On the Stability of Bifurcation Diagrams of Vanishing Flattening Points

On a smooth surface in Euclidean 3-space, we consider vanishing curves whose projections on a given plane are small circles centered at the origin. The bifurcations diagram of a parameter-dependent surface is the set of parameters and radii of the circles corresponding to curves with degenerate flattening points. Solving a problem due to Arnold, we find a normal form of the first nontrivial example of a flattening bifurcation diagram, which contains one continuous invariant.     

Fixed-point theorems for families of weakly non-expansive maps

In this paper, we present some fixed-point theorems for families of weakly non-expansive maps under some relatively weaker and more general conditions. Our results generalize and improve several results due to Jungck [G. Jungck, Fixed points via a generalized local commutativity, Int. J. Math. Math. Sci. 25 (8) (2001) 497-507], Jachymski [J. Jachymski, A generalization of the theorem by Rhoades and Watson for contractive type mappings, Math. Japon. 38 (6) (1993) 1095-1102], Guo [C. Guo, An extension of fixed point theorem of Krasnoselski, Chinese J. Math. (P.O.C.) 21 (1) (1993) 13-20], Rhoades [B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257-290], and others.     

Linearisation and potential symmetries of certain systems of diffusion equations

We consider systems of two pure one-dimensional diffusion equations that have considerable interest in Soil Science and Mathematical Biology. We construct non-local symmetries for these systems. These are determined by expressing the equations in a partially and wholly conserved form, and then by performing a potential symmetry analysis on those systems that can be linearised. We give several examples of such systems, and in a specific case we show how linearising and hodograph-type mappings can lead to new solutions of the diffusion system.     

单调和Fuzzy单调变分不等式的补偿方法

在实自反Banach空间,用补偿方法研究单调和Fuzzy单调广义非线性变分不等式,得到了几个解的存在性定理.改进和推广了Browder,Pascali-Sburlan,Zhang等人的各种结果.     

Fixed point theorems under the interior condition

We show a fixed point theorem for condensing mappings under a new condition of the Leray-Schauder type. We call it the Interior Condition. We also discuss examples that demonstrate the independence of these two conditions.

     

两类拟共形映射的掩盖定理

In this paper we define two classes of quasiconformal mappings,and study their covering properties by methods of module.We obtain some new results.In the meantime,we give new methods to prove Koebe 1/2 covering theorem on convex conformal mappings.     

Fuzzy集的基数

关于Fuzzy集的基数，[1]中曾对有限支集的Fuzzy集以及极苛刻条件下的无限Fuzzy集给出过一种定义。但是，正如本文将要指出的那样，该定义是不合理的。本文将从研究Fuzzy映射入手，给出Fuzzy集基数的一般性定义。基于这一定义，不但得到有关基数的大部分结论，而且有其自身的特殊性。     

A problem in extremal quasiconformal extensions

A constantK ₀ ^(m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK ₀(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established forK ₀ ^(m) (h) =K ₁(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK ₀ ^(m) (h)<K ₁(h), wherek ₁(h) is the maximal dilatation ofh.