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Given and then
Given the then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.
If V is finite-dimensional, then V * has the same dimension as V. Given a basis
Given a left neutral element and for any given then A4 ’ says there exists an such that.
Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has
Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).
Given that Gable and Cukor had worked together before, in Manhattan Melodrama and Gable had no objection to working with him then, and given Selznick's desperation to get Gable for Rhett Butler, if Gable had any objections to Cukor, certainly they would have been expressed before he signed his contract for the film.
Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.
Given Poseidon's connection with horses as well as the sea, and the landlocked situation of the likely Indo-European homeland, Nobuo Komita has proposed that Poseidon was originally an aristocratic Indo-European horse-god who was then assimilated to Near Eastern aquatic deities when the basis of the Greek livelihood shifted from the land to the sea, or a god of fresh waters who was assigned a secondary role as god of the sea, where he overwhelmed the original Aegean sea deities such as Proteus and Nereus.
Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.
Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.
# Given any elements a and b of G, if a ~ b, then b ~ a ( symmetry );
# Given any elements a, b, and c of G, if a ~ b and b ~ c, then a ~ c ( transitivity );
# Given any elements a, a ', b, and b ' of G, if a ~ a ' and b ~ b ', then a * b ~ a ' * b ' ;
# Given any elements a and a ' of G, if a ~ a ', then a < sup >− 1 </ sup > ~ a ' < sup >− 1 </ sup > ( this can actually be proven from the other four, so is strictly redundant ).
Given an embedding of a rooted tree in the plane, if one fixes a direction of children ( starting from root, then first child, second child, etc.
Given " mappings " of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take, based on a set of " rules ", each of the form:
: Given any set A and any set B, if for every set C, C is a member of A if and only if C is a member of B, then A is equal to B.
: Given any set A and any set B, if A is a nonempty set ( that is, if there exists a member C of A ), then if A and B have precisely the same members, then they are equal.
Given the type of warfare that had been carried out in the conflict — wherein highly mobile groups of soldiers and rebels continued to move to and from on the desert terrain, retreating at a time and then suddenly attacking to regain control of small towns and villages in the Eastern rebel held parts of Libya — had led to the technical becoming a vehicle of choice for both sides.
# Given u and v in W, then they can be expressed as u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) and v = ( v < sub > 1 </ sub >, v < sub > 2 </ sub >, 0 ).
# Given u in W and a scalar c in R, if u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) again, then cu = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, c0 ) = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, 0 ).
Given subspaces U and W of a vector space V, then their intersection U ∩ W :=
Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined to be the curvature of that circle or line.
Given an eclipse, then there is likely to be another eclipse after every period P. This remains true for a limited time, because the relation is only approximate.

Given and normal
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 GG / 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 normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude:
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 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.

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