Algebra Of Complex Numbers Research Articles

Application of complex number algebra in plane geometry

Application of complex number algebra in plane geometry

A Size-Dependent Viscoelastic Model for Microbars with Variable Cross-Section

This paper presents a model for microbars with variable cross-sections using the Kelvin–Voigt model for viscoelastic material, accounting for size-dependent effects based on strain gradient theory. The size-dependent dynamic equations for the rod, which consider the variable cross-sectional area, are obtained through the extended Hamilton’s principle. These equations are then reduced in order using the Galerkin method and solved in the steady state using the harmonic response form and the algebra of complex numbers. To solve the equations from the transient state to the steady state, a combined method is implemented using the Grünwald–Letnikov derivative technique and the Newmark method. Furthermore, a model and analysis based on the finite element method are presented to validate the results. In the results section, various factors such as size-dependent effects, the order of the fractional derivative, the amount of the viscoelastic coefficient, and the shape of the section area are examined through the time history graph, frequency response, and maximum displacement in terms of force. The results demonstrate that the transient response converges to the stable response after a certain period of time. Moreover, it is observed that decreasing the order of the fractional derivative in the pre-resonance range leads to a decrease in response sensitivity, while in the resonance frequency range, the sensitivity increases with the increase in order.

О группе автоморфизмов алгебры плюральных чисел

The algebra of dual numbers was first introduced by V. K. Clifford in 1873. The algebras of plural and dual numbers are analogous to the algebra of complex numbers. Dual numbers form an algebra, but not a field, because only dual numbers with a real part not equal to zero have an inverse element. In this work, automorphisms of algebras of plural numbers, which are a generalization of the algebra of dual numbers, are studied. Algebras of plural numbers were in the center of attention of the professor of Kazan University A. P. Shirokov. Studying the geometry of higher-order tangent bundles, he established that higher-order tangent bundles over smooth manifolds have the structure of a smooth manifold over algebras of plural numbers. This allowed him in the 70s of the twentieth century to construct a theory of lifts of tensor fields and linear connections from a smooth manifold to its tangent bundles of arbitrary order. In this paper, we study automorphisms of the algebra of plural numbers. It is proved that the set of all automorphisms of the algebra of plural numbers forms a group. The structure of this group is described. The groups of automorphisms of the algebra of plural numbers with small dimension are indicated as examples.

Процессная семантика комплексных чисел

Perspective approaches to imitation of natural thinking build on waveand quantumlike models, overcoming the limitations of Boolean logic and classical probability calculus. Further development of these methods requires an explanation of the reasons for these advantages. This paper shows that such a reason is the affinity of fundamental principles of natural cognition to the algebra of complex numbers. The vector structure of these numbers, in particular, matches the universal cognitive model of processes in nature. First, different phases of the complex number correspond to the phases "novelty", "goal", "action", and "result" of the life cycle of the considered activity. Real and imaginary axes of the complex plane then describe the factual and potential dimensions of the process's result. Simplicity and universality allow this model to avoid the major difficulties of modern approaches to predictive coding of cognitive information. These features open ways to algorithmic imitation of process thinking and the development of naturelike computational systems.

Gysin sequences and SU(2)‐symmetries of C∗‐algebras

Motivated by the study of symmetries of C*-algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz-Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K-theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK-equivalent to the algebra of complex numbers. In this way, we obtain a six term exact sequence of K-groups containing a noncommutative analogue of the Euler class.

Butterfly spirals

In this article special sequences involving the Butterfly theorem are defined. The Butterfly theorem states that if M is the midpoint of a chord PQ of a circle, then following some definite instructions, it is possible to get two other points X and Y on PQ, such that M is also the midpoint of the segment XY. The convergence investigation of those sequences reveals bounded piecewise spirals, and is done by using the free dynamic mathematics software GeoGebra. It is also supported with rigorous reasoning concerning the convergence of a special sequence. Moreover, it raises related problems that encourage further investigation. This article suggests a content that can be useful for students at secondary as well as undergraduate levels since it incorporates principles taught to students during their first courses in Algebra, Calculus and Geometry especially those related to convergent sequences, the algebra of complex numbers, plane isometries and basic Euclidean geometry.

