By using the index theory for linear bounded self-adjoint operators in a Hilbert space related to a fixed self-adjoint operator A with compact resolvent, the authors discuss the existence and multiplicity of solutions for (nonlinear) operator equations, and give some applications to some boundary value problems of first order Hamiltonian systems and second order Hamiltonian systems.

In the presence of applied magnetic fields H such that | lnɛ| ≪ H ≪ $\tfrac{1}{{\varepsilon ^2 }}$, the author evaluates the minimal Ginzburg-Landau energy with discontinuous constraint. Its expression is analogous to the work of Sandier and Serfaty.

In this paper, the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied. It is proved that the flow converges to a unique geodesic sphere, i.e., every principle curvature of the hypersurfaces converges to a same constant under the flow.

The authors study the existence of homoclinic type solutions for the following system of diffusion equations on ℝ × ℝ ^N$\left\{ \begin{gathered} \partial _t u - \Delta _x u + b \cdot \Delta _x u + au + V(t,x)v = H_v (t,x,u,v), \hfill \\ - \partial _t v - \Delta _x v - b \cdot \Delta _x v + av + V(t,x)u = H_u (t,x,u,v), \hfill \\ \end{gathered} \right.$ where z = (u, v): ℝ × ℝ ^N → ℝ ^m × ℝ ^m, a > 0, b = (b ₁, …, b _N) is a constant vector and V ε C(ℝ × ℝ ^N, ℝ), H ε C ¹ (ℝ × ℝ ^N × ℝ^2m, ℝ). Under suitable conditions on V(t,x) and the nonlinearity for H(t, x, z), at least one non-stationary homoclinic solution with least energy is obtained.

The authors prove the certain de Leeuw type theorems on some non-convolution operators, and give some applications on certain known results.

Let E be a Hilbert C*-module, and S be an orthogonally complemented closed submodule of E. The authors generalize the definitions of S-complementability and S-compatibility for general (adjointable) operators from Hilbert space to Hilbert C*-module, and discuss the relationship between each other. Several equivalent statements about S-complementability and S-compatibility, and several representations of Schur complements of S-complementable operators (especially, of S-compatible operators and of positive S-compatible operators) on a Hilbert C*-module are obtained. In addition, the quotient property for Schur complements of matrices is generalized to the quotient property for Schur complements of S-complementable operators and S*-complementable operators on a Hilbert C*-module.

The complexity of decoding the standard Reed-Solomon code is a well-known open problem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, Li and Wan showed that the error distance can be determined when the degree of the received word as a polynomial is small. In the first part, the result of Li and Wan is improved. On the other hand, one of the important parameters of an error-correcting code is the dimension. In most cases, one can only get bounds for the dimension. In the second part, a formula for the dimension of the generalized trace Reed-Solomon codes in some cases is obtained.

Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose (see [1–4, 7, 8, 1, 11, 16]). Kurosawa, Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying PC(l) of order k from binary linear or nonlinear codes. In this paper, the algebraic-geometric codes over GF(2 ^m) are used to modify the Carlet and Kurosawa-Satoh’s construction for giving vectorial resilient Boolean functions satisfying PC(l) of order k criterion. This new construction is compared with previously known results.

The authors study the compressible limit of the nonlinear Schrödinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singularities in the limit system. On the one hand, the existence and uniqueness of the classical solution are proved for the dispersive perturbation of the quasi-linear symmetric system corresponding to the initial value problem of the above nonlinear Schrödinger equation. On the other hand, in the limit system, it is shown that the density converges to the solution of the compressible Euler equation and the validity of the WKB expansion is justified.

By using the solution to the Helmholtz equation Δu − λu = 0 (λ ≥ 0), the explicit forms of the so-called kernel functions and the higher order kernel functions are given. Then by the generalized Stokes formula, the integral representation formulas related with the Helmholtz operator for functions with values in C(V _3,3) are obtained. As application of the integral representations, the maximum modulus theorem for function u which satisfies H u = 0 is given.

The author first introduces the notion of affine structures on a ringed space and then obtains several related properties. Affine structures on a ringed space, arising from complex analytical spaces of algebraic schemes, behave like differential structures on a smooth manifold.

As one does for differential manifolds, pseudogroups of affine transformations are used to define affine atlases on a ringed space. An atlas on a space is said to be an affine structure if it is maximal. An affine structure is said to be admissible if there is a sheaf on the underlying space such that they are coincide on all affine charts, which are in deed affine open sets of a scheme. In a rigour manner, a scheme is defined to be a ringed space with a specified affine structure if the affine structures make a contribution to the cases such as analytical spaces of algebraic schemes. Particularly, by the whole of affine structures on a space, two necessary and sufficient conditions, that two spaces are homeomorphic and that two schemes are isomorphic, coming from the main theorems of the paper, are obtained respectively. A conclusion is drawn that the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively.