The Necessity of Relay Selection

We determine necessary conditions on the structure of symbol error rate (SER) optimal quantizers for limited feedback beamforming in wireless networks with one transmitter-receiver pair and R parallel amplify-and-forward relays. We call a quantizer codebook "small" if its cardinality is less than R, and "large" otherwise. A "d-codebook" depends on the power constraints and can be optimized accordingly, while an "i-codebook" remains fixed. It was previously shown that any i-codebook that contains the single-relay selection (SRS) codebook achieves the full-diversity order, R. We prove the following: Every full-diversity i-codebook contains the SRS codebook, and thus is necessarily large. In general, as the power constraints grow to infinity, the limit of an optimal large d-codebook contains an SRS codebook, provided that it exists. For small codebooks, the maximal diversity is equal to the codebook cardinality. Every diversity-optimal small i-codebook is an orthogonal multiple-relay selection (OMRS) codebook. Moreover, the limit of an optimal small d-codebook is an OMRS codebook. We observe that SRS is nothing but a special case of OMRS for codebooks with cardinality equal to R. As a result, we call OMRS as "the universal necessary condition" for codebook optimality. Finally, we confirm our analytical findings through simulations.Comment: 29 pages, 4 figure

Systematics of Aligned Axions

We describe a novel technique that renders theories of $N$ axions tractable, and more generally can be used to efficiently analyze a large class of periodic potentials of arbitrary dimension. Such potentials are complex energy landscapes with a number of local minima that scales as $\sqrt{N!}$, and so for large $N$ appear to be analytically and numerically intractable. Our method is based on uncovering a set of approximate symmetries that exist in addition to the $N$ periods. These approximate symmetries, which are exponentially close to exact, allow us to locate the minima very efficiently and accurately and to analyze other characteristics of the potential. We apply our framework to evaluate the diameters of flat regions suitable for slow-roll inflation, which unifies, corrects and extends several forms of "axion alignment" previously observed in the literature. We find that in a broad class of random theories, the potential is smooth over diameters enhanced by $N^{3/2}$ compared to the typical scale of the potential. A Mathematica implementation of our framework is available online.Comment: 68 pages, 17 figure

Noncommutative Bundles and Instantons in Tehran

We present an introduction to the use of noncommutative geometry for gauge theories with emphasis on a construction of instantons for a class of four dimensional toric noncommutative manifolds. These instantons are solutions of self-duality equations and are critical points of an action functional. We explain the crucial role of twisted symmetries as well as methods from noncommutative index theorems.Comment: 64 pages; v2: major changes, parts rewritten, references added. Based on lectures delivered by GL at the ``International Workshop on Noncommutative Geometry NCG2005'', Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, September 200