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Trivial may also refer to any easy case of a proof, which for the sake of completeness cannot be ignored. For instance, proofs by mathematical induction have two parts: the "base case" which shows that the theorem is true for a particular initial value (such as n = 0 or n = 1), and the inductive step which shows that if the theorem is true for a certain value of n, then it is also true for the ...
The unique pair of values a, b satisfying the first two equations is (a, b) = (1, 1); since these values also satisfy the third equation, there do in fact exist a, b such that a times the original first equation plus b times the original second equation equals the original third equation; we conclude that the third equation is linearly ...
LiveMath is a computer algebra system available on a number of platforms including Mac OS, macOS (), Microsoft Windows, Linux (x86) and Solaris (SPARC). It is the latest release of a system that originally emerged as Theorist for the "classic" Mac in 1989, became MathView and MathPlus in 1997 after it was sold to Waterloo Maple, [1] and finally LiveMath after it was purchased by members of its ...
Solving an equation symbolically means that expressions can be used for representing the solutions. For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement.
The s-step Adams–Bashforth method has order s, while the s-step Adams–Moulton method has order + (Hairer, Nørsett & Wanner 1993, §III.2). These conditions are often formulated using the characteristic polynomials ρ ( z ) = z s + ∑ k = 0 s − 1 a k z k and σ ( z ) = ∑ k = 0 s b k z k . {\displaystyle \rho (z)=z^{s}+\sum _{k=0}^{s-1 ...
The recipe is relatively simple: one to two tablespoons of chia seeds mixed into water. While some people like to add honey and lemon to give it some flavor, others just keep it plain with water ...
If this was just five years ago, let alone 10 or 20, the prospect of 72-year-old Bill Belichick as a college football coach would have been more about a splashy hire than the promise of great success.
The Heaviside step function is an often-used step function.. A constant function is a trivial example of a step function. Then there is only one interval, =. The sign function sgn(x), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
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