Dual Numbers Research Articles

Variety of General Position Problems in Graphs

Let X be a vertex subset of a graph G. Then u,v∈V(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u, v\\in V(G)$$\\end{document} are X-positionable if V(P)∩X⊆{u,v}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(P)\\cap X \\subseteq \\{u,v\\}$$\\end{document} holds for any shortest u, v-path P. If each two vertices from X are X-positionable, then X is a general position set. The general position number of G is the cardinality of a largest general position set of G and has been already well investigated. In this paper a variety of general position problems is introduced based on which natural pairs of vertices are required to be X-positionable. This yields the total (resp. dual, outer) general position number. It is proved that the total general position sets coincide with sets of simplicial vertices, and that the outer general position sets coincide with sets of mutually maximally distant vertices. It is shown that a general position set is a dual general position set if and only if its complement is convex. Several sufficient conditions are presented that guarantee that a given graph has no dual general position set. The total general position number, the outer general position number, and the dual general position number of arbitrary Cartesian products are determined.

Pricing in spatial classification system in non-symmetric market demand based on the calculations of double interval grey numbers

PurposeThe contemporary landscape of supply chains necessitates a comprehensive integration of multiple components encompassing production, distribution and customer engagement. The pursuit of supply chain harmony underscores the significance of pricing strategies within the framework of dual-channel distribution, particularly when confronted with the dynamics of asymmetric demand performance.Design/methodology/approachThis paper delves into a nuanced decision-making challenge anchored in a dual-channel distribution context featuring a retailer and two distinct products. Notably, the retailer’s decision-making process employs the computational framework of dual grey numbers, a robust tool for handling uncertainty.FindingsThis study revolves around applying game theory to manufacturers. Each manufacturer presents its aggregated price proposition to the retailer. Subsequently, the retailer identifies the optimal pricing configuration among the manufacturers' aggregate prices while adhering to constraints such as spatial classification and inventory costs. This article’s contribution extends to delineating the retailer’s capacity to discern the influence of product market potential and the aggregate product cost on the overall demand.Originality/valueThe model’s innovation lies in its harmonious fusion of spatial classification, pricing strategies and inventory control. Notably, this novel integration provides a platform for unraveling the intricate interplay between non-symmetric market potential, production costs and cross-sensitivity. The investigation is underscored by the utilization of the double interval grey numbers, a powerful computational approach that accommodates the inherent uncertainty pervasive in the domain. This study fills a gap in the existing literature by offering an integrated framework unifying spatial allocation, pricing decisions and inventory optimization.

Taylor's formula on the separating complex plane and its applications

This article begins by exploring split-complex numbers, also known as dual or double numbers, a mathematical concept that extends the real number system by creating a commutative ring with a zero divisor. Furthermore, the paper extends the Taylor series theory from the real analysis domain to the domain of split-complex numbers, thereby establishing the corresponding Taylor formula on the split-complex plane. The introduction of split-complex numbers opens up new perspectives and provides new tools for the modeling and analysis of quantum systems. In the field of electromagnetism, the application of split-complex numbers also demonstrates its potential, contributing to a more precise description of the behavior of electromagnetic fields under specific conditions. These theoretical advancements are expected to further promote research in the field of split-complex analysis, playing a more critical role in important branches of physics such as quantum mechanics and electromagnetism, thereby facilitating the development of related theories and the emergence of innovative practical applications.

Automatic differentiation for ML-family languages: Correctness via logical relations

Abstract We give a simple, direct, and reusable logical relations technique for languages with term and type recursion and partially defined differentiable functions. We demonstrate it by working out the case of automatic differentiation (AD) correctness: namely, we present a correctness proof of a dual numbers style AD code transformation for realistic functional languages in the ML-family. We also show how this code transformation provides us with correct forward- and reverse-mode AD. The starting point is to interpret a functional programming language as a suitable freely generated categorical structure. In this setting, by the universal property of the syntactic categorical structure, the dual numbers AD code transformation and the basic $\boldsymbol{\omega } \mathbf{Cpo}$ -semantics arise as structure preserving functors. The proof follows, then, by a novel logical relations argument. The key to much of our contribution is a powerful monadic logical relations technique for term recursion and recursive types. It provides us with a semantic correctness proof based on a simple approach for denotational semantics, making use only of the very basic concrete model of $\omega$ -cpos.

Automatic differential kinematics of serial manipulator robots through dual numbers

Dual Numbers are an extension of real numbers known for its capability of performing exact automatic differentiation of one-valued functions theoretically without error approximation. Also, Differential Kinematics of robots involves the computation of the Jacobian, which is a matrix of partial derivatives of the Forward Kinematic equations with respect to the robot’s joints. Thus, to perform the automatic calculation of the Jacobian matrix, this paper presents an extension of dual numbers composed of a scalar real part and a vector dual part, where the real part represents the angular value of the robot joint, and the dual part represents the direction of the corresponding partial derivative for each joint. The presented method was implemented in Matlab through Object Orientes Programming (OOP), and the results for calculating the Jacobian of the KUKA KR 500 robot model for 1000 random postures were subsequently compared in terms of execution time and Mean Squared Error (MSE) with other conventional methods: the geometric method, the symbolic method, and the finite difference method. The results showed a significant improvement in the computing time for calculating the Jacobian of the robotic model compared to the other methods, as well as a minimum MSE having as reference the numerical value of the symbolic calculations.

