04351 Summary -- Spatial Representation: Discrete vs. Continuous Computational Models

Topological notions and methods are used in various areas of the physical sciences and engineering, and therefore computer processing of topological data is important. Separate from this, but closely related, are computer science uses of topology: applications to programming language semantics and computing with exact real numbers are important examples. The seminar concentrated on an important approach, which is basic to all these applications, i.e. spatial representation

What bandwidth do I need for my image?

Computer representations of real numbers are necessarily discrete, with some finite resolution, discreteness, quantization, or minimum representable difference. We perform astrometric and photometric measurements on stars and co-add multiple observations of faint sources to demonstrate that essentially all of the scientific information in an optical astronomical image can be preserved or transmitted when the minimum representable difference is a factor of two finer than the root-variance of the per-pixel noise. Adopting a representation this coarse reduces bandwidth for data acquisition, transmission, or storage, or permits better use of the system dynamic range, without sacrificing any information for down-stream data analysis, including information on sources fainter than the minimum representable difference itself.Comment: submitted to PAS

Good rotations

Numerical integrations in celestial mechanics often involve the repeated computation of a rotation with a constant angle. A direct evaluation of these rotations yields a linear drift of the distance to the origin. This is due to roundoff in the representation of the sine s and cosine c of the angle theta. In a computer, one generally gets c^2 + s^2 1, resulting in a mapping that is slightly contracting or expanding. In the present paper we present a method to find pairs of representable real numbers s and c such that c^2 + s^2 is as close to 1 as possible. We show that this results in a drastic decrease of the systematic error, making it negligible compared to the random error of other operations. We also verify that this approach gives good results in a realistic celestial mechanics integration.Comment: 24 pages, 3 figure

Advancements in number representation for high-precision computing

Efficient representation of data is a fundamental prerequisite for addressing computational problems effectively using computers. The continual improvement in methods for representing numbers in computers serves as a critical step in expanding the scope and capabilities of computing systems. In this research, we conduct a comprehensive review of both fundamental and advanced techniques for representing numbers in computers. Additionally, we propose a novel model capable of representing rational numbers with absolute precision, catering to specific high precision applications. Specifically, we adopt fractional positional notation coupled with explicit codification of the periodic parts, thereby accommodating the entire rational number set without any loss of accuracy. We elucidate the properties and hardware representation of this proposed format and provide the results of extensive experiments to demonstrate its expressiveness and minimal codification error when compared to other real number representation formats. This research contributes to the advancement of numerical representation in computer systems, empowering them to handle complex computations with heightened accuracy, making them more reliable and versatile in a wide range of applications.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was supported by the Spanish Research Agency (AEI) (https://doi.org/10.13039/501100011033) under project HPC4Industry PID2020-120213RB-I00

Presence, in computerized environments

International audienceThe distinction of what is real and what is non-real is an usual and long lasting question of philosophy, as well as of physics. Since the 50's, with the demonstration of Shanon's theorem and its implementation in digital to analog converters, real sensorial data has begun to be producible ex nihilo, i.e. without any real objects, by abstract and symbolic entities such as numbers and algorithms. Indeed, a new problem of presence appears when human beings are (more and more frequently) called upon to perceive and act on spaces that are increasingly distant or different from our current physical world, by means of new instruments as tools for telecommunication, teleoperation, and computer representation, These new tools raise with growing urgency the question of the presence of these distant spaces

A Complex-Envelope FDTD Formulation Using Real-Valued Field-Variables

Use of the complex-envelope (CE) representation of band-pass limited sources and their resulting fields increases the allowable time-step in finite-difference time-domain (FDTD) simulations. The complex envelope representation transforms band-pass limited fields and sources to complex-valued low-pass limited form and Maxwells equations from real-valued partial differential equations (PDEs) to complex-valued PDEs. Previous CE FDTD schemes have used complex valued difference equations in terms of complex valued field quantities to approximate these complex PDEs. This choice requires the use of complex numbers and complex operations in the computer program implementing the solution. An alternative CE FDTD scheme using only real numbers and operations can be derived from the real valued PDEs obtained by substituting the rectangular form of the complex field and source quantities into the complex PDEs and then separating each resulting complex PDE into its two equivalent real valued PDEs. The formulation of the CE FDTD using real values is demonstrated for a two-dimensional geometry where the electric field has only a z component. This implicit formulation requires only the solution of tridiagonal matrices. Results are presented for a 2D cavity problem with an electric current source. A reference solution for this problem is obtained by first solving the problem in the frequency domain and then transform it to the time-domain using the inverse fast Fourier transform (IFFT). Comparison of the two solutions demonstrates the accuracy of the new formulation