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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]

Let ζ = (ζn) be an arbitrary complex sequence such that 0 < |ζ1| ≤ |ζ2| ≤... and ζn → ∞ as n → ∞, let nζ(r) and Nζ(r) be the counting function and the integrated counting function of this sequence, respectively. By Eζ we denote the class of all entire functions whose zeros are precisely the ζn, where a complex number that occurs m times in the sequence ζ corresponds to a zero of multiplicity m. Suppose that Φ is a convex function on R such that Φ(σ)/σ → +∞ as σ → +∞. It is proved that there exists an entire function f ∈ Eζ such that... where Mf (r) denotes the maximum modulus of the function f, and it is shown that the above inequality implies the inequality... The formulated result is a consequence of the following more general statement: if the righthand derivative Φ′+ of the function Φ assumes only integer values and... then there exists an entire function f ∈ Eζ such that lnMf(r) = o(eΦ(ln r)) as r → +∞. [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]

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]

Copyright of Journal of Cryptologic Research (2097-4116) is the property of Editorial Board of Journal of Cryptologic Research and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A complex intuitionistic fuzzy (CIF) set contains the membership and non-membership in the shape of a complex number whose amplitude term and phase term are covered in the unit interval. Moreover, Hamacher interaction aggregation operators are the combination of two major operators, called Hamacher aggregation operators and interaction aggregation operators, and they are used to aggregate the collection of information into one value. In this manuscript, we present the concept of Hamacher interaction operational laws for CIF sets (CIFSs). Further, we develop the CIF Hamacher interaction weighted averaging (CIFHIWA) operator, CIF Hamacher interaction ordered weighted averaging (CIFHIOWA) operator, CIF Hamacher interaction weighted geometric (CIFHIWG) operator, and CIF Hamacher interaction ordered weighted geometric (CIFHIOWG) operator. For these operators, we also discuss some basic properties, such as idempotency, monotonicity, and boundedness. Additionally, we develop a MADM method based on the developed operators and apply it to solve the green supply chain management problems, which can implement environmentally friendly practices to minimize carbon emissions, resource consumption, and waste generation while promoting long-term sustainability. Finally, we verify the superiority and effectiveness of the proposed method based on a comparative analysis between the proposed techniques and existing methods. [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]

Copyright of Evaluation & Analysis of Drug-Use in Hospitals of China is the property of Evaluation & Analysis of Drug-Use in Hospitals of China Editorial Board and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Chinese Mass Spectrometry Society is the property of Journal of Chinese Mass Spectrometry Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

Let L(s, χ) be a Dirichlet L-function associated with a primitive character χ mod q and a be a non zero complex number. We denote by Δ(s, χ) the function which appears in the functional equation L(s, χ) = Δ(s, χ)L(1 - s,χ) and δa,χ = βa,χ + iγa,χ the solutions of the equation Δ(s, χ) = a which are called a-points ofΔ(s,χ). In this note, we will prove that for every complex number a = 0 the mean of the values L(δα,χ,χ) on the sequence of a-points δa,χ of the function Δ(s, χ) exists and equals a + 1. [ABSTRACT FROM AUTHOR]

Numeracy and digital skills are among the most crucial skills that any graduate should possess before entering the workforce, regardless of their field of study. Therefore, both skills should not be treated as an “add on” skill but viewed as essential graduate attributes needing to be purposefully incorporated into the curriculum instead of mere chance. By understanding the learners’ numeracy and digital skills abilities, educators can create suitable learning outcomes, activities, and assessments, enabling them to acquire the skills. The objective of this study is to create a framework for testing numeracy (NSI) and digital skills (DSI) using a cross-sectional design and quantitative methodology. The research entails generating a questionnaire, verifying it via a rigorous procedure of expert content validation, and focusing on sufficiency, clarity, coherence, and relevance. A pilot study involving 218 students from various disciplines was conducted to measure the reliability of the framework using Cronbach Alpha. Feedback from experts was then used to enhance the questionnaire, resulting in the finalisation of five components for the Digital Skills Instrument (DSI): information literacy, computer and technology literacy, digital communication and collaboration skills, digital identity and well-being, and digital ethics. Meanwhile, the five components of the Numeracy Skills Instrument (NSI) encompassed operation and calculation, graphical representation, quantitative reasoning and logical thinking, complex number (advanced concept), spatial visualisation and geometric reasoning. The results indicated a strong internal consistency across all components for both tools, with alpha values ranging from 0.847 to 0.958 for DSI and 0.916 to 0.964 for NSI. Corrected item-total correlations also depicted intercorrelation between items for both instruments. To sum up, the findings demonstrate that the DSI and NSI have high reliability and validity. Because both tools are reliable and valid for measuring digital and numeracy skills in the Malaysian context, they can be confidently used for future research. [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]

