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

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.

If a bialgebra B admits an antipode S, then S is unique (" a bialgebra admits at most 1 Hopf algebra structure ").

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 you can't tell, get help from your county agricultural agent or other local specialist.

If BCG is accidentally given subcutaneously, then a local abscess may form ( a BCG-oma ) that can sometimes ulcerate, and may require treatment with antibiotics.

If given subcutaneously, it may induce local infection and spread to the regional lymph nodes, causing either suppurative and nonsuppurative lymphadenitis.

If the eigenvalues are all positive, then the frequencies are all real and the stationary point is a local minimum.

If it just includes any PUBLIC identifier, it may be processed as standalone, if the XML processor knows this PUBLIC identifier in its local catalog from where it can retrieve an associated DTD entity.

If the low-order bit is 0, the remaining 7 bits of the DSAP specify an individual address, which refers to a single local service access point ( LSAP ) to which the packet should be delivered.

If the cities were treated as relatively autonomous local administrative areas, it would simplify the problems of imperial administration, which as far as Julian was concerned, should be focused on the administration of the law and defense of the empire's vast frontiers.

If set up properly, sent mail can be saved to the server also, in contrast with POP mail, where sent messages exist only in the local MUA and are not visible by other MUAs accessing the same account.

If the key signature indicates G-sharp, a local flat before a G makes it G-flat ( not G natural ), though often this type of rare accidental is expressed as a natural, followed by a flat () to make this clear.

If an unmarried couple change residence to different countries, then the local law on where the couple is last domiciled is applied to them.

If the local court is actually giving extraterritorial effect to a foreign law, it is less than sovereign and so acting in a way that is potentially unconstitutional.

If the case has been submitted to arbitration rather than a national court, say because of a forum selection clause, an arbitrator may decide not to apply local mandatory policies in the face of a choice of law by the parties if this would defeat their commercial objectives.

If the congregation does not have an installed pastor, the Presbytery appoints a minister member or elected member of the presbytery as moderator with the concurrence of the local church session.

If other products are brought inside along with pests and diseases, it would damage the ecosystem seriously and add millions of costs in the local agricultural businesses.

If the two devices are far enough apart or on separate power systems, the local ground connections at either end of the cable will have differing voltages ; this difference will reduce the noise margin of the signals.

If φ: M → N is a local diffeomorphism at x in M then dφ < sub > x </ sub >: T < sub > x </ sub > M → T < sub > φ ( x )</ sub > N is a linear isomorphism.

If the route was learned from an external peer the per-neighbor BGP process computes a LOCAL_PREFERENCE value from local policy rules and then compares the LOCAL_PREFERENCE of all routes from the neighbor.

If the universe is described by an effective local quantum field theory down to the Planck scale, then we would expect a cosmological constant of the order of.

If the object Servant exists on the same server where the method is invoked, they also mediate the local calls.

If Bell's inequalities are violated, either local realism or counterfactual definiteness must be incorrect ; but some physicists dispute that experiments have demonstrated Bell's violations, on the grounds that the sub-class of inhomogeneous Bell inequalities has not been tested or due to experimental limitations in the tests.

( If utility U = f ( x, y ), U, in the third dimension, does not have a local maximum for any x and y values.

Since a one sided maximal ideal A is not necessarily two-sided, the quotient R / A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J ( R ).

If local governments and private organizations are to carry out decentralized functions effectively, they must have an adequate level of revenues, either raised locally or transferred from the central government, as well as the authority to make decisions about expenditures.

If the people of this nation wish to deprive the states of their sovereign rights to determine what is a fair and just tax upon corporations doing a purely local business within their own state boundaries, there is a way provided by the Constitution to accomplish this purpose.

If f is a differentiable function on R ( or an open interval ) and x is a local maximum or a local minimum of f, then the derivative of f at x is zero ; points where are called critical points or stationary points ( and the value of f at x is called a critical value ).