郭店简《老子》校释剩稿

郭店简《老子》自公布以来,学人们考证之作颇多,解决了不少疑难问题,但“《诗》无达诂,《易》无达占”,仍有一些字词文句的释读值得讨论。本文从拙作(郭店楚简〈老子〉校释》中选取未刊之作七条,以作引玉之砖。不妥之处,敬请同道批评。为行文方便,凡非直接讨论的简文,一概以释文代之;后面列出《郭店楚墓竹简》一书《老子》图版的原简号。

On Some Fundamental Theorems in Stability Theory

This paper gives several fundamental theorems for the stability, uniform stability, asymptotic stability and uniform asymptotic stability. Those theorems allow the derivative of Lyapunov functions to be positive on certain sets,relax the restriction about the rate of change of state variable in a system to be bounded in Marachkov's theorem and extend the related results in [4—7].

Oscillations in Systems of Linear Neutral Differential Equations

Algebraic sufficient conditions are obtained for all nontrivial solutions of the linear system d d t[x(t)+cx(t-τ)]+A(t)x(t-σ(t))=0to be oscillatory.

NONTRIVIAL SOLUTIONS FOR A CLASS OF NON-DIVERGENCE EQUATIONS ON POLARIZABLE CARNOT GROUP

Some new properties of polarizable Carnot group are given.By choosing a proper constant a nontrivial solution of a class of non-divergence Dirichlet problem on the polarizable Carnot group is constructed.Thus the multi-solution property of corresponding non-homogeneous Dirichlet problem is proved and the best possible of LQ norm in the famous Alexandrov-Bakelman-Pucci type estimate is discussed.

Global Existence of Positive Solutions for Volterra Integral Equations

This paper considers the global existence and nonexistence of positive solutions for the following volterra integral equations wbers Matrix B is called a positive definite one, if all the principal minors have positive detechants. By considering the existence of positivve solutions for algebra equations, it is proved that if I-A is a positive definite matrix,where I is an identity matrix, then (I) bas global positive solution 1 Otherwise, (I)has no continous nbndeereasing positive solution.

The Osgood Integral or the Cauchy-Osgood Integral？

Initially, Osgood used the integral ∫dr/f（r）for an unicity crite, rion to the differential equation y＇ = f（y）, f （0） = 0. The trivial solution is unique iff this integral goes to the infinite at the origin. Then he can prove the unicity of the trivial solution of y＇ = y Ln｜Y｜, although the second member is not lipschitzian. Later, Bernfeld [1] shows that all the solutions of y＇ = f（y） do not explose iffthe same integral goes to the infinite at the infinite. Finally, we can adapt a result from the Cauchy works as follows： the trivial solution is a singular solution iffthe same integral vanishes at the origin. Using non standard analysis, we present the proofs of the different criterions and show that the Osgood integral was used by Cauchy before in the similar purpose.

Sub-ODE's New Solutions and Their Applications to Two Nonlinear Partial Differential Equations with Higher-Order Nonlinear Terms

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of the first-order sub-ODE F12 = AF2 ＋ BF2＋p ＋ CF2＋2p （where F1= dF/dε, p 〉 0） are obtained. To our best knowledge, these nontrivial solutions have not been found in [X.Z. Li and M.L. Wang, Phys. Lett. A 361 （2007） 115] and IS. Zhang, W. Wang, and J.L. Tong, Phys. Lett. A 372 （2008） 3808] and other existent papers until now. Using these nontrivial solutions, the sub-ODE method is described to construct several kinds of exact travelling wave solutions for the generalized KdV-mKdV equation with higher-order nonlinear terms and the generalized ZK equation with higher-order nonlinear terms. By means of this method, many other physically important nonlinear partial differential equations with nonlinear terms of any order can be investigated and new nontrivial solutions can be explicitly obtained with the help of symbolic computation system Maple or Mathematics.

A Comparative Study on the Nonspecific Immunity of Juvenile Litopenaeus vannamei ever Inhabiting Freshwater and Seawater

A study on the nonspecific immunity of Litopenaeus vannamei ever inhabiting freshwater and seawater was carried out at different molt stages by comparing their total hemocyte count(THC) and respiratory burst(RB) and activity of phenol oxidase(PO), nitric oxide synthase(NOS) and lysozyme(LY). Two-way ANOVA showed that salinity and molt stage independently affected THC and RB and the activity of PO, NOS and LY of juvenile L. vannamei significantly(P < 0.05). The THC and RB and the activity of NOS gradually increased from the post-molt stages(A and B) to the pre-molt stages(D0–D3), which were common in shrimps inhabiting freshwater and seawater. The activity of PO peaked at the inter-molt stage(C), and touched the lowest at the post-molt stage in freshwater and pre-molt stage in seawater. The activity of LY was stable over the molt cycle. The RB and the activity of PO, NOS and LY of juvenile L. vannamei were significantly lower in freshwater than in seawater; whereas THC was significantly higher in freshwater than in seawater(P < 0.05). It was concluded that the post-molt stage(especially stage A) was critical to shrimp culture, which should be intensively attended when L. vannamei was cultured in freshwater.

