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Gender is not only a spectrum; it is a complex number. Each individual has their own unique multidimensional number of gender. Gender as a complex number has developed from the gender algorithm. This conceptualization of gender is seated in theorizing gender from a Bayesian brain foundation, specifically the Free Energy Principle. The phenomenon of phantoms in transgender, transsexual, and nonbinary (TGNB) people has only recently been a subject of inquiry. This article builds on ideas developed through clinical practice and harvested from multidisciplinary theory and research. It applies my theories of interoceptive gender and the gender algorithm to formulations of gender. It will demonstrate how these relate to the theory of trans phantoms, which has important clinical implications and applications for TGNB subjectivity and embodiment. [ABSTRACT FROM AUTHOR]

In this paper, we focus on generalized fuzzy complex numbers and propose a straightforward matrix method to solve the dual rectangular fuzzy complex matrix equations C · Z ˜ + L ˜ = R · Z ˜ + W ˜ , in which C and R are crisp complex matrices and Z ˜ , L ˜ and M ˜ are fuzzy complex number matrices. The existing methods for solving fuzzy complex matrix equations involve separately calculating the extended solution and the corresponding parameters of the real and imaginary parts, whereby we obtain the algebraic solution of the equations. By means of the interval arithmetic and embedding approach, the n × n dual rectangular fuzzy complex linear systems could be converted into 2n × 2n fuzzy linear systems, which are also equivalent to the 4n × 4n real linear systems. By directly solving the 4n × 4n real linear systems, the algebraic solutions can be obtained. The general dual rectangular fuzzy complex matrix equations and dual rectangular fuzzy complex linear systems are investigated by the generalized inverses of matrices. Finally, some examples are given to illustrate the effectiveness of method. [ABSTRACT FROM AUTHOR]

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]

Most any students can explain the meaning of a b , for a ∈ R and for b ∈ Z . And some students may be able to explain the meaning of (a + b i) c , for a , b ∈ R and for c ∈ Z . However, few may be able to explain a d i or a (c + d i) let alone (a + b i) (c + d i) for a , b , c , d ∈ R . Through both mathematical explanation and reader investigations via a dynamic graphing applet, we will investigate the meaning of (a + b i) (c + d i) . Additionally, we provide some course-appropriate investigations which can be used by instructors in various grades 11–16 courses. [ABSTRACT FROM AUTHOR]

This article introduces the space of A-linearly correlated fuzzy complex numbers. Using this space, we study the stationary Schrödinger equation with boundary conditions are given by fuzzy complex numbers. This equation plays an special role in Quantum Mechanics describing the state of the system. We apply the formalism to the step potential, generating quantum results consistent with traditional quantum results. [ABSTRACT FROM AUTHOR]

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

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

The fixed-time synchronization (FTS) of complex-valued memristive competitive neural networks (CVMCNNs) with mixed delays is the main topic of this paper. Firstly, two new fixed-time (FT) stability criteria are obtained. In the FT stability criterion, the exponent is usually positive, but the exponent can be less than zero and varies with the error state in this paper. Then, based on two new stability theorems, the FTS of CVMCNNs is studied in the sense of 1-norm and 2-norm, respectively, without dividing the complex-valued state variables into the real part (RP) and the imaginary part (IP). Finally, two numerical simulation examples are given to further demonstrate the validity and superiority of our results. [ABSTRACT FROM AUTHOR]

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

It plays an important role in the calculation of fuzzy numbers using some special fuzzy numbers to approximate general fuzzy numbers in fuzzy analysis, similar to fuzzy complex analysis. A method is proposed to seek an approximation operator that produces a special kind of fuzzy complex number which is the nearest one to a given fuzzy complex number in this paper. The concept of prismoid fuzzy complex number is introduced, and the nearest prismoid fuzzy complex number of fuzzy complex number which preserve the centroid of the core for the fuzzy complex number with respect to the weighted pseudometric are discussed. Finally, some properties of the approximation operator including scale invariance, identity and nearness are presented. [ABSTRACT FROM AUTHOR]

