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... In 1843, quaternions were first discovered by W. R. Hamilton as an extension of complex numbers. It plays a crucial role in various research areas such as mathematics [1,2], spacecraft attitude control [3][4][5], robot motion control [6][7][8], quantum physics [9][10][11], computer graphic [12,13], signal processing [14][15][16], integrated navigation system [17][18][19]. Nowadays, more and more scholars have paid attention to the problems related to quaternions, especially the quaternion least squares (QLS) problem. ...
... It has become a very practical research tool in many fields such as color image processing B Fengxia Zhang zhangfengxia@lcu.edu.cn 1 College of Mathematical Science, Liaocheng University, Liaocheng 252059, China [20,21], quantum physics [22][23][24], control theory [25]. Many scholars have devoted themselves to the study of the QLS problem. ...
... where Q (0) , Q (1) are quaternion J-unitary matrices by Theorem 3.3. ...
This paper concentrates on the quaternion indefinite least squares (QILS) problem. Firstly, we define the quaternion J-unitary matrix and the quaternion hyperbolic Givens rotation, and study their properties. Then, based on these, we investigate the quaternion hyperbolic QR factorization, and purpose its real structure-preserving (SP) algorithm by the real representation (Q-RR) matrix of the quaternion matrix. Immediately after, we explore the solution of the QILS problem, and give a real SP algorithm of solving the QILS problem. Eventually, to illustrate the effectiveness of proposed algorithms, we offer numerical examples.
... Note that a real quaternion differentiation procedure was empirically proposed and discussed by Deavours [29], Jack [30], and Dunning-Davies [31], which resulted in a close resemblance of the quaternion expressions to the modified version of Maxwell's equations. ...
... A frequently used quaternion derivative was introduced by Fueter [38] [39] using a quaternion version of the Cauchy-Riemann condition. Similarly, a quaternion gradient, its conjugate, and Laplace operators were proposed by Devours [29], who attempted to derive Maxwell's equations using the quaternion approach. ...
... Next, we define the quaternion Laplacian operator multiplying the quaternion derivative by its conjugate [29], ...
Originally, Maxwell attempted to express his electromagnetic theory using four-dimensional mathematics of quaternions. Maxwell's equations were later re-written in a three-dimensional real vector form, which is how the theory is presented today. Thus, an interesting question remains whether we can derive electromagnetic equations analytically from the basic mathematical principles of quaternion algebra and calculus, resulting in general and analytic matter equations. This question seems highly intriguing.
Previously, we developed a mathematical theory of time using a normed division algebra of real quaternions. In this study, we extend the theory of time by presenting a new analytical derivation of electromagnetic matter equations using the calculus of real quaternions, as originally intended by Maxwell.
Therefore, we propose a novel mathematical definition of the quaternion path derivative using the properties of quaternion division. We then apply the quaternion derivative to an external electromagnetic potential and assume that the first quaternion derivative represents the quaternion electromagnetic force. Next, we assume that the second derivative, or quaternion Laplacian operator, applied to an external electromagnetic potential leads to the quaternion electromagnetic current density.
The new analytical expressions are similar to the original empirical Maxwell equations, except for an additional scalar electric field, which allows for a novel formulation of Ohm's conductivity law. We demonstrate that the resulting analytical equations can be written equivalently using either electromagnetic potentials or fields.
Finally, we summarise the key postulates and equations of the new electromagnetic matter theory, which were based on normed division algebra and the calculus of quaternions. The resulting theory appears to be a useful analytical enhancement of the original Maxwell equations, and therefore, seems highly comprehensive, logical, and compelling.
... He famously impressed the fundamental formula for quaternions onto a stone bridge in Dublin. Since then, quaternion algebra has been extensively studied, and its applications have been explored [25][26][27]. In [28], Ozen and Guzeltepe constructed cyclic codes over certain finite quaternion integer rings. ...
... Nonlinearity is a critical concept in many fields, including physics, engineering, economics, and biology, and it plays a crucial role in understanding the behavior of complex systems and designing effective control strategies. The upper bound of NL for the S-box is ( ) = 2 −1 − 2 2 −1 [26]. The optimal value of the NL of the S-box is 120. ...
... Next, the number of pairs of inputs that satisfy the differential characteristic is counted, and the difference between this value and the expected value is measured. If this difference is close to zero, the S-box is said to satisfy the bit-independent criterion [26]. The average value of BIC is 1 2 . ...
In the field of cryptography, block ciphers are widely used to provide confidentiality and integrity of data. One of the key components of a block cipher is its nonlinear substitution function. In this paper, we propose a new design methodology for the nonlinear substitution function of a block cipher, based on the use of Quaternion integers (QI). Quaternions are an extension of complex numbers that allow for more complex arithmetic operations, which can enhance the security of the cipher. We demonstrate the effectiveness of our proposed design by implementing it in a block cipher and conducting extensive security analysis. Quaternion integers give pair of substitution boxes (S-boxes) after fixing parameters but other structures give only one S-box after fixing parameters. Our results show that the proposed design provides superior security compared to existing designs, two making on a promising approach for future cryptographic applications.
... Theorem 7. Two quadratic Riccati systems (5) are diffeomorphic (linearly equivalent) if and only if the algebras associated with them are isomorphic. Two Dirac systems (3) are equivalent [13,14] if and only if the algebras associated with them are isotopic. ...
... Moreover, P can indicate the pairs of obtained singularities that can coalesce (independently of other singularities) and those that cannot coalesce. For example, deformation (14) shows that a k = a + εke 1 can coalesce with a k+1 , whereas a k cannot coalesce with a k+2 (provided m ≥ 3). These observations can be given a more formal description. ...
... Theorem 15 ([28]). Take (13) and (14) and substitute ...
This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between these theories is the nonlinear spectral theory developed for algebra and homogeneous differential equations. A nonlinear spectral method is used to prove the existence of an algebraic first integral, interpretations of various phase zones, and the separatrices construction for ODEs. In algebra, the same methods exploit subalgebra construction and explain fusion rules. In conclusion, perturbation methods may also be interpreted for near-Jordan algebra construction.
... Supposing that all three objects A, ξ and χ are transformed by a priori unrelated unit quaternions α 1 , α 2 and α 3 and still insisting that A is made from ξ and χ, these transformations would have to be related. This relation is exactly the condition that defines triality (32) and leads to the conclusion that the triality transformations is the largest group that preserves (33). However, these transformations are exactly of the form used in supersymmetry (see, for example [31]), so it is only natural that the overall symmetry of these theories is given by the triality algebras. ...
... In physical applications it is important to define the quaternionic gradient operator. Although there have been some derivations of this operator in the literature with different level of details (see, [32][33][34][35][36][37][38][39][40][41] and references therein), it is still not fully clear how this operator can be written in the most general case and how it can be applied to various quaternion-valued functions. ...
... The main obstacle in physical applications of quaternions is that the real-valued functions of quaternion variables Φ(q) are not analytic according to quaternion analysis [32][33][34][35][36][37][38][39][40][41]. ...
The properties of spinors and vectors in (2 + 2) space of split quaternions are studied. Quaternionic representation of rotations naturally separates two SO(2,1) subgroups of the full group of symmetry of the norms of split quaternions, SO(2,2). One of them represents symmetries of three-dimensional Minkowski space-time. Then, the second SO(2,1) subgroup, generated by the additional time-like coordinate from the basis of split quaternions, can be viewed as the internal symmetry of the model. It is shown that the analyticity condition, applying to the invariant construction of split quaternions, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that the derivatives by extra time-like coordinate generate triality (supersymmetric) rotations, the analyticity equation is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time.
... El Análisis Cuaterniónico fue iniciado por Rudolf Fueter en [10]. Luego, otros autores se interesaron en establecer nuevos enfoques para desarrollar las ideas básicas de esta teoría [11][12][13]. ...
... Tomando como base el sistema anterior, podemos obtener las ecuaciones de ondas [9], para ello el operador rotacional es aplicado en ambos lados de las ecuaciones (7) (11)(12) De acuerdo con las ecuaciones (9) y (10) la divergencia de E y B es cero, y sustituyendo las ecuaciones (7) y (8) en (11)(12), obtenemos las ecuaciones de ondas para los campos eléctrico y magnético. ...
... Tomando como base el sistema anterior, podemos obtener las ecuaciones de ondas [9], para ello el operador rotacional es aplicado en ambos lados de las ecuaciones (7) (11)(12) De acuerdo con las ecuaciones (9) y (10) la divergencia de E y B es cero, y sustituyendo las ecuaciones (7) y (8) en (11)(12), obtenemos las ecuaciones de ondas para los campos eléctrico y magnético. ...
RESUMEN En el presente artículo se estudian las ecuaciones de ondas electromagnéticas y la condición de Lorentz, como propiedades emergentes del sistema de Maxwell en el contexto de la Teoría de Sistemas. Para este fin, se deducen las ecuaciones de ondas y la ecuación de Helmholtz. Haciendo uso del operador de Dirac desplazado y su estrecha relación con los principales operadores del cálculo vectorial, es posible establecer una conexión directa entre las soluciones del sistema de Maxwell tiempo-armónico y dos ecuaciones cuaterniónicas. Además, se expone la aplicación de la condición de Lorentz para transformar el sistema de Maxwell tiempo-armónico en una simple ecuación cuaterniónica en función de los potenciales escalar y vectorial. Palabras clave: Sistema de Maxwell, propiedad emergente, ecuación de ondas, condición de Lorentz, operador de Dirac. ABSTRACT This article deals with the study of electromagnetic waves equations and the Lorentz condition, as emergent properties of Maxwell's system in the context of systems theory. To do this, the wave equations and the Helmholtz equation are first deduced. Using the displaced Dirac operator, which is closely related to the main vector calculation operators, it is possible to establish a direct connection between the solutions of the Maxwell time-harmonic system and two quaternion equations. Also, the application of the Lorentz condition to transform the time-harmonic Maxwell system into a simple quaternion equation based on the scalar and vector potentials is exposed. INTRODUCCIÓN El concepto de emergencia tiene su origen en 1862, cuando en el libro "A system of logic", del filósofo inglés John Stuart Mill, aparece la idea de que la interacción y yuxtaposición de las partes que conforman un sistema, resulta insuficiente para entender y explicar las propiedades del sistema. El nombre del concepto se deriva de "emergere", que en latín significa "salir de", y se atribuye a los
... El Análisis Cuaterniónico fue iniciado por Rudolf Fueter en [10]. Luego, otros autores se interesaron en establecer nuevos enfoques para desarrollar las ideas básicas de esta teoría [11][12][13]. ...
