On the second order topological asymptotic expansion

The topological derivative provides the sensitivity of a given shape functional with respect to an infinitesimal (non smooth) domain perturbation at an arbitrary point of the domain. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal perturbations. However, for practical applications, we need to insert perturbations of finite size. Therefore, we consider one more term in the expansion which is defined as the second order topological derivative. In order to present these ideas, in this work we calculate first as well as second order topological derivatives for the total potential energy associated to the Laplace’s equation, when the domain is perturbed with a hole. Furthermore, we also study the effects of different boundary conditions on the hole: Neumann and Dirichlet (both homogeneous). In the Neumann’s case, the second order topological derivative depends explicitly on higher-order gradients of the state solution and also implicitly on the point where the hole is nucleated through the solution of an auxiliary problem. On the other hand, in the Dirichlet’s case, the first order topological derivative depends explicitly on the state solution as well as implicitly through the solution of an auxiliary problem, and the second order topological derivative depends only explicitly on the solution associated to the original problem. Finally, we present two simple examples showing the influence of both terms in the second order topological asymptotic expansion for each case of boundary condition on the hole.     

The topological basis realization and the corresponding Heisenberg model of spin-1 chain

In this paper, it is shown that the Heisenberg model of spin-1 chain can be constructed from the Birman–Wenzl algebra generator while we have got that the Heisenberg model of spin- $\frac{1}{2}$ chain can be constructed from the Temperley–Lieb algebra generator in our previous work (Sun et al. in EPL 94:50001, 2011). Here, we investigate the topological space, we find that the number of topological basis states raise from the previous two to three, and they are also the three eigenstates of a closed four-qubit Heisenberg model of spin-1 chain. Specifically, all the topological basis states are also the spin single states and one of them is the energy single state of the system. It is worth noting that all conclusions we get in this paper are consistent with our previous work (Sun et al. in EPL 94:50001, 2011). These just indicate that the topological basis states have particular properties in the system.     

Topological Features in 2D Symmetric Higher‐Order Tensor Fields

The topological structure of scalar, vector, and second‐order tensor fields provides an important mathematical basis for data analysis and visualization. In this paper, we extend this framework towards higher‐order tensors. First, we establish formal uniqueness properties for a geometrically constrained tensor decomposition. This allows us to define and visualize topological structures in symmetric tensor fields of orders three and four. We clarify that in 2D, degeneracies occur at isolated points, regardless of tensor order. However, for orders higher than two, they are no longer equivalent to isotropic tensors, and their fractional Poincaré index prevents us from deriving continuous vector fields from the tensor decomposition. Instead, sorting the terms by magnitude leads to a new type of feature, lines along which the resulting vector fields are discontinuous. We propose algorithms to extract these features and present results on higher‐order derivatives and higher‐order structure tensors.     

Tracking topological entity changes in 3D collaborative modeling systems

One of the key problems in collaborative geometric modeling systems is topological entity correspondence when topological structure of geometry models on collaborative sites changes.In this article,we propose a solution for tracking topological entity alterations in 3D collaborative modeling environment.We firstly make a thorough analysis and detailed categorization on the alteration properties and causations for each type of topological entity,namely topological face and topological edge.Based on collaborative topological entity naming mechanism,a data structure called TEST (Topological Entity Structure Tree) is introduced to track the changing history and current state of each topological entity,to embody the relationship among topological entities.Rules and algorithms are presented for identification of topological entities referenced by operations for correct execution and model consistency.The algorithm has been verified within the prototype we have implemented with ACIS.     

The topological basis expression of Heisenberg spin chain

In this paper, it is shown that the Heisenberg XY, XXZ, XXX, and Ising model all can be constructed from the Braid group algebra generator and the Temperley–Lieb algebra generator. And a new set of topological basis expression is presented. Through acting on the different subspaces, we get the new nontrivial six-dimensional and four-dimensional Braid group matrix representations and Temperley–Lieb matrix representations. The eigenstates of Heisenberg model can be described by the combination of the set of topological bases. It is worth mentioning that the ground state is closely related to parameter q which is the meaningful topological parameter.     

Structural Equivalence of Topological Machines

Recently, there has been a fair amount of interest in the study of topological machines, e.g. Day [2], Norris [4] and Sikdar [5]. Due to the topological structures endowed in the input, output, and state spaces of the topological machines, most of the results of conventional deterministic machines do not admit of an immediate generalization to the topological case. In this paper, we shall study the structural equivalences of topological machines. Various concepts are defined including quotient submachines, irreducibility, reduced forms, homomorphisms, minimality, and minimal forms. It is shown that every topological machine has a unique reduced form and has a minimal form which is unique up to iseomorphism. Moreover, it is shown that a topological machine is irreducible if and only if it is minimal.     

Topological lines in 3D tensor fields and discriminant Hessian factorization

This paper addresses several issues related to topological analysis of 3D second order symmetric tensor fields. First, we show that the degenerate features in such data sets form stable topological lines rather than points, as previously thought. Second, the paper presents two different methods for extracting these features by identifying the individual points on these lines and connecting them. Third, this paper proposes an analytical form of obtaining tangents at the degenerate points along these topological lines. The tangents are derived from a Hessian factorization technique on the tensor discriminant and leads to a fast and stable solution. Together, these three advances allow us to extract the backbone topological lines that form the basis for topological analysis of tensor fields.     

RPR网络中的亚可靠传输

为了增强处于保护状态下的RPR环网带宽的合理利用,本文提出亚可靠传输机制,并利用拓扑矩阵来制定如何对亚可靠传输业务进行界定的算法。之后还进一步探讨了引入该机 制后RPR公平算法公平帧的运行方式。     

A topological approach for surface reconstruction from sample points

Most algorithms for surface reconstruction from sample points rely on computationally demanding operations to derive the reconstruction. In this paper we introduce an innovative approach for generating 3D piecewise linear approximations from sample points that relies strongly on topological information, thus reducing the computational cost and numerical instabilities typically associated with geometric computations. Discrete Morse theory provides the basis for a topological framework that supports a robust reconstruction algorithm capable of handling multiple components and has low computational cost. We describe the proposed approach and introduce the reconstruction algorithm, called TSR – topological surface reconstructor. Some reconstruction results are presented and the performance of TSR is compared with that of other reconstruction approaches for some standard point sets.     

Ontology-based topological representation of remote-sensing images

This article proposes an ontology-based topological representation of remote-sensing images. Semantics, especially related to the topological relationships between the objects represented, are not explicit in remote-sensing images and this fact limits spatial analysis. Our aim is to provide an explicit ontological definition of the topological relations between objects in the image using the Quadtree data structure for spatial indexing. This structure is explicitly defined in an ontology allowing the automatic interpretation of the representations obtained, taking into account the topological relations and increasing the spatial analytical capabilities. This representation has been validated by a case study of semantic retrieval based on the normalized difference vegetation index (NDVI), taking into account the topological relations between NDVI regions in images. In the experiments, we compare the effectiveness of results from eight queries using four traditional supervised image classification algorithms and the proposal representation. The experimental results show the feasibility of the proposal, supporting the concept of the image retrieval process providing a semantic complement to remote-sensing images. The proposed representation contributes to incorporation of semantics into geographical data, especially to remote-sensing images, and it can be used to develop applications in the Geospatial Semantic Web.