Dirac operator on the sphere with attached wires

An explicitly solvable model for tunnelling of relativistic spinless particles through a sphere is suggested. The model operator is constructed by an operator extensions theory method from the orthogonal sum of the Dirac operators on a semi- axis and on the sphere. The transmission coefficient is obtained. The dependence of the transmission coefficient on the particle energy has a resonant character. One observes pairs of the Breit-Wigner and the Fano resonances. It correlates with the corresponding results for a non-relativistic particle.

一种基于Dirac半金属的可调控极化偏转器件的仿真分析

本文提出了一种基于体Dirac半金属(Bulk Dirac Semimetal,BDS)的动态超材料结构极化偏转器.与传统的极化偏转器不同,该偏转器可以通过改变费米能级来动态控制偏振转换率和不对称传输.模拟结果表明,当BDS的费米能级调整为90 meV时,偏转器的偏振转换率在1.085~1.872 THz频率范围内都保持在90%以上(带宽为0.787 THz),在1.159 THz处偏振转换率达到峰值98%.该偏振转换器的不对称传输参数在1.229~1.831 THz大带宽范围内大于60%.本文还研究了偏振旋转角、椭圆度角、表面电流和电场分布情况以阐明该器件的极化转换特性和物理机制;通过施加栅极电压控制BDS费米能级,实现了对该极化偏转器的偏振转换率以及不对称传输的动态调谐.该极化偏转器的可调谐不对称传输特性,为基于动态不对称传输所设计的多路复用器、太赫兹二极管等设备的制造提供了新的思路.

含有Dirac位势的非线性S-P方程的约束极小元

该文研究了一类含有Dirac位势的非线性Schrödinger-Poisson方程的约束变分问题,在一定的参数和指标假设条件下,证明了约束极小元的存在性,推广了文献[2]中的相关结论.

Eigenvalues of Spinc Dirac Operator

In this paper we research the lower bound of the eigenvalue of Spinc Dirac operator on the Spinc manifold. By the Weisenbock formula, we get an estimate of it, then following the idea of Th Friedrich [2] and X Zhang [6]. We get a finer estimate of it. As an application, we give a condition when the Seiberg-Witten equation only has 0 solution.

共形扭化Dirac算子与非交换留数

考虑了一类共形扭化Dirac算子,给出了其Lichnerowicz公式.利用Laplace算子的局部表示,在法坐标系下计算出了无边流形与共形扭化Dirac算子相关的非交换留数.

具有手性边界条件的耦合Dirac系统

本文研究具有手性边界条件的Dirac系统解的存在性.通过在恰当的分数阶索伯列夫乘积空间上建立解析框架,给出了具有超二次增长的非线性项的Dirac系统解的存在性,把GONG和LU(2017)研究的结果推广到手性边界条件下的情形.

紧Spin流形上Dirac-Laplace方程解的存在性

本研究利用对称形式的山路定理阐释了紧Spin流形上Dirac-Laplace方程解的存在性,改进了文献[2]中的条件,得到了Dirac-Laplace方程具有无穷多对不同的解,这为高阶Dirac方程解的存在性研究提供了重要的参考.

Study of Vacuum Structure by Low-Lying Eigenmodes of Overlap-Dirac Operator

APE smearing and overlap-Dirac operator are combined to filter vacuum configurations. The structures of vacuum are studied by low-lying eigenmodes of the overlap-Dirac operator, which exhibits that instanton liquid model can be used.

Dirac method for nonlinear and non-homogenous boundary value problems of plates

The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.

Semi-Commutative Differential Operators Associated with the Dirac Opetator and Darboux Transformation

In the present paper, the semi-commutative differential oparators associated with the 1-dimensional Dirac operator are constructed. Using this results, the hierarchy of the mKdV (-) polynomials are expressed in terms of the KdV polynomials. These formulas give a new interpretation of the classical Darboux transformation and the Miura transformation. Moreover, the recursion operator associated with the hierarchy of the mKdV (-) polynomials is constructed by the algebraic method.

一类具有复权函数和两个不连续点的Dirac算子的逆问题

本文考虑了一类具有转移条件和复值权函数的Dirac算子的逆问题,采用了谱映射的方法得到了具有两个不连续点的Dirac算子的势函数在整个区间能被两组谱或Weyl函数唯一确定。

ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR

In this paper,we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the formL(f)=zfz-zfz,where f represents normalized harmonic mappings with bounded dilation.Then,using these results,we present better estimations for the Bloch constants of certain harmonic mappings L(f),where f is a K-quasiregular harmonic or open harmonic.Finally,we establish three versions of BlochLandau type theorem for biharmonic mappings of the form L(f).These results are sharp in some given cases and improve the related results of earlier authors.

Fractal Fractional Order Operators in Computational Techniques for Mathematical Models in Epidemiology

New fractional operators, the COVID-19 model has been studied in this paper. By using different numericaltechniques and the time fractional parameters, the mechanical characteristics of the fractional order model areidentified. The uniqueness and existence have been established. Themodel’sUlam-Hyers stability analysis has beenfound. In order to justify the theoretical results, numerical simulations are carried out for the presented methodin the range of fractional order to show the implications of fractional and fractal orders.We applied very effectivenumerical techniques to obtain the solutions of themodel and simulations. Also, we present conditions of existencefor a solution to the proposed epidemicmodel and to calculate the reproduction number in certain state conditionsof the analyzed dynamic system. COVID-19 fractional order model for the case of Wuhan, China, is offered foranalysis with simulations in order to determine the possible efficacy of Coronavirus disease transmission in theCommunity. For this reason, we employed the COVID-19 fractal fractional derivative model in the example ofWuhan, China, with the given beginning conditions. In conclusion, again the mathematical models with fractionaloperators can facilitate the improvement of decision-making for measures to be taken in the management of anepidemic situation.

