Chinese Annals of Mathematics

This paper constructs a polyconvex stored energy function, satisfying the null condition, for isotropic compressible elastic materials with given Lamé constants. The difference between this stored energy function and St Venant-Kirchhoff's is a three order term.

The author gets a blow-up result of C¹ solution to the Cauchy problem for a first order quasilinear non-strictly hyperbolic system in one space dimension.

This paper gives a necessary and sufficient condition for a matrix in SL(2, Z) to be conjugate to its inverse. This condition reduces the determination of the conjugation to solving some indeterminate equation of second degree. It yields an algorithm to determine this conjugation in finite steps based on the elementary number theory.

I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.

The authors introduce concepts of even and odd additive functionals and prove that an even martingale continuous additive functional of a symmetric Markov process vanishes identically. A representation for symmetric super-martingale multiplicative functionals are also given.

The determinant representation of the gauge transformation operators is establised. In this process, the generalized Wronskian determinant is introduced. As a simple application, the authors present a construction of the special τ-function obtained firstly by Chau et al. (Commun. Math. Phys., 149(1992), 263), which involves the generalized Wronskian determinant. Also, some properties of this determinant are given.

The authors discuss the partition of a finite set of points in the plane into empty convex polygons, and improve some upper bound and lower bound in the related enumeration problems.

This paper studies the parameter estimation of one dimensional linear errors-in-variables (EV) models in the case that replicated observations are available in some experimental points. Asymptotic normality is established under mild conditions, and the parameters entering the asymptotic variance are consistently estimated to render the result useable in construction of large-sample confidence regions.

The authors generalize the works in [5] and [6] to prove a Hopf index theorem associated to a smooth section of a real vector bundle with non-isolated zero points.

The Nagumo equation

Let X be an arbitrary smooth irreducible complex projective curve, E ↦ X a rank two vector bundle generated by its sections. The author first represents E as a triple {D₁, D₂, f}, where D₁, D₂ are two effective divisors with d = deg(D₁) + deg(D₂), and f ∈ H⁰(X, [D₁] |_D2) is a collection of polynomials. E is the extension of [D₂] by [D₁] which is determined by f. By using f and the Brill-Noether matrix of D₁ + D₂, the author constructs a 2g × d matrix W_E whose zero space gives Im{H⁰(X, [D₁]) ↦ H⁰(X, [D₁] |_D1)}⊕ Im{H⁰(X, E) ↦ H⁰(X, [D₂]) ↦ H⁰(X, [D₂] |_D2)}. From this and H⁰(X, E) = H⁰(X, [D₁]) ⊕ Im{H⁰(X, E) ↦ H⁰(X, [D₂])}, it is got in particular that dimH⁰(X, E) = deg(E) - rank(W_E) + 2.

The generalized noncommutative torus of rank n was defined in [4] by the crossed product , where the actions α_i of ℤ on the fibre of a rational rotation algebra are trivial, and C*(kℤ × kℤ) ×_α3 ℤ ×_α4 ⋯ ×_αn ℤ is a completely irrational noncommutative torus A_ρ of rank n. It is shown in this paper that is strongly Morita equivalent to A_ρ, and that is isomorphic to if and only if the set of prime factors of k is a subset of the set of prime factors of p.

The authors first give a necessary and sufficient condition for some solvable Lie algebras with l-step nilpotent radicals to be complete, and then construct a new class of infinite dimensional complete Lie algebras by using the modules of simple Lie algebras. The quotient algebras of this new constructed Lie algebras are non-solvable complete Lie algebras with l-step nilpotent radicals.

This paper constructs several classes of new wavelet bases, which are unconditional bases for related operator spaces. Using these bases, the author analyzes non-homogeneous symbolic space and two related kernel-distribution spaces, and characterizes them in two wavelet coefficients spaces. Besides, some properties for singular integral operators are studied.