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Any quadruple of natural numbers is called a Pythagorean quadruple if it satisfies the relationship . This Pythagorean quadruple can always be identified with a rectangular box of dimensions in which is identifiable with the length of its diagonal. The circumscribing sphere of this rectangular box has an integral diameter length corresponding to the Pythagorean quadruple .This result extends the well-known ‘inscribed circle theorem’ for any Pythagorean triple of natural numbers satisfying . This above-mentioned theorem asserts the positive integer nature of the radius of the inscribed circle, that is associated with any right triangle with hypotenuse length , and leg lengths corresponding to any Pythagorean triple of natural numbers.

The sequence $$3, 5, 9, 11, 15, 19, 21, 25, 29, 35,\ldots $$ consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the Erdős–Ford–Tenenbaum constant, to be sharp up to a double-logarithmic factor. We also provide a nontrivial lower bound. Our techniques involve sieve methods, the distribution of Gaussian primes in narrow sectors, and the Hardy–Ramanujan inequality.

Accurate diagnosis is the prerequisite for effective treatment of diseases. Various biostatistics indices have been used to evaluate the performance of diagnostic tests. The distribution of test results commonly shows a relatively fixed overlap between the diseased and non-diseased populations, and the two main properties of laboratory tests sensitivity (Se) and specificity (Sp) usually demonstrate an inverse relationship of each other. An inverse relationship is also seen in the two sides of a right triangle where the hypotenuse serves as an integrated functional index of the two sides, as reflected in the geometric equation Pythagorean Theorem (PT). We have developed a novel test performance index named diagnostic availability (DA) based on the theory of PT, which integrates and reflects the combined efficiency of Se and Sp. When Se and Sp were changing inversely as seen in the vast majority of clinical settings, DA was superior to the other indices diagnostic efficiency (DE), Youden index, and Kappa coefficient (К) in reflecting the overall test performance of a test. It was also the only one responsive to the reciprocal changes of Se and Sp. Therefore, we propose the additional consideration of DA besides Se and Sp during the determination of the optimal test result cut-off point. In the rare scenarios that Se and Sp are changing in the same direction, DA, J and К outperforms DE in reflecting test performance. Furthermore, DA can also help identify the better test when two tests exhibit similar areas under the curve (AUC) in the receiver operating characteristic curve (ROC) analysis. In conclusion, DA is a novel index for overall performance evaluation of diagnostic tests reflecting the combined efficiency of Se and Sp. It shows superiority over other commonly used indices in the assessment of overall test performance and serves as a valuable addition to the existing indices.

This paper presents a compact conical beam antenna based on quarter-mode substrate integrated waveguide (QMSIW). The antenna array is consisted of six isosceles right-angled triangle metallic patch printed on the upper side of the substrate in a windmill-shape and diagonal periodic metallic via holes drilled along the hypotenuse of the triangle patch. The planer structure exhibits conical beam radiation. The typical radiation is obtained at 5.2 GHz with conical beam radiation pattern. The measured gain is about 5.36 dBi and the radiation efficiency is around 50%. Published version

Without reference to the Pythagorean theorem, we show that a right triangle with legs 3 and 4 has hypotenuse 5. The figure can be modified for other integer right triangles.

We consider the diophantine equation x2 + y6 = ze, e ≥ 4. We show that, when e is a multiple of 4 or 6, this equation has no solutions in positive integers with x and y relatively prime. As a corollary, we show that there exists no primitive Pythagorean triangle one of whose leglengths is a perfect cube, while the hypotenuse length is an integer square.

The clinical observation of buccally blocked out canines hypothesize the theory of space opening between the lateral incisor and the premolar. We therefore conduct a study in the course of which we performed a dynamic measurement. Due to the binding, a right triangle with a hypotenuse of 74 mm = 8 .6 mm was constructed. This hypotenuse is actually based on the Pythagoras theorem c2 = a2 + b2, where as a represents the inter-bracket distance of 5 mm and b the height of the cuspid of 7 mm. The positions of the brackets between the premolar and the lateral incisor are changed in steps of 0.5 mm thus reaching 8.50 mm. Subsequently, the cuspid bracket is moved downwards by 0.61 mm, 0.84 mm, 1.37 mm, 2.00 mm, 2.79 mm, 3.84 mm, and 5.68 mm. The results show a gradually decreasing horizontal (101–13 cN) and increasing vertical force (10–51 cN) as well as more as less constant moment (378 nN mm–310 cN mm) on the premolar and lateral incisor bracket. It also shows that the horizontal force and moment on the canine bracket is inconsiderable, whereas the vertical force on the canine bracket (19–108 cN) is twice as strong as the vertical force on the premolar and lateral incisor bracket. In the course of crowding, these horizontal and vertical forces as well as the moments can be useful for levelling dental arches – particularly in the case of high canines.

