Iterative Connections and Abhyankar's Conjecture

Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations. This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, that is, a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra. A Cayley map can be thought of as a partial algebraic analog of the exponential map. A prototypical example is the classical “Cayley transform” for the special orthogonal group SOn defined by Arthur Cayley in 1846. A linear algebraic group G is called stably Cayley if is Cayley for some r≥0. Here denotes the split r-dimensional k-torus. These notions were introduced in 2006 by Lemire, Popov, and Reichstein, who classified Cayley and stably Cayley simple groups over an algebraically closed field of characteristic zero. In this paper, we study reductive Cayley groups over an arbitrary field k of characteristic zero. Our main results are a criterion for a reductive group G to be stably Cayley, formulated in terms of its character lattice, and a classification of stably Cayley simple groups.

This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these modules with iterative connection (and also the modules with integrable iterative connection) form a Tannakian category, assuming some nice properties for the underlying ring, and we show how this generalises to modules over schemes. We also relate these notions to stratifications on modules, as introduced by A. Grothendieck in order to extend integrable (ordinary) connections to finite characteristic. Over smooth rings, we obtain an equivalence of stratifications and integrable iterative connections. Furthermore, over a regular ring in positive characteristic, we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by D. Gieseker. In the second part of this article, we set up a Picard-Vessiot theory for fields of solutions. For such a Picard-Vessiot extension, we obtain a Galois correspondence, which takes into account even nonreduced closed subgroup schemes of the Galois group scheme on one hand and inseparable intermediate extensions of the Picard-Vessiot extension on the other hand. Finally, we compare our Galois theory with the Galois theory for purely inseparable field extensions.

Co-design of an mHealth application for family carers of people with dementia to address needs related to the functional disability of their care recipients

We have developed a new derivative-free algorithm based on Radial Basis Functions (RBFs). Derivative-free optimization is an active field of research and several algorithms have been proposed recently. Problems of this nature in the industrial setting are quite frequent. The reason is that in a number of applications the optimization process contains simulation packages which are treated as black boxes. The development of our own algorithm was originally motivated by an application in biomedical imaging: the medical image registration problem. The particular characteristics of this problem have incited us to develop a new optimization algorithm based on trust-region methods. However it has been designed to be generic and to be applied to a wide range of problems. The main originality of our approach is the use of RBFs to build the models. In particular we have adapted the existing theory based on quadratic models to our own models and developed new procedures especially designed for models based on RBFs. We have tested our algorithm called BOOSTERS against state-of-the-art methods (UOBYQA, NEWUOA, DFO). On the medical image registration problem, BOOSTERS appears to be the method of choice. The tests on problems from the CUTEr collection show that BOOSTERS is comparable to, but not better than other methods on small problems (size 2-20). It is performing very well for medium size problems (20-80). Moreover, it is able to solve problems of dimension 200, which is considered very large in derivative-free optimization. We have also developed a new class of algorithms combining the robustness of derivative-free algorithms with the faster rate of convergence characterizing Newtonlike-methods. In fact, they define a new class of algorithms lying between derivative-free optimization and quasi-Newton methods. These algorithms are built on the skeleton of our derivative-free algorithm but they can incorporate the gradient when it is available. They can be interpreted as a way of doping derivative-free algorithms with derivatives. If the derivatives are available at each iteration, then our method can be seen as an alternative to quasi-Newton methods. At the opposite, if the derivatives are never evaluated, then the algorithm is totally similar to BOOSTERS. It is a very interesting alternative to existing methods for problems whose objective function is expensive to evaluate and when the derivatives are not available. In this situation, the gradient can be approximated by finite differences and its costs corresponds to n additional function evaluations assuming that Rn is the domain of definition of the objective function. We have compared our method with CFSQP and BTRA, two gradient-based algorithms, and the results show that our doped method performs best. We have also a theoretical analysis of the medical image registration problem based on maximization of mutual information. Most of the current research in this field is concentrated on registration based on nonlinear image transformation. However, little attention has been paid to the theoretical properties of the optimization problem. In our analysis, we focus on the continuity and the differentiability of the objective function. We show in particular that performing a registration without extension of the reference image may lead to discontinuities in the objective function. But we demonstrate that, under some mild assumptions, the function is differentiable almost everywhere. Our analysis is important from an optimization point of view and conditions the choice of a solver. The usual practice is to use generic optimization packages without worrying about the differentiability of the objective function. But the use of gradient-based methods when the objective function is not differentiable may result in poor performance or even in absence of convergence. One of our objectives with this analysis is also that practitioners become aware of these problems and to propose them new algorithms having a potential interest for their applications.

Decomposition based approaches are recalled from primal and dual point of view. The possibility of building partially disaggregated reduced master problems is investigated. This extends the idea of aggregated-versus-disaggregated formulation to a gradual choice of alternative level of aggregation. Partial aggregation is applied to the linear multicommodity minimum cost flow problem. The possibility of having only partially aggregated bundles opens a wide range of alternatives with different trade-offs between the number of iterations and the required computation for solving it. This trade-off is explored for several sets of instances and the results are compared with the ones obtained by directly solving the natural node-arc formulation. An iterative solution process to the route assignment problem is proposed, based on the well-known Frank Wolfe algorithm. In order to provide a first feasible solution to the Frank Wolfe algorithm, a linear multicommodity min-cost flow problem is solved to optimality by using the decomposition techniques mentioned above. Solutions of this problem are useful for network orientation and design, especially in relation with public transportation systems as the Personal Rapid Transit. A single-commodity robust network design problem is addressed. In this, an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. A set of new instances that are computationally hard for the natural flow formulation are solved by means of a new heuristic algorithm. Finally, an efficient decomposition-based heuristic approach for a large scale stochastic unit commitment problem is presented. The addressed real-world stochastic problem employs at its core a deterministic unit commitment planning model developed by the California Independent System Operator (ISO).

Both lifetime and efficiency of photovoltaic (PV) modules are negatively affected by high and/or strongly fluctuating temperatures. An approach to improve the thermal performance is to add a phase change material (PCM) to the layer composition of PV modules. In order to be able to determine design parameters of the PCM, e.g. layer thickness of the PCM, melting point or chemical composition, a mathematical 1D-model describing the transient temperature distribution across the module was developed. It considers all relevant heat transfer mechanisms – conduction, convection and radiation – as well as the melting-solidification processes of the PCM. The model is solved numerically by an iterative, implicit algorithm implemented in MATLAB. Input factors include geometrical parameters and material properties as well as time-dependent weather effects (irradiance, ambient temperature, wind speed and direction). To determine these factors, either locally measured data or values derived by empirical models from literature can be employed. The simulated temperature distribution was validated against measured temperatures with an average deviation of ±1.7 K and used as an input for a module lifetime model. The latter model showed that the integration of a PCM into the layer composition could increase the lifetime of a PV module by up to 10 y, yet at the cost of a lower energy generation which is reduced by about 12 % compared to PV modules without PCM.

DISPAR, proposed by Miikkulainen and Dyer (1991), consists of layered neural networks named FGREP modules and executes the task of parsing a story and generating a paraphrase. Patterns in input and output layers of FGREP modules are sequences of word representations, and FGREP modules communicate with each other using word representations which are updated through training. This paper makes it clear that the update equation of the FGREP module has theoretical problems, and the learning algorithm does not follow the steepest descent method. By revising the algorithm and performing an experiment, the noun vectors converge to the same vector and this solution becomes meaningless for the task. CAN has already been proposed as the model which can avoid this problem. This paper moreover describes the modified model of the original FGREP module, then shows comparisons among CAN, the modified FGREP module, and the original FGREP module. The goal of the original FGREP module is to connect several PDP modules to construct an architecture for higher level cognitive tasks. This paper discusses the use of CAN and the modified FGREP module as an element module of the connections.

