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Recently, multi-view representation learning has gained rapid growth in various fields. Most of previous multi-view learning methods rely on strong notions of distances that often provide no useful gradients in deep network training, which greatly degrades the performance in merging complementary information of views. To address this challenge, a multi-view representation learning network with dual optimal transportation (MDOT-Net) is proposed to capture fusion representations embedded in a common manifold with the weak topology. In MDOT-Net, the multi-view representation learning is modelled as an optimal transportation (OT) problem in manifold fitting, which is further decomposed into the intra-view OT and the inter-view OT. The intra-view OT is implemented by a view-specific adversarial variational network, which accurately captures local manifold structures within views by leveraging fusion knowledge. The inter-view OT is implemented by a view-fusion adversarial inference network, which models fusion representations compatible with diversities of sub-manifolds by utilizing view-specific knowledge. The two OTs boost mutually by providing prior knowledge to each other in multi-view representation learning. Finally, numerous experiments are conducted on four benchmark datasets, and the results demonstrate MDOT-Net is competitive against state-of-the-art algorithms.

Description, Strongly confined zero-dimensional localization with weak topological design is verified by near-field optical microscopy., Topological photonic crystals (TPhCs) provide robust manipulation of light with built-in immunity to fabrication tolerances and disorder. Recently, it was shown that TPhCs based on weak topology with a dislocation inherit this robustness and further host topologically protected lower-dimensional localized modes. However, TPhCs with weak topology at optical frequencies have not been demonstrated so far. Here, we use scattering-type scanning near-field optical microscopy to verify mid-bandgap zero-dimensional light localization close to 100 THz in a TPhC with nontrivial Zak phase and an edge dislocation. We show that because of the weak topology, differently extended dislocation centers induce similarly strong light localization. The experimental results are supported by full-field simulations. Along with the underlying fundamental physics, our results lay a foundation for the application of TPhCs based on weak topology in active topological nanophotonics, and nonlinear and quantum optic integrated devices because of their strong and robust light localization.

We consider Mergelyan sets and Farrell sets for $$H^p$$ $$(1\le p < \infty )$$ spaces in the unit disc for both the weak topology and the norm topology, and give a short proof of a theorem of Perez-Gonzalez which answers a question proposed by Rubel and Stray (J Approx Theory 37:44–50, 1983).

In 1976 during a conference in Halifax, A. Lau raised the question on whether left amenability property for a semitopological semigroup is sufficient to ensure the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex set in a dual Banach space. In this paper we establish a positive answer for discrete semigroups and for strongly left amenable semitopological semigroups.

In a dual pair 〈E, F〉 one wants to define topologies on E associated with collections of suitable subsets of F. (This generalises the definition of the norm topology on the dual E′ of a Banach space E, in this case for the dual pair 〈E′, E〉.) Such a collection \(\mathcal M\) defines a ‘polar topology’ on E, where the corresponding neighbourhoods of zero in E are polars of the members of \(\mathcal M\). Examples of such topologies are the weak topology and the strong topology. In the first part of the chapter we define polars and investigate some of their properties.

We generalize Brazas' topology on the fundamental group to the whole universal path space X ˜ , i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action.

The aim of this paper is to introduce a single structure which carries the subsets of X as well as the subsets of Y for studying the information about the ordered pair (A, B) of subsets of X and Y. Such a structure is called a binary structure from X to Y. Mathematically a binary structure from X to Y is defined as a set of ordered pairs (A, B) where AX and BY. The purpose of this paper is to introduce a new topology between two sets called a binary topology and investigate its basic properties where a binary topology from X to Y is a binary structure satisfying certain axioms that are analogous to the axioms of topology. MSC 2010: 54A05, 54A99.

We show that the Mackey topologies exist in the class LIN 2 of linearly topologized Hausdorff groups and we give a complete description of these topologies in terms of B -embedded subgroups. As a consequence, we obtain also a complete description of the Mackey topologies in the class of locally quasi convex bounded abelian groups. This gives as a corollary the following result recently proved in [3] : every metrizable locally quasi convex bounded abelian group is Mackey. Further, we show that a locally quasi-convex group topology on a bounded abelian group is the Mackey topology if every countable subgroup H satisfies | H ∧ | c .

We present a further generalization of the 𝒯AI -density topology introduced in [W. Wojdowski, A category analogue of the generalization of Lebesgue density topology, Tatra Mt. Math. Publ. 42 (2009), 11–25] as a generalization of the I-density topology. We construct an ascending sequence {𝒯 A I(n) } n ∈ℕ of density topologies which leads to the 𝒯 A I(ω) -density topology including all previous topologies. We examine several basic properties of the topologies.

A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion-Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle's causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this 'weak adapted topology' is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions., Minor clarifying changes; 37 pages

In this paper, we examine properties of the division topology on the set of positive integers introduced by Rizza in 1993. The division topology on [Formula: see text] with the division order is an example of [Formula: see text]-Alexandroff topology. We mainly concentrate on closures of arithmetic progressions and connected and compact sets. Moreover, we show that in the division topology on [Formula: see text], the continuity is equivalent to the Darboux property.

Some kind of abstract density topology in a topological Baire space is considered. The semiregularization of this type of topology on the real line in many cases is the coarsest topology for which real functions continuous with respect to the abstract density topology are continuous.

Let k be a ring, X be a k -scheme and R be a k -algebra endowed with an arbitrary topology. In this text, we introduce the fine topology on X ( R ) , which is based on Grothendieck's definition of a topology for affine k -schemes. We prove that the fine topology is functorial in both X and R and that it coincides with Grothendieck's topology for affine k -schemes, with the strong topology for k -varieties over topological fields k and with the adelic topology for k -varieties over a global field k . In some concluding remarks, we explain how properties of the topology of R are reflected in geometric properties of the fine topology, and discuss a possible application to higher local fields.

We discuss the relationships between ? -strongly Chebyshev, ? -approximative compactness, ? -strong proximinality, and proximinality, where ? stands for the norm topology or weak topology. We consider four examples which illustrate the differences, and a counterexample that shows the question regarding the approximative n -compact Chebyshev subset, raised by Bandyopadhyay et al., is negative.

We consider T0-topologies on n-element set that contain more than 2n elements. We solve a problem of listing and counting of such topologies. For this purpose we introduce the notions of an index of the topology and a vector of the topology. We study their properties and single out all possible classes of the topologies under the consideration. We formulate and prove a theorem related to the number of the topologies in each of the classes.

Summary In view of the requirement for instantaneity on power system topology analysis, a novel power system topology analysis method is proposed in this paper. For the plant-station topology method, the nodes that represent the circuit breakers are numbered according to the graph theory. On this basis, combining the main wiring characteristics and the switch status information, the plant-station topology is analyzed using the topological base theorem. When the switch status changes, the affected part of topology is modified using the radius search method, so that reforming of the topology is avoided and fast update of topology is guaranteed. When dealing with inter-plant-station topology, the incidence matrix notation method is used to judge the connectivity of the graph and mark the tree branches and link branches in the network, thus conducting static topology analysis of the power network. And then, when the network topology changes, the property of the changing branch is determined with the help of the loop matrix. On this basis, the local network topology and the loop matrix are updated by means of the broken circle method and the radius search method. Calculation examples verify the advantage of the proposed methods over traditional methods in calculation speed. With the proposed topology analysis methods, fast tracking of the switch status can be realized. Copyright © 2015 John Wiley & Sons, Ltd.

In two-dimensions, a fracture network consist of a system of branches and nodes that can be used to define both geometrical features, such as length and orientation, and the relationship between elements of the network – topology. Branch lengths are preferred to trace lengths as they can be uniquely defined, have less censoring and are more clustered around a mean value. Many important properties of networks are more related to topology than geometry. The proportions of isolated (I), abutting (Y) and crossing (X) nodes provide a basis for describing the topology that can be easily applied, even with limited access to the network as a whole. Node counting also provides an unbiased estimate of frequency and can be used in conjunction with fracture intensity to estimate the characteristic length and dimensionless intensity of the fractures. The nodes can be used to classify branches into three types – those with two I-nodes, one I-node and no I-nodes (or two connected nodes). The average number of connections per branch provides a measure of connectivity that is almost completely independent of the topology. We briefly discuss the extension of topological concepts to 3-dimensions.

