: For example, graph 1 above expresses the abstraction " agent sits on location ".

For curves given by the graph of a function, horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to Vertical asymptotes are vertical lines near which the function grows without bound.

For example, the graph contains the points ( 1, 1 ), ( 2, 0. 5 ), ( 5, 0. 2 ), ( 10, 0. 1 ), ... As the values of x become larger and larger, say 100, 1000, 10, 000 ..., putting them far to the right of the illustration, the corresponding values of y,. 01,. 001,. 0001, ..., become infinitesimal relative to the scale shown.

For example, if a graph has 17 valid 3-colorings, the SAT formula produced by the reduction will have 17 satisfying assignments.

For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point.

For example, they allow representation of a directed graph with trees on the nodes.

For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G. As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well.

For the case of power-of-two, Papadimitriou ( 1979 ) argued that the number of complex-number additions achieved by Cooley – Tukey algorithms is optimal under certain assumptions on the graph of the algorithm ( his assumptions imply, among other things, that no additive identities in the roots of unity are exploited ).

For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries-Wong graph.

For any positive integers and, there exists a graph with girth at least and chromatic number at least ; for instance, the Grötzsch graph is triangle-free and has chromatic number 4, and repeating the Mycielskian construction used to form the Grötzsch graph produces triangle-free graphs of arbitrarily large chromatic number.

For example if a graph represents a road network, the weights could represent the length of each road.

For a planar graph, the crossing number is zero by definition.

For contraint frameworks which are strictly compositional, graph unification is the sufficient satisfiability and combination function.

For instance, if one graphs a set of solutions of an equation in some higher dimensional space, he may ask about the geometric properties of the graph.

For example, a similar geometric system was published in 1987 by Kjell Gustafson, whose method represents a rhythm as a two-dimensional graph.

For example, the figure shows sin θ for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle ( top ) and as a graph ( bottom ).

For example, the minimum spanning tree of the graph associated with an instance of the Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O ( n log n ) time for n points ( considerably less than the number of edges ).

For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function ; in other words, the function's graph.

For a discussion of tree structures in specific fields, see Tree ( data structure ) for computer science: insofar as it relates to graph theory, see tree ( graph theory ), or also tree ( set theory ).

For instance, if I is the category of the directed graph, then C < sup > I </ sup > has as objects the morphisms of C, and a morphism between and in C < sup > I </ sup > is a pair of morphisms and in C such that the " square commutes ", i. e..

For example, a poker hand can be described as a 5-combination ( k = 5 ) of cards from a 52 card deck ( n = 52 ).

For example, if the last player to draw wants three replacements but there are only two cards remaining in the deck, the dealer gives the player the one top card he can give, then shuffles together the bottom card of the deck, the burn card, and the earlier players ' discards ( but not the player's own discards ), and finally deals two more replacements to the last player.

For example, if an ace is drawn from a deck and not reinserted, the next draw is less likely to be an ace and more likely to be of another rank.

For most games, the cards are assembled into a deck, and their order is randomized by shuffling.

For example, the Australian game of Manila uses a piquet deck, and Mexican stud is played with the 8s, 9s, and 10s removed from the deck ( and a joker added ).

For example in draw poker, a hand with four diamonds has nine outs to make a flush: there are 13 diamonds in the deck, and four of them have been seen.

For a deck of given size, the number of Mongean shuffles that it takes to return a deck to starting position, is known.

For example, seven shuffles of a new deck leaves an 81 % probability of winning New Age Solitaire where the probability is 50 % with a uniform random deck ( Mann, especially section 10 ).

For example, in a 10 card deck, if a 7 card cut and a 4 card cut are made, that is, 7 cards are moved from the top of the deck to the bottom and then the resulting top 4 cards are also moved to the bottom, then those two consecutive cuts are equivalent to a cut the size of ( 7 + 4

For 1998 – 99 the Hurricanes curtained off most of the upper deck, lowering the Coliseum's listed capacity to about 12, 000, but attendance continued to lag.

For 1969, Pontiac moved the Grand Prix from the full-sized lineup into a G-body model of its own based on the A-body intermediate chassis, but with distinctive styling and long hood / short deck proportions to create yet another niche product-the intermediate-sized personal-luxury car that offered the luxury and styling of the higher priced personal cars such as the Buick Riviera and Ford Thunderbird but for a much lower pricetag.

For instance, the Rider-Waite tarot deck, currently the widest-circulated deck in existence, depicts The Hierophant or pope as wearing a papal tiara and carrying a papal cross.

For stability, the lowest gundeck had to be very close to the water, and in anything but calm water the gunports had to be kept closed, rendering the entire deck useless.

For playback, the tape would be inserted into a Beta-format deck.

For the naval cannon from the figure above, it will roll backwards and the recoil energy will be mostly absorbed by the friction forces in the wheel axles and between the wheel and the ship deck and this energy is again converted to heat.

For that reason, the seats down the left field line were typically among the last ones sold ; the ( less expensive ) outfield lower deck seating tended to fill up sooner.

For operation in high temperatures, the engine's ventilation was improved by creating slits over the engine deck to the rear of the chassis, and cold weather performance was boosted by adding a device to heat the engine's coolant, as well as a starter fluid injector.

For example, songs used to accompany the work of holystoning the deck have been attested.

For loan to the Government of India under subsection ( G ) of Section 104 of the Act, the rupee equivalent of not more than $538 million for financing such projects to promote balanced economic development as may be mutually agreed.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

For each element a of a group G, conjugation by a is the operation φ < sub > a </ sub >: G → G given by ( or a < sup >− 1 </ sup > ga ; usage varies ).

For example the United States uses NTSC-M, the UK uses PAL-I, France uses SECAM-L, much of Western Europe and Australia uses PAL-B / G, most of Eastern Europe uses PAL-D / K or SECAM-D / K and so on.

For example " GOOD DOG " can be encrypted as " PLLX XLP " where " L " substitutes for " O ", " P " for " G ", and " X " for " D " in the message.

For any topological space X the ( Alexandroff ) one-point compactification αX of X is obtained by adding one extra point ∞ ( often called a point at infinity ) and defining the open sets of the new space to be the open sets of X together with the sets of the form G < font face =" Arial, Helvetica "> U </ font >

For elements g and h of a group G, the commutator of g and h is.

* For any homomorphism f: G → H, f () =.

for some natural number n. Moreover, since, the commutator subgroup is normal in G. For any homomorphism f: G → H,

groups ( G, *) and ( H, ●), denoted by G × H. For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by.

For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.

For their part, the Italians developed several monoplanes such as the Fiat G. 50, but being short on funds, were forced to continue operating obsolete Fiat CR. 42 biplanes.

For example, John G. Lake was present during the years of the Azusa Street revival.