24.128.97.194: /* Hydraulics */

2014-02-03T05:17:05Z

Hydraulics

← Older revision		Revision as of 07:17, 3 February 2014
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	~~My name~~ is ~~Frances~~ (~~22 years old~~) and ~~my hobbies~~ are ~~Freerunning and Basket Weaving~~.<br><br>~~Also visit my blog ::~~ [~~http~~:/~~/www~~.~~palpakchurch~~.~~org~~/~~?document_srl=2429190 wordpress backup~~]		In [[mathematics]], a [[system of linear equations]] or a [[system of polynomial equations]] is considered '''underdetermined''' if there are fewer equations than unknowns (in contrast to an [[overdetermined system]], where there are more equations than unknowns). The terminology can be explained using the concept of constraint counting. Each [[Variable (mathematics)\|unknown]] can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom.

			Therefore the critical case (between overdetermined and underdetermined) occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint removing a degree of freedom. The '''underdetermined''' case, by contrast, occurs when the system has been underconstrained—that is, when the unknowns outnumber the equations.

			== Solutions of undetermined systems ==

			An underdetermined linear system has either no solution or infinitely many solutions.

			For example
			:<math>\begin{align}
			x+y+z&=1\\
			x+y+z&=0
			\end{align}</math>
			is an underdetermined system without any solution; any system of equations having no solution is said to be [[System of linear equations#Consistency\|inconsistent]]. On the other hand, the system

			:<math>\begin{align}
			x+y+z&=1\\
			x+y+2z&=3
			\end{align}</math>

			is consistent and has an infinitude of solutions, such as (''x'', ''y'', ''z'') = (1, −2 ,2), (2 , −3 ,2), and (3, −4 ,2).

			More specifically, according to the [[Rouché–Capelli theorem]], any system of linear equations (underdetermined or otherwise) is inconsistent if the [[rank (linear algebra)\|rank]] of the [[augmented matrix]] is greater than the rank of the [[coefficient matrix]]. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution; since in an underdetermined system this rank is necessarily less than the number of unknowns, there are indeed an infinitude of solutions, with the general solution having ''k'' free parameters where ''k'' is the difference between the number of variables and the rank.

			There are [[algorithm]]s to decide if an underdetermined system has solutions, and if it has any, to express all solutions as linear functions of ''k'' of the variables (same ''k'' as above). The simplest one is [[Gaussian elimination]]. See [[System of linear equations]] for more details.

			== Underdetermined polynomial systems ==
			The main property of linear underdetermined systems, of having either no solution or infinitely many, extends to [[systems of polynomial equations]] in the following way.

			A system of polynomial equations which has fewer equations than unknowns is said to be '''underdetermined'''. It has either infinitely many complex solutions (or, more generally, solutions in an [[algebraically closed field]]) or is inconsistent. It is inconsistent if and only if {{nowrap\|1=0 = 1}} is a linear combination (with polynomial coefficients) of the equations (this is [[Hilbert's Nullstellensatz]]). If an underdetermined system of ''t'' equations in ''n'' variables (''t'' < ''n'') has solutions, then the set of all complex solutions is an [[algebraic set]] of [[dimension of an algebraic variety\|dimension]] at least {{nowrap\|''n'' - ''t''}}. If the underdetermined system is chosen at random the dimension is equal to {{nowrap\|''n'' - ''t''}} with probability one.

			==Underdetermined systems with other constraints and in optimization problems==

			In general, an underdetermined system of linear equations has an infinite number of solutions, if any. However, in [[mathematical optimization\|optimization problems]] that are subject to linear equality constraints, only one of the solutions is relevant, namely the one giving the highest or lowest value of an objective function.

			Some problems specify that one or more of the variables are constrained to take on integer values. An integer constraint leads to [[integer programming]] and [[Diophantine equations]] problems, which may have only a finite number of solutions.

			Another kind of constraint, which appears in [[coding theory]], especially in [[error correcting codes]] and [[signal processing]] (for example [[compressed sensing]]), consists in an upper bound on the number of variables which may be different from zero. In error correcting codes, this bound corresponds to the maximal number of errors that may be corrected simultaneously.

			==See also==

			*[[Overdetermined system]]
			*[[Regularization (mathematics)]]

			==References==

			<references/>

			[[Category:Linear algebra]]
			[[Category:Equations]]

en>Solomonfromfinland: Added Category:Acceleration

2012-05-31T22:02:56Z

Added Category:Acceleration

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My name is Frances (22 years old) and my hobbies are Freerunning and Basket Weaving.<br><br>Also visit my blog :: [http://www.palpakchurch.org/?document_srl=2429190 wordpress backup]

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24.128.97.194: /* Hydraulics */

en>Solomonfromfinland: Added Category:Acceleration