Area Formula Research Articles

Preoperative estimation of hamstring autograft graft diameter in anterior cruciate ligament reconstruction with a new simple method using magnetic resonance imaging.

The aim of this study was to predict intraoperative graft diameter with our new method by evaluating the cross-sectional areas (CSAs) of the hamstrings in axial sections of MRI. This study included 78 patients who underwent single-bundle ACLR between 2022 and 2023. MRIs of the patients were evaulated preoperatively and four CSAs of the hamstring tendons were measured in two different regions by two participants. CSA measurements were calculated manually and the total area was measured by adding up these cross-sectional areas separately. This value was considered as the area of a single circle and the radius was calculated using the area formula of the circle (A = πr2) RESULTS: For participant I, perfect agreement was found in semitendinosus measurement (κ=0.849), moderate agreement was detected at the medial condyle level (κ=0.520). For participant II, perfect agreement was found in the medial condyle region (κ=0.849), substantial agreement was obtained at the joint level (0.631). In the gracilis tendon evaluation of both participants, substantial and moderate agreement was found at the joint level and at the medial condyle level, respectively. CSA of hamstring tendons measured at the joint level by both participants were found to be positively correlated with the intraoperative graft diameter (r=0.51) (P<0.05) CONCLUSIONS: Preoperative MRI assesment could be a effective tool for determing preliminary information about the hamstring graft diameters. Moreover, it is observed that quadruple hamstring graft sizes were more correlated with MRI measurements at the joint level than medial condyle level by using circle area formula.

Symmetry results for the area formula in homogeneous groups

Symmetry results for the area formula in homogeneous groups

Mathematical modeling and CFD investigation of the novel oscillating shear valve generating sinusoidal signals by fluid pressure fluctuation

The oscillating shear valve pulser is a critical component of advanced downhole high-speed transmission systems, playing a vital role in efficient drilling operations. It is highly valued for its flexible and accurate source signal modulation capabilities, as well as its ability to operate safely without blockages. Specifically, how to ensure that the oscillatory shear valve pulse generator produces high-quality sinusoidal differential pressure source signals is particularly crucial. Consequently, this paper initially considers the motion characteristics of the shear valve in acceleration-uniform speed-deceleration, derives the calculation formula of the throttle area of the valve hole based on the fan-arc-fillet-line, and adopts the latest unsteady-liquid thin-walled cutting-edge throttling mechanism (UTCTM) we discovered, innovatively proposing the mechanism model of the differential pressure signal generation of the oscillatory shear valve. Based on this model, a typical optimized four-blade oscillatory shear valve was obtained. Through a series of CFD computational fluid dynamics studies, the amplitude variation characteristics and sinusoidal morphology evolution laws of the differential pressure signal at different frequencies were revealed, forming a unique design method for oscillatory shear valves, which provides a solid theoretical basis and technical support for the development of a new generation of pulsers.

Area of Images of Measurable Sets on Depth 2 Carnot Manifolds with Sub-Lorentzian Structure

The paper is devoted to analysis of metric properties of images of measurable sets in sub-Lorentzian geometry introduced on Carnot manifolds. The current research continues the results obtained earlier for classes of compact sets on Carnot groups. The main difference is that, firstly, the mapping is defined on a measurable set (not necessarily compact), and, secondly, the preimage and image of the mapping do not have a group structure. Also, the definition of sub-Lorentzian analog of Hausdorff measure (which is not a measure in general) is modified: in contrast to earlier research, it does not require ``uniform'' sub-Riemannian differentiability. One of results is the property of quasi-additivity of this sub-Lorentzian analog. The latter enables to derive its parameterization by sub-Riemannian Hausdorff measure. In turn, this property means that the sub-Lorentzian analog of Hausdorff measure has classical properties of measure on certain class of sets. The sub-Lorentzian area formula on Carnot manifold is the main result of the paper. We also demonstrate the main ideas of its proof and show their specificity.

