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However, they are still used on a black board for indicating relationships between formulas. , Used for marking the end of a proof and separating it from the current text. The initialism Q.E.D. or QED (Latin: quod erat demonstrandum, "as was to be shown") is often used for the same purpose, either in its upper-case form or in lower case. ☡
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form =, where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates , the equation is represented by a hyperbola ; solutions occur wherever the curve passes through a point whose x and y ...
The diagonalization process involves solving the Bogoliubov-de Gennes equations, which are a set of self-consistent equations for the coefficients , , and the pairing amplitude . A detailed discussion of the Bogoliubov-de Gennes equations can be found in de Gennes' book on superconductivity.
The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers in the form of a square root or fourth root as solutions to quadratic equations or as coefficients in an equation. [76] He was also the first to solve three non-linear simultaneous equations with three unknown variables. [77]
The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation (often in the form of square roots, cube roots, and fourth roots). [10]
Together with the equation defining K a, there are now three equations in three unknowns. When an acid is dissolved in water C A = C H = C a , the concentration of the acid, so [A] = [H]. After some further algebraic manipulation an equation in the hydrogen ion concentration may be obtained.
The above approach is another form of SGM and allows the obtaining of two independent systems of non-linear algebraic equations that can be solved by any standard iteration procedure. The less Gaussian curvature of the surface is the higher the accuracy of the plane mapping is. As a rule, the plane mapping allows to obtain a pattern with linear ...
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