Showing papers in "Acta Mathematica Sinica in 1992"
TL;DR: In this paper, the authors considered the nonlinear periodic differential equation and proved that all solutions of the above-mentioned equation are bounded in R and there are infinitely many quasi-periodic solutions and an infinity of periodic solutions of minimal periodm, for each positive integerm.
Abstract: We consider the nonlinear periodic differential equation
$$\frac{{d^2 x}}{{dt^2 }} + \beta x^{2n + 1} + a(t)x = p(t), n \geqslant 1,$$
wherea(t) andp(t) are continuous and 1-periodic, β is a positive constant. The purpose of this paper is to prove that all solutions of the above-mentioned equation are bounded fort∈R and there are infinitely many quasi-periodic solutions and an infinity of periodic solutions of minimal periodm, for each positive integerm.
29 citations
TL;DR: In this article, the polynomial time hierarchy is separated from the probabilistic complexity class in relativization by constructing an oracle such that π π 2 √ P, A √ π √ PP √ A ∆ π + π P,A ∆ = PP^A.
Abstract: We construct an oracleA such that\(\Sigma _2^{P,A}
subseteq PP^A \). So the polynomial time hierarchy is separated from the polynomial time probabilistic complexity class in relativization.
21 citations
TL;DR: In this article, a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed, and the constants of the motion for the algebraic system are presented.
Abstract: Under the constraint determined by a relation (a
n
,b
n
)T={f(ϕ)}
n
between the reflectionless potentials and the eigenfunctions of the general discrete Schrodinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution varietyN of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented.f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.
9 citations
TL;DR: In this article, the authors considered the problem of finding the unique connecting orbit for a cooperative vector field in the retangular box, where the Jacobian matrices at O and P are irreducible.
Abstract: This paper considers the problem of the existence and uniqueness of the connecting orbit for a cooperative vector fieldF in the retangular boxB. Suppose that the origin O and the pointP with positive components are vertices ofB, only containing the equilibria O andP, and that the Jacobian matrices ofF at O andP are irreducible. Then there is a unique orbit ofF contained inB which joins O andP.
7 citations
TL;DR: In this paper, the functional equations arising from the differential embedding problem were solved by a method for solving the functional equation arising from differential embeddings, and conditions for embedding one-dimensional diffeomorphisms into differential flows were obtained.
Abstract: In this paper we give a method for solving the functional equations arising from the differential embedding problem. We also obtain the conditions for embedding one-dimensional diffeomorphisms into differential flows.
6 citations
TL;DR: In this paper, the authors established a sharp asymptotic formula for √ √ q, √ t, √ T with an error term O(T^{\frac{1}{3} - \delta + e} ), which is a suitable positive constant.
Abstract: In the present paper we establish a sharp asymptotic formula for\(\sum\limits_{x\bmod q} {\smallint _0^T } |L(\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} + it,x)|^2 dt\) with an error term\(O(T^{\frac{1}{3} - \delta + e} )\) (σ is a suitable positive constant) uniformly inq andT, which improves the best result hitherto given.
5 citations
TL;DR: In this paper, a special Nielsen number, SN(f|ψ), was introduced, which is a lower bound for the number of fixed points on a given map ψ:A→A, where (X,A) is a pair of compact polyhedra.
Abstract: Letf:(X,A)→(X,A) be an extension of a given map ψ:A→A, where (X,A) is a pair of compact polyhedra. We shall introduce a special Nielsen number,SN(f|ψ), which is a lower bound for the number of fixed points onX-A for all extensions in the homotopy class off. It is shown that for many space pairs this lower bound is the best possible one, and that it can be realized without the by-passing condition.
5 citations
TL;DR: In this paper, the authors discuss the concept of generalized exponential dichotomy and give the existence of k invariant manifolds for abstract nonautonomous differential equations in Banach or Hilbert spaces.
Abstract: In this paper we discuss the concept ‘generalized exponential dichotomy’ and give the existence ofCk invariant manifolds for abstract nonautonomous differential equations in Banach or Hilbert spaces. Also we give a classification of invariant manifolds and an estimate of the locality radius of invariant manifolds.