This paper aims to bring together quaternions and generalized complex numbers. Generalized quaternions with generalized complex number components are expressed and their algebraic structures are examined. Several matrix representations and computational results are introduced. An alternative approach for a generalized quaternion matrix with elliptic number entries has been developed as a crucial part. [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]

We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra gl n admits an interpolation to any complex number n. We do this using the Deligne’s category D t , which is a formal way to define the category of finite-dimensional representations of the group G L n , when n is not necessarily a natural number. We also obtain interpolations to any complex number n of the no-monodromy conditions on a space of differential operators of order n, which are considered to be a modern form of the Bethe ansatz equations. We prove that the relations in the algebra of higher Gaudin Hamiltonians for complex n are generated by our interpolations of the no-monodromy conditions. Our constructions allow us to define what it means for a pseudo-differential operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the Gaudin model associated with the Lie superalgebra gl n | n ′ , we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy. [ABSTRACT FROM AUTHOR]

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

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

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

Let H be the Radford Hopf algebra of dimension twelve over a field k . In this paper, we investigate the automorphism groups of the Green ring r(H) and the Green algebra ℂ(H) ≔ ℂ ⊗ℤr(H) over the complex number field ℂ. The ring automorphisms of r(H) and the algebra automorphisms of ℂ(H) are all described. It is shown that the ring automorphism group of r(H) is isomorphic to the Klein group K4, and the algebra automorphism group of ℂ(H) is isomorphic to the direct product ℂx × S4, where ℂx is the multiplicative group of all nonzero complex numbers and S4 is the symmetric group of degree 4. [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]

An elliptic curve E can be immersed in P N - 1 as a curve of degree N by means of the linear system of |NO|, where O is the origin of E. Well-known classical results going back to Bianchi and Klein say that if N is odd, this immersion is uniquely determined by specifying a full-level N structure. In this paper we show that if N is even, uniqueness of immersion is ensured by specifying a level structure associated with a certain congruence subgroup between Γ (N) and Γ (2 N) . Moreover, we construct, over the complex number field, an immersion by means of suitably chosen theta functions, and write down the quadratic equations satisfied by them. [ABSTRACT FROM AUTHOR]

Independent component analysis (ICA), as a statistical and computational approach, has been successfully applied to digital signal processing. Performance analysis for the ICA approach is perceived as a challenging task to work on. This contribution concerns the complex-valued FastICA algorithm in the range of ICA over the complex number domain. The focus is on the robust and equivariant behavior analysis of the complex-valued FastICA estimator. Although the complex-valued FastICA algorithm as well as its derivatives have been widely used methods for approaching the complex blind signal separation problem, rigorous mathematical treatments of the robust measurement and equivariance for the complex-valued FastICA estimator are still missing. This paper strictly analyzes the robustness against outliers and separation performance depending on the global system. We begin with defining the influence function (IF) of complex-valued FastICA functional and followed by deriving its closed-form expression. Then, we prove that the complex-valued FastICA algorithm based on the optimizing cost function is linear-equivariant, depending only on the source signals. [ABSTRACT FROM AUTHOR]

