Right Triangle Research Articles

Natural convection magneto hydrodynamic filled with TiO2 nanofluid with a right-angled triangle obstacle within square cavity

The goal of this study is to examine the result of natural convection, magnetohydrodynamics, and nanofluids within a square cavity with an obstacle of a right-angle triangle within it. Among the various nanoparticles, TiO2 nanoparticles blended with water have shown promise in improving heat transfer. The Finite Element Method is used for the simulation. By utilizing the commercial software 'COMSOL Multiphysics 6.1' graphical illustrations, the influence of prominent parameters on streamline, isotherm contours and the local Nusselt number is effectively portrayed. The key factors that play a dominant role include the Prandtl number (Pr = 6.2) and Rayleigh number (104 <= Ra <= 106), Hartmann number (Ha = 0, 30, 50), and the heat source/sink (Q = -6, 0, 6) employed. The result confirms that the decline in fluid velocity and convective heat transfer is due to the enhancement of the Lorentz force. This outcome of the study affirms that the appropriate amalgamation of nanoparticles significantly improves the transmission of heat properties of fundamental fluids. The discoveries of this investigation could aid in comprehending the hydrothermal aspects of thermal systems, including electronic cooling, solar power and the utilization of nanofluid to regulate liquid movement and thermal exchange attributes within the annular space.

Improving Student Learning Outcomes Through the Probing Prompting Learning Model on Trigonometric Ratios in Right-Angled Triangles in Class X-MPLB-A at SMK Negeri 1 Suwawa

Improving Student Learning Outcomes Through the Probing Prompting Learning Model on Trigonometric Ratios in Right-Angled Triangles in Class X-MPLB-A at SMK Negeri 1 Suwawa

Intermediate spectral statistics of rational triangular quantum billiards.

Triangular billiards whose angles are rational multiples of π are one of the simplest examples of pseudo-integrable models with intriguing classical and quantum properties. We perform an extensive numerical study of spectral statistics of eight quantized rational triangles, six belonging to the family of right-angled Veech triangles and two obtuse rational triangles. Large spectral samples of up to one million energy levels were calculated for each triangle, which permits one to determine their spectral statistics with great accuracy. It is demonstrated that they are of the intermediate type, sharing some features with chaotic systems, like level repulsion, and some with integrable systems, like exponential tails of the level spacing distributions. Another distinctive feature of intermediate spectral statistics is a finite value of the level compressibility. The short-range statistics such as the level spacing distributions, and long-range statistics such as the number variance and spectral form factors were analyzed in detail. An excellent agreement between the numerical data and the model of gamma distributions is revealed.

Performance Simulation of Muon Detectors Based on Structural Design and Array Layout of Plastic Scintillators

The performances of muon detectors have prominent effects on the accuracy and application scenario of muon imaging. Previous studies have paid more attention to the performance improvements of muon detectors in the assembly of imaging systems and optimization of imaging algorithms, while, the structural design and array layout of plastic scintillators in muon detectors have received less attention. In this work, the simulation models of plastic scintillator, wavelength-shifting (WLS) fibers, and muon source are constructed in the Geant4 software. On this basis, different factors affecting light collection efficiency (LCE) have been investigated, including muon hitting position, plastic scintillator cross-sectional shape and size, and WLS fiber size and position. Meanwhile, the influences of scintillator array layout, average width, and muon energy on the position resolution performance of the detector are investigated. The constructive results have been listed as follows: (a) the longitudinal length of the plastic scintillator unit, the shape and size of the WLS fiber, and the position of the WLS fiber have a large impact on the LCE. (b) The Right-angled triangle staggered layout is suitable for small-sized scintillators with single-energy muon hitting, and the Rhombus staggered layout is suitable for large-sized scintillators with multiple-energy muon hitting. This work provides theoretical support for the structural design and array layout of plastic Scintillators, and it has an important significance as a guide for the design of the subsequent muography system.

A Study of Movable Singularities in Non-Algebraic First-Order Autonomous Ordinary Differential Equations

We studied the movable singularities of solutions of autonomous non-algebraic first-order ordinary differential equations in the form of y′=I(y(t)) and y′=I1(y(t))+I2(y(t))+⋯+In(y(t)), aiming to prove that all movable singularities of all complex solutions of these equations are at most algebraic branch points. This study explores the use of the constructing triangle method to analyze complex solutions of autonomous non-algebraic first-order ordinary differential equations. For complex solutions in the form of y=w+iv, we treat the constructing triangle method as a way to construct a right-angled triangle in the complex plane, with the lengths of the adjacent sides being w and v. We use the definitions of the trigonometric functions sin and cos (the ratio of the adjacent side to the hypotenuse) to represent the trigonometric functions of complex solutions y=w+iv. Since the movable singularities of the inverse functions of trigonometric functions are easy to analyze, the properties of the movable singularities of the complex solutions are then easy to deal with.

