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Structures of Inquiry and Corpus-Relevant Skills
Published in Stephen Hester, David Francis, Eric Livingston, Ethnographies of Reason, 2016
Stephen Hester, David Francis, Eric Livingston
At one point in my studies I came upon, to me, a distinctive proof that the medians of a triangle intersect in a common point. A median is the line segment that joins a vertex of a triangle and the midpoint of the opposite side. In Figure 1, the three medians are AA′¯, BB′¯, and CC′¯. Since any two lines in the Euclidean plane are either parallel or intersect, any two medians will meet in a common point.1 A remarkable theorem of Euclidean geometry, illustrated by Figure 1, is that all three of the medians of a triangle intersect at the same point.
Fundamentals of integration
Published in Alan Jeffrey, Mathematics, 2004
Geometrical reasoning, which we will omit, shows the point G is located on any median of the triangle, one third of the way along the median measured from its base. Remember, a median of a triangle is the line drawn from a vertex to the center of the opposite side of the triangle which forms its base, and that all three medians intersect at the same point. Thus G is located at the point where the medians intersect. This result is illustrated in Fig. 7.14 for a triangle with its vertices at the points (6, 0), (0, 6) and (0, -3). In terms of Fig. 7.13(b) this corresponds to a = 6, b = 6 and –c = –3, so c = 3. Thus x¯=a/3=2 and y¯=13(b−c)=1. ■
Geometry of areas
Published in Alberto Carpinteri, Structural Mechanics– A Unified Approach, 2017
Consider the right triangle having base b and height h (Figure 2.12). As is well-known, its centroid coincides with the point of intersection of the three medians, which are at the same time axes of oblique symmetry. The moment of inertia with respect to the axis XC of the triangle MOP is equal to the moment of inertia with respect to the axis Xc of rectangle NOPQ. The latter, in fact, is obtained from the former by suppressing triangle MNC and adding triangle CQP. These two triangles are equal and arranged symmetrically with respect to the axis XC. Hence
Hyperelliptic Curve Diffie–Hellman-Based Two-Server Password-Only Authenticated Key Exchange Protocol for Edge Computing Systems
Published in IETE Journal of Research, 2021
K. Anitha Kumari, T. P. Kamatchi, R. Senthil Prabha, Bharath K. Samanthula
The proposed system presents the mandate authentication between edge machines and cloud servers for further communication. Both the servers jointly authenticate the edge devices/machines and if any one of the servers is compromised by active/passive adversary, it results in authentication failure. Proposed work applies Diffie–Hellman (DH) algorithm for efficient key exchange and HECC point mapping algorithm with the virtues of tetrahedron properties to store password in a secure manner. The protocol works in three phases, namely, initialization, registration, and authentication. During the initialization phase, curve parameters is initialized and during registration phase, password-based information is stored as tetrahedron shape-based properties – circumcenter(ω) and the angle between the medians (θ). And authentication phase is to validate the edge machines/devices based upon the credentials. To augment the speed of the protocol and security, in the proposed methodology, HECC encryption technique is used along with DH key exchange and tetrahedron properties. HECC produces smaller keys when compared to ECC, and it is applicable for resource-constrained environments that require maximum security with less key size [29]. It is an unconditionally secure methodology in NP execution with g ≥ 2, where genus g with value 1 is an elliptic curve. The HECC encryption scheme is proved to be secure under discrete logarithm problem. In addition, it is a known fact that hyperelliptic curve with genus g ≥ 1is supersingular curve, and every supersingular curve is secure against quantum attacks.
Magic of centroids
Published in International Journal of Mathematical Education in Science and Technology, 2018
Daniela Ferrarello, Maria Flavia Mammana, Mario Pennisi
We call median of Q or T the segment joining a vertex with the centroid of the opposite face. The three medians of Q or T are concurrent in the centroid of Q or T that divides each median into two parts that are one (the one containing the vertex) three times the other [13,p.57; 15,p.467].
Association of maternal intake of nitrate and risk of birth defects and preterm birth: a systematic review and dose-response meta-analysis
Published in Archives of Environmental & Occupational Health, 2022
Nader Rahimi Kakavandi, Motahareh Hashemi Moosavi, Tayebeh Asadi, Morteza Abyadeh, Habib Yarizadeh, Ahmad Habibian Sezavar, Mohammad Abdollahi
The effect size of studies was reported by odd ratio (OR) and 95% confidence intervals. Risk estimations and 95% Cls that indicate the correlation between maternal intake of nitrate and potential risk of preterm birth and birth defects were calculated by the fixed-effects or random-effects model for high versus low, linear and non-linear dose-response meta-analyses using pooled ORs and their 95% Cls obtained from first and last quartiles or tertiles of each study.20,21 Also, the chi-square test was applied to find heterogeneity of data, and I2 more than 50% was considered the heterogenic.22,23 Sensitivity analysis was estimated to evaluate the influence of each research on the final results.16 Publication bias was assessed by the application of Egger's test.24 Dose-response meta-analysis between maternal dietary nitrate intake and risk of heart defects was done through Longnecker, and Greenland method using continuous ORs and CIs obtained from all quartiles or tertiles.25,26 Performing dose-response meta-analyses for other birth defects and preterm birth was impossible due to insufficient information about different maternal dietary nitrate intake. The unit of exposure (maternal dietary intake of nitrate) was defined as 0.5 mg/day. The adjusted odds ratio of different categories of maternal dietary intake of nitrate is required for this method. Results of each study were combined using a random-effect model. The median point in each category of dietary nitrate intake was determined. Moreover, we considered the mean as the same as the median if the mean was reported. Furthermore, if median or mean were not reported, the midpoint of the upper and lower bounds was used to estimate the approximate medians. If the highest category was open-ended, we considered it to have the same width as the closest category. If the lowest category was open-ended, the lower bound was determined as equal to zero.27,28 The non-linear associations of maternal dietary nitrate intake with risk of heart defects were carried out non-parametrically, in 5%, 35%, 65%, and 95% distributions; a restricted cubic spline with four nodes was used.29 A P-value for the nonlinearity of the meta-analysis was calculated by testing the null hypothesis that the coefficient of the second spline was equal to zero.16,30