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This problem is known as the primitive circle problem, as it involves searching for primitive solutions to the original circle problem. [9] It can be intuitively understood as the question of how many trees within a distance of r are visible in the Euclid's orchard , standing in the origin.
Edwin Ray Guthrie (/ ˈ ɡ ʌ θ r i /; January 9, 1886 – April 23, 1959), a behavioral psychologist, began his career in mathematics and philosophy in 1917. He spent most of his career at the University of Washington, where he was a full-time professor and later became an emeritus professor in psychology.
Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone. The class of conic optimization problems includes some of the most well known classes of convex optimization problems, namely linear and semidefinite programming .
Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack n unit circles into the smallest possible equilateral triangle. Optimal solutions are known for n < 13 and for any triangular number of circles, and conjectures are available for n < 28 .
One such extension is to find the maximum possible density of a system with two specific sizes of circle (a binary system). Only nine particular radius ratios permit compact packing , which is when every pair of circles in contact is in mutual contact with two other circles (when line segments are drawn from contacting circle-center to circle ...
The problem addressed by the circle method is to force the issue of taking r = 1, by a good understanding of the nature of the singularities f exhibits on the unit circle. The fundamental insight is the role played by the Farey sequence of rational numbers, or equivalently by the roots of unity :
Illustration of Poncelet's porism for n = 3, a triangle that is inscribed in one circle and circumscribes another.. In geometry, Poncelet's closure theorem, also known as Poncelet's porism, states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same ...
Van Hamme and Wasserman have extended the original Rescorla–Wagner (RW) model and introduced a new factor in their revised RW model in 1994: [3] They suggested that not only conditioned stimuli physically present on a given trial can undergo changes in their associative strength, the associative value of a CS can also be altered by a within-compound-association with a CS present on that trial.