Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

Given their sting, however, they must always be treated with caution, and the discovery of men o ' war washed up on a beach may lead to the closure of the whole beach.

Given a separable extension K ′ of K, a Galois closure L of K ′ is a type of splitting field, and also a Galois extension of K containing K ′ that is minimal, in an obvious sense.

Given a graph G with n vertices, the closure cl ( G ) is uniquely constructed from G by repeatedly adding a new edge uv connecting a nonadjacent pair of vertices u and v with until no more pairs with this property can be found.

* Given a field K, the multiplicative group ( K < sup > s </ sup >)< sup >×</ sup > of a separable closure of K is a Galois module for the absolute Galois group.

# Given that the system is in a correct state, it is guaranteed to stay in a correct state, provided that no fault happens ( closure ).

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 a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interior and closure.

Given the celebrations of 40 years of Maltese independence, the 25 years of the closure of the British Base in Malta, and the anticipated admission of Malta into the European Union, the three Lodges of the Irish Constitution met on 5 September 2003 and resolved to form themselves into a Sovereign Grand Lodge.

Given a linear operator, not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of, and we say that is closable.

Given any expression involving complex numbers, bras, kets, inner products, outer products, and / or linear operators ( but not addition ), written in bra-ket notation, the parenthetical groupings do not matter ( i. e., the associative property holds ).

Given a color name to match the name of amber mutants, ochre mutant viruses had a similar property in that they recovered infectious ability within certain suppressor strains of bacteria.

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 pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis.

Given a set of n + 1 data points ( x < sub > i </ sub >, y < sub > i </ sub >) where no two x < sub > i </ sub > are the same, one is looking for a polynomial p of degree at most n with the property

The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H *( BG ) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H *( X ) in the same dimensions.

Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → U < sub > L </ sub >, ( notation as above ) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → A < sub > L </ sub > there exists a unique unital algebra homomorphism g: U → A such that: f (-) = g < sub > L </ sub > ( h (-)).

Given this property, the lifetime distribution function and event density ( F and f below ) are well-defined.

On the issue of the second home he wrote: " I have concluded that Mr MacKay was wrong to designate the Bromsgrove property as his main home Given that, it would be unreasonable to hold that Ms Kirkbride could not make legitimate claims from parliamentary resources for the home in her constituency which she had identified as her second home ".

Given a map, and a space, one says that has the homotopy lifting property, or that has the homotopy lifting property with respect to, if:

Given additionally a map, one says that has the homotopy lifting extension property if:

Given a graph property P, an invariant u, and a set of graphs H, we wish to find the minimum value of m such that every graph in H which has u larger than m possess property P. In the example above, H was the set of n-vertex graphs, P was the property of being cyclic, and u was the number of edges in the graph.

Given the preceding jurisprudence regarding due process, Chief Justice Taney was not entirely breaking ground in his Dred Scott opinion when, without elaboration, he pronounced the Missouri Compromise unconstitutional because an " act of Congress that deprived a citizen of his liberty or property merely because he came himself or brought his property into a particular territory of the United States, and who had committed no offence against the laws, could hardly be dignified with the name of due process of law.

Given an exterior connection D satisfying this compatibility property, there exists a unique extension of D:

Given these definitions, the physical property is:

Given the previous conclusion and noting that depends only on y, the non-decreasing property of R < sup > 2 </ sup > follows directly from the definition above.

Given the new importance that IBD alleles are assuming in genetic association studies, a new definition of IBD that would better suit the underlying inheritance property would be that two or more alleles are identical by descent ( IBD ) if they have been inherited from the same ancestral allele without recombination events.

Given the value of riverside property now that flooding is better managed, factories are now being sold to make way for upscale waterfront apartments.

Given scalar functions f and h, and a real number a, the Laplacian has the property:

Given a small ball or marbles, he would invent games and play by himself for hours.

Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

Given the absolute magnitude, for objects within our galaxy you can also calculate the apparent magnitude from any distance ( in parsecs ):

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

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 more time to prepare for trial, Kidd likely would have been able to find the deposition Palmer gave when he was captured in Rhode Island two years earlier.

Given their speed and maneuverability, clippers frequently mounted cannon or carronades and were used for piracy, privateering, smuggling, or interdiction service.

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 a particular query, the DBMS ' query optimizer is responsible for devising the most efficient strategy for finding matching data.

Given the limited success of the Charity School, however, Wheelock intended his new College as one primarily for whites.

Given a particular query, the DBMS ' query optimizer is responsible for devising the most efficient strategy for finding matching data.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given the magnitude of the disaster, reconstruction and economic recovery will remain the primary focus of the Salvadoran Government for some time to come.

Given a left neutral element and for any given then A4 ’ says there exists an such that.

Given the limited budget for schools, it is claimed that a voucher system would weaken public schools while at the same time not necessarily providing enough money for people to attend private schools.

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 the wide disparity of concepts and methods across domains, 4GL limitations lead to recognition of the need for the 5GL.

Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").