Controlling the path of discretized light in waveguide lattices

A general method for flexible control of the path of discretized light beams in homogeneous waveguide lattices, based on longitudinal modulation of the coupling constant, is theoretically proposed. As compared to beam steering and refraction achievable in graded-index waveguide arrays, the proposed approach enables to synthesize rather arbitrary target paths

Automation of Determination of Optimal Intra-Compute Node Parallelism

Maximizing the productivity of modern multicore and manycore chips requires optimizing parallelism at the compute node level. This is, however, a complex multi-step process. It is an iterative method requiring determining optimal degrees of parallel scalability and optimizing memory access behavior. Further, there are multiple cases to be considered, programs which use only MPI or OpenMP and hybrid (MPI +OpenMP) programs. This paper presents a set of three coordinated workflows for determining the optimal parallelism at the program level for MPI programs and at the loop level for hybrid (MPI+OpenMP) cases. The paper also details mostly automated implementations of these workflows using the PerfExpert infrastructure. Finally the paper presents case studies demonstrating both the applicability and the effectiveness of optimizing parallelism at the compute node level. The results shown in the paper will provide valuable information to further advance in the full automation of the workflows. The software implementing the parallelism scalability optimization is open source and available for download.Texas Advanced Computing Center (TACC)Computer Science

On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2

For an integer $k\geq 2$, let $\{F^{(k)}_{n} \}_{n\geq 0}$ be the $k$--generalized Fibonacci sequence which starts with $0, \ldots, 0, 1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all integers $c$ having at least two presentations as a difference between a $k$--generalized Fibonacci number and a powers of 2 for any fixed $k \geqslant 4$. This paper extends previous work from [9] for the case $k=2$ and [6] for the case $k=3$

On Integrable Quantum Group Invariant Antiferromagnets

A new open spin chain hamiltonian is introduced. It is both integrable (Sklyanin`s type $K$ matrices are used to achieve this) and invariant under ${\cal U}_{\epsilon}(sl(2))$ transformations in nilpotent irreps for $\epsilon^3=1$. Some considerations on the centralizer of nilpotent representations and its representation theory are also presented.Comment: IFF-5/92, 13 pages, LaTex file, 8 figures available from author

A conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials

The Galactic Branches as a Possible Evidence for Transient Spiral Arms

With the use of a background Milky-Way-like potential model, we performed stellar orbital and magnetohydrodynamic (MHD) simulations. As a first experiment, we studied the gaseous response to a bisymmetric spiral arm potential: the widely employed cosine potential model and a self-gravitating tridimensional density distribution based model called PERLAS. Important differences are noticeable in these simulations, while the simplified cosine potential produces two spiral arms for all cases, the more realistic density based model produces a response of four spiral arms on the gaseous disk, except for weak arms -i.e. close to the linear regime- where a two-armed structure is formed. In order to compare the stellar and gas response to the spiral arms, we have also included a detailed periodic orbit study and explored different structural parameters within observational uncertainties. The four armed response has been explained as the result of ultra harmonic resonances, or as shocks with the massive bisymmetric spiral structure, among other. From the results of this work, and comparing the stellar and gaseous responses, we tracked down an alternative explanation to the formation of branches, based only on the orbital response to a self-gravitating spiral arms model. The presence of features such as branches, might be an indication of transiency of the arms.Comment: 17 pages, 9 figures. Accepted for publication in MNRA

Exceptional orthogonal polynomials and the Darboux transformation

We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.Comment: corrected abstract, added references, minor correction