The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length ≤3 and the diagonal lengths ≤3; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group G and the Iwahori-Hecke algebra of type A on the permutation module of G by its Borel subgroup.     

碳纳米管增强型复合材料功能梯度板的自由振动模型与尺度效应

基于一种新的各向异性修正偶应力理论，建立了碳纳米管增强复合材料功能梯度板的自由振动模型。该模型能够描述尺度效应，且仅包含一个尺度参数。基于一阶剪切变形理论和哈密顿原理推演了板的运动微分方程，并以四边简支板为例给出了自振频率的解析解。讨论了板的几何尺寸、碳纳米管体分比含量和分布方式等因素对板的自振频率的影响。结果表明，本文模型所预测的板的自振基频总是高于经典弹性理论的Mindlin板模型的预测结果，两者间的差异在板的几何尺寸接近尺度参数的值时非常明显，且会随着板的几何尺寸的增大而逐渐消失。     

Recursions associated to trapezoid,symmetric and rotation symmetric functions over Galois fields

Rotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients. In this work, a generalization of this result is proved over any Galois field. That is, exponential sums over Galois fields of some rotation symmetric polynomials satisfy linear recurrences with integer coefficients. In the particular case of ${\mathbb{F}}_2$, an elementary method is used to obtain explicit recurrences for exponential sums of some of these functions. The concept of trapezoid Boolean function is also introduced and it is showed that the linear recurrences that exponential sums of trapezoid Boolean functions satisfy are the same as the ones satisfied by exponential sums of the corresponding rotations symmetric Boolean functions. Finally, it is proved that exponential sums of trapezoid and symmetric polynomials also satisfy linear recurrences with integer coefficients over any Galois field ${\mathbb{F}}_q$. Moreover, the Discrete Fourier Transform matrix and some Complex Hadamard matrices appear as examples in some of our explicit formulas of these recurrences.     

Ultradistributional boundary values of harmonic functions on the sphere

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.     

Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators

We give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state. These sets are not necessarily weakly closed as in [3], [4], while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for a-Lyapunov pairs of lower semi-continuous (not necessarily weakly-lsc) functions associated to these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.     

Wave turbulence and intermittency in directional wave fields

The evolution of surface gravity waves is driven by nonlinear interactions that trigger an energy cascade similarly to the one observed in hydrodynamic turbulence. This process, known as wave turbulence, has been found to display anomalous scaling with deviation from classical turbulent predictions due to the emergence of coherent and intermittent structures on the water surface. In the ocean, waves are spread over a wide range of directions, with a consequent attenuation of the nonlinear properties. A laboratory experiment in a large wave facility is presented to discuss the sensitivity of wave turbulence on the directional properties of model wave spectra. Results show that the occurrence of coherent and intermittent structures become less likely with the broadening of the wave directional spreading. There is no evidence, however, that intermittency completely vanishes.