Resolutions and cohomology over complete intersections

This chapter contains a new proof and new applications of a theorem of Shamash and Eisenbud, providing a construction of projective resolutions of modules over a complete intersection. The duals of these infinite projective resolutions are finitely generated differential graded modules over a graded polynomial ring, so they can be represented in the computer, and can be used to compute Ext modules simultaneously in all homological degrees. It is shown how to write Macaulay 2 code to implement the construction, and how to use the computer to determine invariants of modules over complete intersections that are difficult to obtain otherwise

Graph products of spheres, associative graded algebras and Hilbert series

Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Non-commutative Grobner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment-angle complex. We apply our result to the loop-space homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more citations, to appear in Mathematische Zeitschrif

A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation

A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, ..., x_n \}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$ holds in $X^3.$ A non-degenerate involutive solution $(X,r)$ satisfying $r(xx)=xx$, for all $x \in X$, is called \emph{square-free solution}. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions $(X,r)$ and the associated Yang-Baxter algebraic structures -- the semigroup $S(X,r)$, the group $G(X,r)$ and the $k$- algebra $A(k, X,r)$ over a field $k$, generated by $X$ and with quadratic defining relations naturally arising and uniquely determined by $r$. We study the properties of the associated Yang-Baxter structures and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial type, are equivalent. This implies that the Yang-Baxter algebra $A(k, X,r)$ is Poincar\'{e}-Birkhoff-Witt type algebra, with respect to some appropriate ordering of $X$. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof-Schedler.Comment: 34 page

An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota

We sketch the outlines of Gian Carlo Rota's interaction with the ideas that Hermann Grassmann developed in his Ausdehnungslehre of 1844 and 1862, as adapted and explained by Giuseppe Peano in 1888. This leads us past what Rota variously called 'Grassmann-Cayley algebra', or 'Peano spaces', to the Whitney algebra of a matroid, and finally to a resolution of the question "What, really, was Grassmann's regressive product?". This final question is the subject of ongoing joint work with Andrea Brini, Francesco Regonati, and William Schmitt. The present paper was presented at the conference "The Digital Footprint of Gian-Carlo Rota: Marbles, Boxes and Philosophy" in Milano on 17 Feb 2009. It will appear in proceedings of that conference, to be published by Springer Verlag.Comment: 28 page

High sensitivity (1)H-NMR spectroscopy of homeopathic remedies made in water

BACKGROUND: The efficacy of homeopathy is controversial. Homeopathic remedies are made via iterated shaking and dilution, in ethanol or in water, from a starting substance. Remedies of potency 12 C or higher are ultra-dilute (UD), i.e. contain zero molecules of the starting material. Various hypotheses have been advanced to explain how a UD remedy might be different from unprepared solvent. One such hypothesis posits that a remedy contains stable clusters, i.e. localized regions where one or more hydrogen bonds remain fixed on a long time scale. High sensitivity proton nuclear magnetic resonance spectroscopy has not previously been used to look for evidence of differences between UD remedies and controls. METHODS: Homeopathic remedies made in water were studied via high sensitivity proton nuclear magnetic resonance spectroscopy. A total of 57 remedy samples representing six starting materials and spanning a variety of potencies from 6 C to 10 M were tested along with 46 controls. RESULTS: By presaturating on the water peak, signals could be reliably detected that represented H-containing species at concentrations as low as 5 μM. There were 35 positions where a discrete signal was seen in one or more of the 103 spectra, which should theoretically have been absent from the spectrum of pure water. Of these 35, fifteen were identified as machine-generated artifacts, eight were identified as trace levels of organic contaminants, and twelve were unexplained. Of the unexplained signals, six were seen in just one spectrum each. None of the artifacts or unexplained signals occurred more frequently in remedies than in controls, using a p < .05 cutoff. Some commercially prepared samples were found to contain traces of one or more of these small organic molecules: ethanol, acetate, formate, methanol, and acetone. CONCLUSION: No discrete signals suggesting a difference between remedies and controls were seen, via high sensitivity (1)H-NMR spectroscopy. The results failed to support a hypothesis that remedies made in water contain long-lived non-dynamic alterations of the H-bonding pattern of the solvent

Finite Gr\"obner--Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids

This paper shows that every Plactic algebra of finite rank admits a finite Gr\"obner--Shirshov basis. The result is proved by using the combinatorial properties of Young tableaux to construct a finite complete rewriting system for the corresponding Plactic monoid, which also yields the corollaries that Plactic monoids of finite rank have finite derivation type and satisfy the homological finiteness properties left and right $FP_\infty$. Also, answering a question of Zelmanov, we apply this rewriting system and other techniques to show that Plactic monoids of finite rank are biautomatic.Comment: 16 pages; 3 figures. Minor revision: typos fixed; figures redrawn; references update

Pharmacogenomics of the efficacy and safety of Colchicine in COLCOT

© 2021 The Authors. Circulation: Genomic and Precision Medicine is published on behalf of the American Heart Association, Inc., by Wolters Kluwer Health, Inc. This is an open access article under the terms of the Creative Commons Attribution Non-Commercial License, which permits use, distribution, and reproduction in any medium, provided that the original work is properly cited and is not used for commercial purposes.Background: The randomized, placebo-controlled COLCOT (Colchicine Cardiovascular Outcomes Trial) has shown the benefits of colchicine 0.5 mg daily to lower the rate of ischemic cardiovascular events in patients with a recent myocardial infarction. Here, we conducted a post hoc pharmacogenomic study of COLCOT with the aim to identify genetic predictors of the efficacy and safety of treatment with colchicine. Methods: There were 1522 participants of European ancestry from the COLCOT trial available for the pharmacogenomic study of COLCOT trial. The pharmacogenomic study's primary cardiovascular end point was defined as for the main trial, as time to first occurrence of cardiovascular death, resuscitated cardiac arrest, myocardial infarction, stroke, or urgent hospitalization for angina requiring coronary revascularization. The safety end point was time to the first report of gastrointestinal events. Patients' DNA was genotyped using the Illumina Global Screening array followed by imputation. We performed a genome-wide association study in colchicine-treated patients. Results: None of the genetic variants passed the genome-wide association study significance threshold for the primary cardiovascular end point conducted in 702 patients in the colchicine arm who were compliant to medication. The genome-wide association study for gastrointestinal events was conducted in all 767 patients in the colchicine arm and found 2 significant association signals, one with lead variant rs6916345 (hazard ratio, 1.89 [95% CI, 1.52-2.35], P=7.41×10-9) in a locus which colocalizes with Crohn disease, and one with lead variant rs74795203 (hazard ratio, 2.51 [95% CI, 1.82-3.47]; P=2.70×10-8), an intronic variant in gene SEPHS1. The interaction terms between the genetic variants and treatment with colchicine versus placebo were significant. Conclusions: We found 2 genomic regions associated with gastrointestinal events in patients treated with colchicine. Those findings will benefit from replication to confirm that some patients may have genetic predispositions to lower tolerability of treatment with colchicine.info:eu-repo/semantics/publishedVersio