Algebraic Geometry methods associated to the one-dimensional Hubbard model

In this paper we study the covering vertex model of the one-dimensional Hubbard Hamiltonian constructed by Shastry in the realm of algebraic geometry. We show that the Lax operator sits in a genus one curve which is not isomorphic but only isogenous to the curve suitable for the AdS/CFT context. We provide an uniformization of the Lax operator in terms of ratios of theta functions allowing us to establish relativistic like properties such as crossing and unitarity. We show that the respective $\mathrm{R}$-matrix weights lie on an Abelian surface being birational to the product of two elliptic curves with distinct $\mathrm{J}$-invariants. One of the curves is isomorphic to that of the Lax operator but the other is solely fourfold isogenous. These results clarify the reason the $\mathrm{R}$-matrix can not be written using only difference of spectral parameters of the Lax operator.Comment: 24 page

Beyond shared memory loop parallelism in the polyhedral model

2013 Spring.Includes bibliographical references.With the introduction of multi-core processors, motivated by power and energy concerns, parallel processing has become main-stream. Parallel programming is much more difficult due to its non-deterministic nature, and because of parallel programming bugs that arise from non-determinacy. One solution is automatic parallelization, where it is entirely up to the compiler to efficiently parallelize sequential programs. However, automatic parallelization is very difficult, and only a handful of successful techniques are available, even after decades of research. Automatic parallelization for distributed memory architectures is even more problematic in that it requires explicit handling of data partitioning and communication. Since data must be partitioned among multiple nodes that do not share memory, the original memory allocation of sequential programs cannot be directly used. One of the main contributions of this dissertation is the development of techniques for generating distributed memory parallel code with parametric tiling. Our approach builds on important contributions to the polyhedral model, a mathematical framework for reasoning about program transformations. We show that many affine control programs can be uniformized only with simple techniques. Being able to assume uniform dependences significantly simplifies distributed memory code generation, and also enables parametric tiling. Our approach implemented in the AlphaZ system, a system for prototyping analyses, transformations, and code generators in the polyhedral model. The key features of AlphaZ are memory re-allocation, and explicit representation of reductions. We evaluate our approach on a collection of polyhedral kernels from the PolyBench suite, and show that our approach scales as well as PLuTo, a state-of-the-art shared memory automatic parallelizer using the polyhedral model. Automatic parallelization is only one approach to dealing with the non-deterministic nature of parallel programming that leaves the difficulty entirely to the compiler. Another approach is to develop novel parallel programming languages. These languages, such as X10, aim to provide highly productive parallel programming environment by including parallelism into the language design. However, even in these languages, parallel bugs remain to be an important issue that hinders programmer productivity. Another contribution of this dissertation is to extend the array dataflow analysis to handle a subset of X10 programs. We apply the result of dataflow analysis to statically guarantee determinism. Providing static guarantees can significantly increase programmer productivity by catching questionable implementations at compile-time, or even while programming

Counting walks in a quadrant: a unified approach via boundary value problems

The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations as well as on the sign of the covariance of the walkComment: 28 pages; 6 figure

Non-uniform dependences partitioned by recurrence chains

Non-uniform distance loop dependences are a known obstacle to find parallel iterations. To find the outermost loop parallelism in these �irregular� loops, a novel method is presented based on recurrence chains. The scheme organizes non-uniformly dependent iterations into lexicographically ordered monotonic chains. While the initial and final iteration of monotonic chains form two parallel sets, the remaining iterations form an intermediate set that can be partitioned further. When there is only one pair of coupled array references, the non-uniform dependences are represented by a single recurrence equation. In that case, the chains in the intermediate set do not bifurcate and each can be executed as a WHILE loop. The independent iterations and the initial iterations of monotonic dependence chains constitute the outermost parallelism. The proposed approach compares favorably with other treatments of nonuniform dependences in the literature. When there are multiple recurrence equations, a dataflow parallel execution can be scheduled using the technique extensively to find maximum loop parallelism

Dynamic Homotopy and Landscape Dynamical Set Topology in Quantum Control

We examine the topology of the subset of controls taking a given initial state to a given final state in quantum control, where "state" may mean a pure state |\psi>, an ensemble density matrix \rho, or a unitary propagator U(0,T). The analysis consists in showing that the endpoint map acting on control space is a Hurewicz fibration for a large class of affine control systems with vector controls. Exploiting the resulting fibration sequence and the long exact sequence of basepoint-preserving homotopy classes of maps, we show that the indicated subset of controls is homotopy equivalent to the loopspace of the state manifold. This not only allows us to understand the connectedness of "dynamical sets" realized as preimages of subsets of the state space through this endpoint map, but also provides a wealth of additional topological information about such subsets of control space.Comment: Minor clarifications, and added new appendix addressing scalar control of 2-level quantum system

Surface Comparison with Mass Transportation

We use mass-transportation as a tool to compare surfaces (2-manifolds). In particular, we determine the "similarity" of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global M\"obius transformations. Our approach provides a constructive way of defining a metric in the abstract space of simply-connected smooth surfaces with boundary (i.e. surfaces of disk-type); this metric can also be used to define meaningful intrinsic distances between pairs of "patches" in the two surfaces, which allows automatic alignment of the surfaces. We provide numerical experiments on "real-life" surfaces to demonstrate possible applications in natural sciences

Computing canonical heights using arithmetic intersection theory

For several applications in the arithmetic of abelian varieties it is important to compute canonical heights. Following Faltings and Hriljac, we show how the canonical height on the Jacobian of a smooth projective curve can be computed using arithmetic intersection theory on a regular model of the curve in practice. In the case of hyperelliptic curves we present a complete algorithm that has been implemented in Magma. Several examples are computed and the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections. Added new Example