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Journal articles on the topic 'Functions of hypercomplex variable'

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Author: Grafiati

Published: 10 March 2023

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1

Catoni, Francesco, Roberto Cannata, Enrico Nichelatti, and Paolo Zampetti. "Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study." Advances in Applied Clifford Algebras 15, no. 2 (October 2005): 183–212. http://dx.doi.org/10.1007/s00006-005-0011-2.

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2

Gori, Anna, and Fabio Vlacci. "Starlikeness for functions of a hypercomplex variable." Proceedings of the American Mathematical Society 145, no. 2 (August 23, 2016): 791–804. http://dx.doi.org/10.1090/proc/13256.

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3

Dou, Xinyuan, Guangbin Ren, Irene Sabadini, and Ting Yang. "Weak Slice Regular Functions on the n-Dimensional Quadratic Cone of Octonions." Journal of Geometric Analysis 31, no. 11 (May 12, 2021): 11312–37. http://dx.doi.org/10.1007/s12220-021-00682-5.

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AbstractIn the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only the second approach has been considered, i.e. the one based on stem functions. In this paper, we use instead the first definition on the so-called n-dimensional quadratic cone of octonions. These two approaches yield the same class of slice regular functions on axially symmetric slice-domains, however, they are different on other types of domains. We call this new class of functions weak slice regular. We show that there exist weak slice regular functions which are not slice regular in the second approach. Moreover, we study various properties of these functions, including a Taylor expansion.

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4

Ulrych, S. "Conformal relativity with hypercomplex variables." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2168 (August 8, 2014): 20140027. http://dx.doi.org/10.1098/rspa.2014.0027.

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Majorana's arbitrary spin theory is considered in a hyperbolic complex representation. The underlying differential equation is embedded into the gauge field theories of Sachs and Carmeli. In particular, the approach of Sachs can serve as a unified theory of general relativity and electroweak interactions. The method is extended to conformal space with the intention to introduce the strong interaction. It is then possible to use the wave equation, operating on representation functions of the conformal group, to describe the dynamics of matter fields. The resulting gauge groups resemble closely the gauge symmetries of Glashow–Salam–Weinberg and the standard model.

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5

Gryshchuk, S. V. "Hypercomplex monogenic functions of the biharmonic variable in some problems of plane elasticity theory." Reports of the National Academy of Sciences of Ukraine, no. 6 (June 20, 2015): 7–12. http://dx.doi.org/10.15407/dopovidi2015.06.007.

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6

Yefremov, Alexander P. "Physical theories in hypercomplex geometric description." International Journal of Geometric Methods in Modern Physics 11, no. 06 (July 2014): 1450062. http://dx.doi.org/10.1142/s0219887814500625.

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Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.

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7

Park, Su Hyun, and Kwang Ho Shon. "SEDENION FUNCTIONS OF HYPERCOMPLEX VARIABLES IN THE SENSE OF CLIFFORD ANALYSIS." East Asian mathematical journal 29, no. 5 (November 30, 2013): 521–27. http://dx.doi.org/10.7858/eamj.2013.036.

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8

Zakarya, Mohammed, Mahmoud A. Abd-Rabo, and Ghada AlNemer. "Hypercomplex Systems and Non-Gaussian Stochastic Solutions with Some Numerical Simulation of χ-Wick-Type (2 + 1)-D C-KdV Equations." Axioms 11, no. 11 (November 21, 2022): 658. http://dx.doi.org/10.3390/axioms11110658.

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In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the χ-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefficients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively.

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9

Bin Dehaish, Buthinah A. "Exponentially Convex Functions on Hypercomplex Systems." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/290403.

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A hypercomplex system (h.c.s.)L1(Q,m)is, roughly speaking, a space which is defined by a structure measure(c(A,B,r),(A,B∈ℬ(Q))), such space has been studied by Berezanskii and Krein. Our main result is to define the exponentially convex functions (e.c.f.) on (h.c.s.), and we will study their properties. The definition of such functions is a natural generalization of that defined on semigroup.

