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This study aims to describe both fact and concept error of student s in solving space analytic geometry problems. The subjects of this research are two students of Mathematics Department of Universitas Negeri Makassar. Each of them represent s for each error (fact and concept errors). The collecting data were employed by using space analytic geometric tests and depth-interview. Interview guidelines and researcher were as research instruments. Data were qualitatively analyzed, using three stages of analysis: data reduction, data displa y and concluding. The main findings of this research are (1) the subject of fact error made mistakes in writing vector symbol s, (2) the subjects of concept error made mistakes in identifying an equation (plane equation or line equation), since his focus was only in the number of variables of the equation. Keywords: fact error, concept error, space analytic geometry

We propose an end-to-end learned bit-wise autoen-coder neural network (NN) for open loop multiple-input multiple-output (MIMO) antenna based communication systems. The optimized transmit vector constellations are learned based on the number of transmit and receive antennas, the number of bits conveyed per channel use and the signal-to-noise ratio. By training through an i.i.d. complex Gaussian (i.e., Rayleigh ergodic) matrix channel, the system implicitly picks up constellation shaping gains “along the way”. We evaluate and analyze these gains in comparison to the single-input single-output (SISO) results shown in [1] for different symmetric and asymmetric MIMO configurations. We first show that the NN-based receiver is able to compete with the a posteriori probability (APP) receiver performance up to certain configuration settings. Then we perform an end-to-end optimization of the transmit vector constellations using the conventional APP receiver as the decoder part. Thereby, we also investigate an edge case where the number of bits per vector symbol is not a multiple of the number of transmit antennas. Finally, we examine the performance of the system if a non-optimal zero forcing (ZF) receiver is used for inference, while the vector constellation was optimized for the APP receiver during training. We show that constellations optimized for the APP receiver are not necessarily optimal for such sub-optimal receivers that operate with reduced complexity. To fix this, we propose a detector-aware training scheme to learn constellations that have been optimized for a specific sub-optimal receiver and, thus, achieve higher performance in inference.

This research aimed to identify students' mathematical thinking style in mathematics education to resolve the higher-order thinking problem. The type of study was descriptive using qualitative data based on the vector analysis test. The vector analysis was identified by mathematical thinking style that divided into visual, analytic, and integration. This thinking style formed the qualitative data in the form of a description of the suitability of the student answer using indicators of any mathematical thinking styles. The subject whom students in the course of Vector Analysis Fourth Semester was interviewed after got mathematical thinking style data. The results showed that a lot of students has mathematical integration, that is, both visually and analytic. Students who have mathematical thinking style visually answered the question using a sketch of parallel vectors, perpendicular vectors, vector tangent, but they do not put the vector notation. Students who have analytical thinking styles differentiated vectors relationship using the verbal description of the appropriate definition that has been studied when they were in high school level. Students who have integrated mathematical thinking styles draw vectors using vector notation and explain verbally.

This paper proposes the use of systematic LDPC codes for hard decision Message Passing (hMP), which is a predecoder to Vector Symbol Decoding (VSD) for non-binary codes. VSD is a verification-based decoder whose outputs are decoded code words. LDPC are randomly generated and by nature, they are non-systematic codes. Using systematic instead of non-systematic codes allows the decoder to obtain the decoded data directly without a troublesome mapping step. In the simulations, 15 regular parity check matrices H are randomly generated and converted to their systematic forms. Results in a Gilbert-Elliot 2-state fading channel model with 6 different channel conditions show that hMP prefers codes with lower- weight H. For column weight of 9 bits and row weight of 18 bits, systematic H have lower weight after row operation and are better than the non-systematic ones for all channel conditions.

Body-diagonal diffusion couples provide a systematic way to investigate the diffusivity of high entropy alloys, a.k.a. HEAs. By definition these alloys contain 5-elements or more. All independent concentration differences in a body-diagonal couple are equal, except for being positive or negative. They are similar to diagonal diffusion couples for 3-element alloys, but the term body-diagonal applies only to couples with 4-elements or more. Composition vectors for these couples are aligned with body-diagonals in cubes located in the multi-element composition space of HEAs. The body-diagonals can be represented with a simple vector notation. The notation and other properties of body-diagonals are given in this work, including the number of diagonals and the angles between them. An advantage for HEAs is the fact that there are more diagonal directions and, therefore, more potential diffusion couples than needed to measure or validate diffusivity databases. That advantage gives flexibility to avoid 2-phase regions and, when validating databases, to avoid eigenvector directions of the diffusivity matrix. When body-diagonal diffusion couples are compact, their diffusion paths cross at the same composition, making it possible to measure an HEA diffusivity using the Boltzmann–Matano–Kirkaldy method.

The notation for vector analysis has a contentious nineteenth century history, with many different notations describing the same or similar concepts competing for use. While the twentieth century h...

International audience; In this paper, we study the anisotropy classes of the fourth order elastic tensors of the relaxed micro-morphic model, also introducing their second order counterpart by using a Voigt-type vector notation. In strong contrast with the usual micromorphic theories, in our relaxed micromorphic model only classical elasticity-tensors with at most 21 independent components are studied together with rotational coupling tensors with at most 6 independent components. We show that in the limit case Lc → 0 (which corresponds to considering very large specimens of a microstructured metamaterial the meso-and micro-coefficients of the relaxed model can be put in direct relation with the macroscopic stiffness of the medium via a fundamental homogenization formula. We also show that a similar homogenization formula is not possible in the case of the standard Mindlin-Eringen-format of the anisotropic micromorphic model. Our results allow us to forecast the successful short term application of the relaxed micromorphic model to the characterization of anisotropic mechanical metamaterials.

This paper presents a novel linear solar concentrating system named BDLFR, acronym of Beam-Down Linear Fresnel Reflector. A BDLFR system consists of a primary LFR array reflecting sunlight into a secondary hyperbolic –or elliptic– cylinder mirror that beams-downs reflected rays to its secondary focal line at ground level, where the receiver is placed. A model based on vector notation –and validated against SolTrace– predicts the optical behavior of BDLFR and generates layouts to avoid blockings. The edge ray approach is utilized to determine receiver aperture widths. As a function of beam-down curvature ratio, the BDLFR configurations that maximize solar collection are found out. Concentration ratios about 80 are reached with a tertiary re-concentrator (CPC) coupled to the receiver aperture.

There have been some important methods of combining a recursive path ordering and Tait-Girard’s computability argument to provide an ordering for termination proofs of higher-order rewrite systems. The higher-order recursive path ordering HORPO by Jouannaud and Rubio and the computability path ordering CPO by Blanqui, Jouannaud and Rubio are examples of such an ordering. In this paper, we give a case study of yet another direction of such extension of recursive path ordering, avoiding Tait-Girard’s computability method plugged in the above mentioned works. This motivation comes from Levy’s question in the RTA open problem 19, which asks for a reasonably straightforward interpretation of simply typed \(\lambda \)-calculus \(\lambda _{\rightarrow }\) in a certain well founded ordering. As in the cases of HORPO and CPO, the addition of \(\lambda \)-abstraction and application into path orderings might be considered as one solution, but the following question still remains; can the termination of \(\lambda _{\rightarrow }\) be proved by an interpretation in a first-order well founded ordering in the sense that \(\lambda \)-abstraction/application are not directly built in the ordering? Reconsidering one of Howard’s works on proof-theoretic studies, we introduce the path ordering with Howard algebra as a case study towards further studies on Levy’s question.

Poincare wrote out Cartesian components as separate variables and did not use vector notation or vector differential operators such as dell and curl. This leads to less notational conciseness and may cause generations of physicists who learned from works like Jackson (Classical Electrodynamics, 1999) (or other editions) to pause and think a few minutes before recognizing what would otherwise be a familiar equation.

A theorem concerning the Gram matrix of a general set of vectors: a vector polyhedron, is described. Generally speaking, the demonstration is performed by means of a non-negative vector definition: the variance vector, which can be associated to any vector polyhedron belonging to a vector space where a scalar product can be defined. The complete sum of the variance vector, the condensed variance index of any polyhedron, permits to construct the Gram matrix theorem.

In this chapter, we establish some notation and conventions to be followed throughout the book regarding a body and its motion. We rely on three different types of notation to describe concepts and solve problems in this book: vector notation, generally useful in both two- and three-dimensional contexts; complex notation, for problems in the plane; and a less familiar notation called Plucker notation that facilitates our analyses of rigid bodies. These are described, and related to each other, in the following sections. First, we discuss the reference frames we will utilize in the book.

To have the burst error correcting feature of nonbinary codes with low complexity decoding, Vector Symbol Decoder (VSD) uses non-binary codes with the structure of binary $G$ and $H$ matrix. VSD can always correct if the number of error symbols is at most two fewer than the minimum distance $(d)$ . For more errors, weight distribution plays an important role in the ability to correct. This paper aims to analytically compare the performance of different codes with the same $d$ using their weight distributions. The goal is to select the nonbinary codes with better VSD decoding performance without simulations. For example, the (27,14,7) code has better correction capability than the (23,12,7) code from the proposed formula and confirmed by the simulation results. Additional simulation results in Rician fading channel with Doppler Effect also confirm that the first code is superior.

