Mapping Class Research Articles

Eventually entanglement breaking divisible quantum dynamics

Abstract It is shown that a large class of quantum dynamical maps on complex matrix algebras governed by time-local Master Equations tend to become entanglement breaking in the course of time. Such situation seems to be generic for quantum evolution and in particular, completely positive dynamical semigroups with a unique faithful stationary state enjoy this property. Inspired by this observation, we propose a new concept of eventually entanglement breaking divisible (eEB-divisible) dynamics. A dynamical map is eEB-divisible if any propagator becomes entanglement breaking in finite time. It turns out that eEB-divisibility is quite general and holds for a large class of quantum evolutions.

Mapping class groups of exotic tori and actions by 𝑆𝐿_{𝑑}(𝐙)

Mapping class groups of exotic tori and actions by 𝑆𝐿_{𝑑}(𝐙)

Novel Fuzzy Ostrowski Integral Inequalities for Convex Fuzzy-Valued Mappings over a Harmonic Convex Set: Extending Real-Valued Intervals Without the Sugeno Integrals

This study presents new fuzzy adaptations of Ostrowski’s integral inequalities through a novel class of convex fuzzy-valued mappings defined over a harmonic convex set, avoiding the use of the Sugeno integral. These innovative inequalities generalize the recently developed interval forms of real-valued Ostrowski inequalities. Their formulations incorporate integrability concepts for fuzzy-valued mappings (FVMs), applying the Kaleva integral and a Kulisch–Miranker fuzzy order relation. The fuzzy order relation is constructed via a level-wise approach based on the Kulisch–Miranker order within the fuzzy number space. Additionally, numerical examples illustrate the effectiveness and significance of the proposed theoretical model. Various applications are explored using different means, and some complex cases are derived.

Effective count of square-tiled surfaces with prescribed real and imaginary foliations in connected components of strata

Abstract We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most L squares. This result strengthens asymptotic counting formulas in the work of Delecroix, Goujard, Zograf, Zorich, and the author.

On convergence of linear response formulae in some piecewise hyperbolic maps

Abstract When high-dimensional non-uniformly hyperbolic chaotic systems undergo dynamical perturbations, their long-time statistics are generally observed to respond differentiably with respect to the perturbation. Although important in applications, this differentiability, which is thought to be connected to the dimensionality of the system, has remained resistant to rigorous study outside of the one-dimensional setting. To model non-uniformly hyperbolic systems in multiple dimensions, we consider a family of the mathematically tractable class of piecewise smooth hyperbolic maps, the Lozi maps. For these maps, we prove that the existence of a formal derivative of the response reduces to conditional mixing of the SRB measure on the singularity set. Heuristically, this suggests that Lozi maps, and piecewise uniformly expanding maps more generally, should have linear response.

Some New Types of Fuzzy Closed Sets, SeparationAxioms, and Compactness via Double Fuzzy Topologies

In this article, we first defined a stronger form of (r,s)-generalized fuzzy semi-closed sets (briefly, (r,s)-gfsc sets) called (r,s)-g*fsc sets and investigated some of its features. Moreover, we showed that (r,s)-fsc set → (r,s)-g*fsc set → (r,s)-gfsc set, but the converse may not be true. In addition, we explored novel types of fuzzy generalized mappings between double fuzzy topological spaces (U, τ, τ*) and (V, η, η*), and the relationships between these classes of mappings were examined with the help of some illustrative examples. Thereafter, we introduced novel types of higher separation axioms called (r,s)-GFS-regular and (r,s)-GFS-normal spaces with the help of (r,s)-gfsc sets and discussed some topological properties of them. Finally, some novel types of compactness via (r,s)-gfso sets were defined and the relationships between them were introduced.

On the Behavior of One Class of Mappings Acting Upon Domains with Locally Quasiconformal Boundary

On the Behavior of One Class of Mappings Acting Upon Domains with Locally Quasiconformal Boundary

Integral filling volume, complexity, and integral simplicial volume of $3$-dimensional mapping tori

We show that the integral filling volume of a Dehn twist f on a closed oriented surface vanishes, i.e., that the integral simplicial volume of the mapping torus with monodromy f^{n} grows sublinearly with respect to n . We deduce a complete characterization of mapping classes on surfaces with vanishing integral filling volume and, building on results by Purcell and Lackenby on the complexity of mapping tori, we show that, in dimension three, complexity and integral simplicial volume are not Lipschitz equivalent.

On trialities and their absolute geometries

Abstract We introduce the notion of moving absolute geometry of a geometry with triality and show that, in the classical case where the triality is of type (I σ) and the absolute geometry is a generalized hexagon, the moving 5 3 6 absolute geometry also gives interesting flag-transitive geometries with Buekenhout diagram for the groups G 2(k) and 3 D 4(k), for any prime power k ≥ 2. We also classify the absolute geometries for geometries with trialities but no dualities coming from maps of Class III with automorphism group L 2(q 3), where q ≥ 2 is prime power. We then investigate the moving absolute geometries for these geometries, illustrating their interest in this case.

