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For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.

Reciprocal processes, whose concept can be traced back to E. Schrödinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. Léonard.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

Communication networks are vulnerable to natural as well as man-made disasters. The geographical layout of the network influences the impact of these disasters. It is therefore, necessary to identify areas that could be most affected by a disaster and redesign those parts of the network so that the impact of a disaster has least effect on them. In this work, we assume that disasters which have a circular impact on the network. The work presents two new algorithms, namely the WHF-PG algorithm and the WHF-NPG algorithm, designed to solve the problem of finding the locations of disasters that would have the maximum disruptive effect on the communication infrastructure in terms of capacity.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

In a recent paper with N. Tarkhanov, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.

In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.

In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.

<p>Less democratic countries conduct elections under the majoritarian electoral formula more often than under proportional representation by a wide margin. Yet, robust democratic systems utilize both majoritarian and PR electoral formulas with great success. This dissertation approaches this empirical puzzle and tries to unveil what role, if any, electoral formulas play in politics.</p><p> To do so, it focuses on the electoral process exclusively and utilizes Judith Kelley's recently completed comprehensive dataset on election quality to perform some large-sample statistical analyses of the relationship between the electoral formula, ethno-cultural polarization and election quality. Then, it presents three in-depth case studies of Nigeria, Ghana and Indonesia to unveil in more detail institutional origins and the mechanisms of electoral manipulation, as refracted through the electoral formula. </p><p>The conclusions reached are that PR is much better suited for conducting free and fair elections in ethno-culturally polarized countries. Yet, majoritarian and mixed formulas perform just as well when polarization is low. This finding is directly related to an ongoing debate by institutional designers and academics alike and provides systematic quantitative and detailed qualitative support. The study also suggests that PR might not only mediate inter-ethnic differences when disagreement is high, but also reduces the level of polarization if applied over several electoral cycles.</p> / Dissertation