Right Triangle Research Articles

A new approach to the Dirac equation

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.

Modified humeral supracondylar step-cut osteotomy for cubitus varus in children

Objective To investigate the feasibility and clinical effectiveness of modified humeral supracondylar step-cut osteotomy(M-HSSCO) for cubitus varus in children. Methods Between June 2016 and June 2017, the study included 25 children (20 boys and 5 girls) presenting cubitus varus at the age of (8.5±0.3) years old (6-11 years old) in the Children′s Hospital of Nanjing Medical University.M-HSSCO was used in the osteotomy operation.The initial osteotomy was performed as a right triangle osteotomy.The inferior margin of this right triangle was outlined parallel to the joint line 0.5-1.0 cm above the olecranon fossa.Then, they were incised from the medial end of this parallel line, which made an angle between the first and second lines equal to the desired corrective humerus elbow-wrist angle.Next, from the lateral end of the second line, the third line perpendicular to the first line was cut.Finally, desired right triangle was outlined and removed.The second right triangle osteotomy from proximal line of osteotomy was similar as the above.The medial cortex and periosteum was used as a hinge and rotated to make two osteotomy parts combined steadily.The Bellemore evaluation system was used to evaluate the difference in carrying angle degree between the affected side and normal side after osteotomy, and the different degree of flexion and extension in the elbow joint between preoperation and post-operation.Residual varus deformity and other complications were also evaluated. Results All patients were followed up for 6-18 months (mean 12 months). The preoperative ipsilateral carrying angle was-(15°-30°)[-(22.20±1.40)°], the postoperative was 5°-10°[(7.60±0.47)°] and the final follow-up was 5°-9°[(7.30±0.40)°]. There were statistically significant differences in the flexion angles before surgery and after surgery (t=23.34, P=0.000). There were statistically significant differences in the flexion angles before surgery and at final follow-up (t=23.14, P=0.000). The mean range of joint motion of the elbow was -5°-130°[(132.5±0.5)°] before surgery, 0°-125°[(123.6±1.4)°] after surgery and 0°-130°[(126.8±1.3)°] at final follow-up.The carrying angle reduced did not happen by the final follow-up.There was no statistically significant difference between the flexion angles before surgery and after surgery or by final follow-up(P>0.05). Twenty-three (92.0%) patients got an excellent result, 2 cases(18.0%) got a good result and no patient (0) got worse according to the criteria described by Bellemore.The first-class incision healing were achieved in all patients.There was no case with postoperative infection, instabilities, vascular and neural injury or any significant postoperative complications. Conclusions M-HSSCO is an effective osteotomy procedure which shows good feasibility and apparent therapeutic effect in treating cubitus varus in children. Key words: Cubitus varus; Step-cut osteotomy; Child

Predicting \u03b4PMNS, \u03b823PMNS and fermion mass ratios from flavour GUTs with CSD2

Constrained Sequential neutrino Dominance of type 2 (referred to as CSD2) is an attractive building block for flavour Grand Unified Theories (GUTs) because it predicts a non-zero leptonic mixing angle θ13PMNS, a deviation of θ23PMNS from π/4, as well as a leptonic Dirac CP phase δPMNS which is directly linked to the CP violation relevant for generating the baryon asymmetry via the leptogenesis mechanism. When embedded into GUT flavour models, these predictions are modified in a specific way, depending on which GUT operators are responsible for generating the entries of fermion Yukawa matrices. In this paper, we systematically investigate and classify the resulting predictions from supersymmetric SU(5) based flavour models by fitting the known fermion mass and mixing data, in order to provide a roadmap for future model building. Interestingly, the promising models predict the lepton Dirac CP phase δPMNS between 230° and 290°, and the quark CP phase δCKM in accordance with a right-angled unitarity triangle (αUT = 90°). Also, our model setup predicts the quantities θ23PMNS and md/ms with less uncertainty than current experimental precision, and allowing with future sensitivity to discriminate between them.

