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We explore, using numerical simulations, the influence of mass and distance on the evolution of triple black hole systems. Following in the direction of Burrau's famous 3,4,5 problem, black holes are initially placed at the vertices of Pythagorean triangles. Numerical integration of orbits was conducted using relativistic corrections (post-Newtonian) up to the 2.5$^{th}$ order with ARCcode. As a descriptor of the evolution of the systems, the lifetimes, the number of two-body encounters and the number of mergers were all analysed. We found that as the mass unit of the black holes increased there was strong positive correlation with the fraction of mergers (0.9868), strong negative correlation with the average number of two-body encounters (-0.9016) and the average lifetimes of the triple systems decayed exponentially (determination coefficient of 0.9986). Around the mass unit range of 10$^{5.5}$M$_{\odot}$-10$^{5.6}$M$_{\odot}$, there was a transition from escape dominated dynamics to merger dominated dynamics. However, in the mass unit range of 10$^6$ M$_{\odot}$ $-$ 10$^{9}$M$_{\odot}$ with 1 pc distance unit, we find that 25\% of cases resulted in the escape of a supermassive black hole (SMBH) which may be a cause for wandering SMBH's found in some galaxies/galactic merger remnants., Comment: 11 pages, 14 figures

The paper is a commentary to the Polish translation of the Elements Book II, included in this volume.We focus on relations between figures represented and not represented on diagrams and identify rules which enableEuclid to bridge these two kinds of objects. Also, we argue that the main mathematical problem addressed in Book II is constructing a leg of a rightangled triangle, given its hypotenuse and the other leg. In proposition II.14,Euclid solves it through the construction called the geometric mean. We trace the problem in Book III and beyond the Elements: in Heron’s Metrica, Descartes’ La Géométrie, and modern foundations of mathematics. We showthat Descartes, by novel interpretation of the Pythagorean theorem, provides a modern solution to this problem.

Sensitivity and specificity are popular measures described by the receiver operating characteristic (ROC) curve. There are also two other measures such as the positive predictive value (PPV) and negative predictive value (NPV); however, the PPV and NPV cannot be represented by the ROC curve. Based on the total operating characteristic (TOC) curve suggested by Pontius and Si (International Journal of Geographical Information Science, 97, 570-583, 2014), explanatory methods are proposed to geometrically describe the PPV and NPV by the TOC curve. It is found that the PPV can be regarded as the slope of the right-angled triangle connecting the origin to a certain point on the TOC curve, while 1 - NPV can be represented as the slope of the right-angled triangle connecting a certain point to the top right corner of the TOC curve. When the neutral zone exists, the PPV and 1-NPV can be described as the slopes of two other right-angled triangles of the TOC curve. Therefore, both the PPV and NPV can be estimated using the TOC curve, whether or not the neutral zone is present.

To solve the three-phase unbalance and reactive power problems in the electric railway power supply system, this paper proposes a Scott transformer based hypotenuse hybrid traction power system (Scott-HHPS). Compared with other co-phase traction power supply systems, the Scott-HHPS could achieve the same compensation objective with smaller active compensation capacity. The paper analyzes the basic principle and makes a comparison between the Scott-HHPS and typical co-phase power supply system. The mathematical relationships among the load current, power factor and capacity demand in condition of power factor variation and regenerative braking are analyzed. In order to further reduce the compensation capacity of the Scott-HHPS, the partial compensation strategy in respect to the index of power factor and unbalance is proposed. Besides, an instantaneous current detection method based on the theorem of conservation of power is adopted to simplify the control algorithms. Simulations and experiments are both constructed and the results demonstrate that the Scott-HHPS can effectively improve the power quality in the electric railway power supply system.

