THE OPTIMAL GENERALIZED LOGARITHMIC MEAN BOUNDS FOR SEIFFERT’S MEAN

For p ∈ R, the generalized logarithmic mean L p (a, b) and Seiffert’s mean T (a, b) of two positive real numbers a and b are defined in (1.1) and (1.2) below respectively. In this paper, we find the greatest p and least q such that the double-inequality L p (a, b) < T (a, b) < L q (a, b) holds for all a, b > 0 and a ≠b.     

DEFORMING CONVEX HYPERSURFACES BY THE LINEAR COMBINATION OF THE MEAN CURVATURE AND THE N-TH ROOT OF THE GAUSS-KRONECKER CURVATURE

1IntroductionTherehavebeensomeinterestingresultsinstudyingtheflowofconvexhypersurfacesintheEuclideanspacebyfunctionsOftheirprincipalcurvatures.BeingviewedasanextensionOfthetheoremOfGageandHedton[3],Huiskellprovedin16]thatdeformingconvexhypersurforesbytheirmeancurysturefunctiollsconvergetoaroundsphereinasense.FollowingthemethodsOfHuiskell[6]andTso[IOI,Chowshowedin[1]thatthesame.statementasin.[6]reconstrueifthemeancurvatureisreplacedbythen-throotoftheGauss-Kroneckercurvature.FOrgenerality…     

ON IDEAL CLASS GROUPS AND UNITS IN TERMS OF THE QUADRATIC FORM x^2＋32y^2

§1. IntroductionIt is a classical topic to study, for quadratic number ?elds F = Q(√2p) with a prime p,the exact 2-power dividing the narrow class number h (F) and the norm of the fundamentalunit ε of F.The 2-primary subgroup of the narrow class group …     

LOWER BOUNDS FOR TNE FIRST GAP OF EIGENVALUES IN THE SCHRDINGER OPERATOR ON SIMPLE RIEMANNIAN MANIFOLDS

We give a lower bound for the first gap λ_2—λ_1 of the twolowerst eigenvalues of the Schr(o|¨)dinger operator-△+W(p) with the Dirichletboundary condition and a strictly convex potential W(p)on M in which M is acompact simple Riemannian manifold with smooth strictly convex boundary (?)MHere a compact Riemannian manifold M is said to be simple if M～(?)M istopologically R~2.We prove thatλ_2-λ_1≥(π~2)/(d~2)+min{0,-(n-1)K}where d is the diameter of M and-(n-1)K,(K≥0)the lower bound of theRicci curvature of M.This work generalizes the results in the classical Eucli-dean situation due to Singer,Wong and Yau,Yu and Zhong to a kind of curvedRiemannian manifold.     

PROPERTIES IN THE LARGE OF QUADRATIC SYSTEMS IN THE PLANE

The author proves that any quadratic system in the plane can be changed into the quadratic system (E_1) or (E_2) by means of linear transformations, and then gives a necessary and sufficient condition for the systems (E_1) and (E_2) to be bounded for t≥0 and to have precisely one monotone unboundol orbit for t≥0 respectively.     

THE VARIATION OF INVARIANT LINES AND LIMIT CYCLES FOR A QUADRATIC DIFFERENTIAL SYSTEM

１ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｒｅａｌｑｕａｄｒａｔｉｃｄｉｆｅｒｅｎｔｉａｌｓｙｓｔｅｍｘ＝－ｙ＋δｘ－４ｘ２＋３ｘｙ＋１３ｙ２ｙ＝ｘ（１－１３ｘ－ｙ）．（１）Ｆｉｒｓｔ，ｆｏｒｃｏｎｖｅｎｉｅｎｃｅ，ｗｅｄｅｎｏｔｅｔｈｅｔｗｏ...     

GENERALIZATIONS OF THE MEAN VALUES AND THEIR INEQUALITIES

Some generalizations of the classical Mean Values and their inequalities are presented inthis paper,using B-spline functions.The new generalized Means can be applied to nonlineardifference schemes for solving O.D.E.and P.D.E.with singularities([6,7,8]).     

A REMARK ON DUALITY FOR THE SUM OF CONVEX FUNCTIONS IN GENERAL BANACH SPACES

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＥｂｅａｒｅａｌＢａｎａｃｈｓｐａｃｅ，ｎｏｔｎｅｃｅｓａｒｉｌｙｒｅｆｌｅｘｉｖｅ．ＬｅｔΦ，Ψ：Ｅ→］－∞，∞］ｂｅｔｗｏｌｏｗｅｒｓｅｍｉｃｏｎｔｉｎｕｏｕｓｃｏｎｖｅｘｆｕｎｃｔｉｏｎｓ．Ｗｅｄｅｎｏｔｅ...     