This dissertation explores the potential of a virtual reality serious game to help people understand the complexity of a nature-based attraction, and leverage this to cultivate a connectedness to the ecosystem, working towards increasing pro-environmental behaviour. Specifically, this research examines what design considerations are necessary in working towards these goals. To this end, the opportunities in the tourism space for serious games and virtual reality were investigated through a site visit and subsequent pilot application. The opportunities were used to create a set of objectives for a virtual reality serious game artefact. Through several iterations the artefact was developed and modified, and was used to analyse the design considerations relevant to building a videogame around a nature-based tourism attraction. Further to this, design considerations around influencing a player’s connectedness to the attraction were investigated. Iterations One and Two were developed as part of Associate Professor Alexandra Coghlan’s project on VR games and reef conservation through tourism. The final iteration formed part of this dissertation. Utilising a Design Science methodology, the artefact was developed through a series of iterative activities. The Design, Play, Experience (DPE) Framework, a serious game specific extension of the Mechanics-Dynamics-Aesthetics (MDA) Framework, is used to guide the development and evaluation process. While design considerations for creating serious games exists in literature, theory has neglected those specific to nature-based attractions and those around creating a connection between real-world attractions and players. Through the iterative process, a virtual reality serious game is created around the Great Barrier Reef, the chosen nature-based tourism attraction context for the work. Involvement in the project during the Pilot, Iteration One and Iteration Two was as a paid employee for Associate Professor Alexandra Coghlan. Through the first two iterations of design science, the work identifies a series of design considerations for creating virtual reality serious games about nature-based tourism attractions for visitors. From a gameplay perspective, designers can look to engage players with non-typical elements of the attraction, while using both completely player-controlled and completely simulated events and actions from the attraction to show the attraction’s complexity. Towards teaching visitors about the attraction, designers shouldn’t focus on accuracy but interpretation when representing the environment and allow the player to conduct detrimental activities so they can see the consequences of those actions. Designers can utilize virtual reality to showcase unique perspectives, both from a literal vantage point in the attraction, and to help embody the player as the attraction. Finally, designers can create a visual language that separates the videogame components and the simulated real-world components to ensure visitors know how to interpret various elements. Through the final iteration of design science, the work identifies several more design considerations pertaining specifically to creating a sense of connectedness between the visitor and the tourism attraction. Designers should highlight knowledge about the attraction that pertains to its struggles or threats not necessarily to for visitors to remember, but to create emotional moments. Designers should find ways to evoke different emotions from a typical visit to the attraction or find ways to evoke similar emotions towards different elements of the attraction. Designers can consider the emotional journey the visitor goes on while playing the videogame and make clear connections to the real-world attraction through various stages of their journey. Designers can highlight actions that mimic pro-environmental behaviour in the videogame to help visitors continue those actions in the real-world. The findings seek to better enable videogame creators and designers to create systems around complex ecosystems, towards encouraging pro-environmental behaviour in players. The significance of this dissertation is that it gives videogame designers new lenses to look at their designs through, to better capture the complexity of an ecosystem into a simplified, interactive and educational videogame, while ensuring their videogame brings people closer emotionally to the attraction.

An all digital architecture was presented to realize the logarithm modeled automatic gain control(AGC) loop based on field programmable gates array(FPGA) platform.The architecture is mainly made up of four modules,including envelope detector,coordinator rotation digital computer(CORDIC) based iterative realization of effective hyperbolic function,infinite impulse response(IIR) based loop filter,and reconfigurable control logic generator.Envelope detector can meet the filter performance requirement only with finite impulse response(FIR) filter of low order.Symmetric coefficient based transposed architecture reduces the number of multipliers in filter to the half.By approximation to analog integrator and suitable process,IIR based filter can simplify the architecture of digital integrator.The logarithm convertor is easy to be realized by CORDIC based iterative hyperbolic function processor.The architecture can also output with random precision,and realize the compromise of circuit resources and speed,which overcomes the high demand for random access memory(RAM) in traditional algorithms.With fully parallel pipeline architecture,the maximum working clock can achieve 206 MHz.Finally,the realization results based on the FPGA processor and the hardware simulation were given.Experimental results were consistent well with the theoretical analysis.

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G. Let $$ \mathcal{A} $$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth $$ \mathcal{A} $$ -group schemes $$ \mathcal{P} $$ whose generic fibers $$ {{\mathcal{P}} \left/ {K} \right.} $$ coincide with G; these are known as parahoric group schemes. The special fiber $$ {{\mathcal{P}} \left/ {K} \right.} $$ of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that $$ {{\mathcal{P}} \left/ {K} \right.} $$ has a Levi factor, and that any two Levi factors of $$ {{\mathcal{P}} \left/ {K} \right.} $$ are geometrically conjugate.

The demand for 3D models that represent real-world objects such as structures and buildings has increased in recent years. It is becoming increasingly important that the reconstructions are not only visually convincing but also feature high geometric accuracy. This includes, for example, the fields of civil engineering, terrestrial surveying and archeology, where precise measurements are made in the models for documentation and analysis purposes. There are different approaches to create such a reconstruction. The photogrammetric method Structure from Motion and laser scanning are among the most widely used methods here, as they do not require a complicated setup and can be used for scenarios at small to large scale. Recent developments are enabling unmanned robotic systems, especially sensor mounted UAVs, to assist in the recording of areas which are otherwise difficult to observe. The demand for a high geometric accuracy, however, comes at the expense of high computational complexity of up to several days. Hence, especially real-time reconstructions are unfeasible, such that recording and reconstruction procedure must be executed consecutively. The resulting model quality, i.e. completeness and accuracy, is only assessable afterwards. Since it is often difficult or even impossible to improve these models with additional measurements afterwards, methods that ensure a reliable acquisition of sufficient data is required.
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\nIn this thesis we develop new methods and theory that address this problem for the mentioned sensor types. For both, a probabilistic description of the expected surface reconstruction error is maintained cost-efficiently as an estimate for the model quality during the recording procedure. For image sensors this is realized by incrementally constructing confidence ellipsoids that describe the information obtained from all views. With depth sensors the surface quality is described by the variance of a Gaussian process implicit surface regression fit to point cloud data using polyharmonic kernel functions. Sensor poses are then assessed by the information they add to the subsequent reconstruction up to a desired geometric accuracy using a formulation that is motivated from Optimal Experimental Design. This quantity is further used in an iterative next-best-view selection framework as a subproblem of a coverage path planning problem.
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\nThe general formulations presented in this thesis enables a wide range of applications, such as offline and online view planning or various autonomous robot systems under consideration of dynamic and geometric constraints. We present the first multi-view coverage path planning approach, specifically targeted at autonomous Structure from Motion data acquisition. Its correctness is validated in simulation using the physics simulator Gazebo. Furthermore, we lay a foundation for similar applications with depth sensors. All presented algorithms were developed with scalability in mind and show promising results regarding real-time usability.