The triangular plate has widely application in the manipulation of the pr angular plate will often have error. In the former solutions, it is very hard to find one kind of simple and common way to solve the equation of deflection curve of the triangular plate. Under uniform load function, this paper uses the reciprocal theorem method to discuss the bending question of triangular plate with the unfixed hypotenuse and fixed side as well as the other side with simple supported, and the unfixed hypotenuse as well as two fixed sides. Thus the method has overcome complicated calculating process appeared in the classical theory, solved the limitations with difficulties to certain questions and established new way to solved the bending question of triangular plate. In the process of calculating, it will take specific proportion, applying the mixed variables method, carrying the calculation on the Matlab platform, and comparing it with the ANSYS finite element analysis result, to prove the result is correct. Thus the paper gives the equation of deflection curve and the graphs which can be employed in the actual project application. Therefore, the paper has certain theoretical value and project significance.

I n a recent article [2], L. Berrone presented a new approach to the Pythagorean Theorem (PT). The idea is to derive the geometric theorem from analytic and algebraic properties, by methods of functional equations. (So we are not dealing with a method that was an option for the ancients!) I thought about Berrone’s ideas, within his context of functional equations. Some pleasant surprises fell out. Then a surprising gift of functional equations back to geometry closed the circle for me – and will close this article. Let a b denote the hypotenuse of a right triangle with legs a and b. The operation : ð0;1Þ 7!ð0;1Þ is evidently continuous. It is also (a) Homogeneous (of degree one)

for a spherical right triangle with hypotenuse c and legs a and b, is generally presented as the ‘spherical Pythagorean theorem’. Still, it has to be remarked that this formula does not have an immediate meaning in terms of areas of simple geometrical figures, as the Pythagorean theorem does. There is no diagram that can be drawn on the surface of the sphere to illustrate the statement in the spirit of ancient Greek geometry. In this note I reconsider the issue of extending the geometrical Pythagorean theorem to non-Euclidean geometries (with emphasis on the more intuitive spherical geometry). In apparent contradiction with the statement that the Pythagorean proposition is equivalent to Euclid’s parallel postulate, I show that such an extension not only exists, but also yields a deeper insight into the classical theorem. The subject matter being familiar, I can dispense with preliminaries and start right in with Euclid’s Elements [1].

In learning process students are required to play an active role in learning. They do not just accept the concept directly from teachers, but also build their own knowledge so that the learning process becomes more meaningful. Based on the observation, when learning Pythagorean theorem, students got difficulty on determining hypotenuse. One of the solution to solve this problem is using an interactive multimedia learning. This article aims to discuss the interactive multimedia as learning media for students. This was a Research and Development (R&D) by using ADDIE model of development. The results obtained was multimedia which was developed proper for students as learning media. Besides, on Phytagorian theorem learning activity we also compare Discovery Learning (DL) model with interactive multimedia and DL without interactive multimedia, and obtained that DL with interactive gave positive effect better than DL without interactive multimedia. It was also obtainde that interactive multimedia can attract and increase the interest ot the students on learning math. Therefore, the use of interactive multimedia on DL procees can improve student learning achievement.

The architectural differences between Northern and Middle-Southern Vietnam are based on the method for the roof pitch planning. The Northern method is based on the horizon, vertical line and the dimensional unit, the roof line is designed by the horizontal dimension and the height. The Middle-Southern method is not based on the horizon, vertical lines, the plan designed by concrete dimensions, and the roof line fixes one hypotenuse unit as 2 height units. The points are namely the difference of the main design tools, the square tool and the equilateral triangle tool, those can be considered as a gap of the traditional architectural technique and culture between two regions.

The square ground plan of the 8th-century Mosque was divided into two equal rectangles, to form the patio and the oratory. The architect divided the oratory into two squares, took its E. and W. sides as radiuses, and the point where they crossed over gave him the width of the central nave. He divided the remaining areas of each square into five equal parts to obtain the lateral naves. To establish the place of the columns he may have used either a rectangular grid or a rhomboidal grid. With the central axes of the columns known, he elevated the arcades and the ceiling by using a set-square of 90°, 45°, 45° (with hypotenuse √2), and another set-square of 90°, 60°, 30° (with hypotenuse √4). The wall of the qibla ran the length of the 11 naves and contained the niche of the miḥrāb.La planta de la mezquita del siglo VIII era un cuadrado dividido en dos áreas rectangulares iguales, el patio y el oratorio. El alarife dividió este último en dos cuadrados y tomó los lados opuestos de cada uno a E. y O. como radios y, al entrecruzarlos, consiguió el ancho de la nave central. Distribuyó en cinco partes iguales el área restante de cada cuadrado y obtuvo el ancho de las naves laterales. Para hallar la ubicación de las columnas pudo utilizar dos sistemas, o una cuadrícula rectangular o una retícula romboidal. Conocidos los centros de los ejes de las columnas, trazó el alzado de las arquerías y armadura mediante escuadra (=√2), cartabón (=√4) y escuadra menor. El muro de la qibla era corrido en las 11 naves y albergó el nicho del miḥrāb.

The article describes a hypotenuse-height deviation compression method of magnanimous point cloud data based on hierarchy. Firstly the point cloud is stratified into several hierarchies which turn into linear point clouds, then compressed by using hypotenuse-height deviation threshold. By experiment, the compression effect of the method brought up is compared with that of several traditional point cloud compression methods, which verify the capability of realizing high-efficiency compression with abundant detailed features retained.

The sequence of operations for face recognition comprises of the pre-processing techniques, feature extraction, selection and classification for precise identification and substantiation of subject images. Initially, the face images are processed by Gaussian of Laplacian blurring with Median and Weiner filters for the removal of undesirable noise and frequencies, a unique approach in advancing the preprocessing techniques. The pre-processed images are applied to feature extraction transformations, namely, the Discrete Wavelet Transform (DWT) coupled with Slope-form Triangular Discrete Cosine Transform (STDCT) to generate critical essential features from the images. Primarily the feature vector space is searched for optimal selection of feature subset utilizing the Binary Particle Swarm Optimization (BPSO) search algorithm based on mutual behavior of bird flocking or fish schooling. Evaluation of the subset by the Euclidean Classifier produces reliable face recognition rate. The system dependability is attained by processing with standard databases like, the Color Facial Recognition Technology (FERET), Olivetti Research Laboratory (ORL) and Japanese Female Facial Expression (JAFFE). Induction of MATLAB to analyze the imaging methodologies with standard databases, demonstrates computation of slope of the hypotenuse of a right triangle in STDCT for reduced feature extraction. This novel technology transcends other systems by generating enhanced face recognition rate with optimum selected features to achieve on execution with multiple iterations, establishing propriety and reliability of the proposed system during the process of validation.

In this work I propose an analogy between Pythagoras's theorem and the logical-formal structure of Werner Heisenberg's “relations of uncertainty.” The reasons that they have pushed to me to place this analogy have been determined from the following ascertainment: Often, when in exact sciences a problem of measurement precision arises, it has been resolved with the resource of the elevation to the square. To me it seems also that the aporie deriving from the uncertainty principle can find one solution with the resource to this stratagem. In fact, if the first classic example of the argument is the solution of the incommensurability between catheti and the hypotenuse of the triangle rectangle, one of the last cases is that which is represented from Heisenberg's principle of uncertainty.

This paper reports simulation results of a plasmonic wavelength band-pass filter with triangular arrays in a MIM waveguide. Simulation results confirm resonance wavelengths can be easily adjusted by changing length of opposite side to hypotenuse.

In this work we extent our recent results on the stability of rigid systems to cases involving flexible ones. We present closed form analytical expressions that describe the boundaries of the stability regions for digital PD control systems. This is obtained using a newly adopted approach based on the critical stability constraints of Jury test. The considered system consists of a single rigid and a single flexible mode. This simulates many practical systems such as antenna, space shuttle, and robot arm. The obtained closed-form stability criterion shows that the stability region is almost a right triangle in the gain space. The right-angle sides of the proposed triangle coincide exactly with two boundaries of the stability region. The hypotenuse of the triangle is found to well approximate the third boundary. An analytical expression is derived for the slope of the hypotenuse. This approximation results in defining the stability region using only two parameters which is convenient for practical applications. Case study is presented to show the practical usefulness of the proposed close-form stability criteria through investigating the control design of high speed flexible robot arm.

In a right-angled triangle, the hypotenuse is the longest side. So, if all (hypotenuse) vectors from a given set of vectors have the same orthogonal projection onto a certain subspace, we have a lower bound for their lengths. Interpreting the square of such a length as the variance of an unbiased estimator produces an information bound. The Cramér-Rao bound and the van Trees inequality can be seen as consequences of this bound. Another consequence is an inequality for the minimax variance, that is, the maximal variance in shrinking neighbourhoods, minimized over all unbiased estimators. This bound is non-asymptotic and requires almost no regularity conditions.

