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# integer coefficients definition

From Wikipedia

Coefficient

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

7x^2-3xy+1.5+y

the first three terms respectively have the coefficients 7, âˆ’3, and 1.5 (in the third term there are no variables, so the coefficient is the term itself; it is called the constant term or constant coefficient of this expression). The final term does not have any explicitly written coefficient, but is usually 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 in

ax^2+bx+c

when it is understood that these are not considered as variables.

Thus a polynomial in one variable x can be written as

a_k x^k + \cdots + a_1 x^1 + a_0,

for some integer k, where ak, ... a1, a0 are coefficients; to allow this kind of expression in all cases one must allow introducing terms with 0 as coefficient. For the largest i with (if any), ai is called the leading coefficient of the polynomial. So for example the leading coefficient of the polynomial

\, 4x^5 + x^3 + 2x^2

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 (x_1, x_2, ..., x_n) of a vectorv in a vector space with basis \lbrace e_1, e_2, ..., e_n \rbrace , are the coefficients of the basis vectors in the expression

v = x_1 e_1 + x_2 e_2 + ... + x_n e_n .

## Examples of physical coefficients

1. Coefficient of Thermal Expansion(thermodynamics) (dimensionless) - Relates the change in temperature to the change in a material's dimensions.
2. Partition Coefficient(KD) (chemistry) - The ratio of concentrations of a compound in two phases of a mixture of two immiscible solvents at equilibrium.
3. 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.
4. Lift coefficient(CLor CZ) (Aerodynamics) (dimensionless) - Relates the lift generated by an airfoil with the dynamic pressure of the fluid flow around the airfoil, and the planform area of the airfoil.
5. 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.
6. 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.
7. Damping Factora.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

## Chemistry

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 2H_2 + O_2 \rarr 2H_2O, the number 2's in front of H_2 and H_2O are stoichiometric coefficients.

Attenuation coefficient

For "attenuation coefficient" as it applies to electromagnetic theory and telecommunications seepropagation constant. For the "mass attenuation coefficient", see the article mass attenuation coefficient.

The attenuation coefficient is a quantity that characterizes how easily a material or medium can be penetrated by a beam of light, sound, particles, or other energy or matter. A large attenuation coefficient means that the beam is quickly "attenuated" (weakened) as it passes through the medium, and a small attenuation coefficient means that the medium is relatively transparent to the beam. Attenuation coefficient is measured using units of reciprocal length.

The attenuation coefficient is also called linear attenuation coefficient, narrow beam attenuation coefficient, or absorption coefficient. Although all four terms are often used interchangeably, they can occasionally have a subtle distinction, as explainedbelow.

## Overview

The attenuation coefficient describes the extent to which the intensity of an energy beam is reduced as it passes through a specific material. This might be a beam of electromagnetic radiation or sound.

• It is used in the context of X-rays or Gamma rays, where it is represented using the symbol \mu, and measured in cmâˆ’1.
• It is also used for modeling solar and infrared radiative transfer in the atmosphere, albeit usually denoted with another symbol (given the standard use of \mu = \cos(\theta) for slant paths).
• In the case of ultrasound attenuation it is usually denoted as \alpha and measured in dB/cm/MHz.
• The attenuation coefficient is widely used in acoustics for characterizing particle size distribution. A common unit in this contexts is inverse metres, and the most common symbol is the Greek letter \alpha.
• It is also used in acoustics for quantifying how well a wall in a building absorbs sound. Wallace Sabine was a pioneer of this concept. A unit named in his honor is the sabin: the absorption by a 1|m2|sqft|adj=on slab of perfectly-absorptive material (the same amount of sound loss as if there were a 1-square-metre window). Note that the sabin is not a unit of attenuation coefficient; rather, it is the unit of a related quantity.

A small linear attenuation coefficient indicates that the material in question is relatively transparent, while a larger values indicate greater degrees of opacity. The linear attenuation coefficient is dependent upon the type of material and the energy of the radiation. Generally, for electromagnetic radiation, the higher the energy of the incident photons and the less dense the material in question, the lower the corresponding linear attenuation coefficient will be.

## Definitions and formulas

The measured intensity I of transmitted through a layer of material with thickness x is related to the incident intensity I_0 according to the inverse exponential power law that is usually referred to as Beerâ€“Lambert law:

I = I_{0} \, e^{-\alpha \, x},

where x denotes the path length. The attenuation coefficient (or linear attenuation coefficient) is \alpha.