... Tomando como base el sistema anterior, podemos obtener las ecuaciones de ondas [9], para ello el operador rotacional es aplicado en ambos lados de las ecuaciones (7) (11)(12) De acuerdo con las ecuaciones (9) y (10) la divergencia de E y B es cero, y sustituyendo las ecuaciones (7) y (8) en (11)(12), obtenemos las ecuaciones de ondas para los campos eléctrico y magnético. ...
... Tomando como base el sistema anterior, podemos obtener las ecuaciones de ondas [9], para ello el operador rotacional es aplicado en ambos lados de las ecuaciones (7) (11)(12) De acuerdo con las ecuaciones (9) y (10) la divergencia de E y B es cero, y sustituyendo las ecuaciones (7) y (8) en (11)(12), obtenemos las ecuaciones de ondas para los campos eléctrico y magnético. ...
This article deals with the study of electromagnetic waves equations and the Lorentz condition, as emergent properties of Maxwell's system in the context of systems theory. To do this, the wave equations and the Helmholtz equation are first deduced. Using the displaced Dirac operator, which is closely related to the main vector calculation operators, it is possible to establish a direct connection between the solutions of the Maxwell time-harmonic system and two quaternion equations. Also, the application of the Lorentz condition to transform the time-harmonic Maxwell system into a simple quaternion equation based on the scalar and vector potentials is exposed.
... Two of the triplet of functions should be thought of as a single vector valued object, where their vector structure characterises the orientation of the local variations. The monogenic wavelets and wavelet transform are best represented using quaternion [12], rather than, real or complex numbers. Each triplet is therefore considered as a positive real valued amplitude, a pure unit quaternion specifying a direction, and a phase. ...
... We additionally define the two dimensional Fourier transform in terms of any unit quaternion as G eq (f ) = exp(2e q πf x), g , so that the regular Fourier transform corresponds to G j (f ) ≡ G(f ). For more notes on quaternion algebra see [12]. We retain here only the briefest possible usage of the quaternion algebra, necessary for clarity of exposition, and stress that all implementation is discussed in terms of real vector quantities. ...
We define a set of operators that localise a radial image in radial space and radial frequency simultaneously. We find the eigenfunctions of this operator and thus define a non-separable orthogonal set of radial wavelet functions that may be considered optimally concentrated over a region of radial space and radial scale space, defined via a doublet of parameters. We give analytic forms for their energy concentration over this region. We show how the radial function localisation operator can be generalised to an operator, localising any square integrable function in two dimensional Euclidean space. We show that the latter operator, with an appropriate choice of localisation region, approximately has the same eigenfunctions as the radial operator. Based on the radial wavelets we define a set of quaternionic valued wavelet functions that can extract local orientation for discontinuous signals and both orientation and phase structure for oscillatory signals. The full set of quaternionic wavelet functions are component wise orthogonal; hence their statistical properties are tractable, and we give forms for the variability of the estimates of the local phase and orientation, as well as the local energy of the image. By averaging estimates across wavelets, a substantial reduction in the variance is achieved.
... The most known attempt to build a Q-analogue of C-analysis belongs to R. Fueter [17,18], while in [19,20], the authors tried to use the corresponding holomorphic conditions in the framework of chiral and gauge models. Other attempts have also been undertaken, in particular, by A. Deavours [21], A. Yu. Khrennikov [22], Schwartz [23] and others. ...
... vanishes. These correspond to the merging of a pair of branches of the multi-valued field ξ(Z) or, equivalently, to multiple roots of the generating algebraic system (21). At such singular points, the derivatives ∂ D B ξ A , in view of (23), become infinite. ...
We briefly present our version of noncommutative analysis over matrix algebras, the algebra of biquaternions (B) in particular. We demonstrate that any B-differentiable function gives rise to a null shear-free congruence (NSFC) on the B-vector space CM and on its Minkowski subspace M. Making use of the Kerr–Penrose correspondence between NSFC and twistor functions, we obtain the general solution to the equations of B-differentiability and demonstrate that the source of an NSFC is, generically, a world sheet of a string in CM. Any singular point, caustic of an NSFC, is located on the complex null cone of a point on the generating string. Further we describe symmetries and associated gauge and spinor fields, with two electromagnetic types among them. A number of familiar and novel examples of NSFC and their singular loci are described. Finally, we describe a conservative algebraic dynamics of a set of identical particles on the “Unique Worldline” and discuss the connections of the theory with the Feynman–Wheeler concept of “One-Electron Universe”.
... Useful review articles have been published by Deavours (1973), Sudbery (1979), and Fokas and Pinotsis (2007). These discuss the derivation of Fueter's Theorem, as well as the conditions under which the Taylor or Laurent series exist, together with a corresponding residue theorem and a generalisation of Liouville's Theorem. ...
