Dual Numbers Research Articles

Implementation of Dual Number Automatic Differentiation with John Newman’s BAND Algorithm

This paper asserts that the development of continuum-scale mathematical models utilizing John Newman’s BAND subroutine can be simplified through the use of dual number automatic differentiation. This paper covers the salient features of the BAND algorithm as well as dual numbers and how they can be leveraged to algorithmically linearize systems of partial differential equations; these two concepts can be combined to produce accurate and computationally efficient models while significantly reducing the amount of personnel time necessary by eliminating the time-consuming process of equation linearization. As a result, this methodology facilitates more rapid model prototyping and establishes a more intuitive relationship between the numerical model and the differential equations. By utilizing an existing and validated programming module, dnadmod, these advantages are achieved without burdening the general user with significant additional programming overhead.

Optimization of Perturbed Spacecraft Trajectories Using Complex Dual Numbers. Part 2: Numerical Results

Optimization of Perturbed Spacecraft Trajectories Using Complex Dual Numbers. Part 2: Numerical Results

Kinematic Applications of Hyper-Dual Numbers

Hyper-dual numbers are a new number system that is an extension of dual numbers. A hyper-dual number can be written uniquely as an ordered pair of dual numbers. In this paper, some basic algebraic properties of hyper-dual numbers are given using their ordered pair representaions of dual numbers. Moreover, the geometric interpretation of a unit hyper-dual vector is given in module as a dual line. And a geometric interpretation of a subset of unit hyper-dual sphere (the set of all unit hyper-dual vectors) is given as two intersecting perpendicular lines in 3-dimensional real vector space.

Equivalences of (co)module algebra structures over Hopf algebras

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. We apply support equivalence in the study of the asymptotic behaviour of codimensions of H-identities and, in particular, to the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary polynomial identities. As an example we prove this analogue for all unital H-module structures on the algebra $F[x]/(x^2)$ of dual numbers.

Application of Generalized (Hyper-) Dual Numbers in Equation of State Modeling

The calculation of derivatives is ubiquitous in science and engineering. In thermodynamics, in particular, state properties can be expressed as derivatives of thermodynamic potentials. The manual differentiation of complex models can be tedious and error-prone. In this work, we revisit dual and hyper-dual numbers for the calculation of exact derivatives and show generalizations to higher order derivatives and derivatives with respect to vector quantities. The methods described in this paper are accompanied by an open source Rust implementation with Python bindings. Applications of the generalized (hyper-) dual numbers are given in the context of equation of state modeling and the calculation of critical points.

Polynomial functions on rings of dual numbers over residue class rings of the integers

AbstractThe ring of dual numbers over a ringRisR[α] =R[x]/(x2), whereαdenotesx+ (x2). For any finite commutative ringR, we characterize null polynomials and permutation polynomials onR[α] in terms of the functions induced by their coordinate polynomials (f1,f2∈R[x], wheref=f1+αf2) and their formal derivatives onR.We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on ℤpn[α] forn≤p(pprime).

Diagrammatic Differentiation for Quantum Machine Learning

We introduce diagrammatic differentiation for tensor calculus by generalising the dual number construction from rigs to monoidal categories. Applying this to ZX diagrams, we show how to calculate diagrammatically the gradient of a linear map with respect to a phase parameter. For diagrams of parametrised quantum circuits, we get the well-known parameter-shift rule at the basis of many variational quantum algorithms. We then extend our method to the automatic differentation of hybrid classical-quantum circuits, using diagrams with bubbles to encode arbitrary non-linear operators. Moreover, diagrammatic differentiation comes with an open-source implementation in DisCoPy, the Python library for monoidal categories. Diagrammatic gradients of classical-quantum circuits can then be simplified using the PyZX library and executed on quantum hardware via the tket compiler. This opens the door to many practical applications harnessing both the structure of string diagrams and the computational power of quantum machine learning.

De Moivre’s and Euler Formulas for Matrices of Hybrid Numbers

It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the 4×4 matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas.

Hybrid Leonardo numbers

Until today, many researchers have studied related to hybrid numbers which are a generalization of complex, hyperbolic and dual numbers. In this paper, using the Leonardo numbers, we introduce the hybrid Leonardo numbers. Also, we give some algebraic properties of the hybrid Leonardo numbers such as recurrence relation, generating function, Binet’s formula, sum formulas, Catalan’s identity and Cassini’s identity.

