Schur Superpolynomials: Combinatorial Definition and Pieri Rule

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity

The supersymmetric Ruijsenaars-Schneider model

An integrable supersymmetric generalization of the trigonometric Ruijsenaars-Schneider model is presented whose symmetry algebra includes the super Poincar\'e algebra. Moreover, its Hamiltonian is showed to be diagonalized by the recently introduced Macdonald superpolynomials. Somewhat surprisingly, the consistency of the scalar product forces the discreteness of the Hilbert space.Comment: v1: 11 pages, 1 figure. v2: new format, 5 pages, short section added at the end of the article addressing the problem of consistency of the scalar product (e.g., positivity of the weight functions and the normalization of the ground state wave function). To appear in Physical Review Letter

A quartet of fermionic expressions for M(k,2k±1)M(k,2k\pm1) Virasoro characters via half-lattice paths

We derive new fermionic expressions for the characters of the Virasoro minimal models $M(k,2k\pm1)$ by analysing the recently introduced half-lattice paths. These fermionic expressions display a quasiparticle formulation characteristic of the $\phi_{2,1}$ and $\phi_{1,5}$ integrable perturbations. We find that they arise by imposing a simple restriction on the RSOS quasiparticle states of the unitary models $M(p,p+1)$. In fact, four fermionic expressions are obtained for each generating function of half-lattice paths of finite length $L$, and these lead to four distinct expressions for most characters $\chi^{k,2k\pm1}_{r,s}$. These are direct analogues of Melzer's expressions for $M(p,p+1)$, and their proof entails revisiting, reworking and refining a proof of Melzer's identities which used combinatorial transforms on lattice paths. We also derive a bosonic version of the generating functions of length $L$ half-lattice paths, this expression being notable in that it involves $q$-trinomial coefficients. Taking the $L\to\infty$ limit shows that the generating functions for infinite length half-lattice paths are indeed the Virasoro characters $\chi^{k,2k\pm1}_{r,s}$.Comment: 29 pages. v2: minor improvements, references adde

Strategies as Resource Terms, and Their Categorical Semantics

As shown by Tsukada and Ong, simply-typed, normal and η-long resource terms correspond to plays in Hyland-Ong games, quotiented by Melliès' homotopy equivalence. Though inspiring, their proof is indirect, relying on the injectivity of the relational model {w.r.t.} both sides of the correspondence - in particular, the dynamics of the resource calculus is taken into account only via the compatibility of the relational model with the composition of normal terms defined by normalization. In the present paper, we revisit and extend these results. Our first contribution is to restate the correspondence by considering causal structures we call augmentations, which are canonical representatives of Hyland-Ong plays up to homotopy. This allows us to give a direct and explicit account of the connection with normal resource terms. As a second contribution, we extend this account to the reduction of resource terms: building on a notion of strategies as weighted sums of augmentations, we provide a denotational model of the resource calculus, invariant under reduction. A key step - and our third contribution - is a categorical model we call a resource category, which is to the resource calculus what differential categories are to the differential λ-calculus

Synthesis and palladium-catalysed isomerisation of fused polycyclic tetrahydrofurans : efficient and stereoselective one-pot domino construction of functionalised bridged bicyclo[n.2.1] ring systems

International audienceA new one-pot domino reaction for a general entry to functionalised bridged bicyclo[n.2.1] ring systems from α,α'-diactivated cyclic ketones and trans-1,4-dihalides is described. The sequence combines a base promoted C-O cycloalkylation reaction leading to fused polycyclic enol ethers and their in situ palladium-catalysed isomerisation

Cellulose coating and chelation of antibacterial compounds for the protection of flax yarns against natural soil degradation.

