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Journal articles on the topic 'Equation root'

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Published: 8 November 2023

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1

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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2

Assi, Abdullah Dhayea. "ADA Solve the Cubic Equation in a New Method with Engineering Application." Al-Qadisiyah Journal for Engineering Sciences 13, no. 3 (September 30, 2020): 223–31. http://dx.doi.org/10.30772/qjes.v13i3.659.

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Up to date the cubic equation or matrix tensor is consisting of nine values such as stress tensor that turns into the cubic equation which has been used for solving classic method. This is to impose an initial root several times to get it when achieves the equation and any other party is zero. Then dividing the cubic equation on the equation of the root. After that dividing the cubic equation on the equation of the root and using the classical method to find the rest of the roots. This is a very difficult issue, especially if the roots are secret or large for those who are looking in a difficult field or even for those who are in the examination room. In this research, two equations were reached, one that calculates the angle and the other that calculates the three roots at high accuracy without any significant error rate. By taking advantage of the traditional method, not by imposing a value to get the root of that equation, but by imposing an equation to get the solution equation that gives the value of that root. After imposing that equation, the general equation was derived from which that calculated the three roots directly and without any attempts. The angle that was implicitly derived during the derive of the main equation is calculated by taking advantage of the constants that do not change (invariants) for the matrix tensor (T).

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3

KRUGLOV, S. I. ""SQUARE ROOT" OF THE PROCA EQUATION: SPIN-3/2 FIELD EQUATION." International Journal of Modern Physics A 21, no. 05 (February 20, 2006): 1143–55. http://dx.doi.org/10.1142/s0217751x06024980.

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New equations describing particles with spin-3/2 are derived. The nonlocal equation with the unique mass can be considered as "square root" of the Proca equation in the same sense as the Dirac equation is related to the Klein–Gordon–Fock equation. The local equation describes spin-3/2 particles with three mass states. The equations considered involve fields with spin-3/2 and spin-1/2, i.e. multispin 1/2, 3/2. The projection operators extracting states with definite energy, spin, and spin projections are obtained. All independent solutions of the local equation are expressed through projection matrices. The first order relativistic wave equation in the 20-dimensional matrix form, the relativistically invariant bilinear form and the corresponding Lagrangian are given. Two parameters characterizing nonminimal electromagnetic interactions of fermions are introduced, and the quantum-mechanical Hamiltonian is found. It is proved that there is only causal propagation of waves in the approach considered.

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4

Batarius, Patrisius, and Alfri Aristo SinLae. "NILAI AWAL PADA METODE SECANT YANG DIMODIFIKASI DALAM PENENTUAN AKAR GANDA PERSAMAAN NON LINEAR." Jurnal Ilmiah Matrik 21, no. 1 (July 27, 2019): 21–31. http://dx.doi.org/10.33557/jurnalmatrik.v21i1.516.

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Determining the root of an equation means making the equation equal zero, (f (f) = 0). In engineering, there are often complex mathematical equations. With the numerical method approach, the equation can be searching for the value of the equation root. However, to find a double root approach with several numerical methods such as the bisection method, regulatory method, Newton-Raphson method, and Secant method, it is not efficient in determining multiple roots. This study aims to determine the roots of non-linear equations that have multiple roots using the modified Secant method. Besides knowing the effect of determining the initial value for the Secant method that is modifying in determining the non-linear root of persistence that has multiple roots. Comparisons were also make to other numerical methods in determining twin roots with the modified Secant method. A comparison is done to determine the initial value used. Simulations are performing on equations that have one single root and two or more double roots.

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5

Cogan, Brian. "Use of The Analytic Method and Computer Algebra to Plot Root Loci." International Journal of Electrical Engineering & Education 35, no. 4 (October 1998): 350–56. http://dx.doi.org/10.1177/002072099803500406.

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The analytic method for plotting root loci is examined. The basic equation for this method, originally published by Bendrikov and Teodorchik4 in 1959, is derived in a very simple way. A second equation, which complements the original Bendrikov equation, is also derived. An example of how these two equations may be used to plot root loci using Mathematica is presented.

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6

Pulita, Andrea. "p-adic confluence of q-difference equations." Compositio Mathematica 144, no. 4 (July 2008): 867–919. http://dx.doi.org/10.1112/s0010437x07003454.

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AbstractWe develop the theory of p-adic confluence of q-difference equations. The main result is the fact that, in the p-adic framework, a function is a (Taylor) solution of a differential equation if and only if it is a solution of a q-difference equation. This fact implies an equivalence, called confluence, between the category of differential equations and those of q-difference equations. We develop this theory by introducing a category of sheaves on the disk D−(1,1), for which the stalk at 1 is a differential equation, the stalk at q isa q-difference equation if q is not a root of unity, and the stalk at a root of unity ξ is a mixed object, formed by a differential equation and an action of σξ.

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7

Li, Zhong, Werner A. Kurz, Michael J. Apps, and Sarah J. Beukema. "Belowground biomass dynamics in the Carbon Budget Model of the Canadian Forest Sector: recent improvements and implications for the estimation of NPP and NEP." Canadian Journal of Forest Research 33, no. 1 (January 1, 2003): 126–36. http://dx.doi.org/10.1139/x02-165.

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In the Carbon Budget Model of the Canadian Forest Sector (CBM-CFS2), root biomass and dynamics are estimated using regression equations based on the literature. A recent analysis showed that some of these equations might overestimate belowground net primary production (NPPB). The objectives of this study were to update the compilation of root biomass and turnover data, to recalculate the regression equations and to evaluate the impact of the new equations on CBM-CFS2 estimates of net primary production (NPP) and net ecosystem production (NEP). We updated all equations based on 635 pairs of aboveground and belowground data compiled from published studies in the cold temperate and boreal forests. The new parameter for the equation to predict total root biomass for softwood species changed only slightly, but the changes for hardwood species were statistically significant. A new equation form, which improved the accuracy and biological interpretation, was used to predict fine root biomass as a proportion of total root biomass. The annual rate of fine root turnover was currently estimated to be 0.641 of fine root biomass. A comparison of NPP estimates from CBM-CFS2 with results from field measurements, empirical calculations and modeling indicated that the new root equations predicted reasonable NPPB values. The changes to the root equations had little effect on NEP estimates.

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8

Loginov, B. V. "Branching equation in the root subspace." Nonlinear Analysis: Theory, Methods & Applications 32, no. 3 (May 1998): 439–45. http://dx.doi.org/10.1016/s0362-546x(97)00481-1.

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9

Plyushchay, Mikhail S., and Michel Rausch de Traubenberg. "Cubic root of Klein-Gordon equation." Physics Letters B 477, no. 1-3 (March 2000): 276–84. http://dx.doi.org/10.1016/s0370-2693(00)00190-8.

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10

Debiard, Amédée, and Bernard Gaveau. "Analysis on Root Systems." Canadian Journal of Mathematics 39, no. 6 (December 1, 1987): 1281–404. http://dx.doi.org/10.4153/cjm-1987-064-x.

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A great part of mathematical analysis relies directly on the methods of separation of variables and on the successive reduction of several variables problems to one-dimensional equations and to the theory of classical special functions; for example, the theory of elliptic or parabolic equations with regular coefficients (even with non constant coefficients) can be done because we know explicitly the fundamental solutions of the Laplace operator or of the heat equation; these fundamental solutions are functions of one variable; pseudodifferential or parametrices methods are thus basically small perturbations of an explicitly known problem in one variable.On the other hand, there are many problems which are not of this type: they are related to the questions of operators with singular coefficients and to the global behaviour of the solutions; in that case, the local model cannot be reduced to a one variable problem but is fundamentally a several variables problem which cannot be treated in a detailed way by one variable methods or perturbation analysis of a one variable problem.

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11

Wang, Z. H. "Numerical Stability Test of Neutral Delay Differential Equations." Mathematical Problems in Engineering 2008 (2008): 1–10. http://dx.doi.org/10.1155/2008/698043.

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The stability of a delay differential equation can be investigated on the basis of the root location of the characteristic function. Though a number of stability criteria are available, they usually do not provide any information about the characteristic root with maximal real part, which is useful in justifying the stability and in understanding the system performances. Because the characteristic function is a transcendental function that has an infinite number of roots with no closed form, the roots can be found out numerically only. While some iterative methods work effectively in finding a root of a nonlinear equation for a properly chosen initial guess, they do not work in finding the rightmost root directly from the characteristic function. On the basis of Lambert W function, this paper presents an effective iterative algorithm for the calculation of the rightmost roots of neutral delay differential equations so that the stability of the delay equations can be determined directly, illustrated with two examples.

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12

Hisakado, Masato. "Coupled Nonlinear Schrödinger Equation and Toda Equation (The Root of Integrability)." Journal of the Physical Society of Japan 66, no. 7 (July 15, 1997): 1939–42. http://dx.doi.org/10.1143/jpsj.66.1939.

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13

Cui, Yan Mei, Zong De Fang, Jin Zhan Su, and Yuan Peng Liu. "Precise Modeling of Arc Cylinder-Gears with Tooth Root Fillet." Applied Mechanics and Materials 86 (August 2011): 871–74. http://dx.doi.org/10.4028/www.scientific.net/amm.86.871.

