If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

If a union cannot perform this function, then collective bargaining is being palmed off by organizers as a gigantic fraud.

If the Greek letter is used, it is assumed to be a Fourier transform of another function,

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and − 1 indicating perfect anti-correlation.

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 we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If f is not a function, but is instead a partial function, it is called a partial operation.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If your side has two aces and a void, then you are not at risk of losing the first two tricks, so long as ( a ) your void is useful ( i. e., does not duplicate the function of an ace that your side holds ) and ( b ) you are not vulnerable to the loss of the first two tricks in the fourth suit ( because, for instance, one of the partnership hands holds a singleton in that suit or the protected king, giving your side second round control ).

If evolutionary processes are blind to the difference between function F being performed by conscious organism O and non-conscious organism O *, it is unclear what adaptive advantage consciousness could provide.

If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results.

If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained.

If we take the simple valence bond structure and mix in all possible covalent and ionic structures arising from a particular set of atomic orbitals, we reach what is called the full configuration interaction wave function.

If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.

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 we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward.

If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.

If A consisted of three regions, six or more colors might be required ; one can construct maps that require an arbitrarily high number of colors.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

If ƒ maps X to Y, then ƒ < sup >– 1 </ sup > maps Y back to X.

If it does, however, it is unique in a strong sense: given any other inverse limit X ′ there exists a unique isomorphism X ′ → X commuting with the projection maps.

If M is an open subset of R < sup > n </ sup >, then M is a C < sup >∞</ sup > manifold in a natural manner ( take the charts to be the identity maps ), and the tangent spaces are all naturally identified with R < sup > n </ sup >.

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 U is a subset of the metric space M and ƒ: U → R is a Lipschitz continuous function, there always exist Lipschitz continuous maps M → R which extend ƒ and have the same Lipschitz constant as ƒ ( see also Kirszbraun theorem ).

If we let be the inclusion functor from CHaus into Top, maps from to ( for in CHaus ) correspond bijectively to maps from to ( by considering their restriction to and using the universal property of ).

* If a function f: X → Y maps every base element of X into an open set of Y, it is an open map.

* If K is a field and we consider the K-vector space K < sup > n </ sup >, then the endomorphism ring of K < sup > n </ sup > which consists of all K-linear maps from K < sup > n </ sup > to K < sup > n </ sup >.

If a direct visual fix cannot be taken, it is important to take into account the curvature of the Earth when calculating line-of-sight from maps.

If X is a set, a diffeology on X is a set of maps, called plots, from open subsets of R < sup > n </ sup > ( n ≥ 0 ) to X such that the following hold:

Displayed are parts of the ( disjoint ) sets A and B together with parts of the mappings f and g. If the set A ∪ B, together with the two maps, is interpreted as a directed graph, then this bipartite graph has several connected components.

# If C is any small category, then there exists a faithful functor P: Set < sup > C < sup > op </ sup ></ sup > → Set which maps a presheaf X to the coproduct.

If the presheaves or sheaves considered are provided with additional algebraic structure, these maps are assumed to be homomorphisms.

If F ( U ) is a module over the ring O < sub > X </ sub >( U ) for every open set U in X, and the restriction maps are compatible with the module structure, then we call F an O < sub > X </ sub >- module.

* If and are covering maps, then so is the map given by.

If competition beats you to it, this exciting new product era can have real headaches in store.

If the photographically realistic continuity of dreams, however bizarre their combinations, denies that it is purely a composition of the brain, it must be compounded from views of diverse realities, although some of them may never be encountered in what we are pleased to call the real life.

I kept saying, `` If I could just build up a reputation for myself, make some real money, get to be well known as an illustrator -- like Peter Askington, for instance -- then I could take some time off and paint ''.

If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator ; therefore, real scalar fields describe neutral bosons.

Existence is the principle that gives reality to an essence not the same in any way as the existence: " If things having essences are real, and it is not of their essence to be, then the reality of these things must be found in some principle other than ( really distinct from ) their essence.

If X is a real Banach space, then the polarization identity is

If A is a self-adjoint operator, then is always a real number ( not complex ).

If the Czech ethnic origin was to be stressed, combinations like " Bohemian of Bohemian language "( Čech českého jazyka ), " a real Bohemian " ( pravý Čech ) etc.

If the exponent r is even, then the inequality is valid for all real numbers x.

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

If zeal for orthodoxy caused him to overstep the limits of discretion, his real attitude towards Rome is sufficiently clear.

If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin ( i. e. their real and complex component is zero in the continuous time case ).

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.

If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.

If a DBMS system responses users ' request in a given time period, it can be regarded as a real time database.

If this operation has to be done in real time video games there is an easy trick to boost performance.

* If we think of as the set of real numbers, then the direct product is precisely just the cartesian product,.

* If we think of as the group of real numbers under addition, then the direct product still consists of.

* If we think of as the ring of real numbers, then the direct product again consists of.

The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring.