The arithmetic mean of a variable is often denoted by a bar, for example ( read " x bar ") would be the mean of some sample space.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

model includes terms for the main effects ( x, y, z ) and

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

Since p ( x ) is irreducible, this means that p ( x ) = k ( x − a ), for some k ∈ F

Aspect ratios are mathematically expressed as x: y ( pronounced " x-to-y ") and x × y ( pronounced " x-by-y "), with the latter particularly used for pixel dimensions, such as 640 × 480.

for x in data:

for x in data:

for x in data:

for x in data:

for x in data:

for x in data:

for x, weight in dataWeightPairs: # Alternatively " for x, weight in zip ( data, weights ):"

for x in data:

The usual order relation ≤ on the real numbers is antisymmetric: if for two real numbers x and y both inequalities x ≤ y and y ≤ x hold then x and y must be equal.

The coordinate systems chosen for atomic orbitals are usually spherical coordinates ( r, θ, φ ) in atoms and cartesians ( x, y, z ) in poly-atomic molecules.

for all r ∈ R and x, y ∈ A.

for all x, y, and z in A.

This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z ).

These points form a line, and y = x is said to be the equation for this line.

The equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = r < sup > 2 </ sup > is the equation for any circle with a radius of r.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

( pause between x and y, w and x last for two beats )

The oblique asymptote, for the function f ( x ), will be given by the equation y = mx + n.

for all x and y in the algebra.

for all x and y.

Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that ( x, y, z ) belongs to R.

An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I.

For example, the spelling of the Thai word for " beer " retains a letter for the final consonant " r " present in the English word it was borrowed from, but silences it.

EveR-1's name derives from the Biblical Eve, plus the letter r for robot.

The advantage of spherical coordinates ( for atoms ) is that an orbital wave function is a product of three factors each dependent on a single coordinate: ψ ( r, θ, φ ) = R ( r ) Θ ( θ ) Φ ( φ ).

There are typically three mathematical forms for the radial functions R ( r ) which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons.

for all r ∈ R and x ∈ A.

Tradition ( e. g. Donald Knuth's ( 1973 ) hypothetical MIX computer ), for example, uses two instructions called load accumulator from register / memory ( e. g. " LDA r ") and store accumulator to register / memory ( e. g. " STA r ").

The rise time, t < sub > r </ sub >, of an amplifier is the time taken for the output to change from 10 % to 90 % of its final level when driven by a step input.

for every integer r ≥ 0 and every real number x ≥ − 1.

If the exponent r is even, then the inequality is valid for all real numbers x.

for every integer r ≥ 2 and every real number x ≥ − 1 with x ≠ 0.

Now suppose the statement is true for r = k:

However, as 1 + ( k + 1 ) x + kx < sup > 2 </ sup > ≥ 1 + ( k + 1 ) x ( since kx < sup > 2 </ sup > ≥ 0 ), it follows that ( 1 + x )< sup > k + 1 </ sup > ≥ 1 + ( k + 1 ) x, which means the statement is true for r = k + 1 as required.

By induction we conclude the statement is true for all r ≥ 0.

for r ≤ 0 or r ≥ 1, and

for 0 ≤ r ≤ 1.

In Slavic languages, / l, r / as syllabic nuclei are usually the result of the disappearance of,, or ( contrast for example the Russian cognates for vlk and smrt, ' volk ' and ' sm < nowiki >'</ nowiki > ert < nowiki >'</ nowiki >').