We present an oracle factorisation algorithm, which in polynomial deterministic time, finds a nontrivial factor of almost all positive integers n based on the knowledge number points certain elliptic curves residue rings modulo n.

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1, . . . , tk. A formula in this language defines a parametric set St ⊆ Z d as t varies in Z, and we examine the counting function |St| as a function of t. For a single parameter, it is known that |St| can be expressed as an eventual quasi-polynomial (there is a period m such that, for sufficiently la...

A general algorithm for decoding the binary systematic quadratic residue (QR) codes with lookup tables is presented in this study. The algorithm can be applied in decoding the QR codes with either reducible or irreducible generator polynomials. If the generator polynomial of the QR codes is reducible, the number of elements in the Galois field is less than the sum of all correctable error patte...

We obtain a characterization of local Besov spaces of functions on [−1, 1] in terms of algebraic polynomial operators. These operators are constructed using the coefficients in the orthogonal polynomial expansions of the functions involved. The example of Jacobi polynomials is studied in further detail. A by-product of our proofs is an apparently simple proof of the fact that the Cesàro means o...

The “Fibonacci Dichotomy” of Kaiser and Klazar [15] was one of the first general results on the enumeration of permutation classes. It states that if there are fewer permutations of length n in a permutation class than the nth Fibonacci number, for any n, then the enumeration of the class is given by a polynomial for sufficiently large n. Since the Fibonacci Dichotomy was established for permut...