Инженерный расчет гибких оболочек лесосплавных опор

Currently, the issue of delivering wood raw materials from remote forest areas, where the largest volumes are located, is relevant. Often, this delivery is economically justified only when using a network of medium and small rivers with the use of environmentally friendly technologies. On such rivers, it is preferable to use mobile anchors, in particular shore fillable ones, for fixing timber-rafting objects. A description has been given of a new modification of such an anchor, which differs from the prototype by a container having a flexible shell. This allows to significantly reduce the material consumption for manufacturing the container, the dimensions of the anchor in the transport position, and to produce anchors of any realistic size. The choice of the most rational option for such a container implies the scientific justification of its parameters. The aim of the study is to develop a scientific basis for this justification. Analytical formulas have been theoretically obtained for determining the basic geometric characteristics of the cross-section of the container under consideration. These formulas are useful in scientific research, but their application is very problematic in practical engineering calculations. This is due to the dependence of the specified characteristics on elliptic integrals and parameters, which are very difficult to measure in engineering practice. Having performed calculations using these formulas for a single cross-sectional area, we have obtained its specific geometric characteristics for various shapes, i.e. for various width-to-height ratios of a container to be filled. The results of the calculations have been approximating dependencies linking the named characteristics with the specified ratio, which is called the shape factor. Multiplying the specific characteristics by the square root of the cross-sectional area gives the absolute values of the corresponding characteristics. As a result, we have obtained formulas convenient for practical use, allowing to determine the width and height of the container to be filled, the width of its base, the perimeter of the cross-section, the heights of the zero pressure line and the extreme lateral point of the cross-section above the base of the container. The accuracy of calculations based on these and analytical formulas is almost the same. A formula for determining the specific tension of a flexible container shell has been obtained. The nature and degree of influence of the shape factor and length of the container on its other geometric characteristics have been established. An algorithm for substantiating the key parameters of a container to be filled with a flexible shell has been developed using the described results.

A recurrence for the surface area of (n, k)-arrangement graphs

An interesting property of an interconnected network is the number of nodes at distance i from an arbitrary processor, namely the node-centred surface area. This is an important property due to its applications in various fields of study. The ( n , k ) -arrangement graphs, denoted as A ( n , k ) are important class of graphs as they address the scalability issue in simpler topologies such as the hypercube and the star graph. Much work has been done to obtain a closed-form formula for the surface area of this class of graphs, but generally, it is not trivial to find an algorithm to compute the surface area of such graphs in polynomial time or to find an explicit formula with polynomially many terms in regard to the graph's parameters. In this paper, we present a simple recurrence that has linear computational complexity for A ( n , k ) for any arbitrary n and k in their defined range.

Area Formula for Spherical Polygons via Prequantization

Area Formula for Spherical Polygons via Prequantization

Implementasi Augmented Reality untuk Pengenalan Bangun Ruang

Solid Geometry is a part of mathematics learning that has various forms and types. The diversity of spatial shapes has area and volume formulas so that you can know the number of sides and area of each spatial shape. By utilizing Augmented Reality technology to introduce spatial structures, it makes it interesting to study. The aim of this research is to create an AR application for solid geometry in an Android-based 3D model so that it can facilitate learning in a fun way. This AR application was created using Vuforia, Unity and a marker to be able to call up the space that has been created. From this research, it is hoped that the interim results can help the learning and teaching process on spatial building materials. Application research results that can be used for the learning stage.

Analytical solution of nano-concrete-epoxy interaction area considering static equilibrium

This research proposes an analytical solution of the nano-concrete-epoxy interaction area within nano crack region of the reinforced concrete beam by applying Newton’s third law in static equilibrium. For deriving the governing equation, the imaginary beam with free ends (no support) is considered within nano crack region. This imaginary beam is acted along the imaginary line of concrete-epoxy interface. Newton’s third law is applicable for deriving the governing equation because of assuming the absence of frictional and other external forces. The parametric study is performed for implementing the proposed formula of nano interactive area considering variable nano crack depths and thicknesses. The nano interactive area is increased gradually with the increment of depths and thicknesses based on the parametric study because of linear functionality of interactive area and geometry of nano crack region. The maximum interactive area is found to be 314 nm2 at 0.6 ratio of depths and thicknesses of the nano crack. The incremental differences in interactive area between the crack depth or thickness ratios of 0.1 and 0.6 are found to be 25.4% and 1.6% for variations of the crack depth and thickness ratios, respectively. So, the crack depth shows higher impact on the interaction area compared to the thickness of the crack. However, there is a scope for enhancing this research in future by deriving closed-formed analytical formulations to consider appropriate boundary conditions.