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10

Kim, Ji Eun, Su Jin Lim, and Kwang Ho Shon. "Regularity of Functions on the Reduced Quaternion Field in Clifford Analysis." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/654798.

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We define a new hypercomplex structure ofℝ3and a regular function with values in that structure. From the properties of regular functions, we research the exponential function on the reduced quaternion field and represent the corresponding Cauchy-Riemann equations in hypercomplex structures ofℝ3.

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11

Abirov, A. K., N. K. Shazhdekeeva, and T. N. Akhmurzina. "DIFFERENTIAL EQUATIONS IN A HYPERCOMPLEX SYSTEM." BULLETIN Series of Physics & Mathematical Sciences 69, no. 1 (March 10, 2020): 7–11. http://dx.doi.org/10.51889/2020-1.1728-7901.01.

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The article considers the problem of solving an inhomogeneous first-order differential equation with a variable with a constant coefficient in a hypercomplex system. The structure of the solution in different cases of the right-hand side of the differential equation is determined. The structure of solving the equation in the case of the appearance of zero divisors is shown. It turns out that when the component of a hypercomplex function is a polynomial of an independent variable, the differential equation turns into an inhomogeneous system of real variables from n equations and its solution is determined by certain methods of the theory of differential equations. Thus, obtaining analytically homogeneous solutions of inhomogeneous differential equations in a hypercomplex system leads to an increase in the efficiency of modeling processes in various fields of science and technology.

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12

Shpakivskyi, Vitalii. "Hypercomplex method for solving linear PDEs." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 32 (December 28, 2018): 160–81. http://dx.doi.org/10.37069/1683-4720-2018-32-16.

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Algebraic-analytic approach to constructing solutions for given partial differential equations were investigated in many papers. In particular, in papers [1-14]. It involves solving two problems. Problem (P 1) is to describe all the sets of vectors \( e_1, e_2, \ldots, e_d \), which satisfy the characteristic equation (or specify the procedure by which they can be found). And the problem (P 2) is to describe all the components of monogenic (i.e., continuous and differentiable in sense Gateaux) functions. In particular, for the equation (4) we must describe the components of the function \( \Phi(\zeta) = \exp \zeta \). Note that in the papers [15, 16] a constructive description of all analytic functions with values is obtained in an arbitrary finite-dimensional commutative associative algebra over the field \(\mathbb{C}\). The Theorem 5.1 of the paper [17] states that it is enough to limit the study of monogenic functions in algebras with the basis of \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \), where \( \eta_1, \eta_2, \ldots, \eta_{n-1} \) are nilpotents. In addition, in [18] it is showed that in each algebra with a basis of the form \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \) the equation (3) has solutions. That is, the problems (P 1) and (P 2) are completely solved on the classes of commutative associative algebras with the basis \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1}\} \). It is worth noting that in a finite-dimensional algebra a decomposition of monogenic functions has a finite number of components, and therefore, it generates a finite number of solutions of a given partial differential equations. In this paper, we propose a procedure for constructing an infinite number of families of solutions of given linear differential equations with partial derivatives with constant coefficients. We use monogenic functions that are defined on some sequences of commutative associative algebras over the field of complex numbers. To achieve this goal, we first study the solutions of the so-called characteristic equation on a given sequence of algebras. Further, we investigate monogenic functions on the sequence of algebras and study their relation with solutions of partial deferential equations. The proposed method is used to construct solutions of some equations of mathematical physics. In particular, for the three-dimensional Laplace equation and the wave equation, for the equation of transverse oscillations of the elastic rod and the conjugate equation, a generalized biharmonic equation and the two-dimensional Helmholtz equation. We note that this method yields all analytic solutions of the two-dimensional Laplace equation and the two-dimensional biharmonic equation (Goursat formula).