Let L be a complete lattice. On L we define the so called quasi Scott topology, denoted by qSc. This topology is always larger than or equal to the Scott topology and smaller than or equal to the strong Scott topology. Results concerning the above topology are given. Also, we introduce and investigate the notions of q-continuous and q-algebraic complete lattices. Finally, we give and examine the quasi Lawson topology on a complete lattice. 1. Preliminaries Our reference for complete lattices are (2, 3, 8, 9). We shall frequently denote complete lattices with their underlying sets and write L for (L;D). The top element and the bottom element of a complete lattice L will be denoted by 1L and 0L, respectively.

In this paper, we define and study a new topology constructed from any given topology on a set, using irreducible sets. The manner in which this derived topology is obtained is inspired by how the Scott topology on a poset is constructed from its Alexandroff topology. This derived topology leads us to a weak notion of sobriety called k-bounded sobriety. We investigate the properties of this derived topology and k-bounded sober spaces. A by-product of our theory is a novel type of compactness, which involves crucially the Scott irreducible families of open sets. Some related applications on posets are also given.

In this paper, the spectral topology and quasi-spectral topology of proper prime A-ideals in an MV-module are introduced. We show that the spectral topology of proper ⋅-prime ideals of a PMV-algebra with unity for product, is the same as the spectral topology of proper prime ideals in an MV-algebra. Also we show that the set of all prime A-ideals in an MV-module with spectral topology is not T0 and T1 topological spaces but quasi-spectral topology is T0-space and is not T1-space. Finally, we investigate when the set of all prime A-ideals in an MV-module are Hausdorff and disconnected.

This paper addresses the problem of robust topology design of the multi-UAV formation. We use a digraph to model the topology of the multi-UAV formation, and propose the definition of anti-k-communication-destroying topology, meaning the system can still performs normally even when any arbitrary k communicating links are destroyed. By exploring the property of this kind of topology based on graph theory, we propose the algorithm Uniform-Cost Forest Search, UCFS, which is an extension of the classical search strategy uniform-cost search. The proposed algorithm would establish the anti-k-communication-destroying topology for the multi-UAV formation, with k + 1 minimal-cost edge-independent paths found for each UAV. Proof as well as examples validating the algorithm are provided in the paper. Finally, potential research directions are discussed.

Most optimization problems in the field of fluid mechanics can be classified as either topology or shape optimization. Although topology and shape have been considered mutually exclusive optimization methods since their inception, it is conceivable that they will find choicest solutions in tandem, with shape optimization refining a solution found by topology. However, linking the topology optimization problem to that of shape is not trivial and, to the authors' knowledge, has yet to be formally attempted. This paper pro- poses a novel transitional procedure that post-processes 2D adjoint topology solutions,fitting the interface between the solid and fluid topological domains to create a parameterized solution which can be used as either a CAD-compatible representation of the interface or a source for grid generation from which a shape optimization loop can be initialized. The interface to be fit can be extracted from any topological field with distinct fluid and solid domains, meaning that the proposed transition process is independent of the topology approach utilized. To conveniently describe the interface between the solid and fluid topological domains, the topology optimization process employed in this paper is ltered using the level set method. The interface is fit with non-uniform rational B-spline (NURBS) curves through application of sensitivitiesgarnered from the solution of an auxiliary inverse design problem which aims at reducing the difference between signed-distance fields generated about both the NURBS curve being optimized and the section of interface being fit. The geometry defined by the fit NURBS curves is then (optionally) used to build a boundary-fitted grid on which a shape optimization loop is performed. The parameterized result of the topology to shape transition process is compared to that of shape optimization in 2D cases with internal, incompressible fluid flows.

