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This paper gives a concise outline for an extended, multi-dimensional number system from a component perspective. We briefly motivate such a system, sketch the associated algebra and geometry, and conclude with a connection to tensor products that shows it is reasonable, if not desirable, to add tensors of different orders. [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]

The possibility that zitterbewegung opens a window to particle substructure in quantum mechanics is explored by constructing a particle model with structural features inherent in the Dirac equation. This paper develops a self-contained dynamical model of the electron as a lightlike particle with helical zitterbewegung and electromagnetic interactions. The model admits periodic solutions with quantized energy, and the correct magnetic moment is generated by charge circulation. It attributes to the electron an electric dipole moment rotating with ultrahigh frequency, and the possibility of observing this directly as a resonance in electron channeling is analyzed in detail. Correspondence with the Dirac equation is discussed. A modification of the Dirac equation is suggested to incorporate the rotating dipole moment. [ABSTRACT FROM AUTHOR]

Distributed acoustic target tracking is an important application area of wireless sensor networks. In this paper we use algebraic geometry to formally model 2-dimensional acoustic target tracking and then prove its best degree of required sensing coverage. We present the necessary conditions for three sensing coverage to accurately compute the spatio-temporal information of a target object. Simulations show that 3-coverage accurately locates a target object only in 53% of cases. Using 4-coverage, we present two different methods that yield correct answers in almost all cases and have time and memory usage complexity of Θ(1). Analytic 4-coverage tracking is our first proposed method that solves a simultaneous equation system using the sensing information of four sensor nodes. Redundant answer fusion is our second proposed method that solves at least two sets of simultaneous equations of target tracking using the sensing information of two different sets of three sensor nodes, and fusing the results using a new customized formal majority voter. We prove that 4-coverage guarantees accurate 2-dimensional acoustic target tracking under ideal conditions. [ABSTRACT FROM AUTHOR]

Direction-of-arrival (DOA) estimation plays an important role in array signal processing, and the Estimating Signal Parameter via Rotational Invariance Techniques (ESPRIT) algorithm is one of the typical super resolution algorithms for direction finding in an electromagnetic vector-sensor (EMVS) array; however, existing ESPRIT algorithms treat the output of the EMVS array either as a "long vector", which will inevitably lead to loss of the orthogonality of the signal components, or a quaternion matrix, which may result in some missing information. In this paper, we propose a novel ESPRIT algorithm based on Geometric Algebra (GA-ESPRIT) to estimate 2D-DOA with double parallel uniform linear arrays. The algorithm combines GA with the principle of ESPRIT, which models the multi-dimensional signals in a holistic way, and then the direction angles can be calculated by different GA matrix operations to keep the correlations among multiple components of the EMVS. Experimental results demonstrate that the proposed GA-ESPRIT algorithm is robust to model errors and achieves less time complexity and smaller memory requirements. [ABSTRACT FROM AUTHOR]

The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein's Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits. [ABSTRACT FROM AUTHOR]

Abstract: Matrix multiplication was first introduced by Arthur Cayley in 1855 in agreement with the composition of linear transformations. We explore an underlying geometric framework in which matrix multiplication naturally arises from the product of numbers in a geometric (Clifford) algebra. Consequently, all invariants of a linear operator become geometric invariants of the multivectors that they represent. Two different kinds of bases for matrices emerge, a spectral basis of idempotents and nilpotents, and a standard basis of scalars, vectors, bivectors, and higher order k-vectors. The Kronecker product of matrices naturally arises when considering the block structure of a matrix. Conformal geometry of is expressed in terms of the concept of an h-twistor, which is a generalization of a Penrose twistor. [Copyright &y& Elsevier]

A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter. [ABSTRACT FROM AUTHOR]

We define chirality in the context of chiral algebra. We show that it coincides with the more general chirality definition that appears in the literature, which does not require the existence of a quadratic space. Neither matrix representation of the orthogonal group nor complex numbers are used. [ABSTRACT FROM AUTHOR]

