Wiener Index of Hexagonal Systems

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. Hexagonal systems (HS's) are a special type of plane graphs in which all faces are bounded by hexagons. These provide a graph representation of benzenoid hydrocarbons and thus find applications in chemistry. The paper outlines the results known for W of the HS: method for computation of W, expressions relating W with the structure of the respective HS, results on HS's extremal w.r.t. W, and on integers that cannot be the W-values of HS's. A few open problems are mentioned. The chemical applications of the results presented are explained in detail.     

Quadratic variation of functionals of two-parameter Wiener process

The quadratic variation of functionals of the two-parameter Wiener process of the form f(W(s, t)) is investigated, where W(s, t) is the standard two-parameter Wiener process and f is a function on the reals. The existence of the quadratic variation is obtained under the condition that f′ is locally absolutely continuous and f^N is locally square integrable.     

The problem of compensation

Let W₁, W₂ be stationary connected Wiener processes. We investigate the possibility of regular representation for an element of the linear space generated by the processes W₁, W₂. Biliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 144–152. Translated by V. Sudakov.     

A uniformly distributed hitting position for a two-dimensional anisotropic diffusion process: The limit normed curve

Let W₁ and W₂ be two independent Wiener processes on a half-line, and let W^(a)=(W₁, aW₂) (a≥1). We consider open neighborhoods of the initial point with uniform hitting distribution. This property uniquely determines the form of a neighborhood. The main result is as follows: such a neighborhood has a limit form as a→∞. Properties of the limit form are studied. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 333–348.     

Optimal Approximation of the Second Iterated Integral of Brownian Motion

Abstract

In this article, a theorem is proved that describes the optimal approximation (in the L ²(?)-sense) of the second iterated integral of a standard two-dimensional Wiener process, W, by a function of finitely many elements of the Gaussian Hilbert space generated by W. This theorem has some interesting corollaries: First of all, it implies that Euler's method has the optimal rate of strong convergence among all algorithms that depend solely on linear functionals of the Wiener process, W; second, it shows that the approximation of the second iterated integral based on Karhunen–Loève expansion of the Brownian bridge is asymptotically optimal.     

Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L ^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L ^p , L ^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L ^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.     

On the Wiener Index of Graphs

The Wiener index of a graph G is defined as W(G)=∑ _u,v d _G (u,v), where d _G (u,v) is the distance between u and v in G and the sum goes over all the pairs of vertices. In this paper, we first present the 6 graphs with the first to the sixth smallest Wiener index among all graphs with n vertices and k cut edges and containing a complete subgraph of order n−k; and then we construct a graph with its Wiener index no less than some integer among all graphs with n vertices and k cut edges.     

Stochastic anticipative calculus for functionals of a gaussian generalized random field

We give a construction of Skorohod integrals with respect to a Gaussian D'-valued random field W The method is based on the multiple Wiener integral expansion for L ₂-functionals of W We also give a representation of the Malliavin derivative operator of L ²-functionals of W     

Reciprocal complementary Wiener numbers of trees, unicyclic graphs and bicyclic graphs

The reciprocal complementary Wiener number of a connected graph G is defined as

where V(G) is the vertex set, d(u,v|G) is the distance between vertices u and v, d is the diameter of G. We determine the trees with the smallest, the second smallest and the third smallest reciprocal complementary Wiener numbers, and the unicyclic and bicyclic graphs with the smallest and the second smallest reciprocal complementary Wiener numbers.     

Laplacian coefficients of trees with given number of leaves or vertices of degree two

Let G be a simple undirected n-vertex graph with the characteristic polynomial of its Laplacian matrix . It is well known that for trees the Laplacian coefficient c_n-2 is equal to the Wiener index of G, while c_n-3 is equal to the modified hyper-Wiener index of graph. Using a result of Zhou and Gutman on the relation between the Laplacian coefficients and the matching numbers in subdivided bipartite graphs, we characterize the trees with k leaves (pendent vertices) which simultaneously minimize all Laplacian coefficients. In particular, this extremal balanced starlike tree S(n,k) minimizes the Wiener index, the modified hyper-Wiener index and recently introduced Laplacian-like energy. We prove that graph S(n,n-1-p) has minimal Laplacian coefficients among n-vertex trees with p vertices of degree two. In conclusion, we illustrate on examples of these spectrum-based invariants that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution, and pose a conjecture on extremal unicyclic graphs with k leaves.     

The divergence of Banach space valued random variables on Wiener space

The domain of definition of the divergence operator δ on an abstract Wiener space (W,H,μ) is extended to include W–valued and – valued “integrands”. The main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in δ’s extended domain have divergence zero. These results are then applied to the analysis of quasiinvariant flows induced by W-valued vector fields and, among other results, it turns out that these divergence-free vector fields “are responsible” for generating measure preserving flows. Mathematics Subject Classification (2000): Primary 60H07, Secondary 60H05 An erratum to this article is available at .     

Nonlinear Filtering for Diffusions in Random Environments

Suppose that the signal X to be estimated is a diffusion process in a random medium W and the signal is correlated with the observation noise. We study the historical filtering problem concerned with estimating the signal path up until the current time based upon the back observations. Using Dirichlet form theory, we introduce a filtering model for general rough signal X ^W and establish a multiple Wiener integrals representation for the unnormalized pathspace filtering process. Then, we construct a precise nonlinear filtering model for the process X itself and give the corresponding Wiener chaos decomposition.     

Wiener indices of balanced binary trees

We study a new family of trees for computation of the Wiener indices. We introduce general tree transformations and derive formulas for computing the Wiener indices when a tree is modified. We present several algorithms to explore the Wiener indices of our family of trees. The experiments support new conjectures about the Wiener indices.     

Generalized stochastic integrals and the malliavin calculus

Summary The paper first reviews the Skorohod generalized stochastic integral with respect to the Wiener process over some general parameter space T and it's relation to the Malliavin calculus as the adjoint of the Malliavin derivative. Some new results are derived and it is shown that every sufficiently smooth process {u_t, tT} can be decomposed into the sum of a Malliavin derivative of a Wiener functional, and a process whose generalized integral over T vanishes. Using the results on the generalized integral, the Bismut approach to the Malliavin calculus is generalized by allowing non adapted variations of the Wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions.Let e _i be a non-random complete orthonormal system on T, the Ogawa integral u W is defined as _i (e _i u) e _i dW where the integrals are Wiener integrals. Conditions are given for the existence of an intrinsic Ogawa integral i.e. independent of the choice of the orthonormal system and results on it's relation to the Skorohod integral are derived.The transformation of measures induced by (W + u d u non adapted is discussed and a Girsanov-type theorem under certain regularity conditions is derived.The work of M.Z. was supported by the Fund for Promotion of Research at the Technion     

The wiener index of graphs of arbitrary girth and their line graphs

We consider the invariant W(G) of a simple connected undirected graph G which is equal to the sum of distances between all pairs of its vertices in the natural metric (the Wiener index). We show that, for every g ≥ 5, there is a planar graph G of girth g satisfying W(L(G)) = W(G), where L(G) is the line graph of G.     

Degree theory on Wiener space

Summary. Let (W, H, μ) be an abstract Wiener space and let Tw  =  w + u (w), where u is an H-valued random variable, be a measurable transformation on W. A Sard type lemma and a degree theorem for this setup are presented and applied to derive existence of solutions to elliptic stochastic partial differential equations. Received: 19 March 1996 / In revised form: 7 January 1997