The decomposition of level-1 irreducible highest weight modules with respect to the level-0 actions of the quantum affine algebra Uq′(sl^n)U'_q(\hat{sl}_n)

We decompose the level-1 irreducible highest weight modules of the quantum affine algebra $U_q(\hat{sl}_n)$ with respect to the level-0 $U'_q (\hat{sl}_n)$--action defined in q-alg/9702024. The decomposition is parameterized by the skew Young diagrams of the border strip type.Comment: 22 pages, AMSLaTe

Towards an open database of assessment material for STEM subjects: requirements and recommendations from early field trials

If appropriately implemented, open databases of instruction material may help teaching and learning by providing content for teaching activities, scaffolding, and self-assessment. The paper presents the current results of the development and implementation of a database that is expressly built for promoting exchange of questions and exercises, together with the associated solutions among teachers for STEM subjects. Besides presenting and motivating the initiative (together with reporting its current status), the manuscript lists a series of lessons that have been learned while executing the project - including the need for proper management of authorship and version control of the uploaded material. Moreover, the manuscript describes which features any open database of instruction material should implement to aid improved usability, together with a series of nontrivial theoretical and practical problems for future scientific investigations (e.g., developing taxonomies for indexing the difficulty levels of the instruction material uploaded in the database that do not suffer from the subjective interpretability associated with the existing taxonomies)

Sliding drops across alternating hydrophobic and hydrophilic stripes

We perform a joint numerical and experimental study to systematically characterize the motion of 30 μl drops of pure water and of ethanol in water solutions, sliding over a periodic array of alternating hydrophobic and hydrophilic stripes with a large wettability contrast and a typical width of hundreds of microns. The fraction of the hydrophobic areas has been varied from about 20% to 80%. The effects of the heterogeneous patterning can be described by a renormalized value of the critical Bond number, i.e., the critical dimensionless force needed to depin the drop before it starts to move. Close to the critical Bond number we observe a jerky motion characterized by an evident stick-slip dynamics. As a result, dissipation is strongly localized in time, and the mean velocity of the drops can easily decrease by an order of magnitude compared to the sliding on the homogeneous surface. Lattice Boltzmann numerical simulations are crucial for disclosing to what extent the sliding dynamics can be deduced from the computed balance of capillary, viscous, and body forces by varying the Bond number, the surface composition, and the liquid viscosity. Beyond the critical Bond number, we characterize both experimentally and numerically the dissipation inside the droplet by studying the relation between the average velocity and the applied volume forces

Stick-Slip Sliding of Water Drops on Chemically Heterogeneous Surfaces

We present a comprehensive study of water drops sliding down chemically heterogeneous surfaces formed by a periodic pattern of alternating hydrophobic and hydrophilic stripes. Drops are found to undergo a stick-slip motion whose average speed is an order of magnitude smaller than that measured on a homogeneous surface having the same static contact angle. This motion is the result of the periodic deformations of the drop interface when crossing the stripes. Numerical simulations confirm this view and are used to elucidate the principles underlying the experimental observations

Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A

We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out to be expressible only in terms of: - Schensted's classic bumping procedure, - the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to describe, acting on cylindric multipartitions. We explain how this can be seen as an analogue of the bumping algorithm for affine type $A$. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space

Representations of Double Affine Lie algebras

We study representations of the double affine Lie algebra associated to a simple Lie algebra. We construct a family of indecomposable integrable representations and identify their irreducible quotients. We also give a condition for the indecomposable modules to be irreducible, this is analogous to a result in the representation theory of quantum affine algebras. Finally, in the last section of the paper, we show, by using the notion of fusion product, that our modules are generically reducible

Control of HVAC Systems via Scenario-based Explicit MPC

Improving energy efficiency of Heating, Ventilation and Air Conditioning (HVAC) systems is a primary objective for the society. Model Predictive Control (MPC) techniques for HVAC systems have recently received particular attention, since they can naturally account for several factors, such as weather and occupancy forecasts, comfort ranges and actuation constraints. Developing effective MPC based control strategies for HVAC systems is nontrivial, since buildings dynamics are nonlinear and affected by various uncertainties. Further, the complexity of the MPC problem and the burden of on-line computations can lead to difficulties in integrating this scheme into a building management system. We propose to address this computational issue by designing a scenario-based explicit MPC strategy, i.e., a controller that is simultaneously based on explicit representations of the MPC feedback law and accounts for uncertainties in the occupancy patterns and weather conditions by using the scenarios paradigm. The main advantages of this approach are the absence of a-priori assumptions on the distributions of the uncertain variables, the applicability to any type of building, and the limited on-line computational burden, enabling practical implementations on low-cost hardware platforms. We illustrate the practical implementation of the proposed explicit MPC controller on a room of a university building, showing its effectiveness and computational tractability

On minimal affinizations of representations of quantum groups

In this paper we study minimal affinizations of representations of quantum groups (generalizations of Kirillov-Reshetikhin modules of quantum affine algebras introduced by Chari). We prove that all minimal affinizations in types A, B, G are special in the sense of monomials. Although this property is not satisfied in general, we also prove an analog property for a large class of minimal affinization in types C, D, F. As an application, the Frenkel-Mukhin algorithm works for these modules. For minimal affinizations of type A, B we prove the thin property (the l-weight spaces are of dimension 1) and a conjecture of Nakai-Nakanishi (already known for type A). The proof of the special property is extended uniformly for more general quantum affinizations of quantum Kac-Moody algebras.Comment: 38 pages; references and additional results added. Accepted for publication in Communications in Mathematical Physic

Dorey's Rule and the q-Characters of Simply-Laced Quantum Affine Algebras

Let Uq(ghat) be the quantum affine algebra associated to a simply-laced simple Lie algebra g. We examine the relationship between Dorey's rule, which is a geometrical statement about Coxeter orbits of g-weights, and the structure of q-characters of fundamental representations V_{i,a} of Uq(ghat). In particular, we prove, without recourse to the ADE classification, that the rule provides a necessary and sufficient condition for the monomial 1 to appear in the q-character of a three-fold tensor product V_{i,a} x V_{j,b} x V_{k,c}.Comment: 30 pages, latex; v2, to appear in Communications in Mathematical Physic

Trace as an alternative decategorification functor

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an action of the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with arXiv:1405.5920 by other author