Homoclinic Orbits in Several Classes of Three-Dimensional Piecewise Affine Systems with Two Switching Planes

The existence of homoclinic orbits or heteroclinic cycle plays a crucial role in chaos research. This paper investigates the existence of the homoclinic orbits to a saddle-focus equilibrium point in several classes of three-dimensional piecewise affine systems with two switching planes regardless of the symmetry. An analytic proof is provided using the concrete expression forms of the analytic solution, stable manifold, and unstable manifold. Meanwhile, a sufficient condition for the existence of two homoclinic orbits is also obtained. Furthermore, two concrete piecewise affine asymmetric systems with two homoclinic orbits have been constructed successfully, demonstrating the method’s effectiveness.

Analytic Proof of the recent Baseline Primality Conjecture

<div>I have completed the analytic proof of a very important primality conjecture unveiled recently in the Reference, see the url in the Reference field below.</div>

Analytic Proof of the recent Baseline Primality Conjecture

<div>I have completed the analytic proof of a very important primality conjecture unveiled recently in the Reference, see the url in the Reference field below.</div>

Zenali Iteration Method For Approximating Fixed Point of A δZA - Quasi Contractive mappings

This article will introduce a new iteration method called the zenali iteration method for the approximation of fixed points. We show that our iteration process is faster than the current leading iterations like Mann, Ishikawa, oor, D- iterations, and *- iteration for new contraction mappings called quasi contraction mappings. And we proved that all these iterations (Mann, Ishikawa, oor, D- iterations and *- iteration) equivalent to approximate fixed points of quasi contraction. We support our analytic proof by a numerical example, data dependence result for contraction mappings type by employing zenali iteration also discussed.

An ethics-based ‘identity-proof’ of god’s existence. An ontology for philotherapy

A resurgence of scholarly work on proof of God?s existence is noticeable over the past decade, with considerable emphasis on attempts to provide ?analytic proof? based on the meanings and logic of various identity statements which constitute premises of the syllogisms of the ?proof?. Most recently perhaps, Emmanuel Rutten?s ?modal-epistemic proof? has drawn serious academic attention. Like other ?analytic? and strictly logical proofs of God?s existence, Rutten?s proof has been found flawed. In this paper I discuss the possibility of an ?ethics-based? identity proof of God?s existence. Such a proof, the first version of which, I believe, has been offered, indirectly, by Nikolai Lossky, utilizes the form and structure of the analytic proof, but fundamentally rests on the perception of moral values we associate with God and Godliness. The nature of the proof shifts the focus of the very attempt to ?prove? God?s existence from what I believe is an unreasonable standard, unattainable even in ?proving? the existence of the more mundane world, towards a more functional, practical and attainable standard. The proof proposed initially by Lossky, and in a more systematic form here, I believe, shows the indubitable existence of God in the sense of his moral presence in the lives of the faithful, at least with the same degree of certainty as the presence or ?existence? of anything else that can be epistemically proven in principle.

Iterative approximations of common fixed points with simulation results in Banach spaces

In this article, we propose the Abbas-Nazir three step iteration scheme and employ the algorithm to study the common fixed points of a pair of generalized $\alpha$-Reich-Suzuki non-expansive mappings defined on a Banach space. Moreover, we explore a few weak and strong convergence results concerning such mappings. Our findings are aptly validated by non-trivial and constructive numerical examples and finally, we compare our results with that of the other noteworthy iterative schemes utilizing MATLAB $2017$a software. However, we perceive that for a different set of parameters and initial points, the newly proposed iterative scheme converges faster than the other well-known algorithms. To be specific, we give an analytic proof of the claim that the novel iteration scheme is also faster than that of Liu et al.

Stationary currents in long-range interacting magnetic systems

We construct a solution for the 1d integro-differential stationary equation derived from a finite-volume version of the mesoscopic model proposed in Giacomin and Lebowitz (J. Stat. Phys. 87(1), 37–61, 1997). This is the continuous limit of an Ising spin chain interacting at long range through Kac potentials, staying in contact at the two edges with reservoirs of fixed magnetizations. The stationary equation of the model is introduced here starting from the Lebowitz-Penrose free energy functional defined on the interval [−ε− 1, ε− 1], ε > 0. Below the critical temperature, and for ε small enough, we obtain a solution that is no longer monotone when opposite in sign, metastable boundary conditions are imposed. Moreover, the mesoscopic current flows along the magnetization gradient. This can be considered as an analytic proof of the existence of diffusion along the concentration gradient in one-component systems undergoing a phase transition, a phenomenon generally known as uphill diffusion. In our proof uniqueness is lacking, and we have clues that the stationary solution obtained is not unique, as suggested by numerical simulations.