WF-模糊度量空间中的不动点理论

在WF-模糊度量空间中建立压缩型和局部压缩型映射的不动点理论，推广一些重要的不动点定理。     

Topological Fukaya categories of surfaces

We construct the topological Fukaya category of a surface with genus greater than one, making this model intrinsic to the topology of the surface. Instead of using the area form of the surface, we use an admissibility condition borrowed from Heegaard-Floer theory which ensures invariance under isotopy. In this paper we show finiteness of the moduli space using purely topological means and compute the Grothendieck group of the topological Fukaya category. We also show the faithfulness of MCG-action on the topological Fukaya category in this setup.     

Presentation and central extensions of mapping class groups

We give a presentation of the mapping class group of a (possibly bounded) surface, considering either all twists or just non-separating twists as generators. We also study certain central extensions of . One of them plays a key role in studying TQFT functors, namely the mapping class group of a -structure surface. We give a presentation of this extension.

     

Bottle-neck type of neural network as a mapping device towards food specifications

A novel methodology is proposed for food specifications associated with the origin of food. The methodology was tested on honey samples collected within the TRACE EU project. The data were sampled in various regions in Europe and analysed for the trace elements content. The sampling sites were characterized by different geological origins, such as limestone, shale, or magmatic. We have chosen 14 elements, B, Na, Mg, A, K, Ca, Mn, Co, Ni, Cu, Zn, Rb, Sr, and Ba, due to their influence on the separation of samples regarding the geology of the sampling sites. A special architecture of an error back-propagation neural network, so called bottle-neck type of neural network was used to project the data into a 2D plane. The data were fed into the 14-nodes input layer and then transferred through the 2-nodes hidden layer (compared to a bottle-neck) to the 14-nodes output layer. The two hidden nodes representing the two coordinates of the projection plane enable us to map the samples used for training of the bottle-neck network. With the knowledge about the classes of individual samples we determine the clusters in the projection plane and consequently obtain the coordinates of the centroid (gravity point) of a particular cluster. The clusters are characterized with an ellipse shape borders spanning the length of up to 3σ in each dimension. Since the data were classified as regard to the geology, three main clusters were sought: (i) limestone, (ii) shale/mudstone/clay/loess, and (iii) acid-magmatic origin of honey samples. The novel methodology proposed for food specifications was demonstrated on a reduced set of samples, which shows good clustering of all three classes in the projection plane, and on the third class of the original data set.     

Cardy algebras and sewing constraints,II

This is the part II of a two-part work started in [18]. In part I, Cardy algebras were studied, a notion which arises from the classification of genus-0, 1 open–closed rational conformal field theories. In this part, we prove that a Cardy algebra also satisfies the higher genus factorisation and modular-invariance properties formulated in [7] in terms of the notion of a solution to the sewing constraints. We present the proof by showing that the latter notion, which is defined as a monoidal natural transformation, can be expressed in terms of generators and relations, which correspond exactly to the defining data and axioms of a Cardy algebra.     

Alternating mapping functions

We consider functions f from $[n]:=\{1,2,\dots ,n\}$ into itself satisfying that the labels along the iteration orbit of each $i\in [n]$ are forming an alternating sequence, i.e., $i<f(i)>f^2(i)<f^3(i)>?$ or $i>f(i)<f^2(i)>f^3(i)<?$ . We are able to solve the enumeration problem by stating exact and asymptotic formulæ for the number of such so-called alternating n-mapping functions. Furthermore we study the expected component structure of a randomly chosen alternating n-mapping by determining the probability that the underlying mapping graph is connected as well as the limiting distribution of the number of components. Moreover, the corresponding enumeration problem for weakly alternating n-mapping functions has also been solved.     

Mapping of RNA-protein interactions

RNA-protein interactions are important biological events that perform multiple functions in all living organisms. The wide range of RNA interactions demands diverse conformations to provide contacts for the selective recognition of proteins. Various analytical procedures are presently available for quantitative analyses of RNA-protein complexes, but analytical-based mapping of these complexes is essential to probe specific interactions. In this overview, interactions of functional RNAs and RNA-aptamers with target proteins are discussed by means of mapping strategies.     

Corrigendum to “Extending homeomorphisms from punctured surfaces to handlebodies” [Topology Appl. 155 (6) (2008) 610–621]

We correct the statements of Theorems 9 and 10 of [A. Cattabriga, M. Mulazzani, Extending homeomorphisms from punctured surfaces to handlebodies, Topology Appl. 155 (2008) 610–621], by adding missing generators, and improve the statement of Theorem 10, by removing some redundant generators.     

Fuzzy度量空间型映象的不动点定理

本文给出Ｆｕｚｚｙ度量空间一些扩张型映象的不动点定理,这些结果发展和改进了普通度量空间中相应的结果。