Schroder Equation Research Articles

Representation of exact trajectory solutions for chaotic one-dimensional maps in Schroder form

Purpose of the article is to illustrate the genesis, meaning and significance of the functional Schroder equation, introduced in the theory of iterations of rational functions, for the theory of deterministic chaos by analytical calculations of exact trajectory solutions, invariant densities and Lyapunov exponents of one-dimensional chaotic maps. We demonstrate the method for solving the functional Schroder equation for various chaotic maps by passing to a topologically conjugate mappings, for which finding the exact trajectory solution is a simpler mathematical procedure. Results of the analytical solution of the Schroder equation for 12 chaotic mappings of various types and the calculation of the corresponding expressions for exact trajectory solutions, invariant densities and Lyapunov exponents are presented. Conclusion is made about the methodological expediency of formulating and solving the Schroder equations by the study of the dynamics of one-dimensional chaotic mappings.

On simultaneous linearization

Given a uniformly quasiregular mapping, there is typically no reason to assume any relationship between linearizers at different repelling periodic points. However, in the current paper we prove that in the case where the uqr map arises as a solution of a Schroder equation then, with some further natural assumptions, if L is a linearizer at one repelling periodic point, then $$L\circ T$$ is a linearizer at another repelling periodic point, where T is a translation. In this sense we say L simultaneously linearizes f. In the plane, an example would be that $$e^z$$ simultaneously linearizes $$z^2$$ . Our methods utilize generalized derivatives for quasiregular mappings, including a chain rule and inverse derivative formula which may be of independent interest.

On Quasiregular Linearizers

Linearization is a well-known concept in complex dynamics. If $$p$$ is a polynomial and $$z_0$$ is a repelling fixed point, then there is an entire function $$L$$ which conjugates $$p$$ to the linear map $$z\mapsto p^{\prime }(z_0)z$$ . This notion of linearization carries over into the quasiregular setting, in the context of repelling fixed points of uniformly quasiregular mappings. In this article, we investigate how linearizers arising from the same uqr mapping and the same repelling fixed point are related. In particular, any linearizer arising from a uqr solution to a Schroder equation is shown to be automorphic with respect to some quasiconformal group.

SLE for the carbon dioxide +trans-1,3,3,3-tetrafluoropropene

A built experimental set-up was employed for the estimation of the solid-liquid equilibria of alternative refrigerants systems. The behavior of carbon dioxide + 1,3,3,3-tetrafluoropropene (CO 2 + R1234ze(E)) was measured down to temperatures of 160.0 K. The results obtained for the mixtures were interpreted by means of the Schroder equation.

Carbon Dioxide + Fluoromethane and Nitrous Oxide + Fluoromethane: Solid−Liquid Equilibria Measurements

A recently built experimental setup was employed for the estimation of the solid−liquid equilibria (SLE) of alternative refrigerant systems. The behavior of two binaries, that is, carbon dioxide + fluoromethane (CO2 + R41) and nitrous oxide + fluoromethane (N2O + R41), was measured down to temperatures of 126.5 K. To confirm the reliability of the apparatus, the triple points of the pure fluids constituent of the binary systems were measured. All triple-point data measured revealed a generally good consistency with the literature. The results obtained for the mixtures were corrected by the Rossini method and interpreted by means of the Schroder equation.

Solid solutions in binary halide systems having a common anion

Deviations of experimental liquidus curves from ideal curves calculated by the Schroder equation are analyzed for halides that form eutectic systems and systems containing solid solutions. Analysis of phase diagrams shows that the key factors for solid solution formation in binary halide systems having a common anion are as follows: component crystal lattices and close values of the effective radii and polarizing powers of the cations.

Solid–Liquid Equilibria for the CO2 + R143a and N2O + R143a Systems

A recently built experimental setup for determination of solid–liquid equilibria was slightly modified. The action taken on the existing system focused on correcting the drawbacks of the cooling system. In this version of the setup, a direct liquid nitrogen supply was preferred. By means of the modified apparatus, solid–liquid equilibria of the CO2 + R143a and N2O + R143a binary systems were studied. The triple point of R143a was measured to check the reliability of the modified apparatus, revealing good consistency with the literature. The system’s behavior was measured down to temperatures of 148 K. The results obtained for the mixtures were interpreted by means of the Schroder equation.

