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Journal articles on the topic 'Quaternion calculus'
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1
Xu, Dongpo, Cyrus Jahanchahi, Clive C. Took, and Danilo P. Mandic. "Enabling quaternion derivatives: the generalized HR calculus." Royal Society Open Science 2, no. 8 (August 2015): 150255. http://dx.doi.org/10.1098/rsos.150255.
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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.
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Xu, Dongpo, Hua Gao, and Danilo P. Mandic. "A new proof of the generalized Hamiltonian–Real calculus." Royal Society Open Science 3, no. 9 (September 2016): 160211. http://dx.doi.org/10.1098/rsos.160211.
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The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative rules. This further reinforces the theoretical foundation of the GHR calculus and provides a convenient methodology for generic extensions of real- and complex-valued learning algorithms to the quaternion domain.
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Takahashi, Kazuhiko, Eri Tano, and Masafumi Hashimoto. "Feedforward–Feedback Controller Based on a Trained Quaternion Neural Network Using a Generalised HR Calculus with Application to Trajectory Control of a Three-Link Robot Manipulator." Machines 10, no. 5 (May 2, 2022): 333. http://dx.doi.org/10.3390/machines10050333.
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This study derives a learning algorithm for a quaternion neural network using the steepest descent method extended to quaternion numbers. This applies the generalised Hamiltonian–Real calculus to obtain derivatives of a real–valued cost function concerning quaternion variables and designs a feedback–feedforward controller as a control system application using such a network. The quaternion neural network is trained in real-time by introducing a feedback error learning framework to the controller. Thus, the quaternion neural network-based controller functions as an adaptive-type controller. The designed controller is applied to the control problem of a three-link robot manipulator, with the control task of making the robot manipulator’s end effector follow a desired trajectory in the Cartesian space. Computational experiments are conducted to investigate the learning capability and the characteristics of the quaternion neural network used in the controller. The experimental results confirm the feasibility of using the derived learning algorithm based on the generalised Hamiltonian–Real calculus to train the quaternion neural network and the availability of such a network for a control systems application.
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Condurache, Daniel, Mihail Cojocari, and Ionuţ Popa. "Hypercomplex Quaternions and Higher-Order Analysis of Spatial Kinematic Chains." BULETINUL INSTITUTULUI POLITEHNIC DIN IAȘI. Secția Matematica. Mecanică Teoretică. Fizică 69, no. 1-4 (December 1, 2023): 21–34. http://dx.doi.org/10.2478/bipmf-2023-0002.
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Abstract This paper introduces a novel computational method for analyzing the higher-order acceleration field of spatial kinematics chains. The method is based on vector and quaternionic calculus, as well as dual and multidual algebra. A closed-form coordinate-free solution generated by the morphism between the Lie group of rigid body displacements and the unit multidual quaternions is presented. Presented solution is used for higher-order kinematics investigation of lower-pair serial chains. Additionally, a general method for studying the vector field of arbitrary higher-order accelerations is discribed. The method utilizes the “automatic differentiation” feature of multidual and hyper-multidual functions to obtain the higher-order derivative of a rigid body pose without need in further differentiation of the body pose regarding time. Also is proved that all information regarding the properties of the distribution of higher-order accelerations is contained in the specified unit hyper-multidual quaternion.
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Chen, Shenglong, Hong-Li Li, Leimin Wang, Cheng Hu, Haijun Jiang, and Zhiming Li. "Finite-time adaptive synchronization of fractional-order delayed quaternion-valued fuzzy neural networks." Nonlinear Analysis: Modelling and Control 28 (June 19, 2023): 1–20. http://dx.doi.org/10.15388/namc.2023.28.32505.
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Based on direct quaternion method, this paper explores the finite-time adaptive synchronization (FAS) of fractional-order delayed quaternion-valued fuzzy neural networks (FODQVFNNs). Firstly, a useful fractional differential inequality is created, which offers an effective way to investigate FAS. Then two novel quaternion-valued adaptive control strategies are designed. By means of our newly proposed inequality, the basic knowledge about fractional calculus, reduction to absurdity as well as several inequality techniques of quaternion and fuzzy logic, several sufficient FAS criteria are derived for FODQVFNNs. Moreover, the settling time of FAS is estimated, which is in connection with the order and initial values of considered systems as well as the controller parameters. Ultimately, the validity of obtained FAS criteria is corroborated by numerical simulations.
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Li, Yongkun, and Xiaofang Meng. "Existence and Global Exponential Stability of Pseudo Almost Periodic Solutions for Neutral Type Quaternion-Valued Neural Networks with Delays in the Leakage Term on Time Scales." Complexity 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/9878369.
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We propose a class of neutral type quaternion-valued neural networks with delays in the leakage term on time scales that can unify the discrete-time and the continuous-time neural networks. In order to avoid the difficulty brought by the noncommutativity of quaternion multiplication, we first decompose the quaternion-valued system into four real-valued systems. Then, by applying the exponential dichotomic theory of linear dynamic equations on time scales, Banach’s fixed point theorem, the theory of calculus on time scales, and inequality techniques, we obtain some sufficient conditions on the existence and global exponential stability of pseudo almost periodic solutions for this class of neural networks. Our results are completely new even for both the case of the neural networks governed by differential equations and the case of the neural networks governed by difference equations and show that, under a simple condition, the continuous-time quaternion-valued network and its corresponding discrete-time quaternion-valued network have the same dynamical behavior for the pseudo almost periodicity. Finally, a numerical example is given to illustrate the feasibility of our results.
