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 the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Understanding local processes: Contemplating franchisation

Increasing, there is growing acknowledgement of the importance of franchising within all modern global economies. Despite this, little is understood with regards the actual impact of franchising on local economies. This research aims to reframe the contribution of franchising by considering the process of franchisation. This study employed a mixed-method approach, utilizing critical realism to facilitate an outcomes-based explanation of firm survival. The focus of the study was upon generative mechanisms that were assumed to give rise to particular events from which (pizza) firm survival was enhanced vis-à-vis all other community members. A database of 2440 firms (or in excess of 21,000 company years) combined with archival records, interviews and the researcher’s observations provided the researcher with access to the nature of interaction occurring between firms. It was found that the survival of local firms was influenced positively by the day-to-day actions of franchise operators. However, it is argued that to understand how any such advantage my fall to local independent firms, we need too better appreciate the multitude of local processes related to such industries. This research re-examines several ecological concepts with the view of enabling a clearer investigation of underlying local processes. It also represents an authentic autecological approach to the study of firms.

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.

Precise determination of aquatic plant wet mass using a salad spinner

The reliable assessment of macrophyte biomass is fundamental for ecological research and management of freshwater ecosystems. While dry mass is routinely used to determine aquatic plant biomass, wet (fresh) mass can be more practical. We tested the accuracy and precision of wet mass measurements by using a salad spinner to remove surface water from four macrophyte species differing in growth form and architectural complexity. The salad spinner aided in making precise and accurate wet mass with less than 3% error. There was also little difference between operators, with a user bias estimated to be below 5%. To achieve this level of precision, only 10–20 turns of the salad spinner are needed. Therefore, wet mass of a sample can be determined in less than 1 min. We demonstrated that a salad spinner is a rapid and economical technique to enable precise and accurate macrophyte wet mass measurements and is particularly suitable for experimental work. The method will also be useful for fieldwork in situations when sample sizes are not overly large.

Time-domain criteria for theL_{2}-stability of nonstationary feedback systems

Criteria for the L2-stability of linear and nonlinear time-varying feedback systems are given. These are conditions in the time domain involving the solution of certain associated matrix Riccati equations and permitting the use of a very general class of L2-operators as multipliers.

Semi-Classical Mechanics in Phase Space: A Study of Wigner's Function

We explore the semi-classical structure of the Wigner functions ($\Psi $(q, p)) representing bound energy eigenstates $|\psi \rangle $ for systems with f degrees of freedom. If the classical motion is integrable, the classical limit of $\Psi $ is a delta function on the f-dimensional torus to which classical trajectories corresponding to ($|\psi \rangle $) are confined in the 2f-dimensional phase space. In the semi-classical limit of ($\Psi $ ($\hslash $) small but not zero) the delta function softens to a peak of order ($\hslash ^{-\frac{2}{3}f}$) and the torus develops fringes of a characteristic 'Airy' form. Away from the torus, $\Psi $ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when ($\Psi $) is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ($\epsilon $), the system passes through three semi-classical regimes as ($\hslash $) diminishes. (b) For states ($|\psi \rangle $) associated with regions in phase space filled with irregular trajectories, ($\Psi $) will be a random function confined near that region of the 'energy shell' explored by these trajectories (this region has more than f dimensions). (c) For ($\epsilon \neq $0, $\hslash $) blurs the infinitely fine classical path structure, in contrast to the integrable case ($\epsilon $ = 0, where $\hslash $ )imposes oscillatory quantum detail on a smooth classical path structure.

Recapitulation in generating spatial layouts: Representations of embryogenesis in evolutionary design modelling

The noted 19th century biologist, Ernst Haeckel, put forward the idea that the growth (ontogenesis) of an organism recapitulated the history of its evolutionary development. While this idea is defunct within biology, the idea has been promoted in areas such as education (the idea of an education being the repetition of the civilizations before). In the research presented in this paper, recapitulation is used as a metaphor within computer-aided design as a way of grouping together different generations of spatial layouts. In most CAD programs, a spatial layout is represented as a series of objects (lines, or boundary representations) that stand in as walls. The relationships between spaces are not usually explicitly stated. A representation using Lindenmayer Systems (originally designed for the purpose of modelling plant morphology) is put forward as a way of representing the morphology of a spatial layout. The aim of this research is not just to describe an individual layout, but to find representations that link together lineages of development. This representation can be used in generative design as a way of creating more meaningful layouts which have particular characteristics. The use of genetic operators (mutation and crossover) is also considered, making this representation suitable for use with genetic algorithms.

