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Physics and mathematics are interrelated as part of the science, technology, engineering, and mathematics (STEM) disciplines. The learning of science is supported by mathematical skills and knowledge. The aim of this paper is to determine the cognitive obstacles of in-service undergraduate physics student-teachers' understanding of the concept of complex numbers which is part of linear algebra. A case study is presented involving 10 undergraduate student-teachers at a university in Zimbabwe studying for a Bachelor of Science Education Honours Degree in physics. Data were generated from the 10 participants' answers to structured activity sheets and interviews. Action, process, object, schema (APOS) theory was used to explore the possible ways that students may follow to understand the concepts of complex numbers and how they concur with the preliminary genetic decomposition. It was observed that most of the participants were operating at the action level, with a few operating at the process and object levels of understanding. Recommendations are made in this study that instructors should pay more attention to the prerequisite concepts and the "met afters" so that students can encapsulate processes into object understanding of division of complex numbers and polar form.

The following investigation is a cognitive-historical analysis of the conceptual development of complex numbers. The history we explore spans nearly two millennia, from the earliest appearance of the square root of a negative quantity in the calculations of Heron of Alexandra (first century CE) to the full flowering of complex numbers in the first half of the 19th century. The approach we use is Nersessian's, including her formulations of model-based reasoning and mental models. Additional aspects feature the prominent roles played by process representations, including object-process complementarities, and by core numerical systems. Our analysis provides support for mental rotation as a fourth core system and for bootstrapping scenarios in both the senses of Nersessian and of Carey. We use the results of our analysis and discussion to update previous con- ceptual metaphor mapping approaches to complex numbers by Fauconnier and Turner and by Lakoff and Núñez. Finally, we review recent empirical studies of complex number learning and find them to be consistent with the present work. [ABSTRACT FROM AUTHOR]

This study focused on students’ inability to correctly use psychometric tests applied for complex numbers in the context of generating an Algebraic abelian group (closure, commutative, associatively, inverse, and identity). This was done using a psychometric test (Kuder and Richardson formula) and APOS Theory after they had been taught the relevant content. The 41 participants were second-year students in the Department of Statistics at the Federal Polytechnic Ile-Oluji. They were given a class test comprising eleven (11) objective questions, and 4 questions relating to group theory were extracted for this analysis. After that, six (6) students were interviewed to gain insight into their perspectives. It was observed that most of the students had difficulty understanding the concept of algebraic groups about complex numbers, founded on not seeing the relevance in the prerequisite knowledge of an algebraic group. Also, if the reliability test of the question is unacceptable, then a good percentage of the participant lacked the mental construction relating to APOS to answer such a question. Keywords: Complex Number, Algebraic group, Prerequisite, Difficulty, Discrimination index, Psychometrics.

In this study, a classification algorithm based on complex number feature is proposed. Specifically, the SVM framework is reformulated, so each example would be classified in the unitary space. The cost function is redefined by considering the maximum margin of real and imaginary units of the complex number feature at the same time. The cost function is based on the expectation of the hinge loss, and its derivatives can be calculated in closed forms. Using a stochastic gradient descent (SGD) algorithm, this method allows for efficient implementation. For complex number feature, the example uncertainty is modeled by a sample preprocessing method based on within-class Euclidean distance Gaussian distribution sample (DGS). In addition, a complex number feature selection method based on improved hybrid discrimination analysis (HDA) is proposed by considering the correlation between real and imaginary units of complex number feature. The proposed classification algorithm is tested on synthetic data and three publicly available and popular datasets, namely, MNIST, WDBC, and Voc2012. Experimental results verify the effectiveness of the proposed method. The codes are available: https://github.com/luckysomebody/paper-code.