Epistemology is a branch of philosophy concerned with the nature and scope of knowledge. The term was introduced by the Scottish philosopher Ferrier.1 It seeks to address questions like What is knowledge? How do we know that we know something? To what extent is it possible for a particular subject to be known? Broadly speaking, epistemological knowledge can be divided into 2 categories: knowledge that and knowledge how. Most animal behavior, for example, a newborn bird learning to fly or a child learning to ride a bike, are of the knowledge how category. In contrast, most of the subsequent discussion surrounds the knowledge that category. This addresses questions like How do we know that the sum of the squares of the 2 sides of a right-angled triangle equals the square of the hypotenuse? or How do we know that antibiotics work for meningitis? Humans are probably alone in the capacity not only to know something but also to know that they know it, and to consider the foundations of this knowledge. Although most of us know it would be a bad idea to jump from an airplane in flight without a parachute, it is harder to justify how we know that with such certainty. Each of us has clearly not tested that out for ourselves and most would agree it is absurd to test this as a hypothesis. Rather, our knowledge comes from experience in relevant parallel situations (eg, seeing what happens when one jumps from 10 feet and extrapolating it to 10 000 feet), mechanistic understanding of gravity, and inborn instincts for self-preservation. Much of this knowledge is arguably innate: a 4-year-old child knows when it is not safe to jump off a height; so does a baby monkey. But this raises a classic question of particular importance to …

In this note we will show that the inverse image under the stereographic projection of a circular torus of revolution in the 3-dimensional euclidean space has constant mean curvature in the unit 3-sphere if and only if their radii are the catet and the hypotenuse of an appropriate right triangle.Neste artigo mostraremos que a imagem inversa pela projeção estereográfica de um toro circular de revolução no espaço euclidiano de dimensão 3 tem curvatura média constante se e somente se os seus raios são o cateto e a hipotenusa de um triângulo retângulo apropriado.

A scheme to approximate the trimmed surfaces defined by two tensor-product surface patches, intersecting in a smooth curve segment that extends between diametrically opposite patch corners, is formulated. The trimmed surface approximations are specified by triangular Bezier patches, whose tangent planes agree precisely with those of the tensor-product surfaces along the two sides where they coincide. Topological consistency of the two trimmed surfaces is achieved by requiring the "hypotenuse" sides of the triangular patches to be coincident. In the case of bicubic tensor-product patches and quintic triangular trimmed surface approximations, enforcing these conditions entails the solution of a linear system of 30 equations in 32 unknowns. The two remaining scalar freedoms, together with four additional free control points, are employed to enhance the accuracy and/or smoothness properties of the intersection curve and trimmed surface approximations. By means of an appropriate subdivision preprocessing, the trimmed surface scheme may be used on models described by arbitrary bicubic surface patches.

Trigonometry is the connecting link between mathematics and astronomy, between the way calendars are calculated, the gnomon, and the sundial. In the Islamic world, the calculation of spherical triangles was necessary to carry out ritual customs. The qibla, the direction to Mecca, was indicated next to the hour lines on all public sundials. The first trigonometric problems appeared in the field of spherical astronomy. Around the year 773, one of the Indian siddhāntas (astronomy books) was made known in Baghdad. The Indian astronomers Varāhamihira (fifth century) and Brahmagupta (sixth century) solved different problems in spherical astronomy by means of rules equivalent to a general sine theorem for a spherical triangle ABC with sides a, b, c and angles A, B, C (where angle A is opposite to side a, etc.), namely, sin A= sin a ð Þ 1⁄4 sin B= sin b ð Þ 1⁄4 sin C= sin c ð Þ and to the cosine theorem for the same triangle cos a 1⁄4 cos b cos cþ sin b sin c cos A: In the ninth century, Ptolemy’s Almagest and Menelaus’ Spherics were also translated, and commentaries were written to these works. Many trigonometric problems were solved in Ptolemy’s Almagest, in which Menelaus’ theorem on the spherical complete quadrilateral was used. The cases of this theorem used by Ptolemy are equivalent to the sine and tangent theorems for a right‐angled spherical triangle. The Almagest, the Spherics, and the Indian siddhāntas formed the basis on which Arab mathematicians built their trigonometry. The ancient Greek astronomers only used one trigonometric function, the chord of an arc. The “theorem of Ptolemy,” which is equivalent to the formula for the sine of the sum of the angles, forms, together with the formula for the chord of the half arc, the basis for the chord table in the Almagest. The Indian people replaced the chord with the sine, introduced the cosine and the versed sine, and compiled a small table of sine values. The Arabic mathematicians progressively made trigonometry into a science independent of its (astronomical) context. Applications of trigonometry analogous to those in the Indian siddhāntas are found in the astronomical works of al‐Khwārizmī. An analogous geometric construction for finding the azimuth according to the rule formulated in al‐Khwārizmī’s third treatise was provided by al‐Māhānī (ca. 825–888) in his Treatise on the Determination of the Azimuth at Any Time and in Any Place. The rules equivalent to the spherical sine and cosine theorems were also used by Thābit ibn Qurra in his Book on Horary Instruments Called Sundials. With Ḥabash, the applications of the tangent and cotangent functions went beyond the usual applications in the theory of sundials. The introduction of the tangent and cotangent and their application in astronomy was a novelty. The names zill (shadow) and zill maʿqus (reversed shadow) apparently are translations from Sanskrit. In the case of a vertical gnomon, al‐Ḥabas expressed the cosecant as the “diameter of the shadow” for a given height of the sun, i.e., as a hypotenuse. He computed a table for the cosecant with steps of 1 . For a long time, the chord was used along with the sine. A theory of these magnitudes is found in the work of al‐Battānī (ca. 858–929). In his astronomical work Islaḥ al‐Majisṭī (The Perfection of the Almagest), he systematically employed the trigonometric functions sine and versed sine with arguments between 0 and 180 . Since the cosine is defined as the sine of the complement of the angle and since no negative numbers are used, the versed sine is defined in the second quadrant as a sum of two quantities.

