This paper describes the detection and identification of cracking severity in a reinforced concrete beam using modal data. Experimental modal test was performed on the beam before and after application of load of increasing intensity to induce a single crack at a fixed location. Modal parameters extracted were categorised into hvo groups, namely local response parameter (mode shape data) and gl...

We study the ground state solutions for following p&q‐Laplacian equation where is a parameter large enough, with denotes ‐Laplacian operator, and . Under some assumptions periodic potential , weight function nonlinearity without Ambrosetti–Rabinowitz condition, we show above has solution.

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.

In existing anisotropic diffusion-based semi-supervised learning approaches, anisotropic graph Laplacian is estimated based on (potentially noisy) function evaluations. We propose to regularize the graph Laplacian estimates. We develop a framework that regularizes the Laplace-Beltrami operators on Riemannian manifolds, and discretize it to a regularizer on diffusivity operators on graphs. Isotr...

We obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional p−Laplacian operator and nonlinearities at critical growth.

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ±∞ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this...

We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of nonlocal particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasicontinuous linear chain of infinite ...

We present an algorithm for approximating the LaplaceBeltrami operator from an arbitrary point cloud obtained from a k-dimensional manifold embedded in the ddimensional space. We show that this PCD Laplace (PointCloud Data Laplace) operator converges to the LaplaceBeltrami operator on the underlying manifold as the point cloud becomes denser. Unlike the previous work, we do not assume that the ...

Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the non-backtracking walk on the graph, leads to improved performance in detecting...

In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifold-motivated” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between theory and practice for a class of Laplacian-based manifold methods. These methods utilize the graph Laplacia...