Abstract: In this article we propose a method for estimating the camera motion from a video-sequence acquired in the presence of general 3D structures. Solutions to this problem are commonly based on the tracking of point-like features, as they usually back-project onto viewpoint-invariant 3D features. In order to improve the robustness, the accuracy and the generality of the approach, we are interested in tracking and using a wider class of structures. In addition to points, in fact, we also simultaneously consider lines and planes. In order to be able to work on all such structures with a compact and unified formalism, we use here the Conformal Model of Geometric Algebra, which proved very powerful and flexible. As an example of application of our approach, we propose a causal algorithm based on an Extended Kalman Filter, for the estimation of 3D structure and motion from 2D observations of points, lines and coplanar features, and we evaluate its performance on both synthetic and real sequences. [Copyright &y& Elsevier]

Abstract: Spin can be described in the star product formalism by extending the bosonic Moyal product in the fermionic sector. One can then establish the relation to other approaches that describe spin with fermionic variables. The fermionic star product formalism and the fermionic path integral formalism are related in analogy to their bosonic counterparts. [Copyright &y& Elsevier]

We address the problem of finding the correspondences of two point sets in 3D undergoing a rigid transformation. Using these correspondences the motion between the two sets can be computed to perform registration. Our approach is based on the analysis of the rigid motion equations as expressed in the Geometric Algebra framework. Through this analysis it was apparent that this problem could be cast into a problem of finding a certain 3D plane in a different space that satisfies certain geometric constraints. In order to find this plane in a robust way, the Tensor Voting methodology was used. Unlike other common algorithms for point registration (like the Iterated Closest Points algorithm), ours does not require an initialization, works equally well with small and large transformations, it cannot be trapped in “local minima” and works even in the presence of large amounts of outliers. We also show that this algorithm is easily extended to account for multiple motions and certain non-rigid or elastic transformations. [ABSTRACT FROM AUTHOR]

A theory in which four-dimensional space–time is generalized to a larger space, namely a 16-dimensional Clifford space (C-space) is investigated. Curved Clifford space can provide a realization of Kaluza–Klein. A covariant Dirac equation in curved C-space is explored. The generalized Dirac field is assumed to be a polyvector-valued object (a Clifford number) which can be written as a superposition of four independent spinors, each spanning a different left ideal of Clifford algebra. The general transformations of a polyvector can act from the left and/or from the right, and form a large gauge group which may contain the group U(1) × SU(2) × SU(3) of the standard model. The generalized spin connection in C-space has the properties of Yang–Mills gauge fields. It contains the ordinary spin connection related to gravity (with torsion), and extra parts describing additional interactions, including those described by the antisymmetric Kalb–Ramond fields. [ABSTRACT FROM AUTHOR]

Abstract: We describe a geometric structure optimizer that rapidly establishes whether or not SiO4 units in a hypothetical zeolite framework can exist as minimally-deformed regular tetrahedra. The optimizer, SiGH (Silica General Handler), enables an order of magnitude computational speed gain when processing large databases of zeolite graphs through the early rejection of infeasible graphs. [Copyright &y& Elsevier]

Abstract: We use a combination of real-space geometric algebra and reciprocal space dynamical matrix analyses to study the effect of cation substitution on the framework geometries of β-quartz, cordierite and leucite. We show that the geometric stress associated with the substitution in these framework silicates is absorbed by rigid-unit type motion of those coordination polyhedra near the substitution site. We find that the inherent flexibility of these structures enables screening of geometric stress, such that the associated energy cost is minimal and unlikely to influence substitution patterns. [Copyright &y& Elsevier]

In this work, I present a practical way to obtain the vibration-rotation kinetic energy operator for an N-atomic molecule in an arbitrary body-frame $\{\mathbf{e}_{1}^{\prime},\mathbf{e}_{2}^{\prime},\mathbf{e}_{3}^{\prime}\}$. The body-frame need not be orthogonal or rigid. In practice, I derive the explicit form of the measuring vectors associated with the body-frame components of the internal angular momentum. Their inner products with the vector derivatives of the shape coordinates give the "Coriolis" part of the metric tensor appearing in the Hamiltonian, and their inner products among themselves give the "rotational" part. As a simple example, the measuring vectors are explicitly derived in an oblique bond-vector body-frame. The metric tensor elements are also derived for a tetra-atomic pyramidal molecule, whose shape is parametrized in bond-angle coordinates. [ABSTRACT FROM AUTHOR]