Dimensional Synthesis of Watt-I Metal Mechanism based on Motion Generation

In industrial automation, there is always increasing demand of generating desired motion with accuracy and precision. It is possible when the mechanism generates motion for large number of precision points. Therefore, in present work, the dimensional synthesis of a 6-bar, Watt-I mechanism has been carried for twenty coupler positions. The single degree of freedom mechanism has revolute pairs at each of its joint and is able to generate required motion. All the links of the mechanism are made of metal. The dyad-triad design equations are formed in terms of complex number algebra. To solve these equations, a MATLAB code is generated and utilized to yield the dimensional parameters i.e. lengths and angular orientations of all links of the mechanism. The complete synthesis methodology is modeled and demonstrated on a numerical problem. To verify the outcomes, SAM 7.0 application software has been incorporated and applied on the given example.

The role and significance of computer animation in school algebra of complex numbers

The article discusses computer technologies in teaching mathematics, which is relevant in the light of the digitalization of education. The role and importance of computer animation in algebra lessons in the study of complex numbers in the tenth grade in accordance with the current school textbooks are presented. Animation drawings can be used as demonstration drawings when studying the relevant topic, as a tool for testing knowledge, they provide an opportunity for experimentation, support the research style of developing learning. The use of animated drawings effectively replenishes the arsenal of teaching aids for a practicing mathematics teacher. This allows you to eliminate unnecessarily burdensome computational difficulties, to visualize mathematical concepts and statements, turning some of them into obviously correct ones, to facilitate the teacher's work in compiling tasks of the same type with different data. The management of the student's educational and research work on the manufacture of animation-geometric models of objects and phenomena contributes to the successful assimilation of knowledge by students, stimulates an experimental research approach to teaching.

On the invertibility of solutions of first order linear homogeneous differential equations in Banach algebras

This work is devoted to the study of some properties of linear homogeneous differential equations of the first order in Banach algebras. It is found (for some types of Banach algebras), at what right-hand side of such an equation, from the invertibility of the initial condition it follows the invertibility of its solution at any given time. Associative Banach algebras over the field of real or complex numbers are considered. The right parts of the studied equations have the form [F(t)](x(t)), where {F(t)} is a family of bounded operators on the algebra, continuous with respect to t∈R. The problem is to find all continuous families of bounded operators on algebra, preserving the invertibility of elements from it, for a given Banach algebra. In the proposed article, this problem is solved for only three cases. In the first case, the algebra consists of all square matrices of a given order. For this algebra, it is shown that all continuous families of operators, preserving the invertibility of elements from the algebra at zero must be of the form [F(t)](y)=a(t)⋅y+y⋅b(t), where the families {a(t)} and {b(t)} are also continuous. In the second case, the algebra consists of all continuous functions on the segment. For this case, it is shown that all families of operators, preserving the invertibility of elements from the algebra at any time must be of the form [F(t)](y)=a(t)⋅y, where the family {a(t)} is also continuous. The third case concerns those Banach algebras in which all nonzero elements are invertible. For example, the algebra of complex numbers and the algebra of quaternions have this property. In this case, any continuous families of bounded operators preserves the invertibility of the elements from the algebra at any time. The proposed study is in contact with the research of the foundations of quantum mechanics. The dynamics of quantum observables is described by the Heisenberg equation. The obtained results are an indirect argument in favor of the fact, that the known form of the Heisenberg equation is the only correct one.

A Matrix-Vector Construction of the Algebra of Complex Numbers

The aim of this paper is to describe an alternative way to think about the algebra of complex numbers that may be of pedagogical value for introducing related concepts such as linear transformations and convolutions. The method is to define a fixed linear transformation of complex numbers represented in vector form so that products can be evaluated elementwise in the transformed space. The principal results are concrete demonstrations that this can in fact be accomplished.

