Rectangular Space Research Articles

Fixed Point Results in Neutrosophic Rectangular Extended b-Metric Spaces

In this manuscript, we establish the notion of neutrosophic rectangular extended b-metric spaces and derive some fixed point results for contraction mappings. Also, we provide several non-trivial examples. Our results are more generalized with respect to the existing ones in the literature. At the end of the paper, we provide an application to non-linear fractional differential equations to test the validity of the results.

A Common Fixed Point Theorem in Cone Rectangular Metric Space Under Expansive Type Condition

A Common Fixed Point Theorem in Cone Rectangular Metric Space Under Expansive Type Condition

Some fixed point results in rectangular metric spaces

After motivation from Geraghty-type contractions and of Farhan, et al. we define α-admissible mappings and demonstrate the fixed point theorems for the above-mentioned contractions in rectangular metric space in this study. In the end, we discuss some consequences of our results as corollaries. 2010 MSC: 47H10, 54H25.

Fixed Point Theory on Triple Controlled Metric-like Spaces with a Numerical Iteration

Fixed point theory is a versatile mathematical theory that finds applications in a wide range of disciplines, including computer science, engineering, fractals, and even behavioral sciences. In this study, we propose triple controlled metric-like spaces as a generalization of controlled rectangular metric-like spaces. By examining the Θ-contraction mapping within these spaces, we extend and enhance the existing literature to establish significant fixed point results. Utilizing these findings, we demonstrate the existence of solutions to a Fredholm integral equation and provide an example of a numerical iteration method applicable to a specific case of this Fredholm integral equation.

Optimizing Asymmetric Native Knee Flexion Gap Balance Promotes Superior Outcomes in Primary Total Knee Arthroplasty.

Replicating native knee kinematics remains the ultimate goal of total knee arthroplasty (TKA). Technology, such as robotics, provides robust intraoperative data; however, no evidence-based targets currently exist for improved clinical outcomes. Furthermore, some surgeons target a rectangular flexion space in TKA unlike the native knee. This study evaluated the effect of in vivo flexion gap asymmetry on patient-reported outcome measures (PROMs) in contemporary TKA. In vivo tibiofemoral joint space dimensions were measured during 129 TKAs using a calibrated tension device before and after complete posterior cruciate ligament resection. PROMs were compared based on the final dimensions and the change in flexion gap dimensions at 90° of flexion: (1) equal laxity, (2) lateral laxity, and (3) medial laxity. Groups did not differ by demographics ( P ≥ 0.347), clinical follow-up ( P = 0.134), tibiofemoral alignment ( P = 0.498), or preoperative PROMs ( P ≥ 0.093). Mean follow-up for the cohort was 1.5 years (range, 1-3). Pain with climbing stairs, pain while standing upright, and knees "always feeling normal" scores were superior for patients with equal or lateral laxity compared with medial laxity ( P ≤ 0.064). Pain with level walking, University of California Los Angeles activity level, KOOS JR, and satisfaction scores also tended to be superior for patients with equal or lateral laxity, although it lacked statistical significance ( P ≥ 0.111). Results of this study suggest that patients with either an equally tensioned rectangular flexion space or with later-flexion lateral laxity after posterior cruciate ligament resection may achieve superior PROMs. Findings support the clinical benefit of facilitating posterolateral femoral roll back in flexion, which mimics native knee kinematics and further helps define targets for advanced technology.

Solving existence results in multi-term fractional differential equations via fixed points

Many researchers are interested in the existence theory of solutions to fractional differential equations. In the literature, existence results have been obtained by using a variety of fixed point problems, including the fixed-point problems of Lefschetz, Kleene, Tychonoff, and Banach. In this article, we propose a generalized version of the contraction principle in the context of controlled rectangular metric space. With this result, we address the existence and uniqueness results for the following fractional-order differential equations.1. The nonlinear multi-term fractional delay differential equation L(D)ζ(ϖ)=σ(ϖ,ζ(ϖ),ζ(ϖ−τ)),ϖ∈J=[0,T],T>0;ζ(ϖ)=σ̄(ϖ),ϖ∈[−τ,0]. where, L(D)=γwcDδw+γw−1cDδw−1+...+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3....w),γw≠0,0≤δ0<δ1<δ2....δw−1<δw<1, and cDδ denotes the Caputo fractional derivative of order δ.2. The Caputo type fractional differential equation cDδζ(ϖ)=σ(ϖ,ζ(ϖ)),5<δ≤6,ϖ∈[0,T], with ζ(0)=−ζ(T),ζ′(0)=−ζ′(T),ζ′′(0)=−ζ′′(T),ζ′′′(0)=−ζ′′′(T),ζ4(0)=−ζ4(T),ζ5(0)=−ζ5(T)

