The Shor algorithm [1] to factor numbers and its NMRimplementation [2] have propelled the field of quantum computation [3]. Recently a different factorization scheme [4] taking advantage of the periodicity properties of Gauss sums [5] has been proposed [6] and verified by two NMR-experiments [7, 8] and one experiment based on short laser pulses [9]. In the present paper we report the first impl...

We factor the number 157573 using an NMR implementation of Gauss sums. Although the current implementation is classical and scales exponentially, we believe that an extension to the quantum regime using entangled states is possible.

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, the Gauss circle problem Dirichlet divisor problem, in an earlier paper, three of present authors derived representations for certain sums products trigonometric functions as double series Bessel [8] . These are generalized paper introducing novel notion balanced derivatives, leading to further the...

Evaluating sums of multivariate Gaussian kernels is a key computational task in many problems in computational statistics and machine learning. The computational cost of the direct evaluation of such sums scales as the product of the number of kernel functions and the evaluation points. The fast Gauss transform proposed by Greengard and Strain (1991) is a ǫ-exact approximation algorithm that re...

Evaluating sums of multivariate Gaussian kernels is a key computational task in many problems in computational statistics and machine learning. The computational cost of the direct evaluation of such sums scales as the product of the number of kernel functions and the evaluation points. The fast Gauss transform proposed by Greengard and Strain (1991) is a 2-exact approximation algorithm that re...

Let p be a prime, Zp the ring of p-adic integers, Qp the field of p-adic numbers, and Qp the algebraic closure of Qp. Let Fp be the algebraic closure of the finite field Fp, and ω : Fp → Qp the Techmüller lifting. Let q = pa, χ a multiplicative character of Fq into Q, and ψ a character of Zp[μq−1] into Q of exact order pn > 1, where μr is the set of r-th roots of unity in Qp. For simplicity, we...

Using the Fiedler-Polyak-Viro Gauß diagram formulas we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to their untwisted Whitehead doubles, previous results on the polynomials and sign...

where p is prime, e(x) = exp(2πix), and ep(x) = e(x/p). In each case we shall assume that p | / a unless the contrary is explicitly stated. Gauss sums arise in investigations into Waring’s problem, and other additive problems involving k-th powers. Although they are amongst the simplest complete exponential sums, the question as to their true order of magnitude is far from being resolved. We re...