Journal articles: 'Curves, Plane. Curves, Quartic'

1

Yoshihara, Hisao. "Families of Galois closure curves for plane quartic curves." Journal of Mathematics of Kyoto University 43, no. 3 (2003): 651–59. http://dx.doi.org/10.1215/kjm/1250283700.

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2

Baker, Matthew, Yoav Len, Ralph Morrison, Nathan Pflueger, and Qingchun Ren. "Bitangents of tropical plane quartic curves." Mathematische Zeitschrift 282, no. 3-4 (November 16, 2015): 1017–31. http://dx.doi.org/10.1007/s00209-015-1576-7.

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3

Watanabe, S. "The genera of Galois closure curves for plane quartic curves." Hiroshima Mathematical Journal 38, no. 1 (March 2008): 125–34. http://dx.doi.org/10.32917/hmj/1207580347.

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4

Fernández, Julio, Josep González, and Joan-C. Lario. "Plane Quartic Twists of X(5, 3)." Canadian Mathematical Bulletin 50, no. 2 (June 1, 2007): 196–205. http://dx.doi.org/10.4153/cmb-2007-021-8.

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AbstractGiven an odd surjective Galois representation ϱ: Gℚ → PGL2(3) and a positive integer N, there exists a twisted modular curve X(N, 3)ϱ defined over ℚ whose rational points classify the quadratic ℚ-curves of degree N realizing ϱ. This paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case N = 5.

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5

Vakil, Ravi. "The Characteristic Numbers of Quartic Plane Curves." Canadian Journal of Mathematics 51, no. 5 (October 1, 1999): 1089–120. http://dx.doi.org/10.4153/cjm-1999-048-1.

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AbstractThe characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen’s prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.

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6

Miura, Kei. "Galois points on singular plane quartic curves." Journal of Algebra 287, no. 2 (May 2005): 283–93. http://dx.doi.org/10.1016/j.jalgebra.2005.02.015.

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7

Miura, Kei. "Field theory for function fields of singular plane quartic curves." Bulletin of the Australian Mathematical Society 62, no. 2 (October 2000): 193–204. http://dx.doi.org/10.1017/s0004972700018669.

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We study the structure of function fields of plane quartic curves by using projections. Taking a point P ∈ ℙ2, we define the projection from a curve C to a line l with the centre P. This projection induces and extension field k (C)/k (ℙ1). By using this fact, we study the field extension k (C)/k (ℙ1) from a geometrical point of view. In this note, we take up quartic curves with singular points.

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8

Bruin, Nils. "The arithmetic of Prym varieties in genus 3." Compositio Mathematica 144, no. 2 (March 2008): 317–38. http://dx.doi.org/10.1112/s0010437x07003314.

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AbstractGiven a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to perform Chabauty- and Brauer–Manin-type calculations for curves of genus 5 with an fixed-point-free involution. As an application, we determine the rational points on a smooth plane quartic and give examples of curves of genus 3 and 5 violating the Hasse principle. The methods are, in principle, applicable to any genus 3 curve with a double cover. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over $\mathbb {Q}(t)$. By specialization, this also gives examples over $\mathbb {Q}$.

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9

Kamel, Alwaleed, and Waleed Khaled Elshareef. "Weierstrass points of order three on smooth quartic curves." Journal of Algebra and Its Applications 18, no. 01 (January 2019): 1950020. http://dx.doi.org/10.1142/s0219498819500208.

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In this paper, we study the [Formula: see text]-Weierstrass points on smooth projective plane quartic curves and investigate their geometry. Moreover, we use a technique to determine in a very precise way the distribution of such points on any smooth projective plane quartic curve. We also give a variety of examples that illustrate and enrich the subject.

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10

Elsenhans, Andreas-Stephan. "Explicit computations of invariants of plane quartic curves." Journal of Symbolic Computation 68 (May 2015): 109–15. http://dx.doi.org/10.1016/j.jsc.2014.09.006.

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11

Liu, Hang, and Shan Chang. "$K_2$ of certain families of plane quartic curves." Proceedings of the American Mathematical Society 146, no. 7 (February 8, 2018): 2785–96. http://dx.doi.org/10.1090/proc/13963.

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12

Stöhr, Karl Otto. "Non-conservative plane quartic curves in characteristic five." Manuscripta Mathematica 66, no. 1 (December 1990): 183–204. http://dx.doi.org/10.1007/bf02568490.

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13

Dixmier, J. "On the projective invariants of quartic plane curves." Advances in Mathematics 64, no. 3 (June 1987): 279–304. http://dx.doi.org/10.1016/0001-8708(87)90010-7.

