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The longest alternating subsequence problem has also been studied in the setting of online algorithms, in which the elements of are presented in an online fashion, and a decision maker needs to decide whether to include or exclude each element at the time it is first presented, without any knowledge of the elements that will be presented in the future, and without the possibility of recalling ...
Let p be an interior point of the disk, and let n be a multiple of 4 that is greater than or equal to 8. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line n / 2 − 1 times by an angle of 2 π / n radians, and slicing the disk on each of the resulting n / 2 lines.
A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21 The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.
A quadrilateral such as BCEF is called an adventitious quadrangle when the angles between its diagonals and sides are all rational angles, angles that give rational numbers when measured in degrees or other units for which the whole circle is a rational number. Numerous adventitious quadrangles beyond the one appearing in Langley's puzzle have ...
For n = 5, the Schur cover of the alternating group is given by SL(2, 5) → PSL(2, 5) ≅ A 5, which can also be thought of as the binary icosahedral group covering the icosahedral group. Though PGL(2, 5) ≅ S 5 , GL(2, 5) → PGL(2, 5) is not a Schur cover as the kernel is not contained in the derived subgroup of GL(2 ,5).
There are 4 dihedral subgroups: Dih 8, Dih 4, Dih 2, and Dih 1, and 5 cyclic subgroups: Z 16, Z 8, Z 4, Z 2, and Z 1, the last implying no symmetry. On the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
According to the Erdős–Szekeres theorem, any sequence of + distinct integers has an increasing or a decreasing subsequence of length + [7] [8] For inputs in which each permutation of the input is equally likely, the expected length of the longest increasing subsequence is approximately . [9] [2]