Let r≥2 $r\ge 2$ be a real number. A complex nowhere‐zero r $r$‐flow on a graph G $G$ is an orientation of G $G$ together with an assignment φ:E(G)→C $\varphi :E(G)\to {\mathbb{C}}$ such that, for all e∈E(G) $e\in E(G)$, the Euclidean norm of the complex number φ(e) $\varphi (e)$ lies in the interval [1,r−1] $[1,r-1]$ and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph G $G$, denoted by ϕC(G) ${\phi }_{{\mathbb{C}}}(G)$, is the minimum of the real numbers r $r$ such that G $G$ admits a complex nowhere‐zero r $r$‐flow. The exact computation of ϕC ${\phi }_{{\mathbb{C}}}$ seems to be a hard task even for very small and symmetric graphs. In particular, the exact value of ϕC ${\phi }_{{\mathbb{C}}}$ is known only for families of graphs where a lower bound can be trivially proved. Here, we use geometric and combinatorial arguments to give a nontrivial lower bound for ϕC(G) ${\phi }_{{\mathbb{C}}}(G)$ in terms of the odd‐girth of a cubic graph G $G$ (i.e., the length of a shortest odd cycle) and we show that this lower bound is tight. This result relies on the exact computation of the complex flow number of the wheel graph Wn ${W}_{n}$. In particular, we show that for every odd n $n$, the value of ϕC(Wn) ${\phi }_{{\mathbb{C}}}({W}_{n})$ arises from one of three suitable configurations of points in the complex plane according to the congruence of n $n$ modulo 6. [ABSTRACT FROM AUTHOR]

A new approach for implementing the fast Fourier transform (FFT) using complex binary number system (CBNS) is presented in this paper. The advantage of using CBNS is that instead of processing the real and imaginary parts of a complex number separately, we can process both as a single entity. A total of four FFT architectures have been developed, which are 16‐, 64‐, 256‐, and 1024‐point FFT architectures. The proposed FFT is designed using Verilog‐HDL and implemented using a Virtex‐7 field‐programmable gate array (FPGA). The maximum clock rate achieved by the proposed FPGA‐based design is 376.1 MHz. The FPGA resource utilization of the FFT is less than the state‐of‐the‐art FFT designs. An application‐specific integrated circuit (ASIC) for the FFT is also designed. The normalized energy per FFT of the proposed ASIC is 0.14 nJ, and the normalized area per FFT of the ASIC is 1.08 mm2 which is less than other existing designs. [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]

The units of a number field k act naturally on the real vector space k ⊗ Q R , and so on open subsets of (k ⊗ Q R) ⁎ that are stable under the units. A Shintani domain for this action consists of a finite number of polyhedral cones, all having generators in k , whose union is a fundamental domain. Aside from the trivial case of imaginary quadratic fields, no practical method for computing Shintani domains for totally complex number fields has been published. Here we give a quick way to compute Shintani domains for totally complex quartic number fields. To construct these domains we exploit the existence of a forward attractor and backward repeller set for this action. We prove that the number of polyhedral cones needed for the Shintani domain is bounded by an absolute constant, a fact previously known only for cubic or quadratic fields. We also show that our algorithm for finding a Shintani domain runs in polynomial time and we give a table describing the results of running the algorithm on more than 168,000 fields. [ABSTRACT FROM AUTHOR]

Assume that is a faithful normal semifinite trace on a von Neumann algebra , is the unit of , is the -algebra of -measurable operators, and is the Banach space of -integrable operators. We present a new proof of the following generalization of Putnam's theorem (1951): No positive self-commutator with is invertible in . If is infinite then no positive self-commutator with can be of the form , where is a nonzero complex number and is a -compact operator. Given with we seek for the conditions that . If and with then . If and then . If a partial isometry lies in and for some integer then is a commutator and implies that . [ABSTRACT FROM AUTHOR]

In practical applications, the temporal completeness of knowledge graphs is of great importance. However, previous studies have mostly focused on static knowledge graphs, generally neglecting the dynamic evolutionary properties of facts. Moreover, the unpredictable and limited availability of temporal knowledge graphs, together with the complex temporal dependency patterns, make current models inadequate for effectively describing facts that experience temporal transitions. To better represent the evolution of things over time, we provide a learning technique that uses quaternion rotation to describe temporal knowledge graphs. This technique describes the evolution of entities as a temporal rotation change in quaternion space. Compared to the Ermitian inner product in complex number space, the Hamiltonian product in quaternion space is better at showing how things might be connected. This leads to a learning process that is both more effective and more articulate. Experimental results demonstrate that our learning method significantly outperforms existing methods in capturing the dynamic evolution of temporal knowledge graphs, with improved accuracy and robustness across a range of benchmark datasets. [ABSTRACT FROM AUTHOR]

Mandelbrot set, which was provided as a highlight in fractal and chaos, is studied by many researchers. With the extension of Mandelbrot set to generalized M set with different kinds of exponent k (k − M set), properties are hard to understand when k is a complex number. In this paper, fractal property of generalized M set with complex exponent z is studied. First, a relation is constructed between generalized M set with complex and real exponent. Then, distribution of z − M set on complex plane is researched. Meanwhile, symmetry of generalized M set is proved. Finally, graphics, generated by escape time algorithm, are the validated results of this paper. [ABSTRACT FROM AUTHOR]

Herein, we present a dynamic analysis on a Whitworth quick return mechanism, which was proposed as a class project to engage Mechanical Engineering students in the Machine Dynamics course. The objectives of the project are to teach (1) kinematic analysis on planar linkage mechanism using complex numbers and (2) kinetic analyses using distributed and lumped mass models. The analytical models were implemented using an engineering math software – Mathcad, and the results were validated by a multibody dynamics simulation software – Inventor. Loop closure equations were first built and solved to find link positions, and then velocities and accelerations were confirmed with numerical differentiation in Mathcad. Based on the solved kinematic quantities, a kinetic analysis was carried out using distributed and lumped mass models as well as in Inventor. The reactions and torques at the input link of the distributed mass model were found identical to those derived from Inventor. It was found the maximum reactions at pin joints were 11% to 100% different between the distributed mass model and the simplified lumped mass model. However, the shaking forces found in both models were nearly identical, and the peak values of the shaking moments were comparable. It is confirmed the lumped mass model can serve as a quick method to find the magnitude of the shaking forces and moments without solving simultaneous equation systems. [ABSTRACT FROM AUTHOR]

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

Any complex-analytic vector bundle E admits naturally defined homotheties ϕ α , α ∈ C ⁎ , i.e. ϕ α is the multiplication of a local section by a complex number α. We investigate the question when such automorphisms can be lifted to a non-split supermanifold corresponding to E. Further, we compute the automorphism supergroup of a Π-symmetric super-Grassmannian Π Gr n , k , and, using Galois cohomology, we classify the real structures on Π Gr n , k and compute the corresponding supermanifolds of real points. [ABSTRACT FROM AUTHOR]

This work presents a novel approach to quantum computing by proposing a customizable hardware design of a dedicated processor that emulates the execution of quantum algorithms. Unlike software-based quantum computation simulators, which run on standard general-purpose computers and suffer from reduced performance, this hardware design, which is based on classical concepts of bits, registers and memories, aims to leverage pure parallelism and pipelined execution for efficient quantum computations via emulation. The architecture includes several key components: memories, computation unit, measurement unit and control unit. The quantum state memory stores the individual and group states of qubits. This memory is crucial for maintaining the quantum information required for quantum operations. Basic operators are stored in dedicated operator memory. Additionally, a scratch memory allows for larger operators to be dynamically built at runtime. The computation unit is responsible for performing complex number multiplications, which form the basis of tensor and matrix products necessary for executing quantum operations. A measurement unit enables quantum state sampling, which is an essential aspect of quantum computation. Furthermore, a control unit is incorporated to ensure the correct operation of the quantum processor's data path. It utilizes a microprogram to manage and coordinate the functional units. All the functional units communicate with each other through dedicated and shared data buses, depending on the frequency of information exchange. This enables efficient data transfer and coordination among the components. The proposed hardware design has been simulated and proved to be effective in executing quantum operations. By exploiting parallelism and employing a pipelined execution, this architecture overcomes the limitations of software-based simulators, delivering improved performance for emulating quantum algorithms. We use Grover's search algorithm as a benchmark to evaluate the performance of the proposed hardware design and compare it to software-based simulation and to hardware-based algorithm-dedicated emulation. [ABSTRACT FROM AUTHOR]

Recent progress has revealed a number of constraints that cosmological correlators and the closely related field-theoretic wavefunction must obey as a consequence of unitarity, locality, causality and the choice of initial state. When combined with symmetries, namely homogeneity, isotropy and scale invariance, these constraints enable one to compute large classes of simple observables, an approach known as (boostless) cosmological bootstrap. Here we show that it is possible to relax the restriction of scale invariance, if one retains a discrete scaling subgroup. We find an infinite class of solutions to the weaker bootstrap constraints and show that they reproduce and extend resonant non-Gaussianity, which arises in well-motivated models such as axion monodromy inflation. We find no evidence of the new non-Gaussian shapes in the Planck data. Intriguingly, our results can be re-interpreted as a deformation of the scale-invariant case to include a complex order of the total energy pole, or more evocatively interactions with a complex number of derivatives. We also discuss for the first time IR-divergent resonant contributions and highlight an inconsequential inconsistency in the previous literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a membership function used to form the fuzzy Mandelbric set and investigate the structural effects of additive and multiplicative dynamic noises on it. The newly defined membership function of this fuzzy set and its perturbations is a generalization of the indicator function for the classical Mandelbric set. We present an algorithm for detecting each complex number's fuzzy membership degree. Through the use of the membership degrees of each complex number and experimental mathematics based on the visualizations of a variety of versions by utilizing computer-aided design, we gain a deep foresight for the structure characteristics of the additive and multiplicative perturbed fuzzy Mandelbric sets. Our novel approach allows us to identify the symmetry states of the Mandelbric set and its perturbations by the membership degrees of complex numbers, unlike the existing methods described in the literature. [ABSTRACT FROM AUTHOR]

We derive computable formulas for the structured backward errors of a complex number λ when considered as an approximate eigenvalue of rational matrix functions that carry a symmetry structure. We consider symmetric, skew-symmetric, Hermitian, skew-Hermitian, ∗ -palindromic, T-even, T-odd, ∗ -even, and ∗ -odd structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different. [ABSTRACT FROM AUTHOR]

In this paper, we associate the TKK algebra G ˆ (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ , the category of restricted G ˆ (J) -modules of level ℓ is canonically isomorphic to the category of σ -twisted V C g ˆ (ℓ , 0) -modules, where V C g ˆ (ℓ , 0) is a vertex algebra arising from the toroidal Lie algebra of type C 2 and σ is an isomorphism of V C g ˆ (ℓ , 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ , the integrable restricted G ˆ (J) -modules of level ℓ are exactly the σ -twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify the irreducible 1 2 N -graded σ -twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

Complex-valued expression models have been widely used in the application of intelligent decision systems. However, there is a lack of entropy to measure the uncertain information of the complex-valued information. Therefore, how to reasonably measure the uncertain information of the complex-valued information is a gap to be filled. In this paper, inspired by the Rényi entropy, we propose the complex-valued Rényi entropy, which measures uncertain information of the complex-valued probability under the framework of complex number, and this is also the first time to measure uncertain information in the complex space. The complex-valued Rényi entropy contains the features of the classical Rényi entropy, i.e., the complex-valued Rényi entropy corresponds to different information functions with different parameters q. Moreover, complex-valued Rényi entropy has some properties, such as non-negativity, monotonicity and etc. Some numerical examples can demonstrate the flexibility and reasonableness of the complex-valued Rényi entropy. [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]