Thermodynamic properties of liquid alloys of the Ni-Hf system

By the method of calorimetry in isoperibolic conditions are determined integral and partial mixing enthalpies of liquid alloys of the Ni-Hf system at 1770 ± 5 K. Defined that liquid Ni-Hf alloys are formed with allocated large quantity of heat. The analysis of own and literary data has allowed to establish for mixing enthalpies of binary Ni-Hf liquid alloys dependence on temperature. With use of the Schroder equation are calculated the activities of studied alloys components from co-ordinates of liquidus line of the phase diagram of this system. Between calculated and experimentally established values of melts components activities of the Ni-Hf system is observed only qualitative consent. Are also calculated ΔG and ΔS of liquid Ni-Hf alloys.

Solid–Liquid Equilibria for the CO2 + R23 and N2O + R23 Systems

A recently built experimental setup was employed for the estimation of the solid–liquid equilibria of alternative refrigerant systems. In this paper two binaries, i.e., carbon dioxide + trifluoromethane (CO2 + R23) and nitrous oxide + trifluoromethane (N2O + R23), were studied down to temperatures of 117 K. In order to check the reliability of the apparatus, the triple points of the pure fluids contained in the mixture were measured, revealing good consistency with the literature. The results obtained for the mixtures were interpreted by means of the Schroder equation.

Solid−Liquid Equilibria for the Carbon Dioxide + 1,1-Tetrafluoroethane and Nitrous Oxide + 1,1-Tetrafluoroethane Systems

A recently built experimental setup was employed for the estimation of the solid−liquid equilibria of alternative refrigerant systems. The behavior of two binaries, i.e., carbon dioxide + 1,1-tetrafluoroethane (CO2 + R134a) and nitrous oxide + 1,1-tetrafluoroethane (N2O + R134a), was measured down to temperatures of 142 K. The triple points of CO2 and N2O were measured to confirm the reliability of the apparatus. The triple point of R134a was also measured; however, the pressure at the triple point for this fluid was too low to be accurately measured with the current apparatus. All triple point data measured revealed a generally good consistency with the literature. The results obtained for the mixtures were interpreted by means of the Schroder equation.

Interaction in binary fluoride systems

A series of factors affecting the interaction between fluorides is considered. Fluorides are distributed by valence and are arranged in accordance with an increase in the polarizing effect of cations. For eutectic-type melting diagrams, deviations of the experimental liquidus from the ideal one calculated by the Schroder equation are analyzed. In many systems of this type, complex formation is noted in the melts. The most informative factors determining the type of melting diagrams of fluoride systems are refined. Interrelation of various types of melting diagrams is considered.

On Continuous and Smooth Solutions of the Schröder Equation in Normed Spaces

The problem of the existence of continuous and smooth solutions of the Schroder equation defined on normed spaces is examined.

Solid−Liquid Equilibria for the CO2 + R152a and N2O + R152a Systems

A recently built experimental setup was employed for the estimation of the solid-liquid equilibria of alternative refrigerant systems. The behavior of two binaries, i.e., carbon dioxide + 1,1-difluoroethane (CO 2 + R152a) and nitrous oxide + 1,1-difluoroethane (N 2 O + R152a), was measured down to temperatures of 132 K. The triple points of C O 2 and N 2 O were measured to confirm the reliability of the apparatus. The triple point of R152a was also measured; however, the pressure at the triple point for this fluid was too low to accurately measure with the current apparatus. All triple-point data measured revealed a generally good consistency with the literature. The results obtained for the mixtures were interpreted by means of the Schroder equation.

Möbius splines are closed under continuous iteration

The continuous or fractional iterates of a piecewise Mobius (linear / linear) function, or Mobius spline, are also piecewise Mobius. Further, they may be calculated exactly via a simple algorithm. The same remarks apply to the related Schroder equation. Some numerical demonstrations are given. It is suggested that this technique could be used to approximate continuous iterates of an arbitrary function, with possible applications in statistics, signal processing and computer graphics. We also suggest that Mobius splines are a useful tool in the numerical solution of more general functional equations.