Pole-placement Predictive Functional Control for under-damped systems with real numbers algebra

This paper presents the new algorithm of PP-PFC (Pole-placement Predictive Functional Control) for stable, linear under-damped higher-order processes. It is shown that while conventional PFC aims to get first-order exponential behavior, this is not always straightforward with significant under-damped modes and hence a pole-placement PFC algorithm is proposed which can be tuned more precisely to achieve the desired dynamics, but exploits complex number algebra and linear combinations in order to deliver guarantees of stability and performance. Nevertheless, practical implementation is easier by avoiding complex number algebra and hence a modified formulation of the PP-PFC algorithm is also presented which utilises just real numbers while retaining the key attributes of simple algebra, coding and tuning. The potential advantages are demonstrated with numerical examples and real-time control of a laboratory plant.

Approximating Planar Conformal Maps Using Regular Polygonal Meshes

AbstractContinuous conformal maps are typically approximated numerically using a triangle mesh which discretizes the plane. Computing a conformal map subject to user‐provided constraints then reduces to a sparse linear system, minimizing a quadratic ‘conformal energy’. We address the more general case of non‐triangular elements, and provide a complete analysis of the case where the plane is discretized using a mesh of regular polygons, e.g. equilateral triangles, squares and hexagons, whose interiors are mapped using barycentric coordinate functions. We demonstrate experimentally that faster convergence to continuous conformal maps may be obtained this way. We provide a formulation of the problem and its solution using complex number algebra, significantly simplifying the notation. We examine a number of common barycentric coordinate functions and demonstrate that superior approximation to harmonic coordinates of a polygon are achieved by the Moving Least Squares coordinates. We also provide a simple iterative algorithm to invert barycentric maps of regular polygon meshes, allowing to apply them in practical applications, e.g. for texture mapping.

Complex Transfinite Barycentric Mappings with Similarity Kernels

AbstractTransfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define interpolants of values given on a smooth planar contour. When the data is two‐dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some of the basic complex‐valued algebra needed to describe these planar maps, and use it to define similarity kernels, a natural alternative to the usual barycentric kernels.We develop the theory behind similarity kernels, explore their properties, and show that the transfinite versions of the popular three‐point barycentric coordinates (Laplace, mean value and Wachspress) have surprisingly simple similarity kernels.We furthermore show how similarity kernels may be used to invert injective transfinite barycentric mappings using an iterative algorithm which converges quite rapidly. This is useful for rendering images deformed by planar barycentric mappings.

Algebra and geometry of Hamilton’s quaternions

Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three-dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.

A geometric algebra reformulation and interpretation of Steinmetz’s symbolic method and his power expression in alternating current electrical circuits

Developed more than a century ago, Steinmetz’s symbolic method is still puzzling us. It puzzles us because, in spite of its theoretical inconsistencies, it is heuristically efficient. However, it remains the dominant method in design, analysis, and operation of electrical power networks. The paper shows that Steinmetz’s mathematical expression for electrical power is based on assumptions inconsistent with the algebra of complex numbers. The paper argues that, although the numbers are correct, the mathematical interpretation of these numbers is not. Steinmetz got empirical right results for wrong conceptual reasons; the success of the symbolic method is based on the fact that, unwittingly, Steinmetz rediscovered Grassmann–Clifford geometric algebra. The paper challenges the dominant paradigm in power theory which represents voltage, current, active, reactive and apparent power as complex numbers and/or vectors (phasors). The author proposes a new paradigm in which these entities are represented as an algebraic group; the group is composed of a scalar, two vectors and a bivector which are residing in a four-dimensional algebraic space and in a two-dimensional Euclidean geometric space. The paper claims that Steinmetz’s symbolic method is the oldest engineering application of Clifford Algebra. The paper provides a strong motivation for a new didactic of power theory based on Geometric Algebra as Physics’ unifying language.

Spherical tensor multipolar electrostatics and smooth particle mesh Ewald summation: a theoretical study.

