Filling up complex spectral regions through non-Hermitian disordered chains

Eigenspectra that fill regions in the complex plane have been intriguing to many, inspiring research from random matrix theory to esoteric semi-infinite bounded non-Hermitian lattices. In this work, we propose a simple and robust ansatz for constructing models whose eigenspectra fill up generic prescribed regions. Our approach utilizes specially designed non-Hermitian random couplings that allow the co-existence of eigenstates with a continuum of localization lengths, mathematically emulating the effects of semi-infinite boundaries. While some of these couplings are necessarily long-ranged, they are still far more local than what is possible with known random matrix ensembles. Our ansatz can be feasibly implemented in physical platforms such as classical and quantum circuits, and harbors very high tolerance to imperfections due to its stochastic nature.

Development of mandelbrot set for the logistic map with two parameters in the complex plane

In this paper, the study of the dynamical behavior of logistic map has been disused with representing fractals graphics of map, the logistic map depends on two parameters and works in the complex plane, the map defined by f(z,α,β)=αz(1–z)β. where and are complex numbers, and β is a positive integers number, the visualization method used in this work to generate fractals of the map and to inspect the relation between the value of β and the shape of the map, this visualization analysis showed also that, as the value of β increasing, as the number of humps in the function also increasing, and it demonstrate that is true also for the function’s first iteration , f2(x0)=f(f(x0)) and the second iteration , f3(x0)=f(f2(x0)), beside that , the visualization technique showed that the number of humps in that fractal is less than the ones in the second iteration of the original function ,the study of the critical points and their properties of the logistic map also discussed it, whereas finding the fixed point led to find the critical point of the function f, in addition , it haven proven for the set of all pointsα∈C and β∈N, the iteration function f(f(z) has an attractive fixed points, and belongs to the region specified by the disc |1–β(α–1)|<1. Also, The discussion of the Mandelbrot set of the function defined by the f(f(z)) examined in complex plans using the path principle, such that the path of the critical point z=z0 is restricted, finally, it has proven that the Mandelbrot set f(z,α,β) contains all the attractive fixed points and all the complex numbers in which α≤(1/β+1) (1/β+1) and the region containing the attractive fixed points for f2(z,α,β) was identified

A pair of rational double sequences

Double sequences appear in a natural way in cases of iteratively given sequences if the iteration allows to determine besides the successors from the predecessors also the predecessors from their followers. A particular pair of double sequences is considered which appears in a parqueting-reflection process of the complex plane. While one end of each sequence is a natural number sequence, the other consists of rational numbers. The natural numbers sequences are not yet listed in OEIS Wiki. Complex versions from the double sequences are provided.

Julia sets of Zorich maps

The Julia set of the exponential family $E_{\kappa }:z\mapsto \kappa e^z$ , $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<\kappa \leq 1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three-dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the whole of $\mathbb {R}^3$ , generalizing Misiurewicz’s result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb {R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.

Numerical solution of generalized fractional Volterra integro-differential equations via approximation the Bromwich integral

In this paper, we consider the generalized fractional Volterra integro-differential equations with the regularized Prabhakar derivative and represent the solution of this type of equations in the form of Bromwich integral in the complex plane. Then we select the hyperbolic contour as an optimal contour to approximate the Bromwich integral. Further, an example to show absolute errors for various parameters by using our numerical scheme on hyperbolic contour is given.