Dual Numbers and Operational Umbral Methods

Behr, Nicolas; Dattoli, G.; Lattanzi, Ambra; Licciardi, Silvia

doi:10.3390/axioms8030077

Cited by 8 publications

(2 citation statements)

References 31 publications

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“…For example, is a derivative "identically zero" or the limit of a tangent with no slope? Alternatively, we might take zero as a sheer tiny gap; mind the linear algebra's dual numbers a + bε to see how to formally extend a number a by adjoining a multiple b of the nonzero differentiation unit ε with vanishing ε n for a natural number n > 1 [11].…”

Section: Scope and Rationalementioning

confidence: 99%

The Zero Delusion

Rives¹

2023

Preprint

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Zero signifies absence or an amount of no magnitude and allegedly exemplifies one of humanity’s most splendid insights. Nonetheless, it is a questionable number. Why did algebra embrace zero and dismiss infinity despite representing symmetric and complementary concepts? Why is zero exceptional in arithmetic? Is zero a “real” point? Has it a geometrical meaning? Is zero naturalistic? Digit 0 is unnecessary in positional notation (e.g., bijective numeration). The uniform distribution is unreachable, transmitting nill bits of information is impossible, and communication is never error-free. Zero is elusive in thermodynam-ics, quantum field theory, and cosmology. A minimal fundamental extent is plausible but hard to accept because of our acquaintance with zero. Mathematical zeroes are semantically void (e.g., empty set, empty sum, zero vector, zero function, unknot). Because “division by zero” and “iden-tically zero” are uncomputable, we advocate for the nonzero algebraic numbers to build new physics that reflects nature’s countable character. In a linear scale, we must handle zero as the smallest possible nonzero rational or the limit of an asymptotically vanishing sequence of rationals. Zero, as such, is a logarithmic scale’s pointer to a being’s property via log(1)). The exponential function, which decodes the encoded data back to the linear scale, is crucial to understanding the Lie algebra-group correspondence , the Laplace transform, linear fractional transformations, and the notion of conformality. Ultimately, we define a “coding space” as a doubly conformal transformation realm of zero-fleeing hyperbolic geometry that keeps universal relationships of structure and scale.

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Section: Scope and Rationalementioning

confidence: 99%

The Zero Delusion

Rives¹

2023

Preprint

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“…Following our previous study [16,17], a Euler's formula for dual numbers (see [22]) could be the following formula: 1 + ax = e ax , where a 2 = 0. The applications of this formula could be of the following type.…”

Section: Euler's Formulas For Dual Numbersmentioning

confidence: 99%

Mathematics and Poetry • Unification, Unity, Union

Nichita

2020

Sci

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We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.

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Automatic differential kinematics of serial manipulator robots through dual numbers

Orbegoso Moreno,

Valverde Ramírez

2024

TESEA

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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.

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Dual Numbers and Operational Umbral Methods

Cited by 8 publications

References 31 publications

The Zero Delusion

The Zero Delusion

Mathematics and Poetry • Unification, Unity, Union

Automatic differential kinematics of serial manipulator robots through dual numbers

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