Search

In this work, we propose a new approach called ``stationary reduction method based on nonisospectral deformation of orthogonal polynomials" for deriving discrete Painlev\'{e}-type (d-P-type) equations. We successfully apply this approach to (bi)orthogonal polynomials satisfying ordinary orthogonality, $(1,m)$-biorthogonality, generalized Laurent biorthogonality, Cauchy biorthogonality and partial-skew orthogonality. As a result, several seemingly novel classes of high order d-P-type equations, along with their particular solutions and respective Lax pairs, are derived. Notably, the d-P-type equation related to the Cauchy biorthogonality can be viewed as a stationary reduction of a nonisospectral generalization involving the first two flows of the Toda hierarchy of CKP type. Additionally, the d-P-type equation related to the partial-skew orthogonality is associated with the nonisospectral Toda hierarchy of BKP type, and this equation is found to admit a solution expressed in terms of Pfaffians., Comment: 88 pages; Several typos are corrected

In the field of crowd counting research, many recent deep learning based methods have demonstrated robust capabilities for accurately estimating crowd sizes. However, the enhancement in their performance often arises from an increase in the complexity of the model structure. This paper discusses how to construct high-performance crowd counting models using only simple structures. We proposes the Fuss-Free Network (FFNet) that is characterized by its simple and efficieny structure, consisting of only a backbone network and a multi-scale feature fusion structure. The multi-scale feature fusion structure is a simple structure consisting of three branches, each only equipped with a focus transition module, and combines the features from these branches through the concatenation operation. Our proposed crowd counting model is trained and evaluated on four widely used public datasets, and it achieves accuracy that is comparable to that of existing complex models. Furthermore, we conduct a comprehensive evaluation by replacing the existing backbones of various models such as FFNet and CCTrans with different networks, including MobileNet-v3, ConvNeXt-Tiny, and Swin-Transformer-Small. The experimental results further indicate that excellent crowd counting performance can be achieved with the simplied structure proposed by us.

Interacting with pedestrians understandably and efficiently is one of the toughest challenges faced by autonomous vehicles (AVs) due to the limitations of current algorithms and external human-machine interfaces (eHMIs). In this paper, we design eHMIs based on gestures inspired by the most popular method of interaction between pedestrians and human drivers. Eight common gestures were selected to convey AVs' yielding or non-yielding intentions at uncontrolled crosswalks from previous literature. Through a VR experiment (N1 = 31) and a following online survey (N2 = 394), we discovered significant differences in the usability of gesture-based eHMIs compared to current eHMIs. Good gesture-based eHMIs increase the efficiency of pedestrian-AV interaction while ensuring safety. Poor gestures, however, cause misinterpretation. The underlying reasons were explored: ambiguity regarding the recipient of the signal and whether the gestures are precise, polite, and familiar to pedestrians. Based on this empirical evidence, we discuss potential opportunities and provide valuable insights into developing comprehensible gesture-based eHMIs in the future to support better interaction between AVs and other road users., Comment: 26 pages,22 figures

We are exploring variations of the Novikov equation that have weak solutions called peakons. Our focus is on a two-component Novikov equation with a non-self-adjoint $4\times 4$ Lax operator for which we examine the related forward and inverse spectral maps for the peakon sector. To tackle the forward spectral problem, we convert it into a matrix eigenvalue problem, for which the original boundary value problem is a two-fold cover. We then use an isospectral deformation to the long-time regime to calculate the relevant eigenvalues. To support the long-term deformation, we prove the global existence of the peakon flows using ideas based on Moser's deformation method, which was used in his study of the finite Toda lattice. We subsequently solve the inverse problem by studying a trio of Weyl functions that can be approximated simultaneously by rational functions that involve tensor products and an additional symmetry condition. This part of our paper is rooted in Krein's solution to the inverse problem for the Stieltjes string., Comment: 52 pages

The accurate evaluation of the quality of driving behavior is crucial for optimizing and implementing autonomous driving technology in practice. However, there is no comprehensive understanding of good driving behaviors currently. In this paper, we sought to understand driving behaviors from the perspectives of both drivers and passengers. We invited 10 expert drivers and 14 novice drivers to complete a 5.7-kilometer urban road driving task. After the experiments, we conducted semi-structured interviews with 24 drivers and 48 of their passengers (two passengers per driver). Through the analysis of interview data, we found passengers' assessing logic of driving behaviors, divers' considerations and efforts to achieve good driving, and gaps between these perspectives. Our research provided insights into a systematic evaluation of autonomous driving and the design implications for future autonomous vehicles., Comment: Part of the content of the paper will be modified. One of the authors has recommended its withdrawal due to personal reasons

