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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]

In this paper, it is shown that the solution vector obtained by the classic Crout decomposition method is not an algebraic solution of a fuzzy complex linear system. Here, we propose a limited version of the mentioned method to obtain an algebraic solution of a fuzzy complex linear system (if it exists). Two numerical examples are presented to show ability and reliability of our method. [ABSTRACT FROM AUTHOR]

In Euclidean space, there exist four theorems which show that Sn sphere is not parallelizable for n 6 = 1, 3, 7. While three of them are shown by using Bott theorem, the last one is shown by using Hurwitz-Radon numbers. In this paper, a theorem and the proof of this theorem about parallelization of spheres in semi-Euclidean space is given. It is presented that some spheres are parallelizable with respect to specific number systems. [ABSTRACT FROM AUTHOR]

The goal of this development is introducing a reader the solution of one class comprising trigonometric equations in the Teaching Course of Mathematics by using trigonometric form of the complex numbers. An exemplary approach to solving these equations, suitable for students from 11th to 12th grade, as well as for pupils participating in mathematical camps, olympiads, mathematical competitions, computer mathematics olympiads is considered. [ABSTRACT FROM AUTHOR]

The paper presents results of the work of an international group of students in the frames of the realization of the research net project “Mathematical mosaic”. The results are obtained during investigations applying software products GeoGebra, Geometer's Sketchpad and Maple. The method of the complex numbers is used for the proofs of the assertions. [ABSTRACT FROM AUTHOR]

q-Difference equations appear in various contexts in mathematics and physics. The "basis" q is sometimes a parameter, sometimes a fixed complex number. In both cases, one classically associates to any series solution of such equations its q-Gevrey order expressing the growth rate of its coefficients : a (nonarchimedean) q^{-1}-adic q-Gevrey order when q is a parameter, an archimedean q-Gevrey order when q is a fixed complex number. The objective of this paper is to relate these two q-Gevrey orders, which may seem unrelated at first glance as they express growth rates with respect to two very different norms. More precisely, let f(q,z) \in \mathbb {C}(q)[[z]] be a series solution of a linear q-difference equation, where q is a parameter, and assume that f(q,z) can be specialized at some q=q_{0} \in \mathbb {C}^{\times } of complex norm >1. On the one hand, the series f(q,z) has a certain q^{-1}-adic q-Gevrey order s_{q}. On the other hand, the series f(q_{0},z) has a certain archimedean q_{0}-Gevrey order s_{q_{0}}. We prove that s_{q_{0}} \leq s_{q} "for most q_{0}". In particular, this shows that if f(q,z) has a nonzero (nonarchimedean) q^{-1}-adic radius of convergence, then f(q_{0},z) has a nonzero archimedean radius converges "for most q_{0}". [ABSTRACT FROM AUTHOR]

AbstractThis paper is devoted to develop the concepts of semi-c-periodicity, c-uniform recurrence and almost automorphy for functions defined on vertical strips in the complex plane, where c is a non-zero complex number. As an extension of the study performed for functions defined on the real axis, this work aims to investigate the main properties of these classes of functions and establish their connections with the more known class of c-almost periodic functions defined on vertical strips. In fact, we resolve an open problem which was raised in 2020 for the real case. Additionally, we also use this approach to introduce an asymptotic version of these new classes of functions. [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]

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]

The method of linear theory for small perturbations is used to study the stability of heavy inclusions in the upper crust. The existence of such inclusions of increased density is associated with chemical heterogeneity or phase transitions. At small deformations considered in the linear stability theory, the material undergoes transient creep which is described by the Andrade rheological model. With this rheology, the convective instability of thickened segments of the lithosphere is oscillatory: the increment is a complex number with a real part close to zero (threshold instability). Heavy inclusions in the upper crust introduce a small complex correction to the increment that determines the convective instability of the lithosphere, slightly increasing the real and imaginary parts of the increment, enhancing the instability and increasing the oscillation frequency. [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]

