Isometry Research Articles

Geometric and isometric properties of two classes of sequence spaces

We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space X n X_n , the completion of c 00 c_{00} relatively to the norm \[ ‖ x ‖ n = sup { ∑ i ∈ F | x i | , with , F of cardinality at most n } . \|x\|_n = \sup \left \{\sum _{i \in F}|x_i|, \text { with }, F \text { of cardinality at most } n\right \}. \] For this space, we show that the isometry group of X n X_n consists of standard isometries. Second, we consider a Tsirelson-type space, T [ θ , A n ] T[\theta , A_n] ( 0 > θ > 1 0>\theta >1 ) associated with the regular family A n A_n , comprised of all subsets of N \mathbb {N} with at most n n elements. For this space, we characterize the extreme points and show that for θ ≤ 1 n \theta \leq \frac {1}{n} , T [ θ , A n ] T[\theta , A_n] supports standard isometries. We also derive the form for the surjective isometries of T [ θ , A 3 ] T[\theta , A_3] .

Isometry groups of inductive limits of metric spectral triples and Gromov–Hausdorff convergence

AbstractIn this paper, we study the groups of isometries and the set of bi‐Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latrémolière. In particular, we prove that these groups and sets are compact in the automorphism group of the spectral triple ‐algebra with respect to the Monge–Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov–Hausdorff propinquity, a noncommutative analogue of the Gromov–Hausdorff distance, when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.

Primary Schoolers' Response to a Multisensory Serious Game on Cartesian Plane Coordinates in Immersive Virtual Reality.

The Cartesian coordinate system is a fundamental concept for mathematics and science and poses a teaching challenge at primary school level. Learning the Cartesian coordinate system has the potential to promote numerical cognition through number-space associations, as well as core geometric concepts, including isometric transformations, symmetry, and shape perception. Immersive virtual reality (VR) facilitates embodied forms of teaching and learning mathematics through whole-body sensorimotor interaction and offers benefits as a platform to learn the Cartesian coordinate system compared with "real world" classroom activities. Our goal was to validate the Cartesian-Garden, a serious game designed to provide an educationally robust but engaging vehicle to teach these concepts in primary-level mathematics in a multisensory VR environment. In the game, the child explores a Cartesian-Garden, that is, a field of flowers in which each flower corresponds to x and y coordinates. Specifically, we tested whether exploring numbers spatially represented improved spatial and numerical skills independently from the use of VR. Children (n = 49; age 7-11 years old) were divided into experimental and age-matched control groups. The experimental group explored the Cartesian-Garden and picked flowers corresponding to target coordinates; the control group played a VR game unrelated to Cartesian coordinates. To quantify potential improvements, children were tested before and after training with perceptual tests investigating number line and spatial thinking. The results point toward differential age-related improvements depending on the tested concept, especially for the number line. This study provides the guidelines for the successful use of the Cartesian-Garden game, beneficial for specific age groups.

Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

For any pseudo-Riemannian hyperbolic space X over R,C,H or O, we show that the resolvent R(z)=(□−zId)−1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators Cc∞(X)→D′(X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D′(X) forms a representation of the isometry group of X, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces.For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R(z) extends, and the residue representations can be infinite-dimensional.

Group Actions on Twisted Sums of Banach Spaces

We study bounded actions of groups and semigroups G on exact sequences of Banach spaces from the point of view of (generalized) quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of G-centralizer and G-equivariant map. We will show that when (A) G is an amenable group and (U) the target space is complemented in its bidual by a G-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of G-centralizers; and that, under (A) and (U), G-centralizers are bounded perturbations of G-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of L_p(0,1), pne 2 on the Kalton–Peck space Z_p, certain non-unitarizable triangular representations of the free group {mathbb {F}}_infty on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of L_p-spaces. In the penultimate section we consider the category of G-Banach spaces and study its exact sequences, showing that, under (A) and (U), G-splitting and usual splitting coincide. The purpose of the final section is to present some applications, showing that several previous result are optimal and to suggest further open lines of research.

Planar In-Hand Manipulation Using Primitive Rotations Based on Isometric Transformations

Achieving in-hand manipulation of a grasped object is challenging in robotic communities. This article proposes a planar intrinsic in-hand manipulation approach based on isometric transformation. We demonstrate that the isometric transformation (excluding reflections) can be reduced to rotations with two fixed centers, if more than two rotations around two different centers are performed. Furthermore, this article applies this theory to reposition and reorient a grasped object by performing three or more primitive rotations sequentially. We further analyze the feasible manipulation space, where the desired position and orientation of a grasped object can be attained. The theoretical results have been verified through simulations and experiments. In addition, we performed several real-world tasks using a novel robotic hand with two rotational degrees of freedom at each finger, demonstrating that the proposed in-hand manipulation method can reposition and reorient different grasped objects in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{SE}(2)$</tex-math></inline-formula> .

