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Page "Groupoid" ¶ 53
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If and group
`` If the day should ever come that foreign invaders swarm ashore along the Gulf Coast '', the account reads, `` they can count on heavy opposition from a group of commando-trained telephone employees -- all girls.
If coffee is sold at the cafeteria, let a few workers in each department get it for the whole group.
If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.
If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.
If the joining of monomers takes place at such a carbon atom, the free hydroxy group of the pyranose or furanose form is exchanged with an OH-side-chain of another sugar, yielding a full acetal.
If the amino group is removed from an amino acid, it leaves behind a carbon skeleton called an α-keto acid.
If a bubble touches identically-colored bubbles, forming a group of three or more, those bubbles — as well as any bubbles hanging from them — are removed from the field of play, and points are awarded.
If the Bastarnae remained an identifiable group, it is highly likely that they participated in the vast Gothic-led migration, driven by Hunnic pressure, that was admitted into Moesia by emperor Valens in 376 and eventually defeated and killed Valens at Adrianople in 378.
* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).
If the carbonyl group is an aldehyde, the monosaccharide is an aldose ; if the carbonyl group is a ketone, the monosaccharide is a ketose.
If there is a tie, we must consider the atoms at distance 2 from the stereocenter — as a list is made for each group of the atoms bonded to the one directly attached to the stereocenter.
If the addition of a functional group takes place at the less substituted carbon atom of the double bond, then the electrophilic substitution with acids is not possible.
If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.
If G is a group, and g is a fixed element of G, then the conjugation map
If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in " standard form " meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1.
If a database cannot tolerate even such smaller probability of data loss, then the RAID group itself is replicated ( mirrored ).
If a database cannot tolerate even such smaller probability of data loss, then the RAID group itself is replicated ( mirrored ).
* If we think of as the group of real numbers under addition, then the direct product still consists of.
If the ether is part of a more complex molecule, it is described as an alkoxy substituent, so-OCH < sub > 3 </ sub > would be considered a " methoxy -" group.
If there is another functional group at a carbon, it may be named with the Greek letter, e. g., the gamma-amine in gamma-aminobutanoic acid is on the third carbon of the carbon chain attached to the carboxylic acid group.
Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.

If and G
If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.
If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:
If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).
If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.
If at least one of a set of configurations must occur somewhere in G, that set is called unavoidable.
If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.
If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K → H such that f = h φ.
If f is an element of G ( x, y ) then x is called the source of f, written s ( f ), and y the target of f ( written t ( f )).
If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

If and acts
If the weight is suspended off the seabed it acts as a spring or shock absorber to dampen the sudden actions that are normally transmitted to the anchor and can cause it to dislodge and drag.
If a similar ball is charged by the same glass rod, it is found to repel the first: the charge acts to force the two balls apart.
If buried where the earth is very cold, the earth acts like a refrigerator.
If the federal government assumed such powers, its acts could be declared unconstitutional by the states.
# If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
If it lapses, the legal basis for enforcing discipline disappears, and soldiers lose their legal indemnity for acts committed under orders.
" If certain acts of violation of treaties are crimes, they are crimes whether the United States does them or whether Germany does them, and we are not prepared to lay down a rule of criminal conduct against others which we would not be willing to have invoked against us ... We must never forget that the record on which we judge these defendants is the record on which history will judge us tomorrow.
Psychological egoism has been accused of being circular: " If a person willingly performs an act, that means he derives personal enjoyment from it ; therefore, people only perform acts that give them personal enjoyment.
If a judge acts against precedent and the case is not appealed, the decision will stand.
If Bob bets $ 10 ( half the pot ), when his opponent acts, the pot will be $ 30 and it will cost $ 10 to call.
If the idea of absolute freedom and entire self-knowledge is absent from this kind of idea of God's acts in time, then God Himself is ( to express the idea anthropomorphically ) becoming something new, or discovering something new about Himself with each new moment, just as we are.
If ρ acts on a finite dimensional Hilbert space and has eigenvalues
If a person occupies land for the required period of time as set out in provincial limitation acts and, during that time, no legal action is taken to evict or in trespass, the ownership in the land goes from the legal owner to the squatter.
If the center of gravity exists within the foundations, the body is stable since no net torque acts on the body.
If the drain is opened and the tumbler is kept in position over the hole, the water in the tumbler drains out leaving the glass filled with air, and this then acts as the plug.
For example, in a negative feedback loop, the transcription factor acts as its own repressor: If the transcription factor protein binds the DNA of its own gene, it will down-regulate the production of more of itself.
# Lubrication: ( for intercourse or used daily to minimize irritation ) If you have problems with yeast, or are worried you will, avoid lubricants with glycerine in them, which acts like a sugar and will only add to your problems.
If the Senate or a later emperor did not like the acts of an individual, they could have his property seized, his name erased and his statues reworked.
" If the Emperor, has no more of mercy in his heart than he reveals in his policies, then I pity Russia ; if, on the other hand, his true sentiments are really superior to his acts, then I pity the Emperor.
If water gets between the friction surfaces and the drum, it acts as a lubricant and reduces braking efficiency.
If, then he should change about, wrong the people, maim its power, and rob it of the privilege of voting, he has by his own acts deprived himself of his honourable office by not fulfilling the conditions on which he received it ; for otherwise there would be no interference with a tribune even though he should try to demolish the Capitol or set fire to the naval arsenal.
If two players have the same high upcard, the one first in clockwise rotation from the dealer acts first.
If Alice just has two pair and Ted acts strongly, Alice may think Ted has three fours and fold if she doesn't improve.
If only two terminals are used, one end and the wiper, it acts as a variable resistor or rheostat.

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