A new look at infinitely large and infinitely small quantities: Methodological foundations and practical calculations with these numbers on a computer

This article describes a recently proposed methodology that allows one to work with infinitely large and infinitely small quantities on a computer. The approach uses a number of ideas that bring it closer to modern physics, in particular, the relativity of mathematical knowledge and its dependence on the tools used by mathematicians in their studies are discussed. It is shown that the emergence of new computational tools influences the way we perceive traditional mathematical objects, and also helps to discover new interesting objects and problems. It is discussed that many difficulties and paradoxes regarding infinity do not depend on its nature, but are the result of the weakness of the traditional numeral systems used to work with infinitely large and infinitely small quantities. A numeral system is proposed that not only allows one to work with these quantities analytically in a simpler and more intuitive way, but also makes possible practical calculations on the Infinity Computer, patented in a number of countries. Examples of measuring infinite sets with the accuracy of one element are given and it is shown that the new methodology avoids the appearance of some well-known paradoxes associated with infinity. Examples of solving a number of computational problems are given and some results of teaching the described methodology in Italy and Great Britain are discussed.

Numerical Expressions in Chinese: Syntax and Semantics

Numerical expressions are linguistic forms related to numbers or quantities, which directly reflect the relationship between linguistic symbols and mathematical cognition. Featuring some unique properties, numeral systems are somewhat distinguished from other language subsystems. For instance, numerals can appear in various grammatical positions, including adjective positions, determiner positions, and argument positions. Thus, linguistic research on numeral systems, especially the research on the syntax and semantics of numerical expressions, has been a popular and recurrent topic. For the syntax of complex numerals, two analyses have been proposed in the literature. The traditional constituency analysis maintains that complex numerals are phrasal constituents, which has been widely accepted and defended as a null hypothesis. The nonconstituency analysis, by contrast, claims that a complex numeral projects a complementative structure in which a numeral is a nominal head selecting a lexical noun or a numeral-noun combination as its complement. As a consequence, additive numerals are transformed from full NP coordination. Whether numerals denote numbers or sets has aroused a long-running debate. The number-denoting view assumes that numerals refer to numbers, which are abstract objects, grammatically equivalent to nouns. The primary issue with this analysis comes from the introduction of a new entity, numbers, into the model of ontology. The set-denoting view argues that numerals refer to sets, which are equivalent to adjectives or quantifiers in grammar. One main difficulty of this view is how to account for numerals in arithmetic sentences.

An FPGA-Based LOCO-ANS Implementation for Lossless and Near-Lossless Image Compression Using High-Level Synthesis

In this work, we present and evaluate a hardware architecture for the LOCO-ANS (Low Complexity Lossless Compression with Asymmetric Numeral Systems) lossless and near-lossless image compressor, which is based on JPEG-LS standard. The design is implemented in two FPGA generations, evaluating its performance for different codec configurations. The tests show that the design is capable of up to 40.5 MPixels/s and 124 MPixels/s per lane for Zynq 7020 and UltraScale+ FPGAs, respectively. Compared to the single thread LOCO-ANS software implementation running in a 1.2 GHz Raspberry Pi 3B, each hardware lane achieves 6.5 times higher throughput, even when implemented in an older and cost-optimized chip like the Zynq 7020. Results are also presented for a lossless only version, which achieves a lower footprint and approximately 50% higher performance than the version that supports both lossless and near-lossless. Interestingly, these great results were obtained applying High-Level Synthesis, describing the coder with C++ code, which tends to establish a trade-off between design time and quality of results. These results show that the algorithm is very suitable for hardware implementation. Moreover, the implemented system is faster and achieves higher compression than the best previously available near-lossless JPEG-LS hardware implementation.

A Formalization of the Concept of a Numeral System

We consider finite and unconditionally convergent infinite expansions of elements of a given topological monoid G in some base B c G as words of the alphabet B, identify insignificantly different words and define a multiplication and a topology on the set of classes of these words. Classical numeral systems are particular cases of this construction. Then we study algebraic and topological properties of the obtained monoid and, for some cases, find conditions under which it is canonically topologically isomorphic to the initial one.

