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

The f-vector of the descent polytope

For a positive integer n and a subset S of [n-1], the descent polytope DP_S is the set of points x_1, ..., x_n in the n-dimensional unit cube [0,1]^n such that x_i >= x_{i+1} for i in S and x_i <= x_{i+1} otherwise. First, we express the f-vector of DP_S as a sum over all subsets of [n-1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5, ...}. We derive a generating function for the f-polynomial F_S(t) of DP_S, written as a formal power series in two non-commuting variables with coefficients in Z[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.Comment: 14 pages; to appear in Discrete & Computational Geometr

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

Optimal Planar Electric Dipole Antenna

Considerable time is often spent optimizing antennas to meet specific design metrics. Rarely, however, are the resulting antenna designs compared to rigorous physical bounds on those metrics. Here we study the performance of optimized planar meander line antennas with respect to such bounds. Results show that these simple structures meet the lower bound on radiation Q-factor (maximizing single resonance fractional bandwidth), but are far from reaching the associated physical bounds on efficiency. The relative performance of other canonical antenna designs is compared in similar ways, and the quantitative results are connected to intuitions from small antenna design, physical bounds, and matching network design.Comment: 10 pages, 15 figures, 2 tables, 4 boxe

Fundamental bounds on transmission through periodically perforated metal screens with experimental validation

This paper presents a study of transmission through arrays of periodic sub-wavelength apertures. Fundamental limitations for this phenomenon are formulated as a sum rule, relating the transmission coefficient over a bandwidth to the static polarizability. The sum rule is rigorously derived for arbitrary periodic apertures in thin screens. By this sum rule we establish a physical bound on the transmission bandwidth which is verified numerically for a number of aperture array designs. We utilize the sum rule to design and optimize sub-wavelength frequency selective surfaces with a bandwidth close to the physically attainable. Finally, we verify the sum rule and simulations by measurements of an array of horseshoe-shaped slots milled in aluminum foil.Comment: 10 pages, 11 figures. Updated Introduction and Conclusion

A second look at the toric h-polynomial of a cubical complex

We provide an explicit formula for the toric $h$-contribution of each cubical shelling component, and a new combinatorial model to prove Clara Chan's result on the non-negativity of these contributions. Our model allows for a variant of the Gessel-Shapiro result on the $g$-polynomial of the cubical lattice, this variant may be shown by simple inclusion-exclusion. We establish an isomorphism between our model and Chan's model and provide a reinterpretation in terms of noncrossing partitions. By discovering another variant of the Gessel-Shapiro result in the work of Denise and Simion, we find evidence that the toric $h$-polynomials of cubes are related to the Morgan-Voyce polynomials via Viennot's combinatorial theory of orthogonal polynomials.Comment: Minor correction

On a Generalization of Zaslavsky's Theorem for Hyperplane Arrangements

We define arrangements of codimension-1 submanifolds in a smooth manifold which generalize arrangements of hyperplanes. When these submanifolds are removed the manifold breaks up into regions, each of which is homeomorphic to an open disc. The aim of this paper is to derive formulas that count the number of regions formed by such an arrangement. We achieve this aim by generalizing Zaslavsky's theorem to this setting. We show that this number is determined by the combinatorics of the intersections of these submanifolds.Comment: version 3: The title had a typo in v2 which is now fixed. Will appear in Annals of Combinatorics. Version. 2: 19 pages, major revision in terms of style and language, some results improved, contact information updated, final versio

Orientations, lattice polytopes, and group arrangements II: Modular and integral flow Polynomials of graphs

We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law on the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers and answers a question by Beck and Zaslavsky.Regal Entertainment Group (Competitive Earmarked Research Grants 600703)Regal Entertainment Group (Competitive Earmarked Research Grants 600506)Regal Entertainment Group (Competitive Earmarked Research Grants 600608

Phenotypic Modulation of Smooth Muscle Cells in Atherosclerosis Is Associated With Downregulation of LMOD1, SYNPO2, PDLIM7, PLN, and SYNM

OBJECTIVE: Key augmented processes in atherosclerosis have been identified, whereas less is known about downregulated pathways. Here, we applied a systems biology approach to examine suppressed molecular signatures, with the hypothesis that they may provide insight into mechanisms contributing to plaque stability. APPROACH AND RESULTS: Muscle contraction, muscle development, and actin cytoskeleton were the most downregulated pathways (false discovery rate=6.99e-21, 1.66e-6, 2.54e-10, respectively) in microarrays from human carotid plaques (n=177) versus healthy arteries (n=15). In addition to typical smooth muscle cell (SMC) markers, these pathways also encompassed cytoskeleton-related genes previously not associated with atherosclerosis. SYNPO2, SYNM, LMOD1, PDLIM7, and PLN expression positively correlated to typical SMC markers in plaques (Pearson r>0.6, P0.8, P<0.0001). By immunohistochemistry, the proteins were expressed in SMCs in normal vessels, but largely absent in human plaques and intimal hyperplasia. Subcellularly, most proteins localized to the cytoskeleton in cultured SMCs and were regulated by active enhancer histone modification H3K27ac by chromatin immunoprecipitation-sequencing. Functionally, the genes were downregulated by PDGFB (platelet-derived growth factor beta) and IFNg (interferron gamma), exposure to shear flow stress, and oxLDL (oxidized low-density lipoprotein) loading. Genetic variants in PDLIM7, PLN, and SYNPO2 loci associated with progression of carotid intima-media thickness in high-risk subjects without symptoms of cardiovascular disease (n=3378). By eQTL (expression quantitative trait locus), rs11746443 also associated with PDLIM7 expression in plaques. Mechanistically, silencing of PDLIM7 in vitro led to downregulation of SMC markers and disruption of the actin cytoskeleton, decreased cell spreading, and increased proliferation. CONCLUSIONS: We identified a panel of genes that reflect the altered phenotype of SMCs in vascular disease and could be early sensitive markers of SMC dedifferentiation

Late Cenozoic tephrostratigraphy offshore the southern Central American Volcanic Arc: 2. Implications for magma production rates and subduction erosion

Pacific drill sites offshore Central America provide the unique opportunity to study the evolution of large explosive volcanism and the geotectonic evolution of the continental margin back into the Neogene. The temporal distribution of tephra layers established by tephrochonostratigraphy in Part 1 indicates a nearly continuous highly explosive eruption record for the Costa Rican and the Nicaraguan volcanic arc within the last 8 M.y. The widely distributed marine tephra layers comprise the major fraction of the respective erupted tephra volumes and masses thus providing insights into regional and temporal variations of large-magnitude explosive eruptions along the southern Central American Volcanic Arc (CAVA). We observe three pulses of enhanced explosive magmatism between 0-1 Ma at the Cordillera Central, between 1-2 Ma at the Guanacaste and at >3 Ma at the Western Nicaragua segments. Averaged over the long-term the minimum erupted magma flux (per unit arc length) is ∼0.017 g/ms. Tephra ages, constrained by Ar-Ar dating and by correlation with dated terrestrial tephras, yield time-variable accumulation rates of the intercalated pelagic sediments with four prominent phases of peak sedimentation rates that relate to tectonic processes of subduction erosion. The peak rate at >2.3 Ma near Osa particularly relates to initial Cocos Ridge subduction which began at 2.91±0.23 Ma as inferred by the 1.5 M.y. delayed appearance of the OIB geochemical signal in tephras from Barva volcano at 1.42 Ma. Subsequent tectonic re-arrangements probably involved crustal extension on the Guanacaste segment that favored the 2-1 Ma period of unusually massive rhyolite production