The Half Value Layer (HVL) signifies the thickness of a material required to reduce the intensity of the emergent radiation to half its incident magnitude. It is from these equations that engineers decide how much protection is needed for "safety" from potentially harmful radiation. The attenuation factor of a material is obtained by the ratio of the emergent and incident radiation intensities I/I_0.

The linear attenuation coefficient and mass attenuation coefficient are related such that the mass attenuation coefficient is simply \alpha/\rho, where \rho is the density in g/cm3. When this coefficient is used in the Beer-Lambert law, then "mass thickness" (defined as the mass per unit area) replaces the product of length times density.

The linear attenuation coefficient is also inversely related to mean free path. Moreover, it is very closely related to the absorption cross section.

## Attenuation versus absorption

The terms "attenuation coefficient" and "absorption coefficient" are generally used interchangeably. However, in certain situations they are distinguished, as follows.

When a narrow (collimated) beam of light passes through a substance, the beam will lose intensity due to two processes: The light can be absorbed by the substance, or the light can be scattered (i.e., the photons can change direction) by the substance. Just looking at the narrow beam itself, the two processes cannot be distinguished. However, if a detector is set up to measure light leaving in different directions, or conversely using a non-narrow beam, one can measure how much of the lost intensity was scattered, and how much was absorbed.

In this context, the "absorption coefficient" measures how quickly the beam would lose intensity due to the absorption alone, while "attenuation coefficient" measures the total loss of narrow-beam intensity, including scattering as well. "Narrow-beam attenuation coefficient" always unambiguously refers to the latter.

Gini coefficient

The Gini coefficient is a measure of statistical dispersion developed by the ItalianstatisticianCorrado Gini and published in his 1912 paper "Variability and Mutability" (VariabilitÃ  e mutabilitÃ )

The Gini coefficient is a measure of the inequality of a distribution, a value of 0 expressing total equality and a value of 1 maximal inequality. It has found application in the study of inequalities in disciplines as diverse as economics, health science, ecology, chemistry and engineering.

It is commonly used as a measure of inequality of income or wealth. Worldwide, Gini coefficients for income range from approximately 0.23 (Sweden) to 0.70 (Namibia) although not every country has been assessed.

## Definition

The Gini coefficient is usually defined mathematically based on the Lorenz curve, which plots the proportion of the total income of the population (y axis) that is cumulatively earned by the bottom x% of the population (see diagram). The line at 45 degrees thus represents perfect equality of incomes. The Gini coefficient can then be thought of as the ratio of the area that lies between the line of equality and the Lorenz curve (marked 'A' in the diagram) over the total area under the line of equality (marked 'A' and 'B' in the diagram); i.e., G=A/(A+B).

The Gini coefficient can range from 0 to 1; it is sometimes multiplied by 100 to range between 0 and 100. A low Gini coefficient indicates a more equal distribution, with 0 corresponding to complete equality, while higher Gini coefficients indicate more unequal distribution, with 1 corresponding to complete inequality. To be validly computed, no negative goods can be distributed. Thus, if the Gini coefficient is being used to describe household income inequality, then no household can have a negative income. When used as a measure of income inequality, the most unequal society will be one in which a single person receives 100% of the total income and the remaining people receive none (G=1); and the most equal society will be one in which every person receives the same income (G=0).

Some find it more intuitive (and it is mathematically equivalent) to think of the Gini coefficient as half of the relative mean difference. The mean difference is the average absolute difference between two items selected randomly from a population, and the relative mean difference is the mean difference divided by the average, to normalize for scale.

## Calculation

The Gini index is defined as a ratio of the areas on the Lorenz curve diagram. If the area between the line of perfect equality and the Lorenz curve is A, and the area under the Lorenz curve is B, then the Gini index is A/(A+B). Since A+B = 0.5, the Gini index, G = A/(0.5) = 2A = 1-2B. If the Lorenz curve is represented by the function Y = L(X), the value of B can be found with integration and:

G = 1 - 2\,\int_0^1 L(X) dX.