VSP-Fuse: Multifocus Image Fusion Model Using the Knowledge Transferred From Visual Salience Priors

Multifocus image fusion (MFIF), as an efficient way to improve the visual effect of images with partial focus defects, is of great significance in the field of image enhancement. According to the imaging principle of the lens, we summarize the visual salience priors (VSP) from the daily photo scene and two relationships from MFIF. Thereby, an edge-sensitive model for MFIF is presented in this study. Supported by VSP, we consider the correlation between salience object detection (SOD) and MFIF, and select the former as a pre-training task. SOD provides the network with realistic depth of field and bokeh effects to learn, and enhances the network’s ability to extract and express the edges of focused objects. Meanwhile, given the scarcity of real multifocus training sets, we propose a randomized approach to generate massive training sets and pseudo-labels based on limited unlabeled data. Besides, two attention modules are designed based on isometric domain transformation (IDT) in the traditional edge-preservation field. IDT removes interference information from feature maps in a low-cost manner, thereby facilitating channel-wise and spatial-wise weight assignments. Experimental results on four datasets show that the performance of our model is superior to that of many supervised models, without the need of any real MFIF training set.

Ladder symmetries of black holes and de Sitter space: love numbers and quasinormal modes

In this note, we present a synopsis of geometric symmetries for (spin 0) perturbations around (4D) black holes and de Sitter space. For black holes, we focus on static perturbations, for which the (exact) geometric symmetries have the group structure of SO(1,3). The generators consist of three spatial rotations, and three conformal Killing vectors obeying a special melodic condition. The static perturbation solutions form a unitary (principal series) representation of the group. The recently uncovered ladder symmetries follow from this representation structure; they explain the well-known vanishing of the black hole Love numbers. For dynamical perturbations around de Sitter space, the geometric symmetries are less surprising, following from the SO(1,4) isometry. As is known, the quasinormal solutions form a non-unitary representation of the isometry group. We provide explicit expressions for the ladder operators associated with this representation. In both cases, the ladder structures help connect the boundary condition at the horizon with that at infinity (black hole) or origin (de Sitter space), and they manifest as contiguous relations of the hypergeometric solutions.

Isometry groups of six-dimensional filiform nilmanifolds

Isometry groups of six-dimensional filiform nilmanifolds

Simultaneous linearization of diffeomorphisms of isotropic manifolds

Suppose that M is a closed isotropic Riemannian manifold and that R_1,\dots ,R_m generate the isometry group of M . Let f_1,\dots ,f_m be smooth perturbations of these isometries. We show that the f_i are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian [Duke Math. J. {136}, 475–505 (2007)] from S^n to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work.

Consistent truncations from the geometry of sphere bundles

In this paper, we present a unified perspective on sphere consistent truncations based on the classical geometric properties of sphere bundles. The backbone of our approach is the global angular form for the sphere. A universal formula for the Kaluza-Klein ansatz of the flux threading the n-sphere captures the full nonabelian isometry group SO(n + 1) and scalar deformations associated to the coset SL(n + 1, ℝ)/SO(n + 1). In all cases, the scalars enter the ansatz in a shift by an exact form. We find that the latter can be completely fixed by imposing mild conditions, motivated by supersymmetry, on the scalar potential arising from dimensional reduction of the higher dimensional theory. We comment on the role of the global angular form in the derivation of the topological couplings of the lower-dimensional theory, and on how this perspective could provide inroads into the study of consistent truncations with less supersymmetry.

A supersymmetric nonlinear sigma model analogue of the ModMax theory

A decade ago, it was shown that associated with any model for U(1) duality-invariant nonlinear electrodynamics there is a unique U(1) duality-invariant supersymmetric nonlinear sigma model formulated in terms of chiral and complex linear superfields. Here we study the mathcal{N} = 1 superconformal σ-model analogue of the conformal duality-invariant electrodynamics known as the ModMax theory. We derive its dual formulation in terms of chiral superfields and show that the target space is a Kähler cone with U(1) × U(1) being the connected component of the isometry group.

Game-Based Learning Kit Method in Isometric Transformations: Usability and Effects on Students’ Achievement and Motivation

&lt;p style="text-align:justify"&gt;This study focused on the development of teaching aids for the topic of Isometric Transformations for Form 2 students in Malaysia. The first objective was to determine the usability of teaching aids in the isometric transformations kit integrates game-based learning (GBL) Method (KitTI Method) and the second objective was to examine the effectiveness of using the KitTI Method on students’ achievement and motivation. This study was directed by the Design and Development Research (DDR) model with survey and quasi-experimental designs. The sample selection was conducted by cluster sampling involving Form 2 students from schools in the district of Muallim and Batang Padang in the state of Perak. The instruments used were KitTI Method Teaching Aids Usability Questionnaire (KMTAUQ), Isometric Transformations Topic Achievement Test (ITTAT), and KitTI Method Motivation Questionnaire (KMMQ). The findings showed that the mean score of KitTI Method‘s usability was high, the treatment group obtained significantly higher improvement in achievement compared to the control group while the mean scores of pre-test achievement as a covariate and students’ motivation in learning Isometric Transformations were high. Therefore, the KitTI Method is suitable to be implemented as a guide for mathematics teachers in designing dynamic teaching and learning and is able to improve students’ achievement and motivation.&lt;/p&gt;