Temporal completion of knowledge graphs is of significant importance in real-world applications. However, previous research has mostly focused on stationary knowledge graphs, largely ignoring the dynamic evolutionary properties of facts. Furthermore, the volatility and scarcity of temporal knowledge graphs, along with the intricate temporal dependency characteristics, render existing models incapable of accurately representing facts that undergo temporal changes. To more accurately represent changes in entities over time, we provide a learning algorithm for representing temporal knowledge graphs using quaternion rotation. This approach characterises the evolution of entities as a temporal rotation transformation in quaternion space. The Hamiltonian product in quaternion space is more effective at capturing potential interdependencies than the Ermitian inner product in complex number space. This results in a learning process that is both more efficient and more expressive. The model shows exceptional performance, as evidenced by experimental results on benchmark datasets. [ABSTRACT FROM AUTHOR]

Biquaternion algebra is an algebraic structure originating from a complex number and has mainly been used in quantum mechanics, special and general relativity, classical, relativistic, and covariant electrodynamics, and signal processing. In this paper, the problem of the diagonalization of a biquaternion matrix is studied, by means of a complex representation of a biquaternion matrix, and an algebraic algorithm for the diagonalization of a biquaternion matrix is presented. In addition, numerical examples demonstrate the effectiveness of the algebraic algorithm. [ABSTRACT FROM AUTHOR]

This work applies the polar coordinates system of advanced calculus in the summation of the Gaussian distribution. In trying to achieve this aim, sub-concepts such as complex variables, gamma function of half, error function, and the relation between the error function and the standard normal distribution were defined and explained at various stages of the work. The embedded theorem which seems to be a new theorem also came up in the body of the work. [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]

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]

In this paper, we consider the spectral problem for the semi-integer Bessel equation with a boundary condition containing the square of the spectral parameter and a complex physical parameter. The system of eigenfunctions of the problem and the characteristic equation for the eigenvalues are derived. The equation for multiple roots of the characteristic equation is derived. The results on the basis properties (Riesz basis) of the system of eigenfunctions at different values of the parameter are obtained. For each case a biorthogonally conjugate system is constructed. At the end of the paper there is an example for the order of Bessel functions equal to . [ABSTRACT FROM AUTHOR]

Estimation of aerosol radiative forcing continues to suffer from large uncertainties, partially from a lack of observations of aerosol optical properties. Limited measurements of the atmospheric aerosol imaginary refractive index (iRI) have been made, especially in some of the world's most polluted regions. In this study, we measured aerosol optical and micro‐physical properties at a regional site, Rohtak, India, representative of polluted cities in the Indo‐Gangetic plains in northern India. The average PM2.5 measured during the campaign was 163 μg/m3 with a single‐scatter albedo of 0.7, indicating the presence of strongly absorbing aerosol components. Measurements of aerosol absorption, scattering, and particle number size distributions were used to estimate the effective refractive index using an established Mie inversion technique. The calculated iRI was spectrally invariant in the visible region with values ranging between 0.076 and 0.145. Brown carbon absorption, estimated using an existing Mie optimization method, ranged 34–88 Mm−1, with strongly absorbing mass absorption cross‐sections (∼1.9 m2/g). Higher iRI were observed during periods with higher brown carbon absorption, which are likely directly emitted from combustion sources. Low volatility organic carbon fractions dominated during these periods, with likely persistence of atmospheric absorption. The iRI values are at the upper end of previously reported ranges of urban aerosol iRI. In a sensitivity analysis to measured parameters, the absorption had the dominant effect on estimated iRI. Measured single scatter albedos, were lower than those from climate model simulations over the region, demonstrating the need for intrinsic property measurements to evaluate and constrain climate models. Plain Language Summary: Particles in the atmosphere, called aerosol, can absorb or reflect/scatter sunlight to heat or cool the atmosphere, depending upon their physical and chemical properties. Climate models try to simulate these properties to understand their effect on the climate. The strength of the absorption is determined by the size of the particle (generally the diameter) and its refractive index (a complex number) which is a material property. In this study, we measure the refractive index of atmospheric aerosol at Rohtak, which lies in the highly polluted Indo‐Gangetic plains in India. We find that the aerosol are very absorbing in nature, that is, they are warming the atmosphere, and that the absorbing/imaginary part of the refractive index is in the higher range of previously reported values. The aerosol at the site are dominated by emissions from fossil fuel and biomass burning sources. On comparing these absorbing properties with those simulated by climate models, we find that the climate models are not able to simulate the absorbing nature of these aerosol well. Hence, there is a need for more such measurements of the aerosol refractive index to improve climate models and ultimately improve our understanding of the effect of aerosol on our climate. Key Points: Estimate the aerosol effective refractive index (RI); imaginary RI ranges 0.076–0.145, higher than previously reported valuesA single scatter albedo of 0.7 reveals strongly absorbing components, with brown carbon absorption of 34–88 Mm−1Imaginary RI correlates well with near‐UV absorption by brown carbon, which has low volatility and is likely emitted from combustion sources [ABSTRACT FROM AUTHOR]

