Mapping Class Research Articles

Random maps with sociological flavor

Abstract A map of a set to itself admits a representation by a graph with vertices being the elements of the set and an edge between every vertex and its image. Communities defined as the maximal connected components are uni-cyclic. The distributions of the sizes of communities and lengths of cycles for unconstrained random maps is a classical subject. We call experts the images and followers the remaining vertexes, and we further define prophets, egocentrics, and introverts. We introduce and analyze classes of random maps with sociological flavor. 

A study on Fibo-Pascal sequence spaces and associated matrix transformations and applications of Hausdorff measure of non-compactness

Abstract In this article, we introduce Fibo-Pascal sequence spaces P p F {P_{p}^{F}} , 0 < p < ∞ {0<p<\infty} , and P ∞ F {P_{\infty}^{F}} through the utilization of the Fibo-Pascal matrix P F {P^{F}} . We establish that both P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} are BK-spaces, enjoying a linear isomorphism with the classical spaces ℓ p {\ell_{p}} and ℓ ∞ {\ell_{\infty}} , respectively. Further contributing to the depth of our investigation, we proceed to derive the Schauder basis of the space P p F {P_{p}^{F}} , alongside an exhaustive computation of the α-, β-, and γ-duals for both spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . Additionally, we undertake the task of characterizing certain classes of matrix mappings pertaining to the spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . The final section of this study is dedicated to the meticulous characterization of compact operators acting in the spaces P 1 F {P_{1}^{F}} , P p F {P_{p}^{F}} , and P ∞ F {P_{\infty}^{F}} by using Hausdorff measure of non-compactness.

Fixed point results for Geraghty–Ćirić-type contraction mappings in b-metric space with applications

In this study, we manifest a new class of mappings that satisfy Geraghty–Ćirić-type contractive conditions in the context of b-metric spaces and prove a theorem on the existence and uniqueness of fixed points. Our results unify and generalize the results of Geraghty; Ćirić; Dukic, Kadelburg, and Radenović; and Shu-fang Li, Fei Hi, and Ning Lu in the setting of b-metric spaces. Furthermore, we provide examples to verify the correctness and applicability of our results. We also utilize our findings to show the existence of a unique solution for a nonlinear integral equation.

Few-Shot Classification of Wafer Bin Maps Using Transfer Learning and Ensemble Learning

Abstract The high cost of collecting and annotating wafer bin maps (WBMs) necessitates few-shot WBM classification, i.e., classifying WBM defect patterns using a limited number of WBMs. Existing few-shot WBM classification algorithms mainly utilize meta-learning methods that leverage knowledge learned in several episodes. However, meta-learning methods require a large amount of additional real WBMs, which can be unrealistic. To help train a network with a few real-WBMs while avoiding this challenge, we propose the use of simulated WBMs to pre-train a classification model. Specifically, we employ transfer learning by pre-training a classification network with sufficient amounts of simulated WBMs and then fine-tuning it with a few real-WBMs. We further employ ensemble learning to overcome the overfitting problem in transfer learning by fine-tuning multiple sets of classification layers of the network. A series of experiments on a real-dataset demonstrate that our model outperforms the meta-learning methods that are widely used in few-shot WBM classification. Additionally, we empirically verify that transfer and ensemble learning, the two most important yet simple components of our model, reduce the prediction bias and variance in few-shot scenarios without a significant increase in training time.

Involution Generators of the Big Mapping Class Group

Involution Generators of the Big Mapping Class Group

Retraction Note: Curves with prescribed symmetry and associated representations of mapping class groups

Retraction Note: Curves with prescribed symmetry and associated representations of mapping class groups

Invariant measures for random piecewise convex maps

We establish the existence of Lebesgue-equivalent conservative and ergodic σ -finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures around a small neighbourhood of a fixed point where the invariant density functions may diverge. Application covers random intermittent maps with critical points or flat points. We also illustrate that the size of invariant measures tends to infinite for random maps whose right branches exhibit a strongly contracting property on average, so that they have a strong recurrence to a fixed point.

On T-continuous map

It is no secret to the observer that the new open sets play a role in general topology. and they represent research topic for many topologists in different parts of the world. In fact, there is an important topics in general topology and real analysis that deals with continuity of maps and their different forms by relying on new open set. In this work, we presented and studied a new class of continuous maps called T-continuous by using the concepts of T-open sets and T-closed sets, where important and basic properties of T-continuous map and its relationship with other maps were verified.  

Sample size adaptation designs and efficiency comparison with group sequential designs.

