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Page "Manifold" ¶ 31
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More and abstractly
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, " 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 and natural
More recent observations made in the laboratory of antibiosis between micro-organisms led to the discovery of natural antibacterials produced by microorganisms.
More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets ( sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers ).
More idealized motion in the " sublunary " realm could only be achieved through artifice, and prior to the 17th century, many did not view artificial experiments as a valid means of learning about the natural world.
More controversially, medical nanorobots might be used to augment natural human capabilities.
More generally, a meteorite on the surface of any celestial body is a natural object that has come from elsewhere in space.
More substantive basis for this designation is that the vocabulary, grammar, and orthography of Interlingua are natural ; they have been standardized and presented by a linguistic research body, but they predated it and are not themselves considered a product of human invention.
This can stem either from natural causes such as in Olaf Stapledon's novel Odd John, and Theodore Sturgeon's More Than Human, or be the result of intentional augmentation such as in A. E. van Vogt's novel Slan.
More likely it is simply the more natural position of the hand when seizing the peak of a cap.
More precisely, if A is a finite set of generators for G then the word problem is the membership problem for the formal language of all words in A and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on A to the group G. If B is another finite generating set for G, then the word problem over the generating set B is equivalent to the word problem over the generating set A.
More sophisticated methods are used in practice for natural language generation.
More generally, the term glacis can denote any slope, natural or artificial, which fulfils the above requirements.
More sophisticated devices which use a graphical screen with touch sensing or screen-edge buttons provide flexibility while minimizing space used: the meaning of the buttons can change with the screen, and selection involves the natural behavior of pointing at what's desired.
More often they are accused of domineering when doing what would be considered natural by a man.
More specifically, let be the set of all polynomials with integer coefficients ( such a polynomial is also referred to as a polynomial over ), with addition and multiplication defined to be natural polynomial addition and multiplication.
O ’ Neill notes that Cavendish ’ s natural philosophy, and writing in general, was criticized by many of her contemporaries as well as by more recent readers, such as Samuel Pepys, Henry More and Virginia Woolf.
More than six natural springs feed Town Run before it enters the south end of town.
More recently, synthetic velvets have been developed, mostly polyester, nylon, viscose, acetate, and mixtures of different synthetics, or synthetics and natural fibers ( for example viscose mixed with silk ).
More than half ( 56 %) the losses of coastal wetlands resulted from dredging for marinas, canals, port development, and, to some extent, from natural shoreline erosion.
More precisely, it shows that the mere assumption of the existence of the set of natural numbers as a totality ( an actual infinity ) suffices to imply the existence of a model ( a world of objects ) of any consistent theory.
More than 1500 years had elapsed since the Greek doctors, under the influence of Hippocrates had examined illness as a natural process.
# More generally, an assignment of elements from a countable mathematical object, such as a countable group, to natural numbers to allow algorithmic manipulation of the mathematical object.
More realistic, and often overlooked, options include long-term crop rotations, returning to natural cycles that annually flood cultivated lands ( returning lost nutrients indefinitely ) such as the Flooding of the Nile, the long-term use of biochar, and use of crop and livestock landraces that are adapted to less than ideal conditions such as pests, drought, or lack of nutrients.
More than any other Spanish dramatist, Alarcón was preoccupied with ethical aims, and his gift of dramatic presentation is as brilliant as his dialogue is natural and vivacious.
More than two-thirds of that growth can be ascribed to a natural increase resulting from high fertility and birthrates.

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