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This paper deals with 1) angle trisection, 2) Bhaskara’s first proof, and 3) Pythagorean theorem. The purpose of this paper is threefold. First, to show a new, direct method of trisecting the 90deg angle using unmarked straight edge and compass; secondly, to show Bhaskara’s first proof of the Pythagorean theorem (c2 = a2 + b2) as embedded in this new, direct trisection of the 90deg angle; lastly, to show the derivation of the Pythagorean theorem from this trisection of the 90deg angle. This paper employs the direct dissection method. It concludes by presenting four points: a) the concept of trisectability as distinct from concept of constructability; b) the trisection of the 90deg angle as really a new, different method; c) Bhaskara’s first proof of the Pythagorean theorem as truly embedded in this trisection of the 90deg angle and; d) another way of deriving Pythagorean theorem from this trisection of the 90deg angle.

Penetration Twins Penetration twins of the sort seen in neptunite are composed of two individuals that are identical (ideally) and cross at an angle that is determined by the twin law and described by the Miller indices of the twin law. Comparison with figure 11 shows that the correct twin plane is (10 Graph HT ht ), not (90 Graph HT ht ). One of these planes is the true twin plane - it has the same Miller indices as the twin law; the other one, at right angles, has different and in most cases irrational indices and is referred to as a contact plane but not a twin plane. The crystallographic "law" of simple rational indices states that major planes on crystals, including both faces and twin planes, typically have Miller indices involving small integer values (usually 5 or less). [Extracted from the article]

Purpose: To carry out the transition from a cylindrical gear in which the centroids are congruent ellipses with centres of rotation in the foci, to a bevel gear on the basic of congruent spherical ellipses. Design/methodology/approach: Congruent ellipses with centres of rotation in the foci serve as centroids for the design of cylindrical gears with non-circular wheels. The article analytically shows that the analogues of ellipses on the plane - congruent spherical ellipses are the basis for the construction of the axoids of the corresponding bevel gears. An analogue of the centreto-centre distance for ellipses in the plane is the angle between the axes of rotation of conical axoids. Findings: Based on the equality of the arcs of ellipses, the dependence of the angle of rotation of one axoid on the angle of rotation of another is found. Graphs of this dependence for separate cases are given. It is shown under what conditions the axes of axoids intersect at right angle. The parametric equations of spherical ellipses and corresponding axoids are given. They were used to construct spherical ellipses and corresponding conical axoids for different cases. For gears with right angle between the axes, separate positions of the axoids with different angles of their rotation around their axes are constructed. Practical implications: Spherical ellipses are directing curves for the construction of the corresponding conical axoids. Originality/value: The paper shows that congruent spherical ellipses act as centroids for the design of axoids of bevel gears. They roll one by one without sliding, rotating around axes that intersect in the centre of the sphere. To design such gears, it is important to know the interdependence between the geometric parameters, especially for common gears with a right angle between the axes. [ABSTRACT FROM AUTHOR]

his numerical study presents a comparison between two different reverse osmosis channel configurations. The physical properties were considered in the computational model as a function of the solute mass fraction. A critical comparison was performed between double-sided membrane channel and single-sided one considering the concentration and flow distribution. Gravitational effect was implemented by introducing the inclination of double membrane geometry for the first time in the literature of reverse osmosis systems. FORTRAN in-house code was developed to resolve conservation equations (mass, momentum, and solute mass fraction) based on the finite volume method. The results of the simulation show that the water recovery factor of double-membrane arrangement is two times higher than the single membrane arrangement. Concentration polarization (CP) can be reduced by both increasing the feed Reynolds number (Re) and decreasing the Aspect Ratio (AR). Considering the cases of low flow rates (up to Re = 40) with the flow orientation in the direction of gravity inducing buoyancy effects. The influence of the inclination showed that the average permeate flux, and the water recovery are proportional to the inclination angle up to the maximum values at the right angle (vertical plane). [ABSTRACT FROM AUTHOR]

Copyright of FME Transactions is the property of University of Belgrade, Faculty of Mechanical Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A graph is fan-crossing free if it has a drawing in the plane so that each edge is crossed by independent edges, that is the crossing edges have distinct vertices. On the other hand, it is fan-crossing if the crossing edges have a common vertex, so that they form a fan. Both are prominent examples for beyond-planar graphs. Further well-known beyond-planar classes are the k -planar, k -gap-planar, quasi-planar, and right angle crossing graphs. We use the subdivision, node-to-circle expansion and path-addition operations to distinguish all these graph classes. In particular, we show that the 2-subdivision and the node-to-circle expansion of any graph is fan-crossing free, which does not hold for fan-crossing and k -(gap)-planar graphs, respectively. Thereby, we obtain graphs that are fan-crossing free and neither fan-crossing nor k -(gap)-planar. Finally, we show that some graphs have a unique fan-crossing free embedding, that there are thinned maximal fan-crossing free graphs, and that the recognition problem for fan-crossing free graphs is NP-complete. [ABSTRACT FROM AUTHOR]