We present a generalization of the pseudoinverse operation to pairs of matrices, as opposed to single matrices alone. We note the fact that the Singular Value Decomposition can be used to compute the ordinary Moore-Penrose pseudoinverse. We present an analogue of the Singular Value Decomposition for pairs of matrices, which we show is inadequate for our purposes. We then present a more sophisticated analogue of the SVD which includes features of the Jordan Normal Form, which we show is adequate for our purposes. This analogue of the SVD, which we call the Jordan SVD, was already presented in a previous paper by us called 'Matrix decompositions over the double numbers'. We adopt the idea presented in that same paper that a pair of matrices is actually a single matrix over the double-complex number system. [ABSTRACT FROM AUTHOR]

In this paper, we propose a methodology for computing the square root of a complex number based on a piecewise linear (PWL) approximation method. The proposed method relies on a software-based segmentor that automatically divides the three real square root functions used in complex square root computation into the fewest segments with a predefined fractional bit width and maximum absolute error (MAE). The coefficients, including the start point, end point, slope and y-intercept of each segment, are stored for use in the implementation of the hardware design. The proposed fully pipelined circuit is coded in the Verilog hardware description language (HDL). The results of synthesis in TSMC (Taiwan Semiconductor Manufacturing Company) 65-nm CMOS technology show that our design achieves savings of 64.21% in area, 16.67% in delay and 65.08% in power compared to the existing methods. Moreover, implementation results on an FPGA (Field-Programmable Gate Array) platform (XC7Z020-CLG400) show that the proposed design reduces the number of LUTs by 29.38%, delay by 28.57% and power consumption by 53.47%. [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]

Generalized complex number systems generalize all unital real algebras of dimension two, including complex numbers, dual numbers, and double numbers. In this article, we have shown that the real part and the imaginary part of a holomorphic function of a generalized complex variable satisfy several forms of mean value theorem individually. In particular, as a corollary, these results hold true for complex holomorphic functions. [ABSTRACT FROM AUTHOR]

We aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values α, β and p. Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given. [ABSTRACT FROM AUTHOR]

We introduce $q$ -stability conditions $(\sigma,s)$ on Calabi–Yau- $\mathbb {X}$ categories $\mathcal {D}_\mathbb {X}$ , where $\sigma$ is a stability condition on $\mathcal {D}_\mathbb {X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname {QStab}_s\mathcal {D}_\mathbb {X}$ is a complex manifold of dimension $n$ for fixed $s$ , where $n$ is the rank of the Grothendieck group of $\mathcal {D}_\mathbb {X}$ over $\mathbb {Z}[q^{\pm 1}]$. When $s=N$ is an integer, we show that $q$ -stability conditions can be identified with the stability conditions on $\mathcal {D}_N$ , provided the orbit category $\mathcal {D}_N=\mathcal {D}_\mathbb {X}/[\mathbb {X}-N]$ is well defined. To attack the questions on existence and deformation along the $s$ direction, we introduce the inducing method. Sufficient and necessary conditions are given, for a stability condition on an $\mathbb {X}$ -baric heart (that is, a usual triangulated category) of $\mathcal {D}_\mathbb {X}$ to induce $q$ -stability conditions on $\mathcal {D}_\mathbb {X}$. As a consequence, we show that the space $\operatorname {QStab}^\oplus \mathcal {D}_\mathbb {X}$ of (induced) open $q$ -stability conditions is a complex manifold of dimension $n+1$. Our motivating examples for $\mathcal {D}_\mathbb {X}$ are coming from (Keller's) Calabi–Yau- $\mathbb {X}$ completions of dg algebras. In the case of smooth projective varieties, the $\mathbb {C}^*$ -equivariant coherent sheaves on canonical bundles provide the Calabi–Yau- $\mathbb {X}$ categories. Another application is that we show perfect derived categories can be realized as cluster- $\mathbb {X}$ categories for acyclic quivers. [ABSTRACT FROM AUTHOR]