Natural cellulosic fibres such as flax fibres present interesting mechanical properties as well as biodegradability, and by-products, such as short flax fibres, could be used to produce geotextiles to stabilise soils. Today, geotextiles are often made of coir fibres, which have a high lignin percentage leading to their slow degradation in soil. Fibres with a high cellulosic content, such as those of flax, exhibit lower resistance to soil degradation. This study investigates solutions to improve this parameter with a view to increasing their service life and therefore their credibility compared to coir fibres for geotextile applications. For this purpose, a cellulose coating of yarns made of short flax fibres was performed and its stability under a water flow was assessed. The ability to form a cellulose sheath was estimated by chromaticity measurements of flax fibres after applying a dye specific to lignin. Infrared spectrometry analysis to monitor the level of protection against degradation by cellulolytic enzymes was also carried out. It appears that the cellulose coating provides an efficient physical protection, preventing access of these enzymes to their fibrous substrate. Then, the possibility of conferring antibacterial properties on the cellulose coating by chelating phytoalexin molecules such as gramine on it was assayed and proven to be effective against soil cellulolytic bacteria such as Cellvibrio fulvus and Cellvibrio vulgaris. This study therefore establishes that coating flax yarns with cellulose associated with antibacterial molecules could contribute to obtaining a longer service life in soil for geotextiles manufactured from flax fibres

Macdonald polynomials in superspace: conjectural definition and positivity conjectures

We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace, and a rather non-trivial expression for their evaluation. We study the limiting cases q=0 and q=\infty, which lead to two families of Hall-Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q=t=0 or q=t=\infty seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall-Littlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q=t=0 family) are polynomials in q and t with nonnegative integer coefficients.Comment: 18 page

Path representation of su(2)_k states II: Operator construction of the fermionic character and spin-1/2--RSOS factorization

This is the second of two articles (independent of each other) devoted to the analysis of the path description of the states in su(2)_k WZW models. Here we present a constructive derivation of the fermionic character at level k based on these paths. The starting point is the expression of a path in terms of a sequence of nonlocal (formal) operators acting on the vacuum ground-state path. Within this framework, the key step is the construction of the level-k operator sequences out of those at level-1 by the action of a new type of operators. These actions of operators on operators turn out to have a path interpretation: these paths are precisely the finitized RSOS paths related to the unitary minimal models M(k+1,k+2). We thus unravel -- at the level of the path representation of the states --, a direct factorization into a k=1 spinon part times a RSOS factor. It is also pointed out that since there are two fermionic forms describing these finite RSOS paths, the resulting fermionic su(2)_k characters arise in two versions. Finally, the relation between the present construction and the Nagoya spectral decomposition of the path space is sketched.Comment: 28 page

Phytochemical Screening of Quaking Aspen (Populus tremuloides) Extracts by UPLC-QTOF-MS and Evaluation of their Antimicrobial Activity

The continual emergence of pathogen resistance is a recurring challenge and pushes for the development of antimicrobial compounds. Here, we investigated compounds from quaking aspen trees (Populus tremuloides) as potential antimicrobial agents. Several extractions using different solvents were realized, and corresponding antimicrobial activity was tested against eight microorganisms. Results revealed that polar extraction solvents including water, ethanol and methanol gave the best extraction yields (\u3e15.07%). Minimal inhibition concentration (MIC) and minimal bactericidal/fungicidal concentration (MBC/MFC) demonstrated that water extracts had the best antimicrobial activity by a weak to moderate inhibition of growth of all eight tested microorganisms in addition to having a bactericidal effect on three of them. The quaking aspen methanol extract also displayed antimicrobial activity but to a lower level than the water extract. Ultra-performance liquid chromatography quadrupole time-of flight mass spectrometry (UPLC-QTOF-MS) analysis led to the identification of 92 compounds, mainly polyphenols in both extracts, with 22 molecules previously known for their antimicrobial properties. According to the relative abundance, 4-hydroxybenzaldehyde (5.44% in methanol extract) and kaempferol (5.03% in water extract) were the most abundant antimicrobial compounds. Among antimicrobial molecules identified, nine were from the flavonoid family. The results of our study demonstrate the interest of using quaking aspen as source of antimicrobial compounds