Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

Any directed graph generates a small category: the objects are the vertices of the graph, and the morphisms are the paths in the graph ( augmented with loops as needed ) where composition of morphisms is concatenation of paths.

Any ten-vertex Hamiltonian 3-regular graph consists of a ten-vertex cycle C plus five chords.

Any undirected graph may be made into a DAG by choosing a total order for its vertices and orienting every edge from the earlier endpoint in the order to the later endpoint.

Any directed graph may be made into a DAG by removing a feedback vertex set or a feedback arc set.

Any square matrix can be viewed as the adjacency matrix of a weighted directed graph, with representing the weight of the arc from vertex i to vertex j.

Any of these equations can be plotted on a graph.

Any linear code can be represented as a graph, where there are two sets of nodes-a set representing the transmitted bits and another set representing the constraints that the transmitted bits have to satisfy.

Any graph is homomorphically equivalent to a unique core.

A graph coloring is an assignment of one of k colors to a graph G so that the endpoints of each edge have different colors, for some number k. Any coloring corresponds to a homomorphism from G to a complete graph K < sub > k </ sub >: the vertices of K < sub > k </ sub > correspond to the colors of G, and f maps each vertex of G with color c to the vertex of K < sub > k </ sub > that corresponds to c. This is a valid homomorphism because the endpoints of each edge of G are mapped to distinct vertices of K < sub > k </ sub >, and every two distinct vertices of K < sub > k </ sub > are connected by an edge, so every edge in G is mapped to an adjacent pair of vertices in K < sub > k </ sub >.

Any induced subgraph of a comparability graph is itself a comparability graph, formed from the restriction of the partial order to a subset of its elements.

Any finite graph has a finite ( though perhaps exponential ) number of distinct simple cycles, and if the graph is embedded into three-dimensional space then each of these cycles forms a simple closed curve.

Any closed curve within the plane bounds a disk below the plane that does not pass through any other graph feature, and any closed curve through the apex bounds a disk above the plane that does not pass through any other graph feature.

Any Bayesian network or Markov random field can be represented as a factor graph.

Any tragedy, he maintains, has six elements: plot, character, and thought ( the objects of imitation ), diction and melody ( the means of imitation ), and spectacle ( the manner of imitation ).

Any street meeting, sacred or secular, which he and his colleagues uneasily cover has as its explicit or implicit burden the cruelty and injustice of the white domination.

Any string s has at least one description, namely the program:

Any player may declare the game over at any time during his turn if either of two conditions is true: one chain has 41 or more tiles, or there is at least one chain on the board and every chain on the board has 11 or more tiles.

Any individual has one of six possible genotypes ( AA, AO, BB, BO, AB, and OO ) that produce one of four possible phenotypes: " A " ( produced by AA homozygous and AO heterozygous genotypes ), " B " ( produced by BB homozygous and BO heterozygous genotypes ), " AB " heterozygotes, and " O " homozygotes.

Any Banach algebra ( whether it has an identity element or not ) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of.

# Any collection of closed subsets of X with the finite intersection property has nonempty intersection.

Any young player who has good control will become a successful curve pitcher long before the pitcher who is endeavoring to master both curves and control at the same time.

Any use of the term " Cathar " to refer to people after the suppression of Catharism in the 14th century is a cultural or ancestral reference, and has no religious implication.

Any rational number with a denominator whose only prime factors are 2 and / or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.

Any configuration of charges or currents has a ' dipole moment ', which describes the dipole whose field is the best approximation, at large distances, to that of the given configuration.

Any such poset has a dual poset.

:* Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y ;

:* Any subset of the identity relation on X has equivalence classes that are the singletons of X.

# Disturbed natural forest – Any forest type above that has in its interior significant areas of disturbance by people, including clearing, felling for wood extraction, anthropogenic fires, road construction, etc.

# Disturbed natural forest – Any forest type above that has in its interior significant areas of disturbance by people, including clearing, felling for wood extraction, anthropogenic fires, road construction, etc.

Any symbolism has been added later to the three colours, although the orange comes from the House of Orange-Nassau.

Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.

For instance, the Archbishop of Wales has criticized " atheistic fundamentalism " broadly and said " Any kind of fundamentalism, be it Biblical, atheistic or Islamic, is dangerous ".

Any attempt to define it ( X is good if it has property Y ) will simply shift the problem ( Why is Y-ness good in the first place?

Any right action has an equivalent left action, thus only left actions can be considered without any loss of generality.

Any one language has only a subset of the aspectual distinctions attested in the world's languages, and some languages ( such as Standard German ; see below ) do not have aspects.

In an inertial frame, Newton's first law ( the law of inertia ) is satisfied: Any free motion has a constant magnitude and direction.

Any conductor has inductance.

* Any images of minors ( under 18 ) or young participants without adequate proof of age, even if not taking part in the scene: in one case an R18 was cut for sight of a photo of a person's children which came into view while they were having sex, and in another shots of children playing on a beach intercut with lawful adult sexual activity were cut.

Any U. S. citizen or lawful permanent resident over the age of 18 can apply for membership, as can minors over the age of 12 with parental or guardian sponsorship.

Any picture, image, graphic image file, or other visual depiction that – ( i ) taken as a whole and with respect to minors, appeals to a prurient interest in nudity, sex, or excretion ; ( ii ) depicts, describes, or represents, in a patently offensive way with respect to what is suitable for minors, an actual or simulated sexual act or sexual contact, actual or simulated normal or perverted sexual acts, or a lewd exhibition of the genitals ; and ( iii ) taken as a whole, lacks serious literary, artistic, political, or scientific value as to minors ” ( Secs.

Any umpire who lasts ten or more years in the minors deserves to be institutionalized.