Isometry Research Articles

Action of ℝ-Fuchsian groups on ℙℂn

Let [Formula: see text] be the group of orientation preserving isometries of the [Formula: see text]-dimensional real hyperbolic space [Formula: see text], and let [Formula: see text] be the group of holomorphic isometries of the [Formula: see text]-dimensional complex hyperbolic space [Formula: see text]. The group [Formula: see text] is naturally a subgroup of [Formula: see text], so it acts on [Formula: see text] preserving a ball that serves as a model for [Formula: see text]. We consider discrete subgroups of [Formula: see text] with limit set an [Formula: see text]-dimensional real sphere and we look at their action on [Formula: see text] for [Formula: see text] via the natural embedding of [Formula: see text] into [Formula: see text]. We describe the Kulkarni limit set of these set of these actions and show that this is a real semi-algebraic set in [Formula: see text]. We also show that the Kulkarni region of discontinuity can only have either one or three connected components. We use Sylvester’s law of inertia when [Formula: see text]. In the other cases, we use some suitable projections of the [Formula: see text]-dimensional complex projective space to the [Formula: see text]-dimensional complex projective space.

BiLipschitz homogeneous hyperbolic nets

We answer a question of Itai Benjamini by showing there is a \(K< \infty\) so that for any \(\epsilon >0\), there exist \(\epsilon\)-dense discrete sets in the hyperbolic disk that are homogeneous with respect to \(K\)-biLipschitz maps of the disk to itself. However, this is not true for \(K\) close to \(1\); in that case, every \(K\)-biLipschitz homogeneous discrete set must omit a disk of hyperbolic radius \(\epsilon(K)>0\). For \(K=1\), this is a consequence of the Margulis lemma for discrete groups of hyperbolic isometries.

Complete systems of invariants of a hyper-surface in a Euclidean space

Let M(n+1) be the group of all isometric transformations of an n+1-dimensional Euclidean space Rn+1, SM(n + 1) is the group of all Euclidean motions of Rn+1, U is the domain f(u1)2 + · · · + (un)2 < 1g in Rn and x(u) is an U-hyper-surface (that is a C1-mapping x : U ! Rn+1) in Rn+1. Let x(u) be a regular hyper-surface, I(x) = Pn i;j=1 gij(x)duiduj and II(x) = Pn i;j=1 Lij(x)duiduj are the first and second fundamental forms of x. Put K = f(i; j) : 1 ≤ i ≤ j ≤ ng. We consider the following problem: find a proper (minimal, it is desirable) subset T of K such that equalities gij(x)(u) = gij(y)(u); Lij(x)(u) = Lij(y)(u) for all (i; j) 2 T and u 2 U implies an existence of F 2 SM(n + 1) such that y(u) = Fx(u) for all u 2 U. For a solution of this problem, we have obtained a generating system of the differential field of all G-invariant differential rational functions of a hyper-surface in Rn+1 for groups G = M(n + 1); SM(n + 1). In our result, jTj = n(n+1) 2 + n, where jTj is the number of elements of T.

Realization of finite groups as isometry groups and problems of minimality

AbstractA finite group is said to be realized by a finite subset of a Euclidean space if the isometry group of is isomorphic to . We prove that every finite group can be realized by a finite subset consisting of points, where is a generating system for . We define as the minimum number of points required to realize in for some . We establish that provides a sharp upper bound for when considering minimal generating sets. Finally, we explore the relationship between and the isometry dimension of , that is, defined as the least dimension of the Euclidean space in which can be realized.

Forecasting age distribution of life-table death counts via α-transformation

We introduce a compositional power transformation, known as an α-transformation, to model and forecast a time series of life-table death counts, possibly with zero counts observed at older ages. As a generalisation of the isometric log-ratio transformation (i.e. α = 0 ), the α transformation relies on the tuning parameter α, which can be determined in a data-driven manner. Using the Australian age-specific period life-table death counts from 1921 to 2020, the α transformation can produce more accurate short-term point and interval forecasts than the log-ratio transformation. The improved forecast accuracy of life-table death counts is of great importance to demographers and government planners for estimating survival probabilities and life expectancy and actuaries for determining annuity prices and reserves for various initial ages and maturity terms.

Random walks and contracting elements II: Translation length and quasi-isometric embedding

Continuing from Choi (2022), we study random walks on metric spaces with contracting elements. We prove that random subgroups of the isometry group of a metric space are quasi-isometrically embedded into the space. We discuss this problem in two ways, namely in the sense of random walks and counting problem. We also establish the genericity of contracting elements and the central limit theorem and its converse for translation length.

Diffeomorphisms of Foliated Manifolds I

The set $Diff(M)$ of all diffeomorphisms of manifold $M$ onto itself is the group related to composition and inverse mapping. The group of diffeomorphisms of smooth manifolds is of great importance in differential geometry and analysis. It is known that the group $Diff(M)$ is topological group in compact open topology.In this paper we investigate the group $Diff_{F}(M)$ of diffeomorphisms foliated manifold $(M,F)$ with foliated compact open topology. In this paper we prove that if all leaves of the the foliation $F$ are closed subsets of $M$ then the foliated compact open topology of the group $Diff_{F}(M)$ coincides with compact open topology. In addition it is studied the question on the dimension of the group of isometries of foliated manifold is studied when foliation generated by riemannian submersion.

