Uma folheação Riemanniana singular em M, variedade Riemanniana completa, é uma folheação singular tal que as folhas são localmente equidistantes. Existe uma folheação singular, chamada de folheação dual a folheação Riemanniana dada, cuja folha passando por p é o conjunto dos pontos em M que são alcançados por alguma geodésica horizontal quebrada partindo de p. Se M possui curvatura seccional positiva, então a folheação dual possui apenas uma folha. Se a curvatura seccional de M é não-negativa e M não coincidi com alguma folha dual, então o fibrado normal de qualquer geodésica horizontal quebrada é gerado por uma família de campos de Jacobi paralelos. Ambos os resultados são conhecidos com Teorema de Dualização. Uma aplicação destes resultados é a prova da suavidade da projeção métrica na alma. Todos estes resultados são devidos a Wilking. O objetivo desta dissertação de mestrado é discutir tais resultados de Wilking, baseado no trabalho do mesmo e em uma abordagem feita por Gromoll e Walschap. / Let M be a Riemanniana manifold with nonnegative sectional curvature. A singular Riemannian foliation in M is a singular foliation with locally equidistant leaves. The dual leaf though p is the collection of the all points q in M such that p and q are connected with a piece-wise horizontal geodesic. The partition of M into the dual leaves is a singular foliation called dual foliation. Wilking proved that if the sectional curveture is positive, then the dual foliation consists of a single leaf. In other words, any two points in M can be connected with a piece-wise horizontal geodesic. In order to prove this result Wilking showed that, if M is nonnegatively curved, the normal bundle of a dual leaf along a piecewise horizontal geodesic is gerated for parallel Jacobi field. These results are used in the proof that the projection metric in the soul is smoth.

Uma folheação Riemanniana singular em M, variedade Riemanniana completa, é uma folheação singular tal que as folhas são localmente equidistantes. Existe uma folheação singular, chamada de folheação dual a folheação Riemanniana dada, cuja folha passando por p é o conjunto dos pontos em M que são alcançados por alguma geodésica horizontal quebrada partindo de p. Se M possui curvatura seccional positiva, então a folheação dual possui apenas uma folha. Se a curvatura seccional de M é não-negativa e M não coincidi com alguma folha dual, então o fibrado normal de qualquer geodésica horizontal quebrada é gerado por uma família de campos de Jacobi paralelos. Ambos os resultados são conhecidos com Teorema de Dualização. Uma aplicação destes resultados é a prova da suavidade da projeção métrica na alma. Todos estes resultados são devidos a Wilking. O objetivo desta dissertação de mestrado é discutir tais resultados de Wilking, baseado no trabalho do mesmo e em uma abordagem feita por Gromoll e Walschap. / Let M be a Riemanniana manifold with nonnegative sectional curvature. A singular Riemannian foliation in M is a singular foliation with locally equidistant leaves. The dual leaf though p is the collection of the all points q in M such that p and q are connected with a piece-wise horizontal geodesic. The partition of M into the dual leaves is a singular foliation called dual foliation. Wilking proved that if the sectional curveture is positive, then the dual foliation consists of a single leaf. In other words, any two points in M can be connected with a piece-wise horizontal geodesic. In order to prove this result Wilking showed that, if M is nonnegatively curved, the normal bundle of a dual leaf along a piecewise horizontal geodesic is gerated for parallel Jacobi field. These results are used in the proof that the projection metric in the soul is smoth.

Muitas relações min-max em otimização combinatória podem ser demonstradas através de total dual integralidade de sistemas lineares. O conceito algébrico de bases de Hilbert foi originalmente introduzido com o objetivo de melhor compreender a estrutura geral dos sistemas totalmente dual integrais. Resultados apresentados posteriormente mostraram que bases de Hilbert também são relevantes para a otimização combinatória em geral e para a caracterização de certas classes de objetos discretos. Entre tais resultados, foram provadas, a partir dessas bases, versões do teorema de Carathéodory para programação inteira. Nesta dissertação, estudamos aspectos estruturais e computacionais de bases de Hilbert e relações destas com programação inteira e otimização combinatória. Em particular, consideramos versões inteiras do teorema de Carathéodory e conjecturas relacionadas. / There are several min-max relations in combinatorial optimization that can be proved through total dual integrality of linear systems. The algebraic concept of Hilbert basis was originally introduced with the objective of better understanding the general structure of totally dual integral systems. Some results that were proved later have shown that Hilbert basis are also relevant to combinatorial optimization in a general manner and to characterize certain classes of discrete objects. Among such results, there are versions of Carathéodory\'s theorem for integer programming that were proved through those basis. In this dissertation, we study structural and computational aspects of Hilbert basis and their relations to integer programming and combinatorial optimization. In particular, we consider integer versions of Carathéodory\'s theorem and related conjectures.