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13

Alesker, Semyon, and Misha Verbitsky. "Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry." Journal of Geometric Analysis 16, no. 3 (September 2006): 375–99. http://dx.doi.org/10.1007/bf02922058.

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14

Cheng, Daizhan, Zhengping Ji, Jun-e. Feng, Shihua Fu, and Jianli Zhao. "Perfect hypercomplex algebras: Semi-tensor product approach." Mathematical Modelling and Control 1, no. 4 (2021): 177–87. http://dx.doi.org/10.3934/mmc.2021017.

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<abstract><p>The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of $ 2 $-dimensional PHAs are investigated. Second, all the $ 3 $-dimensional PHAs are obtained and the corresponding zero sets are calculated. Finally, $ 4 $- and higher dimensional PHAs are also considered.</p></abstract>

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15

Blaya, Ricardo Abreu, Juan Bory Reyes, and Dixan Peña Peña. "Riemann boundary value problem for hyperanalytic functions." International Journal of Mathematics and Mathematical Sciences 2005, no. 17 (2005): 2821–40. http://dx.doi.org/10.1155/ijmms.2005.2821.

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We deal with Riemann boundary value problem for hyperanalytic functions. Furthermore, necessary and sufficient conditions for solvability of the problem are derived. At the end the explicit form of general solution for singular integral equations with a hypercomplex Cauchy kernel in the Douglis sense is established.

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16

Zaripov, R. G. "Representations of a group of difference information functions in the extended parastatistics of non-extensive systems." Izvestiya vysshikh uchebnykh zavedenii. Fizika, no. 7 (2022): 9–15. http://dx.doi.org/10.17223/00213411/65/7/9.

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Algebraic and matrix representations of the group of information functions of the difference of nonextensive systems for three types of conformally generalized hypercomplex numbers are given. The corresponding geometries with metric functions are global Finsler geometries.

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17

Liu, Chungen. "TWO-DIMENSIONAL SINGULAR INTEGRAL EQUATIONS OF THE HYPERCOMPLEX FUNCTIONS." Acta Mathematica Scientia 16, no. 1 (January 1996): 44–50. http://dx.doi.org/10.1016/s0252-9602(17)30777-4.

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18

Gürlebeck, Klaus, Wolfgang Sprößig, and Ulrike Wimmer. "Hypercomplex function theory for non-linear stokes problems with variable viscosity." Complex Variables, Theory and Application: An International Journal 22, no. 3-4 (September 1993): 195–202. http://dx.doi.org/10.1080/17476939308814659.

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19

Bill, Jeremiah, Lance Champagne, Bruce Cox, and Trevor Bihl. "Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks." Mathematics 9, no. 9 (April 23, 2021): 938. http://dx.doi.org/10.3390/math9090938.

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In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued equivalents. Finally, this work introduces a novel training algorithm using a meta-heuristic approach that bypasses the need for analytic quaternion loss or activation functions. This algorithm allows for a broader range of activation functions over current quaternion networks and presents a proof-of-concept for future work.

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20

Gudkov, V. V. "A FAMILY OF THE SPIRAL SOLUTIONS OF THE NONLINEAR KLEIN‐GORDON EQUATION." Mathematical Modelling and Analysis 3, no. 1 (December 15, 1998): 98–103. http://dx.doi.org/10.3846/13926292.1998.9637092.

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A family of the functions, intended for a construction the exact travelling wave solutions of nonlinear partial differential equations, is given. Exact solutions of the Klein‐Gordon equation with a special potential are obtained. The behavior of complex and hypercomplex solutions of the second order is presented.

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21

BERKOVICH, Y., and A. SHENKMAN. "HYPERNION NUMBERS AND THEIR USE IN THE ANALYSIS OF NETWORKS DRIVEN BY NONSINUSOIDAL SOURCES." Journal of Circuits, Systems and Computers 13, no. 01 (February 2004): 65–76. http://dx.doi.org/10.1142/s0218126604001192.

