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Published in Carl W. Hall, Laws and Models, 2018
Keywords: doubled, response, temperature Source: Morris, C. G. 1992. See also METABOLISM; RESPIRATION; VAN'T HOFF; WILHELMY QUADRANTS, LAWS OF, FOR A SPHERICAL RIGHT TRIANGLE; OR RULE OF SPECIES For a spherical right triangle, this defines a relationship concerning the relative sizes of its sides and angles (species). For a spherical right triangle, let C be a right angle, and let a, b, c be the sides opposite vertices A, B, C. 1. Angle A and side a are the same species, and so are B and b. 2. If side c is less than 90, then a and b are of same species. 3. If side c is greater than 90, then a and b are of different species. Any angle and the side opposite it are in the same quadrant, and when two of the sides are in the same quadrant, the third is in the first quadrant, and when two are in different quadrants, the third side is in the second. The quadrants are first, 0 to 90; second, 90 to 180; third, 180 to 270; and fourth, 270 to 360 (Fig. Q.1). Keywords: algebra, angles, quadrant, sides, triangle Sources: James, R. C. and James, G. 1968; Karush, W. 1989. QUADRATIC EQUATION OR FORMULA An equation or formula giving the roots of a quadratic equation in which the highest power of x, a variable, is 2: ax2 + bx + c = 0, a 0 where a, b, and c = real numbers x = [{–b (b2 – 4ac)1/2}/2a] Keywords: equation, quadratic, roots Sources: James, R. C. and James, G. 1968; Mandel, S. 1972. QUADRATIC RECIPROCITY, LAW OF When p and q are distinct odd primes, then (p/q) (q/p) = (–1) [(p – 1)/2][(q – 1)/2] where p/q and g/p = Legendre symbols J. Gauss gave 6 proofs of the law of quadratic reciprocity, and more than 50 proofs have been devised by others. A number of assertions by P. Fermat can be shown to follow the above law.
Solving quadratic equations
Published in John Bird, Bird's Engineering Mathematics, 2021
If the quadratic expression can be factorised this provides the simplest method of solving a quadratic equation. For example, if2x2−5x−3=0,then,by factorising:(2x+1)(x−3)=0Hence either(2x+1)=0i.e.x=−12or(x−3)=0i.e.x=3
Solving quadratic equations
Published in John Bird, Engineering Mathematics, 2017
If the quadratic expression can be factorised this provides the simplest method of solving a quadratic equation. For example, if2x2-5x-3=0,then,by factorising:(2x+1)(x-3)=0 $$ \begin{aligned}&\text{ For} \text{ example,} \text{ if}&2x^{2} - 5x - 3 = 0, \text{ then},\\&\text{ by} \text{ factorising}{:}&(2x + 1)(x - 3) = 0 \end{aligned} $$ Hence either(2x+1)=0i.e.x=-12or(x-3)=0i.e.x=3 $$ \begin{aligned}&\text{ Hence} \text{ either}&(2x + 1) = 0&\text{ i.e.} \quad&x = -{\frac{{1}}{{2}}}\\&\text{ or}&(x - 3) = 0&\text{ i.e.} \quad&x = 3 \end{aligned} $$
Evasion planning for autonomous intersection control based on an optimized conflict point control formulation
Published in Journal of Transportation Safety & Security, 2022
Di Kang, Zhexian Li, Michael W. Levin
Kinematics with constant acceleration lends itself to quadratic equations. Due to the difficulties in solving mixed-integer quadratically-constrained programs, we choose assumptions and overestimate the safety buffer to obtain an MILP. One major assumption in the MILP, but not the simulation, is that vehicle velocity at the initial conflict point is zero. However, when implementing and simulating the extended AIM+, this assumption can be relaxed as shown in Figure 3. Speeds only need to be maintained within the intersection to avoid collisions. Farther back on the incoming link, the speed can be adjusted without safety issues because car-following is used to maintain safe following distances. Consequently, although the MILP assumes that the initial speed is 0, we adjust the speed to avoid requiring all vehicles slow to a stop before entering the intersection. Figure 3a and 3b depict the velocity change for vehicle from entering the incoming link to exiting the first conflict point in simulation and in MILP, respectively. In Figure 3, the area of D2 denotes the distance between and and it is the same in Figure 3a and b. The area of D1 represents the length of the incoming link plus the area of D2.