This is an expository paper which includes several topics related to the Dirichlet form analysis on the Sierpiński gasket. We discuss the analog of the classical Laplacian; approximation by harmonic functions that gives a notion of a gradient; directional energies and an equipartition of energy; analysis with respect to the energy measure; harmonic coordinates; and non self-similar Dirichlet fo...

The centrality of vertices has been a key issue in network analysis. For unweighted networks where edges are just present or absent and have no weight attached, many centrality measures have been presented, such as degree, betweenness, closeness, eigenvector and subgraph centrality. There has been a growing need to design centrality measures for weighted networks, because weighted networks wher...

A compact binary on an eccentric orbit radiates gravitational waves (GWs) at all integer harmonics of its orbital frequency. Thus, the spectral energy distribution, the power and the timescale of GW radiation of a binary on an eccentric orbit are different from those of a binary on a circular orbit. Therefore, in order to predict spectra of gravitational wave background radiation (GWBR) from co...

Let $L(G)$ be the Laplacian matrix of a digraph $G$ and $S_k(G)$ sum $k$ largest absolute values eigenvalues $G$. $C_n^+$ with $n+1$ vertices obtained from directed cycle $C_n$ by attaching pendant arc whose tail is on $C_n$. A $\mathbb{C}_n^+$-free if it contains no $C_{\ell}^+$ as subdigraph for any $2\leq \ell \leq n-1$. In this paper, we present lower bounds $S_n(G)$ digraphs order $n$. We ...

We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Garding inequalities are derived we establish that the obtained Fredholm of index zero. Their finite dimensional kernels characterized show their dimension is equal to number topological inva...

Abstract We establish some existence results for a class of critical $N$ -Laplacian problems in bounded domain $\mathbb {R}^{N}$ . In the absence suitable direct sum decomposition underlying Sobolev space to which classical linking theorem can be applied, we use an abstract based on {Z}_2$ -cohomological index obtain non-trivial point.

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. In this article we consider the problem whether generalized Fibonacci constants $varphi_n$ $(ngeq 2)$ can be the energy of graphs. We show that $varphi_n$ cannot be the energy of graphs. Also we prove that all natural powers of $varphi_{2n}$ cannot be the energy of a matroid.

A real time scenario of dynamic primary user (PU) is considered in additive Laplacian noise. Two transitions or status changes PU the fixed sensing are considered. The last change point (LSCP) estimated with maximum likelihood estimation by using programming. We consider Cumulative Sum (CuSum) based weighted samples for detection. three detection schemes such as sample mean detector, energy and...

The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the cross-section diminishes. We show that spectral properties of the Laplacian are in this limit approximated well by those of the sum of the Dirichlet Laplacian in the ...