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Page "Mac Lane's planarity criterion" ¶ 5
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However and complete
However, he turned out to be a complete failure in his new position.
However, even if the latent demand for demythologization is not nearly as widespread as we are claiming, at least among the cultured elements of the population there tends to be an almost complete indifference to the church and its traditional message of sin and grace.
However, the Hotaki dynasty came to a complete end in 1738 after being toppled by the Afsharids who were led by Nader Shah of Khorasan.
He tinformed Olga Carlisle that, at the end of The Blind Beauty, he wished to depict " the birth of an enlightened and affluent middle class, open to occidental influences, progressive, intelligent, artistic ..." However, Pasternak fell ill with terminal lung cancer before he could complete the first play of the trilogy.
However, Goukouni ’ s January 1981 statement that Chad and Libya had agreed to work for the realization of complete unity between the two countries generated intense international pressure and Goukouni's subsequent call for the complete withdrawal of external forces.
However none of these " alternatives " has the same breadth of explanation as inflation, and still require inflation for a more complete fit with observation ; they should therefore be regarded as adjuncts to inflation, rather than as alternatives.
However the closed interval is complete ; the given sequence does have a limit in this interval and the limit is zero.
However, the number of crafts required to obtain, transmit, and maintain a complete Dyson sphere far exceeds our present-day industrial capabilities.
However, in Cyclops ( the only complete satyr-play that survives ) Euripides structured the entertainment more like a tragedy and introduced a note of critical irony typical of his other work.
However it would never come to a complete rest, and we therefore call that response to the impulse ( striking it with a hammer ) " infinite " in duration.
However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.
However, by 1915 this process was complete, enabling Portuguese colonial rule to progress in a relatively unruffled state-until the emergence of nationalist movements all over Africa in the 1950s.
However, the Central Government has targeted several roads for complete rehabilitation, and already many have been rehabilitated.
However, the compositional style employed in the opera, with few stand-alone arias, was criticized at the time and remains a barrier to the opera's complete acceptance into the standard repertoire.
However, the civil parishes were not a complete third-tier of local government.
However, more than a century before Ebbinghaus, Johann Andreas Segner invented the “ Segner-wheel ” to see the length of after-images by seeing how fast a wheel with a hot coal attached had to move for the red ember circle from the coal to appear complete.
However, later emperors such as Aurangazeb tried to establish complete Muslim dominance, and as a result several historical temples were destroyed during this period and taxes imposed on non-Muslims.
However, the Indonesian Minangkabau people ( from western Sumatra ), despite being Muslim, employ only complete matrilineal succession with property and land passing down from mother to daughter.
However, complete perfection is unattainable in this life, and the believer should expect a continual struggle against sin.
However, it was undertaken on so large a scale as to make it impossible for them to complete it.
However, in the same letter to Vera Zasulich, Marx points out that " at the core of the capitalist system ... lies the complete separation of the producer from the means of production.
However, the rush to complete this rapid expansion scheme resulted in the deaths of 997 personnel and another 700 wounded.
However, on 20 July 2008 he annonced that he would not be able to complete it.
" However within this line of thought " The tension between anti-vanguardism and vanguardism has frequently resolved itself in two diametrically opposed ways: the first involved a drift towards the party ; the second saw a move towards the idea of complete proletarian spontaneity ... The first course is exemplified most clearly in Gramsci and Lukacs ... The second course is illustrated in the tendency, developing from the Dutch and German far-lefts, which inclined towards the complete eradication of the party form.

However and graph
( However, there are other, similar matrices that are also called " Laplacian matrices " of a graph.
However, these other algorithms can also find minimum spanning forests of disconnected graphs, while Prim's algorithm requires the graph to be connected.
However, in practice, RDF data is often persisted in relational database or native representations also called Triplestores, or Quad stores if context ( i. e. the named graph ) is also persisted for each RDF triple.
However, for graphing by hand these formulas are unwieldy ; instead, it is common to use graph paper designed specifically for the task.
However, in the short-run policymakers will face an inflation-unemployment rate tradeoff marked by the " Initial Short-Run Phillips Curve " in the graph.
However, the recursive operator will actually define the graph of f. First, Φ will contain the pair.
However there are many other kinds of directed acyclic graph that are not formed by orienting the edges of an undirected acyclic graph, and every undirected graph has an acyclic orientation, an assignment of a direction for its edges that makes it into a directed acyclic graph.
However, many different DAGs may give rise to this same reachability relation: for example, the DAG with two edges a → b and b → c has the same reachability as the graph with three edges a → b, b → c, and a → c. If G is a DAG, its transitive reduction is the graph with the fewest edges that represents the same reachability as G, and its transitive closure is the graph with the most edges that represents the same reachability.
However, boundedness of the inverse does follow directly from its existence if one introduces the additional assumption that T is closed ; this follows from the closed graph theorem.
However, it has two graph-theoretic interpretations: as the sum of weights of cycle covers of a directed graph, and as the sum of weights of perfect matchings in a bipartite graph.
However, if a graph contains a " negative cycle ", i. e., a cycle whose edges sum to a negative value, then walks of arbitrarily low weight can be constructed by repeatedly following the cycle, so there may not be a shortest path.
However, it is essentially the same as algorithms previously published by Bernard Roy in 1959 and also by Stephen Warshall in 1962 for finding the transitive closure of a graph.
However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality ; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. ( In other words, it is a collection of sets of size k .) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on.
However, if the algorithm terminates without detecting an odd cycle of this type, then every edge must be properly colored, and the algorithm returns the coloring together with the result that the graph is bipartite.
However, the notion of isomorphism may be applied to all other variants of the notion of graph, by adding the requirements to preserve the corresponding additional elements of structure: arc directions, edge weights, etc., with the following exception.
However, if the system is initialized to a perfectly symmetric state, like all philosophers holding their left side forks, then the graph is cyclic at the outset, and their solution cannot prevent a deadlock.
However, as the frequency-domain graph shows, square waves contain a wide range of harmonics ; these can generate electromagnetic radiation or pulses of current that interfere with other nearby circuits, causing noise or errors.
However, the so-called SIR model is being replaced by more sophisticated models using graph theory, since individuals in a human population-unlike molecules in an ideal solution-do not mix homogeneously.
However, nonplanar graphs frequently arise in applications, so graph drawing algorithms must generally allow for edge crossings.

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