Scale heights of 84 northern spiral galaxies

Using the method proposed by Peng (1988) on the basis of density waves theory and the solution of three-dimensional Poisson’s equation for a logarithmic disturbance of density, and analyzing the spiral patterns, the scale heights of 84 northern spiral galaxies, whose images are taken from the Digitized Sky Survey¹⁾ at Xinglong Observational Station of Beijing Observatory, are measured. The spiral arms of all these galaxies have been fitted on their photographs with some logarithmic spiral curves for getting their correct inclinations.     

Thicknesses of 70 southern spiral galaxies

Using the method proposed by Peng (1988) and analyzing the spiral patterns, the thicknesses of 70 southern spiral galaxies whose images were taken from the Digitized Sky Survey at Xinglong Observational Station of Beijing Observatory were measured. The spiral arms of all these galaxies have been fitted on their images with some logarithmic spiral curves to get their optimum inclinations. Project supported by the Yational Natural Science Foundation of China, the National Climbing Project of China and the Doctoral Program Foundation of State Education Cummission.     

Density Wave Theory of Spiral Structure of Galaxies

The density wave theory of spiral structure in galaxies is studied as a dynamical problem: the gravitational instability of a galactic disk with respect to spiral modes. The asymptotic theory for tightly wound spirals is presented in detail, although the problem has been formulated in a more general form to open the way to future research work in other directions. Some preliminary results based on the general formulation are included; they confirm those based on the asymptotic theory in cases where the latter should be applicable. The fluid dynamical model is adopted; reference to the stellar dynamical model is made when necessary. Besides providing a coherent account of the present status of the theory, it is hoped that this article will further stimulate workers in continuum mechanics to play their important role in future research work in the astrophysics of galaxies.     

A certain feature of logarithmic spirals

A curve formed by inversion of a logarithmic spiral is called a double logarithmic spiral. The curves in this family possess the following property: there always exists such a spiral with continuous and monotone curvature satisfying any possible boundary conditions (endpoints, tangents, and curvatures). The problem of constructing a spiral with continuous curvature and prescribed curvature elements at the endpoints is thus solved. Bibliography: 6 titles.     

A Forcing Mechanism for Spiral Density Waves in Galaxies

In this paper, we consider gas dynamical models of spiral galaxies in which there is a rigidly rotating, weakly barlike structure in the central regions. It is found that, in the neighborhood of the corotation circle, this barlike structure forces a trailing spiral wave. Such a driven wave could then propagate inwards to the bar and complete a feedback loop to maintain the spiral structure. For basic distributions typical of our Galaxy, if a ten-per-cent oval distortion is assumed, the strength of the induced spiral field is found to be of the order of a few per cent of the axisymmetric field, in agreement with observational data.     

Particle simulation of spiral galaxy evolution

Spiral galaxies are modelled as n-body systems by means of long-range and molecular-type short-range forces. A relatively massive, elliptic-type core is set into rotational motion in a less dense, tidal-prone medium. Pressure waves result, along whose leading edges there is increased luminosity, which develops in a spiral pattern. Qualitative results are obtained by parameter choices which yield dimensionless dynamical equations.     

对数螺线、黄金分割与斐波那契数列的完美统一的

本文对对数螺线、黄金分割与斐波那契数列之间的关系进行研究,把线段上黄金分割点的定义推广到射线上的黄金分割点列,发现过极轴上任意一点有且仅有一条特殊的对数螺线与极轴的交点所成的点列为黄金分割点列,并把这个点列所对应的坐标定义为黄金分割数列,我们发现首项为1/√15黄金分割数列无限逼近于斐波那契数列。     

虽经沧海 依然故我——雅格布·伯努利与对数螺线

简述了雅格布·伯努利(Jacob Bernoulli)的生平及主要贡献,讨论了对数螺线的性质,并给出了对数螺线的一些应用.     