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Xu, Dongpo, Lina Zhang, and Huisheng Zhang. "LEARNING ALGORITHMS IN QUATERNION NEURAL NETWORKS USING GHR CALCULUS." Neural Network World 27, no. 3 (2017): 271–82. http://dx.doi.org/10.14311/nnw.2017.27.014.
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Pletinckx, Daniel. "Quaternion calculus as a basic tool in computer graphics." Visual Computer 5, no. 1-2 (January 1989): 2–13. http://dx.doi.org/10.1007/bf01901476.
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Danielewski, Marek, and Lucjan Sapa. "Foundations of the Quaternion Quantum Mechanics." Entropy 22, no. 12 (December 17, 2020): 1424. http://dx.doi.org/10.3390/e22121424.
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We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.
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Shen, Shiping, Bing Li, and Yongkun Li. "Anti-Periodic Dynamics of Quaternion-Valued Fuzzy Cellular Neural Networks with Time-Varying Delays on Time Scales." Discrete Dynamics in Nature and Society 2018 (June 28, 2018): 1–14. http://dx.doi.org/10.1155/2018/5290786.
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A class of quaternion-valued fuzzy cellular neural networks with time-varying delays on time scales is proposed. Based on inequality analysis techniques on time scales, a fixed point theorem and the theory of calculus on time scales, the existence, and global exponential stability of anti-periodic solutions for this class of neural networks are established. The obtained results are completely new and supplement to the known results. Finally, a numerical example is given to illustrate the feasibility of our results.
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Arthurs, A. M., and G. R. Walsh. "On Hammersley's minimum problem for a rolling sphere." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 3 (May 1986): 529–34. http://dx.doi.org/10.1017/s0305004100064471.
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AbstractThe problem posed by Hammersley (1983) of finding the shortest path along which a sphere can roll from one prescribed state to another is formulated by using quaternion calculus of variations and optimal control theory. This leads to a system of coupled nonlinear differential equations with prescribed end conditions. From the resulting expression for the curvature, it is shown that the differential equation of the required path in intrinsic coordinates is the same as the equation of motion of a simple pendulum, giving a solution in terms of elliptic integrals.
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Chelaru, Teodor-Viorel, Valentin Pană, and Costin Ene. "Performance Evaluation for Launcher Testing Vehicle." Aerospace 9, no. 9 (September 9, 2022): 504. http://dx.doi.org/10.3390/aerospace9090504.
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The paper’s purpose is to present a calculus model for a testing vehicle that can be used to validate guidance, navigation and control systems for reusable launchers in all flight phases. The technical solution is based on a throttleable engine with thrust vectoring control and a reaction control system (RCS) used for roll. For calculus, we will develop a nonlinear model with six degrees of freedom, based on quaternion, extended with nonlinear equations that use pulse modulation in order to control roll. In order to synthesize the controller, we also develop a linear model similar to the launcher model. The paper analyzes two basic scenarios, first with the ascending and the descending flight phases and the second having a horizontal flight interleaved between ascending and descending flight phases, both scenarios being specific for reusable launchers. Based on these scenarios, the paper evaluates some performances of the proposed vehicle, namely flight envelope and guidance accuracy.
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Mengüç, Engin Cemal. "Novel quaternion‐valued least‐mean kurtosis adaptive filtering algorithm based on the GHR calculus." IET Signal Processing 12, no. 4 (June 2018): 487–95. http://dx.doi.org/10.1049/iet-spr.2017.0340.
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Szelmanowski, Andrzej, Andrzej Pazur, Mariusz Żokowski, Paweł Janik, and Wojciech Paterek. "Computer modeling of electrical coil characteristics dedicated for helmet-mounted cueing systems with magnetic method." AUTOBUSY – Technika, Eksploatacja, Systemy Transportowe 20, no. 1-2 (February 28, 2019): 352–59. http://dx.doi.org/10.24136/atest.2019.065.
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Paper presents the importance of the helmet-mounted cueing systems in the field of flight safety increasing and mission execution effectiveness. There are presented the selected helmet systems used in the foreign and domestic solutions, utilizing the magnetic method. An original method of pilot’s helmet angular position determination utilizing the magnetic field from single flat coil and quaternion calculus is presented. The valuable elements are given mathematical relationships describing the characteristics of the magnetic field generated by selected types of electric coils as well as the results of simulation tests and the laboratory experimental verification of a flat coil model (designed in the Air Force Institute of Technology - AFIT) with a given number of coils.
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CASTELLANI, L., R. CATENACCI, M. DEBERNARDI, and C. PAGANI. "NONCOMMUTATIVE DE RHAM COHOMOLOGY OF FINITE GROUPS." International Journal of Modern Physics A 19, no. 12 (May 10, 2004): 1961–86. http://dx.doi.org/10.1142/s0217751x04018403.
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We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S3, the dihedral group D4 and the quaternion group Q. Poincaré duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold in general. A short review of the bicovariant (noncommutative) differential calculus on finite G is given for selfconsistency. Exterior derivative, exterior product, metric, Hodge dual, connections, torsion, curvature, and biinvariant integration can be defined algebraically. A projector decomposition of the braiding operator is found, and used in constructing the projector on the space of two-forms. By means of the braiding operator and the metric a knot invariant is defined for any finite group.
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Ghamari, Elham, and Dan Kučerovský. "Quaternions and Functional Calculus." Symmetry 11, no. 8 (July 26, 2019): 953. http://dx.doi.org/10.3390/sym11080953.