The point-charge approximation, typically used by classical molecular mechanics force-fields, can be overcome by a multipolar expansion. For decades multipole moments were only used in the context of the rigid body approximation but recently it has become possible to combine multipolar electrostatics with molecular flexibility. The program DL_MULTI, which is derived from DL_POLY_2, includes efficient multipolar Ewald functionality up to the hexadecapole moment but the code is restricted to rigid bodies. The incorporation of flexibility into DL_MULTI would cause too large an impact on its architecture whereas the package DL_POLY_4 offers a more attractive and sustainable route to handle multipolar electrostatics. This package inherently handles molecular flexibility, which warrants sufficiently transferable atoms or atoms that are "knowledgeable" about their chemical environment (as made possible by quantum chemical topology and machine learning). DL_MULTI uses the spherical multipole formalism, which is mathematically more involved than the Cartesian one but which is more compact. DL_POLY_4 uses the computationally efficient method of smooth particle mesh Ewald (SPME) summation, which has also been parallellized by others. Therefore, combining the strengths of DL_POLY_4 and DL_MULTI poses the challenge of merging SPME with multipolar electrostatics by spherical multipole. In an effort to recast as clearly as possible the principles behind DL_MULTI, its key equations have been reformulated by the more streamlined route involving the algebra of complex numbers, and some of these equations' peculiarities clarified. This article explores theoretically the repercussions of the merging of SPME with spherical multipole electrostatics (as implemented in DL_MULTI). Difficulties in design and implementation of possible future code are discussed.

Is the kinematics of special relativity incomplete?

A thorough analysis of composite inertial motion (relativistic sum) within the framework of special relativity leads to the conclusion that every translational motion must be the symmetrically composite relativistic sum of a finite number of quanta of velocity. It is shown that the resulting spacetime geometry is Gaussian and the four-vector calculus to have its roots in the complex-number algebra. Furthermore, this results in superluminality of signals travelling at or nearly at the canonical velocity of light between rest frames even if resting to each other.

Advantages of Geometric Algebra Over Complex Numbers in the Analysis of Networks With Nonsinusoidal Sources and Linear Loads

An alternative circuit analysis technique is used to study networks with nonsinusoidal sources and linear loads. In contrast to the technique developed by Steinmetz, this method is supported by geometric algebra instead of the algebra of complex numbers, uses multivectors in place of phasors and is performed in the GN domain instead of the frequency domain. The advantages of this method over the present technique involve: determining the flow of current and power quantities in the circuit, validating the results using the principle of conservation of energy, discerning and revealing other forms of reactive power generation, and the ability to design compensators with great flexibility. The power equation is composed of the active power and the CN -power representing the nonactive power. All the CN-power terms are sorted into reactive power terms due to phase shift, reactive power terms due to harmonic interactions and degrading power terms which determine the new quantity called degrading power. This decomposition shows that estimating these quantities is intricate. It also displays the power equation's functionality for power factor improvement. The geometric addition of power quantities is not pre-established but results from applying the established norm and yields the new quantity called net apparent power.

On locally A-convex H*-algebras

We endow any proper A-convex H*-algebra (E, τ) with a locally pre-C*-topology. The latter is equivalent to that introduced by the pre C*-norm given by Ptak function when (E, τ) is a Q-algebra. We also prove that the algebra of complex numbers is the unique proper locally A-convex H*-algebra which is barrelled and Q-algebra.

Signature of gravity in conic sedenions

A method was demonstrated earlier on how extension of complex number algebra using nonreal square roots of +1 could potentially aid mathematical description of physical law, by transitioning between different geometries through genuine hypernumber rotation. The Dirac equation in physics can be expressed on hyperbolic octonion algebra and then transformed into a counterpart on circular octonions, by means of conic sedenions as unifying number concept. This paper examines potential applicability of this approach by calculating eigenvectors and Green’s function of the circular octonion counterpart to the classical Dirac equation. The results exhibit behavior that one might expect from a quantum gravitational primitive.