Geometrically modeled derivative feature descriptor aiding supervised shape retrieval

Recent research on shape retrieval render highly acute feature descriptors that are computationally intensive. Hence, a simple approach through a novel tessellated version of the Tetrakis Square tiling scheme for acute feature descriptor aiding supervised shape retrieval is contributed in this paper. The proposed descriptor labeled as Triangulated Second-Order Shape Derivative (TSOSD) performs feature characterization and abstraction by fusing hybrid geometrical concepts with image derivative operators. First, the mechanism tessellates the image into square tiles that are later organized as right-angled triangles. Secondly, the derivatives from the right-angled triangular neighbors interact locally using the trigonometric identities to produce an angle-based feature map. Finally, the feature descriptor is then formulated by local segmentation of the attained feature maps to produce the shape histogram. Experimental results on three standard benchmark databases demonstrate the effectiveness of the proposed approach, particularly rendering a consistent retrieval rate greater than 95% in comparison with the state-of-the-art methods.

Towards Determination of Inductive Resistances of Single-Core Cables with Voltage of up to 1 kV Insulated by Cross-Linked Polyethylene

To calculate short-circuit currents, voltage losses and reactive power in electrical networks, information on inductive resistances of cable lines is required. In the technical literature there are no exact parameters of single-core cables with voltage of up to 1 kV insulated by crosslinked polyethylene. This results in inaccuracies in the calculations of modes of electrical networks. The article considers the issues of determining the inductance and inductive resistance of power lines up with voltage of up to 1 kV made of single-core cables and insulated by crosslinked polyethylene. The analysis of formulas for determining the inductance given in different literature sources was fulfilled, that demonstrated a significant difference in the numerical values of the obtained calculation results. The formula that provides more reliable calculations of the inductance of the cable lines was identified. The influence of the permissible increase in the thickness of the insulation and the sheath of a single-core cable with voltage of up to 1 kV covered with cross-linked polyethylene on the inductive resistance was determined. The inductance and inductive resistance of single-core cables were calculated when the cables were arranged along the vertices of an equilateral and right-angle triangle, as well as in one plane. It was shown that the specific inductances and inductive resistances are 1.7–1.8 times greater in the plane with the distance between adjacent cables equal to the outer diameter of the cable than in the location is close to the triangle. When the cross sections of conductive cores are large and installations are multi-ampere, there is a surface effect and proximity effect, due to which the magnetic field intensity inside the core and its inductance decrease. Inductivities and inductive resistances of singlecore cables with voltage up to 1 kV were determined taking into account the influence of the surface effect and the proximity effect. The calculations that we performed have shown that the nonconsideration of the above-mentioned effects leads to significant errors in the determination of inductive resistances. The influence of one-core cables of armor made of steel tapes on inductive resistances is considered. An illustrative example shows that the inductive resistance of an armored cable with a cross-section area of the current-conducting core equal to 800 mm2 is about twice as large as the unarmored one. The use of single-core cables in three-phase electrical networks, armored with steel straps, ought to be prohibited by regulatory documents, since such cables dramatically worsen the efficiency of power supply systems.

SEPARATOR PROTOTYPE FOR FOUR TYPES OF OBJECTS BASED ON CANNY EDGE DETECTION METHOD

The development of current technology makes easier for human beings in overcoming work. One of them is a selection of objects on industry. On a large-scale industry, the process of selection objects is done continuously, and it requires a high degree of accuracy. The work goes on for 24 hours unceasingly, which is hard for human. Therefore, it needs an instrument that works automatically to ease the selection process of objects. In this research, a prototype that separates objects based on shape is conducted. Shape recognition process utilizes canny edge detection method with microcontroller ATmega16 as a whole system controller. According to the testing result, the prototype can recognize and separate objects a circle, square, rectangular, and equilateral triangles. The ratio roundness (metric) values for circle, square, rectangular, and equilateral triangle are 1.00-0.90, 0.90-0.83, 0.81-0.70, and <0.70, respectively. Moreover, the level of success for separating circle object, square object, rectangular object, equilateral triangle object, and whole system are 100%, 80%, 100%, 100%, and 85%, severally.

Tetrakis square tiling-based triangulated feature descriptor aiding shape retrieval

Ongoing research addressing shape retrieval render highly acute feature descriptors that are computationally intensive. Hence, a simple approach through a novel tessellated version of the Tetrakis square tiling for acute feature characterization aiding shape retrieval is contributed in this paper. The proposed descriptor, named Triangulated Feature Descriptor (TFD), performs feature characterization and abstraction by fusing hybrid geometrical concepts. The mechanism initially tiles the image into square regions that are later tessellated into right-angled triangles. The right-angled triangular neighbors interact locally using the trigonometric identities to produce an angle-based feature map representing an image. The attained feature maps are locally segmented to finally produce a shape histogram. Since, the intended descriptor is modeled based on intrinsic angular interactions between the local neighbors the resulting descriptor is highly invariant to diverse affine transformations. This characteristic is examined by performing rigorous experiments on MPEG-7, TARI-1000, Articulated and PHOS datasets. Competitive analysis reveals a consistent retrieval rate greater than 85% achieved by TFD with increased performance scores.