Objective: To evaluate the lumbar triangular safety zone, its boundaries and its relationship with the dorsal root ganglion through Magnetic Resonance Imaging (MRI). Methods: The boundaries, shape and dimensions of 303 triangular safety zones were analyzed in Tesla 3.0 Magnetic Resonance Imaging (MRI) coronal sections from L2 to L5, including the dorsal root ganglion. Results: The sample consisted of 101 patients with a mean age of 32 years. The height of the triangular safety zone was formed by the lateral edge of the dura mater, the width by the upper plateau of the lower vertebra and the hypotenuse by the corresponding nerve root. The mean dimensions and the area varied according to the level studied. The dorsal root ganglion invaded the dimensions of the triangle in all the images studied. Conclusion: Based on the data and the analyses performed, we concluded that knowledge of the boundaries of the triangular safety zone through MRI increases the safety of minimally invasive procedures in the lumbar spine. Level of evidence I; Diagnostic studies – Investigation of a diagnostic test. RESUMO Objetivo: Avaliar a zona triangular de segurança lombar, seus limites e sua relação com o gânglio da raiz dorsal por meio da ressonância magnética (RM). Métodos: Foram estudados os limites, o formato e as dimensões de 303 zonas triangulares de segurança no corte coronal de RM 3.0 Tesla de L2 a L5, incluindo o gânglio da raiz dorsal. Resultados: A amostra foi composta por 101 pacientes com média de idade de 32 anos. A altura da zona triangular de segurança era formada pela borda lateral da dura-máter; a largura, pelo platô superior da vértebra inferior; e a hipotenusa, pela raiz nervosa correspondente. A média das dimensões, assim como a área, variaram conforme o nível estudado. O gânglio da raiz dorsal invadiu as dimensões do triângulo em todas as imagens estudadas. Conclusão: Baseados nos dados e nas análises realizadas, concluímos que o conhecimento dos limites da zona triangular de segurança, por meio da imagem da RM, aumenta a segurança dos procedimentos minimamente invasivos na coluna lombar. Nível de evidência I; Estudos diagnósticos–Investigação de um exame para diagnóstico. RESUMEN Objetivo: Evaluar la zona triangular de seguridad lumbar, sus límites y su relación con el ganglio de la raíz dorsal a través de la Resonancia Magnética (RM). Métodos: Se estudiaron los límites, el formato y las dimensiones de 303 zonas triangulares de seguridad en el corte coronal de RM 3.0 Tesla de L2 a L5, incluyendo el ganglio de la raíz dorsal. Resultados: La muestra fue compuesta por 101 pacientes, con promedio de edad de 32 años. La altura de la zona triangular de seguridad estaba formada por el borde lateral de la duramadre, el ancho por la meseta superior de la vértebra inferior y la hipotenusa por la raíz nerviosa correspondiente. El promedio de las dimensiones, así como el área, variaron según el nivel estudiado. El ganglio de la raíz dorsal invadió las dimensiones del triángulo en todas las imágenes estudiadas. Conclusión: Basándose en los datos y análisis realizados, concluimos que el conocimiento de los límites de la zona triangular de seguridad a través de la imagen de RM aumenta la seguridad de los procedimientos mínimamente invasivos en la columna lumbar. Nivel de evidencia I; Estudios diagnósticos - Investigación de un examen para diagnóstico.

This research aims to find out the process of concept formation in reinventing Pythagorean Theorem and to analyze the result of that process and the misconceptions that may emerge. The type of this research is qualitative research which is analyzed descriptively. This research applies class setting method in learning activity with the implementation of PMRI approach and it focuses on two subjects. The results shows that the process of concept formation in reinventing the Pythagorean Theorem with PMRI approach can work well. However, the two subjects have not been able to construct the given problem, which is in the form of a question, to be the form of a mathematical modeling. Subject 1 still processes the information rawly. Therefore, the result of concept finding with his/her knowledge used for solving problems by the same model has not been able to reduce data. Subject 2 has been able to process the obtained information and he/she can interpret the problem further. However, both subjects still experience misconceptions not only in their already possessed knowledge but also their newly acquired knowledge. Those misconceptions are misconception in calculating operation of equation system, which should be calculating operation on both segments but it is only carried out in one segment, misconception in describing hypotenuse and misconception in naming edges of a right triangle.

Dirac equation is based on a novel method that expresses the roots of a square-root operation in terms of matrices. Instead of this technique we propose an alternative approach. The square-root expression is represented as a hypotenuse and it is written in terms of a trigonometric summation of the remaining two edges of a right-angle triangle. Both of the positive and negative energy terms are taken into account and a new matrix equation is obtained. The plane wave solutions of the new differential equation are studied.

In geometry, Waclaw Sierpinski described fractals such as Sierpinski triangle, Sierpinski gasket and Sierpinski carpet. In this work, Sierpinski triangle is used to find any equation between the atomic number and the atomic weight of elements in the periodic table. First by using Matlab program, an algorithm is written to create a right angle triangle and an equation between the atomic numbers and atomic weights. Also, the resulted the atomic numbers and atomic weights are compared with the real ones. Then this original triangle is divide to 8 smaller triangles on the hypotenuse of the original triangle to get more accurate results and reduce errors of the atomic numbers and atomic weights which are shown in tables. Finally, the resulted equation of correlation between the atomic numbers and the atomic weights of elements based on Sierpinski triangle is determinate to calculate the atomic weights of the future discovered elements in eighth period. This work is a combination research in chemistry and geometry.