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In this paper, we develop the notion of generalized characters and a corresponding Gelfand theory for quaternionic C * -algebras. These are C*-algebras whose structure permits an action of the quaternions. Applications are made to functional calculus, and we develop an S-functional calculus related to what we term structural regular functions.
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Bolokhov, Pavel A. "Quaternionic wave function." International Journal of Modern Physics A 34, no. 02 (January 20, 2019): 1950001. http://dx.doi.org/10.1142/s0217751x19500015.
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We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.
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GHILONI, RICCARDO, VALTER MORETTI, and ALESSANDRO PEROTTI. "CONTINUOUS SLICE FUNCTIONAL CALCULUS IN QUATERNIONIC HILBERT SPACES." Reviews in Mathematical Physics 25, no. 04 (May 2013): 1350006. http://dx.doi.org/10.1142/s0129055x13500062.
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The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C*-algebras and to define, on each of these C*-algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.
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Popa, Călin-Adrian. "Synchronization of Clifford-valued neural networks with leakage, time-varying, and infinite distributed delays on time scales." AIMS Mathematics 9, no. 7 (2024): 18796–823. http://dx.doi.org/10.3934/math.2024915.
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<abstract><p>Neural networks (NNs) with values in multidimensional domains have lately attracted the attention of researchers. Thus, complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and their generalization, Clifford-valued neural networks (ClVNNs) have been proposed in the last few years, and different dynamic properties were studied for them. On the other hand, time scale calculus has been proposed in order to jointly study the properties of continuous time and discrete time systems, or any hybrid combination between the two, and was also successfully applied to the domain of NNs. Finally, in real implementations of NNs, time delays occur inevitably. Taking all these facts into account, this paper discusses ClVNNs defined on time scales with leakage, time-varying delays, and infinite distributed delays, a type of delays which have been relatively rarely present in the existing literature. A state feedback control scheme and a generalization of the Halanay inequality for time scales are used in order to obtain sufficient conditions expressed as algebraic inequalities and as linear matrix inequalities (LMIs), using two general Lyapunov-like functions, for the exponential synchronization of the proposed model. Two numerical examples are given in order to illustrate the theoretical results.</p></abstract>
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Ghiloni, Riccardo, Valter Moretti, and Alessandro Perotti. "Spectral representations of normal operators in quaternionic Hilbert spaces via intertwining quaternionic PVMs." Reviews in Mathematical Physics 29, no. 10 (November 2017): 1750034. http://dx.doi.org/10.1142/s0129055x17500349.
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The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann’s foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigroups relying upon it. A problem of the quaternionic formulation is the description of composite quantum systems in the absence of a natural tensor product due to non-commutativity of quaternions. A promising tool towards a solution is a quaternionic projection-valued measure (PVM), making possible a tensor product of quaternionic operators with physical relevance. A notion with this property, called intertwining quaternionic PVM, is presented here. This foundational paper aims to investigate the interplay of this new mathematical object and the spherical spectral features of quaternionic generally unbounded normal operators. We discover, in particular, the existence of other spectral notions equivalent to the spherical ones, but based on a standard non-quadratic notion of resolvent operator.
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Salnikov, Nikolay, Serhiy Melnychuk, and Vyacheslav Gubarev. "Ellipsoidal estimation of parameters of rotational and translational motion of a non-cooperative space vehicle from visual information." International Scientific Technical Journal "Problems of Control and Informatics" 68, no. 6 (November 20, 2023): 35–63. http://dx.doi.org/10.34229/1028-0979-2023-6-3.
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The use of near-Earth space is currently complicated by the presence of space debris objects in Earthʼs orbit, which include spent stages of launch vehicles, inoperative spacecraft, and other large and small objects associated with human activity in space. One of the elements of solving the problem of space debris is the docking and capture of an uncontrolled non-cooperative space object or spacecraft by a so-called on-orbit servicing spacecraft to carry out further actions to repair it, refuel or change its orbit. The situation is complicated by the fact that, under the influence of various factors, uncontrolled space objects are in a state of rotation. The parameters of the orbital motion of such objects are known quite accurately from measurements from the Earth. To carry out safe approach and docking, knowledge of the parameters of rotational motion, as well as the parameters of relative motion, is also required. The most general case of motion of a non-cooperative tumbling spacecraft located in an elliptical orbit is considered. It is assumed that the three-dimensional graphic model of such spacecraft is known. The servicing spacecraft (SSC) is equipped with a mono camera that takes pictures of the non-cooperative spacecraft (NSC). Based on a comparison of the characteristic features of photographs and images obtained using the graphical model, the computer vision system (CVS) determines the distance vector to the so-called graphical coordinate system, rigidly fixed on the NSC and the quaternion of its relative attitude. The specific type of CVS is not considered. It is assumed that the SSC carries out some maneuvers near the satellite. All parameters of the SSC angular motion are assumed to be known. This work considers the most general case of the relative motion of SSC and NSC. Using quaternion calculus, all basic kinematic and dynamic equations are obtained. The measured parameters are not enough to ensure safe rendezvous and docking with the NSC. The stochastic characteristics of errors of the CVS measurement are not assumed to be known and, accordingly, are not used. Only their maximum values are specified for them. We consider the use of new dynamic set-membership filter using ellipsoids to solve the problem of determining the parameters of the relative motion of the NSC which is in free uncontrolled motion. The filter can be implemented under conditions of the limited computational capability available on onboard processors. The relative motion parameters include the distance vector between the centers of mass (c.m.) of the NSC and the SSC, the relative velocity vector, the quaternion of the orientation of the main axes of inertia of the satellite relative to the inertial coordinate system, the ratio of the moments of inertia of the satellite, the vector of the position of the NSC c.m. in the graphical coordinate system. The properties of the proposed algorithm are demonstrated using numerical simulation. The results obtained are expected to be used in the development, creation and testing of a navigation system for the rendezvous and docking of the SSC, developed by a group of Ukrainian space industry enterprises under the leadership of the LLC «Kurs-orbital» (https://kursorbital.com/).