This paper introduces a new method of estimating indifference curves in the Marschak-Machina triangle. The method involves posing questions about indifference. Contrary to previous attempts, where subjects were required to identify those lotteries to which they were indifferent vis-à-vis a given lottery, the subjects are here required to determine its certainty equivalent. The procedure is repeated for a large number of lotteries inside the triangle. Simple, linear interpolation of certainty equivalent values between adjacent points representing the lotteries under consideration allows any indifference curve inside the triangle to be plotted. The experimental results presented in the paper shed new light on the shape of indifference curves inside the Marschak-Machina triangle, where curve parallelism, fanning-out, fanning-in and boundary effects, including (possibly discontinuous) jumps, are all common. As shown, those decision-making models, which can predict jumps on the triangle legs, offer the best econometric fit of the indifference curves obtained in the study.

This paper presents a quick, efficient and simple method for drawing straight lines with antialiasing on a discrete display using the DDA algorithm with fixed-point arithmetic. The algorithm permits the drawing of grey tones lines by using a 3-pixels-high and 1-pixel-wide brush. The computing cost is low and only integer arithmetic is used. If a line is understood as a 1-pixel-width (W) and n-pixels-long (L) rectangle, its surface is given by W×L. The real length (L) increases (hypotenuse of a rectangle triangle) as the line slope does, and so, the total radiation emitted that is proportional to its real surface. This algorithm takes into account this intensity increment. The algorithm maintains the width of the line perpendicular to its longitudinal axis independent of the slope.

Input impedance formulas for feed positions first, on the vertical side, and second, on the hypotenuse, are obtained from coplanar microstrip circuit analysis. The formulas that involve both single and double infinite series are economized by introducing closed forms of the infinite series. These new formulas are verified by comparison with practical results and a simulation based on full wave analysis.

Students need to construct strong knowledge of angles as well as relationships between angles and side lengths in a triangle to succeed in geometry. Although many researchers pointed out the importance of angles and angle-related concepts, students and teachers have had limited knowledge of these concepts. This study is a part of a larger study, and examines pre-service secondary mathematics teachers‘ (PSMTs) mental constructions of relationships between angles and side lengths in a right triangle (RASR). The Action-Process-Object-Schema (APOS) learning theory was used as the theoretical lens and clinical interview methodology was used as the methodology in the study. The study was conducted with four PSMTs, but it focuses on one of PSMTs, Linda, who revealed evidence of schema for RASR. As a result of fine-grained analysis of Linda‘s responses to the tasks, this article reports the mental constructions enough to develop a schema for RASR. The model describes that schema for 2-line angles, right triangles and relationships between opposite angles and side lengths and process level for some relationships including ‗Pythagorean Theorem‘, ‗The hypotenuse is always the longest side in a right triangle‘, ‗Special right triangles‘, ‗Complementary Angles‘, and ‗Triangle inequalities‘ are enough to construct a schema for RASR.