Intracellular protein trafficking processes consisting of three intracellular states are described by three differential equations. To solve the equations, a quadratic equation is required, and its roots are generally real or complex. The purpose of the present study is to clarify the meanings of roots of real and complex numbers. To clarify the point, we define that: 1) ‘kI’ is the insertion rate from an insertion state trafficking to the plasma membrane state; 2) ‘kE’, the endocytotic rate from the plasma membrane state trafficking to a recycling state; 3) ‘kR’, the recycling rate from the recycling state trafficking to the insertion state. Amounts of proteins in three states are expressed as αelt+βemt+γ with α,β,γ = constant and l and m are roots of a quadratic equation, r2+kI+kE+kRr+kIkE+kIkR+kEkR=0. When l and m are real kI2+kE2+kR2>2kIkE+kEkR+kRkI, amounts of proteins in three states shows no oscillatory change but a monotonic change after a transient increase (or decrease); when l and m are complex kI2+kE2+kR2

With the rapid development and wide application of the digital media technology,the information security problem is becoming more and more highly attention.Because of the various kinds of medias,previous research works have focused on a specific digital media from,such as text,image,audio and video.As a fact that all the objects were expressed in digital forms under the environment of computer and network,a unified form for all kinds of digital media under the complex number base system was proposed.The algorithm of the digital media signal and the complex number intercom version were given,so the relationship among the digital media could be converted to the corresponding geometric relationship on the complex plane.In the experiments,the proposed method was applied to the application of the digital media information hiding.

Today the world is witnessing very significant developments in the world of digital communications. For that, the subscribers are demanding for quality and speed for the available services. To achieve these objectives, must be accurate and low impurity communication devices available. The orthogonal frequency division multiplexing (OFDM) scheme, is the downlink modulation scheme utilized in the long term evolution (LTE) system. Because of the coherent addition of the subcarriers, the output power variation will show high peaks compared with the average, known as Crest factor, thus, the amplification gain is degraded and intermodulation distortion will appears. Different sorts of schemes have been introduced to mitigate these problems, but either on the expense of complexity or the bit error rate (BER) performance degradation. In this paper, a novel simple to implement algorithm suggested by the authors to mitigate the high Crest factor. The suggested approach is based on complex numbers manipulation, by exchanging the real and imaginary parts, to destroy the coherence order of the OFDM subcarriers, leading to lower crest factor. Results show that the computational complexity and the reduction gain are improved significantly

Complex numbers have been studied in previous papers, but the papers did not deal with the conversion from algebraic form of a neutrosophic complex number form to exponential or trigonometric form and vice versa, and this prompted us to search for a method that facilitates this conversion process, as this paper dealt with how to move from algebraic form of a neutrosophic complex number to exponential or trigonometric form and vice versa, also, we discussed the roots from order 𝑛 of a neutrosophic complex number, whether this number is given in algebraic or trigonometric form.

In this article, we consider the problem of planar graph-based simultaneous localization and mapping (SLAM) that involves both poses of the autonomous agent and positions of observed landmarks. We present complex (CPL)-SLAM, an efficient and certifiably correct algorithm to solve planar graph-based SLAM using the complex number representation. We formulate and simplify planar graph-based SLAM as the maximum likelihood estimation on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to convex complex semidefinite programming (SDP). Furthermore, we simplify the corresponding complex SDP to Riemannian staircase optimization (RSO) on the complex oblique manifold that can be solved with the Riemannian trust region method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through applications of CPL-SLAM and comparisons with existing state-of-the-art methods on planar graph-based SLAM, which indicates that our proposed algorithm is capable of solving planar graph-based SLAM certifiably, and is more efficient in numerical computation and more robust to measurement noise than existing state-of-the-art methods. The C++ code for CPL-SLAM is available at https://github.com/MurpheyLab/CPL-SLAM. [ABSTRACT FROM AUTHOR]

The concept of unity roots plays a central role in the theory of field extensions and polynomials roots' computing. The objective of this paper is to find the algebraic formula for computing the symbolic 2-plithogenic and 3-plithogenic complex roots of unity, where a general formula will be provided with many related examples up to the exponent 3.

We aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values $\alpha$, $\beta$ and $\mathfrak{p}$. Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given.

