A note on Bernoulli polynomials and solitons

Abstract

The dependence on time of the moments of the one-soliton KdV solutions is given by Bernoulli polynomials. Namely, we prove the formula expressing the moments of the one-soliton function sech²(x-t) in terms of the Bernoulli polynomials Bn(x). We also provide an alternative short proof to the Grosset-Veselov formula connecting the one-soliton to the Bernoulli numbers (D?=?d/dx) published recently in this journal.     

A note on q-Bernoulli numbers and polynomials

Abstract

Recently, B. A. Kupershmidt have constructed a reflection symmetries of q-Bernoulli polynomials (see [9]). In this paper we give another construction of a q-Bernoulli polynomials, which form Barnes’ multiple Bernoulli polynomials at q=1, cf. [1, 13, 14]. By using q-Volkenborn integration, we can also investigate the properties of the reflection symmetries of these’ q-Bernoulli polynomials.     

A new generalization of the Bernoulli and related polynomials

In this paper, we introduce and investigate a generalization of the Bernoulli polynomials by means of a suitable generating function. We establish several interesting properties of these general polynomials. Furthermore, we give explicit series representations for these general polynomials in terms of a certain generalized Hurwitz-Lerch zeta function and the familiar Gauss hypergeometric function.     

Zeroes of chromatic polynomials: A new approach to Beraha conjecture using quantum groups

The number of colourings of a graphG withQ or fewer colors is a polynomial inQ known as the chromatic polynomialP _G (Q). It coïncides with the partition functionF _G of theQ state Potts model onG at zero temperature and in the antiferromagnetic regimee ^K =0. In the planar case, the Beraha conjecture particularizes the numbers \(B_n = 4\cos ^2 \frac{\pi }{n}\) as possible accumulation points of real zeroes ofP _G in the infinite graph limit. We suggest in this work an approach based on recent developments of quantum groups to handle this conjecture. For the square, triangular and honeycomb lattices and systems wrapped on a cylinderl×t, we first exhibit in the (Q, e ^K ) Potts parameter space a critical line, whereF _G(Q,e ^K) has real zeroes converging to and only to theB _n 's asl, t→∞. The analysis is based on the vertex representation of theQ state Potts model, quantum algebraU _qSl (2) properties forq a root of unity, and conformal invariance.U _qSl (2) symmetry is present for anye ^K , including the chromatic polynomial casee ^K =0. Using an additional hypothesis on the eigenvalues structure and knowledge of the Potts parameter space, we then argue that forP _G (Q), real zeros occur and converge toB _n 's, 2≦n≦n ₀ only, wheren ₀ depends on the lattice. Extensions to other kinds of graphs and size dependence of the zeros are discussed. Finally physical applications are also mentioned.     

Quantum gravity and matter: counting graphs on causal dynamical triangulations

An outstanding challenge for models of non-perturbative quantum gravity is the consistent formulation and quantitative evaluation of physical phenomena in a regime where geometry and matter are strongly coupled. After developing appropriate technical tools, one is interested in measuring and classifying how the quantum fluctuations of geometry alter the behaviour of matter, compared with that on a fixed background geometry. In the simplified context of two dimensions, we show how a method invented to analyze the critical behaviour of spin systems on flat lattices can be adapted to the fluctuating ensemble of curved spacetimes underlying the causal dynamical triangulations (CDT) approach to quantum gravity. We develop a systematic counting of embedded graphs to evaluate the thermodynamic functions of the gravity-matter models in a high- and low-temperature expansion. For the case of the Ising model, we compute the series expansions for the magnetic susceptibility on CDT lattices and their duals up to orders 6 and 12, and analyze them by ratio method, Dlog Padé and differential approximants. Apart from providing evidence for a simplification of the model’s analytic structure due to the dynamical nature of the geometry, the technique introduced can shed further light on criteria à la Harris and Luck for the influence of random geometry on the critical properties of matter systems.     

