A CLASS OF FDE WITH DEVIATING ARGUMENT WHICH DEPENDS ON STATE AND RATE OF CHANGE OF THE STATE

ＦｏｒｔｈｅＮｏｎ－Ｒ．Ｎ．Ａ．ＦＤＥ，ｔｈｅｒｅａｒｅｗｉｄｅａｐｐｌｉｃａｔｉｏｎｓ［１］．Ｔｈｅｓｅｅｑｕａｔｉｏｎｓｓｈｏｗａｎｉｍｐａｒｔａｎｔｃｈａｒａｃｔｅｒ，ｗｈｉｃｈｂｏｔｈｐｏｓｉｔｉｖｅａｎｄｎｅｇａｔｉｖｅｄｅｌａｙａｃｔａｓ...     

USE OF LIE SYMMETRIES AND THE PAINLEVÉ ANALYSIS IN COSMOLOGY

Abstract

Two techniques for the analysis of differential equations, the Lie method of extended groups and the Painlevé analysis, are reviewed. The implications for integrability in the case of the existence of symmetries in the former or the possession of the Painlevé property in the latter are explored. By way of an example an equation which arises in shear-free flow of a spherically symmetric perfect fluid with conformal symmetry in general relativity is given an extensive treatment with both techniques.     

ON NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF CARATHÉODORY SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

Given two one-sided Markov chains, the authors illustrate a procedure for ascertaining whether they are essentially the same. Precisely, they show how one can determine whether they are block-isomorphic. An application to hydrology is investigated with an example.     

ON DETECTION OF CHANGE POINTS USING MEAN VECTORS

In this paper,the authors consider the problem of change points within the framework of model selection and propose a procedure for estimating the locations of change points when the number of change points is known.The strong consistency of this procedure is also established. The problem of detecting change points is discussed within the framework of the simultaneous test procedure.The case where the number of change points is unknown will be discussed in another paper.     

NONLINEAR HOMOTOPY AND PRE-POINCARÉ LEMMAS

Abstract

Most homotopies considered in the literature are linear homotopies of the form h ⁱ (λ) = λx ⁱ + (1—λ)y ⁱ , 0 ≤ λ ≤ 1. Although these prove to be adequate in most instances, they lack direct geometric significance because {h ⁱ (λ) | 0 ≤ λ ≤ 1} are not orbits of a vector field. On the other hand, the nonlinear homotopy g ⁱ (s) = e ^s x ⁱ + (1—e ^s )y ⁱ ,—∞ ≤ s ≤ 0, are orbits of a vector field (i.e., dg ⁱ /ds = g ⁱ —y ⁱ , g ⁱ (0) = x ⁱ ), and thus have direct geometric significance. This suggests that useful results can be obtained by replacing linear homotopy by transport along flows of smooth vector fields. The purpose of this paper is to elaborate on this simple idea. We define prehomotopy operators induced by vector fields on a manifold. These allow us to obtain finite transport relations and pre-Poincaré lemmas that generalize the classical results. They are shown to reproduce the classical results as asymptotic limits and to obtain representations of all solutions of complete systems of exterior differential equations on a star shaped region of a manifold.     

NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x~2+α_2xy+α_5x~2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx~j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.     

BIFURCATION PROBLEMS OF MULTIPARAMETER AND MULTIBRANCH AND THEIR APPLICATIONS

This paper deals with the bifurcation problems of multiparameter and multibranch and their approximations. It derives general approximation results of solution branches in the neighbourhood of a singular (limit, turning or multibranch) point, and shows that turning and multibranch points are bifurcation points. The abstract theories are applied to a mixed finite element approximation of stationary Navier-Stokes equation.     

THE INVARIANCE OF STRONG AND ALMOST SPIRALLIKE MAPPINGS OF TYPE β AND ORDER α

The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper.From the maximum modulus principle of holomorphic functions,we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallikeness of type β and order α on the unit ball B~n in C~n and on bounded and complete Reinhardt domains.Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirallikeness of type β,strong and almost starlikeness of order α,strong starlikeness on the corresponding domains.Thus we can construct more subclasses of spirallike mappings in several complex variables.     

DENSITY OF MARKOV SYSTEMS AND ZEROS OF CHEBYSHEV POLYNOMIALS

We raise and partly answer the question: whether there exists a Markov system with respectto which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zerofree interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu-tion that a Markov system, under an additional assumption, is dense if and only if the maxi-mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.     

DISCRETE ANALOGUES OF A THEOREM OF WINTNER AND HARTMAN

１ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｓｅｃｏｎｄｏｒｄｅｒｏｒｄｉｎａｒｙｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｘ″＋ｑ（ｔ）ｘ＝０，ｔ≥０．（１．１）ＳｕｐｐｏｓｅｔｈａｔＱ（ｔ）＝∫＋∞ｔｑ（ｓ）ｄｓ（１．２）ｃｏｎｖｅｒｇｅｓ（ｐｏ...     

EXTREMES OF RATIOS OF DETERMINANTS AND CANONICAL CORRELATION VARIABLES

This article develops some extremes of the ratios of determinants. The results are themultivariate extensions of the extremes of quadratic forms, and can be applied to finding thecanonicai correlation variables of two random vectors. Hence a group of canonical correlationvariables is a solution of the extreme of the ratio of determinants.     

