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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 ...

Decomposition influences carbon and nutrient cycling from crop residues. The nylon-mesh-bag technique was implied to study the decomposition N-release dynamics different residues under field conditions. four types of were: maize (lower than 50% below cob), wheat 25% stubbles), a whole mung bean residue, mixture + residue (1:1 ratio) put on soil surface in sub-surface. N release both at-surface-...

Physical, chemical and biological attributes of soil surface are significantly affected by tillage practices and crop rotation. The objective of this study was to determine the short-term (2006-2009) effects of conventional and conservational tillage practices on selected soil physical properties of a heavy clay soil under two wheat-corn and wheatlegume rotation in a semi-arid Mediterranean...

A general type of linear cyclic codes is introduced as a straightforward generalization of quadratic residue codes, e-residue codes, generalized quadratic residue codes and polyadic codes. A generalized version of the well-known squareroot bound for odd-weight words is derived.

Recently, a new algebraic decoding method was proposed by Truong et al. In this paper, three decoders for the quadratic residue codes with parameters (71, 36, 11), (79, 40, 15), and (97, 49, 15), which have not been decoded before, are developed by using the decoding scheme given by Truong et al. To confirm our results, an exhaustive computer simulation was executed successfully.