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We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are wound round of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated in terms of MV-algebras. For example, the study of groups together with a cyclic order allows to get a first-order characterization of groups of unimodular complex numbers and of finite cyclic groups. We deduce a characterization of pseudofinite MV-chains and of pseudo-simple MV-chains (i.e. which share the same first-order properties as some simple ones). We can generalize these results to some non-lineraly ordered MV-algebras, for example hyper-archimedean MV-algebras.

Year 2021 is special. It sees the renaissance of concept of phase and the birth of a phase theory for matters much beyond complex numbers and single-input single-output (SISO) linear time-invariant (LTI) systems while we celebrate the 60th birthday of Lei Guo, an exemplary research leader of our times. Here we give a short tutorial of the newly developed phase theory, as a birthday present.

Given two different complex numbers a and b, a bounded linear operator T acting on an infinite-dimensional complex Hilbert space H is said to be {a, b}-quadratic if (T − a)(T − b) = 0. We provide in this paper a complete description of all surjective maps Φ (not necessarily additive) on the algebra $${\cal B}(H)$$ of all bounded linear operators on H that satisfy S − λT is {a, b}-quadratic if and only if Φ(S) − λΦ(T) is {a, b}-quadratic for every $$S,T \in {\cal B}(H)$$ and λ ∈ ℂ.

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

The concept of complex numbers (CNs) is used in many disciplines. In many cases, students find it difficult to understand the logic behind CNs. Rotations, vibrations, and oscillations result in sine or cosine waves. Mathematical representation of rotation/vibration/oscillation is done in two ways—trigonometry and complex numbers. But the algebraic calculation is easier if CNs are used instead of trigonometric functions. The use of CNs as an effective representation of sinusoidal variations is discussed in this article.

In this study, the polar equations and complex numbers of curves are used to design elliptical gears that require machine tools that are difficult to produce by conventional methods and require special consultation. Using these equations, a package program has been developed by using AtuoLISP programming language in order to realize the desired number of teeth and design of elliptical gears in AutoCAD environment.ss With this package program, elliptical gear design has been realized in 5 different modules and number of teeth and they have been manufactured in CNC Wire EMD. The most important parameter for analyzing the workability of elliptical gears is the distance changes between axes. In order to analyze this experimentally, an elliptical gear analysis mechanism is designed and manufactured. In this study, a pair of elliptical gear driven crank connecting bar mechanism and four bar mechanism were analyzed. In addition, the distribution of tooth stress using the Cosmosworks module of the Solidworks program and frequency analysis using the ANSYS program were performed. The results of this study exhibitioned that elliptical gears can be used in many industrial applications.

Complex fuzzy N-soft set (CFN-SS) is an important technique to manage awkward and unreliable information in realistic decision-making problems. CFN-SS is a blend of two separate theories, called N-soft sets (N-SSs) and complex fuzzy sets (CFSs), which are the modified versions of soft sets (SSs) and fuzzy sets (FSs) to depict vague and uncertain information in daily life problems. In this manuscript, the novel concept of CFN-SS is explored and their fundamental laws are discussed. CFN-SS contains the grade of truth in the form of a complex number whose real and imaginary parts are limited to the unit interval. Besides, we examine some algebraic properties for CFN-SS like union, intersections and justify these properties with the help of some numerical examples. To examine the superiority and effectiveness of the proposed approaches, the special cases of the investigated approaches are also discussed. A decision-making procedure is developed by using the investigated ideas based on CFN-SSs. Further, some numerical examples are also illustrated with the help of explored ideas to find the reliability and effectiveness of the proposed approaches. Finally, the comparative analysis of the investigated ideas with some existing ideas is also demonstrated to prove the quality of the proposed works. The graphical expressions of the obtained results are also discussed.

In this paper we use the q-deformed binary operations to discuss q-derivative, q-integral, q-exponential function, q-Gamma function and q-logarithm and q-deformed complex number. We define the q-binary operation of q-matrix. Using these, we construct the q-boson algebra and suq(2) algebra based on the q-deformed binary operations.

