Rectangular Space Research Articles

Some critical remarks on “Some new fixed point results in rectangular metric spaces with an application to fractional-order functional differential equations”

In this manuscript, we generalize, improve, and enrich recent results established by Budhia et al. [L. Budhia, H. Aydi, A.H. Ansari, D. Gopal, Some new fixed point results in rectangular metric spaces with application to fractional-order functional differential equations, Nonlinear Anal. Model. Control, 25(4):580–597, 2020]. This paper aims to provide much simpler and shorter proofs of some results in rectangular metric spaces. According to one of our recent lemmas, we show that the given contractive condition yields Cauchyness of the corresponding Picard sequence. The obtained results improve well-known comparable results in the literature. Using our new approach, we prove that a Picard sequence is Cauchy in the framework of rectangular metric spaces. Our obtained results complement and enrich several methods in the existing state-ofart. Endorsing the materiality of the presented results, we also propound an application to dynamic programming associated with the multistage process.

The adjustable aiming device for caspar pin insertion in anterior cervical spine surgery.

Creating a rectangular disc space is an important step during anterior cervical discectomy and fusion or cervical total disc replacement. The study aims to determine the accuracy of Caspar pin insertion by using a novel Adjustable Caspar Pin Aiming Device in anterior cervical procedures. Forty Caspar pins were placed using an Adjustable Caspar Pin Aiming Device in 20 human cadaveric cervical vertebral bodies from C3 to C7 after performing anterior discectomies. Accuracy of pin placement was assessed by lateral fluoroscopy, considering superior endplate slope (SE), inferior endplate slope (IE), Caspar pin slope (CP), and endplate-Caspar pin slope difference (SE/CP, IE/CP). The mean superior endplate slope (SE), inferior endplate slope (IE), and Caspar pin slope (CP) were 10.82 ± 2.3°, 10.32 ± 3.2°, and 15.58 ± 7.9°, respectively. The average superior endplate-Caspar pin slope difference (SE/CP) and inferior endplate-Caspar pin slope difference (IE/CP) were 6.6 ± 0.8° and 7.7 ± 0.8°, respectively. The greatest slope difference was observed at the superior and inferior endplates of C3. No cervical endplate violations occurred. Adjustable Caspar Pin Aiming Device allowed for a highly accurate Caspar pin placement with the average endplate-Caspar pin slope difference of less than 7.7°. It results in accurate placement of the superior and inferior Caspar pins parallel to the index vertebral endplates. Furthermore, it appears to facilitate the safe and effective insertion of Caspar pins for anterior cervical procedures.

Certain new aspects in fuzzy fixed point theory

<abstract> <p>We will establish several fixed point results in new introduced spaces in this manuscript known as fuzzy rectangular metric-like spaces and rectangular b-metric-like spaces. These new results and spaces will improve the approach of existing ones in the literature. Few non-trivial examples and an application also verify the uniqueness of solution.</p> </abstract>

Fixed Point Result on Generalized Cone b-Metric Spaces

The purpose of this paper is to prove a fixed point result on contraction mapping in generalized cone b-metric space (in short GCbMS) as a generalization of cone metric space, cone b metric space and rectangular metric space. The conception of generalized metric space is a generalization of that of classical metric space. Several authors have proved fixed point theorems of contractive mappings on generalized metric spaces, which also generalized some corresponding fixed point results in classical metric spaces. In present paper, we prove a result that is extension of the Kannan fixed point theorem proved by Reny George <i>et al</i>. Our result is extend and unify several well known results in the literature available for cone and cone-b metric space.

Numerical investigation of droplets in a cross-ventilated space with sitting passengers under asymptomatic virus transmission conditions.