We prove a necessary and sufficient condition for a given integer to be the hypotenuse of a primitive Pythagorean triangle. This characterization is based on partitions into consecutive parts.

The present article explores the effects of moving lid on the forced convection flow of micropolar fluid inside a right-angle triangular cavity saturated with porous medium. The base and hypotenuse or inclined sides of the cavity are maintained at constant temperatures, while the vertical side of the enclosure is adiabatic and moving with constant velocity in upward or downward direction. The flow equations are simulated by using the robust finite element numerical technique. The pressure term from the momentum equations is eliminated by using the penalty parameter. For a consistent solution, the value of the penalty parameter is selected as 107. The simulations are performed for the cases based on the direction of moving lid. The numerical outcomes are shown in terms of streamlines, temperature contours, and local and average Nusselt numbers for sundry parameters, such as micropolar parameter, Reynolds number, Richardson number, Darcy number, Hartmann number, and Prandtl number. It is observed that the shape of the inner circulating cell is elliptic when the lid is moving in the upward direction and fluid is clear (Newtonian fluid). It is also found that average Nusselt number in both cases increases with increasing Prandtl number, Richardson number, micropolar parameter, and Darcy number, whereas it decreases with increasing Hartmann number. Further, it achieves a maximum when the lid is moving in the downward direction, regardless of the choice of involved parameters. The numerical code is also validated with previous published results. The investigation of the current study is beneficial in porous heat exchangers, construction of triangular-shaped solar collectors, rigid crystal, polymeric fluid transport, etc.

A numerical study was achieved on a new design of storage solar collector. The storage collector is a triangular face and a right triangular pyramid for the volumetric shape. It is obtained by cutting a cube from one upper corner at 45o, down to the opposite hypotenuse of the base of the cube. The numerical study was carried out using the computational fluid dynamics code (ANSYS-Fluent) software with natural convection phenomenon in the pyramid enclosure. The results show that, the temperature and velocity distributions throughout the operating period were obtained. The influence of introducing an internal partition inside the triangular storage collector was investigated. Also the optimum geometry and location for this partition were obtained. The enhancement was best at y= 0.25 m whereas the height of triangular collector was 0.5 m. The hourly system performance was evaluated for all test conditions.

Fabio Lavinsky,1,2 Camila Zanella Benfica,2 Nédio Castoldi,2 Anne Elise Cruz do Carmo Chaves,2 Paulo Augusto de Arruda Mello1 1Department of Ophthalmology, Paulista School of Medicine, São Paulo Hospital, Universidade Federal de São Paulo (UNIFESP), São Paulo, SP, Brazil; 2Department of Ophthalmology, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS, Brazil Purpose: To evaluate the hypotenuse of the vertical optic nerve head cup (HVOC), measured using the length and depth of the cup obtained with enhanced depth imaging spectral-domain optic coherence tomography (SD-OCT), as a biomarker for glaucoma diagnosis. Methods: This was a prospective cross-sectional study of patients with glaucoma and controls. SD-OCT was performed in all participants to assess average circumpapillary retinal nerve fiber layer (RNFL) thickness. A vertical B-scan of the optic nerve head (ONH) was obtained for HVOC measurement. The length and depth of the optic nerve cup formed the sides of a right triangle that were used to calculate the HVOC. Participants also underwent standard automated perimetry. Results: One hundred and fifty-six eyes were divided into three groups: mean deviation (MD)&nbsp

An exotic wingless female insect in mid-Cretaceous Myanmar amber is described as Aethiocarenus burmanicus sp. et gen. nov. in the order Aethiocarenodea ord. nov. and family Aethiocarenidae fam. nov. The distinguishing feature of Aethiocarenus burmanicus sp. et gen. nov. is its unique head, the dorsum of which is shaped like an isosceles right triangle with the hypotenuse at the top and vertex positioned at the base of the neck. While insects with triangular-shaped heads are common today, the hypotenuse of the triangle is always located at the base of the head and attached to the neck, with the vertex at the apex of the head. Other features of the fossil are the long narrow, flat body, long slender legs, especially the hind pair that are twice the length of the abdomen, lack of wings, protruding eyes, paired ocelli, secretory glands located on the dorsum of the neck and swollen abdomen bearing paired segmented cerci.

