Mapping Class Research Articles

Density of periodic measures and large deviation principle for generalised mod one transformations

We introduce a new class of piecewise monotonic maps, called generalised mod one transformations, which include all (α,β) and generalised β -transformations, and prove that all transitive generalised mod one transformations with positive topological entropy satisfy the level-2 large deviation principle with a unique measure of maximal entropy. This is obtained by our result on the density of periodic measures for more general class of piecewise mototonic maps, which is a partial answer to the open problem posed by Hofbauer and Raith (1998 Fund. Math. 157 221–34); Raith (www.math.iupui.edu/~mmisiure/open/PR1.pdf).

Bohr Inequalities for Certain Classes of Harmonic Mappings

Bohr Inequalities for Certain Classes of Harmonic Mappings

Minimality and distributional chaos in triangular maps

The result of this paper contributes to the classification of triangular maps of the square with zero topological entropy stated by A. N. Sharkovsky in the 1980s. The problem was if a triangular map of the square such that its any ω-limit set contains unique minimal set can be distributionally chaotic. So far such result was disproved only for the class of triangular maps non-decreasing on fibres [L. Paganoni, J. Smítal, Strange distributionally chaotic triangular maps, Chaos Solitons Fractals 26(2) (2005), pp. 581–589]. In this paper, we solve the problem in negative for all triangular maps of the square, correcting the original result from Balibrea and Smítal [Strong distributional chaos and minimal sets, Topology appl. 156 (2009), pp. 1673–1678].

THE CERESA CLASS: TROPICAL, TOPOLOGICAL AND ALGEBRAIC

Abstract The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for nonhyperelliptic curves is in general not easy. We present a ‘combinatorialization’ of this problem, explaining how to define a Ceresa class for a tropical algebraic curve and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over $\mathbb {C}(\!(t)\!)$ and show that the Ceresa class in each of these settings is torsion.

TFS-ViT: Token-level feature stylization for domain generalization

Standard deep learning models such as convolutional neural networks (CNNs) lack the ability of generalizing to domains which have not been seen during training. This problem is mainly due to the common but often wrong assumption of such models that the source and target data come from the same i.i.d. distribution. Recently, Vision Transformers (ViTs) have shown outstanding performance for a broad range of computer vision tasks. However, very few studies have investigated their ability to generalize to new domains. This paper presents a first Token-level Feature Stylization (TFS-ViT) approach for domain generalization, which improves the performance of ViTs to unseen data by synthesizing new domains. Our approach transforms token features by mixing the normalization statistics of images from different domains. We further improve this approach with a novel strategy for attention-aware stylization, which uses the attention maps of class (CLS) tokens to compute and mix normalization statistics of tokens corresponding to different image regions. The proposed method is flexible to the choice of backbone model and can be easily applied to any ViT-based architecture with a negligible increase in computational complexity. Comprehensive experiments show that our approach is able to achieve state-of-the-art performance on five challenging benchmarks for domain generalization, and demonstrate its ability to deal with different types of domain shifts. The implementation is available at this repository.

General primitivity in the mapping class group

For [Formula: see text], let [Formula: see text] be the mapping class group of the closed orientable surface [Formula: see text] of genus [Formula: see text]. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping class can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give an algorithm for determining the conjugacy classes of roots of arbitrary mapping classes. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in [Formula: see text], the Torelli group, the level-[Formula: see text] subgroup of [Formula: see text], and the commutator subgroup of [Formula: see text]. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class [Formula: see text] is [Formula: see text], where [Formula: see text] is a unique positive integer associated with the conjugacy class of [Formula: see text]. Moreover, this bound is realized by a root of a power of a Dehn twist about a separating curve of genus [Formula: see text] in [Formula: see text], where [Formula: see text]. Finally, for [Formula: see text], we show that any pseudo-periodic mapping class having a nontrivial periodic component that is not the hyperelliptic involution, normally generates [Formula: see text]. Consequently, we establish that [Formula: see text] is normally generated by a root of bounding pair map or a root of a nontrivial power of a Dehn twist.

Reducible normal generators for mapping class groups are abundant

In this paper, we study the normal generation of the mapping class group. We first show that a mapping class is a normal generator if its restriction on the invariant subsurface normally generates the (pure) mapping class group of the subsurface. As an application, we provided a criterion for reducible mapping classes to normally generate the mapping class groups in terms of its asymptotic translation lengths on Teichmüller spaces. This is an analogue to the work of Lanier–Margalit dealing with pseudo-Anosov normal generators.

Growth rate of Dehn twist lattice points in Teichmüller space

Athreya et al. [Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J. 161 (2012) 1055–1111] have shown the number of mapping class group lattice points intersecting a closed ball of radius [Formula: see text] in Teichmüller space is asymptotic to [Formula: see text], where [Formula: see text] is the dimension of the Teichmüller space. In contrast we show the number of Dehn twist lattice points intersecting a closed ball of radius [Formula: see text] is coarsely asymptotic to [Formula: see text]. Moreover, we show the number multi-twist lattice points intersecting a closed ball of radius [Formula: see text] grows coarsely at least at the rate of [Formula: see text].

