Selective Excitation of Band Extrema in Valley Photonic Crystals

Flexible control of building blocks of photonic crystals enables achieving desirable band structures. Exploration of photonic band extrema has brought many fantastic features to design artificial optical materials, such as Brillouin‐zone‐corner extrema for valley photonic materials and zone‐center extremum for zero‐index metamaterials. However, two such kinds of extrema are always found independently in different photonic crystals. In this work, a kind of valley photonic crystals possessing both zone‐center and zone‐corner band extrema almost at the same frequency is proposed. Inspired by antennas theory, a three‐antenna array (TAA) source is devoted to individually manipulate each extremum. The correlation coefficient is given to determine the coupling efficiency between the TAA source and extrema eigenmodes. By using a source with a high correlation coefficient, these extrema bulk states are selectively excited consistent with their eigenfields. Furthermore, three control cases are shown that multiple extrema points are simultaneously excited, in order to confirm the validity of the correlation coefficient. Finally, a potential application of a beam‐steering device is proposed through selective excitation of ternary extrema. This work develops binary valley states into ternary mix states, rendering more degrees of freedom for on‐chip optical information transport, particularly for beam steering and mode division multiplexing.     

Extremal generalized centralizers in matrix algebras

We describe matrices with extremal generalized centralizers over algebraically closed fields.     

General non local boundary value problem for second order elliptic equation

In this paper we study a class of second order coefficient operators differential equation with general (possibly non local) boundary conditions. We obtain new results extending those given in a previous paper 1 . Existence, uniqueness and optimal regularity of the strict solution are proved in UMD spaces, using the well‐known Dore–Venni theorem.     

Topological Phase Transition of Non-Hermitian Crosslinked Chain

The contents of topological classification of matter are enriched by non-Hermiticity, such as exceptional points, bulk-edge correspondence, and skin effects. Physically, gain and loss can be introduced by imaginary on-site potentials of lattice Hamiltonians, and the topological phase transition for a cross-linked chain in the presence of such non-Hermiticity is investigated. The topological phase diagram in terms of a winding number is obtained analytically with phase boundaries coinciding with the surfaces of exceptional points. The topologically original edge states with distribution mainly at the joints between domains of different phases are protected even for long chains. The non-Hermitian topological feature can also be reflected by vortex structures in the vector fields of complex eigenenergies, expected values of Pauli matrices, and trajectories of these quantities. This model may be implemented in coupled photonic crystals, fermions trapped in optical lattice, or non-Hermitian electrical-circuit lattices, and the edge states are immune to various kinds of disorders until topological phase transition occurs. This work gives insight into the influence of non-Hermiticity on topological phase of matter.     

Computing upward topological book embeddings of upward planar digraphs

We describe a unified approach for studying book, point-set, and simultaneous embeddability problems of upward planar digraphs. The approach is based on a linear time strategy to compute an upward planar drawing of an upward planar digraph such that all vertices are collinear. Besides having impact in relevant application domains of graph drawing and computational geometry, the presented results open new research directions in the area of upward planarity with constraints of the positions of the vertices.     

Prediction of branch on branch and topological characteristics of low‐density polyethylene polymerization by a novel stochastic approach

A novel approach using Monte Carlo method applied to simulation of low‐density polyethylene (LDPE) polymerization in tubular reactor showing topological characteristics, and the comprehensive kinetic mechanism has been taken into consideration. The results show the precise details of the structure of a chain in the three levels of the backbone, the main branches, and branches on branch. The chain types include dead polymer, dead polymer with unsaturated end, and live polymer with primary radical, secondary radical, and tertiary radical. In this work, the branches on branch were identified in terms of number, length, and position of the branch. Sixty percent of branches on branch are 1 to 5 carbons long, and the longest branch on branch is about 50 carbons. Thus, this study provides a tool for more accurately mapping the polymer chains architecture, superior to determine the number, and position of long‐ and short‐chain branches in past researches. Finally, this approach will advance the prediction of microstructure‐related properties of polymer one step further.