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This chapter provides some details about some of the more important mathematics underlying quantum mechanics, including complex numbers, statistics, Fourier transforms and waves.

Proof of Work (PoW) mechanisms used as part of the consensus mechanisms in block chains often have a major drawback namely that resources are only spent on doing the PoW and nothing else. In this paper we propose an adaption of hash based PoW’s. This adaption consists in two aspects, firstly by embedding the space of the hash’s, \(\mathcal{H}\), in the space of complex numbers \(\mathbb {C}\), more concretely in the subset \(\mathcal{B}=\{z\in \mathbb {C}: 0

In this research, the recent development of three dimensional fuzzy sets namely, spherical fuzzy sets is extended to complex spherical fuzzy sets (CSFSs) and the novelty of CSFSs with their operations and properties are investigated. The range of CSFSs membership functions lie in complex plane, not limited to [0, 1]. So, the CSFSs describe complex membership functions in a set in terms of complex numbers which leads to a new mathematical framework and comprehensive study of their properties. Some of the aggregation operators and distance measures are established with their mathematical proof. Further an intelligent system based on EDAS (Evaluation based on Distance from Average Solution) method has been developed using the proposed complex spherical fuzzy aggregation operators. An application is proposed for aviation industry 4.0 in order to assess the performance of catering services. Finally a numerical illustration has been demonstrated for the proposed innovative method.

This chapter discusses the better performing approaches for modular exponential techniques that are adaptable by the different architectures of PKC. The importance of PKC in the field of information security including militarily level security is decisive. The security of these PKCs depends upon their complex arithmetic problems (e.g. IFP and DLP) which include modular exponentiation as a critical operation. Modular exponentiation is an expensive operation, that is comprised of a series of modular multiplications. The overall performance of PKC is affected by the efficient implementation of modular exponentiation. There are different ways of improving the performance of modular exponentiation, either reduce the modular multiplication operation’s internal architecture to reduce the latency, or reduce the number of modular multiplications required for the computation of modular exponentiation. This chapter discusses the techniques of reducing the number of the required modular multiplications by Bit Forwarding (BFW) techniques and how they can be implemented. The Montgomery multiplication algorithm is modified according to the requirements of BFW techniques. There are two varieties of Montgomery multiplication discussed

The frequency does not change in a linear circuit. The voltage and current throughout the circuit have the form of A cos (ωt + φ), which has a short-hand notation A ∠ φ. This short-hand notation is referred to “phasor.” This “phasor” is, in fact, a complex number.

As mathematicians found out in the last century, there are only four normed division algebras1 over ℝ: the real numbers themselves, the complex numbers, the quaternions and the octonions. Whereas the real and complex numbers are very well-known and most of their properties carry over to the quaternions (apart from the fact that these are not commutative), the octonions are very different and harder to handle since they are not even associative. However, they can be used for several interesting topological constructions, often paralleling constructions known for ℝ, ℂ or ℍ.

In this chapter we introduce the reader to auctions where the seller offers more than one unit (multi-unit auctions).

Mathematics modelling education has potential to support pedagogical innovations by fostering coherence within mathematics as well as supporting interdisciplinary connections. Students often struggle to build coherence in trigonometry presented in multiple mathematical frames such as triangle, circle, vector and complex numbers. Design-based research experiments are conducted to extend the modelling of circular motion to the modelling of advanced periodic orbits from a series of trigonometric functions. Inquiry-based orbital modelling allows students experiment with modelling of periodic orbits with technology-rich tasks in interpreting the meaning and connections of periods and amplitudes of circular functions and the emergent patterns. The results show that learners experience coherence while interpreting, comparing and validating their orbital models in circular, functional and complex trigonometry with connections in between.

Whereas a group possesses a single internal binary rule of composition, a ring is a set with two internal binary operations generically referred to as “addition” and “multiplication,” though the precise character of these operations may not align with our everyday understanding of those terms. Fields—of which the real and complex number systems are examples—and skew fields, such as the quaternion number system, are special cases of rings. The development of complex numbers and quaternions is placed in historical context, and we give a brief introduction to quaternion algebra. Substructures, quotients and products of rings are also described.

In this chapter, the problems of the third chapter are fully solved, in detail, step-by-step, and with different methods.

Coquaternions form a four dimensional real algebra generalizing complex numbers and were introduced by James Cockle at about the same time that Hamilton discovered the famous algebra of quaternions. Although not as popular as quaternions, in recent years one can observe an emerging interest among mathematicians and physicists on the study of these numbers. In this work we revisit a Mathematica package for implementing the algebra of coquaternions – Coquaternions – and discuss a set of Mathematica functions – CoqPolynomial – to deal with coquaternionic polynomials. These two sets of functions provide the basic tools necessary for manipulating coquaternions and unilateral coquaternionic polynomials, reflecting, in its present form, the recent interests of the authors in the area.

We study real steady state varieties of the dynamics of chemical reaction networks. The dynamics are derived using mass action kinetics with parametric reaction rates. The models studied are not inherently parametric in nature. Rather, our interest in parameters is motivated by parameter uncertainty, as reaction rates are typically either measured with limited precision or estimated. We aim at detecting toricity and shifted toricity, using a framework that has been recently introduced and studied for the non-parametric case over both the real and the complex numbers. While toricity requires that the variety specifies a subgroup of the direct power of the multiplicative group of the underlying field, shifted toricity requires only a coset. In the non-parametric case these requirements establish real decision problems. In the presence of parameters we must go further and derive necessary and sufficient conditions in the parameters for toricity or shifted toricity to hold. Technically, we use real quantifier elimination methods. Our computations on biological networks here once more confirm shifted toricity as a relevant concept, while toricity holds only for degenerate parameter choices., Comment: Computations available as ancillary files

As was already mentioned in the Chapter 1, complex numbers may appear as matrix elements. Moreover, the characteristics of real matrices (such as eigenvalues, see Chap. “Vector Spaces” on page 226) in some cases appear to be complex. In this connection, let us discuss the methods of algebra of complex numbers.

Let \(\alpha \) and \(\beta \) be two nonnegative integers such that \(\beta < \alpha \). For an arbitrary sequence \(\{a_n\}_{n\geqslant 1}\) of complex numbers, we investigate linear combinations of the form \(\sum _{k\geqslant 1} S(\alpha k-\beta ,n) a_k\), where S(k, n) is the total number of k’s in all the partitions of n into parts not congruent to 2 modulo 4. The general nature of the numbers \(a_n\) allows us to provide new connections between partitions and functions from multiplicative number theory.

We study the distribution of eigenvalues of varying Toeplitz and Hankel matrices such as \(\left [ a_{n+k-j}\right ] _{j,k}\) and \(\left [ a_{n+k+j} \right ] _{j,k}\) where an behaves roughly like nβ for some non- 0 complex number β, and n →∞. This complements earlier work on these matrices when the coefficients \(\left \{ a_{n}\right \} \) arise from entire functions.

The C*-algebra of compact operators K(H) and the C*-algebra C(X) with X contractible both have the same K-theory as the complex numbers, that is, K0(C) = Z and K1(C) = 0.

By investigation of locus equations we sometimes encounter problems with factorization of resulting polynomials. Commands on factorization of polynomials over the field of rational numbers are implemented in most mathematical software usually by the command factor. We can also use commands on factorization of polynomials over some extension of the field of rational numbers, for instance command AFactor in Maple. Factorization over real or complex numbers is much more difficult. In two examples we will show how to make factorization using dynamic geometry systems in such cases when related commands fail.

Bayesian estimators of a time-varying number of objects and their states are developed. The estimators are based on object superposition, a fundamental concept in finite point processes that is adapted to model multiobject state. The concept of superposition is introduced carefully step by step, starting with its application to the JPDA filter for a specified number of objects. This JPDAS filter lays the groundwork for the extension to a random number of objects, leading quickly and painlessly to the CPHD filter. This natural approach shows the many close connections between the CPHD and the JPDAS filters. A numerical device, new to tracking applications, called the complex step method is used to evaluate the intensity function to machine precision “for free” merely by evaluating the GFL of the Bayes posterior process in complex arithmetic. Examples are presented.

The Nonlinear Identification through Feedback of the Output (NIFO) and Conditioned Reverse Path (CRP) methods are a popular family of approaches for nonlinear system identification. They estimate the underlying linear Frequency Response Function (FRF) as well as the parameters describing the mechanical system’s nonlinearities. However, one troubling aspect is that the parameters obtained are complex numbers and typically are found to vary with frequency, so post-processing must be employed to obtain physically reasonable parameters and an accurate estimate of the underlying FRFs. This work proposes two methods (based on the H1 and H2 algorithms) which can be used in the estimation of the linear FRF as well as frequency-independent nonlinear parameters. This paper evaluates the methods numerically using a single degree of freedom system and exploring various methods for determining which nonlinear parameters to include in the model.

The chapter discusses the procedures for the formation and use of real residuals of real numbers on a real module, as well as complex and real residues of an integer complex number on a complex module. The chapter focuses on the processing of complex and real residuals of an integer complex number by a complex module. This procedure is based on using the results of the first fundamental Gauss theorem. The chapter of the proposed procedure provides examples of determining deductions in a complex numerical domain. On the basis of the considered procedure, an algorithm was developed for determining the real deduction of an integral complex number using a complex module in accordance with which the device was synthesized for its technical implementation. The device received a patent of Ukraine for the invention, which confirms the novelty and practical value of research results. The results obtained in the chapter are advisable to be used when implementing tasks and algorithms in real and complex numerical domains. In particular, the use of real numbers for cryptographic applications was considered.

