We consider overdetermined boundary value problems for the ∞-Laplacian in a domain Ω of Rn and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞-Laplacian, the normalized or game-theoretic ∞-Laplacian and the limit of the p-Laplacian as p→∞ are considered and provide different answers. Mathematics Subject Classification (2000). 35R35, 4...

We investigate the Laplacian eigenvalues of a random graphG(n,d) with a given expected degree distribution d. The main result is that w.h.p. G(n,d) has a large subgraph core(G(n,d)) such that the spectral gap of the normalized Laplacian of core(G(n,d)) is ≥ 1− c0d̄ min with high probability; here c0 > 0 is a constant, and d̄min signifies the minimum expected degree. The result in particular appli...

We consider an elliptic pseudo-differential equation with a highly oscillating linear potential modeled as a stationary ergodic random field. The random field is a function composed with a centered long-range correlated Gaussian process. In the limiting of vanishing correlation length, the heterogeneous solution converges to a deterministic solution obtained by averaging the random potential. W...

This paper proposes the novel model for estimation of bounds in digital coding of images. Entropy coding of images is exploited to measure the useful information content of the data. The bit rate achieved by reversible compression using the rate-distortion theory approach takes into account the contribution of the observation noise and the intrinsic information of hypothetical noise-free image....

Semi-supervised learning gets estimated marginal distribution X P with a large number of unlabeled examples and then constrains the conditional probability ) | ( x y p with a few labeled examples. In this paper, we focus on a regularization approach for semi-supervised classification. The label information graph is first defined to keep the pairwise label relationship and can be incorporated wi...

Determining the number of relevant dimensions in the eigen-space of a graph Laplacian matrix is a central issue in many spectral graph-mining applications. We tackle here the sub-problem of finding the “right" dimensionality of Laplacian matrices, especially those often encountered in the domains of social or biological graphs: the ones underlying large, sparse, unoriented and unweighted graphs...

Sparse grids were recently introduced for classification and regression problems. In this article we apply the sparse grid approach to semi-supervised classification. We formulate the semi-supervised learning problem by a regularization approach. Here, besides a regression formulation for the labeled data, an additional term is involved which is based on the graph Laplacian for an adjacency gra...

Suppose μ1, μ2, ... , μn are Laplacian eigenvalues of a graph G. The Laplacian energy of G is defined as LE(G) = ∑n i=1 |μi − 2m/n|. In this paper, some new bounds for the Laplacian eigenvalues and Laplacian energy of some special types of the subgraphs of Kn are presented. AMS subject classifications: 05C50

We consider the Anderson model at large disorder on $${\mathbb {Z}}^2$$ where potential has a symmetric Bernoulli distribution. prove that localization happens outside small neighborhood of finitely many energies. These energies are Dirichlet eigenvalues minus Laplacian restricted some finite subsets {Z}}^{2}$$ .