Showing papers in "Science China-mathematics in 1997"
TL;DR: In this article, it was proved that for any value s between the maximal and minimal values, there exists an element in M { nk } k ≥ 1, { ck }k ≥ 1) such that its Hausdorff dimension is equal to s. The same result holds for packing dimension.
Abstract: Let M ({ nk } k ≥1,{ ck }k≥1) be the collection of homogeneous Moran sets determined by { n k}k≥1and { ck }k≥1, where { nk }k≥1 is a sequence of positive integers and { ck }k≥1 a sequence of positive numbers. Then the maximal and minimal values of Hausdorff dimensions for elements in M are determined. The result is proved that for any value s between the maximal and minimal values, there exists an element in M { nk } k ≥1, { ck } k ≥1) such that its Hausdorff dimension is equal to s. The same results hold for packing dimension. In the meantime, some other properties of homogeneous Moran sets are discussed.
160 citations
TL;DR: In this paper, a weak in variance principle for strictly stationary negatively associated sequences is proved under some general conditions, and a probability inequality for Sn and some p th moment (p ≥ 2) inequalities for | Sn | and | Sk | are established.
Abstract: A probability inequality for Sn and some p th moment ( p ≥2) inequalities for | Sn | and max 1≤k≤n | Sk | are established. Here Sn is the partial sum of a negatively associated sequence Based on these inequalities, a weak in variance principle for strictly stationary negatively associated sequences is proved under some general conditions
129 citations
TL;DR: In this paper, a general formula for the lower bound of the first eigenvalue on compact Riemannian manifolds is presented, based on the probabilistic approach (i.e. the coupling method).
Abstract: A general formula for the lower bound of the first eigenvalue on compact Riemannian manifolds is presented. The formula improves the main known sharp estimates including Lichnerowicz’ s estimate and Zhong-Yang’s estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on the probabilistic approach (i.e. the coupling method).
81 citations
TL;DR: In this paper, the global asympatotic stability and asymptotic stability for Hopfield-type neural networks with delays are investigated. And they show that Hopfield neural networks have better stability than neural networks without delays.
Abstract: The global asympatotic stability and asymptotic stability for Hopfield-type neural networks with delays are investigated.
61 citations
TL;DR: A general method for a homoclinic loop of planar Hamiltonian systems to bifurcate two or three limit cycles under perturbations is established in this article.
Abstract: A general method for a homoclinic loop of planar Hamiltonian systems to bifurcate two or three limit cycles under perturbations is established.Certain conditions are given under which the cyclicity of a homoclinic loop equals 1 or 2.As an application to quadratic systems,it is proved that the cyclicity of homoclinic loops of quadratic in-tegrable and non-Hamiltonian systems equals 2 except for one case.
54 citations
TL;DR: In this article, the flow characteristics of the viscoelastic fluid in double cylinder rheometer are studied and the analytical solution of which the derivative order is 1/2 is derived with the analytical solutions and the reliability of Laplace numerical inversion based on Crump algorithm for the problem is verified, then the characteristics of second-order fluid flow by using Crump method is analyzed.
Abstract: The fractional calculus approach in the constitutive relationship model of second-order fluid is introduced and the flow characteristics of the viscoelastic fluid in double cylinder rheometer are studied. First, the analytical solution of which the derivative order is 1/2 is derived with the analytical solution and the reliability of Laplace numerical inversion based on Crump algorithm for the problem is verified, then the characteristics of second-order fluid flow in the rheometer by using Crump method is analyzed. The results indicate that the more obvious the viscoelastic properties of fluid are, the more sensitive the dependence of velocity and stress on fractional derivative order is.
39 citations
TL;DR: In this paper, the Herz-type Hardy spaces and Bessel potential spaces were introduced and the Sobolev theorem was established, and some regularity results of nonlinear quantities appearing in the compensated compactness theory on Hardy spaces were given.
Abstract: The Herz-type Sobolev spaces are introduced and the Sobolev theorem is established. The Herz-type Bessel potential spaces and the relation between the Herz-type Sobolev spaces and Bessel potential spaces are discussed. As applications of these theories, some regularity results of nonlinear quantities appearing in the compensated compactness theory on Herz-type Hardy spaces are given.
37 citations
TL;DR: In this article, the authors proposed a semiparametric regression model to estimate the patients' survival times, where the observations are randomly censored on the right and the censoring distribution is unknown.
