The convex hull of the vertices of a simplex and a point lying on the hyperplane spanned by the simplex is called a flat cone on the simplex. This paper proposes a natural way to simplicially triangulate flat cones on simplices and develops its applications to PL homotopy methods.

Let L be a self-adjoint linear operator and N be the gradient of a functional ψ, which is almost quadratic. We first give a theorem on the surjectivity of the operator L+N, then apply the result to semilinear elliptic systems:
We get the existence of weak solution in space L²(Ω) for any f₁(x) in L²(Ω), when the function g has the behavior at infinity like au²+bv²+c‖u‖^1/2‖v‖^1/2, etc.

It was proved in [5] that every finite collapsible polyhedron admits of an injective metric. In this paper, by using some new techniques, we extend the conclusions in [5] to the infinite case.

This paper deals with the exponential L²-convergence for jump processes. We introduce some reduction methods and improve some previous results. Then we prove that for birth-death processes, exponential L²-convergence coincides indeed with exponential ergodicity which is widely studied in the Markov chain theory.

In this paper, I study the microlocal hypoellipticity for a class of totally characteristic operators (1.1). My main result is as follows: Under the conditions (Ⅰ), (Ⅱ), if the indicial operator of (1.1) is microlocally hypoelliptic in the complement of WF_x(Pu(t,·)) for any u(t,x)∈C_b^∞ ([0,T], g), t∈[0, T], λ∈Z₊, then the operator (1.1) is microlocally hypoelliptic in the variable x.

In 1985 I. Rival and N. Zaguia conjectured that in a chain complete poset having the finite cutset property every element belongs to an antichain cutset iff there is no alternating-cover cycle. In this paper a counterexample is given to show that the condition that there is no alternating-cover cycle is not sufficient and other two conditions on generalized alternating-cover paths are added to complete the characterization.

By using the classification theorem of finite simple groups, we have shown that "If G is a finite group, H is a coprime operator group of G, C_G(H)≤S(G), then G is solvable." As a direct corollary, we have completely proved the long-standing conjecture on fixed-point-free automorphism group.

In this paper we discuss an amalgam (M₁₁, SL₂ (5)) of rank 2 and characteristic 3. Suppose G is a group generated by its two subgroups P₁, P₂, and suppose P₁, P₂ and their intersection satisfy some local group-theoretic conditions. Then, assuming that O³ (P₁/O₃(P₁)) and O^3′ (P₂/O₃(P₂)) are isomorphic to M₁₁ and SL₂ (5) respectively, we give an explicit description about the structures of O₃ (P₁) and O₃ (P₂) and of the operations of P_i's on them. At this time, they have structures similar to those of some 3-local subgroups of the sporadic simple group Ly.

In this paper we discuss the time dependence of solutions of the equation (2.3). From it we will get the global qualitative analysis of the equation.

Let M be a manifold satisfying certain conditions which are weaker than those of E. Thomas^[12], and f:M→N be a map with codimension one or two. We give necessary and sufficient conditions for f to be homotopic to a map with maximal rank. As an application, we completely determine the codimension one or two immersions of Dold manifolds in real projective spaces.

In this paper the Pontrjagin forms of a closed oriented manifold immersed in Rⁿ are expressed in terms of Plücker coordinates.

The main purpose of this paper is to give a sharper asymptotic formula for the mean square value , where L′(s, χ) denotes the derivative of L (s, χ) with respect to s.

Let f(z) be a transcendental meromorphic function in the plane. Hayman conjectured thatf f' assumes all finite values except possibly zero infinitely many times. In this paper, we solve this conjecture partly.

In the present note the algebraic independence of certain continued fractions is proved. Especially, we prove that the Böhmer-Mahler's series are algebraically independent, where , are some irrational numbers and g₁,…,g_s are distinct positive integers.

In this paper, we shall prove that for a sufficiently large odd number N, the equation has solutions.

We give a new characterization of the paratingent cone in terms of contingent cones, i.e., the paratingent cone to any open set at a boundary point is the upper limit of the contingent cones at the neighboring points. We use this result to characterize the strict differentiability in terms of the contingent directional derivatives. We also define aP-subderivative for continuous functions and develop a subdifferential calculus with applications to optimality conditions in mathematical programming.

We give a new characterization of the paratingent cone in terms of contingent cones, i.e., the paratingent cone to any open set at a boundary point is the upper limit of the contingent cones at the neighboring points. We use this result to characterize the strict differentiability in terms of the contingent directional derivatives. We also define a P-subderivative for continuous functions and develop a subdifferential calculus with applications to optimality conditions in mathematical programming.

This paper studies the optimal controls of stochastic systems of functional type with end constraints. The systems considered may be degenerate and the control region may be nonconvex. A stochastic maximum principle is derived. The method is based on the idea that stochastic systems are essentially infinite dimensional systems.