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A method of using hypercomplex numbers for the analysis of linear electric circuits with nonsinusoidal voltages and currents has been proposed. Similar to the complex number method for circuits with sinusoidal voltages and currents, the proposed method reduces the analysis of nonsinusoidal circuits to the analysis of direct-current circuits. A special system of hypercomplex numbers, called hypernions, has been created in order to obtain a new efficient method for analyzing nonsinusoidal networks. This system is interesting in that it expands the concept of numbers and attaches meaning to equations and transformations involving discontinuous functions and makes it possible to describe various non-Euclidean spaces. It is shown that the proposed method of analysis of linear nonsinusoidal electric networks makes it possible to carry out numerical calculations for complex circuits by using standard software.

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22

Abreu-Blaya, Ricardo, Juan Bory-Reyes, and Jean-Marie Vilaire. "On the Compactness of the Hypercomplex Commutator in Hölder Continuous Functions Spaces." Complex Analysis and Operator Theory 4, no. 2 (March 7, 2009): 133–43. http://dx.doi.org/10.1007/s11785-009-0013-5.

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23

Pukhtaievych, Roman, and Sergiy Plaksa. "On logarithmic residue of monogenic functions in a three-dimensional commutative algebra with one-dimensional radical." Analele Universitatii "Ovidius" Constanta - Seria Matematica 25, no. 3 (December 20, 2017): 167–82. http://dx.doi.org/10.1515/auom-2017-0043.

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Abstract We consider monogenic functions taking values in a three-dimensional commutative algebra A2 over the field of complex numbers with one- dimensional radical. We calculate the logarithmic residues of monogenic functions acting from a three-dimensional real subspace of A2 into A2. It is shown that the logarithmic residue depends not only on zeros and singular points of a function but also on points at which the function takes values in ideals of A2, and, in general case, is a hypercomplex number.

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24

Cieśliński, Jan L., and Dzianis Zhalukevich. "Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case." Symmetry 12, no. 9 (September 20, 2020): 1550. http://dx.doi.org/10.3390/sym12091550.

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Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of holomorphicity. Until now, only a few particular cases of scator holomorphic functions have been found. In this paper we obtain all solutions of the generalized Cauchy–Riemann system which describes analogues of holomorphic functions in the (1+2)-dimensional scator space.

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25

Faustino, N. "Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle." Applied Mathematics and Computation 247 (November 2014): 607–22. http://dx.doi.org/10.1016/j.amc.2014.09.027.

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26

Aguirre-Mesa, Andres M., Daniel Ramirez-Tamayo, Manuel J. Garcia, Arturo Montoya, and Harry Millwater. "A stiffness derivative local hypercomplex-variable finite element method for computing the energy release rate." Engineering Fracture Mechanics 218 (September 2019): 106581. http://dx.doi.org/10.1016/j.engfracmech.2019.106581.

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27

Ueda, Kazuhiro, and Shin-Ichi Takahashi. "Realization of two transfer functions with complex coefficients by a transfer function with hypercomplex coefficients." Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 79, no. 7 (1996): 65–80. http://dx.doi.org/10.1002/ecjc.4430790707.

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28

Messaoudi, Salim A. "On separable-variable functions." International Journal of Mathematical Education in Science and Technology 33, no. 3 (May 2002): 425–27. http://dx.doi.org/10.1080/00207390110120897.

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29

Odersky, Martin. "Programming with variable functions." ACM SIGPLAN Notices 34, no. 1 (January 1999): 105–16. http://dx.doi.org/10.1145/291251.289433.

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30

Rout, Ranjeet Kumar, Pabitra Pal Choudhury, Sudhakar Sahoo, and Camellia Ray. "Partitioning 1-variable Boolean functions for various classification ofn-variable Boolean functions." International Journal of Computer Mathematics 92, no. 10 (November 14, 2014): 2066–90. http://dx.doi.org/10.1080/00207160.2014.975418.

