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For regular continued fraction, if a real number x and its rational approximation p / q satisfying | x − p / q | < 1 / q 2 , then, after deleting the last integer of the partial quotients of p / q , the sequence of the remaining partial quotients is a prefix of that of x. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set E (ψ) of complex numbers which are well approximated with the given bound ψ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound ψ and its Hausdorff dimension is equal to that of the ψ -approximable set W (ψ) of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník theorem in real case. [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]

Split-complex (hyperbolic) numbers are ordered pairs of real numbers, written in the form x + jy with j² = 1, used to describe the geometry of the Lorentzian plane. Since a null split-complex number does not have an inverse, some methods to calculate the exponential of complex matrices are not valid for split-complex matrices. In this paper, we examined the exponential of a 2 × 2 split-complex matrix in three cases: i. Δ = 0, ii. Δ ̸= 0 and Δ is not null split-complex number, iii. Δ ̸= 0 and Δ is a null split-complex number where Δ = (trA)2 - 4 detA. [ABSTRACT FROM AUTHOR]

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is $ (g,gq,gq^{2},\ldots,gq^{n-1}) $ (g , gq , g q 2 , ... , g q n − 1) , where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples. [ABSTRACT FROM AUTHOR]

This is a further discussion of a previous work of the author on tensors with different rank and symmetric rank. We point out several obstructions towards extending a complex number example to the real number setting and discuss several further questions raised in the literature. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic behavior of a class of delayed quaternion-valued neural networks (QVNNs) with impulses and parameter uncertainties. First, we assume that the activation function and connection matrices in the model are defined in the quaternion domain. Without decomposing the QVNNs into four equivalent neural networks in a real number domain or two equivalent neural networks in a complex number domain, the existence and uniqueness of the equilibrium point (EP) are studied based on the M-matrix and homeomorphism mapping theories. By combing the vector Lyapunov function with mathematical induction, theorems are given to ensure robust exponential stability of the system's EP. The results reflect the influence of connection matrices, delays, impulses, and the activation function on the convergence speed of the EP. Finally, the feasibility of the derived results is explained through three examples. [ABSTRACT FROM AUTHOR]

The study reported in this paper is concerned with undergraduates’ images of the root concept with a focus on consistencies and inconsistencies. The data was collected with an online questionnaire from 115 first-year university students. A statistical analysis was applied to capture the variety of students’ responses to questions in which square roots were extracted in the field of complex and real numbers, from positive, negative and non-real radicands. A qualitative analysis of students’ reasoning revealed the complexity of their concept images, which were compartmentalized into distinct domains and connected with other concepts through schemes. The phenomena of real and complex number biases were indicated. [ABSTRACT FROM AUTHOR]

Diversity is supposed to create better groups and societies but sometimes fails. It is explained why the power of diversity may not create better groups in the current diversity prediction theory. Diversity may hurt civic life and introduce distrust. This is because the current diversity prediction theory is based on real numbers that ignore individual abilities. Its diversity prediction theory maximizes performance with infinite population size. Contrary to this, collective intelligence or swarm intelligence is not maximized by infinite population size, but by population size. The extended diversity prediction theory using the complex number allows us to express individual abilities or qualities. The diversity of complex numbers always produces better groups and societies. The wisdom of crowds, collective intelligence, swarm intelligence or nature-inspired intelligence is implemented in the current machine learning or artificial intelligence, called Random Forest. The problem of the current diversity prediction theory is detailed in this paper. • Diversity is supposed to create better groups and societies but sometimes fails. • Diversity in real numbers ignores individual abilities. • Diversity of population size plays a key role in collective intelligence. • Diversity in complex numbers allows us to express individual abilities. • Diversity in complex numbers always produces better groups and societies. [ABSTRACT FROM AUTHOR]