The techniques of massive data capture, as the Terrestrial Laser Scanner and the digital photogrammetry, allow to accurately analyze close objects and others that are not accessible, by statistical methods. Its application has allowed assessing, from the geometric point of view, the sculptures of the figure Clypeus of Tarragona, Jupiter-Ammon (1st century) and the one of the Orvieto’s Cathedral rosette, with the face of Christ (14th century). The Clypeus of Tarragona and Orvieto’s rosette are divided in 22 parts, forming a polygonal shape known as icosakaidigono. This polygon does not appear in the Elementa of Euclides, neither in the Almagesto of Ptolomeo, nor in the rare Practica Geometriae of the time. These polygons can be built by means of angular division inherited from astrolabe and quadrant builders. This polygon can be defined also through the rotation of the square, since the repeated support of the catheti on the hypotenuse allows the rotation of the tool, or other indirect methods. The conclusion of the analysis of these figures is that, statistically, both sculptures have a geometric construction equal to the Reliquiario del Corporale del Miracolo di Bolsena (1337-1338) of Orvieto’s Cathedral.

By drawing a Pythagorean triangle in a quadratic lattice and attaching a congruent lattice at the hypotenuse there will occur a Moire effect with a new quadratic lattice of enlarged scale in the superposition. This new lattice is related to the parameterization of the Pythagorean triangle. A similar effect occurs with triangles with integer side lengths and an angle of 120° in a regular triangular lattice. We work with dot lattices on transparencies to visualize the optical effects.

The hypotenuse function, z = x 2 + y 2 z=\sqrt {x^2+y^2} , is sometimes included in math library packages. Assuming that it is being computed by a straightforward algorithm, in a binary floating point environment, with round to nearest rounding mode, a sharp roundoff error bound is derived, for arbitrary precision. For IEEE single precision, or higher, the bound implies that | z ¯ − z | > 1.222 u l p ( z ) |\overline z-z|>1.222\, ulp(z) and | z ¯ − z | > 1.222 u l p ( z ¯ ) |\overline z-z|>1.222\, ulp(\overline z) . Numerical experiments indicate that this bound is sharp and cannot be improved.

Des scientifiques ont discute en detail les textes chinois datant de l'Antiquite et concernant la demonstration et la preuve du theoreme de Pythagore

As is well known, in the Principles of Mathematics Russell distinguishes between ‘any’ and ‘all’. He uses ‘any’ to understand mathematicians’ inferences using free variables. For example, when one introduces a right-angled triangle xyz in order to prove the Pythagorean theorem, one is saying that each proposition shown is true of any right-angled triangle. In this way, after one concludes that the square of hypotenuse xy is equal to the sum of the square of yz and xz, one can generalize that this is true of all right-angled triangles. The free variables ‘x’, ‘y’, and ‘z’ are used to pick out ‘any triangle’ or an ‘arbitrary triangle’. Russell allows assertions of open formulas in his mathematical logic at least until (and including) the first edition of Principia Mathematica. His discussion of free variables in the first edition of Principia is strikingly similar, in a particular aspect, to the one in Principles.

In discrete topology optimization, material state is either solid or void and there is no topology uncertainty caused by intermediate material state. A common problem of the current discrete topology optimization is that boundaries are unsmooth. Unsmooth boundaries are caused by the corners in topology solutions. Although outer corner cutting and inner corner filling strategy can mitigate corners, it cannot eliminate them. 90-degree corners are usually mitigated to 135-degree corners under the corner handling strategy. The existence of corners in topology solutions is because of the subdivision model. If regular triangles are used to subdivide a design domain, corners are inevitable in topology solutions. To eradicate corner from any topology solution, an innovative subdivision model is introduced in this paper for discrete topology optimization of compliant mechanisms. A design domain is discretized into quadrilateral design cells and every quadrilateral design cell is further subdivided into special triangular analysis cells that have a curved hypotenuse. With the presented subdivision model, all boundaries are smooth in any topology solution. Two discrete topology optimization examples of compliant mechanisms are solved based on the proposed subdivision approach.Copyright © 2013 by ASME

The author has published several papers which are may be hard to understand and stand as a testament to the discovery of 1:3 mathematics which is an absolute accommodation of numbers, prime numbers placement , and precise angles. The author has published a continuous Prime sieve at the divergence of (5/6, 1/6) and (1/3 and 2/3), a final attempt is made here to communicate the simple mathematics of the finite universe of mathematics to the dead current mathematics, that in the authors opinion is dead to the absolute reality of mathematics, as a form of approximate mathematics and has offered this manuscript for review at any mathematical venue. This discussion is about 1:3 Pythagoras (√1+√9=√10 hypotenuse) and points to the clear mathematical relationship between 19 and 3^2. The span of the divergence at 1:3 is clearly2+ 1/ (3^2) from the base of 2. Current mathematics is currently operating in the approximation of their myriad current theory, and they seem content.