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Inspired by a problem found in a textbook of Faddeev and Sominsky we propose a new form of roots of an arbitrary complex number. Our idea uses the zeros of some polynomials connected with the modified Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the weighted composition operators W f , φ on the Hardy space H 2 that are (T u , λ) -Toeplitz, where u is a nonconstant inner function and λ is a complex number with | λ | ≤ 1 ; that is, T u ∗ W f , φ T u = λ W f , φ . In particular, we prove that for an analytic self-map φ of D , not identically zero on D , and f ∈ H ∞ \ { 0 } , if λ ∈ D ¯ \ { 0 } such that 1 λ (u ∘ φ) is a self-map of D , then W f , φ is (T u , λ) -Toeplitz precisely when C φ u = λ u . We also give several applications of this result. [ABSTRACT FROM AUTHOR]

Quaternion is a four-dimensional and an extension of the complex number system. It is often viewed from various fields, such as analysis, algebra, and geometry. Several applications of quaternions are related to an object's rotation and motion in three-dimensional space in the form of a differential equation. In this paper, we do a systematic literature review on the development of quaternion differential equations. We utilize PRISMA (preferred reporting items for systematic review and meta-analyses) framework in the review process as well as content analysis. The expected result is a state-of-the-art and the gap of concepts or problems that still need to develop or answer. It was concluded that there are still some opportunities to develop a quaternion differential equation using a quaternion function domain. [ABSTRACT FROM AUTHOR]

In this paper, we develop the notion of c-almost periodicity for functions defined on vertical strips in the complex plane. As a generalization of Bohr's concept of almost periodicity, we study the main properties of this class of functions which was recently introduced for the case of one real variable. In fact, we extend some important results of this theory which were already demonstrated for some particular cases. In particular, given a non-null complex number c, we prove that the family of vertical translates of a prefixed c-almost periodic function defined in a vertical strip U is relatively compact on any vertical substrip of U, which leads to proving that every c-almost periodic function is also almost periodic and, in fact, c m -almost periodic for each integer number m. [ABSTRACT FROM AUTHOR]

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

To overcome some limitations of existing models that organize knowledge on the world wide web for online learning, this paper proposes a novel method of using graphical process models to organize knowledge on the web and to guide the learning process. Process models can not only visually represent the logical relationships (i.e., if ... then, all, at least one, and exactly one) of knowledge elements, but also display learning paths that consist of logically connected knowledge elements. This paper also presents a design of an innovative learning system that integrates process models with some existing knowledge organization models to facilitate online learning. This paper makes a useful contribution by proposing process-based knowledge organization as well as providing a visual representation of learning paths that explicitly prompt people to follow a logical process for effective learning. [ABSTRACT FROM AUTHOR]

This paper covers the modeling and nonlinear control of the mechanical systems where instead of angle coordinates, unit complex numbers are used as control states. This approach is useful not only to get rid of trigonometric functions, but mainly because it is a specific case of the 3D configuration that utilizes unit hypercomplex numbers (quaternions) as system states, and therefore facilitates its understanding, which represents a contribution to control education. As an example of application, it is considered a particular case of a cube balanced on one of its edges, which is equivalent to a reaction wheel inverted pendulum. The derived nonlinear control law is equivalent to a linear one and is characterized by only three straightforward tuning parameters. Experimental results are presented to validate modeling and control. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number a is defined by Da p(z) = np(z)+(a -z)p'(z). If p(z) has all its zeros in |z| ≥ k, k ≤ 1, then for |a| ≥ k, Aziz and Rather [Math. Inequal. Appl., 1, (1998), 231-238] proved. In this paper, we first improve as well as generalize the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta Math. Hungar., 104, (2004), 115-126] and also prove an inequality for a subclass of polynomials having all its zeros in |z| ≥ k, k ≤ 1. [ABSTRACT FROM AUTHOR]

In this work, we present the historical part of the complex numbers, showing its emergence and its evolution over time. We classify the triangles in the Argand-Gauss plane, considering its vertices and we discut some results involving the calculus of area to convex polygons, in particular, a formula to determine the area of polygons whose vertices correspond to the roots of a complex number. [ABSTRACT FROM AUTHOR]

In this work, we present the historical part of the complex numbers, showing its emergence and its evolution over time. We classify the triangles in the Argand-Gauss plane, considering its vertices and we discut some results involving the calculus of area to convex polygons, in particular, a formula to determine the area of polygons whose vertices correspond to the roots of a complex number. [ABSTRACT FROM AUTHOR]