For a topological space [math] , it is a natural undertaking to compare its topology with the weak topology generated by a family of real-valued continuous functions on [math] . We present a necessary and sufficient condition for the coincidence of these topologies for an arbitrary family [math] . As a corollary, we give a new proof of the fact that families of functions which separate points on a compact space induce topologies that coincide with the original topology.

It is well-known that the Kullback–Leibler support condition implies posterior consistency in the weak topology, but is not sufficient for consistency in the total variation distance. There is a counter–example. Since then many authors have proposed sufficient conditions for strong consistency; and the aim of the present paper is to introduce new conditions with specific application to nonparametric mixture models with heavy–tailed components, such as the Student-$t$. The key is a more focused result on sets of densities where if strong consistency fails then it fails on such densities. This allows us to move away from the traditional types of sieves currently employed.

Two channels are said to be equivalent if they are degraded from each other. The space of equivalent channels with input alphabet $X$ and output alphabet $Y$ can be naturally endowed with the quotient of the Euclidean topology by the equivalence relation. A topology on the space of equivalent channels with fixed input alphabet $X$ and arbitrary but finite output alphabet is said to be natural if and only if it induces the quotient topology on the subspaces of equivalent channels sharing the same output alphabet. We show that every natural topology is $\sigma$-compact, separable and path-connected. On the other hand, if $|X|\geq 2$, a Hausdorff natural topology is not Baire and it is not locally compact anywhere. This implies that no natural topology can be completely metrized if $|X|\geq 2$. The finest natural topology, which we call the strong topology, is shown to be compactly generated, sequential and $T_4$. On the other hand, the strong topology is not first-countable anywhere, hence it is not metrizable. We show that in the strong topology, a subspace is compact if and only if it is rank-bounded and strongly-closed. We introduce a metric distance on the space of equivalent channels which compares the noise levels between channels. The induced metric topology, which we call the noisiness topology, is shown to be natural. We also study topologies that are inherited from the space of meta-probability measures by identifying channels with their Blackwell measures. We show that the weak-* topology is exactly the same as the noisiness topology and hence it is natural. We prove that if $|X|\geq 2$, the total variation topology is not natural nor Baire, hence it is not completely metrizable. Moreover, it is not locally compact anywhere. Finally, we show that the Borel $\sigma$-algebra is the same for all Hausdorff natural topologies., Comment: 43 pages, submitted to IEEE Trans. Inform. Theory and in part to ISIT2017

The most common construction of a generalized covering space is that of a topologized (appropriate version of a) path space X ˜ . There have been three suggested topologies on it, each with its advantages and disadvantages. They are called the Whisker topology, the Lasso topology and the quotient of the compact open topology. In this paper we study the relationship between these topologies. The main result consists of an example demonstrating that the Lasso topology is not finer that the compact open topology. Our results also apply to the topology of the fundamental group which appears naturally as a subspace of X ˜ .

The present paper aims to address a long-standing and challenging problem in structural topology optimization: explicit feature control of the optimal topology. The basic idea is to resort to the level set solution framework and impose constraints on the extreme values of the signed distance level set function used for describing the topology of the structure. Numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach.

We characterize the completely regular Lindelof spaces X such that C ( X ) with the finest splitting topology is Frechet, a topological group, or a k -space.

This paper will discuss about a new topology, obtained from a grill and a filter on the same set. The Characterizations and open base of the new topology are also aim of this paper. The generalized continuity is also a part of this paper.

Mathematical theorems in control theory are only of interest in so far as their assumptions relate to practical situations. The space of systems with transfer functions in , for example, has many advantages mathematically, but includes large classes of non–physical systems, and one must be careful in drawing inferences from results in that setting. Similarly, the graph topology has long been known to be the weakest, or coarsest, topology in which (1) feedback stability is a robust property (i.e. preserved in small neighbourhoods) and (2) the map from open-to-closed-loop transfer functions is continuous. However, it is not known whether continuity is a necessary part of this statement, or only required for the existing proofs. It is entirely possible that the answer depends on the underlying classes of systems used. The class of systems we concern ourselves with here is the set of systems that can be approximated, in the graph topology, by real rational transfer function matrices. That is, lumped parameter...

