Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations

In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painlevé integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665‐669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth‐order Lax pairs instead of the fifth‐order ones. Moreover, the Painlevé integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painlevé integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto‐Bäcklund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd.     

Determinants and Algebraic Identities Associated with the Root Systems of Classical Lie Algebras

Burnside asked questions about periodic groups in his influential paper of 1902. The study of groups with exponent six is a special case of the study of the Burnside questions on which there has been significant progress. It has contributed a number of worthwhile aspects to the theory of groups and in particular to computation related to groups. Finitely generated groups with exponent six are finite. We investigate the nature of relations required to provide proofs of finiteness for some groups with exponent six. We give upper and lower bounds for the number of sixth powers needed to define the largest 2-generator group with exponent six. We solve related questions about other groups with exponent six using substantial computations which we explain.     

Methods of solving spatial problems of the mechanics of a deformable solid in terms of stresses

The formulation in /1/ of a quasistatic problem of the mechanics of a deformable solid in terms of stresses is discussed, including also the variational formulation, which consists of solving six equations in six symmetric stress tensor components when six boundary conditions are satisfied. Methods of successive approximation are proposed for solving this problem and theorems on the convergence of these methods, including a “rapidly converging” method, whose rate of convergence is substantially higher than a geometric progression, are proved.     

Numerical semigroups with embedding dimension three

Every numerical semigroup generated by three elements is determined by six positive integers that are the solution to a system of three polynomial equations. We give formulas of the Frobenius number and the cardinality of the set of gaps in terms of these six parameters.Received: 6 November 2003     

Coloring Eulerian Triangulations of the Klein Bottle

We show that an Eulerian triangulation of the Klein bottle has chromatic number equal to six if and only if it contains a complete graph of order six, and it is 5-colorable, otherwise. As a consequence of our proof, we derive that every Eulerian triangulation of the Klein bottle with face-width at least four is 5-colorable.     

R~3中的一个Heilbronn型问题

本文给出Ｈｅｉｌｂｒｏｎｎ型问题的结果．设Ｓ是Ｒ~３中六点组成的集合．直径为Ｄ．若ｄ表示Ｓ中任意两点距离的最小值,则Ｄ≥２ｄ．等号当且仅当Ｓ是由正八面体的六个顶点或多面体面△×△１的六个顶点组成时才成立（△１,△２分别表示一维、二维正则单形,且其棱长相等）．     

离散数学中关系性质的判定方法

集合与关系是离散数学中非常重要的内容,直接根据定义判定关系的六种性质是比较困难的,尤其是对传递性和反传递性的判定.本文通过引入矩阵的Hadmard乘积的定义分别给出了这六种性质的矩阵判定方法.     

Fibrations on four-folds with trivial canonical bundles

Four-folds with trivial canonical bundles are divided into six classes according to their holonomy group. We consider examples that are fibred by abelian surfaces over the projective plane. We construct such fibrations in five of the six classes, and prove that there is no such fibration in the sixth class. We classify all such fibrations whose generic fibre is the Jacobian of a genus two curve.     

On the Normal Structure of NonCommutative Association Schemes of Rank 6

The concept of an association scheme is a far-reaching generalization of the notion of a group. Many group theoretic facts have found a natural generalization in scheme theory. One of these generalizations is the observation that, similar to groups, association schemes of finite order are commutative if they have at most five elements and not necessarily commutative if they have six elements. While there is (up to isomorphism) only one noncommutative group of order 6, there are infinitely many pairwise non-isomorphic noncommutative association schemes of finite order with six elements. (Each finite projective plane provides such a scheme, and non-isomorphic projective planes yield non-isomorphic schemes.) In this note, we investigate noncommutative schemes of finite order with six elements which have a symmetric normal closed subset with three elements. We take advantage of the classification of the finite simple groups.     

弹塑性状态下Ⅰ型裂纹前缘应力应变场的奇异性分析

本文从弹塑性力学的三维基本方程出发,分析了幂硬化材料Ⅰ型裂纹前缘应力、应变场的奇异性,发现,裂尖附近诸应力、应变分量的奇异性沿厚度不变;六个应力分量的奇异性不完全相同,六个应变分量的奇异性也不完全相同.     

Cohomological properties of unimodular six dimensional solvable Lie algebras

In the present paper we study six dimensional solvable Lie algebras with special emphasis on those admitting a symplectic structure. We list all the symplectic structures that they admit and we compute their Betti numbers finding some properties about the codimension of the nilradical. Next, we consider the conjecture of Guan about step of nilpotency of a symplectic solvmanifold finding that it is true for all six dimensional unimodular solvable Lie algebras. Finally, we consider some cohomologies for symplectic manifolds introduced by Tseng and Yau in the context of symplectic Hogde theory and we use them to determine some six dimensional solvmanifolds for which the Hard Lefschetz property holds.     

