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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.
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Equivalently and family
Equivalently, a family of closed sets forms a base for the closed sets if for each closed set A and each point x not in A there exists an element of F containing A but not containing x.
Equivalently, every family of graphs that is closed under minors can be defined by a finite set of forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph K < sub > 5 </ sub > and the complete bipartite graph K < sub > 3, 3 </ sub > as minors.
A nonempty family Δ of finite subsets of a universal set S is an abstract simplicial complex if, for every set X in Δ, and every subset Y ⊂ X, Y also belongs to Δ. Equivalently, it is an abstract simplicial complex if and only if there do not exist two sets Y ⊂ X such that X belongs to Δ but Y does not.
Equivalently, the scale-space family can be defined as the solution of the diffusion equation ( for example in terms of the heat equation ),
Equivalently, it is given by the family of seminorms
Equivalently and is
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 and antichain
Equivalently, that every tree of height ω < sub > 1 </ sub > either has a branch of length ω < sub > 1 </ sub > or an antichain of cardinality
Equivalently and inclusion
If X is an affine algebraic set ( irreducible or not ) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some Equivalently, it can be checked that:
Equivalently and lattice
Equivalently a Heyting algebra is a residuated lattice whose monoid operation a • b is a ∧ b ; yet another definition is as a posetal cartesian closed category with all finite sums.
Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice.
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 and over
Equivalently, all the back edges, which DFS skips over, are part of cycles.
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.
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.
Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N which is its inverse ( these are equivalent under Cramer's rule ).
Equivalently, in a system using SS7 over IP ( for example, SIGTRAN ), the result from Global Title Translation may be to a route to an IP server, though the exact details depend greatly on which variant of SS7 over IP is being used.
A rationally connected variety V is a projective algebraic variety over an algebraically closed field such that through every two points there passes the image of a regular map from the projective line into V. Equivalently, a variety is rationally connected if every two points are connected by a rational curve contained in the variety.
Equivalently, V is a rank n real vector bundle over M,
Equivalently ( see bundle map ), φ < sub >*</ sub > = dφ is a bundle map from TM to the pullback bundle φ < sup >*</ sup > TN over M, which may in turn be viewed as a section of the vector bundle Hom ( TM, φ < sup >*</ sup > TN ) over M.
Equivalently, if we define decision problems as sets of finite strings, then the complement of this set over some fixed domain is its complement problem.
Equivalently and power
Equivalently, C shatters A when the power set P ( A ) is the set
Equivalently and set
Equivalently, a preordered set P can be defined as a category with objects the elements of P, and each hom-set having at most one element ( one for objects which are related, zero otherwise ).
Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement.
Equivalently, a set is bounded if it is contained in some open ball of finite radius.
Equivalently, a set is closed if and only if it contains all of its limit points.
More formally, a set R X Y is called a ( combinatorial ) rectangle if whenever R and R then R. Equivalently, R can also be viewed as a submatrix of the input matrix A such that R = M N where M X and N Y.
Equivalently ( and with no need to arbitrarily choose two points ) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points.
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, a set is recursively enumerable if and only if it is the range of some computable function.
Equivalently X is arithmetical if X is or for some integer n. A set X is arithmetical in a set Y, denoted, if X is definable a some formula in the language of Peano arithmetic extended by a predicate for membership in Y. Equivalently, X is arithmetical in Y if X is in or for some integer n. A synonym for is: X is arithmetically reducible to Y.
Equivalently, every dependent set contains a finite dependent set.
Equivalently, a function is convex if its epigraph ( the set of points on or above the graph of the function ) is a convex set.
Equivalently, one can cover the domain of this function by open sets, such that f restricted to each such open set is a homeomorphism onto its image.
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