Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Integral Transformations of Volterra Gaussian Processes

We study integral representations of Gaussian processes with a pre-specified law in terms of other Gaussian processes. The dissertation consists of an introduction and of four research articles. In the introduction, we provide an overview about Volterra Gaussian processes in general, and fractional Brownian motion in particular. In the first article, we derive a finite interval integral transformation, which changes fractional Brownian motion with a given Hurst index into fractional Brownian motion with an other Hurst index. Based on this transformation, we construct a prelimit which formally converges to an analogous, infinite interval integral transformation. In the second article, we prove this convergence rigorously and show that the infinite interval transformation is a direct consequence of the finite interval transformation. In the third article, we consider general Volterra Gaussian processes. We derive measure-preserving transformations of these processes and their inherently related bridges. Also, as a related result, we obtain a Fourier-Laguerre series expansion for the first Wiener chaos of a Gaussian martingale. In the fourth article, we derive a class of ergodic transformations of self-similar Volterra Gaussian processes.

Beasts on Fields. Human-Wildlife Conflicts in Nature-Culture Borderlands

Human-wildlife conflicts are today an integral part of the rural development discourse. In this research, the main focus is on the spatial explanation which is not a very common approach in the reviewed literature. My research hypothesis is based on the assumption that human-wildlife conflicts occur when a wild animal crosses a perceived borderline between the nature and culture and enters into the realms of the other. The borderline between nature and culture marks a perceived division of spatial content in our senses of place. The animal subject that crosses this border becomes a subject out of place meaning that the animal is then spatially located in a space where it should not be or where it does not belong according to tradition, custom, rules, law, public opinion, prevailing discourse or some other criteria set by human beings. An appearance of a wild animal in a domesticated space brings an uncontrolled subject into that space where humans have previously commanded total control of all other natural elements. A wild animal out of place may also threaten the biosecurity of the place in question. I carried out a case study in the Liwale district in south-eastern Tanzania to test my hypothesis during June and July 2002. I also collected documents and carried out interviews in Dar es Salaam in 2003. I studied the human-wildlife conflicts in six rural villages, where a total of 183 persons participated in the village meetings. My research methods included semi-structured interviews, participatory mapping, questionnaire survey and Q- methodology. The rural communities in the Liwale district have a long-history of co-existing with wildlife and they still have traditional knowledge of wildlife management and hunting. Wildlife conservation through the establishment of game reserves during the colonial era has escalated human-wildlife conflicts in the Liwale district. This study shows that the villagers perceive some wild animals differently in their images of the African countryside than the district and regional level civil servants do. From the small scale subsistence farmers point of views, wild animals continue to challenge the separation of the wild (the forests) and the domestics spaces (the cultivated fields) by moving across the perceived borders in search of food and shelter. As a result, the farmers may loose their crops, livestock or even their own lives in the confrontations of wild animals. Human-wildlife conflicts in the Liwale district are manifold and cannot be explained simply on the basis of attitudes or perceived images of landscapes. However, the spatial explanation of these conflicts provides us some more understanding of why human-wildlife conflicts are so widely found across the world.

Inverse Problem for Fractional Brownian Motion with Discrete Data

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.

Homogeneous model theory of metric structures

This thesis studies homogeneous classes of complete metric spaces. Over the past few decades model theory has been extended to cover a variety of nonelementary frameworks. Shelah introduced the abstact elementary classes (AEC) in the 1980s as a common framework for the study of nonelementary classes. Another direction of extension has been the development of model theory for metric structures. This thesis takes a step in the direction of combining these two by introducing an AEC-like setting for studying metric structures. To find balance between generality and the possibility to develop stability theoretic tools, we work in a homogeneous context, thus extending the usual compact approach. The homogeneous context enables the application of stability theoretic tools developed in discrete homogeneous model theory. Using these we prove categoricity transfer theorems for homogeneous metric structures with respect to isometric isomorphisms. We also show how generalized isomorphisms can be added to the class, giving a model theoretic approach to, e.g., Banach space isomorphisms or operator approximations. The novelty is the built-in treatment of these generalized isomorphisms making, e.g., stability up to perturbation the natural stability notion. With respect to these generalized isomorphisms we develop a notion of independence. It behaves well already for structures which are omega-stable up to perturbation and coincides with the one from classical homogeneous model theory over saturated enough models. We also introduce a notion of isolation and prove dominance for it.

