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Chromatic sums of singular maps on some surface
Zhaoxiang Li,Yanpei Liu 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.15 No.-
A map is singular if each edge is on the same face on a surface (i.e., those have only one face on a surface). Because any map with loop is not colorable, all maps here are assumed to be loopless. In this paper povides the explicit expression of chromatic sum functions for rooted singular maps on the projective plane, the torus and the Klein bottle. From the explicit expression of chromatic sum functions of such maps, the explicit expression of enumerating functions of such maps are also derived.
Saddle point and generalized convex duality for multiobjective programming
Zhaoxiang Yan,Shizheng Li 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.15 No.-
In this paper we consider the dual problems for multiobjec-tive programming with generalized convex functions. We obtain the weakduality and the strong duality. At last,we give an equivalent relationshipbetween saddle point and ecient solution in multiobjective programming.
A new class of generalized convex programming
Zhaoxiang Yan,Shizheng Li 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.17 No.1-2
This paper finds a new class of generalized convex function which satisfies the following properties : It’s level set is -convex set; Every feasible Kuhn-Tucker point is a global minimum; If Slater’s constraint qualification holds, then every minimum point is Kuhn-Tucker point; Weak duality and strong duality hold between primal problem and it’s Mond-Weir dual problem
Chromatic sums of rooted triangulations on the projective plane$^{**}$
Zhaoxiang Li,Yanpei Liu 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.18 No.1-2
In this paper we study the chromatic sum functions for rooted nonseparable near-triangular maps on the projective plane. A chromatic sum equation for such maps is obtained.
ZHAOXIANG LI,YANPEI LIU 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
A map is bisingular if each edge is either a loop or an isthmus (i.e., on the boundary of the same face). In this paper we study the number of rooted bisingular maps on the sphere and the torus, and we also present formula for such maps with four parameters: the root-valency,the number of isthmus, the number of planar loops and the number of essential loops.
Preparation and Characterization of Cellulose Nanocrystals from Wheat Straw and Corn Stalk
Zhaoxiang Liu,Ming He,Guangrui Ma,Guihua Yang,Jiachuan Chen 한국펄프·종이공학회 2019 펄프.종이기술 Vol.51 No.2
Wheat straw and corn stalk are agricultural residues which are abundant, inexpensive, and readily available source of renewable lignocellulosic biomass. The aim of this study was to prepare cellulose nanocrystals (CNCs) from wheat straw and corn stalk, and to compare their characteristics. To achieve these goals, the CNCs obtained from hydrolysis were characterized in terms of physical, structural and morphological properties. Particle size distribution and atomic force microscopy (AFM) images showed that the CNC from corn stalk (CNC-CS) was more uniform than that from wheat straw (CNC-WS); FTIR spectra implied that acid hydrolysis treatment had no effect on the chemical structure of CNCs. Moreover, the crystallinity of CNC-CS was higher than that of CNC-WS and thermal stability of CNC-WS was observed to be similar to CNC-CS.
CHROMATIC SUMS OF ROOTED TRIANGULATIONS ON THE PROJECTIVE PLANE
LI, ZHAOXIANG,LIU, YANPEI 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.18 No.1
In this paper we study the chromatic sum functions for rooted nonseparable near-triangular maps on the projective plane. A chromatic sum equation for such maps is obtained.