On the variety of triangles for a hyper-Kähler fourfold constructed by Debarre and Voisin

We study the similarities between the Fano varieties of lines on a cubic fourfold, a hyper-Kähler fourfold studied by Beauville and Donagi, and the hyper-Kähler fourfold constructed by Debarre and Voisin in [3]. We exhibit an analog of the notion of “triangle” for these varieties and prove that the 6-dimensional variety of “triangles” is a Lagrangian subvariety in the cube of the constructed hyper-Kähler fourfold.     

Linear codes associated to skew-symmetric determinantal varieties

In this article we consider linear codes coming from skew-symmetric determinantal varieties, which are defined by the vanishing of minors of a certain fixed size in the space of skew-symmetric matrices. In odd characteristic, the minimum distances of these codes are determined and a recursive formula for the weight of a general codeword in these codes is given.     

Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let k be an algebraically closed field and A the polynomial algebra in r variables with coefficients in k. In case the characteristic of k is 2, Carlsson [9] conjectured that for any DG-A-module M of dimension N as a free A-module, if the homology of M is nontrivial and finite dimensional as a k-vector space, then $2^r\le N$. Here we state a stronger conjecture about varieties of square-zero upper triangular $N\times N$ matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N<8$ or $r<3$ without any restriction on the characteristic of k. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N>4$ and $r=2$.     

Assessment of conventional and microwave heating effects on the variation of the bioactive compounds of Chétoui VOO using HPLC-DAD-ESI-TOF-MS

The goal of this work was to monitor the hydrophilic profile of the virgin olive oil (VOO) of the second main Tunisian variety, “Chétoui”, in order to achieve a better understanding of the behavior of the bioactive phenolic compounds during 0, 2, 5, 10, and 15 min of microwave heating at medium power 800 W, and 0, 2.5, and 5 h of conventional heating at 180 °C. The extent of the oxidative and hydrolytic degradation of the different phenolic subclasses was evaluated using high-performance liquid chromatography (HPLC) coupled to electrospray time-of-flight mass spectrometry (TOF-MS) method. During heating process, the most represented component in Chétoui VOO was found to be isomer 1 and 2 of deacetoxy oleuropein aglycone, and hydroxy decarboxy oleuropein aglycon. These compounds may be considered as direct markers for the degree of transformation of secoiridoids during heating process. Among the studied phenolic compounds, hydroxytyrosol, tyrosol, luteolin and apigenin displayed the highest heating resistance in the whole time range of microwave and conventional applications. However, the main secoiridoids quantified in the fresh VOO, isomer 2 of oleuropein aglycone, and ligstroside aglycone, decreased in concentration with the thermal treatment and this decrease was drastic under conventional heating.     

Semilattice transversals of regular bands I

Every regular band can be isomorphically embedded into a regular band which has a semilattice transversal. The latter constitute the variety of regular split bands, whose subvariety lattice is isomorphic to the lattice of regular band varieties.     

Factoriality for the reductive Zassenhaus variety and quantum enveloping algebra

Let U(g)$U(\mathfrak{g})$ be the enveloping algebra of a finite dimensional reductive Lie algebra g$\mathfrak{g}$ over an algebraically closed field of prime characteristic. Let _U?,P(s:)$U_{?,P}(\mathfrak{s}:)$ be the simply connected quantum enveloping algebra at the root of unity ?  , of a complex semi-simple finite dimensional Lie algebra s:$\mathfrak{s}:$. We show, by similar proofs, that the centers of both are factorial. While the first result was established by R. Tange [32] (by different methods), the second one confirms a conjecture in [4]. We also provide a general criterion for the factoriality of the centers of enveloping algebras in prime characteristic.     

在模情况时二面体群$D_{2p}$作用下的transfer理想

In this paper, we determine the structures of the transfer ideal and its radical ideal for the ring of polynomials F p[x, y] under the action of dihedral group D2 p in the modular case. We mainly use Transfer variety, p order elements, and Hilbert’s Nullstellensatz Theorem.