In this paper, we investigate the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations $$\begin{aligned} A_{k}(z)f(z+c_{k})+\cdots +A_{1}(z)f(z+c_{1})+A_{0}(z)f(z)= & {} 0, \\ A_{k}(z)f(z+c_{k})+\cdots +A_{1}(z)f(z+c_{1})+A_{0}(z)f(z)= & {} F, \end{aligned}$$ A k ( z ) f ( z + c k ) + ⋯ + A 1 ( z ) f ( z + c 1 ) + A 0 ( z ) f ( z ) = 0 , A k ( z ) f ( z + c k ) + ⋯ + A 1 ( z ) f ( z + c 1 ) + A 0 ( z ) f ( z ) = F , where $$A_{k}\left( z\right) ,\ldots ,A_{0}\left( z\right) ,$$ A k z , … , A 0 z , $$F\left( z\right) $$ F z are meromorphic functions and $$c_{j}$$ c j $$\left( 1,\ldots ,k\right) $$ 1 , … , k are non-zero distinct complex numbers. Under some conditions on the coefficients, we extend early results due to Zhou and Zheng, Belaïdi and Benkarouba.

Recently, some uniqueness theorems about meromorphic functions f(z) concerning their derivatives $$f'(z)$$ and shifts $$f(z+c)$$ with three CM sharing values have been obtained. In this paper, we continue to study this topic. We consider not only high order derivatives instead of just $$f'(z)$$ , but also IM sharing value instead of CM sharing value. In fact, we mainly prove that for a non-constant meromorphic function f(z) of hyper order strictly less than 1, if $$f^{(k)}(z)$$ and $$f(z+c)$$ share $$0,\infty $$ CM and 1 IM, then $$f^{(k)}(z)\equiv f(z+c)$$ , where c is a non-zero finite complex number. Our main theorem generalizes and greatly improves the related result due to Qi–Li–Yang. In addition, we give some discussion of this issue and obtain a uniqueness theorem concerning defective values in Sect. 3.

In this paper we present a simple spectral theory of polynomially bounded functions on the half line, and then apply it to study the asymptotic behavior of solutions of fractional differential equations of the form $$D^{\alpha }_Cu(t)=Au(t)+f(t)$$ , where $$D^{\alpha }_Cu(t)$$ is the derivative of the function u in Caputo’s sense, A is generally an unbounded closed operator, f is polynomially bounded. Our main result claims that if u is a mild solution of the Cauchy problem such that $$\lim _{h\downarrow 0} \sup _{t\ge 0} \Vert u(t+h)-u(t)\Vert /(1+t)^n=0$$ , and $$\sup _{t\ge 0} \Vert u(t)\Vert /(1+t)^n

We obtain $q$-analogues of several series for powers of $\pi$. For example, the identity $$\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^3}=\frac{\pi^3}{32}$$ has the following $q$-analogue: \begin{equation*} \sum_{k=0}^\infty(-1)^k\frac{q^{2k}(1+q^{2k+1})}{(1-q^{2k+1})^3}=\frac{(q^2;q^4)_{\infty}^2(q^4;q^4)_{\infty}^6} {(q;q^2)_{\infty}^4}, \end{equation*} where $q$ is any complex number with $|q, Comment: 10 pages. Add Theorem 1.2

In this paper, the concepts of con-numbers and con-matrices are proposed by the introduction of conjugate operators into the field of complex numbers, and some properties of these two concepts are derived. In addition, two potential applications of these two concepts are addressed in details. Specifically, a class of quantum systems is expressed in terms of the proposed con-numbers, and the ℝ-linear mapping is represented by con-matrices. These two applications imply the importance of the con-numbers and con-matrices. In addition, some methods are discussed on the construction of the conjugate-operator-induced numbers. Also, some further research directions are provided on these extended numbers and matrices with their applications to systems control.

