Specifically, suppose that is the projection of the bundle.

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

Specifically, suppose R is a ring, and denote by R-Mod the category of left R-modules and by Mod-R the category of right R-modules ( if R is commutative, the two categories coincide ).

Specifically, suppose that

Specifically, suppose that M and N are a pair of manifolds each carrying a G-structure for a structure group G. This amounts to giving a special class of coframes on M and N. Cartan's method addresses the question of whether there exists a local diffeomorphism φ: M → N under which the G-structure on N pulls back to the given G-structure on M. An equivalence problem has been " solved " if one can give a complete set of structural invariants for the G-structure: meaning that such a diffeomorphism exists if and only if all of the structural invariants agree in a suitably defined sense.

Specifically, suppose that one has a current density that produces an electric field and a magnetic field, where all three are periodic functions of time with angular frequency ω, and in particular they have time-dependence.

Specifically, " every man heard them speak in his own language " and wondered " how hear we every man in our own tongue, wherein we were born?

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, we perceive there to be a group of 36 circles on the left side of the image, and three groups of 12 circles on the right side of the image.

Specifically, given two quantities, p and q, and a rational number m / n we can say that the ratio of p to q is less than, equal to, or greater than m / n when np is less than, equal to, or greater than mq respectively.

Specifically, if we want to solve Ax

Specifically, say we have experimental subjects, rats, in an operant chamber and we require them to press a lever to receive a reward.

Specifically, if a function yields any of different outputs with equal probability and is sufficiently large, then we expect to obtain a pair of different arguments and with after evaluating the function for about different arguments on average.

Specifically, in 1851 William Thomson outlined the view, as based on recent experiments on the dynamical theory of heat, that " heat is not a substance, but a dynamical form of mechanical effect, we perceive that there must be an equivalence between mechanical work and heat, as between cause and effect.

Specifically, assume we have a nonsingular variety X, and n hypersurfaces Z < sub > 1 </ sub >, ..., Z < sub > n </ sub > which have local equations f < sub > 1 </ sub >, ..., f < sub > n </ sub > near x for polynomials f < sub > i </ sub >( t < sub > 1 </ sub >, ..., t < sub > n </ sub >), such that the following hold:

Specifically, let χ be a character modulo k. Then we can write its Dirichlet L-function as

Specifically exemplifying this theory we find Asimov's Solarians: humans evolved from an old Spacer world who had genetically modified themselves into hermaphrodites with telekinetic mental powers.

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

Specifically, we show that

Specifically, we have

Specifically, emotions such as joy and grief tell us about the " good or bad fortune " of the person we are observing them in, whereas anger tells us about the bad fortune with respect to another person.

Specifically, if we sympathize with the feelings of another we judge that their feelings are just, and if we do not sympathize we judge that their feelings are unjust.

Specifically, although we sympathize with the offended person, we fear that the offended person may do harm to the offender, and thus also fear for and sympathize with the danger that faces the offender.

Specifically, we say that a set of clauses of the form is stratified if and only if

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

Specifically, they asked for the Pep Band's return to athletic events, especially those where the marching band does not perform, but the Athletics Department has not acquiesced.

Specifically, US Code Title 44, Chapter 36, defines it as a ' strategic information base ' that defines the mission of an agency and describes the technology and information needed to perform that mission, along with descriptions of how the architecture of the organization should be changed in order to respond to changes in the mission.

Specifically, the CNE training program is intended to train a system administrator on how to use management tools, perform server installations, manage the eDirectory, printing, allocate storage space, email and instant messaging services, and managing security.

Specifically pertaining to CROs providing clinical-trials services, the ICH-GCP ( E6 1. 20 ) defines a Contract Research Organization ( CRO ) as: " A person or an organization ( commercial, academic, or other ) contracted by the sponsor to perform one or more of a sponsor's trial-related duties and functions.

Specifically, in quantum mechanics, the state of an atom, i. e. an eigenstate of the atomic Hamiltonian, is approximated by an expansion ( see configuration interaction expansion and basis set ) into linear combinations of anti-symmetrized products ( Slater determinants ) of one-electron functions.