It is common to need to estimate the frequency response of a system from observed input-output data. This paper uses integral constraints to characterise the undermodelling-induced errors involved in solving this problem via parametric least-squares methods. This is achieved by exploiting the Hilbert-space structure inherent in the least-squares solution in order to provide a geometric interpretation of the nature of frequency-domain errors. The result is that an intuitive process can be applied in which, for a given data collection method and model structure, one identifies the sides of a right triangle, and then, by noting the hypotenuse to be the longest side, integral constraints on the magnitude estimation error are obtained. By also noting that the triangle sides both lie in a particular plane, integral constraints on phase estimation error are derived. This geometric approach is in contrast to earlier work in this area, which has relied on algebraic manipulation.

Mapping of the two-dimensional isosceles triangle billiard onto the circular one-dimensional motion of two mass points is described. The singular nature of trajectories directly incident on an acute vertex is discussed in the framework of the present mapping. For an obtuse-angled isosceles triangle, dynamical equations in two-particle space applied to an orbit along a hypotenuse incident on the obtuse vertex suggests irreversible behavior at the critical angle ψ=2π/3. Thus it is found that the nonsingular motion of a finite smooth-walled disk on this trajectory exhibits irreversibility. A finite spherical smooth-walled particle moving in a uniform right cylinder whose cross section includes this critical vertex angle likewise exhibits irreversibility. Each such example comprises an irreversible orbit for a single-particle Hamiltonian.

This paper describes the design, fabrication, and testing of two hollow prisms. One is a prism with a grating glued to its hypotenuse. This ensemble, prism + grating, is called a grism. It can be applied as an on-axis tunable spectrometer. The other hollow prism is a constant deviation one called a Pellin-Broca. It can be used as a tunable dispersive element in a spectrometer with no moving parts. The application of prisms as temperature sensors is shown.

Cellular structured negative index metamaterials in the form of a right triangular prism have often been tested by observing the refraction of a beam across the prism hypotenuse which is serrated in order to conform to the cell walls. We show that not only can this negative index effect be obtained from a homogeneous dielectric prism having a positive index of refraction, but in addition, for sampling at the walls of the cellular structure, the phase in the material has the illusory appearance of moving in a negative direction. Although many previous reports relied on refraction direction and phase velocity of prism structures to verify negative index design, our investigation indicates that to unambiguously demonstrate material negativity additional empirical evidence is required.

Volume 10 is named Algorithms for Appearances [形法 keihō] as the first of five volumes on geometry. It consists of four chapters. Chapter 1 Algorithms for Squares [方法 hōhō] deals with squares. Chapter 2 is entitled Algorithms for Rectangles [直法 chokuhō]. The longer side of a rectangle [直 choku] is called [縦 tate] or length [長 chō] and the shorter side [横 yoko] or [平 hei] or width [闊 katsu]. Their product is the area [積 seki]. In the following Chapter 3 Algorithms for Right Triangles [勾股法 kōko hō] this is used to prove the Pythagorean theorem. The shorter leg hook [勾 kō] and the longer leg leg [股 ko] of a right triangle stand for the right triangle itself. The hypothenuse is called chord [弦 gen]. There are no formal treatments of proportions as in Euclid’s Elements but it is remarked that rectangles or right triangle with a fixed ratio of heights to widths make a straight line. There are two appendices to Chapter 3. The first one deals with Pythagorean triplets, i. e. the integral solutions of x 2 +y 2 =z 2. The second appendix is a brief introduction to the methods of survey described in Sea Islands Mathematics [海島算經 Haidao Suanjing] by Liu Hui. In the last Chapter 4 Algorothms for Polygons [斜法 shahō] the authors develop a general theory of triangles [三斜 sansha], quadrilaterals [四 斜 shisha], pentagons [五斜] and hexagons [六斜] with the use of their elimination theory

Donetsk Institute for Physics and Technology, Donetsk 83 114, Ukraine(Dated: March 13, 2012)Quantum state transfer in a triangular domain of a two-dimensional, equally-spaced, spin lat-tice with non-homogeneous nearest-neighbor couplings is analyzed. An exact solution of the one-excitation dynamics is provided in terms of 2-variable Krawtchouk orthogonal polynomials thathave been recently defined. The probability amplitude for an excitation to transit from one site toanother is given. For some values of the parameters, perfect transfer is shown to take place fromthe apex of the lattice to the boundary hypotenuse.

