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Yibo Gao

Publications and source records attributed to Yibo Gao.

67 records · Page 4Linked to original sources
arXiv2020

Separable elements in Weyl groups

We define the notion of a separable element in a finite Weyl group, generalizing the well-studied class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat order generated by a separable element are rank-symmetric and rank-unimodal, and that the product of their rank generating functions gives that of the whole group, answering an open problem of Fan Wei. We also prove that separable elements are characterized by pattern avoidance in the sense of Billey and Postnikov.

math.CO↗
arXiv2020

A combinatorial duality between the weak and strong Bruhat orders

In recent work, the authors used an order lowering operator $\nabla$, introduced by Stanley, to prove the strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt interpreted $\nabla$ as a differential operator on Schubert polynomials and used this to prove a new identity for Schubert polynomials and a determinant conjecture of Stanley. In this paper we study a raising operator $Δ$ for the \emph{strong} Bruhat order, which is in many ways dual to $\nabla$. We prove a Schubert identity dual to that of Hamaker et al. and derive formulas for counting weighted paths in the Hasse diagrams of the strong order which agree with path counting formulas for the weak order. We also show that powers of $\nabla$ and $Δ$ have the same Smith normal forms, which we describe explicitly, answering a question of Stanley.

math.CO↗
arXiv2019

Principal specializations of Schubert polynomials and pattern containment

We show that the principal specialization of the Schubert polynomial at $w$ is bounded below by $1+p_{132}(w)+p_{1432}(w)$ where $p_u(w)$ is the number of occurrences of the pattern $u$ in $w$, strengthening a previous result by A. Weigandt. We then make a conjecture relating the principal specialization of Schubert polynomials to pattern containment. Finally, we characterize permutations $w$ whose RC-graphs are connected by simple ladder moves via pattern avoidance.

math.CO↗
arXiv2019

Arithmetic of weighted Catalan numbers

In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the $2$-adic valuations of the weighted Catalan numbers are equal to the $2$-adic valutations of the Catalan numbers. We obtain the same result under weaker conditions by considering a map from a class of functions to $2$-adic integers. These methods are also extended to $q$-weighted Catalan numbers, strengthening a previous result by Konvalinka. Finally, we prove some results on the periodicity of weighted Catalan numbers modulo an integer and apply them to the specific case of the number of combinatorial types of Morse links. Many open questions are mentioned.

math.CO↗
arXiv2019

Decoherence-free propagation and ramification of a solitary pulse

Using a microscopic master equation to account for the environmental effects, we compute the decoherence culminated during the propagation of a microwave pulse of arbitrary shape through a superconducting qubit. It is shown that the qubit decoherence vanishes and the pulse shape remains absorption-free when the latter adopts a soliton shape with $nπ$ area. Otherwise, the environmental feedback decelerates the velocity of the soliton envelop and induces an monotonic increase of phase in the microwave. A pulse of non-$nπ$ area thus ramifies into a transparent part that travels decoherence-free at incident velocity and a slowing part that decays through space. The ramification explains the environmental origin of pulse splitting observed in self-induced transparency.

quant-ph↗
arXiv2018

On the genus of a quotient of a numerical semigroup

We find a relation between the genus of a quotient of a numerical semigroup $S$ and the genus of $S$ itself. We use this identity to compute the genus of a quotient of $S$ when $S$ has embedding dimension $2$. We also exhibit identities relating the Frobenius numbers and the genus of quotients of numerical semigroups that are generated by certain types of arithmetic progressions.

math.NT↗
arXiv2018

Plantbot: A New ROS-based Robot Platform for Fast Building and Developing

In recent years, the Robot Operating System (ROS) is developing rapidly and has been widely used in robotics research because of its flexible, open source, and extensive advantages. In scientific research, the corresponding hardware platform is indispensable for the experiment. In the field of mobile robots, PR2, Turtlebot2, and Fetch are commonly used as research platforms. Although these platforms are fully functional and widely used, they are expensive and bulky. What\u2019s more, these robots are not easily redesigned and expanded according to requirements. To overcome these limitations, we propose Plantbot, an easy-building, low-cost robot platform that can be redesigned and expanded according to requirements. It can be applied to not only fast algorithm verification, but also simple factory inspection and ROS teaching. This article describes this robot platform from several aspects such as hardware design, kinematics, and control methods. At last two experiments, SLAM and Navigation, on the robot platform are performed. The source code of this platform is available.

cs.RO↗
arXiv2018

Diagonal Form of the Varchenko Matrices

Varchenko defined the Varchenko matrix associated to any real hyperplane arrangement and computed its determinant. In this paper, we show that the Varchenko matrix of a hyperplane arrangement has a diagonal form if and only if it is semigeneral, i.e., without degeneracy. In the case of semigeneral arrangement, we present an explicit computation of the diagonal form via combinatorial arguments and matrix operations, thus giving a combinatorial interpretation of the diagonal entries.

math.CO↗
arXiv2016

Toric Mutations in the dP$_2$ Quiver and Subgraphs of the dP$_2$ Brane Tiling

Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we examine the del Pezzo 2 (dP$_2$) quiver and its brane tiling, which arise from the physics literature, in terms of toric mutations on its corresponding cluster. Specifically, we give explicit formulas for all cluster variables generated by toric mutation sequences. Moreover, for each such variable, we associate a subgraph of the dP$_2$ brane tiling to it such that its weight matches the variable.

math.CO↗
arXiv2016

Jacobi-Trudi determinants over finite fields

In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_λ$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in $\mathbb{F}_q$.

math.CO↗
arXiv2016

The Largest Pure Partial Planes of Order 6 Have Size 25

In this paper, we prove that the largest pure partial plane of order 6 has size 25. At the same time, we classify all pure partial planes of order 6 and size 25 up to isomorphism. Our major approach is computer search. The search space is very large so we use combinatorial arguments to rule out some of the cases. For the remaining cases, we subdivide each search by phases and use multiple checks to reduce search space via symmetry.

math.CO↗