The Pythagorean theorem is the most appropriate starting point for a book on mathematics and its history. It is not only the oldest mathematical theorem, but also the source of three great streams of mathematical thought numbers, geometry, and infinity. The number stream begins with Pythagorean triples; triples of integers \((a, b, c)\) such that \(a^2 + b^2 = c^2\). The geometry stream begins with the interpretation of \(a^2, b^2,\ \rm{and}\ c^2\) as squares on the sides of a right-angled triangle with sides a, b, and hypotenuse c. The infinity stream begins with the discovery that \(\sqrt{2}\), the hypotenuse of the right-angled triangle whose other sides are of length 1, is an irrational number. These three streams are followed separately through Greek mathematics in Chapters 2, 3, and 4. The geometry stream resurfaces in Chapter 7, where it takes an algebraic turn. The basis of algebraic geometry is the possibility of describing points by numbers—their coordinates—and describing each curve by an equation satisfied by the coordinates of its points. This fusion of numbers with geometry is briefly explored at the end of this chapter, where we use the formula \(a^2 + b^2 = c^2\) to define the concept of distance in terms of coordinates.

Background and aims Selecting the ideal needle length is an important factor in successful ultrasound guided procedures: too long and the needle will be difficult to control, too short and there is a risk that the target cannot be reached. Educational materials often recommend needle lengths for procedures but obviously patients vary in size and shape. We describe a mathematical method for selecting the optimum needle length for any ultrasound transducer/target structure combination. Methods Pythagorean theorem states that for a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. The needle length can be calculated from the target depth and width, measured from the centre of the ultrasound probe to needle entry point, see figure 1. Results In general, it is best to use the shortest needle that will allow the furthest part of the target to be reached. Have in mind the maximum target depth that can be reached with a 50mm needle with the equipment you are using. You can easily calculate a table for your own equipment by measuring the distance from the centre of the probe to the edge of the probe plus 4mm working distance for needle insertion. Use this figure for value b in the formula, see table 1. Conclusions The application of Pythagorean theorem and production of reference table allows quick and easy selection of optimum needle length. This is useful for new trainees and in departments where a range of makes/models of ultrasound machine are used.

Given a leg of a right-angled triangle a, this formula gives the other leg b and the hypotenuse c by the usage of a pattern observed in Pythagorean triples. This is different as compared to Euclids method since Euclids method takes two arbitrary numbers as the input while this uses the side of the right-angled triangle as the input.

Pythagoras is one of the mathematicians who developed the basic theories of mathematics. One of his taunts that are well-known even by primary school students is a Pythagorean Theorem. This theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of each other sides square. There are many proofs which have been developed by a scientist, we have estimated up to 370 proofs of the Pythagorean Theorem. In this paper, we are trying to develop five new proofs of Pythagorean Theorem by using algebraic-geometric proof. The first proof is proven by the trapezoidal shape constructed by five right triangles. The second and third Proofs are proven by using the constructed parallelograms consisting four right triangles and two isosceles trapezoids. The fourth proof is proven by trapezoidal shape constructed of three pieces of a congruent trapezoid, and the fifth proof is proven by using a rectangle constructed by congruent square. Thus, we conclude that the proof of the Pythagorean Theorem can be proven by using the construction of flat trapezoid, parallelogram, square, and rectangular by means of a right-angle triangle.

The tallest tower in Mexico is a right triangle in plan, its hypotenuse a glass curtain wall supported by visible V-shaped hangers. Supporting floors from a central spine was a challenge, but not n...

This paper exposes an alternative and juxtaposed interdisciplinary view of the Pythagorean Triangle, from a mathematical point of view to ethical applicability. Pythagorean Theorem is understood as a mathematical principle (a2+b2=c2), where the sum of the square of the shorter legs, a and b, is equal to the square of the most extended leg, the hypotenuse, c, resulted in the equation of the right triangle (Pythagorean Triangle). As an antediluvian mathematical figure, the Pythagorean Triangle's beauty and intricacy still amazed and provoked present-day thoughts. It is indubitable that the Theorem intrigues humanity's curiosity to provide proofs of the hypothesis, as well as its application. From such a viewpoint, this paper looks into the interdisciplinary applicability of the mathematical figure of the right triangle (Pythagorean Triangle). Utilizing analogical investigation, as the prime method of this research, the paper argues that the concept of the 'right triangle' could be an analogy for 'right living.' The logic behind the Pythagorean Triangle posits inherent beauty and a transcendent possibility beyond mathematics to other disciplines like ethics. The paper offers an analogical investigation of an alternative insight for the ethical problem of 'right living.' By reflecting on the presence of the existing similar feature, the right triangle (Pythagorean Triangle) could be analogically applied to 'right living.' From extant literature, the paper concludes that uprightness and balance are two essential concepts that could propose alternative ethical views by analogy to understanding what it means for 'right living.' Received: 12 October 2020 / Accepted: 12 December 2020 / Published: 17 January 2021

