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In this study, a classification algorithm based on complex number feature is proposed. Specifically, the SVM framework is reformulated, so each example would be classified in the unitary space. The cost function is redefined by considering the maximum margin of real and imaginary units of the complex number feature at the same time. The cost function is based on the expectation of the hinge loss, and its derivatives can be calculated in closed forms. Using a stochastic gradient descent (SGD) algorithm, this method allows for efficient implementation. For complex number feature, the example uncertainty is modeled by a sample preprocessing method based on within-class Euclidean distance Gaussian distribution sample (DGS). In addition, a complex number feature selection method based on improved hybrid discrimination analysis (HDA) is proposed by considering the correlation between real and imaginary units of complex number feature. The proposed classification algorithm is tested on synthetic data and three publicly available and popular datasets, namely, MNIST, WDBC, and Voc2012. Experimental results verify the effectiveness of the proposed method. The codes are available: https://github.com/luckysomebody/paper-code.

Human being live in three-dimensional space; they can accurately visualize processes taking place in one, two and three dimensions. Although the set of bi-complex numbers and quaternion have attracted attention of many researchers in physics and related branches, they do not really represent processes taking place in the space where human being are located. We suggested a new set of complex number called “the Trinition”. The new set is comprised between complex number with one imaginary part and complex number with three imaginary parts called quaternion/bi-complex numbers. We established a bijection between the new set and the three-dimensional space. We presented some important properties of the new set. We showed that all chaotic attractors in three dimension are simply three-dimensional mapping in the new set. Fewer examples of mapping in such set were presented. A new methodology that can be used to obtain more strange attractors are equally suggested. The methodology combines fractional chaotic models and some fractal mapping within the new set. Some illustrative figures are presented.

The objective of this paperwork is to implement a watermarking algorithm using complex number theory. Here, each pixel of the original image is considered as the real part of each complex number, and the imaginary part corresponds to the pixel of the watermark. The matrix consists of the absolute value of the complex number treated as the watermarked image. The real and imaginary components of each complex number cannot be retrieved without the phase angle, which is treated as the key to a secured watermarked image. The phase angle of each complex number transmitted through a secured channel. At the receiving end, when both the magnitude and phase angle of each pixel estimated properly, then the real and imaginary components of each complex number corresponding to the pixel of original and watermarked images are determined with some accuracy.

Most any students can explain the meaning of ab, for a∈R and for b∈Z. And some students may be able to explain the meaning of (a+bi)c, for a,b∈R and for c∈Z. However, few may be able to explain adi...

As the existing complex number Nth root computation methods are relatively discrete, we propose a general method and architecture based on coordinate rotation digital computer (CORDIC) to compute arbitrary complex number Nth root for the first time. Our method performs the tasks of computing complex modulus, complex phase angle, real Nth root, sine function and cosine function, which can be implemented by circular CORDIC, linear CORDIC and hyperbolic CORDIC. Based on these CORDICs, our proposed architecture can not only improve the hardware efficiency just through shift-add operations, but also flexibly adjust the precision and the input range of complex number Nth root. To prove its feasibility, we conduct a software simulation and implement an example circuit in hardware. Under the TSMC 28nm CMOS technology, we synthesize it and get the report that it has the area of 6561μm2 and the power of 3.95mW at the frequency of 1.5GHz.

In this paper, the image Hash authentication algorithm for orthogonal moments of fractional order chaotic scrambling coupling hyper-complex number Tchebichef is proposed. The improved PDE function and Ring segmentation mechanism are used to process the initial image and output the secondary image to improve the robustness of Hash’s scaling and rotation; and by taking full account into the color information of the image, the image R, G and B position is regarded as its imaginary part. In addition, the invariant moment of the hyper-complex number Tchebichef is desgined, and the robust characteristics of the secondary image is fully extracted to greatly improve its ability to recognize the rotation tampering. At the same time, in order to improve the security of the Hash sequence, the fractional order Logistic mapping and chaotic scrambling technique is constructed to encrypt the eigenvector; and the image information is authenticated by the Hamming distance. Finally, the robustness and tampering recognition accuracy of the proposed Hash algorithm are tested.

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we proove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

The subject of complex numbers issue is very significant because of its wide utility, especially in the engineering circuits representation. In this paper, a modern method to find the square root of the complex number has been analyzed, and some examples on the subject were presented.