... The latter establishes that any regular quaternionic function which is bounded across the whole of quaternion ℝ 4 space is a constant. For further details on these matters see Deavours (1973) or Fokas and Pinotsis (2007). ...
It is well known that there is an integral theorem for quaternion-valued functions analogous to Cauchys Theorem for complex-valued functions, namely Fueters Theorem. The class of quaternionic functions for which this applies are generally referred to as regular functions, and these provide the most productive means of generalising the class of holomorphic complex functions. This paper derives a second integral theorem, also analogous to Cauchys Theorem, and which is believed to be quite distinct from that of Fueter, despite appearances. The paper takes the opportunity to present the basis of the derivation of both theorems, and also their extension to the associated classes of right-regular and conjugate regular functions. Both theorems can also be extended into the biquaternionic domain in which the four quaternion coordinates may be complex valued. This is of interest in physics as the Hermitian biquaternions have a natural norm which is Minkowskian and provide an elegant formalism for Lorentz transformations.
... The roots of quaternionic analysis come from the works of Moisil and Théodoresco (see [1]), and Fueter (see [2,3]). Deavours published the first survey [4] of Fueter's works at the beginning of 70s. The idea is to found a well-defined and explicit counter-part for complex function theory on the plane. ...
... indeed SU (2), Spin(3), Spin(4), SO(3) and SO (4). In the spirit of Clifford analysis, Cauchy-Riemann operators are Spin(4) invariant under L and H actions (see Section 3). ...
This survey-type paper deals with the symmetries related to quaternionic analysis. The main goal is to formulate an SU (2) invariant version of the theory. First, we consider the classical Lie groups related to the algebra of quaternions. After that, we recall the classical Spin(4) invariant case, that is Cauchy–Riemann operators, and recall their basic properties. We define the SU (2) invariant operators called the Coifman– Weiss operators. Then we study their relations with the classical Cauchy–Riemann operators and consider the factorization of the Laplace operator. Using SU (2) invariant harmonic polynomials, we obtain the Fourier series representations for quaternionic valued functions studying in detail the matrix coefficients.
... In the next twelve years Fueter and his collaborators developed the theory of quaternionic analysis. A complete bibliography of this work is contained in ref. [6], and a simple account (in English) of the elementary parts of the theory has been given by Deavours [7]. ...
... Proof [7] The argument of theorem 8 shows that ...
This is the original version of the paper Quaternionic Analysis which was published in Math. Proc.Camb. Phil. Soc. in 1979. It was edited for publication, but the original version contains details which readers might find helpful.
... For more details on Quaternionic calculus in Hilbert spaces, the reader can refer to [1], [8], [13] and [18]. ...
This paper aims to explore the concept of continuous K -frames in quaternionic Hilbert spaces. First, we investigate K -frames in a single quaternionic Hilbert space , where K is a right -linear bounded operator acting on . Then, we extend the research to two quaternionic Hilbert spaces, and , and study -frames for the super quaternionic Hilbert space , where and are right -linear bounded operators on and , respectively. We examine the relationship between the continuous -frames and the continuous -frames for and the continuous -frames for . Additionally, we explore the duality between the continuous -frames and the continuous - and -frames individually.
... The properties of the associated operators of a frame will also be provided. For more details on Quaternionic calculus on Hilbert spaces, the reder can refer to [1], [9], [11], [15]. Furthermore, it turns out that √ L commutes with every operator that commutes with L. ...
The aim of this paper is to study K-frames for quaternionic Hilbert spaces. First, we present the quaternionic version of Douglas's theorem and then investigate K-frames for a quaternionic Hilbert space , where . Given two quaternionic Hilbert spaces and , along with two right -linear bounded operators and , we study the -frames for the super space and their relationship with -frames and -frames for and , respectively. We also explore the -duality in relation to -duality and -duality.
... Remark 1. Unlike to ordinary frames, the pre-frame operator of a K-frame is not surjective, the transform opeartor is not injective with closed range and the frame operator is not invertible. For more details on Quaternionic calculus in Hilbert spaces, the reader can refer to [1], [7], [12] and [17]. ...
The aim of this work is to study frame theory in quaternionic Hilbert spaces. We provide a characterization of frames in these spaces through the associated operators. Additionally, we examine frames of the form , where L is a right -linear bounded operator and is a frame.
... In addition, it is desirable that the activation function were analytic so that gradient descendant techniques can be applied in the training stage. However, the non-analytic condition can only be satisfied by some linear and constant functions [40,41]. A typical way to circumvent this problem is the use of quaternion splits functions, i.e. a mapping f : H → H, such that: ...
In recent years, several models using Quaternion-Valued Convolutional Neural Networks (QCNNs) for different problems have been proposed. Although the definition of the quaternion convolution layer is the same, there are different adaptations of other atomic components to the quaternion domain, e.g., pooling layers, activation functions, fully connected layers, etc. However, the effect of selecting a specific type of these components and the way in which their interactions affect the performance of the model still unclear. Understanding the impact of these choices on model performance is vital for effectively utilizing QCNNs. This paper presents a statistical analysis carried out on experimental data to compare the performance of existing components for the image classification problem. In addition, we introduce a novel Fully Quaternion ReLU activation function, which exploits the unique properties of quaternion algebra to improve model performance.