When designer meets local culture: The promising learning trajectory on the surface area of polyhedron

Polyhedrons present a challenging topic in elementary geometry education. The Indonesian Realistic Mathematics Education (IRME) approach, when contextualized within the Bukit Sulap tourism area, offers a promising solution to this challenge. This study aims to design a learning trajectory for teaching the surface area of polyhedron using IRME in the context of Bukit Sulap tourism. Employing a research design with a validation study type, the study was conducted in three phases: preparation and design, experimental teaching, and retrospective analysis. The participants were 27 elementary school students from Lubuklinggau City, Indonesia. Data collection instruments included surface area problem-solving worksheets, video recordings of in-depth learning interviews, field notes, and observation sheets. The findings suggest that the Bukit Sulap tourism context significantly enhanced students' understanding of surface area problems, specifically those involving combinations of cuboids and cubes. Additionally, IRME effectively facilitated students' comprehension of abstract mathematical concepts, particularly the surface area formulas for cubes and cuboids. The study concludes that using the Bukit Sulap tourism park context as a starting point for teaching the surface area of polyhedron aligns well with the principles of IRME.

볼록 육각형의 나폴레옹의 정리 확장에 대한 연구

This study was based on the research results conducted as a R&amp;E project for the gifted students with a financial support from the Korea Foundation for the Advancement of Science and Creativity. In this study, the Napoleon hexagon ofhexagons was explored by expanding the Fermat-Torricelli point defined in a triangle and the Napoleon triangle into a convex hexagon. The results of this study are as follows. First, the equivalence condition for convex hexagons in which the Napoleon hexagon exists was discovered. Second, the area formula for Napoleon’s hexagon was shown. Third, we found various regular hexagons related to the Napoleon-containing hexagon and found that the centers of each regular hexagon were identical. Through this study, the concepts of the Fermat-Torricelli point and Napoleon triangle of triangles were expanded to convex hexagons, and it is expected that the expansion of mathematical concepts willcontribute to the development of mathematics. In addition, it is expected that additional research will be conducted in the future by exploring attractive regular hexagons whose centers coincide with the searched hexagons including Napoleon.

Evolutionary game analysis and efficiency test of water pollution control driven by emission trading: Evidence from Zhejiang Province, China

Presently, China is actively endorsing the pilot initiative for the remunerative use and trading of emission. By examining the operation and efficacy of emission trading in the context of water pollution control, one can contribute to the advancement and refinement of this system, thereby facilitating the attainment of regional pollution reduction, carbon reduction, and high-quality development objectives. In pursuit of this objective, we develop a theoretical framework for the local government and sewage enterprises evolutionary game of water pollution control, which includes two scenarios without considering and considering emission trading for studying the influencing factors and evolution trajectory of the game subject's. Through the stability analysis, the game interactive mechanism, the difference in evolutionary trajectory, and the response logic of the decision-making body in different situations become clearly visible. Further, the system sensitivity factors are analyzed by solving the partial derivation of the area formula of the phase diagram. And the efficacy of the sewage trading system in water pollution control in Zhejiang Province is empirically examined at the micro level by adopting the trading data of the first pilot area of sewage trading in the country and the case of pollution control in Jinhua City. The research reveals the following conclusions: Under specific circumstances, emission trading can incentivize businesses and even industries to enhance pollution control measures as a whole. The performance and degree of sensitivity factors vary across gaming systems, with public reputation evaluation and central government inspection serving as positive constraints. The initial cost of paid use of emission permits, as a fixed cost component for firms to address pollution, has no effect on the enterprises' behavioral actions to satisfy emission regulations. The findings can furnish local governments with a theoretical foundation and decision support in order to optimize regulatory strategies and enhance pollution control policies.

The Alpha Distance Formulae

In this study, we define the concept of tangent in two and three-dimensional alpha spaces concerning the alpha circles and the alpha spheres. Then using this concept, we derive the alpha distance formulae between points, a point and a line, between two lines and a point and a plane of the alpha spaces. Finally, we give simple area and volume formulas in the three dimensional space in terms of the alpha distances.

