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Journal articles on the topic 'Sextic Equations'

Author: Grafiati

Published: 29 July 2024

Last updated: 30 July 2024

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1

Gaál, István, Borka Jadrijević, and László Remete. "Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields." International Journal of Number Theory 15, no. 01 (February 2019): 11–27. http://dx.doi.org/10.1142/s1793042118501695.

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The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see G. Lettl, A. Pethő and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999) 1871–1894, On the arithmetic of simplest sextic fields and related Thue equations, in Number Theory: Diophantine, Computational and Algebraic Aspects, eds. K. Győry et al. (de Gruyter, Berlin, 1998), pp. 331–348. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper, we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.

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2

Hurley, A. C., and A. K. Head. "Explicit Galois resolvents for sextic equations." International Journal of Quantum Chemistry 31, no. 3 (March 1987): 345–59. http://dx.doi.org/10.1002/qua.560310306.

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3

Lee, Yang-Hi. "On the Hyers-Ulam-Rassias Stability of a General Quintic Functional Equation and a General Sextic Functional Equation." Mathematics 7, no. 6 (June 4, 2019): 510. http://dx.doi.org/10.3390/math7060510.

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The general quintic functional equation and the general sextic functional equation are generalizations of many functional equations such as the additive function equation and the quadratic function equation. In this paper, we investigate Hyers–Ulam–Rassias stability of the general quintic functional equation and the general sextic functional equation.

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4

Gaál, István, and László Remete. "Power integral bases in a family of sextic fields with quadratic subfields." Tatra Mountains Mathematical Publications 64, no. 1 (September 1, 2015): 59–66. http://dx.doi.org/10.1515/tmmp-2015-0041.

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Abstract Let M = Q(i √d) be any imaginary quadratic field with a positive square-free d. Consider the polynomial f(x) = x3 − ax2 − (a + 3)x − 1 with a parameter a ∈ ℤ. Let K = M(α), where α is a root of f. This is an infinite parametric family of sextic fields depending on two parameters, a and d. Applying relative Thue’s equations we determine the relative power integral bases of these sextic fields over their quadratic subfields. Using these results we also determine generators of (absolute) power integral bases of the sextic fields.

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5

Togbé, Alain. "A parametric family of sextic Thue equations." Acta Arithmetica 125, no. 4 (2006): 347–61. http://dx.doi.org/10.4064/aa125-4-4.

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6

Hagedorn, Thomas R. "General Formulas for Solving Solvable Sextic Equations." Journal of Algebra 233, no. 2 (November 2000): 704–57. http://dx.doi.org/10.1006/jabr.2000.8428.

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7

Hu, Shuangnian, and Rongquan Feng. "On the number of solutions of two-variable diagonal sextic equations over finite fields." AIMS Mathematics 7, no. 6 (2022): 10554–63. http://dx.doi.org/10.3934/math.2022588.

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<abstract><p>Let $ p $ be a prime, $ k $ a positive integer, $ q = p^k $, and $ \mathbb{F}_q $ be the finite field with $ q $ elements. In this paper, by using the Jacobi sums, we give an explicit formula for the number of solutions of the two-variable diagonal sextic equations $ x_1^6+x_2^6 = c $ over $ \mathbb{F}_q $, with $ c\in\mathbb{F}_q^* $ and $ p\equiv1({\rm{mod}} \ 6) $. Furthermore, by using the reduction formula for Jacobi sums, the number of solutions of the diagonal sextic equations $ x_1^6+x_2^6+\cdots+x_n^6 = c $ of $ n\geq3 $ variables with $ c\in\mathbb{F}_q^* $ and $ p\equiv1({\rm{mod}} \ 6) $, can also be deduced.</p></abstract>

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8

Bilu, Yuri, István Gaál, and Kálmán Győry. "Index form equations in sextic fields: a hard computation." Acta Arithmetica 115, no. 1 (2004): 85–96. http://dx.doi.org/10.4064/aa115-1-7.

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9

Hoshi, Akinari. "On the simplest sextic fields and related Thue equations." Functiones et Approximatio Commentarii Mathematici 47, no. 1 (September 2012): 35–49. http://dx.doi.org/10.7169/facm/2012.47.1.3.

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10

Kulkarni, Raghavendra G. "Insert a Root to Extract a Root of Quintic Quickly." Annales Mathematicae Silesianae 33, no. 1 (September 1, 2019): 153–58. http://dx.doi.org/10.2478/amsil-2018-0013.

