If A is the major axis of an ellipsoid and B and C are the other two axes, the radius of curvature in the ab plane at the end of the axis Af, and the difference in pressure along the A and B axes is Af.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

** 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 the valid element indices begin at 0, the constant B is simply the address of the first element of the array.

If the numbering does not start at 0, the constant B may not be the address of any element.

If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

If and we have for all v, w in V, then we say that B is symmetric.

In addition to acting and occasionally directing, Campbell has become a writer, starting with an autobiography, If Chins Could Kill: Confessions of a B Movie Actor published on August 24, 2002.

If the sets A and B are equal, this is denoted symbolically as A = B ( as usual ).

If the convention B < sub > 1 </ sub >=− is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS ); with the convention B < sub > 1 </ sub >=+ is known as the second Bernoulli numbers ( / in OEIS ).

* Indicative conditional, a conditional in the form of " If A then B " in natural languages

If an atom A is double-bonded to an atom B, A is treated as being singly bonded to two atoms: B and a ghost atom that has the same atomic number as B but is not attached to anything except A.

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 A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.

If two cofinal subsets of B have minimal cardinality ( i. e. their cardinality is the cofinality of B ), then they are order isomorphic to each other.

If every object X < sub > i </ sub > of C admits a initial morphism to U, then the assignment and defines a functor V from C to D. The maps φ < sub > i </ sub > then define a natural transformation from 1 < sub > C </ sub > ( the identity functor on C ) to UV.

If a set is well-ordered ( or even if it merely admits a wellfounded relation ), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.

He admits, " If there's a bigger influence on Buffy than Kitty, I don't know what it was ... She was an adolescent girl finding out she has great power and dealing with it.

If the accused admits to having a motive consistent with the elements of foresight and desire, this will add to the level of probability that the actual outcome was intended ( it makes the prosecution case more credible ).

If the patient admits to hair pulling, diagnosis is not difficult ; if patients deny hair pulling, a differential diagnosis must be pursued.

If a given manifold admits a geometric structure, then it admits one whose model is maximal.

If such an operation admits an identity element o ( such that for all ), then this element is unique ( ).

In 1743 he published the paper On the structure and diseases of articulating cartilages – which is often cited – especially the following sentence: “ If we consult the standard Chirurgical Writers from Hippocrates down to the present Age, we shall find, that an ulcerated Cartilage is universally allowed to be a very troublesome Disease ; that it admits of a Cure with more Difficulty than carious Bone ; and that, when destroyed, it is not recovered ”.

If the accused admits to the offence, or there are reliable witness statements, the chair will call for proposals.

When Johnny admits himself into a hospital late at night, he is met by Mr. Carlson, a devoted Christian, who tells Johnny that there is nothing wrong with hearing God's voice, but doubted it was God, saying " If God had something He wanted to say, you'd hear it.

If the multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations.

If a differential operator L admits a set of eigenvectors ( i. e., a set of functions and scalars such that ) that is complete, then it is possible to construct a Green's function from these eigenvectors and eigenvalues.

Ure is a recovering alcoholic ; something he openly admits and discusses in his autobiography If I Was.

The length of a finite resolution is the subscript n such that P < sub > n </ sub > is nonzero and P < sub > i </ sub >= 0 for i greater than n. If M admits a finite projective resolution, the minimal length among all finite projective resolutions of M is called its projective dimension and denoted pd ( M ).

Let P be a principal H-bundle on M, equipped with a Cartan connection η: TP → g. If g is a reductive module for H, meaning that g admits an Ad ( H )- invariant splitting of vector spaces g = h ⊕ m, then the m-component of η generalizes the solder form for an affine connection.

If a vector field v admits a vector potential A, then from the equality

If ( and only if ) the victim is more than 1 – 2 hours away from a medical facility, it is recommended to place a lightly constricting band ( that admits one finger beneath it ) above the bitten area to prevent the systemic spread of the venom.

The length of a finite flat resolution is the first subscript n such that F < sub > n </ sub > is nonzero and F < sub > i </ sub >= 0 for i greater than n. If a module M admits a finite flat resolution, the minimal length among all finite flat resolutions of M is called its flat dimension and denoted fd ( M ).

If the restriction of this action to H is trivial, then H is said to be normal, and the quotient scheme admits a natural group law.

If M admits an almost complex structure, it must be even-dimensional.

If M admits local holomorphic coordinates for J around every point then these patch together to form a holomorphic atlas for M giving it a complex structure, which moreover induces J. J is then said to be ' integrable '.

If there is such a map S, then it is called an antipode, and H is a Hopf algebra.

* 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.

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