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

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

For regular continued fraction, if a real number x and its rational approximation p / q satisfying | x − p / q | < 1 / q 2 , then, after deleting the last integer of the partial quotients of p / q , the sequence of the remaining partial quotients is a prefix of that of x. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set E (ψ) of complex numbers which are well approximated with the given bound ψ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound ψ and its Hausdorff dimension is equal to that of the ψ -approximable set W (ψ) of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník theorem in real case. [ABSTRACT FROM AUTHOR]

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

In this article, 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 complex (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 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 SDP to Riemannian staircase optimization (RSO) on the complex oblique manifold that can be solved with the Riemannian trust region 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. [ABSTRACT FROM AUTHOR]

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

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 : C → 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. [ABSTRACT FROM AUTHOR]

In this paper, it is presented two math teachers' discourse analysis results, to identify the possible incidence of the metaphorical language used by them, in the learning of the concept of a complex number with ninth grade students. The analysis of the metaphorical language of the teacher is done in the light of the Cognitive Theory of Mathematics. It seeks the understanding of mathematics in the metaphorical thinking of people, and not in the rigorous demonstrations of books, emphasizing the connection that has the body with the cognitive processes, inducing the main key for the reform of teaching and learning processes focused on mathematical communication. [ABSTRACT FROM AUTHOR]

In this study, we compute the unitarily invariant norms of the matrices, Az = (zizj)ni, j = 1, Bz = (zi - zj)ni, j = 1,and Cz = (zi/zj)n1j = 1, where zis are ith components of any complex sequence (zn). Moreover, we give some corollaries and numerical examples related to norms of these matrices. [ABSTRACT FROM AUTHOR]

• The image Hash authentication algorithm for orthogonal moments of fractional order chaotic is proposed. • The image information is authenticated by the Hamming distance. • The robustness and tampering recognition accuracy of the proposed method is tested. 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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

Retailers are facing with a variety of product information, how to help retailers access to effective commodity sales strategy effectively and accurately is an urgent problem. Based on complex network theory, this paper constructed a complex weighted network model of retail product association. Beginning with the relationship of retail product purchase in the consumer purchase record, retail products wrere nodes and the relationship of different product purchase wrere directed edge, and product sales volume wrere weights. And then, it calculated the node strength. Finally, it used the sales bill data from the K-commerce sites and supermarket to verify the algorithm. This study provided a reference basis for bundling sales, discount promotions and display layout strategies for E-commerce websites or large retailers. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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

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

Abstract: A family of new complex number chaotic maps based image encryption algorithm is proposed in the paper. A family of maps is constructed and proved to be chaotic in the complex number field, and its characteristics are analyzed. Two maps are selected from the chaotic maps family and are utilized to construct pseudorandom keystream sequence. In the proposed encryption algorithm, the pseudorandom keystream sequences are used to scramble and diffuse the plain image data and two entropy coding methods are used to reduce the correlation among the signals. Both theoretical analysis and experimental tests show that the proposed algorithm is secure and efficient. [Copyright &y& Elsevier]

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

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]

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

Abstract: Functional Magnetic Resonance Imaging (fMRI), measuring Blood Oxygen Level-Dependent (BOLD), is a widely used tool to reveal spatiotemporal pattern of neural activity in human brain. Standard analysis of fMRI data relies on a general linear model and the model is constructed by convolving the task stimuli with a hypothesized hemodynamic response function (HRF). To capture possible phase shifts in the observed BOLD response, the informed basis functions including canonical HRF and its temporal derivative, have been proposed to extend the hypothesized hemodynamic response in order to obtain a good fitting model. Different t contrasts are constructed from the estimated model parameters for detecting the neural activity between different task conditions. However, the estimated model parameters corresponding to the orthogonal basis functions have different physical meanings. It remains unclear how to combine the neural features detected by the two basis functions and construct t contrasts for further analyses. In this paper, we have proposed a novel method for representing multiple basis functions in complex domain to model the task-driven fMRI data. Using this method, we can treat each pair of model parameters, corresponding respectively to canonical HRF and its temporal derivative, as one complex number for each task condition. Using the specific rule we have defined, we can conveniently perform arithmetical operations on the estimated model parameters and generate different t contrasts. We validate this method using the fMRI data acquired from twenty-two healthy participants who underwent an auditory stimulation task. [Copyright &y& Elsevier]

