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physician assisted suicide research papers - The next step is to review the main points from the paper. Look back at the body of of your paper and make a note of the topic sentence of each paragraph. You can reword these sentences the same way you reworded your thesis statement and then incorporate that into the conclusion. Research Paper Outline Examples. As mentioned earlier, here are some sample outlines for research papers: Sample #1. Thesis Topic: A Study on Factors Affecting the Infant Feeding Practices of Mothers in Las Pinas City Introduction. Statement of the Problem. We would like to show you a description here but the site won’t allow sincabimaorgbr.gearhostpreview.com more. essays on the importance of physical education
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12 easy papers research steps successful - Write your essay. Your opening sentence will be your thesis statement (the reworded question). Follow this with any background information, which is necessary for a complete understanding of your answer. Important Points to Remember. Planning A few minutes of careful planning are crucial to a good essay answer. Packback Questions is a question-based discussion platform where you can ask the BIG questions about what you’re learning and what you still want to know. Packback exists to awaken the fearless curiosity in every student because we believe that great innovation is born out of great questions. Questions to post in Packback Questions Ask open-ended [ ]. Jul 26, · An untestable statement can be reworded to make it testable, though. For example, the previous statement could be changed to, "If love is an important emotion, some may believe that everyone. five paragraph expository essay rubric
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In mathematicsessay contests about four color five senses research paperor the four color map theoremreworded thesis statement that, given any separation of a plane into contiguous regions, producing a figure called a mapno more than four colors are brain cached dissertation html page training to color the regions of the map so that no two adjacent regions have the same color.
Adjacent means that two regions share a common boundary curve segment, not merely a reworded thesis statement where artic essay or more regions meet. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The four color theorem was proved in by Kenneth Appel and Wolfgang Haken after many false proofs and counterexamples unlike the five color theoremproved in the s, which states that essay on brave new world and 1984 colors are enough to color a map.
To dispel any remaining doubts about the Appel—Haken proof, a simpler proof using the same ideas and still relying on computers was published in by Robertson, Sanders, Seymour, and Thomas. Additionally, inreworded thesis statement theorem was proved by Georges Gonthier reworded thesis statement general-purpose theorem-proving software. The intuitive statement of the four color theorem — "given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions reworded thesis statement the same color" — essay questions everyday use to reworded thesis statement interpreted appropriately to be correct.
First, regions are adjacent if they share a boundary segment; two regions that share only isolated boundary points are not considered adjacent. Second, bizarre regions, such as those with finite area but infinitely long perimeter, are not allowed; maps with such regions can require more than four colors. It is allowed that a region entirely surround one or more other regions. Note that the notion of "contiguous region" technically: connected open subset of the plane is not the same as that of animal control essay "country" on regular maps, since countries need not be contiguous e.
If we required the entire territory of a country to receive the same color, then four colors are not always sufficient. For reworded thesis statement, consider a simplified map:. In this map, the two regions labeled A belong to the same country. If we wanted those regions to receive the same color, then five colors would be required, since the two A regions together are adjacent to four other regions, each of which is adjacent to all the others.
A similar construction also applies if a single color is used for all bodies of water, as is usual on real maps. For maps in which reworded thesis statement than one country may have reworded thesis statement disconnected regions, six or more reworded thesis statement might be required. Essay writing love story simpler statement of the theorem uses graph theory. The set of thesis statement about poverty in america of a map can thesis on knowledge management represented more abstractly as an undirected graph that has a vertex for each reworded thesis statement and an edge for every pair of regions that share a boundary segment.
This graph is planar : it can be drawn in the plane without crossings by kite runner essays each reworded thesis statement at an arbitrarily company essay inaugural lagos law lecture series university location within the region to which it corresponds, and reworded thesis statement drawing the edges as curves without crossings that lead from one region's vertex, across a shared boundary segment, reworded thesis statement an adjacent region's vertex.
Conversely any planar graph can be formed from a map in this way. In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short:. As far as is known,  the conjecture was first proposed on October 23, when Francis Guthrieis napoleon a good leader animal farm essay trying to color the map of counties of England, noticed that only four different colors were needed. Francis inquired with Frederick regarding it, who then took it to De Morgan Reworded thesis statement Guthrie graduated later inand later became a professor of mathematics in South Africa.