Spectral Properties of Dual Unit Gain Graphs

In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too.

An intelligent decision method of quality risk in electromagnet assembly process based on digital twin and q-rung dual hesitant fuzzy number

In order to solve the problem of online risk analysis of assembly process, an intelligent decision-making method of electromagnet assembly risk based on Digital Twin (DT) was proposed. Firstly, a DT model of electromagnet assembly process is constructed, and the potential failure modes are collected in real time at the physical layer. Secondly, in the virtual layer, real-time potential failure modes are obtained through information synchronization and transmitted to the application layer. Then, in the application layer, the optimized Failure Mode and Effects Analysis (FMEA) method is used to analyze the risk of failure mode, and the risk value of failure mode is fed back to the virtual layer. In the improved FMEA method, the q-Rung Dual Hesitant Fuzzy Number (q-RDHFN) is used as the evaluation language. The weight of experts is calculated by Bidirectional Projection Method (BPM), and the comprehensive weight of risk factors is calculated by Best and Worst Method (BWM) method, entropy weight method and Criteria Importance Through Intercriteria Correlation (CRITIC) method. Finally, the physical layer is adjusted to reduce the probability and risk of failure mode occurrence. An example of risk intelligent decision making is given to verify the objectivity and practicability of the new method.

A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers

A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers

On tertions and dual numbers

On tertions and dual numbers

Dual Markov Chain and Dual Number Matrices with Nonnegative Standard Parts

Dual Markov Chain and Dual Number Matrices with Nonnegative Standard Parts

A Review on the Applications of Dual Quaternions

This work explores dual quaternions and their applications. First, a theoretical construction begins at dual numbers, extends to dual vectors, and culminates in dual quaternions. The physical foundations behind the developed theory lie in two important fundamentals: Chasles’ Theorem and the Transference Principle. The former addresses how to represent rigid-body motion whereas the latter provides a method for operating on it. This combination presents dual quaternions as a framework for modeling rigid mechanical systems, both kinematically and kinetically, in a compact, elegant and performant way. Next, a review on the applications of dual quaternions is carried out, providing a general overview of all applications. Important subjects are further detailed, these being the kinematics and dynamics of rigid bodies and mechanisms (both serial and parallel), control and motion interpolation. Discussions regarding dual quaternions and their applications are undertaken, highlighting open questions and research gaps. The advantages and disadvantages of using dual quaternions are summarized. Lastly, conclusions and future directions of research are presented.

QR decomposition of dual matrices and its application

Dual number matrix decompositions have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices. When dealing with large-scale problems, we present the thin QR decomposition of dual number matrices, along with its algorithm with column pivoting. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Finally, we employ the QR decomposition of dual matrices to compute the DMPGI, attaining results of higher precision.

Mechanism and Application of Attitude and Orbit Coupling Dynamics for Spacecraft Proximity Relative Motion

This paper analyzes the root causes of attitude-orbit coupling effects of spacecraft proximity relative motion in space precision collaborative tasks from three aspects: mathematical representation, physical definition, and engineering applications. At first, taking mathematical representation as the context, spacecraft proximity relative motion representations such as particle relative dynamic model, extended particle relative dynamic model, and dual-spiral-based relative dynamic model are investigated in detail. On this basis, the mechanism of attitude-orbit coupling effects originating from different mathematical representations is further investigated. Second, spiral theory–based attitude-orbit coupling relative dynamics is developed. The innovation of this work is extending the dual number representation from rigid body to flexible body, which makes it possible to describe the proximity relative motion between two rigid-flexible coupling spacecraft. Third, the application value of attitude-orbit coupling relative dynamic model in precision collaborative mission such as precision formation, rendezvous and docking, space manipulation, and on-orbit assembly is provided. Finally, simulation results verify the engineering significance of the attitude-orbit coupling relative dynamic model.

The theory of screws derived from a module over the dual numbers

AbstractThe theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator $$\mathcal {E}$$ E , $$\mathcal {E}^2=0$$ E 2 = 0 , that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product ($$\mathbb {D}$$ D -module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view.

