Complex Numbers Research Articles

On Solving a Quadratic Diophantine Equation Involving Odd Powers of 17

Diophantine Equations named after ancient Greek mathematician Diophantus, plays a vital role not only in number theory but also in several branches of science. In this paper, we will solve one of the quadratic Diophantine equations where the right hand side are odd positive integral powers of 17 and provide its complete solutions. The method adopted to solve the given equation is using the concept of polar form of a particular complex number. This concept can be generalized for solving similar equations.

Kinematic Analysis of the Draglink Mechanism Using the Newton Rhapson Iteration Method

Position, speed, and acceleration analysis are all included in kinematic analysis. Vectors of location, velocity, and acceleration can be expressed analytically in the form of complex numbers using the Newton Rhapson Iteration Method. The location vector of the point to be investigated must be found first, and then the acceleration and velocity vectors must be found by differentiating against time. This study analyzes the relationship between the θ1 value and the θ2, θ3, θ4, θ5, and x2 values in a drag-link mechanism. The Newton-Raphson iteration method is employed for kinematic analysis to solve nonlinear equations. In this research, the analysis results indicate a correlation between the θ1 value and the θ2, θ3, θ4, θ5, and x2 values. As θ1 increases, the θ2, θ3, θ4, and θ5 values also proportionally increase. The x2 value is also influenced by changes in θ1. This study combines the Newton-Raphson iteration method with graphical analysis using CAD software. The conclusion of this research highlights a significant relationship between the θ1 value and the θ2, θ3, θ4, θ5, and x2 values in the drag-link mechanism. The analysis provides valuable insights for modeling, designing, or controlling systems involving these variables.

SPN based RGB image encryption over Gaussian integers

This research paper proposes a novel approach for constructing substitution boxes (S-boxes) over Gaussian integers, which are complex numbers with integer coefficients. The proposed method is based on the properties of the Gaussian integers and their arithmetic operations and ensures the S-boxes exhibit strong cryptographic properties. Furthermore, the paper demonstrates how these S-boxes can be utilized for image encryption through a substitution-permutation network (SPN) over Gaussian integers. The SPN involves iteratively applying the S-box and a permutation layer to the input image, which effectively scrambles the image data. Experimental results show that the proposed method achieves high security and robustness against various attacks while providing efficient encryption and decryption performance. This research thus provides a promising avenue for developing secure image encryption schemes based on Gaussian integers.

Necessary condition for information transfer under simulated parity-time-symmetric evolution

Abstract Parity-time $(\mathcal{PT})$ symmetric quantum theory can broaden the scope of quantum dynamics beyond unitary evolution which may lead to numerous counter-intuitive phenomena, including single-shot discrimination of non-orthogonal states, faster evolution of state than the standard quantum speed limit, and violation of no-signaling principle. On the other hand, $\mathcal{PT}$-symmetric evolution can be realized as reduced dynamics of a subsystem in real experiments within the scope of standard QT. In this experimental setup, if one side of a composite system is evolved according to a $\mathcal{PT}$-symmetric way, a non-trivial information transfer can happen, i.e., the operation performed at one side can be gathered by the other side. By considering an arbitrary shared state between two parties situated in two distant locations and arbitrary measurements, we show that the $\mathcal{PT}$-symmetric evolution of the reduced subsystem at one side is not sufficient for this information transfer to occur. Specifically, we prove that the information transfer can only happen when the density matrix and the corresponding measurements contain complex numbers. Moreover, we connect the entanglement content of the shared state with the efficacy of information transfer. We find evidence that the task becomes more efficient with the increase of dimension. 




Fast Eigenvalue Decomposition of Arrowhead and Diagonal-Plus-Rank-k Matrices of Quaternions

Quaternions are a non-commutative associative number system that extends complex numbers, first described by Hamilton in 1843. We present algorithms for solving the eigenvalue problem for arrowhead and DPRk (diagonal-plus-rank-k) matrices of quaternions. The algorithms use the Rayleigh Quotient Iteration with double shifts (RQIds), Wielandt’s deflation technique and the fact that each eigenvector can be computed in O(n) operations. The algorithms require O(n2) floating-point operations, n being the order of the matrix. The algorithms are backward stable in the standard sense and compare well to the standard QR method in terms of speed and accuracy. The algorithms are elegantly implemented in Julia, using its polymorphism feature.

A new teaching-learning sequence to promote secondary school students’ learning of quantum physics using Dirac notation

This paper describes the design of a new teaching-learning sequence on quantum physics aimed at upper secondary school students. In this teaching-learning sequence, GeoGebra simulations and interactive screen experiments are used to investigate the behaviour of a single photon at beam splitter and single photon interference in a Michelson interferometer. We propose a minimal formalism using Dirac notation, which avoids complex numbers and elaborate vector calculus, to make a quantitative description of the quantum optics experiments accessible to secondary school students. With this new educational pathway, we take into account findings from physics education research, which suggest that the introduction of a mathematical formalism tailored to students’ abilities might help them to overcome naive-realist views of quanta or space-time descriptions of quantum phenomena, while at the same time facilitating a transition to a functional understanding of quantum models.

