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Numerical analysis
Published in Alan Jeffrey, Mathematics, 2004
A problem that occurs frequently in mathematics, and which is of fundamental importance, is the numerical computation of the zeros of an nth-degree polynomial P(x), where P(x)=a0xn+a1xn−1+…+an. The so-called zeros of a polynomial P(x) are those values of x which make P(x) = 0 and so, expressed differently, they are the roots of the equation P(x) = 0. A similar problem that is also of importance is the numerical computation of the roots of an equation that is not algebraic but involves trigonometric, hyperbolic, logarithmic and other mathematical functions. These equations are called transcendental equations, and a typical example is sinx−coshx+1=0.
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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.
Analytical Solutions for Laminated Plates
Published in Manoj Kumar Buragohain, Composite Structures, 2017
The auxiliary equation of the fourth-order ordinary differential equation (Equation 7.164) is D11(mπa)4−2(D12+2D66)(mπa)2λ2+D22λ4=0 and the homogeneous solution is Wmh(y)=Ceλy where λ is the root of the auxiliary equation (Equation 7.166). The nature of the roots leads to different possible cases—roots are real and unequal, roots are real and equal and roots are complex. Our objective here is only to demonstrate the method of solution and we shall restrict our discussion only to the first case, that is, roots are real and unequal, as follows.
A complex Fourier series solution for free vibration of arbitrary straight-sided quadrilateral laminates with variable angle tows
Published in Mechanics of Advanced Materials and Structures, 2022
Guojun Nie, Han Hu, Zheng Zhong, Xiaodong Chen
According to Eqs. (31) and (32), we can obtain the approximate values of the characteristic roots, and And the four characteristic roots (or ) may be two pairs of conjugate complex number or a pair of real number and a pair of conjugate complex number. In view of different types of roots, Eq. (28) can be expressed as where the expressions of and are given as follows according to the types of roots.
Combine effects of square root functional response and prey refuge on predator–prey dynamics
Published in International Journal of Modelling and Simulation, 2021
Hence, the theorem. Theorem 5. The interior equilibrium point E* is locally asymptotically stable if .Proof. The Jacobian matrix of system (1) at E* is given by Then the characteristic equation of is given by where The roots of the Equation (6) are negative real number or complex number with negative real part if and holds. So, our proposed system is locally stable around the interior equilibrium point if .
Control of non-minimum phase systems with dead time: a fractional system viewpoint
Published in International Journal of Systems Science, 2020
Shaival Hemant Nagarsheth, Shambhu Nath Sharma
The stability of the fractional quasi-characteristic polynomial (44) can be rephrased in the sense of the stability of the associated natural degree quasi-characteristic polynomial (45). The location of the roots of the fractional quasi-characteristic polynomial is mapped in the plane, where . The left half-plane location of the roots of the fractional quasi-characteristic polynomial implies the condition for the roots of the associated natural degree quasi-characteristic polynomial. Note that where Three stability conditions of Theorem 3.3 are the direct consequence of the above mapping relation. Since they are straightforward, we omit the proof.