We show that for any fixed ε > 0, there are numbers δ > 0 and p 0 2 with the following property: for every prime p p 0 , there is an integer p δ < N p 1/(4 √ e)+ε such that the sequence 1, 2,. .. , N contains at least δN quadratic non-residues modulo p. We then apply this result to obtain a new estimate on the smallest quadratic nonresidue in a Beatty sequence.

Plots of quadratic residues display some visual features that are analyzed mathematically in this paper. The graphical patterns of quadratic residues modulo m depend only on the residues of m modulo the lowest positive integers.

The projective special linear group PSL2(n) is 2-transitive for all primes n and 3-homogeneous for n≡ 3 (mod 4) on the set {0,1, · · · ,n−1,∞}. It is known that the extended odd-like quadratic residue codes are invariant under PSL2(n). Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes n≡ 1 (mod 4), an infinite family of 3-designs for primes n≡ 3 (mod 4)...

The super-Poincaré covariant formalism for the superstring is used to compute mass-less four-point two-loop amplitudes in ten-dimensional superspace. The computations are much simpler than in the RNS formalism and include both external bosons and fermions.

(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...

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...