geometric_algebra
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geometric_algebra [2021/10/20 07:43] – [Articles] pbk | geometric_algebra [2021/11/03 14:55] – [Videos] pbk | ||
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The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation. This imposes a graded structure on Lie groups, which is not evident in their matrix representation. By embracing this graded structure, the invariant decomposition theorem was proven: any composition of k linearly independent reflections can be decomposed into ⌈k/2⌉ commuting factors, each of which is the product of at most two reflections. This generalizes a conjecture by M. Riesz, and has e.g. the Mozzi-Chasles' | The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation. This imposes a graded structure on Lie groups, which is not evident in their matrix representation. By embracing this graded structure, the invariant decomposition theorem was proven: any composition of k linearly independent reflections can be decomposed into ⌈k/2⌉ commuting factors, each of which is the product of at most two reflections. This generalizes a conjecture by M. Riesz, and has e.g. the Mozzi-Chasles' | ||
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+ | In this paper we introduce and motivate geometric algebra as a better representation for word embeddings. Next we describe how to implement the geometric product and interestingly show that neural networks can learn this product. We then introduce a model that represents words as objects in this algebra and benchmark it on large corpuses; our results show some promise on traditional word embedding tasks. Thus, we lay the groundwork for further investigation of geometric algebra in word embeddings. | ||
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+ | Recently, adaptive filtering algorithms have attracted much more attention in the field of signal processing. By studying the shortcoming of the traditional real-valued fixed step size adaptive filtering algorithm, this paper proposed the novel approach to adaptive filtering with variable step size based on Sigmoid function and geometric algebra (GA). | ||
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Geometric Algebra (GA) has proven to be an advanced language for mathematics, | Geometric Algebra (GA) has proven to be an advanced language for mathematics, | ||
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+ | The aim of this thesis will be to study the Clifford algebras that appear in the derivation of the Dirac equation and investigate alternative formulations of the Dirac equation using (complex) quaternions. To this end, we will first look at the symmetries of the Dirac equation and some of the additional insights that follow from the Dirac equation. We will also give a derivation of the Dirac equation starting from the Schrödinger equation, in which we will come across the gamma matrices. | ||
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+ | This paper presents and discusses the usage of Geometric Algebra (GA) for the analysis of electrical alternating current (AC) circuits. The potential benefits of this novel approach are highlighted in the study of linear and nonlinear circuits with sinusoidal and non-sinusoidal sources in the frequency domain, which are important issues in electrical engineering undergraduate courses. | ||
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geometric_algebra.txt · Last modified: 2023/12/30 00:23 by pbk