向量场的Nielsen数

对于紧致流形M上的任意一个向量场X，定义了一个由向量场X确定的自映射fX：M→M，使得向量场X的奇异点均为fX的不动点．证明了向量场的Nielsen数是不依赖于向量场选取的量．     

一类扩展Euler和的表示问题

应用Parseval定理和Nielsen广义多重对数函数的性质,给出了非线性扩展Euler和的Riemann Zeta函数表示.对来自于实验数学中的扩展Euler和∑n=1∞H2n/n2的经验公式给出了严格的理论证明.此方法也适用于求其它扩展Euler和的计算问题.     

Coincidence Wecken Property for Nilmanifolds

Let f, g: X → Y be maps from a compact infra-nilmanifold X to a compact nilmanifold Y with dim X ≥ dim Y. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number N(f, g) vanishes then f and g are deformable to be coincidence free. We also show that if X is a connected finite complex X and the Reidemeister coincidence number R(f, g) = ∞ then f ～ f' so that C(f', g) = {x ∈ X | f'(x) = g(x)} is empty.     

关于映射同伦类的拓扑熵

设ｆ：Ｔ＾ｍ→Ｔ＾ｍ为ｍ维环面自映射，Ｎ＾∞（ｆ）是ｆ的渐近Ｎｉｅｌｓｅｎ数，本文应用Ｎｉｅｌｓｅｎ不动点理论，给出了ｌｏｇＮ＾∞（ｆ）是ｆ的同伦类的拓扑熵的最好下界的一个充要条件；并通过在齐性空间上引入等价度量，将此结论推广到了幂零流形自映射的情形。     

Wecken property for roots

We show that maps from a compact space into a topological manifold which have geometric Nielsen root number zero satisfy the Wecken property, i.e., such that .

     

Coincidences of projections and linear n-valued maps of tori

We prove that the Nielsen fixed point number N(φ) of an n-valued map φ:X?X of a compact connected triangulated orientable q-manifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of φ, a subset of X×X, to the two factors. For certain q×q integer matrices A, there exist “linear” n-valued maps Φn,A,σ:Tq?Tq of q-tori that generalize the single-valued maps fA:Tq→Tq induced by the linear transformations TA:Rq→Rq defined by TA(v)=Av. By calculating the Nielsen coincidence number of the projections of its graph, we calculate N(Φn,A,σ) for a large class of linear n-valued maps.     

Nielsen numbers of n-valued fiber maps

The Nielsen number for n-valued multimaps, defined by Schirmer, has been calculated only for the circle. A concept of n-valued fiber map on the total space of a fibration is introduced. A formula for the Nielsen numbers of n-valued fiber maps of fibrations over the circle reduces the calculation to the computation of Nielsen numbers of single-valued maps. If the fibration is orientable, the product formula for single-valued fiber maps fails to generalize, but a “semi-product formula" is obtained. In this way, the class of n-valued multimaps for which the Nielsen number can be computed is substantially enlarged. Dedicated, with gratitude, to Felix Browder who, long ago, encouraged and supported a young topologist’s interest in fixed point theory     

Self-maps of <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> homotopic to a smooth map with a single <Emphasis Type="Italic">n</Emphasis>-periodic point

We show for which (d,n) ∈ Z×N there exists a smooth self-map f:S² → S² so that deg(f)=d and Fix(fⁿ) is a point.     

Mei Symmetry and Mei Conserved Quantity for a Non-holonomic Systemof Non-Chetaev's Type with Unilateral Constraints in Nielsen Style

Mei symmetry and Mei conserved quantity for a non-holonomic system of non-Chetaev's type with unilateral constraints in the Nielsen style are studied. The differential equations of motion for the system above are established. The definition and the criteria of Mei symmetry, conditions, and expressions of Mei conserved quantity deduced directly from the Mei symmetry are given. An example is given to illustrate the application of the results.     

Form invariance and conserved quantities of Nielsen equations of relativistic variable mass nonholonomic systems

In this paper, the definition and criterion of the form invariance of Nielsen equations for relativistic variable mass nonholonomic systems are given. The relation between the form invariance and the Noether symmetry is studied.Finally, we give an example to illustrate the application of the result.