Equivalently, one can take the Stone space of the complete Boolean algebra of all subsets of X as the

The closure of the image of π ( A ) will be a C *- algebra of operators called the C *- enveloping algebra of A. Equivalently, we can define the

Equivalently,, where C ( SU < sub > μ </ sub >( 2 )) is the C *- algebra generated by α and β, subject to

Let X be an orientable manifold, and let be a measure on X. Equivalently, let be a probability space on, with sigma algebra and probability.

Equivalently, a representation of L is a Z < sub > 2 </ sub >- graded representation of the universal enveloping algebra of L which respects the third equation above.

Equivalently, is the Lie algebra of a compact semisimple Lie group.

Equivalently, a plane-wave is a pp-wave with at least a five dimensional Lie algebra of Killing vector fields, including

With this definition, it is necessary to consider the direction of p ( pointed clockwise or counter-clockwise ) to figure out the sign of L. Equivalently:

Equivalently, a set is countable if it has the same cardinality as some subset of the set of natural numbers.

Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.

Equivalently, the dyne is defined as " the force required to accelerate a mass of one gram at a rate of one centimetre per second squared ":

Equivalently, the DFT is often thought of as a matched filter: when looking for a frequency of + 1, one correlates the incoming signal with a frequency of − 1.

Equivalently, inverting is an involution.

Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable.

Equivalently, it is the lowest wage at which workers may sell their labor.

A submonoid of a monoid M is a subset N of M containing the unit element, and such that, if x, y ∈ N then x · y ∈ N. It is then clear that N is itself a monoid, under the binary operation induced by that of M. Equivalently, a submonoid is a subset N such that N = N *, where the superscript * is the Kleene star: the set is closed under composition or concatenation of its elements.

Equivalently, a perfect number is a number that is half the sum of all of its positive divisors ( including itself ) i. e. σ < sub > 1 </ sub >( n ) = 2n.

Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.

Equivalently, one can define a profinite group to be a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups.

Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors ( the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice ).

Equivalently, an ideal of R is a sub-R-bimodule of R.

Equivalently, a right ideal of is a right-submodule of.

Equivalently, a left ideal of is a left-submodule of.

The row rank of a matrix A is the maximum number of linearly independent row vectors of A. Equivalently, the column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A.

Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice.

Equivalently, a Sperner family is an antichain in the inclusion lattice over the power set of E. A Sperner family is also sometimes called an independent system or, if viewed from the hypergraph perspective, a clutter.

Equivalently, if P is a lattice, p ≠ top, and for all a, b in P,

Equivalently, Γ < sub > 8 </ sub > is self-dual, meaning it is equal to its dual lattice.

Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to R whose restriction to each fibre is a linear functional on the tangent space.

Equivalently, it is a complete separable metric space whose metric has been " forgotten ".

Equivalently, one may define a superalgebra over R as a superring A together with an superring homomorphism R → A whose image lies in the supercenter of A.

A subset S of a metric space X is totally bounded if and only if, given any positive real number E, there exists a finite cover of S by subsets of X whose diameters are all less than E. ( In other words, a " size " here is a positive real number, and a subset is of size E if its diameter is less than E .) Equivalently, S is totally bounded if and only if, given any E as before, there exist elements a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub > of < i > X < i /> such that S is contained in the union of the n open balls of radius E around the points a < sub > i </ sub >.

Generalising the above definitions, a subset S of a uniform space X is totally bounded if and only if, given any entourage E in X, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. ( In other words, a " size " here is an entourage, and a subset is of size E if its Cartesian square is a subset of E .) Equivalently, S is totally bounded if and only if, given any E as before, there exist subsets A < sub > 1 </ sub >, A < sub > 2 </ sub >, ..., A < sub > n </ sub > of X such that S is contained in the union of the A < sub > i </ sub > and, whenever the elements x and y of X both belong to the same set A < sub > i </ sub >, then ( x, y ) belongs to E ( so that x and y are close as measured by E ).

Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers.

Equivalently, H is a matrix whose null space is C. If C is a code with a generating matrix G in standard form, G

Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination ( under the group operation ) of finitely many elements of the subset and their inverses.

Equivalently, for all a, b and c in A:

b. Equivalently, if a ≠ b, then f ( a ) ≠ f ( b ).

Equivalently, we can define π < sub > n </ sub >( X ) to be the group of homotopy classes of maps g: < sup > n </ sup > → X from the n-cube to X that take the boundary of the n-cube to b.

For a non-negative integer k, the kth Betti number b < sub > k </ sub >( X ) of the space X is defined as the rank of the abelian group H < sub > k </ sub >( X ), the kth homology group of X. Equivalently, one can define it as the vector space dimension of H < sub > k </ sub >( X ; Q ), since the homology group in this case is a vector space over Q.

A powerful number is a positive integer m such that for every prime number p dividing m, p < sup > 2 </ sup > also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a < sup > 2 </ sup > b < sup > 3 </ sup >, where a and b are positive integers.

Equivalently, the geometry is S < sup > 2 </ sup >× R if the manifold is non-orientable or if b + Σb < sub > i </ sub >/ a < sub > i </ sub >= 0, and spherical geometry otherwise.