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Let R be an effective commutative ring.. There is an algorithm for testing if an element a is a zero divisor: this amounts to solving the linear equation ax = 0.; There is an algorithm for testing if an element a is a unit, and if it is, computing its inverse: this amounts to solving the linear equation ax = 1.
A schematic diagram of the Blasius flow profile. The streamwise velocity component () / is shown, as a function of the similarity variable .. Using scaling arguments, Ludwig Prandtl [1] argued that about half of the terms in the Navier-Stokes equations are negligible in boundary layer flows (except in a small region near the leading edge of the plate).
In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A (with 1), the projective line P 1 (A) over A consists of points identified by projective coordinates. Let A × be the group of units of A; pairs (a, b) and (c, d) from A × A are related when there is a u in A × such ...
In mathematics, the method of matched asymptotic expansions [1] is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used when solving singularly perturbed differential equations. It involves finding several different approximate solutions, each of which is valid (i ...
For an arbitrary system of ODEs, a set of solutions (), …, are said to be linearly-independent if: + … + = is satisfied only for = … = =.A second-order differential equation ¨ = (,, ˙) may be converted into a system of first order linear differential equations by defining = ˙, which gives us the first-order system:
Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.
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The solution for is not unique since the arctangent function is multivalued, however it is required that the solution for be continuous over the angles of interest. For example, the following explicit solution using the atan2 ( y , x ) function will be valid for − π < γ 1 < π {\displaystyle -\pi <\gamma _{1}<\pi } :
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