Stochastic Wiener Filter in the White Noise Space

In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this spaces in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions

Eccentric connectivity index

The eccentric connectivity index $\xi^c$ is a novel distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as $\xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \epsilon (v)$\,, where $deg (v)$ and $\epsilon (v)$ denote the vertex degree and eccentricity of $v$\,, respectively. We survey some mathematical properties of this index and furthermore support the use of eccentric connectivity index as topological structure descriptor. We present the extremal trees and unicyclic graphs with maximum and minimum eccentric connectivity index subject to the certain graph constraints. Sharp lower and asymptotic upper bound for all graphs are given and various connections with other important graph invariants are established. In addition, we present explicit formulae for the values of eccentric connectivity index for several families of composite graphs and designed a linear algorithm for calculating the eccentric connectivity index of trees. Some open problems and related indices for further study are also listed.Comment: 25 pages, 5 figure

De Branges spaces and Krein's theory of entire operators

This work presents a contemporary treatment of Krein's entire operators with deficiency indices $(1,1)$ and de Branges' Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected and the interplay between them has had an impact in both spectral theory and the theory of functions. This work exhibits the interrelation between Krein's and de Branges' theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.Comment: 37 pages, no figures. The abstract was extended. Typographical errors were corrected. The bibliography style was change

Modeling complex metabolic reactions, ecological systems, and financial and legal networks with MIANN models based on Markov-Wiener node descriptors

[Abstract] The use of numerical parameters in Complex Network analysis is expanding to new fields of application. At a molecular level, we can use them to describe the molecular structure of chemical entities, protein interactions, or metabolic networks. However, the applications are not restricted to the world of molecules and can be extended to the study of macroscopic nonliving systems, organisms, or even legal or social networks. On the other hand, the development of the field of Artificial Intelligence has led to the formulation of computational algorithms whose design is based on the structure and functioning of networks of biological neurons. These algorithms, called Artificial Neural Networks (ANNs), can be useful for the study of complex networks, since the numerical parameters that encode information of the network (for example centralities/node descriptors) can be used as inputs for the ANNs. The Wiener index (W) is a graph invariant widely used in chemoinformatics to quantify the molecular structure of drugs and to study complex networks. In this work, we explore for the first time the possibility of using Markov chains to calculate analogues of node distance numbers/W to describe complex networks from the point of view of their nodes. These parameters are called Markov-Wiener node descriptors of order kth (Wk). Please, note that these descriptors are not related to Markov-Wiener stochastic processes. Here, we calculated the Wk(i) values for a very high number of nodes (>100,000) in more than 100 different complex networks using the software MI-NODES. These networks were grouped according to the field of application. Molecular networks include the Metabolic Reaction Networks (MRNs) of 40 different organisms. In addition, we analyzed other biological and legal and social networks. These include the Interaction Web Database Biological Networks (IWDBNs), with 75 food webs or ecological systems and the Spanish Financial Law Network (SFLN). The calculated Wk(i) values were used as inputs for different ANNs in order to discriminate correct node connectivity patterns from incorrect random patterns. The MIANN models obtained present good values of Sensitivity/Specificity (%): MRNs (78/78), IWDBNs (90/88), and SFLN (86/84). These preliminary results are very promising from the point of view of a first exploratory study and suggest that the use of these models could be extended to the high-throughput re-evaluation of connectivity in known complex networks (collation)

Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space \eufrak{h} and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over \eufrak{h}. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For \eufrak{h}=L^2(\mathbb{R}_+), the divergence operator is shown to coincide with the Hudson-Parthasarathy quantum stochastic integral for adapted integrable processes and with the non-causal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.Comment: 28 pages, amsart styl