We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial p of degree d with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for multiple roots of a polynomial given by a black box for the approximation of its coefficients, and their complexity decreases at least proportionally to the number of roots in a region of interest (ROI) on the complex plane, such as a disc or a square, but we greatly strengthen the main ingredient of the previous algorithms. We build the foundation for a new counting test that essentially amounts to the evaluation of a polynomial p and its derivative \(p'\), which is a major benefit, e.g., for sparse polynomials p. Moreover with evaluation at about \(\log (d)\) points (versus the previous record of order d) we output correct number of roots in a disc whose contour has no roots of p nearby. Our second and less significant contribution concerns subdivision algorithms for polynomials with real coefficients. Our tests demonstrate the power of the proposed algorithms.

A complex number is an ordered pair of real numbers. If each of (a, b) and (c, d) is a complex number then addition and multiplication are defined on the set of complex numbers in the following way: $$\displaystyle \begin {array}{c} (a,b) + (c,d) = (a+c,b+d)\\ (a,b) (c,d)=(a c-b d, a d+b c) \end {array} $$

Many equations in economics and finance are very complex. As a result, existing methods of solving these equations are very complicated and time-consuming. In many practical situations, more efficient algorithms for solving new complex equations appear when it turns out that these equations can be reduced to equations from other application areas—equations for which more efficient algorithms are already known. It turns out that some equations in economics and finance can be reduced to equations from physics—namely, from quantum physics. The resulting approach for solving economic equations is known as quantum econometrics. In quantum physics, the main objects are described by complex numbers; so, to have a reduction, we need to come up with an economic interpretation of these complex numbers. It turns out that in many cases, the most efficient interpretation comes when we separately interpret the absolute value (modulus) and the phase of each corresponding quantum number; the resulting techniques are known as Bohmian econometrics. In this paper, we use an algebraic approach—namely, the idea of invariance and symmetries—to explain why such an interpretation is empirically the best.

The sequence (xn)n≥0 = (xn(a, b;p, q))n≥0 defined by $$\displaystyle x_{n+2} = px_{n+1} + qx_n, \quad x_0 = a, x_1 = b, $$ with a, b, p, and q arbitrary complex numbers is called a Horadam sequence. For simplicity, we shall denote (xn(a, b;p, q))n≥0 by (xn)n≥0 hereafter.

It is the belief that imaginary numbers appeared for the first time around 1540 when the mathematicians Tartaglia and Cardano represented real roots of a cubic equation in terms of conjugated complex numbers. A Norwegian surveyor, Caspar Wessel, was in 1798 the first one to represent complex numbers by points on a plane with its vertical axis imaginary and horizontal axis real. This diagram was later known as the Argand diagram, although the true Aragand’s achievement was an interpretation of \(i=\sqrt{({-}1)}\) as a rotation by a right angle in the plane. Complex numbers received their name by Gauss, and their formal definition as pair of real numbers was introduced by Hamilton in 1835.

Throughout these notes V will be a finite dimensional K vector space and I will denote the identity operator. For all our applications there is no loss in specializing K to be the field of rational numbers, but at some point in our development we will need to assume that K is the field of complex numbers.

In this work, we assume that a brain in the visual cortex (VC) operates with Clifford numbers when it calculates hypercomplex-valued invariants of an image as it recognizes it. Clifford algebras generalize the algebras of complex numbers, quaternions and octonions. Of course, the algebraic nature of hypercomplex numbers must correspond to the spaces with respect to geometrically perceivable properties. For recognition of 2-D (bichromatic), 3-D (color), and n-D (multi-channel) images, we turn the perceptual spaces into corresponding Clifford algebras (and call them the VC-perceptual algebras). This approach gives full representation of how algebraic structures can possess image features and how algebraic structures can be used in different visual systems. It is our aim to show that the use of Clifford algebras fits more naturally to the tasks of recognition of multicolor patterns than does the use of color vector spaces. One can argue that nature has, through evolution, also learned to utilize properties of Cliffordean numbers.

Full-duplex (FD) wireless and phased arrays are both promising techniques that can significantly improve data rates in future wireless networks. However, integrating FD with transmit (Tx) and receive (Rx) phased arrays is extremely challenging, due to the large number of self-interference (SI) channels. In this paper, a model is proposed to minimize the self-interference without large TxBF and RxBF gain losses through optimizing the TxBF and RxBF weights. As the model involves complex numbers and is non-convex, an iterative algorithm by solving series convex model is given to obtain an approximate solution. Meanwhile, in each step of the iterative algorithm, the sub model is translated into real number model. Then a penalty term is added into the sub model of each step of the iterative algorithm. Simulation results show that, the iterative algorithm is effective and can get good results. The penalty term can improve the performance of the algorithm.