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For centuries, the decimal number system always was dominant until the advent of digital computers, which brought the binary and other number systems into the spotlight. Except them, other numeral systems for image representation are also worth exploring. Quaternion, as a hyper-complex number mathematical tool, is widely concerned and used in image representation. However, the current mainstream quaternion representation is only applicable to color image representation, and its applicability is limited. Hence, the paper introduces a novel quaternion system-based for image representation, which is suitable not for color image but also for gray-scale image. The representation scheme is a numeral positional system which uses a quaternion as the base, and the method converts the traditional two-dimensional array representation of one image into a higher four-dimensional space. Then, some properties of the quaternion can be applied to image analysis and processing. Experiments show that the proposed representation method can retain the essence of the original image reversibly and losslessly. Furthermore, some potential image applications, such as image encryption,image scrambling and image secret sharing, are demonstrated that the proposed method has an practical research value in image security. [ABSTRACT FROM AUTHOR]

How to automatically generate short and easy-to-understand proofs for geometric theorems has long been an issue of concern in mathematics education. A novel automated geometric theorem proving method based on complex number identities is proposed in this paper, which acts as a bridge between geometry and algebra. According to the proposed method, the geometric relations in the given proposition are first transformed into a complex number expression, then the complex number identity is generated by the elimination method; finally, the closure property under all four operations of real numbers is employed to prove the proposition. A test on more than 300 geometric problems shows that the proposed method is highly effective, and the corresponding proofs are short, with obvious geometric meaning. [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]

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]

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]

Focuses on two simple properties of the complex number i for the characterization of the complex number. Information on the non-associative Euclidean nearring; Theorem and proof.

In this paper, we introduce one-parameter homothetic motions in the generalized complex number plane ( $${\mathfrak{p}}$$ -complex plane) where 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. [ABSTRACT FROM AUTHOR]

In this study, we firstly give the basic notations of the generalized complex number plane ( $${\mathfrak{p}}$$ -complex plane) $${\mathbb{C}_{\mathfrak{p}}}$$ . Then, we introduce the one-parameter planar motions in $${\mathfrak{p}}$$ -complex plane $${\mathbb{C}_{J}}$$ such that $${\mathbb{C}_{J} \subset \mathbb{C}_{\mathfrak{p}}}$$ . These motions correspond the one-parameter motions in affine Cayley-Klein planes []. We examine this motion theory with aspects of complex motions as given in []. Besides, we discuss the relations between absolute, relative, sliding velocities (accelerations) and pole curves under the motions $${\mathbb{C}_{J}/{\mathbb{C}_{J}^{\prime}}}$$ . [ABSTRACT FROM AUTHOR]

This paper presents an optimized implementation on FPGA of digit-serial Complex-Number Multipliers (CMs) using Booth recoding techniques and tree adders based on Carry Save (CS) and Ripple Carry Adders (RCA). This kind of Complex-Number multipliers can be pipelined at the same level independent of the digit-size. Variable and fixed coefficient CMs have been considered. In the first case an efficient mapping of the modified Booth recoding and the partial product generation is presented which results in a logic depth reduction. The combination of 5:3 and 4:3 converters in the CS structure and the utilization of RCA trees lead to a minimum area requirement. In the case of fixed coefficient CMs, partial products generator is based on look-up tables and multi-bit Booth recoding is used to reduce the area and increase the performance of the circuit. The study reveals that efficient mapping of the 5-bit Booth recoding to generate the partial products is the optimum multibit recoding when Xilinx FPGA devices are used. [ABSTRACT FROM AUTHOR]

The double complex number imaginary unit J D is treated as a special matrix M(J D ); by considering the matrix M(ε) corresponding to the hyperbolic complex imaginary unit ε, a supermatrix solution to the Yang-Baxter equation is constructed. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a uniqueness question of entire functions sharing a nonzero complex number with their difference operators. The results in this paper improve Theorem 1.1 in Liu and Yang (Arch. Math. 92 (), 270-278) and deal with Question 1 in Liu and Yang (), where the entire functions are of finite orders. [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]

We discuss the nature of complex number and its effect on complex-valued neural networks (CVNNs). After we review some examples of CVNN applications, we look back at the mathematical history to elucidate the features of complex number, in particular to confirm the importance of the phase-and-amplitude viewpoint for designing and constructing CVNNs to enhance the features. This viewpoint is essential in general to deal with waves such as electromagnetic wave and lightwave. Then, we point out that, although we represent a complex number as an ordered pair of real numbers for example, we can reduce ineffective degree of freedom in learning or self-organization in CVNNs to achieve better generalization characteristics. This merit is significantly useful not only for waverelated signal processing but also for general processing with frequency-domain treatment through Fourier transform. [ABSTRACT FROM AUTHOR]

Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α n + ν, n = 0, 1, 2, . . . , modulo $${\mathbb{Z}[i],}$$ where $${i=\sqrt{-1}}$$ and $${\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}}$$ is the ring of Gaussian integers. For any $${z\in \mathbb{C},}$$ one may naturally call the quantity z modulo $${\mathbb{Z}[i]}$$ the fractional part of z and write { z} for this, in general, complex number lying in the unit square $${S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) <1 \}}$$. We first show that if α is a complex non-real number which is algebraic over $${\mathbb{Q}}$$ and satisfies | α| > 1 then there are two limit points of the sequence { ξ α n + ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over $${\mathbb{Q}(i)}$$ all lie in the unit disc | z| ≤ 1 and $${\xi\in\mathbb{Q}(\alpha,i).}$$ Then we prove a result in the opposite direction which implies that, for any fixed $${\alpha\in\mathbb{C}}$$ of modulus greater than 1 and any sequence $${z_n\in\mathbb{C},n=0,1,2,\dots,}$$ there exists $${\xi \in \mathbb{C}}$$ such that the numbers ξ α n− z n, n = 0, 1, 2, . . . , all lie ‘far’ from the lattice $${\mathbb{Z}[i]}$$. In particular, they all can be covered by a union of small discs with centers at $${(1+i)/2+\mathbb{Z}[i]}$$ if | α| is large. [ABSTRACT FROM AUTHOR]

The level- m scaled circulant factor matrix over the complex number field is introduced. Its diagonalization and spectral decomposition and representation are discussed. An explicit formula for the entries of the inverse of a level- m scaled circulant factor matrix is presented. Finally, an algorithm for finding the inverse of such matrices over the quaternion division algebra is given. [ABSTRACT FROM AUTHOR]

Given a bounded operator Q on a Hilbert space H , a pair of bounded operators (T 1 , T 2) on H is said to be Q-commuting if one of the following holds: T 1 T 2 = Q T 2 T 1 or T 1 T 2 = T 2 Q T 1 or T 1 T 2 = T 2 T 1 Q. We give an explicit construction of isometric dilations for pairs of Q-commuting contractions for unitary Q, which generalizes the isometric dilation of Ando (Acta Sci Math (Szeged) 24:88–90, 1963) for pairs of commuting contractions. In particular, for Q = q I H , where q is a complex number of modulus 1, this gives, as a corollary, an explicit construction of isometric dilations for pairs of q-commuting contractions, which are well studied. There is an extended notion of q-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an n-tuple of q-commuting contractions, where n ≥ 3 . Generalizing the class of commuting contractions considered by Brehmer (Acta Sci Math (Szeged) 22:106–111, 1961), we construct a class of n-tuples of q-commuting contractions and find isometric dilations explicitly for the class. [ABSTRACT FROM AUTHOR]

Erdös–Lax inequality relates the sup norm of the derivative of a polynomial along the unit circle to that of the polynomial itself (on the unit circle). This paper aims to extend the classical Erdös–Lax inequality to the polar derivative of a polynomial by using the extreme coefficients of the given polynomial. The obtained results not only enrich the realm of Erdös–Lax-type inequalities but also offer a promising avenue for diverse applications where these inequalities play a pivotal role. To illustrate the practical significance of our results, we present a numerical example. It vividly demonstrates that our bounds are considerably sharper than the existing ones in the extensive literature on this captivating subject. [ABSTRACT FROM AUTHOR]

Views are expressed on the use of the complex number notation with applications to well known geodetic transformations and relations. [ABSTRACT FROM AUTHOR]

Quaternion (Q-) mathematics formally containsmany fragments of physical laws; in particular, the Hamiltonian for the Pauli equation automatically emerges in a space with Q-metric. The eigenfunction method shows that any Q-unit has an interior structure consisting of spinor functions; this helps us to represent any complex number in an orthogonal form associated with a novel geometric image (the conicgearing picture). Fundamental Q-unit-spinor relations are found, revealing the geometric meaning of the spinors as Lamé coefficients (dyads) locally coupling the base and tangent surfaces. [ABSTRACT FROM AUTHOR]