The new modular approach to constrained control of higher-order processes with dominant first-order dynamics using generalized controllers with automatic resets (ARCs) is addressed. The controller design is based on the multiple real dominant pole (MRDP) method for the integrator plus dead time (IPDT) process models. The controller output constraints are taken into account by inserting the smallest numerator time constant of the controller transfer function into the positive feedback loop representing the automatic reset (integral) term. In the series realization of the proportional–integral–derivative–acceleration (PIDA) controller (and other controllers with even derivative degree), the time constant mentioned is complex, so only the real part of the time constant has been used so far. Other possible conversions of a complex number to a real number, such as the absolute value (modulus), can be covered by introducing a tuning parameter that modifies the calculated real time constant and generalizes the mentioned conversion when designing controllers with constraints. In this article, the impact of the tuning parameter on the overall dynamics of the control loop is studied by simulation. In addition, an evaluation of the stability of the closed-loop control system is performed using the circle criterion in the frequency domain. The analysis has shown that the approximation of the complex zero by its real part and modulus leads to a near optimal response to the set point tracking. The disturbance rejection can be significantly improved by increasing the tuning parameter by nearly 50%. In general, the tuning parameter can be used to find a compromise between servo and regulatory control. The robustness and applicability of the proposed controller is evaluated using a time-delayed process with first-order dominant dynamics when the actual transfer function is much more complicated than the IPDT model. A comparison of the proposed MRDP-PIDA controller with series PI, PID and PIDA controllers based on a modified SIMC method has shown that the MRDP-PIDA controller performs better than the SIMC method, although the SIMC uses a more complex process model. [ABSTRACT FROM AUTHOR]

In this paper, we classify the family of entire functions { ρ , σ } such that the induced weighted composition map A ρ , σ , is a conjugation namely J b , α , where b is an arbitrary complex number and α is a real number such that 0 < α < 1. In the last section, we characterize the J b , α -symmetric maximal differential operators on the Fock space α 2 . [ABSTRACT FROM AUTHOR]

For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

Complex fuzzy set, as an extension of classical fuzzy sets, could describe the fuzzy characters of things more detail and comprehensively and is very useful in dealing with vagueness and uncertainty of problems that include the periodic or recurring phenomena. Note that a complex fuzzy set is different from the fuzzy complex set introduced and discussed by many scholars, since the membership degree of a complex fuzzy set is a complex number with length less than or equal to 1 while a fuzzy complex set is a real number with membership degree less than or equal to 1, and the universe is the complex plane. As the mathematical theoretical basis of fuzzy mathematics, fuzzy set and its mapping, corresponding fuzzy complex set and its mapping have been investigated in depth because they integrate and cross the methods and results of classical real analysis and complex analysis. However, there is no comprehensive investigation on complex fuzzy set and its corresponding mathematical theory, even include decomposition theorems, extension principles and the basic operations of the complex fuzzy set. As is well known, the cut set of fuzzy sets is the bridge between fuzzy sets and classical sets, which plays a significant role in fuzzy sets and fuzzy systems. In this paper, the concept of (r, θ)-cut sets of complex fuzzy sets is proposed and their properties are discussed. Meanwhile, the decomposition theorems and the extension principles of complex fuzzy set based on (r, θ)-cut sets are deduced and corresponding properties are investigated. All these conclusions not only deeply enrich the fundamental theory of complex fuzzy set, but also provide a powerful tool to investigate complex fuzzy set. Finally, an example application of signal detection demonstrates the utility of the (r, θ)-cut sets of complex fuzzy sets in practice. [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. [ABSTRACT FROM AUTHOR]

Consider orbits (z , κ) of the fractal iterator f κ (z) : = z 2 + κ , κ ∈ ℂ , that start at initial points z ∈ K ̂ κ (m) ⊂ ℂ ̂ , where ℂ ̂ is the set of all rational complex numbers (their real and imaginary parts are rational) and K ̂ κ (m) consists of all such z whose complexity does not exceed some complexity parameter value m (the complexity of z is defined as the number of bits that suffice to describe the real and imaginary parts of z in lowest form). The set K ̂ κ (m) is a bounded-complexity approximation of the filled Julia set K κ . We present a new perspective on fractals based on an analogy with Chaitin's algorithmic information theory, where a rational complex number z is the analog of a program p , an iterator f κ is analogous to a universal Turing machine U which executes program p , and an unbounded orbit (z , κ) is analogous to an execution of a program p on U that halts. We define a real number Υ κ which resembles Chaitin's Ω number, where, instead of being based on all programs p whose execution on U halts, it is based on all rational complex numbers z whose orbits under f κ are unbounded. Hence, similar to Chaitin's Ω number, Υ κ acts as a theoretical limit or a "fractal oracle number" that provides an arbitrarily accurate complexity-based approximation of the filled Julia set K κ . We present a procedure that, when given m and κ , it uses Υ κ to generate K ̂ κ (m) . Several numerical examples of sets that estimate K ̂ κ (m) are presented. [ABSTRACT FROM AUTHOR]