108.30 Nearly isosceles right-angled triangles and square triangular numbers

108.30 Nearly isosceles right-angled triangles and square triangular numbers

Right-angled triangles with almost prime hypotenuse

Right-angled triangles with almost prime hypotenuse

A triangle decomposition method for the mobility control of mecanum wheel-based robots

Mobile robots are used in a variety of applications including research, education, healthcare, customer service, security and so on. Based upon the application, the robots employ different locomotion systems for their mobility. When it comes to rolling locomotion, the wheels used to provide mobility to robots can be categorized as: tracks, omnidirectional wheels, and unidirectional wheels with a steering system. The ability of omnidirectional wheels to drive machines in small spaces makes them interesting to use. Among the types of omnidirectional wheels, mecanum wheels are widely used due to their inherent benefits. With the right control strategy, robots equipped with mecanum wheels can move freely, in all possible directions. In this study, a triangle decomposition approach is employed for controlling omnidirectional mecanum wheel-based robots. The method consists of breaking down any path into a set of linear motions that can be horizontal, vertical, or oblique. Furthermore, the oblique paths are divided into smaller segments that can be resolved into a horizontal and vertical component in a right-angle triangle. The suggested control method is tested and proved on a simple scenario using Webots simulation software.

Triangle area model (TAM) for predicting germination: An approach to enhance hydrothermal time model applications

A thorough examination of assumptions in hydrothermal time models revealed areas for enhancing model performance. We introduce the Triangle Area Model (TAM), which uses the area of right-angled triangles to calculate hydrothermal time for predicting population germination fractions (g). TAM is characterized by its depiction of triangles, considering insightful parameters such as the distance of germination temperature (T) to the base (Tb), optimal (To), and ceiling (Tc) temperatures, the range of Tc – To, To – Tb, and the germination water potential (Ψ), i.e. mean base water potential (Ψb(g)), along with potential g that may occur with T and Ψ combinations within Tc – Tb when Ψ > Ψb(g). Applied to germination data from Ambrosia psilostachya L., Cynanchum acutum L., and Bidens pilosa L., TAM achieves an RMSE of 0.03 for A. psilostachya and C. acutum, and 0.05 for B. pilosa. Moreover, TAM demonstrates an R2 of 0.96, 0.97, and 0.98 for the respective species. TAM significantly outperforms earlier models through a comparison with varying T and Ψ. TAM determined Tb for A. psilostachya, C. acutum, and B. pilosa as 0.19, 14.57, and 5.76 °C; To as 25.1, 39.9, and 29.8 °C; and Tc as 46.7, 53, and 41°C, for the respective species. It also estimates Ψb(g) as -1.48 for A. psilostachya, -0.98 for C. acutum, and -0.97 for B. pilosa. The TAM approach deepens our understanding of temperature-moisture processes influencing plant survival, colonization, and habitat expansion for these three invasive alien species. Furthermore, it can be more widely applied for estimating TT and HTT across different growth stages, enhancing the prediction accuracy of plant phenological development.

Multi-directional strain sensor based on carbon nanotube array for human motion monitoring and gesture recognition

Flexible strain sensors have enormous application demands in health monitoring, soft robotics, and artificial skin. Traditional flexible strain sensors mainly focus on identifying strain amplitude. However, to determine complex movements, it is necessary for sensors to recognize not only the amplitude but also the direction of strain. Here, we propose a multi-directional strain sensor constructed from three flexible strain sensor units, which are based on a carbon nanotube (CNT) array, and configured in a right-angled triangle arrangement. Each strain sensor unit exhibited a strain detection range of over 90%, durability of over 5000 cycles, and fast response time (<150 ms). The multi-directional strain sensor can accurately identify the amplitude and direction (up to 180°) of the strain simultaneously. By optimizing the height and the size of the CNT array strips within the strain sensor unit, the preferred parameters for the sensor unit were determined. Further, by incorporating a neural network algorithm, the multi-directional strain sensor achieved synchronous detection of strain amplitude and direction with an accuracy rate exceeding 98%. Therefore, this work provides a new strategy for the design and construction of multi-directional strain sensors, underscoring their broad application potential in the fields of human motion monitoring and gesture recognition.

