If this strategy prevented Vincentius from receiving reinforcements, it led to the convergence of both Gothic armies against Gaul.

) If c is not the only convergent point, then there is always a number r with 0 < r ≤ ∞ such that the series converges whenever | x − c | < r and diverges whenever | x − c | > r. The number r is called the radius of convergence of the power series ; in general it is given as

If we studied this as a power series, its properties would include, for example, that its radius of convergence is 1.

If neither convergence nor divergence is required in the prescription, " plano " is used to denote a refractive power of zero.

If the standard is parallel to the wall convergence is entirely eliminated.

If this revolutionary digital paradigm shift presumed that old media would be increasingly replaced by new media, the convergence paradigm that is currently emerging suggests that new and old media would interact in more complex ways than previously predicted.

If the area convergence is great enough that the speed of sound is reached, a phenomenon known as " choking " occurs.

If they are not, a run of " bad luck " could lead to the wider interval being used many times, thus slowing down the rate of convergence.

* If the Hilbert space is finite-dimensional, i. e. a Euclidean space, then the concepts of weak convergence and strong convergence are the same.

If you were a man '' --

If she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.

If someone were to drop a match in here, this place would go up like a haystack ''.

If the other pilots were worried, they did not show it.

If it were the enemy, tactically his position was correct.

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

If it were not that I knew who it was I could have mistaken it for my Aunt so well did her clothes fit him.

If an automobile were approaching him, he would know what was required of him, even though he might not be able to act quickly enough.

If there were only darkness, all would be clear.

If there were only the mess, all would be clear ; ;

If these people were denied a voice ( do they have a moral right to a voice??

If we were creating a wholly new society, we could insist that our social, political, economic and philosophic institutions foster rather than hamper man ; ;

If living Jews were unavailable for study, the Bible was at hand.

If his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.

If she were not at home, Mama would see to it that a fresh white rose was there.

If the Stearns were not there, grace would be omitted.

If anything, the conservative Democrats were more opposed to Hearst than the Republicans.

If the hardships of the winter at Valley Forge were trying for healthy men, they were, of course, much more so for those not in good health.

If his circumspection in regard to Philip's sensibilities went so far that he even refused to grant a dispensation for the marriage of Amadee's daughter, Agnes, to the son of the dauphin of Vienne -- a truly peacemaking move according to thirteenth-century ideas, for Savoy and Dauphine were as usual fighting on opposite sides -- for fear that he might seem to be favoring the anti-French coalition, he would certainly never take the far more drastic step of ordering the return of Gascony to Edward, even though, as he admitted to the English ambassadors, he had been advised that the original cession was invalid.

If there were no West Berlin problem, imperialist quarters would have invented an excuse for stepping up the armaments race to try to solve the internal and external problems besetting the United States and its NATO partners.

If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.

If the Hessian troops sent here willy-nilly by the Hessian Government to fight for England in the 1770's were mercenaries, what shall we call the UN troops sent to the Congo willy-nilly by their governments to fight for the United Nations??

If the UN troops are not mercenaries then the Hessians were not mercenaries either.

If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.

If the puppets are of uniform size, you can change them in accord with your child's whims.

If the source emits light uniformly in all directions, the flux can be found by multiplying the intensity by 4π: a uniform 1 candela source emits 12. 6 lumens.

If one watches a bacterium swimming in a uniform environment, its movement will look like a random walk with relatively straight swims interrupted by random tumbles that reorient the bacterium.

The evidence was compiled by W de Sitter ( 1927 ) who wrote " If we accept this hypothesis, then the ' astronomical time ', given by the earth's rotation, and used in all practical astronomical computations, differs from the ' uniform ' or ' Newtonian ' time, which is defined as the independent variable of the equations of celestial mechanics ".

If a game has a board, the board is usually irregular rather than uniform or symmetric ( like Risk rather than chess or Scrabble ); the board is often random ( like Settlers of Catan ) or has random elements ( like Tikal ).

" If it had not been for Apollinaire in uniform ," wrote Cocteau, " with his skull shaved, the scar on his temple and the bandage around his head, women would have gouged our eyes out with hairpins.

If the beam enters the sample perpendicular to the surface, then the activated region is uniform about the axis of the beam and a certain number of electrons " escape " from within the sample.

If X is Tychonoff, then the uniform structure can be chosen so that βX becomes the completion of the uniform space X.

⟨ H ⟩, be the group generated by H. Then the word problem in H < sup >*</ sup > is solvable: given two words h, k in the generators H of H < sup >*</ sup >, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution the word problem for the class K ( say ) of finitely generated groups that can be embedded in G. If this were the case the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers.

If two objects are similar, each is congruent to the result of a uniform scaling of the other.

If the conductor is of a uniform impedance and is properly terminated, the entire transmitted pulse will be absorbed in the far-end termination and no signal will be reflected toward the TDR.

If the transmission line is uniform along its length, then its behaviour is largely described by a single parameter called the characteristic impedance, symbol Z < sub > 0 </ sub >.

If we integrate over a surface of uniform intensity I, for instance over a sphere centered around a point source radiating equally in all directions, the equation becomes:

If the rider is a member of the police or military, they are permitted to wear their respective uniform during competition.

with the element of path along the integration of electric field vector E. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field ( see figure ), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation:

If the existence of the crime is not in doubt, only the identity of the culprit, it has been suggested that the prior should be uniform over the qualifying population.

If the field is uniform in space, the magnet is subject to no net force, although it is subject to a torque.

If a nurse is seen outside in his or her uniform, this is a reprimandable offence and the nurse may be subjected to a review.

If sand particles in a fluidized state is heated to the ignition temperatures of coal, and coal is injected continuously into the bed, the coal will burn rapidly and bed attains a uniform temperature.

If a physical object has uniform density, then its center of mass is the same as the centroid of its shape.

If the gauges are many and evenly distributed over an area of uniform precipitation, using the arithmetic mean method will give good results.

If instead of a uniform downwards gravitational force we consider

If the atmosphere had a uniform density, it would terminate abruptly at an altitude of.

If one took a uniform prior, then the maximum likelihood estimate for the population probability is the observed probability, and one may compute a credible region around this or another estimate.