Classification of the factorial functions of Eulerian binomial and Sheffer posets

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with $n!$, the factorial function of the infinite Boolean algebra, or $2^{n-1}$, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function $B(n) = n!$ has Sheffer factorial function D(n) identical to that of the infinite Boolean algebra, the infinite Boolean algebra with two new coatoms inserted, or the infinite cubical poset. Moreover, we are able to classify the Sheffer factorial functions of Eulerian Sheffer posets with binomial factorial function $B(n) = 2^{n-1}$ as the doubling of an upside down tree with ranks 1 and 2 modified. When we impose the further condition that a given Eulerian binomial or Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite Boolean algebra $B_X$ or the infinite cubical lattice $C_X^{< \infty}$. We also include several poset constructions that have the same factorial functions as the infinite cubical poset, demonstrating that classifying Eulerian Sheffer posets is a difficult problem.Comment: 23 pages. Minor revisions throughout. Most noticeable is title change. To appear in JCT

Professor Andrew David Hamilton Wyllie – Biographical Memoir

Andrew Wyllie graduated from Aberdeen University, becoming an academic pathologist in Aberdeen, Edinburgh and Cambridge. He was the co-discoverer of apoptotic cell death having observed single cells dying following carcinogen exposure. Together with Alastair Currie and John Kerr, he realised the profound importance of this novel mode of cell death that showed a distinctive series of morphological changes, which he first described as a new cell death process. Wyllie and Currie introduced the term “apoptosis” for this cell death process in a seminal paper in 1972. Another landmark discovery was of chromatin fragmentation in apoptosis, due to activation of an endogenous endonuclease that caused internucleosomal DNA cleavage (“chromatin laddering”), which was the first biochemical mechanism of apoptosis. He further characterised chromatin fragmentation in the 1980s, followed by investigations of cell surface changes to produce “eat-me” signals to trigger rapid phagocytosis of the apoptotic cells and bodies, intracellular calcium ion signalling, caspase activation and other mechanisms of apoptosis. His cancer research helped identify the location of APC and generated his demonstration that apoptosis was regulated by oncogenes, MYC and RAS, and tumour suppressor genes, such as TP53. He showed how apoptosis occurred in response to DNA damage and was a key process influencing both carcinogenesis and tumour growth. Andrew made a major scientific observation that changed the understanding of how cells die in health and disease, although it took time for the scientific establishment to understand its fundamental importance. Andrew Wyllie is widely known as the ‘Father of Apoptosis’

Cytokine Profile of Mouse Vaginal and Uterus Lymphocytes at Estrus and Diestrus

It is known that sex hormones regulate IgA and IgG levels in the female reproductive tract. Moreover, antigen presentation by uterine and vaginal epithelial cells is also under strict hormonal control. The effect of the estrous cycle on cytokine secretion by vaginal and uterine lymphoid cells has been examined in mice using simultaneous staining for cytoplasmic cytokines and surface markers after ex vivo culture with PMA/ionomycin in the presence of Brefeldin A, and flow cytometry analysis. Two different mice strains, BALB/c and C57BL/6 mice, were used. The most relevant finding was the increase in the proportion of vaginal cells secreting IFN-γ at diestrus in both strains of mice. Other cytokines (IL-2 and IL-4) as well as some T cell subsets seemed to be modified in a strain dependent fashion. Data also suggest that NK cells are at least partially responsible for IFN-γ secretion. Our data indicate that vaginal and uterus lymphoid cells isolated at diestrus were in vivo activated to secrete cytokines after ex vivo culture. IFN-γ seems to be the key cytokine, since it increases in both strains of mice

HPV infection, anal intra-epithelial neoplasia (AIN) and anal cancer: current issues.

BACKGROUND: Human papillomavirus (HPV) is well known as the major etiological agent for ano-genital cancer. In contrast to cervical cancer, anal cancer is uncommon, but is increasing steadily in the community over the last few decades. However, it has undergone an exponential rise in the men who have sex with men (MSM) and HIV + groups. HIV + MSM in particular, have anal cancer incidences about three times that of the highest worldwide reported cervical cancer incidences. DISCUSSION: There has therefore traditionally been a lack of data from studies focused on heterosexual men and non-HIV + women. There is also less evidence reporting on the putative precursor lesion to anal cancer (AIN - anal intraepithelial neoplasia), when compared to cervical cancer and CIN (cervical intraepithelial neoplasia). This review summarises the available biological and epidemiological evidence for HPV in the anal site and the pathogenesis of AIN and anal cancer amongst traditionally non-high risk groups. SUMMARY: There is strong evidence to conclude that high-grade AIN is a precursor to anal cancer, and some data on the progression of AIN to invasive cancer.RIGHTS : This article is licensed under the BioMed Central licence at http://www.biomedcentral.com/about/license which is similar to the 'Creative Commons Attribution Licence'. In brief you may : copy, distribute, and display the work; make derivative works; or make commercial use of the work - under the following conditions: the original author must be given credit; for any reuse or distribution, it must be made clear to others what the license terms of this work are

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

Enumerative and asymptotic analysis of a moduli space

We focus on combinatorial aspects of the Hilbert series of the cohomology ring of the moduli space of stable pointed curves of genus zero. We show its graded Hilbert series satisfies an integral operator identity. This is used to give asymptotic behavior, and in some cases, exact values, of the coefficients themselves. We then study the total dimension, that is, the sum of the coefficients of the Hilbert series. Its asymptotic behavior involves the Lambert W function, which has applications to classical tree enumeration, signal processing and fluid mechanics.Comment: 14 page

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

Decoupling of net community and export production on submesoscales in the Sargasso Sea

Author Posting. © American Geophysical Union, 2015. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Global Biogeochemical Cycles 29 (2015): 1266–1282, doi:10.1002/2014GB004913.Determinations of the net community production (NCP) in the upper ocean and the particle export production (EP) should balance over long time and large spatial scales. However, recent modeling studies suggest that a horizontal decoupling of flux-regulating processes on submesoscales (≤10 km) could lead to imbalances between individual determinations of NCP and EP. Here we sampled mixed-layer biogeochemical parameters and proxies for NCP and EP during 10, high-spatial resolution (~2 km) surface transects across strong physical gradients in the Sargasso Sea. We observed strong biogeochemical and carbon flux variability in nearly all transects. Spatial coherence among measured biogeochemical parameters within transects was common but rarely did the same parameters covary consistently across transects. Spatial variability was greater in parameters associated with higher trophic levels, such as chlorophyll in >5.0 µm particles, and variability in EP exceeded that of NCP in nearly all cases. Within sampling transects, coincident EP and NCP determinations were uncorrelated. However, when averaged over each transect (30 to 40 km in length), we found NCP and EP to be significantly and positively correlated (R = 0.72, p = 0.04). Transect-averaged EP determinations were slightly smaller than similar NCP values (Type-II regression slope of 0.93, standard deviation = 0.32) but not significantly different from a 1:1 relationship. The results show the importance of appropriate sampling scales when deriving carbon flux budgets from upper ocean observations.NASA Ocean Carbon and Biogeochemistry program Grant Number: NNX11AL94G; WHOI Postdoctoral Scholar fellowship; NASA ACE Grant Number: NNX12AJ25G; NSF Grant Number: OCE-07523662016-02-2