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Journal articles on the topic 'Dual-complex numbers'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

Prasad, B. "Dual complex Fibonacci p-numbers." Chaos, Solitons & Fractals 145 (April 2021): 109922. http://dx.doi.org/10.1016/j.chaos.2020.109922.

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2

Aydın, Fügen Torunbalcı. "Dual-complex k-Fibonacci numbers." Chaos, Solitons & Fractals 115 (October 2018): 1–6. http://dx.doi.org/10.1016/j.chaos.2018.08.015.

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3

Fjelstad, Paul, and Sorin G. Gal. "n-Dimensional dual complex numbers." Advances in Applied Clifford Algebras 8, no. 2 (1998): 309–22. http://dx.doi.org/10.1007/bf03043102.

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4

KÖME, Sure, Cahit KÖME, and Yasin YAZLİK. "Dual-complex generalized k-Horadam numbers." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 70, no. 1 (2021): 117–29. http://dx.doi.org/10.31801/cfsuasmas.780861.

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5

Shuster, John A., and Jens Köplinger. "Elliptic complex numbers with dual multiplication." Applied Mathematics and Computation 216, no. 12 (2010): 3497–514. http://dx.doi.org/10.1016/j.amc.2010.04.069.

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6

Bród, Dorota, Anetta Szynal-Liana, and Iwona Włoch. "On a new one-parameter generalization of dual-complex Jacobsthal numbers." Acta Universitatis Sapientiae, Mathematica 13, no. 1 (2021): 127–44. http://dx.doi.org/10.2478/ausm-2021-0007.

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Abstract In this paper we define dual-complex numbers with generalized Jacobsthal coefficients. We introduce one-parameter generalization of dual-complex Jacobsthal numbers - dual-complex r-Jacobsthal numbers. We investigate some algebraic properties of introduced numbers, among others Binet type formula, Catalan, Cassini, d’Ocagne and Honsberger type identities. Moreover, we present the generating function, summation formula and matrix generator for these numbers. The results are generalization of the properties for the dual-complex Jacobsthal numbers.

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7

Güngör, Mehmet Ali, and Ayşe Zeynep Azak. "Investigation of Dual-Complex Fibonacci, Dual-Complex Lucas Numbers and Their Properties." Advances in Applied Clifford Algebras 27, no. 4 (2017): 3083–96. http://dx.doi.org/10.1007/s00006-017-0813-z.

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8

Bród, Dorota, Anetta Szynal-Liana, and Iwona Włoch. "Two generalizations of dual-complex Lucas-balancing numbers." Acta Universitatis Sapientiae, Mathematica 14, no. 2 (2022): 220–30. http://dx.doi.org/10.2478/ausm-2022-0014.

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Abstract In this paper, we study two generalizations of dual-complex Lucas-balancing numbers: dual-complex k-Lucas balancing numbers and dual-complex k-Lucas-balancing numbers. We give some of their properties, among others the Binet formula, Catalan, Cassini, d’Ocagne identities.

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9

Ait-amrane, Nacima Rosa, and Elif Tan. "On unrestricted dual-generalized complex Horadam numbers." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 73, no. 2 (2024): 517–28. http://dx.doi.org/10.31801/cfsuasmas.1406328.

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This research introduces a novel category of dual-generalized complex numbers, with components represented by unrestricted Horadam numbers. We present various recurrence relations, summation formulas, the Binet formula, and the generating function associated with these numbers. Additionally, a comprehensive bilinear index-reduction formula is derived, which encompasses Vajda’s, Catalan’s, Cassini’s, D’Ocagne’s, and Halton’s identities as specific cases.

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Şentürk, Gülsüm Yeliz, and Nurten Gürses. "Dual quaternion theory over HGC numbers." Journal of Discrete Mathematical Sciences & Cryptography 27, no. 1 (2024): 117–42. http://dx.doi.org/10.47974/jdmsc-1611.