... It turns out that regular functions yield appropriate function spaces for extending complex analysis in the quaternion setting. The articles by H. Haefli, Haefli [29], 1947, V. Iftimie, Iftimie [34], 1965, C. A. Deavours, Deavours [15], 1973, and the already mentioned article by A. Sudbery, Sudbery [67], 1979, contributed over the years to the dissemination of Fueter's work and the development of quaternionic analysis. ...
This article is an updated version of a chapter with the same title published in the first edition of the handbook Operator Theory, Springer Verlag, 2015. The new version, prepared for the second edition, includes new proofs and additional references.
The first part of the article is a brief report on the history of quaternionic and Clifford analysis, two research fields that generalize single variable complex analysis to higher dimension. Appropriate counterparts of the Cauchy-Riemann operator and the Cauchy kernel defined in the frameworks of Hamilton quaternions and Clifford algebras yield spaces of regular and monogenic functions that extend the concept of holomorphic functions.
Part two introduces Cauchy-Pompeiu and Bochner-Martinelli-Koppelman integral representation formulas with remainders for operator-kernel couples consisting of a first order differential operator on a Euclidean space with coefficients in a unital Banach algebra, and a smooth homogeneous algebra valued kernel. The general formulas underscore the role played by Dirac, Cauchy-Riemann, and Laplace operators in Clifford analysis.
The third part of the article is concerned with sharp estimates of fractional integral transforms in a Banach algebra setting, and quantitative Hartogs-Rosenthal theorems on uniform approximation of continuous functions on compact sets of Euclidean spaces by solutions of operator-kernel couples which, in particular include regular and monogenic functions.
... Quaternion(Deavours 1973) forms a four-dimensional union algebra over the real numbers belonging to the class of Clifford algebras. A quaternion Q Î is expressed by one real scalar component and three 0 , m 1 , m 2 , m 3 ä. ...
Objective: Due to non-invasive imaging and the multimodality of Magnetic Resonance Imaging (MRI) images, MRI-based multi-modal brain tumor segmentation (MBTS) studies are attracting more and more attention in recent years. With the great success of convolutional neural networks (CNN) in various computer vision tasks, lots of MBTS models have been proposed to address the technical challenges of MBTS. However, the problem of limited data collection usually exists in MBTS tasks, making existing studies usually have difficulty in fully exploring the multi-modal MRI images to mine complementary information among different modalities.
Approach: We propose a novel Quaternion Mutual Learning Strategy (QMLS), which consists of a voxel-wise lesion knowledge mutual learning mechanism (VLKML mechanism) and a quaternion multimodal feature learning module (QMFL module). Specifically, VLKML mechanism allows the networks to converge to a robust minimum so that aggressive data augmentation techniques can be applied to fully expand the limited data. In particular, quaternion-valued QMFL module treats different modalities as components of quaternions to sufficiently learn complementary information among different modalities on the hypercomplex domain while significantly reduces the number of parameters by about 75\%.
Main results: Extensive experiments on the dataset BraTS 2020 and BraTS 2019 indicate that QMLS achieves superior results to current popular methods with less computational cost.
Significance: We propose a novel algorithm to brain tumor segmentation task and achieves better performance with fewer parameters, which helps the clinical application of automatic brain tumor segmentation.
... Quaternions are a typical example of hypercomplex number systems, which were first discovered by Hamilton in 1843 [18]. Since then, the quaternions have been extensively studied and applied in various fields of mathematics including contact and spin geometry [3,26], function theory [11,38], non-commutative algebra [6,7,24], linear algebra [47], and algebraic topology [20]. On the other side, since the group of unit quaternions is isomorphic to the group SU(2) consisting of 2 × 2 special unitary matrices, they provide us a natural way to represent spatial rotations. ...
In this paper, we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient than other existing methods. Moreover, our algorithm is optimal in the sense of the least number of complex inversions. On the practice side, our algorithm outperforms existing algorithms on randomly generated matrices. We argue that this algorithm can be used to improve the practical utility of recursive Strassen-type algorithms by providing the fastest possible base case for the recursive decomposition process when applied to quaternion matrices.
... Accordingly, some common real-valued functions, like the hyperbolic tangent and sigmoid, will diverge to infinity when they are extended to the complex domain [89]; this makes them unsuitable for representing the behavior of a biological neuron. A similar problem arises in the quaternion domain: "the only quaternion function regular with bounded norm in E 4 is a constant" [91], where E 4 stands for a 4dimensional Euclidean space. ...
Since their first applications, Convolutional Neural Networks (CNNs) have solved problems that have advanced the state-of-the-art in several domains. CNNs represent information using real numbers. Despite encouraging results, theoretical analysis shows that representations such as hyper-complex numbers can achieve richer representational capacities than real numbers, and that Hamilton products can capture intrinsic interchannel relationships. Moreover, in the last few years, experimental research has shown that Quaternion-Valued CNNs (QCNNs) can achieve similar performance with fewer parameters than their real-valued counterparts. This paper condenses research in the development of QCNNs from its very beginnings. We propose a conceptual organization of current trends and analyze the main building blocks used in the design of QCNN models. Based on this conceptual organization, we propose future directions of research.
