On the Duals of Segre Varieties

The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M.M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?     

On Positive and Weakly Positive Implicative BCI-Algebras

Prove that the notion of positive implicative BCI-algebras coincides with that of weakly positive implicative BCI-algebras, thus the whole results in the latter are still true in the former, in particular, one of these results answers definitely the first half of J. Meng and X.L. Xin’s open problem: Does the class of positive implicative BCI-algebras form a variety? The second half of the same problem is: What properties will the ideals of such an algebra have? Here, some further properties are obtained.     

Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations π such that the elements in the closed lower Bruhat interval [id,π] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id,π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner.Our characterisation connects the Poincaré polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups An and Bn. The expressions involve q-Stirling numbers of the second kind, and for the group An putting q=1 yields the poly-Bernoulli numbers defined by Kaneko.     

南充地区桃李资源初报

1988～1989年对南充地区12县(市)开展了全面地桃李资源调查。结果表明,该区桃李栽培历史近千年;桃李各有一个种,64个桃品种(地方品种27个,引进品种37个),30个李品种(地方品种23个,引进品种7个)发现最优良的地方桃品种有白花桃,岳池伏桃2号(伏桃)和红花桃;最优良的地方李品种有营山李、南充桐子李、王黄李、青皮李和黄蜡李。     

On images of affine spaces

We prove that every non-degenerate toric variety, every homogeneous space of a connected linear algebraic group without non-constant invertible regular functions, and every variety covered by affine spaces admit a surjective morphism from an affine space.     

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.     

Laurent polynomials and Eulerian numbers

Duistermaat and van der Kallen show that there is no nontrivial complex Laurent polynomial all of whose powers have a zero constant term. Inspired by this, Sturmfels poses two questions: Do the constant terms of a generic Laurent polynomial form a regular sequence? If so, then what is the degree of the associated zero-dimensional ideal? In this note, we prove that the Eulerian numbers provide the answer to the second question. The proof involves reinterpreting the problem in terms of toric geometry.     

湖南半蒴苣苔属一新变种：鬼溪半蒴苣苔

描述了新发现于湖南的苦苣苔科一新变种——鬼溪半蒴苣苔（Hemiboea gracilis Franch. var. guixiensis G. X. Chen, X. M. Xiang et L. T var. nov.）,新变种与原变种纤细半蒴苣苔（Hemiboea gracilis Franch.）在形态上相近,但通过野外调查以及室内鉴定发现,新变种全株茎疏生柔毛,叶两面和叶柄密生硬毛,苞片、花序梗、花梗、花萼和花冠密生柔毛,果尖端疏生柔毛,二者易于区别.     

Characterization of Basil Volatile Fraction and Study of Its Agronomic Variation by ASCA

Basil is a plant known worldwide for its culinary and health attributes. It counts more than a hundred and fifty species and many more chemo-types due to its easy cross-breeds. Each species and each chemo-type have a typical aroma pattern and selecting the proper one is crucial for the food industry. Twelve basil varieties have been studied over three years (2018–2020), as have four different cuts. To characterize the aroma profile, nine typical basil flavour molecules have been selected using a gas chromatography–mass spectrometry coupled with an olfactometer (GC–MS/O). The concentrations of the nine selected molecules were measured by an ultra-fast CG e-nose and Principal Component Analysis (PCA) was applied to detect possible differences among the samples. The PCA results highlighted differences between harvesting years, mainly for 2018, whereas no observable clusters were found concerning varieties and cuts, probably due to the combined effects of the investigated factors. For this reason, the ANOVA Simultaneous Component Analysis (ASCA) methodology was applied on a balanced a posteriori designed dataset. All the considered factors and interactions were statistically significant (p < 0.05) in explaining differences between the basil aroma profiles, with more relevant effects of variety and year.