Specifically, commodities such as sugar are heavily distorted by subsidies on behalf of powerful economies ( the United States, Europe, and Japan ), who have a disproportionate influence in the WTO.

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

Specifically, after acknowledging the various popular theories in vogue at the time, of how atoms were reasoned to attach to each other, i. e. " hooked atoms ", " glued together by rest ", or " stuck together by conspiring motions ", Newton states that he would rather infer from their cohesion, that " particles attract one another by some force, which in immediate contact is exceedingly strong, at small distances performs the chemical operations, and reaches not far from the particles with any sensible effect.

Specifically, to minimize the cutting down of trees for fuel, kerosene is being subsidized, and efforts are being made to replace the loss of the forest cover caused by ylang-ylang distillation for perfume.

Specifically, a male secondary school enrollment 10 % above the average reduced the chance of a conflict by about 3 %, while a growth rate 1 % higher than the study average resulted in a decline in the chance of a civil war of about 1 %.

Specifically, neocolonialism refers to the theory that former or existing economic relationships, such as the General Agreement on Tariffs and Trade and the Central American Free Trade Agreement, created by former colonial powers were or are used to maintain control of their former colonies and dependencies after the colonial independence movements of the post – World War II period.

Specifically, this demand for political democracy implies complete freedom to agitate for secession and for a referendum on secession by the seceding nation.

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 law makes it illegal to publicly " deny, play down, justify or approve of the genocide committed by the German National Socialist regime during the Second World War ".

Specifically, the process was first used by the English " Good Parliament " against Baron Latimer in the second half of the 14th century.

Specifically, they require that a patient must have pain associated with the bladder, accompanied by one other urinary symptom.

Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v ^< sub > g </ sub > =

Specifically, the Massiah rule applies to the use of testimonial evidence in criminal proceedings deliberately elicited by the police from a defendant after formal charges have been filed.

Specifically, he participated in the attempt to take Fushimi castle by assault in July 1600, in the defense of the besieged Gifu Castle in August of the same year, and finally in the Battle of Sekigahara.

Specifically, it is the worldwide, publicly accessible network of interconnected computer networks that transmit data by packet switching using the standard Internet Protocol ( IP ).

According to Stephen Suleyman Schwartz, in his book " The Two Faces of Islam ", “ some say that during this vagabondage Ibn Abdul Wahhab came into contact with certain Englishmen who encouraged him to personal ambition as well as to a critical attitude about Islam .” Specifically, Mir ’ at al Harramin, a Turkish work by Ayyub Sabri Pasha, written in 1888, states that in Basra, Abdul Wahhab had come into contact with a British spy by the name of Hempher, who “ inspired in him the tricks and lies that he had learned from the British Ministry of the Commonwealth .”

Specifically, the nomenklatura consisted of two separate lists: one was for key positions, appointments to which were made by authorities within the party ; the other was for persons who were potential candidates for appointment to those positions.

Specifically, in 1895 Pierre-Hector Coullié, archbishop of Lyon, voiced his official decision on the reinstatement of sports to Pierre de Coubertin, the founder of the Modern Olympic Games, by stating " Nous acceptons tout, sauf pankration " meaning " We accept all to be reinstated, except pankration ".

Specifically, in an environment where it is considered important to know the probability of a fraudulent login in order to accept the risk, one can ensure that the total number of possible passwords multiplied by the time taken to try each one ( assuming the greatest conceivable computing resources ) is much greater than the password lifetime.

Specifically, the campaign raised $ 253. 7 million for undergraduate financial aid, as well as $ 2. 33 billion for its initiative in " Seeking Solutions " to global problems, $ 1. 61 billion for " Educating Leaders " by improving K-12 education, and $ 2. 11 billion for " Foundation of Excellence " aimed at providing academic support for Stanford students and faculty.

Specifically, for sailing boats, red and green sidelights and a white sternlight are required, although for vessels under 7 metres ( 23. 0 ft ) in length, these may be substituted by a torch or white all-round lantern.