This study is to give a systematic account of sample size adaptation designs (SSADs) and to provide direct proof of the efficiency advantage of general SSADs over group sequential designs (GSDs) from a different perspective. For this purpose, a class of sample size mapping functions to define SSADs is introduced. Under the two-stage adaptive clinical trial setting, theorems are developed to describe the properties of SSADs. Sufficient conditions are derived and used to prove analytically that SSADs based on the weighted combination test can be uniformly more efficient than GSDs in a range of likely values of the true treatment difference . As shown in various scenarios, given a GSD, a fully adaptive SSAD can be obtained that has sufficient statistical power similar to that of the GSD but has a smaller average sample size for all in the range. The associated sample size savings can be substantial. A practical design example and suggestions on the steps to find efficient SSADs are also provided.

Generalized second order vectorial ∞-eigenvalue problems

We consider the problem of minimizing the $L^\infty$ norm of a function of the hessian over a class of maps, subject to a mass constraint involving the $L^\infty$ norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the “hinged” and the “clamped” cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^\infty$ minimizer, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem. Furthermore, we establish upper and lower bounds for the eigenvalue.

Fekete and Szegö Inequalities for a Class of Holomorphic Mappings

Fekete and Szegö Inequalities for a Class of Holomorphic Mappings

Meta-analysis Influence of Integrated Mind Mapping Inquiry Based Learning Model on Student Problem Solving Skills

A This study aims to determine the effect of inquiry-based learning model integrated mind mapping on students' problem-solving skills. This type of research is quantitative research with meta-analysis methods.  Data comes from international journals or proceedings published 2015-2023. Data search through ScienceDirect, Springer, ProQuest, ERIC and Taylor of Francis databases. Research should be selected for data in meta-analyses. The inclusion criteria are that research must be experimental methods or quasi-experiments, research indexed by Scopus, Sinta, Web of Science, Microsoft Academic and EBSCO, Research has an experimental class inquiry learning model integrated mind mapping and conventional model control class, research has a number of samples (N), correlation values (r), t, and F and a sample consisting of 1021 participants. Data analysis with the help of JASP to determine the value of effect size, heterogeneity test and publication bias. The results showed a positive influence of minde mapping-based inquiry learning model on students' problem solving skills ( z = 7.019; p < 0.001; 95% CI [0.61; 1.26]. This finding shows that the influence of mind mapping-based inquiry learning models is included in the very high category (rRE = 1.31).

Variational-like inclusion problems involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p. and fixed point problems

This paper is devoted to the introduction of a new class of accretive mappings in the framework of q-uniformly smooth Banach spaces called A-maximal m-relaxed η-accretive mappings in the sense of semi-inner product of type (p) (for short, s.i.p.(p)). The resolvent operator associated with such a mapping is defined; its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed under some suitable conditions. At the same time, the problem of finding a common element of the set of fixed points of a given nearly asymptotically nonexpansive mapping and the set of solutions of a variational-like inclusion problem involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is investigated. To achieve this goal, a new iterative algorithm is constructed. Using the notions of graph convergence and resolvent operator associated with an A-maximal m-relaxed η-accretive mapping in the sense of s.i.p.(p), a new equivalence relationship between the graph convergence and the resolvent operator convergence of a sequence of A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is established. As an application of this equivalence, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is demonstrated under some suitable assumptions imposed on the parameters. The results presented in this paper unify, improve and generalize some recent works in this field.

On fixed point convergence results for class of nonexpansivemappings in hyperbolic spaces via PJ iteration process

In this paper, we provide certain fixed point results for a mean nonexpansive mapping, as well as a new iterative algorithm called PJ-iterationfor approximating the fixed point of this class of mappings in the setting of hyperbolic spaces. Furthermore, we establish strong and∆-convergence theorem for mean nonexpansive mapping in hyperbolic space. Finally, we present a numerical example to illustrate ourmain result and then display the efficiency of the proposed algorithm compared to different iterative algorithms in the literature. Our resultsobtained in this paper improve, extend and unify some related results in the literature.

Remarks on Collino cycles and hyperelliptic Johnson homomorphisms

A Collino cycle is a higher cycle on the Jacobian of a hyperelliptic curve. The universal family of Collino cycles naturally gives rise to a normal function, whose induced monodromy relates to the hyperelliptic Johnson homomorphism. Colombo computed this monodromy explicitly and made this relation precise. We recast this in the perspective of relative completion. In particular, we use Colombo's result to construct Collino classes, which are cohomology classes of hyperelliptic mapping class groups with coefficients in a certain symplectic representation. We also determine the dimension of their span in the case of the level two hyperelliptic mapping class group.