Negative observations pose a problem in econometric models that apply log-transformation to the data. We propose a simple yet effective solution to this problem by extending the domain of numbers to the set of complex numbers. In particular, this approach suggests that we can replace the negative values with their absolute values and estimate this transformed model with conventional estimation methods. Moreover, we extended this approach to logs of independent variables with negative observations as well. Using our method, we estimated the profit efficiencies of the US banks and illustrated that different treatments for observations with negative profits (loss) may lead to substantially different efficiency estimates. We also showed the importance of controlling bank heterogeneity when estimating efficiency.

We suggest an algorithm for finding the group of birational automorphisms of a hyperelliptic curve y2 = p(x), p ∈ ℚ[x], over the field of complex numbers based on power series expansion. We present an implementation of this algorithm in the computer algebra system Sage and examples of its application. Computer experiments show that the algorithm does not lead to unexpectedly cumbersome calculations. The format of the group representation allows one to use the embedded Sage tools for dealing with finite groups of low order. Bibliography: 20 titles.

In this paper, we investigate the growth of solutions of the differential equation: $$\begin{aligned} f^{\left( k\right) }+A_{k-1}\left( z\right) \exp \left\{ \frac{a_{k-1}}{ \left( z_{0}-z\right) ^{n}}\right\} f^{\left( k-1\right) }+\cdots +A_{0}\left( z\right) \exp \left\{ \frac{a_{0}}{\left( z_{0}-z\right) ^{n}}\right\} f=0, \end{aligned}$$ where $$A_{j}\left( z\right) $$ are analytic functions in the closed complex plane except at $$z_{0}$$ and $$a_{j}\left( j=0,\ldots ,k-1\right) $$ are distinct complex numbers. Other linked cases have been studied.

The main attention of this study is focused on the interaction dynamics of breather, and hybrid solitons and breather for an extended generalization of Vakhnenko equation. The general form of the N-order auxiliary function is first derived by the Hirota bilinear method. Then, the N-order solutions can be obtained. When $$N\ge 2$$ , the loop-like breather may emerge by taking the dispersion coefficients as conjugate complex number. The breather is like a magical spring with good elasticity, changeable period and thickness. Furthermore, novel interaction features between breathers, between soliton/solitons and breather/breathers, are observed by visualizing method. The results show that there are abundant dynamics during the interactions, such as elastic collision, phase shift, amplitude amplification, collapse and bulge effect.

In this paper, we show that for a twice differentiable function g having countable zeros and for Lebesgue almost every $$\beta >1$$ β > 1 , the sequence $$(e^{2\pi i \beta ^ng(\beta )})_{n\in {\mathbb {N}}}$$ ( e 2 π i β n g ( β ) ) n ∈ N is orthogonal to all topological dynamical systems of zero entropy. To this end, we define the Chowla property and the Sarnak property for numerical sequences taking values 0 or complex numbers of modulus 1. We prove that the Chowla property implies the Sarnak property and show that for Lebesgue almost every $$\beta >1$$ β > 1 , the sequence $$(e^{2\pi i \beta ^n})_{n\in {\mathbb {N}}}$$ ( e 2 π i β n ) n ∈ N shares the Chowla property. It is also discussed whether the samples of a given random sequence have the Chowla property almost surely. Some dependent random sequences having almost surely the Chowla property are constructed.

The article delves into the realm of application of complex numbers and matrix transformations in pentagonal tiling. A basic understanding of the concepts in these two branches helps to plot the tiles in any computer algebra system (CAS). The application of the concepts has been demonstrated for the special case of Type 13 pentagonal tiling.

The unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.

