Topological structure of the many vortices solution in Jackiw-Pi model

By using the gauge potential decomposition, we discuss the self-dual equation and its solution in Jackiw-Pi model. We obtain a new concrete self-dual equation and find relationship between Chern-Simons vortices solution and topological number which is determined by Hopf indices and Brouwer degrees of Psi-mapping. To show the meaning of topological number we give several figures with different topological numbers. In order to investigate the topological properties of many vortices, we use five parameters (two positions, one scale, one phase per vortex and one charge of each vortex) to describe each vortex in many vortices solutions in Jackiw-Pi model. For many vortices, we give three figures with different topological numbers to show the effect of the charge on the many vortices solutions. We also study the quantization of flux of those vortices related to the topological numbers in this case.

Topological structure of vortex solution in Jackiw-Pi model

By using phi-mapping method, we discuss the topological structure of the self-duality solution in Jackiw-Pi model in terms of gauge potential decomposition. We set up relationship between Chern-Simons vortex solution and topological number, which is determined by Hopf index and Brouwer degree. We also give the quantization of flux in this case. Then, we study the angular momentum of the vortex, which can be expressed in terms of the flux.

Topological structure of the solitons solution in SU(3) Dunne-Jackiw-Pi-Trugenberger model

By using phi-mapping topological current theory and gauge potential decomposition, we discuss the self-dual equation and its solution in the SU(N) Dunne-Jackiw-Pi-Trugenberger model and obtain a new concrete self-dual equation with a 6 function. For the SU(3) case, we obtain a new self-duality solution and find the relationship between the soliton solution and topological number which is determined by the Hopf index and Brouwer degree of phi-mapping. In our solution, the flux of this soliton is naturally quantized.

Continuously deformable topological structure

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Fuzzy Topological Games and Related Topics

The main purpose of study is to extend the concept of the topological game G(K, X) and some other kinds of games into fuzzy topological games and to obtain some results regarding them. Owing to the fact that topological games have plenty of applications in covering properties, it made an attempt to explore some inter relations of games and covering properties in fuzzy topological spaces. Even though the main focus is on fuzzy para-meta compact spaces and closure preserving shading families, some brief sketches regarding fuzzy P-spaces and Shading Dimension is also provided. In a topological game players choose some objects related to the topological structure of a space such as points, closed subsets, open covers etc. More over the condition on a play to be winning for a player may also include topological notions such as closure, convergence, etc. It turns out that topological games are related to the Baire property, Baire spaces, Completeness properties, Convergence properties, Separation properties, Covering and Base properties, Continuous images, Suslin sets, Singular spaces etc.

Boundaries and Topological Algorithms

This thesis develops a model for the topological structure of situations. In this model, the topological structure of space is altered by the presence or absence of boundaries, such as those at the edges of objects. This allows the intuitive meaning of topological concepts such as region connectivity, function continuity, and preservation of topological structure to be modeled using the standard mathematical definitions. The thesis shows that these concepts are important in a wide range of artificial intelligence problems, including low-level vision, high-level vision, natural language semantics, and high-level reasoning.

ALGEBRAIC AND GEOMETRIC THEORY OF THE TOPOLOGICAL RING OF COLOMBEAU GENERALIZED FUNCTIONS

We continue the investigation of the algebraic and topological structure of the algebra of Colombeau generalized functions with the aim of building up the algebraic basis for the theory of these functions. This was started in a previous work of Aragona and Juriaans, where the algebraic and topological structure of the Colombeau generalized numbers were studied. Here, among other important things, we determine completely the minimal primes of (K) over bar and introduce several invariants of the ideals of 9(Q). The main tools we use are the algebraic results obtained by Aragona and Juriaans and the theory of differential calculus on generalized manifolds developed by Aragona and co-workers. The main achievement of the differential calculus is that all classical objects, such as distributions, become Cl-functions. Our purpose is to build an independent and intrinsic theory for Colombeau generalized functions and place them in a wider context.

Pre-weighted homogeneous map germs finite determinacy and topological triviality

In this paper we introduce the notion of G-pre-weighted homogeneous map germ, (G is one of Mather's groups A or K.) and show that any G-pre-weighted homogeneous map germ is G-finitely determined. We also give an explicit order, based on the Newton polyhedron of a pre-weighted homogeneous germ of function, such that the topological structure is preserved after perturbations by terms of higher order.