Iteration of a class of multi-plateau mappings

Iteration of a class of multi-plateau mappings

Multi-modal Land Cover Classification of Historical Aerial Images and Topographic Maps: A Comparative Study

Multi-modal Land Cover Classification of Historical Aerial Images and Topographic Maps: A Comparative Study

Relaxed inertial self-adaptive algorithm for the split feasibility problem with multiple output sets and fixed-point problem in the class of demicontractive mappings

The purpose of this paper is to study the split feasibility problem with multiple output sets and fixed point problem in the class of demicontractive mappings and propose relaxed inertial self-adaptive algorithms that do not use the least squares method. Under some appropriate assumptions, we establish a strong convergence result for the sequence generated by the proposed algorithm. Our result generalizes and extends several results existing in the literature. Finally, we illustrate the convergence of the proposed algorithm by using a numerical example.

The action of mapping class groups on de Rham quasimorphisms

We study the action of the mapping class group on the subspace of de Rham classes in the degree-two bounded cohomology of a hyperbolic surface. In particular, we show that the only fixed nontrivial finite-dimensional subspace is the one generated by the Euler class. As a consequence, we get that the action of the mapping class group on the space of de Rham quasimorphisms has no fixed points.

Improved Breast Cancer Detection using Modified ResNet50-Based on Gradient-Weighted Class Activation Mapping

Breast cancer is a prevalent and serious disease that affects many women around the world every year. Detecting breast cancer early is crucial for improving survival rates and treating the illness effectively. Researchers are exploring various methods, including neural networks and machine learning, to assist in detecting the disease. However, due to limited data availability, leveraging pre-trained models trained on diverse image datasets has become a common practice. This article introduces a novel approach to identifying breast cancer that involves the utilization of a deep learning model utilizing the ResNet50 framework, coupled with heat mapping and gradient-weighted class activation mapping (Grad-CAM). The suggested method was primarily assessed using the FDDM dataset of Subtracted Contrast Enhanced Spectral Mammography (CDD-CESM) images. The outcomes from this model were then contrasted with those of five other well-known models: VGG16, VGG19, MobileNetV2, EfficientNet-B7, and standard ResNet50. The newly proposed model yielded an accuracy of 0.8920, which was better than the other models. Additionally, Grad-CAM showed nearly flawless feature extraction in a breast cancer classification assignment. In the discussion section, the suggested method was utilized with the MIAS dataset to ensure thoroughness and scalability, and to allow for comparison with prior research. The results demonstrated the effectiveness of the suggested approach, with an accuracy of 0.9830 achieved on the MIAS dataset, surpassing previous works. This study significantly enhances the improvement of breast cancer detection through the integration of deep learning, the ResNet50 architecture, and visualization methods including heat maps and Grad-CAM.

Existence Results of Some Nonlinear Minimization Problems with Application to a System of PDE

A new class of non-self mappings, called proximal condensing operators, is introduced and used to investigate the existence of best proximity points for non-self mappings in reflexive Banach spaces by applying an appropriate measure of noncompactness. In this way, we present generalizations of well-known Schauder and Darbo fixed point problems. We then define the notion of best proximity points for multiplication of two non-self mappings in the setting of Banach algebras and present some sufficient conditions to ensure the existence of such points. Finally, we renorm a Banach space consists of all real valued and continuous functions with two variables to obtain the strict convexity condition and then we survey the existence of an optimal solution for a system of partial differential equations.

Abelian TQFTS and Schrödinger local systems

In this paper, we construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a q -deformation of U(1) for any root of unity q . We prove that, for 3-cobordisms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger bimodule action that reproduces the abelian TQFT.The full abelian TQFT provides a projective representation of the mapping class group \mathrm{Mod}(\Sigma) on the Schrödinger representation, which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of \mathrm{Mod}(\Sigma) on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle.

A classification of regular maps with Euler characteristic a negative prime cube

A classification of regular maps with Euler characteristic a negative prime cube

On the homology of big mapping class groups

AbstractWe prove that the mapping class group of the one‐holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once‐punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite‐type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of dissipated groups; the other is a general homological stability result for mapping class groups of infinite‐type surfaces.

Some refinements of the Fekete and Szegö inequalities for a class of holomorphic mappings associated with quasi-convex mappings

Some refinements of the Fekete and Szegö inequalities for a class of holomorphic mappings associated with quasi-convex mappings

The Borsuk–Ulam property for homotopy classes on bundles, parametrized braids groups and applications for surfaces bundles

Let M and N be fiber bundles over the same base B, where M is endowed with a free involution τ over B. A homotopy class δ∈[M,N]B (over B) is said to have the Borsuk–Ulam property with respect to τ if for every fiber-preserving map f:M→N over B which represents δ there exists a point x∈M such that f(τ(x))=f(x). In the cases that B is a K(π,1)-space and the fibers of the projections M→B and N→B are K(π,1) closed surfaces SM and SN, respectively, we show that the problem of decide if a homotopy class of a fiber-preserving map f:M→N over B has the Borsuk-Ulam property is equivalent of an algebraic problem involving the fundamental groups of M, the orbit space of M by τ and a type of generalized braid groups of N that we call parametrized braid groups. As an application, we determine the homotopy classes of fiber-preserving self maps over S1 that satisfy the Borsuk-Ulam property, with respect to all involutions τ over S1, for the torus bundles over S1 with M=N=MA and A=[1n01].