Quotient Metrics Research Articles

Group-Invariant Max Filtering

AbstractGiven a real inner product space V and a group G of linear isometries, we construct a family of G-invariant real-valued functions on V that we call max filters. In the case where $$V={\mathbb {R}}^d$$ V = R d and G is finite, a suitable max filter bank separates orbits, and is even bilipschitz in the quotient metric. In the case where $$V=L^2({\mathbb {R}}^d)$$ V = L 2 ( R d ) and G is the group of translation operators, a max filter exhibits stability to diffeomorphic distortion like that of the scattering transform introduced by Mallat. We establish that max filters are well suited for various classification tasks, both in theory and in practice.

Curvature of the space of stability conditions

Motivated by the study of the autoequivalence group of triangulated categories via isometric actions on metric spaces, we consider curvature properties (CAT(0), Gromov hyperbolic) of the space of Bridgeland stability conditions with the canonical metric defined by Bridgeland. We then prove that the metric is neither CAT(0) nor hyperbolic, and the quotient metric by the natural $\mathbb{C}$-action is not CAT(0) in case of the Kronecker quiver. Moreover, we also show the hyperbolicity of pseudo-Anosov functors defined by Dimitrov-Haiden-Katzarkov-Kontsevich, which yields the lower-bound of entropy by the translation length. Finally, pseudo-Anosov functors in case of curves have been completely classified.

Psychological assessment of individuals with Mal de Débarquement Syndrome.

To report on the psychological, personality, and behavioral profiles of individuals with persistent Mal de Débarquement Syndrome (MdDS). Individuals with MdDS who participated in neuromodulation clinical trials between May 2013 and June 2019 completed a series of standardized psychological questionnaires and underwent the Structural Clinical Interview for DSM-IV-TR (SCID) for specific psychiatric diagnoses. All data reported are from baseline assessments prior to any study interventions. Scores were compared to population norms for adult women. Complete datasets were available for 55 women. Mean age of onset of MdDS was 49.0 ± 11.9 years (range 22-69years) and median duration of illness of 22months (6months-20years). SCID results were as follows: healthy (48.1%), any lifetime Major Depressive Disorder (35.2%, 7.4% current); any lifetime history of anxiety disorder (11.1%); any lifetime substance use disorders (18.5%, 0% current). Compared to population norms, the MdDS group scored significantly higher on the Patient Health Questionnaire-9 depression scale and the Generalized Anxiety Disorder 7 (GAD-7) anxiety scale, but only the GAD-7 correlated with symptom severity. The NEO-Five Factor Inventory for personality, Positive and Negative Affect Schedule, Behavioral Inhibition System/Behavioral Activation System Scale, and the Empathy Quotient metrics did not correlate with duration of illness. Disability assessed by the 12-item World Health Organization Disability Assessment Schedule 2.0 was 25.7 ± 6.7, comparable to reports for concussion. Disability correlated with severity of depression, anxiety, neuroticism, and affect but not to severity of MdDS. Psychological profiles of MdDS relate to disability but not to duration of illness.

Erratum to: Quotient metrics with applications in convex geometry

Erratum to: Quotient metrics with applications in convex geometry

Quotient metrics with applications in convex geometry

Burago et al. (Graduate studies in mathematics, AMS, 2001) considered quotient metric spaces consisting of orbits with respect to some isometry groups. We extend their approach over some semigroups of transformations. We are concerned with quotient semi-metrics in spaces of generalized orbits and give conditions sufficient for these semi-metrics to be metrics. We apply our approach to hyperspaces of compact convex subsets of Euclidean n-space and to that of convex bodies, endowed, respectively, with the Hausdorff metric and with the symmetric difference metric.

Flat deformation of a spacetime admitting two commuting Killing fields

It is shown that, given an analytic Lorentzian metric on a 4-manifold, gab, which admits two Killing vector fields, there exists a local deformation law ηab = a gab + b Hab, where Hab is a two-dimensional projector, such that ηab is flat and admits the same Killing vectors. We also characterize the particular case when the projector Hab coincides with the quotient metric. We apply some of our results to general stationary axisymmetric spacetimes.

Homogeneous manifolds from noncommutative measure spaces

Let M be a finite von Neumann algebra with a faithful normal trace τ. In this paper we study metric geometry of homogeneous spaces O of the unitary group U M of M , endowed with a Finsler quotient metric induced by the p-norms of τ, ‖ x ‖ p = τ ( | x | p ) 1 / p , p ⩾ 1 . The main results include the following. The unitary group carries on a rectifiable distance d p induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance d ˙ p that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance d O , p . We prove that the distances d ˙ p and d O , p coincide. Based on this fact, we show that the metric space ( O , d ˙ p ) is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O . We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of U M with the p-norm.

Know your corporate IQ

PurposeMost business executives understand the importance of developing new products to support future growth. But they often fail to actively monitor their performance on this metric relative to their peers. Furthermore, many executives do not understand how innovation is critical to driving margins as well as growth. Tracking a company's Innovation Quotientsm, enables executives to assess how they are doing versus competitors in successfully developing new products and services. Examining a company's development pipeline enables executives to forecast their future Innovation Quotient and provides insights into the future performance of the business.Design/methodology/approachIn this article, Jackson considers a number of case examples of businesses that actively monitor their innovation performance and use it as an indicator of future success and value creation. Examples industries cited in the article include consumer products, electronics, industrial products and investment banking. The author then draws lessons that can be applied broadly across any business seeking to improve future growth and profitability through greater innovation.FindingsCompanies should proactively monitor their relative innovation performance compared to competitors using the Innovation Quotient metric. Measuring and monitoring a company's innovation rate and Corporate IQ can help management teams to take the tough choices, and build value for the long term when faced with the pressures of day to day performance. The companies that do the best job of building innovation capabilities and a strong development pipeline in tough times will be rewarded by even higher growth and margins when economic conditions improve.Originality/valueBusiness managers have always recognized that innovation is important to growth. This article extends that thinking to reinforce how a company's innovation performance relative to peers can drive not just future growth but higher future profitability as well.

An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach

Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1 . We investigate several Riemannian metrics on shape space: L 2 -metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.

Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces

The shape-space ∑ m k whose points σ represent the shapes of not totally degenerate k-ads in Rm is introduced as a quotient space carrying the quotient metric. When m = 1, we find that ∑ 1 k = S K − 2 when m ⩾ 3, the shape-space contains singularities. This paper deals mainly with the case m = 2, when the shape-space ∑ 2 k can be identified with a version of CPk−2. Of special importance are the shape-measures induced on CPk−2 by any assigned diffuse law of distribution for the k vertices. We determine several such shape-measures, we resolve some of the technical problems associated with the graphic presentation and statistical analysis of empirical shape distributions, and among applications we discuss the relevance of these ideas to testing for the presence of non-accidental multiple alignments in collections of (i) neolithic stone monuments and (ii) quasars. Finally the recently introduced Ambartzumian density is examined from the present point of view, its norming constant is found, and its connexion with random Crofton polygons is established.