In this article we discuss the 2D-3D pose estimation problem of 3D free-form contours. In our scenario we observe objects of any 3D shape in an image of a calibrated camera. Pose estimation means to estimate the relative position and orientation (containing a rotation and translation) of the 3D object to the reference camera system. The fusion of modeling free-form contours within the pose estimation problem is achieved by using the conformal geometric algebra. The conformal geometric algebra is a geometric algebra which models entities as stereographically projected entities in a homogeneous model. This leads to a linear description of kinematics on the one hand and projective geometry on the other hand. To model free-form contours in the conformal framework we use twists to model cycloidal curves as twist-depending functions and interpretn-times nested twist generated curves as functions generated by 3D Fourier descriptors. This means, we use the twist concept to apply a spectral domain representation of 3D contours within the pose estimation problem. We will show that twist representations of objects can be numerically efficient and easily be applied to the pose estimation problem. The pose problem itself is formalized as implicit problem and we gain constraint equations, which have to be fulfilled with respect to the unknown rigid body motion. Several experiments visualize the robustness and real-time performance of our algorithms. [ABSTRACT FROM AUTHOR]

Abstract: The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficients of superanalysis one obtains quantum mechanics for systems with spin. This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle classical and quantum mechanics in a consistent manner. [Copyright &y& Elsevier]

Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA)Cl1,3 of the underlying Minkowski vector space, and in the algebra of physical space (APS)Cl3. STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA+ of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA+ are all that is needed to calculate physically measurable quantities (calledmeasurables) because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, andvice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In theabsoluteversion, both the mapping to STA+ and hermitian conjugation are observer dependent, and the proper basis vectors of any observer are persistent vectors that sweep out time-like planes in spacetime. To compare measurements by different inertial observers in APS, we express them in the proper algebraic basis of a single observer. This leads to therelativeversion of APS, which can be related to STA by assigning every inertial observer in STA to a single absolute frame in STA. The equivalence of inertial observers makes this permissible. The mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors in APS. We relate the two versions of APS as consistent models within the same algebra. [ABSTRACT FROM AUTHOR]

The language of geometric algebra can be used in the development of computer graphics applications. This paper proposes a method to describe a 3D polygonal mesh model using a representation technique based on geometric algebra and the conformal model of the 3D Euclidean space. It describes also the stages necessary to develop an application that uses this formalism. The current application was used to validate the implementation of the main abstract operations characteristic to a geometric algebra computational environment (programming module GAP). The data structures that characterize this geometric algebra based modeling approach as well as the implementation of geometric algebra based methods for model visualization/transformation are developed in detail. The paper emphasizes the elegance and generality of the geometric algebra approach referring also to the necessary computational resources. [Copyright &y& Elsevier]

We examine the dynamic and geometric phases of the electron in quantum mechanics using Hestenes' spacetime algebra formalism. First the standard dynamic phase formula is translated into the spacetime algebra. We then define new formulas for the dynamic and geometric phases that can be used in Hestenes' formalism. [ABSTRACT FROM AUTHOR]

By means of a novel classical limiting method we derive classical Liouville equations for particles with spin 0 and 1 from the Klein–Gordon and the Duffin–Kemmer–Petiau equations in relativistic quantum phase space within a geometric algebra structure. [Copyright &y& Elsevier]

A new method is proposed for estimating the parameters of ball joints, also known as spherical or revolute joints and hinge joints with a fixed axis of rotation. The method does not require manual adjustment of any optimisation parameters and produces closed form solutions. It is a least squares solution using the whole 3D motion data set. We do not assume strict rigidity but only that the markers maintain a constant distance from the centre or axis of rotation. This method is compared with other methods that use similar assumptions in the cases of random measurement errors, systematic skin movements and skin movements with random measurement noise. Simulation results indicate that the new method is superior in terms of the algorithm used, the closure of the solution, consistency and minimal manual parameter adjustment. The method can also be adapted to joints with translational movements. [Copyright &y& Elsevier]

