" Given its general release in January 1931, City Lights proved to be a popular and financial success — eventually grossing over $ 5 million.

Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Given a general statement such as all ravens are black, a form of the same statement that refers to a specific observable instance of the general class would typically be considered to constitute evidence for that general statement.

Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).

Given local and transient influences on surface height, the values defined below are based on a " general purpose " model, refined as globally precisely as possible within 5 m of reference ellipsoid height, and to within 100 m of mean sea level ( neglecting geoid height ).

Given the parallels between visual and musical arts, it is possible that general acceptance of the value of digital art will progress in much the same way as the increased acceptance of electronically produced music over the last three decades.

Given the general rule in British law that a wife automatically takes her husband's rank upon marriage, Mrs Simpson consequently should have been addressed, upon marriage to the duke, as Her Royal Highness.

Given the extensive linguistic, religious and ethnic diversity of the Indian population, nationalism in India in general does not fall within the purview of a solitary variant of nationalism.

Given this outlook, whatever the level of affection Whistler may have felt for his own mother, one finds an even more divergent use of the image in the Victorian era and later, especially in the United States, as an icon for motherhood, affection for parents, and " family values " in general.

De Tactionibus embraced the following general problem: Given three things ( points, straight lines, or circles ) in position, describe a circle passing through the given points and touching the given straight lines or circles.

Given the similarities with readings from people living on loess soils, the general direction of the local movement according to Price et al., is from the northeast to the southwest.

Given Ríos Montt's staunch anticommunism and ties to the United States, the Reagan administration continued to support the general and his regime, paying a visit to Guatemala City in December 1982.

Given the affine group of an affine space A, the stabilizer of a point p is isomorphic to the general linear group of the same dimension ( so the stabilizer of a point in Aff ( 2, R ) is isomorphic to GL ( 2, R )); formally, it is the general linear group of the vector space: recall that if one fixes a point, an affine space becomes a vector space.

Given any subgroup < math > G < GL ( V )</ math > of the general linear group,

Given their general lack of interest in the area, the British decided to abandon the Sudan in December 1883, holding only several northern towns and Red Sea ports, such as Khartoum, Kassala, Sannar, and Sawakin.

Given the widely different national provisions, this " general regime " gives much more power to the European Commission ( compared to the special regimes ) to ensure a harmonized protection across the European Union.

A more general class are flat G-bundles with for a manifold F. Given a representation, the flat-bundle with monodromy is given by, where acts on the universal cover by deck transformations and on F by means of the representation.

Given four points in general linear position ( no three collinear ; in particular, no two coincident ), there are exactly three pairs of lines ( degenerate conics ) passing through them, which will in general be intersecting, unless the points form a trapezoid ( one pair is parallel ) or a parallelogram ( two pairs are parallel ).

The general scenario is the following: Given a class S of computable functions, is there a learner ( that is, recursive functional ) which for any input of the form ( f ( 0 ), f ( 1 ),..., f ( n )) outputs a hypothesis ( an index e with respect to a previously agreed on acceptable numbering of all computable functions ; the indexed function should be consistent with the given values of f ).

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively.

* Given a triangle OAB in which O is the center of a circle k, and points A ' and B ' inverses of A and B with respect to k, then

Given a fundamental domain for the group action ( for the full, orientation-reversing symmetry group, a ( 2, 3, 7 ) triangle ), the reflection domains ( images of this domain under the group ) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group ( reflections in the lines of a given fundamental triangle give a set of 3 generating reflections ).

Given a triangle ABC, and a triangulation T of the triangle.

Given a counterfactual conditional, e. g., ' If there had been a circle on the blackboard then there would have been a triangle ', and the subsequent information ' in fact there was no triangle ', participants make the modus tollens inference ' there was no circle ' more often than they do from an indicative conditional ( Byrne and Tasso, 1999 ).

Given a unit sphere, a " triangle " on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere.

