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, by the Casorati – Weierstrass theorem, for any transcendental entire function f and any complex w there is a sequence with, is necessarily a polynomial, of degree at least n.

Specifically, the hash function is used to map the search key to the hash.

Specifically, the results of this study suggest that this phenomenon occurs in a manner similar to the way that light passes through fiber optic tubes ( which do not function as effectively when kinked or sharply curved or coiled ).

Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces.

Specifically in the context of data transmission, an abort is a function invoked by a sending station to cause the recipient to discard or ignore all bit sequences transmitted by the sender since the preceding flag sequence.

Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.

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, the entire wave function is treated as a single abstract vector:

Specifically, it is a function

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, a space velocity along the line-of-sight means that the transformation from tangential linear velocity to ( angular ) proper motion is a function of time.

Specifically, the timer. c :: calc_load () function will run the algorithm every 5 * Hz, or roughly every five seconds.

Specifically, this is the scope of potential uses of an object outside of the limits prescribed by culture or convention, and is instead simply the empirical spectrum of function for a technological artifact.

Specifically, Michael Luby and Charles Rackoff analyzed the Feistel cipher construction, and proved that if the round function is a cryptographically secure pseudorandom function, with K < sub > i </ sub > used as the seed, then 3 rounds is sufficient to make the block cipher a pseudorandom permutation, while 4 rounds is sufficient to make it a " strong " pseudorandom permutation ( which means that it remains pseudorandom even to an adversary who gets oracle access to its inverse permutation ).

Specifically, comparative researchers encounter problems associated with individual differences, differences in motivation, differences in reinforcement, differences in sensory function, differences in motor capacities, and species-typical preparedness ( i. e. some species have evolved to acquire some behaviors quicker than other behaviors ).

Specifically, the Legendre transform of a convex function ƒ is the function ƒ < sup >∗</ sup > defined by

Specifically, suppose that X and Y are metric spaces and ƒ is a function from X to Y.

Specifically we were given a boolean function whose input is 1 bit, and asked if it is constant.

Specifically, the active amplification function of the outer hair cells is very sensitive to damage from exposure to trauma from overly-loud sounds or to certain ototoxic drugs.