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

Isometries of Lipschitz‐free Banach spaces

AbstractWe describe surjective linear isometries and linear isometry groups of a large class of Lipschitz‐free spaces that includes, for example, Lipschitz‐free spaces over any graph. We define the notion of a Lipschitz‐free rigid metric space whose Lipschitz‐free space only admits surjective linear isometries coming from surjective dilations (i.e., rescaled isometries) of the metric space itself. We show that this class of metric spaces is surprisingly rich and contains all 3‐connected graphs as well as geometric examples such as nonabelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz‐free rigid space that has only three more points.

Polyhedra with Hexagonal and Triangular Faces and Three Faces Around Each Vertex

AbstractWe analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call “trihexes”. Trihexes are analogous to fullerenes, which are 3-regular planar graphs whose faces are all hexagons and pentagons. Every trihex can be represented as the quotient of a hexagonal tiling of the plane under a group of isometries generated by $$180^\circ $$ 180 ∘ rotations. Every trihex can also be described with either one or three “signatures”: triples of numbers that describe the arrangement of the rotocenters of these rotations. Simple arithmetic rules relate the three signatures that describe the same trihex. We obtain a bijection between trihexes and equivalence classes of signatures as defined by these rules. Labeling trihexes with signatures allows us to put bounds on the number of trihexes for a given number of vertices v in terms of the prime factorization of v and to prove a conjecture concerning trihexes that have no “belts” of hexagons.

Groundwater quality assessment using revised classical diagrams and compositional data analysis (CoDa): Case study of Wadi Ranyah, Saudi Arabia

This research aims to assess the status and chemical properties of groundwater in the shallow aquifer of the Wadi Ranyah region, Saudi Arabia, using robust statistical methods (CoDA) that take into account the unique characteristics of chemical analysis data from forty-five (45) groundwater samples collected in the Wadi Ranyah valley. A centered logarithmic transformation (clr) approach and isometric logarithmic transformation (ilr) plots for compositional data were utilized to determine the different water types and to understand the hydrogeochemical processes influencing groundwater chemistry in the region. The results of principal component analysis and K-means clustering reveal the existence of three groundwater groupings: (i) The first group occupies the recharge region with a relationship between trace metal elements (F, Pb, Mn, Zn), pH, and TDS, indicating the effect of alteration of igneous and metamorphic rock minerals. (ii) The second group characterizes the downstream part of the Al-Hujrah area with a relationship between calcium and nitrates, indicating the influence of agricultural activities. (iii) The third group, located in the Ranyah area, shows a relationship between major ions (Na+, SO42−, Mg2+, HCO3–, Cl−, K+), which explains water mineralization influenced by silicate weathering and evaporation phenomena. According to the modified Piper diagram (ilr Piper), two hydrochemical facies are identified. The upstream region is categorized as Ca-HCO3– while the downstream region is classified as Na-K-HCO3–. Calculated water quality index (WQI) values indicate that all Wadi Ranyah groundwater samples are of very poor quality (WQI > 300). This is due to higher concentrations of polluting elements (Pb, Mn and F), in particular the high lead concentrations which vary between 0.24 and 0.4 mg/L, and are well above the WHO limit (0.01 mg/L) for consumption of drinking water. The application of CoDA for the first time in the study region has provided a more detailed and reliable assessment of the mechanisms governing groundwater quality, permitting suggestions for the management of groundwater resources and defining future research needs.

Gh-convergence of CAT(0)-spaces: stability of the Euclidean factor

AbstractWe prove that if a sequence of geodesically complete CAT(0)-spaces $$X_j$$ X j with uniformly cocompact discrete groups of isometries converges in the Gromov-Hausdorff sense to $$X_\infty $$ X ∞ , then the dimension of the maximal Euclidean factor splitted off by $$X_\infty $$ X ∞ and $$X_j$$ X j is the same, for j big enough. In other words, no additional Euclidean factors can appear in the limit.

Bishop–Jones’ theorem and the ergodic limit set

Abstract For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop–Jones theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

Isometric rigidity of Wasserstein spaces over Euclidean spheres

We study the structure of isometries of the quadratic Wasserstein space W2(Sn,‖⋅‖) over the sphere endowed with the distance inherited from the norm of Rn+1. We prove that W2(Sn,‖⋅‖) is isometrically rigid, meaning that its isometry group is isomorphic to that of (Sn,‖⋅‖). This is in striking contrast to the non-rigidity of its ambient space W2(Rn+1,‖⋅‖) but in line with the rigidity of the geodesic space W2(Sn,∢). One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in W2(Sn,‖⋅‖). A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of Wp(S1,‖⋅‖) for 1≤p<2.

Optimal pinwheel partitions for the Yamabe equation

Abstract We establish the existence of an optimal partition for the Yamabe equation in R N made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to − ∞ , the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation in R N that are different from those previously found by Ding, and del Pino, Musso, Pacard and Pistoia.