The efficiency of flow energy reduction past emergent vegetation has been typically studied assuming a right angle of the vegetated corridor to the flow direction. However, in many real‐world cases the riparian zones of natural, restored, or engineered rivers and waterways, are found at an oblique angle to the flood flow direction. In the current study, the effect of vegetation angle with respect to the flow direction is investigated experimentally in an open channel rectangular flume. The experiments are conducted under a range of subcritical steady flow conditions, with varying Froude number (Fro). The vegetation cover is placed at various angles to the flow direction (90°, 45°, and 30°), for a sparse and intermediate vegetation density, defined from the ratio of spacing of each vegetation element in the cross stream direction (B), and the diameter of vegetation element (d) (B/d = 2.13 and 1.09, respectively). Detailed water surface profiles are obtained for all those cases, demonstrating a considerable backwater rise, increasing with increasing vegetation density, Froude number, and flow approach angle. The energy reduction decreased by increasing the Froude number for the perpendicular (90°) and increased for oblique vegetation (45° and 30°). For the perpendicular vegetation, the average energy reduction rate for sparse (90VS) and intermediate (90VI) vegetation densities are 25.44% and 31.44%, respectively. The range of average energy reduction for sparse vegetation at 45° (45VS) and at 30° (30VS) are 18.3–19.8% and 18.7–19.7%, respectively. Similarly, range of average energy reduction for intermediate vegetation at 45° (45VI) and at 30° (30VI) are 21.4–22% and 18.8–22%, respectively. [ABSTRACT FROM AUTHOR]

An eight-coordinate (8C) Cu(II) heterometallic complex, displaying a distorted dodecahedral structure with an [(O2)4] donor set, has been synthesized by programmable self-assembly. Two 4-(4-carboxylatophenyl)pyrazolate di-anions coordinate to protected-Pd(II) units forming an extended right angle metalloligand as a building block, which links a Cu(II) atom to four short Cu–O(carboxy) bonds [2.052 Å] and four long Cu–O(carboxy) bonds [2.766 Å] in an eight-coordinated environment forming a unique 8C heteromultimetallic complex. [ABSTRACT FROM AUTHOR]

We apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under minimal assumptions for all admissible dimensions. The general (not necessarily radial) case, but for j + k = n − 1, n odd, was studied by S. Helgason [8] and F. Gonzalez [4-5] on smooth compactly supported functions. [ABSTRACT FROM AUTHOR]

This Letter reports the observation of a rotational Doppler shift on reflected beams carrying Orbital Angular Momentum (OAM). More precisely, we study the beat frequency of two optical beams carrying OAM with opposite signs, reflected on a right angle prism. We show that the interference of the two beams leads to a daisy-like pattern that rotates at twice the rotating frequency of the prism. The rotational Doppler frequency shift is equal to the OAM topological charge change times the rotational frequency. Possible applications in the positioning and detection of rotation of objects are then discussed. [ABSTRACT FROM AUTHOR]

An analytical solution for the diffraction of plane waves by a right angle impedance wedge which is lying between double negative and positive materials is investigated. In order to satisfy the boundary conditions and shifted shadow boundary, the diffracted fields are divided to subfields. As a result, an anomalous shift of shadow boundary is observed towards to lower surface of the wedge. In addition, geometric optics fields are determined by using the geometry of the problem. Also, the behavior of the fields is investigated numerically. [ABSTRACT FROM AUTHOR]

The defect opening profile recognition is of great concern in the magnetic flux leakage (MFL) measurement technique. The detected spatial MFL signal has three components: horizontal, vertical, and normal components. Horizontal and normal component signals are commonly used to estimate the defect profile, while the vertical component has always been neglected. With the development of the high resolution and the 3-D MFL testing techniques, the vertical component signal is becoming more available. This paper analyzes the essential right-angle features of the vertical component signal, which is useful for the defect opening profile recognition. After obtaining the initial profile from the horizontal or normal component, the types of the right angle is identified from the vertical component, and the opening profile is further optimized based on these right-angle features. The opening profile recognition method is put forward in this paper to improve the accuracy of the recognition result of the defect. Both simulation and experimental tests are conducted to verify the good performance of the proposed method. Compared with the opening profiles recognized merely by the horizontal component signal, the proposed method shows better recognition results, which also validates that the vertical component signal can also be a useful information for the defect estimation. [ABSTRACT FROM AUTHOR]

In this paper, we realized to produce the propagation-invariant vector beams with square array by use of one 2D grating and right angle prism. The vector beams have the square shape with the square hollow center. The simulation and experimental results demonstrate the feasibility of this method. [ABSTRACT FROM AUTHOR]