Revolutionary Strategy for Depicting Knowledge Graphs with Temporal Attributes

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.

Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field

In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure.

Dual rectangular fuzzy complex matrix equations: extended solution, algebraic solution, solution and their calculating1

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

Diagonalizably realizable implies universally realizable

A spectrum $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix $A$. The spectrum $\Lambda$ is diagonalizably realizable ($\mathcal{DR}$) if the realizing matrix $A$ is diagonalizable, and $\Lambda$ is universally realizable ($\mathcal{UR}$) if it is realizable for each possible Jordan canonical form allowed by $\Lambda.$ In 1981, Minc proved that if $\Lambda$ is the spectrum of a diagonalizable positive matrix, then $\Lambda$ is universally realizable. One of the main open questions about the problem of universal realizability of spectra iswhether $\mathcal{DR}$ implies $\mathcal{UR}$. Here, we prove a surprisingly simple result, which shows how diagonalizably realizable implies universally realizable.

A Birth–Death Toy Model for a Measure of Consciousness

The ancient Ouroboros symbolism (one who eats oneself) is here integrated into a simple birth–death clustering process that needed nothing but itself for a transition from indistinguishable phases to a sort of higher-level “conscious” phases. Birth and death coefficients are formulated in terms of odd and even exponentials used to represent a suitable form for conscious states via the internal transfer of information. This toy model may ideally quantify conscious states having inner causes via an Ouroboros index 0 < [Formula: see text] < 1. The value [Formula: see text] = 0 lends to infinite loops and the limiting values [Formula: see text] →1 disclose transformation into awareness. Relationships with physics theories of consciousness and the use of the Ouroboros index to discern about conscious-like states in Artificial Intelligence systems are discussed. Consciousness and freedom of will may go side-by-side in the model when [Formula: see text] is extended to be a complex number modulus.

Bridging the p-Special Functions between the Generalized Hyperbolic and Trigonometric Families

Here, we study the extension of p-trigonometric functions sinp and cosp family in complex domains and p-hyperbolic functions sinhp and the coshp family in hyperbolic complex domains. These functions satisfy analogous relations as their classical counterparts with some unknown properties. We show the relationship of these two classes of special functions viz. p-trigonometric and p-hyperbolic functions with imaginary arguments. We also show many properties and identities related to the analogy between these two groups of functions. Further, we extend the research bridging the concepts of hyperbolic and elliptical complex numbers to show the properties of logarithmic functions with complex arguments.

L-functions and meromorphic functions satisfying the same Riemann-type functional equation and sharing sets

In 2023, Li–Du–Yi [Complex Var. Elliptic Equ., 68(10)2023, 1653–1677] proved that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and E L 1 ( S ) = E L 2 ( S ) for a finite set S = { c 1 , c 2 , c 3 } , where c 1 , c 2 and c 3 are three distinct finite complex values, then L 1 = L 2 . We prove that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and share a set { α 1 , α 2 , … , α t } CM, where t ≥ 1 is a positive integer, and α 1 , α 2 , ··· , α t are t distinct finite complex numbers, then L 1 = L 2 . The result also improves the corresponding results from Khoai–An [The Ramanujan J., 58(1)(2022),253-267]. We also prove that if a non-constant meromorphic function f with finitely many poles in the complex plane shares a set { α 1 , α 2 , … , α t } CM and some finite value c IM with a non-constant L-function L having a positive degree, then f = L. Some examples are provided to show that the main results obtained in this paper, in a sense, are best possible.

On the irreducibility of local representations of the Braid group $$B_n$$

AbstractWe prove that any homogeneous local representation $$\varphi :B_n \rightarrow GL_n(\mathbb {C})$$ φ : B n → G L n ( C ) of type 1 or 2 of dimension $$n\ge 6$$ n ≥ 6 is reducible. Then, we prove that any representation $$\varphi :B_n \rightarrow GL_n(\mathbb {C})$$ φ : B n → G L n ( C ) of type 3 is equivalent to a complex specialization of the standard representation $$\tau _n$$ τ n . Also, we study the irreducibility of all local linear representations of the braid group $$B_3$$ B 3 of degree 3. We prove that any local representation of type 1 of $$B_3$$ B 3 is reducible to a Burau type representation and that any local representation of type 2 of $$B_3$$ B 3 is equivalent to a complex specialization of the standard representation. Moreover, we construct a representation of $$B_3$$ B 3 of degree 6 using the tensor product of local representations of type 2. Let $$u_i$$ u i , $$i=1,2$$ i = 1 , 2 , be non-zero complex numbers on the unit circle. We determine a necessary and sufficient condition that guarantees the irreducibility of the obtained representation.

