Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

Mathematical models on the impact of noise and dyadic molecular structures on the properties of a cardiac myocyte

In cardiac myocytes (heart muscle cells), coupling of electric signal known as the action potential to contraction of the heart depends crucially on calcium-induced calcium release (CICR) in a microdomain known as the dyad. During CICR, the peak number of free calcium ions (Ca) present in the dyad is small, typically estimated to be within range 1-100. Since the free Ca ions mediate CICR, noise in Ca signaling due to the small number of free calcium ions influences Excitation-Contraction (EC) coupling gain. Noise in Ca signaling is only one noise type influencing cardiac myocytes, e.g., ion channels playing a central role in action potential propagation are stochastic machines, each of which gates more or less randomly, which produces gating noise present in membrane currents. How various noise sources influence macroscopic properties of a myocyte, how noise is attenuated and taken advantage of are largely open questions. In this thesis, the impact of noise on CICR, EC coupling and, more generally, macroscopic properties of a cardiac myocyte is investigated at multiple levels of detail using mathematical models. Complementarily to the investigation of the impact of noise on CICR, computationally-efficient yet spatially-detailed models of CICR are developed. The results of this thesis show that (1) gating noise due to the high-activity mode of L-type calcium channels playing a major role in CICR may induce early after-depolarizations associated with polymorphic tachycardia, which is a frequent precursor to sudden cardiac death in heart failure patients; (2) an increased level of voltage noise typically increases action potential duration and it skews distribution of action potential durations toward long durations in cardiac myocytes; and that (3) while a small number of Ca ions mediate CICR, Excitation-Contraction coupling is robust against this noise source, partly due to the shape of ryanodine receptor protein structures present in the cardiac dyad.

Mathematical Aspects of Financial Markets with Frictions

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property.

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.

Using Mathematical Morphology for Geometric Music Retrieval

The usual task in music information retrieval (MIR) is to find occurrences of a monophonic query pattern within a music database, which can contain both monophonic and polyphonic content. The so-called query-by-humming systems are a famous instance of content-based MIR. In such a system, the user's hummed query is converted into symbolic form to perform search operations in a similarly encoded database. The symbolic representation (e.g., textual, MIDI or vector data) is typically a quantized and simplified version of the sampled audio data, yielding to faster search algorithms and space requirements that can be met in real-life situations. In this thesis, we investigate geometric approaches to MIR. We first study some musicological properties often needed in MIR algorithms, and then give a literature review on traditional (e.g., string-matching-based) MIR algorithms and novel techniques based on geometry. We also introduce some concepts from digital image processing, namely the mathematical morphology, which we will use to develop and implement four algorithms for geometric music retrieval. The symbolic representation in the case of our algorithms is a binary 2-D image. We use various morphological pre- and post-processing operations on the query and the database images to perform template matching / pattern recognition for the images. The algorithms are basically extensions to classic image correlation and hit-or-miss transformation techniques used widely in template matching applications. They aim to be a future extension to the retrieval engine of C-BRAHMS, which is a research project of the Department of Computer Science at University of Helsinki.

ORP1L, a new Rab7 effector and regulator of late endosome functions

Oxysterol binding protein (OSBP) homologues have been found in eukaryotic organisms ranging from yeast to humans. These evolutionary conserved proteins have in common the presence of an OSBP-related domain (ORD) which contains the fully conserved EQVSHHPP sequence motif. The ORD forms a barrel structure that binds sterols in its interior. Other domains and sequence elements found in OSBP-homologues include pleckstrin homology domains, ankyrin repeats and two phenylalanines in an acidic tract (FFAT) motifs, which target the proteins to distinct subcellular compartments. OSBP homologues have been implicated in a wide range of intracellular processes, including vesicle trafficking, lipid metabolism and cell signaling, but little is known about the functional mechanisms of these proteins. The human family of OSBP homologues consists of twelve OSBP-related proteins (ORP). This thesis work is focused on one of the family members, ORP1, of which two variants were found to be expressed tissue-specifically in humans. The shorter variant, ORP1S contains an ORD only. The N-terminally extended variant, ORP1L, comprises a pleckstrin homology domain and three ankyrin repeats in addition to the ORD. The two ORP1 variants differ in intracellular localization. ORP1S is cytosolic, while the ankyrin repeat region of ORP1L targets the protein to late endosomes/lysosomes. This part of ORP1L also has profound effects on late endosomal morphology, inducing perinuclear clustering of late endosomes. A central aim of this study was to identify molecular interactions of ORP1L on late endosomes. The morphological changes of late endosomes induced by overexpressed ORP1L implies involvement of small Rab GTPases, regulators of organelle motility, tethering, docking and/or fusion, in generation of the phenotype. A direct interaction was demonstrated between ORP1L and active Rab7. ORP1L prolongs the active state of Rab7 by stabilizing its GTP-bound form. The clustering of late endosomes/lysosomes was also shown to be linked to the minus end-directed microtubule-based dynein-dynactin motor complex through the ankyrin repeat region of ORP1L. ORP1L, Rab7 and the Rab7-interacting lysosomal protein (RILP) were found to be part of the same effector complex recruiting the dynein-dynactin complex to late endosomes, thereby promoting minus end-directed movement. The proteins were found to be physically close to each other on late endosomes and RILP was found to stabilize the ORP1L-Rab7 interaction. It is possible that ORP1L and RILP bind to each other through their C-terminal and N-terminal regions, respectively, when they are bridged by Rab7. With the results of this study we have been able to place a member of the uncharacterized OSBP-family, ORP1L, in the endocytic pathway, where it regulates motility and possibly fusion of late endosomes through interaction with the small GTPase Rab7.

