Quaternionic Space Research Articles

Quaternionic essential numerical range of complex operators

We study the essential numerical range of complex operators on a quaternionic Hilbert space and its relation with the essential S-spectrum. We give a new characterization of the essential numerical range relating it to the complex essential numerical range. Moreover, we show that the quaternionic essential numerical range of a normal operator is the convex hull of the essential S-spectrum.

Improving quaternion neural networks with quaternionic activation functions

In this paper, we propose novel quaternion activation functions where we modify either the quaternion magnitude or the phase, as an alternative to the commonly used split activation functions. We define criteria that are relevant for quaternion activation functions, and subsequently we propose our novel activation functions based on this analysis. Instead of applying a known activation function like the ReLU or Tanh on the quaternion elements separately, these activation functions consider the quaternion properties and respect the quaternion space H. In particular, all quaternion components are utilized to calculate all output components, carrying out the benefit of the Hamilton product in e.g. the quaternion convolution to the activation functions. The proposed activation functions can be incorporated in arbitrary quaternion valued neural networks trained with gradient descent techniques. We further discuss the derivatives of the proposed activation functions where we observe beneficial properties for the activation functions affecting the phase. Specifically, they prove to be sensitive on basically the whole input range, thus improved gradient flow can be expected. We provide an elaborate experimental evaluation of our proposed quaternion activation functions including comparison with the split ReLU and split Tanh on two image classification tasks using the CIFAR-10 and SVHN dataset. There, especially the quaternion activation functions affecting the phase consistently prove to provide better performance.

Chen-Type Inequality for Generic Submanifolds of Quaternionic Space Form and Its Application

In 1993, the theory of Chen invariants started when Chen wrote basic inequalities for submanifolds in space forms. This inequality is called Chen’s first inequality. Afterward, many geometers studied many papers dealing with this new inequality. The present paper aims to establish a Chen inequality for quaternionic generic submanifolds in a quaternionic space form and obtain this inequality for real hypersurfaces.

Rational homotopy type and nilpotency of mapping spaces between Quaternionic projective spaces

The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models.

On frame diagonalization of square matrices

In this paper, we introduce a frame diagonalization of matrices in M n ( C ) , called QR-frame diagonalization, by using QR-factorization. Furthermore, we show that every matrix A in M n ( C ) is QR-frame diagonalizable. Also we introduce another frame diagonalization via Hadamard matrices in M n ( R ) , called Hadamard frame diagonalization, by using Hadamard matrices for n = 2, 4 or multiple of 4. We show that every matrix A in M n ( R ) which n = 2, 4 or multiple of 4 is Hadamard frame diagonalizable. In this frame diagonalization, we can find entries on main diagonal Δ by using inner product. Moreover we introduce frame diagonalization of matrices on left quaternionic Hilbert spaces M n ( H ) .

Gauge fields and four interactions in the trigintaduonion spaces

The paper aims to apply the trigintaduonion spaces to explore the physical properties of four interactions simultaneously, including the electromagnetic fields, gravitational fields, weak nuclear fields, and strong nuclear fields. J. C. Maxwell first applied the algebra of quaternions to study the physical properties of electromagnetic fields. It inspired some subsequent scholars to introduce the quaternions, octonions, sedenions, and trigintaduonions to research the electromagnetic fields, gravitational fields, weak nuclear fields, strong nuclear fields, quantum mechanics, gauge fields, and curved spaces and so forth. The algebra of trigintaduonions is able to discuss the physical quantities of four interactions, including the field potential, field strength, field source, linear momentum, angular momentum, torque, and force. In the field theories described with the algebra of trigintaduonions, the weak nuclear field is composed of three types of fundamental fields. These three fundamental fields, related to weak nuclear fields, can describe the physical properties of weak nuclear fields collectively. This is consistent with the conclusion of the electroweak theory. Meanwhile the strong nuclear field consists of three types of fundamental fields. These three fundamental fields relevant to strong nuclear fields may investigate the physical properties of strong nuclear fields mutually. It is coincident with the deduction of quark theory. According to the properties of trigintaduonions, one can deduce the Yang‐Mills equation related to the gauge fields. It means that the electromagnetic field occupies a quaternion space. The gravitational field owns one different quaternion space. The weak nuclear fields occupy three mutually independent quaternion spaces. The properties of weak nuclear fields are different from those of electromagnetic fields or gravitational fields. According to the multiplication table of trigintaduonion spaces, the strong nuclear fields own three conjugate quaternion spaces independent of each other. These explorations further deepen the understanding of the physical properties of weak and strong nuclear fields.

