Quotient Space Research Articles

Nasal similarity measure of 3D faces based on curve shape space

We propose a novel method for measuring the nasal similarity among 3D faces. Firstly, we construct a representation for the nose shape, which is composed of a set of geodesic curves, each crosses the bridge of the nose. Next, using these geodesic curves, we formulate a similarity measure to compare among noses in the curve shape space. Under the Riemannian framework, the shape space is a quotient space for which the scaling, translation and rotation are removed. Since the nose similarity measure is based on the shape comparison, the proposed method has the following advantages: (1) the similarity measure is robust to facial expressions since the nose is not affected by facial expressions; (2) the geometric features of the nose shape match well with the human perception; (3) the similarity measure is independent of the mesh grid because the chosen nose curves are not sensitive to the triangular mesh model. We construct a nasal hierarchical structure for noses organization which is based on nose similarity measure results. In our experiments, we evaluate the performance of the proposed method and compare it with competing methods on three public face databases namely, FRGC2.0, Texas3D and BosphorusDB. The results show superiority of the proposed method in terms of both the speed and the accuracy when the nasal measurements are processed in the nasal hierarchical structure and the nasal samples with low sampling rate (5%-25% of original point cloud).

Relational granulation method based on Quotient Space Theory for maximum flow problem

Granular computing (GrC) is a problem-solving concept deeply rooted in human thinking. GrC, as a new and rapidly growing paradigm of information processing, has attracted the attention of many researchers and practitioners. GrC is related to granulation, i.e., a process of drawing a set of objects (or points) together based on their indiscernibility, similarity, proximity, or functionality. In general, two types of granulation processes exist: functional granulation and relational granulation. If the process is based entirely on the attributes of the objects, it is known as functional granulation, whereas if the granulation process is based on the relationship between objects, it is known as relational granulation. This paper proposes a novel method, called the maximum flow based on Quotient Space Theory (MF−QST), for solving the maximum flow problems based on Quotient Space Theory for relational granulation. Using the method MF−QST, substructures are first detected, and the community is described as a substructure. Next, the relational granulation criterion is discussed in detail. The substructure that satisfies the relational granulation criterion is regarded as a coarse-grained node. Subsequently, the construction of a quotient network that is coarser than the original network is described. Finally, the maximum flow algorithm is used to compute the maximum flow on the quotient network as the approximated maximum flow on the original network within a much shorter period of time. Experimental results demonstrate that the novel method MF−QST reduces the cumulative running time after simplifying the network structure with a low error rate. The size of the quotient network is significantly reduced, and the node and edge scales are reduced to 20.59% and 21.62% on average, respectively.

A Letter Concerning Leonetti’s Paper ‘Continuous Projections Onto Ideal Convergent Sequences’

Leonetti proved that whenever $${\mathcal {I}}$$ is an ideal on $${\mathbb {N}}$$ such that there exists an uncountable family of sets that are not in $${\mathcal {I}}$$ with the property that the intersection of any two distinct members of that family is in $${\mathcal {I}}$$ , then the space $$c_{0,{\mathcal {I}}}$$ of sequences in $$\ell _\infty $$ that converge to 0 along $${\mathcal {I}}$$ is not complemented. We provide a shorter proof of a more general fact that the quotient space $$\ell _\infty / c_{0,{\mathcal {I}}}$$ does not even embed into $$\ell _\infty $$ .

Granule structures, distances and measures in neighborhood systems

High-dimensional, quantity, uncertain and diverse data sets bring serious challenges to the development of intelligent systems. Granular computing is a theoretical approach to deal with uncertain and massive data, including rough sets, fuzzy sets, quotient spaces, covering rough sets, neighborhood rough sets and etc. In this paper, by introducing the neighborhood rough set model, some structured data named neighborhood granules are formed to achieve the cognition of a neighborhood system. Then, a three-level structure of granules in the neighborhood system is proposed: the neighborhood granule, the neighborhood granule swarm and the neighborhood granule library. The size measures of neighborhood granules and neighborhood granule swarms are also presented. Furthermore, we define a variety of distance measures for the neighborhood granules and the neighborhood granule swarms, and discuss their properties and relationships. Finally, considering the uncertainties of neighborhood systems, we propose the uncertainty measures of various neighborhood granules from the perspectives of algebra and entropy, and prove the monotonicity principle of these measures. Theoretical analysis and examples show that the granule structures, distances and measures in neighborhood systems are effective tools for complex data measuring and classifying.