QUADRATIC INTEGRABILITY AND BOUNDEDNESS OF THE SOLUTIONS FOR SECOND ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS

In this paper, by means of auxiliary functions and inequalities, we obtain some sufficient conditions of second order nonhomogeneous delay differential equation belonging to the L.S. or L.S. ∩L.C..     

PRACTICAL STABILITY IN TERMS OF TWO MEASURES FOR DISCRETE SYSTEMS

Practical stability in terms of two measures for difference equations is discussed by using the Lyapunov function and a new comparison theorem.     

THE SHARP ESTIMATE OF THE THIRD HOMOGENEOUS EXPANSION FOR A CLASS OF STARLIKE MAPPINGS OF ORDER α ON THE UNIT POLYDISK IN Cn

In this article,first,a sufficient condition for a starlike mapping of order α f(x) defined on the unit ball in a complex Banach space is given. Second,the sharp estimate of the third homogeneous expan...     

ON THE THEOREM OF ARROW-BARANKIN-BLACKWELL FOR WEAKIY COMPACT CONVEX SET

ＯＮＴＨＥＴＨＥＯＲＥＭＯＦＡＲＲＯＷ－ＢＡＲＡＮＫＩＮ－ＢＬＡＣＫＷＥＬＬＦＯＲＷＥＡＫＩＹＣＯＭＰＡＣＴＣＯＮＶＥＸＳＥＴ￥ＦＵＷＡＮＴＡＯＡｂｓｔｒａｃｔ：ＴｈｉｓｐａｐｅｒｓｔｕｄｉｅｓｔｈｅｋｎｏｗｎｄｅｎｓｉｔｙｔｈｅｏｒｅｍｏｆＡｒｒｏ...     

ON THE SPECTRUM OF THE TRANSPORT OPERATOR FOR A BOUNDED CONVEX BODY

The eigenvalue problem of the transport operator for a general bounded convex body V is studied mathematically. Where either V or the eigenfunction of the operator is not necessarily of spherical symmetry.It is shown that the spectrum of the operator consists of pure eigenvalues, possibly plusthe point of negative infinity.There is a countable infinity of real eigenvalues accumulating at minus infinity. Each eigenvalue, especially one in the left half-plane, is of index one, There is no complex (non-real) eigenvalue in the right half-plane Re λ>Σ. The solution of the corresponding initial-value problem is also discussed.     

ELIMINATION CONDITIONS AND LOWER BOUNDS FOR THE PERMUTATION FLOW-SHOP SEQUENCING PROBLEM

We give some new elimination conditions for permutation flow-shop problem and prove that thereis one among them equivalent to Szwarc's elimination condition.Next we propose some new lowerbounds for a branch-and-bound method and design an algorithm.Computational results indicate thatthe proposed algorithm is more efficient than those we found in literature.     

THE CLASSIFICATION AND PERIOD OF STATES FOR MARKOV CHAIN IN RANDOM ENVIRONMENT

This paper is a continuation of [8]. In Section 1, three kinds of communication are introdnced for two states and the relations among them are investigated. In Section 2, two kinds of period of a state are introdnced and it is obtained that the period is a “class property” ,i.e. two states x and y belong to same class implies the period of x is equal to the period of y.     

OPERATOR NORM AND ERROR BOUNDS FOR INTERPOLATING QUADRATIC SPLINES ON A NON-UNIFORM TYPE-2 TRIANGULATION OF A RECTANGULAR DOMAIN

The aim of this paper is to give upper bounds of the norm of the operator and associated error for aLagrange interpolation problem by C~1 quadratic splines. The domain is rectangular and the type-2triangulation is non--uniform. Moreover the location of data points allows a very simple compation of theinterpolant.     

ON THE EMPTY CONVEX PARTITION OF A FINITE SET IN THE PLANE

The authors discuss the partition of a finite set of points in the plane into empty convex polygons, and improve some upper bound and lower bound in the related enumeration problems.     

SHARP CONDITIONS FOR THE UNIFORM STABILITY OF NONAUTONOMOUS HUTCHINSON''''S EQUATION

In this paper we obtain new criteria for the positive equilibrium K of thenonautonomous Hutchinson's equationN'(t) = r(t)N(t)[1-[N(t-τ)]/K], t≥0,to be uniformly stable. Examples illustrating the significance of the results arealso given.     

CONVEXITY AND SYMMETRY OF TRANSLATING SOLITONS IN MEAN CURVATURE FLOWS

This paper proves that any rotationally symmetric translating soliton of mean curvature flow in R3 is strictly convex if it is not a plane and it intersects its symmetric axis at one point. The authors also study the symmetry of any translating soliton of mean curvature flow in Rn.