ON EULER CHARACTERISTIC OF MODULES~(**)

This paper gives a characteristic property of the Euler characteristic for IBN rings. The following results: are proved. (1) If R is a commutative ring, M, N are two stable free R-modules, then χ(MN)=χ(M)χ(N), where χ denotes the Euler characteristic. (2) If f: K_0(R)→Z is a ring isomorphism, where K_0(R) denotes the Grothendieck group of R, K_0(R) is a ring when R is commutative, then f([M])=χ(M) and χ(MN)=χ(M)χ(N) when M, N are finitely generated projective R-modules, where.the isomorphism class [M] is a generator of K_0(R). In addition, some applications of the results above are also obtained.     

ON THE DIOPHANTINE EQUATION $\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$

In this paper,, the author proves the following result: Let $\[{E_{a,k}}(N)\]$ denote the number of natural numbers $\[n \le N\]$ for which equation $$\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$$ is insolable in positive integers $\[{x_i}(i = 0,1, \cdots ,k)\]$.Then $$\[{E_{a,k}}(N) \ll N\exp \{ - C{(\log N)^{1 - \frac{1}{{k + 1}}}}\} \]$$ where the implied constant depends on a and K.     

SECTIONAL CURVATURE OF KAEHLER SUBMANIFOLDS OF A COMPLEX PROJCETIVE SPACE

Let M~n (n≥2) be a complex Kaehler submanifold immersed in the complex projective space CP~m(1). Let K be the sectional curvature of M~n. Then K≥1/8 if and only if M~n is an imbedding submanifold congruent to the standard imbedding CP~n (1) or CP~n(1/2).     

MULTIPLIERS OF SEGAL ALGEBRAS A_p(G)

Let G be a locally compact but non-compact abelian group,It is proved thatM(A_p(G),L_1(G))=M(G)and M(A_p(G),L_1(G)∩C_0(G))=M(L_1(G),L_1(G)∩C_0(G)).If G is discrete,then M(A_p(G),L_1(G))=A_p(G),M(A_p,(G),L_1(G)∩C_0(G))=A_p(G).     

SEGAL ALGEBRA $\[{A_{1,p}}(G)\]$ AND ITS MULTIPLIERS

Let G be a locally compactb abelian group and $\[{A_p}(G)\]$ the p-Fourier algeba of Herz. This pepar studies the space $\[{A_{1,p}}(G) = {L_1}(G) \cap {A_p}(G)\]$ with convolution product. It is proved that $\[{A_{1,p}}(G)\]$ is a character Segal algebra. Moreover, for the multipliers of $\[{A_{1,p}}(G)\]$ the author proves that $\[M({A_{1,p}}(G),{L_1}(G)) = M(G)\]$ and $\[M({A_{1,p}}(G),{A_{1,p}}(G)) = M(G)\]$ provided G is noncompact. If G is discrete, then $\[M({A_{1,p}}(G),{L_1}(G)) = {A_{1,p}}(G)\$ and $\[M({A_{1,p}}(G),{A_{1,p}}(G)) = {A_{1,p}}(G)\]$     

ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION FOR PARAMETERS OF TWO-SIDED TRUNCATION DISTRIBUTION FAMILIES

Consider the two-sided truncation distrbution families written in the formf(x,θ)dx=w(θ_1, θ_2)h(x)I_([θ_1,θ_2])(x)dx, where θ=(θ_1,θ_2).T(x)=(t_1(x), t_2(x))=(min(x_1,…,x_m), max(x_1, …,x_m))is a sufficient statistic and its marginal density is denoted by f(t)dμ~T. The prior distribution of θ belongs to the familyF={G:∫‖θ‖~2dG(θ)<∞}.In this paper, the author constructs the empirical Bayes estimator (EBE) of θ, φ_n (t), by using the kernel estimation of f(t). Under a quite general assumption imposed upon f(t) and h(x), it is shown that φ_n(t) is an asymptotically optimal EBE of θ.     

DEGREE OF COPOSITIVE POLYNOMIAL APPROXIMATION

Denoteby _n(f) the degree of copositive approximation to f(x) by polynomials of degree≤n. For function f(x) ∈ C~k[-1, 1] which alternates in sign finitely many timesin [-1, 1], the author obtains the following Jackson type estimates_n(f)≤Cn~(-k)w(f~(k), 1/n)foa any positive integer k.     

