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The instantaneous reactive power theory (IRPT) has long been utilized for load compensation through active power line conditioners (APLCs). IRPT effectively divides the current vector into two components: the instantaneous power current and the instantaneous reactive current. Among these components, only the former is necessary to transfer instantaneous real power to the load. However, the commonly adopted approach faces challenges when extended to multi-phase systems and their compensation strategy design. This challenge is particularly pertinent today due to the consensus that treating a three-phase system with a neutral wire as a four-conductor system is more appropriate. This paper introduces a formulation of IRPT tailored for multi-phase systems within the framework of geometric algebra (GA). GA is a mathematical structure that defines a single power variable, the instantaneous power multivector, encompassing both instantaneous real power and instantaneous reactive power within a unified mathematical entity. The current components can be directly derived from this power multivector. Additionally, this paper establishes a connection with the original p-q formulations and lays the foundations for time-instantaneous compensation (TIC) and time-average compensation (TAC). Finally, to validate the proposed model, simulation and experimental results from a three-phase four-wire industrial system using a novel approach are presented.

This paper proposes a systematic method for the identification of the load circuit parameters (say the $\boldsymbol{R}$ , $\boldsymbol{L}$ , and $\boldsymbol{C}$ elements) based only on the information of the instantaneous voltage and current measured at the point of common coupling (pcc). Geometric Algebra (GA) and concepts of differential geometry are used to produce a rigorous mathematical framework. The identification is formulated as a multidimensional geometrical problem that is solved conveniently by means of GA. Once the passive elements of the load have been identified, the active and reactive powers can be computed from first electromagnetic principles (Maxwell Equations). The theory is general and is verified with linear and nonlinear circuits. The paper shows single-phase circuits but the theory can be extended to three-phase circuits. The method is easy to program and has shown to be very robust for all tested cases. Because of its generality, the method presented will find applications beyond electric circuits. [ABSTRACT FROM AUTHOR]

Arm-and-hand tracking by technological means allows gathering data that can be elaborated for determining gesture meaning. To this aim, machine learning (ML) algorithms have been mostly investigated looking for a balance between the highest recognition rate and the lowest recognition time. However, this balance comes mainly from statistical models, which are challenging to interpret. In contrast, we present $\mu C^{1}$ and $\mu C^{2}$ , two geometric model-based approaches to gesture recognition which support the visualization and geometrical interpretation of the recognition process. We compare $\mu C^{1}$ and $\mu C^{2}$ with respect to two classical ML algorithms, k-nearest neighbor (k-NN) and support vector machine (SVM), and two state-of-the-art (SotA) deep learning (DL) models, bidirectional long short-term memory (BiLSTM) and gated recurrent unit (GRU), on an experimental dataset of ten gesture classes from the Italian Sign Language (LIS), each repeated 100 times by five inexperienced non-native signers, and gathered with wearable technology (a sensory glove and inertial measurement units). As a result, we achieve a compromise between high recognition rates ($>90\%$) and low recognition times ($ < 0.1 {\mathrm{ s}}$) that is adequate for human–computer interaction. Moreover, we elaborate on the algorithms’ geometric interpretation based on geometric algebra, which supports some understanding of the recognition process. [ABSTRACT FROM AUTHOR]

This paper presents a new framework for a systematic and thorough generalization of the most well-known instantaneous transformations used in electrical engineering for power systems analysis and computing through geometric principles based on the language of Geometric Algebra. By introducing the concepts of Kirchhoff Vector and Kirchhoff Subspace, a new generalized transformation is presented. Thus, it is shown how the Clarke, Park or Hyper-Space vector transformations (widely used in electrical engineering) are particular cases of this unifying framework. Moreover, a generalization to an arbitrary number of phases is achieved. In order to be as close as possible to the geometrical intuition, all the underlying ideas are presented by means of spatial-like conceptualizations, substantiated by their corresponding algebraic formulation. This proposal has potential uses in a wide range of power system applications such as electrical machines, current compensation, power quality, electronic converters or transmission lines. Preliminary results show the superior efficiency of the method compared to matrix methods. Some real-world examples have been included to highlight the potential use of the method.