... With no claim of completeness, for a more extended introduction to quaternionic analysis we refer to the books [17][18][19]32,33] and the references therein. However, the classical works on the subject go back as far as [8,12,21,38,50]. ...
The main goal of this paper is to construct a proportional analogues of the quaternionic fractional Fueter-type operator recently introduced in the literature. We start by establishing a quaternionic version of the well-known proportional fractional integral and derivative with respect to a real-valued function via the Riemann-Liouville fractional derivative. As a main result, we prove a quaternionic proportional fractional Borel-Pompeiu formula based on a quaternionic proportional fractional Stokes formula. This tool in hand allows us to present a Cauchy integral type formula for the introduced quater-nionic proportional fractional Fueter-type operator with respect to a real-valued function.
... Quaternions are a typical example of hypercomplex number systems, which was first discovered by Hamilton in 1843 [18]. Since then, the quaternions have been extensively studied and applied in various fields of mathematics including contact and spin geoemtry [3,26], function theory [38,11], non-commutative algebra [24,6,7], linear algebra [47] and algebraic topology [20]. On the other side, since the group of unit quaternions is isomorphic to the group SU(2) consisting of 2 × 2 special unitary matrices, they provide us a natural way to represent spatial rotations. ...
In this paper we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient that other existing methods. Moreover, our algorithm is optimal in the sense of the least number of complex inversions. On the practice side, our algorithm outperforms existing algorithms on randomly generated matrices. We argue that this algorithm can be used to improve the practical utility of recursive Strassen-type algorithms by providing the fastest possible base case for the recursive decomposition process when applied to quaternion matrices.
... Thus, consider the Hamilton's biquaternions, the complexification of the quaternions. Complexification of the quaternions appears in Devours [1] and Imaeda [2] for Maxwell's equations. The biquaternions lose the use of Fueter's form of Cauchy's integral formula [3,4] (due to lack of generic biquaternion inverses), but this is offset by an alternate scheme, resulting in linear combinations of the original, complex Cauchy's integral formula demonstrated here. ...
This paper presents a three-step program for extension of functions of complex analysis to the biquaternions by means of Cauchy’s integral formula: I. Investigate biquaternion bases consisting of roots of . A complex valued standard function (standardization factor) determines roots of . A root of with a non-zero imaginary part, can uniquely determine a biquaternion ortho-standard basis. II. A single reference basis element determines two subspaces, one the span of scalars and the reference element, the other pure vector biquaternions orthogonal to the reference. The subspaces represent the distinct parts of the generalized Cayley-Dickson form. The Peirce decomposition projects into two subspaces: one is the span of the related idempotents and the other of the nilpotents. III. Using invertible elements in each of these subspaces, biquaternion functional extensions of holomorphic functions follow by Cauchy’s integral formula. Extensions retain analyticity in each biquaternion component. Cauchy integral formula uses separate idempotent and nilpotent representations of biquaternion reciprocals to define holomorphic function extensions. The Peirce projections allow extension to all viable biquaternions.
... First of all, we review some facts of quaternions, which are required throughout the paper. Relevant knowledge can be found in [17][18][19]. Write: ...
Let L2(R,H) denote the space of all square integrable quaternionic-valued functions. In this article, let Φ∈L2(R,H). We consider the perturbation problems of wavelet frame {Φm,n,a0,b0,m,n∈Z} about translation parameter b0 and dilation parameter a0. In particular, we also research the stability of irregular wavelet frame {SmΦ(Smx−nb),m,n∈Z} for perturbation problems of sampling.
... 9-11 So far, motivated by these works, mathematicians have been interested in developing various approaches along classical lines in the study of this theory, see for example, other studies. [12][13][14] The Moisil-Teodorescu operator (a determined first-order elliptic operator that can be expressed in terms of the usual divergence, gradient and curl operators, see Moisil 15 ) is nowadays considered to be a good analogue of the usual Cauchy-Riemann operator of complex analysis in the framework of quaternionic analysis, and it is a square root of the scalar Laplace operator in R 3 . For a close relationship of the Moisil-Teodorescu operator with many spatial models of mathematical physics we refer the reader to other studies. ...
The Moisil‐Teodorescu operator is considered to be a good analogue of the usual Cauchy–Riemann operator of complex analysis in the framework of quaternionic analysis and it is a square root of the scalar Laplace operator in . In the present work, a general quaternionic structure is developed for the local fractional Moisil–Teodorescu operator in Cantor‐type cylindrical and spherical coordinate systems. Furthermore, in order to reveal the capacity and adaptability of the methods, we show two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets by making use of the local fractional Moisil–Teodorescu operator.
... So far, motivated by these works, mathematicians became interested in developing various approaches along classical lines in the study of this theory, see e.g. [12][13][14]. ...
In this paper, general quaternionic structure are developed for the local fractional Moisil-Teodorescu operator in Cantor-type cylindrical and spherical coordinate systems. Two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets are shown by making use of this local fractional Moisil-Teodorescu operator.