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AbstractThe usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up, leading to the solutions. In this paper, a relation among the coefficients of a principal quintic equation is established by effecting a change of variable and inserting a root to the quintic equation, and then equating odd-powers of the resulting sextic equation to zero. This leads to an even-powered sextic equation, or equivalently a cubic equation; thus one needs to solve the cubic equation.We break from this tradition, rather factor the even-powered sextic equation in a novel fashion, such that the inserted root is identified quickly along with one root of the quintic equation in a quadratic factor of the form, u2− g2 = (u + g)(u − g). Thus there is no need to solve any cubic equation. As an extra benefit, this root is a function of only one coefficient of the given quintic equation.

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11

Rashidinia, J., and R. Mohammadi. "Sextic spline solution of variable coefficient fourth-order parabolic equations." International Journal of Computer Mathematics 87, no. 15 (December 2010): 3443–54. http://dx.doi.org/10.1080/00207160903085820.

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12

BRIHAYE, Y., N. DEBERGH, and J. NDIMUBANDI. "ON A LIE ALGEBRAIC APPROACH OF QUASI-EXACTLY SOLVABLE POTENTIALS WITH TWO KNOWN EIGENSTATES." Modern Physics Letters A 16, no. 19 (June 21, 2001): 1243–51. http://dx.doi.org/10.1142/s0217732301004479.

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We compare two recent approaches of quasi-exactly solvable Schrödinger equations, the first one being related to finite-dimensional representations of sl (2, R) while the second one is based on supersymmetric developments. Our results are then illustrated on the Razavy potential, the sextic oscillator and a scalar field model.

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13

Orevkov, Stepan. "Parametric equations of plane sextic curves with a maximal set of double points." Journal of Algebra and Its Applications 14, no. 09 (July 10, 2015): 1540013. http://dx.doi.org/10.1142/s0219498815400137.

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We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field of the minimal possible degree and try to choose coordinates so that the coefficients are as small as we can do.

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14

Hammad, Hasanen A., Hassen Aydi, and Manuel De la Sen. "Refined stability of the additive, quartic and sextic functional equations with counter-examples." AIMS Mathematics 8, no. 6 (2023): 14399–425. http://dx.doi.org/10.3934/math.2023736.

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<abstract><p>In this study, we utilize the direct method (Hyers approach) to examine the refined stability of the additive, quartic, and sextic functional equations in modular spaces with and without the $ \Delta _{2} $-condition. We also use the direct approach to discuss the Ulam stability in $ 2 $-Banach spaces. Ultimately, we ensure that stability of above equations does not hold in a particular scenario by utilizing appropriate counter-examples.</p></abstract>

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15

Gaál, István, László Remete, and Tímea Szabó. "Calculating Power Integral bases by Solving Relative Thue Equations." Tatra Mountains Mathematical Publications 59, no. 1 (June 1, 2014): 79–92. http://dx.doi.org/10.2478/tmmp-2014-0020.

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Abstract In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10500 for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.

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16

Nam, Le Dai, Phan Anh Luan, Luu Phong Su, and Phan Ngoc Hung. "A family of analytically solvable Schrödinger equations related by Levi-Civita transformation." Tạp chí Khoa học 16, no. 3 (September 24, 2019): 103. http://dx.doi.org/10.54607/hcmue.js.16.3.2456(2019).

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Some two-dimensional problems in non-relativistic quantum mechanics can connect to each other by certain spatial transformations such as Levi-Civita transformation. This property allows forming a series of two-dimensional problems into an interrelated family. Starting from two related problems namely Coulomb plus harmonic oscillator and sextic double-well anharmonic oscillator potentials, such family is constructed via repeatedly applying Levi-Civita transformations. Obviously, this family contains various of exactly analytically solvable problems. The quasi-exact solution for each unknown member of this family is also obtained and systematically investigated.

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17

Choudhry, Ajai. "On the Solvability of Two Simultaneous Symmetric Cubic Diophantine Equations with Applications to Sextic Diophantine Equations." Rocky Mountain Journal of Mathematics 32, no. 1 (March 2002): 91–104. http://dx.doi.org/10.1216/rmjm/1030539609.

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18

Rassias, John Micheal, Elumalai Sathya, and Mohan Arunkumar. "Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces." Malaya Journal of Matematik 9, no. 1 (2021): 217–43. http://dx.doi.org/10.26637/mjm0901/0038.

