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A two-column proof published in 1913. A particular way of organising a proof using two parallel columns is often used as a mathematical exercise in elementary geometry classes in the United States. [29] The proof is written as a series of lines in two columns.
In 1981, Gardner's column alternated with a new column by Douglas Hofstadter called "Metamagical Themas" (an anagram of "Mathematical Games"). [1] The table below lists Gardner's columns. [2] Twelve of Gardner's columns provided the cover art for that month's magazine, indicated by "[cover]" in the table with a hyperlink to the cover. [3]
Talk pages (the "Discussion" tab at the top of article pages) are the best way to raise queries about the content of a particular article. The mathematics reference desk is useful if you have a question and don't know where to look up the answer. Explore the category system. The mathematics portal is a good "way in" to mathematics articles on ...
Gorenstein and Lyons's proof for the case of rank at least 4 was 731 pages long, and Aschbacher's proof of the rank 3 case adds another 159 pages, for a total of 890 pages. 1983 Selberg trace formula. Hejhal's proof of a general form of the Selberg trace formula consisted of 2 volumes with a total length of 1322 pages. Arthur–Selberg trace ...
A 3-coloring of the vertices of a triangulated polygon. The blue vertices form a set of three guards, as few as is guaranteed by the art gallery theorem. However, this set is not optimal: the same polygon can be guarded by only two guards. Steve Fisk's proof is so short and elegant that it was chosen for inclusion in Proofs from THE BOOK. [4]
The normal equations can be derived directly from a matrix representation of the problem as follows. The objective is to minimize = ‖ ‖ = () = +.Here () = has the dimension 1x1 (the number of columns of ), so it is a scalar and equal to its own transpose, hence = and the quantity to minimize becomes
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