We give an algebraic proof for which log del Pezzo pairs of Maeda type are K-polystable or not. If the base field is the complex number field, then the result is already known by Li and Sun., 30 pages

In this paper, we study the fattening effect of points over the complex numbers for del Pezzo surfaces $$\mathbb {S}_r$$ S r arising by blowing-up of $$\mathbb {P}^2$$ P 2 at r general points, with $$ r \in \{1, \dots , 8 \}$$ r ∈ { 1 , ⋯ , 8 } . Basic questions when studying the problem of points fattening on an arbitrary variety are what is the minimal growth of the initial sequence and how are the sets on which this minimal growth happens characterized geometrically. We provide a complete answer for del Pezzo surfaces.

International audience; Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

Let $$p\ge 2$$ be an integer, $$M>0$$ be a real number and $$\begin{aligned} {\mathcal {C}}(p,M)= & {} \Bigl \{ z^n + a_{n-p} z^{n-p} + \cdots + a_1 z +a_0 \, \Big | \,\\&\max _{0\le j\le n-p} |a_j| =M, \, n=p, p+1, \ldots \Bigr \}, \end{aligned}$$ where the coefficients $$a_j$$ $$(j= 0, 1,\ldots ,n-p)$$ are complex numbers. Guggenheimer (Am Math Mon 71:54–55, 1964) and Aziz and Zargar (Proc Indian Acad Sci 106:127–132, 1996) proved that if $$P\in {\mathcal {C}}(p,M)$$ , then all zeros of P lie in the disk $$|z

We present a full list of all representations of the special linear group SLnover the complex numbers with complete intersection invariant ring of homological dimension greater than or equal to two, completing the classification of Shmelkin. For this task, we combine three techniques. Firstly, the graph method for invariants of SLndeveloped by the author to compute invariants, covariants and explicit forms of syzygies. Secondly, a new algorithm for finding a monomial order such that a certain basis of an ideal is a Gröbner basis with respect to this order, in between usual Gröbner basis computation and computation of the Gröbner fan. Lastly, a modification of an algorithm by Xin for MacMahon partition analysis to compute Hilbert series.

In 2011, Heittokangas et al. (Complex Var Ellipt Equat 56(1–4):81–92, 2011) proved that if a non-constant finite order entire function f(z) and $$f(z+\eta )$$ share a, b, c IM, where $$\eta $$ is a finite non-zero complex number, while a, b, c are three distinct finite complex values, then $$f(z)=f(z+\eta )$$ for all $$z\in \mathbb {C}$$ . We prove that if a non-constant finite order entire function f and its n-th difference operator $$\Delta ^n_{\eta }f$$ share $$a_1$$ , $$a_2$$ , $$a_3$$ IM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, while $$a_1$$ , $$a_2$$ , $$a_3$$ are three distinct finite complex values, then $$f=\Delta ^n_{\eta }f$$ . The main results in this paper also improve Theorems 1.1 and 1.2 from Li and Yi (Bull Korean Math Soc 53(4):1213–1235, 2016).

International audience; Rigid graph theory is an active area with many open problems, especially regarding embeddings in R^d or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system's complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bézout (m-Bézout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C^d and S^d. The first relates the number of graph orientations to the m-Bézout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d ≥ 5, both in the Euclidean and spherical case. Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in S^2 and C36. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-Bézout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall.

Let H be a Hilbert $$C^*$$ -module, and let $$H_M$$ be the indefinite inner space induced by a self-adjointable and invertible operator M on H. Given weighted projections P and Q on $$H_M$$ , let $$S_{\lambda ,k}=(PQ)^k-\lambda (QP)^k$$ for a pair $$(k, \lambda )$$ , where k is a natural number and $$\lambda $$ is a complex number. It is proved that $$PQ-QP$$ is weighted Moore–Penrose invertible if and only if $$S_{\lambda ,k}$$ is weighted Moore–Penrose invertible for every pair $$(k, \lambda )$$ .