Flutter is one of the most known instability phenomena. This condition occurs when a given structure exhibits sustained, harmonic oscillations, sometimes leading to catastrophic events. The prediction of flutter represents a crucial point for a correct and safe design. When fluid structure interactions produce dynamic instability, flutter analyses require accurate descriptions of body deformations and aerodynamic loads. To this end, aerodynamic theories have been coupled with structural models to develop aeroelastic analysis tools, whose reliability is the results of a trade-off between the accuracy and the computational efficiency. From a computational point of view, the most efficient formulation is based on the 1D assumption, where the problem is reduced to a set of variables that only depends on the beam-axis coordinate. Besides the well-known classical beam theories, several refined kinematic models have been proposed, to study the stability of rotating blades and shafts. However, when these structures are highly deformable or the material distribution involves non-classical structural couplings, 2D and 3D solutions are still required. Within this work, we propose an advanced 1D formulation to analyse the stability of rotating structures. The higher-order beam theories are obtained using the Carrera Unified Formulation (CUF), which allows to derive, at least theoretically, an infinite number of kinematic models. The Equations of Motion (EoM) for shafts and blades include the Coriolis term and the centrifugal effects (spin softening and geometrical stiffening). For the subsonic flow regime, aerodynamic loads are defined following the unsteady strip theories proposed by Theodorsen and Loewy. For the supersonic regime, the linear Piston theory is extended to structures rotating in compressed air flow. The Finite Element Method (FEM) is used to solve the weak form of the EoM. Firstly, to evaluate the accuracy of 1D CUF elements, static and free-vibration analysis are carried out on compact and thin-walled structures of isotropic, orthotropic and functionally graded materials. Then, higher-order elements are used to study the dynamics of laminated shafts, thin cylinders, discs and blades, which rotate about the longitudinal and transverse axis. Results show the improved performance of the 1D CUF theories compared to 2D and 3D solutions. In order to evaluate the proposed aeroelastic formulation, we test different wing configurations, including thin-walled box beams. The effects of the sweep angle and the lamination scheme on flutter conditions are evaluated, and results are compared with plate solutions, experimental tests and aeroelastic analysis carried out with the Doublet Lattice Method (DLM). Moreover, comparisons between Theodorsen and Loewy aerodynamic theories are presented for a realistic rotary-wing model. In the last numerical examples, the linear Piston Theory is used to describe the dynamics of thin plates with different aspect-ratio surrounded by compressed air. For this cases, results are compared with an existing solution based on a non-linear plate theory

In this paper, we further investigate the local Hermitian and skew-Hermitian splitting (LHSS) iteration method and the modified LHSS (MLHSS) iteration method for solving generalized nonsymmetric saddle point problems with nonzero (2,2) blocks. When A is non-symmetric positive definite, the convergence conditions are obtained, which generalize some results of Jiang and Cao [M.-Q. Jiang and Y. Cao, On local Hermitian and Skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 2009(231): 973-982] for the generalized saddle point problems to generalized nonsymmetric saddle point problems with nonzero (2,2) blocks. Numerical experiments show the effectiveness of the iterative methods.

Let K be a differential field with algebraically closed field of constants CK . Let K P V ∞ be the (iterated) Picard-Vessiot closure of K. Let G be a linear differential algebraic group over K, and X a differential algebraic torsor for G over K. We prove that X ( K P V ∞ ) is Kolchin dense in X. In the special case that G is finite-dimensional we prove that X ( K P V ∞ ) = X ( K diff ) (where Kdiff is the differential closure of K). We also give close relationships between Picard-Vessiot extensions of K and the category of torsors for suitable finite-dimensional linear differential algebraic groups over K.