Analyzing dynamic intellectual capital: System-based theory and application

This research report presents an application of systems theory to evaluating intellectual capital (IC) as organization's ability for self-renewal. As renewal ability is a dynamic capability of an organization as a whole, rather than a static asset or an atomistic competence of separate individuals within the organization, it needs to be understood systemically. Consequently, renewal ability has to be measured with systemic methods that are based on a thorough conceptual analysis of systemic characteristics of organizations. The aim of this report is to demonstrate the theory and analysis methodology for grasping companies' systemic efficiency and renewal ability. The volume is divided into three parts. The first deals with the theory of organizations as self-renewing systems. In the second part, the principles of quantitative analysis of organizations are laid down. Finally, the detailed mathematics of the renewal indices are presented. We also assert that the indices produced by the analysis are an effective tool for the management and valuation of knowledge-intensive companies.

Fuzzy subgroups, algebraic and topological points of view and complex analysis

Fuzzy subsets and fuzzy subgroups are basic concepts in fuzzy mathematics. We shall concentrate on fuzzy subgroups dealing with some of their algebraic, topological and complex analytical properties. Explorations are theoretical belonging to pure mathematics. One of our ideas is to show how widely fuzzy subgroups can be used in mathematics, which brings out the wealth of this concept. In complex analysis we focus on Möbius transformations, combining them with fuzzy subgroups in the algebraic and topological sense. We also survey MV spaces with or without a link to fuzzy subgroups. Spectral space is known in MV algebra. We are interested in its topological properties in MV-semilinear space. Later on, we shall study MV algebras in connection with Riemann surfaces. In fact, the Riemann surface as a concept belongs to complex analysis. On the other hand, Möbius transformations form a part of the theory of Riemann surfaces. In general, this work gives a good understanding how it is possible to fit together different fields of mathematics.

Mathematics inspired by Darwin. Adaptive dynamics of dispersal and cooperation

In 1859, Charles Darwin published his theory of evolution by natural selection, the process occurring based on fitness benefits and fitness costs at the individual level. Traditionally, evolution has been investigated by biologists, but it has induced mathematical approaches, too. For example, adaptive dynamics has proven to be a very applicable framework to the purpose. Its core concept is the invasion fitness, the sign of which tells whether a mutant phenotype can invade the prevalent phenotype. In this thesis, four real-world applications on evolutionary questions are provided. Inspiration for the first two studies arose from a cold-adapted species, American pika. First, it is studied how the global climate change may affect the evolution of dispersal and viability of pika metapopulations. Based on the results gained here, it is shown that the evolution of dispersal can result in extinction and indeed, evolution of dispersalshould be incorporated into the viability analysis of species living in fragmented habitats. The second study is focused on the evolution of densitydependent dispersal in metapopulations with small habitat patches. It resulted a very surprising unintuitive evolutionary phenomenon, how a non-monotone density-dependent dispersal may evolve. Cooperation is surprisingly common in many levels of life, despite of its obvious vulnerability to selfish cheating. This motivated two applications. First, it is shown that density-dependent cooperative investment can evolve to have a qualitatively different, monotone or non-monotone, form depending on modelling details. The last study investigates the evolution of investing into two public-goods resources. The results suggest one general path by which labour division can arise via evolutionary branching. In addition to applications, two novel methodological derivations of fitness measures in structured metapopulations are given.

Spectral Analysis of Symmetric and Anti-Symmetric Pairwise Kernels

This work investigates theoretical properties of symmetric and anti-symmetric kernels. First chapters give an overview of the theory of kernels used in supervised machine learning. Central focus is on the regularized least squares algorithm, which is motivated as a problem of function reconstruction through an abstract inverse problem. Brief review of reproducing kernel Hilbert spaces shows how kernels define an implicit hypothesis space with multiple equivalent characterizations and how this space may be modified by incorporating prior knowledge. Mathematical results of the abstract inverse problem, in particular spectral properties, pseudoinverse and regularization are recollected and then specialized to kernels. Symmetric and anti-symmetric kernels are applied in relation learning problems which incorporate prior knowledge that the relation is symmetric or anti-symmetric, respectively. Theoretical properties of these kernels are proved in a draft this thesis is based on and comprehensively referenced here. These proofs show that these kernels can be guaranteed to learn only symmetric or anti-symmetric relations, and they can learn any relations relative to the original kernel modified to learn only symmetric or anti-symmetric parts. Further results prove spectral properties of these kernels, central result being a simple inequality for the the trace of the estimator, also called the effective dimension. This quantity is used in learning bounds to guarantee smaller variance.