Related papers: NMR implementations of Gauss sums
An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown…
Power transforms are popular parametric methods for making data more Gaussian-like, and are widely used as preprocessing steps in statistical analysis and machine learning. However, we find that direct implementations of power transforms…
We present some properties of measures (q-Gaussian) that orthogonalize the set of q-Hermite polynomials. We also present an algorithm for simulating i.i.d. sequences of random variables having q-Gaussian distribution.
An algorithm for integration of polynomial functions with variable weight is considered. It provides extension of the Gaussian integration, with appropriate scaling of the abscissas and weights. Method is a good alternative to usually…
Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.
This paper describes recent progress using nuclear magnetic resonance (NMR) as a platform for implementing quantum information processing (QIP) tasks. The basic ideas of NMR QIP are detailed, examining the successes and limitations of…
This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+\epsilon})$. Strassen's well known factoring approach is…
Certain new inequalities for the sums of factorials are presented.
Gaussian Process (GP) models are often used as mathematical approximations of computationally expensive experiments. Provided that its kernel is suitably chosen and that enough data is available to obtain a reasonable fit of the simulator,…
One-way quantum computing is an important and novel approach to quantum computation. By exploiting the existing particle-particle interactions, we report the first experimental realization of the complete process of deterministic one-way…
We briefly review numerical methods for calculations beyond one loop and then describe new developments within the method of sector decomposition in more detail. We also discuss applications to two-loop integrals involving several mass…
Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show…
Nonnegative matrix factorization (NMF) has an established reputation as a useful data analysis technique in numerous applications. However, its usage in practical situations is undergoing challenges in recent years. The fundamental factor…
Gaussian filters have applications in a variety of areas in computer science, from computer vision to speech recognition. The collapsing sum is a matrix operator that was recently introduced to study Gaussian filters combinatorially. In…
This paper presents a new distance metric to compare two continuous probability density functions. The main advantage of this metric is that, unlike other statistical measurements, it can provide an analytic, closed-form expression for a…
This paper gives new explicit formulas for sums of powers of integers and their reciprocals.
The numerical integration of an analytical function $f(x)$ using a finite set of equidistant points can be performed by quadrature formulas like the Newton-Cotes. Unlike Gaussian quadrature formulas however, higher-order Newton-Cotes…
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be…
A survey paper on some recent results on additive problems with prime powers.
A factor-graph representation of quantum-mechanical probabilities is proposed. Unlike standard statistical models, the proposed representation uses auxiliary variables (state variables) that are not random variables.