In this paper, I respond to a critique of one of my papers previously published in this journal, entitled “Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space.” That paper presents a local-realistic model of quantum correlations based on a quaternionic 3-sphere, taken as a physical space in which we are confined to perform all our experiments. The critique, on the other hand, considers two entirely different models within a flat Euclidean space, neither related to my quaternionic 3-sphere model. It then criticizes its own flat space models and claims that it has thereby criticized the model presented in my paper. Along the way, without providing evidence or proof, it claims that the results in my paper are based on mistakes. I demonstrate that there are no mistakes of any kind in my paper. On the contrary, I bring out a number of elementary mathematical and conceptual mistakes from the critique and the critiques it relies on.

I point out critical errors in the article “Dr. Bertlmann’s Socks in a Quaternionic World of Ambidextral Reality” by J. Christian, published in IEEE ACCESS. Christian’s model does not generate the singlet correlations but in fact simply reproduces the Bertlmann effect. John Bell’s colleague Reinhold Bertlmann of CERN, in his younger days, always wore one pink and one blue sock, at random. The moment you saw his left foot, you knew what colour sock would be on his right foot. Action at a distance? As John Bell liked to explain, quantum entanglement cannot be explained away in such an easy way. Yet Christian’s model assigns the two particles of the EPR-B experiment an equal and opposite spin at the source, the choice being determined by a fair coin toss. However they are measured, these spins are recovered. Christian’s computer simulation works by not actually simulating his model at all but by almost directly tracing the negative cosine built into his computer algebra package. Bell’s theorem has not been disproved. Debate as to what it means for the foundations of physics as well as for quantum information engineering (quantum communication, computation) is more lively today than ever before. A possible role for Geometric Algebra is still wide open and deserves further investigation, informed by a proper understanding of the mathematical content of Bell’s theorem.

In this paper, I respond to a critique of one of my papers previously published in this journal, entitled ‘Dr. Bertlmann’s socks in a quaternionic world of ambidextral reality.’ The geometrical framework presented in my paper is based on a quaternionic 3-sphere, or $S^{3}$ , taken as a model of the physical space in which we are inescapably confined to perform all our experiments. The framework intrinsically circumvents Bell’s theorem by reproducing the singlet correlations local-realistically, without resorting to backward causation, superdeterminism, or any other conspiracy loophole. In this response, I demonstrate point by point that, contrary to its claims, the critique has not found any mistakes in my paper, either in the analytical model of the singlet correlations or in its event-by-event numerical simulation based on Geometric Algebra.

Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020. After a brief introduction of GA, the applications of GA are reviewed across many fields. We discuss the characteristics of the applications of GA to various problems of computer science and engineering. In addition, the challenges and prospects of various applications proposed by many researchers are analyzed. We analyze the developments using GA in image processing, computer vision, neurocomputing, quantum computing, robot modeling, control, and tracking, as well as improvement of computer hardware performance. We believe that up to now GA has proven to be a powerful geometric language for a variety of applications. Furthermore, there is evidence that this is the appropriate geometric language to tackle a variety of existing problems and that consequently, step-by-step GA-based algorithms should continue to be further developed. We also believe that this extensive review will guide and encourage researchers to continue the advancement of geometric computing for intelligent machines.

In this paper, critical errors found in the paper “Bell’s theorem versus local realism in a quaternionic model of physical space,” by J. Christian, published in IEEE Access, are pointed out. The above paper, in fact, contains several conflicting models. None of them form counterexamples to Bell’s theorem. Most of Christian’s paper is devoted to a model based on the detection loophole due to Pearle (1970).

For color image fusion, existing traditional algorithms regard each color image pixel as a scalar which may lose the relationships among spectral channels. In this paper, a novel multi-focus image fusion algorithm using geometric algebra based Discrete Fourier Transform (GA-DFT) is proposed. First, a novel Discrete Fourier Transform based on geometric algebra (GA) is provided. Second, the proposed algorithm represents the color image as a GA multivector by mapping the RGB channels into the GA space. Third, the variance is selected as a contrast measure to fuse the source images in GA space. The experimental results demonstrate that the proposed algorithm can achieve the comparable performance with its counterparts in terms of subjective quality evaluation, and outperform most of the traditional algorithms in terms of objective quality metrics, indicating good performance in image fusion applications.

