The kth power of a graph G, denoted by Gk , is a graph with the same vertex set as G such that two vertices are adjacent in Gk if and only if their distance is at most k in G. The Wiener index is a distance-based topological index defined as the sum of distances between all pairs of vertices in a graph. In this note, we give the bounds on the Wiener index of the graph Gk . The Nordhaus–Gaddum-t...

We define an analogue of the Paley-Wiener space in the context of the Askey-Wilson function transform, compute explicitly its reproducing kernel and prove that the growth of functions in this space of entire functions is of order two and type ln q−1, providing a Paley-Wiener Theorem for the Askey-Wilson transform. Up to a change of scale, this growth is related to the refined concepts of expone...

The eccentric sequence of a connected graph \(G\) is the nondecreasing eccentricities its vertices. Wiener index sum distances between all unordered pairs vertices \(G\). unique trees that minimise among with given were recently determined by present authors. In this paper we show these results hold not only for index, but large class distance-based topological indices which term Wiener-type in...

1 Basic Setting Here, we will only address the discrete time formulation of the Wiener filter. Consider a widesense stationary random process x[n] in the presence of additive noise n[n] independent of x[n]. so that we have u[n] = x[n] + n[n] sent through a time invariant linear system whose output is denoted by y[n]. The signal y[n] will approximate the output of a linear operator L on x[n] by ...

We systematically develop real Paley–Wiener theory for the Fourier transform on Rd for Schwartz functions, Lp-functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley–Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use fo...

For the specification of phase-noise requirements for the front-end of a HiperLAN/2 system we investigated available literature on the subject. Literature differed in several aspects. One aspect is in the type of phase-noise used (Wiener phase-noise or small-angle phase noise). A Wiener phase-noise based analysis leads to contradictions with the type of analysis normally used in the solid state...

We de ne a covariance-type operator on Wiener space: for F and G two random variables in the Gross-Sobolev space D of random variables with a square-integrable Malliavin derivative, we let ΓF,G:= ⟨ DF,−DL−1G ⟩ , where D is the Malliavin derivative operator and L−1 is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. We use Γ to extend the notion of covariance and canonica...

We define a covariance-type operator on Wiener space: for F and G two random variables in the Gross-Sobolev space D of random variables with a square-integrable Malliavin derivative, we let ΓF,G:= 〈 DF,−DL−1G 〉 where D is the Malliavin derivative operator and L−1 is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. We use Γ to extend the notion of covariance and canonical...