etd@IISc Collection:
http://hdl.handle.net/2005/41
Sun, 18 Feb 2018 00:45:26 GMT2018-02-18T00:45:26ZA Formal Proof of Feit-Higman Theorem in Agda
http://hdl.handle.net/2005/3128
Title: A Formal Proof of Feit-Higman Theorem in Agda
Authors: Rao, Balaji R
Abstract: In this thesis we present a formalization of the combinatorial part of the proof of Feit-Higman theorem on generalized polygons. Generalised polygons are abstract geometric structures that generalize ordinary polygons and projective planes. They are closely related to finite groups.
The formalization is carried out in Agda, a dependently typed functional programming language and proof assistant based on the intuitionist type theory by Per Martin-Löf.Sat, 17 Feb 2018 18:30:00 GMThttp://hdl.handle.net/2005/31282018-02-17T18:30:00ZCompactness Theorems for The Spaces of Distance Measure Spaces and Riemann Surface Laminations
http://hdl.handle.net/2005/3131
Title: Compactness Theorems for The Spaces of Distance Measure Spaces and Riemann Surface Laminations
Authors: Divakaran, D
Abstract: Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the Gromov-Hausdorﬀ distance, is a theorem with many applications. In this thesis, we give a generalisation of this landmark result, more precisely, we give a compactness theorem for the space of distance measure spaces equipped with the generalised Gromov-Hausdorﬀ-Levi-Prokhorov distance. A distance measure space is a triple (X, d,µ), where (X, d) forms a distance space (a generalisation of a metric space where, we allow the distance between two points to be infinity) and µ is a finite Borel measure.
Using this result we prove that the Deligne-Mumford compactiﬁcation is the completion of the moduli space of Riemann surfaces under the generalised Gromov-Hausdorﬀ-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactiﬁcation of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov compactness theorem for J-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov compactness theorem for J-holomorphic curves in symplectic manifolds, is an important tool in symplectic topology, its applicability is limited by the lack of general methods to construct pseudo-holomorphic curves. One hopes that considering a more general class of objects in place of pseudo-holomorphic curves will be useful. Generalising the domain of pseudo-holomorphic curves from Riemann surfaces to Riemann surface laminations is a natural choice. Theorems such as the uniformisation theorem for surface laminations by Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and topological classiﬁcation of “almost all" leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations, as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation, we prove a compactness theorem analogous to the Deligne-Mumford compactiﬁcation for the space of Riemann surface laminations.Sat, 17 Feb 2018 18:30:00 GMThttp://hdl.handle.net/2005/31312018-02-17T18:30:00ZA Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities
http://hdl.handle.net/2005/3107
Title: A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities
Authors: Porwal, Kamana
Abstract: The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very pop-ular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different ele-ments. Moreover they are high order accurate and stable methods. Adaptive algorithms reﬁne the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh reﬁnement.
The solution of linear elliptic problem exhibits singularities due to change in boundary con-ditions, irregularity of coefﬁcients and reentrant corners in the domain. Apart from this, the solu-tion of variational inequality exhibits additional irregular behaviour due to occurrence of the free boundary (the part of the domain which is a priori unknown and must be found as a component of the solution). In the lack of full elliptic regularity of the solution, uniform reﬁnement is inefﬁcient and it does not yield optimal convergence rate. But adaptive reﬁnement, which is based on the residuals ( or a posteriori error estimator) of the problem, enhance the efﬁciency by reﬁning the mesh locally and provides the optimal convergence. In this thesis, we derive a posteriori error estimates of the DG methods for the elliptic variational inequalities of the ﬁrst kind and the second kind.
This thesis contains seven chapters including an introductory chapter and a concluding chap-ter. In the introductory chapter, we review some fundamental preliminary results which will be used in the subsequent analysis. In Chapter 2, a posteriori error estimates for a class of DG meth-ods have been derived for the second order elliptic obstacle problem, which is a prototype for elliptic variational inequalities of the ﬁrst kind. The analysis of Chapter 2 is carried out for the general obstacle function therefore the error estimator obtained therein involves the min/max func-tion and hence the computation of the error estimator becomes a bit complicated. With a mild assumption on the trace of the obstacle, we have derived a signiﬁcantly simple and easily com-putable error estimator in Chapter 3. Numerical experiments illustrates that this error estimator indeed behaves better than the error estimator derived in Chapter 2. In Chapter 4, we have carried out a posteriori analysis of DG methods for the Signorini problem which arises from the study of the frictionless contact problems. A nonlinear smoothing map from the DG ﬁnite element space to conforming ﬁnite element space has been constructed and used extensively, in the analysis of Chapter 2, Chapter 3 and Chapter 4. Also, a common property shared by all DG methods allows us to carry out the analysis in uniﬁed setting. In Chapter 5, we study the C0 interior penalty method for the plate frictional contact problem, which is a fourth order variational inequality of the second kind. In this chapter, we have also established the medius analysis along with a posteriori analy-sis. Numerical results have been presented at the end of every chapter to illustrate the theoretical results derived in respective chapters. We discuss the possible extension and future proposal of the work presented in the Chapter 6. In the last chapter, we have documented the FEM codes used in the numerical experiments.Wed, 14 Feb 2018 18:30:00 GMThttp://hdl.handle.net/2005/31072018-02-14T18:30:00ZCentral and Peripheral Correlates of Motor Planning
http://hdl.handle.net/2005/3092
Title: Central and Peripheral Correlates of Motor Planning
Authors: Rungta, Satya Prakash
Abstract: A hallmark of human behaviour is that we can either couple or decouple our thoughts, decision and motor plans from actions. Previous studies have reported evidence of gating of information between intention and action that can happen at different levels in the central nervous system (CNS) involving the motor cortex, subcortical structures such as the basal ganglia and even in the spinal cord. In my research I examine the extent of this gating and its modulation by task context. I will present results obtained by data collected from (a) neck muscles and neural recording from frontal eye field (FEF) in macaque monkeys and (b) putative motor units (MUs) from high density electrode arrays using surface EMG signals in human to delineate the type of information that leaks into muscles in the periphery when subjects are involved in preparing eye and hand movements, respectively, and its modulation by task context Overall, my results reveal that we can assess some aspects of central planning in the activity of motor units Further, the recruitment of these motor units depend on task context.Sat, 10 Feb 2018 18:30:00 GMThttp://hdl.handle.net/2005/30922018-02-10T18:30:00Z