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, 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, the integral can be written as the cap product of the homology class of Z and the cohomology class represented by α.

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.

* Change of any variable quantity, in mathematics and the sciences ( More specifically, the difference operator.

: More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator normalized to be of trace 1.

More generally, if a linear operator on a vector space ( possibly infinite-dimensional ) has finite-dimensional range ( e. g., a finite-rank operator ), then the rank of the operator is defined as the dimension of the range.

More than 2, 700 wireless operator / air-gunners, 1, 800 navigators, and 500 bombardiers passed through the Initial Training Wing before proceeding to Canada.

More precisely, the vorticity of a flow is a vector field, equal to the curl ( rotational ) of its velocity field v. It can be expressed by the vector analysis formula, where is the nabla operator.

More examples of the use of the empty product in mathematics may be found in the binomial theorem, factorial, fundamental theorem of arithmetic, birthday paradox, Stirling number, König's theorem, binomial type, difference operator, Pochhammer symbol, proof that e is irrational, prime factor, binomial series, and multiset.

More generally, a partially defined linear operator A from a topological vector space E into its continuous dual space E < sup >∗</ sup > is said to be symmetric if

* More generally, unitary matrices are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalization of the notion of a unitary matrix.

More recently, Direct operator billing is being deployed in an in-app environment, where mobile application developers are taking advantage of the one-click payment option that Direct operator billing provides for monetising mobile applications.

More generally, the cokernel of a morphism f: X → Y in some category ( e. g. a homomorphism between groups or a bounded linear operator between Hilbert spaces ) is an object Q and a morphism q: Y → Q such that the composition q f is the zero morphism of the category, and furthermore q is universal with respect to this property.

More generally, for electrons and positrons with spin oriented in the ( a, b, c ) direction, the projection operator is

) More generally, the symbol of a differential operator between two vector bundles E and F is a section of the pullback of the bundle Hom ( E, F ) to the cotangent space of X.

More precisely, an elliptic operator D on a compact manifold has a ( non-unique ) parametrix ( or pseudoinverse ) D ′ such that DD ′− 1 and D ′ D − 1 are both compact operators.

Analogues of Heisenberg groups over finite fields of odd prime order p are called extra special groups, or more properly, extra special groups of exponent p. More generally, if the derived subgroup of a group G is contained in the center Z of G, then the map from G / Z × G / Z → Z is a skew-symmetric bilinear operator on abelian groups.

* 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 advanced functions are available in " Scientific " mode, including logarithms, numerical base conversions, some logical operators, operator precedence, radian, degree and gradians support as well as simple single-variable statistical functions.

More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the shape operator ( denoted by ) of a hypersurface,

More generally, for a linear operator T on a Hilbert space and an isometry V on a subspace of, define the compression of T to by