This paper examines motion around equilibrium points (EPs) of an oblate body having a third body in the PR3BP with the disc. The problem is photogravitational, such that the third body gravitates under the influence of two primaries, which emit radiation pressure and the Poynting–Robertson (P–R) drag. Further, the primaries are enclosed by a disc. The mathematical analysis (equations of motion) and the locations of EPs have been studied. There exist six collinear EPs in the line joining the primaries. The first is an additional EP, which depends on parameters, $$\mu$$ and $$W_{i} \,(i = 1,2)$$ of the primaries, while the three are analogous to the collinear EPs of the classical R3BP. The last two are the Jiang–Yeh EPs, which exist due to the mass parameter and the disc. Further, two triangular EPs are found and are characterized by the disc, oblateness of the third body, radiation pressure, P–R drag and mass parameters of the primaries. The stability of the EPs is examined, and it is that they are unbounded due to either a positive root or a positive real part of the complex root. In particular, the numerical exploration is being performed using the binary Achird to compute numerically the locations/or positions of the EPs and the roots of their corresponding characteristic equations. In the case of the collinear point, the roots contain a positive real root and a complex root with positive real part. In view of the triangular points, the roots contain positive real parts of the complex roots. Hence, we conclude that for the R3BP when the primaries emit radiation pressure and P–R drag with disc and the third body is in shape of a sphere, eight EPs exist and are unstable due to radiational forces which reveal positive real part of the complex root.

We introduce a concept of A-strongly statistical convergence for sequences of complex numbers, where A = (ank)n,k∈ℕ is an infinite matrix with nonnegative entries. A sequence (xn) is called strongly convergent to L if $$ \underset{n\to \infty }{\lim }{\sum}_{k=1}^{\infty }{a}_{nk}\left|{x}_k\right.-L\left|=0\right. $$ in the ordinary sense. In the definition of A-strongly statistical limit, we use the statistical limit instead of the ordinary limit with a convenient density. We study some densities and show that the (ank)-strongly statistical limit is an ( $$ {a}_{m_nk} $$ )-strong limit, where the density of the set {mn ∈ ℕ : n ∈ ℕ} is equal to 1. We introduce the notion of dense positivity for nonnegative sequences. A nonnegative sequence (rn) is dense positive provided the limit superior of a subsequence ( $$ {r}_{m_n} $$ ) is positive for all (mn) with density equal to 1. We show that the dense positivity of (rn) is a necessary and sufficient condition for the uniqueness of A-strongly statistical limit, where A = (ank) and rn = $$ {\sum}_{k=1}^{\infty }{a}_{nk} $$ . Furthermore, necessary conditions for the regularity, linearity and multiplicativity of the A-strongly statistical limit are established.

To a pair (X, f), where X is a compact ANR and f : X → 𝕊1 is a continuous angle valued map, a field κ, and a nonnegative integer r, one assigns a finite configuration of complex numbers z with multiplicities δfr (z) and a finite configuration of free κ[t−1, t]-modules $$ {\hat{\delta}}_r^f $$ of rank $$ {\delta}_r^f $$ (z) indexed by the same numbers z. This is in analogy with the configuration of eigenvalues and of generalized eigenspaces of a linear operator in a finite-dimensional complex vector space. The configuration $$ {\delta}_r^f $$ refines the Novikov–Betti number in dimension r, and the configuration $$ {\hat{\delta}}_r^f $$ refines the Novikov homology in dimension r associated with the cohomology class defined by f. In the case of the field κ = C, the configuration $$ {\hat{\delta}}_r^f $$ provides by “von-Neumann completion” of a configuration $$ {\hat{\hat{\delta}}}_r^f $$ of mutually orthogonal closed Hilbert submodules of the L2-homology of the infinite cyclic cover of X determined by the map f, which is an L∞(𝕊1)-Hilbert module.

We introduce and study the Hyers–Ulam stability (HUS) of a Cayley quantum (q-difference) equation of first order, where the constant coefficient is allowed to range over the complex numbers. In particular, if this coefficient is non-zero, then the quantum equation has Hyers–Ulam stability for certain values of the Cayley parameter, and we establish the best (minimal) HUS constant in terms of the coefficient only, independent of q and the Cayley parameter. If the Cayley parameter equals one half, then there is no Hyers–Ulam stability for any coefficient value in the complex plane.

