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

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

If the relative priorities of these substituents need to be established, R takes priority over S. When this happens, the descriptor of the stereocenter is a lowercase letter ( r or s ) instead of the uppercase letter normally used.

If u ( t ) is the control signal sent to the system, y ( t ) is the measured output and r ( t ) is the desired output, and tracking error, a PID controller has the general form

If the weight of the roadway per unit length is w and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable from c to r is wx where x is the horizontal distance between c to r. Proceeding as before gives the differential equation

If F ( r ) represents gravity, it is a negative term proportional to 1 / r < sup > 2 </ sup >, so the net acceleration in r in the rotating frame depends on a difference of reciprocal square and reciprocal cube terms, which are in balance in a circular orbit but otherwise typically not.

If the source of Φ ( r ) is a dipole, as it is assumed here, this term is the only non-vanishing term in the multipole expansion of Φ ( r ).

If it is generally by temporary uninhabitability, for example a nuclear winter, then n < sub > r </ sub > may be relatively high.

*( EF1 ) If a and b are in R and b is nonzero, then there are q and r in R such that and either r = 0 or.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

If R is an integral domain and f and g are polynomials in R, it is said that f divides g or f is a divisor of g if there exists a polynomial q in R such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R and r is an element of R such that f ( r ) = 0, then the polynomial ( X − r ) divides f. The converse is also true.

If F is a field and f and g are polynomials in F with g ≠ 0, then there exist unique polynomials q and r in F with

If a root r < sub > 1 </ sub >

If an object moves with angular velocity ω around a circle of radius r centered at the origin of the x-y plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.

Consider now the acceleration due to the sphere of mass M experienced by a particle in the vicinity of the body of mass m. With R as the distance from the center of M to the center of m, let ∆ r be the ( relatively small ) distance of the particle from the center of the body of mass m. For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass M. If the body of mass m is itself a sphere of radius ∆ r, then the new particle considered may be located on its surface, at a distance ( R ± ∆ r ) from the centre of the sphere of mass M, and ∆ r may be taken as positive where the particle's distance from M is greater than R. Leaving aside whatever gravitational acceleration may be experienced by the particle towards m on account of ms own mass, we have the acceleration on the particle due to gravitational force towards M as:

If through lack of issue, marriage or both, eventually only one person represents the claims of all the sisters, he or she can claim the dignity as a matter of right, and the abeyance is said to be terminated.

If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid.

If a party represents more than 4 states then such parties are considered as national parties.

If the Mythological Cycle represents a Golden Age, the Ulster Cycle is Ireland's Heroic Age.

If negation is cyclic and "∨" is a " max operator ", then the law can be expressed in the object language by ( P ∨ ~ P ∨ ~~ P ∨ ... ∨ ~...~ P ), where "~...~" represents n − 1 negation signs and "∨ ... ∨" n − 1 disjunction signs.

If the alphabet consists of alternative symbols, each symbol represents a message consisting of N bits.

If λ represents wavelength and f represents frequency ( note, λf = c where c is the speed of light ), then z is defined by the equations:

If the secondary winding is made open-circuit, the current I < sub > 0 </ sub > taken by the magnetizing branch represents the transformer's no-load current.

If one of the two events represents the sending of a signal from one location and the second event represents the reception of the same signal at another location, then as long as the signal is moving at the speed of light or slower, the mathematics of simultaneity ensures that all reference frames agree that the transmission-event happened before the reception-event.

If the vector represents a directed distance or displacement from a point A to a point B ( see figure ), it can also be denoted as or < u > AB </ u >.

If the delta function is conceptualized as modeling an idealized point mass at 0, then δ ( A ) represents the mass contained in the set A.

If a signal is present, then this factor represents the decrease in signal to noise ratio incurred using the receiver system with a noise temperature of.

( If the object represents a static field then the parameter is ignored and may be.

( If the object represents a static method then the first parameter is ignored and may be.

If f represents a very small but intense source of heat, then the preceding integral can be approximated by the delta distribution, multiplied by the strength of the source.

) If the analemma is drawn in a diagram, tilted at the appropriate angle for an observer's latitude ( as described above ), and if a horizontal line is drawn to pass through the position of the Sun on the analemma on any given date ( interpolating between the date markings as necessary ), then at sunrise this line represents the horizon.

If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy.

If represents the number of dollars you have in chips after n tosses, with, then the sequence is a Markov process.

If represents the total value of the coins set on the table after n draws, with, then the sequence is not a Markov process.

Pettibon stated " If a white flag means surrender, a black flag represents anarchy.

If the number has a line above it, then it represents a unit fraction ( that is, would be ), whereas a line below the number indicates an integer.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If this seems arbitrary, its effect was to treat citizens of the District of Columbia equally with citizens of the states -- at the expense of expanding a troublesome jurisdiction.

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

If the source is located at an arbitrary source point, denoted by the vector and the field point is located at the point, then we may represent the scalar Green's function ( for arbitrary source location ) as:

If the subject could so judge his own superior, then all lawful superior authority could lawfully be overthrown by the arbitrary judgement of an inferior, and thus all law was under constant threat.

If the former, then justice is arbitrary ; if the latter, then morality exists on a higher order than God, who becomes little more than a passer-on of moral knowledge.

One can use that instruction, which is usually faster than a comparison against some arbitrary value ( requiring a subtraction ), to implement the command " If i ≤ 0, then exit the loop ".

If the inclination is zero or 180 degrees ( π radians ), the azimuth is arbitrary.

If the radius is zero, both azimuth and inclination are arbitrary.

" If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations.

If the strategy ever called for moving on the square already chosen, the first player can then make another arbitrary move.

If a feature was intended to be located from the center of the part, the operator needs to locate it from the center of the model, not, perhaps, from a more convenient edge or an arbitrary point, as he could when using " dumb " solids.

If we consider an arbitrary constant, then average availability is represented as

If arbitrary values and / or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction.

If one allows arbitrary temperature dependence of the prefactor, the Arrhenius description becomes overcomplete, and the inverse problem ( i. e., determining the prefactor and activation energy from experimental data ) becomes singular.

If ( for some arbitrary reason ) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio k,

If the arbitrary area for excavation is wisely chosen, the sequence should be revealed and excavation can return to a truly stratigraphic method.

The reasoning behind existential elimination (∃ E ) is as follows: If it is given that there exists an element for which the proposition function is true, and if a conclusion can be reached by giving that element an arbitrary name, that conclusion is necessarily true, as long as it does not contain the name.

must be true for all values of c over the same domain X ; else, the logic does not follow: If c is not arbitrary, and is instead a specific element of the domain of discourse, then stating P ( c ) might unjustifiably give more information about that object.

If data of arbitrary length is to be encrypted, a simple strategy is to split the data into blocks each matching the cipher's block size, and encrypt each block separately using the same key.

If a function is defined on an interval and is an antiderivative of, then the set of all antiderivatives of is given by the functions, where C is an arbitrary constant.

If n is arbitrary, i. e., if the system is insensitive to any rotation, then every component of L is conserved ; in short, angular momentum is conserved.

* If four arbitrary points A, B, C, D are given that do not form an orthocentric system, then the nine-point circles of ABC, BCD, CDA and DAB concur at a point.