The circle as an attractor of an iterated function system on the plane

We give an iterated function system (IFS) on the plane with the circle as attractor. In doing this, we also give a sufficient condition for radially contracting functions on the plane (or on Rn) to be a contraction. A counterexample shows that radial contractiveness is not enough to be a contraction.     

The ideal center of the dual of a Banach lattice

Let E be a Banach lattice. Its ideal center Z(E) is embedded naturally in the ideal center Z(E′) of its dual. The embedding may be extended to a contractive algebra and lattice homomorphism of Z(E)ʺ into Z(E′). We show that the extension is onto Z(E′) if and only if E has a topologically full center. (That is, for each x ? E{x\in E}, the closure of Z(E)x is the closed ideal generated by x.) The result can be generalized to the ideal center of the order dual of an Archimedean Riesz space and in a modified form to the orthomorphisms on the order dual of an Archimedean Riesz space.     

Korovkin type approximation theorem for BBH type operators via <Emphasis Type="Italic">J</Emphasis>-convergence

This paper provides a Korovkin type approximation theorem for a class of positive linear operators including Bleimann-Butzer and Hahn operators via J-convergence.     

General properties of Bertrand curves in Riemann–Otsuki space

In this paper we studied the general properties of Bertrand curves and their characterizations in Riemann–Otsuki space.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

Numerical solutions of integro-differential equations and application of a population model with an improved Legendre method

In this paper, an improved Legendre collocation method is presented for a class of integro-differential equations which involves a population model. This improvement is made by using the residual function of the operator equation. The error differential equation, gained by residual function, has been solved by the Legendre collocation method (LCM). By summing the approximate solution of the error differential equation with the approximate solution of the problem, a better approximate solution is obtained. We give the illustrative examples to demonstrate the efficiency of the method. Also we compare our results with the results of the known some methods. In addition, an application of the population model is made.     

A numeric–analytic method for time-fractional Swift–Hohenberg (S–H) equation with modified Riemann–Liouville derivative

In this paper, the fractional variational iteration method (FVIM) was applied to obtain the approximate solutions of time-fractional Swift–Hohenberg (S–H) equation with modified Riemann–Liouville derivative. A new application of fractional variational iteration method (FVIM) was extended to derive analytical solutions in the form of a series for these equations. Numerical results showed the FVIM is powerful, reliable and effective method when applied strongly nonlinear equations with modified Riemann–Liouville derivative.     

Chaotic dynamics of resistively coupled DC-driven distinct Josephson junctions and the effects of circuit parameters

Two distinct Josephson junctions (JJs) connected with a constant coupling resistance R_cp are theoretically considered to investigate the overall dynamics below and above the critical current I_c. The circuit model of the device is driven by two DC current sources, I₁ and I₂. Each junction is characterized by a nonlinear resistive–capacitive junction (NRCSJ). Having constructed the circuit model, time-dependent simulations are carried out for a variety of control parameter sets. Common techniques such as bifurcation diagrams, two-dimensional attractors and Lyapunov exponents are applied for the determination of chaotic as well as periodic dynamics of the superconducting junction devices. According to the findings, two states (namely superconducting and ordinary conducting) are determined as functions of the source currents. The chaotic current which flows through R_cp exhibits a very rich behavior depending on the source currents I₁ and I₂ and junction capacitances C₁ and C₂. The device characteristics are summarized by a number of three-dimensional phase diagrams in the parameter space. In addition, for certain parameters, hyper-chaotic cases with two positive Lyapunov exponents are encountered. In contrast to earlier studies claiming the need for a sinusoidal feeding current for generating a chaotic signal, our circuitry can easily generate one via a DC source.     

New Korovkin Type Theorem for Non-Tensor Meyer–König and Zeller Operators

In this paper, we introduce a certain class of linear positive operators via a generating function, which includes the non-tensor MKZ operators and their non-trivial extension. In investigating the approximation properties, we prove a new Korovkin type approximation theorem by using appropriate test functions. We compute the rate of convergence of these operators by means of the modulus of continuity and the elements of modified Lipschitz class functions. Furthermore, we give functional partial differential equations for this class. Using the corresponding equations, we calculate the first few moments of the non-tensor MKZ operators and investigate their approximation properties. Finally, we state the multivariate versions of the results and obtain the convergence properties of the multivariate Meyer–König and Zeller operators.