This article introduces the idea of bicomplex valued S-metric space and deduces some of its features. Additionally, for bicomplex valued S-metric spaces, some fixed point results of contraction maps are shown to meet various categories of rational inequalities. Moreover, these results generalize certain significant, well-known results. An example is provided to highlight our major result. Furthermore, a theorem guaranteeing the existence of the one and only solution to the linear system of equations was developed using our main result.

The objective of this paper is to study for the first time the concept of square real matrices with symbolic 2-plithogenic split-complex entries. Many of their algebraic properties will be discussed and handled, where we find the formula of computing inverses, exponents, and powers of these matrices by building a ring isomorphism between the ring of split-complex symbolic 2-plithogenic matrices and the direct product of the symbolic 2-plithogenic matrices with itself. Also, we give the interested reader many related examples to clarify the validity of our work.

In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too.

In this paper, we introduce the concept of generalized Fourier series, generated by the p-trigonometric functions, namely cosp and sinp, recently introduced related to the generalized complex numbers systems. The aim of this study is to represent a periodic signal as a sum of p-sine and p-cosine functions. In order to achieve this, we first present the integrals of the product of the same or different family of p-trigonometric functions over the full period of these functions to understand the orthogonality properties. Next, we use these integrals to derive the coefficients of the generalized p-Fourier series along with a few examples. The generalized Fourier series can be used to expand an arbitrary forcing function in the solution of a non-homogeneous linear ordinary differential equation (ODE) with constant coefficients.

In this study, we compute the unitarily invariant norms of the matrices, Az = (zizj)ni, j = 1, Bz = (zi - zj)ni, j = 1,and Cz = (zi/zj)n1j = 1, where zis are ith components of any complex sequence (zn). Moreover, we give some corollaries and numerical examples related to norms of these matrices. [ABSTRACT FROM AUTHOR]

Based upon the vector representation of complex numbers and the vector exponential function, we introduce the vector representation of characteristic functions and consider some of its elementary properties such as its polar representation and a vector power expansion.

The aim of this paper is to study the neutrosophic complex finite rings 𝐶(𝑍𝑛 ) 𝑎𝑛𝑑 𝐶(< 𝑍𝑛 ∪ 𝐼 >), and to give a classification theorem of these rings. Also, this work introduces full solutions for 12 Kandasamy-Smarandache open problems concerning these structures of generalized rings modulo integers. Also, a necessary and sufficient condition of invertibility in 𝐶(𝑍𝑛 ) 𝑎𝑛𝑑 𝐶(< 𝑍𝑛 ∪ 𝐼 >) is presented as a partial solution of the famous group of units problem.

This paper discusses incidence structures and their rank. The aim of this paper is to prove that there exists a regular decomposable incidence structure J = (P, B) of maximum degree depending on the size of the set and a predetermined rank. Furthermore, an algorithm for construction of this structures is given. In the proof of the main result, the points of the set P are shown by Euler's formula of complex number. Two examples of construction the described incidence structures of maximum degree 6 and maximum degree 30 are given.

The aim of this paper is to construct dual-generalized complex Fibonacci and Lucas quaternions. It examines the properties both as dual-generalized complex number and as quaternion. Additionally, general recurrence relations, Binet's formulas, Tagiuri's (or Vajda's like), Honsberger's, d'Ocagne's, Cassini's and Catalan's identities are obtained. A series of matrix representations of these special quaternions is introduced. Finally, the multiplication of dual-generalized complex Fibonacci and Lucas quaternions are also expressed as their different matrix representations.

Flipped Learning is one of the alternatives of teaching and learning approach in mathematics classroom. The objective of this study is exploring students’ conceptual understanding about complex number using flipped learning with handout. The subject of the study are the students in 5th semester students of mathematics education department in 2019/2020. The study used qualitative approach to describe the implementation of flipped learning. There are 31,6% of 19 respondents give score very satisfied. This result then observed by using the test with all the students understand with the definition of complex numbers. Besides they can adapt their learning activity using flipped learning with complex analysis handout. As legibility aspect of the handout, there are 52,6% of the respondents gives score satisfied and 26,3% of the respondents are very satisfied. The score indicates that the flipped learning with handout helps students to understand about the complex number concepts.