A core decomposition of compact sets in the plane

A Peano continuum means a locally connected continuum. A compact metric space is called a Peano compactum if all its components are Peano continua and if for any constant C>0 all but finitely many of its components are of diameter less than C. Given a compact set K⊂C, there usually exist several upper semi-continuous decompositions of K into subcontinua such that the quotient space, equipped with the quotient topology, is a Peano compactum. We prove that one of these decompositions is finer than all the others and call it the core decomposition of K with Peano quotient. This core decomposition gives rise to a metrizable quotient space, called the Peano model of K, which is shown to be determined by the topology of K and hence independent of the embedding of K into C. We also construct a concrete continuum K⊂R3 such that the core decomposition of K with Peano quotient does not exist. For specific choices of K⊂C, the above mentioned core decomposition coincides with two models obtained recently, namely the locally connected model for unshielded planar continua (like connected Julia sets of polynomials) and the finitely Suslinian model for unshielded planar compact sets (like polynomial Julia sets that may not be connected). The study of such a core decomposition provides partial answers to several questions posed by Curry in 2010. These questions are motivated by other works, including those by Curry and his coauthors, that aim at understanding the dynamics of a rational map f:Cˆ→Cˆ restricted to its Julia set.

Perfect subspaces of quadratic laminations

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary is defined as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady (1984) and, separately, by Thurston (1985) who used quadratic invariant geolaminations as a major tool. We showed earlier that the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant geolaminations with the Hausdorff distance topology. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that used for the Mandelbrot set. The resulting quotient space is obtained from the Mandelbrot set by ipinching the transitions between adjacent hyperbolic components. In the second case we identify renormalizable geolaminations that can be irenormalized to the same hyperbolic geolamination while no two non-renormalizable geolaminations are identified.

Galois coverings of the product of the projective lines by abelian surfaces

We consider the quotient space of an abelian surface by a finite subgroup of the automorphism group. We classify the analytic representation of the group and the branch divisor of the natural projection to the quotient space, where the quotient space is isomorphic to the product of the projective lines.

On the essential spectrum of the diagonal of an operator

ABSTRACT Let T be a bounded linear operator on a Banach space X, and let M be a closed T-invariant subspace of X. In this paper, we investigate the relationships between the essential spectrum and the Browder spectrum of T, and those of the operators and , where is the restriction of T to M, and is the operator induced by T on the quotient space X/M.

Equiuniform Quotient Spaces

The notion of a quotient space of a G-Tychonoff space is introduced. The universal property of this space is established.

Cellularity in quotient spaces of topological groups

Abstract Let G be an arbitrary topological group. We prove that the cellularity of G is equal to the cellularity of the quotient space G / K {G/K} for every compact subgroup K of G.

Product evolutionary design driven by environmental performance

This article is in terms of product environmental performance demand and proposes four structure evolutionary operation modes which include combined evolutionary method, decomposition evolutionary method, replacement evolutionary method, and material-changing evolutionary method to express the structure evolutionary process of products. Through the quotient space theory and proposed method combined with probability statistics, probability mapping from environmental performance to product structure is established and the evolutionary individuals with outstanding environmental performance are listed. Through the analysis to the specific conditions of the evolutionary individuals, the design constraints are extracted, and the objective function of environmental performance is established. This article presents an interactive genetic algorithm as evolutionary algorithm and combines it with four structure evolutionary operation modes to conduct corresponding gene manipulation and generates evolutionary product. Finally, the proposed methodology is successfully applied to engine gear chamber and the environmental impact is found to be better than before evolution.

