On positivity of Ehrhart polynomials

Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic Combinatorics, a volume of the Association for Women in Mathematics Series, Springer International Publishin

Markov basis and Groebner basis of Segre-Veronese configuration for testing independence in group-wise selections

We consider testing independence in group-wise selections with some restrictions on combinations of choices. We present models for frequency data of selections for which it is easy to perform conditional tests by Markov chain Monte Carlo (MCMC) methods. When the restrictions on the combinations can be described in terms of a Segre-Veronese configuration, an explicit form of a Gr\"obner basis consisting of moves of degree two is readily available for performing a Markov chain. We illustrate our setting with the National Center Test for university entrance examinations in Japan. We also apply our method to testing independence hypotheses involving genotypes at more than one locus or haplotypes of alleles on the same chromosome.Comment: 25 pages, 5 figure

Unimodality Problems in Ehrhart Theory

Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart $h^*$-vector. Ehrhart $h^*$-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart $h^*$-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This version updated October 2017 to correct an error in the original versio

A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.Comment: Published in the proceedings of 2017 Southern Regional Algebra Conferenc

Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$Comment: 31 pages, 15 figure

Blockade of T-cell activation by dithiocarbamates involves novel mechanisms of inhibition of nuclear factor of activated T cells.

Dithiocarbamates (DTCs) have recently been reported as powerful inhibitors of NF-kappaB activation in a number of cell types. Given the role of this transcription factor in the regulation of gene expression in the inflammatory response, NF-kappaB inhibitors have been suggested as potential therapeutic drugs for inflammatory diseases. We show here that DTCs inhibited both interleukin 2 (IL-2) synthesis and membrane expression of antigens which are induced during T-cell activation. This inhibition, which occurred with a parallel activation of c-Jun transactivating functions and expression, was reflected by transfection experiments at the IL-2 promoter level, and involved not only the inhibition of NF-kappaB-driven reporter activation but also that of nuclear factor of activated T cells (NFAT). Accordingly, electrophoretic mobility shift assays (EMSAs) indicated that pyrrolidine DTC (PDTC) prevented NF-kappaB, and NFAT DNA-binding activity in T cells stimulated with either phorbol myristate acetate plus ionophore or antibodies against the CD3-T-cell receptor complex and simultaneously activated the binding of AP-1. Furthermore, PDTC differentially targeted both NFATp and NFATc family members, inhibiting the transactivation functions of NFATp and mRNA induction of NFATc. Strikingly, Western blotting and immunocytochemical experiments indicated that PDTC promoted a transient and rapid shuttling of NFATp and NFATc, leading to their accelerated export from the nucleus of activated T cells. We propose that the activation of an NFAT kinase by PDTC could be responsible for the rapid shuttling of the NFAT, therefore transiently converting the sustained transactivation of this transcription factor that occurs during lymphocyte activation, and show that c-Jun NH2-terminal kinase (JNK) can act by directly phosphorylating NFATp. In addition, the combined inhibitory effects on NFAT and NF-KB support a potential use of DTCs as immunosuppressants

High-density information storage in an absolutely defined aperiodic sequence of monodisperse copolyester

Synthesis of a polymer composed of a large discrete number of chemically distinct monomers in an absolutely defined aperiodic sequence remains a challenge in polymer chemistry. The synthesis has largely been limited to oligomers having a limited number of repeating units due to the difficulties associated with the step-by-step addition of individual monomers to achieve high molecular weights. Here we report the copolymers of ??-hydroxy acids, poly(phenyllactic-co-lactic acid) (PcL) built via the cross-convergent method from four dyads of monomers as constituent units. Our proposed method allows scalable synthesis of sequence-defined PcL in a minimal number of coupling steps from reagents in stoichiometric amounts. Digital information can be stored in an aperiodic sequence of PcL, which can be fully retrieved as binary code by mass spectrometry sequencing. The information storage density (bit/Da) of PcL is 50% higher than DNA, and the storage capacity of PcL can also be increased by adjusting the molecular weight (~38???kDa)

