Derived invariance of the cap product in Hochschild theory

We prove derived invariance of the cap product for associative algebras projective over a commutative ring.Comment: 4 page

Orbifolds of Lattice Vertex Operator Algebras at d=48d=48 and d=72d=72

Motivated by the notion of extremal vertex operator algebras, we investigate cyclic orbifolds of vertex operator algebras coming from extremal even self-dual lattices in $d=48$ and $d=72$. In this way we construct about one hundred new examples of holomorphic VOAs with a small number of low weight states.Comment: 18 pages, LaTe

Characterization of co-blockers for simple perfect matchings in a convex geometric graph

Consider the complete convex geometric graph on $2m$ vertices, $CGG(2m)$, i.e., the set of all boundary edges and diagonals of a planar convex $2m$-gon $P$. In [C. Keller and M. Perles, On the Smallest Sets Blocking Simple Perfect Matchings in a Convex Geometric Graph], the smallest sets of edges that meet all the simple perfect matchings (SPMs) in $CGG(2m)$ (called "blockers") are characterized, and it is shown that all these sets are caterpillar graphs with a special structure, and that their total number is $m \cdot 2^{m-1}$. In this paper we characterize the co-blockers for SPMs in $CGG(2m)$, that is, the smallest sets of edges that meet all the blockers. We show that the co-blockers are exactly those perfect matchings $M$ in $CGG(2m)$ where all edges are of odd order, and two edges of $M$ that emanate from two adjacent vertices of $P$ never cross. In particular, while the number of SPMs and the number of blockers grow exponentially with $m$, the number of co-blockers grows super-exponentially.Comment: 8 pages, 4 figure

On Convex Geometric Graphs with no k+1k+1 Pairwise Disjoint Edges

A well-known result of Kupitz from 1982 asserts that the maximal number of edges in a convex geometric graph (CGG) on $n$ vertices that does not contain $k+1$ pairwise disjoint edges is $kn$ (provided $n>2k$). For $k=1$ and $k=n/2-1$, the extremal examples are completely characterized. For all other values of $k$, the structure of the extremal examples is far from known: their total number is unknown, and only a few classes of examples were presented, that are almost symmetric, consisting roughly of the $kn$ "longest possible" edges of $CK(n)$, the complete CGG of order $n$. In order to understand further the structure of the extremal examples, we present a class of extremal examples that lie at the other end of the spectrum. Namely, we break the symmetry by requiring that, in addition, the graph admit an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull. We show that such graphs exist as long as $q \leq n-2k$ and that this value of $q$ is optimal. We generalize our discussion to the following question: what is the maximal possible number $f(n,k,q)$ of edges in a CGG on $n$ vertices that does not contain $k+1$ pairwise disjoint edges, and, in addition, admits an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull? We provide a complete answer to this question, determining $f(n,k,q)$ for all relevant values of $n,k$ and $q$.Comment: 17 pages, 9 figure

Giving credit to reforestation for water quality benefits.

While there is a general belief that reforesting marginal, often unprofitable, croplands can result in water quality benefits, to date there have been very few studies that have attempted to quantify the magnitude of the reductions in nutrient (N and P) and sediment export. In order to determine the magnitude of a credit for water quality trading, there is a need to develop quantitative approaches to estimate the benefits from forest planting in terms of load reductions. Here we first evaluate the availability of marginal croplands (i.e. those with low infiltration capacity and high slopes) within a large section of the Ohio River Basin (ORB) to assess the magnitude of the land that could be reforested. Next, we employ the Nutrient Tracking Tool (NTT) to study the reduction in N, P and sediment losses from converting corn or corn/soy rotations to forested lands, first in a case study and then for a large region within the ORB. We find that after reforestation, N losses can decrease by 40 to 80 kg/ha-yr (95-97% reduction), while P losses decrease by 1 to 4 kg/ha-yr (96-99% reduction). There is a significant influence of local conditions (soils, previous crop management practices, meteorology), which can be considered with NTT and must be taken into consideration for specific projects. There is also considerable interannual and monthly variability, which highlights the need to take the longer view into account in nutrient credit considerations for water quality trading, as well as in monitoring programs. Overall, there is the potential for avoiding 60 million kg N and 2 million kg P from reaching the streams and rivers of the northern ORB as a result of conversion of marginal farmland to tree planting, which is on the order of 12% decrease for TN and 5% for TP, for the entire basin. Accounting for attenuation, this represents a significant fraction of the goal of the USEPA Gulf of Mexico Hypoxia Task Force to reduce TN and TP reaching the dead zone in the Gulf of Mexico, the second largest dead zone in the world. More broadly, the potential for targeted forest planting to reduce nutrient loading demonstrated in this study suggests further consideration of this approach for managing water quality in waterways throughout the world. The study was conducted using computational models and there is a need to evaluate the results with empirical observations

Cauchy conformal fields in dimensions d>2

Holomorphic fields play an important role in 2d conformal field theory. We generalize them to d>2 by introducing the notion of Cauchy conformal fields, which satisfy a first order differential equation such that they are determined everywhere once we know their value on a codimension 1 surface. We classify all the unitary Cauchy fields. By analyzing the mode expansion on the unit sphere, we show that all unitary Cauchy fields are free in the sense that their correlation functions factorize on the 2-point function. We also discuss the possibility of non-unitary Cauchy fields and classify them in d=3 and 4.Comment: 45 pages; v2: references adde