Asymptomatic virus transmission in public transportation is a complex process that is difficult to analyze computationally and experimentally. We present a high-resolution computational study for investigating droplet dynamics under a speech-like exhalation mode. A large eddy simulation coupled with Lagrangian tracking of drops was used to model a rectangular space with sitting thermal bodies and cross-ventilated with a multislot diffuser. Release of drops from different seat positions was evaluated to analyze the decontamination performance of the ventilation system. The results showed an overall good performance, with an average of 24.1% of droplets removed through the exhaust in the first 40 s. The droplets' distribution revealed that higher concentrations were less prevalent along the center of the domain where the passengers sit. Longitudinal contamination between rows was noted, which is a negative aspect for containing the risk of infection in a given row but has the benefit of diluting the concentration of infectious droplets. Droplets from the window seat raised more vertically and invaded the space of other passengers to a lesser extent. In contrast, droplets released from the middle seat contaminated more the aisle passenger's space, indicating that downward flow from personal ventilation could move down droplets to its breathing region. Droplets released from the aisle were dragged down by the ventilation system immediately. The distance of drops to the mouth of the passengers showed that the majority passed at a relatively safe distance. However, a few of them passed at a close distance of the order of magnitude of 1 cm.

TEA-SEA: Tiling and scheduling of non-uniform two-level perfectly nested loops using an evolutionary approach

Nowadays, in scientific and computational programs, increasing the execution speed of programs is very important. In implementing scientific programs, loops are dedicating a large part of time to themselves. Therefore, the parallel execution of the iteration loop reduces the runtime of all programs. This paper has been working on the parallelism of two-level perfectly nested loops with non-uniform dependencies and rectangular iteration space, and then their scheduling is based on the scheduling of task graphs. For this purpose, the iteration space of non-uniform loop is partitioned using a tiling transformation, and then the tiled space is modelled as a task, graph for scheduling. In the task graph, vertices expressive tiles and edges expressive data dependencies inter-tiles. It should be noted that the computational load of vertices is not the same, and the vertices corresponding imperfect tiles have a less computational load. The scheduling is done on both types of homogeneous and heterogeneous multiprocessor systems. The tiling of iteration space loop and the scheduling of task graph is arranged in the category of NP-Hard problems that using evolutionary algorithms can be efficient to discover a suitable solution for them. The results of the proposed approach show the performance of this method better than previous methods in the tiling and scheduling of non-uniform loops. According to the examinations performed on uniform loops, this algorithm also performs better than previous algorithms on these types of loops.

Some New Results on Equivalent Cauchy Sequences and Their Applications to Meir-Keeler Contraction in Partial Rectangular Metric Space

The study of fixed points in the metric spaces plays a crucial role in the development of Functional Analysis. It is evolved by generalizing the metric space or improving the contractive conditions. Recently, the partial rectangular metric space and its topology have been the center of study for many researchers. They have defined open and closed balls the equivalent Cauchy sequences and Cauchy sequences, convergent sequences which are used as tools in many achieved results. In this paper, two facts for equivalent Cauchy sequences in a partial rectangular metric space are provided by using an ultra - altering distance function. Furthermore, some results of Cauchy sequences in a partial rectangular metric space are highlighted. There is proved that under some conditions the equivalent Cauchy sequences are Cauchy sequences in a partial rectangular metric space. Some fixed point results have been taken as applications of our new conditions of Cauchy sequences and equivalent Cauchy sequences in a partial rectangular metric space <img src=image/13424354_01.gif> for orbitally continuous functions <img src=image/13424354_02.gif>. To illustrate the obtained results some examples are given.

On Fuzzy Extended Hexagonal b-Metric Spaces with Applications to Nonlinear Fractional Differential Equations

The focus of this research article is to investigate the notion of fuzzy extended hexagonal b-metric spaces as a technique of broadening the fuzzy rectangular b-metric spaces and extended fuzzy rectangular b-metric spaces as well as to derive the Banach fixed point theorem and several novel fixed point theorems with certain contraction mappings. The analog of hexagonal inequality in fuzzy extended hexagonal b-metric spaces is specified as follows utilizing the function b(c,d): mhc,d,t+s+u+v+w≥mhc,e,tb(c,d)∗mhe,f,sb(c,d)∗mhf,g,ub(c,d)∗mhg,k,vb(c,d)∗mhk,d,wb(c,d) for all t,s,u,v,w&gt;0 and c≠e,e≠f,f≠g,g≠k,k≠d. Further to that, this research attempts to provide a feasible solution for the Caputo type nonlinear fractional differential equations through effective applications of our results obtained.

