Related papers: Revisiting "What Every Computer Scientist Should K…
Counting integer points in large convex bodies with smooth boundaries containing isolated flat points is oftentimes an intermediate case between balls (or convex bodies with smooth boundaries having everywhere positive curvature) and cubes…
Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing. Julia is designed to be easy and fast. Julia…
Soft sets, as a mathematical tool for dealing with uncertainty, have recently gained considerable attention, including some successful applications in information processing, decision, demand analysis, and forecasting. To construct new soft…
Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new…
The design of embedded control systems is mainly done with model-based tools such as Matlab/Simulink. Numerical simulation is the central technique of development and verification of such tools. Floating-point arithmetic, that is well-known…
Floating-point arithmetic performance determines the overall performance of important applications, from graphics to AI. Meeting the IEEE-754 specification for floating-point requires that final results of addition, subtraction,…
This paper provides a survey of spherical designs and their applications, with a particular emphasis on the perspective of ``numerical analysis''. A set \(X_N\) of \(N\) points on the unit sphere \(\mathbb{S}^d\) is called a…
Mathematical Selection is a method in which we select a particular choice from a set of such. It have always been an interesting field of study for mathematicians. Accordingly, Combinatorial Optimization is a sub field of this domain of…
The exact computation of orbits of discrete dynamical systems on the interval is considered. Therefore, a multiple-precision floating point approach based on error analysis is chosen and a general algorithm is presented. The correctness of…
We use simple equations in order to compare the basins of attraction on the complex plane, corresponding to a large collection of numerical methods, of several order. Two cases are considered, regarding the total number of the roots, which…
Quantifying errors and losses due to the use of Floating-Point (FP) calculations in industrial scientific computing codes is an important part of the Verification, Validation and Uncertainty Quantification (VVUQ) process. Stochastic…
Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels…
The main goal of numerical relativity is the long time simulation of highly nonlinear spacetimes that cannot be treated by perturbation theory. This involves analytic, computational and physical issues. At present, the major impasses to…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
Over the last few years, neural networks have started penetrating safety critical systems to take decisions in robots, rockets, autonomous driving car, etc. A problem is that these critical systems often have limited computing resources.…
In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical…
Symbolic equations are one of the many representations used in physics. Understanding these representations is important for students because they are how students access knowledge in physics. In this paper I build off of the work by Redish…
This paper proposes the problem of point-and-count as a test case to break the what-and-where deadlock. Different from the traditional detection problem, the goal is to discover key salient points as a way to localize and count the number…
We consider the problem of accurate computation of the finite difference $f(\x+\s)-f(\x)$ when $\Vert\s\Vert$ is very small. Direct evaluation of this difference in floating point arithmetic succumbs to cancellation error and yields 0 when…
This paper presents an analysis on the effects of floating-point arithmetic on the constructing bifurcation diagram of the quadratic map. More precisely, we are interested in showing the dependence of initial conditions to obtain some…