Let Γ be a graph with adjacency matrix A. The transition matrix of Γ corresponding to A is defined by H (t) := exp ⁡ (− ι t A) , where ι = − 1 and t ∈ R . The graph is said to exhibit pretty good state transfer between a pair of vertices u and v if there exists a sequence of real numbers { t k } and a complex number γ with unit norm such that lim k → ∞ H (t k) e u = γ e v. In this paper, we explore pretty good state transfer on Cayley graphs over semi-dihedral groups by using the representations of such groups. We show that graphs Cay (S D 8 n , S) have pretty good state transfer for some suitable subsets S if n is a power of 2. Moreover, we present a sufficient and necessary condition for a non-integral graph Cay (S D 8 n , S) to admit pretty good state transfer. Some concrete constructions of Cayley graphs over semi-dihedral groups having PGST are also presented. [ABSTRACT FROM AUTHOR]

Let R be an associative and commutative ring with unity 1 and consider k ∈ N such that 1 + 1 + ⋯ + 1 = k is invertible. Let U T ∞ (k) (R) be the group of upper triangular infinite matrices whose diagonal entries are kth roots of 1. We show that every element of the group U T ∞ (R) can be expressed as a product of 4k−6 commutators all depending on the powers of elements in U T ∞ (k) (R) of order k. If R is the complex number field or the real number field we prove that, in S L n (R) and in the subgroup S L V K (∞ , R) of the Vershik–Kerov group over R, each element in these groups can be decomposed into a product of at most 4k−6 commutators, all depending on two elements of order k. [ABSTRACT FROM AUTHOR]

The time-varying complex quadratic programming problem plays an important role in scientific research and engineering applications, and has received extensive attention. In this paper, by solving the problem in different number domains, two types of novel anti-noise integral enhanced recurrent neural networks (IERNNs) are proposed and analyzed. Specifically, the IERNN-A model solves the problems in the complex number domain, while the IERNN-B model solves the problems by converting the problems from the complex number domain into the real number domain. Both the IERNN-A and IERNN-B have the same design paradigm, i.e., an integral-type error function is first defined to obtain good anti-noise capability, and then substitute it into a differential-driven design formula to ensure convergence. Theory analysis and simulation results verify the global convergence and robustness of the proposed IERNNs. Furthermore, if a linear activation is used, the exponential convergence rate is obtained. Meanwhile, when time-varying bounded noise exists, the IERNNs are input-to-state stable. In addition, compared with other art-of-the-state methods, the proposed methods have faster convergence speed, stronger noise-suppression ability, and no overshoot, which are verified by the simulation and application of manipulator trajectory tracking. [ABSTRACT FROM AUTHOR]

We study the complexity of approximating the value of the independent set polynomial ZG(\lambda) of a graph G with maximum degree \Delta when the activity \lambda is a complex number. When \lambda is real, the complexity picture is well understood, and is captured by two real-valued thresholds \lambda \ast and \lambda c, which depend on \Delta and satisfy 0 < \lambda \ast < \lambda c. It is known that if \lambda is a real number in the interval (\lambda \ast, \lambda c) then there is a fully polynomial time approximation scheme (FPTAS) for approximating ZG(\lambda) on graphs G with maximum degree at most \Delta. On the other hand, if \lambda is a real number outside of the (closed) interval, then approximation is NP-hard. The key to establishing this picture was the interpretation of the thresholds \lambda \ast and \lambda c on the \Delta -regular tree. The ''occupation ratio"" of a \Delta -regular tree T is the contribution to ZT (\lambda) from independent sets containing the root of the tree, divided by ZT (\lambda) itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if \lambda \in [ \lambda \ast, \lambda c]. Unsurprisingly, the case where \lambda is complex is more challenging. It is known that there is an FPTAS when \lambda is a complex number with norm at most \lambda \ast and also when \lambda is in a small strip surrounding the real interval [0, \lambda c). However, neither of these results is believed to fully capture the truth about when approximation is possible. Peters and Regts identified the complex values of \lambda for which the occupation ratio of the \Delta -regular tree converges. These values carve a cardioid-shaped region \Lambda \Delta in the complex plane, whose boundary includes the critical points \lambda \ast and \lambda c. Motivated by the picture in the real case, they asked whether \Lambda \Delta marks the true approximability threshold for general complex values \lambda. Our main result shows that for every \lambda outside of \Lambda \Delta, the problem of approximating ZG(\lambda) on graphs G with maximum degree at most \Delta is indeed NP-hard. In fact, when \lambda is outside of \Lambda \Delta and is not a positive real number, we give the stronger result that approximating ZG(\lambda) is actually \#P-hard. Further, on the negative real axis, when \lambda < \lambda \ast, we show that it is \#P-hard to even decide whether ZG(\lambda) > 0, resolving in the affirmative a conjecture of Harvey, Srivastava, and Vondrak. Our proof techniques are based around tools from complex analysis--specifically the study of iterative multivariate rational maps. [ABSTRACT FROM AUTHOR]

