A construction of real weight functions for certain orthogonal polynomials in two variables

AbstractH.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are(x2+y)uxx+2xyuxy+y2uyy+gxux+g(y−1)uy=λu, andx2uxx+2xyuxy+(y2−y)uyy+g(x−1)ux+g(y−γ)uy=λu. Even though they showed that these equations have a sequence of weak orthogonal polynomial solutions, they were unable to show that these polynomials were, in fact, orthogonal. The orthogonality of these two polynomial sequences was recently established by Kwon, Littlejohn, and Lee solving an open problem from 1967.In this paper, we construct explicit weight functions for these two orthogonal polynomial sequences, using a method first developed by Littlejohn and then further developed by Han, Kim, and Kwon. Moreover, two additional partial differential equations were found by Kwon, Littlejohn, and Lee that have sequences of orthogonal polynomial solutions. These equations are given by(x2−x)uxx+2xyuxy+y2uyy+(dx+e)ux+(dy+h)uy=λu,xuxx+2yuxy+(dx+e)ux+(dy+h)uy=λu. In each of these examples, we also produce explicit orthogonalizing weight functions

Efficient low-threshold lasers based on an erbium-doped holey fiber

We report experimental results on the continuous-wave lasers based on a small core erbium-doped holey fiber. In a simple Fabry-Perot-type cavity with high output coupling, we demonstrate low-threshold (0.55 mW) high slope-efficiency (57.3%) operation confirming both the quality and exceptionally high gain efficiency of the fiber. In an all-fiber ring cavity where the cavity loss is reduced, we show that it is possible to achieve a low-threshold laser with extremely wide tunability (>100 nm around 1550 nm). Our results illustrate some of the unique opportunities provided by active small core holey fibers

Holstein model in infinite dimensions at half-filling

The normal state of the Holstein model is studied at half-filling in infinite dimensions and in the adiabatic regime. The dynamical mean-field equations are solved using perturbation expansions around the extremal paths of the effective action for the atoms. We find that the Migdal-Eliashberg expansion breaks down in the metallic state if the electron-phonon coupling $\lambda$ exceeds a value of about 1.3 in spite of the fact that the formal expansion parameter $\lambda \omega_0/E_F$ ($\omega_0$ is the phonon frequency, $E_F$ the Fermi energy) is much smaller than 1. The breakdown is due to the appearance of more than one extremal path of the action. We present numerical results which illustrate in detail the evolution of the local Green's function, the self-energy and the effective atomic potential as a function of $\lambda$.Comment: Revtex + 17 postscript figures include

Orthogonal polynomials in two variables and second-order partial differential equations

AbstractWe study the second-order partial differential equations L[u] = Auxx +22Buxy + Cuyy + Dux + Euy = λnu, which have orthogonal polynomials in two variables as solutions. By using formal functional calculus on moment functionals, we first give new simpler proofs and improvements of the results by Krall and Sheffer and Littlejohn. We then give a two-variable version of Al-Salam and Chihara's characterization of classical orthogonal polynomials in one variable. We also study in detail the case when L[·] belongs to the basic class, that is, Ay = Cx = 0. In particular, we characterize all such differential equations which have a product of two classical orthogonal polynomials in one variable as solutions

Structures for Interacting Composite Fermions: Stripes, Bubbles, and Fractional Quantum Hall Effect

Much of the present day qualitative phenomenology of the fractional quantum Hall effect can be understood by neglecting the interactions between composite fermions altogether. For example the fractional quantum Hall effect at $\nu=n/(2pn\pm 1)$ corresponds to filled composite-fermion Landau levels,and the compressible state at $\nu=1/2p$ to the Fermi sea of composite fermions. Away from these filling factors, the residual interactions between composite fermions will determine the nature of the ground state. In this article, a model is constructed for the residual interaction between composite fermions, and various possible states are considered in a variational approach. Our study suggests formation of composite-fermion stripes, bubble crystals, as well as fractional quantum Hall states for appropriate situations.Comment: 16 pages, 7 figure

Impacts of Problem-Based Instruction on Students’ Beliefs about Physics and Learning Physics

To help prepare students to address future challenges in Science, Technology, Engineering, and Mathematics (STEM), they need to develop 21st-century skills. These skills are mediated by their beliefs about the nature of scientific knowledge and practices, or epistemological beliefs. One approach shown to support students’ development of these beliefs and skills is problem-based instruction (PBI), which encourages collaborative self-directed learning while working on open-ended problems. We used a mixed-method qualitative approach to examine how implementing PBI in a physics course taught at a Dutch university affected students’ beliefs about physics and learning physics. Analysis of the responses to the course surveys (41–74% response rates) from the first implementation indicated students appreciated opportunities for social interactions with peers and use of scientific equipment with PBI but found difficulties connecting to the Internet given the COVID-19 restrictions. The Colorado Learning Attitudes towards Science Survey (CLASS), a validated survey on epistemological beliefs about physics and learning physics, was completed by a second cohort of students in a subsequent implementation of PBI for the same course; analysis of the students’ pre- and post-responses (28% response rate) showed a slight shift towards more expert-like perspectives despite challenges (e.g., access to lab). Findings from this study may inform teachers with an interest in supporting the development of students’ epistemological beliefs about STEM and the implementation of PBI in undergraduate STEM courses.</p

Skyrmions in Higher Landau Levels

We calculate the energies of quasiparticles with large numbers of reversed spins (``skyrmions'') for odd integer filling factors 2k+1, k is greater than or equals 1. We find, in contrast with the known result for filling factor equals 1 (k = 0), that these quasiparticles always have higher energy than the fully polarized ones and hence are not the low energy charged excitations, even at small Zeeman energies. It follows that skyrmions are the relevant quasiparticles only at filling factors 1, 1/3 and 1/5.Comment: 10 pages, RevTe

Direct Single-Molecule Observation of Mode and Geometry of RecA-Mediated Homology Search

Genomic integrity, when compromised by accrued DNA lesions, is maintained through efficient repair via homologous recombination. For this process the ubiquitous recombinase A (RecA), and its homologues such as the human Rad51, are of central importance, able to align and exchange homologous sequences within single-stranded and double-stranded DNA in order to swap out defective regions. Here, we directly observe the widely debated mechanism of RecA homology searching at a single-molecule level using high-speed atomic force microscopy (HS-AFM) in combination with tailored DNA origami frames to present the reaction targets in a way suitable for AFM-imaging. We show that RecA nucleoprotein filaments move along DNA substrates via short-distance facilitated diffusions, or slides, interspersed with longer-distance random moves, or hops. Importantly, from the specific interaction geometry, we find that the double-stranded substrate DNA resides in the secondary DNA binding-site within the RecA nucleoprotein filament helical groove during the homology search. This work demonstrates that tailored DNA origami, in conjunction with HS-AFM, can be employed to reveal directly conformational and geometrical information on dynamic protein-DNA interactions which was previously inaccessible at an individual single-molecule level

Partially spin polarized quantum Hall effect in the filling factor range 1/3 < nu < 2/5

The residual interaction between composite fermions (CFs) can express itself through higher order fractional Hall effect. With the help of diagonalization in a truncated composite fermion basis of low-energy many-body states, we predict that quantum Hall effect with partial spin polarization is possible at several fractions between $\nu=1/3$ and $\nu=2/5$. The estimated excitation gaps are approximately two orders of magnitude smaller than the gap at $\nu=1/3$, confirming that the inter-CF interaction is extremely weak in higher CF levels.Comment: 4 pages, 3 figure