Quotient Space Research Articles

On Parametrization of the Symplectic Quotient of the Cartesian Product of Coadjoint Orbits of the Complex General Linear Group with Respect to its Diagonal Action

We consider the problem of coordinatizing the manifold constructed via the Marsden–Weinstein quotient. Rational canonical coordinates on the symplectic reduction with respect to the diagonal action of the general linear group on the Cartesian product of coadjoint orbits in the case of the complex general linear group are constructed on an algebraically open subset of the quotient space. The method is based on an iterative process of constructing projection-flag coordinates, and works if the matrices constituting the orbits have a sufficiently rich set of invariant subspaces.

The separable quotient problem for topological groups

The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces.For a topological group G there are four natural analogous problems: Does G have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. Positive answers to all four questions are proved for groups G which belong to the important classes of (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups.However, a surprising example of an uncountable precompact group G is produced which has no non-trivial separable quotient group other than the trivial group. Indeed Gτ has the same property, for every cardinal number τ ≥ 1.

Topologies on quotient space of matrices via semi‐tensor product

AbstractAn equivalence of matrices via semi‐tensor product (STP) is proposed. Using this equivalence, the quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices lead to the product topology and quotient topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the quotient space. This metric leads to a metric topology. A comparison for these three topologies is presented. Some topological properties are revealed.

Algorithms for quantum control without discontinuities: Application to the simultaneous control of two qubits

We propose a technique to design control algorithms for a class of finite dimensional quantum systems so that the control law does not present discontinuities. The class of models considered admits a group of symmetries, which allows us to reduce the problem of control to a quotient space where the control system is “fully actuated.” As a result, we can prescribe a desired trajectory, which is, to some extent, arbitrary, and derive the corresponding control. We illustrate this technique with examples and focus on the application to the simultaneous control of two non-interacting spin 12 particles with different gyromagnetic ratios in zero field nuclear magnetic resonance (NMR). Our method provides a flexible toolbox for the design of control algorithms to drive the state of finite dimensional quantum systems to any desired final configuration, with smooth controls.

VGHC: a variable granularity hierarchical clustering for community detection

Hierarchical clustering is an effective method for community detection. This kind of method usually selects clustering threshold for layering, which will affect the performance of community detection. Most of them select uniform clustering threshold. It means that the hierarchical structure of the community is obtained by merging synchronously. Actually, the communities are merged asynchronously . So, how to capture the adaptive hierarchical structure of the community is a challenge. In this paper, we propose a variable granularity method VGHC to construct the hierarchical structure based on quotient space theory (QST). QST uses cut set to form a hierarchical structure which represents multi-granular spaces. Firstly, initial granules are built with the important nodes as the granule center. Secondly, we choose the median clustering coefficient of the lower layer as the cut set, and then lower layer can be clustering to higher layer. Finally, the extended modularity (EQ) or modularity (Q) is taken as the evaluation criterion to select a certain layer in the hierarchical structure as the community detection result. Some overlapping nodes exist because of clustering mechanism. To get a non-overlapping community structure, a local modularity optimization approach (LMO) is used to deal with overlapping nodes. Experiments on seven real-world networks demonstrate that our method is effective for community detection in networks compared with the state-of-the-art algorithms.

A multi-granular network representation learning method

Granular computing (GrC) as a problem-solving concept and new information processing paradigm is deeply rooted in human thinking, which has attracted many researchers to study it theoretically, and has gradually applied to data-driven problems. Network embedding, as known as network representation learning, aiming to map nodes in network into a low-dimensional representation, is a data-driven problem. Most existing methods are based on a single granular, which learn representations from local structure of nodes. But global structure is important information on the network and has been proven to facilitate several network analysis tasks. Therefore, how to introduce GrC into network embedding to obtain a multi-granular network representation that preserves the global and local structure of nodes is a meaningful and tough task. In this paper, we introduce Quotient Space Theory, one of the GrC theories into network embedding and propose a Multi-Granular Network Representation Learning method based on Quotient Space Theory (MG_NRL, for short), which can preserve global and local structure at different granularities. Firstly, we granulate the network repeatedly to obtain a multi-granular network. Secondly, the embedding of the coarsest network is computed using any existing embedding method. Finally, the network representation of each granular layer is learned by recursively refining method from the coarsest network to original network. Experimental results on multi-label classification task demonstrate that MG_NRL significantly outperforms other state-of-the-art methods.