Erdős proved that every real number is the sum of two Liouville numbers. A set W of complex numbers is said to have the Erdős property if every real number is the sum of two members of W. Mahler divided the set of all transcendental numbers into three disjoint classes S , T and U such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set U and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if $m\in [0,\infty)$ , then there exist $2^{\mathfrak {c}}$ dense subsets W of S each of Lebesgue measure m such that W has the Erdős property and no two of these W are homeomorphic. It is also proved that there are $2^{\mathfrak {c}}$ dense subsets W of S each of full Lebesgue measure, which have the Erdős property. Finally, it is proved that there are $2^{\mathfrak {c}}$ dense subsets W of S such that every complex number is the sum of two members of W and such that no two of these W are homeomorphic. [ABSTRACT FROM AUTHOR]

This paper considers the multidigit multiplication operation, on whose speed the speed of asymmetric cryptographic software and hardware depends. Algorithms for implementation of the multiplication of two -digit numbers based on discrete cosine and sine transforms (DCT and DST) are proposed. Due to the use of DCT and DST, the calculations for the real and imaginary parts of the discrete Fourier transform (DFT) of a real even-length signal are separated, which allows translating the complex number calculations to real number calculations. Algorithm operations are replaced to preserve symmetry in real or imaginary parts of multidigit numbers, which allows the use of DCT and DST of lower length N /2 1 and increases the possibility of parallelism in the implementation of multidigit multiplication. [ABSTRACT FROM AUTHOR]

The q-rational numbers and the q-irrational numbers were introduced by Morier-Genoud and Ovsienko. In this paper, we focus on q-real quadratic irrational numbers, especially q-metallic numbers and q-rational sequences which converge to q-metallic numbers, and consider the radiuses of convergence of them when we assume that q is a complex number. We construct two sequences given by recurrence formula as a generalization of the q-deformation of the Fibonacci and Pell numbers which are introduced by Morier-Genoud and Ovsienko. For these two sequences, we prove a conjecture of Leclere, Morier-Genoud, Ovsienko and Veselov concerning the expected lower bound of the radiuses of convergence. In addition, we obtain a relationship between the radius of convergence of these two sequences in two special cases. [ABSTRACT FROM AUTHOR]

In this paper, a new lemma is proposed to study the stability of a fractional order complex chaotic system without dividing the complex number into real and imaginary parts. The proving process of the new lemma combines the fundamental properties of the complex field and the fractional order extension of the Lyapunov direct method. It extends the fractional order extension of the Lyapunov direct method from the real number field to the complex number field. Based on the new lemma, we propose a new impulsive synchronization scheme for fractional order complex chaotic systems. The numerical simulation results also show the validity of our conclusion. [ABSTRACT FROM AUTHOR]

Let the base β be a complex number, |β|>1, and let A⊂C be a finite alphabet of digits. The A-spectrum of β is the set SA(β)={Σk=0nakβk|n∈N,ak∈A}. We show that the spectrum SA(β) has an accumulation point if and only if 0 has a particular (β,A)-representation, said to be rigid.The first application is restricted to the case that β>1 and the alphabet is A = {−M,..., M}, M≥1 integer. We show that the set Zβ,M of infinite (β,A)-representations of 0 is recognizable by a finite Büchi automaton if and only if the spectrum SA(β) has no accumulation point. Using a result of Akiyama-Komornik and Feng, this implies that Zβ,M is recognizable by a finite Büchi automaton for any positive integer M≥⌈β⌉-1 if and only if β is a Pisot number. This improves the previous bound M≥⌈β⌉.For the second application the base and the digits are complex. We consider the on-line algorithm for division of Trivedi and Ercegovac generalized to a complex numeration system. In on-line arithmetic the operands and results are processed in a digit serial manner, starting with the most significant digit. The divisor must be far from 0, which means that no prefix of the (β,A)-representation of the divisor can be small. The numeration system (β,A) is said to allow preprocessing if there exists a finite list of transformations on the divisor which achieve this task. We show that (β,A) allows preprocessing if and only if the spectrum SA(β) has no accumulation point. [ABSTRACT FROM AUTHOR]