利用泽尼克系数求取衍射光栅的分辨本领

光栅分辨本领检测设备的焦距通常达几米甚至十几米,采用直接测量法难度大、成本高,利用衍射波前间接求取光栅分辨本领是解决该问题的有效途径之一。在光栅光谱成像傅里叶变换理论基础上,建立了利用泽尼克多项式拟合系数求解衍射光栅分辨本领的归一化模型,揭示了光栅衍射波前与分辨本领的求取关系,提出了依据泽尼克多项式拟合系数求取衍射光栅分辨本领的新方法。根据该方法实测了一块衍射光栅的分辨本领,并与直接测量法进行对比测试。结果表明该方法误差小于4.42%,降低了分辨本领的测试难度,是衍射光栅分辨本领求取的有效手段,应用于ZYGO干涉仪等仪器中,通过简单的波前测试即可得到定量的衍射光栅分辨本领指标。     

Geometry of the space of triangulations of a compact manifold

In this paper we study the spaceT _M of triangulations of an arbitrary compact manifoldM of dimension greater than or equal to four. This space can be endowed with the metric defined as the minimal number of bistellar operations required to transform one of two considered triangulations into the other. Recently, this space became and object of study in Quantum Gravity because it can be regarded as a toy discrete model of the space of Riemannian structures onM.Our main result can be informally explained as follows: LetM be either any compact manifold of dimension greater than four or any compact four-dimensional manifold from a certain class described in the paper. We prove that for a certain constantC>1 depending only on the dimension ofM and for all sufficiently largeN the subsetT _M(N) ofT _M formed by all triangulations ofM with N simplices can be represented as the union of at least [C ^N] disjoint non-empty subsets such that any two of these subsets are very far from each other in the metric ofT _M. As a corollary, we show that for any functional from a very wide class of functionals onT _M the number of its deep local minima inT _M(N) grows at least exponentially withN, whenN.This work was partially supported by the New York University Research Challenge Fund Grant, by Grant ARO-DAAL-03-92-G-0143 and by NSERC Grant OGP0155879.     

Experimental analysis of shape deformation of evaporating droplet using Legendre polynomials

Experiments involving heating of liquid droplets which are acoustically levitated, reveal specific modes of oscillations. For a given radiation flux, certain fluid droplets undergo distortion leading to catastrophic bag type breakup. The voltage of the acoustic levitator has been kept constant to operate at a nominal acoustic pressure intensity, throughout the experiments. Thus the droplet shape instabilities are primarily a consequence of droplet heating through vapor pressure, surface tension and viscosity. A novel approach is used by employing Legendre polynomials for the mode shape approximation to describe the thermally induced instabilities. The two dominant Legendre modes essentially reflect (a) the droplet size reduction due to evaporation, and (b) the deformation around the equilibrium shape. Dissipation and inter-coupling of modal energy lead to stable droplet shape while accumulation of the same ultimately results in droplet breakup.     

A comparison study of scratch and wear properties using atomic force microscopy

The patterning technique that uses an AFM (atomic force microscopy) tip as a scratch tool, also known as AFM scratching, has been a vital technique for nanofabrication because of its low cost and potential to reach a resolution into the sub-nanometer domain. The AFM scratching technique was first used to study the scratch characteristics of silicon, with an emphasis on establishing its scratchability or the nanoscale machinability. The effects of the scratch parameters, including the applied tip force and number of scratches, on the size of the scratched geometry were specifically evaluated. The primary property that measures the scratchability was identified and assessed. To illustrate its suitability and reliability, the value of the scratchability, based on the present Si scratching experiments, was compared with the values based on the data available in the literatures for different scratching conditions or for materials other than Si. Since AFM scratching is in some aspects similar to the nanoscale wear test, the scratchability property identified is also compared with two major wear resistance indicators, wear coefficient and hardness. All comparison results indicate that the scratchability property identified, the scratch ratio, is an appropriate manufacturability indicator for measuring the degree of the ease or difficulty of a material scratched by an AFM tip and more suitable than the wear coefficient and hardness to gauge the nanoscale AFM scratchability.     

The strong topology on the algebra of polynomials

It is shown that under a quite general condition on the operator T (unbounded, symmetric) and on the domain $\text{D}$ for the representation x → T of the algebra $\text{P}$(x) on $\text{D}$ in $\text{P}$(T) the strongest locally convex topology τ_∞ coincides with the strong topology σ^$\text{D}$.     

A superintegrable discrete harmonic oscillator based on bivariate Charlier polynomials

Physics of Atomic Nuclei - A simple discrete model of the two-dimensional isotropic harmonic oscillator is presented. It is superintegrable with su(2) as its symmetry algebra. It is constructed...     

