HARMONIC ANALYSIS ON THE QUOTIENT k~x非汉字符号kA~1非汉字符号A~1

0 IntroductionThe group of norm-1 ideles A1 over a global field lt acts by 1111lltip1ication on t11e additivegroup of adeIes A. Let G = A1 be A be tl1eir semidirect product. Let r be its arithnletic sub-group kx oc k. For the case k = Q Dorian Goldfeld gavc ill [2] a Schwartz kernel on rG whosegeometrical trace equals the prime 11un1ber side of the explicit fOr1llulas of A. Weil. A spectralillterpretatiou in the seuse of the Selberg trace formula is cal1ed for. In order that a geolllet…     

A REMARK ON THE UNIQUENESS OF THE HARMONIC MAPS

This paper studies the uniqueness of the harmonic maps from R~n into S~(n-1). At first it is proved that the map ψ(x)=x|x|:B~n→S~(n-1) is the unique energy minimizing harmonic map. Then the uniqueness of the harmonic maps with identity boundary value from B~n to S~(n-1) follows.     

ON THE RAYLEIGH QUOTIENT FOR SINGULAR VALUES

In this paper, the theoretical analysis for the Rayleigh quotient matrix is studied, some results of the Rayleigh quotient （matrix） of Hermitian matrices are extended to those for arbitrary matrix on one hand. On the other hand, some unitarily invariant norm bounds for singular values are presented for Rayleigh quotient matrices. Our results improve the existing bounds.     

THE FOURIER TRANSFORM AND THE PARSEVAL FORMULA IN HARMONIC ANALYSIS FOR OPERATORS

In this paper,integrable operators with respect to right tranlation are defined and theirelementary properties are discussed.The main results of Fourier transform for opertors and theParseval formula for operators are also proved.     

THE PARSEVAL FORMULA IN THE HARMONIC ANALYSIS FOR OPERATOR AND THE SUPPORT FOR OPERATORS

In this paper a harmonic analysis for operators on a homogeneous Banach space on the additional group of real numbers is discussed. Main resu lts are as following:1. The Parseval formula for integrable operators with respect right translation holds under some conditions; i.e., the Fourier transform for the product of opreator T and T~* at zero is equal to the integral for the product of Fourier transform (γ) and (γ)~* in the strong operator topelogy.2. The finite support and finite cosupport for any beunded linear operator are both unique except the having Haar measure zero.3. The following are equivalent: (1) The measurable set M is the supporting set for operator T; (2). The measursble set M is the cosupporting set for T~*.     

ON THE QUOTIENT RING OF COMMUTATIVE RINGS WITH ACC⊥ON ANNIHILATOR IDEALS

The author concludes that every commutative ring with ascending chain conditionan annihilator ideals has a Kasch quotient ring,which generalizes the Theorem thatevery commutative noetherian ring has a Kasch quosient ring.If follows that if R is acommutative ring with acc~⊥,then that Q(R)is semiprimary is equivalent to that it isperfect,or to that R satisfies regular condition.Besides,that Q(R)is quasi-frobeniusequals that Q(R)is FPF or PF,and that Q(R)is artinian equals that B/N,are of finitedimension,i=1,2,…,n.N_i,=J~i∩R.     

ON THE RELIABILITY BOUNDS IN κ-HNBUE AND κ-HNWUE CLASSES

k-HNBUE class and its dual were introduced by Basu and Ebrahimi [2]. In this paper wegive a generalized definition, and discuss their reliability bounds based on the known mean.     

A VANISHING THEOREM ON L~2 HARMONIC FORMS

This paper is concerned with the L2 harmonic forms of a complete noncompact Riemannian manifold, i.e. If M has a pole Q, let 0 < p c >,then Hp={0}. If M has a soul, then similar result is obtained.     

STOCHASTIC ANALYSIS OF THE STOCHASTIC FUNCTIONAL ON THE BASIC SPACE

The concept of tho basic space is introduced in this paper, which contains tho white noise space and tho abstract Wiener space. Many results of stochastic analysis of the stochastic functional on white noiso space ([4]) and abstract Wiener space ([12]) are generalisod into this space, for example, Fourier transformation, Fourier-Wiener transformation and Malliavin's calculus, etc.     