Dynamic properties for the induced maps on symmetric product suspensions of a topological space

Given a nondegenerate compact perfect and Hausdorff topological space X, n∈N and a function f:X→X, we consider the n-fold symmetric product of X, Fn(X), and the induced function Fn(f):Fn(X)→Fn(X). In this paper, if n≥2, we begin the study of the n-fold symmetric product suspension of the topological space X, SFn(X). We study the relationships between the following statements: (1) f∈M, (2) Fn(f)∈M, and (3) SFn(f)∈M, where M is one of the following classes of maps: almost transitive, exact, mixing, transitive, totally transitive, strongly transitive, exactly Devaney chaotic, orbit-transitive, an F-system, scattering, TT++, Touhey, backward minimal, totally minimal, Property P, strong property P or two-sided transitive.

On the approximation of fixed points for the class of mappings satisfying $(CSC)$-condition in Hadamard spaces

In this paper, we consider the class of mappings satisfying $(CSC)$-condition. Further, we prove the strong and $\triangle$-convergence theorems of the $JF$-iteration process for this class of mappings in Hadamard spaces. At the end, we give a numerical example to show that the $JF$-iteration process is faster than some well known iterative processes. Our results improve and extend the corresponding recent results of the current literature.

The Mapping Class Group of Manifolds Which Are Like Projective Planes

Abstract In this paper, we compute the mapping class group of simply-connected closed smooth manifolds $M$ with integral homology $H_{*}(M) \cong {\mathbb {Z}} \oplus {\mathbb {Z}} \oplus {\mathbb {Z}}$ provided that $\dim M \ne 4$.

Simultaneous classification and out-of-distribution detection for wafer bin maps

Defect pattern classification of wafer bin maps (WBMs) assists in identifying the causes of semiconductor manufacturing process failures, thus contributing to the search for appropriate solutions. However, new previously unobserved defect patterns may be generated in real applications, which require classification methods with out-of-distribution (OOD) pattern detection capabilities. In this paper, we propose a novel method called a spectral-normalized neural radial basis function (SNRBF) network to classify defect patterns in WBMs while simultaneously detecting OOD defect patterns by quantifying the predictive uncertainty of a deep neural network. The SNRBF network adds spectral normalization to the weights in each layer of a deep neural network and replaces the output layer with a radial basis function kernel, facilitating high-quality uncertainty quantification and efficient computation. Additionally, we propose a novel regularization method based on singular values of the representation matrix to improve uncertainty quantification. Furthermore, we propose a new noise-filtering method capable of effectively eliminating random defects in WBMs by considering various defect pattern sizes. The proposed method is evaluated on a real WBM dataset and is demonstrated to outperform other competing methods in OOD detection while presenting competitive classification performance.

Generalized Enriched Nonexpansive Mappings and Their Fixed Point Theorems

This paper introduces a novel category of nonlinear mappings and provides several theorems on their existence and convergence in Banach spaces, subject to various assumptions. Moreover, we obtain convergence theorems concerning iterates of α -Krasnosel’skiĭ mapping associated with the newly defined class of mappings. Further, we present that α -Krasnosel’skiĭ mapping associated with b -enriched quasinonexpansive mapping is asymptotically regular. Furthermore, some new convergence theorems concerning b -enriched quasinonexpansive mappings have been proved.

Mapping Class and Electoral Participation in India from 1996 to 2019

This article studies the relation between class and electoral participation. While the relation between political participation and many demographic variables such as caste, gender, age and location has been well researched in India, the same is not the case for the relation between class and electoral participation. Multiple measures of class (income, asset-wealth, occupation and education) are explored and conceptualized in this article, following which these measures of class are operationalized using the National Election Study datasets covering a twenty-three-year period (1996–2019). Each of these measures is used to trace the relation of class with two outcomes of electoral participation (turnout and party vote share) over time. Disaggregation by gender, locality and caste is provided. Finally, regression analysis to study the impact of these variables on turnout and vote share reveals the complexity of class. We find a complex picture of turnout and party choice with variation across different class measures. More significantly, variations in results raise questions about the usefulness of existing class indices. Further, we find that the type of measure being used affects different outcomes differently. For turnout, income and wealth seem to be better predictors, and for party vote share, subjective class is a better fit, whereas asset-wealth displays opposite patterns to income and subjective class in some instances.

On discrete Lorenz-like attractors in three-dimensional maps with axial symmetry.

We describe a class of three-dimensional maps with axial symmetry {x→-x,y→-y,z→z} and the constant Jacobian. We study bifurcations and chaotic dynamics in quadratic maps from this class and show that these maps can possess discrete Lorenz-like attractors of various types. We give a description of bifurcation scenarios leading to such attractors and show examples of their implementation in our maps. We also describe the main geometric properties of the discrete Lorenz-like attractors including their homoclinic structures.