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Study on theory of generation processing arc cylinder-gears, machining cutter was designed by the blade fillet and the top tooth edge generating the tooth root fillet of arc cylinder-gears, to build the linear tooth-surface equation of cutter-gear and the position vector of blade fillet curvature center. The linear tooth surface equations and fillet tooth surface equation were derived based on coordinate transformation principle, meshing equation, meshing geometry principle and kinetic theory. The precise 3D model was established. It verifies the feasibility of generation processing the arc cylinder-gears and achieves smooth transition both working and fillet tooth surface of arc cylinder. It provides theoretical basis for tooth contact analysis, tooth bending stress and kinetic analysis, further offers accurate data to numerical control machining and systematically guides the experimental research and practical application.

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14

Filanovsky, I. M. "The Root Loci Equation and Its Application." International Journal of Electrical Engineering & Education 29, no. 2 (April 1992): 133–38. http://dx.doi.org/10.1177/002072099202900207.

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15

Rezaei, H. "Square Root Functional Equation on Positive Cones." Analysis in Theory and Applications 29, no. 4 (June 2013): 333–41. http://dx.doi.org/10.4208/ata.2013.v29.n4.2.

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16

S. Iwueze, Iheanyi, and Johnson Ohakwe. "Square Root Transformation of the Quadratic Equation." Asian Journal of Mathematics & Statistics 4, no. 4 (September 15, 2011): 186–99. http://dx.doi.org/10.3923/ajms.2011.186.199.

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17

Sabarinathan, S., Hemen Dutta, and B. V. Senthil Kumar. "Approximation of a third root functional equation." Proceedings of the Indian National Science Academy 87, no. 1 (March 2021): 48–56. http://dx.doi.org/10.1007/s43538-021-00004-x.

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18

Lin, S. M. "Dynamic Analysis of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root." Journal of Applied Mechanics 66, no. 3 (September 1, 1999): 742–49. http://dx.doi.org/10.1115/1.2791698.

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A systematic solution procedure for studying the dynamic response of a rotating nonuniform Timoshenko beam with an elastically restrained root is presented. The partial differential equations are transformed into the ordinary differential equations by taking the Laplace transform. The two coupled governing differential equations are uncoupled into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The general solution and the generalized Green function of the uncoupled system are derived. They are expressed in terms of the four corresponding linearly independent homogenous solutions, respectively. The shifting relations of the four homogenous solutions of the uncoupled governing differential equation with constant coefficients are revealed. The generalized Green function of an nth order ordinary differential equation can be obtained by using the proposed method. Finally, the influence of the elastic root restraints, the setting angle, and the excitation frequency on the steady response of a beam is investigated.

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19

Cingara, Aleksandar. "New simple algebraic root locus method for design of feedback control systems." Chemical Industry 62, no. 5 (2008): 269–74. http://dx.doi.org/10.2298/hemind0805269c.

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New concept of algebraic characteristic equation decomposition method is presented to simplify the design of closed-loop systems for practical applications. The method consists of two decompositions. The first one, decomposition of the characteristic equation into two lower order equations, was performed in order to simplify the analysis and design of closed loop systems. The second is the decomposition of Laplace variable, s, into two variables, damping coefficient, ?, and natural frequency, ? n. Those two decompositions reduce the design of any order feedback systems to setting of two complex dominant poles in the desired position. In the paper, we derived explicit equations for six cases: first, second and third order system with P and PI. We got the analytical solutions for the case of fourth and fifth order characteristic equations with the P and PI controller; one may obtain a complete analytical solution of controller gain as a function of the desired damping coefficient. The complete derivation is given for the third order equation with P and PI controller. We can extend the number of specified poles to the highest order of the characteristic equation working in a similar way, so we can specify the position of each pole. The concept is similar to the root locus but root locus is implicit, which makes it more complicated and this is simpler explicit root locus. Standard procedures, root locus and Bode diagrams or Nichol Charts, are neither algebraic nor explicit. We basically change controller parameters and observe the change of some function until we get the desired specifications. The derived method has three important advantage over the standard procedures. It is general, algebraic and explicit. Those are the best poles design results possible; it is not possible to get better controller design results.

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20

Saburov, Mansoor, Mohd Ahmad, and Murat Alp. "The study on general cubic equations over p-adic fields." Filomat 35, no. 4 (2021): 1115–31. http://dx.doi.org/10.2298/fil2104115s.

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A Diophantine problem means to find all solutions of an equation or system of equations in integers, rational numbers, or sometimes more general number rings. The most frequently asked question is whether a root of a polynomial equation with coefficients in a p-adic field Qp belongs to domains Z*p, Zp \ Z*p, Qp \ Zp, Qp or not. This question is open even for lower degree polynomial equations. In this paper, this problem is studied for cubic equations in a general form. The solvability criteria and the number of roots of the general cubic equation over the mentioned domains are provided.

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21

Lee, Myung W., and Sang Y. Suh. "Optimization of one‐way wave equations." GEOPHYSICS 50, no. 10 (October 1985): 1634–37. http://dx.doi.org/10.1190/1.1441853.

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The theory of wave extrapolation is based on the square‐root equation or one‐way equation. The full wave equation represents waves which propagate in both directions. On the contrary, the square‐root equation represents waves propagating in one direction only.

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22

FUCITO, F., and M. MARTELLINI. "LOOP EQUATIONS AND KDV HIERARCHY IN 2-D QUANTUM GRAVITY." International Journal of Modern Physics A 07, no. 10 (April 20, 1992): 2285–93. http://dx.doi.org/10.1142/s0217751x92001022.

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A derivation of the loop equation for two-dimensional quantum gravity from the KdV equations and the string equation of the one-matrix model has been recently given. The loop equation was found to be equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. Starting from the equations expressing these constraints, we are able to rederive the equations of the KdV hierarchy using the vertex operator construction of the [Formula: see text] infinite dimensional twisted Kac-Moody algebra. From these considerations it follows that the solutions of the string equation of the one-matrix model are given by a subset of the solutions of the KdV hierarchy.

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23

Komiyama, Akira, Sasitorn Poungparn, and Shogo Kato. "Common allometric equations for estimating the tree weight of mangroves." Journal of Tropical Ecology 21, no. 4 (June 27, 2005): 471–77. http://dx.doi.org/10.1017/s0266467405002476.

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Inventory data on tree weights of 104 individual trees representing 10 mangrove species were collected from mangrove forests in South-East Asia to establish common allometric equations for the trunk, leaf, above-ground and root weight. We used the measurable tree dimensions, such as dbh (trunk diameter at breast height), DR0.3 (trunk diameter at 30 cm above the highest prop root of Rhizophora species), DB (trunk diameter at lowest living branch), and H (tree height) for the independent variable of equations. Among the mangrove species studied, the trunk shape was statistically identical regardless of site and species. However, ρ (wood density of tree trunk) differed significantly among the species. A common allometric equation for trunk weight was derived, when dbh2H or DR0.32H was selected as the independent variable and wood density was taken into account. The common allometric equations for the leaf and the above-ground weight were also derived according to Shinozaki's pipe model and its extended theory. The common allometric relationships for these weights were attained with given ρ of each species, when DB2 or dbh2 or DR0.32 was selected as the independent variable. For the root weight, the common equation was derived from the allometric relationship between root weight and above-ground weight, since these two partial weights significantly correlated with each other. Based on these physical and biological parameters, we have proposed four common allometric equations for estimating the mangrove tree weight of trunk, leaf, above-ground part and root.

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24

Junjua, Moin-ud-Din, Fiza Zafar, and Nusrat Yasmin. "Optimal Derivative-Free Root Finding Methods Based on Inverse Interpolation." Mathematics 7, no. 2 (February 12, 2019): 164. http://dx.doi.org/10.3390/math7020164.

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Finding a simple root for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest due to its wide applications in many fields of science and engineering. Newton’s method is usually applied to solve this kind of problems. In this paper, for such problems, we present a family of optimal derivative-free root finding methods of arbitrary high order based on inverse interpolation and modify it by using a transformation of first order derivative. Convergence analysis of the modified methods confirms that the optimal order of convergence is preserved according to the Kung-Traub conjecture. To examine the effectiveness and significance of the newly developed methods numerically, several nonlinear equations including the van der Waals equation are tested.

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25

Nasution, Irma Novalinda. "DUAL RECIPROCITY BOUNDARY ELEMENT METHOD (DRBEM) DENGAN PREDIKTOR-KOREKTOR UNTUK MASALAH INFILTRASI STASIONER PADA SALURAN DENGAN LAPISAN IMPERMEABEL DENGAN ROOT WATER UPTAKE." EKSAKTA : Jurnal Penelitian dan Pembelajaran MIPA 4, no. 1 (January 27, 2019): 19. http://dx.doi.org/10.31604/eksakta.v4i1.19-27.

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This study aims to discussed about Dual Reciprocity Boundary Element Method (DRBEM) with a predictor-corrector for steady infiltration problems with impermeabel layer with root water uptake in homogeneous soils. Steady infiltration in homogeneous soils is governed by Richard equation. This equation is transformed using a set of transformation including Khirchoff, dimensionless variabels and dimensionless parameters into a type of modified Helmholtz equation. Furthermore with DRBEM, numerical solution of modified Helmholtz equation obtained. The proposed method is tested on problem involved infiltration from rectangular and trapezoidal channels with impermeabel layer with root water uptake. Keyword : infiltration, root water uptake, modified Helmholtz equation, Richard equation, DRBEM

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26

Jha, Navnit, Venu Gopal, and Bhagat Singh. "Geometric grid network and third-order compact scheme for solving nonlinear variable coefficients 3D elliptic PDEs." International Journal of Modeling, Simulation, and Scientific Computing 09, no. 06 (December 2018): 1850053. http://dx.doi.org/10.1142/s1793962318500538.

