Finite groups with Frobenius normalizer condition for non-normal primary subgroups

A finite group $P$ is said to be \emph{primary} if $|P|=p^{a}$ for some prime $p$. We say a primary subgroup $P$ of a finite group $G$ satisfies the \emph{Frobenius normalizer condition} in $G$ if $N_{G}(P)/C_{G}(P)$ is a $p$-group provided $P$ is $p$-group. In this paper, we determine the structure of a finite group $G$ in which every non-subnormal primary subgroup satisfies the Frobenius normalized condition. In particular, we prove that if every non-normal primary subgroup of $G$ satisfies the Frobenius condition, then $G/F(G)$ is cyclic and every maximal non-normal nilpotent subgroup $U$ of $G$ with $F(G)U=G$ is a Carter subgroup of $G$.Comment: arXiv admin note: text overlap with arXiv:1801.0923

On the Integrability of the One-Dimensional Open XYZ Spin Chain

The Lax pair for the one-dimensional open XYZ spin chain is constructed, this shows that the system is completely integrable .Comment: 6 pages,latex,no figure

Practical Design and Implementation of Metamaterial-Enhanced Magnetic Induction Communication

Although wireless communications in complex environments, such as underground, underwater, and indoor, can enable a large number of novel applications, their performances are constrained by lossy media and complicated structures. Magnetic Induction (MI) has been proved to be an efficient solution to achieve reliable communication in such environments. However, due to the small coil antenna's physical limitation, MI's communication range is still very limited if devices are required to be portable. To this end, Metamaterial-enhanced Magnetic Induction (M$^2$I) communication has been proposed and the theoretical results predict that it can significantly increase the communication performance, namely, data rate and communication range. Nevertheless, currently, the real implementation of M$^2$I is still a challenge and there is no guideline on design and fabrication of spherical metamaterials. In this paper, a practical design is proposed by leveraging a spherical coil array to realize M$^2$I. We prove that the effectively negative permeability can be achieved and there exists a resonance condition where the radiated magnetic field can be significantly amplified. The radiation and communication performances are evaluated and full-wave simulation is conducted to validate the design objectives. By using the spherical coil array-based M$^2$I, the communication range can be significantly extended, exactly as we predicted in the ideal M$^2$I model. Finally, the proposed M$^2$I antenna is implemented and tested in various environments.Comment: arXiv admin note: text overlap with arXiv:1510.0846

On an open problem of Skiba

Let $\sigma=\{\sigma_{i}|i\in I\}$ be some partition of the set $\mathbb{P}$ of all primes, that is, $\mathbb{P}=\bigcup_{i\in I}\sigma_{i}$ and $\sigma_{i}\cap \sigma_{j}=\emptyset$ for all $i\neq j$. Let $G$ be a finite group. A set $\mathcal {H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every non-identity member of $\mathcal {H}$ is a Hall $\sigma_{i}$-subgroup of $G$ and $\mathcal {H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma_{i}\in \sigma(G)$. $G$ is said to be a $\sigma$-group if it possesses a complete Hall $\sigma$-set. A $\sigma$-group $G$ is said to be $\sigma$-dispersive provided $G$ has a normal series $1 = G_1<G_2<\cdots< G_t< G_{t+1} = G$ and a complete Hall $\sigma$-set $\{H_{1}, H_{2}, \cdots, H_{t}\}$ such that $G_iH_i = G_{i+1}$ for all $i= 1,2,\ldots t$. In this paper, we give a characterizations of $\sigma$-dispersive group, which give a positive answer to an open problem of Skiba in the paper

Towards High-quality Visualization of Superfluid Vortices

Superfluidity is a special state of matter exhibiting macroscopic quantum phenomena and acting like a fluid with zero viscosity. In such a state, superfluid vortices exist as phase singularities of the model equation with unique distributions. This paper presents novel techniques to aid the visual understanding of superfluid vortices based on the state-of-the-art non-linear Klein-Gordon equation, which evolves a complex scalar field, giving rise to special vortex lattice/ring structures with dynamic vortex formation, reconnection, and Kelvin waves, etc. By formulating a numerical model with theoretical physicists in superfluid research, we obtain high-quality superfluid flow data sets without noise-like waves, suitable for vortex visualization. By further exploring superfluid vortex properties, we develop a new vortex identification and visualization method: a novel mechanism with velocity circulation to overcome phase singularity and an orthogonal-plane strategy to avoid ambiguity. Hence, our visualizations can help reveal various superfluid vortex structures and enable domain experts for related visual analysis, such as the steady vortex lattice/ring structures, dynamic vortex string interactions with reconnections and energy radiations, where the famous Kelvin waves and decaying vortex tangle were clearly observed. These visualizations have assisted physicists to verify the superfluid model, and further explore its dynamic behavior more intuitively.Comment: 14 pages, 15 figures, accepted by IEEE Transactions on Visualization and Computer Graphic

Tuning Topological Phase Transitions in Hexagonal Photonic Lattices Made of Triangular Rods