The main purpose of this chapter is to introduce the maximum principle as a necessary condition that must be satisfied by any optimal control for the basic problem specified in Sect. 2.1. Although vector notation is used, the reader can consider the problem as one with only a single state variable and a single control variable on the first reading. In Sect. 2.2, the method of dynamic programming is used to derive the maximum principle. We use this method because of the simplicity and familiarity of the dynamic programming concept. The derivation also yields significant economic interpretations. In Appendix C, the maximum principle is also derived by using a more general method similar to that of Pontryagin et al. (1962) , but with certain simplifications. In Sect. 2.3, we apply the maximum principle to solve a number of simple, but illustrative, examples. In Sect. 2.4, the maximum principle is shown to be sufficient for optimal control under an appropriate concavity condition, which holds in many management science applications. Finally, Sect. 2.5 illustrates the use of Excel spreadsheet software to solve an optimal control problem.

This paper proposes a new decoding technique called “MP-VSD” for LDPC codes with large nonbinary symbols. It combines Hard Decision Message Passing (HDMP) with Vector Symbol Decoding (VSD). VSD usually uses very short block codes to limit the number of error symbols, which limits the size of matrix inversion required in the VSD decoding step. MP-VSD can correct more than 60% of the correctable error patterns of VSD for an (60, 30) LDPC code with no matrix inversions in a 2-state fading channel model. Thus, longer block codes may be used for nonbinary symbols.

In this chapter we review ordinary differential equations (ODEs) as a tool to model dynamics. We present examples of how to formulate them based on the dynamical system that needs to be modeled, and demonstrate the mathematical techniques one can employ to solve the equation analytically. We show how to solve linear differential equations with and without a forcing term, the so-called inhomogeneous and homogeneous ODEs, respectively. To illustrate the analysis of these equations, ODEs with first-order derivatives (e.g., d c / d t ) and second-order derivatives (e.g., d 2 c / d t 2 ) are used in the examples. Next, we show how higher-order ODEs can be represented as a set of first-order ones, and how this leads to a formalism in matrix/vector notation that can be efficiently analyzed using techniques from linear algebra. To complete the overview of the available tools for solving ODEs, the final part of this chapter briefly refers to application of Laplace and Fourier transforms (see also Chapter 12 ) to solve them.

Buchholz's apparent power and its derived unbalanced power, determined from the quadratic difference between the apparent power and the positive-sequence apparent power, are formulated in this paper in vector notation for three-phase sinusoidal systems. The proposed unbalanced power vector holds three components that measure the imbalance effects caused by the active and reactive currents, and the voltage imbalance effects, respectively. The total apparent and unbalanced powers of several subsystems (sources or loads) can be respectively obtained as the norm of the vector sums of their apparent and unbalanced power vectors, according to these new formulations. The correctness of the apparent and unbalanced power vector formulations are verified in a three-phase sinusoidal installation formed by two linear loads supplied from a 400 kVA delta/wye distribution transformer, as an application example.

[EN] In this paper we provide two representation theorems for two relevant classes of operators from any p-convex order continuous Banach lattice with weak unit into a Banach space: the class of continuous operators and the class of cone absolutely summing operators. We prove that they can be characterized as spaces of vector measures with finite p-semivariation and p-variation, respectively, with respect to a fixed vector measure. We give in this way a technique for representing operators as integrals with respect to vector measures., J.M. Calabuig and O. Blasco were supported by Ministerio de Economia y Competitividad (Spain) (project MTM2011-23164). E.A. Sanchez-Perez was supported by Ministerio de Economia y Competitividad (Spain) (project MTM2012-36740-C02-02).

Vector projection and vector rejection are highly common and useful operations in mathematics, information theory, and signal processing. In this paper, we find the distribution of the norm of projection and rejection vectors when the original vectors are standard complex normally distributed.

In 1864, the Scottish physicist James Clerk Maxwell undertook the task of combining all that was then known about electric and magnetic phenomena into a single, encompassing theory. Maxwell used as a template the mathematical structure of the well-known theory of elastic media. In this regard, Maxwell followed in the long tradition of applying established mathematical tools to a new problem. Maxwell’s original papers are somewhat difficult to read because they do not incorporate the modern vector notation invented by Oliver Heaviside. In this regard, it makes Maxwell’s achievement even more compelling as he was able to understand the mathematical structure present in the equations even without the notational support that makes it more evident.

