The Volterra theory gives a characterization of nonlinear systems in terms of polynomial functionals. This theory was analysed in comparison to algorithms for the construction of invariant image features in 1] and conditions were derived so that these features remain constant if the images are transformed according to the action of a transformation group. However in general one can observe a hi...

In [1], Hamzeh, Iranmanesh Hossein-Zadeh and M. V. Diudea recently introduced the generalized degree distance of graphs. In this paper, we present explicit formulas for this new graph invariant of the Cartesian product, composition, join, disjunction and symmetric difference of graphs and introduce generalized and modified generalized degree distance polynomials of graphs, such that their first...

We present a new approach to obtain solutions for general random oscillators using a broad class of polynomial chaos expansions, which are more efficient than the classical Wiener–Hermite expansions. The approach is general but here we present results for linear oscillators only with random forcing or random coefficients. In this context, we are able to obtain relatively sharp error estimates i...

Here, Jc F[x]w(dx) denotes the Wiener integral, and / F[0(u, ■ )]v(du) denotes an integral over some Euclidean space. In [1] Cameron determined a pair (v, 0) by imposing on (1.2) the condition that the formula be exact for polynomial functionals of degree ^3. Imposing the same requirement, Vladimirov [5] constructed a family of pairs (v, 6). In this paper we shall develop a class of approximati...

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method...

In this paper the Wiener and hyper Wiener index of two kinds of dendrimer graphs are determined. Using the Wiener index formula, the Szeged, Schultz, PI and Gutman indices of these graphs are also determined.

We consider stationary processes with long memory which are non–Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long–memory parameter. We show that the limit is not Gaussian but can be expressed using the non–Gaus...

In Gaussian process regression, both the basis functions and their prior distribution are simultaneously specified by the choice of the covariance function. In certain problems one would like to choose the covariance independently of the basis functions (e. g., in polynomial signal processing or Wiener and Volterra analysis). We propose a solution to this problem that approximates the desired c...

Many successful tools have been proposed by several researchers for safety analysis, stability analysis, and control/scheduling synthesis for hybrid systems. However, one important area which has not received much attention in the hybrid systems community is the identi cation of piecewise aÆne systems. As a step in that direction, we present an algorithm for identifying piecewise aÆne Wiener mo...

We consider an infinite-dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of Wiener processes. By studying how randomness is spread by the system we develop a counterpart of Hörmander’s classical theory in this setting. We study the distributions of finite-dimensional projections of the solutions and give conditions that provide existence and ...