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19

Kumar, Manoj, and Pankaj Kumar Srivastava. "Computational Techniques for Solving Differential Equations by Cubic, Quintic, and Sextic Spline." International Journal for Computational Methods in Engineering Science and Mechanics 10, no. 1 (February 12, 2009): 108–15. http://dx.doi.org/10.1080/15502280802623297.

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20

Kelil, Abey S., and Appanah R. Appadu. "On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight." Mathematics 8, no. 8 (July 31, 2020): 1250. http://dx.doi.org/10.3390/math8081250.

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Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight |x|αexp(−cx6),c>0,α>−1. Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat’s quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara’s symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials.

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21

Choudhry, Ajai. "A new method of solving certain quartic and higher degree diophantine equations." International Journal of Number Theory 14, no. 08 (August 22, 2018): 2129–54. http://dx.doi.org/10.1142/s1793042118501282.

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In this paper, we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be applied to some diophantine systems in five or more variables. Under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, two examples being a sextic equation in four variables and two simultaneous equations of degrees four and six in six variables. We also simultaneously obtain arbitrarily many rational solutions of certain related nonhomogeneous equations of high degree. We obtain these solutions without finding a curve of genus 0 or 1 on the variety defined by the equations concerned. It appears that there exist projective varieties on which there are an arbitrarily large number of rational points and which do not contain a curve of genus 0 or 1 with infinitely many rational points.

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22

MACFARLANE, A. J. "LIE ALGEBRA AND INVARIANT TENSOR TECHNOLOGY FOR g2." International Journal of Modern Physics A 16, no. 18 (July 20, 2001): 3067–97. http://dx.doi.org/10.1142/s0217751x01004335.

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Proceeding in analogy with su (n) work on λ matrices and f- and d-tensors, this paper develops the technology of the Lie algebra g2, its seven-dimensional defining representation γ and the full set of invariant tensors that arise in relation thereto. A comprehensive listing of identities involving these tensors is given. This includes identities that depend on use of characteristic equations, especially for γ, and a good body of results involving the quadratic, sextic and (the nonprimitivity of) other Casimir operators of g2.

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23

Mijiddorj, Renchin-Ochir, and Tugal Zhanlav. "Algorithm to construct integro splines." ANZIAM Journal 63 (November 16, 2021): 359–75. http://dx.doi.org/10.21914/anziamj.v63.15855.

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We study some properties of integro splines. Using these properties, we design an algorithm to construct splines \(S_{m+1}(x)\) of neighbouring degrees to the given spline \(S_{m}(x)\) with degree \(m\). A local integro-sextic spline is constructed with the proposed algorithm. The local integro splines work efficiently, that is, they have low computational complexity, and they are effective for use in real time. The construction of nonlocal integro splines usually leads to solving a system of linear equations with band matrices, which yields high computational costs. doi:10.1017/S1446181121000316

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24

Mohammed, Pshtiwan Othman, Manar A. Alqudah, Y. S. Hamed, Artion Kashuri, and Khadijah M. Abualnaja. "Solving the Modified Regularized Long Wave Equations via Higher Degree B-Spline Algorithm." Journal of Function Spaces 2021 (March 3, 2021): 1–10. http://dx.doi.org/10.1155/2021/5580687.

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The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave ( MRLW ) equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms L 2 and L ∞ to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.

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25

Eshaghi Gordji, M., Y. J. Cho, M. B. Ghaemi, and H. Majani. "Approximately Quintic and Sextic Mappings Formr-Divisible Groups into Ŝerstnev Probabilistic Banach Spaces: Fixed Point Method." Discrete Dynamics in Nature and Society 2011 (2011): 1–16. http://dx.doi.org/10.1155/2011/572062.

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Using the fixed point method, we investigate the stability of the systems of quadratic-cubic and additive-quadratic-cubic functional equations with constant coefficients formr-divisible groups into Ŝerstnev probabilistic Banach spaces.

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26

Morales, D. A., and Z. Parra-Mejías. "On the relationship between anharmonic oscillators and perturbed Coulomb potentials in N dimensions." Canadian Journal of Physics 77, no. 11 (February 18, 2000): 863–71. http://dx.doi.org/10.1139/p99-063.

