The many proofs of an identity on the norm of oblique projections

Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P ²=P, which is neither null nor the identity, it holds that ||P|| = ||I –P||. This useful equality, while not widely-known, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler ones are presented.     

Patterns and Symmetries in the Visual Cortex and in Natural Images

As borders between different entities, lines are an important element of natural images. Indeed, the neurons of the mammalian visual cortex are tuned to respond best to lines of a given orientation. This preferred orientation varies continuously across most of the cortex, but also has vortex-like singularities known as pinwheels. In attempting to describe such patterns of orientation preference, we are led to consider underlying rotation symmetries: Oriented segments in natural images tend to be collinear; neurons are more likely to be connected if their preferred orientations are aligned to their topographic separation. These are indications of a reduced symmetry requiring joint rotations of both orientation preference and the underlying topography. This is verified by direct statistical tests in both natural images and in cortical maps. Using the statistics of natural scenes we construct filters that are best suited to extracting information from such images, and find qualitative similarities to mammalian vision. PACS84.35+i 89.70.+c 87.57.Nk     

连铸坯角部传热过程的数值模拟

在连铸理论与实践研究的基础上，提出了系数法铸坯角部热通量模型，为准确研究铸坯角部传热、应力和角部缺陷的产生提供了条件。     

不同入射角度下强流脉冲电子束能量沉积剖面和束流传输系数模拟计算

 强流脉冲电子束在材料中的能量沉积剖面、能量沉积系数和束流传输系数受其入射角的影响很大，理论计算了0.5~2.0MeV的电子束以不同的入射角在Al材料中的能量沉积剖面和能量沉积系数，并且还计算了0.4~1.4MeV电子束以不同入射角穿透不同厚度C靶的束流传输系数。计算结果表明，随着入射角的增大，靶材表面层单位质量中沉积的能量增大，电子在靶材料中穿透深度减小，能量沉积系数减小，相应的束流传输系数也减小；能量为0.5~2.0MeV的电子束当入射角在60°~70°时在材料表面层单位质量中沉积的能量较大。     

强流脉冲高斯分布束在磁四极透镜中的非线性传输

用李代数方法分析了高斯分布下强流脉冲束在磁四极透镜中的非线性传输.在高斯分布下,束流的空间电荷势可利用Green函数算出,进而可以得到包含束流自场的粒子运动的Hamilton函数.再施加李变换,就可以得到粒子运动的各级近似解.本文给出二级近似下的结果,根据需要,还可以扩展到更高级近似.计算过程需要进行迭代,即根据每次算出的轨迹值,确定束团在三维实空间中的大小,然后再进行迭代,直到满足精度要求为止.     

Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application

Different versions of the Darboux–Weinstein theorem guarantee the existence of action–angle-type variables and the harmonic-oscillator variables in a neighborhood of isotropic tori in the phase space. The procedure for constructing these variables is reduced to solving a rather complicated system of partial differential equations. We show that this system can be integrated in quadratures, which permits reducing the problem of constructing these variables to solving a system of quadratic equations. We discuss several applications of this purely geometric fact in problems of classical and quantum mechanics.     

ON THE ASYMPTOTIC BEHAVIOUR OF THE STEADY SUPERSONIC FLOWS AT INFINITY

This paper studies the asymptotic behaviour of steady supersonic flow past a piece-wise smooth corner or bend. Under the hypothese that both vertex angle and the total variation of tangent along the boundary are small, it is shown that the solution can be obtained by a modified Glimm scheme, and that the asymptotic behaviour of the solution is determined by the velocity of incoming flow and the limit of the tangent of the boundary at infinity.     

Cyclic permutations and evolutionary trees

Given a tree with leaf set X, there are certain ways of arranging the elements of X in a circular order so that can be embedded in the plane and ‘preserve’ this ordering. We investigate some new combinatorial properties of these ‘circular orderings.’ We then use these properties to establish two results concerning dissimilarity maps on X that are induced by edge-weighted trees with leaf set X.     

Regular Balanced Cayley Maps for Cyclic, Dihedral and Generalized Quaternion Groups

A Cayley map is a Cayley graph embedded in an orientable surface such that. the local rotations at every vertex are identical. In this paper, balanced regular Cayley maps for cyclic groups, dihedral groups, and generalized quaternion groups are classified.     

Dynamics of the periodic type-K competitive Kolmogorov systems

For time-periodic dissipative and irreducible type-K competitive Kolmogorov systems, it is proved that there is a canonically defined countable family F of unordered, disjoint invariant sets with the property that, for every persistent trajectory whose ω-limit set is not a cycle, there exists a unique trajectory in some element of F such that these two trajectories are asymptotic and the corresponding points in these two trajectories are K-related.