The (m,n)(m,n)-rational q,tq, t-Catalan polynomials for m=3m=3 and their q,tq,t-symmetry

We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area,skip,dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q,t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area,skips,dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q,t$-symmetry of $(m,n)$-rational $q, t$-Catalan polynomials for $m=3$.Comment: 11 pages, 4 figure

Rational Convolution Roots of Isobaric Polynomials

In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first kind and second kind. Hence we call them Stirling operators. To construct matrix representations of the roots of GFPs, we use the Stirling operators of the first kind. We give explicit examples to show how the Stirling operators of the second kind appear in the low degree cases for the WIP-roots. As a consequence of the matrix construction, we have matrix representations of multiplicative arithmetic functions under the Dirichlet product into its divisible closure.Comment: 13 page

DNA Methylation Changes in Response to Sulfur Dioxide Stress in Arabidopsis Plants

AbstractSulfur dioxide (SO2) exposure causes alterations of DNA cytosine methylation in Arabidopsis genome. Analysis of methylation-sensitive amplified polymorphism (MSAP) revealed that SO2 induced methylation changes of cytosine in CCGG sites, in which hypermethylation is more than hypomethylation. Bisulfite sequencing showed that the methyaltion levels of cytosine in promoter regions of Arabidopsis genes encoding 1-aminocyclopropane-1-carboxylic acid synthase 6 and nitrilase 2 decreased accompanying with enhanced transcript levels. The global genome hypermethylation and locus-specific hypomethylation in Arabidopsis cells, which contribute to genome stability and gene regulation, can be considered as a very important regulatory mechanism for plants adaption to environmental stress

Theoretical Analysis of “Eco-Man” in Sight of Ecological Civilization

The emergence of a new sort of civilization—ecological civilization, is definitely becoming the historical inevitability symbolizing that human beings have stepped into the era of ecological civilization. As a totally brand new type of civilization, ecological civilization is also in need of a new subject carrier—eco-man. Consequently, the development of ecological civilization is accelerating the establishment of the new type eco-man with personality model. This paper defines the concept and characteristics of “eco-man”. The settlement of “eco-man” theory has significant theoretical importance and practical meaning, and it’s especially important and urgent to establish “eco-man” in face of current situation

A dual approach to structure constants for K-theory of Grassmannians

International audienceThe problem of computing products of Schubert classes in the cohomology ring can be formulated as theproblem of expanding skew Schur polynomial into the basis of ordinary Schur polynomials. We reformulate theproblem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to itsbasis of Schubert structure sheaves in a similar way; we address the problem of expanding the generating functions forskew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials. From this point of view, we produce a chain of bijections leading to Buch’s K-theoretic Littlewood-Richardson rule

Feasibility study for downscaling seasonal tropical cyclone activity using the NCEP regional spectral model

The potential use of the National Centers for Environmental Prediction (NCEP) Regional Spectral Model (RSM) for downscaling seasonal tropical cyclone (TC) activity was analyzed here. The NCEP RSM with horizontal resolution of 50 km, was used to downscale the ECHAM4.5 Atmospheric General Circulation Model (AGCM) simulations forced with observed sea surface temperature (SST) over the western North Pacific. An ensemble of ten runs for June–November 1994 and 1998 was studied. The representation of the TCs is much improved compared to the low-resolution forcing AGCM, but the TCs are not as intense as observed ones, as the RSM horizontal resolution is not sufficiently high. The large-scale fields of the RSM are examined and compared to both the AGCM and the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis. The large-scale fields of RSM characteristics are in general similar to those of the reanalysis. Various properties of the TCs in the RSM are also examined such as first positions, tracks, accumulated cyclone energy (ACE) and duration. While the RSM does not reproduce the higher number of TCs in 1994 than in 1998, other measures of TC activity (ACE, number of cyclone days) in the RSM are higher in 1994 than in 1998

The (m,n)(m, n)-rational q,tq, t-Catalan polynomials for m=3m=3 and their q,tq, t-symmetry

We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area; skip; dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q; t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area; skip; dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q; t$-symmetry of $(m,n)$-rational $q; t$-Catalan polynomials for $m=3$.