This paper introduces one unique price motion process with fractional Brownian motion. We introduce the imaginary number into the agent's subjective probability for the reason of convergence; further, the result similar to Ito Lemma is proved. As an application, this result is applied to Merton's dynamic asset pricing framework. We find that the four order moment of fractional Brownian motion is entered into the agent's decision-making. The decomposition of variance of economic indexes supports the possibility of the complex number in price movement. [ABSTRACT FROM AUTHOR]

In this work, we introduce a nonstandard algebra of q-fermions where q is a nonzero complex deformation parameter for the algebra of the commuting fermions. In order to show that q-fermions provides a proper generalization of the algebra of usual commuting fermions, we prove that there is an inhomogeneous quantum structure associated with q-fermions for a complex number q with |q| = 1. [ABSTRACT FROM AUTHOR]

Let |$f$| be a finite Blaschke product with |$f(0)=0$|⁠ , which is not a rotation and let |$f^{n}$| be its |$n$| -th iterate. Given a sequence |$\{a_{n}\}$| of complex numbers consider |$F= \sum a_n f^{n}$|⁠. If |$\{a_n\}$| tends to |$0$| but |$\sum |a_n| = \infty $|⁠ , we prove that for any complex number |$w$| there exists a point |$\xi $| in the unit circle such that |$\sum a_{n}f^{n}(\xi)$| converges and its sum is |$w$|⁠. If |$\sum |a_n| < \infty $| and the convergence is slow enough in a certain precise sense, then the image of the unit circle by |$F$| has a non-empty interior. The proofs are based on inductive constructions which use the beautiful interplay between the dynamics of |$f$| as a self-mapping of the unit circle and those as a self-mapping of the unit disk. [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]

The development of algorithmic differentiation (AD) tools focuses mostly on handling floating point types in the target language. Taping optimizations in these tools mostly focus on specific operations like matrix vector products. Aggregated types like std::complex are usually handled by specifying the AD type as a template argument. This approach provides exact results, but prevents the use of expression templates. If AD tools are extended and specialized such that aggregated types can be added to the expression framework, then this will result in reduced memory utilization and improve the timing for applications where aggregated types such as complex number or matrix vector operations are used. Such an integration requires a reformulation of the stored data per expression and a rework of the tape evaluation process. We will demonstrate the overheads on a synthetic benchmark and show the improvement when aggregated types are handled properly by the expression framework of the AD tool. [ABSTRACT FROM AUTHOR]

This paper demonstrates how a classical detector that collects non-interacting individual classical massive free particles can generate a quantum interference pattern. The proposed classical picture requires that particles carry the information of a phase equal to an action integral along their trajectory. At the point of their detection, a special type of detector collects the phases from all individual particles reaching it, adds them up over time as complex numbers, and divides them by the square root of their number. The detector announces a number of detections equal to the square of the amplitude of the resulting complex number. An interference pattern is gradually built from the collection of particle phases in the detection bins of the detector after several repetitions of the experiment. We obtain perfect agreement with three solutions of the Schrödinger equation for free particles: a Gaussian wavepacket, two Gaussian wavepackets approaching each other, and a Gaussian wavepacket reflecting off a wall. The main conclusion of the present work is that the interference of quantum mechanics is basically due to the detectors that collect the particles when there are macroscopic detectors operating as proposed. Finally, a simple physical experiment with a single-photon detector is proposed that will be able to test our theory. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space and B s (H) the Jordan algebra of all bounded self-adjoint linear operators on H. Assume that k is any positive integer with k < dim H and ξ is any complex number. In this paper, we first give some useful properties about k-dimensional numerical range of ξ -Lie product [ A , B ] ξ = A B - ξ B A for A , B ∈ B s (H) ; and then, based on these properties, all surjective maps preserving k-dimensional numerical ranges of ξ -Lie product on B s (H) are completely characterized. [ABSTRACT FROM AUTHOR]

Moreover, for some "mysterious" reasons, the new sequence HT ht turns out to be an integer sequence! Indeed, it follows directly from (4) that (5) can be rewritten without division as HT ht . If HT ht is a square root of -1, then I A i is a complex number, called a Gaussian integer. But if HT ht satisfies the condition HT ht then I A i is called a dual number. [Extracted from the article]