Euclidean Space

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euclidean space

[yü′klid·ē·ən ′spās]
(mathematics)
A space consisting of all ordered sets (x1, …, xn ) of n numbers with the distance between (x1, …, xn ) and (y1, …, yn ) being given by the number n is called the dimension of the space.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Euclidean Space

 

in mathematics, a space whose properties are described by the axioms of Euclidean geometry. In a more general sense, a Euclidean space is an n-dimensional vector space, into which several special Cartesian coordinates can be introduced so that its metric is defined in the following manner: If point M has the coordinates (x1x2, …, xn and point M* has the coordinates (x1*, x2*, …, xn*), then the distance between these points is

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
The updated results by using GA show that only the error of first-order natural frequency is over 10%, and the updated results have been improved a lot (its error 2-norm is smallest).
In the next three sections, we apply QSip in turn to problems with g equal to the 1-norm, the group LASSO (i.e., sum of 2-norm functions), and total variation.
where [parallel]{m}[parallel] is the 2-norm of vector {m}.
At long last, the assaulter checks whether the 2-Norm of the redesigned tis short of what [[tau].sub.a] or not.
A trivial example of an 2-normed space is X = [R.sup.2], equipped with the Euclidean 2-norm [[parallel][x.sub.1],[x.sub.2][parallel].sub.E] = the volume of the parallellogram spanned by the vectors [x.sub.1], [x.sub.2] which may be given expicitly by the formula
2-norm distance = [n.summation over (i=1)][[absolute value of [x.sub.i] - [y.sub.i]]
We had to face the fact that the 2-norm Cartesian distance does not correctly describe the situation because the airplane cannot move on the shortest path.
The concept of a linear 2-normed space was introduced as a natural 2-metric analogue of that of a normed space.
Without considering the transient responses in the first 0.03 second, the designed nozzle pressure virtual sensor achieved, on average, a normalized 2-norm of the estimation error at 6.4% and an infinite-norm at 14.5%, which represent 39% and 25% improvement over corresponding performance measures of model prediction.
The denominator consists simply of "the 2-norm of X" times "the 2-norm of [upsilon]." If [REL.sub.2] is rewritten so as to keep the scaling factor entirely in the denominator, it turns out that Lewin is in fact himself finding the cosine of some angle, as his function can be generalized to cos [theta]: [11]
where [[(I - A)]] is the 2-norm of the (I - A) matrix, [[[(I - A).sup.-1]]] is the 2-norm of the [(I - A).sup.-1] (the Leontief inverse matrix), and [[Delta].sub.max] (I - A) is the largest singular value and [[Delta].sub.min] (I - A) is the smallest singular value of the (I - A) matrix, respectively (Golub and Van Loan 1983).(6)
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