Topological invariance and spatial scaling of surface roughness in two highly eroded zones of Mexico: a comparative study

The Fractal Image Informatics toolbox (Oleschko et al., 2008 a; Torres-Argüelles et al., 2010) was applied to extract, classify and model the topological structure and dynamics of surface roughness in two highly eroded catchments of Mexico. Both areas are affected by gully erosion (Sidorchuk, 2005) and characterized by avalanche-like matter transport. Five contrasting morphological patterns were distinguished across the slope of the bare eroded surface of Faeozem (Queretaro State) while only one (apparently independent on the slope) roughness pattern was documented for Andosol (Michoacan State). We called these patterns ?the roughness clusters? and compared them in terms of metrizability, continuity, compactness, topological connectedness (global and local) and invariance, separability, and degree of ramification (Weyl, 1937). All mentioned topological measurands were correlated with the variance, skewness and kurtosis of the gray-level distribution of digital images. The morphology0 spatial dynamics of roughness clusters was measured and mapped with high precision in terms of fractal descriptors. The Hurst exponent was especially suitable to distinguish between the structure of ?turtle shell? and ?ramification? patterns (sediment producing zone A of the slope); as well as ?honeycomb? (sediment transport zone B) and ?dinosaur steps? and ?corals? (sediment deposition zone C) roughness clusters. Some other structural attributes of studied patterns were also statistically different and correlated with the variance, skewness and kurtosis of gray distribution of multiscale digital images. The scale invariance of classified roughness patterns was documented inside the range of five image resolutions. We conjectured that the geometrization of erosion patterns in terms of roughness clustering might benefit the most semi-quantitative models developed for erosion and sediment yield assessments (de Vente and Poesen, 2005).

Find my office: Navigating real space from semantic descriptions

This paper shows that by using only symbolic language phrases, a mobile robot can purposefully navigate to specified rooms in previously unexplored environments. The robot intelligently organises a symbolic language description of the unseen environment and “imagines” a representative map, called the abstract map. The abstract map is an internal representation of the topological structure and spatial layout of symbolically defined locations. To perform goal-directed exploration, the abstract map creates a high-level semantic plan to reason about spaces beyond the robot’s known world. While completing the plan, the robot uses the metric guidance provided by a spatial layout, and grounded observations of door labels, to efficiently guide its navigation. The system is shown to complete exploration in unexplored spaces by travelling only 13.3% further than the optimal path.

Parallel Computation of 3D Morse-Smale Complexes

The Morse-Smale complex is a topological structure that captures the behavior of the gradient of a scalar function on a manifold. This paper discusses scalable techniques to compute the Morse-Smale complex of scalar functions defined on large three-dimensional structured grids. Computing the Morse-Smale complex of three-dimensional domains is challenging as compared to two-dimensional domains because of the non-trivial structure introduced by the two types of saddle criticalities. We present a parallel shared-memory algorithm to compute the Morse-Smale complex based on Forman's discrete Morse theory. The algorithm achieves scalability via synergistic use of the CPU and the GPU. We first prove that the discrete gradient on the domain can be computed independently for each cell and hence can be implemented on the GPU. Second, we describe a two-step graph traversal algorithm to compute the 1-saddle-2-saddle connections efficiently and in parallel on the CPU. Simultaneously, the extremasaddle connections are computed using a tree traversal algorithm on the GPU.

Approximate knowledge of rationality and correlated equilibria

We extend Aumann's [3] theorem deriving correlated equilibria as a consequence of common priors and common knowledge of rationality by explicitly allowing for non-rational behavior. We replace the assumption of common knowledge of rationality with a substantially weaker notion, joint p-belief of rationality, where agents believe the other agents are rational with probabilities p = (pi)i2I or more. We show that behavior in this case constitutes a constrained correlated equilibrium of a doubled game satisfying certain p-belief constraints and characterize the topological structure of the resulting set of p-rational outcomes. We establish continuity in the parameters p and show that, for p su ciently close to one, the p-rational outcomes are close to the correlated equilibria and, with high probability, supported on strategies that survive the iterated elimination of strictly dominated strategies. Finally, we extend Aumann and Dreze's [4] theorem on rational expectations of interim types to the broader p-rational belief systems, and also discuss the case of non-common priors.

Where Tori fear to tread : hypermassive neutron star remnants and absolute event horizons Or topics in computational general relativity

Computational general relativity is a field of study which has reached maturity only within the last decade. This thesis details several studies that elucidate phenomena related to the coalescence of compact object binaries. Chapters 2 and 3 recounts work towards developing new analytical tools for visualizing and reasoning about dynamics in strongly curved spacetimes. In both studies, the results employ analogies with the classical theory of electricity and magnitism, first (Ch. 2) in the post-Newtonian approximation to general relativity and then (Ch. 3) in full general relativity though in the absence of matter sources. In Chapter 4, we examine the topological structure of absolute event horizons during binary black hole merger simulations conducted with the SpEC code. Chapter 6 reports on the progress of the SpEC code in simulating the coalescence of neutron star-neutron star binaries, while Chapter 7 tests the effects of various numerical gauge conditions on the robustness of black hole formation from stellar collapse in SpEC. In Chapter 5, we examine the nature of pseudospectral expansions of non-smooth functions motivated by the need to simulate the stellar surface in Chapters 6 and 7. In Chapter 8, we study how thermal effects in the nuclear equation of state effect the equilibria and stability of hypermassive neutron stars. Chapter 9 presents supplements to the work in Chapter 8, including an examination of the stability question raised in Chapter 8 in greater mathematical detail.