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Alpay, Daniel, Fabrizio Colombo, Jonathan Gantner, and David P. Kimsey. "Functions of the infinitesimal generator of a strongly continuous quaternionic group." Analysis and Applications 15, no. 02 (January 25, 2017): 279–311. http://dx.doi.org/10.1142/s021953051650007x.
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The quaternionic analogue of the Riesz–Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that [Formula: see text] is the quaternionic infinitesimal generator of a strongly continuous group of operators [Formula: see text] and we show how we can define bounded operators [Formula: see text], where [Formula: see text] belongs to a class of functions that is larger than the one to which the quaternionic functional calculus applies, using the quaternionic Laplace–Stieltjes transform. This class includes functions that are slice regular on the [Formula: see text]-spectrum of [Formula: see text] but not necessarily at infinity. Moreover, we establish the relation between [Formula: see text] and the quaternionic functional calculus and we study the problem of finding the inverse of [Formula: see text].
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Enslin, J. H. R. "Ortogonale voorstelling van drywing deur middel van kwaternione." Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 9, no. 1 (July 5, 1990): 11–14. http://dx.doi.org/10.4102/satnt.v9i1.433.
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The increased use of power control equipment resulted in the distortion of the excitation and response functions in power systems from simple sinusoids. These systems resulted in discrepancies in the general definition of power. This distortion has a negative effect on the accurate definition and representation of power in a contaminated power system. The mathematical representation of power with the aid of the theory of quaternions in vector calculus is investigated to obtain a generalized definition of power in all power systems, especially in power systems where the excitation and response functions show non-sinusoidal characteristics. The general description of power is illustrated with the aid of electrical power systems, but is however proposed to be beneficial in all power systems, being mechanical, thermal or chemical. Power is divided into different orthogonal components which describes the energy transfer through a system. The quaternion theory has orthogonal properties which can be used together with the Cauchy-Schwarz inequality to describe the power components using the excitation and respective response functions.
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Schwartz, Charles. "Calculus with a quaternionic variable." Journal of Mathematical Physics 50, no. 1 (January 2009): 013523. http://dx.doi.org/10.1063/1.3058642.
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Alpay, Daniel, Fabrizio Colombo, and Irene Sabadini. "Perturbation of the generator of a quaternionic evolution operator." Analysis and Applications 13, no. 04 (April 28, 2015): 347–70. http://dx.doi.org/10.1142/s0219530514500249.
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The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz–Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille–Phillips–Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille–Phillips–Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator [Formula: see text] we study under which condition a closed operator P is such that T + P generates the evolution operator [Formula: see text]. This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory.
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Colombo, Fabrizio, and Jonathan Gantner. "Formulations of the -functional calculus and some consequences." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 3 (May 5, 2016): 509–45. http://dx.doi.org/10.1017/s0308210515000645.
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In this paper we introduce the two possible formulations of the -functional calculus that are based on the Fueter–Sce mapping theorem in integral form and we introduce the pseudo--resolvent equation. In the case of dimension 3 we prove the -resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the -functional calculus.
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Vasilescu, Florian-Horia. "Spectrum and analytic functional calculus in real and quaternionic frameworks: An overview." AIMS Mathematics 9, no. 1 (2023): 2326–44. http://dx.doi.org/10.3934/math.2024115.
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<abstract><p>An approach to the elementary spectral theory for quaternionic linear operators was presented by the author in a recent paper, quoted and discussed in the Introduction, where, unlike in works by other authors, the construction of the analytic functional calculus used a Riesz-Dunford-Gelfand type kernel, and the spectra were defined in the complex plane. In fact, the present author regards the quaternionic linear operators as a special class of real linear operators, a point of view leading to a simpler and a more natural approach to them. The author's main results in this framework are summarized in the following, and other pertinent comments and remarks are also included in this text. In addition, a quaternionic joint spectrum for pairs of operators is discussed, and an analytic functional calculus which uses a Martinelli type kernel in two variables is recalled.</p></abstract>
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Ramesh, G., and P. Santhosh Kumar. "Borel functional calculus for quaternionic normal operators." Journal of Mathematical Physics 58, no. 5 (May 2017): 053501. http://dx.doi.org/10.1063/1.4982047.
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Colombo, Fabrizio, and Irene Sabadini. "On the formulations of the quaternionic functional calculus." Journal of Geometry and Physics 60, no. 10 (October 2010): 1490–508. http://dx.doi.org/10.1016/j.geomphys.2010.05.014.
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Colombo, Fabrizio, and Irene Sabadini. "On Some Properties of the Quaternionic Functional Calculus." Journal of Geometric Analysis 19, no. 3 (March 14, 2009): 601–27. http://dx.doi.org/10.1007/s12220-009-9075-x.
