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Isotropy

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The shelters and some of the staff homes were lost.
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Efforts were made to restore the park from 1980, including publication of the management plan and purchase of land to regularize the park's tenure.
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Decree 90023 of 2 August 1984 delimited the area of the park as .
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In the 1990s the municipality of Guapimirim was split off from Magé, and also contains the biggest part of the park.
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From the 1990s the old buildings have been restored and new ones built.
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The park was included in the Central Rio de Janeiro Atlantic Forest Mosaic, created in 2006.
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The climate is tropical superhumid, with 80% to 90% relative humidity caused by moist air from the Atlantic most of the year.
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Average temperatures range from , but may reach and may fall below freezing in the highest parts.
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Average rainfall is , with more rain in the summer (December to March) and a dry season in the winter from June to August.
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The south east side facing the ocean receives more rain than the north west side.
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The park is in the Atlantic Forest biome, and due to the high rainfall has rich vegetation, much of it unique to this biome.
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More than 2,800 species of plant have been recorded including 360 of orchids and over 100 bromeliads.
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Up to the lower slopes are covered by typical lowland rainforest.
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From the vegetation is montane forest, with significant variations depending on the conditions in each area.
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In many places the upper canopy is with emergent trees reaching up to .
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From there is cloud forest, typically trees of with crooked trunks covered in epiphytic moss and plants such as bromeliads and orchids.
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The understory has shrubs and the outcrops are populated by ferns and mosses.
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There are various endemic species.
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Above the vegetation is high montane, with open fields and small woody shrubs.
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347 species have been found in this environment of which 66 are endemic to this ecosystem.
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The park is one of the few natural habitats of species of "Schlumbergera", which were developed into the colourful "Thanksgiving Cactus" and "Christmas Cactus", widely grown as house plants.
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Hickory Dickory Dock
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"Hickory Dickory Dock" or "Hickety Dickety Dock" is a popular English language nursery rhyme.
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It has a Roud Folk Song Index number of 6489.
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The most common modern version is:
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<poem>
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Hickory, dickory, dock.
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The mouse ran up the clock.
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The clock struck one,
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The mouse ran down,
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Hickory, dickory, dock.</poem>
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Other variants include "down the mouse ran" or "down the mouse run" or "and down he ran" or "and down he run" in place of "the mouse ran down".
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<score vorbis="1">
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\new Staff «
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\clef treble \key d \major {
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%\new Lyrics \lyricmode {
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»
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</score>
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The earliest recorded version of the rhyme is in "Tommy Thumb's Pretty Song Book", published in London in about 1744, which uses the opening line: 'Hickere, Dickere Dock'.
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The next recorded version in "Mother Goose's Melody" (c. 1765), uses 'Dickery, Dickery Dock'.
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The rhyme is thought by some commentators to have originated as a counting-out rhyme.
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Westmorland shepherds in the nineteenth century used the numbers "Hevera" (8), "Devera" (9) and "Dick" (10) which are from the language Cumbric.
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The rhyme is thought to have been based on the astronomical clock at Exeter Cathedral.
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The clock has a small hole in the door below the face for the resident cat to hunt mice.
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Extremal graph theory
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Extremal graph theory is a branch of mathematics that studies how global properties of a graph influence local substructure.
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It encompasses a vast number of results that describe how do certain graph properties - number of vertices (size), number of edges, edge density, chromatic number, and girth, for example - guarantee the existence of certain local substructures.
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One of the main objects of study in this area of graph theory are extremal graphs, which are maximal or minimal with respect to some global parameter, and such that they contain (or do not contain) a local substructure- such as a clique, or an edge coloring.
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Extremal graph theory can be motivated by questions such as the following:
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Question 1.
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What is the maximum possible number of edges in a graph on formula_1 vertices such that it does not contain a cycle?
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If a graph on formula_1 vertices contains at least formula_1 edges, then it must also contain a cycle.
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Moreover, any tree with formula_1 vertices contains formula_5 edges and does not contain cycles; trees are the only graphs with formula_5 edges and no cycles.
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Therefore, the answer to this question is formula_5, and trees are the extremal graphs.
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Question 2.