Construction of spirals with prescribed boundary conditions

Spirality, regarded as monotonicity of curvature, is preserved under inversions. This property is used for constructing a spiral transition curve with predefined curvature elements at the endpoints. These boundary conditions define two invariant values: Coxeter’s inversive distance and the width of the lens. In order to solve the problem, it suffices to realize the corresponding values on two curvature elements of any known spiral. The rest is achieved by inversion. In particular, any boundary conditions compatible with spirality can be satisfied by inverting an arc of the logarithmic spiral. Bibliography: 9 titles.     

Order and Chaos in Dynamical Systems

We consider the characteristics of order and chaos in dynamical systems, with emphasis on the orbits in astronomical systems. Celestial mechanics deals with orbits in the solar system, which are mainly ordered. On the other hand the orbits of stars in galaxies were considered to be chaotic. However numerical experiments have shown that in general a system contains both ordered and chaotic orbits. Thus a new classification of dynamical systems has been established. We describe ordered and chaotic orbits in galaxies and in mappings. Some ordered orbits appear even in strongly perturbed systems. The transition from order to chaos is due to resonance overlapping. Then we describe some recent developments concerning order and chaos in the solar system and in galaxies. The outer spiral arms in strong barred galaxies are composed mainly of sticky chaotic orbits. Ordered and chaotic orbits appear also in Bohmian quantum mechanics. If the initial probability p is not equal to the square of the wave function |ψ|², then in the case of ordered orbits p never approaches |ψ|², while in the case of chaotic orbits p → |ψ|² after a time interval called “quantum Nekhoroshev time”.     

星系密度波的线性增长

本文以线性密度波局部渐近解为初值,求解二维非定常流体力学方程组和泊松方程,研究星系密度波的线性增长.数值计算结果表明,线性密度波将在几千万年时间内增长到与基态同量级,形成在中心区域具有棒形结构的螺旋图样.螺旋结构的图样速度及扰动密度的增长率随空间位置及时间变化.讨论了准稳螺旋结构假设的近似性.     

Spirals in Hilbert space: With an application in information theory

A (logarithmic) spiral of order is defined as a continuous path tx(t) in a real Hilbert space such that

For α=0 the spiral becomes a helix. The elegant proof by P. Masani of the spectral characterization of helices, due to Kolmogorov and to von Neumann and Schoenberg, is adapted here to spirals. As an application a conjecture by F. Topsøe that certain kernels on considered in information theory are negative definite, and hence are squares of metrics on , is confirmed.     

Promoting convergence: The Phi spiral in abduction of mouse corneal behaviors

Why do mouse corneal epithelial cells display spiraling patterns? We want to provide an explanation for this curious phenomenon by applying an idealized problem solving process. Specifically, we applied complementary line‐fitting methods to measure transgenic epithelial reporter expression arrangements displayed on three mature, live enucleated globes to clarify the problem. Two prominent logarithmic curves were discovered, one of which displayed the ? ratio, an indicator of an optimal configuration in phyllotactic systems. We then utilized two different computational approaches to expose our current understanding of the behavior. In one procedure, which involved an isotropic mechanics‐based finite element method, we successfully produced logarithmic spiral curves of maximum shear strain based pathlines but computed dimensions displayed pitch angles of 35° (? spiral is ～17°), which was altered when we fitted the model with published measurements of coarse collagen orientations. We then used model‐based reasoning in context of Peircean abduction to select a working hypothesis. Our work serves as a concise example of applying a scientific habit of mind and illustrates nuances of executing a common method to doing integrative science. © 2014 Wiley Periodicals, Inc. Complexity 20: 22–38, 2015     

Limit cycles for two families of cubic systems

In this paper we study the number of limit cycles of two families of cubic systems introduced in previous papers to model real phenomena. The first one is motivated by a model of star formation histories in giant spiral galaxies and the second one comes from a model of Volterra type. To prove our results we develop a new criterion on the non-existence of periodic orbits and we extend a well-known criterion on the uniqueness of limit cycles due to Kuang and Freedman. Both results allow to reduce the problem to the control of the sign of certain functions that are treated by algebraic tools. Moreover, in both cases, we prove that when the limit cycles exist they are non-algebraic.     