This is the post-print version of the Article. The official published version can be accessed from the links below - Copyright @ 2012 Elsevier An analytic mode-matching method suitable for the solution of problems involving scattering in three-dimensional waveguides with flexible walls is presented. Prerequisite to the development of such methods is knowledge of closed form analytic expressions for the natural fluid–structure coupled waveforms that propagate in each duct section and the corresponding orthogonality relations. In this article recent theory [J.B. Lawrie, Orthogonality relations for fluid–structural waves in a 3-D rectangular duct with flexible walls, Proc. R. Soc. A. 465 (2009) 2347–2367] is extended to construct the non-separable eigenfunctions for acoustic propagation in a three-dimensional rectangular duct with four flexible walls. For the special case in which the duct cross-section is square, the symmetrical nature of the eigenfunctions enables the eigenmodes for a right-angled, isosceles triangular duct with flexible hypotenuse to be deduced. The partial orthogonality relation together with other important properties of the triangular modes are discussed. A mode-matching solution to the scattering of a fluid–structure coupled wave at the junction of two identical semi-infinite ducts of triangular cross-section is demonstrated for two different sets of “junction” conditions.

What first comes to mind? For you is 2 a number? Is it a radical? Is it a measure? Is it the length of the diagonal in a unit square? Is it the length of the hypotenuse of a unit-length right triangle? Maybe it is the secant of 45o? How you imagine 2 is dependent on the experiences you have had with it. The more limited our experiences with a mathematical object the more limited our understanding of it. The broader our experiences with the object the greater our understanding of it. 2 is an interesting object. What is it? What does it look like? How does it behave? What might a mathematics teacher do to provide opportunities for learner activity that broadens the learner’s relationship with 2, making possible greater understanding of it and greater understanding in general of radicals?

Nonlinear preferences models are theories for choice under risk which have weakened in one way or another the Independence Axiom of Expected Utility. In this paper we present an experiment designed to discriminate between some basic ideas in this literature. We tested the Betweenness axiom, a weaker form of Independence, against an alternative which predicts that indifference curves in the probabilities simplex are convex along the lower edge and concave along the hypotenuse. We also checked whether preferences among single-stage lotteries, as those revealed by our test of Betweenness, should be extended to preferences among two-stage lottery via the classical Reduction of Compound Lottery axiom or via the Certainty Equivalent Mechanism. Copyright 1994 by Royal Economic Society.

The bhujājyā and koṭijyā of the difference between the longitudes of the Sun and the Moon is multiplied by half of the koṭijyā of the difference between the longitudes of the Sun and the mandocca of the Moon and divided by the trijyā. Of these two, the one obtained from the koṭijyā is applied to 10 times the hypotenuse of the Moon in minutes, positively or negatively depending on whether the ayanas are the same or differenṭ The square root of the sum of the squares of that and the bāhuphala is the distance of separation between theMoon and the Earth (in yojanas).Whatever is obtained from the bhujājyā [or bāhuphala] has to be multiplied by trijyā and divided by that [dvit¯iya-sphuṭa-karṇa]. The result obtained must be applied positively or negatively to the true (manda-corrected) Moon, depending upon whether the longitude of Sun minus the mandocca of the Moon lies within the six signs beginning with Karka or Mṙga in a bright fortnight, and reversely in a dark fortnight.

Based on the analysis of inertia weight of the standard PSO, a PSO method is described with self-adaptive stochastic inertia weight based on diversity of individual location and fitness value. Position and fitness value correspond to the axis, based on the difference of location and fitness value from the generation and the current generation to construct a right triangle. It is to modify the inertia weight by change of hypotenuse. By the experiments of six functions, compared with standard PSO and algorithm from the literature, experimental result show that the new algorithm cost lower running time and faster convergence, improved the overall performance.

Well known relationships exist between thicknesses and indices of a stack of optical layers which permit the definition of very good polarizers of light. One of the most famous types of polarizer is the MacNeille cube. In this case the hypotenuse face of an isosceles right angled prism is coated with a polarizing deposit and then cemented to the hypotenuse face of an identical prism. This polarizer can be used over a large spectral range located in the s vibration stop band. Unfortunately, the insertion losses for thep vibration are large when the angular field is larger than +/-2 degrees . Using very simple optical considerations, we design suitable coating structures that allow the use of cube polarizers over a wide angular range, typically +/-10 degrees in air, when T(p)0.97 and T(s)/= 10(-3). However, the spectral range is reduced. Diagrams are given to calculate the thickness of the layers according to the substrate and the indices of the evaporated materials. The prism angle is also determined to have a symmetric angular field in air. Such polarizers are suitable for semiconductor lasers because they can be used without a collimating lens in spite of their large divergence. Good optical characteristics up to T(p) approximately 0.95 and T(s) approximately 10(-4) over the range of +/-5 degrees have been measured for these polarizers manufactured in the Laboratoires de Marcoussis.

For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent exceeding 1/2. We find infinitely many Pythagorean triples in O whose hypotenuse, area, and product of side lengths have few prime factors, where "few" is explicitly quantified. We also compute the asymptotic of the number of such Pythagorean triples of norm at most T, up to bounded constants., 37 pages, 8 figures, 1 table