Composition operators and a pull-back measure formula

A pull-back measure formula obtained in some particular cases by E. A. Nordgren and this author is generalized in the framework of boundary measures for zero-free Nevanlinna class fuctions on the unit polydisk. The formula is used to characterize the zero-free Nevanlinna class functions which are solutions of Schroder's equation induced by a polydisk automorphism ϕ (i.e. to determine the zero-free functionsf belonging to the Nevanlinna class which are solutions of the functional equationf ° π=λf, for some constant λ), thus generalizing earlier results obtained by R. Mortini and this author.

Recent results on functional equations in a single variable, perspectives and open problems

In 1990 Cambridge University Press published the by now well-known monograph Iterative Functional Equations by Marek Kuczma, Bogdan Choczewski and Roman Ger. In the paper we will pay attention mainly on, we think, important papers published in the last decade or not discussed in this book. Of course, the selection we made reflects our personal preferences only.¶The topics of this paper are: 1. A view of the theory developed up to now; 2. Iterative roots; 3. Functional equations with superpositions of the unknown function; 4. Some linear and convolution equations; 5. Schilling's problem; 6. Daroczy's equation; 7. Simultaneous functional equations; 8. Schroder's equation; 9. Abel's equation; 10. Miscellaneous results and problems.

The Symmetry Structure of the Heavenly Equation

We show that excitations of physical interest for the heavenly equation are generated by symmetry operators which yield two reduced equations with different characteristics. One equation is of the Liouville type and gives rise to gravitational instantons, including those found by Eguchi–Hanson and Gibbons–Hawking. The second equation appears for the first time in the theory of heavenly spaces and provides meron-like configurations endowed with a fractional topological charge. A link is also established between the heavenly equation and the so-called Schroder equation, which plays a crucial role in the bootstrap model and in renormalization theory.

On continuous solutions of a functional equation

This paper discusses continuous solutions of the functional equation φ[f(x)] = g(x, φ(x)) in topological spaces. Let us consider the equation (1) φ[f(x)] = g(x, φ(x)) with φ : X → Y as unknown function. In order to obtain a solution of equation (1), it is enough to extend a function defined on a set which for every x contains exactly one element of the form f(x), where k = 0,±1,±2, . . . and f(x) denotes the kth iterate of the function f (cf. [3] and [4]). In the case when X is an open interval and Y is a Banach space, it is well known under what conditions these extensions are continuous (cf. [5]). Paper [6] by M. Sablik brings theorems which answer the above question for X and Y contained in some Banach spaces ([6, Th. 2.1, Th. 2.2]). In the case when X and Y are locally convex vector spaces the continuity of similar extensions was examined by W. Smajdor in [7] but for the Schroder equation (i.e. φ[f(x)] = sφ(x), 0 < |s| < 1). We are going to adopt the method given in that paper to the more general situation. We shall employ Baron’s Extension Theorem proved in [1] (cf. also [2]). This theorem concerns extending solutions of functional equations from a neighbourhood of a distinguished point (Lemma 7). We shall deal with the following hypotheses: (i) X is a Hausdorff topological space; ξ is a given (and fixed) point of X; Y is a topological space. (ii) The function f maps X into X in such a manner that (2) f is homeomorphism of X onto f(X); (3) ξ ∈ int f(X); 1991 Mathematics Subject Classification: Primary 39B52.

The solubility of tungsten in molten iron

The solubility of tungsten in iron has been determined at 1560–1620°C by successively saturating molten iron with tungsten while ensuring a uniform distribution of the content over the height. The temperature dependence of the solubility over that temperature range is described by a Schroder equation: Cs=(3.94±0.11)×103e-(84.4±0.4)/RT.

On the determination of a distribution by its median residual life function: a functional equation

In reliability studies, it is well known that the mean residual life function determines the distribution function uniquely. In this paper we show how closely we can determine a distribution when its median residual life function M[S | t] is known. This amounts to solving the functional equation , where R is the reliability function. We actually study a more general functional equation f(φ(t)) = sf(t) called Schroder's equation. It is shown that, under mild assumptions on φ, the solution is of the form f(t) = f 0(t)k(log f 0 (t)), where f 0 is a well-behaved particular solution which can be constructed and k is a periodic function; thus the solution is not unique. Two examples are solved to illustrate the method. Finally, these examples are used to solve the problem of linear M[S | t] studied by Schmittlein and Morrison. As an extra benefit, all of our results hold equally well for the more general sth percentile residual life function.