Related papers: Gauss-Type Quadrature Rules Based on Identity-Type…
We consider $q$-analytic derivations of the $q$-Gauss summation formula for a $\, _2\phi _1$ that respect the symmetry in its upper parameters.
Motivated by classical results of Aubry, Davenport and Cassels, we define the notion of a Euclidean quadratic form over a normed integral domain and an ADC form over an integral domain. The aforementioned classical results generalize to:…
We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < \epsilon$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the…
We characterize the generating function of the number of representations described in the title in terms of the theory of modular forms. Appealing to this characterization we obtain explicit formulas for the representation numbers as…
It is shown that quadrature formulas in many different applications can be derived from rational approximation of the Cauchy transform of a weight function. Since rational approximation is now a routine technology, this provides an easy new…
In this paper we study the set of values of quadratic form at points of a cut and project set. We will establish conditions which ensure that the set of values is dense. Our methods involve homogeneous dynamics and we will prove a orbit…
A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds…
Gaussian states -- or, more generally, Gaussian operators -- play an important role in Quantum Optics and Quantum Information Science, both in discussions about conceptual issues and in practical applications. We describe, in a tutorial…
A straightforward 3-point quadrature formula of closed type is derived that improves on Simpson's rule. Just using the additional information of the integrand's derivative at the two endpoints we show the error is sixth order in grid…
We study cliques in graphs arising from quadratic forms where the vertices are the elements of the module of the quadratic form and two vertices are adjacent if their difference represents some fixed scalar. We determine structural…
We offer some partition functions related to ternary quadratic forms, and note on their upper bounds and related properties. We offer these results as an application of a simple method related to conjugate Bailey pairs presented in a prior…
Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over C, this important data coincides with the topological class. In this paper, we characterise a family of singularities,…
A sum rule is an identity connecting the entropy of a measure with coefficients involved in the construction of its orthogonal polynomials (Jacobi coefficients). Our paper is an extension of Gamboa, Nagel and Rouault (2016), where we have…
A Gauss equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic curvature less than or equal to A, defined by a cubic inequality on the difference of arc and chord, has…
We investigate in the context of the scattering equations, how one-loop linear propagator integrands in gauge theories can be linked to integrands with quadratic propagators using a double forward limit. We illustrate our procedure through…
In this short note, we collect some facts on the weighted Gau{\ss}--Radau quadrature. In particular, we focus on the location of the Gau\ss--Radau points being a continuous function of the $L^1$-weighting function.
The convolution of a function with an isotropic Gaussian appears in many contexts such as differential equations, computer vision, signal processing, and numerical optimization. Although this convolution does not always have a closed form…
We show that various identities from [1] and [3] involving Gould-Hopper polynomials can be deduced from the real but also complex orthogonal invariance of multivariate Gaussian distributions. We also deduce from this principle a useful…
We employ a multivariate extension of the Gauss quadrature formula, originally due to Berens, Schmid and Xu [BSX95], so as to derive cubature rules for the integration of symmetric functions over hypercubes (or infinite limiting…
Integrable difference equations commonly have more low-order conservation laws than occur for nonintegrable difference equations of similar complexity. We use this empirical observation to sift a large class of difference equations, in…