More abstractly, one talks about the product in category theory, which formalizes these notions.

More abstractly, Campbell's idea that myths are how we shape our lives deeply informs the picture of life in Glorantha throughout the game world's publication history.

More abstractly, the convex-hull operator Conv () has the characteristic properties of a closure operator:

More abstractly, one can define the dicyclic group Dic < sub > n </ sub > as any group having the presentation

More abstractly, when using infix notation

More abstractly, the symplectic group can be defined as the set of linear transformations of a 2n-dimensional vector space over F that preserve a nondegenerate, skew-symmetric, bilinear form.

More abstractly, in classical mechanics phase space is the cotangent space of configuration space, and in this interpretaton the procedure above expresses that a choice of local coordinates on configuration space induces a choice of natural local Darboux coordinates for the standard symplectic structure on a cotangent space.

More abstractly and more precisely, it may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square.

More abstractly, it can be defined as an incidence structure ( P, L, I ) consisting of a set P of points, a set L of lines, and an incidence relation I stating which points lie on which lines.

More generally and abstractly, given any group G and a representation of G on a vector space V,

More abstractly, it means that the cohomology class in the classifying space for pulls back from the cohomology class in under the inclusion ( which corresponds to the inclusion and similar ).

so too is the dual group More abstractly, these are both examples of representable functors, being represented respectively by K and T.

The order of PSL ( n, q ) is the above, divided by, the number of scalar matrices with determinant 1 – or equivalently dividing by, the number of classes of element that have no nth root, or equivalently, dividing by the number of nth roots of unity in .< ref group =" note "> These are equal because they are the kernel and cokernel of the endomorphism formally, More abstractly, the first realizes PSL as SL / SZ, while the second realizes PSL as the kernel of

More abstractly, a conservative vector field is an exact 1-form.

More generally, if W is a linear subspace of a ( possibly infinite dimensional ) vector space V then the codimension of W in V is the dimension ( possibly infinite ) of the quotient space V / W, which is more abstractly known as the cokernel of the inclusion.

More abstractly, it studies how global properties of a graph influence local substructures of the graph.

More abstractly, the mean curvature is the trace of the second fundamental form divided by n ( or equivalently, the shape operator ).

More abstractly, some aspects of the narrative appear in the crucifixion story of Jesus found in the gospels.

More abstractly: If X < sub > 1 </ sub >, ..., X < sub > n + 1 </ sub > are conditionally independent random variables that each can assume the value 0 or 1, then, if we know nothing more about them,

More abstractly and generally, we note that the two quantities asserted to be equal count the subsets of size k and n − k, respectively, of any n-element set S. There is a simple bijection between the two families F < sub > k </ sub > and F < sub > n − k </ sub > of subsets of S: it associates every k-element subset with its complement, which contains precisely the remaining n − k elements of S. Since F < sub > k </ sub > and F < sub > n − k </ sub > have the same number of elements, the corresponding binomial coefficients must be equal.

More abstractly, the " pay-off set " ( i. e., the set of all plays in which the angel wins ) is a closed set ( in the natural topology on the set of all plays ), and it is known that such games are determined.

More abstractly, f is proper if for any space Z the map

More abstractly, " G-bundles over X " is a functor in G: given a map H → G, one gets a map from H-bundles to G-bundles by inducing ( as above ).

More abstractly, a natural class of objects to study in topology are objects that are homogeneous ( all points are topologically the same: the group of self-homeomorphisms acts transitively ) and " finite type " or " tame " ( to rule out spaces such as the Cantor set, where each open set contains uncountably many connected components ); more generally, a space of " finite type " where the self-homeomorphism group has finitely many orbits, forming the strata.

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral.

More generally, the net change in charge density within a volume of integration is equal to the area integral over the current density through the closed surface, which is in turn equal to the net current:

More recent philosophy of art, however, considers interaction with tradition as integral to the development of new artistic expression.

More specifically, the energy Eigen states form a basis-any wavefunction may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure.

More general formulations of integration by parts exist for the Riemann – Stieltjes integral and Lebesgue – Stieltjes integral.

More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface.

More precisely, ζ ( X, s ) can be written as a finite alternating product < dl >< dd ></ dl > where each P < sub > i </ sub >( T ) is an integral polynomial.

An integral part of the Black Arts movement of the 1970s, Evans had his first plays, the one acts Orrin and Sugarmouth Sam Don ’ t Dance No More performed in 1972 at the Crossroads Theatre, a professional playhouse in New Brunswick, New Jersey.

More generally, if L is any positive definite unimodular lattice of dimension 25 with at least 4 vectors of norm 1, then the Weyl vector of its norm 2 roots has integral length, and there is a similar construction of the Leech lattice using L and this Weyl vector.

More generally, if F is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the nth moment of the probability distribution is given by the Riemann – Stieltjes integral

More generally, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space R < sup > n </ sup >.

* More generally, every integral domain is a reduced ring since a nilpotent element is a fortiori a zero divisor.

More precisely, for a given orbital momentum quantum number ℓ ( representing the azimuthal quantum number associated with angular momentum ), there are 2ℓ + 1 integral magnetic quantum numbers m ranging from-ℓ to ℓ, which restrict the fraction of the total angular momentum along the quantization axis so that they are limited to the values m. This phenomenon is known as space quantization.

* More generally, if Ω is any domain in R < sup > n </ sup > and the integral kernel k: Ω × Ω → R is a Hilbert — Schmidt kernel, then the operator T on L < sup > 2 </ sup >( Ω ; R ) defined by

More generally, the theorem follows immediately from the Crofton formula in integral geometry according to which the length of any curve equals the measure of the set of lines that cross the curve, multiplied by their numbers of crossings.

More specifically, the basic concepts of calculus such as continuity, derivative, and integral can be defined using infinitesimals without reference to epsilon, delta.

More explicitly, let be a dominant integral weight ; then it is still true that for all, but it is no longer true that this G-module is simple in general, although it does contain the unique highest weight module of highest weight as a G-submodule.

More generally for R an integral domain contained in a field K we define to be an R-order in a K-algebra A if it is a subring of A which is a full R-lattice.

More generically, can be calculated by evaluating the following surface integral:

More recent ecological research has shown, however, that fire is an integral component in the function and biodiversity of many natural habitats, and that the organisms within these communities have adapted to withstand, and even to exploit, natural wildfire.

" More than ever, black culture is an integral part of society.

More than just a club, the FFA is an integral part of the program of agricultural education in many high schools as a result of Public Law 740 in 1950 ( Currently revised as Publication 105-225 of the 105th Congress of the United States ), with 500, 823 FFA members ( 2007 – 2008 ).