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In this paper, we study an inverse spectral problem for the fourth-order differential equation $y^{(4)} - (p y')' + q y = \lambda y$ with real-valued coefficients $p$ and $q$ of $L^2(0,1)$. We prove that, for near-constant coefficients, the two spectra corresponding to the Dirichlet and the Dirichlet-Neumann boundary conditions uniquely determine either $p$ or $q$. The result extends the Borg theorem of the second-order case to the fourth-order case.

We introduce and develop equivalent spectral graph theory for several fundamental graph cut problems including maxcut, mincut, Cheeger cut, anti-Cheeger cut, dual Cheeger problem and their useful variants. The finer structure of the eigenvectors, the Courant nodal domain theorem and the graphic feature of eigenvalues are studied systematically in the setting of these new nonlinear eigenproblems. A specified strategy for achieving an equivalent eigenproblem is proposed for a general graph cut problem via the Dinkelbach scheme. We also establish several multi-way inequalities relating the variational eigenvalues and min-max $k$-cut problems, which closely resembles the higher-order Cheeger inequality on graph 1-Laplacian.

This report introduces the first-place winning solution for the Autonomous Grand Challenge 2024 - Mapless Driving. In this report, we introduce a novel online mapping pipeline LGmap, which adept at long-range temporal model. Firstly, we propose symmetric view transformation(SVT), a hybrid view transformation module. Our approach overcomes the limitations of forward sparse feature representation and utilizing depth perception and SD prior information. Secondly, we propose hierarchical temporal fusion(HTF) module. It employs temporal information from local to global, which empowers the construction of long-range HD map with high stability. Lastly, we propose a novel ped-crossing resampling. The simplified ped crossing representation accelerates the instance attention based decoder convergence performance. Our method achieves 0.66 UniScore in the Mapless Driving OpenLaneV2 test set.

The existing inverse power ($\mathbf{IP}$) method for solving the balanced graph cut lacks local convergence and its inner subproblem requires a nonsmooth convex solver. To address these issues, we develop a simple inverse power ($\mathbf{SIP}$) method using a novel equivalent continuous formulation of the balanced graph cut, and its inner subproblem allows an explicit analytic solution, which is the biggest advantage over $\mathbf{IP}$ and constitutes the main reason why we call it $\mathit{simple}$. By fully exploiting the closed-form of the inner subproblem solution, we design a boundary-detected subgradient selection with which $\mathbf{SIP}$ is proved to be locally converged. We show that $\mathbf{SIP}$ is also applicable to a new ternary valued $\theta$-balanced cut which reduces to the balanced cut when $\theta=1$. When $\mathbf{SIP}$ reaches its local optimum, we seamlessly transfer to solve the $\theta$-balanced cut within exactly the same iteration algorithm framework and thus obtain $\mathbf{SIP}$-$\mathbf{perturb}$ -- an efficient local breakout improvement of $\mathbf{SIP}$, which transforms some ``partitioned" vertices back to the ``un-partitioned" ones through the adjustable $\theta$. Numerical experiments on G-set for Cheeger cut and Sparsest cut demonstrate that $\mathbf{SIP}$ is significantly faster than $\mathbf{IP}$ while maintaining approximate solutions of comparable quality, and $\mathbf{SIP}$-$\mathbf{perturb}$ outperforms $\mathtt{Gurobi}$ in terms of both computational cost and solution quality., Comment: 24 pages, 10 figures

In this work, we study the inverse spectral problems for the Sturm-Liouville operators on [0,1] with complex coefficients and a discontinuity at $x=a\in(0,1)$. Assume that the potential on (a,1) and some parameters in the discontinuity and boundary conditions are given. We recover the potential on (0,a) and the other parameters from the eigenvalues. This is the so-called partial inverse problem. The local solvability and stability of the partial inverse problems are obtained for $a\in(0,1)$, in which the error caused by the given partial potential is considered. As a by-product, we also obtain two new uniqueness theorems for the partial inverse problem., Comment: 26 pages

In this paper, we consider the recovery of third-order differential operators from two spectra, as well as fourth-order or fifth-order differential operators from three spectra, where these differential operators are endowed with complex-valued distributional coefficients. For the case of multiple spectra, we first establish the relationship between spectra and the Weyl-Yurko matrix. Secondly, we prove the uniqueness theorem for the solution of the inverse problems. Our approach allows us to obtain results for the general case of complex-valued distributional coefficients.

We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph uniquely determines the potentials. We obtain a reconstruction procedure, which is based on the reduction of the differential Schr\"odinger operator to a discrete one. As a corollary of the main results, it is proved that the S-matrix for all energies in any given open set in the continuous spectrum uniquely specifies the potentials on the square lattice.

We consider the Sturm-Liouville operator on the lasso graph with a segment and a loop joined at one point, which has arbitrary length. The Ambarzumyan's theorem for the operator is proved, which says that if the eigenvalues of the operator coincide with those of the zero potential, then the potential is zero.

We consider the vector-impulsive Sturm-Liouville problem with Neumann conditions. The Ambarzumyan$^{\textbf{,}}$s theorem for the problem is proved, which states that if the eigenvalues of the problem coincide with those of the zero potential, then the potential is zero.

In this work, we consider Dirac-type operators with a constant delay less than two-fifths of the interval and not less than one-third of the interval. For our considered Dirac-type operators, an incomplete inverse spectral problem is studied. Specifically, when two complex potentials are known a priori on a certain subinterval, reconstruction of the two potentials on the entire interval is studied from complete spectra of two boundary value problems with one common boundary condition. The uniqueness of the solution of the inverse problem is proved. A constructive method is developed for the solution of the inverse problem., Comment: arXiv admin note: substantial text overlap with arXiv:2305.10752

In this work, we consider Dirac-type operators with a constant delay less than half of the interval and not less than two-fifths of the interval. For our considered Dirac-type operators, two inverse spectral problems are studied. Specifically, reconstruction of two complex $L_{2}$-potentials is studied from complete spectra of two boundary value problems with one common Dirichlet boundary condition and Neumann boundary condition, respectively. We give answers to the full range of questions usually raised in the inverse spectral theory. That is, we give uniqueness, necessary and sufficient conditions of the solvability, reconstruction algorithm and uniform stability for our considered inverse problems., Comment: arXiv admin note: text overlap with arXiv:2204.08259 by other authors