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## Dubrovin Valuation Skew Group Rings

### Acta Mathematica Sinica (2002-04-01) 18: 339-346 , April 01, 2002

Some equivalent characterizations for a skew group ring to be a Dubrovin valuation ring are given. Among them all the prime ideals of a Dubrovin valuation skew group ring are characterised.

## On “Problems on von Neumann Algebras by R. Kadison, 1967”

### Acta Mathematica Sinica (2003-07-01) 19: 619-624 , July 01, 2003

A brief summary of the development on Kadison's famous problems (1967) is given. A new set of problems in von Neumann algebras is listed.

## Lattice Structure for Paraunitary Linear–phase Filter Banks with Accuracy

### Acta Mathematica Sinica (2006-05-01) 22: 679-688 , May 01, 2006

Multivariate filter banks with a polyphase matrix built by matrix factorization (lattice structure) were proposed to obtain orthonormal wavelet basis. On the basis of that, we propose a general method of constructing filter banks which ensure second and third accuracy of its corresponding scaling function. In the last part, examples with second and third accuracy are given.

## Coupling for jump processes

### Acta Mathematica Sinica (1986-06-01) 2: 123-136 , June 01, 1986

## Bifurcation of nonlinear problems modeling flows through porous media

### Acta Mathematica Sinica (1998-01-01) 14: 125-134 , January 01, 1998

This paper deals with a class of nonlinear boundary value problems which appears in the study of models of flows through porous media. Existence results of asymptotic bifurcation and continua are reported both for operator equations and for boundary value problems.

## A Note on the Mean Value of Numbers of the Solutions of xα ≡ 1(mod n)

### Acta Mathematica Sinica (2004-11-01) 20: 1095-1102 , November 01, 2004

Let *T* = *T*(*p*, *q*, α) be the number of solutions of the congruence *x*^{α} ≡ 1 (mod *p*^{η}*q*^{θ}). Let *A*
and *B* be sets of primes satisfying *x*_{1} < *p* ≤ *x*_{2} and *y*_{1} < *q* ≤ *y*_{2}, respectively. A mean value estimation
of
$$
\frac{1}
{{\left| {A\left\| {B\left| {} \right.} \right.} \right.}}{\sum\nolimits_{p \in A} {{\sum\nolimits_{q \in B} {\log \;T{\left( {p,q,\alpha } \right)}} }} }
$$
is given.

## Area of Julia sets of holomorphic self-maps onC *

### Acta Mathematica Sinica (1993-06-01) 9: 160-165 , June 01, 1993

It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps on*C*^{*}. We'll prove that*J(f)* has positive area, where*f:C*^{*}→*C*^{*},*f(z)=z*^{m}*c*^{P(z)+Q(1/z)},*P(z)* and*Q(z)* are monic polynomials of degree*d*, and*m* is an integer.

## Bessel (Riesz) potentials on banach function spaces and their applications I theory

### Acta Mathematica Sinica (1998-07-01) 14: 327-340 , July 01, 1998

In this paper, we shall introduce the concept of the Bessel (Riesz) potential Köthe function spaces*X*^{s} (*X*^{s}) and give some dual estimates for a class of operators determined by a semi-group in the spaces*L*^{q} (*−T, T; X*^{s}) (*L*^{q}(*−T, T; X*^{s})). Moreover, some time-space*L*^{p}*−L*^{p′} estimates for the semi-group exp(*i*t(-Δ)^{m/2}) and the operator*A*:=∫
_{0}^{t}
exp(i(*t*-τ)(-Δ)^{m/2})·*d*τ in the Lebesgue-Besov spaces*L*^{q}*(−T,T;B*_{p,2}^{s}
are given. On the basis of these results, in a subsequent paper we shall present some further applications to a class of nonlinear wave equations.

## The Fefferman-Stein-Type Inequality for the Kakeya Maximal Operator II

### Acta Mathematica Sinica (2002-07-01) 18: 447-454 , July 01, 2002

Let *K*_{δ}, 0 < *δ*≪1, be the Kakeya maximal operator defined as the supremum of averages over tubes of the eccentricity *δ*. The (so-called) Fefferman-Stein-type inequality:
$$
{\left\| {K_{\delta } f} \right\|}_{{L^{d} {\left( {{\text{R}}^{d} ,w} \right)}}} \leqslant C_{d} {\left( {\frac{1}
{\delta }} \right)}^{{{{\left( {d - 2} \right)}} \mathord{\left/
{\vphantom {{{\left( {d - 2} \right)}} {2d}}} \right.
\kern-\nulldelimiterspace} {2d}}} {\left( {\log {\left( {\frac{1}
{\delta }} \right)}} \right)}^{{a_{d} }} {\left\| f \right\|}_{{L^{d} {\left( {{\text{R}}^{d} ,K_{\delta } w} \right)}}}
$$
is shown, where *C*_{d} and *α*_{d} are constants depending only on the dimension *d* and *w* is a weight. The result contains the exponent (*d*−2)/2*d* which is smaller than the exponent (*d*−2)(*d*−1)/*d*(2*d*−3) obtained in [7].

## A Proof of the Jacobian Conjecture on Global Asymptotic Stability

### Acta Mathematica Sinica (2001-01-01) 17: 119-132 , January 01, 2001

Let ƒ∈*C*^{1} (*R*^{1}, *R*^{2}), ƒ(0) = 0. The Jacobian Conjecture states that if for any *x*∈*R*^{2}, the eigenvalues of the Jacobian matrix *D*ƒ(*x*) have negative real parts, then the zero solution of the differential equation *x* = ƒ(*x*) is globally asymptotically stable. In this paper we prove that the conjecture is true.