Traditional electrical power theories and one of their most important concepts—apparent power—are still a source of debate, because they present several flaws that misinterpret the power-transfer and energy-balance phenomena under distorted grid conditions. In recent years, advanced mathematical tools such as geometric algebra (GA) have been introduced to address these issues. However, the application of GA to electrical circuits requires more consensus, improvements and refinement. In this paper, electrical power theories for single-phase systems based on GA were revisited. Several drawbacks and inconsistencies of previous works were identified, and some amendments were introduced. An alternative expression is presented for the electric power in the geometric domain. Its norm is compatible with the traditional apparent power defined as the product of the RMS voltage and current. The use of this expression simplifies calculations such as those required for current decomposition. This proposal is valid even for distorted currents and voltages. Concepts are presented in a simple way so that a strong background on GA is not required. The paper included some examples and experimental results in which measurements from a utility supply were analysed. [ABSTRACT FROM AUTHOR]

This paper presents a new framework based on geometric algebra (GA) to solve and analyse three-phase balanced electrical circuits under sinusoidal and non-sinusoidal conditions. The proposed approach is an exploratory application of the geometric algebra power theory (GAPoT) to multiple-phase systems. A definition of geometric apparent power for three-phase systems, that complies with the energy conservation principle, is also introduced. Power calculations are performed in a multi-dimensional Euclidean space where cross effects between voltage and current harmonics are taken into consideration. By using the proposed framework, the current can be easily geometrically decomposed into active- and non-active components for current compensation purposes. The paper includes detailed examples in which electrical circuits are solved and the results are analysed. This work is a first step towards a more advanced polyphase proposal that can be applied to systems under real operation conditions, where unbalance and asymmetry is considered. [ABSTRACT FROM AUTHOR]

In the field of building constructions, there is undeniably a growing need to optimize the energy systems which are a key target in new modern constructions and industrial buildings. In this sense, energy systems are being traced for the development of energy distribution networks that are increasingly smart, efficient, and sustainable. Modern generation and distribution energy systems, such as microgrids control systems, are being affected by the presence of linear and nonlinear loads, resulting a distorted voltage and current waveforms. Thus, it is stated that industrial and residential building heating and cooling loads behave essentially like sources of harmonics. This paper presents a new framework based on geometric algebra (GA) to the definition of a multivectorial distortion power concept, which is represented by a bivector that is geometrically interpreted to distinguish the rotated distortion and distortion power bivectors in these kinds of loads. Both bivectors, and their relations to the phase angles of distorted voltage are the main subject of this paper to interpret an optimal control of building energy. Numerical examples are used to illustrate of the suggested distortion power concept, as well as the information it provides for energy control in new buildings in a more sustainable way. [ABSTRACT FROM AUTHOR]

• Upgraded geometric algebra power theory is presented for single phase circuits. • Geometric Power is redefined as the product of voltage and current reverse. • Several additions and modifications are introduced to reflect interharmonics and active current based on Fryze proposal. • This new method is validated through some real examples with linear and non-linear loads. The aim of this work is to present major upgrades to existing power theories based on geometric algebra for single-phase circuits in the frequency domain. It also embodies an interesting new approach with respect to traditionally accepted power theories, revisiting power concepts in both sinusoidal and non-sinusoidal systems with linear and nonlinear loads for a proper identification of its components to achieve passive compensation of true non-active current. Moreover, it outlines traditional power theories based on the apparent power S and confirms that these should definitively be reconsidered. It is evidenced that traditional proposals based on the concepts of Budeanu, Fryze and others fail to identify the interactions between voltage and current harmonics. Based on the initial work of Castro-Núñez and others, new aspects not previously included are detailed, modified and reformulated. As a result, it is now possible to analyze non sinusoidal electrical circuits, establishing power balances that comply with the principle of energy conservation, and achieving optimal compensation scenarios with both passive and active elements in linear and non-linear loads. [ABSTRACT FROM AUTHOR]

A design for a truss mechanism of a secondary mirror based on robotics is proposed. This design would allow for the construction of larger vehicle-mobile telescopes. As the new truss mechanism combines the original support structure and adjustment mechanism, the problems in designing new structures needs to be overcome. In this paper, the basic form of the truss mechanism is determined by finite element method, and the number of limbs meeting the requirements of resonance frequencies and stiffness is obtained. Degrees-of-freedom of the new truss mechanism is calculated by motion space based on geometry algebra and screw theory, It can provide more accurate and specific results compared with the G-K formula. The optimal structure is calculated to meet the requirement in degrees-of-freedom with the minimum possible limbs and kinematic pairs. After the form and the value of joints are determined, the deformations are calculated by stiffness evaluation index. Wavefront aberrations simulated with Zernike polynomials are used to verify the structure. [ABSTRACT FROM AUTHOR]

This paper presents a geometric construction for fitting planar motions to three control poses within a particular geometric algebra. The immediate impact of the geometric construction along with the control pose representation is the provision of simple, usable tools for the design and manipulation of motions in a similar way to the highly successful B-spline approach for curves and surfaces in CAD. • Free-form motions using geometric algebra; • use of the slerp construction; • fitting of planar motion through three control poses. [ABSTRACT FROM AUTHOR]

• Skeleton sequence space as a subset of Geometric Algebra is constructed to represent each skeleton sequence along both spatial and temporal dimensions. • Rotor-based view transformation overcomes the view variation challenge and reserves relative motions among skeletons. • Spatio-temporal view invariant model is constructed to model spatial configuration and temporal dynamics of skeleton joints and bones. • Four skeleton sequence shape and motion representations are learned to comprehensively describe skeleton-based actions, which are fed to a selected multi-stream convolutional neural network for action recognition. Skeleton-based action recognition is an increasing attentioned task that analyses spatial configuration and temporal dynamics of a human action from skeleton data, which has been widely applied in intelligent video surveillance and human-computer interaction. How to design an effective framework to learn discriminative spatial and temporal characteristics for skeleton-based action recognition is still a challenging problem. The shape and motion representations of skeleton sequences are the direct embodiment of spatial and temporal characteristics respectively, which can well address for human action description. In this work, we propose an original unified framework to learn comprehensive shape and motion representations from skeleton sequences by using Geometric Algebra. We firstly construct skeleton sequence space as a subset of Geometric Algebra to represent each skeleton sequence along both the spatial and temporal dimensions. Then rotor-based view transformation method is proposed to eliminate the effect of viewpoint variation, which remains the relative spatio-temporal relations among skeleton frames in a sequence. We also construct spatio-temporal view invariant model (STVIM) to collectively integrate spatial configuration and temporal dynamics of skeleton joints and bones. In STVIM, skeleton sequence shape and motion representations which mutually compensate are jointly learned to describe skeleton-based actions comprehensively. Furthermore, a selected multi-stream Convolutional Neural Network is employed to extract and fuse deep features from mapping images of the learned representations for skeleton-based action recognition. Experimental results on NTU RGB+D, Northwestern-UCLA and UTD-MHAD datasets consistently verify the effectiveness of our proposed method and the superior performance over state-of-the-art competitors. [ABSTRACT FROM AUTHOR]

Abstract: Deforming the algebraic structure of geometric algebra on the phase space with a Moyal product leads naturally to supersymmetric quantum mechanics in the star product formalism. The supersymmetric Hamiltonian emerges then from the classical one by the transition from commutative to noncommutative geometry on the phase space. [Copyright &y& Elsevier]

• Geometric algebra framework is applied to solve non sinusoidal and nonlinear circuits. • A new definition of non-active power based on geometric algebra is presented. • New current decomposition is introduced to minimize line looses and supply active power. • This new method is validated through some examples with linear and non-linear loads. A new approach for the definition of non-active power in electrical systems is presented in this paper. Through the use of geometric algebra, it is possible to define a new term called geometric non-active power, which is applicable to both sinusoidal and non-sinusoidal systems, and to both linear and nonlinear loads. The classic definitions of distortion and reactive power are compared and discussed in our proposal. We verify how geometric non-active power can appear in both purely resistive and purely reactive systems. The superiority of geometric algebra is revealed through several examples of electrical circuits previously analysed in specialised literature. Furthermore, a new geometrical current decomposition is proposed, for the first time, to provide a greater physical sense to existing geometric power. The results obtained confirm that classic concepts based on apparent power S are based on a lack of physical meaning, which is why geometric algebra theory should be adopted instead. [ABSTRACT FROM AUTHOR]