THE AUTOMORPHISMS OF NON—DEFECTIVE ORTHOGONAL GROUPS $\[{\Omega _S}(V)\]$ AND $\[O_s^''\]$ IN CHARACTERISTIC 2

Let $V$ be a non-defective S-dimensional quadratic space over a field $F$ of characteristic 2, $\[F \ne {F_2}\]$. We prove that if there is an exceptional automorphism of either $\[{\Omega _S}(V)\]$ or $\[O_S^''(V)\]$ then $\[{V^\alpha }\]$ has a Cayley algebra structure for some $\[\alpha \]$ in F. Moreover, every exceptional automorphism of $\[O_S^''(V)\]$ has exactly one of the following forms: $$\[{\varphi _1} \circ {\Phi _g}or{\varphi _2} \circ {\Phi _g}\]$$ where $\[{\Phi _g}\]$ is an automorphism of $\[O_S^''(V)\]$ given by conjugation by a semilinear automorphism of V which preserves the quadratic structure, and $\[{\varphi _1}\]$ and $\[{\varphi _2}\]$ are the automorphisms induced by triality principle. Every exceptional automorphism of $\[{\Omega _S}(V)\]$ is the restriction of a unique exceptional automorpliism of $\[O_S^''(V)\]$.     

ON A CONJECTURE OF F. NEVANLINNA CONCERNING DEFICIENT FUNCTION (II)

The paper proves on the basis of [1] the following theorem: Let $\[f(z)\]$ be an entire function of lower order $\[\mu < \infty \]$, and $\[{a_i}(z)(l = 1,2, \cdots ,k)\]$ be meromorphic functions which satisfy $\[T(r,{a_i}(z)) = o\{ T(r,f)\} \]$. If $$\[\sum\limits_{i = 1}^k {\delta ({a_i}(z),f) = 1\begin{array}{*{20}{c}} {({a_i}(z) \ne \infty )}&{(1)} \end{array}} \]$$ then the deficiencies $\[\delta ({a_i}(z),f)\]$ are equal to $\[\frac{{{n_1}}}{\mu }\]$, where $\[{n_i}\]$ is an integer,$\[l = 1,2, \cdots ,k\]$.     

INVERSE THEOREMS IN Lp FOR SOME MULTIDIMENSIONAL POSITIVE LINEAR OPERATORS

Let {L_n}_(n∈s) be positive linear operators in L_(I), I=[0, 1] or [0, ∞). This paper considers their variants in L_p(I×I)L_(n,m)(F; x, y)=L_n(L_m(F(u, v); y); x)=L_m(L_n(F(u, v); x); y), n, m∈N.The characterization problem for these operators is solved which gives the inverse theorems in L_p for multidimensional Bernstein type operators.     

A SORT OF POLYNOMIAL IDENTITIES OF $\[{M_n}(F)\]$ WITH CHAR $\[F \ne 0\]$

Let $F$ denote a field, finite or infinite, with characteristic $\[p \ne 0\]$. In this paper, the author obtains the following result: The symmetric polynomial on $t$ letters $$\[{S_{sym(t)}}({x_1},{x_2}, \cdots ,{x_t}) = \sum\limits_{x \in sym(t)} {{X_{\pi 1}}{X_{\pi 2}} \cdots {X_{\pi t}}} \]$$ is a polynomial identity of $\[{M_n}(F)\]$ when $\[t \ge pn\]$, and this is sharp in the sense that if $\[t \le pn - 1\]$,it is not a polynomial identity of $\[{M_n}(F)\]$.     

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.     