A comparison of glycans and polyglycans using solid-state NMR and X-ray powder diffraction

Individual polyglycans and their corresponding monomers have been studied separately for several decades. Attention has focused primarily on the modifications of these polyglycans instead of the simple relationship between the polyglycans themselves and their corresponding monomers. Two polyglycans, chitin and chitosan, were examined along with their respective monomeric units, N-acetyl-D-glucosamine (GlcNAc) and (+)D-glucosamine (GlcN) using solid-state proton decoupling Magic Angle Turning (MAT) techniques and X-Ray Powder Diffraction (XRPD). A down-field shift in isotropic (13)C chemical shifts was observed for both polymers in Cross Polarization/Magic Angle Spinning (CP/MAS) spectra. An explanation of misleading peak assignments in previous NMR studies for these polyglycans was determined by comparing sideband patterns of the polymers with their corresponding monomers generated in a 2D FIve pi REplicated Magic Angle Turning (FIREMAT) experiment processed by Technique for Importing Greater Evolution Resolution (TIGER). Structural changes in the crystalline framework were supported by XRPD diffraction data.     

A solution to Bloch NMR flow equations for the analysis of hemodynamic functions of blood flow system using m-Boubaker polynomials

This paper proposes a solution to Bloch NMR flow equations in biomedical fluid dynamics using a new set of real polynomials. In fact, the authors conjugated their efforts in order to take benefit from similarities between independent Bloch NMR flow equations yielded by a recent study and the newly proposed characteristic differential equation of the m-Boubaker polynomials. The main goal of this study is to establish a methodology of using mathematical techniques so that the accurate measurement of blood flow in human physiological and pathological conditions can be carried out non-invasively and becomes simple to implement in medical clinics. Specifically, the polynomial solutions of the derived Bloch NMR equation are obtained for use in biomedical fluid dynamics. The polynomials represent the T₂-weighted NMR transverse magnetization and signals obtained in terms of Boubaker polynomials, which can be an attractive mathematical tool for simple and accurate analysis of hemodynamic functions of blood flow system. The solutions provide an analytic way to interpret observables made when the rF magnetic fields are designed based on the Chebichev polynomials. The representative function of each component is plotted to describe the complete evolution of the NMR transverse magnetization component for medical and biomedical applications. This mathematical technique may allow us to manipulate microscopic blood (cells) at nano-scale. We may be able to theoretically simulate nano-devices that may travel through tiny capillaries and deliver oxygen to anemic tissues, remove obstructions from blood vessels and plaque from brain cells, and even hunt down and destroy viruses, bacteria, and other infectious agents.     

Schubert polynomials and the Littlewood-Richardson rule

The decomposition of a product of two irreducible representations of a linear group Gl(N, ) is explicitly given by the Littlewood-Richardson rule, which amounts to finding how many Young tableaux satisfy certain conditions. We obtain more general multiplicities by generating vexillary permutations and by using partially symmetrical polynomials (Schubert polynomials).A la mémoire de S. Ulam, exemple et ami.     

Single-clipped correlation and the bell polynomials

A closed form for the infinite sum occurring in the expression of the single-clipped correlation is derived and its relation with the “exponential Bell polynomials” is established.     

A comparison of optical modulator structures using a matrix simulation approach

The high operating frequency, small dimensions and complex nature of advanced integrated photonics structures lead to a need for full 3D simulations in order to obtain accurate propagation characteristics. However, 3D simulations are very computationally expensive, especially for design optimization. Therefore a matrix analysis has been employed to model the propagation characteristics and analyze structure variations without the need to perform a separate simulation for each design iteration. Designs investigated include two that are promising for silicon-based integrated dense wavelength division multiplexing applications: the resonant cavity modulator and the microring resonator. The ability to accurately model and quickly optimize these electro-optic devices will be important for future large-scale integrated optics systems.     

The uniform and the strong topology on realizations of the algebra of polynomials

It is shown that under quite general assumptions on the operators A₁,…,A_n (unbounded, symmetric) and on the domain $\text{D}$ on the realization $\text{P}\text{(A}{}_1\text{,…,A}{}_{\mathrm{n}}\text{)}$ of the algebra of polynomials $\text{P}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)}$, the strongest locally convex topology τ_st coincides with the uniform topology τ_$\text{D}$ as well as with the strong operator topology τ_s. In the case n = 2 some conditions are given, under which these general assumptions are fulfilled.     

q-Orthogonal polynomials and the oscillator quantum group

The oscillator quantum algebra is shown to provide a group-theoretic setting for the q-Laguerre and q-Hermite polynomials.On leave from Laboratoire de Physique Nucléaire, Université de Montréal, Montréal, Canada H3C 3J7.     

Filtrations on Springer fiber cohomology and Kostka polynomials

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.