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A classical example of a word equation is the commutation equation =, in which is an unknown and is a constant word. It is well-known [ 4 ] that the solutions of the commutation equation are exactly those morphisms h {\displaystyle h} mapping x {\displaystyle x} to some power of w {\displaystyle w} .
If each unknown appears at most twice, then a word equation is called quadratic; in a quadratic word equation the graph obtained by repeatedly applying Levi's lemma is finite, so it is decidable if a quadratic word equation has a solution. [2] A more general method for solving word equations is Makanin's algorithm. [3] [4]
In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence f ( n ) {\displaystyle f(n)} of non-negative real numbers , the series ∑ n = 1 ∞ f ( n ) {\textstyle \sum \limits _{n=1}^{\infty }f(n)} converges if and only if the "condensed ...
The right-hand side of this equation minus ( + ) = is the approximation by the trapezoid rule of the integral (! ) − 1 2 ln n ≈ ∫ 1 n ln x d x = n ln n − n + 1 , {\displaystyle \ln(n!)-{\tfrac {1}{2}}\ln n\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1,}
The logarithmic decrement can be obtained e.g. as ln(x 1 /x 3).Logarithmic decrement, , is used to find the damping ratio of an underdamped system in the time domain.. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5; it does not apply at all for a damping ratio greater than 1.0 because the system is overdamped.
is a function space.Its elements are the essentially bounded measurable functions. [2]More precisely, is defined based on an underlying measure space, (,,). Start with the set of all measurable functions from to which are essentially bounded, that is, bounded except on a set of measure zero.
In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Statement [ edit ]
LHS – left-hand side of an equation. Li – offset logarithmic integral function. li – logarithmic integral function or linearly independent. lim – limit of a sequence, or of a function. lim inf – limit inferior. lim sup – limit superior. LLN – law of large numbers. ln – natural logarithm, log e. lnp1 – natural logarithm plus 1 ...