Here, we discuss the generalized Bernstein-Vazirani algorithm for determining a complex number string. The generalized algorithm presented here has the following structure. Given the set of complex values {a1, a2, a3,…, aN} and a special function $$ g:\mathbf{C}\to \mathbf{C} $$, we determine N real parts of values of the function l(a1), l(a2), l(a3),…, l(aN) and N imaginary parts of values of the function h(a1), h(a2), h(a3),…, h(aN) simultaneously. That is, we determine the N complex values g(aj) = l(aj) + ih(aj) simultaneously. We mention the two computing can be done in parallel computation method simultaneously. The speed of determining the string of complex values is shown to outperform the best classical case by a factor of N. Additionally, we propose a method for calculating many different matrices A, B, C,... into g(A), g(B), g(C),... simultaneously. The speed of solving the problem is shown to outperform the classical case by a factor of the number of the elements of them. We hope our discussions will give a first step to the quantum simulation problem.

Google PageRank is widely used by search engines and known for its high calculation cost. This study proposes the usage of complex numbers in node ranking calculations for a network. The node scores are calculated in the graph using the Hermitian adjacency matrix and the complex plane through an algorithm, called HermitianStatus. The result of the proposed method is then compared to that of Google PageRank. In conclusion, HermitianStatus has advantages over PageRank.

For regular continued fraction, if a real number [Formula: see text] and its rational approximation [Formula: see text] satisfying [Formula: see text], then, after deleting the last integer of the partial quotients of [Formula: see text], the sequence of the remaining partial quotients is a prefix of that of [Formula: see text]. 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 [Formula: see text] of complex numbers which are well approximated with the given bound [Formula: see text] 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 [Formula: see text] and its Hausdorff dimension is equal to that of the [Formula: see text]-approximable set [Formula: see text] of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník theorem in real case.

This study is focused on a proposed alternative algorithm for Google's PageRank, named Hermitian centrality score, which employs complex numbers for scoring a node of the network to overcome the issues of PageRank’s link analysis. This study presents the Hermitian centrality score as a solution for the problems of PageRank, which are associated with the damping factor of Google’s algorithm. The algorithm for Hermitian centrality score is designed to be free from a damping factor, and it reproduces PageRank results well. Moreover, the proposed algorithm can mathematically and systematically change the point of a node of a network.

Determination of complex number modulus and argument is commonly encountered task in digital signal processing. According to definition, this task requires evaluation of square root and inverse tangent functions. When computing hardware resources are limited, e.g. in real-time applications, an approximation by basic arithmetic and logical operations are of interest. We propose method for modulus approximation that uses addition, scaling and comparison, and provides also argument information. At each additional level of our algorithm, modulus approximation error decreases by factor of 4 and argument error by factor of 8.

In this paper, we consider the problem of planar graph-based simultaneous localization and mapping (SLAM) that involves both poses of the autonomous agent and positions of observed landmarks. We present CPL-SLAM, an efficient and certifiably correct algorithm to solve planar graph-based SLAM using the complex number representation. We formulate and simplify planar graph-based SLAM as the maximum likelihood estimation (MLE) on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to convex complex semidefinite programming (SDP). Furthermore, we simplify the corresponding complex semidefinite programming to Riemannian staircase optimization (RSO) on the complex oblique manifold that can be solved with the Riemannian trust region (RTR) method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through applications of CPL-SLAM and comparisons with existing state-of-the-art methods on planar graph-based SLAM, which indicates that our proposed algorithm is capable of solving planar graph-based SLAM certifiably, and is more efficient in numerical computation and more robust to measurement noise than existing state-of-the-art methods. The C++ code for CPL-SLAM is available at https://github.com/MurpheyLab/CPL-SLAM.

In this paper, we present CPL-Sync, a certifiably correct algorithm to solve planar pose graph optimization (PGO) using the complex number representation. We formulate planar PGO as the maximum likelihood estimation (MLE) on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to complex semidefinite programming (SDP). Furthermore, we simplify the corresponding semidefinite programming to Riemannian staircase optimization (RSO) on complex oblique manifolds that can be solved with the Riemannian trust region (RTR) method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through comparisons with existing methods as well as applications on planar PGO in simultaneous localization and mapping (SLAM), which indicates that the proposed algorithm is capable of solving planar PGO certifiably, and is more efficient in numerical computation and more robust to measurement noises than existing state-of-the-art methods. The C++ code for CPL-Sync is available at https://github.com/fantaosha/CPL-Sync.

