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Given two intersecting lines, the two nonadjacent angles with the same vertex are said to be vertical angles. One can easily prove that vertical angles are congruent. Some ...

The system of partial differential equations (partial_t+partial_z^3+partial_(z^_)^3)v+partial_z(uv)+partial_(z^_)(vw) (1) partial_(z^_)u=3partial_zv (2) ...

Solution of a system of second-order homogeneous ordinary differential equations with constant coefficients of the form (d^2x)/(dt^2)+bx=0, where b is a positive definite ...

An auction in which the highest bidder wins but pays only the second-highest bid. This variation over the normal bidding procedure is supposed to encourage bidders to bid the ...

The substitution of x=w-p/(3w) (1) into the standard form cubic equation x^3+px=q. (2) The result reduces the cubic to the equation w^3-(p^3)/(27w^3)-q=0, (3) which is easily ...

An abstract machine that is implemented in software is termed a virtual machine.

The Banach space L^1([0,1]) with the product (fg)(x)=int_0^xf(x-y)g(y)dy is a non-unital commutative Banach algebra. This algebra is called the Volterra algebra.

A triple integral over three coordinates giving the volume within some region G, V=intintint_(G)dxdydz.

If the top and bottom bases of a solid are equal in area, lie in parallel planes, and every section of the solid parallel to the bases is equal in area to that of the base, ...

The W-transform of a function f(x) is defined by the integral where Gamma[(beta_m)+s, 1-(alpha_n)-s; (alpha_p^(n+1))+s, 1-(beta_q^(m+1))-s] =Gamma[beta_1+s, ..., beta_m+s, ...