Optimization of Perturbed Spacecraft Trajectories Using Complex Dual Numbers. Part 1: Theory and Method

Optimization of Perturbed Spacecraft Trajectories Using Complex Dual Numbers. Part 1: Theory and Method

A novel stochastic group decision-making framework with dual hesitant fuzzy soft set for resilient supplier selection

Suppliers can be regarded as unavoidable sources of external risks in modern supply chains, which may cause disruption of supply chains. A resilient supplier usually has a high adaptive ability to reduce the vulnerability against disruptions and recover from disruption to keep continuity in operations. This paper develops an effective multi-attribute group decision-making (MAGDM) framework for resilient supplier selection. Because of the many uncertainties in resilient supplier selection, the dual hesitant fuzzy soft sets (DHFSSs), a very flexible tool to express uncertain and complex information of decision-makers, is utilized to cope with it. In order to obtain the resilient supplier’s partial order relationship and consider the psychological behavior of decision-makers, this paper proposes the MAGDM framework with DHFSSs based on the TOPSIS method and prospect theory for resilient supplier selection. Furthermore, we consider the consensus level among experts of different backgrounds and experiences and propose a consensus measure method based dual hesitant fuzzy soft numbers (DHFSNs) before selecting a resilient supplier. The expert weights are calculated by the group consensus level between the expert and the group opinions. Meanwhile, we define the entropy of DHFSSs to determine the attribute weights objectively in the decision-making process. Based on this, the proposed method is applied to a practical manufacturing enterprise with an international supply chain for a resilient supplier selection problem. Finally, by performing a sensitivity analysis and a comparative analysis, the results demonstrate the robustness and validity of the proposed method.

A combined approach to Perrin and Padovan hybrid sequences

Recently, there has been huge interest to a new numeric set, which brings together three numerical systems: complex, hyperbolic and dual numbers, called as hybrid number. Motivated and based on the importance of this system, we apply the definition of hybrid numbers to Perrin numbers and introduce Perrin hybrid numbers and examine some properties of them. In other words, we obtain their some properties such as Binet-like-formula, partial sums, generating functions, etc. Also we investigate some similar properties with Padovan hybrid sequence. Finally, we give a source Maple 13 code.

Effects Of Dual Task And Repetition On Five-times Sit-to-Stand Task In Healthy Young Adults

The Sit-to-Stand (STS) task is an essential activity of independent daily living. It is commonly transferred into the clinic as five-times STS task (FTSTS) for assessment and prognosis of balance and muscle efficiency in the lower extremities of various populations. Research has shown that for activities such as the STS task, a certain level of attentional demand is required to perform the multiple components of a transitional movement. However, to our knowledge, the changes that occur in body mechanics with greater repetitions and the effects of dual tasking while performing FTSTS are currently unknown. PURPOSE: To determine the effects of dual tasking and multiple repetitions on the FTSTS task in healthy, young adults. METHODS: 10 healthy adults (mean age 24 (4.1) years) participated. Each participant stood up and sat down five times as fast as possible as part of FTSTS assessment. Three trials were completed without a cognitive task (SingleTask) and then with a concurrent cognitive task of counting backwards by 3 starting from a random number (DualTask). Time to complete FTSTS was measured. Impulse (Ns/BW), peak force (N/BW), and power (Nm/BW.s) variables were calculated using ground reaction force measurements. A 2-way ANOVA and paired samples t-test were conducted. RESULTS: Participants took significantly longer to complete FTSTS while dual tasking (8.16 [1.77]s) compared to SingleTask (7.38 [1.08]; p = .05). Concentric impulse significantly increased from 0.55 (0.02) during SingleTask to 0.59 (0.03) during DualTask (p = .022), however, power significantly decreased from 0.99 (0.04) during SingleTask to 0.92 (0.05) during DualTask (p = .017). FTSTS concentric, eccentric, and total impulse significantly increased from 1st to 5th repetition respectively: 0.56 (0.03) to 0.59 (0.03; p = .005), 0.49 (0.03) to 0.56 (0.04; p = .013), 1.05 (0.05) to 1.15 (0.06; p = .006). Also, standing peak force significantly decreased from 1st repetition (1.39 [0.03]) to 5th repetition (1.34 [0.03]; p = .004). The mean peak force standing decreased more from 1st to 5th repetition under SingleTask (1.39 [0.04] to 1.32 [0.03]) compared to DualTask (1.39 [0.03] to 1.35 [0.03]; p = .044). CONCLUSION: The force characteristics are altered by both dual tasking and number of repetitions during the FTSTS task in healthy, young adults.

Decision-making model of machine tool remanufacturing alternatives based on dual interval rough number clouds

Remanufacturing of machine tools has become one of the crucial topical subjects as it can effectively reduce the manufacturing cost compared with traditional new product development. The assessment and selection of different remanufacturing alternatives are vital for successfully implementing the remanufacturing of machine tools. Although numerous fuzzy theory-based methods have been developed to evaluate and select the reasonable remanufacturing alternatives in the machine tool remanufacturing process, they still have some drawbacks, such as requiring extra assumptions, ignoring the impact of internal relationships among distinct criteria, and lacking the mechanism of manipulating diverse uncertain information. To cover such limitations, this study develops an applicable decision-making method to assist managers in extracting the essential remanufacturing alternatives for product improvement. First, a new concept named dual interval rough number clouds (DIRNCs) is developed by combining the merit of interval rough numbers (IRNs) and interval cloud model in handling uncertain information. Then, an applicable decision-making support model of remanufacturing alternatives is constructed based on DIRNCs, two non-linear weighting methods, and a technique for order performance by similarity to ideal solution (TOPSIS). Finally, an assessment of remanufacturing alternatives of machine tools is presented to illustrate the model, whose effectivity is demonstrated by comparing extant models. Results display that the developed DIRNC-based model is more advantageous than triangular fuzzy-based, cloud model-based, fuzzy rough-based, and IRN-based methods.

