Specifically, if A is a linear operator and is a bra, then is another bra defined by the rule

Specifically, a topological space is compact if, whenever a collection of open sets covers the space, some subcollection consisting only of finitely many open sets also covers the space.

Specifically, if p is the natural length of a section of spring, then the length of the spring with tension T applied has length

Specifically, if one or more of the produced neutrons themselves interact with other fissionable nuclei, and these also undergo fission, then there is a possibility that the macroscopic overall fission reaction will not stop, but continue throughout the reaction material.

Specifically, if a judge consistently gives low scores for all divers, or consistently gives high scores for the same divers, the judging will yield fair relative results and will cause divers to place in the correct order.

Specifically, if he / she chooses to determine the path, then he / she has no influence whatsoever over which of the two paths, the left one or the right one, nature will tell him / her is the one in which the particle is found.

Specifically, if hair texture were simply the result of adaptively arbitrary human aesthetic preferences, one would expect that the global distribution of the various hair textures would be fairly random.

Specifically, if ƒ is an invertible function with domain X and range Y, then its inverse ƒ < sup >− 1 </ sup > has domain Y and range X, and the inverse of ƒ < sup >− 1 </ sup > is the original function ƒ.

Specifically, if the cylinders are oriented parallel to one another, there is very little volume that is excluded from the center-of-mass of the approaching cylinder ( it can come quite close to the other cylinder ).

Specifically, umpires were now given the power — and the responsibility — to intervene if they considered a bowler was deliberately aiming at a batsman with intent to injure.

Specifically, if the angular momentum is high enough the event horizons will disappear.

Specifically, some cardrooms only grant the right to view a mucked losing hand if the requesting player articulates a concern about possible collusion.

Specifically, if L / K is a Galois extension, we consider the group G = Gal ( L / K ) consisting of all field automorphisms of L which keep all elements of K fixed.

Specifically, extra management is needed if multiple threads are allowed to perform find operations concurrently.

Specifically, a construct in a language is called syntactic sugar if it can be removed from the language without any effect on what the language can do: functionality and expressive power will remain the same.

Specifically, if v is a tangent vector of M at a point x ( thought of as a derivation ) then define the directional derivative in the direction v by

This can be shown using the concept of left cosets of H in G. The left cosets are the equivalence classes of a certain equivalence relation on G and therefore form a partition of G. Specifically, x and y in G are related if and only if there exists h in H such that x

Specifically, it is unique up to a unique isomorphism: if ( A ′, φ ′) is another such pair, then there exists a unique isomorphism k: A → A ′ such that φ ′

Specifically, if F is a vector space of linear functionals on X which separates points of X, then the continuous dual of X with respect to the topology σ ( X, F ) is precisely equal to F.

Specifically, if the vacuum energy is negative and its absolute value is substantially larger than it appears to be in the observed universe ( say, a factor of 10 larger ), holding all other variables ( e. g. matter density ) constant, that would mean that the universe is closed ; furthermore, its lifetime would be shorter than the age of our universe, possibly too short for the intelligent life to form.

Specifically, the above argument would become a fallacious ad baculum if the conclusion stated:

Specifically, when the scale of the universe doubles, the density of matter is reduced by a factor of 8, but the density of dark energy is nearly unchanged ( it is exactly constant if the dark energy is a cosmological constant ).

A category C is said to have limits of type J if every diagram of type J has a limit in C. Specifically, a category C is said to

Specifically, if one freezes a right-handed wave in time, when one curls the fingers of one ’ s right hand around the helix, the thumb will point in the direction which the helix progresses given that sense of rotation.

Specifically, the authors proved that the positivity required of an analytic function F ( z ) which de Branges would use to construct his proof would also force it to assume certain inequalities that, according to them, the functions actually relevant to a proof do not satisfy.

Specifically, f (' F ', ' F ', ..., ' F ') = ' F ', and these functions correspond to linear maps over the Boolean vector space.

Specifically, the symbol F in a formal language is a functional symbol if, given any symbol X representing an object in the language, F ( X ) is again a symbol representing an object in that language.

Specifically, if you can prove that for every X ( or every X of a certain type ), there exists a unique Y satisfying some condition P, then you can introduce a function symbol F to indicate this.

Specifically, if F has domain type T and codomain type U, then it can be replaced with a predicate P of type ( T, U ).

Specifically, the similarity between the fiber bundle E and a product space B × F is defined using a continuous surjective map

Specifically, if F is a diagram in Top and ( L, φ ) is a limit of UF in Set, the corresponding limit of F in Top is obtained by placing the initial topology on ( L, φ ).

Specifically, the burglars were interested in information they thought was held by Lawrence F. O ' Brien, head of the DNC.

Specifically, a straight line on a log-log plot containing points ( F < sub > 0 </ sub >, x < sub > 0 </ sub >) and ( F < sub > 1 </ sub >, x < sub > 1 </ sub >) will have the function:

Specifically, if H is a subgroup of Gal ( E / F ), then | H | = and | Gal ( E / F )/ H | =.

Specifically, the air temperature at the altitude where the air pressure is 850 millibars ( or altitude ) should be 13 ° C ( 24 ° F ) lower than the temperature of the air at the surface.

The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism is isomorphic to where ker ( φ ) denotes the kernel of φ.

Specifically, the adjacency matrix of a finite graph G on n vertices is the n × n matrix where the non-diagonal entry a < sub > ij </ sub > is the number of edges from vertex i to vertex j, and the diagonal entry a < sub > ii </ sub >, depending on the convention, is either once or twice the number of edges ( loops ) from vertex i to itself.

Specifically, let G be a simply-connected Lie group with Lie algebra.

Specifically, if is an n-dimensional representation of G, then u < sub > ij </ sub > ∈ C ( G ) for all i, j, and for all i, j.

Specifically, given g ∈ G, ρ ( g ) is the linear map on V determined by its action on the basis by right translation by g < sup >− 1 </ sup >, i. e.

In general, a different choice of origin o will lead to a quotient of G by a different subgroup H < sub > o ′</ sub > which is related to H < sub > o </ sub > by an inner automorphism of G. Specifically,

Specifically, it is a single-subunit small GTPase, which is related in structure to the G < sub > α </ sub > subunit of heterotrimeric G proteins ( large GTPases ).

Specifically, let G be a group and let X and Y be two associated G-sets.

Specifically, if X and Y are the representation spaces of two linear representations of G then a linear map f: X → Y is called an intertwiner of the representations if it commutes with the action of G. Thus an intertwiner is an equivariant map in the special case of two linear representations / actions.

Specifically, GSM uses a regular pulse excited-long term prediction ( RPE-LTP ) coder for voice data between the mobile device and the BSS, but pulse code modulation ( A-law or μ-law standardized in ITU G. 711 ) upstream of the BSS.

Specifically, the G. R. E. A. T.

If H is a Lie subgroup of G, then there is a natural one-to-one correspondence between reductions of a G-bundle B to H and global sections of the fiber bundle B / H obtained by quotienting B by the right action of H. Specifically, the fibration B → B / H is a principal H-bundle over B / H.