Geometrical optics accepts certain convenient simplifications. One of these approaches is to treat the refractive index as a fixed, real number. This simplification is adopted despite strong indications that the refractive index is wavelength dependent. What more would we need than the analysis of colors through a prism or the chromatic aberrations in lenses? In addition, birefringence indicates that the refractive index in optical anisotropic materials is not just a number, but a 3×3 tensor; in other words, it is not one, but nine numbers, which, in principle, means that the numbers are complex! However, this simplification is quite a useful approach.

A complete invariant defined for (closed, connected, orientable) 3-manifolds is an invariant defined for the 3-manifolds such that any two 3-manifolds with the same invariant are homeomorphic. Further, if the 3-manifold itself is reconstructed from the data of the complete invariant, then it is called a characteristic invariant defined for the 3-manifolds. In previous papers by the first author, a characteristic lattice point invariant and a characteristic rational invariant defined for the 3-manifolds were constructed which also produced a smooth real function with the definition interval ( − 1 , 1 ) as a characteristic invariant defined for the 3-manifolds. In this paper, a complex number-valued characteristic invariant for the 3-manifolds whose norm is smaller than or equal to one half is introduced by using an embedding of a set of lattice points called the ADelta set, distinct from the PDelta set, into the set of complex numbers. The distributive situation for the invariants of the 3-manifolds of lengths up to 10 is plotted in the complex number plane with radius smaller than or equal to one half. By using this complex number-valued characteristic invariant, a holomorphic function with the unit open disk as the definition domain which is called the characteristic quantity function is constructed as a characteristic invariant defined for the 3-manifolds.

The paper deals with a new fuzzy version of the Lee-Carter (LC) mortality model, in which mortality rates as well as parameters of the LC model are treated as triangular fuzzy numbers. As a starting point, the fuzzy Koissi-Shapiro (KS) approach is recalled. Based on this approach, a new fuzzy mortality model – CNMM – is formulated using orthonormal expansions of the inverse exponential membership functions of the model components. The paper includes numerical findings based on a case study with the use of the new mortality model compared to the results obtained with the standard LC model.

The transition from the He atom to the complex atoms description in the method of interacting configurations in the complex number representation has been presented. As a first step the problem of ionization of H- and Li+ ions is considered. The spectroscopic characteristics of the Be, Mg and Ca atoms in the problem of the electron-impact ionization of these atoms are investigated. Few results in the photoionization problem on the 1P autoionizing states above the n=2 threshold of helium-like Be++ ion are presented. The energies and the widths of the lowest 1S; 1P; 1D; and 1F autoionizing states of the Be, Mg atoms, and the lowest (1P) autoionizing states of Ca atom, are calculated. We consider briefly both a review of our previous results (together with results of other authors) and new calculations of our group. A brief review of the methods of the quasi-stationary states calculation is given.

本文给出了非均匀复数的定义，非均匀复解析函数的定义和性质，建立了非均匀复解析函数与偏微分方程的关系。 In this paper, we introduce the definitions of heterogeneous complex number and heterogeneous complex function, and then discuss the property of heterogeneous complex function. We establish the relationship between heterogeneous complex analytic function and partial differential equation system.

Network security is a general idea to ensure information transmission over PC and portable systems. Elliptic curve cryptosystems are nowadays widely used in public communication channels for network security. Their security relies upon the complexity of clarifying the elliptic curve discrete alogarithm issue. But, there are several general attacks in them. Elliptic bend number juggling is actualized over complex fields to enhance the security of elliptic curve cryptosystems. This paper starts with the qualities of elliptic curve cryptosystems and their security administrations. At that point we talk about limited field number-crunching and its properties, prime field number-crunching, twofold field math and complex number-crunching, and elliptic bend number-crunching over prime field and parallel field. This paper proposes how to execute the unpredictable number of math under prime field and double field utilizing java BigInteger class. also, we actualize elliptic bend math and elliptic bend cryptosystems utilizing complex numbers over prime field and double field and talk about our trials that got from the usage.