Abstract: Suppose that the patients’ survival times.Y, are random variables following the semiparametric regression modelY = Xβ +g(T) + e, where (X,T) is a radom vector taking values inR×[0,1],βis an unknown parameter,g (*) is an unknown smooth regression function andE is the random error with zero mean and variance σ2. It is assumed that (X,T) is independent of E. The estimators\(\hat \beta _n \) andgn(*) of P andg(*) are defined, respectively, when the observations are randomly censored on the right and the censoring distribution is unknown. Moreover, it is shown that\(\hat \beta _n \) is asymptotically normal andgn (*) is weak consistence with rateOp(n-1/3).
26 citations
TL;DR: By forming a sequence of coverings of the Sierpinski gasket, a descending sequence of the upper limits of Hausdorff measure is obtained in this paper, which is the best upper limit known so far.
Abstract: By forming a sequence of coverings of the Sierpinski gasket, a descending sequence of the upper limits of Hausdorff measure is obtained. The limit of the sequence is the best upper limit of the Hausdorff measure known so far.
18 citations
TL;DR: In this paper, the dependence of the microstructural change and lattice space symmetry of nano-SnO2 on the annealing temperature has been studied systematically using Raman spectroscopy and X-ray diffraction.
Abstract: The dependence of the microstructural change and lattice space symmetry of nano-SnO2 on the annealing temperature has been studied systematically using Raman spectroscopy and X-ray diffraction. Comparing the results of nano-SnO2 with the results of amorphous film and single crystal of SnO2, it is found that the new Raman peaks N1 and N2 are in accordance with Matossi’s force constant model completely. When the annealing temperature is near 673K, the local lattice disorders and the density of vacant lattice decrease rapidly in the nano-SnO2 grains. The lattice distortion and the new Raman peaks disappear almost at the same time. The possible mechanisms of the microstructural change and the new Raman peaks N1 and N2 are discussed.
17 citations
TL;DR: In this paper, a global measure one solution of the transportation equations is constructed with the help of convex hull of a potential function. But the solution is not a global one.
Abstract: The transportation equations are a mathematical model of zero-pressure flow in gas dynamics and the adhesion particle dynamics system to explain the formation of large scale structures in the universe. With the help of convex hull of a potential function, the solution is explicitly constructed here. It is straightforward to prove that the solution is a global measure one. And Dirac delta-shocks explained as the concentration of particles may develop in the solution.
TL;DR: The boundary value problem for nonlinear parabolic systems is solved by the finite difference method with intrinsic parallelism as discussed by the authors, and the existence of the discrete vector solution for the general finite difference schemes with intrinsic parism is proved by the fixed-point technique in finite-dimensional Euclidean space.
Abstract: The boundary value problem for nonlinear parabolic system is solved by the finite difference method with intrinsic parallelism. The existence of the discrete vector solution for the general finite difference schemes with intrinsic parallelism is proved by the fixed-point technique in finite-dimensional Euclidean space. The convergence and stability theorems of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. The limitation vector function is just the unique generalized solution of the original problem for the parabolic system.
TL;DR: In this article, an automated reasoning method based on Wu's method and calculus of differential forms is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry, which has been used to simplify one of Chem's theorems: "The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces".
Abstract: An automated reasoning method, based on Wu’s method and calculus of differential forms, is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry. The method has been used to simplify one of Chem’s theorems: “The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces.” Some other theorems are also tested by this method. The proofs are generally simpler than those in differential geometry textbooks.
TL;DR: Dimensions of the Hochschild cohomology groups of a truncated algebra of a basic cycle are given in this paper, where they are shown to be invariant to the number of vertices.
Abstract: Dimensions of the Hochschild cohomology groups of a truncated algebra of a basic cycle are explicitly given.
TL;DR: In this paper, it was proved that the dynamics on the Fatou sets will influence the topological complexity of the Julia sets of rational and entire functions and the complexity of rational functions has been described.
Abstract: The topological structures of the Julia sets of rational and entire functions have been investigated and the complexity of the Julia sets of rational functions has been described. For entire functions, it is proved that the dynamics on the Fatou sets will influence the topological complexity of Julia sets.
TL;DR: In this paper, the first boundary problem of the quasilinear parabolic system is considered and the convergence of the difference solution for the iterative difference schemes to the unique solution of the problem is proved.