The critical analysis of the starting point of the theory of complex numbers is proposed. The unity of formal logic and rational dialectics is methodological basis of the analysis. The analysis leads to the following main results: (1) the definition of a complex number contradicts to the laws of formal logic, because this definition is the union of two contradictory concepts: the concept of a real number and the concept of a non-real (imaginary) number - an image. The concepts of a real number and a non-real (imaginary) number are in logical relation of contradiction: the essential feature of one concept completely negates the essential feature of another concept. These concepts have no common feature (i.e. these concepts have nothing in common with each other), therefore one cannot compare these concepts with each other. Consequently, the concepts of a real number and a non-real (imaginary) number cannot be united and contained in the definition of a complex number. The concept of a complex number is a gross formal-logical error; (2) the real part of a complex number is the result of a measurement. But the non-real (imaginary) part of a complex number is not the result of a measurement. The non-real (imaginary) part is a meaningless symbol, because the mathematical (quantitative) operation of multiplication of a real number by a meaningless symbol is a meaningless operation. This means that the theory of complex number is not a correct method of calculation. Consequently, mathematical (quantitative) operations on meaningless symbols are a gross formal-logical error; (3) a complex number cannot be represented (interpreted) in the Cartesian geometric coordinate system, because the Cartesian coordinate system is a system of two identical scales (rulers). The standard geometric representation (interpretation) of a complex number leads to the logical contradictions if the scales (rulers) are not identical. This means that the scale of non-real (imaginary) numbers cannot exist in the Cartesian geometric coordinate system. Consequently, the theory of complex numbers and the use of the theory of complex numbers in mathematics and physics (electromagnetism and electrical engineering, fluid dynamics, quantum mechanics, relativity) represent a gross methodological error and lead to gross errors in mathematics and physics. [ABSTRACT FROM AUTHOR]

Background: A pandemic outbreak of severe acute respiratory syndrome coronavirus 2 (COVID 19) incidence data are largely available online. Until August 17, COVID 19 has hit more than 22 million individuals all over the globe. So, it is urged to get clear information about the prevalence of the virus. Therefore, one can manipulate easily a suitable mathematical model to fit these published data. Methods: We propose a mathematical model that considers the total population, in 25 countries, either infected by COVID 19 or confined (safe) during the period from November 17, 2019, to August 17, 2020. The model considers the total population as a complex number; the imaginary part is the number of infected individuals and the real part is the number of confined individuals. This classification combined with mathematical treatments leads to a transmission dynamics of the virus to be as wave-like motion. The virus can hit any country either by one wave or by successive waves (up to 11 waves). Findings: We find net discrimination between the 25 countries investigated in this report. The immediate response to the first attack is a substantial parameter to determine whether the epidemic attack will be in one wave or it can be in successive waves. For example, the best case was such as individuals in China hit by one wave while the individuals in the USA were attacked by nine waves; it is the worst case all over the globe. In addition, the model differentiates between the daily reproduction numbers (Rd0) and the median reproduction number (R0). We have found that Rd0 decreases exponentially with time from high values down to zero at the wave maximum point; and R0 varies from a country to another. For example, the virus hit individuals in Germany in R0 = 1.39 (96% CI 1.01–3.87) and in the USA R0 = 3.81 (91% CI 1.71–5.15). We have found that twice the virus has hit both the USA and Iran. The great protestation of black matter lives in the USA and the great assemblage of the new Iranian year, on March 21, 2020, have been the cause of the second epidemic attack in both countries. Interpretation: Our results show that COVID 19 transmission depends on the prompt reaction against the first viral-wave. The reaction depends on both the social behaviour of individuals and on the swift system-decision by the governmental decision-maker(s). The Chinese strictly follow the decision-maker and therefore the virus hit by only one wave; while in the USA, the system-decision was different and the American-responses were different, therefore ten waves followed the first wave. [ABSTRACT FROM AUTHOR]

Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences (log |ζ (ρ + z)|) and (arg ζ (ρ +z)). Here ρ = ½ + iγ runs over the nontrivial zeros of the zeta-function, 0 < γ ≤ T, T is a large real number, and z = u + iv is a nonzero complex number of modulus ≪ 1log T. Our approach proceeds via a study of the integral moments of these sequences. If we let z tend to 0 and further assume that all the zeros ρ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence (log (|ζ' (ρ)|log T)) has an approximate Gaussian distribution with mean 0 and variance ½ log log T. This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution. [ABSTRACT FROM AUTHOR]

We provide a complete classification of three-dimensional associative algebras over the real and complex number fields based on a completely elementary proof. We list up all the multiplication tables of the algebras up to isomorphism. We compare our results with those given by mathematicians in the 19th century and this century. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to determine all matrix representations of the real numbers. It is shown that every such representation is completely reducible, while all non-trivial irreducible representations must be of 2-dimensional and can be expressed in a unique form. It is found that those representations are essentially determined by the ways of embedding the real numbers into the complex numbers. This results in a one-to-one correspondence between the equivalent classes of irreducible representations and the equivalent classes of homomorphisms from the real number field to the complex number field. The matrix representations of the complex numbers are also determined. [ABSTRACT FROM AUTHOR]

Complex conductivity imaging (also called induced polarization imaging or spectral induced polarization imaging when conducted at multiple frequencies) involves estimating the frequency-dependent complex electrical conductivity distribution of the subsurface. The superior diagnostic capabilities provided by complex conductivity spectra have driven advancements in mechanistic understanding of complex conductivity as well as modelling and inversion approaches over the past several decades. In this work, we demonstrate the theory and application for an approach to 3-D modelling and inversion of complex conductivity data in the real number domain. Beginning from first principles, we demonstrate how the equations for the real and imaginary components of the complex potential may be decoupled. This leads to a description of the real and imaginary source current terms, and a corresponding assessment of error arising from an assumption necessary to complete the decoupled modelling.We show that formost earth materials,which exhibit relatively small phases (e.g. less than 0.2 radians) in complex conductivity, these errors become insignificant. For higher phase materials, the errors may be quantified and corrected through an iterative procedure. We demonstrate the accuracy of numerical forward solutions by direct comparison to corresponding analytic solutions. We demonstrate the inversion using both synthetic and field examples with data collected over a waste infiltration trench, at frequencies ranging from 0.5 to 7.5 Hz. In addition to insight provided by the decoupled equations,we see two primary advantages of the decoupled inversion in comparison to fully coupled inversion in complex number domain. First, memory requirements are reduced by a factor of two because it is unnecessary to simultaneously store both the real and complex components and associated Jacobian matrices. Second, and most importantly, data noise is specified separately for the real and imaginary components of potential (or magnitude and phase), which removes the ambiguity in data misfit residuals that exists in the complex number domain, thereby enabling the inversion to appropriately fit the data. Additionally, the modelling and inversion algorithms are equivalent to the direct current electrical resistivity tomography counterpart. This enables direct application of advancements and approaches developed originally for electrical resistivity tomography. [ABSTRACT FROM AUTHOR]

Complex fuzzy set, as an extension of classical fuzzy sets, could describe the fuzzy characters of things more detail and comprehensively and is very useful in dealing with vagueness and uncertainty of problems that include the periodic or recurring phenomena. Note that a complex fuzzy set is different from the fuzzy complex set introduced and discussed by many scholars, since the membership degree of a complex fuzzy set is a complex number with length less than or equal to 1 while a fuzzy complex set is a real number with membership degree less than or equal to 1, and the universe is the complex plane. As the mathematical theoretical basis of fuzzy mathematics, fuzzy set and its mapping, corresponding fuzzy complex set and its mapping have been investigated in depth because they integrate and cross the methods and results of classical real analysis and complex analysis. However, there is no comprehensive investigation on complex fuzzy set and its corresponding mathematical theory, even include decomposition theorems, extension principles and the basic operations of the complex fuzzy set. As is well known, the cut set of fuzzy sets is the bridge between fuzzy sets and classical sets, which plays a significant role in fuzzy sets and fuzzy systems. In this paper, the concept of (r, θ)-cut sets of complex fuzzy sets is proposed and their properties are discussed. Meanwhile, the decomposition theorems and the extension principles of complex fuzzy set based on (r, θ)-cut sets are deduced and corresponding properties are investigated. All these conclusions not only deeply enrich the fundamental theory of complex fuzzy set, but also provide a powerful tool to investigate complex fuzzy set. Finally, an example application of signal detection demonstrates the utility of the (r, θ)-cut sets of complex fuzzy sets in practice. [ABSTRACT FROM AUTHOR]

