The eigenvalue problem for a thin linearly elastic shell, of thickness 2ε, clamped along its lateral surface is considered. Under the geometric assumption on the middle surface of the shell that the space of inextensional displacements is non-trivial, the authors obtain, as ε → 0, the eigenvalue problem for the two-dimensional “flexural shell” model if the dimension of the space is infinite. If the space is finite dimensional, the limits of the eigenvalues could belong to the spectra of both flexural and membrane shells. The method consists of rescaling the variables and studying the problem over a fixed domain. The principal difficulty lies in obtaining suitable a priori estimates for the scaled eigenvalues.
Extensions of the Hardy and the Bergman modules over the disc algebra are studied. The author relates extensions of these canonical modules to the symbol spaces of corresponding Hankel operators. In the context of function theory, an explicit formula of Ext(L α 2(D), H 2(D)) is obtained. Finally, it is also proved that Ext(L α 2(D), L α 2(D)) ≠ 0. This may be the essential difference between the Hardy and the Bergman modules over the disk algebra.
With the method of inverse limit, the author obtains several criteria of chaos of piecewise monotone continuous maps on finite graphs.
Let p be an odd prime and let n ≥ 1, k ≥ 0 and r be integers. Denote by B k the k-th Bernoulli number. It is proved that (i) If r ≥ 1 is odd and suppose p ≥ r + 4, then ${άthop♪m⌜mits_{j=1}^{p-1}} {1⩈er{(np+j)^r}}≡uiv-{(2n+1)r(r+1)⩈er{2(r+2)}}B_{p-r-2}p^2 ({⤪ mod} p^3)$. (ii) If r ≥ 2 is even and suppose p ≥ r + 3, then ${άthop♪m⌜mits_{j=1}^{p-1}} {1⩈er{(np+j)^r}}≡uiv-{r⩈er{r+1}}B_{p-r-1}p ({⤪ mod} p^2)$. (iii) ${άthop♪m⌜mits_{j=1}^{p-1}} {1⩈er{(np+j)}^{p-2}}≡uiv-(2n+1)p ({⤪ mod} p^2)$. This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolsten-holme’s theorem is obtained.
The authors obtain a new property of the n-dimensional binary undirected de Bruijn graph UB(n) for n ≥ 4, namely, there is a vertex x such that for any other vertex y there exist at least two internally disjoint paths of length at most n - 1 between x and y in UB(n). The result means that the (n - 1, 2)-dominating number of UB(n) is equal to one if n ≥ 4.
Conditions for existence of positive periodic solutions of nonautonomous, nonconvoluton type predator-prey system with infinite delay are given.
Some sufficient conditions for oscillation of the generalized Liénard equations $${⩀t x}=h(y)-F(x),$$ $${⩀t y}=-g(x)$$ are given, which generalize the results of [1–7].
The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping $$u_{tt}+u_t-{\rm div}(a(\nabla_u)\nabla_u)=0,$$ and show that, at least when n ≤ 3, they tend, as t → +∞, to those of the nonlinear parabolic equation $$v_t-{\rm div}(a(\nabla_v)\nabla_v)=0,$$ in the sense that the norm $\|u(.,t)-v(.,t)\|{_L\infty}({\rm R}^n)$ of the difference u - v decays faster than that of either u or v. This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves, first observed by Hsiao, L. and Liu Taiping (see [1, 2]).
When the rotatory inertia is taken into account, vibrations of a linear plate can be described by the Kirchhoff plate equation. Consider this equation with locally distributed control forces and some boundary condition which is the simply supported boundary condition for a rectangular plate. In this paper, the authors establish exact controllability of the system in terms of the equivalence to exact internal controllability of the wave equation, by means of a frequency domain characterization of exact controllability introduced recently in [11].
It is known (for example see [2]) that the maximum genus of a graph is mainly determined by the Betti deficiency of the graph. In this paper, the authors establish an upper bound on the Betti deficiency in terms of the independence number as well as the girth of a graph, and thus use the formulation in [2] to translate this result to lower bound on the maximum genus. Meantime it is shown that both of the bounds are best possible.
Under some conditions on probability, this note discusses the equivalence between the complete convergence and the law of large number for B-valued independent random elements. The results of [10] become a simple corollary of the results here. At the same time, the author uses them to investigate the equivalence of strong and weak law of large numbers, and there exists an example to show that the conditions on probability are weaker.
Based on the spectral flow and the stratification structures of the symplectic group Sp(2n, C), the Maslov-type index theory and its generalization, the ω-index theory parameterized by all ω on the unit circle, for arbitrary paths in Sp(2n, C) are established. Then the Bott-type iteration formula of the Maslov-type indices for iterated paths in Sp(2n, C) is proved, and the mean index for any path in Sp(2n, C) is defined. Also, the relation among various Maslov-type index theories is studied.
The necessary and sufficient conditions are given so that a non-anticipative transformation in Hilbert space is isometric. In terms of second order Wiener process, these conditions assure that a non-anticipative transformation of Wiener process is a Wiener process, too.
The authors show that a symplectically embedded surface in a symplectic 4-manifold with b 2 + greater than one minimizes genus in its homology class.
The authors consider the global existence of the heat flow of harmonic maps from noncompact manifolds while imposing restrictions on the initial data.