Long nn-zero-free sequences in finite cyclic groups

A sequence in the additive group ${\mathbb Z}_n$ of integers modulo $n$ is called $n$-zero-free if it does not contain subsequences with length $n$ and sum zero. The article characterizes the $n$-zero-free sequences in ${\mathbb Z}_n$ of length greater than $3n/2-1$. The structure of these sequences is completely determined, which generalizes a number of previously known facts. The characterization cannot be extended in the same form to shorter sequence lengths. Consequences of the main result are best possible lower bounds for the maximum multiplicity of a term in an $n$-zero-free sequence of any given length greater than $3n/2-1$ in ${\mathbb Z}_n$, and also for the combined multiplicity of the two most repeated terms. Yet another application is finding the values in a certain range of a function related to the classic theorem of Erd\H{o}s, Ginzburg and Ziv.Comment: 11 page

Dynamics of D3-D7 Brane Inflation in Throats

Dynamics of D3-branes in models of warped D3-D7 inflationary set up is studied where perturbative $\alpha'$ correction to the K\"ahler potential and the nonperturbative corrections to the superpotential are included. It is shown that a dS minimum can be obtained without introducing anti-branes. Some specific configurations of D7-branes embedding were studied. After stabilizing the angular directions, it is shown that the resulting D3-D7 potential of the radial position of the D3-brane is too steep to allow slow-roll inflation. Depending on D7-branes embedding and the stabilized angular directions, the mobile D3-brane can move either towards the tip of the throat or towards the D7-branes.Comment: minor changes, to appear in JHE

New classes of topological crystalline insulators with unpinned surface Dirac cones

We theoretically predict two new classes of three-dimensional topological crystalline insulators (TCIs), which have an odd number of unpinned surface Dirac cones protected by crystal symmetries. The first class is protected by a single glide plane symmetry; the second class is protected by a composition of a twofold rotation and time-reversal symmetry. Both classes of TCIs are characterized by a quantized $\pi$ Berry phase associated with surface states and a $Z_2$ topological invariant associated with the bulk bands. In the presence of disorder, these TCI surface states are protected against localization by the average crystal symmetries, and exhibit critical conductivity in the universality class of the quantum Hall plateau transition. These new TCIs exist in time-reversal-breaking systems with or without spin-orbital coupling, and their material realizations are discussed.Comment: 4 pages plus supplementary material

Co-existence of Weyl Fermion and Massless Triply Degenerate Nodal Points

By using first-principles calculations, we propose that WC-type ZrTe is a new type of topological semimetal (TSM). It has six pairs of chiral Weyl nodes in its first Brillouin zone, but it is distinguished from other existing TSMs by having additional two paris of massless fermions with triply degenerate nodal points as proposed in the isostructural compounds TaN and NbN. The mirror symmetry, three-fold rotational symmetry and time-reversal symmetry require all of the Weyl nodes to have the same velocity vectors and locate at the same energy level. The Fermi arcs on different surfaces are shown, which may be measured by future experiments. It demonstrates that the "material universe" can support more intriguing particles simultaneously.Comment: 16 pages and 9 figure

Cooperative Pursuit with Multi-Pursuer and One Faster Free-moving Evader

This paper addresses a multi-pursuer single-evader pursuit-evasion game where the free-moving evader moves faster than the pursuers. Most of the existing works impose constraints on the faster evader such as limited moving area and moving direction. When the faster evader is allowed to move freely without any constraint, the main issues are how to form an encirclement to trap the evader into the capture domain, how to balance between forming an encirclement and approaching the faster evader, and what conditions make the capture possible. In this paper, a distributed pursuit algorithm is proposed to enable pursuers to form an encirclement and approach the faster evader. An algorithm that balances between forming an encirclement and approaching the faster evader is proposed. Moreover, sufficient capture conditions are derived based on the initial spatial distribution and the speed ratios of the pursuers and the evader. Simulation and experimental results on ground robots validate the effectiveness and practicability of the proposed method