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The mean square displacement (MSD) of a set of N displacements x_n is given by <|x|^2>=sum_(k=1)^N|x_k|^2. It arises particularly in Brownian motion and random walk problems. ...

The mean tetrahedron volume V^_ is the average volume of a tetrahedron in tetrahedron picking within some given shape. As summarized in the following table, it is possible to ...

The dual of the rhombidodecadodecahedron U_(38) and Wenninger dual W_(76).

The 30-faced dual of the truncated dodecadodecahedron and Wenninger dual W_(98).

The dual of the snub icosidodecadodecahedron U_(44) and Wenninger dual W_(112).

The dual of the icosidodecadodecahedron and Wenninger dual W_(83).

The dual of the snub dodecadodecahedron U_(40) and Wenninger dual W_(111).

P_n(cosalpha)=(sqrt(2))/piint_0^alpha(cos[(n+1/2)phi])/(sqrt(cosphi-cosalpha))dphi, where P_n(x) is a Legendre polynomial.

The hypergeometric orthogonal polynomials defined by P_n^((lambda))(x;phi)=((2lambda)_n)/(n!)e^(inphi)_2F_1(-n,lambda+ix;2lambda;1-e^(-2iphi)), (1) where (x)_n is the ...

A_m(lambda)=int_(-infty)^inftycos[1/2mphi(t)-lambdat]dt, (1) where the function phi(t)=4tan^(-1)(e^t)-pi (2) describes the motion along the pendulum separatrix. Chirikov ...