Cross-modal learning representation using new margin combination for speech recognition task

Cross-modal retrieval aims to elucidate the fusion of information, mimic human learning, and advance the field. The main challenge in cross-modal matching is to build a shared subspace reflecting semantic proximity. Previous works fail to capture asymmetric relevance by adopting symmetric similarity computations. To overcome these shortcomings, an efficient approach called quaternion representation learning (QRL) is introduced for better cross-modal matching. Thus, a better representation of the shared semantics is offered by virtue of its richer representation capacity of the quaternionic space and its strong expressive power. Transfer learning is a crucial aspect in this context. By leveraging pre-trained models, the knowledge gained from one task or domain can be effectively transferred to another, allowing for improved performance and generalization. In this study, transfer learning is employed to enhance the cross-modal retrieval system. Specifically, a pre-trained ResNet-512 model is utilized in conjunction with the proposed total margin (TM) loss function, which combines the QRL approach with the novel adaptive mean margin (AMM) methodology. The TM loss function, coupled with the pre-trained ResNet-512 model, is evaluated on the Audio-Visual Arabic Speech Database (AVAS) and the Arabic Visual Speech Database (AVSD), along with other audio-visual datasets. Experimental results demonstrate the effectiveness of the TM loss function in consistently improving performance on both databases. The recall scores (R@k) and mean average precision (mAP) values achieved on the AVAS Database are as follows: R@1: 42.1±0.7, R@2: 70.2±0.1, R@5: 78.5±1.0, and mAP: 53.0±1.1. Similarly, on the AVSD Database, the results are R@1: 41.7±0.3, R@2: 69.2±1.1, R@5: 78.0±0.3, and mAP: 52.7±0.5. By incorporating transfer learning and the TM loss function into the cross-modal retrieval framework, this study demonstrates the potential for improving clustering efficiency and enhancing visual and speech understanding. The combination of pre-trained models and the TM loss function offers a promising avenue for advancing crossmodal matching techniques and achieving state-of-the-art performance.

Characterising trees and hyperbolic spaces by their boundaries

We use the language of proper CAT(-1) spaces to study thick, locally compact trees, the real, complex and quaternionic hyperbolic spaces and the hyperbolic plane over the octonions. These are rank 1 Euclidean buildings, respectively rank 1 symmetric spaces of non-compact type. We give a uniform proof that these spaces may be reconstructed using the cross ratio on their visual boundary, bringing together the work of Tits and Bourdon.

Circle Actions on Oriented Manifolds With 3 Fixed Points

Abstract Let the circle group act on a compact oriented manifold $M$ with a non-empty discrete fixed point set. Then the dimension of $M$ is even. If $M$ has one fixed point, $M$ is the point. In any even dimension, such a manifold $M$ with two fixed points exists, a rotation of an even dimensional sphere. Suppose that $M$ has three fixed points. Then the dimension of $M$ is a multiple of 4. Under the assumption that each isotropy submanifold is orientable, we show that if $\dim M=8$, then the weights at the fixed points agree with those of an action on the quaternionic projective space $\mathbb{H}\mathbb{P}^{2}$, and show that there is no such 12-dimensional manifold $M$.

On the convexity of the quaternionic essential numerical range

AbstractThe numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.