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By using nonuniform (geometric) grid network, a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type. Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions. The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values. As an experiment, applications of the compact scheme to Schrödinger equations, sine-Gordon equations, elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values. The results corroborate the reliability and efficiency of the scheme.

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27

Wang, Liang Wen, Wei Gang Tang, Xin Jie Wang, and Xue Wen Chen. "Computer Aided Geometric Method of Forward Kinematics Analysis of Multi-Legged Walking Robots." Advanced Materials Research 317-319 (August 2011): 829–34. http://dx.doi.org/10.4028/www.scientific.net/amr.317-319.829.

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The high order equation causing by analytic method in multi-legged robot forward kinematics analysis may have imaginary root, repeated root, extraneous root or even lost solution. A system based on the theory of computer aided geometric method is proposed. Consideration with the internal structural constraint relations of multi-legged walking robots, the solidworks model was constructed and Visual Basic develop platform was adopted to fulfill the secondary development of solidworks. A system of forward kinematics analysis of multi-legged walking robots is established. The example validates that the system is simple and effective for all reptiles-like quadruped walking robot.

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28

Zuev, Sergei. "A finite difference approach to find exact solution of differential equations." International Journal of Modern Physics: Conference Series 38 (January 2015): 1560080. http://dx.doi.org/10.1142/s2010194515600800.

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This paper contains the background and samples of an approach to construct exact solutions of a wide range of differential equations (DEs). This approach is based on the finite difference equation which corresponds to the given DE. There are three cases considered: linear partial differential equation (PDE) with constant coefficients and at least one non-zero root of characteristic equation, linear PDE with constant coefficients and completely zero roots of the characteristic equation, and a case of nonlinear autonomous dynamical system of second order. Each of these cases is illustrated by an example.

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29

Yeniçerioğlu, Ali Fuat, Vildan Yazıcı, and Cüneyt Yazıcı. "Asymptotic Behavior and Stability in Linear Impulsive Delay Differential Equations with Periodic Coefficients." Mathematics 8, no. 10 (October 16, 2020): 1802. http://dx.doi.org/10.3390/math8101802.

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We study first order linear impulsive delay differential equations with periodic coefficients and constant delays. This study presents some new results on the asymptotic behavior and stability. Thus, a proper real root was used for a representative characteristic equation. Applications to special cases, such as linear impulsive delay differential equations with constant coefficients, were also presented. In this study, we gave three different cases (stable, asymptotic stable and unstable) in one example. The findings suggest that an equation that is in a way that characteristic equation plays a crucial role in establishing the results in this study.

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30

Junkins, D. R., and R. R. Steeves. "ACCUMULATION OF THE CHOLESKY SQUARE ROOT IN HELMERT BLOCKING." Canadian Surveyor 40, no. 3 (September 1986): 297–314. http://dx.doi.org/10.1139/tcs-1986-0024.

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The Helmert blocking method is being used in the present effort to readjust North American geodetic networks. Combining this method with the Cholesky computational method enables the efficient solution of very large systems of linear equations. A by-product of this approach is a “partial” Cholesky square root for each Helmert block. This paper demonstrates that the Cholesky square root for the entire system of normal equations can be constructed from partial Cholesky square root blocks that are produced during the Helmert block adjustment, even though various reorderings of the unknown parameters are necessary throughout the computations. The entire Cholesky square root can be used to compute the inverse of the normal equation coefficient matrix, which is needed for post-adjustment statistical analyses.

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31

Greenwood, W. H., and K. W. Chase. "Root Sum Squares Tolerance Analysis with Nonlinear Problems." Journal of Engineering for Industry 112, no. 4 (November 1, 1990): 382–84. http://dx.doi.org/10.1115/1.2899604.

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Elementary tolerance analysis with nonlinear assembly equations or functions can lead to significant errors when either Worst Limits or Root Sum Squares are applied. A previous paper [1] presented the errors and a corrective design method for Worst Limit elementary tolerance analysis. In this paper, Root Sum Squares tolerance analysis is applied to a nonlinear assembly equation, which gives significant errors in the assembly acceptance fraction. Advanced statistical analysis can be used in an iterative design procedure to specify the independent variables which meet the specified assembly in spite of the nonlinearities.

