Related papers: On a new general method of summation

We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…

Gaussian Quadrature is a well known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest. In this paper we apply these ideas to…

It is mathematical folklore that 1 + 2 + 3 + 4 + ... = --1/12. This result is usually achieved using elaborate analytical methods, such as zeta function regularization or Ramanujan summation. However, in its notebooks, Ramanujan has also…

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…

A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a…

We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to…

The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…

A new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption. The main tool is a result on fibers in the spectrum of algebra of essentially bounded functions established recently…

Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) a role which can be played by a number.…

A method for selecting solution constructors in narrowing is presented. The method is based on a sort discipline that describes regular sets of ground constructor terms as sorts. It is extended to cope with regular sets of ground…

We define a general notion of "summability" of a set $I\subseteq\mathbb{C^{N}}$ and show that some trivial condition necessary for a set to be summable, is also sufficient. We deduce some intresting corollaries.

We present Korovkin approximation theorems that incorporate summability methods. These result allows us to obtain a unified treatment of several previous results, focusing on the underlying structure and the properties that a summability…

Regular resolution is a refinement of the resolution proof system requiring that no variable be resolved on more than once along any path in the proof. It is known that there exist sequences of formulas that require exponential-size proofs…

We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences which satisfy a p-summability condition and for integration of functions from Lebesgue…

A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…

This paper introduces a reformulation of the classical convergence theorem for spectral sequences of filtered complexes which provides an algorithm to effectively compute the induced filtration on the total (co)homology, as soon as the…

While advances in computing resources have made processing enormous amounts of data possible, human ability to identify patterns in such data has not scaled accordingly. Efficient computational methods for condensing and simplifying data…

In this paper we study the aggregation problem that can be formulated as follows. Assume that we have a family of estimators $\mathcal{F}$ built on the basis of available observations. The goal is to construct a new estimator whose risk is…

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…

The continuous and rapid growth of highly interconnected datasets, which are both voluminous and complex, calls for the development of adequate processing and analytical techniques. One method for condensing and simplifying such datasets is…