24 papers · ranked by Valyu relevance
Johann Brehmer, Pim de Haan, Sönke Behrends, Taco Cohen
Problems involving geometric data arise in physics, chemistry, robotics, computer vision, and many other fields. Such data can take numerous forms, for instance points, direction vectors, translations, or rotations, but to date there is no single architecture that can be applied to such a wide variety of geometric…
Charles Gunn, Steven De Keninck
∗ Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components…
Eckhard Hitzer
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this approach to elementary representations of arbitrary points, point pairs, lines…
Grégoire Veyrac, Zeno Toffano, Lu Wei
Quantum simulation qubit models of electronic Hamiltonians rely on specific transformations in order to take into account the fermionic permutation properties of electrons. These transformations (principally the Jordan-Wigner transformation (JWT) and the Bravyi-Kitaev transformation) correspond in a quantum circuit to…
Tobias Löw, Mariana de Paula Assis Fonseca, Vitalii Pruks, Graham Deacon + 2 more
We propose a task-space admittance controller for dual-arm robotic systems using conformal geometric algebra. The controller is a reinterpretation of a previous work using dual quaternion algebra. By introducing conformal geometric algebra, we aim to enhance the geometric expressiveness, which simplifies the modeling…
Pablo Bañón Pérez, Maarten DeKieviet
geometric language Authors: ['Pablo Bañón Pérez' 'Maarten DeKieviet'] | 1 | Introduction | | 2 | | --- | --- | --- | --- | | 2 | | Theory of geometric calculus | 3 | | | 2.1 | Geometric algebra | 3 | | | 2.2 | Complex numbers and rotor techniques | 5 | | | 2.3 | The vector derivative | 6 | | | 2.4 | Fundamental theorem…
Garret Sobczyk
Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in mathematics is the Periodic Table of Geometric Numbers discussed here. In 1878 William…
Richard David Gill
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…
Charles Gunn
The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief review of PGA, the article focuses on P(R ∗ 2,0,1), the PGA for euclidean plane…
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…
Xiaomin Jiang, Yangfei Huang, Feng Zhang, Ding Ma + 1 more
The study of spatial geometric similarity plays a significant role in spatial data retrieval. Many researchers have examined spatial geometric similarity, which is useful for spatial analysis and data retrieval. However, the majority of them focused on objects of the same type. Methods to support the spatial geometric…
Charles Gunn
The discussion of how to apply geometric algebra to euclidean n-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from 19th century mathematics. We then introduce the dual projectivized Clifford…
Basel Sharaf, Seth Lewis, David Choung, Abhinav Goyal + 12 more
'Kristen M. Scheitler' 'Lydia S. Hong' 'Charles D. Blaha' 'Barbara Hanna' 'Kyungwon Chang' 'Jason Yuen' 'Yoonbae Oh' 'Hojin Shin' 'Sanjeet Grewal' 'Jin Woo Chang' 'Kai Miller' 'Kendall H. Lee'] Stereotactic systems have traditionally used Cartesian coordinate combined with linear algebraic mathematical models to…
Authors not listed
Atomic electrons exist under the central Coulomb potential of the nucleus, a constraint that mandates their wave functions be described by spherical harmonics. While standard quantum mechanics treats these functions as static geometric solutions, the empirical structure of the periodic table (2, 8, 8, 18, 18, ...)…
Authors not listed
Building on prior mathematical formulations of the periodic law, this paper demonstrates that the algebraic formula S(k) = 2 * (floor(k/2) + 1)^2 accurately predicts the period lengths (2, 8, 8, 18, 18, 32, ...) of the periodic table. This pattern is not a numerical coincidence but a direct consequence of the…
Yuri Dabaghian
In the mammalian brain, many neuronal ensembles are involved in representing spatial structure of the environment. In particular, there exist cells that encode the animal’s location and cells that encode head direction. A number of studies have addressed properties of the spatial maps produced by these two populations…
Stefan Vuckovic
Nearly all electronic structure simulations begin with obtaining approximate geometries, making a systematic quantification of errors in approximate molecular structures of key importance. Recently, the geometric energy offset (GEO) framework based on a single and natural measure for quantifying and analysing these…
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…
Chenchao Zhao, Jun S. Song
In the statistical learning language, samples are snapshots of random vectors drawn from some unknown distribution. Such vectors usually reside in a high-dimensional Euclidean space, and thus, the “curse of dimensionality” often undermines the power of learning methods, including community detection and clustering…
Authors not listed
This article is the second in a two-part tutorial review on electronic spin-dependent dynamics. In Part I, we presented the fundamental theory within the adiabatic Born– Huang framework that describes the interaction between nuclear motion and the elec- tronic (spin and spatial) degrees of freedom. In particular, we…
Yuansheng Zhou, Tatyana Sharpee
Understanding the patterns of gene expression is key to elucidating the differences between cell types and across disease conditions. The overwhelmingly large number of genes involved generally makes this problem intractable. Yet, we find that gene expression patterns in five different data datasets can all be…
Nao Takashina, Buntarou Kusumoto, Yasuhiro Kubota, Evan P. Economo
Understanding macroecological patterns across scales is a central goal of ecology and a key need for conservation biology. Much research has focused on quantifying and understanding macroecological patterns such as the species-area relationship (SAR), the endemic-area relationship (EAR) and relative species abundance…
Akhil Shajan, Madushanka Manathunga, Andreas Goetz, Kenneth Merz
Based on a series of energy minimizations with starting structures obtained from the Baker test set of 30 organic molecules, a comparison is made between various open source geometry optimization codes that are interfaced with the open-source QUantum Interaction Computational Kernel (QUICK) program for gradient and…
Arturo Tozzi
A wide range of biological and evolutionary processes is determined not merely by the occurrence of specific events, but by the exact order in which those events unfold. Gene regulation, developmental pathways, metabolic cascades and genetic evolution often display non-commutative behaviour, in which reversing the…