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result

p63 expression in normal skin and usual cutaneous carcinomas

Background: p63 is a p53 homologue that is mapped to chromosome 3q27. This gene encodes six different isoforms, which have either transactivating or dominant negative effects on p53-reporter genes. It has been described that in contrast to p53, p63 seems not to be associated with tumor predisposition, as neither p63 knockout mouse models nor germline p63 mutations are related to an increased risk of tumorigenesis. It has been demonstrated that p63 is a reliable keratinocyte stem cell marker and that it is involved in the maintenance of the stem cell population. Scant data on p63 expression in normal skin, basal cell carcinomas (BCCs), keratoacanthomas and squamous cell carcinomas (SCCs) have been reported. We herein evaluated p63 expression in 16 BCCs, one keratoacanthoma and 13 SCCs. Methods: Immunohistochemistry according to the streptavidinbiotin-peroxidase technique, using the antibody 4A4 raised against all p63 isoforms, was performed. p63 expression was evaluated in epidermal cells and skin appendages. Semi-quantitative evaluation (–, π, ππ, πππ) of p63 expression in BCCs, keratoacanthoma and SCCs was carried out. Only nuclear expression was considered as specific. Results: p63 was expressed in the nuclei of epidermal basal and suprabasal cells, in the cells of the germinative hair matrix and the external root sheath of hair follicles, in the basal cells of the sebaceous gland and in the myoepithelial/basal cells of the sweat glands. All terminally differentiated cells were negative for p63. All BCCs showed ππto πππ immunoreactivity. At variance, keratoacanthomas and grade I and II SCCs showed variable p63 reactivity in a basal layerlike distribution, whereas undifferentiated cells of grade III SCCs showed ππto πππ positivity. A grade IV spindle SCC showed π immunoreactivity. The SCCs in situ showed remarkable expression of p63 in all cell layers. Terminally differentiated squamous cells were either negative or showed only focal immunoreactivity in the carcinomas. Conclusions: p63 is consistently expressed in the basal cells of epidermis and cutaneous appendages, including the basal/ myoepithelial cells of sweat glands. Based on our findings, the balance of probabilities favors that p63 might play a role in the pattern of differentiation and in the oncogenesis of usual carcinomas of the skin.Fundação para a Ciência e a Tecnologia (FCT) - SFRH/BD/5386/2001. Fundação para a Ciência e a Tecnologia (FCT) – Programa Operacional “Ciência, Tecnologia, Inovação” (POCTI)

Inducible colitis-associated glycome capable of stimulating the proliferation of memory CD4+ T cells

Immune responses are modified by a diverse and abundant repertoire of carbohydrate structures on the cell surface, which is known as the glycome. In this study, we propose that a unique glycome that can be identified through the binding of galectin-4 is created on local, but not systemic, memory CD4+ T cells under diverse intestinal inflammatory conditions, but not in the healthy state. The colitis-associated glycome (CAG) represents an immature core 1–expressing O-glycan. Development of CAG may be mediated by down-regulation of the expression of core-2 β1,6-N-acetylglucosaminyltransferase (C2GnT) 1, a key enzyme responsible for the production of core-2 O-glycan branch through addition of N-acetylglucosamine (GlcNAc) to a core-1 O-glycan structure. Mechanistically, the CAG seems to contribute to super raft formation associated with the immunological synapse on colonic memory CD4+ T cells and to the consequent stabilization of protein kinase C θ activation, resulting in the stimulation of memory CD4+ T cell expansion in the inflamed intestine. Functionally, CAG-mediated CD4+ T cell expansion contributes to the exacerbation of T cell–mediated experimental intestinal inflammations. Therefore, the CAG may be an attractive therapeutic target to specifically suppress the expansion of effector memory CD4+ T cells in intestinal inflammation such as that seen in inflammatory bowel disease