* 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 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 we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.

If pressed by the sitter for more detail, she may be able to bring the picture more into focus and see more sharply, almost as if she were physically going closer.

If Bultmann's own definition of myth is strictly adhered to ( and it is interesting that this is almost never done by those who make such pronouncements ), the evidence is overwhelming that he does not at all exaggerate the extent to which the mythological concepts of traditional theology have become incredible and irrelevant.

If the disk was not otherwise prepared with a custom format, ( e. g. for data disks ), 664 blocks would be free after formatting, giving 664 × 254 = 168, 656 bytes ( or almost 165 kB ) for user data.

If the glass is cooled quickly in the absence of a strong light source by putting ice on the glass or placing it in the freezer with the door almost closed, it turns backwards ( i. e. the silver sides trail ).

If is much smaller than, the mole ratio is almost identical to the mole fraction.

If is much smaller than, the mass ratio is almost identical to the mass fraction.

If the checksums match, the data was almost certainly not altered.

* If ƒ ( z ) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy – Riemann equations weakly, then ƒ agrees almost everywhere with an analytic function in Ω.

If the ticking rate is everywhere almost this same, the fractional rate of change is the same as the absolute rate of change, so that:

If Polybius is correct in his figure for the number of troops he commanded after the crossing of the Rhone, this would suggest that he had lost almost half of his force.

If one SD is 15 points, as is common in almost all modern tests, then 95 % of the population are within a range of 70 to 130, and 98 % are below 131.

If the applied voltage is large enough, the liquid crystal molecules in the center of the layer are almost completely untwisted and the polarization of the incident light is not rotated as it passes through the liquid crystal layer.

If the list is stored as an ordered array, then binary search is almost always more efficient than linear search as with n > 8, say, unless there is some reason to suppose that most searches will be for the small elements near the start of the sorted list.

:# If a general medical condition or another mental disorder is present, the fear in Criterion A ( Exposure to the social or performance situation almost invariably provokes an immediate anxiety response ) is unrelated to it, e. g., the fear is not of Stuttering, trembling in Parkinson's disease, or exhibiting abnormal eating behavior in Anorexia Nervosa or Bulimia Nervosa.

If the transmitter's output is coherent ( phase synchronized ), there is another effect that can be used to make almost instant speed measurements ( no memory is required ), known as the Doppler effect.

If they had any significant level of non-organic parts then something as simple as an x-ray scan would show them not to be human ( unless their creators designed the non-organic parts to have almost exactly the same structure and density as the organic components they were replacing, which would be both infeasible and highly improbable but not completely out of the question ).

If a tumor is causing syringomyelia, removal of the tumor is the treatment of choice and almost always eliminates the syrinx.

If the relations between letters and sounds are similar in both languages, a transliteration may be ( almost ) the same as a transcription.

If I speak in an almost worshipful sense about this music, it is not because I have lost perspective, it is precisely because I have found it, within music, yes, that was made five years ago.

If a whisky is labelled " pure malt " or just " malt " it is almost certain to be a blended malt whisky.

If such sources are removed from residential areas, almost the only mosquitoes able to attack people are those that fly in or are blown in from where they have bred.

If it was really a vassal state, it was contrary to the policy of the Reconquista to allow it to flourish for almost two centuries and a half after the fall of Sevilla in 1248.

If his plans failed, it was almost always out of luck on behalf of the other characters.

If the sun-facing surface is an almost perfect reflector, the force would approach double that ( 7. 22 µPa ) depending on how close to an ideal reflector the surface is polished.