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

Pure O-sequences and matroid h-vectors

We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley's conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley's conjecture in full generality. Our technical work on pure O-sequences also uses very recent results of the third author and collaborators.Comment: Contains several changes/updates with respect to the previous version. In particular, a discussion of a possible approach to the general case is included at the end. 13 pages. To appear in the Annals of Combinatoric

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

Mechanism-Based Treatment Strategies for IBD: Cytokines, Cell Adhesion Molecules, JAK Inhibitors, Gut Flora, and More

Background Although TNF inhibitors revolutionized the therapy of inflammatory bowel disease (IBD), we have been reaching a point where other therapies with different mechanisms of action are necessary. A rising number of elderly IBD patients with contraindications to established therapies and a growing group of patients losing response to anti-TNF therapy compel us to find safer, better-tolerated, and, ideally, personalized treatment options. However, in order to choose the right drug to fit a patient, it is indispensable to understand the pathomechanism involved in IBD. Summary The aim of this review is to explain the inflammatory signaling pathways in IBD and how to inhibit them with current and future therapeutic approaches. Next to biologic agents targeting inflammatory cytokines (anti-TNF agents, anti-IL-12/-23 agents, and specific inhibitors of IL-23), biologics blocking leukocyte trafficking to the gut (anti-integrin antibodies) are available nowadays. More recently, small molecules inhibiting the JAK-STAT pathway (JAK inhibitors) or preventing lymphocyte trafficking (sphingosine-1-phosphate modulators) have been approved or are under investigation. Furthermore, modifying the microbiota has potential therapeutic effects on IBD, and autologous hematopoietic or mesenchymal stem cell transplantation may be considered for a highly selected group of IBD patients. Key Message Physicians should understand the different mechanisms of action of the potential therapies for IBD to select the right drug for the right patient

Backbone and side-chain 1H, 15N and 13C resonance assignments of S18Y mutant of ubiquitin carboxy-terminal hydrolase L1

Ubiquitin carboxy-terminal hydrolase L1 (UCH-L1), also known as PGP9.5, is a protein of 223 amino acids. Although it was originally characterized as a deubiquitinating enzyme, recent studies indicate that it also functions as a ubiquitin (Ub) ligase and a mono-Ub stabilizer. It is highly abundant in brain, constituting up to 2% of total brain proteins. Down-regulation and extensive oxidative modification of UCH-L1 have been observed in the brains of Alzheimer’s disease (AD) and Parkinson’s disease (PD) patients. Mutations in the UCH-L1 gene have been reported to be linked to Parkinson’s disease, in particular, the I93 M variant is associated with a higher susceptibility of PD in contrast to a higher protection against PD for the S18Y variant. Hence, the structure of UCH-L1 and the underlying effects of disease associated mutations on the structure and function of UCH-L1 are of considerable interest. Here, we report the NMR spectral assignments of the S18Y human UCH-L1 mutant with the aim to obtain better understanding about the risk of Parkinson’s disease against structural and dynamical changes induced by this mutation on UCH-L1

Monomial ideals whose depth function has any given number of strict local maxima

We construct monomial ideals with the property that their depth function has any given number of strict local maxima

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

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

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)