Origins and spread of formal ceremonial complexes in the Olmec and Maya regions revealed by airborne lidar.

City plans symbolizing cosmologies have long been recognized as a defining element of Mesoamerican civilizations. The origins of formal spatial configurations are thus the key to understanding early civilizations in the region. Assessment of this issue, however, has been hindered by the lack of systematic studies of site plans over broad areas. Here, we report the identification of 478 formal rectangular and square complexes, probably dating from 1,050 to 400 BC, through a lidar (laser imaging, detection and ranging) survey across the Olmec region and the western Maya lowlands. Our analysis of lidar data also revealed that the earlier Olmec centre of San Lorenzo had a central rectangular space, which possibly provided the spatial template for later sites. This format was probably formalized and spread after the decline of San Lorenzo through intensive interaction across various regions. These observations highlight the legacy of San Lorenzo and the critical role of inter-regional interaction.

An Application of an Unequal-Area Facilities Layout Problem with Fixed-Shape Facilities

The unequal-area facility layout problem (UA-FLP) is the problem of locating rectangular facilities on a rectangular floor space such that facilities do not overlap while optimizing some objective. The objective considered in this paper is minimizing the total distance materials travel between facilities. The UA-FLP considered in this paper considers facilities with fixed dimension and was motivated by the investigation of layout options for a production area at the Toyota Motor Manufacturing West Virginia (TMMWV) plant in Buffalo, WV, USA. This paper presents a mathematical model and a genetic algorithm for locating facilities on a continuous plant floor. More specifically, a genetic algorithm, which consists of a boundary search heuristic (BSH), a linear program, and a dual simplex method, is developed for an UA-FLP. To test the performance of the proposed technique, several test problems taken from the literature are used in the analysis. The results show that the proposed heuristic performs well with respect to solution quality and computational time.

Some Fixed Point Results in Partial Rectangular Metric-Like Spaces

In this paper, we introduce a new concept of partial rectangular metric-like space and prove some results on the existence and uniqueness of a fixed point of a functionT:X⟶X, defined on a partial rectangular metric-like spaceX, which fulfills a nonlinear contractive condition using a comparison function and the diameter of the orbits. The obtained results generalize some previously acknowledged results in partial metric spaces, partial rectangular metric spaces, and rectangular metric-like spaces. The examples presented prove the usefulness of the introduced generalizations.

A Path Planning Strategy for Multi-Robot Moving with Path-Priority Order Based on a Generalized Voronoi Diagram

This paper proposes a new path planning strategy called the navigation strategy with path priority (NSPP) for multiple robots moving in a large flat space. In the space, there may be some static or/and dynamic obstacles. Suppose we have the path-priority order for each robot, then this article aims to find an efficient path for each robot from its starting point to its target point without any collision. Here, a generalized Voronoi diagram (GVD) is used to perform the map division based on each robot’s path-priority order, and the proposed NSPP is used to do the path planning for the robots in the space. This NSPP can be applied to any number of robots. At last, there are several simulations with a different number of robots in a circular or rectangular space to be shown that the proposed method can complete the task effectively and has better performance in average trajectory length than those by using the benchmark methods of the shortest distance algorithm (SDA) and reciprocal orientation algorithm (ROA).

F-contractions Fixed point results in Extended Rectangular b-metric space

In the present paper, we prove common fixed point theorem for F-contraction by considering one map is orbitally continuous map in $Omega-$Extended rectangular b-metric space. In addition, we find fixed point for Banach and Kannan type contraction inequality without consideration of orbitally continuity. Also at the fixed point maps does not force to be continuous. An examples is also provided for the utility of our results.