On Complex Pisot Numbers That Are Roots of Borwein Trinomials

Let n>m be positive integers. Polynomials of the form zn±zm±1 are called Borwein trinomials. Using an old result of Bohl, we derive explicit formulas for the number of roots of a Borwein trinomial inside the unit circle |z|<1. Based on this, we determine all Borwein trinomials that have a complex Pisot number as a root. There are exactly 29 such trinomials.

Navigating Number Systems for Computational Precision: An Analytical Exploration

The main objective of this article is to study all numbers that have occurred so far in the theoretical discussions were real (or complex) numbers in the strict mathematical sense. That is, they were to be conceived as infinite decimal fractions, or as Dedekind cuts. For the purposes of computation such numbers have to be approximated by real numbers of a rather special type, such as terminating decimal fractions, or other rational numbers. The present article is devoted to a study of the number systems that can be used for the purpose of computation.

Machine learning unveils the action of different endogenous phenolic compounds present or formed along the rooting development in olive stem cuttings

Changes in the endogenous phenolic content of plant rooting could be influenced by genotype, hormones, or the interaction of both factors, thus modifying the plant rooting capacity/ability along the rooting development. In this study, the combined effects of IBA treatment and cultivar capacity on the rooting development of olive cuttings were modeled by the temporary assessment of simple and total phenolic content (TPC). Rooting ability and phenolic profile were monitored. Results indicated the lowest cumulative rooting in `Lechin-de-Sevilla´, and the highest rooting speed during the first days of the trial in high-rooting cultivars (‘Verdial-de-Huévar’ and ‘Arbequina’). A total of 14 phenolic compounds comprising phenolic acids and aldehyde (8), flavonoids (5), and a glycosylated seco-iridoid were quantified in rooted olive cuttings. Most of the phenolics presented a significant higher concentration in the ‘Lechin-de-Sevilla’ cultivar, except for the ferulic acid and luteolin, as well the TPC that varied as did the rooting parameter. At the end of the rooting trial (60 days), luteolin in `Verdial de Huévar’ and `Gordal sevillana’ cultivars and ferulic acid in `Lechin de Sevilla’ were the phenolics exclusively found in the highest concentrations. Time did not affect vanillin and oleuropein, although in the case of high-rooting cultivars, values for these compounds were almost constant, while in low- or medium-rooting cultivars, by general, increased or decreased, respectively. The Random Forest algorithm allowed to predict from a complex number of variables, which affected the rooting ability of olive cuttings independently of the cultivar and auxin treatment, finding that luteolin was the best indicator over time.

ANALISIS KEMAMPUAN SISWA SMPN 2 KOTA TAMBOLAKA DALAM MENYELESAIKAN SOAL FISIKA PADA MATERI HUKUM NEWTON DENGAN MENGGUNAKAN TEORI POLYA

The research conducted aims to determine the ability of students in solving physics problems using Polya's theory and the difficulties experienced in solving problems. The method used is quantitative with a descriptive approach. The research subjects were 15 students of class VIII SMP Negeri 2 Tambolaka City. Data collection using test instruments in the form of essay questions and interviews to students. The results showed that, the ability of students in solving physics problems was in the high category 26%, moderate 7%, low 47% and very low 20% while aspects of student ability based on Polya's stages are students who understand the problem in the problem 10 students, plan the problem in the problem 9 students, carry out planning in the problem 8 students, look back at the answers in the problem 6 students. Thus the ability of students in grade VIII SMP Negeri 2 Tambolaka city is in the low category in solving physics problems. At the stage of understanding the problem and planning the problem some students have been able to solve it, but at the stage of carrying out the planning and rechecking the answers there are still many students who experience errors in solving it. The difficulties experienced by students are in determining equations, remembering magnitude and units, and calculating complex numbers.

Application of complex number algebra in plane geometry

Application of complex number algebra in plane geometry

Inverse of \(\alpha\)-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph

Let \(X\) be bipartite mixed graph and for a unit complex number \(\alpha\), \(H_\alpha\) be its \(\alpha\)-hermitian adjacency matrix. If \(X\) has a unique perfect matching, then \(H_\alpha\) has a hermitian inverse \(H_\alpha^{-1}\). In this paper we give a full description of the entries of \(H_\alpha^{-1}\) in terms of the paths between the vertices. Furthermore, for \(\alpha\) equals the primitive third root of unity \(\gamma\) and for a unicyclic bipartite graph \(X\) with unique perfect matching, we characterize when \(H_\gamma^{-1}\) is \(\pm 1\) diagonally similar to \(\gamma\)-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a new construction for the \(\pm 1\) diagonal matrix.