Muitas relações min-max em otimização combinatória podem ser demonstradas através de total dual integralidade de sistemas lineares. O conceito algébrico de bases de Hilbert foi originalmente introduzido com o objetivo de melhor compreender a estrutura geral dos sistemas totalmente dual integrais. Resultados apresentados posteriormente mostraram que bases de Hilbert também são relevantes para a otimização combinatória em geral e para a caracterização de certas classes de objetos discretos. Entre tais resultados, foram provadas, a partir dessas bases, versões do teorema de Carathéodory para programação inteira. Nesta dissertação, estudamos aspectos estruturais e computacionais de bases de Hilbert e relações destas com programação inteira e otimização combinatória. Em particular, consideramos versões inteiras do teorema de Carathéodory e conjecturas relacionadas. / There are several min-max relations in combinatorial optimization that can be proved through total dual integrality of linear systems. The algebraic concept of Hilbert basis was originally introduced with the objective of better understanding the general structure of totally dual integral systems. Some results that were proved later have shown that Hilbert basis are also relevant to combinatorial optimization in a general manner and to characterize certain classes of discrete objects. Among such results, there are versions of Carathéodory\'s theorem for integer programming that were proved through those basis. In this dissertation, we study structural and computational aspects of Hilbert basis and their relations to integer programming and combinatorial optimization. In particular, we consider integer versions of Carathéodory\'s theorem and related conjectures.

My PhD-project has two main research directions. The first direction is on partial regularities which we define as refinements of the Castelnuovo-Mumford regularity. Main results are: relationship of partial regularities and related invariants, like the a-invariants or the Castelnuovo-Mumford regularity of the syzygy modules; algebraic properties of partial regularities via a filter-regular sequence or a short exact sequence; generalizing a well-known result for the Castelnuovo-Mumford regularity to the case of partial regularities of stable and squarefree stable monomial ideals; finally extending an upper bound proven by Caviglia-Sbarra to partial regularities. The second direction of my project is to develop a theory on monomial preorders. Many interesting statements from the classical theory of monomial orders generalize to monomial preorders. Main results are: a characterization of monomial preorders by real matrices, which extends a result of Robbiano on monomial orders; secondly, leading term ideals with respect to monomial preorders can be studied via flat deformations of the given ideal; finally, comparing invariants of the given ideal and the leading term ideal with respect to a monomial preorder.

Traditionally, source coding (data compression) and channel coding (error protection) are performed separately and sequentially, resulting in what we call a tandem (separate) coding system. In practical implementations, however, tandem coding might involve a large delay and a high coding/decoding complexity, since one needs to remove the redundancy in the source coding part and then insert certain redundancy in the channel coding part. On the other hand, joint source-channel coding (JSCC), which coordinates source and channel coding or combines them into a single step, may offer substantial improvements over the tandem coding approach. This thesis deals with the fundamental Shannon-theoretic limits for a variety of communication systems via JSCC. More specifically, we investigate the reliability function (which is the largest rate at which the coding probability of error vanishes exponentially with increasing blocklength) for JSCC for the following discrete-time communication systems: (i) discrete memoryless systems; (ii) discrete memoryless systems with perfect channel feedback; (iii) discrete memoryless systems with source side information; (iv) discrete systems with Markovian memory; (v) continuous-valued (particularly Gaussian) memoryless systems; (vi) discrete asymmetric 2-user source-channel systems. For the above systems, we establish upper and lower bounds for the JSCC reliability function and we analytically compute these bounds. The conditions for which the upper and lower bounds coincide are also provided. We show that the conditions are satisfied for a large class of source-channel systems, and hence exactly determine the reliability function. We next provide a systematic comparison between the JSCC reliability function and the tandem coding reliability function (the reliability function resulting from separate source and channel coding). We show that the JSCC reliability function is substantially larger than the tandem coding reliability function for most cases. In particular, the JSCC reliability function is close to twice as large as the tandem coding reliability function for many source-channel pairs. This exponent gain provides a theoretical underpinning and justification for JSCC design as opposed to the widely used tandem coding method, since JSCC will yield a faster exponential rate of decay for the system error probability and thus provides substantial reductions in complexity and coding/decoding delay for real-world communication systems. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-05-13 22:31:56.425