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A challenging goal for both the game industry and the research community of computer graphics is the generation of 3D virtual avatars that automatically perform realistic human motions with high speed at low monetary cost. So far, full body motion estimation of human complexity remains an important open problem. We propose a realistic motion estimation framework to control the animation of 3D avatars. Instead of relying on a motion capture device as the control signal, we use low-cost and ubiquitously available 3D accelerometer sensors. The framework is developed in a data-driven fashion, which includes two phases: model learning from an existing high quality motion database, and motion synthesis from the control signal. In the phase of model learning, we built a high quality motion model of less complexity that learned from a large motion capture database. Then, by taking the 3D accelerometer sensor signal as input, we were able to synthesize high-quality motion from the motion model we learned. In this thesis, we present two different techniques for model learning and motion synthesis, respectively. Linear and nonlinear reduction techniques for data dimensionality are applied to search for the proper low dimensional representation of motion data. Two motion synthesis methods, interpolation and optimization, are compared using the 3D acceleration signals with high noise. We evaluate the result visually compared to the real video and quantitatively compared to the ground truth motion. The system performs well, which makes it available to a wide range of interactive applications, such as character control in 3D virtual environments and occupational training. / Master of Science

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc. In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that: (G has finitely many k-conjugacy classes) implies (G is finite)? Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have. We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc. In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that: (G has finitely many k-conjugacy classes) implies (G is finite)? Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have. We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.

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In Geostatistical modelling of the spatial distribution of rock attributes, the multivariate distribution of a Random Function defines the range of possible values and the spatial relationships among them. Under a decision of stationarity, the Random Function distribution and its statistics are inferred from data within a spatial domain deemed statistically homogenous. Assuming stationary multiGaussianity allows spatial prediction techniques to take advantage of this simple parametric distribution model. These techniques compute the local distributions with surrounding data and global spatially invariant statistics. They often fail to reproduce local changes in the mean, variability and, particularly, the spatial continuity, that are required for geologically realistic modelling of rock attributes. The proposed alternative is to build local Random Function models that are deemed stationary only in relation to the locations where they are defined. The corresponding location-dependent distributions and statistics are inferred by weighting the samples inversely proportional to their distance to anchor locations. These distributions are locally Gaussian transformed. The transformation models carry information on the local histogram. The distance weighted experimental measures of spatial correlation are able to adapt to local changes in the spatial continuity and are semi-automatically fitted by locally defined variogram models. The fields of local variogram and transformation parameters are used in locally stationary spatial prediction algorithms. The resulting attribute models are rich in non-stationary spatial features. This process implies a higher computational demand than the traditional techniques, but, if data is abundant enough to allow a reliable inference of the local statistics, the proposed locally stationary techniques outperform their stationary counterparts in terms of accuracy and precision. These improved models have the potential of providing better decision support for engineering design. / Mining Engineering

In the first part of this thesis, suitable function spaces for analysis of partial differ- ential equations in unbounded domains are introduced and studied. The results are then applied in the second part on semilinear wave equation in Rd with non- linear source term and nonlinear damping. The source term is supposed to be bounded by a polynomial function with a subcritical growth. The damping term is strictly monotone and satisfying a polynomial-like growth condition. Global existence is proved using finite speed of propagation. Dissipativity in locally uni- form spaces and the existence of a locally compact attractor are then obtained after additional conditions imposed on the damping term.

Let K be a number field with ring of integers O_K and let G be a finite group.By a result of E. Noether, the ring of integers of a tame Galois extension of K with Galois group G is a locally free O_K[G]-module of rank 1.Thus, to any tame Galois extension L/K with Galois group G we can associate a class [O_L] in the locally free class group Cl(O_K[G]). The set of all classes in Cl(O_K[G]) which can be obtained in this way is called the set of realizable classes and is denoted by R(O_K[G]).In this dissertation we study different problems related to R(O_K[G]).The first part focuses on the following question: is R(O_K[G]) a subgroup of Cl(O_K[G])? When the group G is abelian, L. McCulloh proved that R(O_K[G]) coincides with the so-called Stickelberger subgroup St(O_K[G]) of Cl(O_K[G]). In Chapter 2, we give a detailed presentation of unpublished work by L. McCulloh that extends the definition of St(O_K[G]) to the non-abelian case and shows that R(O_K[G]) is contained in St(O_K[G]) (the opposite inclusion is still not known in the non-abelian case).Then, just using its definition and Stickelberger's classical theorem, we prove in Chapter 3 that St(O_K[G]) is trivial if K=Q and G is either cyclic of order p or dihedral of order 2p, where p is an odd prime number. This, together with McCulloh's results, allows us to have a new proof of the triviality of R(O_K[G]) in the cases just considered.The main original results are contained in the second part of this thesis. In Chapter 4, we prove that St(O_K[G]) has good functorial behavior under restriction of the base field. This has the interesting consequence that, if N/L is a tame Galois extension with Galois group G, and St(O_K[G]) is known to be trivial for some subfield K of L, then O_N is stably free as an O_K[G]-module.In the last chapter, we prove an equidistribution result for Galois module classes amongst tame Galois extensions of K with Galois group G in which a given prime p of K is totally split.

We say a graph is locally P if the induced graph on the neighbourhood of every vertex has the property P. Specically, a graph is locally traceable (LT) or locally hamiltonian (LH) if the induced graph on the neighbourhood of every vertex is traceable or hamiltonian, respectively. A locally locally hamiltonian (L2H) graph is a graph in which the graph induced by the neighbourhood of each vertex is an LH graph. This concept is generalized to an arbitrary degree of nesting, to make it possible to work with LkH graphs. This thesis focuses on the global cycle properties of LT, LH and LkH graphs. Methods are developed to construct and combine such graphs to create others with desired properties. It is shown that with the exception of three graphs, LT graphs with maximum degree no greater than 5 are fully cycle extendable (and hence hamiltonian), but the Hamilton cycle problem for LT graphs with maximum degree 6 is NP-complete. Furthermore, the smallest nontraceable LT graph has order 10, and the smallest value of the maximum degree for which LT graphs can be nontraceable is 6. It is also shown that LH graphs with maximum degree 6 are fully cycle extendable, and that there exist nonhamiltonian LH graphs with maximum degree 9 or less for all orders greater than 10. The Hamilton cycle problem is shown to be NP-complete for LH graphs with maximum degree 9. The construction of r-regular nonhamiltonian graphs is demonstrated, and it is shown that the number of vertices in a longest path in an LH graph can contain a vanishing fraction of the vertices of the graph. NP-completeness of the Hamilton cycle problem for LkH graphs for higher values of k is also investigated. / Mathematical Sciences / D. Phil. (Mathematics)

Conner and Spencer used ultrafilters to construct homomorphisms between fundamental groups that could not be induced by continuous functions between the underlying spaces. We use methods from Shelah and Pawlikowski to prove that Conner and Spencer could not have constructed these homomorphisms with a weak version of the Axiom of Choice. This led us to define and examine a class of pathological objects that cannot be constructed without a strong version of the Axiom of Choice, which we call the class of inscrutable objects. Objects that do not need a strong version of the Axiom of Choice are scrutable. We show that the scrutable homomorphisms from the fundamental group of a Peano continuum are exactly the homomorphisms induced by a continuous function.We suspect that any proposed theorem whose proof does not use a strong Axiom of Choice cannot have an inscrutable counterexample.