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Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 114-117. / by Hans Kleppe. / Ph.D.

Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions. We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kos\c{c}az and Vafa. Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on $\C^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.

The objective of this dissertation is to investigate incremental redundancy low-density parity-check (IR-LDPC) codes for hybrid forward error correction / automatic repeat request (HybridARQ) schemes. Powerful capacity-approaching IR-LDPC codes are one of the key functional elements in high-throughput HybridARQ schemes and provide a flexible rate-compatible structure, which is necessary for low-complexity HybridARQ schemes. This dissertation first studies the design and performance evaluation of IR-LDPC codes, which have good error rate performance at short block lengths. The subset codes of the IR-LDPC codes are compared to conventional random punctured codes and multiple dedicated codes. As a system model for this work, an adaptive LDPC coded system is presented. This adaptive system can confront the nature of time-varying channels and approach the capacity of the system with the aid of LDPC codes. This system shows remarkable throughput improvement over a conventional punctured system and, for systems that use multiple dedicated codes, provides comparable performance with low-complexity at every target error rate. This dissertation also focuses on IR-LDPC codes with a wider operating code range because the previous IR-LDPC codes exhibited performance limitation related to the maximum achievable code rate. For this reason, this research proposes a new way to increase the maximum code rate of the IR-LDPC codes, which provides throughput improvement at high throughput regions over conventional random punctured codes. Also presented is an adaptive code selection algorithm using threshold parameters. This algorithm reduces the number of the unnecessary traffic channels in HybridARQ schemes. This dissertation also examines how to improve throughput performance in HybridARQ schemes with low-complexity by exploiting irregular repeat accumulate (IRA) codes. The proposed adaptive transmission method with adaptive puncturing patterns of IRA codes shows higher throughput performance in all of operating code ranges than does any other single mode in HybridARQ schemes.

In this thesis we study moduli stacks \calM_\Phi^n, indexed by an integer n>0 and a CM-type (K,\Phi), which parametrize abelian schemes equipped with action by \OK and an \OK-linear principal polarization, such that the representation of \OK on the relative Lie algebra of the abelian scheme consists of n copies of each character in \Phi. We do this by systematically applying Serre's tensor construction, and for that we first establish a general correspondence between polarizations on abelian schemes M\otimes_R A arising from this construction and polarizations on the abelian scheme A, along with positive definite hermitian forms on the module M. Next we describe a tensor product of categories and apply it to the category \Herm_n(\OK) of finite non-degenerate positive-definite \OK-hermitian modules of rank n and the category fibred in groupoids \calM_\Phi^1 of principally polarized CM abelian schemes. Assuming n is prime to the class number of K, we show that Serre's tensor construction provides an identification of this tensor product with a substack of the moduli space \calM_\Phi^n, and that in some cases, such as when the base is finite type over \CC or an algebraically closed field of characteristic zero, this substack is the entire space. We then use this characterization to describe the Galois action on \calM_\Phi^n(\overline{\QQ}), by using the description of the action on \calM_\Phi^1(\overline{\QQ}) supplied by the main theorem of complex multiplication.

In this thesis we study moduli stacks \calM_\Phi^n, indexed by an integer n>0 and a CM-type (K,\Phi), which parametrize abelian schemes equipped with action by \OK and an \OK-linear principal polarization, such that the representation of \OK on the relative Lie algebra of the abelian scheme consists of n copies of each character in \Phi. We do this by systematically applying Serre's tensor construction, and for that we first establish a general correspondence between polarizations on abelian schemes M\otimes_R A arising from this construction and polarizations on the abelian scheme A, along with positive definite hermitian forms on the module M. Next we describe a tensor product of categories and apply it to the category \Herm_n(\OK) of finite non-degenerate positive-definite \OK-hermitian modules of rank n and the category fibred in groupoids \calM_\Phi^1 of principally polarized CM abelian schemes. Assuming n is prime to the class number of K, we show that Serre's tensor construction provides an identification of this tensor product with a substack of the moduli space \calM_\Phi^n, and that in some cases, such as when the base is finite type over \CC or an algebraically closed field of characteristic zero, this substack is the entire space. We then use this characterization to describe the Galois action on \calM_\Phi^n(\overline{\QQ}), by using the description of the action on \calM_\Phi^1(\overline{\QQ}) supplied by the main theorem of complex multiplication.

Automatic Repeat-reQuest (ARQ) schemes are extensively used in communication systems and computer networks to achieve reliable transmission. Using space-time codes (STCs) with multiple input multiple output (MIMO) or multiple input single output (MISO) systems is an effective way to combat multipath fading, which is the most severe impairment for wireless communication systems. STCs are designed to use the rich scattering multipath environment provided by using multiple transmit antennas. The work done in this thesis focuses on the use of ARQ schemes with non-orthogonal space-time block codes (NOSTBCs) based on Reed Solomon codes. The truncated-selective ARQ (TS-ARQ) scheme is considered and three novel hybrid ARQ (HARQ) schemes are proposed. Simulation results reveal that, compared to using TS-ARQ with orthogonal space-time block codes (OSTBCs), using NOSTBCs with any of the three proposed HARQ schemes can provide significant gains in terms of dropped packet rate and spectral efficiency at the cost of increased decoding complexity. The performance can be further improved by using the water filling principle to adaptively allocate transmit power among transmit antennas.

This thesis investigates investing in pyramid schemes. Study 1 explored the relationships between people s perceptions of investment options and their investment decisions. These options included a bank, a pyramid scheme, stock market and a safe. In this study, participants imagined they could invest money in any of the options and rated their perceptions of each option on various scales. When investing money, participants invested larger amounts in the options that they rated more positively. Compared to other investors, pyramid investors had higher positive correlations between their ratings of the pyramid scheme and how much money they invested. In Study 2 participants indicated how much money they would invest in each option and how risky they perceived the investment. As the perceived risk of an investment option increased, people invested less money. However, participants did not identify the pyramid scheme as the most risky option and rated it as being no more or less risky than the stock market. In both Studies 1 and 2 about half of the participants were willing to invest in the pyramid scheme. In Studies 3 and 4, participants imagined they had invested money in a pyramid scheme and were recruiting new target investors. Two experimental conditions were devised. In the first condition, participants were not informed of the potential for monetary loss, whereas in the second condition, monetary loss was made explicit. Potential target investors varied in the closeness of their rated relationship to the participant. When in the early non-loss condition, participants selected targets that were close to themselves, but in the loss condition they favoured targets that were less close. Furthermore, when in the non-loss condition, participants persuaded those targets they were closer to invest, whereas in the loss condition they persuaded them not to invest. Studies 5 and 6 found that there was no difference in sensation seeking propensities or intellect between pyramid scheme investors and non-investors. One clear finding for the research is that many people did not select the pyramid scheme as the poor investment that it is, a result which indicates its present illegal status is justified.

Thesis (M.S.)--University of Michigan, 1968.