The biological functions of mouse twinfilin isoforms

The actin cytoskeleton is essential for many cellular processes, including motility, morphogenesis, endocytosis and signal transduction. Actin can exist in monomeric (G-actin) or filamentous (F-actin) form. Actin filaments are considered to be the functional form of actin, generating the protrusive forces characteristic for the actin cytoskeleton. The structure and dynamics of the actin filament and monomer pools are regulated by a large number of actin-binding proteins in eukaryotic cells. Twinfilin is an evolutionarily conserved small actin monomer binding protein. Twinfilin is composed of two ADF/cofilin-like domains, separated by a short linker and followed by a C-terminal tail. Twinfilin forms a stable, high affinity complex with ADP-G-actin, inhibits the nucleotide exchange on actin monomers, and prevents their assembly into filament ends. Twinfilin was originally identified from yeast and has since then been found from all organisms studied except plants. Not much was known about the role of twinfilin in the actin dynamics in mammalian cells before this study. We set out to unravel the mysteries still covering twinfilins functions using biochemistry, cell biology, and genetics. We identified and characterized two mouse isoforms for the previously identified mouse twinfilin-1. The new isoforms, twinfilin-2a and -2b, are generated from the same gene through alternative promoter usage. The three isoforms have distinctive expression patterns, but are similar biochemically. Twinfilin-1 is the major isoform during development and is expressed in high levels in almost all tissues examined. Twinfilin-2a is also expressed almost ubiquitously, but at lower levels. Twinfilin-2b turned out to be a muscle-specific isoform, with very high expression in heart and skeletal muscle. It seems all mouse tissues express at least two twinfilin isoforms, indicating that twinfilins are important regulators of actin dynamics in all cell and tissue types. A knockout mouse line was generated for twinfilin-2a. The mice homozygous for this knockout were viable and developed normally, indicating that twinfilin-2a is dispensable for mouse development. However, it is important to note that twinfilin-2a shows similar expression pattern to twinfilin-1, suggesting that these proteins play redundant roles in mice. All mouse isoforms were shown to be able to sequester actin filaments and have higher affinity for ADP-G-actin than ATP-G-actin. They are also able to directly interact with heterodimeric capping protein and PI(4,5)P2 similar to yeast twinfilin. In this study we also uncovered a novel function for mouse twinfilins; capping actin filament barbed ends. All mouse twinfilin isoforms were shown to possess this function, while yeast and Drosophila twinfilin were not able to cap filament barbed ends. Twinfilins localize to the cytoplasm but also to actin-rich regions in mammalian cells. The subcellular localizations of the isoforms are regulated differently, indicating that even though twinfilins biochemical functions in vitro are very similar, in vivo they can play different roles through different regulatory pathways. Together, this study show that twinfilins regulate actin filament assembly both by sequestering actin monomers and by capping filament barbed ends, and that mammals have three biochemically similar twinfilin isoforms with partially overlapping expression patterns.