From the perspective of rock mechanics, transmission of fluid pressure that drives groundwater flow is not instantaneous; instead it is a slow compressional (P) wave in porous rocks saturated with a fluid. In this paper, the approach of monochromatic wave is used to analyze the slow P‐wave. Permeability is derived as a function of the frequency of the wave. The permeability appears to be a complex number which asymptotes to Darcy permeability at the low frequency end but tends to vanish at the high frequency end. With Biot theory, the permeability predicts phase velocity and the quality factor of the monochromatic wave. Using Berea sandstone as an illustrative example, the complex‐number permeability can yield a lower phase velocity (within frequency of 105–108 Hz) and a larger attenuation (for frequency above 106 Hz) than Darcy permeability does. This study extends the recent research of the low‐frequency waves of fluid pressure to the regime of very high frequency. [ABSTRACT FROM AUTHOR]

Matrix factorizations in dual number algebra, a hypercomplex number system, have been applied to kinematics, spatial mechanisms, and other fields recently. We develop an approach to identify spatiotemporal patterns in the brain such as traveling waves using the singular value decomposition (SVD) of dual matrices in this paper. Theoretically, we propose the compact dual singular value decomposition (CDSVD) of dual complex matrices with explicit expressions as well as a necessary and sufficient condition for its existence. Furthermore, based on the CDSVD, we report on the optimal solution to the best rank-k approximation under a newly defined quasi-metric in the dual complex number system. The CDSVD is also related to the dual Moore--Penrose generalized inverse. Numerically, comparisons with other available algorithms are conducted, which indicate less computational costs of our proposed CDSVD. In addition, the infinitesimal part of the CDSVD can identify the true rank of the original matrix from the noise-added matrix, but the classical SVD cannot. Next, we employ experiments on simulated time-series data and a road monitoring video to demonstrate the beneficial effect of the infinitesimal parts of dual matrices in spatiotemporal pattern identification. Finally, we apply this approach to the large-scale brain functional magnetic resonance imaging data, identify three kinds of traveling waves, and further validate the consistency between our analytical results and the current knowledge of cerebral cortex function. [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]

Representing one of the last populations of an IUCN Critically Endangered subspecies, estimates suggest the Kordofan giraffe of Cameroon's Bénoué Complex number fewer than 300 individuals, threatened by poaching and habitat degradation. This study investigated the viability of Bénoué National Park's Kordofan giraffe and efficacy of available interventions through a population viability analysis using VORTEX. The relative impacts of anti‐poaching activity, population supplementation, habitat protection and vaccination were explored, alongside a sensitivity analysis investigating parameter uncertainty. The baseline model projects a 78.6% probability of extinction (PE) within 100 years. Poaching of just two individuals every 5 years results in a 98.1% PE, highlighting anti‐poaching activity as a vital intervention. Movement of a female‐biased group into the park proved effective, with supplementation of 24 females reducing the PE by over 35%, sustaining a raised population size above the baseline for over 60 years on average. Annual carrying capacity reduction of 2% year−1 for 20 years resulted in a 97.8% PE. Conversely, habitat restoration had little consequence unless combined with further interventions. These findings underscore the magnitude of the threat facing Bénoué National Park's Kordofan giraffe. Conservation management should prioritise strengthening existing anti‐poaching activity in conjunction with protecting wildlife corridors to aid dispersal. [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]

Electrical impedance tomography (EIT) is a promising method to image the frequency-dependent complex electrical conductivity distribution of the subsurface in the mHz to kHz frequency range. In contrast to the well-developed electrical resistivity tomography (ERT) method, the inversion approach for EIT data is less established. Different inversion strategies have been proposed, but the implications of the differences between these methods have not been investigated yet. In this study, we aim to compare four different inversion strategies for EIT measurements. The first strategy (CVI) formulates the inverse problem in the complex number domain and is mathematically the most elegant method. The second strategy (RVI) is the established real-valued inversion method, which decouples the inversion of the real and imaginary parts and completely ignores the complex nature. The third strategy (ALT) is very similar to the RVI strategy in case of small phase angles, but it considers the complex coupling in the forward operator and alternately updates the real and imaginary parts of the model in the case of large phase angles. The fourth and final strategy (CVI+) was newly formulated in this study. It fully considers the complex nature of EIT measurements but separates the treatment of the real and imaginary part in terms of the data weighting and regularization. The different inversion strategies were tested with two synthetic models. The first model has a small phase contrast and the second model has a large phase contrast. In the case of a small phase contrast, the CVI strategy was able to resolve the distribution of electrical conductivity amplitude, but the inversion result for the phase angle was less reliable. The other three strategies presented similar results and the models were well resolved within the expected data misfit. In the case of a model with large phase contrast, only the newly formulated CVI + strategy was able to produce reliable results. It was found that the extremely large phase angle can have a significant influence on the modelled amplitude of data. The cross-sensitivity (i.e. the imaginary part of the sensitivity) that describes the influence on the real part of data due to a change in the imaginary part of model, or that on the imaginary part of data due to a change in the real part of model, provided unique information during the inversion. It was concluded that the CVI + strategy is theoretically the most comprehensive and correct approach for EIT inversion, but that in the case of small phase angles the RVI strategy has the practical advantage that no complex calculations are required, which substantially reduces the required computational effort. [ABSTRACT FROM AUTHOR]

In this paper, we study meromorphic solutions of the following nonlinear complex difference equation f n (z) + P (z) Δ η l f (z) = H 0 (z) + H 1 (z) e p 1 (z) + ⋯ + H m (z) e p m (z) , where n, l, m, q are positive integers with n ≥ 2 , η is a nonzero complex number satisfying Δ η l f (z) ≢ 0 , p 1 , ... , p m are polynomials of degree q, whose leading coefficients α 1 , ... , α m are distinct nonzero complex numbers, and P , H 0 , H 1 , ... , H m are meromorphic functions of order less than q such that P H 1 ... H m ≢ 0. In addition, we analyze meromorphic solutions of the above equation for n = 1 . Our proofs depend on Cartan's theorem and the variation of Nevanlinna's theorem regarding a set of meromorphic functions. Several examples are provided to demonstrate our results. These results generalize some very recent known results. [ABSTRACT FROM AUTHOR]

The dynamical stability of strange-dwarf hybrid stars and strange planets, constituted by strange-quark-matter cores and dilute-nuclear-matter crusts, is revisited by analyzing the fundamental mode eigenfrequencies of the radial oscillation equations with boundary conditions for slow (rapid) conversions originating at the density-discontinuous interface characterizing extremely large (small) microscopic timescales compared to the oscillation periods. For the hadronic crust we used an analytic fit of the BPS results matched to the massless MIT bag model. For the rapid case, our calculations indicate that the zero mode is the so-called reaction mode whose frequency is a complex number, thus ruling out the existence of strange dwarfs (planets) in nature. On the other hand, slow conversions still provide a sizeable stability window which, interestingly, also reproduces the Glendenning-Kettner-Weber results. The robustness of our findings is demonstrated for different transition densities and using an equation of state from perturbative QCD for the ultra-dense core. [ABSTRACT FROM AUTHOR]

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

We introduce the class of analytic and univalent functions of the form f(z) = z + ∑ az ∞ j j=2 for z ∈ △:= {z: |z| < 1} that satisfies 1 + 1 μ {(Sλ tf(z)) ′ − 1} ≺ (1+Lz) (1+Mz), z∈△, such that L and M are fixed numbers where satisfy −1 ≤ M < L ≤ 1, T ≥ 0, λ > −1 and μ is arbitrary complex number. We denote this subclass by Qμ (τ, λ, L, M). In this paper, we determine coefficients estimates, distortion theorem and maximization theorem for the new subclass Qμ (τ, λ, L, M). [ABSTRACT FROM AUTHOR]

We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type I. In our framework, each theta relation is controlled by combinatorial properties of a pair (T , P) of a regular matrix T over an imaginary quadratic field and a positive-definite Hermitian matrix P over the complex number field. [ABSTRACT FROM AUTHOR]