Let Tc := D(x - c)((x - c)D + 2II) be a second-order linear differential operator, where c is an arbitrary complex number, $$D: = \frac{d}{{dx}}$$ and II represents the identity on the linear space of polynomials with complex coefficients. The aim of this paper is to describe all of the Tc-classical orthogonal polynomials. Two canonical situations appear: the Laguerre $$\{L_n^{(2)}\}_{n\geq0}$$ and the Jacobi $$\{P_n^{(\alpha-2,2)}\}_{n\geq0}$$

Dempster–Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. However, the existing evidence theory is insufficient to consider the situations where it has no capability to express the fluctuations of data at a given phase of time during their execution, and the uncertainty and imprecision which are inevitably involved in the data occur concurrently with changes to the phase or periodicity of the data. In this paper, therefore, a generalized Dempster–Shafer evidence theory is proposed. To be specific, a mass function in the generalized Dempster–Shafer evidence theory is modeled by a complex number, called as a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster’s combination rule is exploited. In contrast to the classical Dempster’s combination rule, the condition in terms of the conflict coefficient between the evidences is released in the generalized Dempster’s combination rule. Hence, it is more general and applicable than the classical Dempster’s combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences is less than 1. In a word, this generalized Dempster–Shafer evidence theory provides a promising way to model and handle more uncertain information. Thanks to this advantage, an algorithm for decision-making is devised based on the generalized Dempster–Shafer evidence theory. Finally, an application in a medical diagnosis illustrates the efficiency and practicability of the proposed algorithm.

The complex roots of the chromatic polynomial $$P_{G}(x)$$ of a graph G have been well studied, but the p-adic roots have received no attention as yet. We consider these roots, specifically the roots in the ring $$\mathbb{Z}_p$$ of p-adic integers. We first describe how the existence of p-adic roots is related to the p-divisibility of the number of colourings of a graph—colourings by at most k colours and also ones by exactly k colours. Then we turn to the question of the circumstances under which $$P_{G}(x)$$ splits completely over $$\mathbb{Z}_p$$, giving some generalities before considering in detail an infinite family of graphs whose chromatic polynomials have been discovered, by Morgan (LMS J Comput Math 15, 281–307, 2012), to each have a cubic abelian splitting field.

We consider a class of plane orthotropic deformations of the form ex = σx + a12σy, γxy = 2(p − a12)Txy, ey = a12σx + σy, where σx, Txy, σy and$$ {\upvarepsilon}_x\frac{\upgamma_{xy}}{2},{\upvarepsilon}_Y $$ are components of the stress tensor and the deformation tensor, respectively, real parameters p and a12 satisfy the inequalities: -1 < p < 1, -1 < a12 < p. A class of solutions of the Lame equilibrium system for displacements is built in the form of linear combinations of components of “analytic” functions which take values in commutative and associative two-dimensional algebras with unity over the field of complex numbers.

This note is devoted to a weaker version of Bishop property $$\beta$$ at a given complex number. We show in particular that this notion is a regularity and hence the induced spectrum satisfies all classical properties of the spectrum.

A complex fuzzy set, the generalization of fuzzy set provides a powerful mathematical framework whose membership degrees are in the form of complex numbers in the unit disc. The averaging operators consisting of the properties of both t-norm and t-conorm are of great importance in complex fuzzy environment. In this paper, we present certain quadratic averaging operators, including complex fuzzy weighted quadratic averaging, complex fuzzy ordered weighted quadratic averaging, complex fuzzy Einstein weighted quadratic averaging and complex fuzzy Einstein ordered weighted quadratic averaging operators. These operators are used to study many different issues of periodic nature. We apply these models to the multi-attribute decision-making problems and wireless detection of target location. Conclusively, we can choose the best opinion by the ranking of the aggregated outputs and detect the position and direction of a target. Moreover, we describe these models through numerical examples to check their validity and importance in real life problems. To explain the consistency and authenticity of our model, we examine a comparative analysis with existing aggregation techniques.

