Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

" Given the elimination of these pernicious influences, Marx allowed that " normal conditions of spontaneous development " of the rural commune could exist.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given their vertical-cavity geometry, VCSOAs are resonant cavity optical amplifiers that operate with the input / output signal entering / exiting normal to the wafer surface.

Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

Given that any proposition containing conjunction, disjunction, and negation can be equivalently rephrased using conjunction and negation alone ( the conjunctive normal form ), we can now handle any compound proposition.

Given a ring R and a unit u in R, the map ƒ ( x ) = u < sup >− 1 </ sup > xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

Given two intersecting planes described by and, the dihedral angle between them is defined to be the angle between their normal directions:

Given the great depth of Lake Tahoe, and the locations of the normal faults within the deepest portions of the lake, modeling suggests that earthquakes on these faults can trigger tsunamis.

Given 100 balls, some of them red and some blue, a human with normal color vision can easily separate them into red ones and blue ones.

Given the observed percentage difference p − q ( 2 % or 0. 02 ) and the standard error of the difference calculated above (. 03 ), any statistical calculator may be used to calculate the probability that a sample from a normal distribution with mean 0. 02 and standard deviation 0. 03 is greater than 0.

Given that it would be complicated to perform this test on every patient with sleep-related breathing problems, some suggest that measuring bicarbonate levels in normal ( venous ) blood would be a reasonable screening test.

Given then a normal extension L of K, with automorphism group Aut ( L / K ) = G, and containing α, any element g ( α ) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates.

Given our current estimate of the parameters θ < sup >( t )</ sup >, the conditional distribution of the Z < sub > i </ sub > is determined by Bayes theorem to be the proportional height of the normal density weighted by τ:

Given a perfect normal subgroup of the fundamental group of a connected CW complex, attach two-cells along loops in whose images in the fundamental group generate the subgroup.

Given an incident direction from the surface to the light source and the surface normal direction the specularly reflected direction ( all unit vectors ) is:

Given that the angle of incidence is larger than the critical angle ( with respect to the normal of the surface ) and the refractive index is lower on the other side of the boundary relative to the incident ray, TIR will occur and no light will be able to pass through.

Given that ( A ) the Georges are among the most rational, principled and educated persons on the planet, and ( B ) their differences from normal humans are purely physical, they conclude that in any situation where the Three laws would come into play, their own orders should take priority over that of a regular human.

Given the low accelerations placed on the spacecraft while in the outer solar system, MOND may be in effect, modifying the normal gravitational equations.

Given only the artwork and outline to work with during production, writer Yoshinori Kitase felt that one early scene of the character approaching a castle was too " boring to make completely normal ", and decided to ad-lib a scene with one of his accompanying soldiers dusting off his boots in the middle of a desert.

Given an arbitrary square matrix, the elementary divisors used in the construction of the Jordan normal form do not exist over F, so the invariant factors a < sub > i </ sub >( x ) as given above must be used instead.

Given the predictable timeline of childhood behavior, he proposed " libido development " as a model of normal childhood sexual development, wherein the child progresses through five psychosexual stages — ( i ) the oral, ( ii ) the anal, ( iii ) the phallic, ( iv ) the latent, and ( v ) the genital — in which the source pleasure is in a different erogenous zone.

Given that sound levels at such events usually exceed recommended safe levels of exposure, this is a problem which is probably showing up variations between people, which may be genetic, or the result of stress or ill-health, or it may be caused by abnormal response in the tensor tympani and stapedius muscles which function in the normal acoustic reflex response that protects the inner ear from loud sounds.

Given these, a new term Anthropocene, is specifically proposed and used informally for the latest part of this epoch since approximately synchronous lithospheric evidence, or more recently atmospheric evidence, of human impacts have been found on the Earth and its ecosystems ; these impacts may be considered of global significance for future evolution of living species.

Given his training in atmospheric physics, McDonald was able to examine UFO reports in greater detail than most other scientists, and was able to offer explanations for some previously unexplained reports.

Given the lack of atmospheric pollution in this area, it is particularly suitable for astronomical observations.

Given the abundance of spacecraft data for other planets in the Solar System, data assimilation is now also applied beyond the Earth to obtain re-analyses of the atmospheric state of extraterrestrial planets.

Given these conditions, Lovelock expects human civilization will be hard pressed to survive.

Given the properties of a concept, these are necessary and sufficient conditions to classify the concept.

Given the fluid flow conditions and the desired shaft output speed, the specific speed can be calculated and an appropriate turbine design selected.

Given good viewing conditions, Thuban is relatively easy to spot in the night sky, due to its location in relation to the Big Dipper asterism of Ursa Major.

Given a ring R and a proper ideal I of R ( that is I ≠ R ), I is a maximal ideal of R if any of the following equivalent conditions hold:

Given the harsh conditions of the retreat, the infant was left with a local family ( Two Europeans retracing the Long March route in 2003 met a woman in rural Yunnan province, said by local officials to be Mao and He Zizhen's long-lost daughter ).

Given the many presentations and the relatively low occurrence of porphyria, the patient may initially be suspected to have other, unrelated conditions.

Given that " production could be carried on without any organization is, firm at all ", Coase asks, why and under what conditions should we expect firms to emerge?

Given a general triangle ABC, the following conditions would need to be fulfilled for the case to be ambiguous:

Given the weather conditions close to the summit, cold weather injuries are very common.

Given the desperate economic conditions in Switzerland, 193 volunteers decided to leave their homeland to start anew in America.

Given a manifold M representing ( continuous / smooth / with certain boundary conditions / etc.

Given the landscape changes ( e. g., land development, urbanization, clearcutting in forested areas ) in the watersheds of many freshwater bodies, returning to pristine conditions would be a significant challenge.

Given the right reaction conditions ring-opening metathesis polymerization ( ROMP ) can be rendered living.

Given a set with three binary operations and, and two distinguished elements 0 and 1, then is a Heyting algebra for these operations ( and the relation defined by the condition that when ) if and only if the following conditions hold for any elements and of:

Given the right overall conditions of the compressed fuel — high enough density and temperature — this heating process will result in a chain reaction, burning outward from the center where the shock wave started the reaction.

Given the conditions of the experiment, the Canterbury group conclude that if any such ' Tajmar ' effect exists, it is at least 22 times smaller than predicted by Tajmar in 2006.

Given good conditions, a bud can produce a shoot with up to ten or more leaves.

Given sufficient time under fire or heat conditions, these products char away, crumble, and disappear.

Given the origins of the cardinal tetra, namely blackwater rivers whose chemistry is characterised by an acidic pH, low mineral content and the presence of humic acids, the species is adaptable to a wide range of conditions in captivity, though deviation from the soft, acidic water chemistry of their native range will impact severely upon breeding and fecundity.

Given the same energy source and operating conditions, a higher COP heat pump will consume less purchased energy than one with a lower COP.

Given such conditions, the chances are good that the whales can be driven fully ashore or close enough to the shore that they can be killed from land.

Given the difficult topographic and climatic conditions, construction units were prefabricated.