Collision of Almost Parallel Vortex Filaments

We investigate the occurrence of collisions in the evolution of vortex filaments through a system introduced by Klein, Majda, and Damodaran and Zakharov . We first establish rigorously the existence of a pair of almost parallel vortex filaments, with opposite circulation, colliding at some point in finite time. The collision mechanism is based on one of the self‐similar solutions of the model, described by the authors in an earlier work. In the second part of this paper we extend this construction to the case of an arbitrary number of filaments, with polygonal symmetry, that are perturbations of a configuration of parallel vortex filaments forming a polygon, with or without its center, rotating with constant angular velocity.© 2016 Wiley Periodicals, Inc.     

Probabilistic Models of Vortex Filaments

A model of vortex filaments based on stochastic processes is presented. In contrast to previous models based on semimartingales, here processes with fractal properties between 1/2 and 1 are used, which include fractional Brownian motion and similar non-Gaussian examples. Stochastic integration for these processes is employed to give a meaning to the kinetic energy.     

(1+2)-Dimensional Radially Symmetric Wave Maps Revisit

The author gives an alternative and simple proof of the global existence of smooth solutions to the Cauchy problem for wave maps from the (1+2)-dimensional Minkowski space to an arbitrary compact smooth Riemannian manifold without boundary, for arbitrary smooth, radially symmetric data. The author can also treat non-compact manifold under some additional assumptions which generalize the existing ones.     

Dynamics of Axially Symmetric Perturbed Hamiltonians in 1:1:1 Resonance

We study the dynamics of a family of perturbed three-degree-of-freedom Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry, and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by normalisation up to terms of degree 4 in rectangular coordinates; after truncation of higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction in the two symmetries leads to a one-degree-of-freedom system that is completely analysed in the twice reduced space. All the relative equilibria together with the stability and parametric bifurcations are determined. Moreover, the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM 3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduced space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system.     

Regular Space-like Hypersurfaces in S_1~(m+1)with Parallel Para-Blaschke Tensors

In this paper, we give a complete conformal classification of the regular space-like hypersurfaces in the de Sitter Space S_1~(m+1) with parallel para-Blaschke tensors.     

Sn+1中M(o)ebius形式平行的超曲面

如果x:M→Sn+1是不含脐点的超曲面,且M的Moebius形式 =0和Blaschke张量A=λg,就称M为Moebius迷向超曲面,如果x:M→Sn+1 是不合脐点的超曲面,且M的Moebius形式 平行( =0)和Blaschke张量A=λg,就称M为Moebius拟迷向超曲面,这里g是M上的Moebius度量,λ:M→R是M上的光滑函数,本文证明了如下结果: (1)设x:M→Sn+1(n 3)是不含脐点的超曲面,则M是拟迷向超曲面当且仅当M是迷向超曲面,(2)设x:M→Sn+1(n 3)是不合脐点的超曲面,且M的Moebius形式 平行和Blaschke张量A也平行( A=0),则 =0.     

The exponential Diophantine equation x2 + (3n 2 + 1) y = (4n 2 + 1) z

Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x ² + (3n ² + 1) ^y = (4n ² + 1) ^z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n ³ + 3n, 1, 3).     

Approximate double-periodic solutions in (1+1)-dimensionalϗ 4-theory

Double-periodic solutions of the Euler-Lagrange equation for the (1+1)-dimensional scalarϗ ⁴-theory are considered. The nonlinear term is assumed to be small, and the Poincaré method is used to seek asymptotic solutions in the standing-wave form. The principal resonance problem, which arises for zero mass, is resolved if the leading-order term is taken in the form of a Jacobi elliptic function. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 2, pp. 182–192, August, 1998.     

一类新的关于(4+4+1)-体共轭套中心构型

本文研究R3中一类(4+4+1)-体中心构型.利用中心构型等价类的性质及代数、分析方法,得到了该构型构成中心构型的充分和必要条件,证明了对任意给定的质量比这类中心构型存在的结论,解决了给定不同质量比范围该类中心构型是否唯一的问题,推广了文[16]的结论.     

Dynamics of kink-soliton solutions of the $$(2+1)$$ -dimensional sine-Gordon equation

Theoretical and Mathematical Physics - We study the dynamics of explicit solutions of the $$(2+1)$$ -dimensional (2D) sine-Gordon equation. The Darboux transformation is applied to the associated...     

Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.     

The cardinality of FM(1+1+n)

The aim of this note is to develop a counting formula for the modular lattice FM(1+1+n) freely generated by two single elements and an n-element chain. This answers Problem 44 in Birkhoff [1] which asks one to determine FM(1+1+n). The proof of our recursive formula is based on the scaffolding method developed by R. Wille.     

On the equationx(x +d 1)…(x + (k - 1)d 1) =y(y +d 2)…(y + (mk - 1)d 2)

For given positive integersm ≥ 2,d ₁ andd ₂, we consider the equation of the title in positive integersx, y andk ≥ 2. We show that the equation implies thatk is bounded. For a fixedk, we give conditions under which the equation implies that max(x, y) is bounded. Dedicated to the memory of Professor K G Ramanathan     

Hadamard matrices of order 4(2p + 1)

It is shown in this paper that if p is a prime and q = 2p ? 1 is a prime power, then there exists an Hadamard matrix of order 4(2p + 1).     

Nonisomorphic solutions of (4t+3, 2t+1, t) designs

A technique has been developed to get a large number of nonisomorphic solutions of a (4t+3, 2t+1, t) design. Some results are proved on the conjecture for Hadamard matrices.     

Linear (q+1)-fold Blocking Sets in PG(2, q4)

A (q+1)-fold blocking set of size (q+1)(q⁴+q²+1) in PG(2, q⁴) which is not the union of q+1 disjoint Baer subplanes, is constructed     

具有性质cr+1=3,ar+1=3a+1或 4a的(a+1,3)型距离正则图

设г是直径为d且型为(a 1,3)的距离正则图,其中a≥2.用l(c,a,b)表示交叉阵列ι(г)中列(c,a,6)t的个数,记r=r(г)=l(c1,a1,b1),8=8(г)=l(Cr 1,ar 1,br 1)及t=t(г)=l(cr s 1,ar s 1,br s 1).那末,若Cr 1=3,ar 1=4a或3a 1,则d=r t 2.     

LUMP AND INTERACTION SOLUTIONS TO LINEAR (4+1)-DIMENSIONAL PDES

Taking a class of linear(4+1)-dimensional partial differential equations as examples, we would like to show that there exist lump solutions and interaction solutions in(4+1)-dimensions. We will compute abundant lump solutions and interaction solutions to the considered linear(4+1)-dimensional partial differential equations via symbolic computations,and plot three specific solutions with Maple plot tools, which supplements the existing literature on lump, rogue wave and breather solutions and their interaction solutions in soliton theory.     

Über die Funktionalgleichungf (1 + x) + f (1 + f (x)) = 1

Ohne Zusammenfassung