An approach to stability analyses in general relativity via symplectic geometry

We begin with a review of the statements of non-linear, linear and mode stability of autonomous dynamical systems in classical mechanics, using symplectic geometry. We then discuss what the Arnowitt–Deser–Misner (ADM) phase space and the ADM Hamiltonian of general relativity are, what constitutes a dynamical system, and subsequently present a nascent attempt to draw a formal analogy between the notions of stability in these two theories. We wish to note that we have not discussed here the construction of the reduced phase space by forming the quotient space of the constrained phase space with the gauge orbits. Our approach here is pedagogical and geometric, and the motivation is to unify and simplify a formal understanding of the statements regarding the stability of stationary solutions of general relativity. That is, typically the governing equations of motion of a Hamiltonian dynamical system are simply the flow equations of the associated symplectic Hamiltonian vector field, defined on phase space, and the non-linear stability analysis of its critical points have simply to do with the divergence of its flow there. Further, the linear stability of a critical point is related to the properties of the tangent flow of the Hamiltonian vector field. In the second half of this work, we posit that a study of the genericity of a particular black hole or naked singularity spacetime forming as an endstate of gravitational collapse is equivalent to an inquiry of how sensitive the orbits of the symplectic Hamiltonian vector field of general relativity are to changes in initial data. We demonstrate this by conducting a restricted non-linear stability analysis of the formation of a Schwarzschild black hole, working in the usual initial value formulation of general relativity.

Moduli Spaces of Framed G–Higgs Bundles and Symplectic Geometry

Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal $G$--bundle is a pair $(E_G,\, \phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $\phi$ assigns to each $x\, \in\, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,\, \phi)$ together with a section $\theta\, \in\, H^0(X,\, \text{ad}(E_G)\otimes K_X\otimes{\mathcal O}_X(D))$ such that $\theta(x)$ is compatible with the framing $\phi$ for every $x\, \in\, D$. We construct a holomorphic symplectic structure on the moduli space $\mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $\mathcal{M}_{FH}(G)$ to the moduli space $\mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,\, \theta)$ that forgets the framing, is Poisson. These results generalize \cite{BLP} where $(G,\, \{H_x\}_{x\in D})$ is taken to be $(\text{GL}(r,{\mathbb C}),\, \{\text{I}_{r\times r}\}_{x\in D})$. We also investigate the Hitchin system for $\mathcal{M}_{FH}(G)$ and its relationship with that for $\mathcal{M}_{H}(G)$.

Coset spaces and cardinal invariants

We present an example of a closed subgroup H of a topological group G such that the quotient space G/H is compact, first-countable, and satisfies $${\pi}w(G/H) < w(G/H)$$ . We show that if H is a closed neutral subgroup of a topological group G such that G/H has countable character (pseudocharacter), then G/H is metrizable (submetrizable). We also prove that if G is an $$\omega$$ -balanced paratopological group and H is a closed neutral subgroup of G such that G/H is Hausdorff and first-countable, then G/H is submetrizable. Assuming $$MA+{\neg}CH$$ , we construct a topological group G with a closed subgroup H such that the quotient space G/H is compact and satisfies $${\pi}{\chi}(G/H) < {\chi}(G/H)$$ . We also show that if H is a closed neutral subgroup of a topological group G, then the $$\pi$$ -character ( $$\pi$$ -weight) and the character (weight) coincide in the quotient space G/H. The same conclusion is valid if H is an arbitrary closed subgroup of a feathered (equivalently, almost metrizable) topological group G.

