Hamiltonicity of edge chromatic critical graphs

Vizing conjectured that every edge chromatic critical graph contains a 2-factor. Believing that stronger properties hold for this class of graphs, Luo and Zhao (2013) showed that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{6n}{7}$ is Hamiltonian. Furthermore, Luo et al. (2016) proved that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{4n}{5}$ is Hamiltonian. In this paper, we prove that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{3n}{4}$ is Hamiltonian. Our approach is inspired by the recent development of Kierstead path and Tashkinov tree techniques for multigraphs.     

Bounded stationary reflection II

Bounded stationary reflection at a cardinal λ is the assertion that every stationary subset of λ reflects but there is a stationary subset of λ that does not reflect at arbitrarily high cofinalities. We produce a variety of models in which bounded stationary reflection holds. These include models in which bounded stationary reflection holds at the successor of every singular cardinal $\mu >{\mathrm{?}}_{\omega }$ and models in which bounded stationary reflection holds at ${\mu }^{+}$ but the approachability property fails at μ.     

Classification of edges using compactly supported shearlets

We analyze the detection and classification of singularities of functions $f={\chi }_S$, where $S?{\mathbb{R}}^d$ and $d=2,3$. It will be shown how the set ?S can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if ?S is a $d?1$ dimensional piecewise smooth manifold with $d=2$ or 3, we will classify smooth and non-smooth components of ?S. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in 3 dimensions by compactly supported shearlets. Finally, geometric properties of ?S such as curvature are described in terms of the continuous shearlet transform of f.     

Edge agreement of second-order multi-agent system with dynamic quantization via the directed edge Laplacian

This work explores the edge agreement problem of second-order multi-agent systems with dynamic quantization under directed communication. To begin with, by virtue of the directed edge Laplacian, we propose a model reduction representation of the closed-loop multi-agent system depending on the spanning tree subgraph. Considering the limitations of the finite bandwidth channels, the quantization effects of second-order multi-agent systems under directed graph are considered. The static quantizers generally contain a fixed quantization interval and infinite quantization level, which are, to some extent, inefficient and impractical. To further reduce the bit depth (number of bits available) and to obtain better precision, the dynamic quantized communication strategy referring to zooming in-zooming out scheme is required. Based on the reduced model associated with the essential edge Laplacian, the asymptotic stability of second-order multi-agent systems under dynamic quantized effects with only finite quantization level can be guaranteed. Finally, the simulation of altitude alignment of micro air vehicles is provided to verify the theoretical results.     

Light‐Directing Omnidirectional Circularly Polarized Reflection from Liquid‐Crystal Droplets

Constructing and tuning self‐organized three‐dimensional (3D) superstructures with tailored functionality is crucial in the nanofabrication of smart molecular devices. Herein we fabricate a self‐organized, phototunable 3D photonic superstructure from monodisperse droplets of one‐dimensional cholesteric liquid crystal (CLC) containing a photosensitive chiral molecular switch with high helical twisting power. The droplets are obtained by a glass capillary microfluidic technique by dispersing into PVA solution that facilitates planar anchoring of the liquid‐crystal molecules at the droplet surface, as confirmed by the observation of normal incidence selective circular polarized reflection in all directions from the core of individual droplet. Photoirradiation of the droplets furnishes dynamic reflection colors without thermal relaxation, whose wavelength can be tuned reversibly by variation of the irradiation time. The results provided clear evidence on the phototunable reflection in all directions.     

Mach reflection of a plane shock wave passing a mountain of 45° inclination

The transition from regular reflection (RR) to Mach reflection (MR) as a plane shock wave diffracts around a triangular mountain of 45° inclination is analysed in this paper, both by optical measurement in a shock tube and by numerical simulation the numerical method developed by Li Yingfan^[1] is of the FLIC type with triangular mesh. The dependence of the critical transition point Lk ofRR→MR on shock Mach numberM _i is analyzed and the variations of the incidence angle ω _i of the impinging shock and the reflection angle ω _r with the distanceL ^* are investigated. Our experimental and numerical results agree well with the theoretical results of Iton and Italya.     

Edge waves in asymmetrically laminated plates

We study edge waves propagating along the edge of an asymmetrically laminated elastic plate for which the out-of-plane component of displacement is coupled with the in-plane components. A Stroh-like formulation is used to show that such a plate can support at most two edge waves. An efficient method for computing the edge-wave speeds is proposed and explained through examples.     

Interaction between an edge dislocation and a circular inclusion with an inhomogeneously imperfect interface

This research presents an analytical study of the interaction problem of an edge dislocation with a circular inclusion with a circumferentially inhomogeneously imperfect interface. The interface, which is modeled as a spring (interphase) layer with vanishing thickness, is characterized by that in which there is a displacement jump across the interface in the same direction as the corresponding tractions, and the same degree of imperfection is realized in both the normal and tangential directions. Furthermore, the interface parameter is nonuniform along the interface. In order to arrive at an elementary form solution, we introduce a conformal mapping function. Then the stress field as well as the Peach–Koehler force acting on the edge dislocation can be obtained from the derived complex potentials. Calculations demonstrate that the nonuniform interface parameter has a significant influence on the stress field.     

Deformation and reverse snapping of a circular shallow shell under uniform edge tension

In this paper we study the deformation and stability of a shallow shell under uniform edge tension, both theoretically and experimentally. Von Karman’s plate model is adopted to formulate the equations of motion. For a shell with axisymmetrical initial shape, the equilibrium positions can be classified into axisymmetrical and unsymmetrical solutions. While there may exist both stable and unstable axisymmetrical solutions, all the unsymmetrical solutions are unstable. Since the unsymmetrical solutions will not affect the stability of the axisymmetrical solutions, it is concluded that for quasi-static analysis, there is no need to include unsymmetrical assumed modes in the calculation. If the shell is initially in the unstrained configuration, it will only be flattened smoothly when the edge tension is applied. No snap-through buckling is possible in this case. On the other hand, if the shell is initially in the strained position, it will be snapped back to the stable position on the other side of the base plane when the edge tension reaches a critical value. Experiment is conducted on several free brass shells of different initial heights to verify the theoretical predictions. Generally speaking, for the range of initial height H < 10 the experimental measurements of the deformation and the reverse snapping load agree well with theoretical predictions.