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The square knot, also called the reef knot, is a composite knot of six crossings consisting of a knot sum of a trefoil knot and its mirror image (Rolfsen 1976, p. 220). The ...

A determinant used to determine in which coordinate systems the Helmholtz differential equation is separable (Morse and Feshbach 1953). A determinant S=|Phi_(mn)|=|Phi_(11) ...

In nonstandard analysis, the limitation to first-order analysis can be avoided by using a construction known as a superstructure. Superstructures are constructed in the ...

For a scalar function f over a surface parameterized by u and v, the surface integral is given by Phi = int_Sfda (1) = int_Sf(u,v)|T_uxT_v|dudv, (2) where T_u and T_v are ...

Prellberg (2001) noted that the limit c=lim_(n->infty)(T_n)/(B_nexp{1/2[W(n)]^2})=2.2394331040... (OEIS A143307) exists, where T_n is a Takeuchi number, B_n is a Bell number, ...

Let T(x,y,z) be the number of times "otherwise" is called in the TAK function, then the Takeuchi numbers are defined by T_n(n,0,n+1). A recursive formula for T_n is given by ...

The contraction of a tensor is obtained by setting unlike indices equal and summing according to the Einstein summation convention. Contraction reduces the tensor rank by 2. ...

A tesseral harmonic is a spherical harmonic of the form cos; sin(mphi)P_l^m(costheta). These harmonics are so named because the curves on which they vanish are l-m parallels ...

Toroidal functions are a class of functions also called ring functions that appear in systems having toroidal symmetry. Toroidal functions can be expressed in terms of the ...

A special function psi_1(z) corresponding to a polygamma function psi_n(z) with n=1, given by psi_1(z)=(d^2)/(dz^2)lnGamma(z). (1) An alternative function ...