Multivariate statistical analysis and bespoke deviation network modeling for geochemical anomaly detection of rare earth elements

Rare earth elements (REEs), a significant subset of critical minerals, play an indispensable role in modern society and are regarded as “industrial vitamins,” making them crucial for global sustainability. Geochemical survey data proves highly effective in delineating metallic mineral prospects. Separating geochemical anomalies associated with specific types of mineralization from the background reflecting geological processes has long been a significant subject in exploration geochemistry. The processing of high-dimensional, non-linear geochemical survey data necessitates a systematic framework to address common issues, including missing values, the closure effect, the selection of appropriate multivariate analysis methods, and anomaly detection techniques in order to detect geochemical anomalies associated with mineral occurrences. The Curnamona Province in South Australia is considered an emerging REE province with significant REE mineralization potential. In this study, we use data from this region to evaluate the performance of a novel machine learning-based framework that incorporates data pre-processing, multivariate statistical analysis, and anomaly recognition to address challenges such as missing data, noise interference, data imbalance and high non-linearity. We utilize lithogeochemical data to map potential greenfield regions of REE mineralization. The primary advantages of our framework lie in its provision of an effective random forest-based data imputation method, utilization of isometric log-ratio transformation to eliminate the closure effect, and reduction of the impact of outliers on data interpretation through robust principal component analysis. Additionally, the framework utilizes a deviation network to learn anomaly scores from complex, non-linear data under imbalanced data conditions, identifying geochemical anomalies associated with REE occurrences by leveraging prior knowledge rather than those caused by data noise or anthropogenic factors. The anomalous areas identified by this framework delineate all known REE deposits and extend to the surrounding regions. Furthermore, a close spatial coupling relationship exists between these strongly anomalous areas and the felsic granite intrusions. The comprehensive workflow for processing geochemical data proposed in this study can effectively address common challenges in the geochemical exploration of critical minerals. The identified geochemical anomalies can provide important clues for subsequent exploration.

The Symmetry Group of the Grand Antiprism

The grand antiprism A is an outlier among the uniform 4-polytopes, since it is not obtainable from Wythoff’s construction. Its symmetry group G(A) has been incorrectly described as [[10,2+,10]] or even as an ‘ionic diminished Coxeter group’. In fact, G(A) is another group of order 400, namely the group ±[D10×D10]·2, in the notation of Conway and Smith. We explain all this and so correct a persistent error in the literature. This fresh look at the beautiful geometry of the polytope A is also a fine opportunity to introduce the reader to the elegance of Wythoff’s construction and to the less familiar use of quaternions to classify the finite 4-dimensional isometry groups.

Geometry of plastic deformation in metals as piecewise isometric transformations

Deformation mechanisms of crystalline solids has been the subject of research for more than two centuries. The theory of dislocations dominates modern views but still has significant gaps demanding the introduction of additional concepts for the coherent quantitative description of physical phenomena. In this work, we propose a coherent geometric description of motion and deformation in crystalline solids as piecewise isometric transformations (PWIT). The latter only includes operations that, similar to interatomic spacing in crystalline lattice, do not alter distances between reference points, i.e. translations, rotations and mirror reflections. The difference between solid-body translations and plastic deformations is that the isometric transformations have discontinuities that in real-life materials realise through dislocations (termination of shifts), disclinations (termination of rotations), and twins (mirror reflections). The conceptual description of plastic deformations as PWIT can be useful for the better description of physical phenomena, proposing new hypothesis, and for developing predictive analytical models. In this paper, the use of this conceptual description enables proposing new hypothesis about the nature of such interesting phenomena in severe plastic deformation as (i) stationary ‘solid state turbulence’ stage in high pressure torsion, and (ii) rate of mass transfer (mechanically assisted diffusion) in simple-shear deformation.

Isometry Groups of Formal Languages for Generalized Levenshtein Distances

Isometry Groups of Formal Languages for Generalized Levenshtein Distances

Fuzzy Gravity: Four‐Dimensional Gravity on a Covariant Noncommutative Space and Unification with Internal Interactions

AbstractIn the present work, an extended description of the covariant noncommutative space is presented, which accommodates the Fuzzy Gravity model constructed previously. It is based on the historical lesson that the use of larger algebras containing all generators of the isometry of the continuous one helped in formulating a fuzzy covariant noncommutative space. Specifically a further enlargement of the isometry group leads the authors, in addition to the construction of the covariant noncommutative space, also to the suggestion of the group that should be gauged on such a space in order to construct a Fuzzy Gravity theory. As a result, two Fuzzy Gravity models are obtained, one in de Sitter and one in anti‐de Sitter space, depending on the extension of the isometry group, and their spontaneous symmetry breaking leading to fuzzy versions of the noncommutative gravity are discussed. In addition, how to introduce fermions in the fuzzy gravity is discussed for the first time, and even more importantly, how to unify the constructed noncommutative‐fuzzy gravity with internal interactions based on or as grand unified theories.