Affine Weyl group symmetries of Frobenius Painlevé equations

In this paper, we introduce a Frobenius Painlevé IV equation and the corresponding Hamilton system, and we give the symmetric form of the Frobenius Painlevé IV equation. Then, we construct the Lax pair of the Frobenius Painlevé IV equation. Furthermore, we recall the Frobenius modified KP hierarchy and the Frobenius KP hierarchy by bilinear equations, then we show how to get Frobenius Painlevé IV equation from the Frobenius modified KP hierarchy. In order to study the different aspects of the Frobenius Painlevé IV equation, we give the similarity reduction and affine Weyl group symmetry of the equation. Similarly, we introduce a Frobenius Painlevé II equation and show the connection between the Frobenius modified KP hierarchy and the Frobenius Painlevé II equation.     

GNSS辅助捷联惯导行进间对准自适应滤波方法

针对车载武器系统快速发射需求,提出一种基于GNSS辅助的捷联惯导行进间对准自适应滤波方法。该方法把行进间传递对准分为粗对准与精对准两个阶段。粗对准阶段以GNSS为观测基准完成对捷联惯导姿态的粗捕获,降低初始偏差不确定性对于精对准阶段的影响。在精对准阶段,考虑到车载系统的运动特性,提出一种"水平+方位"行进间对准双滤波器并行的设计思路,利用车载系统在不同时间段的动力学特性,对三轴姿态估计进行分时解耦,实现初始姿态的高精度估计;与此同时,引入协方差成形自适应调节过程,以最小化Frobenius范数为优化指标,实现对行进间对准卡尔曼滤波器的自适应调节,增强系统鲁棒性。数值仿真表明,协方差成形自适应卡尔曼滤波方法能够有效保证系统在全运动剖面内的稳定,结合双滤波器并行方案能够有效解决行进间对准精度不高与稳定性欠佳等问题,水平对准精度优于1.5′(1σ),方位对准精度优于6′(1σ)。     

Bounding the Fitting height of a solvable group by the number of zeros in a character table

In this paper, we bound the Fitting height of a solvable group by the number of zeros in a character table.

     

Congruences modulo powers of 3 for 3- and 9-colored generalized Frobenius partitions

Let $c{?}_k\left(n\right)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $c{?}_3\left(n\right)$ and $c{?}_9\left(n\right)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\ge 3$ and $n\ge 0$, we prove that

$c{?}_3\left(3^{2k}n+\frac{7?3^{2k}+1}{8}\right)\equiv 0(mod{3}^{4k+5}).$

We give two different proofs to the congruences satisfied by $c{?}_9\left(n\right)$. One of the proofs uses a relation between $c{?}_9\left(n\right)$ and $c{?}_3\left(n\right)$ due to Kolitsch, for which we provide a new proof in this paper.     

Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula

We study asymptotics of an irreducible representation of the symmetric group Sn corresponding to a balanced Young diagram λ (a Young diagram with at most rows and columns for some fixed constant C) in the limit as n tends to infinity. We show that there exists a constant D (which depends only on C) with a property that