具线性退化特征场对角型方程组Goursat问题的奇性形成机制

For an inhomogeneous quasilinear hyperbolic system of diagonal form, under the assumptions that the system is linearly degenerate and the C^1 norm of the boundary data is bounded, we show that the mechanism of the formation of singularities of C^1 classical solution to the Goursat problem with C^1 compatibility conditions at the origin must be an ODE type. The similar result is also obtained for the weakly discontinuous solution with C^0 compatibility conditions at the origin.     

双孔干涉产生偏振奇点分析

在近轴近似的条件下,采用标量衍射积分的方法,对电磁场的双孔干涉进行了研究,并对千涉场中产生偏振奇点的情况进行了推广。在研究中通过双孔的光电场为线偏振状态,具有不同的偏振方向。数值计算结果表明,即使通过双孔的光电场的相位相同,只要双孔的尺寸存在一定的差异,干涉场中除了能够产生普通的的线偏振外,圆偏振奇点也会出现。     

Vortex structures with complex points singularities in two-dimensional Euler equations. New exact solutions

In this work we present a new class of exact stationary solutions for two-dimensional (2D) Euler equations. Unlike already known solutions, the new ones contain complex singularities. We consider point singularities which have a vector field index greater than 1 as complex. For example, the dipole singularity is complex because its index is equal to 2. We present in explicit form a large class of exact localized stationary solutions for 2D Euler equations with a singularity whose index is equal to 3. The solutions obtained are expressed in terms of elementary functions. These solutions represent a complex singularity point surrounded by a vortex satellite structure. We also discuss the motion equation of singularities and conditions for singularity point stationarity which provide the stationarity of the complex vortex configuration.     

Lightcone dualities for hypersurfaces in the sphere

In this paper, we consider hypersurfaces in the unit lightlike sphere. The unit sphere can be canonically embedded in the lightcone and de Sitter space in Minkowski space. We investigate these hypersurfaces in the framework of the theory of Legendrian dualities between pseudo‐spheres in Minkowski space.     

LIMIT CYCLES BIFURCATION FOR A CLASS OF DEGENERATE SINGULARITY

In this paper, bifurcation of limit cycles for the degenerate equilibrium to a three- dimensional system is investigated. Firstly, we use formal series to calculate the focal values at the high-order critical point on center manifold. Then an example is studied, and the existence of 3 limit cycles on the center manifold is proved. In terms of high- order singularities in high-dimensional systems, our results are new.     

An explicit recursive formula for computing the normal forms associated with semisimple cases

This paper presents an explicit, computationally efficient, recursive formula for computing the normal forms, center manifolds and nonlinear transformations for general n-dimensional systems, associated with semisimple singularities. Based on the formula, we develop a Maple program, which is very convenient for an end-user who only needs to prepare an input file and then execute the program to “automatically” generate the result. Several examples are presented to demonstrate the computational efficiency of the algorithm.     

一个求留数方法的注记

对解析函数f（z）＝p（z）/q（z）,当z=a分别是p（z）和q（z）的m级零点和n级零点,且0≤m＜n时,特别地n＝m＋2时,讨论了求f（z）在极点z=a处留数的方法并举例说明其应用.     

Psi‐series method for equality of random trees and quadratic convolution recurrences

An unusual and surprising expansion of the form as , is derived for the probability p_n that two randomly chosen binary search trees are identical (in shape, hence in labels of all corresponding nodes). A quantity arising in the analysis of phylogenetic trees is also proved to have a similar asymptotic expansion. Our method of proof is new in the literature of discrete probability and the analysis of algorithms, and it is based on the logarithmic psi‐series expansions for nonlinear differential equations. Such an approach is very general and applicable to many other problems involving nonlinear differential equations; many examples are discussed in this article and several attractive phenomena are discovered.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 67–108, 2014     

Simulation of oscillatory flow in an aortic bifurcation using FVM and FEM: A comparative study of implementation strategies

Cardiovascular diseases are one of the major causes of long‐term morbidity and mortality in human beings. The nearly epidemic increase in prevalence of such diseases poses a serious threat to public health and calls for efficient methods of diagnosis and treatment. Non‐invasive diagnostic procedures such as MRI are often used in this context; however, these are limited in terms of spatial and temporal resolution and do not provide information on time‐dependent pressures and wall shear stresses—key quantities considered to be partially responsible for the formation and development of related pathologies. The present study is concerned with the numerical simulation of oscillatory flow through the abdominal aortic bifurcation. Computational fluid dynamics simulation of oscillatory flow in a branched geometry at high Reynolds numbers poses considerable challenges. The present study reports a detailed comparison of simulations performed with a finite volume and a finite element method, two approaches with significant differences in their discretization strategy, treatment of boundary conditions and other numerical aspects. Both solvers were parallelized, using loop parallelization of the BiCGStab linear solver for the finite volume and domain decomposition based on the Schur complement method for the finite element technique. The experience gained with these two approaches for the solution of flow in a bifurcation forms the focus of this study. Although similar results were obtained for both methods, the computation time required for convergence was found to be significantly smaller for the finite element approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders

The velocity field and the associated shear stress corresponding to the longitudinal oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and at t = 0＋ both cylinders suddenly begin to oscillate along their common axis with simple harmonic motions having angular frequencies Ω1 and Ω2. The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, the effect of different parameters on the flow of ordinary second grade and generalized second grade fluid are investigated graphically by plotting velocity profiles.