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31

Ludkowski, Sergey Victor. "Noncommutative Integration of Generalized Diffusion PDE." Symmetry 14, no. 10 (October 1, 2022): 2049. http://dx.doi.org/10.3390/sym14102049.

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The article is devoted to the noncommutative integration of a diffusion partial differential equation (PDE). Its generalizations are also studied. This is motivated by the fact that many existing approaches for solutions of PDEs are based on evolutionary operators obtained as solutions of the corresponding stochastic PDEs. However, this is restricted to PDEs of an order not higher than 2 over the real or complex field. This article is aimed at extending such approaches to PDEs of an order higher than 2. For this purpose, measures and random functions having values in modules over complexified Cayley–Dickson algebras are investigated. Noncommutative integrals of hypercomplex random functions are investigated. By using them, the noncommutative integration of the generalized diffusion PDE is scrutinized. Possibilities are indicated for a subsequent solution of higher-order PDEs using their decompositions and noncommutative integration.

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32

Ramirez-Tamayo, Daniel, Matthew Balcer, Arturo Montoya, and Harry Millwater. "Mixed-mode stress intensity factors computation in functionally graded materials using a hypercomplex-variable finite element formulation." International Journal of Fracture 226, no. 2 (November 2, 2020): 219–32. http://dx.doi.org/10.1007/s10704-020-00489-5.

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33

Ma‘u, Sione. "Plurisubharmonic functions characterized by one-variable extremal functions." Arkiv för Matematik 44, no. 1 (April 2006): 111–31. http://dx.doi.org/10.1007/s11512-005-0011-4.

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34

Buescu, Jorge, and A. C. Paixão. "Complex Variable Positive Definite Functions." Complex Analysis and Operator Theory 8, no. 4 (July 18, 2013): 937–54. http://dx.doi.org/10.1007/s11785-013-0319-1.

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35

Barnsley, M. F., J. Elton, D. Hardin, and P. Massopust. "Hidden Variable Fractal Interpolation Functions." SIAM Journal on Mathematical Analysis 20, no. 5 (September 1989): 1218–42. http://dx.doi.org/10.1137/0520080.

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36

Blackorby, Charles, Walter Bossert, and David Donaldson. "Rationalizable variable-population choice functions." Economic Theory 19, no. 2 (February 2002): 355–78. http://dx.doi.org/10.1007/pl00004217.

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37

Oláh, Judit, József Popp, Szabolcs Duleba, Anna Kiss, and Zoltán Lakner. "Positioning Bio-Based Energy Systems in a Hypercomplex Decision Space—A Case Study." Energies 14, no. 14 (July 20, 2021): 4366. http://dx.doi.org/10.3390/en14144366.

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The optimization of the energy portfolio for a small, open, landlocked economy with rather limited fossil resources is a complex task because it must find a long-range, sustainable balance between the various goals of society under the constant pressure of different interest groups. The opinions of independent, informed experts could be an essential input in the decision-making process. The goal of this research was to determine the relative importance of the values and goals potentially accompanying projects, based on the utilization of bioenergy. The current research is based on a wide-ranging survey of 65 non-partisan experts, applying the Pareto analytic hierarchy process to ensure the unbiased prioritization of project segments. The results of the survey put a spotlight on the importance of the economic role of bioenergy projects. Contrary to previous expectations and considerations, the social functions of these projects have hitherto been given relatively little importance. The results highlight the importance of bioenergy in increasing the income-generating capacity of agricultural producers by optimal utilization of natural resources for agricultural production. This can be achieved without considerable deterioration of the natural environment. Modern agricultural production is characterized by high levels of mechanization and automatization. Under these conditions, the social role of bioenergy projects (job creation) is rather limited.

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38

G. Otiniano, Cira, and Yuri Maluf. "Random variable functions used in hydrology." Selecciones Matemáticas 6, no. 2 (December 30, 2019): 178–88. http://dx.doi.org/10.17268/sel.mat.2019.02.04.

