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Page "Polynomial" ¶ 93
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If and R
If the rake angle **yc of the knife is high enough and the friction angle **yt between the front of the knife and the back of the chip is low enough to give a positive value for Af, the resultant vector R will lie above the plane of the substrate.
If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.
If Af denotes the net profit from stage R and Af, then the principle of optimality gives Af.
* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =
* If the balance factor of P is-2 then the right subtree outweighs the left subtree of the given node, and the balance factor of the right child ( R ) must be checked.
* If the balance factor of R is-1, a single left rotation ( with P as the root ) is needed ( Right-Right case ).
* If the balance factor of R is + 1, two different rotations are needed.
If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and 1 indicating perfect anti-correlation.
If the thumb points in the direction of the 4th substitutent, the enantiomer is R. Otherwise, it's S.
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 a is a point in R < sup > n </ sup >, then the higher dimensional chain rule says that:
If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane ( see caption at right ), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R < sup > 3 </ sup > ( right-hand rule ).
If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.
If, i. e., it has a large norm with each value of s, and if, then Y ( s ) is approximately equal to R ( s ) and the output closely tracks the reference input.
If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).
* If x < sub > 0 </ sub > is a real number, we can turn the set R
( If X is also empty then R is reflexive.
*( 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.
If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b ( that is, if there are elements x and y in R such that d · x = a and d · y = b ).
If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

If and is
If the circumstances are faced frankly it is not reasonable to expect this to be true.
If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.
If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.
If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;
If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.
If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.
If it is an honest feeling, then why should she not yield to it??
If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.
If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.
If he is good, he may not be legal ; ;
If the man on the sidewalk is surprised at this question, it has served as an exclamation.
If the existent form is to be retained new factors that reinforce it must be introduced into the situation.
If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??
If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.
If he is a traditionalist, he is an eclectic traditionalist.
If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.
If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If and integral
If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.
If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.
If φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,
If m and n are natural numbers and f ( x ) is a smooth ( meaning: sufficiently often differentiable ) function defined for all real numbers x in the interval, then the integral
If and then the limit is where is the complete elliptic integral of the first kind
If the random variable is real-valued ( or more generally, if a total order is defined for its possible values ), the cumulative distribution function gives the probability that the random variable is no larger than a given value ; in the real-valued case it is the integral of the density.
If they are participant in the integral physical expression of a living human being who has absorbed and metabolized them, or if they are now the physical remains of a once-living human being, the substance of what they actually are is human, hence, a human body.
If the liquid is incompressible the molar volume is constant and the integral becomes.
If the baseline value is large or if it is not subtracted out of the image, then the computed D4σ value will be larger than the actual value because the baseline value near the edges of the sensor are weighted in the D4σ integral by x < sup > 2 </ sup >.
If g is large and b is small so that the angle is not negligible one can write the integral for the on-axis case ( P < sub > 1 </ sub > is at the center of the shadow ) as ( see ):
Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is " If the sequence f < sub > n </ sub >( x ) increases to the limit f ( x ), the integral of f < sub > n </ sub >( x ) tends to the integral of f ( x ).
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 r < 1, the derivatives of φ may be computed by differentiating under the integral sign, and one can verify that φ is analytic, even if u is continuous but not necessarily differentiable.
If R = K is a field, then K (( X )) is in fact a field, which may alternatively be obtained as the field of fractions of the integral domain K < nowiki ></ nowiki > X < nowiki ></ nowiki >.
If R possesses no zero divisors, it is called an integral domain since it closely resembles the integers in some ways.
* If R is an integral domain, the units of the Laurent polynomial ring RX < sup >− 1 </ sup > have the form uX < sup > k </ sup >, where u is a unit of R and k is an integer.
If we take such a tiling and enlarge it so that the formerly smallest tile now has the size of the square S we started out from, then we see that we obtain from this a tiling of the plane with integral squares, each having a different size.
If the index is over, then the sum is not defined and is replaced by an integral instead.
If the boundary is piecewise smooth, then we interpret the integral as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the angles by which the smooth portions turn at the corners of the boundary.
If the situation is as above but the extension L of K is algebraic of infinite degree, then it is still possible for the integral closure S of R in L to be a Dedekind domain, but it is not guaranteed.

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