Decimal Digits Research Articles

Rapid computation of the plasma dispersion function: Rational and multi-pole approximation, and improved accuracy

The plasma dispersion function Z(s) is a fundamental complex special integral function widely used in the field of plasma physics. The simplest and most rapid, yet accurate, approach to calculating it is through rational or equivalent multi-pole expansions. In this work, we summarize the numerical coefficients that are practically useful to the community. In addition to the Padé approximation to obtain coefficients, which are accurate for both small and large arguments, we also employ optimization methods to enhance the accuracy of the approximation for the intermediate range. The best coefficients provided here for calculating Z(s) can deliver 12 significant decimal digits. This work serves as a foundational database for the community for further applications.

Enhanced and generalized one–step Neville algorithm: Fractional powers and access to the convergence rate

The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular choice of the abscissas used for the extrapolation, one can replace the recursive Neville scheme by a simple one-step transformation, while also obtaining access to subleading terms for the transformed series after convergence acceleration. The matrix-based, unified formulas allow one to estimate the rate of convergence of the partial sums of the input series to their limit. In particular, Bethe logarithms for hydrogen are calculated to 100 decimal digits. Generalizations of the method to series whose remainder terms can be expanded in terms of inverse factorial series, or series with half-integer powers, are also discussed.

On the Square Root Computation in Liber Abaci and De Practica Geometrie by Fibonacci

We study the square root computation by Leonardo Fibonacci (or Leonardo of Pisa) in his MSS Liber Abaci from c1202 and c1228 and De Practica Geometrie from c1220. In this MSS, Fibonacci systematically describes finding the integer part of the square root of an integer in numerous examples with three to seven decimal digits. The results of these examples are summarized in a table in the paper. Liber Abaci also describes in detail finding an approximation to the fractional part of the square root. However, in other examples in Liber Abaci and De Practica Geometrie, only the approximate values of the fractional part of the square roots are stated. This paper further explores these approximate values using techniques like reverse engineering. Contrary to many claims that Fibonacci also used other methods or approximations, we show that all examples can be explained using one digit-by-digit method to compute the integer part of the square root and one approximation scheme for the fractional part. Further, it is shown that the approximation scheme is tied to the method to compute the integer part of the square root.

Ground-state energy of H−: a critical test of triple basis sets

We report an improved variational upper bound for the ground state energy of H− using Hylleraas-like wave functions in the form of a triple basis set having three distinct distance scales. The extended precision DQFUN of Bailey, allowing for 70 decimal digit arithmetic, is implemented to retain sufficient precision. Our result exceeds the previous record [A. M. Frolov, Euro. J. Phys. D 69, 132 (2015)], indicating that the Hylleraas triple basis set exhibits comparable convergence to the widely used pseudorandom all-exponential basis sets, but the numerical stability against roundoff error is much better. It is argued that the three distance scales have a clear physical interpretation. The new variational bound for infinite nuclear mass is –0.527 751 016 544 377 196 590 814 478 a.u. New variational bounds are also presented for the finite mass cases of the hydrogen, deuterium, and tritium negative ions H−, D−, and T−.

A new semiring and its cryptographic applications

This paper introduced a novel semiring structure involving nonnegative integers, where operations depended on the comparison of the magnitudes of decimal digit sums. Consequently, a corresponding matrix semiring can be established on this commutative semiring. We showed that the 3-satisfiability problem can be polynomial-time reduced to solving systems of quadratic polynomial equations over this semiring. We proposed a key exchange protocol based on this matrix semiring, with its security relying on the two-sided digital circulant matrix action problem over this semiring. This scheme provides a novel cryptographic primitive for post-quantum cryptography.

