The logic of preference and decision supporting systems

In this thesis we are exploring some models for von Wright's preference logic. Given (initial) set of axioms and a set of formulae, some of them valid, some of them problematic (in the sense that it is not always intuitively clear should they be valid or not), we investigated some matrix semantics for those formulae including semantics in relevance logics (first degree entailment and RM3), various many--valued (Kleene's, {\L}ukasiewicz's, \dots) and/or paraconsistent logics, in Sugihara matrix, and one interpretation for preference relation using modal operators L and M. In each case, we also investigated dependence results between various formulae. Opposite problem (i.e.\ searching for a logic that satisfies given constraints) is also addressed. At the end, a {\tt LISP} program is presented that implements von Wright's logic as a decision supporting system, i.e.\ that decides for a given set of preferences, what alternatives (world--situation) should we choose, according to von Wright's preference logic system

Kato type decompositions and generalizations of Drazin invertibility

The main objective of this dissertation is to give necessary and sufficient conditions under which a bounded linear operator T can be represented as the direct sum of a nilpotent (quasinilpotent, Riesz) operator TN and an operator TM which belongs to any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators, bounded below operators, surjective operators and invertible operators. These results are applied to different types of spectra. In addition, we introduce the notions of the generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators. Moreover, we study the generalized Drazin spectrum of an upper triangular operator matrix acting on the product of Banach or separable Hilbert spaces. Further, motivated by the Atkinson type theorem for B-Fredholm operators, we introduce the notion of a B-Fredholm Banach algebra element. These objects are characterized and their main properties are studied. We also extend some results from the Fredholm theory to unbounded closed operators

Spectral Analysis of Protein-Protein Interactions in Drosophila melanogaster

Within a case study on the protein-protein interaction network (PIN) of Drosophila melanogaster we investigate the relation between the network's spectral properties and its structural features such as the prevalence of specific subgraphs or duplicate nodes as a result of its evolutionary history. The discrete part of the spectral density shows fingerprints of the PIN's topological features including a preference for loop structures. Duplicate nodes are another prominent feature of PINs and we discuss their representation in the PIN's spectrum as well as their biological implications.Comment: 9 pages RevTeX including 8 figure

Fermion spectrum and localization on kinks in a deconstructed dimension

We study the deconstructed scalar theory having nonlinear interactions and being renormalizable. It is shown that the kink-like configurations exist in such models. The possible forms of Yukawa coupling are considered. We find the degeneracy in mass spectrum of fermions coupled to the nontrivial scalar configuration.Comment: 19pages, 39figures, revised versio

Universality in Complex Networks: Random Matrix Analysis

We apply random matrix theory to complex networks. We show that nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks, namely scale-free, small-world and random networks follow universal Gaussian orthogonal ensemble statistics of random matrix theory. Secondly we show an analogy between the onset of small-world behavior, quantified by the structural properties of networks, and the transition from Poisson to Gaussian orthogonal ensemble statistics, quantified by Brody parameter characterizing a spectral property. We also present our analysis for a protein-protein interaction network in budding yeast.Comment: 4+ pages, 4 figures, to appear in PRE, major change in the paper including titl

Note on Branching

It has been demonstrated that the spectrum of the molecular graph contains information about the extent of branching of the molecular skeleton. In particular, the largest eigenvalue, xi, in the spectrum has been shown to be closely related to the total number of walks in the graph (eqs. (11) and (15)). Thus, a justification of the recent empirical finding that x1 is a measure of branching9,in has been obtained