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Knowing the applications of quaternions in various fields, such as robotics, navigation, computer visualization and animation, in this study, we give the theory of dual quaternions considering Hyperbolic-Generalized Complex (HGC) numbers as coefficients via generalized complex and hyperbolic numbers. We account for how HGC number theory can extend dual quaternions to HGC dual quaternions. Some related theoretical results with HGC Fibonacci/Lucas numbers are established, including their dual quaternions. Given HGC Fibonacci/Lucas numbers, their special matrix correspondences have been identifie

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11

Cheng, H. H., and S. Thompson. "Singularity Analysis of Spatial Mechanisms Using Dual Polynomials and Complex Dual Numbers." Journal of Mechanical Design 121, no. 2 (1999): 200–205. http://dx.doi.org/10.1115/1.2829444.

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Complex dual numbers wˇ = x + iy + εu + iεv which form a commutative ring are introduced in this paper to solve dual polynomial equations numerically. It is shown that the singularities of a dual input-output displacement polynomial equation of a mechanism correspond to its singularity positions. This new method of identifying singularities provides clear physical insight into the geometry of the singular configurations of a mechanism, which is illustrated through analysis of special configurations of the RCCC spatial mechanism. Numerical solutions for dual polynomial equations and complex dua

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12

GÖKBAŞ, Hasan. "Dual-Gaussian Pell and Pell-Lucas numbers." Cumhuriyet Science Journal 43, no. 4 (2022): 665–71. http://dx.doi.org/10.17776/csj.1067983.

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In this study, we define a new type of Pell and Pell-Lucas numbers which are called dual-Gaussian Pell and dual-Gaussian Pell-Lucas numbers. We also give the relationship between negadual-Gaussian Pell and Pell-Lucas numbers and dual-complex Pell and Pell-Lucas numbers. Also, some sum ve product properties of Pell and Pell-Lucas numbers are given. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sum formulas for these new type numbers. Some algebraic proporties of dual-Gaussian Pell and Pell-Lucas numbers are inv

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13

Dagdeviren, Ali. "A generalization of complex, dual and hyperbolic quaternions: Hybrid quaternions." Filomat 37, no. 25 (2023): 8441–54. http://dx.doi.org/10.2298/fil2325441d.

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Hybrid numbers are a new non-commutative number system which is a generalization of the complex (i2 = ?1), dual (?2 = 0), and hyperbolic numbers (h2 = 1). In this article, firstly we define a new quaternion system called hybrid quaternions by taking the coefficients of real quaternions as hybrid numbers. This new quaternion system is a combination of complex quaternions (biquaternions), hyperbolic (perplex) quaternions, and dual quaternions, and it can be viewed as a generalization of these quaternion systems. Then, we present the basic properties of hybrid quaternions including fundamental op

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14

Aymaz, İsmail, and Mustafa Emre Kansu. "Dual-complex quaternion representation of gravitoelectromagnetism." International Journal of Geometric Methods in Modern Physics 18, no. 11 (2021): 2150178. http://dx.doi.org/10.1142/s0219887821501784.

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In this paper, we propose the generalized description of electromagnetism and linear gravity based on the combined dual numbers and complex quaternion algebra. In this approach, the electromagnetic and gravitational fields can be considered as the components of one combined dual-complex quaternionic field. It is shown that all relations between potentials, field strengths and sources can be formulated in the form of compact quaternionic differential equations. The alternative reformulation of equations of gravitoelectromagnetism based on formalism of [Formula: see text] matrices is also discus

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15

GÜRSES, NURTEN, and GÜLSÜM YELİZ ŞENTÜRK. "MATRIX THEORY OVER DGC NUMBERS." Journal of Science and Arts 23, no. 1 (2023): 209–28. http://dx.doi.org/10.46939/j.sci.arts-23.1-a17.

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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

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16

Szynal-Liana, Anetta, and Iwona Włoch. "On Special Spacelike Hybrid Numbers." Mathematics 8, no. 10 (2020): 1671. http://dx.doi.org/10.3390/math8101671.