Forecasts of female and male mortality that are conducted independently run the risk of projecting implausible sex differentials and fail to exploit correlations that are known to exist between the sexes. We present a new model for the simultaneous modeling of female and male mortality. The model casts mortality as a complex-valued process where the real and imaginary parts correspond to female and male mortalities, respectively. Calculations proceed similarly to the usual Lee–Carter model, via the singular value decomposition, albeit in complex form. Initial applications suggest that the complex Lee–Carter gives fits that are broadly comparable to independent real fits, while offering the advantage of explicit modeling of the relationship between the sexes. Furthermore, model parameters are informative and easily-interpretable.

This qualitative research, drawing on the theoretical frameworks by Even (1990, 1993) and Sfard (2007), investigated five high school mathematics teachers' geometric interpretations of complex number multiplication along with the roots of unity. The main finding was that mathematics teachers constructed the modulus, the argument, and the conjugate of a complex number along with the roots of unity through a series of discursive transformations without specifying the common terminology. While teachers exhibited a variety of visualizations, each founded in a diversity of approaches on the dynamic geometry software, writing mathematical expressions and equations proved challenging. Construction of roots of unity required teachers' mathematical proficiency—in particular, strategic competence in simultaneously coordinating various interpretations of complex numbers, and representational fluency in analytic geometrical and transformational reasoning.

In this study, the combining strategies are considered to be effective tools to improve the performance of the image quality assessment (IQA) metrics. A new metric is proposed to evaluate the quality of test images of combined degradation and individual degradation. The complex numbers are used to describe the image structure in the proposed method. On the basis of that, the properties of the classical IQA method based on singular value decomposition are analysed. The difference between energy visual map and structure visual map is shown. The complex-number-based approach is different from the classical scalar-based techniques, which are insufficient to describe image structure. The proposed C_SVDQ metric can be considered as a vectorial expansion of structure similarity. In the experiments, an extensive comparison between the proposed C_SVDQ and other IQA metrics on image quality database was performed. Both the overall tests and the individual distortion tests show the superiority of this new approach in IQA.

This criterion by Macha has been repeatedly verified, analyzed and special cases of this criterion have been taken into account, reducing the complex stress to its uniaxial equivalent. Since both normal and shear stresses are vectors, we encounter the mathematical problem of adding these vectors, and the question arises how to consider the obtained equivalent stress, because two perpendicular vectors are added with weighting factors. Therefore, in this work it is proposed to adopt a system of complex numbers. Normal stress is defined here as a real part and shear stress as an imaginary part. As a result, on the basis of the defined complex number and based on pendulum bending and double-sided torsion, after transformations, the expression for equivalent stress proves to be identical to those previously defined on the basis of the concept of Macha.

Arithmetic operations involving complex numbers are widely used in the signal processing functions in the physical layer of modern wireless and wireline communication systems, electronic instrumentation and control systems. With the ever increasing throughput requirements of such systems, the power consumption of the hardware realization is increasing beyond the allowed budget. Arithmetic circuits based on binary numeral system that have been optimized rigorously over the past few decades are currently being used for the computation involving complex numbers. In this paper, we present the potential of arithmetic circuits for complex number computations based on the Quater-imaginary (QI) base numeral system to reduce power consumption. We show that for a simple multiplier implementation in the QI base, the savings in power and area consumption could be up to 40% when synthesized in 28nm TSMC standard cell technology node.

In this paper, we introduce one-parameter homothetic motions in the generalized complex number plane ( $${\mathfrak{p}}$$ -complex plane) $$\mathbb{C}_{J}=\left\{x+Jy:\,\,\, x,y \in \mathbb{R},\quad J^2=\mathfrak{p},\quad \mathfrak{p} \in \{-1,0,1\} \right\} \subset \mathbb{C}_\mathfrak{p}$$ where $$\mathbb{C}_\mathfrak{p}=\{x+Jy:\,\,\, x,y \in \mathbb{R}, \quad J^2=\mathfrak{p}\}$$ such that $${-\infty < \mathfrak{p} < \infty}$$ . The velocities, accelerations and pole points of the motion are analysed. Moreover, three generalized complex number planes, of which two are moving and the other one is fixed, are considered and a canonical relative system for one-parameter planar homothetic motion in $${\mathbb{C}_{J}}$$ is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, during the one-parameter homothetic motions, is obtained with the aim of this canonical relative system.