Mathematical frameworks for representing fields and waves and expressing Maxwell's equations of electromagnetism include vector calculus, differential forms, dyadics, bivectors, tensors, quaternions, and Clifford algebras. Vector notation is by far the most widely used, particularly in applications. Of the more sophisticated notations, differential forms stand out as being close enough to vectors that most practitioners can readily understand the notation, yet at the same time offering unique visualization tools and graphical insight into the behavior of fields and waves. We survey recent papers and book on differential forms and review the basic concepts, notation, graphical representations, and key applications of the differential forms notation to Maxwell's equations and electromagnetic field theory.

This letter shows how to extend packet/vector symbol burst error correction capability beyond prior known limits with the aid of soft information interaction with the inner code of a concatenated code. If the errors prove not to be in a burst, further correction still is possible with the aid of soft inner code information. Also, the soft information provides strong additional error detection.

In this chapter we discuss a wide range of basic topics related to vectors of real numbers. Some of the properties carry over to vectors over other fields, such as complex numbers, but the reader should not assume this. Occasionally, for emphasis, we will refer to “real” vectors or “real” vector spaces, but unless it is stated otherwise, we are assuming the vectors and vector spaces are real. The topics and the properties of vectors and vector spaces that we emphasize are motivated by applications in the data sciences.

Vectors and matrices are useful in representing multivariate numeric data, and they occur naturally in working with linear equations or when expressing linear relationships among objects. Numerical algorithms for a variety of tasks involve matrix and vector arithmetic. An optimization algorithm to find the minimum of a function, for example, may use a vector of first derivatives and a matrix of second derivatives; and a method to solve a differential equation may use a matrix with a few diagonals for computing differences.

In Chap. 3 we studied the properties of the space \(\mathbb{R}^{n}\) and extended the definition of its algebraic structure to a general vector space (Definition 3.2.2).

The system variable vector \( {\bar{\text{X}}}_{{\text{sys}}} \) of an optical system is, nearly always, different from the boundary variable vector \( {\bar{\text{X}}}_{\text{i}} \) of a boundary surface. Furthermore, changes in the system variable vector may have a profound effect on the behavior of the rays as they propagate through the system. Therefore, the Jacobian matrix \( \text{d}\bar{\text{X}}_{\text{i}} /\text{d}\bar{\text{X}}_{{\text{sys}}} \) of the boundary variable vector \( {\bar{\text{X}}}_{\text{i}} \) with respect to the system variable vector \( {\bar{\text{X}}}_{{\text{sys}}} \) is of crucial concern to optical systems designers.

An efficient technique is developed to simplify computations in the field of vector analysis. The evaluation of vector algebraic and differential operations becomes more simple and straightforward by simply transforming the vector operations into matrix operations. The matrix operations are especially useful when there are mixed coordinate basis involved in the vector operations.

On the basis of analyzing the relations between vector operation and properties of multiplication operation of quaternions, the geometric representation and mapping of quaternions and vectors defined in three-dimensional space are established. The mathmatic operation of space vector, in addition to any space unit vector, rotating around three coordinate axises is carried based on the utilization of quaternions.The main purpose of this paper is to solve the issues of mathmatic tools introduced in spatial mechanism analysises and integrations by making use of quatnions.

In geophysical fluids like the ocean and atmosphere, the horizontal plane and vertical direction have much different properties from each other, partly because of a small aspect ratio and partly because of stable density stratification and rotation of the earth. A useful system of vector notations is proposed for describing geophysical fluid dynamics in a concise manner. We first introduce "¬", a vector operator to be called strophe or turn. It rotates a horizontal vector clockwise at right angle. Likewise "◁ ≡ ¬∇" (blana or alongent) is defined. When operated on a scalar field ψ, blana yields a vector that is as large as ||∇ψ||, but parallel to the isoline of ψ with higher ψ on the left-hand side. A three-dimensional vector is written as u__- = u + w, where u in boldface indicates the horizontal components and w is the vertical component of three-dimensional vector u__-. Examples and applications are presented to show the utility of the present system of notations called GFDVN (Geophysical Fluid Dynamics Vector Notation), which not only simplifies the description, but also gives a clear geometrical image of vectors in oceanography or meteorology. We observe quite symmterical relations between the inner and outer product of two vectors, divergence and rotation of a vector field, through ¬ or ◁. As a focused application, an investigatios is made of the geometrical implication of Lagrange's formula, or the formula of triple vector product. By using GFDVN we find that the formula is none other than a representation of skew coordinates on a plane. Also that formula in two-dimension turned out to have a simple relation with the mixing ratio of three water types on a T-S diagram used in oceanography.