Given a point inside a triangle it is also desirable to obtain the barycentric coordinates, and at this point.

Given a triangle ABC and its three Malfatti circles, let D, E, and F be the points where two of the circles touch each other, opposite vertices A, B, and C respectively.

Given a triangle ABC, and a transversal line that crosses BC, AC and AB at points D, E and F respectively, with D, E, and F distinct from A, B and C, then

Given An equilateral triangle inscribed on a circle and a point on the circle.

The chords are of considerable historical importance because, along with the sides of the triangle and tetragon ( square ), they enable the generation of a table of half chords ( effectively sine values ) Given the unit circle with sector ABO subtending an arc of, we may write − a relationship expressed in words by Copernicus:

Given any natural numbers l, m, n > 1 exactly one of the classical two-dimensional geometries ( Euclidean, spherical, or hyperbolic ) admits a triangle with the angles ( π / l, π / m, π / n ), and the space is tiled by reflections of the triangle.

Given any Euclidean triangle ABC and an arbitrary point P let d ( P ) = PA + PB + PC.

Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle.

The has a crossing point at the origin if there is a triangle with sides a, b and c. Given the previous conditions, this means that the curve crosses the origin if and only if b < a + c.

: Given three points A, B and C on a circle with center O, the angle AOC is twice as large as the angle ABC.

Given Carson's undisputed status as the dean of late-night television, NBC heard his request as an ultimatum, fearing he might use the issue as grounds to defect to either ABC or CBS.

The choir have recorded three CDs-Christmas at St James ( 2003 ), No Ordinary Sunday ( 2004 ) and Any Given Sunday ( 2007 )-and broadcasts regularly on ABC Radio, both in their own right as well as with leading ensembles such as Australian Baroque Brass.

Given Nexstar's termination of the Fox affiliations of several of its stations during the past two years, it is ironic that KLRT is one of two Fox affiliates that are being picked up by either Nexstar or Mission ; the other is WJKT in Jackson, Tennessee ( although that station is controlled by and based out of Memphis, Tennessee's ABC affiliate WPTY-TV, which is also being sold to Nexstar ).

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):

Given the apparent magnitude (> 20 ) of all but the biggest trans-Neptunian objects, the physical studies are limited to the following:

Given these facts, algorithm defined by the following pseudocode:

Informally it solves the following problem: Given an integer N, find its prime factors.

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that

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 following XML document:

Given these assumptions, the following forces should be taken into account:

Given the slow speed and the coarse resolution this was not a feasible technique for printing large images, but could usefully print a small logo onto a letterhead and then the following letter, all in a single unattended print run without changing the print element.

Given all of the above premises are true, the angle B may be acute or obtuse ; meaning, one of the following is true:

Given that the UK Government opted not to pursue a tidal scheme in the Severn Estuary following the completion of the feasibility study, the chances of a third crossing anytime in the near future seem doubtful.

Given an SVD of M, as described above, the following two relations hold:

Given a natural number x, h outputs the index of the partial computable function that performs the following computation:

Given the fact that the following Popes John, from John XV ( 985 – 996 ) until John XIX ( 1024 – 1034 ), seemed to have neglected the existence of John XIV " bis ", Petrus Hispanus meant to correct this error by electing for himself the name John XXI.

Given a set S with three subsets, J, K, and L, the following holds:

Given this definition, the following are all equivalent:

Given a non-empty totally ordered set R with the following four properties:

Given the following:

Given Vanier's physical state, Pearson was hesitant to advise the Queen to act along those lines, but his worry was short lived, as the following day, after hearing the Holy Sacrifice of the Mass and receiving Holy Communion in the chapel, the Governor General died.

Given the preceding information, the knowledge gap hypothesis can be expressed using the following set of related propositions:

Given the angular velocity vector and velocity of the object, the resulting force can be calculated using the following formula:

Given a bounded lattice with largest and smallest elements 1 and 0, and a binary operation, these together form a Heyting algebra if and only if the following hold: