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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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 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]

The photonic spin Hall effect (PSHE) holds great potential for applications in the fields of precision measurement and sensing. Due to its splitting shift usually at the sub-wavelength level, the current observation and application of PSHE usually rely on weak measurement amplification technology. However, the current optimized weak measurement models are only for the out-of-plane splitting shift of PSHE, and are only applicable to certain specific situations. This greatly limits the application of PSHE in many fields such as precision measurement and sensing, and to some extent hinders the experimental research on PSHE. Based on this issue, by changing the polarization state from 0° or 90° to arbitrary linear polarization incidence, and extending the reflection coefficient from real number to complex number, a set of optimized weak measurement model with wider applicability for both in-plane and out-of-plane splitting shifts of PSHE is established in this article. It is not only applicable to the case where the pre-selection and post-selection states are arbitrary linearly polarized states, but also to the case where the reflection interface is both absorbent medium and non-absorbent medium. In addition, the correctness of this model is demonstrated through comparative verification. This work not only help to conduct more comprehensive and in-depth research on the PSHE and will also further promotes the application and development of PSHE in the fields of precision measurement and high-sensitivity sensing. • A more versatile weak measurement model for PSHE is established. • It is applicable to both in-plane and out-of-plane spin splitting. • It is suitable for pre-selected states at arbitrary polarization angles. • It is applicable to both non-absorbent and absorbent media interfaces. [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]

• A method to simultaneously predict both intensity and phase information from a speckle image employing a deep despeckle complex neural network (DespeckleNet). • A new complex network structure that realizes the mapping from single-channel real number field to the dual-channel complex number field. • We achieved simultaneous prediction of grayscale intensity and quantitative phase that can generalize to the unseen objects through unseen diffuser patterns. • We achieved high contrast and real-time observation of unlabeled living Hela cells. • Our solution breaks the limitation of the memory effect to achieve biological tissue imaging through scattering media. The existence of a scattering medium causes the degeneration of intensity and phase information, especially in biological imaging. The present techniques to address this challenge only focus on the reconstruction of intensity information, yet few attempts have tried to recover the phase information. We propose a method to simultaneously predict both intensity and phase information from a speckle image employing a deep despeckle complex neural network (DespeckleNet). By combining the advantages of both the complex network and the generative adversarial network framework, our method enables the high contrast single-shot imaging of complicated biological samples through scattering media without labeling. Various experiments demonstrate the superior reconstruction and generalization performance of our method under multiple types of biological samples with different scattering media. We also provide the real-time observation of living cellular activities without any contaminations or damages to the cells. Our method offers simple yet effective imaging through scattering media and paves the way for real-time unlabeled biological imaging. [ABSTRACT FROM AUTHOR]

Magneto-optical effects are caused by parameters of the off-diagonal components in the dielectric tensor of materials, which is a complex number. Because the real parts of off-diagonal components rotate the photo-induced polarization in the materials from the direction of the incident light electric field, the parts must become one of the effective parameters for light energy flow control in photonic crystals. In this study, we theoretically proposed a mechanism to realize light intensity imbalance to the left and right by inverting the sign of the real parts for 2D magnetic photonic crystals based on a finite-difference time-domain simulation and photonic band calculations. Our analyzes reveal that the maximal intensity peaks of light magnetic field distributions rotate ∼ 6 0 ∘ around the hole of the crystal at the Γ point in the Brillouin zone, and these rotations are caused by the real parts. The imbalance is caused by the deflections of light energy flow owing to the rotation, which are explained based on the following two mechanisms: the walk-off phenomenon and uneven bifurcation. Furthermore, the proposed mechanism was experimentally confirmed by observing the imbalance in the scattered light intensity from the surface of rough-cutting magnetic photonic crystals fabricated from a cerium-substituted yttrium iron garnet. The results of this study serve as a foundation for the development of light energy flow control by engineering photonic crystals using the real part of the off-diagonal components in dielectric tensors. • Light energy flow deflects in magnetic photonic crystals. • The real parts of off-diagonal components of dielectric tensors rotate polarizations. • The rotation angle is enhanced in photonic crystals. • The deflection is observed in wide wavelength region. • A new principle for light energy flow control using the real parts. [ABSTRACT FROM AUTHOR]

• Combining algorithms of Abrarov / Quine and Kuntz using method by Schreier. • Use Abrarov / Quine and Kuntz algorithms in place of those by Weideman and Humlíček. • Voigt profile algorithm for lower processing power hardware targeting HITRAN. In this note it is suggested that the complex number rational approximations of Weideman and Humlíček may be replaced with the real number approximations of Abrarov / Quine and Kuntz respectively in the technique suggested by F. Schreier (JQSRT 112(6) (2011) 1010–1025) to provide an accurate Voigt function calculation algorithm with reduced calculation time targeting HITRAN database calculations. The resulting algorithm may be useful in applications using electronics with lower processing power. [ABSTRACT FROM AUTHOR]