If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

If the master of scops who was most responsible for the poem ever used kennings that were traditional, he was at least partly deprived of free will and not inclined towards shrewd and sophisticated misuse of speech elements.

If elements in the sample space increase arithmetically, when placed in some order, then the median and arithmetic average are equal.

If so, they may have originally comprised residual Celtic elements in central eastern Europe such as the Cotini, who formed a Celtic enclave in the Germanic-speaking zone and are described by Tacitus as iron-ore miners working as tributaries of the powerful Quadi Germanic people.

More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.

If so, it is unclear how much Tatian may have borrowed from this previous author in determining his own narrative sequence of Gospel elements.

If ~ is an equivalence relation on X, and P ( x ) is a property of elements of X, such that whenever x ~ y, P ( x ) is true if P ( y ) is true, then the property P is said to be well-defined or a class invariant under the relation ~.

If is an equivalence relation on, and is a property of elements of such that whenever, is true if is true, then the property is said to be an invariant of, or well-defined under the relation.

If found, it removes it by concatenating all the elements prior to it with all the elements after it.

) If a set is finite, its elements may be written as a sequence:

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 the Gospel of Barnabas is seen as an attempted synthesis of elements from both Christianity and Islam, then 16th and 17th century parallels can be suggested in Morisco and anti-Trinitarian writings.

If so, these huge stars would have quickly consumed their supply of fuel and became supernovae, releasing heavy elements into the interstellar medium.

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 the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional ( grassman -) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups ( F < sub > 4 </ sub >, E < sub > 6 </ sub >, E < sub > 7 </ sub > or E < sub > 8 </ sub >) depending on the details.

If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are " spread apart " from each other while leaving the arithmetic mean unchanged — then the geometric mean always decreases.

If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are " spread apart " from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.

If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible.

If R is the direct sum of the rings R < sub > 1 </ sub >,..., R < sub > n </ sub >, then the identity elements of the rings R < sub > i </ sub > are central idempotents in R, pairwise orthogonal, and their sum is 1.

If the figure comprises more than three elements, it is said to be rekit “ extended ”.

If a particular KR is inadequate in some way, that set of problematic KR elements may be transformed by importing them into a KBS, modified and operated on to eliminate the problematic elements or augmented with additional knowledge imported from other sources, and then exported into a different, more appropriate KR.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

* If S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )

* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =

* If a set S is in M and S is congruent to T then T is also in M and a ( S )

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

* 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.

If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

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 I even hint at it do you think it will matter that you are his nephew -- and not even a blood nephew ''??

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 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 to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.

If Jews are identified as a religious body in a controversy that comes before a national or international tribunal, it is obviously compatible with the goal of human dignity to protect freedom of worship.

If they are right, they will prevail of and by themselves.

Without preliminaries, Esther asked him, `` If you are a world citizen, will you take Garry Davis' place in his tent while he goes to the hospital ''??

If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.

If the would-be joiner asks these questions he is not likely to be duped by extremists who are seeking to capitalize on the confusions and the patriotic apprehensions of Americans in a troubled time.

If we break the minister to our bit, we are buying back our own sins.

If the record buyer's tastes are somewhat eclectic or even the slightest bit esoteric, he will find them satisfied on educational records.

If Daddy's books are out of bounds his own picture books are not.

If it will simply delay the debates until the qualifications are closed next spring, and then carry all the candidates on a tour of debates, it can provide a service to the state.

If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??

If the Communists are sincere in wanting a united, neutral and disarmed Germany, it might well be advantageous for the German people in this nuclear age.

If only this could be done more often -- with such heartening results -- many of the earth's `` big problems '' would shrink to the insignificances they really are.

If any are left, presently, we may expect to see signs specifically prohibiting the feeding of them too.

If the raw population figures are crucially relevant, then it is idle to think of liberation, as idle as to suppose that Poland might liberate Russia.

If it is not enough that all of our internationalist One Worlders are advocating that we join this market, I refer you to an article in the New York Times' magazine section ( Nov. 12, 1961 ), by Mr. Eric Johnston, entitled `` We Must Join The Common Market ''.

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