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The relation between the perturbed Coulomb problem in N dimensionsand the sextic anharmonic oscillator in N' dimensionsis presented and generalized in this work.We show that by performing a transformation, containing a free parameter, on the equations for the two problems we can relate the two systems in dimensions that have not been previously linked. Exact solutions can be obtained for the N-dimensional systems from knownthree-dimensional solutions of the two problems. Using the known ground-state wave functions for these systems, we construct supersymmetric partner potentials that allow us to apply the supersymmetric large-Nexpansion to obtain accurate approximate energy eigenvalues.PACS Nos.: 03.65.Ge, 03.65.Fd, 11.30.Na

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27

Cho, Yeol Je, Mohammad Bagher Ghaemi, Mehdi Choubin, and Madjid Eshaghi Gordji. "On the Hyers-Ulam stability of sextic functional equations in β-homogeneous probabilistic modular spaces." Mathematical Inequalities & Applications, no. 4 (2013): 1097–114. http://dx.doi.org/10.7153/mia-16-85.

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28

GAÁL, ISTVÁN, and MICHAEL POHST. "On the Resolution of Index Form Equations in Sextic Fields with an Imaginary Quadratic Subfield." Journal of Symbolic Computation 22, no. 4 (October 1996): 425–34. http://dx.doi.org/10.1006/jsco.1996.0060.

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29

Wang, Xu, and Peter Schiavone. "Green’s functions for an anisotropic half-space and bimaterial incorporating anisotropic surface elasticity and surface van der Waals forces." Mathematics and Mechanics of Solids 22, no. 3 (August 6, 2016): 557–72. http://dx.doi.org/10.1177/1081286515598826.

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In this paper we derive explicit expressions for the Green’s functions in the case of an anisotropic elastic half-space and bimaterial subjected to a line force and a line dislocation. In contrast to previous studies in this area, our analysis includes the contributions of both anisotropic surface elasticity and surface van der Waals interaction forces. By means of the Stroh sextic formalism, analytical continuation and the state-space approach, the corresponding boundary value problem is reduced to a system of six (for a half-space) or 12 (for a bimaterial) coupled first-order differential equations. By employing the orthogonality relations among the corresponding eigenvectors, the coupled system of differential equations is further decoupled to six (for a half-space) or 12 (for a bimaterial) independent first-order differential equations. The latter is solved analytically using exponential integrals. In addition, we identify four and seven non-zero intrinsic material lengths for a half-space and a bimaterial, respectively, due entirely to the incorporation of the surface elasticity and surface van der Waal forces. We prove that these material lengths can be only either real and positive or complex conjugates with positive real parts.

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30

Blechschmidt, J. L., and J. J. Uicker. "Linkage Synthesis Using Algebraic Curves." Journal of Mechanisms, Transmissions, and Automation in Design 108, no. 4 (December 1, 1986): 543–48. http://dx.doi.org/10.1115/1.3258767.

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A method to snythesize four-bar linkages using the algebraic curve of the motion of the coupler point on the coupler link of the four-bar linkage is developed. This method is a departure from modern synthesis methods, most of which are based upon Burmester theory. This curve, which is a planar algebraic polynomial in two variables for the four-bar linkage, is a trinodal tricircular sextic (sixth order). These properties are used to determine the coefficients of the curve given a set of points that the coupler point of the coupler link is to pass through. The coefficients of this curve are nonlinear functions of the linkage parameters. The resulting set of nonlinear equations are solved using iterative/optimization techniques for the linkage parameters.

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31

Barnett, D. M., and H. O. K. Kirchner. "A proof of the equivalence of the Stroh and Lekhnitskii sextic equations for plane anisotropic elastostatics." Philosophical Magazine A 76, no. 1 (July 1997): 231–39. http://dx.doi.org/10.1080/01418619708209971.

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32

Akram, Ghazala, Muhammad Abbas, Hira Tariq, Maasoomah Sadaf, Thabet Abdeljawad, and Manar A. Alqudah. "Numerical Approximations for the Solutions of Fourth Order Time Fractional Evolution Problems Using a Novel Spline Technique." Fractal and Fractional 6, no. 3 (March 19, 2022): 170. http://dx.doi.org/10.3390/fractalfract6030170.

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Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline technique (SPST). These equations are frequently applied in a wide variety of real-world applications, such as strain gradient elasticity, phase separation in binary mixtures, and modelling of thin beams and plates, all of which are key parts of mechanical engineering. The SPST can be used for space discretization, whereas the backward Euler formula can be used for time discretization. For the temporal discretization, the method’s convergence and stability are assessed. To show the accuracy and applicability of the proposed technique, numerical simulations are employed.