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32

Li, Ling, Yun Jiang Miao, Zhong Bin Wang, and Xiong Bing Li. "Kinematics Modeling on CFRP Curved Part Ultrasonic Test." Advanced Materials Research 186 (January 2011): 136–40. http://dx.doi.org/10.4028/www.scientific.net/amr.186.136.

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Aimed at inner flaw in CFRP（carbon fiber reinforced plastic）curved part, build the ultrasonic test technological process. Based on five-freedom CFRP curved part robot, the mechanics structure model is set up. And then, by basic principle of robot kinematics, the kinematics equation of five-freedom ultrasonic test system is derived. Finally, through solving the direct root and converse root, the mathematics relation expression between the movement variable of servo motors and ultrasonic probe coordinate is obtained.

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33

McBride, Adam. "Remarks on Pell's Equation and Square Root Algorithms." Mathematical Gazette 83, no. 496 (March 1999): 47. http://dx.doi.org/10.2307/3618682.

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34

Milinazzo, Fausto A., Cedric A. Zala, and Gary H. Brooke. "Rational square-root approximations for parabolic equation algorithms." Journal of the Acoustical Society of America 101, no. 2 (February 1997): 760–66. http://dx.doi.org/10.1121/1.418038.

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35

武, 松. "Interval Algorithm of Nonlinear Equation with Multiple Root." Advances in Applied Mathematics 07, no. 08 (2018): 1020–27. http://dx.doi.org/10.12677/aam.2018.78119.

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36

Abadir, Karim M. "A DISTRIBUTION GENERATING EQUATION FOR UNIT-ROOT STATISTICS." Oxford Bulletin of Economics and Statistics 54, no. 3 (August 1992): 305–23. http://dx.doi.org/10.1111/j.1468-0084.1992.tb00004.x.

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37

Bzdak, A., and J. Szwed. "The “square root” of the interacting Dirac equation." Europhysics Letters (EPL) 69, no. 2 (January 2005): 189–92. http://dx.doi.org/10.1209/epl/i2004-10328-9.

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Juhari, Juhari. "On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of Trigonometric Function." CAUCHY 7, no. 1 (November 12, 2021): 84–96. http://dx.doi.org/10.18860/ca.v7i1.12934.

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This study discusses the analysis of the modification of Newton-Secant method and solving nonlinear equations having a multiplicity of by using a modified Newton-Secant method. A nonlinear equation that has a multiplicity is an equation that has more than one root. The first step is to analyze the modification of the Newton-Secant method, namely to construct a mathematical model of the Newton-Secant method using the concept of the Newton method and the concept of the Secant method. The second step is to construct a modified mathematical model of the Newton-Secant method by adding the parameter . After obtaining the modified formula for the Newton-Secant method, then applying the method to solve a nonlinear equations that have a multiplicity . In this case, it is applied to the nonlinear equation which has a multiplicity of . The solution is done by selecting two different initial values, namely and . Furthermore, to determine the effectivity of this method, the researcher compared the result with the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified. The obtained results from the analysis of modification of Newton-Secant method is an iteration formula of the modified Newton-Secant method. And for the result of using a modified Newton-Secant method with two different initial values, the root of is obtained approximately, namely with less than iterations. whereas when using the Newton-Raphson method, the Secant method, and the Newton-Secant method, the root is also approximated, namely with more than iterations. Based on the problem to find the root of the nonlinear equation it can be concluded that the modified Newton-Secant method is more effective than the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified

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Lin, S. M. "The Instability and Vibration of Rotating Beams With Arbitrary Pretwist and an Elastically Restrained Root." Journal of Applied Mechanics 68, no. 6 (August 23, 2000): 844–53. http://dx.doi.org/10.1115/1.1408615.

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The governing differential equations and the boundary conditions for the coupled bending-bending-extensional vibration of a rotating nonuniform beam with arbitrary pretwist and an elastically restrained root are derived by Hamilton’s principle. The semianalytical solution procedure for an inextensional beam without taking account of the coriolis forces is derived. The coupled governing differential equations are transformed to be a vector characteristic governing equation. The frequency equation of the system is derived and expressed in terms of the transition matrix of the vector governing equation. A simple and efficient algorithm for determining the transition matrix of the general system with arbitrary pretwist is derived. The divergence in the Frobenius method does not exist in the proposed method. The frequency relations between different systems are revealed. The mechanism of instability is discovered. The influence of the rotatory inertia, the coupling effect of the rotating speed and the mass moment of inertia, the setting angle, the rotating speed and the spring constants on the natural frequencies, and the phenomenon of divergence instability are investigated.

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ZHU, JIANXIN, and YA YAN LU. "VALIDITY OF ONE-WAY MODELS IN THE WEAK RANGE DEPENDENCE LIMIT." Journal of Computational Acoustics 12, no. 01 (March 2004): 55–66. http://dx.doi.org/10.1142/s0218396x0400216x.

