Mapping Class Research Articles

The coarse geometry of hexagon decomposition graphs

Abstract We define and study graphs associated to hexagon decompositions of surfaces by curves and arcs. One of the variants is shown to be quasi-isometric to the pants graph, whereas the other variant is quasi-isometric to (a Cayley graph of) the mapping class group.

The Bogomolny–Carioli Twisted Transfer Operators and the Bogomolny–Gauss Mapping Class Group

The twisted reflection operators are defined on the hyperbolic plane. They are then specialized in hyperbolic reflections, according to which the desymmetrized PSL (2,Z) group is rewritten. The Bogomolny–Carioli transfer operators are newly analytically expressed in terms of the Dehn twists. The Bogomolny–Gauss mapping class group of the desymmetrized PSL (2,Z) domain is newly proven. The paradigm to apply the Hecke theory on the CAT spaces on which the Dehn twists act is newly established. The Bogomolny–Gauss map is proven to be one of infinite topological entropy.

Orbits of families of discrete dynamical systems evolving in the natural numbers

In this paper, we give a class of one-dimensional discrete dynamical systems with state space N+. This class of systems is defined by two parameters: one of them sets the number of nearest neighbors that determine the rule of evolution, and the other parameter determines a segment of natural numbers Λ={1,2,…,b}. In particular, we investigate the behavior of a class of one-dimensional maps where an integer moves to an other integer given by the sum of the nearest neighbors minus a multiple of b∈N+. We find the coexistence of fixed points and periodic cycles. Two single parameter families of maps are introduced and their dynamics in the segment of natural sequence Λ. Furthermore, an order of the numbers of the set Λ−b is given by these families. Last, we present a connection of the N+ generated by the orbits of a particular case.

The Language of Infographics: Toward Understanding Conceptual Metaphor Use in Scientific Storytelling.

We apply an approach from cognitive linguistics by mapping Conceptual Metaphor Theory (CMT) to the visualization domain to address patterns of visual conceptual metaphors that are often used in science infographics. Metaphors play an essential part in visual communication and are frequently employed to explain complex concepts. However, their use is often based on intuition, rather than following a formal process. At present, we lack tools and language for understanding and describing metaphor use in visualization to the extent where taxonomy and grammar could guide the creation of visual components, e.g., infographics. Our classification of the visual conceptual mappings within scientific representations is based on the breakdown of visual components in existing scientific infographics. We demonstrate the development of this mapping through a detailed analysis of data collected from four domains (biomedicine, climate, space, and anthropology) that represent a diverse range of visual conceptual metaphors used in the visual communication of science. This work allows us to identify patterns of visual conceptual metaphor use within the domains, resolve ambiguities about why specific conceptual metaphors are used, and develop a better overall understanding of visual metaphor use in scientific infographics. Our analysis shows that ontological and orientational conceptual metaphors are the most widely applied to translate complex scientific concepts. To support our findings we developed a visual exploratory tool based on the collected database that places the individual infographics on a spatio-temporal scale and illustrates the breakdown of visual conceptual metaphors.

Some reformulations and extensions of the theory of rhythmic canons

The algebraic theory of periodic rhythmic canons was developed by the author of the present paper in connection with the study of a special class of rhythmic canons, nowadays referred as “Vuza canons”. A basic concept in the theory is the outer rhythm attached to a canon. In the original paper, for any given canon an auxiliary canon was constructed and the outer rhythm was expressed as an object attached to the latter canon. In the present paper we show that the outer rhythm can be expressed directly in terms of the given canon, without the need of an auxiliary construction. Strongly related to the outer rhythm is the canon category, which is a numerical measure of the periodic symmetry of the outer rhythm relative to the canon inner rhythm. We give a new definition of the category in terms of the stability groups associated to a canon. Many interesting results due to various authors have been obtained for Vuza canons, which by their definition must have maximal category. We show here that interesting facts can also be said about canons whose category is not maximal. We describe partitions of a canon into subcanons of minimal and maximal category and we discuss the relation between outer rhythm, category and a class of maps that can be regarded as natural morphisms between canons.

The radial symmetry of minimizers to the p$p$ weighted Dirichlet energy in R3$\mathbb {R}^3$

AbstractLet and be annuli in . Let , and assume that is the class of Sobolev homeomorphisms of onto . Then, we consider the following Dirichlet‐type energy of : We prove that this energy integral attains its minimum for and , and the minimum is a certain radial diffeomorphism .For general , we minimize the Dirichlet‐type integral throughout the class of radial mappings between given annuli, and this minimum always exists for . For , the image annulus cannot be too thick, which is opposite to the Nitsche‐type phenomenon known for the standard Dirichlet energy, where the image annulus cannot be too thin.

