The pre-WDVV ring of physics and its topology

We show how a simplicial complex arising from the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations of string theory is the Whitehouse complex. Using discrete Morse theory, we give an elementary proof that the Whitehouse complex $\Delta_n$ is homotopy equivalent to a wedge of $(n-2)!$ spheres of dimension $n-4$. We also verify the Cohen-Macaulay property. Additionally, recurrences are given for the face enumeration of the complex and the Hilbert series of the associated pre-WDVV ring.Comment: 13 pages, 4 figures, 2 table

On the non-existence of an R-labeling

We present a family of Eulerian posets which does not have any R-labeling. The result uses a structure theorem for R-labelings of the butterfly poset.Comment: 6 pages, 1 figure. To appear in the journal Orde

The Tchebyshev transforms of the first and second kind

We give an in-depth study of the Tchebyshev transforms of the first and second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform (of the first kind) preserves desirable combinatorial properties, including Eulerianess (due to Hetyei) and EL-shellability. It is also a linear transformation on flag vectors. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg and Readdy omega map of oriented matroids. One consequence is that nonnegativity of the cd-index is maintained. The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on the space of quasisymmetric functions QSym. It coincides with Stembridge's peak enumerator for Eulerian posets, but differs for general posets. The complete spectrum is determined, generalizing work of Billera, Hsiao and van Willigenburg. The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's classical quasisymmetric function of a poset, this map is a comodule morphism with respect to the quasisymmetric functions QSym. Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps. One such occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and Sottile's result on the terminal object in the category of combinatorial Hopf algebras. In contrast, the chain map of the first kind is both an algebra map and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page

Level Eulerian Posets

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem--Mahler--Lech theorem, the ${\bf ab}$-series of a level poset is shown to be a rational generating function in the non-commutative variables ${\bf a}$ and ${\bf b}$. In the case the poset is also Eulerian, the analogous result holds for the ${\bf cd}$-series. Using coalgebraic techniques a method is developed to recognize the ${\bf cd}$-series matrix of a level Eulerian poset

Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure

Working Group for the Bay of Biscay and the Iberian Waters Ecoregion (ICES WGBIE) (2-9 May 2109)

The ICES Working Group for the Bay of Biscay and the Iberian Waters Ecoregion (WGBIE) assesses the status of 23 stocks distributed from ICES Divisions 3.a–4.a though to Subarea 9, mostly distributed in Subareas 7, 8 and 9. The group was tasked with conducting assessments of stock status for 23 stocks using analytical, forecast methods or trends indicators to provide catch forecasts and a first draft of the ICES advice for 2019. For two of the Nephrops stocks updates were provided on catch data with the advice release delayed until October after the completion of the surveys used for the assessment. Analytical assessments using age-structured models were conducted for the northern stock of white anglerfish, the northern and southern stocks of megrim, four-spot megrim and sole in the Bay of Biscay. The two hake stocks and one southern stock of anglerfish were assessed using models that allow the use of length-structured data (no age data). A surplus-production model, without age or length structure, was used to assess the second southern stock of anglerfish and an age-length structure model was used for the European seabass in the Bay of Biscay. The state of stocks for which no analytical assessment could be performed was inferred from examination of catch, commercial LPUE or CPUE data and from survey information, where available. The northern stock of hake was benchmarked this year to incorporate discards into the model that were previously omitted. New reference points with the accepted benchmark assessment were proposed by the group and new proxy biomass reference points where proposed for black anglerfish in Division’s 7b-k, 8abd. A recurrent issue significantly constrained the group’s ability to fully address the terms of reference this year. Despite an ICES data call with a deadline of six weeks before the meeting, data for most stocks were submitted to ICES only two days before the start of the meeting and in one case 2 days after the meeting commenced. This delayed the process of having the data quality checked and the assessment completed before the start of the working group. This is an important matter of concerns for the working group members. The structure of the report is set out with section 1 presenting a summary of each stock, discussing general issues and conclusions. Section 2 provides descriptions of the relevant fishing fleets and surveys used in the assessment of the stocks. Sections 3–18 contains the single stock assessments