Boundedness and convergence of singular integrals on fractal type sets

The topic of this dissertation lies in the intersection of harmonic analysis and fractal geometry. We particulary consider singular integrals in Euclidean spaces with respect to general measures, and we study how the geometric structure of the measures affects certain analytic properties of the operators. The thesis consists of three research articles and an overview. In the first article we construct singular integral operators on lower dimensional Sierpinski gaskets associated with homogeneous Calderón-Zygmund kernels. While these operators are bounded their principal values fail to exist almost everywhere. Conformal iterated function systems generate a broad range of fractal sets. In the second article we prove that many of these limit sets are porous in a very strong sense, by showing that they contain holes spread in every direction. In the following we connect these results with singular integrals. We exploit the fractal structure of these limit sets, in order to establish that singular integrals associated with very general kernels converge weakly. Boundedness questions consist a central topic of investigation in the theory of singular integrals. In the third article we study singular integrals of different measures. We prove a very general boundedness result in the case where the two underlying measures are separated by a Lipshitz graph. As a consequence we show that a certain weak convergence holds for a large class of singular integrals.

On Games on Non-Wellfounded Sets and Stationary Sets

In this thesis we study a few games related to non-wellfounded and stationary sets. Games have turned out to be an important tool in mathematical logic ranging from semantic games defining the truth of a sentence in a given logic to for example games on real numbers whose determinacies have important effects on the consistency of certain large cardinal assumptions. The equality of non-wellfounded sets can be determined by a so called bisimulation game already used to identify processes in theoretical computer science and possible world models for modal logic. Here we present a game to classify non-wellfounded sets according to their branching structure. We also study games on stationary sets moving back to classical wellfounded set theory. We also describe a way to approximate non-wellfounded sets with hereditarily finite wellfounded sets. The framework used to do this is domain theory. In the Banach-Mazur game, also called the ideal game, the players play a descending sequence of stationary sets and the second player tries to keep their intersection stationary. The game is connected to precipitousness of the corresponding ideal. In the pressing down game first player plays regressive functions defined on stationary sets and the second player responds with a stationary set where the function is constant trying to keep the intersection stationary. This game has applications in model theory to the determinacy of the Ehrenfeucht-Fraisse game. We show that it is consistent that these games are not equivalent.

Data mining for telecommunications network log analysis

Telecommunications network management is based on huge amounts of data that are continuously collected from elements and devices from all around the network. The data is monitored and analysed to provide information for decision making in all operation functions. Knowledge discovery and data mining methods can support fast-pace decision making in network operations. In this thesis, I analyse decision making on different levels of network operations. I identify the requirements decision-making sets for knowledge discovery and data mining tools and methods, and I study resources that are available to them. I then propose two methods for augmenting and applying frequent sets to support everyday decision making. The proposed methods are Comprehensive Log Compression for log data summarisation and Queryable Log Compression for semantic compression of log data. Finally I suggest a model for a continuous knowledge discovery process and outline how it can be implemented and integrated to the existing network operations infrastructure.