香港地区湍蛙遗传多样性研究

依据线粒体上ND2和CO1两个变异较大的基因序列分析了香港地区香港湍蛙7种群、华南湍蛙1种群，以及大陆其他地区华南湍蛙7种群，戴云湍蛙1种群，武夷湍蛙1种群的系统发育关系，进而探讨香港湍蛙的遗传多样性、香港湍蛙特有性、如何确定香港湍蛙最佳保护单元以及这四种湍蛙的物种分类地位。 1． 香港湍蛙保护遗传学研究 香港湍蛙核苷酸传多样性较低，从其遗传多样性信息、单倍型网络分析、中性检验值以及岐点分布结果一致显示香港湍蛙很可能经历了瓶颈后的扩张，种群正在由一个较小的有效种群大小迅速增长, 有足够的时间通过变异用于积累单倍型的多态性, 而对于提高核苷酸多样化而言, 时间尚短（Nei M et al，1975，Avise J C，2000；李明等，2003）。 分子变异分析结果显示香港湍蛙种群间存在较多的基因交流，且系统发育树上各种群间交叉在一起，没有形成与地理单元相关的分支，而从其单倍型网络看，他们源于共同的祖先，是一个单系群，与地理单元间没有形成显著的遗传分化。因此应作为一个进化显著单元（ESU）。结合其与其他湍蛙发育关系及遗传距离以及野外采集信息认为香港湍蛙只在香港地区有分布，属于香港特有种。该物种内遗传多样性较低，又属于世界自然保护联盟红皮书中的近危种，同时也是《野生动物保护条例》中的受保护野生动物，且由于香港城市建设等使得其栖息环境受到威胁，因此在香港特别行政区应该受到重点保护。 从单倍型分布和核苷酸多样性可以看出大榄涌种群和城门种群具有较高的单倍型多样性和核苷酸多样性，应该作为保护的重点区域。 2. 华南湍蛙东、南沿海种群间系统关系 华南湍蛙分布广，各种群存在着丰富的遗传多样性信息且中部种群广西龙胜和湖南张家界种群核苷酸多样性明显高于其他边缘种群华南湍蛙。种群间几乎没有基因交流，且各种群间无共享单倍型，可见已形成了显著的遗传分化。各种群间遗传距离都较远，其中广东南昆山种群以及福建三港种群与其他种群距离最远，因此可以推测其他种群（广东深圳、香港大屿山、广西龙胜和防城以及湖南张家界种群）可能为独立进化的种群。但是否是一新种或一隐存种，还需要结合形态学进行更深入的研究。 本研究中无论从系统关系看还是从遗传距离看，大屿山种群与深圳种群最近，支持陈坚峰等将其定为华南湍蛙，即华南湍蛙新增一个分布点：香港大屿山。 系统树上广西防城种群（支B）与龙胜和湖南种群（支A）形成姐妹群。香港大屿山种群与深圳种群先形成姐妹群（支C），但却没有与其距离很近的广东南岭及南昆山种群（支D）形成姐妹群，可能粤北和粤中的环境及气候较复杂因此与粤南其他种群形成了明显的隔离。同时可以看出华南湍蛙种群遗传分化与地理距离没有显著的相关性。 3. 四种湍蛙间的系统关系 根据线粒体CO1基因建立四种湍蛙间的系统关系及其遗传距离，很清楚地看到，香港湍蛙与戴云湍蛙关系很近，而华南湍蛙则与武夷湍蛙较近。然而，戴云湍蛙同一个种群内部共有两个单倍型DY1和DY2，且两个单倍型间遗传距离大于DY1与香港湍蛙间遗传距离，更远远大于香港湍蛙种群内部的距离，即戴云湍蛙内部两个单倍型间遗传距离达到了种级水平，同样在系统发育树上这两个单倍型与香港湍蛙形成并系。但是，戴云湍蛙种内在形态上差异不显著。因此，其是否属于萌芽物种分化形成（budding speciation）或已经完全分化为两个不同的种值得进一步研究？ 与戴云湍蛙香港湍蛙关系类似，从系统树上看华南湍蛙不形成单系，而是分成两个大支，与武夷湍蛙形成并系，且福建和南昆山的华南湍蛙与武夷湍蛙遗传距离远大于武夷湍蛙种内福建种群与浙江种群的遗传距离，达到了种级分化水平。由此，可以推断武夷湍蛙是有效种。系统树上广东深圳、香港大屿山、广西防城和龙胜以及湖南张家界种群与华南湍蛙福建及南昆山各种群间遗传距离已超出了种内各种群间的遗传距离，但是至于这一支是否应为另外一个种，有必要扩大采样，并结合核基因及形态信息进行进一步研究。 MtDNA of ND2 and CO1 gene were used to investigate genetic diversity of Amolops in Hongkong .We collected seven populations of A. hongkongensis,，one population of A.ricketti from Hong Kong and other seven populations of A.ricketti from East and South of Chinese mainland. As well as one population of A. daiyunensis and one population of A.wuyiensis Phylogenetic relationship were analyzed of four species. Discussed whether A.hongkongensis is an endemic species and how can we make the conservation and management decisions. 1. Conservation Genetics of A. hongkongensis A. hongkongensis has a low nucleotide diversity, the results of genetic diversity, haplotype network, neutrality test and the mismatch distributions indicate that A. hongkongensis experienced a recent expansion after a bottle neck. They had enough time to accumulated haplotype diversity, but it’s too short to have a high nucleotide diversity（Nei M et al，1975，Avise J C，2000；Li et al，2003）. The result of AMOVA reveals that it has much gene exchange among the populations of A. hongkongensis. The clades of the phylogenetic tree were mixed together, no significant genetic differentiation among 8 populations and they share the same ancestor from the network analysis, these indicate that they are monophyly and should be protected as one ESU. Combined with the information of relationships of interspecies, genetic distance and distribution investigate, We conclude that A. hongkongensis is an endemic species of Hong Kong. Considering on the status of low genetic diversity in A.hongkongensis, and this species was listed in the IUCN red list as near threatened, as well as listed in the . Furthermore, it’s habitat loss and degradation more rapidly as the human activity got higher and higher. So it’s urgent to protect them in Hong Kong. Our results suggest that Tai Lam Wu and TAI MO Shan -Shing Mun populations have the higher priority to be protected because their higher genetic diversity. 