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If the total group of the canonical series is divided into two parts, the difference between the number of points in each part and the double of the dimension of the complete ...

Let M be a finitely generated module over a commutative Noetherian ring R. Then there exists a finite set {N_i|1<=i<=l} of submodules of M such that 1. intersection ...

If two curves phi and psi of multiplicities r_i!=0 and s_i!=0 have only ordinary points or ordinary singular points and cusps in common, then every curve which has at least ...

Any irreducible curve may be carried by a factorable Cremona transformation into one with none but ordinary singular points.

An extremely powerful theorem in physics which states that each symmetry of a system leads to a physically conserved quantity. Symmetry under translation corresponds to ...

If two projective pencils of curves of orders n and n^' have no common curve, the locus of the intersections of corresponding curves of the two is a curve of order n+n^' ...

An entire Cremona transformation is a birational transformation of the plane. Cremona transformations are maps of the form x_(i+1) = f(x_i,y_i) (1) y_(i+1) = g(x_i,y_i), (2) ...

An intrinsic property of a mathematical object which causes it to remain invariant under certain classes of transformations (such as rotation, reflection, inversion, or more ...

A quaternion with complex coefficients. The algebra of biquaternions is isomorphic to a full matrix ring over the complex number field (van der Waerden 1985).

A Dedekind ring is a commutative ring in which the following hold. 1. It is a Noetherian ring and a integral domain. 2. It is the set of algebraic integers in its field of ...