A versatile interaction framework for robot programming based on hand gestures and poses

This paper proposes a framework for industrial and collaborative robot programming based on the integration of hand gestures and poses. The framework allows operators to control the robot via both End-Effector (EE) and joint movements and to transfer compound shapes accurately to the robot. Seventeen hand gestures, which cover the position and orientation controls of the robotic EE and other auxiliary operations, are designed according to cognitive psychology. Gestures are classified by a deep neural network, which is pre-trained for two-hand pose estimation and fine-tuned on a custom dataset, achieving a test accuracy of 99%. The index finger’s pointing direction and the hand’s orientation are extracted via 3D hand pose estimation to indicate the robotic EE’s moving direction and orientation, respectively. The number of stretched fingers is detected via two-hand pose estimation to represent decimal digits for selecting robot joints and inputting numbers. Finally, we integrate these three manners seamlessly to form a programming framework. We conducted two interaction experiments. The reaction time of the proposed hand gestures in indicating randomly given instructions is significantly less than that of other gesture sets, such as American Sign Language (ASL). The accuracy of our method in compound shape reconstruction is much better than that of hand movement trajectory-based methods, and the operating time is comparable with that of teach pendants.

ωA-IvE methodology: Admissible interleaving entropy methods applied to video streaming traffic classification

This work focuses on the width-based interval fuzzy entropy notion, considering the interval data diameter as a measure of the lack of knowledge and uncertainty related to the precise membership degrees of elements in an interval-valued fuzzy set. The w-preserving notion relates the uncertainty from input to output in system information. Such a concept generates a new entropy by applying width-based average functions and admissible order to compare interval data and define width-based fuzzy connectives in data fuzzy computations. A new admissible total order is introduced, requesting just one injective and increasing function, illustrated by a Decimal Digit Interleaving (DDI) function. The proposal methods are based on the admissible interleaving order and related expressions for width-based interval entropy considering different conditions for composition among width-based interval fuzzy operators, as negations with equilibrium and average functions, also including idempotent aggregation function and restricted equivalence functions. Finally, we illustrate the application of the proposed methods for solving a video streaming traffic classification problem. The admissible interleaving entropy analysis is performed over the attributes modeling the FuzzyNetClass approach, a computational model for traffic classification related to video streaming, exploring the integration of inference systems based on interval-valued fuzzy logic and machine learning algorithms. The results by comparison with other width-based interval fuzzy entropy methods were promising and pointed to continuing study and research efforts.

Numerical integration of the Cauchy problem with non-singular special points

Solutions of many applied Cauchy problems for ordinary differential equations have one or more multiple zeros on the integration segment. Examples are the equations of special functions of mathematical physics. The presence of multiples of zeros significantly complicates the numerical calculation, since such problems are ill-conditioned. Round-off errors may corrupt all decimal digits of the solution. Therefore, multiple zeros should be treated as special points of the differential equations. In the present paper, a local solution transformation is proposed, which converts the multiple zero into a simple one. The calculation of the latter is not difficult. This makes it possible to dramatically improve the accuracy and reliability of the calculation. Illustrative examples have been carried out, which confirm the advantages of the proposed method.

High-speed binary coded decimal digit multipliers with multiple error detection

Decimal arithmetic in the form of binary coded decimal (BCD) numbers is preferred in many financial, commercial and scientific applications. BCD multipliers are introduced as a key hardware unit to support both integer and floating-point decimal arithmetic operations. However, due to the increasing sensitivity of VLSI-based digital designs to the environmental effects, BCD multipliers are also prone to faults and errors similar to other arithmetic circuits. In addition, multiple error occurrence is possible in current digital systems especially in space applications which motivates to reach multiple error detection/correction in addition to single errors. In this paper, digit-by-digit BCD multipliers are introduced capable of multiple error detection with low delay overheads. To show the effectiveness of the proposed combined method, a 4-digit BCD multiplier is presented. Experimental results based on analysis and error injection-based simulations show that in addition to 100% single error detection, multiple errors can be detected with at least 99.6% probability in the 4-digit BCD multiplier as the implemented architecture.