Based on the novel prescription for the power function, (x+iy)m, the new expression for Ψ(x,y|m), the eigenfunction of the operator of the third component of the angular momentum, M^z, is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x+iy)m=ρmeimϕ, are invariant under the rotations at 2π for any, not necessarily integer, m—the eigenvalue of M^z. For any real m the functions Ψ(x,y|m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of M^z. The eigenfunctions and eigenvalues of the angular momentum operator squared, M^2, derived for the two different prescriptions for the square root, (m2)1/2, (m2)1/2=|m| and (m2)1/2=±m, are reported. The normalizable eigenfunctions of M^2 are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the M^2.

This article identifies the general form of matrix power and trace of an integer power of Special Form of the 3 × 3 Complex Circulant matrix. The research begins to determine the general form of an integer power of Special Form of the 3 × 3 Complex Circulant matrix, followed by determining the general form trace of an integer power of Special Form of the 3 × 3 Complex Circulant matrix. The proof is done by using mathematical induction. The final result of this article is to obtain the general form of the matrix A^n and tr(A^n) for n integers in special form of the 3 × 3 complex Circulant matrix.

In this study, we generalize the well-known formulae of De-Moivre and Euler of complex numbers to dual-complex numbers. Furthermore, we investigate the roots and powers of a dual-complex number by using these formulae. Consequently, we give some examples to illustrate the main results in this paper.

Youth employment in labour force has a critical importance in socio-economic planning. It is expected that the use of dynamic and able-bodied work force can increase the quality of the industrial products and it is also necessary to strengthen the economy. More importantly, the contribution to labour force has a crucial importance for public welfare. At this stage, since the role of young women in the total labour force is not considered sufficiently, this study concentrates on the determination of the relative effect of women labour force. For this purpose, first the youth employment is considered as a complex quantity; along with the real component (young men), the contribution of young women is treated as the imaginary component. By using the data derived from 20 different domains (sectors), the problem is stated as a complex value problem and a measurement uncertainty analysis is utilized. The measurement uncertainty of the complex quantity (employment) is expressed by a region in the complex plane. Finally, a confidence ellipse at 95% confidence level is produced. The phase diagrams produced by statistical learning have provided some abnormalities and also potentials.

The concept of unity roots plays a central role in the theory of field extensions and polynomials roots' computing. The objective of this paper is to find the algebraic formula for computing the symbolic 2-plithogenic and 3-plithogenic complex roots of unity, where a general formula will be provided with many related examples up to the exponent 3. [ABSTRACT FROM AUTHOR]

This paper introduces one unique price motion process with fractional Brownian motion. We introduce the imaginary number into the agent’s subjective probability for the reason of convergence; further, the result similar to Ito Lemma is proved. As an application, this result is applied to Merton’s dynamic asset pricing framework. We find that the four order moment of fractional Brownian motion is entered into the agent’s decision-making. The decomposition of variance of economic indexes supports the possibility of the complex number in price movement.

Since the product of complex numbers and rectangular fuzzy complex numbers (RFCN) is not necessarily a RFCN in the former fuzzy complex linear system (FCLS), the scalar multiplication and addition operations of complex numbers and fuzzy complex numbers (FCN) based on a new representation of FCN are proposed. We also introduce a new method for solving FCLS, which can convert FCLS into two distinct linear systems. One is an n×n complex linear system, and the other is an (mn)×(mn) real linear system, where n is the number of unknown variables, and m is the number of substitutional cyclic sets composed of coefficients of FCLS. In particular, using this method to solve one-dimensional fuzzy linear systems, a (2n)×(2n) RLS is obtained, which is consistent with Friedman’s method. Finally, FCLS based on the RFCN as a special case are also investigated.