On the positivity and stability of non-stationary systems

The positivity of operators in Hilbert spaces is an important concept finding wide application in various branches of Mathematical System Theory. A frequency- domain condition that ensures the positivity of time-varying operators in L2 with a state-space description, is derived in this paper by using certain newly developed inequalities concerning the input-state relation of such operators. As an interesting application of these results, an L2 stability criterion for time-varying feedback systems consisting of a finite-sector non-linearity is also developed.

Human error risk management methodology for rail crack incidents

The paper presents an innovative approach to modelling the causal relationships of human errors in rail crack incidents (RCI) from a managerial perspective. A Bayesian belief network is developed to model RCI by considering the human errors of designers, manufactures, operators and maintainers (DMOM) and the causal relationships involved. A set of dependent variables whose combinations express the relevant functions performed by each DMOM participant is used to model the causal relationships. A total of 14 RCI on Hong Kong’s mass transit railway (MTR) from 2008 to 2011 are used to illustrate the application of the model. Bayesian inference is used to conduct an importance analysis to assess the impact of the participants’ errors. Sensitivity analysis is then employed to gauge the effect the increased probability of occurrence of human errors on RCI. Finally, strategies for human error identification and mitigation of RCI are proposed. The identification of ability of maintainer in the case study as the most important factor influencing the probability of RCI implies the priority need to strengthen the maintenance management of the MTR system and that improving the inspection ability of the maintainer is likely to be an effective strategy for RCI risk mitigation.

On Multi-Dimensional Packets of Surface Waves

By using the method of operators of multiple scales, two coupled nonlinear equations are derived, which govern the slow amplitude modulation of surface gravity waves in two space dimensions. The equations of Davey and Stewartson, which also govern the two-dimensional modulation of the amplitude of gravity waves, are derived as a special case of our equations. For a fully dispersed wave, symmetric about a point which moves with the group velocity, the coupled equations reduce to a nonlinear Schrödinger equation with extra terms representing the effect of the curvature of the wavefront.

Workers are doing it for themselves? Local labor solidarity in precarious times

Historically, organized labor has played a fundamental role in guaranteeing basic rights and privileges for screen media workers and defending union and guild members (however unevenly) from egregious abuses of power. Yet, despite the recent turn to labor in media and cultural studies, organized labor today has received only scant attention, even less so in locations outside Hollywood. This presentation thus intervenes in two significant ways: first, it acknowledges the ongoing global ‘undoing’ of organized labor as a consequence of footloose production and conglomeration within the screen industries, and second, it examines a case example of worker solidarity and political praxis taking shape outside formal labor institutions in response to those structural shifts. Accordingly, it links an empirical study of individual agency to broader debates associated with the spatial dynamics of screen media production, including local capacity, regional competition, and precariousness. Drawing from ethnographic interviews with local film and television workers in Glasgow, Scotland, I consider the political alliance among three nascent labor organizations in the city: one for below-the-line crew, one for facility operators, and (oddly enough) one for producers. Collectively, the groups share a desire to transform Glasgow into a global production hub, following the infrastructure developments in nearby cities like Belfast, Prague, and Budapest. They furthermore frame their objectives in political terms: establishing global scale is considered a necessary maneuver to improve local working conditions like workplace safety, income disparity, skills training, and job access. Ultimately, I argue these groups are a product of an inadequate union structure and outdated policy vision for the screen sector , once-supportive institutions currently out of sync with the global realities of media production. Furthermore, the groups’ advocacy efforts reveal the extent to which workers themselves (in additional to capital) can seek “spatial fixes” to suture their prospects to specific political and economic goals. Of course, such activities manifest under conditions outside of the workers’ control but nevertheless point to an important tension within capitalist social relations, namely that the agency to reshape the spatial relationships in their own lives recasts the geography of labor in terms that aren’t inherent or exclusive to the interests of global capital.