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Numerical solutions of the Helmholtz equation and the one-way Helmholtz equation are compared in the weak range dependence limit, where the overall range distance is increased while the range dependence is weakened. It is observed that the difference between the solutions of these two equations persists in this limit. The one-way Helmholtz equation involves a square root operator and it can be further approximated by various one-way models used in underwater acoustics. An operator marching method based on the Dirichlet-to-Neumann map and a local orthogonal transform is used to solve the Helmholtz equation.

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Feng, Jing-Mei, and San-Yang Liu. "A Three-Step Iterative Method for Solving Absolute Value Equations." Journal of Mathematics 2020 (July 25, 2020): 1–7. http://dx.doi.org/10.1155/2020/8531403.

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In this paper, we transform the problem of solving the absolute value equations (AVEs) Ax−x=b with singular values of A greater than 1 into the problem of finding the root of the system of nonlinear equation and propose a three-step algorithm for solving the system of nonlinear equation. The proposed method has the global linear convergence and the local quadratic convergence. Numerical examples show that this algorithm has high accuracy and fast convergence speed for solving the system of nonlinear equations.

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Bychkov, Yuri, Elena Solovyeva, and Sergei Scherbakov. "Analytical-Numerical Calculation Algorithm of Algebraic Equations Roots with Specified Limits of Errors." SPIIRAS Proceedings 18, no. 6 (November 29, 2019): 1491–514. http://dx.doi.org/10.15622/sp.2019.18.6.1491-1514.

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This paper proposes an algorithm for calculating approximate values of roots of algebraic equations with a specified limit of absolute errors. A mathematical basis of the algorithm is an analytical-numerical method of solving nonlinear integral-differential equations with non-stationary coefficients. The analytical-numerical method belongs to the class of one-step continuous methods of variable order with an adaptive procedure for choosing a calculation step, a formalized estimate of the error of the performed calculations at each step and the error accumulated during the calculation. The proposed algorithm for calculating the approximate values of the roots of an algebraic equation with specified limit absolute errors consists of two stages. The results of the first stage are numerical intervals containing the unknown exact values of the roots of the algebraic equation. At the second stage, the approximate values of these roots with the specified limit absolute errors are calculated. As an example of the use of the proposed algorithm, defining the roots of the fifth-order algebraic equation with three different values of the limiting absolute error is presented. The obtained results allow drawing the following conclusions. The proposed algorithm enables to select numeric intervals that contain unknown exact values of the roots. Knowledge of these intervals facilitates the calculation of the approximate root values under any specified limiting absolute error. The algorithm efficiency, i.e., the guarantee of achieving the goal, does not depend on the choice of initial conditions. The algorithm is not iterative, so the number of calculation steps required for extracting a numerical interval containing an unknown exact value of any root of an algebraic equation is always restricted. The algorithm of determining a certain root of the algebraic equation is computationally completely autonomous.

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Xing, Hai Jun. "Damping Force Analysis of Magnetorheological (MR) Dampers." Applied Mechanics and Materials 187 (June 2012): 311–14. http://dx.doi.org/10.4028/www.scientific.net/amm.187.311.

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In this paper, utilizing Herschel-Bulkley model, the equation of MR fluid pressure gradient is derived in order to predict MR damper’s force-velocity behavior. The equation, showing as a complicated nonlinear algebraic expression including various parameters, is then simplified to a nondimensional equation. This is followed by the analysis of the root of this nondimensional equation and an approximate root closely corresponding to numerical result is given.

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Goel, Sudhir Kumar, and Denise T. Reid. "Activities for Students: A Graphical Approach to Understanding the Fundamental Theorem of Algebra." Mathematics Teacher 94, no. 9 (December 2001): 749–56. http://dx.doi.org/10.5951/mt.94.9.0749.

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The fundamental theorem of algebra states, Every polynomial equation of degree n ≥ 1 with complex coefficients has at least one complex root. This fact implies that these equations have exactly n roots, counting multiple roots, in the set of complex numbers.

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Rinaldo, Natascia, Ilaria Saguto, Federica De Luca, Margherita Neri, Paolo Frisoni, and Emanuela Gualdi-Russo. "Estimation of Age in Humans Using Dental Translucency of Permanent Teeth: An Experimental Study." Applied Sciences 13, no. 10 (May 21, 2023): 6289. http://dx.doi.org/10.3390/app13106289.

