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, 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, 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.

Thus the classical solutions to G Chern – Simons theory are the flat connections of principal G-bundles on M. Flat connections are determined entirely by holonomies around noncontractible cycles on the base M. More precisely, they are in one to one correspondence with equivalence classes of homomorphisms from the fundamental group of M to the gauge group G up to conjugation.

More than anything, it is the therapist's intuitive sensing of these latent meanings in the stereotype which helps these meanings to become revealed, something like a spread-out deck of cards, on sporadic occasions over the passage of the patient's and his months of work together.

More time was spent in trying to marry these incompatibles than over any subject discussed at Yalta.

More and more, these days, she'd been driving that pretty little mare that looked like her, over to Tillie's and Nick's -- his own old square frame box on posts, chickens and cats and pups under the house, everybody friendly inside, making a to-do over the babies dressed like dollies.

More complex assistive technology devices have been developed over time, and as a result, sports for people with disabilities " have changed from being a clinical therapeutic tool to an increasingly competition-oriented activity ".

More recently, disagreements over homosexuality have strained the unity of the Communion as well as its relationships with other Christian denominations, leading to another round of withdrawals from the Anglican Communion.

More recent researchers, in particular Ronald Willis and Joy Munns have studied the tour in detail and concluded that the presentation was made after a private cricket match played over Christmas 1882 when the English team were guests of Sir William Clarke, at his property " Rupertswood ", in Sunbury, Victoria.

More redeployments are expected since the states of Rio de Janeiro, Minas Gerais and Espírito Santo still concentrate over 49, 000 soldiers.

More than one-third of them died or left the service before they were disbanded along with the rest of the RIC in 1922, an extremely high wastage rate, and well over half received government pensions.

More controversial than the role of technology is the argument over the role of politics in the origins of the region's decline.

More dramatically, in July 1958, he sent 15, 000 Marines and soldiers to Lebanon as part of Operation Blue Bat, a non-combat peace-keeping mission to stabilize the pro-Western government and to prevent a radical revolution from sweeping over that country.

More generally one is interested in properties and invariants of smooth manifolds which are carried over by diffeomorphisms, another special kind of smooth mapping.

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 than 1500 students over four years were compared to an equal number of carefully matched students at conventional schools.

More often, an electoral suite or embassy was sent to cast the vote ; the credentials of such representatives were verified by the Archbishop of Mainz, who presided over the ceremony.

More recent contributors to the loss of forest cover include the growth in cattle herd size since their introduction around 1000 years ago, a continued reliance on charcoal as a fuel for cooking, and the increased prominence of coffee as a cash crop over the past century.

More than 90 % of the population lives in the southern part of the country, especially in Bamako, which has over 1 million residents.

* James M. O ' Fallon, The Case of Benjamin More: A Lost Episode in the Struggle over Repeal of the 1801 Judiciary Act, 11 43 ( 1993 ).

More formally, the environment is modeled as a Markov decision process ( MDP ) with states and actions with the following probability distributions: the instantaneous cost distribution, the observation distribution and the transition, while a policy is defined as conditional distribution over actions given the observations.

More than 2 million species of plant and animal life have been identified to date, and estimates of the actual number of existing species range from several million to well over 50 million.

More than one-half of the summits are over 4, 500 meters, and more than fifty peaks reach above 6, 500 meters.

More skilled shooters can use the RPG self-destruct feature to make it explode over the enemy at closer range.

More practical applications of social epistemology can be found in the areas of library science, academic publishing, knowledge policy and debates over the role over the Internet in knowledge transmission and creation.

More members were ousted over the years for a variety of infractions, both political and personal, while others left of to pursue creativity of their own style.