2．Phylogenetic relationships among populations of Amolops ricketti from the Southern and eastern China A. ricketti has the considerable genetic diversity of mitochondrial haplotypes within and among populations, and Mitochondrial DNA diversity was higher in populations at the central area of the present distribution range of the frog，i. e. the Longsheng population and Zhangjiajie population, than at the edges of their distribution range. They have no share haplotype among populations, and have a significant genetic differentiation. Genetic distance is high among the populations, especially the distance of Nankun and Sangang group with others, which suggested that they evolved independently. May be there is a cryptic species or a new species, a further study is needed. The results of gene tree and the genetic distance clearly demonstrate that the population from LanTau island is A. ricketti, so we agree with Chen et al(2005) . That means A.ricketti have a new distribution site: LanTau island, HongKong. Phylogenetic relationships were analyzed through NJ and Mrbayes methods and got a consistent topological structure, the structure indicated that the ingroup were comprised four groups. Populations Longsheng and Zhangjiajie were first clustered as clade A; Populations Fangcheng was clustered together (clade B) as a sister group to clade A；Populations Shenzhen and Lantau island were sister groups and clustered as clade C；Then the clade D included populations Nankunshan and Nanling in Guangdong province and Sangang in Fujian province. 3. Phylogenetic Relationships among these four specises Phylogenetic relationships based on 1503bp CO1 gene and the genetic distance show that A. hongkongensis close to A. daiyunensis whereas A.ricketti near to A.wuyiensis. Nevertheless, there are two haplotypes in A.daiyunensis and the genetic distance between them higher than the distance between DY1 with A. hongkongensis. A. hongkongensis is nested in the paraphyletic ancestral species A. daiyunensis. Without significant difference in the morphological characters, So, we considered both A.daiyunensis and A.hongkongensis are valid species, may be this represents a case of ‘budding speciation’ like Batrachuperus pinchonii(Fu and Zeng,2008) in the population of A. daiyunensis. Just like two species above A. wuyiensis and A. ricketti are not monophyly, instead, A.wuyiensis is nested in the paraphyletic ancestral species A.ricketti. We need do more research to make sure whether they are new species.

Decomposition of Noncommutative U(1) Gauge Potential

We investigate the decomposition of noncommutative gauge potential (A) over cap (i), and find that it has inner structure, namely, (A) over cap (i) can he decomposed in two parts, (b) over cap (i) and (a) over cap (i), where (b) over cap (i) satisfies gauge transformations while (a) over cap (i) satisfies adjoint transformations, so close the Seiberg-Witten mapping of noncommutative, U(1) gauge potential. By, means of Seiberg-Witten mapping, we construct a mapping of unit vector field between noncommutative space and ordinary space, and find the noncommutative U(1) gauge potential and its gauge field tensor can be expressed in terms of the unit vector field. When the unit vector field has no singularity point, noncommutative gauge potential and gauge field tensor will equal ordinary gauge potential and gauge field tensor