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A finite or infinite square matrix with rational entries. (If the matrix is infinite, all but a finite number of entries in each row must be 0.) The sum or product of two ...

The parametric equations for a catenary are x = t (1) y = cosht, (2) giving the involute as x_i = t-tanht (3) y_i = secht. (4) The involute is therefore half of a tractrix.

A circle bundle pi:E->M is a fiber bundle whose fibers pi^(-1)(x) are circles. It may also have the structure of a principal bundle if there is an action of SO(2) that ...

f(x)=1/x-|_1/x_| for x in [0,1], where |_x_| is the floor function. The natural invariant of the map is rho(y)=1/((1+y)ln2).

The curve y=1-e^(ax), illustrated above.

When a point P moves along a line through the circumcenter of a given triangle Delta, the pedal circle of P with respect to Delta passes through a fixed point (the Griffiths ...

alpha(x) = 1/(sqrt(2pi))int_(-x)^xe^(-t^2/2)dt (1) = sqrt(2/pi)int_0^xe^(-t^2/2)dt (2) = 2Phi(x) (3) = erf(x/(sqrt(2))), (4) where Phi(x) is the normal distribution function ...

An epicycloid with n=5 cusps, named after the buttercup genus Ranunculus (Madachy 1979). Its parametric equations are x = a[6cost-cos(6t)] (1) y = a[6sint-sin(6t)]. (2) Its ...

In the illusion illustrated above, when the concentric square borders with rounded edges are rotated slowly, the entire pattern appears to pulsate radially.

For a semicubical parabola with parametric equations x = t^2 (1) y = at^3, (2) the involute is given by x_i = (t^2)/3-8/(27a^2) (3) y_i = -(4t)/(9a), (4) which is half a ...