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The Klein bottle crossing number of a graph G is the minimum number of crossings possible when embedding G on a Klein bottle (cf. Garnder 1986, pp. 137-138). While the ...

A limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. For example, the point is a degenerate case of the circle as ...

The j-function is the modular function defined by j(tau)=1728J(tau), (1) where tau is the half-period ratio, I[tau]>0, ...

The Pippenger product is an unexpected Wallis-like formula for e given by e/2=(2/1)^(1/2)(2/34/3)^(1/4)(4/56/56/78/7)^(1/8)... (1) (OEIS A084148 and A084149; Pippenger 1980). ...

A polyhedral graph is completely regular if the dual graph is also regular. There are only five types. Let rho be the number of graph edges at each node, rho^* the number of ...

An n-persistent number is a positive integer k which contains the digits 0, 1, ..., 9 (i.e., is a pandigital number), and for which 2k, ..., nk also share this property. No ...

Let two spheres of radii R and r be located along the x-axis centered at (0,0,0) and (d,0,0), respectively. Not surprisingly, the analysis is very similar to the case of the ...

There are two curves known as the butterfly curve. The first is the sextic plane curve given by the implicit equation y^6=x^2-x^6 (1) (Cundy and Rollett 1989, p. 72; left ...

Let f(x,y)=u(x,y)+iv(x,y), (1) where z=x+iy, (2) so dz=dx+idy. (3) The total derivative of f with respect to z is then (df)/(dz) = ...

Cauchy's integral formula states that f(z_0)=1/(2pii)∮_gamma(f(z)dz)/(z-z_0), (1) where the integral is a contour integral along the contour gamma enclosing the point z_0. It ...