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If a graph on formula_1 vertices does not contain any triangle subgraph, what is the maximum number of edges it can have?
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Mantel's Theorem answers this question – the maximal number of edges is formula_9.
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The corresponding extremal graph is a complete bipartite graph on formula_10 vertices, i.e., the two parts differ in size by at most 1.
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A generalization of Question 2 follows:
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Question 3.
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Let formula_11 be a positive integer.
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How many edges must there be in a graph formula_12 on formula_1 vertices in order to guarantee that, no matter how the edges are arranged, the clique formula_14 can be found as a subgraph?
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The answer to this question is formula_15 and it is answered by Turán's Theorem.
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Therefore, if a graph formula_12 on formula_1 vertices is formula_18-free, it can have at most formula_19
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many edges; the corresponding extremal graph with that many edges is the Turán graph, shown in the figure above.
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It is the complete join of formula_20 independent sets (with sizes as equal as possible – such a partition is called "equitable").
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The following question is a generalization of Question 3, where the complete graph formula_21 is replaced by any graph formula_22:
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Question 4.
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Let formula_11 be a positive integer, and formula_22 any graph on formula_11 vertices.
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How many edges must there be in a graph formula_12 on formula_1 vertices in order to guarantee that, no matter how the edges are arranged, formula_22 is a subgraph of formula_12?
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This question is mostly answered by the Erdős–Stone theorem.
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The main caveat is that for bipartite formula_22, the theorem does not satisfactorily determine the asymptotic behavior of the extremal edge count.
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For many particular (classes of) bipartite graphs, determining the asymptotic behavior remains an open problem.
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Several foundational results in extremal graph theory answer questions which follow this general formulation:
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Question 5.
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Given a graph property formula_31, a parameter formula_32 describing a graph, and a set of graphs formula_33, we wish to find the minimal possible value formula_34 such that every graph formula_35 with formula_36 has property formula_31.
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Additionally, we might want to describe graphs formula_12 which are extremal in the sense of having formula_39 close to formula_34 but which do not satisfy the property formula_31.
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In Question 1, for instance, formula_33 is the set of formula_1-vertex graphs, "formula_31" is the property of containing a cycle, formula_45 is the number of edges, and the cutoff is formula_46.
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The extremal examples are trees.
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Mantel's Theorem (1907) and Turán's Theorem (1941) were some of the first milestones in the study of Extremal graph theory.
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In particular, Turán's theorem would later on become a motivation for the finding of results such as the Erdős-Stone-Simonovits Theorem (1946).
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This result is surprising because it connects the chromatic number with the maximal number of edges in an formula_22-free graph.
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An alternative proof of Erdős-Stone-Simonovits was given in 1975, and utilised Szemerédi's Theorem, an essential technique in the resolution of extremal graph theory problems.
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A global parameter which has an important role in extremal graph theory is subgraph density; for a graph formula_48 and a graph formula_22, its subgraph density is defined as
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formula_50.
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In particular, edge density is the subgraph density for formula_51:
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formula_52
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The theorems mentioned above can be rephrased in terms of edge density.
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For instance, Mantel's Theorem implies that the edge density of a triangle-free subgraph is at most formula_53.
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Turán's theorem implies that edge density of formula_14-free graph is at most formula_55.
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Moreover, the Erdős-Stone-Simonovits theorem states that
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formula_56
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where formula_57 is the maximal number of edges that an formula_22-free graph on formula_1 vertices can have, and formula_60 is the chromatic number of formula_22.
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An interpretation of this result is that the edge density of an formula_22-free graph is asymptotically formula_63.
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Another result by Erdős, Reyni and Sós (1966) shows that graph on formula_1 vertices not containing formula_65 as a subgraph has at most the following number of edges.
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formula_66
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The theorems stated above give conditions for a small object to appear within a (perhaps) very large graph.
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Another direction in extremal graph theory is looking for conditions that guarantee the existence of a structure that covers every vertex.
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Note that it is even possible for a graph with formula_1 vertices and formula_68 edges to have an isolated vertex, even though almost every possible edge is present in the graph.
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Edge counting conditions give no indication as to how the edges in the graph are distributed, leading to results which only find bounded structures on very large graphs.