Characteristic Numbers of Matrix Lie Algebras

A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.

Every Graph Embedded on the Surface with Euler Characteristic Numberε=-1 is Acyclically 11-choosable

A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G.A graph G is acyclically k-choosable if for any list assignment L={L(v):v∈V(G)}with|L(v)|≥k for each vertex v∈V(G),there exists an acyclic proper vertex coloringφof G such thatφ(v)∈L(v)for each vertex v∈V(G).In this paper,we prove that every graph G embedded on the surface with Euler characteristic numberε=-1 is acyclically 11-choosable.

Unified Characteristic Numbers and Solutions of Equations for Birth and Death Processes with Barriers

The state 0 of a birth and death process with state space E = {0, 1, 2,....} is a barrier which can be classified into four kinds： reflection, absorption, leaping reflection, quasi-leaping reflection. For the first, second and fourth barriers, the characteristic numbers of different forms have been introduced. In this paper unified characteristic numbers for birth and death processes with barriers were introduced, the related equations were solved and the solutions were expressed by unified characteristic numbers. This paper concerns work solving probability construction problem of birth and death processes with leaping reflection barrier and quasi-leaping reflection barrier.

Lyapunov Criteria for Structural Stability of Supply Chain System

In this paper, based on Cobb-Douglas production function, the structural stability of the supply chain system are analyzed by employing Lyapunov criteria. That the supply chain system structure, with the variance of the rate of re-production input funding, becomes unstable is proved. Noticeably, the solutions shows that when the optimal combination of input parameter element, the qualitative properties of supply chain system change and the supply chain system becomes unstable.

COBORDISM CLASSES OF FIBERINGS OVER PRODUCT OF PROJECTIVE SPACES

Let RP（k） denote the k-dimensional real projective space. This article determines which cobordism classes are represented by the total space of a fibering with prescribed base space RP（3）× RP（1）, RP（2） × RP（1）, RP（2）× RP（1）× RP（1） or RP（3）× RP（2）.

Formation and harmfulness of high fluoride-bearing ground-water

According to the distribution of high fluoride\|bearing groundwater in Liaoning Province, China, the cause of formation and hydrogeochemical characteristics as well as its relationship with human health and illness rate were discussed. Strategies to prevent and control fluoride pollution have also been outlined.

From a projective invariant to some new properties of algebraic hypersurfaces

Projective invariants are not only important objects in mathematics especially in geometry,but also widely used in many practical applications such as in computer vision and object recognition. In this work,we show a projective invariant named as characteristic number,from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of the hypersurface and some lines that constitute a closed loop. From this property,two high-dimensional generalizations of Pascal's theorem are given,one establishing the connection of hypersurfaces of distinct degrees,and the other concerned with the intersections of a hypersurface and a simplex.

On Singularity of Spline Space Over Morgan-Scott's Type Partition

Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.

Two Commuting Involutions Fixing RP_(1)(2m+1)∪RP_(2)(2m+1)

Let Z2 denote a cyclic group of 2 order and Z_(2)^(2)=Z_(2)×Z_(2)the direct product of groups.Suppose that(M,Φ)is a closed and smooth manifold M with a smooth Z_(2)^(2)-action whose fixed point set is the disjoint union of two real projective spaces with the same dimension.In this paper,the authors give a sufficient condition on the fixed data of the action for(M,Φ)bounding equivariantly.

Fiberings with Fiber RP(k)

Let κ be non-negative integer. The unoriented bordism classes, which can be represented as [RP（ξ^κ）] where ξ^κ is a k-plane bundle, form an ideal of the unoriented bordism ring MO.. A group of generators of this ideal expressed by a base of MO. and a necessary and sufficient condition for a bordism class to belong to this ideal are given.

ETA INVARIANTS, DIFFERENTIAL CHARACTERS AND FLAT VECTOR BUNDLES

The purpose of this paper is to give a refinement of the Atiyah-Singer families index theorem at the level of differential characters. Also a Riemann-Roch-Grothendieck theorem for the direct image of flat vector bundles by proper submersions is proved,with Chern classes with coefficients in C/Q. These results are much related to prior work of Gillet-Soule, Bismut-Lott and Lott.