Some fixed point theorems on a C∗-algebra valued rectangular quasi-metric spaces

In this work, we discuss the existence and uniqueness of fixed points for a self-mapping defined on a C∗-algebra valued rectangular quasi-metric space. Our results extend and supplement several recent results in the literature. Some examples are provided to illustrate our results.

Fixed point theorems for ψ−contractive mapping in C∗−algebra valued rectangular b-metric spaces

In this paper, we present a new insight of C∗−algebra valued rectangular b−metric spaces in the perspective of the fixed point theory using contractive mapping. Using contractive mapping in the rectangular b−metric spaces, we discussed the existence and the uniqueness of the fixed point with mapping satisfying a contractive condition. As a result, we obtained an interesting and important result for the general case of C∗−algebra valued metric spaces. In particular, we study some fixed point theorems in the C∗−algebra valued rectangular b−metric spaces using a positive function.

Stochastic Fixed Point Theorems in Rectangular Metric Spaces

In this paper, we present the random (Stochastic) version of Banach contraction principle and some of its generalizations in setting of rectangular metric spaces.

Exact optimization and decomposition approaches for shelf space allocation

Shelf space is one of the scarcest resources, and its effective management to maximize profits has become essential to gain a competitive advantage for retailers. We consider the shelf space allocation problem with additional features (e.g., integer facings, rectangular arrangement restrictions) motivated by literature and our interactions with a local bookstore. We determine optimal number of facings of all products in two aspects (width and height of a rectangular arrangement space for each product), and allocate them as contiguous rectangles to maximize profit. We first develop a mixed-integer linear mathematical programming model (MIP) for our problem and propose a solution method based on logic-based Benders decomposition (LBBD). Next, we construct an exact 2-stage algorithm (IP1/IP2), inspired by LBBD, which can handle larger and real-world size instances. To compare performances of our methods, we generate 100 test instances inspired by real-world applications and benchmarks from the literature. We observe that IP1/IP2 finds optimal solutions for real-world instances efficiently and can increase the local bookstore’s profit up to 16.56%. IP1/IP2 can provide optimal solutions for instances with 100 products in minutes and optimally solve up to 250 products (assigned to 8 rows x 160 columns) within a time limit of 1800 s. This exact 2-stage IP1/IP2 solution approach can be effective in solving similar problems such as display problem of webpage design, allocation of product families in grocery stores, and flyer advertising.

Fixed Point Results via α-Admissibility in Extended Fuzzy Rectangular b-Metric Spaces with Applications to Integral Equations

In this article, the concept of extended fuzzy rectangular b-metric space (EFRbMS, for short) is initiated, and some fixed point results frequently used in the literature are generalized via α-admissibility in the setting of EFRbMS. For the illustration of the work presented, some supporting examples and an application to the existence of solutions for a class of integral equations are also discussed.

Some fixed point theorems for \(F\)-expansive mapping in generalized metric spaces

In this paper, we present the notion of generalized \(F\)-expansive mapping incomplete rectangular metric spaces and study various fixed point theorems for such mappings. The findings of this paper, generalize and improve many existing results in the literature.

At the Edges of Unmeeting: Geometries of Sea and Land in Marianne Moore’s Seascapes

AbstractThis article theorizes the edge between the land and the sea in Marianne Moore’s poetry and in Jeff Wall’s photograph “The Flooded Grave.” For these artists, the edge is constituted as an array of fluctuating temporal rhythms created by living organisms and dynamic abiotic elements, such as changing tides and the rocky or sandy substrates that make up the geological formation of the coastline. The edge appears in a number of Moore’s poems, including “The Fish,” which depicts the intertidal zone populated by injured “crow-blue mussel-shells,” “ink- / bespattered jelly fish,” and sea-stars, which resemble “pink / rice-grains.” Wall’s photograph, in turn, portrays a transposition between an aquatic and a terrestrial environment by placing an intertidal pool filled with star fish and sea anemones into the rectangular space of a grave. This article reads such moments as experiments in the textured edges of displacement. In these instances of intertidal encounter, organisms find themselves in unlikely configurations of space and time as a consequence of geophysical and ecological redistributions associated with effects of climate change.