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Hybrid numbers are generalizations of complex, hyperbolic and dual numbers. A hyperbolic complex structure is frequently used in both pure mathematics and numerous areas of physics. In this paper we introduce a special kind of spacelike hybrid number, namely the F(p,n)-Fibonacci hybrid numbers and we give some of their properties.

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17

Atanassov, Krassimir T. "On tertions and dual numbers." Notes on Number Theory and Discrete Mathematics 30, no. 2 (2024): 443–52. http://dx.doi.org/10.7546/nntdm.2024.30.2.443-452.

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In a previous author’s paper [1], the mathematical object called “tertion” was discussed. Some operations over tertions were introduced and their properties were studied. There, it was showed that the complex numbers and quaternions can be represented by tertions. Here, we show that the dual numbers also are representable by tertions. The concept of a “0-quaternion” is introduced and its representation by tertions is given. Ideas for future research are described.

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18

BISWAS, DEBAPRIYA, and SANDIPAN DUTTA. "Geometric Invariants Under the Möbius Action of the Group SL(2;R)." Kragujevac Journal of Mathematics 45, no. 6 (2021): 925–41. http://dx.doi.org/10.46793/kgjmat2106.925b.

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In this paper we have introduced new invariant geometric objects in the homogeneous spaces of complex, dual and double numbers for the principal group SL(2; ℝ), in the Klein’s Erlangen Program. We have considered the action as the Möbius action and have taken the spaces as the spaces of complex, dual and double numbers. Some new decompositions of SL(2; ℝ) have been used.

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19

Szynal-Liana, Anetta, and Iwona Włoch. "On generalized Mersenne hybrid numbers." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 74, no. 1 (2020): 77. http://dx.doi.org/10.17951/a.2020.74.1.77-84.

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The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we consider a special kind of hybrid numbers, namely the Mersenne hybrid numbers and we give some of their properties.

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20

Tan, Elif, and Umut Öcal. "On a generalization of dual-generalized complex Fibonacci quaternions." Notes on Number Theory and Discrete Mathematics 29, no. 4 (2023): 635–46. http://dx.doi.org/10.7546/nntdm.2023.29.4.635-646.

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21

Alhasan, Yaser Ahmad, Lee Xu, Raja Abdullah Abdulfatah, and Abuobida M. Ahmed Alfahal. "The Geometrical Characterization for The Solutions of a Vectorial Equation By Using Weak Fuzzy Complex Numbers and Other Generalizations Of Real Numbers." International Journal of Neutrosophic Science 21, no. 4 (2023): 155–59. http://dx.doi.org/10.54216/ijns.210415.

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The main goal of this paper is to study the geometrical characterization of the solutions for a vectorial equation defined in the twothree dimensional Euclidean spaces. The geometrical characterization of the solutions for the desired vectorial equation is obtained for many different values of t based on the circles and spheres in some generalizations of the real field, especially dual numbers, weak fuzzy complex numbers split-complex numbers, and complex numbers.

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22

Szynal-Liana, Anetta, and Iwona Włoch. "On Jacobsthal and Jacobsthal-Lucas Hybrid Numbers." Annales Mathematicae Silesianae 33, no. 1 (2019): 276–83. http://dx.doi.org/10.2478/amsil-2018-0009.

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AbstractThe hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we consider special kinds of hybrid numbers, namely the Jacobsthal and the Jacobsthal-Lucas hybrid numbers and we give some their properties.

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23

Szynal-Liana, Anetta, and Iwona Włoch. "Generalized Fibonacci-Pell hybrinomials." Online Journal of Analytic Combinatorics, no. 15 (December 31, 2020): 1–12. https://doi.org/10.61091/ojac-1514.

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Hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we introduce and study Fibonacci-Pell hybrinomials, i.e. polynomials, which are a generalization of hybrid numbers of the Fibonacci type.

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24

Rubajczyk, Mariola, and Anetta Szynal-Liana. "Cobalancing hybrid numbers." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 78, no. 1 (2024): 87–95. https://doi.org/10.17951/a.2024.78.1.87-95.

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Hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper, we define and study hybrid numbers with cobalancing and Lucas-cobalancing coefficients. We derive some fundamental identities for these numbers, among others the Binet formulas and the general bilinear index-reduction formulas which imply the Catalan, Cassini, Vajda, d’Ocagne and Halton identities. Moreover, the generating functions for cobalancing and Lucas-cobalancing hybrid numbers are presented.

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25

Torres del Castillo, G. F. "Some applications in classical mechanics of the double and the dual numbers." Revista Mexicana de Física E 65, no. 2 Jul-Dec (2019): 152. http://dx.doi.org/10.31349/revmexfise.65.152.

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26

Dündar, Furkan Semih. "Parabolic Numbers: A New Perspective." Journal of New Theory, no. 49 (December 31, 2024): 7–15. https://doi.org/10.53570/jnt.1526699.

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Thus far, many studies have been conducted on $p$-complex numbers. Depending on the sign of $p$, there are three cases: hyperbolic, dual, and elliptic. In the literature, dual numbers are called parabolic numbers, but they do not parameterize parabolas. Therefore, a number system that parameterizes parabolas is worth studying. This paper defines $p$ as a function of the coordinate $y$ and obtains a number system named parabolic numbers whose circles are parabolas. These parabolic numbers complete the set of number systems where circles are conic sections. Finally, this paper discusses the pros

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27

Sultanov, A. Ya, G. A. Sultanova, and O. A. Monakhova. "On the group of automorphisms of the algebra of plural numbers." Differential Geometry of Manifolds of Figures, no. 54(2) (2023): 63–70. http://dx.doi.org/10.5922/0321-4796-2023-54-2-6.

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The algebra of dual numbers was first introduced by V. K. Clifford in 1873. The algebras of plural and dual numbers are analogous to the algebra of complex numbers. Dual numbers form an algebra, but not a field, because only dual numbers with a real part not equal to zero have an inverse element. In this work, automorphisms of algebras of plural numbers, which are a generalization of the algebra of dual numbers, are studied. Algebras of plural numbers were in the center of attention of the professor of Kazan University A. P. Shirokov. Studying the geometry of higher-order tangent bundles, he e

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28

Bród, Dorota, and Anetta Szynal-Liana. "A new hybrid generalization of Fibonacci and Fibonacci-Narayana polynomials." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 77, no. 1 (2023): 1–12. http://dx.doi.org/10.17951/a.2023.77.1.1-12.

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The hybrid numbers are generalization of complex, hyperbolic and dual numbers. The hybrinomials are polynomials which generalize hybrid numbers. In this paper, we introduce and study the distance Fibonacci hybrinomials, i.e. hybrinomials with coefficients being distance Fibonacci polynomials.

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29

Kürüz, Ferhat, Ali Dağdeviren, and Paula Catarino. "On Leonardo Pisano Hybrinomials." Mathematics 9, no. 22 (2021): 2923. http://dx.doi.org/10.3390/math9222923.

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A generalization of complex, dual, and hyperbolic numbers has recently been defined as hybrid numbers. In this study, using the Leonardo Pisano numbers and hybrid numbers we investigate Leonardo Pisano polynomials and hybrinomials. Furthermore, we also describe the basic algebraic properties and some identities of the Leonardo Pisano polynomials and hybrinomials.

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Pavlov, Stanislav, Dmitry Kozlov, Mikhail Bakulin, Aleksandr Zuev, Andrey Latyshev, and Alexander Beliaev. "Generalization of Neural Networks on Second-Order Hypercomplex Numbers." Mathematics 11, no. 18 (2023): 3973. http://dx.doi.org/10.3390/math11183973.

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The vast majority of existing neural networks operate by rules set within the algebra of real numbers. However, as theoretical understanding of the fundamentals of neural networks and their practical applications grow stronger, new problems arise, which require going beyond such algebra. Various tasks come to light when the original data naturally have complex-valued formats. This situation is encouraging researchers to explore whether neural networks based on complex numbers can provide benefits over the ones limited to real numbers. Multiple recent works have been dedicated to developing the

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31

Cui, Chunfeng, Yong Lu, Liqun Qi, and Ligong Wang. "Spectral Properties of Dual Unit Gain Graphs." Symmetry 16, no. 9 (2024): 1142. http://dx.doi.org/10.3390/sym16091142.

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In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gai

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32

Doğan Yazıcı, Bahar, and Murat Tosun. "Multicomponent hybrid numbers: On algebraic properties and matrix representations of hybrid-hyperbolic numbers." Notes on Number Theory and Discrete Mathematics 28, no. 1 (2022): 26–40. http://dx.doi.org/10.7546/nntdm.2022.28.1.26-40.

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In this study, the hybrid-hyperbolic numbers are introduced. This number system is a more general form of the hybrid number system, which is an interesting number system, as well as a number system that includes multicomponent number systems (i.e., complex-hyperbolic, dual-hyperbolic and bihyperbolic numbers). In this paper, we give algebraic properties of hybrid-hyperbolic numbers. In addition, 2 × 2 and 4 × 4 hyperbolic matrix representations of hybrid-hyperbolic numbers are given and some properties of them are examined.

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33

Wang, Qing-Wen, Lv-Ming Xie, and Zi-Han Gao. "A Survey on Solving the Matrix Equation AXB=C with Applications." Mathematics 13, no. 3 (2025): 450. https://doi.org/10.3390/math13030450.

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This survey provides a comprehensive overview of the solutions to the matrix equation AXB=C over real numbers, complex numbers, quaternions, dual quaternions, dual split quaternions, and dual generalized commutative quaternions, including various special solutions. Additionally, we summarize the numerical algorithms for these special solutions. This matrix equation plays an important role in solving linear systems and control theory. We specifically explore the application of this matrix equation in color image processing, highlighting its unique value in this field. Taking the dual quaternion

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34

Li, Zehui, Yitong Fan, and Qianhao Sun. "Taylor's formula on the separating complex plane and its applications." Highlights in Science, Engineering and Technology 115 (October 28, 2024): 81–88. http://dx.doi.org/10.54097/wbq4x885.

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This article begins by exploring split-complex numbers, also known as dual or double numbers, a mathematical concept that extends the real number system by creating a commutative ring with a zero divisor. Furthermore, the paper extends the Taylor series theory from the real analysis domain to the domain of split-complex numbers, thereby establishing the corresponding Taylor formula on the split-complex plane. The introduction of split-complex numbers opens up new perspectives and provides new tools for the modeling and analysis of quantum systems. In the field of electromagnetism, the applicat

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35

Akbıyık, Mücahit, Seda Yamaç Akbıyık, Emel Karaca, and Fatih Yılmaz. "De Moivre’s and Euler Formulas for Matrices of Hybrid Numbers." Axioms 10, no. 3 (2021): 213. http://dx.doi.org/10.3390/axioms10030213.

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It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the 4×4 matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas.

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36

Xie, Yaen, Xianliang Zhang, Xiangshuai Song, Xiaobin Lian, and Jian Zhang. "Integrated Dynamics of Space Rigid-Flex Combination System with Time-Varying Configuration." International Journal of Aerospace Engineering 2023 (September 13, 2023): 1–16. http://dx.doi.org/10.1155/2023/9980780.

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Dual numbers were applied to the dynamics of a rigid-flexible combination system (RFCS) with time-varying configuration in this paper. The six-dimensional spinor form of the motion of flexible modules, including the dual vector, dual momentum, dual inertia operator, dual coupling coefficient operator, and dual-modal coordinates, was derived using the dual numbers that could represent spiral motion in a compact form. On this basis, the integrated dynamic model of a rigid-flexible combination system with a time-varying configuration was proposed. And then, the relative dynamics equations between

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37

Torres del Castillo, G. F. "Applications of the double and the dual numbers. The Bianchi models." Revista Mexicana de Física E 17, no. 2 Jul-Dec (2020): 146. http://dx.doi.org/10.31349/revmexfise.17.146.

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We show that by using complex, double, and dual numbers one can find the invariant one-forms employed in the metrics of the Bianchi cosmological models. The result is equivalent to find, locally, all the Lie groups of dimension three.

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38

Mangueira, Milena Carolina dos Santos, Renata Passos Machado Vieira, Francisco Régis Vieira Alves, and Paula Maria Machado Cruz Catarino. "The Hybrid Numbers of Padovan and Some Identities." Annales Mathematicae Silesianae 34, no. 2 (2020): 256–67. http://dx.doi.org/10.2478/amsil-2020-0019.

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AbstractIn this article, we will define Padovan’s hybrid numbers, based on the new noncommutative numbering system studied by Özdemir ([7]). Such a system that is a set involving complex, hyperbolic and dual numbers. In addition, Padovan’s hybrid numbers are created by combining this set, satisfying the relation ih = −hi = ɛ + i. Given this, some properties and identities are shown for these numbers, such as Binet’s formula, generating matrix, characteristic equation, norm, and generating function. In addition, these numbers are extended to the integer field and some identities are made.

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Alp, Yasemin. "Hybrid hyper-Fibonacci and hyper-Lucas numbers." Notes on Number Theory and Discrete Mathematics 29, no. 1 (2023): 154–70. http://dx.doi.org/10.7546/nntdm.2023.29.1.154-170.

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Different number systems have been studied lately. Recently, many researchers have considered the hybrid numbers which are generalization of the complex, hyperbolic and dual number systems. In this paper, we define the hybrid hyper-Fibonacci and hyper-Lucas numbers. Furthermore, we obtain some algebraic properties of these numbers such as the recurrence relations, the generating functions, the Binet’s formulas, the summation formulas, the Catalan’s identity, the Cassini’s identity and the d’Ocagne’s identity.

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KÖME, Sure, and Zeynep KUMTAS. "Generalized bivariate conditional Fibonacci and Lucas hybrinomials." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 73, no. 1 (2023): 37–63. http://dx.doi.org/10.31801/cfsuasmas.1249576.

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The Hybrid numbers are generalizations of complex, hyperbolic and dual numbers. In recent years, studies related with hybrid numbers have been increased significantly. In this paper, we introduce the generalized bivariate conditional Fibonacci and Lucas hybrinomials. Also, we present the Binet formula, generating functions, some significant identities, Catalan’s identities and Cassini’s identities of the generalized bivariate conditional Fibonacci and Lucas hybrinomials. Finally, we give more general results compared to the previous works.

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41

Petukhov, V. G., and Sung Wook Yoon. "Optimization of Perturbed Spacecraft Trajectories Using Complex Dual Numbers. Part 2: Numerical Results." Cosmic Research 59, no. 6 (2021): 517–28. http://dx.doi.org/10.1134/s0010952521060083.

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42

Kyrov, V. A., and G. G. Mikhailichenko. "Solving Three Systems of Functional Equations Associated with Complex, Double, and Dual Numbers." Russian Mathematics 67, no. 7 (2023): 34–42. http://dx.doi.org/10.3103/s1066369x23070058.

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43

PETROUDI, SEYYED HOSSEIN JAFARI, and ARZU ÖZKOÇ ÖZTÜRK. "MORE SUMMATION FORMULAS ON HYBRINOMIAL SEQUENCE OF VAN DER LAAN." Journal of Science and Arts 24, no. 4 (2024): 797–814. https://doi.org/10.46939/j.sci.arts-24.4-a03.

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Özdemir [1] introduced Hybrid numbers as a generalization of complex, hyperbolic and dual numbers. Hybrinomial sequence is the combination of hybrid numbers and polynomial sequence. In this paper we consider special kind of hybrinomial sequence, namely the Van Der Laan hybrinomial sequence. Binet-Like Formula, generating function and exponential generating function of this sequence are shown in this paper. Properties and some summation identities for Van Der Laan polynomial sequence and hybrinomial sequence are represented in this paper. In addition, some interesting summation identities of Va

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44

Herzog, Jürgen, and Takayuki Hibi. "Componentwise linear ideals." Nagoya Mathematical Journal 153 (1999): 141–53. http://dx.doi.org/10.1017/s0027763000006930.