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33

Chang, Ick-Soon, Yang-Hi Lee, and Jaiok Roh. "Nearly General Septic Functional Equation." Journal of Function Spaces 2021 (December 13, 2021): 1–7. http://dx.doi.org/10.1155/2021/5643145.

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If a mapping can be expressed by sum of a septic mapping, a sextic mapping, a quintic mapping, a quartic mapping, a cubic mapping, a quadratic mapping, an additive mapping, and a constant mapping, we say that it is a general septic mapping. A functional equation is said to be a general septic functional equation provided that each solution of that equation is a general septic mapping. In fact, there are a lot of ways to show the stability of functional equations, but by using the method of G a ˘ vruta, we examine the stability of general septic functional equation ∑ i = 0 8 C 8 i − 1 8 − i f x + i − 4 y = 0 which considered. The method of G a ˘ vruta as just mentioned was given in the reference Gavruta (1994).

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34

Rashidinia, J., and R. Mohammadi. "Numerical Methods Based on Non-Polynomial Sextic Spline for Solution of Variable Coefficient Fourth-Order Wave Equations." International Journal for Computational Methods in Engineering Science and Mechanics 10, no. 4 (June 19, 2009): 266–76. http://dx.doi.org/10.1080/15502280902939445.

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35

Xu, TianZhou, JohnMichael Rassias, MatinaJohn Rassias, and WanXin Xu. "A Fixed Point Approach to the Stability of Quintic and Sextic Functional Equations in Quasi--Normed Spaces." Journal of Inequalities and Applications 2010, no. 1 (2010): 423231. http://dx.doi.org/10.1155/2010/423231.

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36

Choudhry, Ajai. "On the Solvability of Quintic and Sextic Diophantine Equations of the Type f(x, y)=f(u, v)." Journal of Number Theory 88, no. 2 (June 2001): 225–40. http://dx.doi.org/10.1006/jnth.2000.2603.

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37

Fu, Y. B. "Hamiltonian interpretation of the Stroh formalism in anisotropic elasticity." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2088 (September 11, 2007): 3073–87. http://dx.doi.org/10.1098/rspa.2007.0093.

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Stroh's sextic formalism for static problems or steady motions in anisotropic elasticity is a formulation in which the equation of equilibrium/motion is written as a system of first-order differential equations for the displacement and traction in terms of one of the spatial variables. The so-called fundamental elasticity matrix N appearing in this formulation has the property that, when partitioned as a 2×2 block matrix, its 12- and 21-blocks are symmetric matrices and its 11-block is the transpose of its 22-block. This property gives rise to a large number of orthogonality and closure relations and is fundamental to the success of the Stroh formalism in solving a large variety of problems in general anisotropic elasticity. First, we show that the matrix N is guaranteed to have the above property by the fact that the Stroh formulation is in fact a Hamiltonian formulation with one of the spatial variables acting as the time-like variable. This interpretation provides a much desired guide in dealing with other problems for which the governing equations are different, such as incompressible elasticity and problems associated with anisotropic elastic plates as described by the Kirchhoff plate theory. We show that for the last two problems the Hamiltonian interpretation simplifies the derivations significantly, leading to a Stroh formulation in each case which is equivalent to, but much simpler than, what is available in the existing literature.

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38

Bender, Carl M., and Javad Komijani. "Addendum: Painlevé transcendents and PT -symmetric Hamiltonians (2015 J. Phys. A: Math. Theor. 48 475202)." Journal of Physics A: Mathematical and Theoretical 55, no. 10 (March 8, 2022): 109401. http://dx.doi.org/10.1088/1751-8121/ac4fa7.

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Abstract This paper is an Addendum to reference Bender and Komijani (2015 J. Phys. A: Math. Theor. 48 475202) (which stems from an earlier paper Bender et al (2014 J. Phys. A: Math. Theor. 47 235204)), where it was demonstrated that unstable separatrix solutions to the Painlevé equations I and II are determined by PT -symmetric Hamiltonians. Here, unstable separatrix solutions of the fourth Painlevé transcendent are studied numerically and analytically. It is shown that for a fixed initial value such as y(0) = 1 a discrete set of initial slopes y′(0) = b n give rise to separatrix solutions. Similarly, for a fixed initial slope such as y′(0) = 0 a discrete set of initial values y(0) = c n give rise to separatrix solutions. For Painlevé IV the large-n asymptotic behavior of b n is b n ∼ B IV n 3/4 and that of c n is c n ∼ C IV n 1/2. The constants B IV and C IV are determined both numerically and analytically. The analytical values of these constants are found by reducing the nonlinear Painlevé IV equation to the linear eigenvalue equation for the sextic PT -symmetric Hamiltonian H = 1 2 p 2 + 1 8 x 6 .