ON RUBIN’S HARMONIC ANALYSIS AND ITS RELATED POSITIVE DEFINITE FUNCTIONS

In this paper,a new formulation of the Rubin’s q-translation is given,which leads to a reliable q-harmonic analysis.Next,related q-positive definite functions are introduced and studied,and a Bochner’s theorem is proved.     

ON THE HYPERSPHERICAL HARMONIC METHODTO THREE AND FOUR BODY NUCLEAR PROBLEMS

In this paper, the hyperspherical harmonics used for solving the three and four body problems in nuclear physics are given. The equations of the adiabatic approximation to hyperspherical harmonic method are derived. Some properties of the hypernuclous Heare given as the illustrative example of the application of this method.     

ON HEYDE’S THEOREM ON THE GROUP $$\mathbb{R}$$ × $$\mathbb{T}$$

Doklady Mathematics - According to the well-knows Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of...     

QUALITATIVE ANALYSIS ON A CLASS OF ECOSYSTEM

In this paper, we give complete qualitative analysis on a class of biological system and prove the nonexistence, existence and uniqueness of a limit cycle for the system.     

ON THE ANISOTROPIC ACCURACY ANALYSIS OF ACMES NONCONFORMING FINITE ELEMENT

The main aim of this paper is to study the superconvergence accuracy analysis of the famous ACM＇s nonconforming finite element for biharmonic equation under anisotropic meshes. By using some novel approaches and techniques, the optimal anisotropic interpolation error and consistency error estimates are obtained. The global error is of order O（h^2）. Lastly, some numerical tests are presented to verify the theoretical analysis.     

ON THE ANISOTROPIC ACCURACY ANALYSIS OF ACM''''S NONCONFORMING FINITE ELEMENT

The main aim of this paper is to study the superconvergence accuracy analysis of thefamous ACM's nonconforming finite element for biharmonic equation under anisotropicmeshes. By using some novel approaches and techniques, the optimal anisotropic inter-polation error and consistency error estimates are obtained. The global error is of orderO(h~2). Lastly, some numerical tests are presented to verify the theoretical analysis.     

COMPLETE CONVERGENCE THEORY OF THE CONTACT PROCESS ON Td × Z1

The author considers the contact process on a branching plane T_d × Z, which is the product of a regular tree T_d and the line Z. It is shown that above the second critical point, the complete convergence theory holds.     

ON THE DERIVATIVE OF A POLYNOMIAL

For an entire function f(z), let M(f,r) = max is a polynomial of degree n, then, ingeneral, it is difficult to obtain a lower bound far M (p',1). But if the zeros of the polynomial are close to the origin, then various lower bounds for M(p' ,1) have been obtained in the past. In this paper, we have considered polynomials having all their zeros in , with a possible zero of order m(m>0) at the origin and have obtained a lower bound for M(p', 1), which is better than most of the known lower bounds. Our bound is sharp for m=0.     

HARMONIC FUNCTIONS ON A COMPLETE NONCOMPACT MANIFOLD WITH ASYMPTOTICALLY NONNEGATIVE CURVATURE

The authors prove the space of harmonic functions with polynomial growth of a fixed rate on a complete noncompact Riemannian manifold with asymptotically nonnegative curvature is finite dimensional.     

QUANTUM PHENOMENON OF THE ENERGY DENSITY OF A HARMONIC MAP TO A SPHERE

This paper proves that if the energy density of a harmonic map to a unit sphere varies between two successive half eigenvalues, then it must be one of them. Applying this result to the Gaussian maps of some submanifolds, the quantum phenomena of the square length of the second fundamental forms of these submanifolds is obtained. Some related topics are discussed in this note.     

ON THE CONVERGENCE OFSPLITTINGS FOR A Z-MATRIX

In this paper, first we present a characterization of semiconvergence for nonnegative splittings of a singular Z-mattix, which generalizes the corresponding result of . Second, a characterization of convergence for L1-regular splittings of a singular E-matrix is given, which im-prove the resuh of [3]. Third, convergence of weak nonnegative splittings and regular splittings isdiscussed, and we obtain some necessary and sufficient conditions such that the splittings of a Z-matrix converge,