(here b and a run over the quadratic nonresidues and quadratic residues, respectively, that lie between 0 and p) is a unit of the real quadratic field R(y/p), and that 77> 1. The fact that -n > 1 is usually deduced from the theory of the classnumber of quadratic fields. We present a short proof independent of the theory of the class-number. As in the paper of Chowla and Mordell [Note on the non...

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Residues of antibiotic drugs in food and products have been received much attention recent years because growing safety concerns for public health. There are serious effects residues meat human consumption (e.g., increasing antimicrobial resistance, carcinogenicity, mutagenicity, hypersensitivity). In this study, we used broiler chicks as a laboratory animal to detect the colistin sulfate resid...

Binary optimal codes often contain optimal or near-optimal subcodes. In this paper we show that this is true for the family of self-dual codes. One approach is to compute the optimum distance profiles (ODPs) of linear codes, which was introduced by Luo, et. al. (2010). One of our main results is the development of general algorithms, called the Chain Algorithms, for finding ODPs of linear codes...

Binary optimal codes often contain optimal or near-optimal subcodes. In this paper we show that this is true for the family of self-dual codes. One approach is to compute the optimum distance profiles (ODPs) of linear codes, which was introduced by Luo, et. al. (2010). One of our main results is the development of general algorithms, called the Chain Algorithms, for finding ODPs of linear codes...

In concert hall acoustics, the reflection characteristics of the ceiling and the walls are important for minimizing the interaural cross correlation. Many design methods have been presented so far in order to design highly diffusing surfaces. This paper presents a two-dimensional digital waveguide mesh having a highly diffusing boundary using quadratic residue sequences, and illustrates its ref...

It is argued that gravitational descendants in the theory of topological gravity coupled to topological Landau-Ginzburg theory can be constructed from matter fields alone (without metric fields and ghosts).In this sense topological gravity is ”induced”. We discuss the mechanism of this effect(that turns out to be connected with K.Saito’s higher residue pairing: K(σi(Φ1),Φ2) = K(Φ1,Φ2) , and dem...

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual...

(where χ (mod q) is a Dirichlet character) arise naturally in many classical problems of analytic number theory, from estimating the least quadratic nonresidue (mod p) to bounding L-functions. Recall that for any character χ (mod q) , |Sχ(x)| is trivially bounded above by φ(q). A folklore conjecture (which is a consequence of the Generalized Riemann Hypothesis) predicts that for nonprincipal ch...