Wideband CPW-Fed Flexible Bow-Tie Slot Antenna for WLAN/WiMax Systems

A bow-tie antenna based on a slot configuration in a single metal sheet on top of a very thin flexible substrate is introduced. The antenna is constructed from two slotted right-angle triangles fed by a coplanar waveguide transmission line. The topology is very simple and extremely easy to tune in order to reach the proper characteristics after mounting on a supporting structure. A prototype is designed, fabricated, and characterized experimentally. The tunability is proven by considering both the version in free space and the version for use on a brick wall. Measurements demonstrate good agreement with simulations. Both versions cover the wireless local area network (2.4 and 3.65 GHz) and WiMax (2.3, 2.5, and 3.5 GHz) spectra, with an overall impedance bandwidth of 1.79 GHz (57.7%) and 1.46 GHz (49.7%), respectively. The radiation of the antenna is bidirectional with maximum gains of 6.30 and 5.09 dBi for the free space and brick wall versions, respectively.

New moduli for Banach spaces

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 Banaś.

Hermitian matrices: Spectral coupling, plane geometry/trigonometry and optimisation

The paper presents the information processing that can be performed by a general hermitian matrix when two of its distinct eigenvalues are coupled, such as λ<λ′, instead of considering only one eigenvalue as traditional spectral theory does. Setting a=λ+λ′2≠0 and e=λ′−λ2>0, the information is delivered in geometric form, both metric and trigonometric, associated with various right-angled triangles exhibiting optimality properties quantified as ratios or product of |a| and e. The potential optimisation has a triple nature which offers two possibilities: in the case λλ′>0 they are characterised by e|a| and |a|e and in the case λλ′<0 by |a|e and |a|e. This nature is revealed by a key generalisation to indefinite matrices over R or C of Gustafson's operator trigonometry.

Population variance in prey, diets and their macronutrient composition in an endangered marine predator, the Franciscana dolphin

Disentangling the intricacies governing dietary breadth in wild predators is important for understanding their role in structuring ecological communities and provides critical information for the management and conservation of ecologically threatened species. Here we combined dietary analysis, nutritional composition analysis of prey, literature data and nutritional geometry (right-angled mixture triangle models -RMT-) to examine the diet of the most threatened small cetacean in the western South Atlantic Ocean, the Franciscana dolphin (Pontoporia blainvillei). We applied a recently developed extension of niche theory based on the RMT to help understand the dietary strategies of this species. Our results showed that across their range the Franciscanas consumed prey with variable protein-to-lipid energy ratios (LMM, p<0.001). In an intensive study of one area, FMA IV, we found that dolphins sub-populations, which recent genetic evidence suggest should be differentiated into three management units, have diets with different protein energy and water mass compositions, but similar protein-to-lipid energy ratios. Furthermore, dolphins from the three areas mixed different combinations of prey in their diets to achieve the observed macronutrient ratios. These results suggest that the different habitats that each sub-population occupies (estuarine, north marine area and south marine) might be associated with different prey composition niches, but similar realized nutritional niches. Future priorities are to better comprehend possible geographical and long-term seasonal effects on prey consumption and dietary breadth of the different Franciscana populations to identify potential impacts (environmental and human-related), enhance the current management strategies to protect this endangered marine predator.

Graphical Methodology of Global Pollution Index for the Environmental Impact Assessment Using Two Environmental Components

One of the applied methods for environmental impact assessment is the index of global pollution (IGP) proposed by Rojanschi in 1991. This methodology enables the global estimation for the ecosystem state affected more or less by human activities. Unfortunately, Rojanschi’s method has a limitation; it can be applied only if at least three environmental components are considered. Frequently, many environmental impact assessment applications rely on analysis of only two environmental components. Therefore, this work aimed to develop a new graphical method to extend Rojanschi’s approach for the case of two environmental components. The proposed method avoids the average value of evaluation grades and uses only the graphical correspondence for calculation of the index of global pollution. A right-angle triangle graph methodology was proposed, where bases represented the values of evaluation grades. Thus, for the case of two environmental components, the index of global pollution was calculated as the relation between the ideal and real ecosystem states represented by the ratio between areas of external and enclosed right triangles. The developed graphical method was tested and validated for real case studies: the environmental impact assessment from a refinery located on the Romanian Black Sea Coast considering Air and Water environmental components and from a coal-fired thermoelectric power plant from Eastern Romania regarding Air and Soil environmental components. In this way, it was provided a reliable and faster tool to be used for the pollution characterization of human-derived chemicals for better decisions in risk management.

