arXiv subjects

Shunyu Yao

Publications and source records attributed to Shunyu Yao.

98 records · Page 6Linked to original sources

arXiv2020

k-Ary spanning trees contained in tournaments

A rooted tree is called a $k$-ary tree, if all non-leaf vertices have exactly $k$ children, except possibly one non-leaf vertex has at most $k-1$ children. Denote by $h(k)$ the minimum integer such that every tournament of order at least $h(k)$ contains a $k$-ary spanning tree. It is well-known that every tournament contains a Hamiltonian path, which implies that $h(1)=1$. Lu et al. [J. Graph Theory {\bf 30}(1999) 167--176] proved the existence of $h(k)$, and showed that $h(2)=4$ and $h(3)=8$. The exact values of $h(k)$ remain unknown for $k\geq 4$. A result of Erdős on the domination number of tournaments implies $h(k)=Ω(k\log k)$. In this paper, we prove that $h(4)=10$ and $h(5)\geq13$.

math.CO↗

arXiv2020

Integer Linear Programming Formulations for Double Roman Domination Problem

For a graph $G= (V,E)$, a double Roman dominating function (DRDF) is a function $f : V \to \{0,1,2,3\}$ having the property that if $f (v) = 0$, then vertex $v$ must have at least two neighbors assigned $2$ under $f$ or {at least} one neighbor $u$ with $f (u) = 3$, and if $f (v) = 1$, then vertex $v$ must have at least one neighbor $u$ with $f (u) \ge 2$. In this paper, we consider the double Roman domination problem, which is an optimization problem of finding the DRDF $f$ such that $\sum_{v\in V} f (v)$ is minimum. We propose {five integer linear programming (ILP) formulations and one mixed integer linear programming formulation with polynomial number of constraints for this problem. Some additional valid inequalities and bounds are also proposed for some of these formulations.} Further, we prove that {the first four models indeed solve the double Roman domination problem, and the last two models} are equivalent to the others regardless of the variable relaxation or usage of a smaller number of constraints and variables. Additionally, we use one ILP formulation to give an $H(2(Δ+1))$-approximation algorithm. All proposed formulations and approximation algorithm are evaluated on randomly generated graphs to compare the performance.

math.CO↗

arXiv2019

Non-Hermitian Topological Invariants in Real Space

The topology of non-Hermitian systems is drastically shaped by the non-Hermitian skin effect, which leads to the generalized bulk-boundary correspondence and non-Bloch band theory. The essential part in formulations of bulk-boundary correspondence is a general and computable definition of topological invariants. In this paper, we introduce a construction of non-Hermitian topological invariants based directly on real-space wavefunctions, which provides a general and straightforward approach for determining non-Hermitian topology. As an illustration, we apply this formulation to several representative models of non-Hermitian systems, efficiently obtaining their topological invariants in the presence of non-Hermitian skin effect. Our formulation also provides a dual picture of the non-Bloch band theory based on the generalized Brillouin zone, offering a unique perspective of bulk-boundary correspondence.

cond-mat.mes-hall↗

arXiv2019

Non-Hermitian skin effect and chiral damping in open quantum systems

One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian skin effect, namely that the eigenstates are exponentially localized at the boundary of the system. For open quantum systems, a short-time evolution can often be well described by the effective non-Hermitian Hamiltonians, while long-time dynamics calls for the Lindblad master equations, in which the Liouvillian superoperators generate time evolution. In this Letter, we find that Liouvillian superoperators can exhibit the non-Hermitian skin effect, and uncover its unexpected physical consequences. It is shown that the non-Hermitian skin effect dramatically shapes the long-time dynamics, such that the damping in a class of open quantum systems is algebraic under periodic boundary condition but exponential under open boundary condition. Moreover, the non-Hermitian skin effect and non-Bloch bands cause a chiral damping with a sharp wavefront. These phenomena are beyond the effective non-Hermitian Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged open quantum dynamics.

cond-mat.quant-gas↗

arXiv2018

3D-Aware Scene Manipulation via Inverse Graphics

We aim to obtain an interpretable, expressive, and disentangled scene representation that contains comprehensive structural and textural information for each object. Previous scene representations learned by neural networks are often uninterpretable, limited to a single object, or lacking 3D knowledge. In this work, we propose 3D scene de-rendering networks (3D-SDN) to address the above issues by integrating disentangled representations for semantics, geometry, and appearance into a deep generative model. Our scene encoder performs inverse graphics, translating a scene into a structured object-wise representation. Our decoder has two components: a differentiable shape renderer and a neural texture generator. The disentanglement of semantics, geometry, and appearance supports 3D-aware scene manipulation, e.g., rotating and moving objects freely while keeping the consistent shape and texture, and changing the object appearance without affecting its shape. Experiments demonstrate that our editing scheme based on 3D-SDN is superior to its 2D counterpart.

cs.CV↗

arXiv2018

Non-Hermitian Chern bands

The relation between chiral edge modes and bulk Chern numbers of quantum Hall insulators is a paradigmatic example of bulk-boundary correspondence. We show that the chiral edge modes are not strictly tied to the Chern numbers defined by a non-Hermitian Bloch Hamiltonian. This breakdown of conventional bulk-boundary correspondence stems from the non-Bloch-wave behavior of eigenstates (non-Hermitian skin effect), which generates pronounced deviations of phase diagrams from the Bloch theory. We introduce non-Bloch Chern numbers that faithfully predict the numbers of chiral edge modes. The theory is backed up by the open-boundary energy spectra, dynamics, and phase diagram of representative lattice models. Our results highlight a unique feature of non-Hermitian bands and suggest a non-Bloch framework to characterize their topology.

cond-mat.mes-hall↗

arXiv2018

Edge states and topological invariants of non-Hermitian systems

The bulk-boundary correspondence is among the central issues of non-Hermitian topological states. We show that a previously overlooked `non-Hermitian skin effect' necessitates redefinition of topological invariants in a generalized Brillouin zone. The resultant phase diagrams dramatically differ from the usual Bloch theory. Specifically, we obtain the phase diagram of non-Hermitian Su-Schrieffer-Heeger model, whose topological zero modes are determined by the non-Bloch winding number instead of the Bloch-Hamiltonian-based topological number. Our work settles the issue of the breakdown of conventional bulk-boundary correspondence and introduces the non-Bloch bulk-boundary correspondence.

cond-mat.mes-hall↗

arXiv2017

Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects

Periodically driven (Floquet) systems have been under active theoretical and experimental investigations. This paper aims at a systematic study in the following aspects of Floquet systems: (i) A systematic formulation of topological invariants of Floquet systems based on the cooperation of topology and symmetries. Topological invariants are constructed for the ten symmetry classes in all spatial dimensions, for both homogeneous Floquet systems (Floquet topological insulators and superconductors) and Floquet topological defects. Meanwhile, useful representative Dirac Hamiltonians for all the symmetry classes are obtained and studied. (ii) A general theory of Floquet topological defects, based on the proposed topological invariants. (iii) Models and proposals of Floquet topological defects in low dimensions. Among other defect modes, we investigate Floquet Majorana zero modes and Majorana Pi modes in vortices of topologically trivial superconductors under a periodic drive. In addition, we clarified several notable issues about Floquet topological invariants. Among other issues, we prove the equivalence between the effective-Hamiltonian-based band topological invariants and the frequency-domain band topological invariants.

cond-mat.str-el↗