23 papers · ranked by Valyu relevance
James M. Chappell, Azhar Iqbal, Lachlan J. Gunn, Derek Abbott + 1 more
'Cheng-Yi Xia'] As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field…
Ahmad H. Eid
Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying algebraic system to describe geometric models, the use of software abstractions…
Timothy D. Barfoot
We retrace Davenport's solution to Wahba's classic problem of aligning two pointclouds using the formalism of Geometric Algebra (GA). GA proves to be a natural backdrop for this problem involving three-dimensional rotations due to the isomorphism between unit-length quaternions and rotors. While the solution to this…
Michał Lipiński, Jacek Kubica, Marian Mrozek, Thomas Wanner
We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in Mrozek (Found Comput Math 17(6):1585-1633, 2017). The generalization is threefold. First, we drop the restraining assumption in Mrozek (Found Comput Math 17(6):1585-1633, 2017) that every multivector must have a…
A. Dargys, A. Acus
The problem of multivector (MV) multiple square roots in real geometric Clifford algebras Clp,q with symbolic coefficients is considered. The method to find multiple MV square roots that is based on R. Bott's periodicity table and matrix eigensystem in Clp,q is proposed. The method can be applied to MV having both…
Venkatraman Gopalan
It is shown that there are 41 types of multivectors representing physical quantities in nonrelativistic physics in arbitrary dimensions within the formalism of Clifford Algebra. The classification is based on the action of three symmetry operations on a general multivector: spatial inversion, 1", time-reversal, 1# …
Wilder B. Lopes, Cássio G. Lopes
—This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language wellsuited for the…
James M. Chappell, Azhar Iqbal, Nicolangelo Iannella, Derek Abbott + 1 more
'Enrico Scalas'] Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension , with the unit imaginary producing the correct spacetime distance , and the results of Einstein’s then recently developed theory of special relativity…
Eduardo A. Notte-Cuello, Waldyr A. Rodrigues
This paper presents a thoughful review of: (a) the Clifford algebra Cl(HV ) of multivecfors which is naturally associated with a hyperbolic space HV ; (b) the study of the properties of the duality product of multivectors and multiforms; (c) the theory of k multivector and l multiform variables multivector extensors…
Manuel-V. Castilla
Many definitions have been formulated to represent nonactive power for distorted voltages and currents in electronic and electrical systems. Unfortunately, no single universally suitable representation has been accepted as a prototype for this power component. This paper defines a nonactive power multivector from the…
Arturo Tozzi, Ksenija Jaušovec
Understanding intelligence-related variations in electroencephalographic (EEG) activity requires advanced mathematical approaches capable of capturing geometric transformations and long-range dependencies in neural dynamics. These approaches may provide methodological advantages over conventional spectral and…
Will Penny, Tom Sambrook, Louis Renoult
Factorial designs are a mainstay of the scientific paradigm, allowing the effects of multiple experimental factors and their interactions to be efficiently studied within a single experiment. In brain imaging, however, multivariate data analyses commonly proceed using multivariate decoding and we argue that the…
Alejandro Santos-Mayo, Faith Gilbert, Laura Ahumada, Caitlin Traiser + 4 more
Neuroscience research has increasingly used decoding techniques, in which multivariate statistical methods identify patterns in neural data that allow the classification of experimental conditions or participant groups. Typically, the features used for decoding are spatial in nature, including voxel patterns and…
Jun Li, Wanrong Hong, Yusheng Xiang
Machine learning methods based on statistical principles have proven highly successful in dealing with a wide variety of data analysis and analytics tasks. Traditional data models are mostly concerned with independent identically distributed data. The recent success of end-to-end modelling scheme using deep neural…
José L. Medina-Franco, Ana L. Chávez-Hernández, Edgar López-López, Fernanda I. Saldívar-González
Technological advances and practical applications of the chemical space concept in drug discovery, natural product research, and other research areas have attracted the scientific community´s attention. The large- and ultra-large chemical spaces are associated not only with the significant increase in the number of…
Yuanning Li, R. Mark Richardson, Avniel Singh Ghuman
The lack of multivariate methods for decoding the representational content of interregional neural communication has left it difficult to know what information is represented in distributed brain circuit interactions. Here we present Multi-Connection Pattern Analysis (MCPA), which works by learning mappings between the…
William G. Faris
This paper is a modern exposition of old ideas. The setting is a Euclidian space E of dimension n with associated vector space V of dimension n. A (non-zero) sliding vector is a vector in V that is free to move, but only within a line L of E. The set of sliding vectors has dimension 2n−1. This set is naturally embedded…
Jixin Chen
I have been teaching physical chemistry to undergraduate students for several years now. I have observed that my interpretation of vector-matrix calculations and complex calculations has not been well received by the chemistry students. The traditional vector notation of adding an arrow on top of a variable and the…
Stefan Canzar, Van Hoan Do, Slobodan Jelić, Sören Laue + 2 more
Metric multidimensional scaling is one of the classical methods for embedding data into low-dimensional Euclidean space. It creates the low-dimensional embedding by approximately preserving the pairwise distances between the input points. However, current state-of-the-art approaches only scale to a few thousand data…
H. Robert Frost
We present an approach for modeling single cell RNA-sequencing (scRNA-seq) data using quaternions. Quaternions are four dimensional hypercomplex numbers that, along with real numbers, complex numbers and octonions, represent one of the four normed division algebras. Quaternions have been most widely employed to…
Authors not listed
We present a unified, set–theoretic framework that extends molecular graphs to hypergraphs and superhypergraphs via iterated power sets. We define Molecular Graphs, Molecular HyperGraphs, and Molecular SuperHyperGraphs, and develop four complements over them: Weighted, Rough, Neural, and Multipolar frameworks. We prove…
Authors not listed
Hyperstructures and their hierarchical extensions [1]—SuperHyperStructures—provide a versatile formalism for modeling multi-layered and complex phenomena [2, 3]. A MicroStructure is a measurable mapping that assigns to each material point a single local state, representing attributes such as phase, crystal orientation…
Authors not listed
NMR chemical shifts depend on the applied magnetic flux density, and this becomes more and more important as stronger and stronger magnetic fields are becoming available. Herein, we develop a theory of the field dependence of NMR shifts of paramagnetic molecules in solution. Our derivation leads to two distinct…