We study the Gel'fand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: X = B ∪ G, where B is called the set of the boundary vertices and G is called the set of the interior vertices. We consider the case where the vertices in the set G and the edges connecting them are unknown. Assume that we are given the set B and the pairs (λj, φj|B), where λj are the eigenvalues of the graph Laplacian and φj|B are the values of the corresponding eigenfunctions at the vertices in B. We show that the graph structure, namely the unknown vertices in G and the edges connecting them, along with the weights, can be uniquely determined from the given data, if every boundary vertex is connected to only one interior vertex and the graph satisfies the following property: any subset S ⊆ G of cardinality |S| 2 contains two extreme points. A point x ∈ S is called an extreme point of S if there exists a point z ∈ B such that x is the unique nearest point in S from z with respect to the graph distance. This property is valid for several standard types of lattices and their perturbations.

We prove an explicit estimate on the stability of the unique continuation for the wave operator on compact Riemannian manifolds with smooth boundary. Our estimate holds on domains arbitrarily close to the optimal domain, and is uniform in a class of Riemannian manifolds with bounded geometry. As an application, we obtain a quantitative estimate on the stability of the Gel'fand inverse boundary problem.

Rapid and nondestructive evaluation of pavement crack depths is a major challenge in pavement maintenance and rehabilitation. This article presents a computer-based methodology with which one can estimate the actual depths of shallow, surface-initiated fatigue cracks in asphalt pavements based on rapid measurement of their surface characteristics. It is shown that the complex overall relationship among crack depths, surface geometrical properties of cracks, pavement properties, and traffic characteristics can be learnt effectively by a neural network (NN). The learning task is facilitated by a database that includes relevant traffic and pavement characteristics of Florida's state highway network. In addition, the specific data used for the NN model development also contained laser-scanned microscopic surface geometrical properties of cracks in 95 pavement sections and pavement core samples scattered within five counties of Florida. Relatively advanced training algorithms were investigated in addition to the Standard Backpropagation algorithm to determine the optimal NN architecture. In terms of optimizing the NN training process, the "early stopping method" was found to be effective. The crack depth evaluation model was validated based on an unused portion of the database and fresh core samples. The results indicate the promise of NN usage in

I prove that the spectrum of the Laplace-Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued out of compact Riemannian manifolds of the same dimension.