一类边界条件中含谱参数的Dirac算子的谱性质

本文考虑了一类内部具有两个不连续点且边界条件依赖谱参数的Dirac算子的谱性质。首先通过引入适当的Hilbert空间并在其上定义新的自伴算子,使得所考虑问题的特征值与该算子的特征值一致。然后通过构造基本解得到了特征值的一些性质。最后给出了问题的Green函数和预解算子。In this paper, we consider the spectral properties of a class of Dirac operators with two internal discontinuities and spectral parameter-dependent boundary conditions. First, the eigenvalues of the problem under consideration are made to coincide with the eigenvalues of the operator by introducing a suitable Hilbert space and defining a new self-adjoint operator on it. Then some properties of the eigenvalues are obtained by constructing the basic solution. Finally, Green’s function and the resolvent operator of the problem are given.

Novel Investigation of Stochastic Fractional Differential Equations Measles Model via the White Noise and Global Derivative Operator Depending on Mittag-Leffler Kernel

Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.

AN EXPLANATION ON FOUR NEW DEFINITIONS OF FRACTIONAL OPERATORS

Fractional calculus has drawn more attentions of mathematicians and engineers in recent years.A lot of new fractional operators were used to handle various practical problems.In this article,we mainly study four new fractional operators,namely the CaputoFabrizio operator,the Atangana-Baleanu operator,the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the k-Prabhakar fractional integral operator.Usually,the theory of the k-Prabhakar fractional integral is regarded as a much broader than classical fractional operator.Here,we firstly give a series expansion of the k-Prabhakar fractional integral by means of the k-Riemann-Liouville integral.Then,a connection between the k-Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown,respectively.In terms of the above analysis,we can obtain this a basic fact that it only needs to consider the k-Prabhakar fractional integral to cover these results from the four new fractional operators.

From Dirac’s Aether to the Dirac Equation

In 1951, Dirac proposed a formalism for a Lorentz invariant Aether with a vacuum state that contains all possible velocity states at each space-time point. Dirac showed no explicit path from the Aether towards the Quantum Mechanics. In this paper, we demonstrate that Dirac’s proposed Aether can be described by a lattice of possible events in space-time built in the local Lorentz frame. The idealised case of single velocity state leads to the famous Dirac equation for a plane wave state and is compatible with quantum statistics. On the lattice, possible space-time events are connected by the Dirac spinors which provide the probability of observing an event. The inertial mass of a particle is shown to be equivalent to the density of possible events on the lattice. Variation of the lattice density of events modifies the metric and provides a space-time curvature leading to the Hilbert action associated with general relativity. In classical limit, the perturbation in the density of possible events of the Aether is proportional to the Newtonian gravitational potential.

Magnetic Topological Dirac Semimetal Transition Driven by SOC in EuMg_(2)Bi_(2)

Magnetic topological semimetals have been at the forefront of condensed matter physics due to their ability to exhibit exotic transport phenomena.Investigating the interplay between magnetic and topological orders in systems with broken time-reversal symmetry is crucial for realizing non-trivial quantum effects.We delve into the electronic structure of the rare-earth-based antiferromagnetic Dirac semimetal EuMg_(2)Bi_(2) using first-principles calculations and angle-resolved photoemission spectroscopy.Our calculations reveal that the spin-orbit coupling(SOC)in EuMg_(2)Bi_(2) prompts an insulator to topological semimetal transition,with the Dirac bands protected by crystal symmetries.The linearly dispersive states near the Fermi level,primarily originating from Bi 6p orbitals,are observed on both the(001)and(100)surfaces,confirming that EuMg_(2)Bi_(2) is a three-dimensional topological Dirac semimetal.This research offers pivotal insights into the interplay between magnetism,SOC and topological phase transitions in spintronics applications.

GENERALIZED FORELLI-RUDIN TYPE OPERATORS BETWEEN SEVERAL FUNCTION SPACES ON THE UNIT BALL OF C^(N)

In this paper,we investigate sufficient and necessary conditions such that generalized Forelli-Rudin type operators T_(λ,τ,k),S_(λ,τ,k),Q_(λ,τ,k)and R_(λ,τ,k)are bounded between Lebesgue type spaces.In order to prove the main results,we first give some bidirectional estimates for several typical integrals.

Single pair of charge-2 Dirac and charge-2Weyl phonons in GeO_(2)

The presence of a pair of Weyl and Dirac points(WP-DP)in topological semimetal states is intriguing and sought after due to the effects associated with chiral topological charges.However,identifying these states in real materials poses a significant challenge.In this study,by means of first-principles calculations we predict the coexistence of charge-2 Dirac and charge-2 Weyl phonons at high-symmetry points within a noncentrosymmetric P4_(1)2_(1)2 space group.Furthermore,we propose GeO_(2)as an ideal candidate for realizing these states.Notably,we observe two distinct surface arcs that connect the Dirac and Weyl points across the entire Brillouin zone,which could facilitate their detection in future experimental investigations.This study not only presents a tangible material for experimentalists to explore the topological properties of WP-DP states but also opens up new avenues in the quest for ideal platforms to study chiral particles.