Representation Of Real Numbers Research Articles

Physicochemical reaction mechanism study and model optimization based on big data analysis

The huge amount of data generated by information technology, intelligence and automation has serious redundancy and duplication, which becomes a hidden cost of consuming resources. In the field of physicochemical analysis, for the explanation of mechanism, the analysis of big data can provide reference and integration. The purpose of big data-based analysis is to remove redundancies and inconsistencies in the given raw characterization data to better improve its completeness and accuracy. The paper takes the density matrix of quantum bits, which is different from the classical real number representation, etc. as the representation of feature data, and investigates the cleaning of feature data based on quantum representation from the perspective of the intersection of theoretical physics and computer science. This paper comprehensively describes the basic theories and methods of data mining. On the basis of understanding and analyzing a variety of data mining techniques, it focuses on the BP neural network-based data mining techniques to provide in-depth analysis and elaboration. Further, this paper proposes an improved algorithm with multi-algorithm advantage integration and joint optimization for the deficiencies in the BP neural network algorithm, which is fully demonstrated in the nonlinear function simulation. In addition, based on the HA-BP algorithm, this paper has designed anomaly data detection model and variable factor analysis model, and their reliability and practicality are also fully demonstrated in the nonlinear function simulation in the analytical study of physicochemical reaction mechanism.

Representation of Real Numbers by Perron Series, Their Geometry, and Some Applications

Representation of Real Numbers by Perron Series, Their Geometry, and Some Applications

Exploring Hardware Fault Impacts on Different Real Number Representations of the Structural Resilience of TCUs in GPUs

The most recent generations of graphics processing units (GPUs) boost the execution of convolutional operations required by machine learning applications by resorting to specialized and efficient in-chip accelerators (Tensor Core Units or TCUs) that operate on matrix multiplication tiles. Unfortunately, modern cutting-edge semiconductor technologies are increasingly prone to hardware defects, and the trend to highly stress TCUs during the execution of safety-critical and high-performance computing (HPC) applications increases the likelihood of TCUs producing different kinds of failures. In fact, the intrinsic resiliency to hardware faults of arithmetic units plays a crucial role in safety-critical applications using GPUs (e.g., in automotive, space, and autonomous robotics). Recently, new arithmetic formats have been proposed, particularly those suited to neural network execution. However, the reliability characterization of TCUs supporting different arithmetic formats was still lacking. In this work, we quantitatively assessed the impact of hardware faults in TCU structures while employing two distinct formats (floating-point and posit) and using two different configurations (16 and 32 bits) to represent real numbers. For the experimental evaluation, we resorted to an architectural description of a TCU core (PyOpenTCU) and performed 120 fault simulation campaigns, injecting around 200,000 faults per campaign and requiring around 32 days of computation. Our results demonstrate that the posit format of TCUs is less affected by faults than the floating-point one (by up to three orders of magnitude for 16 bits and up to twenty orders for 32 bits). We also identified the most sensible fault locations (i.e., those that produce the largest errors), thus paving the way to adopting smart hardening solutions.

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.

Drawing on a computer algorithm to advance future teachers’ knowledge of real numbers: A case study of task design

In our investigation of university students’ knowledge about real numbers in relation to computer algebra systems (CAS) and how it could be developed in view of their future activity as teachers, we used a computer algorithm as a case to explore the relationship between CAS and the knowledge of real numbers as decimal representations. Our work was carried out in the context of a course for university students who aim to become mathematics teachers in high schools. The main data consists of students’ written responses to an assignment of the course and interviews to clarify students’ perspectives in relation to the responses. The analysis of students’ work is based on the anthropological theory of the didactic (ATD). Our results indicate that simple CAS-routines have a potential to help university students (future teachers) to apply their university knowledge on certain problems related to the decimal representation of real number which are typically encountered but not well explained in high school.

