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1.
Abstract We identify ℝ7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for
the unit sphere S6. It is known that a cone over a surface M in S6 is an associative submanifold of ℝ7 if and only if M is almost complex in S6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S6 are the equation for primitive maps associated to the 6-symmetric space G2=T2, and use this to explain some of the known results. Moreover, the equation for S1-symmetric almost complex curves in S6 is the periodic Toda lattice, and a discussion of periodic solutions is given.
(Dedicated to the memory of Shiing-Shen Chern)
* Partially supported by NSF grant DMS-0529756. 相似文献
2.
给定广义自反矩阵R,S,即R=R=R-1,S=S=S-1,若复矩阵X满足条件RXS=X(或RXS=X),则称其为(R,S)-对称矩阵(或(R,S)-斜对称矩阵).分别讨论了线性流形上(R,S)-对称矩阵和(R,S)-斜对称矩阵约束下矩阵方程MZN=E的最小二乘问题,得到了通解表达式. 相似文献
3.
本文考虑 Lienard方程 x″+f (x) x′+g(x) =e(t) ,我们得到 :当 -∞ 0且 0 相似文献
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In terms of the almost complex affine connection and moving unitary frames,all totally rael minimal immersions from R^2 into the nearly Kahler S^6 are determine explicitly. Moreover, the complete flat almost complex curves in the nearly Kahler S^6 aredetermined completely. 相似文献
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In this work,by virtue of the properties of weakly almost periodic points of a dynamical system (X,T) with at least two points,the authors prove that,if the measure center M(T) of T is the whole space,... 相似文献
7.
In this paper, we study the Logistic type equation x= a(t)x -b(t)x^2+ e(t). Under the assumptions that e(t) is small enough and a(t), b(t) are contained in some positive intervals, we prove that this equation has a positive bounded solution which is stable. Moreover, this solution is a periodic solution if a(t), b(t) and e(t) are periodic functions, and this solution is an almost periodic solution if a(t), b(t) and e(t) are almost periodic functions. 相似文献
8.
Shen Yibing 《数学年刊B辑(英文版)》1998,19(1):87-96
THENEARLYK¨AHLERSTRUCTUREANDMINIMALSURFACESINS6SHENYIBINGManuscriptreceivedFebruary11,1995.RevisedOctober17,1996.Departme... 相似文献
9.
We consider spaces of immersed (pseudo-)holomorphic curves in an almost complex manifold of dimension four. We assume that they are either closed or compact with boundary in a fixed totally real surface, so that the equation for these curves is elliptic and has a Fredholm index. We prove that this equation is regular if the Chern class is ≥ 1 (in the case with boundary, if the ambient Maslov number is ≥ 1). Then the spaces of holomorphic curves considered will be manifolds of dimension equal to the index. 相似文献
10.
Yongge Tian 《Advances in Applied Clifford Algebras》2000,10(1):61-90
As is well-known, the real quaternion division algebra ℍ is algebraically isomorphic to a 4-by-4 real matrix algebra. But
the real division octonion algebra
can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because
is a non-associative algebra over ℝ. However since
is an extension of ℍ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations
of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete
investigation to real matrix representations of octonions, and consider their various applications to octonions as well as
matrices of octonions. 相似文献
11.
Lienard方程周期解、概周期解的存在性 总被引:20,自引:2,他引:18
本文考虑Lienard方程x”十f(x)x’+g(x)=e(t),我们得到:当且时,对于任意周期或概周期。数e(t),它有周期或概周期解.而对于Lienard方程x”+f(x)x’+cx=e(t),我们得到:当c>0且时,对于任意周期、或概周期函数e(t),它有周期或概周期解. 相似文献
12.
In this paper we prove the fundamental theorem of algebra for polynomials with coefficients in the skew field of Hamilton
numbers (quaternions) and in the division algebra of Cayley numbers (octonions). The proof, inspired by recent definitions
and results on regular functions of a quaternionic and of a octonionic variable, follows the guidelines of the classical topological
argument due to Gauss.
G. Gentili and F. Vlacci are partially supported by G.N.S.A.G.A. of the I.N.D.A.M. and by M.I.U.R. 相似文献
13.
设S={x1,x2,…xn}是不同正整数的集合。已经知道当n≤7时在最大公因数封闭集S上的LCM矩阵是可逆的;也知道当n≥9时有无限多个包含整数1的最大公因数封闭集它们的LCM矩阵是奇异的;这篇文章的主要结果是证明当n=8且包含整数1时,除了20个最大公因数封闭集外,其余所有最大公因数封闭集上的LCM矩阵都是可逆的,而这归结为解一个不定方程。 相似文献
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对摆型方程x+Gx(x,t)=p(t),其中G(x,t)∈C1(R2)关于变量x是1周期的,并且sup(x,t)∈R2|Gx(x,t)|<+∞,limsupt→∞{supx∈R}=0,p(t)是平均值非零的概周期函数,证明了在柱面S1×R上方程具有无穷多的无界解. 相似文献
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G. J. Tee 《Numerische Mathematik》1964,6(1):142-159
Summary Finite-difference equations for the steady-state solution of a parabolic equation with periodic boundary conditions producep-cyclic matrices, wherep can be arbitrarily large. The periodic solution can be found by applying S.O.R. to the equations with thep-cyclic matrix, and this procedure is more economical than the standard step-by-step procedures for solving the parabolic equation. Bounds for the discretization error are found in terms of bounds for the known second derivatives of the boundary values. 相似文献
18.
We discuss the problem of the existence of almost periodic in distribution solutions of affine stochastic differential equations with almost periodic coefficients. We prove that if the linear part of the affine equation is exponentially stable in mean square then the unique continuous L2 -bounded solution of the affine system has the onedimensional distributions almost periodic. An analogous result is shown for the asymptotic almost periodic case 相似文献
19.
P.E. Kloeden H.M. Rodrigues 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(7):2695-2719
A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C0-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations. 相似文献
20.
Goda Hiroshi; Hirasawa Mikami; Yamamoto Ryosuke 《Proceedings London Mathematical Society》2001,83(2):472-492
Let R be a Seifert surface obtained by applying Seifert's algorithmto a connected diagram D for a link L. In this paper, lettingD be almost alternating, we give a practical algorithm to determinewhether L is a fibered link and R is a fiber surface. We furthershow that L is a fibered link and R is a fiber surface for Lif and only if R is a Hopf plumbing, that is, a successive plumbingof a finite number of Hopf bands. It has been known for sometime that this is true if D is alternating, and we show thatit is not always true if D is 2-almost alternating. In the appendix,we partially answer C. Adams's open question concerning almostalternating diagrams. 2000 Mathematical Subject Classification:57M25. 相似文献