At present, regions of the same class determined by Support Vector Machines (SVM) classifier, Support Vector Domain Description (SVDD) classifier and Deep Learning (DL) classifier may occupy regions of other classes or unknown classes in feature space. There exists a risk that samples of other classes or unknown classes are wrongly classified as a known class. In this paper, the Support Vector Domain Tightly Wrapping Description Design (SVDTWDD) method with appropriate rejection regions and the corresponding incremental learning algorithm are proposed to overcome the above problem. The main work includes: (1) We develop a construction algorithm of the tightly wrapping set for the homogeneous feature set; (2) Based on the homogeneous feature set and tightly wrapping set, a novel algorithm is presented for obtaining the tightly wrapping surface of the homogeneous feature region; (3) The method for constructing all the public regions outside of the tightly wrapping surface and the intersections of wrapping regions in two different tightly wrapping surfaces, as the rejection region of the classifier; (4) An incremental algorithm is also presented based on the SVD-TWDD method. The experimental results with UCI data sets show that the correct recognition rate of our proposed method is nearly100% even if with a low rejection rate.

Adaptive filtering algorithms based on higher-order statistics are proposed for multi-dimensional signal processing in geometric algebra (GA) space. In this paper, the proposed adaptive filtering algorithms utilize the advantage of GA theory in multi-dimensional signal processing to represent a multi-dimensional signal as a GA multivector. In addition, the original least-mean fourth (LMF) and least-mean mixed-norm (LMMN) adaptive filtering algorithms are extended to GA space for multi-dimensional signal processing. Both the proposed GA-based least-mean fourth (GA-LMF) and GA-based least-mean mixed-norm (GA-LMMN) algorithms need to minimize cost functions based on higher-order statistics of the error signal in GA space. The simulation results show that the proposed GA-LMF algorithm performs better in terms of convergence rate and steady-state error under a much smaller step size. The proposed GA-LMMN algorithm makes up for the instability of GA-LMF as the step size increases, and its performance is more stable in mean absolute error and convergence rate.

With the increasing demand for multidimensional data processing, Geometric algebra (GA) has attracted more and more attention in the field of geographical information systems. GA unifies and generalizes real numbers and complex, quaternion, and vector algebra, and converts complicated relations and operations into intuitive algebra independent of coordinate systems. It also provides a solution for solving multidimensional information processing with a high correlation among the dimensions and avoids the loss of information. Traditional methods of computer vision and artificial intelligence (AI) provide robust results in multidimensional processing after being combined with GA and give additional feature analysis facility to remote sensing images. In this paper, we provide a detailed review of GA in different fields of AI and computer vision regarding its applications and the current developments in geospatial research. We also discuss the Clifford-Fourier transform (CFT) and quaternions (sub-algebra of GA) because of their necessity in remote sensing image processing. We focus on how GA helps AI and solves classification problems, as well as improving these methods using geometric algebra processing. Finally, we discuss the issues, challenges, and future perspectives of GA with regards to possible research directions.

Three-dimensional image modeling is essential in many scientific disciplines, including computer vision and precision agriculture. So far, various methods of creating three-dimensional (3D) models have been considered. However, the processing of transformation matrices of each input image data is not controlled. Site-specific crop mapping is essential because it helps farmers determine yield, biodiversity, energy, crop coverage, etc. Clifford Geometric Algebraic understanding of signaling and image processing has become increasingly important in recent years. Geometric Algebraic treats multi-dimensional signals in a holistic way to maintain relationship between side sizes and prevent loss of information. This article has used agricultural images acquired by unmanned aerial vehicles (UAVs) to construct three-dimensional models using Clifford geometric algebra. The qualitative and quantitative performance evaluation results show that Clifford geometric algebra can generate a three-dimensional geometric statistical model directly from drones' RGB images. Through peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and visual comparison, the proposed algorithm's performance is compared with latest algorithms. Experimental results show that proposed algorithm is better than other leading 3D modeling algorithms.

Unlike our basic theories of space and time, quantum mechanics is not a locally causal theory. Moreover, it is widely believed that any hopes of restoring local causality within any realistic theory have been undermined by Bell's theorem and the experimental investigations it has inspired. In this pedagogical paper, John S. Bell's amusing example of Dr. Bertlmann's socks to illustrate the results of these experiments is reconsidered, first within a toy model of a two-dimensional one-sided world of a non-orientable Möbius strip, and then within a real world of three-dimensional quaternionic sphere, S3, which results from an addition of a single point to IR3 at infinity. In the latter quaternionic world, which happens to be the spatial part of a solution of Einstein's field equations of general relativity, the singlet correlations between a pair of entangled fermions can be understood as classically as those between Dr. Bertlmann's colorful socks.

In this paper, we extend the original Normalized Least Mean Fourth (NLMF) and Normalized Least Mean Square (NLMS) adaptive filtering algorithms into Geometric Algebra (GA) space to enable them to process multidimensional signals. We redefine the cost functions and propose the GA based NLMF and NLMS algorithms (GA-NLMF & GA-NLMS). We take full advantage of the ability of GA to represent multidimensional signals in GA space. GA-NLMS minimizes the cost function of the normalized mean square of the error signal, and remain stable as the input signal of the filter increases. GA-NLMS has fast convergence rate but higher steady-state error. The GA-NLMF algorithm minimizes the cost function of the normalized mean fourth of the error signal. Simulation results show that our proposed GA-NLMS adaptive filtering algorithm outperforms original NLMS algorithm in terms of convergence rate and steady-state error, and GA-NLMF outperforms both NLMF and GA-NLMS algorithms. GA-NLMF has faster convergence rate and lower steady state error, which is proved in the experiments.

Recently, a new class of multivector-valued adaptive filters, called geometric-algebra adaptive filters (GA-AFs), have been proposed and applied to 3D registration of point clouds, rotation estimation in computer vision, and so on. To offer a complete theoretical foundation for GA-AFs, we present a transient behavior evaluation of the geometric algebra least mean square (GA-LMS) algorithm. Specifically, the transient mean square deviation (MSD) is obtained by using the results of the mean square theoretical behavior of the GA-LMS. Furthermore, the transient excess mean square error (EMSE) is also given by using the results of MSD. The analytical results rely on the independence theory which has been commonly used in the convergence analysis of real-valued adaptive filter. Considering the non-commutative geometric algebra, the conventional method of analysis is not suitable for the GA-LMS algorithm. Therefore, we propose a novel method to analyze the GA-LMS algorithm under white noise assumption by separating the weight-error array. The obtained theoretical results can be used to accurately predict the transient performance of the GA-LMS. Moreover, the stability condition and steady-state performance are also analyzed. Our proposed steady-state model is more accurate than the previous model. Finally, numerical experiments are presented to confirm the accuracy of the theoretical analysis results. [ABSTRACT FROM AUTHOR]

A novel least-mean kurtosis adaptive filtering algorithm based on geometric algebra (GA-LMK) is proposed for multidimensional signal processing. First, taking advantage of geometric algebra (GA) in terms of the representation of multidimensional signal, the GA-LMK algorithm represents a multidimensional signal as a GA multivector. Second, we extend the original least mean kurtosis (LMK) algorithm in GA space for multidimensional signal processing. The proposed GA-LMK algorithm minimizes the cost function of negated kurtosis of the error signal in GA space, and provides a way to make tradeoff problem between convergence rate and steady-state error. Third, we study the steady-state behavior of the GA-LMK algorithm under Gaussian noises to acquire conditions of misadjustment. The simulation results show that our proposed GA-LMK adaptive filtering algorithm can outperform significantly existing the state-of-the-art algorithms in terms of convergence rate and steady-state error.

Image segmentation is widely used in the field of agriculture to improve the yields and protecting them from pests, herbs, shrubs, and weeds. Precision agriculture is also contributing to the inter and intra crop monitoring. Recently, unmanned aerial vehicles are used widely for acquiring images. In this paper, we purpose Clifford geometric algebra to enhance the segmented images acquired from the UAVs of different agricultural fields. The Clifford geometric algebra is also sometimes used as a collective term for the diverse range of mathematical fields, both classical and modern algebraic mathematics. Previous image segmentation approaches depend upon the intensity of red, green, and blue colors; but the complete perspective could not be obtained from these approaches. Geometric algebra overcomes this limitation and leads to a genuine color space image processing. It is mainly used in the processing of medical images. Subalgebra of the Clifford algebra is Quaternions. We have used this approach in agricultural images. The image segmentation of foreground and background is enhanced using Clifford geometric algebra; hence, the results obtained are fine-tuned segmented images. The anticipated result of our research would have a positive impact on the amelioration of the condition of the farmers and their livelihood.

In the context of EPR-Bohm type experiments and spin detections confined to spacelike hypersurfaces, a local, deterministic and realistic model within a Friedmann-Robertson-Walker spacetime with a constant spatial curvature (S3) is presented that describes simultaneous measurements of the spins of two fermions emerging in a singlet state from the decay of a spinless boson. Exact agreement with the probabilistic predictions of quantum theory is achieved in the model without data rejection, remote contextuality, superdeterminism or backward causation. A singularity-free Clifford-algebraic representation of S3 with vanishing spatial curvature and non-vanishing torsion is then employed to transform the model in a more elegant form. Several event-by-event numerical simulations of the model are presented, which confirm our analytical results with the accuracy of 4 parts in 104. Possible implications of our results for practical applications such as quantum security protocols and quantum computing are briefly discussed.

In this paper, we propose a novel human body posture representation based on Geometric Algebra to extract the angles and orientations of the most informative body joints to describe human body postures. As a motion usually consists of a number of postures, which are different even in the same type of motion. We treat the postures of a motion independently. For each posture, a new Geometric Algebra based skeleton posture descriptor is used to construct the feature vectors as the input for the Support Vector Machine classifier to decide its motion type. To get the type of the whole motion, we choose the most frequent class from the sequence of predictions of the motion postures using a simple voting scheme. We have tested the method on a public benchmark SYSU-3D-HIO and an in-house dataset of human exercises. The results have demonstrated the effectiveness of our method.

The speeded-up robust features (SURFs) algorithm is the best and most efficient local invariant feature algorithm for application to 2-D images and is widely applied in the fields of 2-D image processing and computer vision. Compared to 2-D images, a video has motion information in addition to its appearance information. Here, to make full use of the video appearance and motion information, we use geometric algebra as the mathematical calculation and analysis framework to obtain the embedded appearance and motion information on a local area of a video. In our proposed model of appearance and motion variation (UMAMV), we developed SURF feature detection and description algorithms operating on the spatio-temporal domain with video appearance and motion information. First of all, a model of appearance and motion variation, which contains video appearance and motion information in the framework of geometric algebra, is proposed. Then, based on this model, we propose a novel detection algorithm, the UMAMV-SURF detector, which mainly contains Hessian matrix construction, Hessian matrix determinant approximation calculation, and non-maximal suppression determination feature points as its key steps. Then, we introduce the UMAMV-SURF description algorithm, which mainly includes determining the dominant orientation of UMAMV-SURF feature points and generating the UMAMV-SURF feature descriptors. Finally, by experimenting with the Weizman and UCF101 datasets, the experimental results show that the proposed UMAMV-SURF algorithm can detect those SURF feature points which can have unique appearance information in the spatial domain and reflect motion change in the temporal domain. Moreover, it offers a higher accuracy than other spatio-temporal interest point algorithms in human behavior recognition of video footage.

Multi-modal image registration in medical image analysis is very challenging as the appearance of body structures in images from different imaging devices can be very different. In this paper, we propose a new method [GA-speeded up robust features (SURF)], which incorporates the geometric algebra (GA) into SURF framework, to detect features from images. We model the volumetric data and register the multimodal medical images using feature spheres formulated in conformal geometric algebra (CGA). Specifically, we first extract features from medical images using GA-SURF. Second, we construct the feature spheres using the feature points and find the correspondence of feature spheres in the two images using CGA. With that, we can register the images based on the correspondence of the feature spheres. The experimental results evaluated by RIRE have shown that our method can register the multi-modal images with high accuracy. The maximum registration error is less than 4mm.

Correntropy is an efficient tool for analyzing higher order statistical moments in non-Gaussian noise environments. Correntropy has been used with real, complex, and quaternion data. However, there is no literature studying the correntropy with geometric algebra data. The geometric algebra has been widely used in the engineering applications including the signal processing and image processing. From the probabilistic view, this brief presents a novel geometric algebra correntropy (GAC) which represents the similarity measure between two random geometric algebra (multivector valued) variables. Then, we develop the adaptive filter based on the maximum GAC criteria (MGACC), which is robust against the non-Gaussian noise. Simulation is conducted to verify the advantages of the proposed adaptive algorithm under the non-Gaussian environment. [ABSTRACT FROM AUTHOR]

Geometric algebra has been called a “unified language for mathematics and physics.” Sometimes known as Clifford algebra, it is based on the notion of an invertible product of vectors that captures the geometric relationship between two vectors, i.e., their relative magnitudes and the angle between them. This seemingly simple concept leads to a rich system of algebra and calculus that encompasses the diverse areas of complex numbers, quaternions, vectors, tensors, spinors, and differential forms. This tutorial provides a basic introduction to geometric algebra and presents formulations of known electrical engineering and signal processing concepts to illustrate some inherent advantages of geometric algebra for formulating and solving problems involving vectors. Being introductory, the goal of the tutorial is to introduce this emerging area that, although old as a mathematics discipline, has only recently started to garner significant attention in engineering communities. Geometric algebra should give another potentially powerful tool for pursuing research in any area that uses vectors. [ABSTRACT FROM PUBLISHER]

Maxwell's equations have undergone several changes of form in the 150 years since they first emerged. This has been due not only to changes in their mathematical expression and physical interpretation, but also to historical accidents and trends. This article examines: what they were in the beginning; how they evolved into their customary form over the course of 40 years; the significant variations there have been since then; who have been the major proponents of these changes; and finally, what they did, or did not, contribute. Brief explanations of the main mathematical variations are included. [ABSTRACT FROM PUBLISHER]

The perfect electromagnetic conductor (PEMC) was introduced as an observer-independent “axion medium” that generalizes the concepts of perfect electric conductor (PEC) and perfect magnetic conductor (PMC). Following the original boundary definition, its 3-D medium definition corresponds to a 4-D representation that is, actually, observer-dependent (i.e., it is not isotropic for the whole class of inertial observers), leading to a nonunique characterization of the electromagnetic field inside. This characterization of the PEMC, then, violates the boundary conditions—unless some extraneous waves, called “metafields,” are surgically extracted from the final solution. In this paper, using spacetime algebra, we define the PEMC as the unique limit of the most general class of isotropic media in Minkowskian spacetime, which we call Minkowskian isotropic media (MIM). An MIM is actually a “dilaton-axion medium.” Its isotropy is a Lorentz invariant characterization: It is an observer-independent property, contrary to isotropy in 3-D Gibbsian characterization. Hence, a more natural definition of a PEMC is herein presented: It leads to a unique electromagnetic field in its interior; it corresponds, though, to the same original boundary definition. This new approach is applied to the analysis of an air–MIM interface that, as a particular case, reduces to an air–PEMC interface. [ABSTRACT FROM PUBLISHER]

A new mathematical tool, the geometric algebra of Euclidean 3-space (G3), is introduced for electromagnetic vector-sensor array processing herein. This paper focuses on modeling the six-component outputs of a vector-sensor holistically by an entry called as a multivector in G3. A compact polarized model for the array, termed as a geometric algebra model (G-MODEL), is then presented. Using the G-MODEL, a novel data covariance matrix model is defined by the geometric products in G3 and then analyzed. The analytical results show that the six-component measurement noise of a vector-sensor can naturally be whitened if the noise cross-correlations between the different axial electric and magnetic components are equal to one another. Compared with the known best quad-quaternion model, the new covariance matrix model results in a reduction of half memory requirements while the amount of divisions is reduced to 1/2, multiplications and additions reduced to almost 1/7. [ABSTRACT FROM AUTHOR]

Geometric algebra is used in this paper for a rigorous mathematical treatment of power in single-phase circuits under nonsinusoidal conditions, as complex algebra for sinusoidal conditions. This framework clearly displays the multidimensional nature of power, which is represented by a multivector. The power multivector with its three attributes (magnitude, direction and sense) provides the means to encode all the necessary information in a single entity. This property, in conjunction with the fact that there is a one-to-one correspondence between the terms of this multiveclor, the instantaneous and the apparent power equation, distinguishes it as a highly efficient mathematical tool. In this way one can successfully describe power phenomena and handle practical problems (e.g., power factor improvement). Two simple examples show some of these features. In short, the power multivector under nonsinusoidal situations can be perceived as the generalization of the complex power under sinusoidal situations. [ABSTRACT FROM AUTHOR]