A generalized Wintgen inequality in quaternion Kähler geometry

In this paper, we establish a generalized Wintgen inequality for quaternionic bi-slant submanifolds and QR-submanifolds (with minimal codimension) in quaternion space forms. We also aim to characterize the second fundamental form of those submanifolds for which the equality cases can hold. Finally, we provide examples of submanifolds embedded in quaternion space forms to support our results.

Revolutionary Strategy for Depicting Knowledge Graphs with Temporal Attributes

In practical applications, the temporal completeness of knowledge graphs is of great importance. However, previous studies have mostly focused on static knowledge graphs, generally neglecting the dynamic evolutionary properties of facts. Moreover, the unpredictable and limited availability of temporal knowledge graphs, together with the complex temporal dependency patterns, make current models inadequate for effectively describing facts that experience temporal transitions. To better represent the evolution of things over time, we provide a learning technique that uses quaternion rotation to describe temporal knowledge graphs. This technique describes the evolution of entities as a temporal rotation change in quaternion space. Compared to the Ermitian inner product in complex number space, the Hamiltonian product in quaternion space is better at showing how things might be connected. This leads to a learning process that is both more effective and more articulate. Experimental results demonstrate that our learning method significantly outperforms existing methods in capturing the dynamic evolution of temporal knowledge graphs, with improved accuracy and robustness across a range of benchmark datasets.

Spatial Numerical range of bounded operators on right quaternionic Banach spaces

Spatial Numerical range of bounded operators on right quaternionic Banach spaces

Rank one quaternionic operators and additive preservers

In this paper, we completely describe all additive surjective maps, on the set of all bounded finite rank right linear operators acting on a right quaternionic Banach space, that preserve the set of all bounded rank one (resp. rank one idempotent, rank one nilpotent) right linear operators, in both directions.

Generalized fusion frame in quaternionic Hilbert spaces

The notion of a generalized fusion frame in quaternionic Hilbert space is introduced. A characterization of generalized fusion frame in quaternionic Hilbert space with the help of frame operator is being discussed. Finally, g-fusion frame in quaternionic Hilbert space using invertible bounded right Q-linear operator on quaternionic Hilbert space is constructed.

Browder operators on quaternionic Hilbert space

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications

Abstract We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space H p {H^{p}} on quaternionic Siegel upper half space for 2 3 < p ≤ 1 {\frac{2}{3}<p\leq 1} . Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

Global Stabilization of Antipodal Points on $n$-Sphere With Application to Attitude Tracking

Existing approaches to robust global asymptotic stabilization of a pair of antipodal points on unit $n$-sphere $\mathbb{S}^n$ typically involve the non-centrally synergistic hybrid controllers for attitude tracking on unit quaternion space. However, when switching faults occur due to parameter errors, the non-centrally synergistic property can lead to the unwinding problem or in some cases, destabilize the desired set. In this work, a hybrid controller is first proposed based on a novel centrally synergistic family of potential functions on $\mathbb{S}^n$, which is generated from a basic potential function through angular warping. The synergistic parameter can be explicitly expressed if the warping angle has a positive lower bound at the undesired critical points of the family. Next, the proposed approach induces a new quaternion-based controller for global attitude tracking. It has three advantageous features over existing synergistic designs: 1) it is consistent, i.e., free from the ambiguity of unit quaternion representation; 2) it is switching-fault-tolerant, i.e., the desired closed-loop equilibria remain asymptotically stable even when the switching mechanism does not work; 3) it relaxes the assumption on the parameter of the basic potential function in literature. Comprehensive simulation confirms the high robustness of the proposed centrally synergistic approach compared with existing non-centrally synergistic approaches.

The L^P-spaces of functions from the n-dimensional real space to the N-dimensional quaternionic space

The L^P-spaces of functions from the n-dimensional real space to the N-dimensional quaternionic space

The string topology coproduct on complex and quaternionic projective space

On the free loop space of compact symmetric spaces Ziller introduced explicit cycles generating the homology of the free loop space. We use these explicit cycles to compute the string topology coproduct on complex and quaternionic projective space. The behavior of the Goresky-Hingston product for these spaces then follows directly.