Assessing last decimal digits in retracted research articles: accounting for missing zeroes

Assessing last decimal digits in retracted research articles: accounting for missing zeroes

Information guided adaptation of complex biological systems

Introduction: Functional information transmission through a complex biological system is introduced as a method for biological response (bioresponse) adaptation using qualified biological marker (biomarker) data.Methods: This information guided adaptation methodology traverses the series of complex connections, defined by disparate bioresponse and biomarkers data sets, by placing both data sets on the common platiorm defined by information. The absolute uncertainty associated with these data sets can be defined in terms of decimal digits of information. This relationship between the data’s absolute uncertainty and its information entropy is used to decompose information entropy into functional and relative uncertainty components, where the functional component quantifies the function or meaning of a data set in units of information.Results: Application of functional information to adapt patient medical treatments using the information values for the bioresponse model and the biomarker model are outlined in detail and presented tabularly.Discussion: Functional information provides a mathematical connection between a bioresponse model and a biomarker model by quantifying both data sets in the units of information and thereby providing the means to implement precision therapeutic plans by quantitatively adapting patient treatments using their biomarker measurements.

Developmental trajectories of numerical magnitude representations of fractions and decimals.

We examined the development of numerical magnitude representations of fractions and decimals from fourth to 12th grade. In Experiment 1, we assessed the rational number magnitude knowledge of 200 Chinese fourth, fifth, sixth, eighth, and 12th graders (92 girls and 108 boys) by presenting fraction and decimal magnitude comparison tasks as well as fraction and decimal 0-1 and 0-5 number line estimation tasks. Magnitude representations of decimals became accurate earlier, improved more rapidly, and reached a higher asymptotic accuracy than magnitude representations of fractions. Analyses of individual differences revealed positive relations between the accuracy of decimal and fraction magnitude representations at all ages. In Experiment 2, we presented an additional set of 24 fourth graders (14 girls and 10 boys) with the same tasks but with the decimals that were being compared varying in the number of decimal digits. The decimal advantage continued to be present for both magnitude comparison and estimation tasks, indicating that the greater accuracy with decimals was not limited to decimals with equal numbers of decimal digits, though unequal numbers of decimal digits did impact performance with decimals on both magnitude comparison and number line estimation tasks. Implications for understanding numerical development and education are discussed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

A Multi-Key with Partially Homomorphic Encryption Scheme for Low-End Devices Ensuring Data Integrity

In today’s hyperconnected world, the Internet of Things and Cloud Computing complement each other in several areas. Cloud Computing provides IoT systems with an efficient and flexible environment that supports application requirements such as real-time control/monitoring, scalability, fault tolerance, and numerous security services. Hardware and software limitations of IoT devices can be mitigated using the massive on-demand cloud resources. However, IoT cloud-based solutions pose some security and privacy concerns, specifically when an untrusted cloud is used. This calls for strong encryption schemes that allow operations on data in an encrypted format without compromising the encryption. This paper presents an asymmetric multi-key and partially homomorphic encryption scheme. The scheme provides the addition operation by encrypting each decimal digit of the given integer number separately using a special key. In addition, data integrity processes are performed when an untrusted third party performs homomorphic operations on encrypted data. The proposed work considers the most widely known issues like the encrypted data size, slow operations at the hardware level, and high computing costs at the provider level. The size of generated ciphertext is almost equal to the size of the plaintext, and order-preserving is ensured using an asymmetrical encryption version.

