Let R be a closed rectangle of sides 1 and 2 in the plane,S be a finite set of points in R,f(S) denote the minimum distance between pairs of points of S,and fR(n) = . Inthis paper,the exact values of fR(n) for 2≤n≤6 and the corresponding configurations aregiven.

An analogue of the multi wavelet bases which lie in the space of functions defined on a discrete set of points {x 1,...,x n}×{y 1,...,y n}R 2 is taken into account.

Let G be a double-outer plane graph, V(G), E(G), F(G) be the set of points, edge and face, a plane graph G is k-total colorable, if the elements of V(G) E(G) can be colored with K-colors, any two distinct adjacent or incident elements receive different colors.

For any ergodic measure,we considered the set of points which satisfy the Brin-Katok theorem and Birkhoff ergodic theorem with respect to this measure.

In this paper, the writes proves that: Let X be a regular space satisfying axiom A and Y be a topological space, for any y∈Y, is a nonempty subset in X and multivalued mapping F is upper semicontinuous on Y, then the set of points where F is lower semicontinuous is a residual set in Y.

Based on binary image made of image segmentation pretreatment, the recursion marking was assigned to the corresponding target. The algorithm avoids pixel "1" being rescanned in detection of the connected set of points, which realized the requirements of only considering abstracting the target characteristics without obtaining final marked image itself.

Closed orbits are described and a set of points of dense orbit is explicitly given.

In particular, the results in this article show that the oscillations of a function at large scale are comparable to the oscillations of its samples on an appropriate discrete set of points.

In the application of CAD/CAM, the target form of a curve, which is used for plotting or as the data supplied for CAM, is a set of points on (or near by) the curve.

The author proves that the set of points where the Chung type LIL fails for the path of the infinite series of independent Ornstein-Uhlenbeck processes is a random fractal, and eval-uates its Hausdorff dimension.

At each iterate, by reduplication, crossover and mutation, a finite set of points can be used.

Consider the differential equation =f(t,x) (1)where x and f(t,x) are n-vectors,f(t, x)∈C[I×R~n, R~n], I denotes thekalf line:0≤t<∞, and R~n denotes n-Euclidean space. Let g(t,x) be a h-vector (h≤n) and g(t, x)∈C[I×R~n, R~n]. The set of points (t,x)∈I×R~n satisfying the relation g(t, x)=0 defines an (n-h+1)-manifold St, Suppose thet (t_0, x_0)∈S_(t0) implies (t,x(t; t_0,x_0)∈S_t for t≥t_0, than S_t is a posititively invariant manifold of equation (1). In this paper we consider some new types of stability...

Consider the differential equation =f(t,x) (1)where x and f(t,x) are n-vectors,f(t, x)∈C[I×R~n, R~n], I denotes thekalf line:0≤t<∞, and R~n denotes n-Euclidean space. Let g(t,x) be a h-vector (h≤n) and g(t, x)∈C[I×R~n, R~n]. The set of points (t,x)∈I×R~n satisfying the relation g(t, x)=0 defines an (n-h+1)-manifold St, Suppose thet (t_0, x_0)∈S_(t0) implies (t,x(t; t_0,x_0)∈S_t for t≥t_0, than S_t is a posititively invariant manifold of equation (1). In this paper we consider some new types of stability and boundedness of solutions of (1) with respect to S_t, and obtain some theorems more general than that of [3]、[5]、[7].

In paper(1),a class of geometric inequalities have been constructed on the bounded set of points,and some relations of the invariants for the k-ciimcnsional simplex in the Euclidean space have been given by the algebric methods. The well-known Neuberg-Pcodc inequality concerning two triangles has been extended to any two simplices in the n-dimensional Euclidean space in(2). This paper obtains two inequalities ( Theorem 1 and Theorem 2) concerning any two simplices from another viewpoint. As the special...

In paper(1),a class of geometric inequalities have been constructed on the bounded set of points,and some relations of the invariants for the k-ciimcnsional simplex in the Euclidean space have been given by the algebric methods. The well-known Neuberg-Pcodc inequality concerning two triangles has been extended to any two simplices in the n-dimensional Euclidean space in(2). This paper obtains two inequalities ( Theorem 1 and Theorem 2) concerning any two simplices from another viewpoint. As the special cases of these two inequalities,we may deduce some other relations among the invariants,which differ from that in (1)

Given a set of points in plane:P_i(x_(?),y_i) (i=0, 1, …, n;X_0points P_(i-1), P_i, P_(i+1)~- lie on a minor arc of a circle and any four consecutive points P_(i-1), P_i, P_(i+1) P_(i+2) are not concyclic. A curve is said to be a fine shape curve of the given set of points {P_i}, if it is a curve of class C2 passing through these points, the number v of its Points of inflection is minimum, the number μ of its points of extreme...

Given a set of points in plane:P_i(x_(?),y_i) (i=0, 1, …, n;X_0points P_(i-1), P_i, P_(i+1)~- lie on a minor arc of a circle and any four consecutive points P_(i-1), P_i, P_(i+1) P_(i+2) are not concyclic. A curve is said to be a fine shape curve of the given set of points {P_i}, if it is a curve of class C2 passing through these points, the number v of its Points of inflection is minimum, the number μ of its points of extreme curvature is minimum and it satisfies certain additional conditions which will fix the intervals to which certain points of inflection or points of extreme curvature will belong.How can we interpolate points into the given set of points so that every fine shape curve of the new set of points will always be a fine shape curve of the original set of points? The present paper gives a solution to the case μ =v= 0 and indicates how to get an heuristic algorithm in general to make the new set of points to have a fine shape curve with v' points of inflection and with μ' points of extreme curvature so that v' =v and the difference μ'-μ is small. At the end of this paper some remarks are made to account for why this problem was raised and how it is related to the method of sequence of cicular rates which is applied to the Hudon Hull Construction System.