Dual-complex quaternion representation of gravitoelectromagnetism

In this paper, we propose the generalized description of electromagnetism and linear gravity based on the combined dual numbers and complex quaternion algebra. In this approach, the electromagnetic and gravitational fields can be considered as the components of one combined dual-complex quaternionic field. It is shown that all relations between potentials, field strengths and sources can be formulated in the form of compact quaternionic differential equations. The alternative reformulation of equations of gravitoelectromagnetism based on formalism of [Formula: see text] matrices is also discussed. The results reveal the similarity and isomorphism of distinctive algebraic structures.

Multidual Sensitivity Method in Ray-Tracing Transport Simulations

The multidual differentiation method has been implemented in a ray-tracing transport simulation for the purpose of calculating arbitrary-order sensitivities of the uncollided particle leakage. This method extends dual number differentiation by perturbing variables along multiple nonreal axes to calculate arbitrary-order derivatives. Numerical results of first-through third-order multidual sensitivities of the uncollided particle leakage with respect to isotope densities, microscopic cross sections, source emission rates, and material interface locations (including the outer boundary) are shown for a two-region sphere. The relative error of first and second partial derivatives with respect to isotopic parameters and first partial derivatives of the leakage with respect to interface locations are within 9.8E−10% of existing adjoint-based sensitivities. Higher-order multidual-based derivatives that are not available with the adjoint method are in excellent agreement with central difference approximations.

Dual-complex generalized k-Horadam numbers

The purpose of this paper is to provide a broad overview of the generalization of the various dual-complex number sequences, especially in the disciplines of mathematics and physics. By the help of dual numbers and dual-complex numbers, in this paper, we define the dual-complex generalized k-Horadam numbers. Furthermore, we investigate the Binet formula, generating function, some conjugation identities, summation formula and a theorem which is generalization of the Catalan's identity, Cassini's identity and d'Ocagne's identity.

Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras

We prove a recent conjecture by Gyenge, Némethi, and Szendrői giving a formula of the generating function of Euler numbers of Hilbert schemes of points HilbN(C2∕Γ) on a simple singularity C2∕Γ, where Γ is a finite subgroup of SL(2). We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with Γ at ζ=exp(2π −1 2(h∨+1)) are always 1, which is a special case of an earlier conjecture by Kuniba. Here h∨ is the dual Coxeter number. We also prove the claim, which was not known for E7, E8 before.

On the group of unit-valued polynomial functions

Let R be a finite commutative ring. The set {{mathcal{F}}}(R) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units {{mathcal{F}}}(R)^times is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on R[x]/(x^2)=R[alpha ], the ring of dual numbers over R, and show that the group {mathcal{P}}_{R}(R[alpha ]), consisting of those polynomial permutations of R[alpha ] represented by polynomials in R[x], is embedded in a semidirect product of {{mathcal{F}}}(R)^times by the group {mathcal{P}}(R) of polynomial permutations on R. In particular, when R={mathbb{F}}_q, we prove that {mathcal{P}}_{{mathbb{F}}_q}({mathbb{F}}_q[alpha ])cong {mathcal{P}}({mathbb{F}}_q) ltimes _theta {{mathcal{F}}}({mathbb{F}}_q)^times . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo {p^n} and obtain canonical representations for these functions.

Open momentum space method for the Hofstadter butterfly and the quantized Lorentz susceptibility

We develop a generic $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ open momentum space method for calculating the Hofstadter butterfly of both continuum (moir\'e) models and tight-binding models, where the quasimomentum is directly substituted by the Landau level (LL) operators. By taking a LL cutoff (and a reciprocal lattice cutoff for continuum models), one obtains the Hofstadter butterfly with in-gap spectral flows. For continuum models such as the moir\'e model for twisted bilayer graphene, our method gives a sparse Hamiltonian, making it much more efficient than existing methods. The spectral flows in the Hofstadter gaps can be understood as edge states on a momentum space boundary, from which one can determine the two integers (${t}_{\ensuremath{\nu}},{s}_{\ensuremath{\nu}}$) of a gap $\ensuremath{\nu}$ satisfying the Diophantine equation. The spectral flows can also be removed to obtain a clear Hofstadter butterfly. While ${t}_{\ensuremath{\nu}}$ is known as the Chern number, our theory identifies ${s}_{\ensuremath{\nu}}$ as a dual Chern number for the momentum space, which corresponds to a quantized Lorentz susceptibility ${\ensuremath{\gamma}}_{xy}=eB{s}_{\ensuremath{\nu}}$.