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ on $n$ vertices. In this paper, we prove this conjecture, which also confirm a conjecture for the upper bound of $S_2(G)$ proposed by Ashraf et al., Comment: 15 pages, 5 figures

Motivated by the Novikov equation and its peakon problem, we propose a new mixed type Hermite--Pad\'{e} approximation whose unique solution is a sequence of polynomials constructed with the help of Pfaffians. These polynomials belong to the family of recently proposed partial-skew-orthogonal polynomials. The relevance of partial-skew-orthogonal polynomials is especially visible in the approximation problem germane to the Novikov peakon problem so that we obtain explicit inverse formulae in terms of Pfaffians by reformulating the inverse spectral problem for the Novikov multipeakons. Furthermore, we investigate two Hermite--Pad\'{e} approximations for the related spectral problem of the discrete dual cubic string, and show that these approximation problems can also be solved in terms of partial-skew-orthogonal polynomials and nonsymmetric Cauchy biorthogonal polynomials. This formulation results in a new correspondence among several integrable lattices., Comment: 37 pages; to appear in Advances in Mathematics

On error of value function inevitably causes an overestimation phenomenon and has a negative impact on the convergence of the algorithms. To mitigate the negative effects of the approximation error, we propose Error Controlled Actor-critic which ensures confining the approximation error in value function. We present an analysis of how the approximation error can hinder the optimization process of actor-critic methods.Then, we derive an upper boundary of the approximation error of Q function approximator and find that the error can be lowered by restricting on the KL-divergence between every two consecutive policies when training the policy. The results of experiments on a range of continuous control tasks demonstrate that the proposed actor-critic algorithm apparently reduces the approximation error and significantly outperforms other model-free RL algorithms.

In this article, we consider the generalised two-parameter Cauchy two-matrix model and corresponding integrable lattice equation. It is shown that with parameters chosen as $1/k_i$ when $k_i\in\mathbb{Z}_{>0}$ ($i=1,\,2$), the average characteristic polynomials admit $(k_1+k_2+2)$-term recurrence relations, which provide us spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest example with exact solvability is demonstrated., Comment: 15 pages; comments are welcome

The experimental determination of the superconducting transition requires the observation of the emergence of zero-resistance and perfect diamagnetism state. Based on the close relationship between superconducting transition temperature (Tc) and electron density of states (DOS), we take two typical superconducting materials Hg and ZrTe3 as samples and calculate their DOS versus temperature under different pressures by using the first-principle molecular dynamics simulations. According to the analysis of the calculation results, the main contributors that induce superconducting transitions are deduced by tracing the variation of partial density of states near Tc. In particular, the microscopic mechanism of pressure increasing Tc is further analyzed., Comment: 18 pages, 4 figures

The isospectral deformations of the Frobenius-Stickelberger-Thiele (FST) polynomials introduced in [32](Spiridonov et al. Commun. Math. Phys. 272:139--165, 2007 ) are studied. For a specific choice of the deformation of the spectral measure, one is led to an integrable lattice (FST lattice), which is indeed an isospectral flow connected with a generalized eigenvalue problem. In the second part of the paper the spectral problem used previously in the study of the modified Camassa-Holm (mCH) peakon lattice is interpreted in terms of the FST polynomials together with the associated FST polynomials, resulting in a map from the mCH peakon lattice to a negative flow of the finite FST lattice. Furthermore, it is pointed out that the degenerate case of the finite FST lattice unexpectedly maps to the interlacing peakon ODE system associated with the two-component mCH equation studied in [17](Chang et al. Adv. Math. 299:1--35, 2016)., Comment: 29 pages. online in Commun. Math. Phys

Vibrations of an elastic rod are described by a Sturm-Liouville system. We present a general discussion of isospectral (spectrum preserving) deformations of such a system. We interpret one family of such deformations in terms of a two-component modified Camassa-Holm equation (2-mCH) and solve completely its dynamics for the case of discrete measures (multipeakons). We show that the underlying system is Hamiltonian and prove its Liouville integrability. The present paper generalizes our previous work on interlacing multipeakons of the 2-mCH and multipeakons of the 1-mCH. We give a unified approach to both equations, emphasizing certain natural family of interpolation problems germane to the solution of the inverse problem for 2-mCH as well as to this type of a Sturm-Liouville system with singular coefficients., Comment: 34 pages, 2 figures. to appear in Nonlinear Systems and Their Remarkable Mathematical Structures, Vol. 2, CRC Press, Boca Raton, FL

We construct new sequence transformations based on Wynn's epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn's epsilon and rho algorithm and of Osada's generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series., Comment: 32 pages, LaTeX2e, 0 figures