Applying parabolic Peterson: affine algebras and the quantum cohomology of the Grassmannian

The Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson’s map is only a surjection, and one needs to quotient by a suitable ideal on the affine side to map isomorphically onto the quantum cohomology. We provide a detailed exposition of this parabolic Peterson isomorphism in the case of the Grassmannian of m-planes in complex n-space, including an explicit recipe for doing quantum Schubert calculus in terms of the appropriate subset of non-commutative k-Schur functions. As an application, we recast Postnikov’s affine approach to the quantum cohomology of the Grassmannian as a consequence of parabolic Peterson by showing that the affine nilTemperley–Lieb algebra arises naturally when forming the requisite quotient of the homology of the affine Grassmannian

Maximal Newton points and the quantum Bruhat graph

We discuss a surprising relationship between the partially ordered set of Newton points associated with an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the unique maximum element in this poset in terms of paths in the quantum Bruhat graph, whose vertices are indexed by elements in the finite Weyl group. Key to establishing this connection is the fact that paths in the quantum Bruhat graph encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson’s isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. One important geometric application of the present work is an inequality which provides a necessary condition for nonemptiness of certain affine Deligne–Lusztig varieties in the affine flag variety

An affine approach to Peterson comparison

The Peterson comparison formula proved by Woodward relates the three-pointedGromov-Witten invariants for the quantum cohomology of partial flag varietiesto those for the complete flag. Another such comparison can be obtained bycomposing a combinatorial version of the Peterson isomorphism with a result ofLapointe and Morse relating quantum Littlewood-Richardson coefficients for theGrassmannian to k-Schur analogs in the homology of the affine Grassmannianobtained by adding rim hooks. We show that these comparisons on quantumcohomology are equivalent, up to Postnikov's strange duality isomorphism.<br

An equivariant quantum Pieri rule for the Grassmannian on cylindric shapes

The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of Gorbounov and Korff in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of Huang and Li and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety

A novel algorithm for dynamic student profile adaptation based on learning styles

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.E-learning recommendation systems are used to enhance student performance and knowledge by providing tailor- made services based on the students’ preferences and learning styles, which are typically stored in student profiles. For such systems to remain effective, the profiles need to be able to adapt and reflect the students’ changing behaviour. In this paper, we introduce new algorithms that are designed to track student learning behaviour patterns, capture their learning styles, and maintain dynamic student profiles within a recommendation system (RS). This paper also proposes a new method to extract features that characterise student behaviour to identify students’ learning styles with respect to the Felder-Silverman learning style model (FSLSM). In order to test the efficiency of the proposed algorithm, we present a series of experiments that use a dataset of real students to demonstrate how our proposed algorithm can effectively model a dynamic student profile and adapt to different student learning behaviour. The results revealed that the students could effectively increase their learning efficiency and quality for the courses when the learning styles are identified, and proper recommendations are made by using our method

Direct observation of photonic Landau levels and helical edge states in strained honeycomb lattices

We report the realization of a synthetic magnetic field for photons and polaritons in a honeycomb lattice of coupled semiconductor micropillars. A strong synthetic field is induced in both the s and p orbital bands by engineering a uniaxial hopping gradient in the lattice, giving rise to the formation of Landau levels at the Dirac points. We provide direct evidence of the sublattice symmetry breaking of the lowest-order Landau level wavefunction, a distinctive feature of synthetic magnetic fields. Our realization implements helical edge states in the gap between n=0 and n=1 Landau levels, experimentally demonstrating a novel way of engineering propagating edge states in photonic lattices. In light of recent advances in the enhancement of polariton-polariton nonlinearities, the Landau levels reported here are promising for the study of the interplay between pseudomagnetism and interactions in a photonic system

T-2 toxin adsorption by hectorite

The adsorption of T-2 toxin by the natural smectite mineral – hectorite at pH 3.0, 7.0 and 9.0 was investigated. The results of T-2 toxin adsorption on hectorite showed that the T-2 adsorption capacity decreased with increasing concentration of adsorbent in the suspension for all the investigated pH values. From the adsorption isotherms, an increase in T-2 toxin adsorption with increasing initial T-2 toxin concentration was observed for all the investigated pH values. The T-2 toxin adsorption by hectorite followed a non-linear (Langmuir) type of isotherm at pH 3.0, 7.0 and 9.0, with correlation coefficients (r2) of 0.943 at pH 3.0, 0.919 at pH 7.0 and 0.939 at pH 9.0. The estimated maximum T-2 toxin adsorption by hectorite based on the Langmuir fit to the data (9.178 mg/g at pH 3.0, 9.930 mg/g at pH 7.0, and 19.341 mg/g at pH 9.0), indicated that the adsorption of T-2 toxin by hectorite is pH dependent. The obtained data suggest the existence of specific active sites in hectorite onto which the T-2 toxin is adsorbed

Chagasic Thymic Atrophy Does Not Affect Negative Selection but Results in the Export of Activated CD4+CD8+ T Cells in Severe Forms of Human Disease

Extrathymic CD4+CD8+ double-positive (DP) T cells are increased in some pathophysiological conditions, including infectious diseases. In the murine model of Chagas disease, it has been shown that the protozoan parasite Trypanosoma cruzi is able to target the thymus and induce alterations of the thymic microenvironment and the lymphoid compartment. In the acute phase, this results in a severe atrophy of the organ and early release of DP cells into the periphery. To date, the effect of the changes promoted by the parasite infection on thymic central tolerance has remained elusive. Herein we show that the intrathymic key elements that are necessary to promote the negative selection of thymocytes undergoing maturation during the thymopoiesis remains functional during the acute chagasic thymic atrophy. Intrathymic expression of the autoimmune regulator factor (Aire) and tissue-restricted antigen (TRA) genes is normal. In addition, the expression of the proapoptotic Bim protein in thymocytes was not changed, revealing that the parasite infection-induced thymus atrophy has no effect on these marker genes necessary to promote clonal deletion of T cells. In a chicken egg ovalbumin (OVA)-specific T-cell receptor (TCR) transgenic system, the administration of OVA peptide into infected mice with thymic atrophy promoted OVA-specific thymocyte apoptosis, further indicating normal negative selection process during the infection. Yet, although the intrathymic checkpoints necessary for thymic negative selection are present in the acute phase of Chagas disease, we found that the DP cells released into the periphery acquire an activated phenotype similar to what is described for activated effector or memory single-positive T cells. Most interestingly, we also demonstrate that increased percentages of peripheral blood subset of DP cells exhibiting an activated HLA-DR+ phenotype are associated with severe cardiac forms of human chronic Chagas disease. These cells may contribute to the immunopathological events seen in the Chagas disease