COMPLETELY POSITIVE MAPS AND $*$-OMORPHISM OF $\[{C^*}\]$-ALGEBRAS

Let $A$, $B$ be unital $\[{C^*}\]$-algebras. $\[{\chi _A} = \{ \varphi |\varphi \]$ are all completely postive linear maps from $\[{M_n}(C)\]$ to $A$ with $\[\left\| {a(\varphi )} \right\| \le 1\]$ $}$. $\[(a(\varphi ) = \left( {\begin{array}{*{20}{c}} {\varphi ({e_{11}})}& \cdots &{\varphi ({e_{1n}})}\{}& \cdots &{}\{\varphi ({e_{n1}})}& \cdots &{\varphi ({e_{nn}})} \end{array}} \right),\]$ where $\[\{ {e_{ij}}\} \]$ is the matrix unit of $\[{M_n}(C)\]$. Let $\[\alpha \]$ be the natural action of $\[SU(n)\]$ on $\[{M_n}(C)\]$ For $\[n \ge 3\]$, if $\[\Phi \]$ is an $\[\alpha \]$-invariant affine isomorphism between $\[{\chi _A}\]$ and $\[{\chi _B}\]$, $\[\Phi (0) = 0\]$, then $A$ and $B$ are $\[^*\]$-isomorphic In this paper a counter example is given for the case $\[n = 2\]$.     

COMPACT HYPERSURFACES IN ELLIPSOLD WITH PRESCRIBED MEAN CURVATURE

$S^{n}(1)$ and $\[{S^1}(1) \times {S^1}(1)\]$ can be embedded into special ellipscid with suitable mean eurvature function.     

UNIFORM STRONG CONVERGENCE RATE OF NEAREST NEIGHBOR DENSITY ESTIMATION

Based on [3] and [4],the authors study strong convergence rate of the k_n-NNdensity estimate f_n(x)of the population density f(x),proposed in [1].f(x)>0 and fsatisfies λ-condition at x(0<λ≤2),then for properly chosen k_nlim sup(n/(logn)~(λ/(1 2λ))丨_n(x)-f(x)丨C a.s.If f satisfies λ-condition,then for propeoly chosen k_nlim sup(n/(logn)~(λ/(1 3λ)丨_n(x)-f(x)丨C a.s.,where C is a constant.An order to which the convergence rate of 丨_n(x)-f(x)丨andsup 丨_n(x)-f(x)丨 cannot reach is also proposed.     

EXISTENCE AND NON-EXISTENCE OF GLOBAL SMOOTH SOLUTIONS FOR QUASILINEAR HYPERBOLIC SYSTEMS~*

Consider initial value probiom v_t-u_x=0, u_t+p(v)_x=0, (E), v(x, 0)=v_0(x), u(x, 0)=u_0(x), (I), where A≥0, p(v)=K~2v~(-γ), K>0, 0<γ<3. As 0<γ≤1, the authors give a sufficient condition for that (E), (I) to have a unique global smooth solution, As 1≤γ<3, a necessary condition is given for that.     

QUOTIENTS OF STRONGLY S-DECOMPOSABLE OPERATORS

This paper shows that if T∈C(X)is strongly S-decomposable,where S is a closedsubset of σ_(T)and S≠C_∞,then for any(e)spectral maximal subspace Y of T,T~Y is aclosed strongly S∩σ_(T~Y)-decomposable operator.     

ESTIMATE OF d_0/d~* FOR STARLIKE FUNCTIONS

Let S~* be the class of functionsf(z)analytic,univalent in the unit disk|z|<1 andmap|z|<1 onto a region which is starlike with respect to w=0 and is denoted as D_f.Letr_0=r_0(f)be the radius of convexity of f(2).In this note,the author proves the following result:(d_0/d~*)≥0.4101492,where d_0= f(z),d~*=|β|.     

THE PERTURBATION OF ALMOST PERIODIC SOLUTION OF ALMOST PERIODIC SYSTEM

By using the exponential dichotomy and the averaging method,a perturbation theoryis established for the almost periodic solutions of an almost differential system.Suppose that the almost periodic differential system(dx)/(dt)=f(x,t) ε~2g(x,t,ε)(1)has an almost periodic solution x=x_0(t,M)for ε=0,where M=(m_1,…,m_k)is theparameter vector.The author discusses the conditions under which(1)has an almostperiodic solution x=x(t,ε)such that x(t,ε)=x_0(t,M)holds uniformly.The results obtained are quite complete.     

AN NONEXISTENCE THEOREM FOR HARMONIC MAPS WITH SLOWLY DIVERGENT ENERGY

It is proved that a harmonic map or a relative harmonic map from Euclidean space $\[{R^n}(n \ne 2)\]$ into an m-dimensional Riemannian manifold $\[{M_m}\]$ with finite energy or slowly divergent energy must be a constant map. Some physical applications are also presented.