Connectivity of pseudomanifold graphs from an algebraic point of view

The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis and Bjorner. In this note, we provide a unifying approach which allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley--Reisner rings.Comment: 4 pages, minor change

Permanent His-bundle pacing maintains long-term ventricular synchrony and left ventricular performance, unlike conventional right ventricular apical pacing

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Anomalous T-dependence of phonon lifetimes in metallic VO2

We investigate phonon lifetimes in VO2 single crystals. We do so in the metallic state above the metal-insulator transition (MIT), where strong structural fluctuations are known to take place. By combining inelastic X-ray scattering and Raman spectroscopy, we track the temperature dependence of several acoustic and optical phonon modes up to 1000 K. Contrary to what is commonly observed, we find that phonon lifetimes decrease with decreasing temperature. Our results show that pre-transitional fluctuations in the metallic state give rise to strong electron-phonon scattering that onsets hundreds of degrees above the transition and increases as the MIT is approached. Notably, this effect is not limited to specific points of reciprocal space that could be associated with the structural transition

Castelnuovo-Mumford Regularity and Powers

This note has two goals. The first is to give a short and self contained introduction to the Castelnuovo-Mumford regularity for standard graded rings R= iEN Ri over general base rings R0. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers Iv of an homogeneous ideal I of R is eventually a linear function in v. Finally we show how the flexibility of the definition of the Castelnuovo-Mumford regularity over general base rings can be used to give a simple proof of a result proved by the authors in “Maximal minors and linear powers”

Gröbner Bases, Initial Ideals and Initial Algebras

The first chapter gives a compact, but quite complete introduction to Gröbner bases and Sagbi bases in general. The focus is on the structural aspects, namely, the use of Gröbner and Sagbi degenerations in the transfer of homological and enumerative information from Stanley-Reisner and/or toric rings to those objects that degenerate to them

Hankel determinantal rings have rational singularities

Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as linear specializations of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel determinantal rings have rational singularities; in the case of positive prime characteristic, we prove that they are F-pure. Independent of the characteristic, we give a complete description of the divisor class groups of these rings, and show that each divisor class group element is the class of a maximal Cohen-Macaulay modul

Algebras Defined by Minors

In Chap. 4 we have studied the Gröbner deformations of determinantal ideals defined by their initial ideals. We now turn to the study of algebras generated by minors through their initial algebras. Since the initial algebras are normal monoid domains, toric algebra can be applied to them. Since normal monoid domains are very well understood, we can draw strong consequences for the algebras defined by minors