A note on exotic integrals

We consider Bernoulli measures μ p \mu _p on the interval [ 0 , 1 ] [0,1] . For the standard Lebesgue measure the digits 0 0 and 1 1 in the binary representation of real numbers appear with an equal probability 1 / 2 1/2 . For the Bernoulli measures, the digits 0 0 and 1 1 appear with probabilities p p and 1 − p 1-p , respectively. We provide explicit expressions for various μ p \mu _p -integrals. In particular, integrals of polynomials are expressed in terms of the determinants of special Hessenberg matrices, which, in turn, are constructed from the Pascal matrices of binomial coefficients. This allows us to find closed-form expressions for the Fourier coefficients of μ p \mu _p in the Legendre polynomial basis. At the same time, the trigonometric Fourier coefficients are values of some special entire functions, which admit explicit infinite product expansions and satisfy interesting properties, including connections with the Stirling numbers and the polylogarithm.

ON SOME PROPERTIES OF THE DIGIT SHIFT OPERATOR Q_s^*-REPRESENTATION OF REAL NUMBERS AND UNIFORMLY DISTRIBUTED SEQUENCES PRODUCED BY IT

The work is devoted to the study of the properties of the left-shift operator $Q_{s}^{*}$-representation of real numbers and the study of the type of distribution of the sequences produced by the corresponding operator. The $Q_{s}^{*}$-representation of real numbers is a natural generalization of the classical s-representation and is topologically similar to the latter. E. Borel's classic result that almost all numbers are s-normal was over time translated into the terms of uniformly distributed sequences produced by the left-shift operator of the digits of the corresponding representation. It was proved that a number is s-normal only when the corresponding sequence generated by this number in the sense of the left shift operator is uniformly distributed. Despite the topological similarity between the $Q_{s}^{*}$-representation of real numbers and the classical s-representation, proving similar results for the former requires fundamentally new approaches that include the use of the apparatus of ergodic theory. The absence of the effect of metric transitivity of the appearance of digits, which is characteristic of the classical s-representation, does not allow the use of appropriate approaches to the $Q_{s}^{*}$-representation. The construction of normal numbers in various representation systems is a separate non-trivial problem and is the subject of many studies. In many cases, criteria for the normality of numbers, which can have a continuous structure (similar to the classical criteria of uniform distribution of the sequence) or a discrete structure, are useful for constructing the corresponding numbers. This paper presents generalizations of discrete criteria for the normality of numbers, which applied both to the classical s-representation and to the $Q_{s}$-representation of real numbers (the latter is a partial case of the $Q_{s}^{*}$-representation).

On the Degree Distribution of Haros Graphs

Haros graphs are a graph-theoretical representation of real numbers in the unit interval. The degree distribution of the Haros graphs provides information regarding the topological structure and the associated real number. This article provides a comprehensive demonstration of a conjecture concerning the analytical formulation of the degree distribution. Specifically, a theorem outlines the relationship between Haros graphs, the corresponding continued fraction of its associated real number, and the subsequent symbolic paths in the Farey binary tree. Moreover, an expression that is continuous and piecewise linear in subintervals defined by Farey fractions can be derived from an additional conclusion for the degree distribution of Haros graphs.

Representations of Real Numbers Induced by Probability Distributions on ℕ

Abstract We observe that a probability distribution supported by ℕ, induces a representation of real numbers in [0, 1) with digits in ℕ. We first study the Hausdorff dimension of sets with prescribed digits with respect to these representations. Then, we determine the prevalent frequency of digits and the Hausdorff dimension of sets with prescribed frequencies of digits. As examples, we consider the geometric distribution, the Poisson distribution and the zeta distribution.

A Lightweight Posit Processing Unit for RISC-V Processors in Deep Neural Network Applications

Nowadays, two groundbreaking factors are emerging in neural networks. First, there is the RISC-V open instruction set architecture (ISA) that allows a seamless implementation of custom instruction sets. Second, there are several novel formats for real number arithmetic. In this work, we combined these two key aspects using the very promising posit format, developing a light Posit Processing Unit (PPU-light). We present an extension of the base RISC-V ISA that allows the conversion between 8 or 16-bit posits and 32-bit IEEE Floats or fixed point formats in order to offer a compressed representation of real numbers with little-to-none accuracy degradation. Then we elaborate on the hardware and software toolchain integration of our PPU-light inside the Ariane RISC-V core and its toolchain, showing how little it impacts in terms of circuit complexity and power consumption. Indeed, only 0.36% of the circuit is devoted to the PPU-light while the full RISC-V core occupies the 33% of the overall circuit complexity. Finally we present the impact of our PPU-light on a deep neural network task, reporting speedups up to 10 on sample inference processing time.

