The polynomial residue number system (PRNS) has been considered as a useful tool for digital signal processing (DSP) since it can support parallel, carry-free, high speed arithmetic with minimum multiplication count provided that an appropriate modular ring is chosen. In this paper, the properties of two-dimensional (2-D) PRNS are investigated in detail. It is shown that in the 2-D PRNS system,...

Let q > 1 be an integer and let a and b be elements of the residue ring ZZq of integers modulo q. We show how, when given a polynomial f ∈ ZZq[X] and approximations to v0, v1 ∈ ZZq such that v1 ≡ f(v0) mod q one can recover v0 and v1 efficiently. This result has direct applications to predicting the polynomial congruential generator: a sequence (vn) of pseudorandom numbers defined by the relati...

a function is said to be bi-univalent on the open unit disk d if both the function and its inverse are univalent in d. not much is known about the behavior of the classes of bi-univalent functions let alone about their coefficients. in this paper we use the faber polynomial expansions to find coefficient estimates for four well-known classes of bi-univalent functions which are defined by subord...

We study n × n symmetric random matrices H, possibly discrete, with iid abovediagonal entries. We show that H is singular with probability at most exp(−nc), and ‖H−1‖ = O(√n). Furthermore, the spectrum of H is delocalized on the optimal scale o(n−1/2). These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of T...

For a triple (n, t,m) of positive integers, we attach to each t-subset S = {a1, . . . , at} ⊆ {1, . . . , n} the sum f(S) = a1 + · · · + at (modulo m). We ask: for which triples (n, t,m) are the ( n t ) values of f(S) uniformly distributed in the residue classes mod m? The obvious necessary condition, that m divides ( n t ) , is not sufficient, but a q-analogue of that condition is both necessa...

In Descriptive Complexity, there is a vast amount of literature on decision problems, and their classes such as P, NP, L and NL. However, research on the descriptive complexity of optimisation problems has been limited. Optimisation problems corresponding to the NP class have been characterised in terms of logic expressions by Papadimitriou and Yannakakis, Panconesi and Ranjan, Kolaitis and Tha...

Based on unique decoding of the polynomial residue code with non-pairwise coprime moduli, a polynomial with degree less than that of the least common multiple (lcm) of all the moduli can be accurately reconstructed when the number of residue errors is less than half the minimum distance of the code. However, once the number of residue errors is beyond half the minimum distance of the code, the ...