Abstract: Iterative difference schemes for the first boundary problem of the quasilinear parabolic system are established and the convergence of the difference solution for the iterative difference schemes to the unique solution of the problem is proved.
TL;DR: In this article, the distribution of 0 and 1 is studied in the highest level of primitive sequences over Z /(2e) and the upper and lower bounds on the ratio of the number of 0 to the number 1 in one period of a primitive sequence over Z/(1) are obtained.
Abstract: The distribution of 0 and 1 is studied in the highest levela
e-1 of primitive sequences overZ /(2e). and the upper and lower bounds on the ratio of the number of 0 to the number of 1 in one period ofa
e-1, are obtained. It is revealed that the largere is, the closer to 1 the ratio will be.
TL;DR: The structure of the WIP sodium fluorescence lidar and the preliminary observation result for the sodium layer are reported in this paper, where the authors also show that the lidar can be used to detect the presence of a small amount of sodium.
Abstract: The structure of the WIP sodium fluorescence lidar and the preliminary observation result for the sodium layer are reported.
TL;DR: In this paper, the heating effect of the system in the microwave field, which was influenced by several factors including dielectric properties of synthesis system and thermal insulate structures, was discussed in detail.
Abstract: Microwave equipment at 2 450 MHz was employed to prepare BaTiO3. The heating effect of the system in the microwave field, which was influenced by several factors including dielectric properties of synthesis system and thermal insulate structures, was discussed in detail. The heating rates of the synthesis system were mainly determined by BaCO3 and TiO2 at low temperature and by TiO2 and BaTiO3 at high temperature. The results show that the heating effects in microwave field are greatly different from those in conventional furnace. The reaction of BaCO3 and TiO2 only lasts for 3 min at 1 100°C, and the fine, narrow-distributed and well-crystallized powders were prepared.
TL;DR: In this paper, the local control of pointed groups is generalized to the concept of relative local control, and it is proved that there exists a lifting for a covering of a block source algebra if the relative control holds.
Abstract: The local control of pointed groups is generalized to the concept of relative local control, and it is proved that there exists a lifting for a covering of a block source algebra if the relative local control holds. As an application, a result is proved on the source algebras of blocks, whose defect groups are direct products of a normal subgroup and a subgroup that gives a relative local control.
TL;DR: The possible high dimensional integrable models are studied in three different aspects: starting from a strong symmetry operator of a known (1+1) -dimensionalintegrable model, and starting from the Schwartz equations which possess conformal invariance, and from every concrete realization of the generalized Virasoro algebra, the models possess generalized VirAsoro symmetry algebra.
Abstract: The possible high dimensional integrable models are studied in three different aspects: (i) starting from a strong symmetry operator of a known (1+1) -dimensional integrable model, we can construct a type of (n+1)-dimensional integrable models, high dimensional breaking soliton equations; (ii) from every concrete realization of the generalized Virasoro algebra, we can get many high dimensional integrable models in the meaning that the models possess generalized Virasoro symmetry algebra; (iii) starting from the Schwartz equations which possess conformal invariance, we can also get various high dimensional integrable models in the meaning that they possess Painleve property.
TL;DR: A new algorithm of sequential systems of linear equations for general nonlinear optimization problems with arbitrary initial point is presented that only needs to solve three systems oflinear equations having the same coefficient matrix per iteration, and has global convergence and local superlinear convergence.
Abstract: For current sequential quadratic programming (SQP) type algorithms, there exist two problems: (i) in order to obtain a search direction, one must solve one or more quadratic programming subproblems per iteration, and the computation amount of this algorithm is very large. So they are not suitable for the large-scale problems; (ii) the SQP algorithms require that the related quadratic programming subproblems be solvable per iteration, but it is difficult to be satisfied. By using e-active set procedure with a special penalty function as the merit function, a new algorithm of sequential systems of linear equations for general nonlinear optimization problems with arbitrary initial point is presented. This new algorithm only needs to solve three systems of linear equations having the same coefficient matrix per iteration, and has global convergence and local superlinear convergence. To some extent, the new algorithm can overcome the shortcomings of the SQP algorithms mentioned above.
TL;DR: In this article, it was proved that a Suzuki-Ree group can be characterized by the bet of its order components, and it was shown that a group can also be characterized based on the order components of its components.
Abstract: It is proved that a Suzuki-Ree group can he characterized by the bet of its order components
TL;DR: In this article, the monotoneity properties of certain functions defined in terms of the η-distortion function ηκ(t) in quasiconformal theory are studied and asymptotically sharp bounds are obtained for η η(t), thus proving some properties of the upper bound function K(t, r) in Schottky's theorem on analytic functions.