Numeral Systems in Various Fula Lects

The paper presents the analysis of Fula numeral system. Fula is an Atlantic language, dispersed throughout a vast territory in Sub-Saharan Africa. It has over 20 differently-sized lects subdivided into three zones - Western, Central and Eastern. The research included collecting the language data on six major lects, two from each zone. Then this data was analyzed from the point of view of internal and external syntax of ordinal, cardinal, distributive and fraction numerals. The comparative analysis of Fula numeral system on inter-lect scale has never been reflected in earlier Fula studies. Apart from a unique collection of numeral data from all language zones of Fula, the paper presents comparative analysis of such data, including, but not limited to, phonetic and contact-induced variabilities. The sources of structural and lexical loans in the system are listed wherever possible, because the disperse nature of Fula lects means that the contact languages and the ensuing changes can be different for each lect. Special attention is paid to the numeral bases, which can be somatic, commercial of a combination of the two. The Fula numeral system has never been analyzed from this point of view, and the contact changes to it are of great linguistic interest. Also cardinal numerals in Fula can change their form depending on the personality parameter of head noun. This system is unique for Fula and, more specifically, to its numeral system, and is properly described for the first time. Some parts of the Fula numeral system, like distributive and fraction numerals for some lects, have been found underrepresented and poorly described, which leaves room for further research, both field one and typological.

Numerical Methods I

Textbook of several universities. 2nd edition. The color edition is also available at Glasstree Bookstore. It is recommended for students. The series of books Numerical Methods is written primarily for students at technical universities, but also as a useful handbook for engineers, PhD students and scientists. This volume introduces the reader into numeral systems and representation of numbers in digital computers. Possibly the most important part of this book are descriptions of differences between constant and random variables, related types of errors and error propagation. These topics are supplemented with various types of regression analyses. Finally, direct and iterative methods for finding roots of polynomials are explained. Practical application is supported by 77 examples and 13 algorithms. For reasons of simplicity, algorithms are written in pseudo-code, so they can easily be included in any computer program.

Numerical Methods I

Textbook, established at several universities. Second edition. *** Written primarily for students at technical studies. Valuable handbook for engineers, PhD students and scientists. *** Published in several variants. *** Seven chapters. In the first chapter, a historical background and basic properties of various numeral systems, as well as conversion of numbers from one system to another are briefly explained. In the second chapter, numbers in digital computers, namely integers and floating point numbers are described. This helps the reader to choose precision and range limits of stored numbers. The third chapter explains constant variables and related numerical errors, including error propagation and algorithm instability. The fourth and fifth chapters explain random variables and related random errors, uncertainty, confidence level, as well as propagation of random errors. Various types of regression analyses of experimental data are described in the sixth chapter. Direct methods for finding roots of the third and fourth degree polynomials are described in the seventh chapter, followed by general iterative methods for polynomials of any degree. *** Why the presented topics are so important? Simply, they are common to all numerical methods. *** Practical application is supported by 84 examples and 17 algorithms. For reasons of simplicity, algorithms are written in pseudo-code, so they can easily be implemented in any computer program. Finally, the given text with 98 figures and 52 tables represents a valuable background for understanding, applying and developing various numerical analyses.

Les systèmes numéraux en quechua et en aymara : une histoire de suffixes

The Quechuan and Aymaran numeral systems show strong similarities, e.g., they have a decimal base and thirteen simple lexemes (simple cardinal numerals). In the case of Quechuan, when forming compound cardinal numerals (namely, numerals with two or more simple lexemes), two distinct sets can be identified: in the first, the compound cardinals are the results of the juxtaposition of simple lexemes. In the second, to this juxtaposition are added nominal suffixes: /-yuq/ or /-n/ or /-wan/. This paper is focused on studying these two numeral Quechua sets involving both diachronic and geographical considerations, and puts forward the hypothesis that nominal suffixes presence in one of them is a consequence of its intense contact with the other great Andean linguistic family, the Aymara. Indeed, Aymaran in all its components, as will be described, adds nominal suffixes to its compound cardinals similarly to the Quechua suffixal numeral set.