农业车辆作业后不同深度的土壤应力分析

In modern agriculture, with the development and widespread use of agricultural mechanization, mechanical compaction of soils has become a growing problem, resulting in soil degradation in the field. Based on the Boussinesq solution, the soil stress formula for the circular load area is derived, and MATLAB is used to simulate the stress-strain relationship of the soil at different depths. The results show that under the same load conditions, as the soil depth increases, the soil stress gradually decreases, with the most significant stress change occurring at 0.2 m depth. Soil compression experiments conducted using a consolidation instrument revealed that the soil void ratio dropped rapidly under loading of 50-200 kPa, and the decline slowed after 400 kPa. When the soil void ratio decreases to 0.2-0.4, the soil stress changes tend to stabilize. Comparison between the theoretical formula and the compression experimental data indicates that the soil stress gradually decreases as the thickness of the soil layer increases and the pressure load increases, verifying the linear relationship predicted by the theoretical formula.

Prediction of formation abrasiveness under the action of a PDC bit based on the fractal dimension of a rock surface

During rock drilling, a drill bit will wear as it breaks the rock. However, there is no uniform grading standard for rock abrasiveness. To solve this problem, the wear mechanisms of a polycrystalline diamond compact (PDC) bit and the formation it is drilling into are analyzed in depth, and an abrasiveness evaluation method based on the fractal dimension of the rock surface topography is established. Initially, a three-dimensional digital model is generated from a scanning electron microscope image of the rock after drilling; next, an evaluation of the irregularities on the rock surface is performed using an adapted Weierstrass–Mandelbrot (W-M) function to ascertain the fractal dimensionality. Then, the microcontact characteristics of the contact surface between the formation and the PDC bit are analyzed, and the distribution of the microconvex contact points of the two-body friction pair in a region is obtained. Because the sliding friction between the drill bit and the rock produces a large amount of heat, according to the contact area formula of the friction surface and heat conduction theory, the temperature rise and overall temperature distribution of the formation and PDC bit under the condition of sliding friction are revealed, and the real contact area between the formation and the drill bit within a certain temperature range is obtained. Finally, the evaluation index of rock abrasiveness under sliding conditions is established by adopting the wear weight loss of the rock cutting tool per unit volume as the index of rock abrasiveness, and the model is verified by a microdrilling experiment. The research in this paper is highly important for improving the rock-breaking efficiency and bit service life during drilling.

Desain Pembelajaran Bangun Ruang Sisi Lengkung Bola Menggunakan PMRI Berbantuan Adobe Animate

Understanding the concept of geometric figures on the curved side of a sphere is an important subject for students in learning mathematics. However, students had difficulties when they were learning about geometric shapes on the curved side of a sphere. This research aimed to createta learning pathway which supports students in understanding the concept of the curved side of a ball through applying the context of Mitoni tradition inlclass IX of junior high school. The research subjects involved students at SMP N 39 Semarang, consisting of 33 students. The method applied was a validation research type design, where consists of three stages; they are initial design, design experiments (pilot experiments and teaching experiments), and retrospective analysis. The arrangement of learning activities was designed and developed based on the PMRI approach with the help of Adobe Animate. The results of this research were three activities involving: observing videos of the Mitoni tradition to identify the characteristics of the sphere; determine the formula for the surface area of a sphere; determine the ball volume formula; as well as solving contextual problems relatedtto sphere. Through the activities carried out, students can increase their understanding oftthe concept of geometric shapes on the curved side of a sphere in class IX of SMP N 39 Semarang. Apart from that, it can increase local cultural alternatives which can be used as context intthe mathematics learning process,respecially in material on the curved side of a sphere.

