Nous décrivons un nouvelle famille d'exemples d'hypersurfaces de la sphère satisfaisant le cas d'égalité de la majoration extrinsèque de C. Bär de la plus petite valeur propre de l'opérateur de Dirac. / We describe a new family of examples of hypersurfaces in the sphere satisfying the limitingcase in C. Bär's extrinsic upper bound for the smallest eigenvalue of the Dirac operator.

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China and India have many in common, are geographically large developing countries with enormous populations, but with notably different economic systems. This paper focus on exploring the relationship between GDP and consumption behaviors in China and India over the period 1978-2006. We use GDP as a proxy to represent income and household final consumption as a proxy to represent consumption. The long run relations are estimated by ARDL (1, 1) model. We find that India’s consumption is in the line of theory. But the relationship between GDP and consumption in China is unique, in terms of a negative intercept, a negative time trend and a larger than one marginal propensity to consumption. This may due to two possible explanations. First, after almost 30 years high growth households in China are optimistic to their future incomes. They prefer to borrowing for smoothing their consumptions. Second, there might have winding income what are not in the statistics. Finally, we adopt a series of diagnostic tests to check if selected models are strong enough and analysis the results.

This thesis is intended to provide several new bounding techniques for channel codes over quasi-static fading channels (QSFC). This type of channel has drawn more and more attention recently with the demanding need for higher capacity and more reliable wireless communication systems. Although there have been some published results on analyzing the performance of channel codes over QSFCs, most of them produced quite loose performance upper bounds. In this thesis, the general Gallager bounding approach which provides convergent upper bounds of coded systems over QSFCs is addressed first. It is shown that previous Gallager bounds employing trivial low SNR bounds tended to be quite loose. Then improved low instantaneous SNR bounds are derived for two classes of convolutional codes including turbo codes. Consequently, they are combined with the classical Union-Chernoff bound to produce new performance upper bounds for simple convolutional and turbo codes over single-input single-output (SISO) QSFCs. The new bound provides a much improved alternative to characterizing the performance of channel codes over QSFCs over the existing ones. Next the new bounding approach is extended to cases of serially concatenated space-time block codes, which show equivalence with SISO QSFCs. Tighter performance bounds are derived for this coding scheme for two specific cases: first a convolutional code, and later a turbo code. Finally, the more challenging cases of multiple-input multiple-output (MIMO) QSFCs are investigated. Several performance upper bounds are derived for the bit error probability of different cases of space-time trellis codes (STTC) over QSFCs using a new and tight low SNR bound. Also included in this work is an algorithm for computing the unusual information eigenvalue spectrum of STTCs.

The most popular class of quantum error correcting codes is stabilizer codes. Binary quantum stabilizer codes have been well studied, and Calderbank, Rains, Shor and Sloane (July 1998) have constructed a table of upper bounds on the minimum distance of these codes using linear programming methods. However, not much is known in the case of nonbinary stabilizer codes. In this thesis, we establish a bridge between selforthogonal classical codes over the finite field containing q2 elements and quantum codes, extending and unifying previous work by Matsumoto and Uyematsu (2000), Ashikhmin and Knill (November 2001), Kim and Walker (2004). We construct a table of upper bounds on the minimum distance of the stabilizer codes using linear programming methods that are tighter than currently known bounds. Finally, we derive code construction techniques that will help us find new codes from existing ones. All these results help us to gain a better understanding of the theory of nonbinary stabilizer codes.

Iterative decoding and codes on graphs were first devised by Gallager in 1960, and then rediscovered by Berrou, Glavieux and Thitimajshima in 1993. This technique plays an important role in modern communications, especially in coding theory and practice. In particular, low-density parity-check (LDPC) codes, introduced by Gallager in the 1960s, are the class of codes at the heart of iterative coding. Since these codes are quite general and exhibit good performance under message-passing decoding, they play an important role in communications research today. A thorough analysis of iterative decoding systems and the relationship between maximum a posteriori (MAP) and belief propagation (BP) decoding was initiated by Measson, Montanari, and Urbanke. This analysis is based on density evolution (DE), and extrinsic information transfer (EXIT) functions, introduced by ten Brink. Following their work, this thesis considers the MAP decoding thresholds of three iterative decoding systems. First, irregular repeat-accumulate (IRA) and accumulaterepeataccumulate (ARA) code ensembles are analyzed on the binary erasure channel (BEC). Next, the joint iterative decoding of LDPC codes is studied on the dicode erasure channel (DEC). The DEC is a two-state intersymbol-interference (ISI) channel with erasure noise, and it is the simplest example of an ISI channel with erasure noise. Then, we introduce a slight generalization of the EXIT area theorem and apply the MAP threshold bound for the joint decoder. Both the MAP and BP erasure thresholds are computed and compared with each other. The result quantities the loss due to iterative decoding Some open questions include the tightness of these bounds and the extensions to non-erasure channels.

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We study fundamental properties of Locally Decodable Codes (LDCs). LDCs are motivated by the intuition that traditional codes do not have a good tradeoff between resistance to arbitrary error and probe complexity. For example, if you apply a traditional code on a database, the resulting codeword can be resistant to error even if a constant fraction of it was corrupted; however, to accomplish this, the decoding procedure would typically have to analyze the entire codeword. For large data sizes, this is considered computationally expensive. This may be necessary even if you are only trying to recover a single bit of the database! This motivates the concept of LDCs, which encode data in such a way that up to a constant fraction of the result could be corrupted; while the decoding procedures only need to read a sublinear, ideally constant, number of codeword bits to retrieve any bit of the input with high probability. Our most exciting contribution is an exponential lower bound on the length of three query LDCs (binary or linear) with high correctness. This is the first strong length lower bound for any kind of LDC allowing more than two queries. For LDCs allowing three or more queries, the previous best lower bound, given by Woodruff, is below [omega](n2). Currently, the best upper bound is sub-exponential, but still very large. If polynomial length constructions exist, LDCs might be useful in practice. If polynomial length constructions do not exist, LDCs are much less likely to find adoption -- the resources required to implement them for large database sizes would be prohibitive. We prove that in order to achieve just slightly higher correctness than the current best constructions, three query LDCs (binary or linear) require exponential size. We also prove several impossibility results for LDCs. It has been observed that for an LDC that withstands up to a delta fraction of error, the probability of correctness cannot be arbitrarily close to 1. However, we are the first to estimate the largest correctness probability obtainable for a given delta. We prove close to tight bounds for arbitrary numbers of queries. / text

We investigate the problem of minimizing a nonconvex function with respect to convex constraints, and we study different techniques to compute a lower bound on the optimal value: The method of using convex envelope functions on one hand, and the method of exploiting nonconvex duality on the other hand. We investigate which technique gives the better bound and develop conditions under which the dual bound is strictly better than the convex envelope bound. As a byproduct, we derive some interesting results on nonconvex duality. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

We derive no-arbitrage bounds for expected excess returns to generate scenarios used in financial applications. The bounds allow to distinguish three regions: one where arbitrage opportunities will never exist, a second where arbitrage may be present, and a third, where arbitrage opportunities will always exist. No-arbitrage bounds are derived in closed form for a given covariance matrix using the least possible number of scenarios. Empirical examples illustrate the practical potential of knowing these bounds. (authors' abstract)