In this communication, simple methods are shown for correcting bursts of large size and bursts combined with random errors using vector symbols and primarily vector XOR and feedback shift register operations. One result is that any (n, k) cyclic code with minimum distance > 2 can correct all full vector symbol error bursts of length n-k-1 or less if the error vectors are linearly independent. If the bursts are not full but contain some error-free components, the capability of correcting bursts up to n-k or less is code dependent. Also, vector symbol decoding with Reed-Solomon component codes can correct, very simply, with probability ges 1- n(n - k)2-r, all cases of e les n - k - 1 r-bit random errors in any cyclic span of length les n - k. The techniques often work when there is linear dependence. In cases where most errors are in a burst but a small number of errors are outside, the solution, given error-correcting capability, can be broken down into a simple solution for the small number of outside errors, followed by a simple subtraction to reveal all the error values in the burst part.

The bulk of this chapter is devoted to homogeneous linear systems of ODEs with real constant coefficients. This means systems of the form $$\displaystyle{ \begin{array}{ccc} x_{1}^{{\prime}}& =& a_{11}x_{1} + a_{12}x_{2} +\ldots +a_{1d}x_{d}, \\ x_{2}^{{\prime}}& =& a_{21}x_{1} + a_{22}x_{2} +\ldots +a_{2d}x_{d},\\ \\ \vdots&\vdots&\vdots\\ \\ x_{d}^{{\prime}}& =&a_{d1}x_{1} + a_{d2}x_{2} +\ldots +a_{dd}x_{d}.\end{array} }$$ (2.1) (From now on, we shall let d be the dimension of our systems, so that the index n is available for other uses.) The written-out system (2.1) is awkward to read or write, and we shall normally use the vastly more compact linear-algebra notation $$\displaystyle{ \mathbf{x}^{{\prime}} = A\mathbf{x}, }$$ (2.2) where x = (x1, x2, …, x d ) is a d-dimensional vector of unknown functions, A is a d × d matrix with real entries, and matrix multiplication is understood in writing Ax. In vector notation, an appropriate initial condition for (2.2) is $$\displaystyle{ \mathbf{x}(0) = \mathbf{b}, }$$ (2.3) where \(\mathbf{b} \in \mathbb{R}^{d}\).

We obtain the biggest advantage of bond graph when a system is composed of different energy domains. In this chapter, we will cover some examples of bond graph modeling in different energy systems with advanced and more practical concepts in addition to the details covered in the last chapter. In the last chapter, basic bond graph formulation and state space representation were discussed. In this chapter, we will discuss the concepts of algebraic loops, additional two port elements known as fields, the effect of derivative causality, and bond graph in the vector notation.

This paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But this is just the "internal" interpretation, which has to be developed simultaneously with, and is justified by, an "external" one, offered by natural deduction: the sequent calculus corresponds to a bi-directional, agnostic (w.r.t. the call strategy), computational lambda-calculus. Next, the duality between control and co-control is studied and proved in the context of classical logic, where one discovers that the classical sequent calculus has a distortion towards control, and that sequent calculus is the de Morgan dual of natural deduction., (undefined)

In this paper, we propose a generalized expression of multi-target probability density in the framework of finite-set statistics (FISST). It differs from the existing expression in that it is completely defined by a set of parameters. The parameterized expression offers three important advantages. Firstly, the statistics of any subset of an random finite set (RFS) can be conveniently extracted by directly computing the marginal distribution of the proposed parameterized probability density. Note that the theoretically sound way of computing the marginal distribution of an RFS probability density is not well defined in current FISST. Secondly, it inherits advantages of the vector notation based density in that the identities of target states are implicitly embedded with no need to augment target state with an extra label. Thirdly, it is a generalized expression capable to describe the statistics of almost all kinds of RFSs and doesn't need to assume a specified measurement model for the development of target tracking filter. As a preliminary study, this paper provides both the derivations of the proposed expression and the resultant tracking filter for a two-target scenario. The aforementioned advantages are clearly highlighted by the numerical results.

