Given a *- ring, there is also the map.

Given a state ρ of A, there is a *- representation π of A with distinguished cyclic vector ξ such that its associated state is ρ, i. e.

Given an operator T, the range of the continuous functional calculus h → h ( T ) is the ( abelian ) C *- algebra C ( T ) generated by T. The Borel functional calculus has a larger range, that is the closure of C ( T ) in the weak operator topology, a ( still abelian ) von Neumann algebra.

Given two reductive groups and a ( well behaved ) morphism between their corresponding L-groups, this conjecture relates their automorphic representations in a way that is compatible with their L-functions.

Given two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ.

Given two representations ρ: G → GL ( n, C ) and τ: G → GL ( m, C ) a morphism between ρ and τ is a linear map T: C < sup > n </ sup > → C < sup > m </ sup > so that for all g in G we have the following commuting relation: T ° ρ ( g ) = τ ( g ) ° T.

* Given two representations ρ < sub > 1 </ sub >, ρ < sub > 2 </ sub > we may construct their direct sum ρ < sub > 1 </ sub > ⊕ ρ < sub > 2 </ sub > by ( ρ < sub > 1 </ sub > ⊕ ρ < sub > 2 </ sub >) ( g )( v, w )

Given two irreducible representations V < sub > i </ sub > and V < sub > j </ sub >, we can construct a G-representation

Given any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations ( to avoid ambiguity, we can specify the square root with complex argument less than π ).

Given a permutation π of m elements,

Given two permutations π and σ of m elements and the corresponding permutation matrices P < sub > π </ sub > and P < sub > σ </ sub >

Given any r on this sphere, and an angle – π < a ≤ π, the versor is on the 3-sphere of H.

Given a fiber bundle π: E → B and a continuous map f: B ′ → B one can define a " pullback " of E by f as a bundle f * E over B ′.

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.

In his 1859 paper On the Number of Primes Less Than a Given Magnitude Riemann found an explicit formula for the normalized prime-counting function π < sub > 0 </ sub >( x ) which is related to the prime-counting function π ( x ) by

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

* Given any combination of complex numbers, bras, kets, inner products, outer products, and / or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

Given is each element's name, atomic number, year of first report, name of the discoverer, and some notes related to the discovery.

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

* Named entity recognition ( NER ): Given a stream of text, determine which items in the text map to proper names, such as people or places, and what the type of each such name is ( e. g. person, location, organization ).

* Part-of-speech tagging: Given a sentence, determine the part of speech for each word.

* Topic segmentation and recognition: Given a chunk of text, separate it into segments each of which is devoted to a topic, and identify the topic of the segment.

# Given an array of values to be sorted, set up an auxiliary array of initially empty " pigeonholes ," one pigeonhole for each key through the range of the original array.

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 an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates.

Given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city.

Given two school classes, one with 20 students, and one with 30 students, the grades in each class on a test were:

Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.

Given the maximum observed frequency ( MOF ) for a mode on each day of the month at a given hour, the MUF is the highest frequency for which an ionospheric communications path is predicted on 50 % of the days of the month.

* Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.

* Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.

The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f ( x ) = a < sup > x </ sup > mod N. Given N, find Q = 2 < sup > q </ sup > such that < math > N ^ 2

Given a fixed physical relationship in space between the different individual microphone transducer array elements, simultaneous DSP ( digital signal processor ) processing of the signals from each of the individual microphone array elements can create one or more " virtual " microphones.

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:

BONUS: " Given a pair of resistors, give their equivalent resistance if they were connected in parallel with each other.

* Given two hypothetical point particles each of Planck mass and elementary charge, separated by any length, α is the ratio of their electrostatic repulsive force to their gravitational attractive force.

Given a testing procedure E applied to each prepared system, we obtain a sequence of values

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.

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 this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

Given the opportunity, adults eat hatchlings, which may be protected by separating the two groups with a net, breeding box or separate tank.

Love and Erlandson continued to pursue with the band, and released the single " Be A Man "— an outtake from the Celebrity Skin sessions — for the soundtrack of the Oliver Stone film Any Given Sunday ( 1999 ).

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

Given the bias of the Victorian age, female students were denied direct study in anatomy and could not attend drawing classes with live models ( who were often prostitutes ) until a decade later.

Given the small population of the region ( compared with the Central Canadian provinces or the New England states ), the regional economy is a net exporter of natural resources, manufactured goods, and services.

Given the rigorous admission process it is not surprising that CUNY BA boasts a 70 % graduation rate within an average of 2. 2 years and that 60 % graduate with academic honors.

Given Chrétien's known tendency to create new stories and characters, being the first to mention the hero Lancelot and his love affair with Queen Guinevere for example, the name might also be entirely invented.

Given that the dispute with Pizarro over Cuzco had kept intensifying, Almagro spent a great deal of time and money equipping a company of 500 men for a new exploration south of Peru.

Given Mountbatten's determination, Nehru and Patel's inability to deal with the Muslim League and lastly Jinnah's obstinacy, all Indian party leaders ( except Gandhi ) acquiesced to Jinnah's plan to divide India, which in turn eased Mountbatten's task.

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

Given limited defense budgets, air forces tended to be conservative in their aircraft purchases, and biplanes remained popular with pilots because of their agility, and remained in service long after they had ceased to be competitive.

Given Guatemala's large expatriate community in the United States, it is the top remittance recipient in Central America, with inflows serving as a primary source of foreign income equivalent to nearly two-thirds of exports.

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 differences of opinion within the Lord Aberdeen cabinet over the direction of foreign policy with regard to relations between Britain and the French under Napoleon III, it is not surprising that debate raged within the government as Louis Bonaparte, now assuming the title of Emperor Napoleon III of France.

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given the limited amount of good quality agricultural land to begin with, that policy has resulted in continual deforestation and subsequent erosion.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.