Using a simple basis of rational polynomial-like functions P0, . . . , Pn−1 for the free module of functions Z/nZ → Z/mZ, we characterize the subfamily of congruence preserving functions as the set of linear combinations of the products lcm(k)Pk where lcm(k) is the least common multiple of 2, . . . , k (viewed in Z/mZ). As a consequence, when n ≥ m, the number of such functions is independent o...

Let (K, 3) be a (commutative) topological field. (We do not require that multiplicative inversion be continuous, i.e. 3 is a ring topology. See [l, p. 274] for the definition of the latter.) Throughout this paper, 11 will denote a basic system of neighborhoods of zero for 3. Let P(X) be a polynomial in K[X] of degree »sS2, and let S= {P(a)\a£EK}. We will be concerned with suitably defining a mu...

Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hubert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer o...

We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gröbner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves sums of squares and semidefinite programming. This opens up the possibility of using semidefinite pr...

Recently progress has been made in the development of algorithms for optimizing polynomials. The main idea being stressed is that of reducing the problem to an easier problem involving semidefinite programming [18]. It seems that in many cases the method dramatically outperforms other existing methods. The idea traces back to work of Shor [16][17] and is further developed by Parrilo [10] and by...

In wireless sensor networks, data aggregation protocols are used to prolong the network lifetime. However, the problem of how to perform data aggregation while preserving data privacy is challenging. This paper presents a polynomial regressionbased data aggregation protocol that preserves the privacy of sensor data. In the proposed protocol, sensor nodes represent their data as polynomial funct...

We prove an approximation result for Lipschitz functions on the quantum sphere $S_q^2$, from which we deduce that two natural metric structures $S_q^2$ have Gromov-Hausdorff distance zero.