Let $$a(\ne 0)$$, b be finite complex numbers, and $$k\ge 3$$ be a positive integer. If f is a meromorphic function of order $$\rho >2$$, then $$f^{\prime }-af^k-b$$ has infinitely many zeros on the complex plane.

Real-valued neural networks or complex-valued neural networks are sometimes inappropriate for some engineering and research problems for instance where the data is multi-dimensional, such as 4-D signals, color images and body images. Hence, researchers explored recently a more general and sophisticated model than the previous one, which is the quaternion-valued neural networks. The quaternions, which can also be defined as $$2\times 2$$ matrix of complex numbers, have the capacity to analyze three or more dimensional signals and represent spatial transformations. In this paper, some sufficient conditions are given for the existence and various kinds of stability for the unique Stepanov-like pseudo almost periodic solution of quaternion-valued fuzzy recurrent neural networks. The results are established by employing Lyapunov functionals, Nemytskii’s operator and Banach fixed point theorem. Also, a new direct method is used to establish our theoretical results in order to avoid the decomposition of the considered model into real-valued or complex-valued system. Finally, a numerical example is given to illustrate the validity of the obtained results.

In this paper, using the first-order Nedelec conforming edge element space of the second type, we develop and analyze a continuous interior penalty finite element method (CIP-FEM) for the time-harmonic Maxwell equation in the three-dimensional space. Compared with the standard finite element methods, the novelty of the proposed method is that we penalize the jumps of the tangential component of its vorticity field. It is proved that if the penalty parameter is a complex number with negative imaginary part, then the CIP-FEM is well-posed without any mesh constraint. The error estimates for the CIP-FEM are derived. Numerical experiments are presented to verify our theoretical results.

Complex numbers are indispensable parts of describing states of quantum systems and their dynamic behavior, and they are widely used in classical and quantum physics. In recent years, some studies and experiments have shown the necessity of complex numbers in quantum physics. In this work, we investigate the quantification of imaginarity and explore connections of several properties of imaginary measures. In addition, we focus on two specific imaginarity measures: the weight of imaginarity and the relative entropy of imaginarity, proving that they have some nice properties. We also study the corresponding semidefinite programming form of the weight of imaginarity and obtain a closed expression of the relative entropy of imaginarity. Moreover, we analyze the quantitative relationships of several imaginarity measures. Our work further studies the quantification of imaginarity, and provides two concrete measures with interesting properties for imaginarity, which will contribute to the development of quantum mechanics and quantum technology.

The classical theorems of Mittag-Leffler and Weierstrass show that when $\{\lambda_n\}$ is a sequence of distinct points in the open unit disk $\D$, with no accumulation points in $\D$, and $\{w_n\}$ is any sequence of complex numbers, there is an analytic function $\phi$ on $\D$ for which $\phi(\lambda_n) = w_n$. A celebrated theorem of Carleson \cite{MR117349} characterizes when, for a bounded sequence $\{w_n\}$, this interpolating problem can be solved with a bounded analytic function. A theorem of Earl \cite{MR284588} goes further and shows that when Carleson's condition is satisfied, the interpolating function $\phi$ can be a constant multiple of a Blaschke product. In this paper, we explore when the interpolating $\phi$ can be an outer function. We then use our results to refine a result of McCarthy \cite{MR1065054} and explore the common range of the co-analytic Toeplitz operators on a model space., Comment: 27 pages

Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. An algebraist’s perspective on the many subgroups and subalgebras of Clifford algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible physical interpretations are indicative only, and may not work. Nevertheless, both the existence of three generations of fermions and the symmetry-breaking of the weak interaction seem to emerge naturally from an extension of the Dirac algebra from complex numbers to quaternions.