Dempster–Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. However, the existing evidence theory is insufficient to consider the situations where it has no capability to express the fluctuations of data at a given phase of time during their execution, and the uncertainty and imprecision which are inevitably involved in the data occur concurrently with changes to the phase or periodicity of the data. In this paper, therefore, a generalized Dempster–Shafer evidence theory is proposed. To be specific, a mass function in the generalized Dempster–Shafer evidence theory is modeled by a complex number, called as a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster's combination rule is exploited. In contrast to the classical Dempster's combination rule, the condition in terms of the conflict coefficient between the evidences is released in the generalized Dempster's combination rule. Hence, it is more general and applicable than the classical Dempster's combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster's combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences is less than 1. In a word, this generalized Dempster–Shafer evidence theory provides a promising way to model and handle more uncertain information. Thanks to this advantage, an algorithm for decision-making is devised based on the generalized Dempster–Shafer evidence theory. Finally, an application in a medical diagnosis illustrates the efficiency and practicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The split and hyperbolic (countercomplex) octonions are eight‐dimensional nonassociative algebras over the real numbers, which are in the form X=x0e0+∑j=17xjej, where em's 0≤m≤7 have different properties for them. The main purpose of this paper is to define the split‐type octonion and its matrix whose inputs are split‐type octonions and give some properties for them by using the real quaternions, split, and hyperbolic (countercomplex) octonions. On the other hand, to make some definitions, we present some operations on the split‐type octonions. Also, we show that every split‐type octonions can be represented by 2 × 2 real quaternion matrix and 4 × 4 complex number matrix. The information about the determinants of these matrix representations is also given. Besides, the main features of split‐type octonion matrix concept are given by using properties of  real quaternion matrices. Then, 8n × 8nreal matrix representations of split‐type octonion matrices are shown, and some algebraic structures are examined. Additionally, we introduce real quaternion adjoint matrices of split‐type octonion matrices. Moreover, necessary and sufficient conditions and definitions are given for split‐type octonion matrices to be special split‐type octonion matrices. We describe some special split‐type octonion matrices. Finally, oct‐determinant of split‐type octonion matrices is defined. Definitive and understandable examples of all definitions, theorems, and conclusions were given for a better understanding of all these concepts. [ABSTRACT FROM AUTHOR]

A numerical algorithm that can quantitatively solve transcendental equation sets with multiple independent variables in the complex number domain is presented in this paper. A strict mathematical proof of solving the unary transcendental equation in the real domain is given in detail to show the process of the algorithm, which can be generalized to the equations with two or more variables. The complex variable concerned in this method is treated as two independent variables in real domain that represent the real part and the imaginary part of the variable, respectively. The criterion for choosing the dimension of scanning elements is given to overcome the difficulty in solving the general multivariate equation. Finally, the algorithm is applied in the problem of plotting 3-D dispersion curves in complex wave number and real frequency domains. Three examples are used to illustrate the correctness and effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Let D be a weighted digraph with weights taken from the set of non-zero real numbers and let be its eigenvalues. The energy of D is defined as , where denotes the real part of the complex number . In this paper, we obtain lower bounds for the energy of weighted digraphs in terms of weights of directed cycles of length 2. We study normal weighted digraphs and give spectral and structural characterizations of normal weighted digraphs. Using these results, we determine unicyclic and bicyclic normal weighted digraphs. For normal weighted digraphs, we obtain an improved and sharp lower bound for the energy. [ABSTRACT FROM AUTHOR]

