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 nonplanar graph has as minors either the complete graph, or the complete bipartite graph, but the Petersen graph has both as minors.

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

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 abilities I may have were achieved in their present shape from experience in sharing in the growth and control of my business, coupled with raising my family.

Any claimant whose claim is denied, or is approved for less than the full amount of such claim, shall be entitled, under such regulations as the Commission may prescribe, to a hearing before the Commission, or its duly authorized representatives, with respect to such claim.

Any example of grand or exquisite diction may have been created by the poet who compiled numerous lays into the two works we possess or may be due to one of his completely unknown fellow-craftsmen.

Any flat-bed press capable of pressing a specimen 22 inches square may be used as an alternative.

Any place called Altenberg may have given rise to Altenberg as a family name, such as:

Any information may be conveyed by an analog signal ; often such a signal is a measured response to changes in physical phenomena, such as sound, light, temperature, position, or pressure.

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 number of players may be replaced by substitutes during the course of the game.

Any of these may be sources of essential amino acids.

Any of these reactions may be used in LPCVD, but the silane reaction is also done in APCVD.

Any player may block another player's passage, so long as he does not hold or trip the player he intends to block.

Any large, smoothbore, muzzle-loading gun — used before the advent of breech-loading, rifled guns — may be referred to as a cannon, though once standardized names were assigned to different sized cannons, the term specifically referred to a gun designed to fire a shot, as opposed to a demi-cannon-, culverin-, or demi-culverin-.

Any string of characters may or may not be a well-formed name for a file or a link depending upon the context of application.

# Any experience which is inconsistent with the organization of the structure of the self may be perceived as a threat, and the more of these perceptions there are, the more rigidly the self structure is organized to maintain itself.

Any suggestion I may have made to you will be without force when I say the word ' cancelled.

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 scalar or other field may have a dipole moment.

Any elected or appointed board, committee, town officer or ten voters, may place an article on the warrant.

Any form of energy may be transformed into another form.

Any consistent set of units may be used, although SI units are preferred.

Any voluntary muscle in the body may be affected, although the tremor is most commonly seen in the hands and arms and slightly less commonly in the neck ( causing the patient's head to shake ), tongue, and legs.

Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra.

Any minister and the cabinet as a whole however must have continuing trust of the parliament and may be voted out, resign or be replaced.

Any posse riding down the street to demand Blue Throat's surrender would be wiped out with one deadly burst of fire.

Any fee so determined shall be entered as a part of such award, and payment thereof shall be made by the Secretary of the Treasury by deducting the amount thereof from the total amount paid pursuant to the award.

Any agreement to the contrary shall be unlawful and void.

Any and all payments required to be made by the Secretary of the Treasury under this title pursuant to any award made by the Commission to the Government of the United States shall be covered into the Treasury to the credit of miscellaneous receipts.

Any person who makes application for any such payment shall be held to have consented to all the provisions of this Title.

Any failure to file timely returns or make estimated tax payments when due must be fully explained.

Any alteration of one of these factors is distortion, although we generally use that word only for effects so pronounced that they can be stated quantitatively on the basis of standard tests.

Any irregularity on the piston heads will make it impossible, with normal means, to determine the final combustion chamber volume because the volume displaced by the piston heads cannot be readily computed.

Any fraction of an inch involved in the measurement must be converted to a decimal equivalent to simplify the mathematics.

Any house built now below that line without air conditioning will be obsolete in 10 years.

Any other point of intersection between C and Af will be called a tangent point.

Any free elections that were to be held in Poland would have to produce a government in which Moscow had complete confidence, and all pressure from the West for free voting by anti-Soviet elements in Poland would be met by restrictions on voting by these elements.

Any retaliatory gas attack would be airborne.

Any animal could when travelin' fast, be sent heels over head by the simple process of overtakin' the brute, seizin' its tail, and givin' the latter a pull to one side.

Any number of other ways exist to be rich.

* Any large snake that " crushes " its prey ( see Constriction ), if applied loosely, could be called anaconda.

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Any historical material on anagrams must always be interpreted in terms of the assumptions and spellings that were current for the language in question.