The well-known quaternion algebra is a four-dimensional natural extension of the field of complex numbers and plays a significant role in various aspects of signal processing, particularly for representing signals wherein several instincts are to be controlled simultaneously. For efficient analysis of such quaternionic signals, we introduce the notion of linear canonical wavelet transform in quaternion domain by invoking the elegant convolution structure associated with the quaternion linear canonical transform. The preliminary analysis encompasses the study of fundamental properties of the proposed linear canonical wavelet transform in quaternion domain including the Rayleigh’s theorem, inversion formula and a characterization of the range. Subsequently, we formulate three uncertainty principles; viz, Heisenberg-type, logarithmic and local uncertainty inequalities associated with the linear canonical wavelet transform in quaternion domain.

Let $$f_\omega (z)=\sum \nolimits _{j=0}^{\infty }\chi _j(\omega ) a_j z^j$$ be a transcendental random entire function, where $$\chi _j(\omega )$$ are independent and identically distributed random variables defined on a probability space $$(\Omega , \mathcal {F}, \mu )$$ . In this paper, we study a family of random entire functions, which includes Gaussian, Rademacher, and Steinhaus entire functions. Then we prove that if two random entire functions in this family share two distinct complex numbers counting multiplicities, then they are identically equal.

Let $${\mathbb {C}}$$ be the set of complex numbers and $$\sigma $$ be a continuous automorphism and $$\tau $$ be a continuous anti-automorphism such that $$\sigma ^{2}=\tau ^{2}=id.$$ The purpose of this paper is to generalize the small-dimension lemma [20, Small Dimension Lemma] and by help of it we find on any compact group G the non-zero continuous solutions $$f:G\rightarrow {\mathbb {C}}$$ of the functional equation $$\begin{aligned} f(x\sigma (y))+f(\tau (y)x)=2f(x)f(y), \ \ \ x,y \in G, \end{aligned}$$ in terms of continuous characters of G.

We analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set ofmsuch ‘chosen’ elements (out of$$n>m)$$n>m). Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change$$\varphi $$φ, all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers$$\varphi $$φand$$\alpha $$αwhich is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on$$\alpha $$α. In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change$$\varphi ,$$φ,and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm ($$m=1,\phi =\pi $$m=1,ϕ=π), which are of the form, the found previously, unitary operators.

Let k be a nonzero complex number. In this paper, we consider a k-circulant matrix whose first row is $$(Q_{1},Q_{2},\ldots ,Q_{n})$$ , where $$Q_{n}$$ is the nth Pell–Lucas number. The formulas for the eigenvalues of such matrix are obtained. Namely, the result which can be obtained from the result of Theorem 7. (Yazlik and Taskara, J Inequal Appl 2013:394, 2013) is improved. The obtained formulas for the eigenvalues of a k-circulant matrix involving the Pell–Lucas numbers show that the result of Theorem 8. (Jing, Li and Shen, WSEAS Trans Math 12(3):341-351, 2013) (i.e. Theorem 8. (Yazlik and Taskara 2013)) is not always applicable. The Euclidean norm of such matrix is determined. The upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is $$(Q_{1}^{-1},Q_{2}^{-1},\ldots ,Q_{n}^{-1})$$ are also investigated. The obtained results are illustrated by examples. As a consequence of the previous results, the eigenvalues, the determinant, the Euclidean norm of a k-circulant matrix whose first row is $$(q_{1},q_{2},\ldots ,q_{n})$$ , where $$q_{n}$$ is the nth modified Pell number, are presented. Also, the upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is $$(q_{1}^{-1},q_{2}^{-1},\ldots ,q_{n}^{-1})$$ are given

In this paper, we transfer the approach of finding the explicit solutions to linear impulsive differential equations (LIDEs) to quaternion-valued case. We develop the similar idea to find the solutions to LIDEs and give the details to derive the representations of the solutions to quaternion-valued LIDEs in the sense of complex numbers and quaternion. In addition, we present that two formulas are equivalent to each other. Finally, some examples are given to illustrate the theory approach.