Compilation method of a catalogue of reasonable worst-case rainfall series for flash flood simulations of short, convective rainstorms

In this study, a categorisation method was developed and applied to identify, select and rank reasonable worst-case rainfall storm events with time-series resolution of one minute and duration of up to one hour. The method yields a catalogue of 24 reasonable worst-case rainfall series, i.e. potentially catastrophic events that could be used for safe-to-fail flash flood model scenarios, thus encouraging the testing of critical infrastructure and pedestrian safety for people trapped in floods. The identified 24 events were shown to be considerably more extreme than corresponding design storms, with a return interval of 100 years. In the selection of the extremes, special care was taken to consider different hazard attributes, which are associated with intrinsically different observed hyetograph shapes, including rectangular, short and long triangle, and double peak shape categories. Hydrodynamic simulations were carried out using the 2D robust shallow water model hms++ to quantify the flooding impact of the 24 rainfall scenarios in a small urbanised catchment, which enabled the ranking of the scenarios based on different hazard attributes. Overall, the rectangular and long-triangle events showed the highest total discharge, and the rectangular events comprise both the longest duration for hazard velocity and deep flooding. The long-triangle events presented the highest peak discharge and peak water depths. The short-triangle ones showed the highest flashiness (shortest time to peak), whereas the double peak events yielded the longest time to peak. The ranking emphasised the importance of the hyetograph shape characteristics of extreme storms and which needs to be taken into account in flash flood management. The novel categorisation method is easily transferable to other regions, if high-resolution rain series are available.

A digital distance on the kisrhombille tiling.

The kisrhombille tiling is the dual tessellation of one of the semi-regular tessellations. It consists of right-angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons.

The Relationship Between the Sum of the Radii of Three Connected Right-Angled Triangles, and the Height of One of Them

The Relationship Between the Sum of the Radii of Three Connected Right-Angled Triangles, and the Height of One of Them

Evaluation of Scapular Dynamics in Sports Trainees

Background: The scapula provides stability and mobility for efficient upper-limb movement. Optimal scapular kinematics are essential for efficient force generation and injury prevention. Sports trainees, particularly those involved in sports requiring repetitive or high-intensity shoulder movements, are susceptible to injuries and performance limitations if associated with aberrant scapular mechanics. Understanding the intricacies of scapular dynamics is essential for designing tailored training interventions that enhance athletic performance. Rationale and objectives: Previous research has emphasized the need for sports-specific training protocols. The measurement of scapular rotation has gained significant attention. Evaluating asymmetries in scapular dynamics across various sports trainee populations is a step forward for the need for tailored interventions and injury prevention strategies in preventing injury and planning better exercise programs. The study uses a PALM meter for the measurement of scapular rotation. Methodology: 40 Subjects 19 to 24 years old from the sports authority of Andhra Pradesh, India, were studied after obtaining written informed consent. Scapular rotation measured in a neutral and with hands on the hip position. Root of the scapula (RSS), inferior angle of the scapula (IAS), and spinous processes (Sp) of C7 to T8 are taken as the anatomical landmarks. Measurements of scapular position taken in two arm positions: shoulder neutral, hands on hip. A right-angle triangle created by dropping a perpendicular line from the scapula's root to intersect the horizontal line between the scapula's inferior angle and the thoracic spine's closest spinous process (IAS-Sp). The hypotenuse represents the distance IAS to RSS, while the side opposite the angle θ is the distance IAS-Sp minus RSS-Sp. A positive result indicates upward scapular rotation, while a negative result indicates downward scapular rotation. Results and Conclusion: This cross-sectional study revealed variations in scapular rotation and the distances from spinous processes in the scapular position in neutral and hands-on Hip positions from right to left and male to female. Further studies are required to understand scapular rotation variations concerning various sports involving overhead activity, heavy weight lifting, endurance sports and anatomical variations, muscle activation patterns, or task demands. KEYWORDS: Scapular Rotation, Sports Trainees, Shoulder Movements, Scapular Mechanics, Athletic Performance.

Levenberg–Marquardt backpropagation neural networking (LMB-NN) analysis of hydrodynamic forces in fluid flow over multiple cylinders

The mathematical formulation of the flowing liquid stream around and through confined multiply connected domains brings a complex differential system. Due to this, one cannot provide a complete description of flow field properties. The current pagination provides a numerical remedy by the use of artificial intelligence in this direction. To be more precise, fluid is considered in the rectangular channel. The circular, square, and triangular-shaped cylinders are positioned as an obstacle to the continuous fluid. The channel’s left wall is viewed as an inlet and two different velocity profiles are introduced at an inlet that are constant and parabolic profile. To discretize the computational domain, hybrid meshing is used. The variance in basic variables, namely, the velocity of the liquid and the distribution of the liquid pressure, is recorded via graphs. The nine different meshed grades are tested for the drag and lift coefficients around the right-angle triangle, square, and circular barrier. The neural networking model is constructed by using 99 datasets of sample values for drag coefficient when characteristic length, the density of fluid, the dynamic viscosity of the fluid, and mean flow velocity are taken as inputs. The training of the neural network takes up 69 slots (about 70%), while the testing and validation of the neural network each take up 15 slots (15%). The Levenberg–Marquardt backpropagation algorithm is used for training. We have observed that for the parabolic profile, the drag coefficient is higher in intensity for each obstacle compared to the constant profile, while the lift coefficient shows opposite patterns.