A graph-theoretic proof of the non-existence of self-orthogonal Latin squares of order 6

The non-existence of a pair of mutually orthogonal Latin squares of order six is a well-known result in the theory of combinatorial designs. It was conjectured by Euler in 1782 and was first proved by Tarry in 1900 by means of an exhaustive enumeration of equivalence classes of Latin squares of order six. Various further proofs have since been given, but these proofs generally require extensive prior subject knowledge in order to follow them, or are ‘blind’ proofs in the sense that most of the work is done by computer or by exhaustive enumeration. In this paper we present a graph-theoretic proof of a somewhat weaker result, namely the non-existence of self-orthogonal Latin squares of order six, by introducing the concept of a self-orthogonal Latin square graph. The advantage of this proof is that it is easily verifiable and accessible to discrete mathematicians not intimately familiar with the theory of combinatorial designs. The proof also does not require any significant prior knowledge of graph theory.     

Motion of a nonlinear viscoelastic liquid in cylindrical channels of triangular cross section

The problem of the nonrectilinear steady-state flow of a nonlinear viscoelastic liquid in cylindrical channels is considered. It is established that in channels whose cross sections form an equilateral triangle or an isosceles right-angled triangle there are six transverse currents (eddies) in a plane perpendicular to the main longitudinal flow. In cylinders with cross sections in the form of an arbitrary triangle there may be four or six eddies, depending on the shape of the triangle.     

On the genus of five- and six-regular graphs

This paper shows how to construct infinitely many regular graphs of degrees five and six having given genus γ > 0, which settles favorably Conjecture 1 stated by T. W. Tucker. Tucker has shown that there are infinitely many regular graphs of degrees four and three of arbitrary given genus (Theorem 1). He also proved that the number of regular graphs of degree greater than six embeddable in a given surface is finite (Corollary to Proposition 1). The case of the regular graphs of degrees six and five was left unanswered (Conjecture 1). This paper also shows a new way of constructing infinitely many regular graphs of degrees three and four of arbitrary genus γ > 0.     

直接最优化方法的收敛性与不动点

最优化方法中的直接法是仅仅利用目标函数值(或者通过试验结果)的信息,试图去寻求最优解.直接法迭代步骤简单,特别当目标函数的解析表达式十分复杂,甚至写不出来时,那些要求计算目标函数的偏导数的方法(称为解析方法)就无能为力了.然而,多因     

基金“拥挤交易”对A股股价泡沫的影响研究

随着基金数量的不断增加,多家基金重仓一只股票所呈现的“拥挤交易”现象日渐明显。本文运用面板数据模型,对基金“拥挤交易”与A股股价泡沫之间的关系进行考察。结果表明,当公司超过6只基金重仓持股时,基金拥挤交易会对股价泡沫形成促进作用;当小于6只基金重仓持股时,这一关系则不显著。即:基金拥挤交易需要在“拥挤”到一定程度后才会促进股价泡沫膨胀。因此,管理层应高度重视6家以上基金重仓持股的股票,这类股票往往会形成较大的泡沫,同时也存在较大的崩溃风险。另外,管理层还应审慎研究政策出台时可能带来的预期,最大限度的避免基金经理产生一致的信念和一致行动,降低基金拥挤交易现象的发生。     

Operations on maps,and outer automorphisms

By representing maps on surfaces as transitive permutation representations of a certain group Γ, it is shown that there are exactly six invertible operations (such as duality) on maps; they are induced by the outer automorphisms of Γ, and form a group isomorphic to S₃. Various consequences are deduced, such as the result that each finite map has a finite reflexible cover which is invariant under all six operations.     

Comparing simulation models for market risk stress testing

The subprime crisis has reminded us that effective stress tests should not only combine subjective scenarios with historical data, but also be probabilistic. In this paper, we combine three hypothetical shocks, of varying degrees, with more than six years of daily data on USD-INR and Euro-INR. Our objective is to compare six simulation-based stress models for foreign exchange positions. We find that while volatility-weighted historical simulation is the best model for volatility persistence, jump diffusion based Monte Carlo simulation is better at capturing correlation breakdown. Loss estimates from very fat-tailed distributions are not sensitive to the severity of stress scenarios.     

On the six conics problem

There do not exist six points in the plane and six distinct congruent hyperbolas, each containing exactly five of the points. The methods are elementary. Part of the discussion includes determining how two congruent hyperbolas may intersect in four points. When combined with a result of Seidel and van Vollenhoven for ellipses, the above solves the six conics problem in the negative.     

六关节机械臂运动路径设计

就关节式机械臂指尖在任意两点间移动、沿固定曲线移动、机械臂绕开障碍物执行任务以及参数优化等问题展开研究.首先确定了自由度组合到指尖空间位置的映射,建立了求解上述问题的最小二乘模型、泛函条件极值模型,并给出了数值解法.最后,结合图像处理等技术,对各参数的优化设计提出了改进措施.