In this pioneering 1954 paper on selection, Bechhofer recognized the importance of considering experiments carried out in some design other than the completely randomized design. However, over the years, little has been done along these lines. This paper's goal is to extend selection procedures to other designs. We first briefly review the existing state of knowledge of selection in designed experiments, and then proceed to new results on complete factorial experiments. In particular, we formulate and study a selection procedure which is valid in the presence of unbounded interaction, yet adapts to the presence/absence of interaction.

G.Shimura introduced a certain kind of differential operators when he studied the arithmetic properties of Eisenstein series of several variables. The invariable differential operators of the classical domain of types Ⅰ and Ⅱ are included in it as its special cases. In this paper the eigenvalues of this kind of differential operators with respect to the eigenfunction (det i((^tZ-Z))² are calculated.

The main purpose of the present paper is to describe the Taylor joint spectra for n-tuples of double commuting hyponormal operators, and to study the representation of the joint spectra in terms of that of n-tuples of commuting normal operators.

In 1965 the author introduced the notion of Chern classes for an algebraic variety with arbitrary singularities. Based on this definition the well-known Miyaoka-Yao inequalities have been proved and extended by quite simple direct computations.

This paper is concerned with the global existence of the second initial boundary value problem for fully nonlinear parabolic equations. It is proved that when the initial data is sufficiently small, the problem admits a unique global solution. Moreover, as t goes to +∞, the solution exponentially decays to zero.

Let R be an arbitrary commutative ring, and n be an integer ≥3. It is proved for any ideal J of R that
ESp_2n(R,J)=[ESp_2n(R), ESp_2n(J)]=[ESp_2n(R), ESp_2n(R,J)] =[ESp_2n(R), GSp_2n(R,J)]=[Sp_2n(R), ESp_2n(R,J)].
Furthermore, the problem of normal subgroups of Sp_2n(R) has an affirmative solution if and only if aR=a²R+2aR for each a in R. This covers the relevant results of [4], [8], [10], [12] and [13].

In this paper we study a sub-class of compact Ricci flat manifolds from certain standard constructions.

The delay differential equations of the form
x'(t)=-a(t)x(t-1),t≥0 are considered, where a(t)≥0
is locally integrable on [0,∞). The main result: Let 0<c(t)≤a(t)≤k(t) for large t ∫^∞, and c(t)≤Mc(t') for t, t'≥T,|t-t'|≤l with some constants l>0,M>1,T≥0. Then the condition with some constant α>0 dependent on l, M, ensures that all solutions of (*) tend to zero as t→∞.

Le A_m ,B_m ,m=1,...,p, be linear operators on an n-dimensional unitary space V.L is a linear operator on ⊕²V, the tensor product space with the customarily induced inner product.

In this paper we will introduce the concept of canonical reducing set of a surface homeomorphism, and prove that it is unique up to an isotopy. As an application, we will give a simple proof of Thurston's theorem on classifying mappings on non-orientable surface, using the techniques of quasiconformal mappings and some known results in the orientable case, especially the Thurston theorem on orientable surfaces.

We prove that for monotone function f(x)∈L_p [-1, 1], 1≤p<∞, there exists a monotone algebraic polynomial L_n (f,x) of degree n such that , where and ω_2,Ψ, is the second-order modulus of smoothness which weighs differently the behavior of f(x) in the middle of the interval and near the endpoints. This estimate improves the known result of Shvedov.

The non-linear differential eguation

is studied. It is the main aim of this paper to show the existence of bifurcation of saddle connection type; and to show the creation of limit cycles under certain conditions of the parameters, together with their biological significance.

Let Q_k (p) be a set consisting of all polynomials of degree k with integral coefficients
f(x)=a_kx^k+...+a₁x,
where p×a_k.

Existence of 2π-periodic solutions to the equation is proved, under sharp nonresonance conditions on the interaction, of g(x)/x with two consecutive eigenvalues m² and(m+1)²: touch with the eigenvalues is allowed.

This article gives a direct proof of the local index theorem for the signature operators.

In paper[1] (On the stability of a saddle-point separatrix loop and analytical criterion for its bifurcation limit cycles Acta Mathematica Sinica Vol. 28. No. 1, 55-70, Bejing China 1985), we considered the problem of generating limit cycles by the bifurcation of a stable or an unstable loop of a saddle-point separatrix. We gave for the first time a criterion for the stability of the loop as following:L₀ is stable (unstable) if wherex=-(t),y=-(t) then a sufficient condition for the bifurcation which generates limit cycles. This paper generalizes the result of[1] to the case where the loop contains a center or the loop tends to an infinite saddle-point, and removes the restriction that the saddle-point should be an elementary singular point. Applying the results of this paper, the author studies a two-parameter system

The results obtained by the author in this in real field coincides with the results given by Prof. Qin Yuanxun by means of the complex qualitative theory in complex field.