Neuromagnetic studies on cortical somatosensory functions in infants and children : Normal development and effect of early brain lesions

Until recently, objective investigation of the functional development of the human brain in vivo was challenged by the lack of noninvasive research methods. Consequently, fairly little is known about cortical processing of sensory information even in healthy infants and children. Furthermore, mechanisms by which early brain insults affect brain development and function are poorly understood. In this thesis, we used magnetoencephalography (MEG) to investigate development of cortical somatosensory functions in healthy infants, very premature infants at risk for neurological disorders, and adolescents with hemiplegic cerebral palsy (CP). In newborns, stimulation of the hand activated both the contralateral primary (SIc) and secondary somatosensory cortices (SIIc). The activation patterns differed from those of adults, however. Some of the earliest SIc responses, constantly present in adults, were completely lacking in newborns and the effect of sleep stage on SIIc responses differed. These discrepancies between newborns and adults reflect the still developmental stage of the newborns’ somatosensory system. Its further maturation was demonstrated by a systematic transformation of the SIc response pattern with age. The main early adult­like components were present by age two. In very preterm infants, at term age, the SIc and SIIc were activated at similar latencies as in healthy fullterm newborns, but the SIc activity was weaker in the preterm group. The SIIc response was absent in four out of the six infants with brain lesions of the underlying hemisphere. Determining the prognostic value of this finding remains a subject for future studies, however. In the CP adolescents with pure subcortical lesions, contrasting their unilateral symptoms, the SIc responses of both hemispheres differed from those of controls: For example the distance between SIc representation areas for digits II and V was shorter bilaterally. In four of the five CP patients with cortico­subcortical brain lesions, no normal early SIc responses were evoked by stimulation of the palsied hand. The varying differences in neuronal functions, underlying the common clinical symptoms, call for investigation of more precisely designed rehabilitation strategies resting on knowledge about individual functional alterations in the sensorimotor networks.

Characterization of LRRTM and NGR gene families: Expression and functions

Some leucine-rich repeat (LRR) -containing membrane proteins are known regulators of neuronal growth and synapse formation. In this work I characterize two gene families encoding neuronal LRR membrane proteins, namely the LRRTM (leucine-rich repeat, transmembrane neuronal) and NGR (Nogo-66 receptor) families. I studied LRRTM and NGR family member's mRNA tissue distribution by RT-PCR and by in situ hybridization. Subcellular localization of LRRTM1 protein was studied in neurons and in non-neuronal cells. I discovered that LRRTM and NGR family mRNAs are predominantly expressed in the nervous system, and that each gene possesses a specific expression pattern. I also established that LRRTM and NGR family mRNAs are expressed by neurons, and not by glial cells. Within neurons, LRRTM1 protein is not transported to the plasma membrane; rather it localizes to endoplasmic reticulum. Nogo-A (RTN4), MAG, and OMgp are myelin-associated proteins that bind to NgR1 to limit axonal regeneration after central nervous system injury. To better understand the functions of NgR2 and NgR3, and to explore the possible redundancy in the signaling of myelin inhibitors of neurite growth, I mapped the interactions between NgR family and the known and candidate NgR1 ligands. I identified high-affinity interactions between RTN2-66, RTN3-66 and NgR1. I also demonstrate that Rtn3 mRNA is expressed in the same glial cell population of mouse spinal cord white matter as Nogo-A mRNA, and thus it could have a role in myelin inhibition of axonal growth. To understand how NgR1 interacts with multiple structurally divergent ligands, I aimed first to map in more detail the nature of Nogo-A:NgR1 interactions, and then to systematically map the binding sites of multiple myelin ligands in NgR1 by using a library of NgR1 expression constructs encoding proteins with one or multiple surface residues mutated to alanine. My analysis of the Nogo-A:NgR1 -interactions revealed a novel interaction site between the proteins, suggesting a trivalent Nogo-A:NgR1-interaction. Our analysis also defined a central binding region on the concave side of NgR1's LRR domain that is required for the binding of all known ligands, and a surrounding region critical for binding MAG and OMgp. To better understand the biological role of LRRTMs, I generated Lrrtm1 and Lrrtm3 knock out mice. I show here that reporter genes expressed from the targeted loci can be used for maping the neuronal connections of Lrrtm1 and Lrrtm3 expressing neurons in finer detail. With regard to LRRTM1's role in humans, we found a strong association between a 70 kb-spanning haplotype in the proposed promoter region of LRRTM1 gene and two possibly related phenotypes: left-handedness and schizophrenia. Interestingly, the responsible haplotype was linked to phenotypic variability only when paternally inherited. In summary, I identified two families of neuronal receptor-like proteins, and mapped their expression and certain protein-protein interactions. The identification of a central binding region in NgR1 shared by multiple ligands may facilitate the design and development of small molecule therapeutics blocking binding of all NgR1 ligands. Additionally, the genetic association data suggests that allelic variation upstream of LRRTM1 may play a role in the development of left-right brain asymmetry in humans. Lrrtm1 and Lrrtm3 knock out mice developed as a part of this study will likely be useful for schizophrenia and Alzheimer s disease research.