We have seen before that Fourier analysis is very useful in the study of signals and linear and time invariant (LTI) systems. The main reason is that a lot of signals can be expressed as a linear combination of complex exponentials of the form e with s = jw. There are many properties that still apply when s is not restricted to be pure imaginary. That is why we introduce a generalization of the...

For safe and sound utilization of the applications made polylactic acid that output an electric signal to intended mechanical input within undesirable thermal environment, transient thermoelectroelastic field is investigated for infinite cylinder with D∞ symmetry subjected shear stress as temperature unfavorable environment. By use analytical technique constructed previously, quantities are rep...

We consider the random Schrödinger operator−ε−2∆(d) +ξ (ε)(x), with ∆(d) the discrete Laplacian on Zd and ξ (ε)(x) are bounded and independent random variables, on sets of the form Dε := {x ∈ Zd : xε ∈ D} for D bounded, open and with a smooth boundary, and study the statistics of the Dirichlet eigenvalues in the limit ε ↓ 0. Assuming Eξ (ε)(x) = U(xε) holds for some bounded and continuous funct...

Let H be the discrete Schrödinger operator Hu(n) := u(n − 1) + u(n + 1) + v(n)u(n), u(0) = 0 acting on l(Z) where the potential v is real-valued and v(n) → 0 as n → ∞. Let P be the orthogonal projection onto a closed linear subspace L ⊂ l(Z). In a recent paper E.B. Davies defines the second order spectrum Spec2(H,L) of H relative to L as the set of z ∈ C such that the restriction to L of the op...

Inequalities on networks have played important roles in the theory of netwoks. We study several famous inequalities on networks such as Wirtinger’s inequality, Hardy’s inequality, Poincaré-Sobolev’s inequality and the strong isoperimetric inequality, etc. These inequalities are closely related to the smallest eigenvalue of weighted discrete Laplacian. We discuss some relations between these ine...

We study discrete Green’s functions and their relationship with discrete Laplace equations. Several methods for deriving Green’s functions are discussed. Green’s functions can be used to deal with diffusion-type problems on graphs, such as chip-firing, load balancing and discrete Markov chains.

Let H be the discrete Schrödinger operator Hu(n) := u(n − 1) + u(n + 1) + v(n)u(n), u(0) = 0 acting on l(Z) where the potential v is real-valued and v(n) → 0 as n → ∞. Let P be the orthogonal projection onto a closed linear subspace L ⊂ l(Z). In a recent paper E.B. Davies defines the second order spectrum Spec2(H,L) of H relative to L as the set of z ∈ C such that the restriction to L of the op...

We study discrete Green’s functions and their relationship with discrete Laplace equations. Several methods for deriving Green’s functions are discussed. Green’s functions can be used to deal with diffusion-type problems on graphs, such as chip-firing, load balancing and discrete Markov chains.

This paper exploits the properties of the commute time to develop a graphspectral method for image segmentation. Our starting point is the lazy random walk on the graph, which is determined by the heat-kernel of the graph and can be computed from the spectrum of the graph Laplacian. We characterise the random walk using the commute time between nodes, and show how this quantity may be computed ...

I present an example of a discrete Schrödinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously iden...