In this paper, we study topological phases in a 2D photonic crystal with broken time ($\mathcal{T}$) and parity ($\mathcal{P}$) symmetries by performing calculations of band structures, Berry curvatures, Chern numbers, edge states and also numerical simulations of light propagation in the edge modes. Specifically, we consider a hexagonal lattice consisting of triangular gyromagnetic rods. Here the gyromagnetic material breaks $\mathcal{T}$ symmetry while the triangular rods breaks $\mathcal{P}$ symmetry. Interestingly, we find that the crystal could host quantum anomalous Hall (QAH) phases with different gap Chern numbers ($C_g$) including $|C_g| > 1$ as well as quantum valley Hall (QVH) phases with contrasting valley Chern numbers ($C_v$), depending on the orientation of the triangular rods. Furthermore, phase transitions among these topological phases, such as from QAH to QVH and vice versa, can be engineered by a simple rotation of the rods. Our band theoretical analyses reveal that the Dirac nodes at the $K$ and $K'$ valleys in the momentum space are produced and protected by the mirror symmetry ($m_y$) instead of the $\mathcal{P}$ symmetry, and they become gapped when either $\mathcal{T}$ or $m_y$ symmetry is broken, resulting in a QAH or QVH phase, respectively. Moreover, a high Chern number ($C_g = -2$) QAH phase is generated by gapping triply degenerate nodal points rather than pairs of Dirac points by breaking $\mathcal{T}$ symmetry. Our proposed photonic crystal thus provides a platform for investigating intriguing topological phenomena which may be challenging to realize in electronic systems, and also has promising potentials for device applications in photonics such as reflection-free one-way waveguides and topological photonic circuits

Optical left-handed metamaterials made of arrays of upright split-ring pairs

Electromagnetic metamaterials are man-made structures that have novel properties such as a negative refraction index, not attainable in naturally occurring materials. Although negative index materials (NIMs) in microwave frequencies were demonstrated in 2001, it has remained challenging to design NIMs for optical frequencies especially those with both negative permittivity and negative permeability [known as left-handed metamaterials (LHMs)]. Here, by going beyond the traditional concept of the combination of artificial electronic and magnetic meta-atoms to design NIMs, we propose a novel LHM composed of an array of simple upright split-ring pairs working in the near infrared region. Our electromagnetic simulations reveal the underlying mechanism that the coupling of the two rings can stimulate simultaneously both the electric and magnetic resonances. The proposed structure has a highest refractive index of -2, a highest figure of merit of 21, good air-matched impedance and 180 nm double negative bandwidth, which excel the performances of many previous proposals. We also numerically demonstrate the negative refraction of this metamaterial in both the single-layer form and wedge-shaped lens

Singular Perturbation of Nonlinear Dynamics by Parasitic Noise

In nonlinear systems analysis, minor fractions of higher-order dynamics are often neglected for simplicity. Here, we show that machine epsilon levels of parasitic higher-order dynamics due to computer roundoff alone can cause divergence of the H\'enon attractor to new attractors or instability. The divergence develops exponentially regardless of whether the original or new attractor is chaotic or not. Such singular perturbation by parasitic higher-order dynamics is a novel property of nonlinear dynamics that is of wide practical significance in dynamical systems modeling, simulation and control.Comment: 14 pages, 4figure

Analyzing and Disentangling Interleaved Interrupt-driven IoT Programs

In the Internet of Things (IoT) community, Wireless Sensor Network (WSN) is a key technique to enable ubiquitous sensing of environments and provide reliable services to applications. WSN programs, typically interrupt-driven, implement the functionalities via the collaboration of Interrupt Procedure Instances (IPIs, namely executions of interrupt processing logic). However, due to the complicated concurrency model of WSN programs, the IPIs are interleaved intricately and the program behaviours are hard to predicate from the source codes. Thus, to improve the software quality of WSN programs, it is significant to disentangle the interleaved executions and develop various IPI-based program analysis techniques, including offline and online ones. As the common foundation of those techniques, a generic efficient and real-time algorithm to identify IPIs is urgently desired. However, the existing instance-identification approach cannot satisfy the desires. In this paper, we first formally define the concept of IPI. Next, we propose a generic IPI-identification algorithm, and prove its correctness, real-time and efficiency. We also conduct comparison experiments to illustrate that our algorithm is more efficient than the existing one in terms of both time and space. As the theoretical analyses and empirical studies exhibit, our algorithm provides the groundwork for IPI-based analyses of WSN programs in IoT environment

Efficient Quantum State Estimation with Over-complete Tomography

It is widely accepted that the selection of measurement bases can affect the efficiency of quantum state estimation methods, precision of estimating an unknown state can be improved significantly by simply introduce a set of symmetrical measurement bases. Here we compare the efficiencies of estimations with different numbers of measurement bases by numerical simulation and experiment in optical system. The advantages of using a complete set of symmetrical measurement bases are illustrated more clearly