On the GIT quotient space of quintic surfaces

We describe the GIT compactification for the moduli space of smooth quintic surfaces in ${\mathbb {P}^{3}}$. In particular, we show that a normal quintic surface with at worst isolated double points or minimal elliptic singularities is stable. We also describe the boundary of the GIT quotient, and we discuss the stability of the nonnormal surfaces.

Various kinds of fuzzy quotient hypervector spaces

Various kinds of fuzzy quotient hypervector spaces

N-Normed Spaces with Norms of Its Quotient Spaces

The concept of n-normed spaces is a generalization of the concept of normed spaces. Some characteristics of n-normed spaces have been discussed by many researchers. These spaces are usually observed using a set of n linear independent vectors. In this paper, we will construct quotient spaces of an n-normed space with respect to n linear independent vectors. We define a norm in each quotient space by using the n-norm that we have. Norms of these quotient spaces will be a new viewpoint in observing characteristics of n-normed spaces.

Bayesian Linear Size-and-Shape Regression with Applications to Face Data

Regression models for size-and-shape analysis are developed, where the model is specified in the Euclidean space of the landmark coordinates. Statistical models in this space (which is known as the top space or ambient space) are often easier for practitioners to understand than alternative models in the quotient space of size-and-shapes. We consider a Bayesian linear size-and-shape regression model in which the response variable is given by labelled configuration matrix, and the covariates represent quantities such as gender and age. It is important to parameterize the model so that it is identifiable, and we use the LQ decomposition in the intercept term in the model for this purpose. Gamma priors for the inverse variance of the error term, matrix Fisher priors for the random rotation matrix, and flat priors for the regression coefficients are used. Markov chain Monte Carlo algorithms are used for sampling from the posterior distribution, in particular by using combinations of Metropolis-Hastings updates and a Gibbs sampler. The proposed Bayesian methodology is illustrated with an application to forensic facial data in three dimensions, where we investigate the main changes in growth by describing relative movements of landmarks for each gender over time.

The Algebraic Structure of Quantity Calculus

Abstract The algebraic structure underlying the quantity calculus is defined axiomatically as an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them bound by algebraic restrictions. Subspaces, tensor product, and quotient spaces are considered, as well as homomorphisms to end with a classification theorem of these structures. The new structure provides an axiomatic foundation of quantity calculus which is centered on the concept of dimension, rather than on the concept of unit, which is regarded as secondary, and uses only integer exponents of the dimensions.

Carleson measures and chord-arc curves

Following Semmes and Zinsmeister, we continue the study of Carleson measures and their invariance under pull-back and push-forward operators. We also study the analogous statements for vanishing Carleson measures. As an application, we show that some quotient space of the space of chord-arc curves has a natural complex structure.

The Gibbs Paradox.

The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs’ notion of ‘generic phase’). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. ‘Distinguishability’, in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.

ON THE COMPLETION OF -METRIC SPACES

Based on the metrisation of$b$-metric spaces of Paluszyński and Stempak [‘On quasi-metric and metric spaces’,Proc. Amer. Math. Soc.137(12) (2009), 4307–4312], we prove that every$b$-metric space has a completion. Our approach resolves the limitation in using the quotient space of equivalence classes of Cauchy sequences to obtain a completion of a$b$-metric space.

General Embedding Theorem

The quotient space of an arbitrary Banach space B by any subspace B1 ⊂ B equipped with the norm \(||b|B/{B_1}|| = \mathop {\inf }\limits_{f \in B/{B_1}} ||f|b||\) is considered. In the case where the infimum is a minimum, i.e., it is attained at some element, a formula for this element is presented. The proof is based on restating the original problem in dual spaces with the help of corresponding Legendre transforms. Although the original problem is nonlinear, it is found that its formulation in dual spaces is always linear and solvable. The results are applied to the general theory of boundary value problems for differential equations of mathematical physics.