In this paper, a method that enables adjustment of the free spectral range (FSR) of a swept-source is presented by introducing a right-angled prism inside a polygon-based filter. Firstly, the feasibility of this method is analyzed in theory. When adopting the prism at two different cases, the theoretical prediction value of FSR changes differently. In Case 1, when the beam incidents from the hypotenuse surface of the prism, a broader FSR is obtained. In Case 2, when the beam incidents from the right-angled surface of the prism, a narrower FSR is achieved. Then, to verify these two different changes, the filters deployed in these two cases, as well as a standard reference filter, are tested and compared on a seed swept-source. There is no prism in the reference configuration which has the FSR of 187.9 nm. Compared with that, in Case 1 and Case 2, the measured FSR increases to be 234.4 nm and decreases to be 156.6 nm, respectively. The experimental results are not only consistent with the predicted changes, but also show the ability of fine and continuous adjustment of the FSR. This method is promising to improve the performance of swept-sources, as well as the imaging systems which utilize swept-sources as the light sources.

Using the bond-propagation algorithm, we study the critical Ising model on the square lattice with the shape of right triangle and free boundaries. The length of short leg is N and that of long leg is M . The aspect ratio between the two legs ρ = M ∕ N is fixed. For six ratios: ρ = 1 , 2 , 3 , 4 , 8 , 16 , the critical free energy density, internal energy density and specific heat on the lattices with 30 ≤ N ≤ 600 are calculated. Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The corner term in free energy density proportional to ln N is consistent to the conformal field theory in the accuracy of 1 0 − 10 at the least. It is found that the edge terms in the internal energy proportional to ln N related to the hypotenuse are the same for ρ = 1 , 2 , 3 , 4 , 8 , 16 , i.e. this term is geometry independent. However the edge terms in the specific heat proportional to ln N are geometry dependent, i.e. they are different for different ρ . The corner terms in the internal energy and specific heat are also obtained.

The present paper refutes the assertion of T. S. Kuppanna Shastri that the use of the hypotenuse in the computation of the equation of the centre under the epicyclic theory is one of the principal characteristics of the school of Āryabhaṭa I. It has been shown that the followers of Āryabhaṭa I, like other Hindu astronomers, did not employ the hypotenuse in calculating the equation of the centre under the epicyclic theory. The reason for not using the hypotenuse is explained and the views of the prominent Hindu astronomers, such as Bhāskara I, Govinda Svāmi, Parameśvara, Nīlakaṇṭha, and others are cited in support.

We examine the nonlinear variability of quasi-periodic oscillations (QPOs) using the bicoherence, a measure of phase coupling at different Fourier frequencies. We analyse several observations of RXTE/PCA archival data of the black hole binary GX 339-4 which show QPOs. In the type C QPOs, we confirm the presence of the 'hypotenuse' pattern where there is nonlinear coupling between low frequencies that sum to the QPO frequency as well as the 'web' pattern where in addition to the hypotenuse, nonlinear coupling between the QPO frequency and the broadband noise is present. We find that type B QPOs show a previously unknown pattern. We also show that the bicoherence pattern changes gradually from 'web' to 'hypotenuse' as the source moves from a hard intermediate state to a soft intermediate state. Additionally, we reconstruct the QPO waveforms from six observations using the biphase. Finally, we present a scenario by which a moderate increase in the optical depth of the hard X-ray emission region can explain the changes in the non-linearity seen during this transition., Comment: 9 pages, 7 figures, 1 table, accepted for publication in MNRAS Main Journal

The cryptologyis consisted of Kryptos ( hidden) and logos (word) terms in the Greek. It alsomeans that “secrecy science” at the communication. In the present days, theexpansion of the electronic comminucation network has more increased theimportance of cryptology. In this work, we have focused on Knapsackcryptosystem. In this purpose, the Bell numbers in the form of ‘super-increasing sequence’, which constitute the hypotenuse of the Bell triangle, aregenerated in the Python programming. The Knapsack encryption and decryption ofthese numbers are modeled using the Python program. Furthermore, the encryptionwas tested on a physical equation and it was observed that the Bell numbers aresuitable for Knapsack encryption., The cryptology is consisted of Kryptos ( hidden)and logos (word) terms in the Greek. It also means that “secrecy science” atthe communication. In the present days, the expansion of the electroniccomminucation network has more increased the importance of cryptology. In thiswork, we have focused on Knapsack cryptosystem. In this purpose, the Bellnumbers in the form of ‘super- increasing sequence’, which constitute thehypotenuse of the Bell triangle, are generated in the Python programming. TheKnapsack encryption and decryption of these numbers are modeled using thePython program. As an example, “ULUDAG UNIVERSITY” was considered, a 12-bitencryption was performed. It was observed that the Bell numbers are suitablefor Knapsack encryption.