A Physical Proof of the Pythagorean Theorem

What proof of the Pythagorean theorem might appeal to a physics teacher? A proof that involved the notion of mass would surely be of interest. While various proofs of the Pythagorean theorem employ the circumcenter and incenter of a right-angled triangle, we are not aware of any proof that uses the triangle’s center of mass. This note details one such proof. Though far from the most elegant approach, we believe it to be novel.

New effective power terms and right-angled triangle (RAT) power theory

This article presents two new electrical power terms called effective active (Pef) power term and effective reactive (Qef) power term. These two terms, can approve the orthogonality relationship between fundamental and distorted components.At the same time, these terms are necessary to complete the form of the right-angled power triangle which is compulsory condition to calculate the total apparent (St) in the non-sinusoidal systems.This paper also, offers a new definition for the total apparent power St or can be called effective apparent power (Sef) in non-sinusoidal situation in order to avoid the misconception of the old definitions. Moreover, it shows a new power diagram represents all power components in one diagram consists of six right-angled triangles called the right-angled triangle (RAT) diagram.

Dalle sagome alle figure geometriche. Un itinerario didattico sui quadrilateri

This paper is addressed to primary school teachers, in order to provide an example of a didactic path which, casually born and developed through simple activities, appeared rich of suggestions and potentiality. The didactic itinerary, experimented with 9-10 years old children, is focused on the manipulation of coloured templates, with the shape of particular right-angled triangles, and on their juxtaposition to represent other geometrical shapes, mainly triangles and quadrilaterals. After the description of the theoretical background, I illustrate its phases, with comments on the students’ tasks and the related results.

Predictive mathematical modeling for EC50 calculation of antioxidant activity and antibacterial ability of Thai bee products

Antioxidant activities of bee products from Thailand (honey, bee pollen and propolis) via the 2,2-diphenyl-1-picrylhydrazyl (DPPH) and 2,2’-azinobis-(3-ethylbenzothiazoline-6-sulphonate) (ABTS) assays were determined. The prediction of the EC50 (the half maximal effective concentration) were studied using the logistic, sigmoidal, dose response, and asymmetric 5 parameters (5P) regression models. The antimicrobial ability was tested against Staphylococcus aureus (TISTR 517), Bacillus cereus (TISTR 687), and Escherichia coli (TISTR 1261). Propolis extract with higher total phenolic content (TPC) exhibited more effective antiradical action against the DPPH and ABTS, followed by bee pollen extract and honey. All four regression models could be used to estimate the EC50 of the bee products. However, the dose-respond and 5P provide the better EC50 prediction for the bee products than the others based on the comparability of their results to those of right-angled triangle method. Thai bee products had effective antimicrobial activities on each test microorganism. The antimicrobial potency of the bee products was ranged in the order: propolis > bee pollen > honey. Results revealed that antioxidant activity and antimicrobial ability of the bee products correlated with the TPC values.

Pythagorean Triangle with (One leg of right triangle p 2 -q 2 )-2 Area/Perimeter = α 3 - β 3

Pythagorean Triangle with (One leg of right triangle p 2 -q 2 )-2 Area/Perimeter = α 3 - β 3

Proof Without Words: A Right Triangle Identity

SummaryWe use a figure to relate the semiperimeter, inradius, and circumradius of a right triangle.

Orthogonal triangulation of polygons

In this study, the question whether any convex polygon can be divided using an orthogonal grid into right-angled triangles is answered in the negative. Moreover, it is demonstrated that there exists a convex pentagon which cannot be even approximated by divisible pentagons.

Integral points in rational polygons: a numerical semigroup approach

In this paper we use an elementary approach by using numerical semigroups (specifically, those with two generators) to give a formula for the number of integral points inside a right-angled triangle with rational vertices. This is the basic case for computing the number of integral points inside a rational (not necessarily convex) polygon.