This paper proposed a robust searching algorithm to generate path planning for mobile robot. The searching algorithm employs Laplacian Behaviour-Based Control (LBBC) during space exploration of the environment. The LBBC make use of the potential function in the configuration space to guide its exploration. Laplace’s equation was used to represent the potential function in the configuration spac...

When attempting to develop wavelet transforms for graphs and networks, some researchers have used graph Laplacian eigenvalues and eigenvectors in place of the frequencies and complex exponentials in the Fourier theory for regular lattices in the Euclidean domains. This viewpoint, however, has a fundamental flaw: on a general graph, the Laplacian eigenvalues cannot be interpreted as the frequenc...

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to th...

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...

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G → M → X so that M is also a complex manifold satisfying a local subelliptic estimate. In this work, we show that if G acts by holomorphic transformations in M , then the Laplacian = ∂̄∗∂̄ + ∂̄∂̄∗ on M has the following properties: The kernel of restricted to the forms Λ with q > 0 i...

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In...

In this paper we introduce a new morphological sampling theory and show some of its applications to image coding. As in previous works on morphological sampling, we establish conditions that the signal should satisfy before (sub)sampling in order to meet a prescribed reconstruction error criteria. In contrast to previous theorems, however, the whole process is self dual. When compared to previo...

We consider two optimization problems on synchronization of oscillator networks: maximization of synchronizability and minimization of synchronization cost. We first develop an extension of the well-known master stability framework to the case of non-diagonalizable Laplacian matrices. We then show that the solution sets of the two optimization problems coincide and are simultaneously characteri...

We consider the oscillatory property of the following p(t)-Laplacian equations −(|u′|p(t)−2u′)′ = 1/tθ(t)g(t,u), t > 0. Since there is no Picone-type identity for p(t)Laplacian equations, it is an unsolved problem that whether the Sturmian comparison theorems for p(x)-Laplacian equations are valid or not. We obtain sufficient conditions of the oscillatory of solutions for p(t)-Laplacian equations.

The trees, respectively unicyclic graphs, on n vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the n-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the n-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian pe...