International audience; We review several ways to split a floating-point number, that is, to decompose it into the exact sum of two floating-point numbers of smaller precision. All the methods considered here involve only a few IEEE floating-point operations, with rounding to nearest and including possibly the fused multiply-add (FMA). Applications range from the implementation of integer functions such as round and floor to the computation of suitable scaling factors aimed, for example, at avoiding spurious underflows and overflows when implementing functions such as the hypotenuse.

We present here a novel method of computing spectral distances in doubled Moyal plane in a noncommutative geometrical framework using Dirac eigen-spinors, while solving the Lipschitz ball condition explicitly through matrices. The standard results of longitudinal, transverse and hypotenuse distances between different pairs of pure states have been computed and Pythagorean equality between them have been re-produced. The issue of non-unital nature of Moyal plane algebra is taken care of through a sequence of projection operators constructed from Dirac eigen-spinors, which plays a crucial role throughout this paper. At the end, a toy model of "Higgs field" has been constructed by fluctuating the Dirac operator and the variation on the transverse distance has been demonstrated, through an explicit computation.

The underlying idea behind the construction of indices of economic inequality is based on measuring deviations of various portions of low incomes from certain references or benchmarks, which could be point measures like the population mean or median, or curves like the hypotenuse of the right triangle into which every Lorenz curve falls. In this paper, we argue that, by appropriately choosing population-based references (called societal references) and distributions of personal positions (called gambles, which are random), we can meaningfully unify classical and contemporary indices of economic inequality, and various measures of risk. To illustrate the herein proposed approach, we put forward and explore a risk measure that takes into account the relativity of large risks with respect to small ones.

In this chapter we determine an asymptotic formula for the number of primitive right triangles with bounded hypotenuse.

This paper is focused on the experimental study and numerical simulation of isolated spread concrete foundation slab with a large width-to-height ratio (in short ISCFS-LWR) to investigate the failure modes and uplift bearing capacity, as well as the design method of uplift capacity. First, a total of 16 isolated spread concrete foundation slabs with the width-to-height ratio varied from 1.5 to 4 and the hypotenuse slope varied from 10° to 30° were tested under uplift load. Based on the test results, effects of the width-to-height ratio and the hypotenuse slope on uplift bearing capacity of ISCFS-LWR were analyzed and discussed. Then, several numerical models were built using the finite element software ABAQUS and the results of numerical analysis agreed well with the test results. Furthermore, the cross-sectional performance of ISCFS-LWR was studied, and the coefficients of internal force arm were also evaluated further using previous validated numerical models. To obtain the suggested design method of uplift capacity for the foundation slab, effective width correction coefficientkand slope correction coefficientjwere introduced to propose a design formula. Finally, the proposed design method was applied to a practical engineering, and the economic indicators obtained from the suggested design method were compared with that from the original design method. The results of this paper showed that the correction coefficientjsksbased on numerical analysis agreed well with the recommended correction coefficientjk, and the error was between 1% and 3.4%, by which the reasonability of the proposed design method of uplift capacity for ISCFS-LWR has been proved. It can also be found that the economic benefits of the practical engineering in this paper were obvious due to the suggested design method, and this paper can provide a reference for other engineering practices and the further research work on ISCFS-LWR.

Within the entire volume of power space, the concept of phasor diagram so we have plays a very important role for the complete description of the physicality of any on the ground of some fundamental laws belong to the same. One of the most common phasor diagram, which we generally deal with for the understanding of the concept of power factor is the power (or impedance) triangle. The main objective of this paper is to investigate the dynamic behaviour on account of bilateral swinging of the hypotenuse arm for a given length of the right-angled (or Pythagorean) triangle employed for the physicality of certain, here, this Pythagorean triangle employed for the portrayal of 3 powers of the power space. Due to the dynamicality of such a trigonometrical identity based on the axiom of sides sum difference for triangle lengths is also achieve.

SummaryBy fixing either one leg or the hypotenuse of a right triangle and letting the other sides vary, we obtain three new proofs of the Pythagorean theorem.