4 citations
TL;DR: A negative answer is given for the theorems on the tractability of random algorithm and on the average area of approximate zeros and discussion about the algorithm of the global Newton's method is devoted in this paper.
Abstract: In his report at ICM, Smale posed a problem whether the inequalityα(t, ψ
d
)≤1 holds for allt∈(0,1), where
$$\psi _d (t) = \left( {\sum\limits_{i = 0}^d {t^{2i} } } \right)^{1/2} $$
. This paper gives a negative answer. In addition, elementary proofs are given for the theorems on the tractability of random algorithm and on the average area of approximate zeros. Meanwhile, discussion about the algorithm of the global Newton's method is devoted in this paper.
4 citations
TL;DR: In this paper, a sequence of characterizations of inner amenable groups is given by developing the well-known Folner's conditions for amenable locally compact groups, where the inner amenability is a considerably weaker condition than amenability, but closely related to the quite deep property of groups.
Abstract: For infinite discrete groups, Effros introduced the notion of inner amenability which gives a new classification of discrete groups. The inner amenability is a considerably weaker condition than amenability, but closely related to the quite deep property Γ of groups. In this paper the author investigates the structures of inner amenable groups by theoretical set theory. A sequence of characterizations of inner amenable groups is given here by developing the well-known Folner's conditions for amenable locally compact groups.
4 citations
TL;DR: In this paper, the authors extend the results of Brezis and Nirenberg in [4] to the problem of finding a positive solution to a lower order perturbation in a bounded smooth domain.
Abstract: In this paper we extend the results of Brezis and Nirenberg in [4] to the problem
$$\left\{ \begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) in\Omega , \hfill \\ p = (n + 2)/(n - 2) \hfill \\ u > 0 in\Omega , u = 0 \partial \Omega , \hfill \\ \end{gathered} \right.$$
whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofup at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofaij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have\(a_{ij} \left( x \right) \in C\left( {\bar \Omega } \right)\) such thatLu=up possesses a positive solution inH01(Ω).
TL;DR: In this article, the large deviation theorem for Hill's estimator was obtained under the assumption that the d.f. F is continuous and the exponent of the estimator is a constant.
Abstract: To estimate the exponent of a regularly varying d.f. F, the asymptotic behaviour of Hill's estimator has been extensively discussed. Under the assumption that the d.f. F is continuous, we obtain the large deviation theorem for Hill's estimator.
TL;DR: In this article, a two-parameter family of systems was studied, in which a contour consisting of a saddle point and two periodic motions of saddle type was considered, and the situation is similar to that described by Lorenz equations for parametersb=8/3, σ=10,r=rl=24.06, and some results concerning bifurcation phenomenon and dynamical behavior of the orbits of the systems in a small neighborhood of the contour for near zero.
Abstract: In this paper, we study a two-parameter family of systemsEc in whichE0 as a contour consisting of a saddle point and two periodic motions of saddle type, i.e., the situation is similar to that described by Lorenz equations for parametersb=8/3, σ=10,r=rl=24.06, and get some results concerning bifurcation phenomenon and dynamical behavior of the orbits ofEc in a small neighborhood of the contour for |e| near zero. Thus, under a few natural assumptions which are verified numerically, we can explain some numerical results of Lorenz equations for parameters near the above values in a mathematically precise way, which is different from the methods of J. Guckenheimer et al.([3], [4]), by considering Lorenz equation as a one—or two—dimensional map.
TL;DR: In this paper, the closedness of generalized vertex operators defined by the central binomial coefficient under Lie bracket has been studied and the formula of Lie product of GVOPs has been obtained.
Abstract: This paper deals with the closedness of generalized vertex operators defined by the central binomial coefficient under Lie bracket. The formula of Lie product of generalized vertex operators has also been obtained.
TL;DR: In this paper, the authors provided two kinds of forbidden configurations for the rectilinear O-embeddability of triangle free planar graphs, and characterizations of the Oembeddingability for outerplanar graphs and Halin graphs are found.
Abstract: This paper provides two kinds of forbidden configurations for the rectilinear O-embeddability of triangle free planar graphs. Meanwhile, the characterizations of the O-embeddability for outerplanar graphs and Halin graphs are found.
