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).     

ON THE TAYLOR'S JOINT SPECTRUM OF 2n-TUPLE (L_A, R_B)

Let A= (A_1, …, A_n) and B=(B_1, …, B_n) be double commuting n-tuples of operators on Hilbert space H and let L_(A_i), and R_(B_j), decode the left and right multiplications induced by A_i and B_j, respectively. The following results are proven: Sp (L_A, R_B)=Sp(A)×Sp(B), Sp_e(L_A, R_B)=Sp_e(A)×Sp(B) ∪ Sp(A)×Sp_e(B).     

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 $\[n = 2\]$.     

ON ADMISSffilLITY OF VARIANCE COMPONENTS ESTIMATES

Suppose that there is a variance components model $$\[\left\{ {\begin{array}{*{20}{c}} {E\mathop Y\limits_{n \times 1} = \mathop X\limits_{n \times p} \mathop \beta \limits_{p \times 1} }\{DY = \sigma _2^2{V_1} + \sigma _2^2{V_2}} \end{array}} \right.\]$$ where $\[\beta \]$,$\[\sigma _1^2\]$ and $\[\sigma _2^2\]$ are all unknown, $\[X,V > 0\]$ and $\[{V_2} > 0\]$ are all known, $\[r(X) < n\]$. The author estimates simultaneously $\[(\sigma _1^2,\sigma _2^2)\]$. Estimators are restricted to the class $\[D = \{ d({A_1}{A_2}) = ({Y^''}{A_1}Y,{Y^''}{A_2}Y),{A_1} \ge 0,{A_2} \ge 0\} \]$. Suppose that the loss function is $\[L(d({A_1},{A_2}),(\sigma _1^2,\sigma _2^2)) = \frac{1}{{\sigma _1^4}}({Y^''}{A_1}Y - \sigma _1^2) + \frac{1}{{\sigma _2^4}}{({Y^''}{A_2}Y - \sigma _2^2)^2}\]$. This paper gives a necessary and sufficient condition for $\[d({A_1},{A_2})\]$ to be an equivariant D-asmissible estimator under the restriction $\[{V_1} = {V_2}\]$, and a sufficient condition and a necessary condition for $\[d({A_1},{A_2})\]$ to equivariant D-asmissible without the restriction.     

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|.     

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.     

ON THE DIOPHANUNE EQUATION $\[\sum\limits_{i = 0}^N {\frac{{{x_i}}}{{{d_i}}}} \equiv 0\]$ (mod 1) AND ITS APPLICATIONS

The number $\[A({d_1}, \cdots ,{d_n})\]$ of solutions of the equation $$\[\sum\limits_{i = 0}^n {\frac{{{x_i}}}{{{d_i}}}} \equiv 0(\bmod 1),0 < {x_i} < {d_i}(i = 1,2, \cdots ,n)\]$$ where all the $\[{d_i}s\]$ are positive integers, is of significance in the estimation of the number $\[N({d_1}, \cdots {d_n})\]$ of solutiohs in a finite field $\[{F_q}\]$ of the equation $$\[\sum\limits_{i = 1}^n {{a_i}x_i^{{d_i}}} = 0,{x_i} \in {F_q}(i = 1,2, \cdots ,n)\]$$ where all the $\[a_i^''s\]$ belong to $\[F_q^*\]$. the multiplication group of $\[F_q^{[1,2]}\]$. In this paper, applying the inclusion-exclusion principle, a greneral formula to compute $\[A({d_1}, \cdots ,{d_n})\]$ is obtained. For some special cases more convenient formulas for $\[A({d_1}, \cdots ,{d_n})\]$ are also given, for example, if $\[{d_i}|{d_{i + 1}},i = 1, \cdots ,n - 1\]$, then $$\[A({d_1}, \cdots ,{d_n}) = ({d_{n - 1}} - 1) \cdots ({d_1} - 1) - ({d_{n - 2}} - 1) \cdots ({d_1} - 1) + \cdots + {( - 1)^n}({d_2} - 1)({d_1} - 1) + {( - 1)^n}({d_1} - 1).\]$$     

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.     

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.     

