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The following investigation is a cognitive-historical analysis of the conceptual development of complex numbers. The history we explore spans nearly two millennia, from the earliest appearance of the square root of a negative quantity in the calculations of Heron of Alexandra (first century CE) to the full flowering of complex numbers in the first half of the 19th century. The approach we use is Nersessian's, including her formulations of model-based reasoning and mental models. Additional aspects feature the prominent roles played by process representations, including object-process complementarities, and by core numerical systems. Our analysis provides support for mental rotation as a fourth core system and for bootstrapping scenarios in both the senses of Nersessian and of Carey. We use the results of our analysis and discussion to update previous con- ceptual metaphor mapping approaches to complex numbers by Fauconnier and Turner and by Lakoff and Núñez. Finally, we review recent empirical studies of complex number learning and find them to be consistent with the present work. [ABSTRACT FROM AUTHOR]

Copyright of Formación Universitaria is the property of Centro de Informacion Tecnologica (CIT) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

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]

• 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]

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]

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]

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]

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]

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]

• We often deal with the combination of mutually perpendicular vectors of the normal and shear stress. • Although this is inconsistent with the principles of mathematics. • Mathematical compatibility is not fully valid in the criterion based on the concept by Macha. • It turns out that treating stress in the critical plane as a complex number results in an expression identical to Macha's criterion. This criterion by Macha has been repeatedly verified, analyzed and special cases of this criterion have been taken into account, reducing the complex stress to its uniaxial equivalent. Since both normal and shear stresses are vectors, we encounter the mathematical problem of adding these vectors, and the question arises how to consider the obtained equivalent stress, because two perpendicular vectors are added with weighting factors. Therefore, in this work it is proposed to adopt a system of complex numbers. Normal stress is defined here as a real part and shear stress as an imaginary part. As a result, on the basis of the defined complex number and based on pendulum bending and double-sided torsion, after transformations, the expression for equivalent stress proves to be identical to those previously defined on the basis of the concept of Macha. [ABSTRACT FROM AUTHOR]

The concept of an ambiguous set (AS) was initially introduced by Singh et al. [P. Singh, Y.-P. Huang and T.-T. Lee — A novel ambiguous set theory to represent uncertainty and its application to brain MR image segmentation, in Proc. IEEE Int. Conf. Systems, Man and Cybernetics (SMC), Bari, Italy, 2019, pp. 2460–2465]. Can we examine the possibility of expanding it to introduce ambiguous complex numbers? This study aims to answer this question by first introducing the concept of an AS and then providing a set of formulas that aid in determining the membership degrees of the set. Then, this study expands the idea of AS to a single-valued ambiguous number (SVAN). Based on SVAN, the concept of single-valued ambiguous complex number (SVACN) is illustrated. Following that, various theories and properties of SVACN are addressed in terms of their complex conjugate, polar, exponential forms, and roots. Furthermore, the study proposes an algorithm tailored to generating the Mandelbrot set using varying numbers of SVACNs. Additionally, this approach furnishes directional vectors for the SVACNs within a two-dimensional plane. The study also deliberates on the manifold advantages of this methodology, particularly emphasizing its capacity to provide directional guidance for SVACNs. [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]

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P ′ (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we prove some Lr inequalities for the polar derivative of a polynomial have all zeros in |z| ≤ 1. Our theorem generalizes a result of Dewan and Mir [K. K. Dewan, A. Mir, Inequalities for the polar derivative of a polynomial, J. Interd. Math. 10 (2007), no. 4, 525–531] and includes as special cases several interesting many known results [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]

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]

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]

The use of registrations is useful in shortening Mizar proofs [1], [2], both in terms of formalization time and article space. The proposed system of classes for complex numbers aims to facilitate proofs involving basic arithmetical operations and order checking. It seems likely that the use of self-explanatory adjectives could also improve legibility of these proofs, which would be an important achievement [3]. Additionally, some potentially useful definitions, following those defined for real numbers, are introduced. [ABSTRACT FROM AUTHOR]

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]

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]

Split-complex (hyperbolic) numbers are ordered pairs of real numbers, written in the form x + jy with j² = 1, used to describe the geometry of the Lorentzian plane. Since a null split-complex number does not have an inverse, some methods to calculate the exponential of complex matrices are not valid for split-complex matrices. In this paper, we examined the exponential of a 2 × 2 split-complex matrix in three cases: i. Δ = 0, ii. Δ ̸= 0 and Δ is not null split-complex number, iii. Δ ̸= 0 and Δ is a null split-complex number where Δ = (trA)2 - 4 detA. [ABSTRACT FROM AUTHOR]