Oka-1 manifolds: New examples and properties

In this paper we investigate Oka-1 manifolds and Oka-1 maps, a class of complex manifolds and holomorphic maps recently introduced by Alarcón and Forstnerič. Oka-1 manifolds are characterised by the property that holomorphic maps from any open Riemann surface to the manifold satisfy the Runge approximation and Weierstrass interpolation conditions, while Oka-1 maps enjoy similar properties for liftings of maps from open Riemann surfaces in the absence of topological obstructions. We also formulate and study the algebraic version of the Oka-1 condition, called aOka-1. We show that it is a birational invariant for compact algebraic manifolds and holds for all rational manifolds. This gives a Runge approximation theorem for maps from compact Riemann surfaces to uniformly rational projective manifolds. Finally, we study a class of complex manifolds with an approximation property for holomorphic sprays of discs. This class lies between the smaller class of Oka manifolds and the bigger class of Oka-1 manifolds and has interesting functorial properties.

Exploring α−ψ−ϕ contractive mapping: novel fixed point theorems in complete b-metric spaces

This paper explores the concept of α-ψ-ϕ contractive mappings, contributing to the advancement of self-map extensions and fixed-point theorems within b-metric spaces. We introduce a new class of contractive mappings and demonstrate how they extend traditional contraction principles, offering a broader framework for analyzing fixed points in non-standard spaces. The main result of this study is a generalization of existing fixed-point theorems, supported by comprehensive corollaries, illustrative examples, and rigorous proofs. These findings provide deeper insights into the structure of b-metric spaces and open avenues for further applications in fields such as optimization and machine learning.

CONSTRUCTION OF SMOOTH “SOURCE-SINK” ARCS IN THE SPACE OF DIFFEOMORPHISMS OF A TWO-DIMENSIONAL SPHERE

It is well known that the mapping class group of the two-dimensional sphere 𝕊2 is isomorphic to the group ℤ2 = {−1, +1}. At the same time, the class +1 (−1) contains all orientation-preserving (orientationreversing) diffeomorphisms and any two diffeomorphisms of the same class are diffeotopic, that is, they are connected by a smooth arc of diffeomorphisms. On the other hand, each class of maps contains structurally stable diffeomorphisms. It is obvious that in the general case, the arc connecting two diffeotopic structurally stable diffeomorphisms undergoes bifurcations that destroy structural stability. In this direction, it is particular interesting in the question of the existence of a connecting them stable arc ”— an arc pointwise conjugate to arcs in some of its neighborhood. In general, diffeotopic structurally stable diffeomorphisms of the 2-sphere are not connected by a stable arc. In this paper, the simplest structurally stable diffeomorphisms (“source-sink” diffeomorphisms) of the 2-sphere are considered. The non-wandering set of such diffeomorphisms consists of two hyperbolic points: the source and the sink. In this paper, the existence of an arc connecting two such orientation-preserving (orientation-reversing) diffeomorphisms and consisting entirely of “source-sink” diffeomorphisms is constructively proved.

A Hierarchy of Covariant Generalized Schwarz Maps in M2(ℂ)

A class of linear maps in M2(C) displaying diagonal unitary and orthogonal symmetries is analyzed. Using a notion of ω-duality, we prove that a map which is ω-dual to a generalized Schwarz map is again generalized Schwarz. We introduce an infinite hierarchy of generalized Schwarz maps and study the property of an asymptotic limiting map. Interestingly, it is shown that the first example of Schwarz but not completely positive map found by Choi is an example of an asymptotic map.

Annual 30-m maps of global grassland class and extent (2000–2022) based on spatiotemporal Machine Learning

The paper describes the production and evaluation of global grassland extent mapped annually for 2000–2022 at 30 m spatial resolution. The dataset showing the spatiotemporal distribution of cultivated and natural/semi-natural grassland classes was produced by using GLAD Landsat ARD-2 image archive, accompanied by climatic, landform and proximity covariates, spatiotemporal machine learning (per-class Random Forest) and over 2.3 M reference samples (visually interpreted in Very High Resolution imagery). Custom probability thresholds (based on five-fold spatial cross-validation) were used to derive dominant class maps with balanced user’s and producer’s accuracy, resulting in f1 score of 0.64 and 0.75 for cultivated and natural/semi-natural grassland, respectively. The produced maps (about 4 TB in size) are available under an open data license as Cloud-Optimized GeoTIFFs and as Google Earth Engine assets. The suggested uses of data include (1) integration with other compatible land cover products and (2) tracking the intensity and drivers of conversion of land to cultivated grasslands and from natural / semi-natural grasslands into other land use systems.

Class number for pseudo-Anosovs

Given two automorphisms of a group G G , one is interested in knowing whether they are conjugate in the automorphism group of G G , or in the abstract commensurator of G G , and how these two properties may differ. When G G is the fundamental group of a closed orientable surface, we present a uniform finiteness theorem for the class of pseudo-Anosov automorphisms. We present an explicit example of a commensurably conjugate pair of pseudo-Anosov automorphisms of a genus 3 3 surface, that are not conjugate in the mapping class group, and we also show that infinitely many pairwise non-commuting pseud-Anosov automorphisms have class number equal to one. In the appendix, we briefly survey the Latimer-MacDuffee theorem that addresses the case of automorphisms of Z n \mathbb {Z}^n , with a point of view that is suited to an analogy with surface group automorphisms.

