normalize

Dictionary: nor·mal·ize (nôr'mə-līz') pronunciation

v., -ized, -iz·ing, -iz·es.

v.tr.

To make normal, especially to cause to conform to a standard or norm: normalize a patient's temperature; normalizing relations with a former enemy nation.
To make (a text or language) regular and consistent, especially with respect to spelling or style.
To remove strains and reduce coarse crystalline structures in (metal), especially by heating and cooling.

v.intr.
To become or return to normal: waiting for diplomatic relations to normalize.

normalization nor'mal·i·za'tion (-mə-lĭ-zā'shən) n.
normalizer nor'mal·iz'er n.

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US Military Dictionary: normalize

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v. bring or return to a normal condition or state: Vietnam and China agreed to normalize diplomatic relations in 1991.

normalization n. normalizer n.

See the Introduction, Abbreviations and Pronunciation for further details.

Measures and Units: normalized

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mathematics Applied to numbers expressed as a mantissa multiplied by an exponential expression, e.g. 8.97~ × 10²⁰, the term applies to having the mantissa within some defined range. The above is identically 0.897~ × 10²¹, 89.7~ × 10¹⁹, 897.~ × 10¹⁸, 0.089 7~ × 10²², and any other of the unlimited equivalent variants. Clearly any non-zero number can be written such that the mantissa is within some pre-set range that spans, multiplicatively, the size of the base. For base 10, as in the above example, we can fix the range to be from 1 (inclusive) to 10 (exclusive), i.e. 1 to 9.999~; just as easily it can be 0.1 to 0.999~, but it could be 0.5 to 4.999~, for example. In computers, where a number in such two-part form is called a floating-point number, a common choice for normalization is to have the mantissa of maximal size less than 1, e.g. from 0.1 to 0.999~ for a decimal-based scheme, from 0.062 5 to 0.999~ for a hexadecimal-based scheme. This results in any multiplication of two mantissas being also less than 1, any addition being less than 2, so accommodated by just one bit of integer place. For desk-top use it is preferable to use 1 as the minimum, giving range 1 to 9.999~ in the familiar decimal context, but the recommendation of the SI system to use only multiples of 3 for an exponent, which effectively makes the SI a millesimal scheme, requires mantissas in the range 1 to 999.999~ (else 0.001 to 0.999~).

physics As applied to measure-unit systems, it refers to a system with natural units in which the base units are chosen to make selected physical constants equal to unity. The pioneer example had the speed of light, the Newtonian constant of gravitation and the elementary charge all equal to 1.
[Stoney G. J. Phil. Mag. (GB) Vol. 36, 138 (1868)] The Planck constant over 2π was substituted for elementary charge by Planck; the Planck length, etc., was the result.

Veterinary Dictionary: normalize

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To convert a set of data by, for example, converting them to logarithms or reciprocals so that their previous non-normal distribution is converted to a normal one.

Wikipedia: Centralizer and normalizer

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Normalizer redirects here. For the process of increasing audio amplitude, see Audio normalization

In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. These subgroups provide insight into the structure of G.

Definitions

The centralizer of an element a of a group G (written as C_G(a)) is the set of elements of G which commute with a; in other words, C_G(a) = {x ∈ G : xa = ax}. If H is a subgroup of G, then C_H(a) = C_G(a) ∩ H. If there is no danger of ambiguity, we can write C_G(a) as C(a). Another, less common, notation is sometimes used when there is no danger of ambiguity, namely, Z(a), which parallels the notation for the center of a group. With this latter notation, one must be careful to avoid confusion between the center of a group G, Z(G), and the centralizer of an element g in G, given by Z(g).

More generally, let S be any subset of G (not necessarily a subgroup). Then the centralizer of S in G is defined as C(S) = {x ∈ G : ∀ s ∈ S, xs = sx}. If S = {a}, then C(S) = C(a).

C(S) is a subgroup of G; since if x, y are in C(S), then xy⁻¹s = xsy⁻¹ = sxy⁻¹ for all s in S.

The center of a group G is C_G(G), usually written as Z(G). The center of a group is both normal and abelian and has many other important properties as well. We can think of the centralizer of a as the largest (in the sense of inclusion) subgroup H of G having a in its center, Z(H).

A related concept is that of the normalizer of S in G, written as N_G(S) or just N(S). The normalizer is defined as N(S) = {x ∈ G : xS = Sx}. Again, N(S) can easily be seen to be a subgroup of G. The normalizer gets its name from the fact that if S is a subgroup of G, then N(S) is the largest subgroup of G having S as a normal subgroup. The normalizer should not be confused with the normal closure.

A subgroup H of a group G is called a self-normalizing subgroup of G if N_G(H) = H.

Properties

If G is an abelian group, then the centralizer or normalizer of any subset of G is all of G; in particular, a group is abelian if and only if Z(G) = G.

If a and b are any elements of G, then a is in C(b) if and only if b is in C(a), which happens if and only if a and b commute. If S = {a} then N(S) = C(S) = C(a).

C(S) is always a normal subgroup of N(S): If c is in C(S) and n is in N(S), we have to show that n⁻¹cn is in C(S). To that end, pick s in S and let t = nsn⁻¹. Then t is in S, so therefore ct = tc. Then note that ns = tn; and n⁻¹t = sn⁻¹. So

(n⁻¹cn)s = (n⁻¹c)tn = n⁻¹(tc)n = (sn⁻¹)cn = s(n⁻¹cn)

which is what we needed.

If H is a subgroup of G, then the N/C theorem states that the factor group N(H)/C(H) is isomorphic to a subgroup of Aut(H), the automorphism group of H.

Since N_G(G) = G, the N/C Theorem also implies that G/Z(G) is isomorphic to Inn(G), the subgroup of Aut(G) consisting of all inner automorphisms of G.

If we define a group homomorphism T : G → Inn(G) by T(x)(g) = T_x(g) = xgx⁻¹, then we can describe N(S) and C(S) in terms of the group action of Inn(G) on G: the stabilizer of S in Inn(G) is T(N(S)), and the subgroup of Inn(G) fixing S is T(C(S)).

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		US Military Dictionary. The Oxford Essential Dictionary of the U.S. Military. Copyright © 2001, 2002 by Oxford University Press, Inc. All rights reserved. Read more
		Measures and Units. A Dictionary of Weights, Measures, and Units. Copyright © Donald Fenna 2002, 2004. All rights reserved. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Centralizer and normalizer". Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

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normalize

Definitions

Properties

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