In the article, we continue [7] the formalization of the work devoted to Tarski's geometry – the book "Metamathematische Methoden in der Geometrie" (SST for short) by W. Schwabhäuser, W. Szmielew, and A. Tarski [14], [9], [10]. We use the Mizar system to systematically formalize Chapter 8 of the SST book. We define the notion of right angle and prove some of its basic properties, a theory of intersecting lines (including orthogonality). Using the notion of perpendicular foot, we prove the existence of the midpoint (Satz 8.22), which will be used in the form of the Mizar functor (as the uniqueness can be easily shown) in Chapter 10. In the last section we give some lemmas proven by means of Otter during Tarski Formalization Project by M. Beeson (the so-called Section 8A of SST). [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider several classical and novel algorithmic problems for right-angled Artin groups, some of which are closely related to graph theoretic problems, and study their computational complexity. We study these problems with a view towards applications to cryptography. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mosaic Research is the property of Uludag University, Mosaic Research Center and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A new approach for the design of a balanced bandpass filter (BPF) is presented in this paper. Stemming from the mode degeneration properties of a square patch resonator, a complete set of resonant modes in a right-angled isosceles triangular patch resonator (RAITPR) is thoroughly investigated so as to construct a modified RAITPR with one electric wall and two magnetic walls along its three sides for filter design. This proposed resonator can support the half-TM10-mode of the RAITPR, and thus, it is named as the half-mode RAITPR here. By properly adopting the relationship between one RAITPR and two coupled half-mode RAITPRs, a second-order coupling topology is presented. Its working principle is then explained by using the field distribution of the desired resonant modes involved. A few high-order balanced filters are in final implemented and fabricated to evidently demonstrate their good differential-mode filtering selectivity and high common-mode suppression. Good agreement is achieved between the simulated and measured results for all the designed three balanced BPFs. To further improve the concerned out-of-band rejection performance, the feeding scheme of the modified structure has been discussed. [ABSTRACT FROM AUTHOR]

Starting from any given rational-sided, right triangle, for example, the (3,4,5)-triangle with area 6, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show further that the set of all such triangles of a given area is finitely generated under our geometric construction. Such areas are known as “congruent numbers” and have a rich history in which all the results in this article have been proved and far more. Yet, as far as we can tell, this seems to be the first exploration using this kind of geometric technique. [ABSTRACT FROM AUTHOR]

We consider relations in the predecessor-successor chain emerged at rotation-similarity transformation of right-angle triangle with an arbitrary leg ratio in 2-dimensional Euclidean space. Following this, we generalize structural ratios obtained earlier. We find conditions favoring to the structural invariance for above triangle in the predecessor-successor pair. Finally, we discuss image of above transformation and compare probability of folding/unfolding development for random transformation conditions. [ABSTRACT FROM AUTHOR]

Dead Theory: Derrida, Death, and the Afterlife of Theory, a collection of essays edited by Jeffrey R. Di Leo, provides a wide range of approaches interrogating the relationship between Theory and death. It considers the relationship through three groupings: the "death of Theory," "death in Theory" and "theory of death." Eleven essays explore this three-layered relationship through a variety of angles, all the while pointing to a future resurrection of Theory. [ABSTRACT FROM AUTHOR]

Helping robots understand indoor scenes has considerable value in computer vision. However, due to the diversity of indoor scenes, understanding them remains a big challenge. There are many spatial right-angles in indoor scenes. These spatial right-angles are projected into diverse 2D projections. These projections can be considered a composition of a pair of lines (line-pairs). Given the vanishing points (VPs), line segments can be assigned to 1 of 3 main orthogonal directions. The line-pairs (intersection of 2 lines), such that each of them converges to a different VP, are likely to be the projection of a spatial right-angle onto the image plane. These projections may enable us to estimate their original orientation and position in 3D scenes. In this paper, we presented a method to efficiently understand indoor scenes from a single image, without training or any knowledge of the camera’s internal calibration. Through geometric inference of line-pairs, it is possible to find these spatial right-angle projections. Then, these projections can be assigned to different clusters, and the line that lies in the neighbor-cluster helps us estimate the layout of the indoor scene. The proposed approach required no prior training. We compared the room layout estimated by our algorithm against the room box ground truth, measuring the percentage of pixels that were correctly classified. These experiments showed that our method estimated not only room layout, but also details of the indoor scene. [ABSTRACT FROM AUTHOR]

Objectives Mandibular third molar extraction is among the most commonly performed dental procedures. Fracture of the angle of mandible after third molar extraction is a rare complication of this procedure. Case Herein, we report fracture of the right angle of mandible immediately after extraction of mandibular right third molar in a 38-year old healthy female patient, which was surgically managed. A review of the relevant literature is also presented. Thirty four cases of mandibular fracture following third molar extraction are reported in the literature. Most of these cases were removed surgically. Conclusion Systematic review of papers showed that angulation of impacted third molar is one of the most related factors in following fractures of mandibular angle.e [ABSTRACT FROM AUTHOR]

