* 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 )

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

* 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:

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

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.

If ( m, n ) is regular and M and N have i and j prime factors respectively, then ( m, n ) is said to be of type ( i, j ).

If X is a set and M is a complete metric space, then the set B ( X, M ) of all bounded functions ƒ from X to M is a complete metric space.

If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.

If used, the word „ gothic “ was used to describe ( mostly early ) works of F. M. Dostoevsky.

If ψ is satisfiable in a structure M, then certainly so is φ and if ψ is refutable, then is provable, and then so is ¬ φ, thus φ is refutable.

If is satisfiable in a structure M, then, considering, we see that is satisfiable as well.

If the password is correct, then M releases the transferred sum to B ( 3b ), usually minus a small commission.

If M is an R module and is its ring of endomorphisms, then if and only if there is a unique idempotent e in E such that and.

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 a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

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 he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

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

If the absorption losses can be neglected, ε is a Hermitian matrix.

If D is Hermitian the effect is dichroism, while a unitary matrix models birefringence.

For a particle which has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is everywhere constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right.

If A has real entries and is symmetric ( or more generally, has complex-valued entries and is Hermitian ) and positive definite, then A can be decomposed as

If H is Hermitian ( the name for self-adjoint in the physics literature ) and f is a Borel function,

An observable is represented by a self-adjoint ( i. e. Hermitian ) operator A on H. If H is finite dimensional, by the spectral theorem, A has an orthonormal basis of eigenvectors.

If the conjugate transpose of a matrix is denoted by, then the Hermitian property can be written concisely as

If X is a Hilbert space and T is a normal operator, then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators ( Hermitian matrices, for example ).

* If 2 is invertible, then and are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of symmetric and anti-symmetric ( Hermitian and skew Hermitian ) elements.

* If A is skew-Hermitian, then both iA and-iA are Hermitian.

* If A is skew-Hermitian, then A < sup > k </ sup > is Hermitian if k is an even integer and skew-Hermitian if k is an odd integer.

If A and H are Hermitian matrices, then

* If f is Hermitian, then

If they are the same then φ is said to be Hermitian.

* If, i. e., is Hermitian, every eigenvalue is real.

If we allow magneto-optic materials, but restrict ourselves to the situation where material absorption is negligible, then ε and μ are in general 3 × 3 complex Hermitian matrices.