Noncommutative Quantum Field Theories and UV/IR Mixing

Our present-day understanding of fundamental constituents of matter and their interactions is based on the Standard Model of particle physics, which relies on quantum gauge field theories. On the other hand, the large scale dynamical behaviour of spacetime is understood via the general theory of relativity of Einstein. The merging of these two complementary aspects of nature, quantum and gravity, is one of the greatest goals of modern fundamental physics, the achievement of which would help us understand the short-distance structure of spacetime, thus shedding light on the events in the singular states of general relativity, such as black holes and the Big Bang, where our current models of nature break down. The formulation of quantum field theories in noncommutative spacetime is an attempt to realize the idea of nonlocality at short distances, which our present understanding of these different aspects of Nature suggests, and consequently to find testable hints of the underlying quantum behaviour of spacetime. The formulation of noncommutative theories encounters various unprecedented problems, which derive from their peculiar inherent nonlocality. Arguably the most serious of these is the so-called UV/IR mixing, which makes the derivation of observable predictions especially hard by causing new tedious divergencies, to which our previous well-developed renormalization methods for quantum field theories do not apply. In the thesis I review the basic mathematical concepts of noncommutative spacetime, different formulations of quantum field theories in the context, and the theoretical understanding of UV/IR mixing. In particular, I put forward new results to be published, which show that also the theory of quantum electrodynamics in noncommutative spacetime defined via Seiberg-Witten map suffers from UV/IR mixing. Finally, I review some of the most promising ways to overcome the problem. The final solution remains a challenge for the future.

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.

Truth, Proof and Gödelian Arguments : A Defence of Tarskian Truth in Mathematics

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion.

Society by Numbers : Studies on Model-Based Explanations in the Social Sciences

The aim of this dissertation is to provide conceptual tools for the social scientist for clarifying, evaluating and comparing explanations of social phenomena based on formal mathematical models. The focus is on relatively simple theoretical models and simulations, not statistical models. These studies apply a theory of explanation according to which explanation is about tracing objective relations of dependence, knowledge of which enables answers to contrastive why and how-questions. This theory is developed further by delineating criteria for evaluating competing explanations and by applying the theory to social scientific modelling practices and to the key concepts of equilibrium and mechanism. The dissertation is comprised of an introductory essay and six published original research articles. The main theses about model-based explanations in the social sciences argued for in the articles are the following. 1) The concept of explanatory power, often used to argue for the superiority of one explanation over another, compasses five dimensions which are partially independent and involve some systematic trade-offs. 2) All equilibrium explanations do not causally explain the obtaining of the end equilibrium state with the multiple possible initial states. Instead, they often constitutively explain the macro property of the system with the micro properties of the parts (together with their organization). 3) There is an important ambivalence in the concept mechanism used in many model-based explanations and this difference corresponds to a difference between two alternative research heuristics. 4) Whether unrealistic assumptions in a model (such as a rational choice model) are detrimental to an explanation provided by the model depends on whether the representation of the explanatory dependency in the model is itself dependent on the particular unrealistic assumptions. Thus evaluating whether a literally false assumption in a model is problematic requires specifying exactly what is supposed to be explained and by what. 5) The question of whether an explanatory relationship depends on particular false assumptions can be explored with the process of derivational robustness analysis and the importance of robustness analysis accounts for some of the puzzling features of the tradition of model-building in economics. 6) The fact that economists have been relatively reluctant to use true agent-based simulations to formulate explanations can partially be explained by the specific ideal of scientific understanding implicit in the practise of orthodox economics.

Concepts and Object-Oriented Knowledge Representation

Tutkielma käsittelee nykyisiä kognitiotieteen teorioita käsitteistä ja niiden mallintamista oliokeskeisillä tietämyksen esittämisen menetelmillä. Käsiteteorioista käsitellään klassinen, määritelmäteoria, prototyyppiteoria, duaaliteoriat, uusklassinen teoria, teoria-teoria ja atomistinen teoria. Oliokeskeiset menetelmät ovat viime aikoina jakautuneet kahden tyyppisiin kieliin: oliopohjaisiin ja luokkapohjaisiin. Uudet olio-pohjaiset olio-ohjelmointikielet antavat käsitteiden representointiin mahdollisuuksia, jotka puuttuvat aikaisemmista luokka-pohjaisista kielistä ja myös kehysmenetelmistä. Tutkielma osoittaa, että oliopohjaisten kielten uudet piirteet tarjoavat keinoja, joilla käsitteitä voidaan esittää symbolisessa muodossa paremmin kuin perinteisillä menetelmillä. Niillä pystytään simuloimaan kaikkea mitä luokkapohjaisilla kielillä voidaan, mutta ne pystyvät lisäksi simuloimaan perheyhtäläisyyskäsitteitä ja mahdollistavat olioiden dynaamisen muuttamisen ilman, että siinä rikotaan psykologisen essentialismin periaatetta. Tutkielma osoittaa lisäksi vakavia puutteitta, jotka koskevat koko oliokeskeistä menetelmää. Avainsanat: käsitteet, käsiteteoriat, tekoäly, komputationaalinen psykologia, olio-ohjelmointi, tiedon esittäminen

Aritmetiikasta algebraan : Muutoksia osaamisessa peruskoulun päättöluokalla 20 vuoden aikana