On the Rademacher maximal function

Tools known as maximal functions are frequently used in harmonic analysis when studying local behaviour of functions. Typically they measure the suprema of local averages of non-negative functions. It is essential that the size (more precisely, the L^p-norm) of the maximal function is comparable to the size of the original function. When dealing with families of operators between Banach spaces we are often forced to replace the uniform bound with the larger R-bound. Hence such a replacement is also needed in the maximal function for functions taking values in spaces of operators. More specifically, the suprema of norms of local averages (i.e. their uniform bound in the operator norm) has to be replaced by their R-bound. This procedure gives us the Rademacher maximal function, which was introduced by Hytönen, McIntosh and Portal in order to prove a certain vector-valued Carleson's embedding theorem. They noticed that the sizes of an operator-valued function and its Rademacher maximal function are comparable for many common range spaces, but not for all. Certain requirements on the type and cotype of the spaces involved are necessary for this comparability, henceforth referred to as the “RMF-property”. It was shown, that other objects and parameters appearing in the definition, such as the domain of functions and the exponent p of the norm, make no difference to this. After a short introduction to randomized norms and geometry in Banach spaces we study the Rademacher maximal function on Euclidean spaces. The requirements on the type and cotype are considered, providing examples of spaces without RMF. L^p-spaces are shown to have RMF not only for p greater or equal to 2 (when it is trivial) but also for 1 < p < 2. A dyadic version of Carleson's embedding theorem is proven for scalar- and operator-valued functions. As the analysis with dyadic cubes can be generalized to filtrations on sigma-finite measure spaces, we consider the Rademacher maximal function in this case as well. It turns out that the RMF-property is independent of the filtration and the underlying measure space and that it is enough to consider very simple ones known as Haar filtrations. Scalar- and operator-valued analogues of Carleson's embedding theorem are also provided. With the RMF-property proven independent of the underlying measure space, we can use probabilistic notions and formulate it for martingales. Following a similar result for UMD-spaces, a weak type inequality is shown to be (necessary and) sufficient for the RMF-property. The RMF-property is also studied using concave functions giving yet another proof of its independence from various parameters.

IP Mobility in Wireless Operator Networks

Wireless network access is gaining increased heterogeneity in terms of the types of IP capable access technologies. The access network heterogeneity is an outcome of incremental and evolutionary approach of building new infrastructure. The recent success of multi-radio terminals drives both building a new infrastructure and implicit deployment of heterogeneous access networks. Typically there is no economical reason to replace the existing infrastructure when building a new one. The gradual migration phase usually takes several years. IP-based mobility across different access networks may involve both horizontal and vertical handovers. Depending on the networking environment, the mobile terminal may be attached to the network through multiple access technologies. Consequently, the terminal may send and receive packets through multiple networks simultaneously. This dissertation addresses the introduction of IP Mobility paradigm into the existing mobile operator network infrastructure that have not originally been designed for multi-access and IP Mobility. We propose a model for the future wireless networking and roaming architecture that does not require revolutionary technology changes and can be deployed without unnecessary complexity. The model proposes a clear separation of operator roles: (i) access operator, (ii) service operator, and (iii) inter-connection and roaming provider. The separation allows each type of an operator to have their own development path and business models without artificial bindings with each other. We also propose minimum requirements for the new model. We present the state of the art of IP Mobility. We also present results of standardization efforts in IP-based wireless architectures. Finally, we present experimentation results of IP-level mobility in various wireless operator deployments.

