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We present an exposition of the basic properties of the Jacobi symbol, with a method of calculating it due to Eisenstein. Fix a prime p. For an integer a relatively prime to p the Legendre symbol is defined by (a/p) = 1 if a is a quadratic residue (mod p) and (a/p) =−1 if a is a quadratic nonresidue (mod p). We recall Euler’s theorem that (a/p) ≡ a(p−1)/2 (mod p). We have the famous Law of Quad...

Duadic codes were introduced by Leon et al. [5] as cyclic codes generalizing quadratic residue codes. Brualdi and Pless generalized them to polyadic cyclic codes [2] and Rushanan did so to duadic Abelian group codes [9]. Theorems concerning the existence of these codes in terms of field and group restrictions have been proved during this development, beginning with the one of Smid for cyclic du...

Wedescribe a polynomial-size conic quadratic reformulation for amachine-job assignment problemwith separable convex cost. Because the conic strengthening is based only on the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation. © 2009 Elsevier B.V. All rights reserved.

= 1 or −1 according as j is or is not a quadratic residue mod p. A multivariable generalization of Theorem 1.1 follows. Theorem 1.1 is a special case of Theorem 1.2 with x3 = · · · = xp = 0. Theorem 1.2. Let p be an odd prime and p = (−1)(p−1)/2p. Then there exist integer polynomials R(x1, x2, . . . , xp) and S(x1, x2, . . . , xp) such that 4 · det(circ(x1, x2, . . . , xp)) x1 + x2 + · · ·+ xp ...