The paper presents a square-root-extended complex Kalman filter (SRECKF) by decomposing covariance matrix with its square-root forms to improve stability of the filter for estimating complex number. αβ transformation is used to map three-phase instantaneous voltages in the abc phases into instantaneous voltages on the αβ axes, and a non-linear state equation and observation equation of the three-phase voltages are built by introducing a complex vector and defining state variables. Positive symmetrical component, negative symmetrical components, and frequency of the three-phase voltages are estimated using traditional extended complex Kalman filter (ECKF), the estimation results show that the method proposed here are superior to traditional extended complex Kalman filter on estimation accuracy and convergence rate.

The objective of this paper is to study for the first time the concept of square real matrices with symbolic 2-plithogenic split-complex entries. Many of their algebraic properties will be discussed and handled, where we find the formula of computing inverses, exponents, and powers of these matrices by building a ring isomorphism between the ring of split-complex symbolic 2-plithogenic matrices and the direct product of the symbolic 2-plithogenic matrices with itself. Also, we give the interested reader many related examples to clarify the validity of our work. [ABSTRACT FROM AUTHOR]

Dispensing with the common property of distributivity and replacing classical trigonometric functions with their l p -counterparts in Euler’s trigonometric representation of complex numbers, classes of l p -complex numbers are introduced and some of their basic properties are proved. The collection of all points that leave the l p -absolute value of each l p -complex number invariant under l p -complex numbers multiplication is shown to be a group of elements that have l p -absolute value one but not the symmetry group.

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

Complex radix (−1 + j) allows the arithmetic operations of complex numbers to be done without treating the divide and conquer rules, which offers the significant speed improvement of complex numbers computation circuitry. Design and hardware implementation of complex radix (−1 + j) converter has been introduced in this paper. Extensive simulation results have been incorporated and an application of this converter towards the implementation of discrete Fourier transformation (DFT) processor has been presented. The functionality of the DFT processor have been verified in Xilinx ISE design suite version 14.7 and performance parameters like propagation delay and dynamic switching power consumption have been calculated by Virtuoso platform in Cadence. The proposed DFT processor has been implemented through conversion, multiplication and addition. The performance parameter matrix in terms of delay and power consumption offered a significant improvement over other traditional implementation of DFT processor.

In the case of many low-cycle multiaxial fatigue criteria, we encounter a mathematical problem of adding vectors of normal and shear strains. Typically, the problem of defining an equivalent strain is solved by weighting factors. Unfortunately, this ignores the fact that these vectors represent other physical quantities: the normal strain is a longitudinal strain, and the shear strain is a rotation angle. Therefore, the goal of the present work was to propose a method of combining different types of strains by adopting a system of complex numbers. The normal strain was defined as the real part and the shear strain was defined as the imaginary part. Using this approach, simple load states, such as pure bending and pure torsion, have been transformed into an expression for equivalent strain identical to the previously proposed criteria defined by Macha.

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P ′ (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we prove some Lr inequalities for the polar derivative of a polynomial have all zeros in |z| ≤ 1. Our theorem generalizes a result of Dewan and Mir [K. K. Dewan, A. Mir, Inequalities for the polar derivative of a polynomial, J. Interd. Math. 10 (2007), no. 4, 525–531] and includes as special cases several interesting many known results [ABSTRACT FROM AUTHOR]

In this study, a different perspective was brought to Narayana sequences and one-, two-, three- and n --dimensional recurrence relations of these sequences were created. Then, some identities ranging from one to n --dimensions of these recurrences were created. [ABSTRACT FROM AUTHOR]