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AbstractA componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal IΔ arising from a simplicial complex Δ is compo

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Filali, M. "Linear equations in B(ℤ)*". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 123, № 6 (1993): 1001–9. http://dx.doi.org/10.1017/s030821050002967x.

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SynopsisLet B(ℤ)* be the Banach dual of the space of all bounded complex-valued functions on ℤ. For each n ε ℤ, let Ln be the translation operator on B(ℤ) and Tn be its adjoint operator on B(ℤ)*. This paper concerns itself with equations of the formwhere (an)nεℤ is a sequence of complex numbers.

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BASEILHAC, PASCAL. "HIDDEN SYMMETRIES IN QUANTUM FIELD THEORIES FROM EXTENDED COMPLEX NUMBERS." International Journal of Modern Physics A 14, no. 26 (1999): 4201–35. http://dx.doi.org/10.1142/s0217751x99001986.

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The two-dimensional space–time sine–Gordon field theory is extended algebraically within the n-dimensional space of extended complex numbers. This field theory is constructed in terms of an adapted extension of standard vertex operators. A whole set of nonlocal conserved charges is constructed and studied in this framework. Thereby, an algebraic nonperturbative description is possible for this n-1 parameters family of quantum field theories. Known results are obtained for specific values of the parameters, especially in relation to affine Toda field theories. Different (dual)-models can then b

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BASEILHAC, P., and D. REYNAUD. "DUAL GENERALIZATIONS OF SINE–GORDON FIELD THEORY AND INTEGRABILITY SUBMANIFOLDS IN PARAMETER SPACE." International Journal of Modern Physics A 15, no. 21 (2000): 3315–40. http://dx.doi.org/10.1142/s0217751x00001063.

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The dual relationship between two n-1 parameter families of quantum field theories based on extended complex numbers is investigated in two dimensions. The nonlocal conserved charges approach is used. The lowest rank affine Toda field theories are generated and identified as integrability submanifolds in parameter space. A truncation of the model leads to a conformal field theory in extended complex space. Depending on the projection over the usual complex space chosen, a parametrized central charge is calculated.

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Babadag, Faik, Mirwais Mansoor Kakar, and Ali Atasoy. "A New Approach to Dual Jacobsthal Split Quaternions with Different Polar Representation." Journal of Advances in Mathematics and Computer Science 39, no. 2 (2024): 52–62. http://dx.doi.org/10.9734/jamcs/2024/v39i21867.

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In this paper, we introduce split quaternions with components including dual Jacobsthal and dual Jacobsthal- Lucas number sequences. By using Binet's formulas of these type split quaternions we give an explicit form of classic polar representations of them, after that we demonstrate a new polar representation by using Cayley-Dikson's notation of split quaternions which is based on two complex numbers. Some fundamental properties and identities for these type of split quaternions are studied. In further the current paper, it would be valuable to replicate similar approaches polar representation

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Minati, Gianfranco. "The Vacuum as Imaginary Space. The Unreasonable Effectiveness of Complex Numbers." PROOF 2 (December 31, 2022): 187–201. http://dx.doi.org/10.37394/232020.2022.2.25.

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The background to the article is the classic and quantum understandings of the vacuum and the use of imaginary numbers in quantum models. The purpose of the article is to outline the possible understanding of the vacuum as imaginary space always coupled with the real space in the complex space of complex numbers. This understanding relates to the duality real-potential, collapsed–collapsible, and superimpositions of waves-phenomena as in quantum mechanics. The incomputability of the imaginary parts may represent the physical meaning of the permanent potential pending nature of the vacuum. The

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Petukhov, V. G., and Sung Wook Yoon. "Optimization of Perturbed Spacecraft Trajectories Using Complex Dual Numbers. Part 1: Theory and Method." Cosmic Research 59, no. 5 (2021): 401–13. http://dx.doi.org/10.1134/s0010952521050099.

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