The goal of this paper is to define for the first time the concept of symbolic 2-plithogenic weak fuzzy complex number as new generalization generated by combining real numbers with symbolic 2-plithogenic numbers. We study the elementary properties of this new class such as Invertibility and nilpotency, with many related examples that explain its novelty. [ABSTRACT FROM AUTHOR]

In recent years, various chaotic equation based pseudorandom number generators have been proposed. However, the chaotic equations are all defined in the real number field. In this paper, an equation is proposed and proved to be chaotic in the imaginary axis. And a pseudorandom number generator is constructed based on the chaotic equation. The alteration of the definitional domain of the chaotic equation from the real number field to the complex one provides a new approach to the construction of chaotic equations, and a new method to generate pseudorandom number sequences accordingly. Both theoretical analysis and experimental results show that the sequences generated by the proposed pseudorandom number generator possess many good properties. [ABSTRACT FROM AUTHOR]

Generation of hidden attractor with complicated phase portrait in chaotic system with no equilibrium has presented a new research focus in the past decade. However, the existing approaches usually follow the rule that you reap what you sow, i.e., taking an no-equilibrium chaotic system as the seed. In this paper, a novel approach to the generation of no-equilibrium multi-fold hidden attractors is presented. The offset boosting and the operation of complex number are applied to several cases, including the seed chaotic systems with no equilibrium, with single unstable node, or with single stable node-focus. In specific, the offset boosting shifts the single non-zero equilibrium to the origin, then the operation of complex number rules out the equilibrium in the folded system and generates multi-fold trajectories. The highlight is that the proposed approach is not limited to the no-equilibrium seed system, but also applicable for different types of single-equilibrium seed systems, even the seed system with a single line of equilibrium. The detailed numerical simulations and experiments confirmed the feasibility of the proposed approach. Finally, an image encryption algorithm and its practical microcontroller-based implementation are presented to support the potential application. • Systematic approach for folding trajectory of chaotic system. • New multi-fold chaotic hidden attractors with no equilibrium. • New chaos-based encryption algorithm for image security. • Dynamic memory scheme for microcontroller implementation. [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]

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. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new generation method of orthogonal periodic complex number sequence sets based on chirp sequences that can derive the orthogonal periodic complex number sequence set having the minimum cross-correlation peak for any period. First, we use the Ohue–Okahisa Conjecture that is proven in a practical range related to the orthogonal periodic polyphase sequence obtained by taking the inverse discrete Fourier transform of a chirp sequence to derive the relationship between the period N having the minimum peak of the cross-correlation and the chirp sequence parameter R, and clearly explain a generation method of an orthogonal periodic complex number sequence pair having the minimum cross-correlation peak for any period. Next, we present a generation method of the orthogonal periodic complex number sequence set having the minimum cross-correlation peak for any two sequences in the set. The dimension of this set is clearly equal to the minimum prime factor of the period N. Furthermore, we present a generation method of the orthogonal periodic polyphase sequence set having the minimum cross-correlation peak for any period and show that the dimension of this set becomes a value 1 less than the minimum prime factor of period N. We show that this generation method can derive the orthogonal periodic complex number sequence set having the minimum cross-correlation peak for periods that could not be generated by conventional techniques and can derive the orthogonal periodic polyphase sequence set demonstrated previously. Finally, we present a generation example of the orthogonal periodic sequence set having the minimum cross-correlation peak and clearly show that the orthogonal periodic complex number sequence set and orthogonal periodic polyphase sequence set having the minimum cross-correlation peak can be generated. © 2004 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(5): 1–11, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10129 [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]