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In forensics, the positive identification of decomposed or skeletonized bodies is a fundamental task, with the age-at-death estimation of adult individuals as one of the main objectives. Among different dental methods, root dentin translucency (RDT) is often applied since it is easy to perform and non-destructive. However, this method has some biases, and several equations have been proposed in the literature. This study aimed to test the performance of the previously published equations in an Italian sample with known age and sex, and to develop an equation specific to the Italian population. In total, we examined a sample of 155 single and multi-rooted teeth from male and female individuals aged 18-85 years. The regression equation developed for Italians was tested on a holdout sample drawn from the same population. Intra- and inter-observer errors were calculated using ICC analysis. Both root length and RDT showed excellent repeatability and reproducibility regardless of tooth type. Two of the seven published equations tested performed better in our sample, but the newly proposed equation performed better than those on the Italian population. In conclusion, RDT has proven to be a reliable indicator for age estimation, and the proposed new formula may be effective in such estimation, especially in individuals aged <40.

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Zhao, Zhi Chao, Sheng Qi Sun, and Xue Bin Li. "The Research of Position Inverse Solution for 6-DOF Manipulator Based on Quaternion Algorithm." Applied Mechanics and Materials 336-338 (July 2013): 1109–13. http://dx.doi.org/10.4028/www.scientific.net/amm.336-338.1109.

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For the problem of 6-DOF serial manipulator position inverse solution, the paper proposed a new method for 6-DOF serial manipulator position inverse solution by introducing the quaternion algorithm. The relative pose transformation expression in quaternion coordinate system is proposed based on the transformation relationship between dual quaternion and double quaternion, based on which the closed equation of 6-DOF serial manipulator position inverse solution times in the form of double quaternion is established. For the obtained four position constraint equations, the Dixon resultant is constructed and a equation of 16 yuan is derived which is neither with increases root nor with the leak root. And this point is proved by the Mathematica practical calculations. The process of the method is simple. And it can not only avoid extraneous roots, but also overcome some singular position caused by the correlation equation in the process of trajectory planning, and further, the real-time performance of control process is improved.

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HIKAMI, KAZUHIRO, and YASUSHI KOMORI. "NOTES ON OPERATOR-VALUED SOLUTIONS OF THE YANG-BAXTER EQUATION AND THE REFLECTION EQUATION." Modern Physics Letters A 11, no. 36 (November 30, 1996): 2861–70. http://dx.doi.org/10.1142/s0217732396002848.

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We reconsider the operator-valued solutions of the Yang-Baxter equation and the reflection equation. We construct quantum Knizhnik-Zamolodchikov type operators, and discussed the relationship with the MacDonald q-polynomial theory associated with the classical root systems.

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Shin, Yong Hyun, and Ho-Seok Lee. "A Regime Switching Model of Schooling Choice as a Job Search Process." Advances in Mathematical Physics 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/475279.

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We propose a regime switching model of schooling choice as a job search process. We adopt a two-state Markov process and the derived coupled Bellman equations are solved by seeking the root of an auxiliary algebraic equation. Some numerical examples are also considered.

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Chavarría-Krauser, Andrés, Willi Jäger, and Ulrich Schurr. "Primary root growth: a biophysical model of auxin-related control." Functional Plant Biology 32, no. 9 (2005): 849. http://dx.doi.org/10.1071/fp05033.

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Plant hormones control many aspects of plant development and play an important role in root growth. Many plant reactions, such as gravitropism and hydrotropism, rely on growth as a driving motor and hormones as signals. Thus, modelling the effects of hormones on expanding root tips is an essential step in understanding plant roots. Here we achieve a connection between root growth and hormone distribution by extending a model of root tip growth, which describes the tip as a string of dividing and expanding cells. In contrast to a former model, a biophysical growth equation relates the cell wall extensibility, the osmotic potential and the yield threshold to the relative growth rate. This equation is used in combination with a refined hormone model including active auxin transport. The model assumes that the wall extensibility is determined by the concentration of a wall enzyme, whose production and degradation are assumed to be controlled by auxin and cytokinin. Investigation of the effects of auxin on the relative growth rate distribution thus becomes possible. Solving the equations numerically allows us to test the reaction of the model to changes in auxin production. Results are validated with measurements found in literature.

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Lin, Yiqin. "Cross-Gramian-Based Model Reduction for Descriptor Systems." Symmetry 14, no. 11 (November 13, 2022): 2400. http://dx.doi.org/10.3390/sym14112400.

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In this paper, we explore model order reduction for large-scale square descriptor systems. A balancing-free square-root method is proposed. The balancing-free square-root method is based on two cross Gramians, one of which is known as the proper cross Gramian and the other as the improper cross Gramian. The proper cross Gramian is the unique solution of a projected generalized continuous-time Sylvester equation, and the improper cross Gramian solves a projected generalized discrete-time Sylvester equation. In order to compute the low-rank factors of these two cross Gramians, we extend the low-rank iteration of the alternating direction implicit method and the Smith method to the projected generalized Sylvester equations. We illustrate the effectiveness of the balance truncation method with one numerical example.

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