Related papers: Hyperreal Numbers for Infinite Divergent Series

Counters that hold natural numbers are ubiquitous in modeling and verifying software systems; for example, they model dynamic creation and use of resources in concurrent programs. Unfortunately, such discrete counters often lead to…

Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been…

The paper is devoted to the development of the theory of inverse problems for evolution equations with terms rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints are…

Under certain circumstances, some of which are made explicit here, one can deduce bounds on the full sum of a perturbation series of a physical quantity by using a variational Borel map on the partial series. The method is illustrated by…

Inverse problems are concerned with the reconstruction of unknown physical quantities using indirect measurements and are fundamental across diverse fields such as medical imaging, remote sensing, and material sciences. These problems serve…

While deep learning methods have achieved state-of-the-art performance in many challenging inverse problems like image inpainting and super-resolution, they invariably involve problem-specific training of the networks. Under this approach,…

Generalized L\"uroth series generalize $b$-adic representations as well as L\"uroth series. Almost all real numbers are normal, but it is not easy to construct one. In this paper, a new construction of normal numbers with respect to…

We give a survey of work on the number of vertices of the convex hull of integer points defined by the system of linear inequalities. Also, we present our improvement of some of these.

The area of inverse problems in mathematics is highly interdisciplinary. In various fields of science, engineering, medicine, and industry, there arises a need to reconstruct information about unknown entities that cannot be directly…

The divergence of the harmonic series is proved by direct comparison with a series whose nth partial sum telescopes to the natural logarithm of n. The key idea is to apply the classical inequality x>=log(1+x) (valid for x>-1) with x=1/k and…

Modelling real world systems frequently requires the solution of systems of nonlinear equations. A number of approaches have been suggested and developed for this computational problem. However, it is also possible to attempt solutions…

This paper introduces the expanded real numbers as an ordered subring of the hyperreal number field that does not contain any infinitesimals, and defines the set of all integrable functions from the real numbers to the expanded real…

The set of answers to a query may be very large, potentially overwhelming users when presented with the entire set. In such cases, presenting only a small subset of the answers to the user may be preferable. A natural requirement for this…

Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying…

Several recent works have identified patterns that must exist in dense subsets of either the vertices or the edges of a large hypercube. We introduce a framework, based on the concept of series-parallel graphs, that unifies and generalizes…

We deal with the solution of a generic linear inverse problem in the Hilbert space setting. The exact right hand side is unknown and only accessible through discretised measurements corrupted by white noise with unknown arbitrary…

There are numerous methods for detecting anomalies in time series, but that is only the first step to understanding them. We strive to exceed this by explaining those anomalies. Thus we develop a novel attribution scheme for multivariate…

Twin prime number problem is mainly the structure of the twin prime numbers and whether there are infinitely many prime twins group. In this paper, by constructing a special cluster number set(see formula(2.3)in the paper), proves that the…

In recent years, there have been significant advances in the use of deep learning methods in inverse problems such as denoising, compressive sensing, inpainting, and super-resolution. While this line of works has predominantly been driven…

Inverse problems are ubiquitous in nature, arising in almost all areas of science and engineering ranging from geophysics and climate science to astrophysics and biomechanics. One of the central challenges in solving inverse problems is…