Isometry groups of six-dimensional nilmanifolds

We determine the 6-dimensional nilpotent metric Lie algebras such that the Lie algebra {mathfrak {n}} has a descending series of ideals invariant under all automorphisms of {mathfrak {n}} and the dimension of the consecutive members of the series decreases by one. We call them metric Lie algebras having a framing determined by ideals. We classify the isometry equivalence classes and determine the isometry groups of connected and simply connected Riemannian nilmanifolds on 6-dimensional nilpotent Lie groups having a Lie algebra {mathfrak {n}} as their Lie algebra.

On emergent conformal symmetry near the photon ring

In this note we revisit the emergent conformal symmetry in the near-ring region of warped spacetime. In particular, we propose a novel construction of the emergent near-ring sl(2, ℝ)QNM symmetry. We show that each eikonal QNM family falls into one highest-weight representation of this algebra, and sl(2, ℝ)QNM can be related to the near-ring isometry group sl(2, ℝ)ISO in a simple way. Furthermore we find that the coherent state space of sl(2, ℝ)QNM can be identified with the phase space of the photon ring.

Superradiance and black resonator strings encounter helical black strings

We construct a cohomogeneity-1 helical black string in six-dimensional Einstein gravity. The helical solution branches from the onset of the gravitational superradiant instability of the equal-spinning Myers-Perry black string. The isometry group of the helical black string is ℝT × U(1)Z × SU(2), where the first two are helical isometries generated by linear combinations of time translation, shifts along the string, and rotation, each of which is individually broken by the superradiant instability. The helical black string is stationary, non-axisymmetric, and has nonzero horizon velocity despite the absence of momentum in the string direction. The entropy of the helical black string is higher than that of the Myers-Perry black string, but lower than cohomogeneity-2 “black resonator strings” (recently found) when the solutions overlap in the microcanonical ensemble. The entropy of the helical black string approaches zero when the horizon velocity along the string reaches its maximum given by the speed of light. Nevertheless, we find no evidence for the existence of regular horizonless solutions in this limit.

Isometry groups with radical, and aspherical Riemannian manifolds with large symmetry, I

Isometry groups with radical, and aspherical Riemannian manifolds with large symmetry, I

The Polish topology of the isometry group of the infinite dimensional hyperbolic space

We consider the isometry group of the infinite-dimensional separable hyperbolic space with its Polish topology. This topology is given by pointwise convergence. For non-locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinite-dimensional groups like \mathcal{S}_\infty , the group of all permutations of a countable set on the other side. Our main results are All along the text, we lead a parallel study with the sibling group \mathrm{Isom}(\mathcal{H}) , where \mathcal{H} is a separable Hilbert space.

Study of ideal gases in curved spacetimes

The influence of a curved spacetime [Formula: see text] on the physical behavior of an ideal gas of [Formula: see text] particles is analyzed by considering the phase space of the system as a region of the cotangent bundle [Formula: see text] and using Souriau’s Lie group thermodynamics to define the corresponding probability distribution function. While the construction of the phase space respects the separability of the system, by forcing each particle to satisfy the so-called mass-shell constraint, the probability distribution is constructed by mixing the natural symplectic structure of the cotangent bundle with a Hamiltonian description of the system. In this way, the spacetime is introduced into the statistics and its isometries turn out to be of special interest because the distributions are parametrized by the elements of the Lie algebra of the isometry group, through the momentum map of the action of the isometries in [Formula: see text]. We find the Gibbs distribution that, in the simplest case of a flat spacetime, reduces to the so-called modified Jüttner distribution, used to describe ideal gases in the regime of special relativity. We also define a temperature-like function using the norm of a Killing vector, which allows us to recover the so-called Tolman–Ehrenfest effect. As a particular example, we study the outer region of a Schwarzschild black hole, for which a power series expansion of the Schwarzschild radius allows us to represent the partition function and the Gibbs distribution in terms of the corresponding quantities of the Minkowski spacetime.

Isometric actions on the four dimensional Minkowski spacetime

In this paper we give representations of all connected Lie groups acting isometrically on the four dimensional Minkowski spacetime, up to conjugacy within the full isometry group of the space. For each obtained group, we study its induced orbits. Then we classify the Lie groups up to orbit equivalence.