In the present paper we obtain new upper bound estimates for the number of solutions of the congruence x ≡ yr (mod p); x, y ∈ N, x, y ≤ H, r ∈ U , for certain ranges of H and |U|, where U is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence x ≡ λ (mod p); x ∈ N, L < x < L+ p/n,...

For Lagrangian subbundles with singularities in symplectic vector bundles, explicit formulas of relation between their residue classes and Maslov classes outside singularities are obtained. Then a Lagrangian subbundle with singularity is constructed where all possible Maslov classes are nonzero but residue classes vanish for dimension > 2 . Moreover, a Lagrangian immersion with singularity is c...

This paper addresses the problem of improving the unknown syndrome representations to develop algebraic decoding of the (17,9,5) and (23,12,7) binary quadratic residue codes up to true minimum distance, respectively. The proposed unknown syndrome representations are expressed as binary polynomials in terms of the single known syndrome, which is different from the known syndrome in [Chang-Lee, A...

We study the quadratic residue problem known as an NP complete problem by way of the prime number and show that a nondeterministic polynomial process does not belong to the class P because of a random distribution of solutions for the quadratic residue problem.

We compute explicit transgression forms for the Euler and Pontrjagin classes of a Riemannian manifold M of dimension 4 under a conformal change of the metric, or a change to a Riemannian connection with torsion. These formulae describe the singular set of some connections with singularities on compact manifolds as a residue formula in terms of a polynomial of invariants. We give some applicatio...