Common Core State Standards F-TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. It would be wonderfully convenient if the following two claims were true: sin ( x + y ) = sin ( x ) + sin ( y ) cos ( x + y ) = cos ( x ) + cos ( y ) . As experimentation on a calculator shows (or just put in x = 30° and y = 60°) these dream formulas do not hold. Can we find expressions for sin( x + y ) and cos( x + y ) nonetheless? You bet! We can use an approach similar to the one we used to prove the Pythagorean Theorem. Draw two copies each of two right triangles, each with hypotenuse 1. Arrange them into a rectangle as shown: The area of the white space is the sum of the areas of two small rectangles: White Space = sin( x ) cos( y ) + cos( x ) sin( y ) . We can also rearrange the four triangles within the large rectangle as The white space is now a rhombus with side length 1. The area of a rhombus (in fact, of any parallelogram) is “base times height.” The base length is 1 and the height is the length h indicated. We see that h is the opposite edge of a right triangle of hypotenuse 1 and angle x + y . Thus: White Space = 1 × h = 1 × sin( x + y ) = sin( x + y ) . It is the same white space. Thus we have proved: sin( x + y ) = sin x cos y + cos x sin y, at least for angles x and y that lie between 0° and 90°. (For visual ease, we've omitted displaying all the parentheses in this equation.) In the same way, we can consider this variation of the proof. (Do you see the change on which angle is called y ?) It establishes cos( x – y ) = cos x cos y + sin x sin y, at least for angles x and y that lie between 0° and 90°.

Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, for example, as lengths of catheti of right-angled triangles (defined via so-called quasiorthogonality). These triangles have two boundary points of the unit ball of a Banach space as endpoints of their hypotenuse, and their third vertex lies in a supporting hyperplane of one of the two other vertices. Among other things, it is our goal to quantify via such triangles the local deviation of the unit sphere from its supporting hyperplanes. We prove respective Day-Nordlander-type results involving generalizations of the modulus of convexity and the modulus of Banas.

The current Federal Communications Commission (FCC) Rules for distance calculations use two methods: flat-earth and spherical-earth. The flat-earth method assumes the distance between two points to be the hypotenuse of a right triangle whose sides are determined by the difference in latitude and longitude of the starting and ending points multiplied by the length per degree of latitude and longitude at the mid-latitude of the two points, as shown in Figure 1.7-1. The “flat-earth” term refers to the use of a right triangle to calculate the distance. Because the lengths of a degree of longitude and a degree of latitude used in the flat-earth method are derived from an ellipsoid rather than a spheroid model of the earth, the flatearth method is actually more accurate than the sphericalearth method for short to moderate distances.

Theorem (Pythagorean Theorem). Let ΔCBA be a right triangle with hypotenuse c and two legs a and b. Then c2=a2+b2 .Proof. A few remarks are in order.• The proof is wordless and it is inspired by Hu...

The main task considered in the article is to find the condition primitive integer solutions of the Diophantine Pithagorean equation x2+y2=z2 It is known that for this purpose it is enough to find primive solution of x, y such that x is even and y is odd. In this paper, in particular, we proved that the z of a primitive solution is a Prime number of the form 4k+1. It is prove in this paper that any right triangle with integer side lengths has a hypotenuse equal to a Prime of the form 4k+1and we show with the help of the descent axiom how to find primitive solutions of x and y in this case. We divide the search for primitive solutions (x, y, z) of right triangles into two cases: 1) the hypotenuse of such triangles is a Prime number of the form 4k+1 and 2) the hypotenuse of such triangles is a composite number. In section 3 we use formulas known to the ancient Hindus to find primitive solutions of Pithagorean equations in cases where m and n

We ask if there is any right isosceles triangle in the hyperbolic plane, constructible with ruler and compass, for which the ratio of the hypotenuse to the side is rational. Although the question remains open, we determine that there are no such triangles with ratio $$1+\frac{m}{n}$$ , with $$n\le 12$$ .

Longer mixing times and higher power consumption are common problems in the design of photobioreactors. In this study, a vertical triangular external airlift loop photobioreactor was designed, constructed and operated for microalgae production studies. Gas feeding was performed by two spargers: one at the bottom of the hypotenuse (downcomer) and another at the bottom of the vertical side (riser). This configuration provided more effective countercurrent liquid–gas flow in the hypotenuse. The mass transfer coefficient, gas hold-up, mixing time, circulation time, dimensionless mixing time, bubble size, and volumetric power consumption were measured and optimized using response surface methodology. Investigations were carried out on the performance of the riser (the vertical side), downcomer (the hypotenuse), and separator. The countercurrent flow in the hypotenuse provided sufficient contact between gas and liquid phases, and increased mixing and mass transfer rates, in contrast to the results of previous studies. The promising results of this geometry were shorter mixing time and a significant decrease in volumetric power consumption in comparison with other configurations for photobioreactors.

