The additive structure of multiplicative semigroup mod (odd prime p) is analysed. Order ( 1)1 of cyclic group of units mod implies product , with cyclic 'core' , and 'extension subgroup' which consists of all units mod , generated by +1. The -th power residues mod in form an order subgroup ,...

A new version of the Bäcklund-Darboux transformation for the matrix Kadomtsev-Petviashvili equation is used to construct and study explicit multi-parameter solutions and wave functions (in terms of the matrix exponents). A class of the self-adjoint non-singular solutions of KP I is introduced...

This note will show that there is a deterministic polynomial time algorithm for computing quadratic nonresidues of absolute values for all primes , and > 0 a fixed constant. Further, the same method is used to determine the least quadratic nonresidues ≤ (log()) of any prime ≠ 8 − 1

In this paper we study the existence of nonzero, axially symmetric solution for a class of gradient systems on strip-like domains where the nonlinearity term is not necessarily continuously differentiable. The proof is based on a non-smooth version of the Mountain Pass Theorem with Cerami...

A directed graph with vertices and e arcs, no self-loops, and no more than one arc directed from any vertex to any vertex , is numbered by assigning to each vertex a distinct element from the set = {1, 2,..., +1}. An arc (, ) from vertex to vertex is numbered (, )= () − () mod ( + 1), where ()...

We classify the path-components of the space of circle-valued Morse functions on compact surfaces: two Morse functions f,g : → belong to same path-component of this space if and only if they are homotopic and have equal numbers of critical points at each index