate the most recent measurements of the equilibrium thermodynamic temperature of the fixed points, to revise the interpolation equations, or to change the specifications of the interpolating measuring devices. The latest of these scales is the international temperature scale of 1990 (ITS-90). It supersedes the earlier international practical temperature scale of 1968 (IPTS-68), along with an interim scale (EPT-76). These temperature scales replaced earlier versions (ITS-48 and ITS-27). 

Temperatures on ITS-90, as on earlier scales, are defined in terms of fixed points, interpolating instruments, and equations that relate the measured property of the instrument to temperature. The report on ITS-90 of the Consultative Committee on Thermometry is published in Metrologia and in the Journal of Research of the National Institute of Standards and Technology The description that follows is extracted from those publications.3 Two additional documents by CCT further describe ITS-90 Supplementary Information for the ITS-90, and Techniques for Approximating the ITS-90.4... 

In summary, to obtain 7% from a platinum resistance thermometer, one selects the range of interest, calibrates the thermometer at the fixed points specified for those ranges, and uses the appropriate function to calculate AW(Tw) to be used in equation (A2.5). Companies are available that perform these calibrations and provide tables of W T<)0) versus 790 that can be interpolated to give 7% for a measured W T90). 

The flux ( J ) is a common measure of the rate of mass transport at a fixed point. It is defined as the number of molecules penetrating a unit area of an imaginary plane in a unit of time, and has the units of mol cm 2 s-1. The flux to the electrode is described mathematically by a differential equation, known as the Nemst-Planck equation, given here for one dimension ... 

Cooper and Child [14] have given an extensive description of the effects of nonzero angular momentum on the nature of the catastrophe map and the quantum eigenvalue distributions for polyads in its different regions. Here we note that the fixed points and relative equilibria, for nonzero L = L/2J, are given by physical roots of the equation... 

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