THE OPTIMAL RATE OF CONVERGENCE OF ERROR FOR kNN MEDIAN REGRESSION ESTIMATES

Let(X,Y),(X_1,Y_1),…,(X_n,Y_n)be iid.random vectors,where Y is one-dimensional.It is desired to estimate the conditional median(X)of Y,by use of Z_n={(X_i,Y_i),i=1,…,n}and X.Denote by(X,Z_n)the kNN estimate of(X),and putH_(nk)(Z_n)=E{|(X,Z_n)-(X)||Z_n},the conditional mean absolute error.This articalestablishes the optimal convergence rate of P(H_(nk_n)(Z_n)>ε),under fairly generalassumptions on(X,Y)and k_n,which tends to ∞ in some suitable way.     

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.     

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 JOINT SPECTRUM FOR N-TUPLE OF HYPONORMAL OPERATORS

Let A=(A_1,…,A,)be an n-tuple of double commuting hyponormal operators.It is-proved that:1.The joint spectrum of A has a Cartesian decomposition:Re[Sp(A)]=S_p(ReA),Im[Sp(A)]=Sp(ImA);2.The.joint resolvent of A satisfies the growth condition:‖()‖=1/(dist(z,Sp(A)));3.If 0σ(A_i),i=1,2,…,n,then‖A‖=γ_(sp)(A).     

TIME OPTIMAL BOUNDARY CONTROL FOR SYSTEMS GOVERNED BY PARABOLIC EQUATIONS

In this paper we consider the systems governed, by parabolioc equations \[\frac{{\partial y}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}} ({a_{ij}}(x,t)\frac{{\partial y}}{{\partial {x_j}}}) - ay + f(x,t)\] subject to the boundary control \[\frac{{\partial y}}{{\partial {\nu _A}}}{|_\sum } = u(x,t)\] with the initial condition \[y(x,0) = {y_0}(x)\] We suppose that U is a compact set but may not be convex in \[{H^{ - \frac{1}{2}}}(\Gamma )\], Given \[{y_1}( \cdot ) \in {L^2}(\Omega )\] and d>0, the time optimal control problem requiers to find the control \[u( \cdot ,t) \in U\] for steering the initial state {y_0}( \cdot )\] the final state \[\left\| {{y_1}( \cdot ) - y( \cdot ,t)} \right\| \le d\] in a minimum, time. The following maximum principle is proved: Theorem. If \[{u^*}(x,t)\] is the optimal control and \[{t^*}\] the optimal time, then there is a solution to the equation \[\left\{ {\begin{array}{*{20}{c}} { - \frac{{\partial p}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}({a_{ji}}(x,t)\frac{{\partial p}}{{\partial {x_j}}}) - \alpha p,} }\{\frac{{\partial p}}{{\partial {\nu _{{A^'}}}}}{|_\sum } = 0} \end{array}} \right.\] with the final condition \[p(x,{t^*}) = {y^*}(x,{t^*}) - {y_1}(x)\], such that \[\int_\Gamma {p(x,t){u^*}} (x,t)d\Gamma = \mathop {\max }\limits_{u( \cdot ) \in U} \int_\Gamma {p(x,t)u(x)d\Gamma } \]     

ON TFE THIRD CONJECTURE OF K.OGIUE

In this paper, the authors prove following result:Let M~n be a complete Bechner-Kaehler submanifold of complex dimension (n≥4) in a complex projective space CP~(n p)(1) of complex dimension n p, endowed with the FubiniStudy metric of constant holomorphic sectional curvature 1. If the sectional curvature K of M~n satisfies K<1, then codimension p of M~n is not less then n(n 1)/2.     

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.     

INITIAL VALUE PROBLEMS FOR NONLINEAR DEGENERATE SYSTEMS OF FILTRATION TYPE

In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type $\[{u_t} = (grad\varphi (u))\]$ are studied, where $\[u = ({u_1}, \cdots ,{u_N})\]$ is an N-dimensional vector valued function, $\[\varphi (u)\]$ is a strict convex function of vector variable $\[u\]$, and its matrix of derivatives of second order is zero-definite at $\[u = 0\]$. This system is degenerate. The definition of the generalized solution of the problem: $\[u(x,t) \in {L_\infty }((0,T);{L_2}(R)),\]$, grad $\[\varphi (u) \in {L_\infty }((0,T);W_2^{(1)}(R)),\]$ and it satisfies appropriate integral relation. The existence and uniqueness of the generalized solution of the problem are proved. When N=1, the system is the commonly so-called degenerate partial differential equation of filtration type.     

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).