Graphs with 3-rainbow index n−1n-1 and n−2n-2

Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the same color. For a vertex set $S\subseteq V(G)$, the tree connecting $S$ in $G$ is called an $S$-tree. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for each $k$-set $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$. In \cite{Zhang}, they got that the $k$-rainbow index of a tree is $n-1$ and the $k$-rainbow index of a unicyclic graph is $n-1$ or $n-2$. So there is an intriguing problem: Characterize graphs with the $k$-rainbow index $n-1$ and $n-2$. In this paper, we focus on $k=3$, and characterize the graphs whose 3-rainbow index is $n-1$ and $n-2$, respectively.Comment: 14 page

Picasso, Matisse, or a Fake? Automated Analysis of Drawings at the Stroke Level for Attribution and Authentication

This paper proposes a computational approach for analysis of strokes in line drawings by artists. We aim at developing an AI methodology that facilitates attribution of drawings of unknown authors in a way that is not easy to be deceived by forged art. The methodology used is based on quantifying the characteristics of individual strokes in drawings. We propose a novel algorithm for segmenting individual strokes. We designed and compared different hand-crafted and learned features for the task of quantifying stroke characteristics. We also propose and compare different classification methods at the drawing level. We experimented with a dataset of 300 digitized drawings with over 80 thousands strokes. The collection mainly consisted of drawings of Pablo Picasso, Henry Matisse, and Egon Schiele, besides a small number of representative works of other artists. The experiments shows that the proposed methodology can classify individual strokes with accuracy 70%-90%, and aggregate over drawings with accuracy above 80%, while being robust to be deceived by fakes (with accuracy 100% for detecting fakes in most settings)

The 3-rainbow index of a graph

Let $G$ be a nontrivial connected graph with an edge-coloring $c: E(G)\rightarrow \{1,2,...,q\},$ $q \in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the same color. For a vertex subset $S\subseteq V(G)$, a tree that connects $S$ in $G$ is called an $S$-tree. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for each $k$-subset $S$ of $V(G)$ is called $k$-rainbow index, denoted by $rx_k(G)$. In this paper, we first determine the graphs whose 3-rainbow index equals 2, $m,$ $m-1$, $m-2$, respectively. We also obtain the exact values of $rx_3(G)$ for regular complete bipartite and multipartite graphs and wheel graphs. Finally, we give a sharp upper bound for $rx_3(G)$ of 2-connected graphs and 2-edge connected graphs, and graphs whose $rx_3(G)$ attains the upper bound are characterized.Comment: 13 page

Adaptive and Robust Fault-Tolerant Tracking Control of Contact force of Pantograph-Catenary for High-Speed Trains

Abstract This paper presents a modified multi-body dynamic model and a linear time-invariant model with actuator faults (loss of effectiveness faults, bias faults) and matched and unmatched uncertainties. Based on the fault model, a class of adaptive and robust tracking controllers are proposed which are adjusted online to tolerate the time-varying loss of effectiveness faults and bias faults, and compensate matched disturbances without the knowledge of bounds. For unmatched uncertainties, optimal control theory is added to the controller design processes. Simulations on a pantograph are shown to verify the efficiency of the proposed fault-tolerant design approach

Non-classical non-Gaussian state of a mechanical resonator via selectively incoherent damping in three-mode optomechanical systems

We theoretically propose a scheme for the generation of a non-classical single-mode motional state of a mechanical resonator (MR) in the three-mode optomechanical systems, in which two optical modes of the cavities are linearly coupled to each other and one mechanical mode of the MR is optomechanically coupled to the two optical modes with the same coupling strength simultaneously. One cavity is driven by a coherent laser light. By properly tuning the frequency of the weak driving field, we obtain engineered Liouvillian superoperator via engineering the selective interaction Hamiltonian confined to the Fock subspaces. In this case, the motional state of the MR can be prepared into a non-Gaussian state, which possesses the sub-Poisson statistics although its Wigner function is positive.Comment: 6 pages, 5 figure