A degree one element of the Orlik-Solomon algebra of a hyperplane arrangement defines a cochain complex known as the Aomoto complex. The Aomoto complex can be considered as the "linear approximation" of the twisted cochain complex with coefficients in a complex rank one local system. In this paper, we discuss q-deformations of the Aomoto complex. The q-deformation is defined by replacing the entries of representation matrices of the coboundary maps with their q-analogues. While the resulting maps do not generally define cochain complexes, for certain special basis derived from real structures, the q-deformation becomes again a cochain complex. Moreover, it exhibits universality in the sense that any specialization of q to a complex number yields the cochain complex computing the corresponding local system cohomology group. [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 the history of written Estonian, three patterns were used for number words ranging from 21 to 99. As a result, complex numerals varied considerably, e.g., the word 21 exhibited unit-before-ten order (üks peale kakskümmend 'one upon twenty'), tenbefore-unit order (kakskümmend peale üks 'twenty upon one' - kakskümmend ja üks 'twenty and one' - kakskümmend üks 'twenty one'), and overcounting (üks kolmatkümmend 'one from the third decade' - üks kolmat 'one from the third'). This variation has been studied through analysis of the earliest extant Estonian texts from the 16th to 19th centuries, as evidenced by the corpus of old written Estonian, the historical concordance of the Estonian Bible translation, and the database of digitized Estonian newspapers. The first two preserved instances of Estonian complex number words, dating back to 1589 and 1601, followed the unit-before-ten order. This pattern was used throughout the 17th century and discarded at the end of the century, possibly due to puristic concerns, as several other German traits were also abandoned during this period. Only sporadic instances of this pattern were found in the 18th century. Overcounting was prevalent in the Bible translations of certain 17th century authors (e.g., J. Gutslaff, A. Virginius, and J. Hornung), while others, including translators who mentioned it in their own grammars (H. Stahl and H. Göseken), did not use the pattern at all. In texts published during the first half of the 18th century, overcounting prevailed. Thereafter, overcounting ceased to be the dominant pattern in books and gradually disappeared. The final instances of overcounted 21-99 appeared in newspapers in the 1820s. In oral usage, however, this pattern persisted well into the 20th century, particularly for counting yarn, fish, years of life, etc. The ten-before-unit order was used in almost all writings from the 1630s until the end of the 19th century. By the mid-19th century, it had become the exclusive pattern in written Estonian. The decline of overcounting and the prevalence of the ten-before-unit order coincided with a period during which the literacy of Estonians increased immensely. Therefore, it is reasonable to assume that the proliferation of literacy influenced the usage of complex numerals. While the logic of overcounting stemmed from oral traditions and finger-counting, the ten-before-unit order was more straightforward in written texts. This shift in number words was facilitated by the transition from Roman to Arabic numerals. [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 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]

In order to further improve the self-learning and noise reduction ability of speech enhancement model, a speech enhancement method based on multi-domain fusion and neural architecture search was proposed. The multi-spatial domain mapping and fusion mechanism of speech signals were designed to realize the mining of real complex number correlation. Based on the characteristics of convolution pooling of the model, a complex neural architecture search mechanism was proposed, and the speech enhancement model was constructed efficiently and automatically through the designed search space, search strategy and evaluation strategy. In the comparison and generalization experiment between the optimal speech enhancement model and the baseline model, the two indexes of PESQ and STOI increase by 5.6% compared with the optimal baseline model, and the number of model parameters is the lowest. [ABSTRACT FROM AUTHOR]

Let L(s, χ) be a Dirichlet L-function associated with a primitive character χ mod q and a be a non zero complex number. We denote by Δ(s, χ) the function which appears in the functional equation L(s, χ) = Δ(s, χ)L(1 − s, χ) and δa, χ = βa, χ + iγa, χ the solutions of the equation Δ(s, χ) = a which are called a-points of Δ(s, χ). In this note, we will prove that for every complex number a ̸= 0 the mean of the values L(δa, χ, χ) on the sequence of a-points δa, χ of the function Δ(s, χ) exists and equals a + 1. [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]

Copyright of Journal of Beijing University of Technology is the property of Journal of Beijing University of Technology, Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

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]

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]

Copyright of Journal of Molecular Science is the property of Journal of Molecular Science Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the degree is allowed to be a complex number instead of a non-negative integer. These functions are referred to as Jacobi functions. In a similar fashion as associated Legendre functions, these break into two categories, functions which are analytically continued from the real line segment (− 1 , 1) and those analytically continued from the real ray (1 , ∞). Using properties of Gauss hypergeometric functions, we derive multi-derivative and multi-integral representations for the Jacobi functions of the first and second kind. [ABSTRACT FROM AUTHOR]

The introduction of degree apprenticeships in England in 2015 has not resulted in a significant increase in mature students accessing higher education provision overall. While there are likely to be a complex number of reasons contributing to this, the extent to which employers understand and value degree apprenticeship provision may be an influencing factor, the terminology may be another. As employers are pivotal in the decision-making process for degree apprenticeship recruitment, their comprehension of the impact that degree apprenticeship graduates have upon business outcomes is essential in developing this aspect of higher education. Success measures within education have usually focused on quantifiable outcomes, such as retention and achievement. Within higher education, undergraduate success is often measured by retention on programme and degree classification. These measures have merit in traditional undergraduate provision, however, the introduction of degree apprenticeships and the influence that employers have on these programmes suggests that new success metrics may be required. This would help to make the measurement of the success of this provision more meaningful to the key stakeholders, and to support employer engagement in recruiting employees on to degree apprenticeships. Having delivered degree apprenticeships since 2016, University Centre Quayside (UCQ) has undertaken a single-institution case study investigation with initial findings suggesting that the use of work-related measures, based on the impact that vocational learning has within the workplace, is a more effective and accurate reflection of graduate effectiveness and could generate more meaning for employers. The other finding suggests that, rather than the degree apprenticeship raising the perception of all apprenticeships, the term 'apprenticeship' may be preventing more degree apprenticeship recruitment. [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]

We begin a study of Schur analysis when the variable is now a matrix rather than a complex number. We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, matrices of split quaternions, and other algebras of hypercomplex numbers. [ABSTRACT FROM AUTHOR]

Hyperdimensional (HD) computing (also referred to as vector symbolic architectures, VSAs) offers a method for encoding symbols into vectors, allowing for those symbols to be combined in different ways to form other vectors in the same vector space. The vectors and operators form a compositional algebra, such that composite vectors can be decomposed back to their constituent vectors. Many useful algorithms have implementations in HD computing, such as classification, spatial navigation, language modeling, and logic. In this letter, we propose a spiking implementation of Fourier holographic reduced representation (FHRR), one of the most versatile VSAs. The phase of each complex number of an FHRR vector is encoded as a spike time within a cycle. Neuron models derived from these spiking phasors can perform the requisite vector operations to implement an FHRR. We demonstrate the power and versatility of our spiking networks in a number of foundational problem domains, including symbol binding and unbinding, spatial representation, function representation, function integration, and memory (i.e., signal delay). [ABSTRACT FROM AUTHOR]

A method is presented to demonstrate the application of computer simulations in the kinematic analysis of planar mechanisms, emphasizing its use in teaching the latter topic in a corresponding undergraduate course. Concepts of rigid-body dynamics are utilized in the kinematics of machines to analyze the motions (and forces in dynamics) transmitted within multiple interconnected links that make a mechanism, such as a car engine, airplane landing gear, press machine, door closer, and so on. Due to the tediousness of the analytical solutions, most textbooks limit the derivation of the equations to four-bar linkages like crank-rocker and crank-slider mechanisms. Benefiting from the advancements in computer programs, such as MATLAB, and their efficiency in solving large systems of linear and nonlinear equations, a method is proposed to facilitate teaching kinematic analysis of multi-bar linkages to undergraduate students while fostering the application of computational engineering via real-life examples. The results obtained from this method are shown to be in excellent agreement with the algebraic solution of the relative motion equations for each element in the mechanism. [ABSTRACT FROM AUTHOR]

Let ℓ be a prime number different from the residue characteristic of a non-archimedean local field F. We give formulations of ℓ -adic local Langlands correspondences for connected reductive algebraic groups over F, which we conjecture to be independent of a choice of an isomorphism between the ℓ -adic coefficient field and the complex number field. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new complexity computation based on the complex function. This measure exploits the dispersion Lempel–Ziv complexity (DLZC) and the dispersion statistical complexity measure based on Jensen–Shannon divergence (DCJS) of the analytic signal and constructs the binary complex function, called complex number complexity (CNC). The statistical measure depicts the complex system from different angle and mines more potential information. Through simulation experiments, we prove that the proposed method is able to detect data fluctuations more sensitively and accurately. For an application, the CNC is applied to fault detection which has different fault diameters and rotational speeds. The results deliver that the CNC measure can well represent the complexity of faulty bearings and has significant differences in any two fault types. The technique is effective to characterize different fault. [ABSTRACT FROM AUTHOR]

In this investigation, using Opoola differential operator (Dm(µ, β, t)f(z)), a new integral operator: I m,σ t,β,µ(f1, ..., fn)(z) : An → A is defined in the unit disk, U = {z ∈ C : |z| < 1}; and we investigated the Univalence conditions of this generalized operator. Finally, a number of corollaries and remarks which show the extension of our results are presented. [ABSTRACT FROM AUTHOR]

In this study, we propose the CVMFCC-DR (Complex-Valued Mel-Frequency Cepstral Coefficients Dimensionality Reduction) algorithm as an efficient method for reducing the dimensionality of speech signals. By utilizing the complex-valued MFCC technique, which considers both real and imaginary components, our algorithm enables dimensionality reduction without information loss while decreasing computational costs. The efficacy of the proposed algorithm is validated through experiments which demonstrate its effectiveness in building a speech recognition model using a complex-valued neural network. Additionally, a complex-valued softmax interpretation method for complex numbers is introduced. The experimental results indicate that the approach yields enhanced performance compared to traditional MFCC-based techniques, thereby highlighting its potential in the field of speech recognition. [ABSTRACT FROM AUTHOR]

The dominant mechanism of P-wave attenuation in fluid-saturated rock is well known to be the pore-scale squirt. However, the S-wave does not change the average fluid pressure in a pore and the pore volume stays unchanged. As such, the S-wave has a mechanism of attenuation distinctively different from the P-wave. This paper attempts to resolve the mechanisms of S-wave attenuation in fluid-saturated grainstones. The first mechanism is the viscous boundary layer in a fracture, the second is fluid acceleration plus a real-number permeability and the third mechanism is fluid acceleration plus a complex-number permeability (which arises from the viscous boundary layer). D'Euville limestone and Berea sandstone are used as illustrative examples. Ultrasonic velocity and the quality factor due to pore fluid (Qs) are compared between the three models and measurement. The results indicate that fluid acceleration plus the viscous boundary layer is the mechanism of S-wave attenuation in the water-saturated grainstones. In addition, the third model reveals a novel convex in the curve of Qs in the frequency range of 107–108 Hz. [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]

The location identification of dynamic load is an important part of load-identification technology. Traditional methods are mostly aimed at the identification of dynamic load's amplitude and phase. A new method for dynamic load location identification is proposed in this paper. An amplitude ratio or a phase difference between structural dynamic response signals is only related to the frequency response function (FRF), which is a complex-valued function that implies the location information of the load to be identified. In this method, the amplitude and phase variables of the excitation can be eliminated with symbolic calculations, and the location variables can be extracted from the FRF in a complex-number field. An excitation location can be identified quickly with parameter optimization using a Genetic Algorithm (GA), avoiding the ill-posed problem caused by matrix inversion. Numerical simulations and experiments demonstrate that this method can realize the fast recognition of several excitation positions, and has a high recognition accuracy, a short calculation time and a strong anti-noise ability. [ABSTRACT FROM AUTHOR]

Let be a polynomial of degree having all its zeros in , , then for every real or complex number with , Govil and McTume [7] showed that the following inequality holds In this paper, we have obtained a generalization of this inequality involving sequence of operators known as polar derivatives. In addition, the problem for the limiting case is also considered. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to analyze the heat transfer mechanism of a new type of regenerator with a low temperature difference and low current resistance under oscillatory flow at room temperature. Taking the single tube of the regenerator as the research object, the exact analytical solution of the radial heat transfer characteristics of the regenerator is obtained by studying its analytical model. The factors affecting the heat transfer characteristics are analyzed, and then the regenerator is optimized to improve the performance and efficiency of the regenerator system. In this study, we systematically analyzed the radial heat transfer characteristics of a regenerator under isochoric process conditions. A closed-system physical model of the incompressible isochoric process under oscillating flow was established. Then, the radial analytical solutions of pressure fluctuation, fluid velocity, fluid-solid temperature, and heat were derived in the complex number field. Furthermore, the fluid velocity, fluid-solid coupling wall temperature, heat, and equivalent heat transfer coefficient were assessed. Furthermore, the influences of frequency, inner diameter R1 of the regenerator, and different working medium and materials on the above parameters were discussed. It was found that the analysis and evaluation of fluid velocity, fluid-structure coupling wall temperature, heat, and equivalent heat transfer coefficient are helpful in understanding the dynamic characteristics of radial heat transfer in a regenerator system. Through the study of radial heat transfer under oscillating flow, it was found that the working medium, frequency, inner diameter of the regenerator, and material quality of the regenerator are helpful for the design optimization of the regenerator. Furthermore, our investigations established that the variation law of wall fluid-solid coupling temperature amplitude could be divided into three parts: the unidirectional flow part; the low-frequency part, where the temperature amplitude falls rapidly with increasing frequency; and the high-frequency part, where the temperature amplitude increases with the frequency. In addition, the variation of radial heat transfer of the fluid-solid coupling surface is similar to the changes in the temperature amplitude. We also discovered that the equivalent heat transfer coefficient of the fluid-solid surface is related to thermal conductivity of the material. Specifically, larger thermal conductivity values result in greater equivalent heat transfer coefficients. Based on the research into the radial heat transfer characteristics, the new regenerator has great application potential in the Stirling air conditioning system at room temperature. [ABSTRACT FROM AUTHOR]

Accurate attitude determination is crucial for satellites and spacecraft. Among attitude determination devices, star sensors are the most accurate. Solving the lost-in-space problem is the most critical function of the star sensor. Our research introduces a novel star-identification system that utilizes a polygon-recognition algorithm to assign a unique complex number to polygons created by stars. This system aims to solve the lost-in-space problem. Our system includes a full solution with a lens, image sensor, processing unit, and algorithm implementation. To test the system's performance, we analyzed 100 night sky images that resembled what a real star sensor in orbit would experience. We used a k-d tree algorithm to accelerate the search in the star catalog of complex numbers. We implemented various verification methods, including internal polygon verification and a voting mechanism, to ensure the system's reliability. We obtained the star database used as a reference from the Gaia DR2 catalog, which we filtered, to eliminate irrelevant stars, and which we arranged by apparent magnitude. Despite manually introducing up to three false stars, the system successfully identified at least one star in 97% of the analyzed images. [ABSTRACT FROM AUTHOR]

Under uneven lighting conditions, power inspection (PI) images captured by drones exhibit low contrast, lost details and severe noise, which affects the efficiency of monitoring. In response to these issues, this paper proposes a new method for enhancing PI images based on discrete fractional wavelet transform (DFRWT). Firstly, according to the image entropy feature, an appropriate fractional order of DFRWT is selected to decompose the PI image. Secondly, we introduce a low-frequency fusion enhancement method based on complex number domain. It employs different mapping functions to enhance the brightness and contrast of PI image, then the results are combined to reconstruct a new low-frequency component. Here, an improved quantum-behaved particle swarm optimization (QPSO) algorithm is proposed to determine the optimal parameters of the functions, preventing over-enhancement. Finally, an adaptive threshold function is applied to handle noise and edge coefficients in high-frequency components. Using 5 quality metrics to evaluate 8 existing algorithms on 9 datasets, the results indicate that the proposed algorithm can effectively enhance the clarity of PI images, performing excellently in noise suppression and edge detail enhancement. [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]

The subject of this paper is the study of some properties of -Stirling numbers of the second kind for a complex or a -adic complex number. In the -adic setting, as we known, the Laplace transform plays an important role in the study of some arithmetic sequences. We remind the definition of the Laplace transform of a -adic measure and its link with the moment of this measure. With the aid of a specific measure we establish some identities and congruences for the -Stirling numbers when is a non zero -adic complex number and for the generalized -Stirling numbers of the second kind attached to a -adic function that is invariant by . Also, we express the generalized -Stirling numbers according to generalized Stirling numbers . [ABSTRACT FROM AUTHOR]

Based on N different coherent states with equal weights and phase-space rotation symmetry, we introduce N-headed incoherent superposition states (NHICSSs) and N-headed coherent superposition states (NHCSSs). These N coherent states are associated with N-order roots of the same complex number. We study and compare properties of NHICSSs and NHCSSs, including average photon number, Mandel Q parameter, quadrature squeezing, Fock matrix elements and Wigner function. The main characteristic of this study is to use the same complex number to express the related states that appear in a large number of literatures. [ABSTRACT FROM AUTHOR]