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39

K, Vedavathi, and A. Swapna. "Deep Learning based Handwritten Polynomial Equation Solver." International Journal for Research in Applied Science and Engineering Technology 10, no. 6 (June 30, 2022): 1880–83. http://dx.doi.org/10.22214/ijraset.2022.44171.

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Abstract: Polynomials are algebraic expressions involving a sum of powers in one or more variables multiplied by its coefficients. If x is a variable, a0x n +a1x n-1 +a2x n-2 +....+an then it's a (n) powers polynomial. Human is capable to solve this type of mathematical problems. In this work, we propose a system in which machines can achieve the cognitional skills that can understand the problem by visual context. By taking an input image of Handwritten polynomial equations and simplifies the problem by generating the answer as an output. Here machine can able to solve quadratic, cubic, quartic, quantic, sextic as well as (n) powers polynomials. This proposed work can be workable in an embedded system as well as a mobile application. In this scope for recognition purposes, we use a CNN model. Robust handwritten character recognition is a tricky job in the area of image processing. Among all the problem handwritten mathematical expression recognition is one of the complicated issue in the area of computer vision research. Segmentation and classification of specific character makes the task more difficult. In this paper a group of handwritten quadratic equation as well as a single quadratic equation are considered to recognize and make a solution for those equations. Horizontal compact projection analysis and combined connected component analysis methods are used for segmentation. For classification of specific character we apply Convolutional Neural Network. Each of the correct detection, character string operation is used for the solution of the equation. The proposed workflow system automatically simplifies the Handwritten polynomial equation and has been done a really good performance. Developing an automatic equation recognizer and solver has been a desire of the researchers who worked in the field of NLP for many years.

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Degtyarev, Alex. "On the Artal–Carmona–Cogolludo construction." Journal of Knot Theory and Its Ramifications 23, no. 05 (April 2014): 1450028. http://dx.doi.org/10.1142/s021821651450028x.

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We derive explicit defining equations for a number of irreducible maximizing plane sextics with double singular points only. For most real curves, we also compute the fundamental group of the complement; all groups found are abelian, which suffices to complete the computation of the groups of all non-maximizing irreducible sextics. As a by-product, examples of Zariski pairs in the strongest possible sense are constructed.

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El-Hawary, H. M., and K. A. El-Shami. "SexticC1-spline collocation methods for solving delay differential equations." International Journal of Computer Mathematics 89, no. 5 (March 2012): 679–90. http://dx.doi.org/10.1080/00207160.2011.648928.

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Grant, P. W., J. A. Sharp, M. F. Webster, and X. Zhang. "Sparse matrix representations in a functional language." Journal of Functional Programming 6, no. 1 (January 1996): 143–70. http://dx.doi.org/10.1017/s095679680000160x.

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AbstractThis paper investigates several sparse matrix representation schemes and associated algorithms in Haskell for solving linear systems of equations arising from solving realistic computational fluid dynamics problems using a finite element algorithm. This work complements that of Wainwright and Sexton (1992) in that a Choleski direct solver (with an emphasis on its forward/backward substitution steps) is examined. Experimental evidence comparing time and space efficiency of these matrix representation schemes is reported, together with associated forward/backward substitution implementations. Our results are in general agreement with Wainwright and Sexton's.

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Rehme, Michael, Stephen Roberts, and Dirk Pflüger. "Uncertainty quantification for the Hokkaido Nansei-Oki tsunami using B-splines on adaptive sparse grids." ANZIAM Journal 62 (June 29, 2021): C30—C44. http://dx.doi.org/10.21914/anziamj.v62.16121.