Mechanisms for alphavirus nonstructural polyprotein processing

Replication and transcription of the RNA genome of alphaviruses relies on a set of virus-encoded nonstructural proteins. They are synthesized as a long polyprotein precursor, P1234, which is cleaved at three processing sites to yield nonstructural proteins nsP1, nsP2, nsP3 and nsP4. All the four proteins function as constitutive components of the membrane-associated viral replicase. Proteolytic processing of P1234 polyprotein is precisely orchestrated and coordinates the replicase assembly and maturation. The specificity of the replicase is also controlled by proteolytic cleavages. The early replicase is composed of P123 polyprotein intermediate and nsP4. It copies the positive sense RNA genome to complementary minus-strand. Production of new plus-strands requires complete processing of the replicase. The papain-like protease residing in nsP2 is responsible for all three cleavages in P1234. This study addressed the mechanisms of proteolytic processing of the replicase polyprotein in two alphaviruses Semliki Forest virus (SFV) and Sindbis virus (SIN) representing different branches of the genus. The survey highlighted the functional relation of the alphavirus nsP2 protease to the papain-like enzymes. A new structural motif the Cys-His catalytic dyad accompanied with an aromatic residue following the catalytic His was described for nsP2 and a subset of other thiol proteases. Such an architecture of the catalytic center was named the glycine specificity motif since it was implicated in recognition of a specific Gly residue in the substrate. In particular, the presence of the motif in nsP2 makes the appearance of this amino acid at the second position upstream of the scissile bond a necessary condition for the cleavage. On top of that, there were four distinct mechanisms identified, which provide affinity for the protease and specifically direct the enzyme to different sites in the P1234 polyprotein. Three factors RNA, the central domain of nsP3 and the N-terminus of nsP2 were demonstrated to be external modulators of the nsP2 protease. Here I suggest that the basal nsP2 protease specificity is inherited from the ancestral papain-like enzyme and employs the recognition of the upstream amino acid signature in the immediate vicinity of the scissile bond. This mechanism is responsible for the efficient processing of the SFV nsP3/nsP4 junction. I propose that the same mechanism is involved in the cleavage of the nsP1/nsP2 junction of both viruses as well. However, in this case it rather serves to position the substrate, whereas the efficiency of the processing is ensured by the capability of nsP2 to cut its own N-terminus in cis. Both types of cleavages are demonstrated here to be inhibited by RNA, which is interpreted as impairing the basal papain-like recognition of the substrate. In contrast, processing of the SIN nsP3/nsP4 junction was found to be activated by RNA and additionally potentiated by the presence of the central region of nsP3 in the protease. The processing of the nsP2/nsP3 junction in both viruses occurred via another mechanism, requiring the exactly processed N-terminus of nsP2 in the protease and insensitive to RNA addition. Therefore, the three processing events in the replicase polyprotein maturation are performed via three distinct mechanisms in each of two studied alphaviruses. Distinct sets of conditions required for each cleavage ensure sequential maturation of P1234 polyprotein: nsP4 is released first, then the nsP1/nsP2 site is cut in cis, and liberation of the nsP2 N-terminus activates the cleavage of the nsP2/nsP3 junction at last. The first processing event occurs differently in SFV and SIN, whereas the subsequent cleavages are found to be similar in the two viruses and therefore, their mechanisms are suggested to be conserved in the genus. The RNA modulation of the alphavirus nonstructural protease activity, discovered here, implies bidirectional functional interplay between the alphavirus RNA metabolism and protease regulation. The nsP2 protease emerges as a signal transmitting moiety, which senses the replication stage and responds with proteolytic cleavages. A detailed hypothetical model of the alphavirus replicase core was inferred from the data obtained in the study. Similar principles in replicase organization and protease functioning are expected to be employed by other RNA viruses.

Playing Games on Sets and Models

The most prominent objective of the thesis is the development of the generalized descriptive set theory, as we call it. There, we study the space of all functions from a fixed uncountable cardinal to itself, or to a finite set of size two. These correspond to generalized notions of the universal Baire space (functions from natural numbers to themselves with the product topology) and the Cantor space (functions from natural numbers to the {0,1}-set) respectively. We generalize the notion of Borel sets in three different ways and study the corresponding Borel structures with the aims of generalizing classical theorems of descriptive set theory or providing counter examples. In particular we are interested in equivalence relations on these spaces and their Borel reducibility to each other. The last chapter shows, using game-theoretic techniques, that the order of Borel equivalence relations under Borel reduciblity has very high complexity. The techniques in the above described set theoretical side of the thesis include forcing, general topological notions such as meager sets and combinatorial games of infinite length. By coding uncountable models to functions, we are able to apply the understanding of the generalized descriptive set theory to the model theory of uncountable models. The links between the theorems of model theory (including Shelah's classification theory) and the theorems in pure set theory are provided using game theoretic techniques from Ehrenfeucht-Fraïssé games in model theory to cub-games in set theory. The bottom line of the research declairs that the descriptive (set theoretic) complexity of an isomorphism relation of a first-order definable model class goes in synch with the stability theoretical complexity of the corresponding first-order theory. The first chapter of the thesis has slightly different focus and is purely concerned with a certain modification of the well known Ehrenfeucht-Fraïssé games. There we (me and my supervisor Tapani Hyttinen) answer some natural questions about that game mainly concerning determinacy and its relation to the standard EF-game

Distributed algorithms for edge dominating sets

An edge dominating set for a graph G is a set D of edges such that each edge of G is in D or adjacent to at least one edge in D. This work studies deterministic distributed approximation algorithms for finding minimum-size edge dominating sets. The focus is on anonymous port-numbered networks: there are no unique identifiers, but a node of degree d can refer to its neighbours by integers 1, 2, ..., d. The present work shows that in the port-numbering model, edge dominating sets can be approximated as follows: in d-regular graphs, to within 4 − 6/(d + 1) for an odd d and to within 4 − 2/d for an even d; and in graphs with maximum degree Δ, to within 4 − 2/(Δ − 1) for an odd Δ and to within 4 − 2/Δ for an even Δ. These approximation ratios are tight for all values of d and Δ: there are matching lower bounds.