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Gover, A. Rod, and Jan Slovák. "Invariant local twistor calculus for quaternionic structures and related geometries." Journal of Geometry and Physics 32, no. 1 (November 1999): 14–56. http://dx.doi.org/10.1016/s0393-0440(99)00018-2.
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Gantner, Jonathan. "Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic S-Functional Calculus and Spectral Operators." Memoirs of the American Mathematical Society 267, no. 1297 (September 2020): 0. http://dx.doi.org/10.1090/memo/1297.
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Hertrich-Jeromin, Udo. "Supplement on curved flats in the space of point pairs and isothermic surfaces: A quaternionic calculus." Documenta Mathematica 2 (1997): 335–50. http://dx.doi.org/10.4171/dm/33.
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Gürlebeck, Klaus, Dmitrii Legatiuk, and Kemmar Webber. "Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures." Mathematics 10, no. 10 (May 13, 2022): 1670. http://dx.doi.org/10.3390/math10101670.
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The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticity, micropolar elasticity and Cosserat elasticity. Evidently, a different macro-behaviour is expected depending on the specific theory used in the background. Although there have been several theoretical studies of different elasticity theories in recent years, there is still a lack of understanding of how modelling assumptions of different elasticity theories influence the modelling results of masonry structures. Therefore, a rigorous approach to comparison of different three-dimensional elasticity models based on quaternionic operator calculus is proposed in this paper. In this way, three elasticity models are described and spatial boundary value problems for these models are discussed. In particular, explicit representation formulae for their solutions are constructed. After that, by using these representation formulae, explicit estimates for the solutions obtained by different elasticity theories are obtained. Finally, several numerical examples are presented, which indicate a practical difference in the solutions.
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Kraußhar, Rolf Sören. "Applications of the Quaternionic Calculus to the Convective Stationary MHD Equations in $${\mathbb{R}^3}$$ R 3." Advances in Applied Clifford Algebras 24, no. 4 (August 10, 2014): 1047–58. http://dx.doi.org/10.1007/s00006-014-0481-1.
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Richards, Joan L. "Generations of Reason: A Family's Search for Meaning in Post-Newtonian England." Perspectives on Science and Christian Faith 75, no. 1 (March 2023): 63–65. http://dx.doi.org/10.56315/pscf3-23richards.
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GENERATIONS OF REASON: A Family's Search for Meaning in Post-Newtonian England by Joan L. Richards. New Haven, CT: Yale University Press, 2021. 456 pages, with 21 b/w illustrations, 1,218 endnotes, and a 35-page index. Hardcover; $45.00. ISBN: 9780300255492. *The title gives no clue who this book is about. Nor does the publisher's description on its website, the abbreviated blurb inside the book jacket, the four endorsements posted on the jacket's back ("beautifully written," "epic masterpiece," "magnificent study," "compelling and wide-ranging"), or even the chapter titles. The reader first learns whom the book is about and how it came into focus in the author's Acknowledgments. In studying the divergent interests of Augustus De Morgan and his wife, Sophia, the importance of De Morgan's father-in-law William Frend's thinking became apparent. This is turn led Richards to delve into the lives and beliefs of two ancestors from the previous generation, Francis Blackburne and Theophilus Lindsey, who felt compelled by their commitment to "reasoned conclusions about matters of faith" (p. x) to move away from orthodox Anglicanism and establish the first Unitarian church in England. Thus the book eventually evolved into chronicling the lives of three generations over a century and a half during (roughly) the Enlightenment era. *A central motif running through the experiences, beliefs, and work of these families was their steadfast commitment to a form of enlightened rationality that provided coherence and foundational meaning for their lives. Reason informed their ecclesiastical commit-ment to Unitarianism, their views of science and mathematics, and their public activity favoring social and educational reforms. But also, paradoxically, their search for reason led to the beliefs and practices (of some family members) that today would be considered pseu-do-scientific--mesmerism, phrenology, and spiritism, among others. *As Richards notes in the book's opening sentence, for her, Generations of Reason is "the culmination of a life devoted to understanding the place of mathematics in modern European cultural and intellectual history." The mathematics and logic of early- to mid-nineteenth-century Britain has been an ongoing research interest for Richards during her forty-year tenure as a historian of mathematics at Brown Universi-ty. It is this that largely drew me to the book and which I will focus on here: it climaxes in a substantive treatment of the progressive mathematics of De Morgan, whose work contributed to transforming British algebra and logic. This is in stark contrast with the radical ideas of Frend, who refused to admit negative numbers into mathematics. *A central figure behind the developments under investigation is John Locke, whose Essay Concerning Human Understanding (1689) and The Reasonableness of Christianity, as Delivered in the Scriptures (1695) exercised a tremendous influence over and challenge for eighteenth- and nineteenth-century British thinkers. Locke's ideas defined and emphasized rationality in relation to knowledge generally and to scientific and religious knowledge in particular, providing dissenters with a rationale for combatting traditional theology and conformist science and philosophy. For Locke, however, a literal reading of scripture was still authoritative for religious beliefs. This was true for Frend and De Morgan also, even though they held tolerant attitudes toward a wide latitude of thinkers. *Locke's view of Reason also affected period reflections on mathematics. Like others in the early modern and Enlightenment eras, Locke had held up mathematics as a model of absolutely certain knowledge because of the clarity of its ideas and the supposed self-evidence of its axiomatic truths. Of course, this characterization applied more to Euclidean geometry than to the burgeoning domains of analytic mathematics, such as calculus, which, as Berkeley charged, still lacked a sound theoretical