WordNet

a constant number that serves as a measure of some property or characteristic
(physics) a coefficient that expresses how much of a specified property is possessed by a specified substance
an integer that can be divided without remainder into the difference between two other integers; "2 is a modulus of 5 and 9"
the absolute value of a complex number

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(数学で)係数,(物理で)係数,率

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/11/02 23:16:09」(JST)

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In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but in any case does not involve any variables of the expression. For instance in

the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term. Often coefficients are numbers as in this example, although they could be parameters of the problem, as a, b, and c, where "c" is a constant, in

when it is understood that these are not considered variables. Thus a polynomial in one variable x can be written as

for some integer , where are coefficients; to allow this kind of expression in all cases one must allow introducing terms with 0 as coefficient. For the largest with (if any), is called the leading coefficient of the polynomial. So for example the leading coefficient of the polynomial

is 4.

Specific coefficients arise in mathematical identities, such as the binomial theorem which involves binomial coefficients; these particular coefficients are tabulated in Pascal's triangle.

Linear algebra

In linear algebra, the leading coefficient of a row in a matrix is the first nonzero entry in that row. So, for example, given

M = \begin{pmatrix} 1 & 2 & 0 & 6\\ 0 & 2 & 9 & 4\\ 0 & 0 & 0 & 4\\ 0 & 0 & 0 & 0 \end{pmatrix}

.

The leading coefficient of the first row is 1; 2 is the leading coefficient of the second row; 4 is the leading coefficient of the third row, and the last row does not have a leading coefficient.

Though coefficients are frequently viewed as constants in elementary algebra, they can be variables more generally. For example, the coordinates of a vector in a vector space with basis , are the coefficients of the basis vectors in the expression

Examples of physical coefficients

Coefficient of Thermal Expansion (thermodynamics) (dimensionless) - Relates the change in temperature to the change in a material's dimensions.
Partition Coefficient (K_D) (chemistry) - The ratio of concentrations of a compound in two phases of a mixture of two immiscible solvents at equilibrium. H2O is a coefficient
Hall coefficient (electrical physics) - Relates a magnetic field applied to an element to the voltage created, the amount of current and the element thickness. It is a characteristic of the material from which the conductor is made.
Lift coefficient (C_L or C_Z) (Aerodynamics) (dimensionless) - Relates the lift generated by an airfoil with the dynamic pressure of the fluid flow around the airfoil, and the plan-form area of the airfoil.
Ballistic coefficient (BC) (Aerodynamics) (units of kg/m²) - A measure of a body's ability to overcome air resistance in flight. BC is a function of mass, diameter, and drag coefficient.
Transmission Coefficient (quantum mechanics) (dimensionless) - Represents the probability flux of a transmitted wave relative to that of an incident wave. It is often used to describe the probability of a particle tunnelling through a barrier.
Damping Factor a.k.a. viscous damping coefficient (Physical Engineering) (units of newton-seconds per meter) - relates a damping force with the velocity of the object whose motion is being damped.

A coefficient is a number placed in front of a term in a chemical equation to indicate how many molecules (or atoms) take part in the reaction. For example, in the formula

,

the number 2's in front of and are stoichiometric coefficients.

References

Sabah Al-hadad and C.H. Scott (1979) College Algebra with Applications, page 42, Winthrop Publishers, Cambridge Massachusetts ISBN 0-87626-140-3 .
Gordon Fuller, Walter L Wilson, Henry C Miller, (1982) College Algebra, 5th edition, page 24, Brooks/Cole Publishing, Monterey California ISBN 0-534-01138-1 .
Steven Schwartzman (1994) The Words of Mathematics: an etymological dictionary of mathematical terms used in English, page 48, Mathematics Association of America, ISBN 0-88385-511-9.

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English Journal

Genetic relatedness of faecal coliforms and enterococci bacteria isolated from water and sediments of the Apies River, Gauteng, South Africa.

Ekwanzala MD1, Abia AL2, Ubomba-Jaswa E3, Keshri J2, Momba NB4.
AMB Express.AMB Express.2017 Dec;7(1):20. doi: 10.1186/s13568-016-0319-4. Epub 2017 Jan 7.
To date, the microbiological quality of river sediments and its impact on water resources are not included in the water quality monitoring assessment. Therefore, the aim of this study was to establish genetic relatedness between faecal coliforms and enterococci isolated from the river water and rive
PMID 28063147

An experimental phantom study of the effect of gadolinium-based MR contrast agents on PET attenuation coefficients and PET quantification in PET-MR imaging: application to cardiac studies.

O' Doherty J1, Schleyer P2.
EJNMMI physics.EJNMMI Phys.2017 Dec;4(1):4. doi: 10.1186/s40658-017-0173-8. Epub 2017 Jan 13.
BACKGROUND: Simultaneous cardiac perfusion studies are an increasing trend in PET-MR imaging. During dynamic PET imaging, the introduction of gadolinium-based MR contrast agents (GBCA) at high concentrations during a dual injection of GBCA and PET radiotracer may cause increased attenuation effects
PMID 28084612

Patient-specific optimisation of administered activity and acquisition times for 18F-FDG PET imaging.

Wickham F1, McMeekin H2, Burniston M3, McCool D3, Pencharz D3, Skillen A3, Wagner T3.
EJNMMI research.EJNMMI Res.2017 Dec;7(1):3. doi: 10.1186/s13550-016-0250-3. Epub 2017 Jan 13.
BACKGROUND: The purpose of this study is to identify a method for optimising the administered activity and acquisition time for 18F-FDG PET imaging, yielding images of consistent quality for patients with varying body sizes and compositions, while limiting radiation doses to patients and staff. Pati
PMID 28091978