Multiple solutions for a Schrdinger-Poisson-Slater equation with external Coulomb potential

Consider the following Schr？dinger-Poisson-Slater system, （P） where ω 〉 0, λ 〉 0 and β 〉 0 are real numbers, p ∈ （1, 2）. For β=0, it is known that problem （P） has no nontrivial solution if λ 〉 0 suitably large. When β 〉 0, -β/｜x｜ is an important potential in physics, which is called external Coulomb potential. In this paper, we find that （P） with β 〉 0 has totally different properties from that of β = 0. For β 〉 0, we prove that （P） has a ground state and multiple solutions if λ 〉 cp,ω, where cp,ω 〉 0 is a constant which can be expressed explicitly via ω and p.

Multiplicity results for the unstirred chemostat model with general response functions

We investigate the multiplicity of positive steady state solutions to the unstirred chemostat model with general response functions. It turns out that all positive steady state solutions to this model lie on a single smooth solution curve, whose properties determine the multiplicity of positive steady state solutions. The key point of our analysis is to study the ＂turning points＂ on this positive steady state solution curve, and to prove that any nontrivial solution to the associated linearized problem is one of sign by constructing a suitable test function. The main tools used here include bifurcation theory, monotone method, mountain passing lemma and Sturm comnarison theorem.

Case Collection of Acupuncture Treatment for the Refractory Eye Diseases by Professor Zhang Ren

Zhang Ren, a master of medicine, professor, chief physician, and State Council expert for Special Allowances. He is the current vice president of Chinese Acupuncture Society, the honorary chairman of Shanghai Acupuncture Society, a member of Shanghai Intangible Cultural Heritage Assessment Committee. He used to be the director of Shanghai Municipal Literature Museum of Traditional Chinese Medicine and the director of the Shanghai Information Institute of Traditional Chinese Medicine. He has been engaged in acupuncture clinical and literature research for more than 40 years. He has been to Europe to give lectures and treat patients in clinic for 3 times, and received a favorable evaluation. And he has independently written and edited more than 60 books on acupuncture and traditional Chinese medicine as a chief editor （including English and Japanese versions）, which were published in Beijing, Shanghai, Chongqing, Taipei and Tokyo. He has published nearly 100 papers in both English and Chinese. He has also chaired Shanghai Municipal Public Health Bureau research projects, and participated in a number of projects such as Shanghai Municipal Science and Technology Commission project, the National 973 project. He has accumulated a wealth of clinical experience on acupuncture treamlent tor multiple modem intractable diseases, especially has unique experience on the acupuncture treammnt for stubborn eye diseases.

Multiple solutions for weighted nonlinear elliptic system involving critical exponents

In this paper, by using the idea of category, we investigate how the shape of the graph of h（x） affects the number of positive solutions to the following weighted nonlinear elliptic system： = （ N-2-2a 2. where 0 is a smooth bounded domain in ]1N （N 〉 3）, A, cr 〉 0 are parameters, 0 ≤ μ 〈 μa a 2 ＇ h（x）, KI（X） and K2（x） are positive continuous functions in , 1 〈 q 〈 2, a, β 〉 1 and a ＋ β = 2＊（a,b） （2＊ （a, b） 2N = N-2（1＋a-b） is critical Sobolev-Hardy exponent）. We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters （）, r） belongs to a certain subset of N2.

Existence of Nontrivial Solutions for p-Laplacian Variational Inclusion Systems in R^N

The authors study the existence of nontrivial solutions to p-Laplacian variational inclusion systems where N ≥ 2, 2 ≤ p ≤ N and F ： R^2 →% is a locally Lipschitz function. Under some growth conditions on F, and by Mountain Pass Theorem and the principle of symmetric criticality, the existence of such solutions is guaranteed.

Recent advances in partially coherent beams with prescribed degrees of coherence

Coherence is one of the most salient features of a laser beam,and laser beams which are not completely coherent,the so-called partially coherent beams,are preferred in many applications.The degrees of coherence of conventional spatial partially coherent beams can be described by Gaussian distributions.Currently,more and more attention is being paid to partially coherent beams with prescribed degrees of coherence due to their extraordinary optical properties,such as self-focusing,self-shaping,selfsplitting,periodicity reciprocity,and super-strong reconstruction. Manipulating the structure of the degree of a partially coherent beam provides a novel way for modulating and controlling its propagation properties and is useful in beam shaping,free-space optical communications,optical trapping,optical encryption and image or information transfer through adverse inhomogeneous environments.In this paper,we present a review on the recent advances in partially coherent beams with prescribed degrees of coherence.

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities ｜u｜t ≥ -△2^u＋｜v｜^q, ｜v｜t ≥-△2v＋｜u｜p^ in S=R^n ×R^＋ withp, q 〉 1, n ≥1. AFujita- Liouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n/4≤ max（ p＋1/pq-1, q＋1/pq-1 ）. Since the general maximum-comparison principle does not hold for the fourth-order problem, the authors use the test function method to get the global non-existence of nontrivial solutions.