Vertex concavity-convexity detection for spatial objects is a basic algorithm of computer graphics, as well as the foundation for the implementation of other graphics algorithms. In recent years, the importance of the vertex concavity-convexity detection algorithm for three-dimensional (3D) spatial objects has been increasingly highlighted, with the development of 3D modeling, artificial intelligence, and other graphics technologies. Nonetheless, the currently available vertex concavity-convexity detection algorithms mostly use two-dimensional (2D) polygons, with limited research on vertex concavity-convexity detection algorithms for 3D polyhedrons. This study investigates the correlation between the outer product and the topology of the spatial object based on the unique characteristic that the outer product operation in the geometric algebra has unified and definitive geometric implications in space, and with varied dimensionality. Moreover, a multi-dimensional unified vertex concavity-convexity detection algorithm framework for spatial objects is proposed, and this framework is capable of detecting vertex concavity-convexity for both 2D simple polygons and 3D simple polyhedrons. [ABSTRACT FROM AUTHOR]

In 2007, and in a series of later papers, Joy Christian claimed to refute Bell's theorem, presenting an alleged local realistic model of the singlet correlations using techniques from geometric algebra (GA). Several authors published papers refuting his claims, and Christian's ideas did not gain acceptance. However, he recently succeeded in publishing yet more ambitious and complex versions of his theory in fairly mainstream journals. How could this be? The mathematics and logic of Bell's theorem is simple and transparent and has been intensely studied and debated for over 50 years. Christian claims to have a mathematical counterexample to a purely mathematical theorem. Each new version of Christian's model used new devices to circumvent Bell's theorem or depended on a new way to misunderstand Bell's work. These devices and misinterpretations are in common use by other Bell critics, so it useful to identify and name them. I hope that this paper can serve as a useful resource to those who need to evaluate new "disproofs of Bell's theorem". Christian's fundamental idea is simple and quite original: he gives a probabilistic interpretation of the fundamental GA equation a · b = (a b + b a) / 2 . After that, ambiguous notation and technical complexity allows sign errors to be hidden from sight, and new mathematical errors can be introduced. [ABSTRACT FROM AUTHOR]

The low-cost, indoor-feasibility, and non-intrusive characteristic of passive infrared sensors (PIR sensors) makes it widely used in human motion detection, but the limitation of its object identification ability makes it difficult to further analyze in the field of Geographic Information System (GIS). We present a template matching approach based on geometric algebra (GA) that can recover the semantics of different human motion patterns through the binary activation data of PIR sensor networks. A 5-neighborhood model was first designed to represent the azimuth of the sensor network and establish the motion template generation method based on GA coding. Full sets of 36 human motion templates were generated and then classified into eight categories. According to human behavior characteristics, we combined the sub-sequences of activation data to generate all possible semantic sequences by using a matrix-free searching strategy with a spatiotemporal constraint window. The sub-sequences were used to perform the matching operation with the generation-templates. Experiments were conducted using Mitsubishi Electric Research Laboratories (MERL) motion datasets. The results suggest that the sequences of human motion patterns could be efficiently extracted in different observation periods. The extracted sequences of human motion patterns agreed well with the event logs under various circumstances. The verification based on the environment and architectural space shows that the accuracy of the result of our method was up to 96.75%. [ABSTRACT FROM AUTHOR]

New-generation power networks, such as microgrids, are being affected by the proliferation of nonlinear electronic systems, resulting in harmonic disturbances both in voltage and current that affect the symmetry of the system. This paper presents a method based on the application of geometric algebra (GA) to the resolution of power flow in nonsinusoidal single-phase electrical systems for the correct determination of its components to achieve passive compensation of true quadrature current. It is demonstrated that traditional techniques based on the concepts of Budeanu, Fryze or IEEE1459 fail to determine the interaction between voltage and current and therefore, are not suitable for being used as a basis for the compensation of nonactive power components. An example is included that demonstrates the superiority of GA method and is compared to previous work where GA approaches and traditional methods have also been used. [ABSTRACT FROM AUTHOR]