In order to reduce the cost of Radio Frequency (RF) chains in the spatial multiplexing systems with Vertical-Bell Labs Layered Space-Time (V-BLAST) nonlinear receiver, a novel transmit antenna selection criterion is proposed with the motivation of minimizing the Vector Symbol Error Rate (VSER). In the proposed scheme, both the number of substreams and the mapping of substreams to antennas are dynamically adjusted based on the knowledge of channel. Simulation results illustrate that the proposed two-step selection criterion outperforms the existing eigenmode based selection criterion by 0.3 dB at a VSER of 10−3.

We derive an effectively three-dimensional relativistic spin precession formalism. The formalism is applicable to any spacetime where an arbitrary timelike reference congruence of worldlines is specified. We employ what we call a stopped spin vector which is the spin vector that we would get if we momentarily make a pure boost of the spin vector to stop it relative to the congruence. Starting from the Fermi transport equation for the standard spin vector we derive a corresponding transport equation for the stopped spin vector. Employing a spacetime transport equation for a vector along a worldline, corresponding to spatial parallel transport with respect to the congruence, we can write down a precession formula for a gyroscope relative to the local spatial geometry defined by the congruence. This general approach has already been pursued by Jantzen et. al. (see e.g. Jantzen, Carini and Bini, Ann. Phys. 215 (1997) 1), but the algebraic form of our respective expressions differ. We are also applying the formalism to a novel type of spatial parallel transport introduced in Jonsson (Class. Quantum Grav. 23 (2006) 1), as well as verifying the validity of the intuitive approach of a forthcoming paper (Jonsson, Am. Journ. Phys. 75 (2007) 463) where gyroscope precession is explained entirely as a double Thomas type of effect. We also present the resulting formalism in explicit three-dimensional form (using the boldface vector notation), and give examples of applications., 27 pages, 8 figures

An efficient technique is developed to simplify the computations in the area of vector analysis. The evaluation of vector algebraic and differential operations becomes more simple and straightforward by simply converting the vector operations into matrix operations. The matrix formulation technique is especially useful when there are mixed coordinate basis involved in the vector operations.

By introducing a kind of maximal operator of fractional order associated with the mean Luxemburg norm and using the technique of Sharp function, multilinear commutators of fractional integral operator with vector symbol b = (b1,...bm)defined by $$I_\alpha ^b f(x) = \int_R {\left[ {\prod\limits_{j = 1}^m {(b_j (x) - b_j (y))} } \right]} \frac{1}{{|x - y|^{n - \alpha } }}f(y)\user2{d}y$$ are considered. The following priori estimates are proved. For 1 0} \frac{1}{{\Phi (\tfrac{1}{t})}} \left| { \left\{ {y \in R^n : \left| { I_a^b f\left( y \right) } \right| > t} \right\}} \right|\tfrac{1}{q} \leqslant \hfill \\ c \mathop {\sup }\limits_{t > 0} \frac{1}{{\Phi (\tfrac{1}{t})}}\left| {\left\{ {y \in R^n :M_{L(logL)} \tfrac{1}{r}_{,a} \left( { \left\| { b } \right\| f} \right)\left( y \right) > t} \right\}} \right|\tfrac{1}{q}, \hfill \\ \end{gathered} $$ where \(\parallel b\parallel = \prod\nolimits_{j - 1}^m {\parallel b_j \parallel } _{Osc_{expL^{rj} } } ,\Phi (1 + log^ + t)^{\tfrac{1}{r}} ,\frac{1}{r} = \frac{1}{{r_1 }} + \cdots + \frac{1}{{r_m }},\)\(M_{L(logL)} \tfrac{1}{{r,\alpha }}\) is an Orlicz type maximal operator.

We have attempted to express the frequency response functions of a linear and a quadratic non-linear system in terms of spectral vectors. These vector notations convey the system characteristics in physically realisable measures. One of the valuable tools to verify the non-linear system features is the expression of a coherence function using vector notation. The visualisation of a coherence function demonstrates the dependency of an output signal on the linear and quadratic non-linear terms in the Volterra model. Multiple coherence functions are also introduced to validate the modelling errors.

Vectors and quaternions are quite different mathematical quantities because they have different symmetry properties. Gibbs and Heaviside created their vector system starting from the quaternion system invented by Hamilton. They identified a pure quaternion as a vector and introduced some changes in the product of two vectors defined by Hamilton without realizing that the scalar product and vector product cannot be interpreted as the scalar part and vector part of the quaternion product. Toward the end of the 19th century some authors realized that there was an incompatibility between the vector and quaternion formalisms, but the central problem was not altogether clear. This paper will show that the main difficulty arose from Hamilton’s contradictory use of i, j, and k both as versors and as vectors.