Haros graphs: an exotic representation of real numbers

AbstractThis article introduces Haros graphs, a construction which provides a graph-theoretical representation of real numbers in the unit interval reached via paths in the Farey binary tree. We show how the topological structure of Haros graphs yields a natural classification of the real numbers into a hierarchy of families. To unveil such classification, we introduce an entropic functional on these graphs and show that it can be expressed, thanks to its fractal nature, in terms of a generalized de Rham curve. We show that this entropy reaches a global maximum at the reciprocal of the Golden number and otherwise displays a rich hierarchy of local maxima and minima that relate to specific families of irrationals (noble numbers) and rationals, overall providing an exotic classification and representation of the reals numbers according to entropic principles. We close the article with a number of conjectures and outline a research programme on Haros graphs.

Limits of real numbers in the binary signed digit representation

We extract verified algorithms for exact real number computation from constructive proofs. To this end we use a coinductive representation of reals as streams of binary signed digits. The main objective of this paper is the formalisation of a constructive proof that real numbers are closed with respect to limits. All the proofs of the main theorem and the first application are implemented in the Minlog proof system and the extracted terms are further translated into Haskell. We compare two approaches. The first approach is a direct proof. In the second approach we make use of the representation of reals by a Cauchy-sequence of rationals. Utilizing translations between the two represenation and using the completeness of the Cauchy-reals, the proof is very short. In both cases we use Minlog's program extraction mechanism to automatically extract a formally verified program that transforms a converging sequence of reals, i.e.~a sequence of streams of binary signed digits together with a modulus of convergence, into the binary signed digit representation of its limit. The correctness of the extracted terms follows directly from the soundness theorem of program extraction. As a first application we use the extracted algorithms together with Heron's method to construct an algorithm that computes square roots with respect to the binary signed digit representation. In a second application we use the convergence theorem to show that the signed digit representation of real numbers is closed under multiplication.

Combining attention with spectrum to handle missing values on time series data without imputation

In the development of predictive models, the problem of missing data is a critical issue that traditionally requires a two-step analysis. Data scientists analyze the patterns of missing values, select variables, impute missing values on the basis of domain knowledge, and then train a model. Models typically have their input sizes hardcoded, and have limitations in handling data with high missing rates or changes in available variables. We propose an attention-based neural network combined with a novel real number representation, which requires little work on manually selecting variables, and in which missing data can be overlooked, making imputation unnecessary. In this proposed model, data analysis can be one step, omitting the first step of imputing missing values. The study included data on 32,709 intensive care unit (ICU) admissions and 60 healthcare variables from the Medical Information Mart for Intensive Care (MIMIC)-IV. The proposed algorithm yielded an area under the receiver operating characteristic curve (AUC) of 0.842 (95% CIs: 0.828–0.856) when predicting prolonged length of stay in the ICU, outperforming current approaches using imputation methods. The proposed algorithm can be applied to a range of problems in data science, as it addresses the issue of incomplete data with automatic variable selection.

Interval-filling sequences of order $N$ and a representation of real numbers in canonical number systems

Interval-filling sequences of order $N$ and a representation of real numbers in canonical number systems

Some representations of real numbers using integer sequences

AbstractThe paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong–Reeb line. Two of the new models, contrary to the Harthong–Reeb line, provide accurate integer “views” on real numbers at a sequence of growing scales $B^n$ ( $B\ge2$ ).

