Eigenvalues of a potential dynamical system with damping forces that are described by an indefinite real symmetric matrix can behave as those of a Hamiltonian system when gain and loss are in a perfect balance. This happens when the indefinitely damped system obeys parity-time ( ) symmetry. How do pure imaginary eigenvalues of a stable -symmetric indefinitely damped system behave when variation...

We examine the PT-symmetry-breaking transition for a magnetic metamaterial of a finite extent, modeled as an array of coupled split-ring resonators in the equivalent circuit model approximation. Small-size arrays are solved completely in closed form, while for arrays larger than N=5 results were computed numerically for several gain and loss spatial distributions. In all cases, it is found that...

The quantum mechanical brachistochrone system with a PT-symmetric Hamiltonian is Naimark-dilated and reinterpreted as a subsystem of a Hermitian system in a higher-dimensional Hilbert space. This opens a way to a direct experimental implementation of the recently hypothesized PT-symmetric ultrafast brachistochrone regime of Bender et al. [Phys. Rev. Lett. 98, 040403 (2007)] in an entangled two-...

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and the average degree of the vertices of G. Motivated by the work from Sharafdini an...

For any simple connected undirected graph, it is well known that the Kirchhoff and multiplicative degree-Kirchhoff indices can be computed using the Laplacian matrix. We show that the same is true for the additive degree-Kirchhoff index and give a compact Matlab program that computes all three Kirchhoffian indices with the Laplacian matrix as the only input.

In this paper an estimator for speech enhancement based on Laplacian Mixture Model has been proposed. The proposed method, estimates the complex DFT coefficients of clean speech from noisy speech using the MMSE  estimator, when the clean speech DFT coefficients are supposed mixture of Laplacians and the DFT coefficients of  noise are assumed zero-mean Gaussian distribution. Furthermore, the MMS...

Let $S(G)$ be the Seidel matrix of a graph $G$ of order $n$ and let $D_S(G)=diag(n-1-2d_1, n-1-2d_2,ldots, n-1-2d_n)$ be the diagonal matrix with $d_i$ denoting the degree of a vertex $v_i$ in $G$. The Seidel Laplacian matrix of $G$ is defined as $SL(G)=D_S(G)-S(G)$ and the Seidel signless Laplacian matrix as $SL^+(G)=D_S(G)+S(G)$. The Seidel signless Laplacian energy $E_{SL^+...

We address a conjecture of D. Applebaum on small time trace asymptotics for subordinate Brownian motion on compact manifolds. In [1], heat trace asymptotics are computed for the semigroup of the square root of the Laplacian (the generator of the Cauchy process) on the n-dimensional torus, SU(2) and SO(3), and a conjecture is made [1, pp. 2493-2494] that such asymptotics should hold for all α-st...

a spanning tree of graph g is a spanning subgraph of g that is a tree. in this paper, we focusour attention on (n,m) graphs, where m = n, n + 1, n + 2 and n + 3. we also determine somecoefficients of the laplacian characteristic polynomial of fullerene graphs.