In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example:

• For a population uniform on the values yi, i = 1 to n, indexed in non-decreasing order ( yiâ‰¤ yi+1):
G = \frac{1}{n}\left ( n+1 - 2 \left ( \frac{\Sigma_{i=1}^n \; (n+1-i)y_i}{\Sigma_{i=1}^n y_i} \right ) \right )
This may be simplified to:
G = \frac{2 \Sigma_{i=1}^n \; i y_i}{n \Sigma_{i=1}^n y_i} -\frac{n+1}{n}
• For a discrete probability functionf(y), where yi, i = 1 to n, are the points with nonzero probabilities and which are indexed in increasing order ( yi&lt; yi+1):
G = 1 - \frac{\Sigma_{i=1}^n \; f(y_i)(S_{i-1}+S_i)}{S_n}
where
S_i = \Sigma_{j=1}^i \; f(y_j)\,y_j\, and S_0 = 0\,
G = 1 - \frac{1}{\mu}\int_0^\infty (1-F(y))^2dy = \frac{1}{\mu}\int_0^\infty F(y)(1-F(y))dy
• Since the Gini coefficient is half the relative mean difference, it can also be calculated using formulas for the relative mean difference. For a random sample S consisting of values yi, i = 1 to n, that are indexed in non-decreasing order ( yiâ‰¤ yi+1), the statistic:
G(S) = \frac{1}{n-1}\left (n+1 - 2 \left ( \frac{\Sigma_{i=1}^n \; (n+1-i)y_i}{\Sigma_{i=1}^n y_i}\right ) \right )
is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like, G, G(S) has a simpler form:
G(S) = 1 - \frac{2}{n-1}\left ( n - \frac{\Sigma_{i=1}^n \; iy_i}{\Sigma_{i=1}^n y_i}\right ) .

There does not exist a sample statistic that is in general an unbiased estimator of the population Gini coefficient, like the relative mean difference.

Sometimes the entire Lorenz curve is not known, and only values at certain intervals are given. In that case, the Gini coefficient can be approximated by using various techniques for interpolating the missing values of the Lorenz curve. If ( X k , Yk ) are the known points on the Lorenz curve, with the X k indexed in increasing order ( X k - 1&lt; X&nbsp;k&nbsp;), so that:

• Xk is the cumulated prop

Question:Find a polynomial with integer coefficients and a leading coefficient of one that satisfies the given conditions. Q has degree 3, and zeros -5 and 1 + i. Q(x) =?

Answers:x^3+ kx^2 + m*x + c (x+5)(x-(1+i))(x-(1-i)) = (x+5)(x^2-x(1+i)-x(1-i)+(1-i^2) = x^3 -2x^2 +2x +5x^2 -10x+ 10 = x^3+3x^2-8x+10 k=3, m=-8, c=10 Q(x)=x^3 + 3x^2 - 8x+ 10

Question:Find a polynomial with integer coefficients and a leading coefficient of one that satisfies the given conditions. P has degree 2, and zeros 1 + i 5 and 1 - i 5. P(x) =?

Answers:If x = a is a zero, then x - a is a factor. Since x = 1 + i 5 and x = 1 - i 5 are solutions, we have the following equation for P(x): P(x) = [x - (1 + i 5)][x - (1 - 5)] ==> P(x) = (x - 1 - i 5)(x - 1 + i 5) You can re-write this and get: P(x) = [(x - 1) + i 5][(x - 1) - i 5] This is a difference of squares, so we can get: P(x) = (x - 1) - (i 5) ==> P(x) = x - 2x + 1 - i (5) ==> P(x) = x - 2x + 1 - (-1)(5) ==> P(x) = x - 2x + 1 + 5 ==> P(x) = x - 2x + 6 Answer Verification: http://www.wolframalpha.com/input/?i=x%C2%B2+-+2x+%2B+6 I hope that helps!

Question:The question is: write the polynomial equation of least degree having the roots 1, -1, and 0.5 with all integer coefficients. what does 'with all integer coefficients' mean?

Answers:x = -1, 1/2, 1 (x + 1)(2x - 1)(x - 1) = 0 2x - x - 2x + 1 = 0 y = 2x - x - 2x + 1 Integer coefficients means you use (2x - 1) as a root instead of (x - 1/2) or (x - 0.5).

Question:Directions say- Write the following equations in standard form with integer coefficients. 1) 5y = 6 Do i need an X? and does C=6? Thank you for the help, i can pretty much get the idea for the rest if someone will help with this one. i thought this was the Ax + By = C thing

Answers:no, you don't need an x if there is not one given, because you wouldn't write '0x' . writing in standard form just means getting the 'y' by itself. so it would be: 5y/5=6/5 and for the final answer: y=6/5 or y=1.2 ok and what is 'C'? edit: oh ok gotcha