Some Combinatorial Properties of Schubert Polynomials

Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schützenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial in terms of the reduced decompositions of the permutation w. Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set of tableaux with w, each tableau contributing a single term to . This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function , where denotes a skew Schur function.Sara C. Billey: Supported by the National Physical Science Consortium. William Jockusch: Supported by an NSF Graduate Fellowship. Richard P. Stanley: Partially supported by NSF grants DMS-8901834 and DMS-9206374     

Multinomial Combinatorial Group Representations of the Octahedral and Cubic Symmetries

We consider the full multinomial combinatorics of all irreducible representations of the octahedral (cubic) symmetry as a function of partitions for vertex, face and edge colorings. Full combinatorial tables for all irreducible representations and all multinomial partitions are constructed. These enumerations constitute multinomial expansions of character-based cycle index polynomials, and grow in combinatorial complexity as a function of edge or vertex coloring partitions.     

Interpreting divergences and Madelung-type sums via extrapolations of sequence generating polynomials: Method of intersection and area rule

Based on explorations in estimating certain Madelung constants, we put forward here two separate strategies to understand the meaning of two distinct classes of divergent non-power-series expansions. One class refers to alternating series representations, the other to monotonic ones. They chiefly rest on precise and approximate polynomial extrapolations, depending on situations. In case of sawtooth sequences, e.g., the partial-sums obtainable from Dirichlet eta or beta function at negative integer arguments, exact sequence-generating polynomials are found. Extrapolations yield a graphical meaning to anti-limit here, along with the exact answer. For staircase sequences, like the ones obtained from partial-sums of series representations for lambda and zeta functions, again at negative integer arguments, anti-limits do not exist. But, correct sequence-generating polynomials are obtained. There, our recipe relies on estimation of specific, finite areas embedded by such polynomials. The schemes put forward here are direct, independent and conceptually appealing. A subsequent extension of the latter strategy to alternating series also lends extra credence. Two new interpretations of summability are gained. Pilot calculations on several types of lattice sums reveal the worth of our endeavor with approximate extrapolations as well.     

New generating function formulae of even- and odd-Hermite polynomials obtained and applied in the context of quantum optics

By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.     

Backward errors for eigenvalues and eigenvectors of Hermitian,skew-Hermitian,H-even and H-odd matrix polynomials

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.     

Asymptotic Results for Semigroups of Order-Preserving Partial Transformations

Let 𝒫  _n be the semigroup of all decreasing and order-preserving partial transformations of an n-element chain, and let E(𝒫  _n ) be its set of idempotents. Among other results, asymptotic formulae for |𝒫  _n | and |E(𝒫  _n )|/|𝒫  _n | are obtained. Similar results for 𝒫  _n the (larger) semigroup of all order-preserving partial transformations of an n-element chain are also obtained.     

An apodized cubic phase mask used in a wavefront coding system to extend the depth of field

The point spread function (PSF) caused by a wavefront coding system with a cubic phase mask has big side-lobes which leads to bad image restoration. This paper proposes a novel apodized cubic phase mask to suppress the side-lobes of the PSF. Simulated annealing algorithm is used to optimize the cubic and the truncation parameter of the phase mask. The system with the novel phase mask has better performance in the modulation transfer function (MTF) especially in low-and-medium spatial frequency region. The simulation results show that the restored images with the novel phase mask are superior to the one with the classic cubic phase mask in contrast and ringing effect. The experimental results show that the side-lobes of the PSF are suppressed by using the apodized cubic phase mask.     

Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category $\mathsf{Sp}(\beta )$, depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta =m?n$ the monoidal category of representations of $U_q({\mathfrak{gl}}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(\beta )$ with a direct limit of quotients of a dual idempotented quantum group ${\dot{\mathbf{U}}}_q({\mathfrak{gl}}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(\beta )$ gives a unified natural setting for defining the colored ${\mathfrak{gl}}_{m|n}$ link invariant (for $\beta =m?n$) and the colored HOMFLY-PT polynomial (for β generic).     

连续可导函数类的最优拉格朗日插值

在构造拉格朗日插值算法时,插值结点的选择是十分重要的.给定一个足够光滑的函数,如果结点选择的不好,当插值结点个数趋于无穷时,插值函数不收敛于函数本身.例如龙格现象:对于龙格函数f(x)=1/1+25x^2,如果拉格朗日插值的结点取[-1,1]上的等距结点,那么逼近的误差会随着结点个数增多而趋于无穷大⑴,由此可知插值结点的选择尤为重要.     

Improvement of random coefficient differential models of growth of anaerobic photosynthetic bacteria by combining Bayesian inference and gPC

The time evolution of microorganisms, such as bacteria, is of great interest in biology. In the article by D. Stanescu et al. [Electronic Transactions on Numerical Analysis, 34, 44–58 (2009)], a logistic model was proposed to model the growth of anaerobic photosynthetic bacteria. In the laboratory experiment, actual data for two species of bacteria were considered: Rhodobacter capsulatus and Chlorobium vibrioforme. In this paper, we suggest a new nonlinear model by assuming that the population growth rate is not proportional to the size of the bacteria population, but to the number of interactions between the microorganisms, and by taking into account the beginning of the death phase in the kinetic curve. Stanescu et al. evaluated the effect of randomness into the model coefficients by using generalized polynomial chaos (gPC) expansions, by setting arbitrary distributions without taking into account the likelihood of the data. By contrast, we utilize a Bayesian inverse approach for parameter estimation to obtain reliable posterior distributions for the random input coefficients in both the logistic and our new model. Since our new model does not possess an explicit solution, we use gPC expansions to construct the Bayesian model and to accelerate the Markov chain Monte Carlo algorithm for the Bayesian inference.