A fixed point result for DND-convex-power condensing mappings

In this paper, based on the so-called degree of nondensifiability (DND), we introduce the concept of DND-convex-power condensing mapping which generalizes that of the DND-condensing one. A fixed point result for this new class of mappings is proved, and with some examples we evidence the differences between it and other fixed point theorems of the same type. As application of our results, we prove under suitable conditions the existence of fixed point for the superposition operator, defined in the Banach spaces of the continuous functions from [Formula: see text] and with values in a Banach space.

From Movements to Metrics: Evaluating Explainable AI Methods in Skeleton-Based Human Activity Recognition.

The advancement of deep learning in human activity recognition (HAR) using 3D skeleton data is critical for applications in healthcare, security, sports, and human-computer interaction. This paper tackles a well-known gap in the field, which is the lack of testing in the applicability and reliability of XAI evaluation metrics in the skeleton-based HAR domain. We have tested established XAI metrics, namely faithfulness and stability on Class Activation Mapping (CAM) and Gradient-weighted Class Activation Mapping (Grad-CAM) to address this problem. This study introduces a perturbation method that produces variations within the error tolerance of motion sensor tracking, ensuring the resultant skeletal data points remain within the plausible output range of human movement as captured by the tracking device. We used the NTU RGB+D 60 dataset and the EfficientGCN architecture for HAR model training and testing. The evaluation involved systematically perturbing the 3D skeleton data by applying controlled displacements at different magnitudes to assess the impact on XAI metric performance across multiple action classes. Our findings reveal that faithfulness may not consistently serve as a reliable metric across all classes for the EfficientGCN model, indicating its limited applicability in certain contexts. In contrast, stability proves to be a more robust metric, showing dependability across different perturbation magnitudes. Additionally, CAM and Grad-CAM yielded almost identical explanations, leading to closely similar metric outcomes. This suggests a need for the exploration of additional metrics and the application of more diverse XAI methods to broaden the understanding and effectiveness of XAI in skeleton-based HAR.

The Hole Argument without the notion of isomorphism

In this paper, I argue that the Hole Argument can be formulated without using the notion of isomorphism, and for this reason it is not threatened by the criticism of Halvorson and Manchak (Br J Philos Sci, 2022. https://doi.org/10.1086/719193). Following Earman and Norton (Br J Philos Sci 38, pp. 515–525, 1987), I divide the Hole Argument into two steps: the proof of the Gauge Theorem and the transition from the Gauge Theorem to the conclusion of radical indeterminism. In the analaysis of the first step, I argue that the Gauge Theorem does not rely on the notion of isomorphism but on the notion of the diffeomorphism-invariance of the equations of local spacetime theories; however, for this approach to work, the definition of local spacetime theories needs certain amendments with respect to Earman and Norton’s formulation. In the analysis of the second step, I postulate that we should use the notion of radical indeterminism instead of indeterminism simpliciter and that we should not decide in advance what kind of maps are to be used in comparing models. Instead, we can tentatively choose some kind of maps for this purpose and check whether a given choice leads to radical indeterminism involving empirically indistinguishable models. In this way, the use of the notion of isomorphism is also avoided in the second step of the Hole Argument. A general picture is that physical equivalence can be established by means of an iterative procedure in which we examine various candidate classes of maps, and, depending on the outcomes, we need to broaden or narrow these classes. The Hole Argument can be viewed as a particular instance of this procedure.

On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces

In this paper, we consider a faster iterative method for approximating the fixed points of generalized α-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results.

A family of robust chaotic S-unimodal maps based on the Gaussian function

This research paper introduces a family of one-dimensional S-unimodal maps based on the Gaussian function, designed to exhibit robust chaos across a wide range of parameters. These maps are developed to display robust chaos by avoiding multiple fixed points that are primarily responsible for the coexisting attractors in 1D maps. The parameter space analysis reveals that chaotic behaviour is sustained across the entire parameter space, except for a very narrow region. The study employs a comprehensive computational approach, including quantitative measures such as sample entropy, Lyapunov exponent, and invariant measures. The uniformly higher values of sample entropy, uniform positive values of the Lyapunov exponent, and the existence of invariant measures in a region of parameter space confirm the presence of robust chaos in these maps. Such a promising class of robust chaotic maps may be potentially used in diverse fields such as chaos-based cryptography, pseudo-random number generation, communication systems, and more.