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39

Braden, Bart. "Picturing Functions of a Complex Variable." College Mathematics Journal 16, no. 1 (January 1985): 63. http://dx.doi.org/10.2307/2686634.

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40

VİJAY and A. K. B. CHAND. "Zipper Fractal Functions with Variable Scalings." Advances in the Theory of Nonlinear Analysis and its Application 6, no. 4 (December 30, 2022): 481–501. http://dx.doi.org/10.31197/atnaa.1149689.

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Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an improved version of iterated function system by using a binary parameter called a signature. The signature allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can be non-differentiable on a dense subset of an interval I. In this paper, we construct k-times continuously differentiable ZFIFs with variable scaling functions on I. Some properties like the positivity, monotonicity, and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space of k-times continuously differentiable functions and the space of p-integrable functions for p ∈ [1,∞) are studied. We introduce the zipper versions of full Müntz theorem for continuous function and p-integrable functions on I for p ∈ [1,∞).

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41

Poliakova, Nadiia Salikhzhanovna. "Differentiation of functions of quaternionic variable." Chebyshevskii sbornik 20, no. 2 (2019): 303–15. http://dx.doi.org/10.22405/2226-8383-2019-20-2-303-315.

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42

Teuscher, Dawn, Kylie Palsky, and Charlie Y. Palfreyman. "Inverse Functions: Why Switch the Variable?" Mathematics Teacher 111, no. 5 (March 2018): 374–81. http://dx.doi.org/10.5951/mathteacher.111.5.0374.

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43

Pu, Deqian. "Monogenic functions of a complete variable." Complex Variables and Elliptic Equations 51, no. 8-11 (August 2006): 999–1019. http://dx.doi.org/10.1080/17476930600709163.

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44

Su, Yong, Aifang Xie, and Hua-Wen Liu. "Generating Implications from One-Variable Functions." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 23, no. 06 (December 2015): 927–47. http://dx.doi.org/10.1142/s0218488515500427.

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In this paper, we firstly introduce two new classes of fuzzy implications generated from one-variable functions, called [Formula: see text]- and [Formula: see text]-implications, respectively. Then we give a series of necessary and sufficient conditions that these implications satisfy: left neutrality property, identity principle, ordering principle, law of contraposition, modus ponens and modus tollens, respectively. We also discuss the relations between [Formula: see text]- implication ([Formula: see text]-implications, respectively) and other known classes of fuzzy implications.

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45

Calabrese, Philip G. "Operating on Functions with Variable Domains." Journal of Philosophical Logic 32, no. 1 (February 2003): 1–18. http://dx.doi.org/10.1023/a:1022822822130.

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46

Braden, Bart. "Picturing Functions of a Complex Variable." College Mathematics Journal 16, no. 1 (January 1985): 63–72. http://dx.doi.org/10.1080/07468342.1985.11972856.

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47

Moya, A. M., V. V. Fernández, and W. A. Rodrigues. "Multivector functions of a real variable." Advances in Applied Clifford Algebras 11, S3 (October 2001): 69–77. http://dx.doi.org/10.1007/bf03219148.

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Moya, A. M., V. V. Fernández, and W. A. Rodrigues. "Multivector functions of a multivector variable." Advances in Applied Clifford Algebras 11, S3 (October 2001): 79–91. http://dx.doi.org/10.1007/bf03219149.

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Golias, Mihalis M., and Hercules E. Haralambides. "Berth scheduling with variable cost functions." Maritime Economics & Logistics 13, no. 2 (May 12, 2011): 174–89. http://dx.doi.org/10.1057/mel.2011.4.

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Lüdecke, Daniel. "sjmisc: Data and Variable Transformation Functions." Journal of Open Source Software 3, no. 26 (June 20, 2018): 754. http://dx.doi.org/10.21105/joss.00754.

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