Nonnegatively curved quotient spaces with boundary

Let M be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group $$\mathsf {G}$$ in a way that the quotient space $$M/\mathsf {G}$$ has nonempty boundary. Let $$\pi : M \rightarrow M/\mathsf {G}$$ denote the quotient map and B be any boundary stratum of $$M/\mathsf {G}$$. Via a specific soul construction for $$M/\mathsf {G}$$, we construct a smooth closed submanifold N of M such that $$M {\setminus } \pi ^{-1}(B)$$ is diffeomorphic to the normal bundle of N. As an application we show that a simply connected torus manifold admitting an invariant metric of nonnegative curvature is rationally elliptic.

Signal reconstruction without phase by norm retrievable frames

ABSTRACT This paper is concerned with the signal reconstruction without phase by norm retrievable frames in finite dimensional real Hilbert space . Specifically, we show that the nonlinear analysis maps are injective, with and , where is a norm retrievable frame for and is the quotient space corresponding to a special equivalence relation on . Using Householder matrices, the members of any are characterized. Also, we show that α has upper Lipschitz bound and local lower Lipschitz bound , with respect to an appropriate metric d on , where A and B are the lower and upper frame bounds of the norm retrievable frame , respectively. Additionally, we show that β has upper Lipschitz bound and local lower Lipschitz bound , with respect to an appropriate metric D on , where and B is the upper frame bound of the norm retrievable frame .

Topologies of the quotient spaces induced by the M-topological plane and the infinite M-topological sphere

The present paper studies two topologies of the quotient spaces related to the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological plane. The topologies are exactly proved to be the cofinite particular point and the excluded two points topology. We also concern separation axioms and nowhere dense subset properties of the quotient spaces of the M-topological plane. Furthermore, we investigate both a particular point and an excluded point topological structure associated with the quotient spaces of the M-topological plane. Finally, we investigate the fixed point property of the compactification of the M-topological plane. Unlike the non-fixed point property of the Hausdorff compactification of the Euclidean topological plane, we prove that every continuous self-bijection of the compactification of the M-topological plane has a fixed point.

A REMARK ON THE SEPARABLE QUOTIENT PROBLEM FOR TOPOLOGICAL GROUPS

The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space $B$ has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when $B$ is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group $G$ have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if $G$ is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if $G$ is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite $k_{\unicode[STIX]{x1D714}}$-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus $\mathbb{T}^{\unicode[STIX]{x1D714}}$, which is an infinite-dimensional metrisable group, as a quotient group.

Granulation of Hypernetwork Models under the q-Rung Picture Fuzzy Environment

In this paper, we define q-rung picture fuzzy hypergraphs and illustrate the formation of granular structures using q-rung picture fuzzy hypergraphs and level hypergraphs. Further, we define the q-rung picture fuzzy equivalence relation and q-rung picture fuzzy hierarchical quotient space structures. In particular, a q-rung picture fuzzy hypergraph and hypergraph combine a set of granules, and a hierarchical structure is formed corresponding to the series of hypergraphs. The mappings between the q-rung picture fuzzy hypergraphs depict the relationships among granules occurring at different levels. The consequences reveal that the representation of the partition of the universal set is more efficient through q-rung picture fuzzy hypergraphs and the q-rung picture fuzzy equivalence relation. We also present an arithmetic example and comparison analysis to signify the superiority and validity of our proposed model.

When is the underlying space of an orbifold a manifold?

We classify orthogonal actions of finite groups on Euclidean vector spaces for which the corresponding quotient space is a topological, homological, or Lipschitz manifold, possibly with boundary. In particular, our results answer the question of when the underlying space of an orbifold is a manifold.

