On the Local Connected and Jordanian Julia Sets

In this article, we investigate the dynamical properties of fλ（z） =λzke2, for λ（≠0） ∈Cand k≥2. We will show that the boundaries of some （or all ） Fatou components are Jordan curves and the Julia sets are Sierpinski carpet, and they are locally connected for some certain λ.

Changes in sensorimotor regions of the cerebral cortex in congenital amusia:a case-control study

Perceiving pitch is a central function of the human auditory system;congenital amusia is a disorder of pitch perception.The underlying neural mechanisms of congenital amusia have been actively discussed.However,little attention has been paid to the changes in the motor rain within congenital amusia.In this case-control study,17 participants with congenital amusia and 14 healthy controls underwent functional magnetic resonance imaging while resting with their eyes closed.A voxel-based degree centrality method was used to identify abnormal functional network centrality by comparing degree centrality values between the congenital amusia group and the healthy control group.We found decreased degree centrality values in the right primary sensorimotor areas in participants with congenital amusia relative to controls,indicating potentially decreased centrality of the corresponding brain regions in the auditory-sensory motor feedback network.We found a significant positive correlation between the degree centrality values and the Montreal Battery of Evaluation of Amusia scores.In conclusion,our study identified novel,hitherto undiscussed candidate brain regions that may partly contribute to or be modulated by congenital amusia.Our evidence supports the view that sensorimotor coupling plays an important role in memory and musical discrimination.The study was approved by the Ethics Committee of the Second Xiangya Hospital,Central South University,China(No.WDX20180101GZ01)on February 9,2019.

REMARKS ON JOHN DISKS

Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.

Local connectivity of Julia sets for a family of biquadratic polynomials

The local connectivity of Julia sets for the family of biquadratic polynomials f_c(z)= (z^2-2c^2)z^2 with a parameter c is discussed.It is proved that for any parameter c,the boundary of the immediately attracting domain of f_c is a Jordan curve.

Two Kinds Equal Frequency Circuits to Achieve Locally Resonant Band Gap of a Circular Plate Attached Alternately by Piezoelectric Unimorphs

A circular thin plate is proposed for vibration attenuation,which is attached alternately by annular piezoelectric unimorphs with resonant shunt circuits.Two kinds of equal frequency resonant shunt circuits are designed to achieve an integrated locally resonant（LR）band gap（BG） with a much smaller transmission factor：（1） the structure is arrayed periodically while the resonant shunt circuits are aperiodic;（2） the resonant shunt circuits are periodic while the structure is aperiodic.The transmission factor curve is calculated,which is validated by the finite element method.Dependences of the LR BG performance upon the geometric and electric parameters are also analyzed.

Bounded critical Fatou components are Jordan domains for polynomials

We prove that any bounded Fatou component of a polynomial of degree at least two, which is not(eventually) a Siegel disk, is a Jordan domain.

Fatou Components and Julia Sets of Singularly Perturbed Rational Maps with Positive Parameter

In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpirlski curve is given.

Claw─centre Independent Graphs

gives that every connected,locally connected K 1,4 free,almost claw free graph on at least three vertices is vertices pancyclic. This paper shows that every connected,locally 2 connected K 1,4 free claw centre independent graph on at least three vertices is vertex pancyclic.