Polymer melt dynamics: Microscopic roots of fractional viscoelasticity

The rheological properties of polymer melts and other complex macromolecular fluids are often successfully modeled by phenomenological constitutive equations containing fractional differential operators. We suggest a molecular basis for such fractional equations in terms of the generalized Langevin equation (GLE) that underlies the renormalized Rouse model developed by Schweizer [J. Chem. Phys. 91, 5802 (1989)]. The GLE describes the dynamics of the segments of a tagged chain under the action of random forces originating in the fast fluctuations of the surrounding polymer matrix. By representing these random forces as fractional Gaussian noise, and transforming the GLE into an equivalent diffusion equation for the density of the tagged chain segments, we obtain an analytical expression for the dynamic shear relaxation modulus G(t), which we then show decays as a power law in time. This power-law relaxation is the root of fractional viscoelastic behavior.

Real-coded genetic algorithm for machining condition optimization

In this paper, we consider the machining condition optimization models presented in earlier studies. Finding the optimal combination of machining conditions within the constraints is a difficult task. Hence, in earlier studies standard optimization methods are used. The non-linear nature of the objective function, and the constraints that need to be satisfied makes it difficult to use the standard optimization methods for the solution. In this paper, we present a real coded genetic algorithm (RCGA), to find the optimal combination of machining conditions. We present various issues related to real coded genetic algorithm such as solution representation, crossover operators, and repair algorithm in detail. We also present the results obtained for these models using real coded genetic algorithm and discuss the advantages of using real coded genetic algorithm for these problems. From the results obtained, we conclude that real coded genetic algorithm is reliable and accurate for solving the machining condition optimization models.

High-quality polar UV measurements : scientific analyses and transfer of the irradiance scale

The Earth's ecosystems are protected from the dangerous part of the solar ultraviolet (UV) radiation by stratospheric ozone, which absorbs most of the harmful UV wavelengths. Severe depletion of stratospheric ozone has been observed in the Antarctic region, and to a lesser extent in the Arctic and midlatitudes. Concern about the effects of increasing UV radiation on human beings and the natural environment has led to ground based monitoring of UV radiation. In order to achieve high-quality UV time series for scientific analyses, proper quality control (QC) and quality assurance (QA) procedures have to be followed. In this work, practices of QC and QA are developed for Brewer spectroradiometers and NILU-UV multifilter radiometers, which measure in the Arctic and Antarctic regions, respectively. These practices are applicable to other UV instruments as well. The spectral features and the effect of different factors affecting UV radiation were studied for the spectral UV time series at Sodankylä. The QA of the Finnish Meteorological Institute's (FMI) two Brewer spectroradiometers included daily maintenance, laboratory characterizations, the calculation of long-term spectral responsivity, data processing and quality assessment. New methods for the cosine correction, the temperature correction and the calculation of long-term changes of spectral responsivity were developed. Reconstructed UV irradiances were used as a QA tool for spectroradiometer data. The actual cosine correction factor was found to vary between 1.08-1.12 and 1.08-1.13. The temperature characterization showed a linear temperature dependence between the instrument's internal temperature and the photon counts per cycle. Both Brewers have participated in international spectroradiometer comparisons and have shown good stability. The differences between the Brewers and the portable reference spectroradiometer QASUME have been within 5% during 2002-2010. The features of the spectral UV radiation time series at Sodankylä were analysed for the time period 1990-2001. No statistically significant long-term changes in UV irradiances were found, and the results were strongly dependent on the time period studied. Ozone was the dominant factor affecting UV radiation during the springtime, whereas clouds played a more important role during the summertime. During this work, the Antarctic NILU-UV multifilter radiometer network was established by the Instituto Nacional de Meteorogía (INM) as a joint Spanish-Argentinian-Finnish cooperation project. As part of this work, the QC/QA practices of the network were developed. They included training of the operators, daily maintenance, regular lamp tests and solar comparisons with the travelling reference instrument. Drifts of up to 35% in the sensitivity of the channels of the NILU-UV multifilter radiometers were found during the first four years of operation. This work emphasized the importance of proper QC/QA, including regular lamp tests, for the multifilter radiometers also. The effect of the drifts were corrected by a method scaling the site NILU-UV channels to those of the travelling reference NILU-UV. After correction, the mean ratios of erythemally-weighted UV dose rates measured during solar comparisons between the reference NILU-UV and the site NILU-UVs were 1.007±0.011 and 1.012±0.012 for Ushuaia and Marambio, respectively, when the solar zenith angle varied up to 80°. Solar comparisons between the NILU-UVs and spectroradiometers showed a ±5% difference near local noon time, which can be seen as proof of successful QC/QA procedures and transfer of irradiance scales. This work also showed that UV measurements made in the Arctic and Antarctic can be comparable with each other.