Knowledge Distance Measure for the Multigranularity Rough Approximations of a Fuzzy Concept

Different rough approximation spaces could be induced for an information system by its different attribute subsets, thus the multigranularity rough approximations of a fuzzy concept could be developed. Research on the uncertainty in multi-granulation spaces becomes a basic issue of uncertainty measure. If the uncertainty measure is not accurate enough, two different rough approximation spaces of a fuzzy concept may have the same uncertainty, and the difference between them for describing a fuzzy concept cannot be reflected. In this case, attribute reduction, granularity selection, and multigranularity measure cannot be conducted effectively. Therefore, establishing an uncertainty measure model with strong distinguishing ability in multi-granulation spaces is a key issue in uncertainty knowledge processing. In this paper, this problem will be solved in the view of knowledge distance. First, a fuzzy knowledge distance measure (FKD) based on the Earth Mover's distance is introduced. Even if two rough approximation spaces possess the same uncertainty when describing a fuzzy concept, they can be discriminated by FKD. Then, by studying the change rules of the FKD in a hierarchical quotient space structure, it is found that the FKD between any two rough approximation spaces in an HQSS is equal to the difference between their granularity measure or information measure. Furthermore, in order to show the applicability of the FKD, the FKD is used in granularity selection, attribute reduct, and multigranularity measure. The experimental results show that the FKD-based attribute significance function has a more powerful ability to obtain shorter reduct and it is more robustness, which show the effectiveness of the FKD.

Vector Bundle of Prym Differentials over Teichm¨uller Spaces of Surfaces with Punctures

In this paper we study multiplicative meromorphic functions and differentials on Riemann surfaces of finite type. We prove an analog of P. Appell’s formula on decomposition of multiplicative functions with poles of arbitrary multiplicity into a sum of elementary Prym integrals. We construct explicit bases for some important quotient spaces and prove a theorem on a fiber isomorphism of vector bundles and n!-sheeted mappings over Teichm¨uller spaces. This theorem gives an important relation between spaces of Prym differentials (Abelian differentials) on a compact Riemann surfaces and on a Riemann surfaces of finite type

An m-Polar Fuzzy Hypergraph Model of Granular Computing

An m-polar fuzzy model plays a vital role in modeling of real-world problems that involve multi-attribute, multi-polar information and uncertainty. The m-polar fuzzy models give increasing precision and flexibility to the system as compared to the fuzzy and bipolar fuzzy models. An m-polar fuzzy set assigns the membership degree to an object belonging to [ 0 , 1 ] m describing the m distinct attributes of that element. Granular computing deals with representing and processing information in the form of information granules. These information granules are collections of elements combined together due to their similarity and functional/physical adjacency. In this paper, we illustrate the formation of granular structures using m-polar fuzzy hypergraphs and level hypergraphs. Further, we define m-polar fuzzy hierarchical quotient space structures. The mappings between the m-polar fuzzy hypergraphs depict the relationships among granules occurring at different levels. The consequences reveal that the representation of the partition of a universal set is more efficient through m-polar fuzzy hypergraphs as compared to crisp hypergraphs. We also present some examples and a real-world problem to signify the validity of our proposed model.

Generalized Open Mapping Theorem for X-Normed Spaces

The theory of X-normed spaces over non-Archimedean valued fields with valuations of higher rank was introduced by H. Ochsenius and W. H. Schikhof in [9] and further developed in [10–12, 16, 17] and [13]. In order to obtain results like the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem, H. Ochsenius and W. H. Schikhof used 1st countability conditions in the value group of the based field. In this article the author develops a new tool to work with transfinite induction simplifying the techniques employed in X-normed spaces, thus accomplishing a Generalized Baire Category Theorem that allows the proof of an Open Mapping Theorem for X-normed spaces without restrictions on the value group of the based field. Additionally, some contributions to the theory of X-normed spaces are presented regarding quotient spaces.

On the Quotients of Regular Operators

We give some results about quotients of regular operators on Banach lattices by the linear span of the positive M-weakly and positive L-weakly compact operators. We also present a representation of the quotient space created by the linear span of the positive L-weakly compact operators.