TL;DR: In this article, the authors considered a class of quasilinear elliptic eigenvalue problems with limiting nonlinearity and used the concentration-compactness principle to get the existence of a minimum solution.
Abstract: In this paper, we consider a class of quasilinear elliptic eigenvalue problems with limiting nonlinearity. First, we use the concentration-compactness principle to get the existence of a minimum ueH
0
1
(ω,R
N
) of the minimization problem
$$I_{\lambda _0 } = \inf \{ \smallint _\Omega (a_{\alpha \beta } (x)g_{ij} (u)D_\alpha u^i D_\beta u^j + h(x)|u|^2 )|u \in H_0^1 (\Omega ,R^N ),\smallint _\Omega |u|^{2n/(n - 2)} = \lambda _0 \} ;$$
then we apply the reverse Holder inequality to prove thatueL
∞
(ω, R
N
).
TL;DR: In this paper, the authors study the stability of non-autonomous retarded difference-differential equations by constructing Lyapunov functionals and conclude that the retarded difference differential equation is uniformly asymptotically stable under the strong diagonal dominance.
Abstract: In this paper, we study the uniformly asymptotic stability of nonautonomous retarded difference-differential equations by constructing Lyapunov functionals We conclude that the retarded difference-differential equation is uniformly asymptotically stable under the strong diagonal dominance
TL;DR: In this article, the authors apply the Malliavin calculus for two-parameter Wiener functionals to show that the solutions of stochastic differential equations in plane have a smooth density under the restricted Hormander's condition.
Abstract: In this paper we apply the Malliavin calculus for two-parameter Wiener functionals to show that the solutions of stochastic differential equations in plane have a smooth density under the restricted Hormander's condition. This answers a question mentioned by Nualart and Sanz in [3].
TL;DR: In this article, a double indexing family of reducible SU(2)-connections over S2 × S2 was constructed in a geometric way, and the spectrum of the Hessian of the Yang-Mills functional at these connections was computed.
Abstract: In this paper, we construct a double indexing family of reducibleSU(2)-connctions overS2 ×S2 in a geometric way. By separation of variables, we compute the spectrum of the Hessian of the Yang-Mills functional at these connections. Using the implicit function theorem, we prove that these connections are isolated non-minimal solutions to the Yang-Mills equations onS2 ×S2.
TL;DR: In this article, the C ∞ regularity for a very strict local minimum of functionals with genuine degenerate quasiconvex integrand was proved for all real ϕ∈c 0 ∞ (K).
Abstract: We prove in this paper theC ∞ regularity for a “very strict” local minimum of classC loc ρ , ρ>3, of functionals with genuine degenerate quasiconvex integrand\(\int_\Omega {F(x,u,Du)dx} \) depending on a vector-valued function u. Such a minimum satisfies the condition: for all x∈Ω, there exists a neighbourhoodK(x) ofx in Ω andC 1 (x)>0,C 2 (x)>0,1≥e(x)>0, such that\(\int_\Omega {F(x,u + \varphi ,Du + D\varphi )dx \geqslant \int_\Omega {F(x,u,Du)dx + C_1 \left\| \varphi \right\|_e^2 - C_2 \left\| \varphi \right\|_0^2 } } \) for all real ϕ∈c 0 ∞ (K).
TL;DR: In this article, the analyticity of Wiener functionals is defined and its properties and applications to oscillatory Wiener functions are studied. But the analysis is restricted to a single function.
Abstract: We define the analyticity of Wiener functionals and study its properties and applications to oscillatory Wiener functionals.
TL;DR: In this article, the authors prove that the Cauchy problem has a global solution and show that the problem can be solved by strictly hyperbolic conservation laws with finite total variation and sufficiently small deviation.
Abstract: Consider a pair of genuinely nonlinear strictly hyperbolic conservation lawsU t +F(U) x =0 with initial dataU(O,X)=U o (X). Suppose that the initial dataU o (X)=U 1 (X)+U 2 (X), whereU 1 (X) will issue rarefaction waves only,U 2 (X) has any finite total variation and sufficiently small deviation. We prove that the Cauchy problem has a global solution.
