# Browsing by Subject "sovellettu matematiikka"

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• (Helsingin yliopisto, 2009)
The problem of recovering information from measurement data has already been studied for a long time. In the beginning, the methods were mostly empirical, but already towards the end of the sixties Backus and Gilbert started the development of mathematical methods for the interpretation of geophysical data. The problem of recovering information about a physical phenomenon from measurement data is an inverse problem. Throughout this work, the statistical inversion method is used to obtain a solution. Assuming that the measurement vector is a realization of fractional Brownian motion, the goal is to retrieve the amplitude and the Hurst parameter. We prove that under some conditions, the solution of the discretized problem coincides with the solution of the corresponding continuous problem as the number of observations tends to infinity. The measurement data is usually noisy, and we assume the data to be the sum of two vectors: the trend and the noise. Both vectors are supposed to be realizations of fractional Brownian motions, and the goal is to retrieve their parameters using the statistical inversion method. We prove a partial uniqueness of the solution. Moreover, with the support of numerical simulations, we show that in certain cases the solution is reliable and the reconstruction of the trend vector is quite accurate.
• (Helsingin yliopisto, 2012)
Speciation theory is undergoing a renaissance period, largely due to the new methods developed in molecular biology as well as advances in the mathematical theory of evolution. In this thesis, I explore mathematical techniques applicable to the evolution of traits relevant to speciation processes. Some of the theory is further developed and is part of a general framework in the research of evolution. In nature, sister species may coexist in close geographical proximity. However, the question as to whether a speciation event has been a local event driven by the interactions (perhaps complex ones) of individuals that affect their survival and reproduction, has not yet been satisfactorily answered. This is the key issue I address in my thesis. The emphasis is given to the role of non-random mating in an environment where individuals experience diversifying ecological selection. Firstly, I investigate the role of assortative mating, and find that assortative mating works against the speciation process in the initial stages of diversification. However, once the population has diversified, ecological and sexual selection drives the population to a state of reproductive isolation. Secondly, I explore a scheme where individuals choose mates according to the level of adaptation of the mate. I find, that when the level of adaptation to the environment depends on the structure of the population in a frequency-dependent manner, the dynamics of the population may be highly complex and even chaotic. Furthermore, this setting does not facilitate reproductive isolation when mating happens across the habitats. However, if mate choice takes into account the survival and reproduction of the progeny, reproductive isolation can be maintained. Finally, some advances are made in the theory of adaptive dynamics, which along with the theory of population genetics, has been a focal tool in this thesis. My contribution to adaptive dynamics is to resolve an open question on the coexistence of similar strategies near so-called singular points. Singular points play a central role in the theory of adaptive dynamics and their existence is essential to a continuous diversification process.
• (Helsingin yliopisto, 2008)
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.
• (Helsingin yliopisto, 2006)