... In 1935, Fueter [2] defined regular quaternionic functions in R 4 . Later Deavours [1] and Subdery [12] developed quaternionic analysis, based on complex analysis. ...
We give the definition of hyperholomorphic pseudocomplex functions, i. e., functions with values in a special form of quaternions, and propose the necessary variables, functions, and Dirac operators to describe the Cauchy integral theorem and the generalized Cauchy-Riemman system. We investigate the properties and corollaries corresponding to the Cauchy integral theorem for the pseudo-complex number system discussed in this paper.
... Fueter and his school were able to obtain a quaternionic counterpart for regular functions of many classical theorems in complex analysis in one variable such as Cauchy's theorem, Liouville's theorem and Laurent series expansion. (See [3] for a survey of these results). Since then, the theory of regular functions has been greately developed, and the parallel with holomorphic functions has been extended to several variables. ...
We prove that a function of several quaternionic variables is regular in the sense of Fueter if and only if it is regular in each variable separately, thus providing a quaternionic analog of a celebrated theorem of Hartogs. We also establish a result similar in spirit to the Hanges and Tr\`eves theorem, showing that a disc contained in the boundary of a domain is a propagator of regular extendibility across the boundary.
In this chapter, firstly, we show that the quaternionic spike neural networks in conjunction with the training algorithm, working on quaternion algebra they outperform the real-valued spike neural networks. They are ideal for applications in the area of neurocontrol of manipulators with angular movements. Secondly, we present an adaptive control in medical robotics for haptic devices using the quaternion wavelet network.
Since their first applications, Convolutional Neural Networks (CNNs) have solved problems that have advanced the state-of-the-art in several domains. CNNs represent information using real numbers. Despite encouraging results, theoretical analysis shows that representations such as hyper-complex numbers can achieve richer representational capacities than real numbers, and that Hamilton products can capture intrinsic interchannel relationships. Moreover, in the last few years, experimental research has shown that Quaternion-valued CNNs (QCNNs) can achieve similar performance with fewer parameters than their real-valued counterparts. This paper condenses research in the development of QCNNs from its very beginnings. We propose a conceptual organization of current trends and analyze the main building blocks used in the design of QCNN models. Based on this conceptual organization, we propose future directions of research.
In this paper we study regular (or holomorphic) functions of several quaternionic variables. We give an integral representation formula of Bochner-Martinelli type, and prove, among others, analogs of a theorem of Hartogs and the Plemelj formulas
The Hayman Theorem of left-monogenic function in a Clifford Analysis Setting is established in this article. A few established conclusions regarding subharmonic functions in Euclidean half space are extended to Clifford half space.
A control strategy containing Lyapunov functions is proposed in this paper. Based on this strategy, the fixed-time synchronization of a time-delay quaternion-valued neural network (QVNN) is analyzed. This strategy is extended to the prescribed-time synchronization of the QVNN. Furthermore, an improved two-step switching control strategy is also proposed based on this flexible control strategy. Compared with some existing methods, the main method of this paper is a non-decomposition one, does not contain a sign function in the controller, and has better synchronization accuracy. Two numerical examples verify the above advantages.
In this study, we propose a non-decomposing method, the implicit Lyapunov function method, for the finite-time stabilization of a class of quaternion-valued neural networks (QVNNs) with time delays. Our approach has certain advantages, including more relaxed constraints on time delays, more flexible controllers without sign functions, and more general activation functions. Specifically, the derivative of an implicit Lyapunov function with quaternions is analyzed. Some properties of the implicit Lyapunov function, such as positive definiteness, monotonicity, and radial unboundedness, are deduced. With regard to two control schemes, two sets of sufficient conditions are provided for the finite-time stabilization of the QVNN, and an improved adaptive control strategy is designed to enhance stabilization efficiency. Two numerical examples illustrate the correctness, applicability and effectiveness of the method.
Various attempts have been made in defining the derivative of a quaternionic function due to the noncommutativity of the product over quaternions. We observe that the difference in the left and right operations caused by the noncommutativity of the quaternion product is determined by the vector part of the quaternion. In this paper, we propose a corresponding derivative to replace the derivative of a quaternion-valued function of a quaternionic variable using the component terms of a quaternion. Further, the analogous constant, product, and quotient rules for the proposed calculations are given. Application of the proposed derivatives is provided to compute the derivatives of elementary functions. Several illustrations are also presented.
In this study, we focused on n-dimensional quaternionic space Hn. To create the module structure, first part is devoted to define a metric depending on the product order relation of Rn. The set of Hn has been rewritten with a different representation of n-vectors. Using this notation, formulations corresponding to the basic operations in Hn are obtained. By adhering these representations, module structure of Hn over the set of real ordered n-tuples is given. Afterwards, we gave limit, continuity and the derivative basics of quaternion valued functions of a real variable.
M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., (8) 23 (1957), 220–225 as well as some comments and historical remarks.
Let [Formula: see text] be a space of square integrable quaternionic-valued functions defined on real line. In this paper, for [Formula: see text] [Formula: see text] and [Formula: see text], if the sequence of functions [Formula: see text] is a wavelet frame of [Formula: see text], we study the stability of the wavelet frame when the sampling [Formula: see text] or the mother wavelet [Formula: see text] has perturbation by using the theory of wavelet analysis.
For functions of two quaternionic variables that are regular in the sense of Fueter, we establish a result similar in spirit to the Hanges and Trèves theorem. Namely, we show that a ball contained in the boundary of a domain is a propagator of regular extendability across the boundary.
Same description as that for the Greek version of the thesis, except in English for help with International Researchers. Cantor Bouquets are exponentially powered Cracks that move unpredictably on the Hilbert Kernel. Analytic profil of the topology in the Complex Plane of the iterated exponential map, with companion Maple code which shows Fatou and Julia domains.
Cantor Bouquets as they result from iterating the exponential function. article analyzing the topology and Maple code for producing the resultant pictures. Thesis submitted in the Mathematics dept. at the Agricultural University of Athens in 2019. Revision submitted in 2020.
Quaternions discovered by W. R. Hamilton made a great contribution to the progress in noncommutative algebra and vector analysis. However, the analysis of quaternion functions has not been duly developed. The matter is that the notion of a derivative of quaternion functions of a quaternion variable has not been known until recently. The author has succeeded in improving the situation. The present work contains an account of the results obtained by him in this direction. The notion of an ℍ-derivative is introduced for quaternion functions of a quaternion variable. The existence of an ℍ-derivative of elementary functions is established retaining the well-known formulas for the corresponding functions from complex (real) analysis. The rules on the ℍ-differentiation of a sum, a product, and an inverse function are formulated and proved. Necessary and sufficient conditions for the existence of an ℍ-derivative are established. The notions of ℂ²-differentiation and ℂ²-holomorphy are introduced for quaternion functions of a quaternion variable. Three equivalent conditions are found, each of them being a necessary and sufficient one for ℂ²-differentiation. Representations by an integral and a power series are given for ℂ²-holomorphic functions. It is proved that the harmonicity of functions f(z), z · f(z), and f(z) · z is the necessary and sufficient condition for a function f to be Fueter-regular.
A formulation of quaternionic quantum mechanics (QM) is presented in terms of a real Hilbert space. Using a physically motivated scalar product, we prove the spectral theorem and obtain a novel quaternionic Fourier series. After a brief discussion on unitary operators in this formalism, we conclude that this quantum theory is indeed consistent, and can be a valuable tool in the search for new physics.
In this work, by reformulating screw theory (generalization of quaternions) in the conformal geometric algebra framework, we address the interpolation, virtual reality, graphics engineering, haptics. We derive intuitive geometric equations to handle surface operations like in kidney surgery. The interpolation can handle the interpolation and dilation in 3D of points, lines, planes, circles and spheres. With this procedure, we interpolate trajectories of surgical instrument. Using quaternions, we formulate the quaternion spike neural network for control. This new neural network structure is based on Spike Neural Networks and developed using the quaternion algebra. The real valued training algorithm was extended so that it could make adjustments of the weights according to the properties and product of the quaternion algebra. In this spike neural network, we are taking into account two relevant ideas the use of Spike neural network which is the best model for oculo-motor control and the role of geometric computing. As illustration. the quaternion spike neural network is applied for control of robot manipulator. The experimental analysis shows promising possibilities for the use of this powerful geometric language to handle multiple tasks in human–machine interaction and robotics.
A novel quaternion-valued least-mean kurtosis (QLMK) adaptive filtering algorithm is proposed for three- and four-dimensional processes by using the recent generalised Hamilton-real (GHR) calculus. The proposed QLMK algorithm based GHR calculus minimises the negated kurtosis of the error signal as a cost function in the quaternion domain, thus provides an elegant way to solve a trade-off problem between the convergence rate and steady-state error. Moreover, the proposed QLMK algorithm has naturally a robust behaviour for a wide range of noise signals due to its kurtosis-based cost function. Furthermore, the steady-state performance of the proposed QLMK algorithm is analysed to obtain convergence and misadjustment conditions. The comprehensive simulation results on benchmark and real-world problems show that the use of this cost function defined by the quaternion statistics in the proposed QLMK algorithm allows us to process quaternion-valued signals and thus, significantly enhances the performance of the adaptive filter in terms of both the steady-state error and the convergence rate, as compared with the quaternion-valued least-mean-square algorithm based on the recent GHR calculus.
Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.
We define the nth root of unity in quaternion space, and then we define the discrete quaternion Fourier transform. We use a first-order quaternion filter for implementing a fourth-order real coefficient filter.
We study the relations between the quaternion H-type group and the boundary of the unit ball on the two-dimensional quaternionic space. The orthogonal projection of the
space of square integrable functions defined on quaternion H-type group into its subspace of boundary values of q-holomorphic functions is considered. The precise form of Cauchy-Szegö kernel and the orthogonal projection operator is obtained.
The fundamental solution for the operator Δλ is found.
We extend previous results on the exponential off-diagonal decay of the
entries of analytic functions of banded and sparse matrices to the case where
the matrix entries are elements of a -algebra.
- Haefeli H.
Über vierfachperiodische Funktionen
- R Fueter