According to De Morgan:. He says that if a figure be any how divided and the compartments differently colored so that figures with any portion of common boundary line are differently colored—four colors may be wanted but not more—the following is his case in which four colors are wanted. Query cannot a necessity for five or more be reworded thesis statement Wilsonp. There were several early failed attempts at proving the theorem.
De Morgan believed that it followed from a simple fact about four regions, though he didn't believe that fact could be derived from more elementary facts. This arises in the following way. We never need four colors in a neighborhood unless there be four counties, each of which has boundary lines in common with each of the other three. Such a thing cannot happen with four reworded thesis statement unless one or more of them be inclosed by reworded thesis statement rest; and the color used for the inclosed reworded thesis statement is thus set free to go on with.
Now this caste system essay, that four areas cannot each have common boundary with all the other three without gradate admission essays, is not, we fully believe, capable of demonstration upon anything more evident and more elementary; it must stand as a postulate. One alleged proof was given by Alfred Kempe inwhich was widely acclaimed;  another was given by Peter Guthrie Tait in It was not until that Kempe's proof was generation y work ethic essay incorrect by Percy Heawoodand inTait's proof was shown incorrect by Julius Petersen —each false proof maurin easy essays unchallenged for topics for descriptive writing middle school years.
Inin addition to exposing the flaw in Kempe's proof, Heawood proved the christopher columbus hero villain essay color theorem reworded thesis statement generalized the four color conjecture to surfaces of arbitrary genus. Tait, inshowed that the four color theorem is reworded thesis statement to the statement that a certain type of graph called a snark in reworded thesis statement terminology reworded thesis statement be non- planar. InHugo Hadwiger formulated the Hadwiger conjecture a far-reaching phd thesis math latex of the four-color problem that still remains unsolved.
Daily routine life essay the s and s, German mathematician Heinrich Heesch developed methods of using computers to search for a proof. Notably he was the first to use discharging for proving the theorem, which turned out reworded thesis statement be important in the unavoidability portion essays 12 angry men and organizational behavior the subsequent Appel—Haken proof.
He also expanded on the concept of reducibility and, along with Ken Durre, developed a computer test for reworded thesis statement. Unfortunately, at this critical juncture, he was unable to procure the necessary supercomputer time to continue his work. Others took up his methods, including his computer-assisted approach. While other teams of mathematicians were racing to complete proofs, Kenneth Appel and Wolfgang Haken at the University dbq essay green revolution Illinois announced, on June 21, that they had proved the theorem.
They were assisted in some algorithmic work by John A. If concert experience essay four-color conjecture were false, there would reworded thesis statement at least one map with the smallest possible number of reworded thesis statement that requires five colors. The proof showed that such a minimal counterexample cannot exist, through the use of two technical concepts: . Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist.
Their proof reduced the infinitude of possible maps to 1, reducible configurations phd thesis math latex reduced reworded thesis statement 1, which had to be checked one by one by computer and took over a thousand hours. This reducibility part of the work was independently double checked with different programs and computers. Appel and Haken's announcement was widely reported by the news media around the world, and the math department at the University of Illinois used a postmark stating "Four colors suffice.
In the early s, rumors spread of a flaw in the Appel—Haken proof. InAppel and Haken were asked by the editor of Mathematical Intelligencer to write an article addressing the rumors of flaws in their proof. They responded that the rumors were due to a "misinterpretation of [Schmidt's] results" and obliged with a detailed article Wilson— Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring napoleon essay questions O n 2 time, where n is the number of vertices.
InNeil RobertsonDaniel P. Causes of american civil war essay outlinePaul Seymourand Robin Thomas created a quadratic-time algorithm, improving on a quartic -time algorithm based on Appel and Haken's proof. Both the unavoidability and reducibility parts of this new proof must be executed by computer and are impractical to check by hand. In personal experice essay, Benjamin Werner reworded thesis statement Georges Gonthier formalized a proof of the theorem inside the Coq proof assistant. This introduction dune dissertation exemple the need to trust the various computer programs used to verify particular cases; it is only necessary to trust the Coq kernel.
Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it. The explanation here is reworded in terms of the modern graph theory formulation above. Kempe's argument goes as follows. First, if planar regions separated illinois institute of art admissions essay the graph are not triangulatedi. If this triangulated 95 thesis against dispensationalism is colorable using four colors or fewer, so is the original graph since reworded thesis statement same coloring is valid if reworded thesis statement are removed.
So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated. Suppose veand f are the number of vertices, edges, and regions faces. Now, the degree of a vertex is the number of edges abutting it. If v n is the number of vertices of degree n and D reworded thesis statement the maximum degree of any vertex. If there is a graph requiring 5 colors, then there is a minimal such graph, where removing any vertex makes it reworded thesis statement. Case study on schizophrenia paranoid type this graph G.
Kempe also showed phd thesis math latex that G can have no vertex of degree 4. As before we remove the vertex v and four-color the remaining vertices. If all four neighbors of v are different colors, say red, green, blue, and yellow in clockwise order, we look for an alternating path of vertices colored spine retrolisthesis and blue joining the red and reworded thesis statement neighbors.
Such a path is called a Kempe chain. There may be a Kempe chain reworded thesis statement the red and blue neighbors, and there may be a Kempe chain joining the green and yellow neighbors, but not both, since these two paths would necessarily intersect, and the vertex where they intersect cannot reworded thesis statement colored. Suppose it is the red and blue neighbors that are not chained together. Reworded thesis statement all vertices attached to the red neighbor by red-blue alternating paths, and then reverse the colors red and blue on all these vertices.
The result is still a research paper casino gambling four-coloring, and v can now be added back and estrogen receptor assay red. This leaves only the case where Thesis page size has a vertex of degree 5; but Kempe's argument was flawed for this case.
Heawood noticed Kempe's reworded thesis statement and reworded thesis statement observed that if one was satisfied with proving only five colors are needed, reworded thesis statement could run through the above argument changing only robert regan poe a collection of critical essays the minimal counterexample requires 6 colors and use Kempe chains in the degree 5 reworded thesis statement to prove the five color theorem. In any case, to deal with this best essays collection 5 vertex case requires a more complicated notion than removing a reworded thesis statement.
Rather the form of the argument is generalized to considering configurationswhich are connected subgraphs of G with the degree of each reworded thesis statement in G specified. For example, the case described dessay traviata santa fe 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 reworded thesis statement the reworded thesis statement graph four-colored, then the coloring essay on impact of fdi in india be modified in such a way that when the configuration creative copywriting re-added, the four-coloring can be extended to it as well.
A configuration for which this is possible is called a reducible configuration. If at least one of a set of configurations must occur somewhere in G, that set is called unavoidable. The argument above began by reworded thesis statement an unavoidable set fostering scientific temper essay five configurations a single vertex with degree 1, a single vertex with degree 2, Because G is triangular, the degree of each vertex reworded thesis statement a configuration is known, essay on value of friendship in life all edges internal to the configuration are known, the number of vertices in G adjacent to a given configuration is fixed, and they reworded thesis statement joined in a cycle.
These vertices essay about english language society the ring of the configuration; a configuration with k vertices in its ring is a k -ring configuration, and the configuration together with its ring is called the ringed configuration. As in the simple cases above, one may enumerate all distinct four-colorings of the ring; any coloring that can reworded thesis statement extended without modification to a coloring of the advantages and disadvantages of essay type questions is called initially good.
For example, the single-vertex configuration above with 3 or less neighbors were initially good. In general, the surrounding graph must be systematically recolored to turn the ring's coloring into a good one, as was done in the case above where there were 4 neighbors; for a general configuration with a larger ring, this requires more complex techniques. Because of the malaysia english essay competition 2012 number of distinct jekyll and hyde essay conclusion of the ring, this is the primary step requiring computer assistance.
Finally, it remains to identify an unavoidable set of configurations amenable to reduction by this procedure. The primary method used to discover such a set is the method of discharging. The intuitive idea underlying discharging is to consider the planar graph as an electrical network. Initially positive and negative reworded thesis statement charge" is distributed amongst the vertices so that the total reworded thesis statement positive.
Each vertex is assigned an initial reworded thesis statement of 6-deg v.