The age of infection approach introduced by Kermack and Mckendrick in 1927 gives a unified way of describing and analyzing a variety of epidemic models, including models with multiple stages, treatment, and heterogeneous mixing. The author gives a description of the main results for such models, emphasizing the use of the final size relation to estimate the size of the epidemic.
The author obtains sharp gradient estimates for the heat kernels in two kinds of higher dimensional Heisenberg groups — the non-isotropic Heisenberg group and the Heisenberg type group ℍ n,m. The method used here relies on the positive property of the Bakry-Émery curvature Γ2 on the radial functions and some associated semigroup technics.
Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the υ-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the υ-function. Under such restrictions, curvature estimates in terms of υ-function composed with the Gauss map can be carried out.
The authors show the existence and uniqueness of solution for a class of stochastic wave equations with memory. The decay estimate of the energy function of the solution is obtained as well.
A Schwarz-Pick estimate of higher order derivative for holomorphic functions with positive real part on B n is presented. This improves the earlier work on Schwarz-Pick estimate of higher order derivatives for holomorphic functions with positive real part on the unit disk in ℂ.
Seventy years ago, Myers and Steenrod showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. In 2007, Bagaev and Zhukova proved the same result for a Riemannian orbifold. In this paper, the authors first show that the isometry group of a Riemannian manifold M with boundary has dimension at most ½ dimM(dimM − 1). Then such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension are completely classified.
An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super Dirac systems. Under the obtained symmetry constraint, the n-th flow of the super Dirac hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold R 4N|2N with the corresponding dynamical variables x and t n. The integrals of motion required for Liouville integrability are explicitly given.
The authors introduce a new idea related to Montel-type theorems in higher dimension and prove some Montel-type criteria for normal families of holomorphic mappings and normal holomorphic mappings of several complex variables into P N(ℂ) for continuously moving hyperplanes in pointwise general position. The main results are also true for continuously moving hypersurfaces in pointwise general position. Examples are given to show the sharpness of the results.
The author uses analytic methods to study the distribution of integral ideals and Hecke Grössencharacters in algebraic number fields. Nowak’s results on the distribution of integral ideals, and Chandrasekharan and Good’s results on the distribution of Hecke Grössencharacters are improved.
The purpose of this paper is to study relations among equivariant operations on 3-dimensional small covers. The author gets three formulas for these operations. As an application, the Nishimura’s theorem on the construction of oriented 3-dimensional small covers and the Lü-Yu’s theorem on the construction of all 3-dimensional small covers are improved. Moreover, for a construction of 3-dimensional 2-torus manifolds, it is shown that all operations can be obtained by using the equivariant surgeries.
The authors establish a general monotonicity formula for the following elliptic system $\Delta u_i + f_i \left( {x,u_1 , \cdots ,u_m } \right) = 0 in \Omega $ where Ω ⊂⊂ ℝ n is a regular domain, (f i(x, u 1, ..., u m)) = ∇$\vec u$ F(x, $\vec u$), F(x, $\vec u$) is a given smooth function of x ∈ ℝ n and $\vec u$ = (u 1, ..., u m) ∈ ℝ m. The system comes from understanding the stationary case of Ginzburg-Landau model. A new monotonicity formula is also set up for the following parabolic system $\partial _t u_i - \Delta u_i - f_i \left( {x,u_1 , \cdots ,u_m } \right) = 0 in \left( {t_1 ,t_2 } \right) \times \mathbb{R}^n $, where t 1 < t 2 are two constants, (f i(x, $\vec u$)), is given as above. The new monotonicity formulae are focused more attention on the monotonicity of nonlinear terms. The new point of the results is that an index β is introduced to measure the monotonicity of the nonlinear terms in the problems. The index β in the study of monotonicity formulae is useful in understanding the behavior of blow-up sequences of solutions. Another new feature is that the previous monotonicity formulae are extended to nonhomogeneous nonlinearities. As applications, the Ginzburg-Landau model and some different generalizations to the free boundary problems are studied.