The complex-number multiplier is one of the key arithmetic components for the baseband signal processing of modern digital communication systems such as channel equalization, timing recovery, modulation and demodulation. This paper presents two algorithms suitable for a high-speed complex-number multiplier, which are based on redundant binary (RB) representation of partial products. The basic idea behind our approach is to convert a pair of binary partial products into a RB form so that the post-addition/subtraction which is inevitable in the conventional methods based on binary multiplication, is eliminated. With the proposed algorithms, the complex-number multiplication is reduced to two RB multiplications, one for the real part and the other for the imaginary part. The RB multiplication is defined by an addition of RB partial products, and is performed in parallel without carry propagation from the least-significant digit to the most-significant digit. This work results not only in simplified arithmetic operations, but also in highly parallel and simple architecture when compared with conventional methods using binary multiplications. To demonstrate the algorithms, two test chips have been implemented using a 0.8 mum CMOS technology. [ABSTRACT FROM AUTHOR]

In this paper, we expand our recently introduced concept of unique word orthogonal frequency division multiplexing (UW-OFDM). In UW-OFDM the cyclic prefixes (CPs) are replaced by deterministic sequences, the so-called unique words (UWs). The UWs are generated by appropriately loading a set of redundant subcarriers. By that a systematic complex number Reed–Solomon (RS) code construction is introduced in a quite natural way, because an RS code may be defined as the set of vectors, for which a block of successive zeros occurs in the other domain w.r.t. a discrete Fourier transform. (For a fixed block different to zero, i.e., a UW, a coset code of an RS code is generated.) A remaining problem in the original systematic coded UW-OFDM concept is the fact that the redundant subcarrier symbols disproportionately contribute to the mean OFDM symbol energy. In this paper we introduce the concept of non-systematic coded UW-OFDM, where the redundancy is no longer allocated to dedicated subcarriers, but distributed over all subcarriers. We derive optimum complex valued code generator matrices matched to the best linear unbiased estimator (BLUE) and to the linear minimum mean square error (LMMSE) data estimator, respectively. With the help of simulations we highlight the advantageous spectral properties and the superior bit error ratio (BER) performance of non-systematic coded UW-OFDM compared to systematic coded UW-OFDM and to CP-OFDM in additive white Gaussian noise (AWGN ) as well as in frequency selective environments. [ABSTRACT FROM PUBLISHER]

Abstract: We studied the sharp asymmetric Fano line shape in fiber ring resonator systems and provided an explicit expression for asymmetry parameters using the physical parameters of the system. The fiber ring system was controllable and reconfigurable, allowing us to produce a variety of Fano line shapes in different configurations. Experimentally observed asymmetric spectral structures were fully reproduced using the complex-number asymmetry parameters, validating the approximations used to reduce the analytical expression for the line shape to the phenomenological Fano formula. The results may be useful in the design of on-chip application systems. [Copyright &y& Elsevier]

The article offers tips on how to construct the square root of a complex number using the Geometer's Sketchpad (GSP). One tip refers to constructing the square root of a positive real number and mentions the geometric principles underlying the construction. Another is the application of the approach to the complex numbers. Moreover, several figures are presented depicting the solution process of the problem.

The classical least-square method was extended from the real number into the complex number domain, which is called the complex least-square method. The mathematical derivation and its applications show that the complex least-square method is different from one that the real number and the imaginary number are separately calculated with the classical least-square, by which the actual least- square estimation cannot be obtained in practice. Applications of this new method to an arbitrarily given series and to the precipitation in rainy season at 160 meteorological stations in China mainland show advantages of this method over other conventional statistical models. [ABSTRACT FROM AUTHOR]