EAST PODILLYA DIALECT IN THE COLLECTIONS OF OLHA PAVLENKO "MALVY NA PRYCHELKU" AND "WHOM GOD LOVES"

The article analyses the typical East Podillya dialect features represented in the collections of the contemporary Ukrainian writer Olha Pavlenko "Malvy na prychelku" and "Whom God Loves". It is noted that, unlike other writers who mainly use only lexical dialectal elements in their works, Olha Pavlenko does not limit herself to the lexical level, but also introduces words that demonstrate phonetic and morphological features of the Eastern Podillya dialects into the canvas of the literary text. In particular, the disappearance of the correlation |р| : |р'| correlation, dislocation of [т'] in verb forms, prototic consonants [г], [й] before vowels [о], [и], epenthetic consonants, compounds [мн'], [вл'] instead of [мй], [вй], realisation of the phoneme |ф| by the sound combination [хв], truncation of word bases or engthening of foreign language bases, etc. The accentuation peculiarities of the Eastern Podillya dialects are traced in the rhyming texts included in the stories of the collections. Among the morphological features of the East Podillya dialects are the remnants of the dual number forms in nouns, the lack of differentiation of feminine and masculine nouns of the first and second cases in the singular ablative into hard, soft and mixed groups, the formation of degrees of comparison of adjectives with the help of suffixes -енн-, -езн-, -ч-, -ецьк-, adverbs of adjectival origin - with the suffix -ч-, the absence of consonant alternation in verbal personal forms, the formation of passive participles and impersonal verbs with the suffix -ян-, the use of numerals єден and штири, etc. The article emphasises that the author uses dialect words only in the speech of the characters. It is concluded that the lexemes that demonstrate the East Podillya dialect features are a linguistic means of creating a special colour of the stories from Olha Pavlenko's collections, convey the expression of the characters' speech and are stylistically motivated.

Geometry over algebras

We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric structures are employed to describe the spaces of oriented geodesics in the hyperbolic plane, the Euclidean plane, and the round 2-sphere. We also introduce a simple and natural geometric transition between these spaces. Finally, we present a projective model for the hyperbolic bidisc, that is, the Riemannian product of two hyperbolic discs.

High throughput dynamic dual entropy source true random number generator based on FPGA

True Random Number Generator (TRNG), a primitive language of hardware security, plays a significant role in key generation, data encryption, initialization vectors, and other situations. To address the problem of incompatibility between low resource overhead and high throughput of TRNG, a dual entropy source TRNG architecture based on Field Programmable Gate Array (FPGA) is proposed, whose entropy sources include clock jitter and metastability. Different from other TRNGs that use multiplexers (MUX), it also uses the ring oscillators and path delay differences to dynamically convert the oscillation frequency of the multiplexer ring oscillator (MRO), thus increasing the randomness of the entropy source. This architecture is implemented on Xilinx Artix-7 and Kintex-7 FPGAs and has been tested with NIST tests, AIS-31 tests, as well as voltage and temperature tests, and has been compared with other up-to-date TRNGs. The experimental results show that the MRO-TRNG only requires 10 LUTs, 2 DFFs, and 1 MUX, providing a throughput of up to 300 Mbps.

Dual quaternion singular value decomposition based on bidiagonalization to a dual number matrix using dual quaternion householder transformations

We propose a practical method for computing the singular value decomposition of dual quaternion matrices. The dual quaternion Householder matrix is first proposed, and by combining the properties of dual quaternions, we can implement the transformation of a dual quaternion matrix to a bidiagonalized dual number matrix. We have proven that the singular values of a dual quaternion matrix are same to the singular values of its bidiagonalized dual number matrix. Numerical experiment demonstrates the effectiveness of our proposed method for computing the singular value decomposition of dual quaternion matrices.

The Interaction Between Morphosyntactic Features and the Performance of Machine Translation Tools: The Case of Google Translate, Systran, and Microsoft Bing in English-Arabic Translation

The present study compares the performance of three machine translation tools in English-Arabic translation to answer the questions of (a) whether the three machine translation tools, Google Translate, Systran and Microsoft Bing can be ordered in a hierarchy of performance, and (b) whether they can handle lexically and structurally ambiguous sentences and garden path sentences. Using a number of constructed and selected English sentences, the morphosyntactic features of number, gender, case, definiteness, and humanness, agreement between cardinal numerals and their head nouns, lexically and structurally ambiguous sentences and garden path sentences are used to test the three machine translation tools for performance. The results show that (a) as far as handling the morphosyntactic features of subject-verb agreement in Standard Arabic, all three machine translation tools perform equally well, and no machine translation tool seems to perform significantly better than the other two (b) some marked features (e.g. dual number and humanness) of SA seem to pose some problems for machine translation tools, and (c) lexically and structurally ambiguous sentences and garden path sentences seem to be the most challenging sentences for the three machine translation tools.

Dual octonions and rigid body kinematics

In this paper, we first give the basic information about octonions and present the Euclidean rotation matrix formed by an octonion in seven‐dimensional Euclidean space. Next, we define and introduce the ‐module and dual vectors using dual numbers. Then, we provide the transformation that maps the points on the unit dual sphere one‐to‐one with the directed lines in . We also define a subset of the unit dual sphere, demonstrating that each element of this subset corresponds to two intersecting perpendicular directed lines in seven‐dimensional Euclidean space. Following that, we introduce dual octonions with their basic algebraic properties and examine rigid body (screw) motions in seven‐dimensional Euclidean space using dual octonions. Finally, we define an operator and express that this operator transforms two perpendicular intersecting directed lines in seven‐dimensional Euclidean space into two perpendicular intersecting directed lines.