From Jack to Double Jack Polynomials via the Supersymmetric Bridge

The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine ${\widehat{\mathfrak {sl}}_2}$ algebra

Jack superpolynomials: physical and combinatorial definitions

Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this quantum-mechanical problem. But Jack superpolynomials can also be defined more combinatorially, starting from the multiplicative bases of symmetric superpolynomials, enforcing orthogonality with respect to a one-parameter deformation of the combinatorial scalar product. Both constructions turns out to be equivalent. This provides strong support for the correctness of the various underlying constructions and for the pivotal role of Jack superpolynomials in the theory of symmetric superpolynomials.Comment: 6 pages. To appear in the proceedings of the {\it XIII International Colloquium on Integrable Systems and Quantum Groups}, Czech. J . Phys., June 17-19 2004, Doppler Institute, Czech Technical Universit

Trace elements in stream bed sediments from agricultural catchments (Gascogne region, S-W France): Where do they come from?

The Gascogne region (SW of France) is cultivated for more than 75% of the area. 83 samples of stream bed sediments were collected in three main Gascogne river basins (Gers, Save and Touch, left tributaries of the Garonne river) to evaluate the impact of agricultural practices on trace elements behavior. Eight potential harmful elements (PHE) (Cr, Co, Ni, Cu, Zn, As, Cd and Pb), four reference elements for normalization (Sc, Cs, Al and Fe) and four major elements (Mn, Ca, Mg and P) were considered. The average trace element concentrations in the fine fractions (b63 μm) are in the decreasing order: ZnNCrNNiNPbNCuNCoNAsNScNCsNCd. Geochemical investigations and an original approach combining regression analysis and chemical sequential extraction allowed to select the most adequate reference material (regional molasse)and reference element (Cs) for normalization procedure. The enrichment factor (EF) is generally lower than 2.5, particularly for Cr, Ni, Cu, As, Zn; however, 23% of the sampling stations are more contaminated (2.5bEFb4.5), particularly for Cd, Pb and Co. The PHE in the Gascogne river sediments are mainly originated from natural weathering processes; nevertheless, anthropogenic contribution could represent up to 34% of the total sediment content. For lead, geochemical and isotopic methods gave very similar anthropogenic contributions (24% and 22%, respectively). The enrichment of Cu, Pb, Zn, Co, As, Ni, Cr was mainly related to global and local atmospheric deposition of industrial emissions and gasoline combustion, and was associated to forested catchments. All PHE's are controlled by clay and oxi-hydroxides minerals. Cdwas the only PHE enriched downstreamcultivated catchments and this enrichmentwas linked to Ca and P. This indicates a major origin of Cd fromfertilizer inputs and a main control by carbonate mineral

Distribution and origin of lead in stream sediments from small agricultural catchments draining Miocene molassic deposits (SW France)

Total Pb concentrations and isotopic composition were determined in stream-bed sediments and bedrock from 29 small agricultural or forested catchments in the Gascogne area (SW France). The contribution of Pb from various natural and anthropogenic sources was investigated in this rural area which is very weakly impacted by industrial or urban emissions. Environmental parameters in catchments (importance of forest cover, organic matter and oxide content in sediments) were considered. A combination of geochemical (enrichment calculation, sequential extraction) and isotopic investigations was performed to constrain the origin of Pb and the distribution of anthropogenic Pb in sediments. Most of the sediments have low total Pb content compared to other agricultural regions more impacted by industrial or urban emissions. The results indicated a moderate but significant Pb enrichment, particularly for catchments draining forested areas. This enrichment was positively related to organic C content in sediment and catchment forest cover, whereas in entirely cultivated catchments it was related to Fe-oxide content. An average anthropogenic end-member was determined using Pb isotopes, and was supposed to be representative of background atmospheric Pb pollution, with a weak influence of Pb from recent gasoline and local fertiliser spreading. The amount of anthropogenic Pb (Pbanthr) in sediments estimated using a geochemical approach (mean 63.7 ± 20.4%) was higher than that estimated using an isotopic approach (mean 36.6 ± 17.8%), but the same trend was observed among the samples, except for low anthropogenic contributions. The distribution of Pbanthr in sediments from weakly forested catchments indicated a strong affinity for carbonates and Fe-oxides. Amorphous Fe-oxides became preferential trapping compounds as soon as Pb enrichment increased and carbonate content decreased. Finally, in cultivated catchments, organic C was not a main trapping component for Pbanthr in sediments

Inhibition of Connexin43 hemichannels impairs spatial short-term memory without affecting spatial working memory

Astrocytes are active players in higher brain function as they can release gliotransmitters, which are essential for synaptic plasticity. Various mechanisms have been proposed for gliotransmission, including vesicular mechanisms as well as non-vesicular ones, for example by passive diffusion via connexin hemichannels (HCs). We here investigated whether interfering with connexin43 (Cx43) HCs influenced hippocampal spatial memory. We made use of the peptide Gap19 that blocks HCs but not gap junction channels and is specific for Cx43. To this end, we microinfused transactivator of transcription linked Gap19 (TAT-Gap19) into the brain ventricle of male NMRI mice and assessed spatial memory in a Y maze. We found that the in vivo blockade of Cx43 HCs did not affect the locomotor activity or spatial working memory in a spontaneous alternation Y maze task. Cx43 blockade did however significantly impair the spatial short-term memory in a delayed spontaneous alternation Y maze task. These results indicate that Cx43 HCs play a role in spatial short-term memory