In this paper, a natural lighting system composed of a tilted prismatic daylight collector, an outdoor reflector and an indoor diffuse reflector is described. The light emerging from the hypotenuse of the prismatic daylight collector is redirected onto the indoor reflector for natural light illumination. The results of measured illuminances are in agreement with the results of calculation. The prismatic daylight collector not only decreases discomfort glare but also collects daylight for the natural lighting system. The open natural lighting system, which uses less material, can be realized by the detailed analysis of the characteristics of the light emerging from the hypotenuse of the prismatic daylight collector for sunlight incident on the collector.

At the majority of signalized intersections along an arterial, the mainline movement passes straight through the intersection. At four-leg intersections where the major arterial movement is not a through movement (i.e., an L-shaped corridor vertex), the intersection can sometimes experience more congestion than adjacent intersections along the corridor because the two minor approaches, which are adjacent conflicting movements, must be served in a split-phase fashion. A new, unconventional intersection design, referred to as a “triangabout,” is proposed to improve mobility and enhance safety for such intersections. The two approaches that are the mainline arterial form two legs of a triangle and a new diagonal section of roadway connecting those two legs is the hypotenuse. Flow within the triangabout is one-way counterclockwise, similar to a roundabout. The three intersections at the vertices of the triangle are signalized to provide a dedicated right-of-way. A case study is performed for the intersection of Van Voorhis Road and Chestnut Ridge Road in Morgantown, West Virginia, along the WV-705 corridor. Simulation results from VISSIM show that the triangabout design can reduce travel delay by 50% compared with the existing configuration. Also, the number of conflict points compared with a conventional intersection is reduced by 34%. There are a number of qualitative benefits of this design compared with conventional intersections at an L-shaped corridor vertex.

[1] The rupture history of the 20 April 2013 Mw 6.6 Lushan (China) earthquake is constrained by inverting waveforms of local strong motion, teleseismic broadband body waves, and long-period surface waves. This earthquake ruptured a blind thrust fault oriented N210°E (along the Longmenshan fault zone) and dipping 40° to the NW. The inverted slip distribution is heterogeneous, dominated by a slip patch with a roughly right triangular shape, which spans a depth range of 5–20 km and accounts for two thirds of the total seismic moment (8.9 × 1018 N m). The rupture initiated roughly at the middle of the triangle's hypotenuse and, during the first 4 s, propagated mainly in along-strike and downdip directions, toward a peak slip of 1.2 m. Despite a large number of fatalities and economic loss, the estimated static and apparent stress drops of the Lushan earthquake are 1.5 MPa and 0.35 MPa, considerably low with respect to other similar intraplate earthquakes.

Analysis of natural convection in porous right angled triangular enclosures with a concave/ convex hypotenuse has been carried out using the Bejan's heatlines approach. A generalized non-Darcy model without Forchheimer term is employed for fluid flow in a porous matrix and the governing equations are solved by the Galerkin finite element method. The cavity is subjected to a thermal boundary condition of an isothermal cold left wall, isothermal hot curved right wall, and adiabatic bottom wall. Due to intense closed loop heatlines, thermal mixing is higher in convex cases compared to the concave case for all parameters. Average heat transfer rate is found to be largest in the concave hypotenuse case. Copyright � Taylor & Francis Group, LLC.

More than 370 years ago the famous French mathematician Pierre de Fermat proposed to solve the following problem: to find a Pythagorean triples whose hypotenuse and the sum of the legs were squares, which, despite its simplicity, has been very difficult. Problems associated with its solution involved many mathematicians such as (Leonhard Euler, Joseph-Louis Lagrange, Ljnggren, Waclaw Sierpinski and etc.) But in the end it did not reach solution. In our article, solution communicated to obtain the equations giving the required values all elements of the Pythagorean triples in positive integers (natural) are co-prime integers, and provides a second solution of this problem (the values of x, y, z of 45 digits), and some consequences. � § 1 Problems associated with the solution of this problem, were engaged Euler, Lagrange, Ljunggren and etc. The first person who show how to obtain first solution of the problem, that resulted by Fermat was the Waclaw Franciszek Sierpinski. He also solved the problem of obtaining all results that include coprime solutions. ("Pythagorean triangle", Uchpedgiz, 1959, §12). However, he has not led the explicit equations for solutions to this problem. Therefore, we believe that it makes sense to bring below it's a solution different from the mentioned, with the corresponding equations and give the value of the elements of the second triangle of the required type.

The results of a new experiment show that the Allais paradox (or, more generally, the common consequence effect) gets reversed, i.e. fanning-in choice patterns significantly outnumber fanning-out choice patterns. Revealed indifference curves fan in along the horizontal axis and hypotenuse of the Marschak–Machina probability triangle. No significant asymmetry between fanning-in and fanning-out choice patterns is observed along the vertical axis of the probability triangle.