basis. As for logic, Locke had an acute antipathy toward traditional argument forms and proposed that one should reason with ideas rather than words, assessing their agreement or disagreement in less convoluted ways than in a syllogism. In expressing such relations with language, though, one should use meaningful and unambiguous terms. This was somewhat problematic in algebra and calculus, where symbolic expressions were manipulated to produce useful and important results, even when their meaning was less than clear. *Around the turn of the nineteenth century, Frend campaigned to bring algebra in line with Lockean reasoning: algebra was conceptualized at that time as universal arithmetic, containing such laws as the transposition rule if a + b = c then a = c - b. Thus, no expression should be employed if its meaning was unintelligible. In the above equations, one must assume the condition b < c to rule out negative values, since numbers, which represent quantities of discrete things, cannot be less than 0. Excising negative quantities from mathematics was extreme but necessary in order to adhere to a literalistic view of rationality. *British mathematicians largely resisted following Frend down this path of purity, though they were unsure how to rationally justify their use of negative and imaginary quantities without going outside mathematics and appealing to things like debts. Robert Woodhouse, in an 1803 work, was one of the first Cambridge mathemati-cians to propose a more formalistic algebraic approach in calculus. This agenda was furthered a decade later by members of Cambridge's Analytical Society, one of whom was George Peacock. His and others' attempts to convert Cambridge analysis from Newtonian to Leibnizian calculus were waged through translating a French textbook and making notational changes in Cambridge's mathematical examinations. *In 1830 Peacock's Treatise on Algebra introduced a more formalistic approach in algebra. Richards argues, drawing upon some fairly recent research, that Peacock's position was grounded in a progressivist view of history: arithmetic developed naturally out of fluency with counting, and algebra out of familiarity with arithmetic. Arithmetic suggests equivalent forms (equations, or symbolic assertions like the above rule) that can also be accepted as equiva-lent/valid in algebra without being constrained by restrictions appropriate to arithmetic. Such transitions, he thought, constitute genuine historical progress. Algebra thus splits into two parts for Peacock, arithmetical algebra and symbolical algebra, the latter based upon his principle of the permanence of equivalent forms, as found in his 1830 A Treatise on Algebra. *Peacock's approach to algebra set the stage for later British mathematicians such as De Morgan (Peacock's student), Boole, and others. Initially inclined to follow his future father-in-law's restrictive approach in algebra, De Morgan was soon won over to Peacock's point of view, even going beyond it in his own work. In a series of articles around 1840, De Morgan identified the basic rules governing ordinary calculations, but he also began entertaining the notion of a symbolical algebra less tightly tied to arithmetical algebra. By more completely separating the interpretation of algebra's operations and symbols from its axioms, symbolical algebra gained further independence from arithmetic. This gave algebra more flexibility, making room for subsequent developments such as the quaternion algebra of William Rowan Hamilton (1843) and Boole's algebra of logic (1847). *After exploring the foundations of algebra, De Morgan turned his attention to analyzing forms of reasoning, a topic made popular by the resurgence of syllogistic logic instigated at Oxford around 1825 by Richard Whately. Traditional Aristotelian logic parsed valid arguments into syllogisms containing categorical statements such as every X is Y. De Morgan treated such sentences exten-sionally, using parentheses to indicate total or partial inclusion between classes X and Y. Thus, every X is Y was symbolized by X)Y since the parenthesis opens toward X; to be more precise, one should indicate whether X and Y are coextensive or X is only a part of Y. By thus quantifying the predicate, as it was called, De Morgan allowed for these two possibilities to be symbolized respectively by X)(Y and X))Y, in compact symbolic form as ')(' and '))'. Combining the two premises of a syllogistic argument using this notation, one could then apply an erasure rule to draw its conclusion. De Morgan enthusiastically elaborated his symbolic logic by adopting an abstract version of algebra that paved the way for operating with formal symbols in logic. De Morgan's symbolism is not as inaccessible as Frege's later two-dimensional concept-writing (though the full version of De Morgan's notation is more complex than indicated here), but it is still rather forbid-ding and failed to find adherents. *In addition to expanding Aristotelian forms by quantifying the predicate, yielding eight basic categorical forms instead of the standard four, by 1860 De Morgan was generalizing the copula "is" in such sentences to other relations, such as "is a brother of" or "is greater than." He began to systematically investigate the formal properties of such relations and the ways in which relations might be compounded. Though intended as a way to generalize categorical statements and expand syllogistic logic, his treatment of relations was later recognized as an important contribution that could be incorporated into predicate logic. Richards's treatment gives the reader a fair sense of what De Morgan's logic was like, and while a detailed comparison is not developed, the reader can begin to see how De Morgan's system compares to Aristotelian logic, Boole's algebra of logic, and contemporary mathematical logic. *However, as indicated at the outset, exploring De Morgan's algebraic and logical work is only a subplot of Richards's story. Her book is principally a brief for how Reason grounded the work and lives of several significant thinkers in an extended family over three generations. As she ties various threads together, the reader occasionally senses that the presentation may be too tidy, drawing parallels between vastly different developments to make them seem of a piece, all motivated by the same driving force of Reason. Nevertheless, Richards's account forces the reader to continually keep the bigger picture in mind and to connect various facets of the actors' lives and work to their deeper commitment to Reason. Her book thus offers a commendable case study for how technical trends in mathematics might be tied to broader cultural and philosophical concerns. *Reviewed by Calvin Jongsma, Professor Emeritus of Mathematics, Dordt University, Sioux Center, IA 51250.