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is $ (g,gq,gq^{2},\ldots,gq^{n-1}) $ (g , gq , g q 2 , ... , g q n − 1) , where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples. [ABSTRACT FROM AUTHOR]

Independent component analysis (ICA), as a statistical and computational approach, has been successfully applied to digital signal processing. Performance analysis for the ICA approach is perceived as a challenging task to work on. This contribution concerns the complex-valued FastICA algorithm in the range of ICA over the complex number domain. The focus is on the robust and equivariant behavior analysis of the complex-valued FastICA estimator. Although the complex-valued FastICA algorithm as well as its derivatives have been widely used methods for approaching the complex blind signal separation problem, rigorous mathematical treatments of the robust measurement and equivariance for the complex-valued FastICA estimator are still missing. This paper strictly analyzes the robustness against outliers and separation performance depending on the global system. We begin with defining the influence function (IF) of complex-valued FastICA functional and followed by deriving its closed-form expression. Then, we prove that the complex-valued FastICA algorithm based on the optimizing cost function is linear-equivariant, depending only on the source signals. [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]

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]

Copyright of Journal of Chinese Mass Spectrometry Society is the property of Journal of Chinese Mass Spectrometry Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

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]

Let L(s, χ) be a Dirichlet L-function associated with a primitive character χ mod q and a be a non zero complex number. We denote by Δ(s, χ) the function which appears in the functional equation L(s, χ) = Δ(s, χ)L(1 - s,χ) and δa,χ = βa,χ + iγa,χ the solutions of the equation Δ(s, χ) = a which are called a-points ofΔ(s,χ). In this note, we will prove that for every complex number a = 0 the mean of the values L(δα,χ,χ) on the sequence of a-points δa,χ of the function Δ(s, χ) exists and equals a + 1. [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]

Since the product of complex numbers and rectangular fuzzy complex numbers (RFCN) is not necessarily a RFCN in the former fuzzy complex linear system (FCLS), the scalar multiplication and addition operations of complex numbers and fuzzy complex numbers (FCN) based on a new representation of FCN are proposed. We also introduce a new method for solving FCLS, which can convert FCLS into two distinct linear systems. One is an n × n complex linear system, and the other is an (m n) × (m n) real linear system, where n is the number of unknown variables, and m is the number of substitutional cyclic sets composed of coefficients of FCLS. In particular, using this method to solve one-dimensional fuzzy linear systems, a (2 n) × (2 n) RLS is obtained, which is consistent with Friedman's method. Finally, FCLS based on the RFCN as a special case are also investigated. [ABSTRACT FROM AUTHOR]

A degree one element of the Orlik-Solomon algebra of a hyperplane arrangement defines a cochain complex known as the Aomoto complex. The Aomoto complex can be considered as the "linear approximation" of the twisted cochain complex with coefficients in a complex rank one local system. In this paper, we discuss q-deformations of the Aomoto complex. The q-deformation is defined by replacing the entries of representation matrices of the coboundary maps with their q-analogues. While the resulting maps do not generally define cochain complexes, for certain special basis derived from real structures, the q-deformation becomes again a cochain complex. Moreover, it exhibits universality in the sense that any specialization of q to a complex number yields the cochain complex computing the corresponding local system cohomology group. [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]

In this paper, we introduce a membership function used to form the fuzzy Mandelbric set and investigate the structural effects of additive and multiplicative dynamic noises on it. The newly defined membership function of this fuzzy set and its perturbations is a generalization of the indicator function for the classical Mandelbric set. We present an algorithm for detecting each complex number's fuzzy membership degree. Through the use of the membership degrees of each complex number and experimental mathematics based on the visualizations of a variety of versions by utilizing computer-aided design, we gain a deep foresight for the structure characteristics of the additive and multiplicative perturbed fuzzy Mandelbric sets. Our novel approach allows us to identify the symmetry states of the Mandelbric set and its perturbations by the membership degrees of complex numbers, unlike the existing methods described in the literature. [ABSTRACT FROM AUTHOR]

Let L(s, χ) be a Dirichlet L-function associated with a primitive character χ mod q and a be a non zero complex number. We denote by Δ(s, χ) the function which appears in the functional equation L(s, χ) = Δ(s, χ)L(1 − s, χ) and δa, χ = βa, χ + iγa, χ the solutions of the equation Δ(s, χ) = a which are called a-points of Δ(s, χ). In this note, we will prove that for every complex number a ̸= 0 the mean of the values L(δa, χ, χ) on the sequence of a-points δa, χ of the function Δ(s, χ) exists and equals a + 1. [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]