A generalization of the Heine--Stieltjes theorem

We extend the Heine-Stieltjes Theorem to concern all (non-degenerate) differential operators preserving the property of having only real zeros. This solves a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question on how to generalize their results to higher degrees. Many of the results are new even for the classical case.Comment: 12 pages, typos corrected and refined the interlacing theorem

Towards Deconstruction of the Type D (2,0) Theory

We propose a four-dimensional supersymmetric theory that deconstructs, in a particular limit, the six-dimensional $(2,0)$ theory of type $D_k$. This 4d theory is defined by a necklace quiver with alternating gauge nodes $\mathrm{O}(2k)$ and $\mathrm{Sp}(k)$. We test this proposal by comparing the 6d half-BPS index to the Higgs branch Hilbert series of the 4d theory. In the process, we overcome several technical difficulties, such as Hilbert series calculations for non-complete intersections, and the choice of $\mathrm{O}$ versus $\mathrm{SO}$ gauge groups. Consistently, the result matches the Coulomb branch formula for the mirror theory upon reduction to 3d

STEAM (SCIENCE, TECHNOLOGY, ENGINEERING, ART, AND MATHEMATICS) EDUCATION AND TEACHERS’ PEDAGOGICAL DISCONTENTMENT LEVELS

Literature focused on the emerging implementation of STEAM (Science, Technology, Engineering, Art, and Mathematics) education within K-12 education programs in the USA emphasizes the need for researchers to address the potential for disrupting the pedagogical contentment of teachers assigned to teach in STEAM disciplines who have no backgrounds in the fields represented by STEAM and/or who have no prior teaching experience in the areas of STEAM. Research objectives for the current study focused on examining the influence of intensive professional development on K-12 teachers’ pedagogical discontentment levels relative to the implementation of STEAM in all classrooms within a rural school district in the southeast region of the United States. Data sources included: (a) pre-post assessments of 93 teachers’ pedagogical discontentment levels; (b) classroom observation data collected by external observers; and (c) teachers’ perceptions of STEAM coaching.  Data retrieved were analyzed using quantitative analyses. Study findings indicated teachers’ pedagogical discontentment levels decreased over time. Future research must support teachers new to STEAM activities

Localization Properties of the Chalker-Coddington Model

The Chalker Coddington quantum network percolation model is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We study the model restricted to a cylinder of perimeter 2M. We prove firstly that the Lyapunov exponents are simple and in particular that the localization length is finite; secondly that this implies spectral localization. Thirdly we prove a Thouless formula and compute the mean Lyapunov exponent which is independent of M.Comment: 29 pages, 1 figure. New section added in which simplicity of the Lyapunov spectrum and finiteness of the localization length are proven. To appear in Annales Henri Poincar