The global EPTO database: Worldwide occurrences of aquatic insects

AbstractMotivationAquatic insects comprise 64% of freshwater animal diversity and are widely used as bioindicators to assess water quality impairment and freshwater ecosystem health, as well as to test ecological hypotheses. Despite their importance, a comprehensive, global database of aquatic insect occurrences for mapping freshwater biodiversity in macroecological studies and applied freshwater research is missing. We aim to fill this gap and present the Global EPTO Database, which includes worldwide geo‐referenced aquatic insect occurrence records for four major taxa groups: Ephemeroptera, Plecoptera, Trichoptera and Odonata (EPTO).Main type of variables containedA total of 8,368,467 occurrence records globally, of which 8,319,689 (99%) are publicly available. The records are attributed to the corresponding drainage basin and sub‐catchment based on the Hydrography90m dataset and are accompanied by the elevation value, the freshwater ecoregion and the protection status of their location.Spatial location and grainThe database covers the global extent, with 86% of the observation records having coordinates with at least four decimal digits (11.1 m precision at the equator) in the World Geodetic System 1984 (WGS84) coordinate reference system.Time period and grainSampling years span from 1951 to 2021. Ninety‐nine percent of the records have information on the year of the observation, 95% on the year and month, while 94% have a complete date. In the case of seven sub‐datasets, exact dates can be retrieved upon communication with the data contributors.Major taxa and level of measurementEphemeroptera, Plecoptera, Trichoptera and Odonata, standardized at the genus taxonomic level. We provide species names for 7,727,980 (93%) records without further taxonomic verification.Software formatThe entire tab‐separated value (.csv) database can be downloaded and visualized at https://glowabio.org/project/epto_database/. Fifty individual datasets are also available at https://fred.igb‐berlin.de, while six datasets have restricted access. For the latter, we share metadata and the contact details of the authors.

Biot-Savart routines with minimal floating point error

A set of routines to compute the magnetic vector potential and magnetic field of two types of current carriers is presented. The (infinitely thin) current carrier types are a straight wire segment and a circular wire loop. The routines are highly accurate and exhibit the correct asymptotic behavior far away from and close to the current carrier. A suitable global set of test points is introduced and the methods presented in this work are tested against results obtained using arbitrary-precision arithmetic on all test points. The results are accurate to approximately 16 decimal digits of precision when computed using 64 bit floating point arithmetic. There are a few exceptions where accuracy drops to 13 digits. These primitive current carrier types can be used to assemble more complex arrangements, such as a current along a polygon (by means of defining straight wire segments from point to point along the polygon) and a multi-winding coil with circular cross-section. Reference data is provided along with the code for benchmarks with other implementations. Program summaryProgram Title: Biot-Savart Routines with Minimal Floating Point ErrorCPC Library link to program files:https://doi.org/10.17632/zcwk7zzt9y.1Developer's repository link:https://github.com/jonathanschilling/abscabLicensing provisions: Apache-2.0Programming language: C, Python, Java, FortranSupplementary material: Reference output data for all methods described in this article.Nature of problem: A common task in computational physics is to compute the magnetic field and magnetic vector potential originating from current-carrying wire arrangements. These current carriers are often approximated for computational simplicity as infinitely thin filaments following the center lines of the real current carrier. Computational methods are thus needed to compute the magnetic field and the magnetic vector potential of a single straight wire segment or a single circular wire loop as basis for modeling more complex current carrier arrangements by superposition of the current carrier primitives. A current carrier path specified by (x,y,z) coordinates can be modeled as a set of straight wire segments from point to point along the polygon describing the current carrier geometry. Closed circular wire loops are also commonly used as a proxy for a physical coil with helical windings.Solution method: Analytical expressions are derived in this work to accurately compute the magnetic field and the magnetic vector potential of a straight wire segment and a circular wire loop. The expressions consist of several special cases, which are automatically switched between depending on the location of the evaluation location in the coordinate system of the current carrier primitive. This approach guarantees that always the most accurate formulation available for a given evaluation location is used, including explicit use of simplifications in certain special cases for speed and accuracy.Additional comments including restrictions and unusual features: For most test cases, the full relative accuracy of the floating point arithmetic chosen for implementation is retained throughout the computations (16 digits of precision in the IEEE-754 binary64 implementation). There are a few exceptions where accuracy drops by up to 3 digits of precision (in the binary64 case). The provided implementations have not been optimized for speed.

Digit Maps

The happy function S of each positive integer x is defined to be the sum of the squares of the decimal digits of x. For example, S(2) = 4 and S(123) = 12 + 22 + 32 = 14. It is well known that for any , there exists such that , where S(n) is the n-fold composition of S. In addition, if and for some , then x is called a happy number.

