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The authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions W_Q on X₀(N) for N ≤ 50 are Weierstrass points or not.

The design is by using Lawittes's and Schoeneberg's theorems.

Finding all Weierstrass points on X₀(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.

The Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X₀(N).

The purpose of this paper is to study the coupled fixed point problem and the coupled best proximity problem for single-valued and multi-valued contraction type operators defined on cyclic representations of the space. The approach is based on fixed point results for appropriate operators generated by the initial problems.

This paper aims to prove some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results of this paper generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. Furthermore, the authors produce an example to demonstrate the applicability of the results.

The results of this paper are theoretical and analytical in nature.

The authors established some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. An example was constructed to demonstrate the applicability of the results.

Analytical and theoretical results.

The results of this paper can be applied in science and engineering.

The results of this paper is applicable in certain social sciences.

The results of this paper are new and will open up new areas of research in mathematical sciences.

In this paper, we use the notion of cyclic representation of a nonempty set with respect to a pair of mappings to obtain coincidence points and common fixed points for a pair of self-mappings satisfying some generalized contraction- type conditions involving a control function in partial metric spaces. Moreover, we provide some examples to analyze and illustrate our main results.

Theoretical method.

We establish some coincidence points and common fixed point results in partial metric spaces.

Results of this study are new and interesting.

This pragmatic research paper aims to unravel the smart vision-based method (SVBM), an AI program to correlate the computer vision (recorded and live videos using mobile and embedded cameras) that aids in manual lifting human pose deduction, analysis and training in the construction sector.

Using a pragmatic approach combined with the literature review, this study discusses the SVBM. The research method includes a literature review followed by a pragmatic approach and lab validation of the acquired data. Adopting the practical approach, the authors of this article developed an SVBM, an AI program to correlate computer vision (recorded and live videos using mobile and embedded cameras).

Results show that SVBM observes the relevant events without additional attachments to the human body and compares them with the standard axis to identify abnormal postures using mobile and other cameras. Angles of critical nodal points are projected through human pose detection and calculating body part movement angles using a novel software program and mobile application. The SVBM demonstrates its ability to data capture and analysis in real-time and offline using videos recorded earlier and is validated for program coding and results repeatability.

Literature review methodology limitations include not keeping in phase with the most updated field knowledge. This limitation is offset by choosing the range for literature review within the last two decades. This literature review may not have captured all published articles because the restriction of database access and search was based only on English. Also, the authors may have omitted fruitful articles hiding in a less popular journal. These limitations are acknowledged. The critical limitation is that the trust, privacy and psychological issues are not addressed in SVBM, which is recognised. However, the benefits of SVBM naturally offset this limitation to being adopted practically.

The theoretical and practical implications include customised and individualistic prediction and preventing most posture-related hazardous behaviours before a critical injury happens. The theoretical implications include mimicking the human pose and lab-based analysis without attaching sensors that naturally alter the working poses. SVBM would help researchers develop more accurate data and theoretical models close to actuals.

By using SVBM, the possibility of early deduction and prevention of musculoskeletal disorders is high; the social implications include the benefits of being a healthier society and health concerned construction sector.

Human pose detection, especially joint angle calculation in a work environment, is crucial to early deduction of muscoloskeletal disorders. Conventional digital technology-based methods to detect pose flaws focus on location information from wearables and laboratory-controlled motion sensors. For the first time, this paper presents novel computer vision (recorded and live videos using mobile and embedded cameras) and digital image-related deep learning methods without attachment to the human body for manual handling pose deduction and analysis of angles, neckline and torso line in an actual construction work environment.

The purpose of this paper is to reduce the difficulty of model predictive control (MPC) deployment on FPGA so that researchers can make better use of FPGA technology for academic research.

In this paper, the MPC algorithm is written into FPGA by combining hardware with software. Experiments have verified this method.

This paper implements a ZYNQ-based design method, which could significantly reduce the difficulty of development. The comparison with the CPU solution results proves that FPGA has a significant acceleration effect on the solution of MPC through the method.

Due to the limitation of practical conditions, this paper cannot carry out a hardware-in-the-loop experiment for the time being, instead of an open-loop experiment.

This paper proposes a new design method to deploy the MPC algorithm to the FPGA, reducing the development difficulty of the algorithm implementation on FPGA. It greatly facilitates researchers in the field of autonomous driving to carry out FPGA algorithm hardware acceleration research.

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases.

The results of this paper are theoretical and analytical in nature.

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases.

The results are theoretical and analytical.

The results were applied to solving nonlinear integral equations.

The results has several social applications.

The results of this paper are new.

The purpose of this paper is to extend the recent results of Okeke et al. (2018) to the class of multivalued ρ-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of T-stability, almost T-stability and summably almost T-stability to modular function spaces and give some results.

In this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map T such that PρT, an associate multivalued map, is a ρ-contractive-like mapping.

The concepts of relative ρ-stability and weak ρ-stability are introduced, and conditions in which these multistep iterations are relatively ρ-stable, weakly ρ-stable and ρ-stable are established for the newly introduced strong ρ-quasi-contractive-like class of maps.

Noor type, Ishikawa type and Mann type iterative sequences are deduced as corollaries in this study.

The results obtained in this work are complementary to those proved in normed and metric spaces in the literature.