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Modeling uncertainties in the input parameters of computer simulations is an established way to account for inevitably limited knowledge. To overcome long run-times and high demand for computational resources, a surrogate model can replace the original simulation. We use spatially adaptive sparse grids for the creation of this surrogate model. Sparse grids are a discretization scheme designed to mitigate the curse of dimensionality, and spatial adaptivity further decreases the necessary number of expensive simulations. We combine this with B-spline basis functions which provide gradients and are exactly integrable. We demonstrate the capability of this uncertainty quantification approach for a simulation of the Hokkaido Nansei–Oki Tsunami with anuga. We develop a better understanding of the tsunami behavior by calculating key quantities such as mean, percentiles and maximum run-up. We compare our approach to the popular Dakota toolbox and reach slightly better results for all quantities of interest. References B. M. Adams, M. S. Ebeida, et al. Dakota. Sandia Technical Report, SAND2014-4633, Version 6.11 User’s Manual, July 2014. 2019. https://dakota.sandia.gov/content/manuals. J. H. S. de Baar and S. G. Roberts. Multifidelity sparse-grid-based uncertainty quantification for the Hokkaido Nansei–Oki tsunami. Pure Appl. Geophys. 174 (2017), pp. 3107–3121. doi: 10.1007/s00024-017-1606-y. H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numer. 13 (2004), pp. 147–269. doi: 10.1017/S0962492904000182. M. Eldred and J. Burkardt. Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. 47th AIAA. 2009. doi: 10.2514/6.2009-976. K. Höllig and J. Hörner. Approximation and modeling with B-splines. Philadelphia: SIAM, 2013. doi: 10.1137/1.9781611972955. M. Matsuyama and H. Tanaka. An experimental study of the highest run-up height in the 1993 Hokkaido Nansei–Oki earthquake tsunami. National Tsunami Hazard Mitigation Program Review and International Tsunami Symposium (ITS). 2001. O. Nielsen, S. Roberts, D. Gray, A. McPherson, and A. Hitchman. Hydrodymamic modelling of coastal inundation. MODSIM 2005. 2005, pp. 518–523. https://www.mssanz.org.au/modsim05/papers/nielsen.pdf. J. Nocedal and S. J. Wright. Numerical optimization. Springer, 2006. doi: 10.1007/978-0-387-40065-5. D. Pflüger. Spatially Adaptive Sparse Grids for High-Dimensional Problems. Dr. rer. nat., Technische Universität München, Aug. 2010. https://www5.in.tum.de/pub/pflueger10spatially.pdf. M. F. Rehme, F. Franzelin, and D. Pflüger. B-splines on sparse grids for surrogates in uncertainty quantification. Reliab. Eng. Sys. Saf. 209 (2021), p. 107430. doi: 10.1016/j.ress.2021.107430. M. F. Rehme and D. Pflüger. Stochastic collocation with hierarchical extended B-splines on Sparse Grids. Approximation Theory XVI, AT 2019. Springer Proc. Math. Stats. Vol. 336. Springer, 2020. doi: 10.1007/978-3-030-57464-2_12. S Roberts, O. Nielsen, D. Gray, J. Sexton, and G. Davies. ANUGA. Geoscience Australia. 2015. doi: 10.13140/RG.2.2.12401.99686. I. J. Schoenberg and A. Whitney. On Pólya frequence functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves. Trans. Am. Math. Soc. 74.2 (1953), pp. 246–259. doi: 10.2307/1990881. W. Sickel and T. Ullrich. Spline interpolation on sparse grids. Appl. Anal. 90.3–4 (2011), pp. 337–383. doi: 10.1080/00036811.2010.495336. C. E. Synolakis, E. N. Bernard, V. V. Titov, U. Kânoğlu, and F. I. González. Standards, criteria, and procedures for NOAA evaluation of tsunami numerical models. NOAA/Pacific Marine Environmental Laboratory. 2007. https://nctr.pmel.noaa.gov/benchmark/. J. Valentin and D. Pflüger. Hierarchical gradient-based optimization with B-splines on sparse grids. Sparse Grids and Applications—Stuttgart 2014. Lecture Notes in Computational Science and Engineering. Vol. 109. Springer, 2016, pp. 315–336. doi: 10.1007/978-3-319-28262-6_13. D. Xiu and G. E. Karniadakis. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24.2 (2002), pp. 619–644. doi: 10.1137/S1064827501387826.

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Acosta-Humánez, Primitivo B., Mourad E. H. Ismail, and Nasser Saad. "Sextic anharmonic oscillators and Heun differential equations." European Physical Journal Plus 137, no. 7 (July 2022). http://dx.doi.org/10.1140/epjp/s13360-022-03029-3.