Let p,x be real numbers, and s be a complex number, with R(s) > 1 - r, p ≥ 1, and x+1>0. The zeta function Zαp (s; x) is defined by ..., where α = (α1, ..., αr) is a r-tuple positive integers, and Liα(z) is the one-variable multiple polylogarithms. Since Zα1 (s; 0) = ξ(α; s), we call this function as a generalized Arakawa-Kaneko zeta function. In this paper, we investigate the properties and values of Zαp (s; x) with different values s, x, and p. We then give some applications on them. [ABSTRACT FROM AUTHOR]

In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra. [ABSTRACT FROM PUBLISHER]

This paper presents the new algorithm of PP-PFC (Pole-placement Predictive Functional Control) for stable, linear under-damped higher-order processes. It is shown that while conventional PFC aims to get first-order exponential behavior, this is not always straightforward with significant under-damped modes and hence a pole-placement PFC algorithm is proposed which can be tuned more precisely to achieve the desired dynamics, but exploits complex number algebra and linear combinations in order to deliver guarantees of stability and performance. Nevertheless, practical implementation is easier by avoiding complex number algebra and hence a modified formulation of the PP-PFC algorithm is also presented which utilises just real numbers while retaining the key attributes of simple algebra, coding and tuning. The potential advantages are demonstrated with numerical examples and real-time control of a laboratory plant. [ABSTRACT FROM AUTHOR]

A complex Polytopic fuzzy set (CPoFS) extends a Polytopic fuzzy set (PoFS) by handling vagueness with degrees that range from real numbers to complex numbers within the unit disc. This extension allows for a more nuanced representation of uncertainty. In this research, we develop Complex Polytopic Fuzzy Sets (CPoFS) and establish basic operational laws of CPoFS. Leveraging these laws, we introduce new operators under a confidence level, including the confidence complex Polytopic fuzzy Einstein weighted geometric aggregation (CCPoFEWGA) operator, the confidence complex Polytopic fuzzy Einstein ordered weighted geometric aggregation (CCPoFEOWGA) operator, the confidence complex Polytopic fuzzy Einstein hybrid geometric aggregation (CCPoFEHGA) operator, the induced confidence complex Polytopic fuzzy Einstein ordered weighted geometric aggregation (I-CCPoFEOWGA) operator and the induced confidence complex Polytopic fuzzy Einstein hybrid geometric aggregation (I-CCPoFEHGA) operator, enhancing decision-making precision in uncertain environments. We also investigate key properties of these operators, including monotonicity, boundedness, and idempotency. With these operators, we create an algorithm designed to solve multiattribute decision-making problems in a Polytopic fuzzy environment. To demonstrate the effectiveness of our proposed method, we apply it to a numerical example and compare its flexibility with existing methods. This comparison will underscore the advantages and enhancements of our approach, showing its efficiency in managing complex decision-making scenarios. Through this, we aim to demonstrate how our method provides superior performance and adaptability across different situations. [ABSTRACT FROM AUTHOR]

Let a 0 , a 1 , ... , a n − 1 be real numbers and let A = C i r c (a 0 , a 1 , ... , a n − 1) be a circulant matrix with f (x) = Σ j = 0 n − 1 a j x j . First, we prove that C i r c (a 0 , a 1 , ... , a n − 1) must be invertible if the sequence a 0 , a 1 , ... , a n − 1 is a strictly monotonic sequence and a 0 + a 1 + ⋯ + a n − 1 ≠ 0. Next, we reduce the calculation of f (ε 0) f (ε) ... f (ε n − 1) for a prime n by using the techniques on finite fields, where ε is a primitive n -th root of unity. Finally, we provide two examples to explain how to use the obtained results to calculate the determinant of a circulant matrix. [ABSTRACT FROM AUTHOR]

In this article I argue that i is a quantity associated with the two-dimensional real number plane, whether as a vector, a bi-vector, a point or a transformation (rotation). This position provides a foundation for the complex numbers and accounts for complex numbers in some equations of applied mathematics and physics. I also argue that complex numbers are fundamentally geometrical and can be described by geometric algebra, and that moreover the meaning of complex numbers in physics varies with dimension and geometry of the manifold. [ABSTRACT FROM AUTHOR]

Suppose that N is a sufficiently large real number. In this paper it is proved that for 2 < c < 990/479, the Diophantine equation ... is solvable in primes p1, p2, p3, p4, p5 such that each of the h numbers pi + 2, i = 1, 2, 3, 4, 5 has at most [6227/3960-1916c] prime factors. [ABSTRACT FROM AUTHOR]