Representation Of Real Numbers Research Articles

Representations of real numbers by alternating Perron series and their geometry

Representations of real numbers by alternating Perron series and their geometry

On the Lebesgue measure of one generalised set of subsums of geometric series

In the present paper, we study a set that can be treated as a generalised set of subsums for a geometric series. This object was discovered independently in various mathematical aspects. For instance, it is closely related to various systems of representation of real numbers. The main object of this paper was particularly studied by R. Kenyon, who brought up a question about the Lebesgue measure of the set and conjectured that it is positive. Further, Z. Nitecki confirmed the hypothesis by using nontrivial topological techniques. However, the aforementioned result is quite limited, as this particular case should satisfy a rigid condition of homogeneity. Despite the limited progress, the problem remained understudied in a general framework. The study of topological, metric, and fractal properties of the set of subsums for a numerical series is a separate research direction in mathematics. On the other hand, the topic is related to another modern mathematical problem, namely, deepening of the Jessen-Wintner theorem for infinite Bernoulli convolutions and their generalisations. The essence of the problem is to reveal the necessary and sufficient conditions for the probability distribution of a random subsum of a geometric series to be absolutely continuous or singular. The Jessen-Wintner theorem guarantees that the distribution is pure (pure discrete, pure singular, or pure absolutely continuous).Meanwhile, the Levy theorem gives us the necessary and sufficient condition for the distribution to be discrete.Since the set of subsums for an absolutely convergent series coincides with the set of possible outcomes of the corresponding probability distribution, under certain conditions, it allows us to apply various probability techniques for its further investigation. In particular, some techniques help us to prove that the above sets have a positive Lebesgue measure and allow to deepen the Jessen-Wintner theorem under certain conditions.

Optimal expansions of Kakeya sequences

AbstractWe investigate optimal expansions of Kakeya sequences for the representation of real numbers. Expansions of Kakeya sequences generalize the expansions in non-integer bases and they display analogous redundancy phenomena. In this paper, we characterize optimal expansions of Kakeya sequences, and we provide conditions for the existence of unique expansions with respect to Kakeya sequences.

Semi-Regular Continued Fractions with Fast-Growing Partial Quotients

In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular counterparts, which are produced from the sequences of alternating plus and minus ones. In this study, we investigate the structure and features of semi-regular continuous fractions through the lens of dimension theory. We prove a primary result about the Hausdorff dimension of number sets whose partial quotients increase more quickly than a given pace. Furthermore, we conduct numerical analyses to illustrate the differences between regular and semi-regular continued fractions, shedding light on potential future directions in this field.

Development of optimization method for truss structure by quantum annealing

In this study, we developed a new method of topology optimization for truss structures by quantum annealing. To perform quantum annealing analysis with real variables, representation of real numbers as a sum of random number combinations is employed. The nodal displacement is expressed with binary variables. The Hamiltonian H is formulated on the basis of the elastic strain energy and position energy of a truss structure. It is confirmed that truss deformation analysis is possible by quantum annealing. For the analysis of the optimization method for the truss structure, the cross-sectional area of the truss is expressed with binary variables. The iterative calculation for the changes in displacement and cross-sectional area leads to the optimal structure under the prescribed boundary conditions.

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.

Novel real number representations in Ising machines and performance evaluation: Combinatorial random number sum and constant division.

Quantum annealing machines are next-generation computers for solving combinatorial optimization problems. Although physical simulations are one of the most promising applications of quantum annealing machines, a method how to embed the target problem into the machines has not been developed except for certain simple examples. In this study, we focus on a method of representing real numbers using binary variables, or quantum bits. One of the most important problems for conducting physical simulation by quantum annealing machines is how to represent the real number with quantum bits. The variables in physical simulations are often represented by real numbers but real numbers must be represented by a combination of binary variables in quantum annealing, such as quadratic unconstrained binary optimization (QUBO). Conventionally, real numbers have been represented by assigning each digit of their binary number representation to a binary variable. Considering the classical annealing point of view, we noticed that when real numbers are represented in binary numbers, there are numbers that can only be reached by inverting several bits simultaneously under the restriction of not increasing a given Hamiltonian, which makes the optimization very difficult. In this work, we propose three new types of real number representation and compared these representations under the problem of solving linear equations. As a result, we found experimentally that the accuracy of the solution varies significantly depending on how the real numbers are represented. We also found that the most appropriate representation depends on the size and difficulty of the problem to be solved and that these differences show a consistent trend for two annealing solvers. Finally, we explain the reasons for these differences using simple models, the minimum required number of simultaneous bit flips, one-way probabilistic bit-flip energy minimization, and simulation of ideal quantum annealing machine.

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