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 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.

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 organism that falters or misperceives the signals or weakens is done.

Any information we have here in Taiwan is at your disposal ''.

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 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 musician playing Beethoven here, where Beethoven was born, is likely to examine his own interpretations with special care.

Any action is understood to create " seeds " in the mind that sprout into the appropriate results ( Pāli vipaka ) when they meet the right conditions.

** Any union of countably many countable sets is itself countable.

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 quantity of energy absorbed by the electron in excess of this amount is converted to kinetic energy according to the conservation of energy.

Any breach of these laws is a criminal offence and can lead to fine of up to £ 5, 000 or six months in prison.

The prohibition against retaliation or coercion applies broadly to any individual or entity that seeks to prevent an individual from exercising his or her rights or to retaliate against him or her for having exercised those rights ... Any form of retaliation or coercion, including threats, intimidation, or interference, is prohibited if it is intended to interfere.

Any word or phrase that exactly reproduces the letters in another order is an anagram.

Any work ( δW ) done must be done at the expense of internal energy U, since no heat δQ is being supplied from the surroundings.

" Any threat of war Clovis could make would only be effective if they were neighbors ; " it is nowhere written that Syagrius was handed over in 486 or 487.

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 person who is seen as a threat to civil aviation is banned from flying.

* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.

* Any ring A is an algebra over its center Z ( A ), or over any subring of its center.

* Any commutative ring R is an algebra over itself, or any subring of R.

Any form of intelligent life would do ; so, specifying carbon-based life, per se, is irrelevant.

* Any base that is water-soluble and forms hydroxide ions or the solution of a base in water.

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

Any units of angular measure can be used for declination, but it is customarily measured in the degrees ( ° ), minutes ( ' ), and seconds ( " ) of sexagesimal measure, with 90 ° equivalent to circle.

* Norton's theorem: Any network of voltage or current sources and resistors is electrically equivalent to an ideal current source in parallel with a single resistor.

* Thévenin's theorem: Any network of voltage or current sources and resistors is electrically equivalent to a single voltage source in series with a single resistor.

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

Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction.

Any units of angular measure can be used for right ascension, but it is customarily measured in hours (< sup > h </ sup >), minutes (< sup > m </ sup >), and seconds (< sup > s </ sup >), with 24 < sup > h </ sup > being equivalent to a full circle.

Any context-free grammar can be transformed into an equivalent grammar that has no left recursion, but removal of left recursion does not always yield an LL ( k ) grammar.

Any such satisficing problem can be formulated as an ( equivalent ) optimization problem using the Indicator function of the satisficing requirements as an objective function.

Any additional shaders, and any deviations from the standard shaders presented in this specification, must be documented by providing the equivalent shader expressed in the RenderMan shading language.

Note that for orientable compact surfaces without boundary, the Euler characteristic equals, where is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and counts the number of handles.

Any self-adjoint operator is unitarily equivalent to a multiplication operator.

Any black box containing only voltage sources, current sources, and other resistors can be converted to a Norton equivalent circuit, comprising exactly one ideal current source and one resistor.

Any Black box ( systems ) | black box containing only voltage sources, current sources, and other resistors can be converted to a Thévenin equivalent circuit, comprising exactly one voltage source and one resistor.

Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence ( hence " the " canonical divisor ).

Any one-dimensional equation of order is equivalent to an-dimensional first-order system ( as described in Ordinary differential equation # Reduction to a first order system ), but not necessarily vice versa.

* Any category is equivalent to its skeleton.

Any convention for proper Euler angles is equivalent to three such rotations, with one axis being repeated ( ZXZ for example ).

Any properly-defined register machine model is Turing equivalent.

Any vector-matrix equation may be transposed to an equivalent equation where the order of the factors is reversed.

Any linear system containing only voltage sources, current sources, and other resistors can be converted to a Thévenin equivalent circuit, comprising exactly one voltage source and one resistor representing " internal resistance ".

Any finite set of lines in the Euclidean plane has a combinatorially equivalent arrangement in the hyperbolic plane ( e. g. by enclosing the vertices of the arrangement by a large circle and interpreting the interior of the circle as a Klein model of the hyperbolic plane ).

Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.

* The quasi-isometry classification of rank one lattices: Any quasi-isometry of a hyperbolic lattice is equivalent to a commensurator.