This paper deals with bifurcations of subharmonic solutions and invariant tori generated from limit cycles in the fast dynamics for a nonautonomous singularly perturbed system. Based on Poincaré map, a series of blow-up transformations and the theory of integral manifold, the conditions for the existence of invariant tori are obtained, and the saddle-node bifurcations of subharmonic solutions are studied.
In this paper, a new approach to analyze synchronization of linearly coupled map lattices (LCMLs) is presented. A reference vector $\ifmmode\expandafter\hat\else\expandafter\^\fi{x}$(t) is introduced as the projection of the trajectory of the coupled system on the synchronization manifold. The stability analysis of the synchronization manifold can be regarded as investigating the difference between the trajectory and the projection. By this method, some criteria are given for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to the eigenvalue "0" of the coupling matrix play key roles in the stability of synchronization manifold for the coupled system. Moreover, it is revealed that the stability of synchronization manifold for the coupled system is different from the stability for dynamical system in usual sense. That is, the solution of the coupled system does not converge to a certain knowable s(t) satisfying s(t+1) = f(s(t)) but to the reference vector on the synchronization manifold, which in fact is a certain weighted average of each x i(t) for i = 1, ⋯ ,m, but not a solution s(t) satisfying s(t + 1) = f(s(t)).
In this paper, the authors prove that if M n is a complete noncompact Kähler manifold with a pole p, and its holomorphic bisectional curvature is asymptotically non-negative to p, then it is a Stein manifold.
Basic facts for Gabor frame {E u(m)b T u(n)a g} m, n∈ρ on local field are investigated. Accurately, that the canonical dual of frame {E u(m)b T u(n)a g} m, n∈ρ also has the Gabor structure is showed; that the product ab decides whether it is possible for {E u(m)b T u(n)a g} m, n∈ρ to be a frame for L 2(K) is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L 2(K) are established. An example is finally given.
Quasi-regression, motivated by the problems arising in the computer experiments, focuses mainly on speeding up evaluation. However, its theoretical properties are unexplored systemically. This paper shows that quasi-regression is unbiased, strong convergent and asymptotic normal for parameter estimations but it is biased for the fitting of curve. Furthermore, a new method called unbiased quasi-regression is proposed. In addition to retaining the above asymptotic behaviors of parameter estimations, unbiased quasi-regression is unbiased for the fitting of curve.
The notion of finite-type open set condition is defined to calculate the Hausdorff dimensions of the sections of some self-similar sets, such as the dimension of intersection of the Koch curve and the line x = a with a ∈ ℚ.
We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361–380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of ℂP 2 × V , where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.
In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n ≥ 4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.
The Bott generator of the homotopy group π 2k−1U(∞) is used to construct an almost complex structure on S 6, which is integrable except a small neighborhood.
The embedding of the Bernoulli shift into the logistic map x → μx(1 − x) for μ > 4 is reinterpreted by the theory of anti-integrability: it is inherited from the anti-integrable limit μ → ∞.
In this paper, based on the Pauli matrices, a notion of augmented spinor space is introduced, and a uniqueness of such augmented spinor space of rank n is proved. It may be expected that this new notion of spaces can be used in mathematical physics and geometry.
In this paper, the authors introduce the concept of h-quasiconvex functions on Carnot groups G. It is shown that the notions of h-quasiconvex functions and h-convex sets are equivalent and the L ∞ estimates of first derivatives of h-quasiconvex functions are given. For a Carnot group G of step two, it is proved that h-quasiconvex functions are locally bounded from above. Furthermore, the authors obtain that h-convex functions are locally Lipschitz continuous and that an h-convex function is twice differentiable almost everywhere.
In this paper, by means of Sadovskii fixed point theorem, the authors establish a result concerning the controllability for a class of abstract neutral functional differential systems where the linear part is non-densely defined and satisfies the Hille-Yosida condition. As an application, an example is provided to illustrate the obtained result.
Consider the stable Steinberg group St(K) over a skew field K. An element x is called an involution if x 2 = 1. In this paper, an involution is allowed to be the identity. The authors prove that an element A of GL n(K) up to conjugation can be represented as BC, where B is lower triangular and C is simultaneously upper triangular. Furthermore, B and C can be chosen so that the elements in the main diagonal of B are β 1, β 2,⋯ , β n, and of C are γ 1, γ 2, ⋯, γ n c n, where c n ∈[K*,K*] and ${\prod\limits_{j = 1}^n {\overline{{\beta _{j} \gamma _{j} }} } } = \det A.$ It is also proved that every element δ in St(K) is a product of 10 involutions.