Search results
Results from the WOW.Com Content Network
Consequently, the solutions in rational numbers are all rescalings of integer solutions. Given an Euler brick with edge-lengths (a, b, c), the triple (bc, ac, ab) constitutes an Euler brick as well. [1]: p. 106 Exactly one edge and two face diagonals of a primitive Euler brick are odd. At least two edges of an Euler brick are divisible by 3.
In the 1950s, Evgenii Landis and Ivan Petrovsky published a purported solution, but it was shown wrong in the early 1960s. [9] In 1954 Zarankiewicz claimed to have solved Turán's brick factory problem about the crossing number of complete bipartite graphs, but Kainen and Ringel later noticed a gap in his proof. Complex structures on the 6-sphere.
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [ 1 ] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences . [ 2 ]
Euler's argument shows that a necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree. This condition turns out also to be sufficient—a result stated by Euler and later proved by Carl Hierholzer. Such a walk is now called an Eulerian trail or Euler walk in his honor ...
You are free: to share – to copy, distribute and transmit the work; to remix – to adapt the work; Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made.
The question of minimizing the number of crossings in drawings of complete bipartite graphs is known as Turán's brick factory problem, and for , the minimum number of crossings is one. K 3 , 3 {\displaystyle K_{3,3}} is a graph with six vertices and nine edges, often referred to as the utility graph in reference to the problem. [ 1 ]
Numerical solution with arbitrary accuracy was given by Eisenberger (1991). [11] 222 years after Euler's mistake in 1778, Elishakoff [12] [13] revisited this problem and derived closed-form solutions for self-buckling problems. [14]
Prouhet used the Thue–Morse sequence to construct a solution with = for any .Namely, partition the numbers from 0 to + into a) the numbers each with an even number of ones in its binary expansion and b) the numbers each with an odd number of ones in its binary expansion; then the two sets of the partition give a solution to the problem. [3]