Q-evankeliumi ja psykohistoria : Uuden testamentin eksegetiikan väitöskirja

In the course of my research for my thesis The Q Gospel and Psychohistory, I moved on from the accounts of the Cynics ideals to psychohistorical explanations. Studying the texts dealing with the Cynics and the Q Gospel, I was amazed by the fact that these texts actually portrayed people living in greater poverty than they had to. I paid particular attention to the fact that the Q Gospel was born in traumatising, warlike circumstances. Psychiatric traumatology helped me understand the Q Gospel and other ancient documents using historical approaches in a way that would comply with modern behavioural science. Even though I found some answers to the questions I had posed in my research, the main result of my research work is the justification of the question: Is it important to ask whether there is a connection between the ethos expressed by means of the religious language of the Q Gospel and the predominantly war-related life experiences typical to Palestine at the time. As has been convincingly revealed by a number of studies, traumatic events contribute to the development of psychotic experiences. I approached the problematic nature, significance and complexity of the ideal of poverty and this warlike environment by clarifying the history of psychohistorical literary research and the interpretative contexts associated with Sigmund Freud, Jacques Lacan and Melanie Klein. It is justifiable to question abnormal mentality, but there is no reliable return from the abnormal mentality described in any particular text to the only affecting factor. The popular research tendency based on the Oedipus complex is just as controversial as the Oedipus complex itself. The sociological frameworks concerning moral panics and political paranoia of an outer and inner danger fit quite well with the construction of the Q Gospel. Jerrold M. Post, M.D., Professor of Psychiatry, Political Psychology and Interna-tional Affairs at George Washington University, and founder and director of the Center for the Analysis of Personality and Political Behavior for the Central Intelligence Agency, has focused on the role played by charisma in the attracting of followers and detailed the psychological styles of a "charismatic" leader. He wrote the books Political Paranoia and Leaders and Their Followers in a Dangerous World: the Psychology of Political Behavior among others. His psychoanalytic vocabulary was useful for my understanding of the minds and motivations involved in the Q Gospel s formation. The Q sect began to live in a predestined future, with the reality and safety of this world having collapsed in both their experience and their fantasies. The deep and clear-cut divisions into good and evil that are expressed in the Q Gospel reveal the powerful nature of destructive impulses, envy and overwhelming anxiety. Responsible people who influenced the Q Gospel's origination tried to mount an ascetic defense against anxiety, denying their own needs, focusing their efforts on another objective (God s Kingdom) and a regressive, submissive earlier phase of development (a child s carelessness). This spiritual process was primarily an ecclesiastic or group-dynamical tactic to give support to the power of group leaders.

Electrocardiographic repolarization variables in detecting myocardial infarction and ischemic injury : From body surface potential mapping to a single lead

The aim of the studies was to improve the diagnostic capability of electrocardiography (ECG) in detecting myocardial ischemic injury with a future goal of an automatic screening and monitoring method for ischemic heart disease. The method of choice was body surface potential mapping (BSPM), containing numerous leads, with intention to find the optimal recording sites and optimal ECG variables for ischemia and myocardial infarction (MI) diagnostics. The studies included 144 patients with prior MI, 79 patients with evolving ischemia, 42 patients with left ventricular hypertrophy (LVH), and 84 healthy controls. Study I examined the depolarization wave in prior MI with respect to MI location. Studies II-V examined the depolarization and repolarization waves in prior MI detection with respect to the Minnesota code, Q-wave status, and study V also with respect to MI location. In study VI the depolarization and repolarization variables were examined in 79 patients in the face of evolving myocardial ischemia and ischemic injury. When analyzed from a single lead at any recording site the results revealed superiority of the repolarization variables over the depolarization variables and over the conventional 12-lead ECG methods, both in the detection of prior MI and evolving ischemic injury. The QT integral, covering both depolarization and repolarization, appeared indifferent to the Q-wave status, the time elapsed from MI, or the MI or ischemia location. In the face of evolving ischemic injury the performance of the QT integral was not hampered even by underlying LVH. The examined depolarization and repolarization variables were effective when recorded in a single site, in contrast to the conventional 12-lead ECG criteria. The inverse spatial correlation of the depolarization and depolarization waves in myocardial ischemia and injury could be reduced into the QT integral variable recorded in a single site on the left flank. In conclusion, the QT integral variable, detectable in a single lead, with optimal recording site on the left flank, was able to detect prior MI and evolving ischemic injury more effectively than the conventional ECG markers. The QT integral, in a single-lead or a small number of leads, offers potential for automated screening of ischemic heart disease, acute ischemia monitoring and therapeutic decision-guiding as well as risk stratification.