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

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 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.

Cult status was established giving some credence to the publicity material-" The world may soon divide into those who have been through THE WARP and those who have not " More recently the cycle was revived in the 1990s in a production directed by Campbell's daughter Daisy.

More generally, the idea of a contractive mapping can be defined for maps between metric spaces.

More modern and sophisticated approaches seek to maintain the idea of the morpheme while accommodating non-concatenative, analogical, and other processes that have proven problematic for Item-and-Arrangement theories and similar approaches.

That is not to put forward the idea that Faith No More was the sole catalyst behind what ultimately morphed into the nu metal of Jonathan Davis ' description.

More recent work on Word Grammar cites neurocognitive linguistics as a source of inspiration for the idea that language is nothing but a network.

Nobel laureate P. W. Anderson used this idea in his 1972 paper Science, ' More is different ' to expose some of the limitations of reductionism.

More importantly, the very idea that formal rules can be applied to areas outside of logic or mathematics, may itself have been catalyzed by Europe's contact with the work of Sanskrit grammarians.

More detailed electron microscopic comparisons between cyanobacteria and chloroplasts ( for example studies by Hans Ris ), combined with the discovery that plastids and mitochondria contain their own DNA ( which by that stage was recognized to be the hereditary material of organisms ) led to a resurrection of the idea in the 1960s.

More recently, the idea has been " borrowed " by Barbara Wallraff for her new book Word Fugitives: In Pursuit of Wanted Words, where " word fugitives " is her term for invented words.

More recently, the idea of " creative destruction " was utilized by Max Page in his 1999 book, The Creative Destruction of Manhattan, 1900 – 1940.

More generally, researchers attacked the idea that “ integrated financial services firms ” had played a role in creating the Great Depression or the collapse of the banking system in the 1930s.

More specifically, this note related to the concept of the eternal recurrence, which is, by Nietzsche's admission, the central idea of Zarathustra ; this idea occurred to him by a " pyramidal block of stone " on the shores of Lake Silvaplana in the Upper Engadine, a high alpine region whose valley floor is at 6, 000 ft. Nietzsche planned to write the book in three parts over several years.

More directly, he promoted the Polaris missile program at a time when others in the Navy were very skeptical of the idea of a missile launched from a submarine.

She told More magazine in 2004, " I'm stuck in this idea that I need to be authentic ... My face needs to look the way I feel.

More importantly, the Act helped to publicise the idea of establishing workhouses to a national audience.

More generally, Fish finds the idea of radical culture as an oppositional ideal in which context is continuously refined or rejected impracticable at best, and impossible at worst.

More recently another possible idea has been put forward called the theory of alternative stable states which suggests that there is not one end point but many which transition between each other over ecological time.

More recently, Fazioli has modified Theodore Steinway's original idea by creating a stainless-steel track, fixed to the cast iron plate, on which aliquots slide.

More recently, former ECKist Ford Johnson formed a spiritual organization based on the idea that one does not need a master or spiritual guru to achieve spiritual enlightenment.

More specifically, the movement replaced sexuality over the traditional Chinese idea of kinship positionality.

More aware of their social position than the idea that his sons could have professions in science, Nardo Antonio was raising the boys more as gentlemen ; very well rounded gentlemen at that.

More recently the same idea has been taken into the theory of optimal control.

More importantly, his council introduced the idea of artists as " creative problem solvers " and by paying artists to " solve problems for the state " rather than make art, they by-passed the objections of many conservative lawmakers.

More recently, some of its supporters have dropped the idea of an icy lunar surface, though they continue to support the view that it was captured and that its capture destroyed Atlantis.