Convergence and gradient algorithm of a class of neural networks based on the polygonal fuzzy numbers representation

As a special case of general fuzzy numbers, the polygonal fuzzy number can describe a fuzzy object by means of an ordered representation of finite real numbers. Different from general fuzzy numbers, the polygonal fuzzy numbers overcome the shortcoming of complex operations based on Zadeh’s traditional expansion principle, and can maintain the closeness of arithmetic operation. Hence, it is feasible to use a polygonal fuzzy number to approximate a general fuzzy number. First, an extension theorem of continuous functions on a real compact set is given according to open set construction theorem. Then using Weierstrass approximation theorem and ordered representation of the polygonal fuzzy numbers, the convergence of a single hidden layer feedforward polygonal fuzzy neural network is proved. Secondly, the gradient vector of the approximation error function and the optimization parameter vector of the network are given by using the ordered representation of polygonal fuzzy numbers, and then the gradient descent algorithm is used to train the optimal parameters of the polygonal fuzzy neural network iteratively. Finally, two simulation examples are given to verify the approximation ability of the network. Simulation result shows that the proposed network and the gradient descent algorithm are effective, and the single hidden layer feedforward network have good abilities in learning and generalization.

Strengthening Quality of Chaotic Bit Sequences

We discuss chaos and its quality as measured through the 0-1 test for chaos. When the 0-1 test indicates deteriorating quality of chaos, because of the finite precision representations of real numbers in digital implementations, then the process may eventually lead to a periodic sequence. A simple method for improving the quality of a chaotic signal is to mix the signal with another signal by using the XOR operation. In this paper, such mixing of weak chaotic signals is considered, yielding new signals with improved quality (with K values from the 0-1 test close to 1). In some sense, such a mixing of signals could be considered as a two-layer prevention strategy to maintain chaos. That fact may be important in those applications when the hardware resources are limited. The 0-1 test is used to show the improved chaotic behavior in the case when a continuous signal (for example, from the Chua, Rössler or Lorenz system) intermingles with a discrete signal (for example, from the logistic, Tinkerbell or Henon map). The analysis is presented for chaotic bit sequences. Our approach can further lead to hardware applications, and possibly, to improvements in the design of chaotic bit generators. Several illustrative examples are included.

Intelligent stability margin improvement using series and shunt controllers

Electric market always prefers to use full capacity of existing power system to control the costs. Flexible alternate current transmission system (FACTS) devices introduced by Electric Power Research Institute (EPRI) to increase the usable capacity of power system. Placement of FACTS controllers in power system is a critical issue to reach their maximum advantages. This article focused on the application of FACTS devices to increase the stability of power system using artificial intelligence. Five types of series and shunt FACTS controllers are considered in this study. Continuation power flow (CPF) analysis used to calculate the collapse point of power systems. Controlling parameters of FACTS devices including their locations are determined using real number representation based genetic algorithm (RNRGA) in order to improve the secure margin of operating condition of power system. The 14 and 118 buses IEEE standard test systems are utilized to verify the recommended method. The achieved results manifestly proved the effectiveness of proposed intelligent method to increase the stability of power system by determining the optimum location and size of each type of FACTS devices.

Fundamentals of metric theory of real numbers in their $\overline{Q_3}$-representation

In the paper we were studied encoding of fractional part of a real number with an infinite alphabet (set of digits) coinciding with the set of non-negative integers. The geometry of this encoding is generated by $Q_3$-representation of real numbers, which is a generalization of the classical ternary representation. The new representation has infinite alphabet, zero surfeit and can be efficiently used for specifying mathematical objects with fractal properties.
 We have been studied the functions that store the "tails" of $\overline{Q_3}$-representation of numbers and the set of such functions,some metric problems and some problems of probability theory are connected with $\overline{Q_3}$-representation.

Intelligent stability margin improvement using series and shunt controllers

Electric market always prefers to use full capacity of existing power system to control the costs. Flexible alternate current transmission system (FACTS) devices introduced by Electric Power Research Institute (EPRI) to increase the usable capacity of power system. Placement of FACTS controllers in power system is a critical issue to reach their maximum advantages. This article focused on the application of FACTS devices to increase the stability of power system using artificial intelligence. Five types of series and shunt FACTS controllers are considered in this study. Continuation power flow (CPF) analysis used to calculate the collapse point of power systems. Controlling parameters of FACTS devices including their locations are determined using real number representation based genetic algorithm (RNRGA) in order to improve the secure margin of operating condition of power system. The 14 and 118 buses IEEE standard test systems are utilized to verify the recommended method. The achieved results manifestly proved the effectiveness of proposed intelligent method to increase the stability of power system by determining the optimum location and size of each type of FACTS devices.