The Rahman polynomials and the Lie algebra sl_3(C)

We interpret the Rahman polynomials in terms of the Lie algebra $sl_3(C)$. Using the parameters of the polynomials we define two Cartan subalgebras for $sl_3(C)$, denoted $H$ and $\tilde{H}$. We display an antiautomorphism $\dagger$ of $sl_3(C)$ that fixes each element of $H$ and each element of $\tilde{H}$. We consider a certain finite-dimensional irreducible $sl_3(C)$-module $V$ consisting of homogeneous polynomials in three variables. We display a nondegenerate symmetric bilinear form on $V$ such that $ = $ for all $\beta \in sl_3(C)$ and $\xi,\zeta \in V$. We display two bases for $V$; one diagonalizes $H$ and the other diagonalizes $\tilde{H}$. Both bases are orthogonal with respect to . We show that when $$ is applied to a vector in each basis, the result is a trivial factor times a Rahman polynomial evaluated at an appropriate argument. Thus for both transition matrices between the bases each entry is described by a Rahman polynomial. From these results we recover the previously known orthogonality relation for the Rahman polynomials. We also obtain two seven-term recurrence relations satisfied by the Rahman polynomials, along with the corresponding relations satisfied by the dual polynomials. These recurrence relations show that the Rahman polynomials are bispectral. In our theory the roles of $H$ and $\tilde{H}$ are interchangable, and for us this explains the duality and bispectrality of the Rahman polynomials. We view the action of $H$ and $\tilde{H}$ on $V$ as a rank 2 generalization of a Leonard pair

A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application

The one variable Krawtchouk polynomials, a special case of the $_2F_1$ function did appear in the spectral representation of the transition kernel for a Markov chain studied a long time ago by M. Hoare and M. Rahman. A multivariable extension of this Markov chain was considered in a later paper by these authors where a certain two variable extension of the $F_1$ Appel function shows up in the spectral analysis of the corresponding transition kernel. Independently of any probabilistic consideration a certain multivariable version of the Gelfand-Aomoto hypergeometric function was considered in papers by H. Mizukawa and H. Tanaka. These authors and others such as P. Iliev and P. Tertwilliger treat the two-dimensional version of the Hoare-Rahman work from a Lie-theoretic point of view. P. Iliev then treats the general $n$-dimensional case. All of these authors proved several properties of these functions. Here we show that these functions play a crucial role in the spectral analysis of the transition kernel that comes from pushing the work of Hoare-Rahman to the multivariable case. The methods employed here to prove this as well as several properties of these functions are completely different to those used by the authors mentioned above

Association between antenatal depression and low birthweight in a developing country

Rahman A, Bunn J, Lovel H, Creed F. Association between antenatal depression and low birthweight in a developing country

STREAM Journal, Vol. 6, No. 1, pp 1-21. Jan-March 2007

CONTENTS: Approaches to understanding pond-dike systems in Asia: the POND-LIVE project approach, by Dave Little, Marc Verdegem and Roel Bosma. The contribution of fish ponds to nutrient cycling in integrated farming systems, by P.N. Muendo, J.J. Stoorvogel and Marc Verdegem. Improving the contribution of fishfarming to livelihoods in Northeast Thailand, by Chittra Arjinkit, Roel Bosma, Danai Turongrouang. Benefits of pond-dike systems in Bangladesh, by M.S. Kabir, M.A. Wahab and Marc Verdegem. Common carp increases rohu production in farmers ponds, by Mohammad Mustafizur Rahman, Md. Abdul Wahab and Marc C.J. Verdegem. Improving pond-dike farming systems in the Mekong delta, Vietnam; the Can Tho approach, by Dang Kieu Nhan, Le Thanh Duong, Le Thanh Phong, Roel H. Bosma and Marc C.J. Verdegem. Fuzzy pathways for farm development in Vietnam, by Roel H. Bosma, Le Thanh Phong, and Dang Kieu Nhan

A Simplified Approach to Optimally Controlled Quantum Dynamics

A new formalism for the optimal control of quantum mechanical physical observables is presented. This approach is based on an analogous classical control technique reported previously[J. Botina, H. Rabitz and N. Rahman, J. chem. Phys. Vol. 102, pag. 226 (1995)]. Quantum Lagrange multiplier functions are used to preserve a chosen subset of the observable dynamics of interest. As a result, a corresponding small set of Lagrange multipliers needs to be calculated and they are only a function of time. This is a considerable simplification over traditional quantum optimal control theory[S. shi and H. Rabitz, comp. Phys. Comm. Vol. 63, pag. 71 (1991)]. The success of the new approach is based on taking advantage of the multiplicity of solutions to virtually any problem of quantum control to meet a physical objective. A family of such simplified formulations is introduced and numerically tested. Results are presented for these algorithms and compared with previous reported work on a model problem for selective unimolecular reaction induced by an external optical electric field.Comment: Revtex, 29 pages (incl. figures

On the Ionization of Luminous WMAP Sources in the Galaxy : Constraints from He Recombination Line Observations with the GBT

The Wilkinson Microwave Anisotropy Probe (WMAP) free-free foreground emission map is used to identify diffuse ionized regions (DIR) in the Galaxy (Rahman & Murray 2010). It has been found that the 18 most luminous WMAP sources produce more than half of the total ionizing luminosity of the Galaxy. We observed radio recombination lines (RRLs) toward the luminous WMAP source G49.75-0.45 with the Green Bank Telescope near 1.4 GHz. Hydrogen RRL is detected toward the source but no helium line is detected, implying that n_He+/n_H+ < 0.024. This limit puts severe constraint on the ionizing spectrum. The total ionizing luminosity of G49 (3.05 x 10^51 s^-1) is ~ 2.8 times the luminosity of all radio HII regions within this DIR and this is generally the case for other WMAP sources. Murray & Rahman (2010) propose that the additional ionization is due to massive clusters (~ 7.5 x10^3 Msun for G49) embedded in the WMAP sources. Such clusters should produce enough photons with energy \geq 24.6 eV to fully ionize helium in the DIR. Our observations rule out a simple model with G49 ionized by a massive cluster. We also considered 'leaky' HII region models for the ionization of the DIR, suggested by Lockman and Anantharamaiah, but these models also cannot explain our observations. We estimate that the helium ionizing photons need to be attenuated by > ~10 times to explain the observations. If selective absorption of He- ionizing photons by dust is causing this additional attenuation, then the ratio of dust absorption cross sections for He- and H- ionizing photons should be > ~6.Comment: Accepted for publication in the ApJ; 14 pages, 3 figure