* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

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.

If we define r < sub > i </ sub > as the displacement of particle i from the center of mass, and v < sub > i </ sub > as the velocity of particle i with respect to the center of mass, then we have

where θ < sub > r, p </ sub > is the angle between r and p measured from r to p ; an important distinction because without it, the sign of the cross product would be meaningless.

Where r < sub > screen </ sub > is the radius of the detection screen from the tip centre, and r < sub > tip </ sub > the tip radius.

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.

t < sub > r </ sub > * BW = 0. 35, where t < sub > r </ sub > is rise time in seconds and BW is bandwidth in Hz.

: r = 9 × 2 < sup > 2n − 1 </ sup > − 1,

where n > 1 is an integer and p, q, and r are prime numbers, then 2 < sup > n </ sup >× p × q and 2 < sup > n </ sup >× r are a pair of amicable numbers.

Angular momenta of a classical object .< p > Left: intrinsic " spin " angular momentum S is really orbital angular momentum of the object at every point ,</ p >< p > right: extrinsic orbital angular momentum L about an axis ,</ p >< p > top: the moment of inertia tensor | moment of inertia tensor I and angular velocity ω ( L is not always parallel to ω )</ p >< p > bottom: momentum p and it's radial position r from the axis .</ p > The total angular momentum ( spin + orbital ) is J.

where n > m > 0 are integers and p, q, and r are prime numbers, then 2 < sup > n </ sup >× p × q and 2 < sup > n </ sup >× r are a pair of amicable numbers.

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

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

For r = 0,

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.

Again, the strict versions of these inequalities require x ≠ 0 and r ≠ 0, 1.

For any real numbers x, r > 0, one has

where h ( 0 ) is the height of the water at r

where h ( 0 ) is the height at r = 0 ( the axis ).

Diagram of forces acting on a segment of a catenary from c to r. The forces are the tension T < sub > 0 </ sub > at c, the tension T at r, and the weight of the chain ( 0, − λgs ).

Finally, weight of the chain is represented by ( 0, − λgs ) where λ is the mass per unit length, g is the acceleration of gravity and s is the length of chain between c and r.

where p is the natural length of the segment from c to r and λ < sub > 0 </ sub > is the mass per unit length of the spring with no tension and g is the acceleration of gravity.

Applying Lagrange's formula with the observation that Ω • r ( t ) = 0 at all times,

And to " measure " is to place a shorter measuring length s successively ( q times ) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q * s, q being the quotient, or remainder r is the " modulus ", the integer-fractional part left over after the division.

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 ).

In most varieties of North American English, the sound corresponding to the letter r is an alveolar approximant or retroflex rather than a trill or a tap.

The loss of syllable-final r in North America is confined mostly to the accents of eastern New England, New York City and surrounding areas and the coastal portions of the South, and African American Vernacular English.

In rural tidewater Virginia and eastern New England, ' r ' is non-rhotic in accented ( such as " bird ", " work ", " first ", " birthday ") as well as unaccented syllables, although this is declining among the younger generation of speakers.

Dropping of syllable-final r sometimes happens in natively rhotic dialects if r is located in unaccented syllables or words and the next syllable or word begins in a consonant.

On the plane the most common alternative is polar coordinates, where every point is represented by its radius r from the origin and its angle θ.

In England, the lost r was often changed into ( schwa ), giving rise to a new class of falling diphthongs.

The radial coordinate is often denoted by r, and the angular coordinate by θ or t.

For any natural number n, an " n-sphere ," often written as S < sup > n </ sup >, is the set of points in ()- dimensional Euclidean space which are at a fixed distance r from a central point of that space, where r is, as before, a positive real number.

The symbol ρ ( rho ) is often used instead of r.

Spherical coordinates ( r, θ, φ ) as often used in mathematics: radial distance r, azimuthal angle θ, and polar angle φ.

In both systems ρ is often used instead of r. Other conventions are also used so great care needs to be taken to check which one is being used.

The standard convention ( r, θ, φ ) conflicts with the usual notation for the two-dimensional polar coordinates, where θ is often used for the azimuth.

A vector may also be multiplied, or re-scaled, by a real number r. In the context of conventional vector algebra, these real numbers are often called scalars ( from scale ) to distinguish them from vectors.

In the second form given above, the phase is often generalized to, by replacing the wavenumber k with a wave vector that specifies the direction and wavenumber of a plane wave in 3-space, parameterized by position vector r. In that case, the wavenumber k, the magnitude of k, is still in the same relationship with wavelength as shown above, with v being interpreted as scalar speed in the direction of the wave vector.

In the US, magazines and newspapers often do not use it, instead printing " family-friendly " censored versions, usually " n * gg * r ", " n ** ger ", " n ——", and " the N-word "; however, historians and social activists, such as Dick Gregory, criticize the euphemisms and their usage as intellectually dishonest, because using the euphemism " the N-word " instead of nigger robs younger generations of Americans of the full history of Black people in America.

The Scots pronunciation of wh as becomes English south of Carlisle but remains in Northumberland, but Northumberland realises “ r ” as, often called the burr, which is not a Scots realisation.

As such they are often classed as opportunistic r-strategists ( i. e. those organisms whose ecology is defined by a high growth rate, r ).

In algebra, the ring K < nowiki ></ nowiki > X < sub > 1 </ sub >, ..., X < sub > r </ sub >< nowiki ></ nowiki > ( where K is a field ) is often used as the " standard, most general " complete local ring over K.

Investment is often modeled as a function of Income and Interest rates, given by the relation < var style =" font-family: serif ; font-size: 1. 1em ;"> I </ var > = < var > f </ var >(< var > Y </ var >, < var > r </ var >).

At the time, however, Eric's younger and most famous half-brother Haakon, often nicknamed Aðalsteinsfóstri, had been staying at the West-Saxon court, having been sent there to be reared as fosterson to King Æthelstan ( r. 924 – 939 ).

: r = the genetic relatedness of the recipient to the actor, often defined as the probability that a gene picked randomly from each at the same locus is identical by descent.

The ' r ' in French is often a uvular trill ( symbolized by ).

The newsgroup alt. religion. scientology ( often abbreviated a. r. s or ARS ) is a Usenet newsgroup started in 1991 to discuss the controversial beliefs of Scientology, as well as the Church of Scientology, which claims exclusive intellectual property rights thereto and is viewed by many as a dangerous cult.

* r has a long, curved shoulder, often connecting with the next letter.

Because of the destruction caused by poorly-built high-rise insulae collapsing, several Roman emperors, beginning with Augustus ( r. 30 BC-14 AD ), set limits of 20 – 25 metres for multi-storey buildings, but met with limited success, as these limits were often ignored despite the likelihood of taller insulae collapsing.

Solutions to such recurrence relations of higher order are found by systematic means, often using the fact that a < sub > n </ sub > = r < sup > n </ sup > is a solution for the recurrence exactly when t = r is a root of the characteristic polynomial.

* Speech sound disorders involve difficulty in producing specific speech sounds ( most often certain consonants, such as / s / or / r /), and are subdivided into articulation disorders ( also called phonetic disorders ) and phonemic disorders.