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 we fail to develop the means to hunt down and destroy the enemy's military force with extreme care and precision, and if war comes in spite of our most ardent desires for peace, our choice of alternatives will be truly frightening.

If, in Larkin's eyes, they are nothing but Piccadilly farmers, he has as much to learn about them as they have to learn about the ways of truly rural living.

Naamani Tarkow has written: " If one is to make sweeping statements, one may say that, save Magna Carta ( more truly, its implications ), the Act of Settlement is probably the most significant statute in English history ".

If the descent were truly unbroken, father-to-son, since Confucius's lifetime, the males in the family would all have the same Y chromosome as their direct male ancestor, with slight mutations due to the passage of time.

:“ If an integer n is greater than 2, then has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain .”

If a criminal were truly aware of the mental and spiritual consequences of his actions, he would neither commit nor even consider committing those actions.

If we wish for nothing but what God wills, we shall be truly free, and all will come to pass with us according to our desire ; and we shall be as little subject to restraint as Zeus himself.

If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which all other particles are made.

The Council of Trent decreed: " If anyone shall say that a man once justified can sin no more, nor lose grace, and that therefore he who falls and sins was never truly justified ; or, on the contrary, that throughout his whole life he can avoid all sins even venial sins, except by a special privilege of God, as the Church holds in regard to the Blessed Virgin: let him be anathema.

If the key is truly random, as large as or greater than the plaintext, never reused in whole or part, and kept secret, the ciphertext will be impossible to decrypt or break without knowing the key.

If they were elected it would pose the dilemma of which House was truly representative of the electorate.

If such a union of human and divine occurred, Nestorius believed that Christ could not truly be con-substantial with God and con-substantial with us because he would grow, mature, suffer and die ( which Nestorius argued God cannot do ) and also would possess the power of God that would separate him from being equal to humans.

Nimoy also played the Mermelstein role and believes: " If every project brought me the same sense of fulfillment that Never Forget did, I would truly be in paradise.

If the central office is not convinced that such a message is truly sent from an authorized source, acting on such a request could be a grave mistake.

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

If such a union of human and divine occurred, Nestorius believed that Christ could not truly be con-substantial with God and con-substantial with us because he would grow, mature, suffer and die ( which he said God cannot do ) and also would possess the power of God that would separate him from being equal to humans.

If there be an object truly ridiculous in nature, it is an American patriot, signing resolutions of independency with the one hand, and with the other brandishing a whip over his affrighted slaves.

If man and he were truly generic, the parallel phrase would have been he has difficulties in childbirth.

If you want to know which, pay attention to what it means to be truly human in a world that half the time we're in love with and half the time scares the hell out of us.

If the British people were ever to ask themselves what power they truly enjoyed under our political system they would be amazed to discover how little it is, and some new Chartist agitation might be born and might quickly gather momentum.

For example, given a graph G =( V, E ), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then the path p < sub > 1 </ sub > from u to w and p < sub > 2 </ sub > from w to v are indeed the shortest paths between the corresponding vertices ( by the simple cut-and-paste argument described in CLRS ).

If he truly was the author of the five or six long works in Scots which different witnesses ascribe to him, then he would have been one of the most voluminous writers of Early Scots, if not the most voluminous of all Scots poets.

If Vedanta truly epitomises the state of learnedness, in achieving this spiritual progress " the first stage for a layman is the external / material worship ; struggling to rise high, mental prayer is the next stage, but the highest stage is when the divine has been realized " Unity in variety is the scheme of nature, and the Hindu has recognized it and practised eversince the yore through his equanimity to all and universal tolerance ".

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

* If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two non-real roots, then the Galois group of f is the full symmetric group S < sub > p </ sub >.

If a decomposition exists with each c < sub > i </ sub > a centrally primitive idempotent, then R is a direct sum of the corner rings c < sub > i </ sub > Rc < sub > i </ sub >, each of which is ring irreducible.

If irreducible complexity is an insurmountable obstacle to evolution, it should not be possible to conceive of such pathways.

If W is irreducible then each type can, through a sequence of mutations ( i. e. powers of W ) mutate into all the other types.

If W isn't irreducible, then the dominant species ( or quasispecies ) that develops can depend on the initial population, as is the case in the simple example given below.

If n is odd, this representation is irreducible.

If n is even, it splits again into two irreducible representations called the half-spin representations.

# If n = 2k + 1, then the action of the unit vector u on the left ideal decomposes the space into a pair of isomorphic irreducible eigenspaces ( both denoted by Δ ), corresponding to the respective eigenvalues + 1 and − 1.

If k is a field and G is a group, then a group representation of G is a left module over the group ring k. The simple k modules are also known as irreducible representations.

If q < sub > 1 </ sub >,..., q < sub > m </ sub > are irreducible elements of R and w is a unit such that

If irreducible, hernias can develop several complications ( hence, they can be complicated or uncomplicated ):

* If P ( x ) is a primitive polynomial which has integer coefficients but is not monic, and P is irreducible over Q, then none of the roots of P are algebraic integers.

If R is an integral domain, an element f of R which is neither zero nor a unit is called irreducible if there are no non-units g and h with f = gh.

If X is an affine algebraic set ( irreducible or not ) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some Equivalently, it can be checked that:

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.

If k is not algebraically closed, for example the field of real numbers, the picture becomes more complicated because of the existence of non-linear irreducible polynomials.

If complex numbers are allowed, only 1st-degree polynomials can be irreducible.

If one allows only rational numbers, or a finite field, then some higher-degree polynomials are irreducible.

* If Q ( x ) contains factors which are irreducible over the given field, then the numerator N ( x ) of each partial fraction with such a factor F ( x ) in the denominator must be sought as a polynomial with deg N < deg F, rather than as a constant.

If, for example, we simply look at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum of m ( m − 1 )/ 2, where m is the multiplicity, over all infinitely near singular points Q lying over the singular point P. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at P. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of where is the local ring at P and is its integral closure.

If there are symmetries, then the correlation function can be broken up into irreducible representations of the symmetries — both internal and spacetime.

If G is a Kac-Moody algebra, then in any irreducible highest weight representation of U < sub > q </ sub >( G ), with highest weight, the multiplicities of the weights are equal to their multiplicities in an irreducible representation of U ( G ) with equal highest weight.