Confidentially Compare Rational Numbers under the Malicious Model

&lt;p&gt;Secure multi-party computation is a hotspot in the cryptography field, and it is also a significant means to realize privacy computation. The Millionaires&amp;rsquo; problem is the most fundamental problem among them, which is the basic module of secure multi-party computation protocols. Although there are many solutions to this problem, there are few anti-malicious adversarial protocols besides protocols based on Yao&amp;rsquo;s garbled circuit. Only a few solutions have low efficiency, and there is no protocol for rational numbers comparison under the malicious model, which restricts the solution of many secure multi-party computation problems. In this paper, the possible malicious behaviors are analyzed in the existing Millionaires&amp;rsquo; problem protocols. These behaviors are discovered and taken precautions against through the triangle area formula, zero-knowledge proof, and cut-and-choose method, so the protocol of comparing confidentially rational numbers is proposed under the malicious model. And this paper adopts the real/ideal model paradigm to prove the security of the malicious model protocol. Efficiency analysis indicates that the proposed protocol is more effective than existing protocols. The protocol of rational numbers comparison under the malicious model is more suitable for the practical applications of secure multi-party computation, which has important theoretical and practical significance.&lt;/p&gt; &lt;p&gt;&amp;nbsp;&lt;/p&gt;

An improvement of body surface area formulas using the 3D scanning technique.

Body surface area (BSA) is one of the major parameters used in several medical fields. However, there are concerns raised about its usefulness, mostly due to the ambiguity of its estimation. Authors have conducted a voluntary study to investigate BSA distribution and estimation in a group of 179 adult people of various sex, age, and physique. Here, there is provided an extended analysis of the majority of known BSA formulas. Furthermore, it was supplement with a comparison with the authors' propositions of enhanced formulas coefficients for known formulas models as well as with new power models based on an increased number of anthropometric data. Introduction of the enhanced formulas coefficients cause a reduction of at least 30.5% in mean absolute error and 21.1% in maximum error in comparison with their known counterparts. In the context of the analysis presented it can be stated that the development of a single universal body surface area formula, based on a small number of state variables, is not possible. Therefore, it is necessary and justified to search for new estimation models that allow for quick and accurate calculation of body surface area for the entire population, regardless of individual body variations. The new formulas presented are such an alternative, which achieves better results than the previously known methods. Int J Occup Med Environ Health. 2024;37(2):205-19.

A surface area formula for compact hypersurfaces in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document}

The classical Cauchy surface area formula states that the surface area of the boundary ∂K=Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\partial K=\\Sigma $\\end{document} of any n-dimensional convex body in the n-dimensional Euclidean space Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document} can be obtained by the average of the projected areas of Σ along all directions in Sn−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{S}^{n-1}$\\end{document}. In this note, we generalize the formula to the boundary of arbitrary n-dimensional submanifold in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document} by introducing a natural notion of projected areas along any direction in Sn−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{S}^{n-1}$\\end{document}. This surface area formula derived from the new notion coincides with not only the result of the Crofton formula but also with that of De Jong (Math. Semesterber. 60(1):81–83, 2013) by using a tubular neighborhood. We also define the projected r-volumes of Σ onto any r-dimensional subspaces and obtain a recursive formula for mean projected r-volumes of Σ.

Hartle-Hawking state and its factorization in 3d gravity

We study 3d quantum gravity with two asymptotically anti-de Sitter regions, in particular, using its relation with coupled Alekseev-Shatashvili theories and Liouville theory. Expressions for the Hartle-Hawking state, thermal 2n-point functions, torus wormhole correlators and Wheeler-DeWitt wavefunctions in different bases are obtained using the ZZ boundary states in Liouville theory. Exact results in 2d Jackiw-Teitelboim (JT) gravity are uplifted to 3d gravity, with two copies of Liouville theory in 3d gravity playing a similar role as Schwarzian theory in JT gravity. The connection between 3d gravity and the Liouville ZZ boundary states are manifested by viewing BTZ black holes as Maldacena-Maoz wormholes, with the two wormhole boundaries glued along the ZZ boundaries. In this work, we also study the factorization problem of the Hartle-Hawking state in 3d gravity. With the relevant defect operator that imposes the necessary topological constraint for contractibility, the trace formula in gravity is modified in computing the entanglement entropy. This trace matches with the one from von Neumann algebra considerations, further reproducing the Bekenstein-Hawking area formula from entanglement entropy. Lastly, we propose a calculation for off-shell geometrical quantities that are responsible for the ramp behavior in the late time two-point functions, which follows from the understanding of the Liouville FZZT boundary states in the context of 3d gravity, and the identification between Verlinde loop operators in Liouville theory and “baby universe” operators in 3d gravity.