Comparison of DenseNet201 and ResNet50 for lip reading of decimal Digits

Comparison of DenseNet201 and ResNet50 for lip reading of decimal Digits

A Fully Passive Ten-Digit Numeric Keypad Sensor Using Chipless RFID Technology

A ten-digit chipless radio frequency identification (CRFID)-based numeric keypad that works on the principle of touch event sensing is presented. The tag is designed to operate using the passive frequency-domain (FD) technique. The proposed tag has ten circular resonator rings for each of the keypad’s corresponding decimal digits. Each circular ring resonates at a particular frequency within the 400-MHz band from 2.7 to 7.6 GHz to uniquely identify a digit. Identification is performed by decoding radar cross Section (RCS) response of the whole keypad tag. A customized human finger using standard tissue characteristics is modeled to verify the keypad’s working mechanism. When the finger is placed on any of the nine resonators, its corresponding resonance disappears from the allocated band. The proposed CRFID keypad sensor tag is built upon a Rogers substrate of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$85\times 95\times0.8$ </tex-math></inline-formula> mm volume. The backside of the proposed tag is fully copper filled for an easy interface with a variety of backgrounds, including metallic walls. The design exactly mimics the shape of ordinary keypad. The finger model used in the simulations realizes the practical scenario. This CRFID keypad is deemed an intriguing solution for future automated cities and will help in the conceptualization of Internet of Things (IoT) systems.

Exact Versus Inexact Decimal Floating-Point Numbers and Arithmetic

The IEEE 754 standard does not distinguish between exact and inexact floating-point numbers. There is no bit or field in the binary encoding that indicates whether a floating-point number is exact or not. This is the case for binary and decimal floats. An inexact operation raises an inexact flag in a floating-point status register. The inexact result is rounded and used in a later operation as if it were exact. The floating-point arithmetic unit treats all the input operands as if there were exact, and hence might produce substantial errors in the final computed results. This paper focuses on making the distinction between exact and inexact decimal numbers and defines arithmetic operations on both types of numbers. If the result of a sequence of operations is exact, the user can trust that every decimal digit in the computed result is correct. On the other hand, if some input operands are inexact or the result cannot be computed exactly, a loss of significant digits occurs. A different representation is used for the inexact computed value. An estimate of the absolute error is also part of the inexact computed result. The decimal numbers and arithmetic operations introduced in this paper produce more accurate results that those computed by the IEEE 754 standard. A simple evaluation is shown in the last section of this paper.

Mass exchange in two-layer medium moving through the narrow cylindrical channel

This work further develops previous studies devoted to numerical modeling of diffuse mass transfer in narrow pore channels. The problem of diffusion in a two-layer liquid moving through narrow cylindrical channel, into which a neutral component, which does not interact with heterogeneous inclusions, diffuses, is considered. The dispersion carrier fluid moves close to the wal, while a reologically complicated two-phase medium occupies most of the channel. During the flow of a fine-dispersed concentrated suspension in a quasi-balanced condition, the reological properties of the medium were accepted as parameters of some homogenous liquid, which can be deemed an adequate approximation. This model can be used for some situations of flow in the chanal of fluid bodies which are capillary-porous and broken, such as those that contain muddy or clay inclusions. Similar mathematical models can be applied to paste sliding flows because of poor capillary fluid fixation. In this paper, two cases are considered. In the first scenario, a portion of the channel midsection is exceeded by the diffusing component. In the second, this component in the same location exits the channel. The non-stable problem is numerically solved before the creation of the stationary state. The installation procedure was monitored up to the fifth decimal digit. The resulting solution determines the concentration fields of the diffuse component. It is demonstrated that distribution of the diffuse component concentration in the working area is influenced by the rate of the medium speed; diffuse flow through the wall, and effective diffusion coefficient. In this work, the case is considered when there is no interaction between the diffusing component and the dispersed medium. However, the interaction of these components of the medium in soils, biological systems, and natural layers containing organic inclusions is of great interest. Therefore, accounting of adsorption properties of the dispersed medium in relation to the elements involved in mass exchange can be in the focus of future study. Taking into account large-scale metabolic processes, such as those in blood in the veins, is crucial in many practically significant circumstances.