SEMI-OPENNESS AND ALMOST-OPENNESS OF INDUCED MAPPINGS

Given a mapping f between continua. Let 2^f and C(f) mean the induced mappings between hyperspaces. Relations are studied under the conditions :f is semi-open (almost open, respectively), 2^f is semi-open (almost open, respectively) and C(f) is semi-open (almost open, respectively).     

FEATURES AND APPLICATIONS OF BERTALANFFY-RICHARDS‘ AND SCHNUTE'S GROWTH EQUATIONS

ABSTRACT. Based on various ranges of the parameter m (or b), the paper analyzes the features and the integral forms of the Schnute and the Bertalanffy-Richards growth equations as well as the two aspects of their special cases (such as Gompertz, logistic and monomolecular models). It is a first attempt to investigate all the corresponding relationships among parameters of the models derived from the Schnute and the Bertalanffy-Richards growth equations. All the models from the two are empirically fitted by different data sets for eucalypt plantations. Unlike earlier papers, the results of this paper show that either of the two growth equations can be considered as a model for estimating forest growth given a parameter range, and both can produce similar growth performances. Some other aspects of the two growth equations are discussed so that the two can be used correctly.     

ON THE DIFFERENTIAL AND DIFFERENCE INDEPENDENCE OF Γ AND ζ

In this paper, we study the algebraic differential and the difference independence between the Riemann zeta function and the Euler gamma function. It is proved that the Riemann zeta function and the Euler gamma function cannot satisfy a class of nontrivial algebraic differential equations and algebraic difference equations.     

ARBORICITY AND COMPLEMENT OF A GRAPH

1.IntroductionAllthegraphsweconsiderareundirectedandsimple.ThecompletegraphofordernisdenotedbyKm,anindependentsetofnvenicesbyK..ThenotationGUHmeans*ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.thedisjointunionoftwographsGandH,andG HthedisjointunionofGandHplusalltheedgesbetweenGandH,HSGindicatesthatHisasubgraphofG.or(G)istheindependentnumberofG,0(G)thecardinalofthemaximumcliqueofG.ThearboricityofagraphGistheminimumnumberofsubsetsilltowhichtheedgesetofGcanbepar…     

QUASI-MODULUS OF CONTINUITY AND BEST APPROXIMATIONS

In this paper a concept of the quasi-modulus of continuity of the functions of L^2(T) on the unit circle T is introduced, and the relations between it and modulus of continuity are discussed. And we give a sufficient condition such that the best uniform approxi marion of the continuous functions on T is continuous.     

NEIGHBORHOOD UNION OF INDEPENDENT SETS AND HAMILTONICITY OF CLAW-FREE GRAPHS

Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u. For any U V(G) ,let N(U)=Uu,∈UN(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K1.3. One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al. : Let G be a 2-connected claw-free graph of order n,and d(u) d(v) d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that, for any three positive integers s ,t and w,such that if G is a (s t w-1)connected claw-free graph of order n,and d(S) d(T) d(W)＞n-(s t w) for every three disjoint independent vertex sets S,T,W with |S |=s, |T|=t, |W|=w,and S∪T∪W is also independent ,then G is Hamiltonian. Other related results are obtained too.     

ON FUNDAMENTAL SOLUTIONS OF OPERATORS OF PRINCIPAL AND NONPRINCIPAL TYPES

Using integrals in complex space,more precisely,inte-grals similar to Hoermander's step integrals,in this paper weconstruct global fundamental solutions for some linear partialdifferential operators(1.p.d.o)with variable coefficients,in-cluding some operators of principal type and non-principal typewith higher multiple characteristics.     

KANTOROVICH-TYPE INEQUALITIES AND THE MEASURES OF INEFFICIENCY OF THE GLSE

In this paper we introduce some Kantorovich inequalities for the Euclidean norm of a matrix,that is, the upper bounds to ‖ (X'B~(-1)X)~(-1)X'B~(-1)AB~(-1)X (X'B~(-1)X)~(-1)A~(-1)X‖~2and to ‖(X'AX)~(-1)X'BX (X'AX)~(-1)X'CX‖~2 are given, where ‖A‖~2=trace(A'A). In terms of these inequalities the upperbounds to the three measures of inefficiency of the generalized least squares estimator (GLSE) ingeneral Gauss-Markov models are also established.     

Φ-INEQUALITIES AND LAWS OF LARGE NUMBERS OF HARDY MARTINGALETRANSFORMS

In[101,GarlinghasintroducedtheAUMDpropertyofComplexBanachspaceusinginequalityofmartingaletransforms,somefactsshowwhichdiffersfromUMDpropertyofrealBanachspaceintroducedbyBurkholder[6].GarlingalsoprovedthatAUMDpropertyimpliessuperreflexivityandARNP.In[13],XuprovedHMT.isequivalenttoAMT.thatistheBanachspacesforwhicllHardymartingalesconvergeunconditiollallyarethoseforwhichanalyticmartingalesconvergeunconditionally.Hardymartingalesandanalyticmartingalesformaspecialclassofmartingalesforwh…