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Braun, A. P., and R. Valandro. "G 4 flux, algebraic cycles and complex structure moduli stabilization." Journal of High Energy Physics 2021, no. 1 (January 2021). http://dx.doi.org/10.1007/jhep01(2021)207.

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Abstract We construct G4 fluxes that stabilize all of the 426 complex structure moduli of the sextic Calabi-Yau fourfold at the Fermat point. Studying flux stabilization usually requires solving Picard-Fuchs equations, which becomes unfeasible for models with many moduli. Here, we instead start by considering a specific point in the complex structure moduli space, and look for a flux that fixes us there. We show how to construct such fluxes by using algebraic cycles and analyze flat directions. This is discussed in detail for the sextic Calabi-Yau fourfold at the Fermat point, and we observe that there appears to be tension between M2-tadpole cancellation and the requirement of stabilizing all moduli. Finally, we apply our results to show that even though symmetric fluxes allow to automatically solve several F-term equations, they typically lead to flat directions.

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El Amine Monir, Mohammed. "Exact solutions of Sixth and Fifth Degree Equations." Journal of Advances in Mathematics and Computer Science, December 27, 2021, 1–6. http://dx.doi.org/10.9734/jamcs/2021/v36i1230420.

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The real problematic with algebraic polynomial equations is how to exactly solve any sixth and fifth degree polynomial equations. In this study, we give a new absolute method that presents a new decomposition to exactly solve a sixth degree polynomial equation, while the corresponding fifth degree equation can be easily transformed into a sixth degree equation of this kind (sixth degree equation solvable by this method), then the sextic equation (sixth degree equation) obtained will be solved by applying the principles of this method; moreover, the solutions of the quintic equation (fifth degree equation) will be easily deduced.

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Miahi, Mina, Farshid Mirzaee, and Hamid Khodaei. "Heat Kernel Method for Quintic and Sextic Equations in Distributions and Hyperfunctions." Qualitative Theory of Dynamical Systems 22, no. 2 (February 19, 2023). http://dx.doi.org/10.1007/s12346-023-00737-8.

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Jin, Sun-Sook, and Yang-Hi Lee. "Hyers-Ulam-Rassias Stability of a General Septic Functional Equation." Journal of Advances in Mathematics and Computer Science, December 19, 2022, 12–28. http://dx.doi.org/10.9734/jamcs/2022/v37i121725.

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In this paper, we investigate the stability of the following general septic functional equation: \(\sum_{i=0}^8{ }_8 C_i(-1)^{8-i} f(x+(i-4) y)=0\)which is a generalization of many functional equations such as the additive functional equation, the quadratic functional equation, the cubic functional equation, the quartic functional equation, the quintic functional equation, and the sextic functional equation. The equation is analysed from the perspective of Hyers-Ulam-Rassias stability.

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Harikumar, E., Suman Kumar Panja, and Partha Guha. "Application of regularization maps to quantum mechanical systems in two and three dimensions." Modern Physics Letters A 37, no. 07 (March 7, 2022). http://dx.doi.org/10.1142/s0217732322500432.

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In this paper, we generalize the application of the Levi-Civita (L-C) and Kustaanheimo–Stiefel (K-S) regularization methods to quantum mechanical systems in two and three dimensions. Schrödinger equations in two and three dimensions, describing a particle moving under the combined influence of [Formula: see text] and [Formula: see text] potentials are mapped to that of a harmonic oscillator with inverted sextic potential, and interactions, in two and four dimensions, respectively. Using the perturbative solutions of the latter systems, we derive the eigen spectrum of the former systems. Using Bohlin–Sundmann transformation, a mapping between the Schrödinger equations describing shifted harmonic oscillator and H-atom is also derived. Exploiting this equivalence, the solution to the former is obtained from the solution of the latter.

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MIJIDDORJ, R., and T. ZHANLAV. "ALGORITHM TO CONSTRUCT INTEGRO SPLINES." ANZIAM Journal, September 13, 2021, 1–17. http://dx.doi.org/10.1017/s1446181121000316.

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Abstract We study some properties of integro splines. Using these properties, we design an algorithm to construct splines $S_{m+1}(x)$ of neighbouring degrees to the given spline $S_m(x)$ with degree m. A local integro-sextic spline is constructed with the proposed algorithm. The local integro splines work efficiently, that is, they have low computational complexity, and they are effective for use in real time. The construction of nonlocal integro splines usually leads to solving a system of linear equations with band matrices, which yields high computational costs.

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