SymTFTs and duality defects from 6d SCFTs on 4-manifolds

In this work we study particular TQFTs in three dimensions, known as Symmetry Topological Field Theories (or SymTFTs), to identify line defects of two-dimensional CFTs arising from the compactification of 6d (2, 0) SCFTs on 4-manifolds M4. The mapping class group of M4 and the automorphism group of the SymTFT switch between different absolute 2d theories or global variants. Using the combined symmetries, we realize the topological defects in these global variants. Our main example is ℙ1 × ℙ1. For N M5-branes the corresponding 2d theory inherits ℤN 0-form symmetries from the SymTFT. We reproduce the orbifold groupoid for theories with ℤN 0-form symmetries and realize the duality defects at fixed points of the coupling constant under elements of the mapping class group. We also study other Hirzebruch surfaces, del Pezzo surfaces, as well as the connected sum of ℙ1 × ℙ1. We find a rich network of global variants connected via automorphisms and realize more interesting topological defects. Finally, we derive the SymTFT on more general 4-manifolds and provide two examples.

Commensurators of abelian subgroups and the virtually abelian dimension of mapping class groups

Let Mod(S) be the mapping class group of a compact connected orientable surface S, possibly with punctures and boundary components, with negative Euler characteristic. We prove that for any infinite virtually abelian subgroup H of Mod(S), there is a subgroup H′ commensurable with H such that the commensurator of H equals the normalizer of H′. As a consequence we give, for each n≥2, an upper bound for the geometric dimension of Mod(S) for the family of abelian subgroups of rank bounded by n. These results generalize work by Juan-Pineda–Trujillo-Negrete and Nucinkis–Petrosyan for the virtually cyclic case.

Comparison of Three Machine Learning Algorithms Using Google Earth Engine for Land Use Land Cover Classification

Google Earth Engine (GEE) is presently the most innovative international open-source platform for the advanced-level analysis of geospatial big data. In this study, we used three machine learning algorithms to apply this cloud platform for Land Use Land Cover (LULC) research in the Mardan, Pakistan. The machine learning algorithm in GEE is the most advanced technique to generate reliable and informative LULC maps from various satellite data to present reliable results. The primary goal of the present study is to compare the performance of various machine learning models (i.e., classification and regression trees [CART], support vector machine [SVM], and random forest [RF]) in GEE for the reliable four classes LULC maps using the Sentinel-2 imageries of 2022. In the current study, three satellite indices like the Normalized Difference Vegetation Index, Modified Normalized Difference Water Index, and Normalized Difference Built Index were applied to detect the features (i.e., vegetation, built, barren land, and water bodies in the study area). The performance of all three models was evaluated by validation and accuracy assessments. The Kappa coefficients of CART, SVM, and RF for Sentinel-2 images were 94%, 95%, and 97%, while the average overall accuracy is 96.25%, 97%, and 98.68%, respectively. The present study illustates that in this classification and comparison, RF performed better than SVM and CART. The current research study revealed that GEE has speedily processed the satellite imageries to develop the four classes of reliable LULC maps of the study area with the best accuracy results and deliver excellent support for further analysis.

Linear fractional self-maps of the unit ball

Abstract Determining the range of complex maps plays a fundamental role in the study of several complex variables and operator theory. In particular, one is often interested in determining when a given holomorphic function is a self-map of the unit ball. In this paper, we discuss a class of maps in $\mathbb {C}^N$ that generalize linear fractional maps. We then proceed to determine precisely when such a map is a self-map of the unit ball. In particular, we take a novel approach, obtaining numerous new results about this class of maps along the way.

Classification of additive mappings on certain rings and algebras

The objective of this research is to prove that an additive mapping Δ:A→A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta :{\\mathcal {A}}\\rightarrow {\\mathcal {A}}$$\\end{document} will be a generalized derivation associated with a derivation ∂:A→A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial :{\\mathcal {A}}\\rightarrow {\\mathcal {A}}$$\\end{document} if it satisfies the following identity Δ(rm+n+p)=Δ(rm)rn+p+rm∂(rn)rp+rm+n∂(rp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (r^{m+n+p})=\\Delta (r^m)r^{n+p}+r^m\\partial (r^{n})r^p+r^{m+n}\\partial (r^p)$$\\end{document} for all r∈A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\in {\\mathcal {A}}$$\\end{document}, where m,n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m, n\\ge 1$$\\end{document} and p≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 0$$\\end{document} are fixed integers and A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} is a semiprime ring. Another analogous has been done where an additive mapping behaves like a generalized left derivation associated with a left derivation on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} satisfying certain algebraic identity. The proofs of these advancements are derived employing algebraic concepts. These theorems have been validated by offering an example that shows they are not insignificant. Furthermore, we provide an application in the framework of Banach algebra.