From Arithmetic to Algebra. Changes in the skills in comprehensive school over 20 years. In recent decades we have emphasized the understanding of calculation in mathematics teaching. Many studies have found that better understanding helps to apply skills in new conditions and that the ability to think on an abstract level increases the transfer to new contexts. In my research I take into consideration competence as a matrix where content is in a horizontal line and levels of thinking are in a vertical line. The know-how is intellectual and strategic flexibility and understanding. The resources and limitations of memory have their effects on learning in different ways in different phases. Therefore both flexible conceptual thinking and automatization must be considered in learning. The research questions that I examine are what kind of changes have occurred in mathematical skills in comprehensive school over the last 20 years and what kind of conceptual thinking is demonstrated by students in this decade. The study consists of two parts. The first part is a statistical analysis of the mathematical skills and their changes over the last 20 years in comprehensive school. In the test the pupils did not use calculators. The second part is a qualitative analysis of the conceptual thinking of pupils in comprehensive school in this decade. The study shows significant differences in algebra and in some parts of arithmetic. The largest differences were detected in the calculation skills of fractions. In the 1980s two out of three pupils were able to complete tasks with fractions, but in the 2000s only one out of three pupils were able to do the same tasks. Also remarkable is that out of the students who could complete the tasks with fractions, only one out of three pupils was on the conceptual level in his/her thinking. This means that about 10% of pupils are able to understand the algebraic expression, which has the same isomorphic structure as the arithmetical expression. This finding is important because the ability to think innovatively is created when learning the basic concepts. Keywords: arithmetic, algebra, competence

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

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.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

Knowledge through Discussion : An Interpretation of Epistemology in Plato's Meno

According to Meno s paradox we cannot inquire into what we do not know because we do not know what we are inquiring into. There are many ways to interpret the paradox but the central issue about our ability to reach truth is a profound one. In the dialogue Meno, Plato presents the paradox and an outline of a solution which enables us to reach knowledge (epistēmē) through philosophical discussion. During the last century Meno has often been considered transitional between Socratic thinking and Plato s own philosophy, and thus the dialogue has not been adequately interpreted as an integrated whole. Therefore the distinctive epistemology of the dialogue has not gained due notice. In this thesis the dialogue is analysed as an integrated whole and the philosophical interpretation also takes into account its dramatic features. The thesis emphasises the role of language and definitions in acquiring knowledge. Among the results concerning these subjects is a new interpretation of Socrates s defintion of shape (schēma). The theory of anamnēsis all learning is recollection in the Meno is argued to answer the paradox philosophically although Plato s presentation also contains playful and ironic elements. The background of the way Plato presents his case is that he appreciated the fact that no argument can plausibly demonstrate that argumentation is able to reach truth. In the Meno, Plato makes the earliest explicit distinction between knowledge and true belief in the history of Western philosophy. He also gives a definition of knowledge which is the basis of the so called classical definition of knowledge as justified true belief. In the Meno, true beliefs become knowledge when someone ties them down by reasoning about the explanation. The analysis of the epistemology of the dialogue from this perspective gives an interpretation which integrates the central concepts of the epistemology in the dialogue elenchos, anamnēsis and hypothetical inquiry into a unified whole which contains a plausible argument according to which the ignorant can reach knowledge through discussion. The conception that emerges by such an analysis is interesting both from the point of view of current interests and that of the history of philosophy. The method of knowledge acquisition in the Meno can, for example, be seen as a predecessor of modern scientific methods. The Meno is the earliest Greek mathematical text that has survived in its original form. The analysis presented in the thesis of the geometric passages in the dialogue provides new results both concerning Socrates s geometry lesson with the slave and the example presenting the hypothetical method. Concerning the latter, a new interpretation is presented. Keywords: anamnēsis, epistēmē, knowledge, Meno s paradox, Plato

Network Knowledge versus Cluster Knowledge- The Gordian Knot of Knowledge Transfer Concepts

Both management scholars and economic geographers have studied knowledge and argued that the ability to transfer knowledge is critical to competitive success. Networks and other forms for cooperation are often the context when analyzing knowledge transfer within management research, while economic geographers focus on the role of the cluster for knowledge transfer and creation. With the common interest in knowledge transfer, few attempts to interdisciplinary research have been made. The aim of this paper is to outline the knowledge transfer concepts in the two strands of literature of management and economic geography (EG). The paper takes an analytical approach to review the existing contributions and seek to identify the benefits of further interaction between the disciplines. Furthermore, it offers an interpretation of the concepts of cluster and network, and suggests a clearer distinction between their respective definitions. The paper posits that studies of internal networks transcending national borders and clusters are not necessarily mutually exclusive when it comes to transfer of knowledge and the learning process of the firm. Our conclusion is that researchers in general seem to increasingly acknowledge the importance of studying both the effect of and the need for geographical proximity and external networks for the knowledge transfer process, but that there exists equivocalness in defining clusters and networks.

Networks, Organisations and Men: Concepts and Interrelations

Research on men’s networks and homosociality in and around organisations can produce knowledge on organisational power relations, and contribute to the efforts to promote equality in working life. The search for a conceptual framework to study these issues arises in this paper from my ongoing work on men's social networks and gendered power in and around organisations. Men give each other social support through networks in which formal and informal relationships intermingle, but networks are also contexts of competition and oppression, and of construction of masculinities that are in hierarchical relations with each other and with femininities. For studying the networks men have with each other in work organisations I suggest a broader starting point that contextualises these homosocial networks with men’s other personal relations, and integrates different perspectives deriving from social network analysis, critical studies on men and organisational studies.