New fixed point theorems for $$(\phi , F)$$-contraction on rectangular b-metric spaces

New fixed point theorems for $$(\phi , F)$$-contraction on rectangular b-metric spaces

CLINICAL PATHOLOGIC CONFERENCE CASE 1: A MULTILOCULAR RADIOLUCENCY OF THE RIGHT MANDIBLE

CLINICAL PATHOLOGIC CONFERENCE CASE 1: A MULTILOCULAR RADIOLUCENCY OF THE RIGHT MANDIBLE

Generative Design of Housing Spatial Layout Based on Rectangular Spaces

The generative spatial layout design process can generate and optimize a wide range of design responses by complying with all desired requirements and criteria and evaluating them based on one or more specific functions. Considering the complexities and diversity of spatial layout responses, it is important to know the various mechanisms of the product design process related to them. Based on this, the aim of this research is to provide a mechanism for designing a generative spatial layout (GSL) based on a housing design problem. The method of this research with a quantitative approach is the simulation and placement of spaces through coding in Grasshopper and Python software under the Grasshopper platform. The main variables of the research are the dimensions of the spaces of the residential unit, the proximity matrix, and the spatial relationships of the residential unit. With the restrictions made, 440 spatial layout responses were produced in four general shapes, including an incomplete square, a rectangle with a one-to-two ratio, an incomplete rectangle with an incomplete one-to-two ratio, and L-shape. The geometrical data of production plans have been subjected to correlation and linear regression tests in the SPSS software. Two models have been developed based on the perimeter of the plan and the area of its peripheral rectangle. Based on the obtained results, GSL design will be able to provide more favorable solutions. The results indicate that, by providing the design constraints in all the results, the area-oriented approach to the productive design of housing configurations can serve as an assistant mechanism for the designer in providing a variety of floor plans in terms of area for the designer.

The Application of The Indigenization Technique in Modern Chinese Church Architecture

"Indigenization" is a process that presents a blend and adaptable feature by combining foreign elements with the local natural environment and cultural factors. Church buildings are the most tangible manifestation of the "indigenization" of Christianity. This paper discusses the application of indigenization techniques in modern Chinese Catholic church architecture from four aspects: layout, structure, materials, and decoration. In terms of layout, indigenization is reflected in space, plan, and orientation. The space mainly imitates traditional Western-style churches and simplifies with the simplification of function. In terms of plan form, the basic rectangular space is retained and combined with local architectural features to evolve into some new variants. Orientation, some churches have absorbed local building orientation patterns. In terms of structural indigenization, the main materials are stone, brick, and soil, and the building structure is mainly supported by load-bearing walls and wooden beams. In terms of material selection, indigenization techniques are mainly based on regional resource conditions and are combined with the characteristics of local residences and climate conditions. The choice of building materials is made by considering the availability, processing difficulty, and physical properties of materials. The application of decorative indigenization often borrows common decoration techniques from local residential buildings and combines them with traditional Chinese culture to create a unique Chinese and Western fusion style building, making the church better integrated into the local architectural community and accelerating local residents' acceptance of Catholicism.

An automated topology optimization interpreter that generates space frames

This paper presents the methodology and theoretical foundation of topology optimization (TO) Results Spaceframe Interpreter, an automatic TO results interpreter that generates a closely associated space frame consisting of welded structural tubing or rectangular bars. TO is a computational technique that uses a finite element (FE) formulation to identify the most weight-efficient structure within a design domain. Density-based TO results in structures that take organic forms and is usually a tedious and cumbersome process to generate a computer-aided-design (CAD) model to manufacture through conventional techniques. The optimal topology frequently resembles a space frame, which is well-known as being a rigid, lightweight structure. The methodology of the TO results interpreter leverages several techniques from volumetric image processing and has four primary processes. First, the results are obtained from commercial FE/TO software and mapped into a cubic grid of voxels. Second, junction locations are extracted and member connectivity that represents a frame is identified. Third, a sizing optimization is incorporated to determine appropriate sectional dimensions of the circular or rectangular space frame members. Fourth, the optimized space frame geometry is imported into a CAD design tool to automatically create a design model. The logic and are implemented within the frame extraction processes. The automated TO interpreter is designed to interact with commercial FE analysis and CAD systems. The interpreter is demonstrated on various spatial examples including aerospace and automotive applications. In each case, the welded space frame closely resembles the TO result, with nearly equivalent stiffness and mass.