The color of a complex number is defined as the number of vertices of the convex hull of powers of that number. This induces a coloring of the unit disk. The structure of the set Λ of points where the color changes is investigated here. It is observed that there is a connection between this fractal set Λ and some family of trinomial equations. Three algorithms for coloring the unit disk are described, the last one (related to the Farey sequence) arising out of a conjecture. This conjecture is formulated and proved in this presentation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a technique to solve LR-bipolar fuzzy linear system(BFLS), LR-complex bipolar fuzzy linear (CBFL) system with real coefficients and LR-complex bipolar fuzzy linear (CBFL) system with complex coefficients of equations. Initially, we solve the LR-BFLS of equations using a pair of positive (∗) and negative (∙) of two n × n LR-real linear systems by using mean values and left-right spread systems. We also provide the necessary and sufficient conditions for the solution of LR-BFLS of equations. We illustrate the method by using some numerical examples of symmetric and asymmetric LR-BFLS equations and obtain the strong and weak solutions to the systems. Further, we solve the LR-CBFL system of equations with real coefficients and LR-CBFL system of equations with complex coefficients by pair of positive (∗) and negative (∙) two n × n real and complex LR-bipolar fuzzy linear systems by using mean values and left-right spread systems. Finally, we show the usage of technique to solve the current flow circuit which is represented by LR-CBFL system with complex coefficients and obtain the unknown current in term of LR-bipolar fuzzy complex number. [ABSTRACT FROM AUTHOR]

Introduction: Multiply-accumulate operation is the most fundamental operation in digital signal processing for image processing, robotics and automatic control. In this paper, a novel architecture of dynamically reconfigurable fused multiply-adder (FMA) is proposed. Methods: Dynamic reconfiguration is a method that can change the circuit configuration without stop of operation. The proposed circuit provides the following four calculation modes by dynamic reconfiguration: (1) complex number FMA mode, (2) real number FMA mode, (3) complex number parallel calculation mode, and (4) real number parallel calculation mode. The data format is single precision floating point format based on IEEE754. The proposed circuit was designed using Verilog-HDL. It was simulated by logic circuit simulator, and implemented on FPGA. Result: As a result of circuit synthesis, we confirmed the reduction of resource in the proposed circuit. Furthermore, we confirmed proper result for each calculation mode by logic simulation and experiment on FPGA. Conclusion: The proposed circuit provides the four calculation modes by dynamic reconfiguration. We confirmed the reduction of resource and proper result for each calculation mode. [ABSTRACT FROM AUTHOR]

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

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

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

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]

We say an integer polynomial p, on Boolean inputs, weakly m-represents a Boolean function f if p is nonconstant and is zero (mod m), whenever f is zero. In this paper we prove that if a polynomial weakly m-represents the Mod q function on n inputs, where q and m are relatively prime and m is otherwise arbitrary, then the degree of the polynomial is $ \Omega(n) $ . This generalizes previous results of Barrington, Beigel and Rudich, and Tsai, which held only for constant or slowly growing m. In addition, the proof technique given here is quite different. We use a method (adapted from Barrington and Straubing) in which the inputs are represented as complex q-th roots of unity. In this representation it is possible to compute the Fourier transform using some elementary properties of the algebraic integers. As a corollary of the main theorem and the proof of Toda's theorem, if q, p are distinct primes, any depth-three circuit that computes the Mod q function, and consists of an exact threshold gate at the output, Mod p gates at the next level, and AND gates of polylog fan-in at the inputs, must be of exponential size. We also consider the question of how well circuits consisting of one exact gate over ACC( p)-type circuits (where p is an odd prime) can approximate parity. It is shown that such circuits must have exponential size in order to agree with parity for more than 1/2 + o(1) of the inputs. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic behavior of a class of delayed quaternion-valued neural networks (QVNNs) with impulses and parameter uncertainties. First, we assume that the activation function and connection matrices in the model are defined in the quaternion domain. Without decomposing the QVNNs into four equivalent neural networks in a real number domain or two equivalent neural networks in a complex number domain, the existence and uniqueness of the equilibrium point (EP) are studied based on the M-matrix and homeomorphism mapping theories. By combing the vector Lyapunov function with mathematical induction, theorems are given to ensure robust exponential stability of the system's EP. The results reflect the influence of connection matrices, delays, impulses, and the activation function on the convergence speed of the EP. Finally, the feasibility of the derived results is explained through three examples. [ABSTRACT FROM AUTHOR]

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

In this paper, we study the uniqueness of entire function and its differential-difference operators. We prove the following result: let f be a transcendental entire function of finite order, let η be a non-zero complex number, n ≥ 1 , k ≥ 0 two integers and let a and b be two distinct finite complex numbers. If f and (Δ η n f) (k) share a CM and share b IM, then f ≡ (Δ η n f) (k) . [ABSTRACT FROM AUTHOR]

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

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]

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]

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]