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Alpay, Daniel, Fabrizio Colombo, Tao Qian, and Irene Sabadini. "The H∞ functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators." Journal of Functional Analysis 271, no. 6 (September 2016): 1544–84. http://dx.doi.org/10.1016/j.jfa.2016.06.009.
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Crilly, Tony. "Real quaternionic calculus handbook by João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig, p. 216, £54.99, ISBN 978-3-0348-0621-3, Birkhäuser (2014)." Mathematical Gazette 105, no. 563 (June 21, 2021): 370–71. http://dx.doi.org/10.1017/mag.2021.90.
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Gudrun Kalmbach HE. "Interactions, space presentations, blocks and cross products." GSC Advanced Research and Reviews 6, no. 2 (February 28, 2021): 061–73. http://dx.doi.org/10.30574/gscarr.2021.6.2.0012.
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Physics counts four basic forces, the electromagnetic EMI, weak WI, strong SI interactions and gravity GR. The first three are provided with a unified theory which partly needs revision and has the symmetry U(1)xSU(2)xSU(3). In this article their space presentations are described in order to inlcude a theory for gravity which cannot be added directly to the standrd model. There are many instances of gravitational actions which are different from the other three interactions. Gravity uses geometrical models beside spactime, often projective, including stereographic and spiralic orthogonal subspace projections. Real and complex cross products, symmetries which belong to the complex Moebius transformation subgroups, complex cross ratios, Gleason frame GF measures, dihedrals nth roots of unity with symmetris are some new tools (figure 14) for a new gravity model. The basic vector space is 8-dimensional, but beside the usual vector addition and calculus there are different multiplications added. The author uses complex multiplications in the complex 4-dimensional space C4 for calculus. The SU (3) multiplication of GellMann 3x3-matrices is used for C³ and its three 4-dimensional C² projections. Projective spaces are CP² for nucleons and a GR Higgs plane P² and projective measuring GF‘s which have 3-dimensional, orthogonal base vectors like spin. The doubling of quaternionic spacetime to octonians has a different multiplication and seven GF‘s which partly occur in physics as cross product equations. Beside the real, the complex cross product extends the spacetime dimensions from 4 to 8. Consequences are that there are many 3-dimensional, many 4-dimensional, some 6-dimensional and also projective 5-dimensional spaces in which the actions of gravity can then be described. Spacetime is for this not sufficient. No symmetry can be muliplied to the standard model since the new symmetries belong to different geometries and are not directly related to a set of field quantums like one photon for EMI, three weak bosons (or four) for WI, eight gluons for SI. GR has graviton waves similar to EMI waves and in quasiparticle form rgb-graviton whirls, for mass Higgs bosons, maybe also solitons (density as mass per volume changing). They attribute to a distance metric between two points (kept fixed) an amplitude density (operator} which changes the metrical diameter of the volume, but not the mass.
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Khedhiri, Hedi, and Taher Mkademi. "Foundational aspects of a new matrix holomorphic structure." Arab Journal of Mathematical Sciences, March 21, 2024. http://dx.doi.org/10.1108/ajms-08-2023-0002.
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PurposeIn this paper we talk about complex matrix quaternions (biquaternions) and we deal with some abstract methods in mathematical complex matrix analysis.Design/methodology/approachWe introduce and investigate the complex space HC consisting of all 2 × 2 complex matrices of the form ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1, (z1,w1,z2,w2)∈C4.FindingsWe develop on HC a new matrix holomorphic structure for which we provide the fundamental operational calculus properties.Originality/valueWe give sufficient and necessary conditions in terms of Cauchy–Riemann type quaternionic differential equations for holomorphicity of a function of one complex matrix variable ξ∈HC. In particular, we show that we have a lot of holomorphic functions of one matrix quaternion variable.
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"Collection of Three Monographs Pertaining to Quaternionic Analysis." Journal of Electrical Electronics Engineering 3, no. 3 (June 14, 2024): 01–42. http://dx.doi.org/10.33140/jeee.03.03.001.
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This particular submission contains (inter alia) copies of three (3) monographs, whose purpose is to further elaborate upon various topics having been enunciated in the author’s previous set of submissions, namely – (a) “A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PARTS 1/10 to 10/10.” (b) “Supplementary Notes pertaining to a Specific Quaternion Analogue of the Cauchy-Goursat Theorem.” Which have been published under the ‘VIXRA’ Mathematics subheading: - ‘Functions and Analysis’. II. Copy of Author’s Original Monograph No.1. (1) Title of Monograph:- “Correlation of Specific Results having been enunciated in Various Expository Articles and Papers.” Re: - Mathematical Paper, thus entitled – “A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions.” (2) Date of Completion: - 14th June 2018. (3) Total number of A4 pages to follow = 22. (4) Referee’s Assessment:- For further details, the reader should accordingly refer to the contents of Appendix A1.
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Li, Qing, and Zhida Ren. "A new fractional-order augmented quaternion-valued approach for degradation prognostics of bearings using generalized Hamilton-real calculus." IEEE Transactions on Instrumentation and Measurement, 2022, 1. http://dx.doi.org/10.1109/tim.2022.3218546.
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Popa, Călin-Adrian. "Stability and synchronization of octonion-valued neural networks with leakage and mixed delays on time scales." Computational and Applied Mathematics 43, no. 5 (June 22, 2024). http://dx.doi.org/10.1007/s40314-024-02820-5.
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AbstractThere has been a great deal of interest in the last few years for neural networks (NNs) with values in multidimensional domains. The most popular models are complex-valued neural networks (CVNNs), followed by quaternion-valued neural networks (QVNNs), and, more recently, by Clifford-valued neural networks (ClVNNs). However, also very recently, a different type of NNs were put forward, namely octonion-valued neural networks (OVNNs). OVNNs are defined on the 8D octonion algebra, and they are not a special type of ClVNNs, because Clifford numbers are associative, whereas octonions are not. Moreover, beside the complex and quaternion algebras, the only other normed division algebra over the reals is the algebra of octonions, which makes OVNNs a direct generalization of CVNNs and QVNNs from this point of view, raising interest for applications handling high-dimensional data. On the other hand, systems defined on time scales were proposed as a generalization of both discrete time and continuous time systems, or any type of hybrid combination between the two. Finally, time delays appear as a consequence of implementing NNs in real life circuits. Taking all these into consideration, this paper studies the fundamental properties of exponential stability and exponential synchronization for OVNNs with leakage and mixed delays defined on time scales. In order to avoid the problems raised by the non-associativity of the octonion algebra, the OVNN model is decomposed into a real-valued one. Then, two different Lyapunov-type functionals are defined and the particularities of time scale calculus are used in order to deduce sufficient conditions expressed as scalar and linear matrix inequalities (LMIs) for the exponential stability of the proposed models, based on Halanay-type inequalities suitable for time scale systems. Afterwards, a state feedback controller is used to deduce sufficient criteria given as scalar inequalities and LMIs for the exponential synchronization of the same type of models. The generality of the model, given by the definition on the algebra of octonions, the use of different types of delays, and the definition on time scales represent an important advantage. Also, for less general models, it is possible to particularize the obtained results. With the aim to demonstrate each of the paper’s four theorems, four numerical examples are provided.
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Colombaro, Ivano. "Time-like definition of quaternions in exterior algebra." Ricerche di Matematica, August 22, 2023. http://dx.doi.org/10.1007/s11587-023-00810-z.
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AbstractA formal description of quaternions by means of exterior calculus is presented. Considering a three-dimensional space-time characterized by three time-like coordinates, we have been able to consistently recover a suitable formulation of quaternions by means of the properties arising from exterior algebra and calculus. As an application, it is also illustrated how rotations may be written in terms of quaternions, in accordance with definition provided in exterior algebra.
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Montgomery-Smith, Stephen. "Functional Calculus for Dual Quaternions." Advances in Applied Clifford Algebras 33, no. 3 (May 24, 2023). http://dx.doi.org/10.1007/s00006-023-01282-y.
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"Real quaternionic calculus handbook." Choice Reviews Online 52, no. 05 (December 18, 2014): 52–2594. http://dx.doi.org/10.5860/choice.185180.
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Ceyhan, Hazal, Zehra Özdemir, and Ismail Gök. "Multiplicative generalized tube surfaces with multiplicative quaternions algebra." Mathematical Methods in the Applied Sciences, April 3, 2024. http://dx.doi.org/10.1002/mma.10065.
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Along with other types of calculus, multiplicative calculus brings an entirely new perspective. Geometry now has a new field as a result of this new understanding. In this study, multiplicative differential geometry was used to explore peculiar surfaces. Multiplicative quaternions are also used to depict surfaces. Additionally, multiplicative differential geometry was used to generate the accretive surface subject, which is a developing subject. The derived surfaces' perspective silhouette curve equation is provided. The Bishop multiplicative frame was also established and applied when expressing surfaces along with these. Finally, the surfaces and perspective silhouette curves were visualized using Maple, and the equations were obtained.
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Cerejeiras, P., U. Kähler, and R. S. Kraußhar. "Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations." Complex Analysis and Operator Theory 18, no. 3 (March 1, 2024). http://dx.doi.org/10.1007/s11785-023-01455-4.
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AbstractIn this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.
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Vasilescu, Florian-Horia. "Quaternionic Regularity via Analytic Functional Calculus." Integral Equations and Operator Theory 92, no. 2 (March 21, 2020). http://dx.doi.org/10.1007/s00020-020-2574-7.
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Colombo, Fabrizio, Stefano Pinton, and Peter Schlosser. "The $$H^\infty $$-Functional Calculi for the Quaternionic Fine Structures of Dirac Type." Milan Journal of Mathematics, March 6, 2024. http://dx.doi.org/10.1007/s00032-024-00392-x.
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AbstractIn recent works, various integral representations have been proposed for specific sets of functions. These representations are derived from the Fueter–Sce extension theorem, considering all possible factorizations of the Laplace operator in relation to both the Cauchy–Fueter operator (often referred to as the Dirac operator) and its conjugate. The collection of these function spaces, along with their corresponding functional calculi, are called the quaternionic fine structures within the context of the S-spectrum. In this paper, we utilize these integral representations of functions to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, by leveraging the aforementioned factorization of the Laplace operator, we identify four distinct classes of functions: slice hyperholomorphic functions (leading to the S-functional calculus), axially harmonic functions (leading to the Q-functional calculus), axially polyanalytic functions of order 2 (leading to the $$P_2$$ P 2 -functional calculus), and axially monogenic functions (leading to the F-functional calculus). By applying the respective product rule, we establish the four different $$H^\infty $$ H ∞ -versions of these functional calculi.