Quantum cohomology via vicious and osculating walkers

We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang–Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged u^(n)k -WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov’s toric Schur functions and can be interpreted as generating functions for Gromov–Witten invariants. We reveal an underlying quantum group structure in terms of Yang–Baxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra

Atmospheric neutrons

Contributions to fast neutron measurements in the atmosphere are outlined. The results of a calculation to determine the production, distribution and final disappearance of atmospheric neutrons over the entire spectrum are presented. An attempt is made to answer questions that relate to processes such as neutron escape from the atmosphere and C-14 production. In addition, since variations of secondary neutrons can be related to variations in the primary radiation, comment on the modulation of both radiation components is made

Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model

Starting from the Verma module of U_q sl(2) we consider the evaluation module for affine U_q sl(2) and discuss its crystal limit (q=0). There exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations. Its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra, an extension of the finite plactic algebra first discussed by Lascoux and Sch\"{u}tzenberger. The corresponding noncommutative elementary symmetric polynomials were recently shown to be generated by the transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin and Kitanine. Here we establish that both generating functions satisfy Baxter's TQ-equation in the crystal limit by tying them to special U_q sl(2) solutions of the Yang-Baxter equation. The TQ-equation amounts to the well-known Jacobi-Trudy formula leading naturally to the definition of noncommutative Schur polynomials. The latter can be employed to define a ring which has applications in conformal field theory and enumerative geometry: it is isomorphic to the fusion ring of the sl(n)_k -WZNW model whose structure constants are the dimensions of spaces of generalized theta-functions over the Riemann sphere with three punctures.Comment: 24 pages, 6 figures; v2: several typos fixe

Non-crystallographic reduction of generalized Calogero-Moser models

We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero–Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is recovered. For the Hamiltonians of trigonometric, hyperbolic and elliptic types, we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio. We explicitly show for the simplest of these non-crystallographic models, how the corresponding classical equations of motion can be derived from a Lie algebraic Lax pair based on the larger, crystallographic Coxeter group

Physical and mental health comorbidity is common in people with multiple sclerosis: nationally representative cross-sectional population database analysis

<b>Background</b> Comorbidity in Multiple Sclerosis (MS) is associated with worse health and higher mortality. This study aims to describe clinician recorded comorbidities in people with MS. <p></p> <b>Methods</b> 39 comorbidities in 3826 people with MS aged ≥25 years were compared against 1,268,859 controls. Results were analysed by age, gender, and socioeconomic status, with unadjusted and adjusted Odds Ratios (ORs) calculated using logistic regression. <p></p> <b>Results</b> People with MS were more likely to have one (OR 2.44; 95% CI 2.26-2.64), two (OR 1.49; 95% CI 1.38-1.62), three (OR 1.86; 95% CI 1.69-2.04), four or more (OR 1.61; 95% CI 1.47-1.77) non-MS chronic conditions than controls, and greater mental health comorbidity (OR 2.94; 95% CI 2.75-3.14), which increased as the number of physical comorbidities rose. Cardiovascular conditions, including atrial fibrillation (OR 0.49; 95% CI 0.36-0.67), chronic kidney disease (OR 0.51; 95% CI 0.40-0.65), heart failure (OR 0.62; 95% CI 0.45-0.85), coronary heart disease (OR 0.64; 95% CI 0.52-0.71), and hypertension (OR 0.65; 95% CI 0.59-0.72) were significantly less common in people with MS. <p></p> <b>Conclusion</b> People with MS have excess multiple chronic conditions, with associated increased mental health comorbidity. The low recorded cardiovascular comorbidity warrants further investigation

Baxter operators for the quantum sl(3) invariant spin chain

The noncompact homogeneous sl(3) invariant spin chains are considered. We show that the transfer matrix with generic auxiliary space is factorized into the product of three sl(3) invariant commuting operators. These operators satisfy the finite difference equations in the spectral parameters which follow from the structure of the reducible sl(3) modules.Comment: 20 pages, 4 figures, references adde

Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators

The integrable quantum models, associated to the transfer matrices of the 6-vertex reflection algebra for spin 1/2 representations, are studied in this paper. In the framework of Sklyanin's quantum separation of variables (SOV), we provide the complete characterization of the eigenvalues and eigenstates of the transfer matrix and the proof of the simplicity of the transfer matrix spectrum. Moreover, we use these integrable quantum models as further key examples for which to develop a method in the SOV framework to compute matrix elements of local operators. This method has been introduced first in [1] and then used also in [2], it is based on the resolution of the quantum inverse problem (i.e. the reconstruction of all local operators in terms of the quantum separate variables) plus the computation of the action of separate covectors on separate vectors. In particular, for these integrable quantum models, which in the homogeneous limit reproduce the open spin-1/2 XXZ quantum chains with non-diagonal boundary conditions, we have obtained the SOV-reconstructions for a class of quasi-local operators and determinant formulae for the covector-vector actions. As consequence of these findings we provide one determinant formulae for the matrix elements of this class of reconstructed quasi-local operators on transfer matrix eigenstates.Comment: 40 pages. Minor modifications in the text and some notations and some more reference adde

Secondary analysis of teaching methods in introductory physics: A 50 k-student study

Citation: Von Korff, J., Archibeque, B., Gomez, K. A., Heckendorf, T., McKagan, S. B., Sayre, E. C., . . . Sorell, L. (2016). Secondary analysis of teaching methods in introductory physics: A 50 k-student study. American Journal of Physics, 84(12), 969-974. doi:10.1119/1.4964354Physics education researchers have developed many evidence-based instructional strategies to enhance conceptual learning of students in introductory physics courses. These strategies have historically been tested using assessments such as the Force Concept Inventory (FCI) and the Force and Motion Conceptual Evaluation (FMCE). We have performed a review and analysis of FCI and FMCE data published between 1995 and 2014. We confirm previous findings that interactive engagement teaching techniques are significantly more likely to produce high student learning gains than traditional lecture-based instruction. We also establish that interactive engagement instruction works in many settings, including those with students having a high and low level of prior knowledge, at liberal arts and research universities, and enrolled in both small and large classes. (C) 2016 Author(s)