Covariant Schwarz maps in M2(ℂ)

Abstract We analyze a class of qubit maps displaying diagonal unitary and orthogonal symmetries. For unital maps we characterize all covariant maps satisfying an operator Schwarz inequality. In particular well known Pauli maps are completely characterized. Going beyond the unital case we consider recently proposed generalizations of Schwarz inequality and provide the corresponding necessary and sufficient conditions for the entire class of covariant maps.

Cubulating surface‐by‐free groups

AbstractLet be an exact sequence where is the fundamental group of a closed surface of genus greater than one, is hyperbolic, and is finitely generated free. The aim of this paper is to provide sufficient conditions to prove that is cubulable and construct examples satisfying these conditions. The main result may be thought of as a combination theorem for virtually special hyperbolic groups when the amalgamating subgroup is not quasiconvex. Ingredients include the theory of tracks, the quasiconvex hierarchy theorem of Wise, the distance estimates in the mapping class group from subsurface projections due to Masur–Minsky, and the model geometry for doubly degenerate Kleinian surface groups used in the proof of the ending lamination theorem. An appendix to this paper by Manning, Mj, and Sageev proves a reduction theorem by showing that cubulability of follows from the existence of an essential incompressible quasiconvex track in a surface bundle over a graph with fundamental g\roup .

Certain Novel Fixed-Point Theorems Applied to Fractional Differential Equations

In this paper, we introduce a new class of contractions in normed spaces, referred to as generalized enriched Kannan contractions. These contractions expand the familiar enriched Kannan contractions to three-point versions, broadening the scope of Kannan contractions. These mappings are typically discontinuous, except at the fixed points, where they exhibit continuity, similar to enriched Kannan mappings. However, through suitable examples, we demonstrate that these two classes of mappings are distinct from one another. We present new results for generalized enriched Kannan contractions. Additionally, by incorporating conditions of continuity and asymptotic regularity, we extend the class of operators to which fixed-point methods can be applied. Additionally, we derive two more results for generalized enriched Kannan contractions in normed spaces, without the requirement that they be Banach spaces. Finally, we use our main result to demonstrate the existence of solutions for a boundary value problem involving a fractional differential equation.

Big mapping class groups are not extremely amenable

This paper uses the renowned Kechris-Pestov-Todorcevic machinery to show that (big) mapping class groups are not extremely amenable unless the underlying surface is a sphere or a once-punctured sphere, or equivalently when the mapping class group is trivial. The same techniques also show that the pure mapping class groups, as well as compactly supported mapping class groups, of a surface with genus at least one can never be extremely amenable.

Classes of maps with continuous selections and homotopy extension

Classes of maps with continuous selections and homotopy extension

Defect relations for holomorphic curves of finite lower order intersecting hypersurfaces

In this paper, we will give a better upper bound for the defect relation for a class of holomorphic maps, this result is generalized to hypersurfaces from the hyperplane case in [9]. More precisely, let D1,D2,...,Dq be hypersurfaces in general position, and let f:C→Pn(C) be holomorphic map of lower order μ, such that f(C)⊄Di for all i=1,2,...,q. If 0<μ≤1/2 then∑i=1qδf(Di)≤n.

Approximation of Time-Frequency Shift Equivariant Maps by Neural Networks

Based on finite-dimensional time-frequency analysis, we study the properties of time-frequency shift equivariant maps that are generally nonlinear. We first establish a one-to-one correspondence between Λ-equivariant maps and certain phase-homogeneous functions and also provide a reconstruction formula that expresses Λ-equivariant maps in terms of these phase-homogeneous functions, leading to a deeper understanding of the class of Λ-equivariant maps. Next, we consider the approximation of Λ-equivariant maps by neural networks. In the case where Λ is a cyclic subgroup of order N in ZN×ZN, we prove that every Λ-equivariant map can be approximated by a shallow neural network whose affine linear maps are simply linear combinations of time-frequency shifts by Λ. This aligns well with the proven suitability of convolutional neural networks (CNNs) in tasks requiring translation equivariance, particularly in image and signal processing applications.

Three dimensional topological quantum field theory from [formula omitted] and U(1|1) Chern–Simons theory

We introduce an unrolled quantization UqE(gl(1|1)) of the complex Lie superalgebra gl(1|1) and use its categories of weight modules to construct and study new three dimensional non-semisimple topological quantum field theories. These theories are defined on categories of cobordisms which are decorated by ribbon graphs and cohomology classes and take values in categories of graded super vector spaces. Computations in these theories are enabled by a detailed study of the representation theory of UqE(gl(1|1)). We argue that by restricting to subcategories of integral weight modules we obtain topological quantum field theories which are mathematical models of Chern–Simons theories with gauge supergroups psl(1|1) and U(1|1) coupled to background flat C×-connections, as studied in the physics literature by Rozansky–Saleur and Mikhaylov. In particular, we match Verlinde formulae and mapping class group actions on state spaces of non-generic tori with results in the physics literature. We also obtain explicit descriptions of state spaces of generic surfaces, including their graded dimensions, which go beyond results in the physics literature.