If Franklin was an authentic genius, then Alexander Hamilton, with his exceptional precocity, consuming energy, and high ambition, was a political prodigy.

If we grasp this opportunity to build an age of productive partnership between the less fortunate nations and those that have already achieved a high state of economic advancement, we will make brighter the outlook for a world order based upon security and freedom.

If the railroads, for example, regularly slaughtered 25,000 passengers each year, the high priests of the cult would have cause to tremble for their personal safety, for such a holocaust would excite demands for the hanging of every railroad president in the United States.

If we wait until children are in junior high or high school, we will never manage it.

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 high accuracy is required in preflight leveling, it is usually necessary to integrate or doubly integrate the accelerometer outputs ( this also minimizes the noise problem ).

If the platform is not too high off the ground, a transit can be mounted on a stand to raise it up to the platform.

If anyone thought of the John Harvey, it was to observe that she was straddled by a pair of ships heavily laden with high explosive and if they were hit the John Harvey would likely be blown up with her own ammo and whatever else it was that she carried.

If the farm is in an industrial area where wages are high, farm labor costs will also be high.

If William wished to continue operations for a year, why not simply leave the Negroes undisturbed and pay them `` as high wages to remain there as are ever paid the labor of persons of their sex & age.

If the student drops out of high school, the break with the school is even more complete.

If the concentration of element or compound in a sample is too high for the detection range of the technique, it can simply be diluted in a pure solvent.

If the brine regenerator uses high heat, they also partially sterilise the air.

" If the remainder of the Conservative Party could muster the electoral support necessary to form a government, then Disraeli was now guaranteed high office.

If the shuttlecock has dropped lower, then a smash is impossible and a full-length, high clear is difficult.

If the risk of asphyxiation is deemed to be high, doctors may use various devices ( such as an oropharyngeal airway, nasopharyngeal airway or endotracheal tube ) to safeguard the airway.

If the glass is damaged, atmospheric pressure can collapse the vacuum tube into dangerous fragments which accelerate inward and then spray at high speed in all directions.

If both high priority substituents are on the same side of the double bond, i. e. in the cis configuration, then the stereoisomer is assigned a Z or Zusammen configuration.

If a naive rendering algorithm is used without any filtering, high frequencies in the image function will cause ugly aliasing to be present in the final image.

If a team places high enough at selected USASF / IASF sanctioned national competitions, they could be included in the Cheerleading Worlds and compete against teams from all over the world, as well as receive money for placing.

If the signal matches the desired user's code then the correlation function will be high and the system can extract that signal.

If SR calcium stores become too high, some ions are released spontaneously through SR ryanodine receptors.

If the level of an attractant remains high, the level of phosphorylation of CheA ( and therefore CheY and CheB ) will remain low, the cell will swim smoothly, and the level of methylation of the MCPs will increase ( because CheB-P is not present to demethylate ).

* 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.

If F ≥ F < sub > Critical </ sub > ( Numerator DF, Denominator DF, α )

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

If ΔS and / or T are small, the condition ΔG < 0 may imply that ΔH < 0, which would indicate an exothermic reaction.

If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

If the first allele is dominant to the second, then the fraction of the population that will show the dominant phenotype is p < sup > 2 </ sup > + 2pq, and the fraction with the recessive phenotype is q < sup > 2 </ sup >.

If activated cytotoxic CD8 < sup >+</ sup > T cells recognize them, the T cells begin to secrete various toxins that cause the lysis or apoptosis of the infected cell.

If ADH production is excessive in heart failure, Na < sup >+</ sup > level in the plasma may fall ( hyponatremia ), and this is a sign of increased risk of death in heart failure patients.

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

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

If A is expressed as an N × N matrix, then A < sup >†</ sup > is its conjugate transpose.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If one statement is true in a category C then its dual will be true in the dual category C < sup > op </ sup >.

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 ).