Let me start this chapter with a simple why-question: Why general topology? What is the main problem it wishes to solve? The answer is deceivingly simple: general topology aims at analyzing and describing topological spaces.

Two prime network interconnection topology used today in High Performance computing (HPC) are the fat tree and the torus topology. But due to the various advantages of torus network over fat tree, currently many HPC networks using fat tree are turning to the torus topology. In fat tree topology the switches have high end functionalities. Suppose a packet is traversing from the source to destination. If it has an address which is outside the range of addresses at a particular cluster head switch, then the switches at higher level with respect to the end nodes are used to route the packets to other clusters. Here checking for the destination address at all clusters is avoided, which is a great advantage of fat tree topology. On the other hand torus topology has no inheritance in the interconnection design while every node possess alike capability. This is the reason interval switching is not easy in torus network. A simple method of labeling is proposed which can be used to overcome this disadvantage of the torus topology and avail torus topology for using interval based switching. This paper shows need for labeling of nodes and four labeling schemes for the torus topology with their comparison performed with the graph theory.

This paper studies structural controllability of multi-agent systems over finite fields, where each agent is governed by general linear dynamics defined over finite fields. For fixed topology, an absolute protocol is proposed, under which we investigate the structural controllability of multi-agent systems over finite fields. It is shown that a multi-agent system is structurally controllable over a finite field if the graph has a spanning forest. For switching topology, an algebraic condition is given for the structural controllability of switched multi-agent systems.

Given an uncountable cardinal ℵ, the product space Iℵ, I=[0,1], is called a Tychonoff cube. A collection of closed subsets of a subspace Y of a Tychonoff cube Iℵ that covers Y determines a weak topology for Y. The collection of compact subsets of Iℵ that have a countable dense subset covers Iℵ. It is known from work of the author and I. Ivanšić that the weak topology generated by this collection is pseudo-compact. We are going to show that it is not compact. The author and I. Ivanšić have also considered weak topologies on some other ``naturally occurring'' subspaces of such Iℵ. The new information herein along with the previous examples will lead to the existence of vast naturally occurring classes of pseudo-compacta any set of which has a pseudo-compact product. Some of the classes consist of Tychonoff spaces, so the product spaces from subsets of these are also Tychonoff spaces.

We define and study the properties of some subsets of X with respect to a weak structure on X and generalize some already established results.

The material distribution based on an element density is adequate for most of 2-D topology optimization problems. However, in 3-D topology optimization it is usually difficult to obtain a smooth topological configuration and a virtual connectivity phenomenon easily appears in a low-density domain. A 3-D structural topology optimization and novel surface-smoothing scheme based on SIMP (solid isotropic microstructure with penalization) and sub-element bilinear interpolation has been developed using node densities and a corresponding computer program was written in order to validate the proposed method. Compared to a common element density method, the proposed method resulted in more enhanced smooth surfaces of optimum topology designs. To show the usefulness of the method, three examples of 3-D structural topology optimizations were illustrated.

In this paper, some properties of order topology and bi-Scott topology on a poset are obtained. Order-convergence in posets is further studied. Especially, a sufficient and necessary condition for order-convergence to be topological is given for some kind of posets.

A Snake is an active contour for representing object contours. Traditional snake algorithms are often used to represent the contour of a single object. However, if there is more than one object in the image, the snake model must be adaptive to determine the corresponding contour of each object. Also, the previous initialized snake contours risk getting the wrong results when tracking multiple objects in successive frames due to the weak topology changes. To overcome this problem, in this paper, we present a new snake method for efficiently tracking contours of multiple objects. Our proposed algorithm can provide a straightforward approach for snake contour rapid splitting and connection, which usually cannot be gracefully handled by traditional snakes. Experimental results of various test sequence images with multiple objects have shown good performance, which proves that the proposed method is both effective and accurate.