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We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript.

We show that the ideal generated by maximal minors (i.e., $(k+1)$-minors) of a $(k+1) \times n$ Vandermonde matrix is radical and Cohen-Macaulay. Note that this ideal is generated by all Specht polynomials with shape $(n-k,1,...,1)$., Comment: 6 pages, simplified the proof of the main result. To appear in Math. Scand

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results in particular give a self-contained proof of Cohen-Macaulayness of certain $h$-equals sets, a result previously obtained by Etingof-Gorsky-Losev over the complex numbers using rational Cherednik algebras., 42 pages, 2 figures

We give a detailed proof for Gordan-Noether’s results in “Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet.” C. Lossen has written a paper in a similar direction as the present paper, but did not provide a proof for every result. In our paper, every result is proved. Furthermore, our paper is independent of Lossen’s paper and includes a considerable number of new observations.

We relate the Weyr structure of a square matrix B to that of the t × t block upper triangular matrix C that has B down the main diagonal and first superdiagonal, and zeros elsewhere. Of special interest is the case t = 2 and where C is the nth Sierpinski matrix B n , which is defined inductively by B 0 = 1 and B n = [ B n − 1 B n − 1 0 B n − 1 ] . This yields an easy derivation of the Weyr structure of B n as the binomial coefficients arranged in decreasing order. Earlier proofs of the Jordan analogue of this had often relied on deep theorems from such areas as algebraic geometry. The result has interesting consequences for commutative, finite-dimensional algebras.

Let $\mathcal {V}=\bigsqcup _{i=0}^n\mathcal {V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\mathbb{F} _q$, and let $\mathcal {A}$ be the graded Gorenstein algebra defined over $\mathbb{Q} $ which has $\mathcal {V}$ as a $\mathbb{Q} $ basis. Let $F$ be the Macaulay dual generator for $\mathcal {A}$. We explicitly compute the Hessian determinant $|{\partial ^2F}/{\partial X_i \partial X_j}|$, evaluated at the point $X_1 = X_2 = \cdots = X_N=1$, and relate it to the determinant of the incidence matrix between $\mathcal {V}_1$ and $\mathcal {V}_{n-1}$. Our exploration is motivated by the fact that both of these matrices naturally arise in the study of the Sperner property of the lattice and the Lefschetz property for the graded Artinian Gorenstein algebra associated to it.

We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules., Comment: 9 pages

BACKGROUND Users of antipsychotics (APs) have a risk of sudden cardiac death (SCD). Sudden cardiac death in such patients is thought to be largely due to drug-induced QT prolongation. It has been reported that many subjects with drug-induced torsades de pointes (TdP) have risk alleles associated with subclinical congenital long QT syndrome. METHODS We investigated the effects of the risk alleles associated with long QT on the QT interval in patients receiving APs using 24-hour Holter electrocardiograms to take into account the circadian fluctuation of QT intervals. We investigated 8 single-nucleotide polymorphisms identified on a GWAS. RESULTS We found that increased numbers of risk alleles at rs7188697 in NDRG4 and rs11970286 in PLN were the major predictors of an increased maximum QT interval over 24 hours in users of APs. CONCLUSIONS It could be useful to perform a DNA-based analysis before the initiation of APs to reduce the risk of drug-induced torsades de pointes and SCD.

Let $F$ be a homogeneous polynomial in $n$ variables of degree $d$ over a field $K$. Let $A(F)$ be the associated Artinian graded $K$-algebra. If $B \subset A(F)$ is a subalgebra of $A(F)$ which is Gorenstein with the same socle degree as $A(F)$, we describe the Macaulay dual generator for $B$ in terms of $F$. Furthermore when $n=d$, we give necessary and sufficient conditions on the polynomial $F$ for $A(F)$ to be a complete intersection.

Association between gastric inhibitory polypeptide receptor polymorphism, rs10423928, and body mass index in olanzapine-treated schizophrenia was examined. Body mass index change for the A/T+A/A genotypes was significantly higher than that for the T/T genotype. rs10423928 may predict weight gain in schizophrenia.

We compute the resultants for quadratic binomial complete intersections. As an application we show that any quadratic binomial complete intersection can have the set of square-free monomials as a vector space basis if the generators are put in a normal form., Comment: 14 pages

Let $A$ be a graded complete intersection over a field and $B$ the monomial complete intersection with the generators of the same degrees as $A$. The EGH conjecture says that if $I$ is a graded ideal in $A$, then there should be an ideal $J$ in $B$ such that $B/J$ and $A/I$ have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property., Comment: 7 pages

We prove that any quadratic complete intersection with a certain action of the symmetric group has the strong Lefschetz property over a field of characteristic zero. Furthermore, we discuss under what conditions its ring of invariants by a Young subgroup is a homogeneous complete intersection with a standard grading.

Background Olanzapine (OLZ) treatment is associated with a high risk of weight gain, and may cause abnormalities in glycolipid metabolism. Therefore, the underlying mechanism of OLZ-related weight gain is needed to clarify but not yet been adequately determined. In recent years, adipocytokines such as leptin, adiponectin, and tumor necrosis factor (TNF)-α, which play important roles in energy homeostasis, have been suggested as biomarkers of weight gain. Here, we determined if baseline plasma concentrations of leptin, adiponectin, and TNF-α predict weight gain following OLZ treatment. Methods We recruited 31 schizophrenia outpatients (12 men and 19 women, 28.8 ± 10.2 years old) that were unmedicated or on another antipsychotic monotherapy medication. Baseline body mass index (BMI) and plasma levels of leptin, adiponectin, and TNF-α were obtained. All patients started or were switched to OLZ monotherapy for a maximum of 1 year. BMI was also obtained at the endpoint. Results Mean BMI change following OLZ treatment was 2.1 ± 2.7 kg/m2. BMI change from baseline to endpoint negatively-correlated with baseline leptin levels in female patients (r = −0.514, P = 0.024), but not male patients. Baseline adiponectin or TNF-α levels were not correlated with BMI change. Conclusion Baseline plasma leptin can have an effect on subsequent weight gain following OLZ treatment in female patients with schizophrenia.