Konvergensi Barisan dan Kelengkapan pada Ruang Metrik Parsial Rectangular

The metric space is one of the objects studied in functional analysis. The metric space has undergone many developments, some eHamples of which are partial metric spaces and rectangular metric spaces. The difference between the metric space and the partial metric space can be seen in the distance of a point from itself. In the metric space, it is always equal to zero, while in the partial metric space it is not equal to zero. On the other hand, the difference between a metric space and a metric rectangular space can be seen in the inequalities used. In the metric space we use triangular inequalities, while in the metric rectangular space we use rectangular inequalities. Shukla in 2014 presents the development of another metric space called rectangular partial metric space, which combines the concept of partial metric space with rectangular metric space. This research we discusses the problem of the properties of the rectangular partial metric space, including convergence sequences, Cauchy sequences, and completeness of space in the rectangular partial metric space.

Fuzzy Triple Controlled Metric like Spaces with Applications

In this article, we introduce the concept of a fuzzy triple controlled metric like space in the sense that the self distance may not be equal to one. We have used three functions in our space that generalize fuzzy controlled rectangular, extended fuzzy rectangular, fuzzy b–rectangular and fuzzy rectangular metric like spaces. Various examples are given to justify our definitions and results. As for the topological aspect, we prove a fuzzy triple controlled metric like space is not Hausdorff. We also apply our main result to solve the uniqueness of the solution of a fractional differential equation.

Chaotic attractors and fixed point methods in piecewise fractional derivatives and multi-term fractional delay differential equations

Using generalized cyclic contractions, we establish some fixed point results in controlled rectangular metric spaces. Some subsequent outcomes are obtained. Moreover, some necessary conditions to demonstrate the existence of solutions for the multi-term fractional delay differential equations with wth order and the piecewise equations under the setting of non-singular type derivative are established in this paper. In order to demonstrate the effectiveness of our results, we provided some numerical examples.

D-optimal designs for two-variable logistic regression model with restricted design space

The problem of constructing locally D-optimal designs for two-variable logistic model with no interaction has been studied in many literature. In Kabera, Haines, and Ndlovu (2015), the model is restricted to have positive slopes and negative intercept for the assumptions that the probability of response increases with doses for both drugs and that the probability of response is less than 0.5 at zero dose level of both drugs. The design space mainly discussed is the set [ 0 , ∞ ) × [ 0 , ∞ ) , while the finite rectangular design space is presented only in scenarios where the results for the unlimited design space are still appropriate. In this paper, we intend to loose these restrictions and discuss the rectangular design spaces for the model where the D-optimal designs can not be obtained. The result can be extended to the models where drugs have negative or opposite effects, or the models with positive intercept, by using translation and reflection in the first quadrant.

Fixed point results on triple controlled quasi rectangular metric like spaces

&lt;abstract&gt;&lt;p&gt;In this article, by utilizing the idea of controlled functions, we present a novel notion of triple controlled quasi rectangular metric like spaces and prove Banach fixed point principal in such spaces. A topology in such spaces and its topological properties have been discussed. The result, presented here is a new contribution to the field of fixed point theory. Examples of this new structure are given.&lt;/p&gt;&lt;/abstract&gt;

Complex-valued controlled rectangular metric type spaces and application to linear systems

&lt;abstract&gt;&lt;p&gt;Fixed point theory can be generalized to cover multidisciplinary areas such as computer science; it can also be used for image authentication to ensure secure communication and detect any malicious modifications. In this article, we introduce the notion of complex-valued controlled rectangular metric-type spaces, where we prove fixed point theorems for self-mappings in such spaces. Furthermore, we present several examples and give two applications of our main results: solving linear systems of equations and finding a unique solution for an equation of the form $ f(x) = 0 $.&lt;/p&gt;&lt;/abstract&gt;

Solving an integral equation vian orthogonal neutrosophic rectangular metric space

&lt;abstract&gt;&lt;p&gt;In this paper, we introduce the notion of an orthogonal neutrosophic rectangular metric space and prove fixed point theorems. We extend some of the well-known results in the literature. As applications of the main results, we apply our main results to show the existence of a unique solution.&lt;/p&gt;&lt;/abstract&gt;

New fixed point theorems for (τ-ψ)-contraction mapping in complete rectangular M-metric space

New fixed point theorems for (τ-ψ)-contraction mapping in complete rectangular M-metric space

FIXED POINT RESULTS IN ORDERED EXTENDED RECTANGULAR b- METRIC SPACE WITH GERAGHTY-WEAK CONTRACTION

In this paper, we prove some ordered-theoretic fixed point results for a Geraghty-weak contraction on an ordered extended rectangular b􀀀metric spaces. Our results generalize several core results of the existing literature especially involving Geraghty-weak contractions and the results proved in extended rectangular b-metric space. Some examples are also furnished to exhibits the utility of our main results.