Abstract: The monotoneity properties of certain functions defined in terms of the η-distortion function ηκ(t) in quasiconformal theory are studied and asymptotically sharp bounds are obtained for ηκ(t), thus proving some properties of the upper bound functionK(t, r) in Schottky’s theorem on analytic functions and improving the known explicit bounds forK (t, r).
TL;DR: In this paper, the authors showed that the reaction dynamics in microwave is suitable for the Carter equation and that the activity energy for reaction of BaCO3 and TiO2 in the microwave field was 4226 kj/mol, which was only one fifth of the conventional reaction.
Abstract: The difference of intermediate products, microstructure and element concentration in the particles between microwave synthesized samples and conventional samples was responsible for the existence of non-thermal effect in the microwave field The diffusions of Ba2+, Ti4+ in the microwave field were enhanced, so that the diffusion of Ti4+ could not be neglected as in the conventional solid state reactions The influences of the microwave field were mainly expressed as diffusion coefficient and the driving force of ionic motion The intermediate phase Ba2TiO4 which occurred in the conventional solid reaction was not found during microwave syntheses The quantity analyses based on XRD experimental data show that the reaction dynamics in microwave is suitable for the Carter equation The activity energy for reaction of BaCO3 and TiO2 in the microwave field was 4226 kj/mol, which was only one fifth of the conventional reaction
TL;DR: In this article, the effect of pressure gradient on coherent structures in a turbulent boundary layer is investigated by using the idea of resonant triad of the theory of hydrodynamic stability.
Abstract: By using the idea of resonant triad of the theory of hydrodynamic stability, the effect of pressure gradient on coherent structures in a turbulent boundary layer is investigated. The favorable pressure gradient suppresses the generation of the coherent structure, while the adverse pressure gradient has the opposite effect. The scale, form, as well as the propagation speed of the coherent structures are different from those with zero pressure gradient. The theoretical results are, in general, in agreement with those found from experiments. From the calculated probability density curve of the circulation differences of the nearly streamwise vortex pairs, it is found that the adverse pressure gradient makes the vortex pair more symmetric.
TL;DR: In this article, an automated reasoning method based on Wu's method and calculus of differential forms is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry, which has been used to simplify one of Chem's theorems: "The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces".
Abstract: An automated reasoning method, based on Wu’s method and calculus of differential forms, is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry. The method has been used to simplify one of Chem’s theorems: “The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces.” Some other theorems are also tested by this method. The proofs are generally simpler than those in differential geometry textbooks.
TL;DR: In this paper, it was shown that for a topological space X satisfying the second axiom of countability and for an outer measurem on a Borel σ-algebra ℬ(X) satisfying the conditions: (i) every non-empty open set of X ism-measurable with positive m-measure; (ii) the restriction ofm on Borel √ X is a probability measure; and (iii) for everyY⊂ X there exists a BoreL setB⊆ X such that B�
Abstract: The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m
i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer
i such that limi→∞
f
r
i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension.
TL;DR: In this article, the uniqueness and existence of restricted Chebyshev center with respect to arbitrary subset is investigated, and the concept of almost Chebysheshev sets is introduced for bounded subsets, and it is proved that each closed subset in a reflexive locally uniformly convex (uniformly convex, respectively) Banach space is an almost Chebyhev subset.
Abstract: The uniqueness and existence of restricted Chebyshev center with respect to arbitrary subset are investigated. The concept of almost Chebyshev sets with respect to bounded subsets is introduced. It is proved that each closed subset in a reflexive locally uniformly convex (uniformly convex, respectively) Banach space is an almost Chebyshev subset with respect to compact convex subsets (bounded convex subsets and bounded subsets, respectively).
TL;DR: In this article, it was proved that if D>0 is not a square, and e =x0 denotes the fundamental solution ofx2−Dy2=−1, thenx2+1=Dy4 is solvable if and only ify0=A2, where A is an integer.
Abstract: Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method. As an application of this result, it is proved that: if D>0 is not a square, and e =x0 denotes the fundamental solution ofx2−Dy2=−1, thenx2+1=Dy4 is solvable if and only ify0=A2, where A is an integer. Moreover, if e>64, thenx2+1=Dy4 has at most one positive integral solution (x, y).