Let p be an odd prime. The authors detect a nontrivial element ã p of order p 2 in the stable homotopy groups of spheres by the classical Adams spectral sequence. It is represented by $a_0^{p - 2} h_1 \in Ext_A^{p - 1,pq + p - 2} (\mathbb{Z}/p,\mathbb{Z}/p)$ in the E 2-term of the ASS and meanwhile p · ã p is the first periodic element α p.
The authors prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions. For the special case of one complex variable, the obtained results give the classic boundary Schwarz lemma.
The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given.
Let Ω be a bounded and connected open subset of ℝ N with a Lipschitz-continuous boundary, the set Ω being locally on the same side of ∂Ω. A vector version of a fundamental lemma of J. L. Lions, due to C. Amrouche, the first author, L. Gratie and S. Kesavan, asserts that any vector field v = (u i) ∈ (D′(Ω)) N, such that all the components $\frac{1}{2}({\partial _j}{v_i} + {\partial _i}{v_j})$, 1 ≤ i, j ≤ N, of its symmetrized gradient matrix field are in the space H−1(Ω), is in effect in the space (L2(Ω)) N. The objective of this paper is to show that this vector version of J. L. Lions lemma is equivalent to a certain number of other properties of interest by themselves. These include in particular a vector version of a well-known inequality due to J. Nečas, weak versions of the classical Donati and Saint-Venant compatibility conditions for a matrix field to be the symmetrized gradient matrix field of a vector field, or a natural vector version of a fundamental surjectivity property of the divergence operator.
The aim of this paper is two-fold. Given a recollement (T′, T, T″, i*, i *, i !, j !, j*, j *), where T′, T, T″ are triangulated categories with small coproducts and T is compactly generated. First, the authors show that the BBD-induction of compactly generated t-structures is compactly generated when i * preserves compact objects. As a con-sequence, given a ladder (T′, T, T″, T, T′) of height 2, then the certain BBD-induction of compactly generated t-structures is compactly generated. The authors apply them to the recollements induced by homological ring epimorphisms. This is the first part of their work. Given a recollement (D(B-Mod),D(A-Mod),D(C-Mod), i*, i *, i !, j !, j*, j *) induced by a homological ring epimorphism, the last aim of this work is to show that if A is Gorenstein, A B has finite projective dimension and j ! restricts to D b(C-mod), then this recollement induces an unbounded ladder (B- G proj,A- G proj, C- G proj) of stable categories of finitely generated Gorenstein-projective modules. Some examples are described.
A subgroup of index p k of a finite p-group G is called a k-maximal subgroup of G. Denote by d(G) the number of elements in a minimal generator-system of G and by δ k(G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G. In this paper, the authors classify the finite p-groups with δ d(G)(G) ≤ p 2 and δ d(G)−1(G) = 0, respectively.
Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold, where the Hamiltonian satisfies the condition: The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit. It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a 1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.
In this paper, the complete convergence and the complete moment convergence for extended negatively dependent (END, in short) random variables without identical distribution are investigated. Under some suitable conditions, the equivalence between the moment of random variables and the complete convergence is established. In addition, the equivalence between the moment of random variables and the complete moment convergence is also proved. As applications, the Marcinkiewicz-Zygmund-type strong law of large numbers and the Baum-Katz-type result for END random variables are established. The results obtained in this paper extend the corresponding ones for independent random variables and some dependent random variables.
The authors compute non-zero structure constants of the full flag manifold M = SO(7)/T with nine isotropy summands, then construct the Einstein equations. With the help of computer they get all the forty-eight positive solutions (up to a scale ) for SO(7)/T, up to isometry there are only five G-invariant Einstein metrics, of which one is Kähler Einstein metric and four are non-Kähler Einstein metrics.
In this paper bifurcations of heterodimensional cycles with highly degenerate conditions are studied in three dimensional vector fields, where a nontransversal intersection between the two-dimensional manifolds of the saddle equilibria occurs. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, the authors construct a Poincaré return map under the nongeneric conditions and further obtain the bifurcation equations. By means of the bifurcation equations, the authors show that different bifurcation surfaces exhibit variety and complexity of the bifurcation of degenerate heterodimensional cycles. Moreover, an example is given to show the existence of a nontransversal heterodimensional cycle with one orbit flip in three dimensional system.
This paper deals with the electrostatic MEMS-device parabolic equation ${u_t} - \Delta u = \frac{{\lambda f(x)}}{{{{(1 - u)}^p}}}$ in a bounded domain Ω of ℝ N, with Dirichlet boundary condition, an initial condition u 0(x) ∈ [0, 1) and a nonnegative profile f, where λ > 0, p > 1. The study is motivated by a simplified micro-electromechanical system (MEMS for short) device model. In this paper, the author first gives an asymptotic behavior of the quenching time T * for the solution u to the parabolic problem with zero initial data. Secondly, the author investigates when the solution u will quench, with general λ, u 0(x). Finally, a global existence in the MEMS modeling is shown.
The main goal of this paper is to approximate the Kuramoto-Shivashinsky (K-S for short) equation on an unbounded domain near a change of bifurcation, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider PDEs with quadratic nonlinearities and derive rigorously the modulation equation, which is called the Ginzburg-Landau (G-L for short) equation, for the amplitudes of the dominating modes.