In this paper, we prove that every strongly continuous semigroup of bounded operators on a Hilbert space may be extended to a strongly continous semigroup of normal operators on a larger Krein space. Several equivalent formulations for the case where the extension space is a Pontrjagin space are given.

Under the constrained condition induced by the eigenfunction expression of the reflectionless potentialu=-2<q,q>≡f(q), the Schr?dinger equation, the spatial part of the Lax pair of theKdV equation, is nonlinearized to be a completely integrable system (R^2N ,dpΛdq, H=F₀):
,
while the nonlinearization of the time part leads to itsN-involutive system {F _m}. The involutive solutions of the compatible system (F₀), (F_m) are mapped byf into the solutions of the m-thKdV equation.

Let X(t) be a symmetric conservative Hunt process. In this paper, we decompose the functional u(X(t), t) into a sum of a square integrable martingale and a regular 0-quadratic variation process.

In this paper we give a condition under which an unstable polynomial A_p -algebra can be obtained from the modular polynomial algebra. From this we car deduce both the Aguades theorem^[2] and Clark's theorem^[4].

H. Crapo raised in 1982 the following problem: If P is a complete lattice, is Retr (P) a complete lattice? here Retr (P) denotes the set of all retraction operators on P with the pointwise order, that is,f≤g in Retr (P) iff f(x)≤g(x) for every x in P. An affirmative answer will be given in the present paper.

Let (M, s,d, x) be a σ-finite measure space with a σ-field s countably generated. We call a linear map T uniformly contractive if which maps measurable functions on M to measurable functions and
‖Tf‖p≤‖f‖p,for p,1≤p≤+∞.
If a linear map T which maps measurable functions on M to measurable functions has positivity property, namely,Tf≧0 forf≧0, we call it a submarkovian operator. In this article we prove.

The well-known Goldback-Vinogradov theorem states that every large odd integer is a sum of three primes. In the present paper it is further proved that every large odd integer N can be represented as

wher c>0 is an absolute constant, and p_i(1≦i≦3) are primes. The result is obtained by means of Hardy-Littlewood circle method, Heath-Brown's identity and power moments of Dirichlet L-functions over short intervals.

In this paper we will study some existence theorems of critical points for periodic functional. As applications, we consider the multiple solutions of conservative pendulum-like equation with periodic boundary conditions and forcing term of mean value zero.

Let q, k be positive integers, (q, k)=1, and P(k, q) be the smallest prime number satisfying p≡k (mod q) In this paper, we prove that forc q sufficiently large, one has P(k,q)<<q⁸.

In this paper,we consider the convergence and saturation problems of the following discrete type interpolation operators:

where f is any bounded function on the real axis R,while U(x) is an appropriate entire function of exponential type 1 and U_σ(x)=:U(σx),we obtain some resuls similar to those in the trigonometric polynomial interpolation operators’ case^[4].As a special case,our results include Dryanov's results in [5].

This paper is devoted to studying the problem of optimal recovery for Sobolev function classes W₂^r(R) in L²(R) by using the information of function values on denumerable points.For r≥1,we determine that the optimal sampling points are arranged equidistantly in a suitable collection of sets of sampling points and find two kinds of cardinal spline interpolation as optimal algorithms.

In this paper,by the method used in [7,9,10],we construct the heat kernels (i.e.the fundamental solutions of the heat equation) of the following spaces: the quaternion hyperbolic space Sp (1,n)/Sp(1)×Sp (n),the quaternion classical domain of type Ⅲ Sp (n,C)/Sp (n),the complex Grassmannian manifold U (m+n)/U(m)×U(n).

In this paper,we prove that on a large class of bounded contractible and noncontractible domains with piecewise smooth boundary,there is at least one positive solution to Δu+a(x)u+u^(N+2)/(N-2)=0 (N≥3) with zero Dirichlet boundary condition.

The present paper discusses,from a viewpoint of homology,the nature of systems (ψ,α,σ) which induce exceptional isomorphisms Φ_α^σ of the three-dimensional linear groups over commutative rings.

In this paper,a characterization of almost periodic strongly continuous Sine operator function is given,and in a Hilbert space,it is shown that the almost periodicity of a Sine operator function implies that of the corresponding Cosine operator function.

An unbounded solution of a Duffing's equation with a smooth superlinear nonlinearity is constructed.

In this paper we introduce the concept of quasinormal and subnormal operators on a Krein space and prove that every quasinormal operator is subnormal.And some conditions for an operator on a Hilbert space to be a subnormal operator in the Krein space sense are obtained.

Applying the generalized maximum principle,a Liouvilletype theorem of subharmonic functions on complete Riemannian manifolds is shown and a Liouville-type differential inequality on properly immersed complete submanifolds is given.