Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra A < sup > op </ sup > ( the opposite ring with the same action by K since the image of K → A is in the center of A ).

Given this, the structures can be shown to coincide, and the resulting monoid demonstrated to be commutative.

Given a regular expression E representing S, it is easy to compute the syntactic monoid of S.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

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 an evaluation e of variables by elements of M < sub > w </ sub >, we

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

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 two complexes M < sub >*</ sub > and N < sub >*</ sub >, a chain map between the two is a series of homomorphisms from M < sub > i </ sub > to N < sub > i </ sub > such that the entire diagram involving the boundary maps of M and N commutes.

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

Given a set M of molecules, chemical reactions can be roughly defined as pairs r =( A, B ) of subsets from M.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given the morphological distinctness of the Cape Verde birds and the fact that the Cape Verde population was isolated from other populations of Red Kites, it cannot be conclusively resolved at this time whether the Cape Verde population was not a distinct subspecies ( as M. migrans fasciicauda ) or even species that frequently absorbed stragglers from the migrating European populations into its gene pool.

Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Given a smooth curve γ on ( M, g ) and a vector field V along γ its derivative is defined by

Given such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.

Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

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 basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

# Given a large enough beta-tester and co-developer base, almost every problem will be characterized quickly and the fix obvious to someone.

: 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 expectations about returns on fixed investment, every level of the real interest rate ( i ) will generate a certain level of planned fixed investment and other interest-sensitive spending: lower interest rates encourage higher fixed investment and the like.

Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.

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.

The law states that " given enough eyeballs, all bugs are shallow "; or more formally: " Given a large enough beta-tester and co-developer base, almost every problem will be characterized quickly and the fix will be obvious to someone.

Given an input consisting of any number of bytes, it produces as output a single byte that is strongly dependent on every byte of the input.

Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin.

for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:

* Given 27 same-size cubes whose nominal values progress from 1 to 27, a 3 × 3 × 3 magic cube can be constructed such that every row, column, and corridor, and every diagonal passing through the center, is composed of 3 cubes whose sum of values is 42.

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 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 that there are around 700, 000 live births in Britain each year, Meadow argued that a double cot death would be expected to occur once every hundred years.

Given an arbitrary ( n ; m ; p ) machine S, such that every two of its states are distinguishable from one another, then there exists an experiment of length

* Given a maximal torus T in G, every element g ∈ G is conjugate to an element in T.

Given a labelled state transition system ( S, Λ, →), a simulation relation is a binary relation R over S ( i. e. R ⊆ S × S ) such that for every pair of elements p, q ∈ S, if ( p, q )∈ R then for all α ∈ Λ, and for all p ' ∈ S,

Given our formula φ, we group strings of quantifiers of one kind together in blocks:

* 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.

Given that this scale is much smaller than any cosmological scale these strings are often studied in the zero width, or Nambu-Goto approximation.

Given two strings, equally taut and heavy, one twice as long as the other, the longer would vibrate with a pitch one octave lower than the shorter.

Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that

Given the context of this conversation ( Blackadder is attempting to get transferred into the Royal Flying Corps ), and Darling's character in general, he has presumably pulled some strings to allow him to remain in the army without having to serve in battle.

Given two strings, of length and of length, find the longest strings which are substrings of both and.

Given the set of strings, where and Σ.

Given a sequence " qrj ," " aty ," " qur ," " dim ," " ofu ," " gcl ," " rhv ," " clq ," " ecd ," " qsu " of randomly generated three character long strings, the following table would be generated ( using Bob Jenkins ' One-at-a-Time hash algorithm ) with a table of size 10:

Given a binary relation R between fixed strings in the alphabet, called rewrite rules, denoted by, an SRS extends the rewriting relation to all strings in which the left-and right-hand side of the rules appear as substrings, that is, where s, t, u, and v are strings.