This paper addresses the uniqueness problem concerning the j-th derivative of a meromorphic function f(z) and the k-th derivative of its shift, f(z + c), where j, k are integers with 0 ≤ j < k. In this regard, our work surpasses the achievements of [2], as we have improved upon the existing results and provided a more refined understanding of this specific aspect. We give some illustrative examples to enhance the realism of the obtained outcomes. Denote by E(a, f) the set of all zeros of f-a, where each zero with multiplicity m is counted m times. In the paper proved, in particular, the following statement: Let f(z) be a non-constant meromorphic function of finite order, c be a non-zero finite complex number and j, k be integers such that 0 ≤ j < k. If f(j)(z) and f(k)(z + c) have the same a-points for a finite value a(̸= 0) and satisfy conditions... [ABSTRACT FROM AUTHOR]

This paper is an extension of the existing works on the frequency-domain-based bit flipping control strategy for stabilizing the single-bit high-order interpolative sigma delta modulator. In particular, this paper proposes the implementation and performs the performance evaluation of the control strategy. For the implementation, a frequency detector is used to detect the resonance frequencies of the input sequence of the sigma delta modulator. Then, a neural-network-based controller is used for finding the solution of the integer programming problem. Finally, the buffers and the combinational logic gates as well as an inverter are used for implementing the proposed control strategy. For the performance evaluation, the stability region in terms of the input dynamical range is evaluated. It is found that the control strategy can significantly increase the input dynamical range from 0.24 to 0.58. Besides, the control strategy can be applied to a wider class of the input signals compared to the clipping method.

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Generation of hidden attractor with complicated phase portrait in chaotic system with no equilibrium has presented a new research focus in the past decade. However, the existing approaches usually follow the rule that you reap what you sow, i.e., taking an no-equilibrium chaotic system as the seed. In this paper, a novel approach to the generation of no-equilibrium multi-fold hidden attractors is presented. The offset boosting and the operation of complex number are applied to several cases, including the seed chaotic systems with no equilibrium, with single unstable node, or with single stable node-focus. In specific, the offset boosting shifts the single non-zero equilibrium to the origin, then the operation of complex number rules out the equilibrium in the folded system and generates multi-fold trajectories. The highlight is that the proposed approach is not limited to the no-equilibrium seed system, but also applicable for different types of single-equilibrium seed systems, even the seed system with a single line of equilibrium. The detailed numerical simulations and experiments confirmed the feasibility of the proposed approach. Finally, an image encryption algorithm and its practical microcontroller-based implementation are presented to support the potential application. • Systematic approach for folding trajectory of chaotic system. • New multi-fold chaotic hidden attractors with no equilibrium. • New chaos-based encryption algorithm for image security. • Dynamic memory scheme for microcontroller implementation. [ABSTRACT FROM AUTHOR]

The use of registrations is useful in shortening Mizar proofs [1], [2], both in terms of formalization time and article space. The proposed system of classes for complex numbers aims to facilitate proofs involving basic arithmetical operations and order checking. It seems likely that the use of self-explanatory adjectives could also improve legibility of these proofs, which would be an important achievement [3]. Additionally, some potentially useful definitions, following those defined for real numbers, are introduced. [ABSTRACT FROM AUTHOR]

The goal of this paper is to define for the first time the concept of symbolic 2-plithogenic weak fuzzy complex number as new generalization generated by combining real numbers with symbolic 2-plithogenic numbers. We study the elementary properties of this new class such as Invertibility and nilpotency, with many related examples that explain its novelty. [ABSTRACT FROM AUTHOR]

Classical matrix theory for real, complex and hypercomplex numbers is a well-known concept. Is it possible to construct matrix theory over dual-generalized complex (DGC) matrices? The answer to this question is given in this paper. The paper is constructed as follows. Firstly, the fundamental concepts for DGC matrices are introduced and DGC special matrices are defined. Then, theoretical results related to eigenvalues/eigenvectors are obtained and universal similarity factorization equality (USFE) regarding to the dual fundamental matrix are presented. Also, spectral theorems for Hermitian and unitary matrices are introduced. Finally, due to the importance of unitary matrices, a method for finding a DGC unitary matrix is stated and examples for spectral theorem are given. [ABSTRACT FROM AUTHOR]