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

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

Tensor data are becoming important recently in various application fields. In this paper, we consider the maximal rank problem of 3-tensors and extend Atkinson and Stephens’ and Atkinson and Lloyd’s results over the real number field. We also prove the assertion of Atkinson and Stephens: $${{\rm max.rank}_{\mathbb{R}}(m,n,p) \leq m+\lfloor p/2\rfloor n}$$, $${{\rm max.rank}_{\mathbb{R}}(n,n,p) \leq (p+1)n/2}$$ if p is even, $${{\rm max.rank}_{\mathbb{F}}(n,n,3)\leq 2n-1}$$ if $${\mathbb{F}=\mathbb{C}}$$ or n is odd, and $${{\rm max.rank}_{\mathbb{F}}(m,n,3)\leq m+n-1}$$ if m < n where $${\mathbb{F}}$$ stands for $${\mathbb{R}}$$ or $${\mathbb{C}}$$. [ABSTRACT FROM AUTHOR]

Using the second order Taylor expansion of a function f(z) in complex space, a polynomial p(z) of degree 2 results. We demonstrate that the smallest root of p(z) is a search direction that always decreases the modulus of f(z). Using this property, a new algorithm is proposed. We prove that the algorithm either finds a root or the estimate goes at infinity, without creating any accumlation points. In the special case of a polynomial, the algorithm allways finds a root. Numerical examples are also given. [ABSTRACT FROM AUTHOR]

Using the second order Taylor expansion of a function f(z) in complex space, a polynomial p(z) of degree 2 results. We demonstrate that the smallest root of p(z) is a search direction that always decreases the modulus of f(z). Using this property, a new algorithm is proposed. We prove that the algorithm either finds a root or the estimate goes at infinity, without creating any accumlation points. In the special case of a polynomial, the algorithm allways finds a root. Numerical examples are also given. [ABSTRACT FROM AUTHOR]

Ralstonia solanacearum is a widely distributed phytopathogenic bacterium that is known to invade more than 200 host species, mainly in tropical areas. Reference strain GMI1000 is naturally transformable at in vitro and also in planta conditions and thus has the ability to acquire free exogenous DNA. We tested the ubiquity and variability of natural transformation in the four phylotypes of this species complex using 55 strains isolated from different hosts and geographical regions. Eighty per cent of strains distributed in all the phylotypes were naturally transformable by plasmids and/or genomic DNA. Transformability can be considered as a ubiquitous physiological trait in the R. solanacearum species complex. Transformation performed with two independent DNA donors showed that multiple integration events occurred simultaneously in two distant genomic regions. We also engineered a fourfold-resistant R. solanacearum GMI1000 mutant RS28 to evaluate the size of DNA exchanged during natural transformation. The results demonstrated that this bacterium was able to exchange large DNA fragments ranging from 30 to 90 kb by DNA replacement. The combination of these findings indicated that the natural transformation mechanism could be the main driving force of genetic diversification of the R. solanacearum species complex. [ABSTRACT FROM AUTHOR]

ADHD (attention-deficit hyperactivity disorder) and academic difficulties are frequently associated, but to date this link is poorly understood. In order to explore which components of number processing and calculation skills may be disturbed in children with ADHD we presented a series of respective tasks to 9- to 12-year-old children with ADHD-combined type and matched children without ADHD (of any type) without concomitant dyscalculia and/or dyslexia. Overall, group differences were not significant regarding overlearned and explicitly trained simple and complex calculation skills. More basic number processing skills are – for instance – the ability to compare one-digit numbers according to their magnitude (so-called magnitude comparison), to count or to transcode numbers, i.e. to write down an Arabic number ‘21’ in verbal dictation. Significant differences favouring control children in basic number processing skills were obtained in a number comparison task and in a dot enumeration task. Importantly, our results cannot be explained by group differences regarding specific working memory and executive function components. Thus, number processing skills and in particular the processing of numerical magnitude should be investigated in children diagnosed with ADHD even when no comorbid learning disabilities are reported. [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]

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]

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

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

For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

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 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]

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]