The weak Lefschetz property of artinian algebras associated to paths and cycles

Given a base field $\Bbbk$ of characteristic zero, for each graph $G$, we associate the artinian algebra $A(G)$ defined by the edge ideal of $G$ and the squares of the variables. We study the weak Lefschetz property of $A(G)$. We classify some classes of graphs with relatively few edges, including paths and cycles, such that its associated artinian ring has the weak Lefschetz property.Comment: 21 pages. Comments are welcome

Weak Lefschetz property of graded Gorenstein algebras associated to the Apéry set of a numerical semigroup

It has been conjectured that all graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras A of the Apéry set of M-pure symmetric numerical semigroups generated by four natural numbers. These algebras are graded Artinian Gorenstein algebras of codimension three

A new proof of Stanley's theorem on the strong Lefschetz property

A standard graded artinian monomial complete intersection algebra $A=\Bbbk[x_1,x_2,\ldots,x_n]/(x_1^{a_1},x_2^{a_2},\ldots,x_n^{a_n})$, with $\Bbbk$ a field of characteristic zero, has the strong Lefschetz property due to Stanley in 1980. In this paper, we give a new proof for this result by using only the basic properties of linear algebra. Furthermore, our proof is still true in the case where the characteristic of $\Bbbk$ is greater than the socle degree of $A$, namely $a_1+a_2+\cdots+a_n - n$

A new record of congrid eel, Bathycongrus bleekeri (Congridae) from Vietnam

Three specimens of Bathycongrus bleekeri were collected during the survey in 2015 and 2017. The newly recorded species is described herein, a small, moderately elongate species of Bathycongrus with head and body compressed; vertical fins without black mark; tail slender, attenuated, but not filiform; trunk length 1.2 times greater than head length; teeth small, conical, in about three-four rows on jaws, in a small triangular patch on vomer; 28–29 preanal vertebrae, total of 111–113 vertebrae; 26–28 preanal lateral-line pores. Additional data on the morphology and distribution of the species increases the total number of known species in the family Congridae in Vietnamese waters to twelve

Interpretation of 1H-NMR spectrum of alginate by 1H-1H TOCSY and COSY spectrum

The sodium alginate was prepared from brown seaweeds in Thua-Thien-Hue coastal area. The obtained alginate was characterized by 1H-NMR, 1H-1H TOCSY and COSY spectra. The  assignment of alginate peaks in 1H-NMR spectrum is 5.09, 4.446, 4.152, 4.318, 4.135 ppm in correspondence to H1, H2, H3, H4, H5 of D-mannuronic acid and 5.473, 4.318, 4.446, 4.571, 4.883 ppm in correspondence to H1, H2, H3, H4, H5 of L-guluronic acid, respectively. The present results are essential for the investigating guidance of alginate and its derivatives