This paper propose a new delayed feedback control (DFC) method which is an extension of the observer-based DFC method to continuous-time cases for the stabilization problem of an unstable periodic orbit (UPO). The proposed method construct a complex-number observer which estimates the tracking error against the UPO. This paper proposes the extension of the author's previous research, and can be used for the time-invariant chaotic systems. The controller tunes the time-shift value of the desired UPO, and overcomes the problem of pole-zero canceling at the origin in the complex plane. The proposed method is applied to the stabilization problem of a UPO in the trajectories of Lorenz model.

DNA Computing, since its inception in 1994, has caught the eyes of researchers due to its massive parallelism and extremely high data density. These powers have given DNA Computer the ability to solve computationally "hard" problems using search over large search space, as well as a powerful data storage technique. This would be much more powerful and general purpose when its ability is increased to solve general-purpose problems too, which can be solved easily on a conventional computer. As an example, DNA Computing falls short in mathematical computations, such as performing simple arithmetic as well as complex number arithmetic etc. In this work, the major purpose is to increase the power of DNA Computing to solve such general purpose problems. This work attempts to perform complex number arithmetic, such as addition, subtraction, multiplication, and division of two complex numbers. True to the best of authors knowledge, no work has been performed to perform representation and arithmetic of complex numbers using DNA molecules. This work has several advantages, such as bio-molecular operations being used here are easy to implement, and operations can be carried out on arbitrarily large numbers.

Recommendation can be reduced to a sub-problem of link prediction, with specific nodes (users and items) and links (similar relations among users/items, and interactions between users and items). However, previous link prediction approaches must be modified to suit recommendation instances because they neglect to distinguish the fundamental relations similar vs. dissimilar and like vs. dislike. Here, we propose a novel and unified way to cope with this deficiency, modeling the relational dualities using complex numbers. Previous works can still be used in this representation. In experiments with the MovieLens dataset and the Android software website AppChina.com, the proposed Complex Representation-based Link Prediction method (CORLP) achieves significant performance in accuracy and coverage compared with state-of-the-art methods. In addition, the results reveal several new findings. First, performance is improved, when the user and item degrees are taken into account. Second, the item degree plays a more important role than the user degree in the final recommendation. Given its notable performance, we are preparing to use the method in a commercial setting, AppChina.com, for application recommendation.

We present a novel complex number formulation along with tight convex relaxations for the aircraft conflict resolution problem. Our approach combines both speed and heading control and provides global optimality guarantees despite non-convexities in the feasible region. As a side result, we present a new characterization of the conflict separation condition in the form of disjunctive linear constraints. Our formulation features one binary variable per pair of aircraft, is free of trigonometric functions, and captures the non-convexity in a set of quadratic concave constraints. Using our approach, we are able to close a number of open instances and reduce computational time by up to two orders of magnitude on standard instances.

[EN] The computation of the arctangent of a complex number, i.e., the atan2 function, is frequently needed in hardware systems that could profit from an optimized operator. In this brief, we present a novel method to compute the atan2 function and a hardware architecture for its implementation. The method is based on a first stage that performs a coarse approximation of the atan2 function and a second stage that improves the output accuracy by means of a lookup table. We present results for fixed-point implementations in a field-programmable gate array device, all of them guaranteeing last-bit accuracy, which provide an advantage in latency, speed, and use of resources, when compared with well-established fixed-point options., This work was supported by the Spanish Ministerio de Economia y Competitividad and FEDER under Grant TEC2015-70858-C2-2-R.

The aim of this study was to examine the abstraction process of tenth grade students' complex number knowledge. A semi-structured interview was held with two students who were successful at mathematics and who voluntarily participated in the study. In this interview, the students faced three different research problems prepared by the researcher in order to reveal the students' knowledge formation processes. The students studied these research problems together. In the meantime, their cognitive processes related to recognizing, building-with, and constructing of discourses was analysed. The analyses indicated that both of the students recognized and built-with the preliminary knowledge necessary for them to construct the knowledge of complex numbers. Moreover, it was understood that one of these students recognized and build-with their previous knowledge of linear functions, coordinate systems, and parallel displacement. In conclusion, it could be said that this student constructed the knowledge of complex plane.