TL;DR: In this paper, the uniform convergence of an empirical Bayes estimator or linear empirical bayes (l.B.) estimator is analyzed under mild assumptions on the conditional density of the sample, and it is proved that the convergence rates and uniform convergence rates of these estimators are all one with respect to the corresponding prior families.
Abstract: In this paper the thought on the uniform convergence of an empirical Bayes estimator or linear empirical Bayes (l.e.B.) estimator is advanced. Under two different models the l.e.B. estimators of the parameter are constructed respectively. It is proved that the convergence rates and uniform convergence rates of these l.e.B. estimators are all one with respect to the corresponding prior families. It is shown that the uniform convergence rate one is the best, under mild assumptions imposed on the conditional density of the sample.
TL;DR: In this paper, the existence of singular directions concerning the differential polynomials was proved for the case of ∞ √ √ n √ f √ k + √ 2 √ 1 √ a √ t √ m √ 0 √ b √ {a_j f * f * - af * n }
Abstract: We prove the existance of a kind of singular directions concerning the differential polynomials\(f^{(k)} + \sum\limits_{j = 0}^{k - 1} {a_j f^{(j)} - af^n } \).
TL;DR: For unital C *-algebras, D A 1 is the set of all completely positive maps ϕ from A to M n (C), with Tr ϕ(I)≤1(n≥3) as discussed by the authors.
Abstract: LetA, B be unitalC *-algebras,D A 1 the set of all completely positive maps ϕ fromA toM n (C), with Tr ϕ(I)≤1(n≥3). If Ψ is an α-invariant affine homeomorphism betweenD A 1 andD B 1 with Ψ (0)=0, thenA is*-isomorphic toB.
Abstract: As we know, B.Sz-Nagy and C.Foins studied systematically contractions on Hilbert spaces and developed the harmonic analysis theory of operators on Hilbert spaces. Since 1950s, people paid great attention to the study of contractions on πk spaces. Only a few results have been obtained until today; in particular, the spectral theory of contractions on πk Spaces and corresponding harmonic analysis theory have left still unexplored. This paper, as a continuation of [1], [2], [6], in which the authors after discussing some problems such as the negative invariant subspaces and unitary dilations of contractions on complete spaces with indefinite metrics, establish the triangle model of contractions on πk spaces and furthermore, apply the triangle model to the study of spectral theory of contractions on πk spaces, which is essential to the harmonic analysis of operators on πk spaces.
TL;DR: In this paper, it was shown that the Hausdorff dimension of the sample path of the three dimensional coordinate process is exactly two a.e.g. −ν(g), where ν(g) is the polymer measure constructed by Westwater.
Abstract: In this paper, we prove that the Hausdorff dimension of the sample path of the three dimensional coordinate process is exactly two a.e. −ν(g), where ν(g) is the polymer measure constructed by Westwater. Furthermore, we prove that the sample path of the Westwater process has also zero Hausdorff 2-measure. This answers a problem suggested by E. Nelson.
TL;DR: In this article, the volume of tubes around complex submanifolds in complex space form in terms of the technique of Jacobi fields is given, where the Jacobi field technique is used to measure the number of tubes.
Abstract: In this paper we give the volume of tubes around complex submanifolds in complex space form in terms of the technique of Jacobi fields.
TL;DR: In this paper, it was shown that the spectrum of every unicellular unilateral weighted shift operator on a symmetric Banach space is the singleton set (SDS) and that this is the answer to Rosenthal-Shields' problem.
Abstract: In this paper it will be shown that the spectrum of every unicellular unilateral weighted shift operator on a symmetric Banach space is the singleton set {0}. From this, we give an affirmative answer to Rosenthal-Shields' problem.
TL;DR: In this paper, a necessary and sufficient condition for the one-parameter families of integral diffeomorphisms onS1 to be stable and a necessary condition for multinomial families onS 1 was given.
Abstract: In this paper, we give a necessary and sufficient condition for the one-parameter families of diffeomorphisms onS1 to be stable and a necessary condition for the multi-parameter families to be stable; and, moreover, we prove that phase-locking is a generic property of the one-parameter families of diffeomorphisms onS1. We also get a necessary and sufficient condition of phase-locking for the one-parameter families of integral diffeomorphisms onS1 which strengthens a result in [2].