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Consider a circle in with center at the origin and radius . Gauss's circle problem asks how many points there are inside this circle of the form ( m , n ) {\displaystyle (m,n)} where m {\displaystyle m} and n {\displaystyle n} are both integers.
The reduction needs to solve twice the similar problem where the center of the sought-after enclosing circle is constrained to lie on a given line. The solution of the subproblem is either the solution of the unconstrained problem or it is used to determine the half-plane where the unconstrained solution center is located.
It is a (non-linear) curve which a circle of radius a rolling on a straight line, with its center at the x axis, intersects perpendicularly at all times. The function admits a horizontal asymptote. The curve is symmetrical with respect to the y-axis. The curvature radius is r = a cot x / y .
A circle containing one acre is cut by another whose center is on the circumference of the given circle, and the area common to both is one-half acre. Find the radius of the cutting circle. The solutions in both cases are non-trivial but yield to straightforward application of trigonometry, analytical geometry or integral calculus.
In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field.A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center.
Consider a solution circle of radius r s and three given circles of radii r 1, r 2 and r 3. If the solution circle is externally tangent to all three given circles, the distances between the center of the solution circle and the centers of the given circles equal d 1 = r 1 + r s, d 2 = r 2 + r s and d 3 = r 3 + r s, respectively.
In particular, for some placements of the two points the optimal solution is generated by a discontinuous function that is nonzero at the two points and zero everywhere else. This function leads to a surface of revolution consisting of two circular disks, one for each point, connected by a degenerate line segment along the axis of revolution.
[4] [5] The form of the equation given here was derived by J. Robert Oppenheimer and George Volkoff in their 1939 paper, "On Massive Neutron Cores". [1] In this paper, the equation of state for a degenerate Fermi gas of neutrons was used to calculate an upper limit of ~0.7 solar masses for the gravitational mass of a neutron star. Since this ...