As a local invariant feature of videos, the spatiotemporal interest point (STIP) has been widely used in computer vision and pattern recognition. However, existing STIP detectors are generally extended from detection algorithms constructed for local invariant features of two-dimensional images, which does not explicitly exploit the motion information inherent in the temporal domain of videos, thus weakening the performance of existing STIP detectors in a video context. To remedy this, we aim to develop an STIP detector that uniformly captures appearance and motion information for video, thus yielding substantial performance improvement. Specifically, under the framework of geometric algebra, we first develop a spatiotemporal unified model of appearance and motion-variation information (UMAMV), and then a UMAMV-based scale space of the spatiotemporal domain is proposed to synthetically analyze appearance information and motion information in a video. Based on this model, we propose an STIP feature of UMAMV-SIFT that embraces both appearance and motion variation information of the videos. Three datasets with different sizes are utilized to evaluate the proposed model and the STIP detector. We present experimental results to show that the UMAMV-SIFT achieves state-of-the-art performance and is particularly effective when dataset is small. [ABSTRACT FROM AUTHOR]

Non-linear loads in circuits cause the appearance of harmonic disturbances both in voltage and current. In order to minimize the effects of these disturbances and, therefore, to control the flow of electricity between the source and the load, passive or active filters are often used. Nevertheless, determining the type of filter and the characteristics of their elements is not a trivial task. In fact, the development of algorithms for calculating the parameters of filters is still an open question. This paper analyzes the use of genetic algorithms to maximize the power factor compensation in non-sinusoidal circuits using passive filters, while concepts of geometric algebra theory are used to represent the flow of power in the circuits. According to the results obtained in different case studies, it can be concluded that the genetic algorithm obtains high quality solutions that could be generalized to similar problems of any dimension. [ABSTRACT FROM AUTHOR]

A method of solution to solve the compressible unsteady 3D Navier-Stokes Equations in cylindrical co-ordinates coupled to the continuity equation in cylindrical coordinates is presented in terms of an additive solution of the three principle directions in the radial, azimuthal and z directions of flow. A dimensionless parameter is introduced whereby in the large limit case a method of solution is sought for in the tube. A reduction to a single partial differential equation is possible and integral calculus methods are applied for the case of a body force in the direction of gravity to obtain an integral form of the Hunter-Saxton equation. [ABSTRACT FROM AUTHOR]

Sustainable smart buildings play an essential role in terms of more efficient energy. However, these buildings as electric loads are affected by an important distortion in the current and voltage waveforms caused by the increasing proliferation of nonlinear electronic devices. Overall, buildings all around the world consume a significant amount of energy, which is about one-third of the total primary energy resources. Optimization of the power transfer process of such amount of energy is a crucial issue that needs specific tools to integrate energy-efficient behaviour throughout the grid. When nonlinear loads are present, new capable ways of thinking are needed to consider the effects of harmonics and related power components. In this manner, technology innovations are necessary to update the power factor concept to a generalized total or a true one, where different power components involved in it calculation, properly reflect each harmonic interaction. This work addresses an innovative theory that applies the Poynting Vector philosophy via Geometric Algebra to the electromagnetic energy transfer process providing a physical foundation. In this framework, it is possible to analyse and detect the nature of disturbing loads in the exponential growth of new globalized buildings and architectures in our era. This new insight is based on the concept of geometric objects with different dimension: vector, bivector, trivector, multivector. Within this paper, these objects are correlated with the electromagnetic quantities responsible for the energy flow supplied to the most common loads in sustainable smart buildings. Besides, it must be considered that these phenomena are characterized by a quality index multivector appropriate even for detecting harmonic sources. A numerical example is used to illustrate the clear capabilities of the suggested index when it applies to industrial loads for optimization of energy control systems and enhance comfort management in smart sustainable buildings. [ABSTRACT FROM AUTHOR]