The galaxies of nonstandard enlargements of infinite and transfinite graphs

The galaxies of the nonstandard enlargements of connected, conventionally infinite graphs as well as of walk-connected transfinite graphs are defined, analyzed, and illustrated by some examples. It is then shown that any such enlargement either has exactly one galaxy, its principal one, or it has infinitely many galaxies. In the latter case, the galaxies are partially ordered by their “closeness” to the principal galaxy. If an enlargement has a galaxy Γ different from its principal galaxy, then it has a two-way infinite sequence of galaxies that contains Γ and is totally ordered according to that “closeness” property. There may be many such totally ordered sequences.Furthermore, a walk-connected graph G¹ of transfinite rank 1 consists in general of connected conventional graphs (graphs of rank 0, called 0-sections) that are walk-connected together at their infinite extremities. The enlargement of G¹ consists of the enlargement of G¹, as well as of the enlargements of its 0-sections. The latter enlargements are all contained within the principal galaxy of . Moreover, may have other galaxies of rank 1; these too are partially and totally ordered as before. These results extend to the enlargements of transfinite graphs of ranks greater than 1.     

Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

We compute the Szegö kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in ${\mathbb C^n}We compute the Szeg? kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in \mathbb Cⁿ{\mathbb C^n} for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szeg? kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.     

Numerical Methods for Stiff Index-3 DAEs

Space semidiscretization of PDAEs, i.e. coupled systems of PDEs and algebraic equations, give raise to stiff DAEs and thus the standard theory of numerical methods for DAEs is not valid. As the study of numerical methods for stiff ODEs is done in terms of logarithmic norms, it seems natural to use also logarithmic norms for stiff DAEs. In this paper we show how the standard conditions imposed on the PDAE and the semidiscretized problem are formally the same if they are expressed in terms of logarithmic norms. To study the mathematical problem and their numerical approximations, this link between the standard conditions and logarithmic norms allow us to use for stiff DAEs techniques similar to the ones used for stiff ODEs. The analysis is done for problems which appear in the context of elastic multibody systems, but once the tools, i.e., logarithmic norms, are developed, they can also be used for the analysis of other PDAEs/DAEs.     

Factorizable ribbon quantum groups in logarithmic conformal field theories

We review the properties of quantum groups occurring as the Kazhdan-Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 510–535, March, 2008.     

On the jump behavior of distributions and logarithmic averages

The jump behavior and symmetric jump behavior of distributions are studied. We give several formulas for the jump of distributions in terms of logarithmic averages, this is done in terms of Cesàro-logarithmic means of decompositions of the Fourier transform and in terms of logarithmic radial and angular local asymptotic behaviors of harmonic conjugate functions. Application to Fourier series are analyzed. In particular, we give formulas for jumps of periodic distributions in terms of Cesàro–Riesz logarithmic means and Abel–Poisson logarithmic means of conjugate Fourier series.     

Super-large-scale structures in the distribution of infrared galaxies

Typical scales in the distribution of IRAS galaxies with the unnormalized pair count method. Samples are those provided by QDOT redshift survey and sorted out from the IRAS faint sources catalog. Analysis is concentrated on the structures at super-large scales. The results show that statistically significant typical scales do exist in the distribution of all these samples from both 2-dimensiortal and 3-dimensional analyses. These scales are consistent with those found from the analyses of galaxies and clusters of galaxies by Mo et al. and also consistent with those found from analysis of quasars by Deng et al. The analysis provides additional evidence for the existence of typical scales in the large-scale structure of the universe. The existence of typical scales challenges all the existing models on the formation of galaxies and structures. Project supported by the National Natural Science Foundation of China (Grant Nos. 19773014 and 19773010).