Density-functional theory for clustering of two-dimensional hard particle fluids

Fluids made of two-dimensional hard particles with polygonal shapes may stabilize symmetries which do not result directly from the particle shape. This is due to the formation of clusters in the fluid. Entropy alone can drive these effects, which represent a challenge for standard theories. In this article we present a general density-functional theory for clustering effects in fluids of hard particles in two dimensions. The theory combines a free-energy functional of the angular distribution function with an association energy term which qualitatively reproduces the clustering tendencies of the particles found in Monte Carlo simulations. Application is made to a fluid of hard right-angled triangles.

Classical Wave Equation: A Gross Error in Mathematics and Physics

A detailed proof of the incorrectness of the classical wave equation is proposed. The correct methodological basis for the proof is the unity of formal logic and rational dialectics. The proof leads to the following irrefutable statement: the classical wave equation and the derivation of the classical wave equation are a gross error in mathematics and physics. The proof of this statement is based on the following irrefutable main results: (1) The first gross error is the following approximate relationship: “sine of angle is approximately equal to tangent of angle; cosine of angle is approximately equal to 1”. This relationship means (implies) that the quantity of the angle and the right-angled triangle do not exist. Consequently, the relationship between the tangent of the angle and the derivative of the displacement with respect to coordinate does not exist; (2) The second gross error is that the second-order derivative of the displacement with respect to coordinate does not exist, because the dimensions of displacement and coordinate are “meter”, the first-order derivative of the displacement with respect to coordinate is dimensionless quantity (i.e. the first-order derivative of the displacement respect to coordinate has no the dimension “meter”); (3) The third gross error is that the first-order derivative of the displacement with respect to coordinate cannot be expanded into the Taylor series, because the second-order derivative of the displacement with respect to coordinate does not exist; (4) The fourth gross error is as follows: (a) the left side of the wave equation contains the condition that time is a variable quantity, and the coordinate is a constant quantity; (b) the right side of the wave equation contains the condition that time is a constant, and the coordinate is a variable. This means (implies) that the equation contains contradictory conditions (propositions). Therefore, the equation represents a violation of the formal-logical law of the lack (absence) of contradiction; (5) The fifth gross error is that the standard derivation of the equation relies on the following false theories: negative number theory, complex numbers theory, trigonometry, vector calculus, differential calculus, and Newton's second law. Thus, the classical wave equation does not satisfy the criterion of truth and is not a scientific achievement.

Limitation of the Pothayanar Formula for Finding the Hypotenuse of a Right-Angle Triangle

A Tamil Mathematician Pothayanar has given a formula for evaluating the hypotenuse of a right-angle triangle when two sides are given. In this short note, it is shown that it works only for limited values of the two sides.

Shape of Pulmonary Artery Doppler Flow Profile and Right Ventricular Hemodynamics in Neonates

To characterize pulmonary artery Doppler flow profile (PAFP) patterns among infants receiving care in neonatal intensive care units (NICUs) and to examine the association of PAFP patterns with pulmonary and right ventricular (RV) hemodynamics. This is a retrospective study at two tertiary NICUs over 4 years that included neonates who demonstrated a complete tricuspid regurgitation envelope on targeted neonatal echocardiography (TNE). Separate personnel reviewed TNEs to characterize PAFP patterns, divide cohort into PAFP groups, and measure quantitative indices of RV hemodynamics (RV systolic pressure [RVSP], pulmonary artery acceleration time [PAAT] and its ratio with RV ejection time [RVET/PAAT], tricuspid annular plane systolic excursion [TAPSE] and RV output [RVO]), for intergroup comparisons. We evaluated TNEs from 186 neonates with median (IQR) gestational age of 28.5 weeks (25.9-35.9) weeks. Four distinct PAFP patterns were identified: A. near-isosceles triangle (22%), B. right-angled triangle (29%), C. notching (40%), and D. low peak velocity (<0.4 m/s; 9%). Groups A-C demonstrated a stepwise worsening in all indices of PH, while pattern D was associated with lower TAPSE and RVO. Using common definitions of pulmonary hypertension (PH), pattern A performed best to rule out PH [sensitivity range 81-90%] and pattern C for diagnosing PH [specificity range 63-78%]. Inspection of PAFP is a simple bedside echocardiography measure that provides clinically meaningful information on underlying RV hemodynamics and may aid in screening and monitoring of patients for PH in NICUs.

Solar air collector with right-angle triangle corrugations for an indirect type solar dryer for enhancement in performance

Solar air collector with right-angle triangle corrugations for an indirect type solar dryer for enhancement in performance