The majority of components used in aeronautic, automotive or transport industries are subjected to fatigue loads. Those elements should be designed considering the fatigue life. In this paper the geometrical design of a right angle bracket has been simulated by FEM, considering the type of material, the applied load and the geometry of the pieces. The geometry of the right angle bracket has been optimized using the Taguchi's robust optimization method. As far as the authors know, there are no publications dealing with the use of the Taguchi's robust optimization method applied to the geometric design of industrial pieces having as output variable the Findley fatigue coefficient. This paper presents a method for determining the critical design parameters to extend the product life of components subject to fatigue loading. This pioneer idea is applicable to any other design or physical or mechanical application so it can be widely used as new methodology to fatigue robust design of mechanical parts and components. [ABSTRACT FROM AUTHOR]

For a finite simplicial graph Γ, let A(Γ) denote the right-angled Artin group on Γ. Recently, Kim and Koberda introduced the extension graph Γe for Γ, and established the Extension Graph Theorem: for finite simplicial graphs Γ1 and Γ2, if Γ1 embeds into Γ2e as an induced subgraph then A(Γ1) embeds into A(Γ2). In this paper, we show that the converse of this theorem does not hold for the case Γ1 is the complement of a tree and for the case Γ2 is the complement of a path graph. [ABSTRACT FROM AUTHOR]

We present a collection of simple construction problems and determine conditions for which the construction can be carried out with compass and straightedge. Since the problems reduce to finding the roots of quartic polynomials, it is not sufficient to show that the polynomials are irreducible to conclude that the roots are not constructible. Some interesting results appear as we discover general conditions that determine when the solutions are or are not constructible. [ABSTRACT FROM AUTHOR]

Abstract: Outer automorphism groups of right‐angled Artin groups, denoted Out ( A Γ ), interpolate between Out ( F n ) and GL n ( Z ). We consider several vastness properties for which Out ( F n ) behaves very differently from GL n ( Z ): virtually mapping onto all finite groups, SQ‐universality, virtually having an infinite dimensional space of homogeneous quasi‐morphisms, and not being boundedly generated. We give a necessary and sufficient condition in terms of the defining graph Γ for each of these properties to hold. Notably, the condition for all four properties is the same, meaning Out ( A Γ ) will either satisfy all four or none. In proving this result, we describe conditions on Γ that imply Out ( A Γ ) is large. Techniques used in this work are then applied to the case of McCool groups, defined as subgroups of Out ( F n ) that preserve a given family of conjugacy classes. In particular, we show that any McCool group that is not virtually abelian virtually maps onto all finite groups, is SQ‐universal, is not boundedly generated, and has a finite index subgroup whose space of homogeneous quasi‐morphisms is infinite dimensional. [ABSTRACT FROM AUTHOR]

Recently Berceanu and Iqbal proved that the growth rate of all the spherical Artin monoids is bounded above by 4. In this paper we compute the Hilbert series of the right-angled spherical Artin monoid M(Dn) and graphically prove that growth rate is bounded by 4. We also discuss its recurrence relations and other main properties. [ABSTRACT FROM AUTHOR]

Polarized Raman scattering and resonance Raman scattering spectra of Cu 2 ZnSiSe 4 crystals measured at temperature 300 and 10 K were investigated. Nine vibrational modes of A 2 symmetry, seven modes of B 2 symmetry and nine modes of B 1 symmetry were determined in Raman spectra taken at right angle configuration from the (2 1 0) crystal plane. A resonance Raman scattering with participation of 2LO, 3LO and more phonons was observed at photon energies higher than the ground state of exciton transition at low temperature. [ABSTRACT FROM AUTHOR]

The theory of quadratic equations (with real coefficients) is an important topic in the secondary school mathematics curriculum. Usually students are taught to solve a quadratic equation ax2 + bx + c = 0 (a ≠ 0) algebraically (by factorisation, completing the square, quadratic formula), graphically (by plotting the graph of the quadratic polynomial y = ax2 + bx + c to find the x-intercepts, if any), and numerically (by the bisection method or Newton-Raphson method). Less well-known is that we can indeed solve a quadratic equation geometrically (by geometric construction tools such as a ruler and compasses, R&C for short). In this article we describe this approach. A more comprehensive discussion on geometric approaches to quadratic equations can be found in [1]. We have also gained much insight from [2] to develop our methods. The tool we use is a set square rather than the more common R&C. But the methods to be presented here can also be carried out with R&C. We choose a set square because it is more convenient (one tool is used instead of two). [ABSTRACT FROM AUTHOR]