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14

Lorenzo García, Elisa. "Twists of non-hyperelliptic curves of genus 3." International Journal of Number Theory 14, no. 06 (July 2018): 1785–812. http://dx.doi.org/10.1142/s1793042118501075.

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In this paper, we compute explicit equations for the twists of all the smooth plane quartic curves defined over a number field [Formula: see text]. Since the plane quartic curves are non-hyperelliptic curves of genus [Formula: see text] we can apply the method developed by the author in a previous paper. The starting point is a classification due to Henn of the plane quartic curves with non-trivial automorphism group up to [Formula: see text]-isomorphism.

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15

Duyaguit, Ma Cristina Lumakin, and Hisao Yoshihara. "Galois Lines for Normal Elliptic Space Curves." Algebra Colloquium 12, no. 02 (June 2005): 205–12. http://dx.doi.org/10.1142/s1005386705000192.

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Let C be a curve, and l and l0 be lines in the projective three space ℙ3. Consider a projection πl: ℙ3 ⋯ → l0 with center l, where l ⋂ l0= ∅. Restricting πl to C, we obtain a morphism πl|C : C → l0 and an extension of fields (πl|C)* : k(l0) ↪ k(C). If this extension is Galois, then l is said to be a Galois line. We study the defining equations, automorphisms and the Galois lines for quartic curves, and give some applications to the theory of plane quartic curves.

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16

Hassett, Brendan. "Stable log surfaces and limits of quartic plane curves." manuscripta mathematica 100, no. 4 (December 1, 1999): 469–97. http://dx.doi.org/10.1007/s002290050213.

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17

Miura, Kei, and Hisao Yoshihara. "Field Theory for Function Fields of Plane Quartic Curves." Journal of Algebra 226, no. 1 (April 2000): 283–94. http://dx.doi.org/10.1006/jabr.1999.8173.

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18

FARAHAT, M. A. "THE LOCUS OF SMOOTH PLANE CURVES WITH A SEXTACTIC POINT." Journal of Algebra and Its Applications 13, no. 01 (August 20, 2013): 1350079. http://dx.doi.org/10.1142/s0219498813500795.

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Let Mg be the moduli space of isomorphism classes of genus g smooth curves over ℂ. We show that the locus S2d-r ⊂ Mg whose general points represent smooth plane curves of degree d ≥ 4 with a sextactic point of sextactic order 2d - r, where r ∈ {0, 1, 2}, is an irreducible and rational subvariety of codimension d(d - 4) + 2 - r of Mg. These results generalize those results introduced by the author in case of quartic curves (see K. Alwaleed and M. Farahat, The locus of smooth quartic curves with a sextactic point, Appl. Math. Inf. Sci.7(2) (2013) 509–513).

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19

Thorne, Jack. "Arithmetic invariant theory and 2-descent for plane quartic curves." Algebra & Number Theory 10, no. 7 (September 27, 2016): 1373–413. http://dx.doi.org/10.2140/ant.2016.10.1373.

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20

Yoshihara, Hisao. "Galois Lines for Normal Elliptic Space Curves, II." Algebra Colloquium 19, spec01 (October 31, 2012): 867–76. http://dx.doi.org/10.1142/s1005386712000739.

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For each linearly normal elliptic curve C ⊂ ℙ3, we determine Galois lines and their arrangement. We prove that the curve C has exactly six V4-lines. In case j(C) = 1, it has eight Z4-lines in addition. The V4-lines form the edges of a tetrahedron. In case j(C) = 1, for each vertex of the tetrahedron, there exist exactly two Z4-lines passing through it. As a corollary we obtain that each plane quartic curve of genus 1 does not have more than one Galois point.

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21

Kamel, Alwaleed, and M. Farahat. "On the moduli space of smooth plane quartic curves with a sextactic point." Applied Mathematics & Information Sciences 7, no. 2 (March 1, 2013): 509–13. http://dx.doi.org/10.12785/amis/070211.

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22

Stöhr, Karl-Otto. "On Bertini's theorem for fibrations by plane projective quartic curves in characteristic five." Journal of Algebra 315, no. 2 (September 2007): 502–26. http://dx.doi.org/10.1016/j.jalgebra.2007.05.027.

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23

SAKAMOTO, MARISA, and KOUKI TANIYAMA. "PLANE CURVES IN AN IMMERSED GRAPH IN ℝ2." Journal of Knot Theory and Its Ramifications 22, no. 02 (February 2013): 1350003. http://dx.doi.org/10.1142/s021821651350003x.

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For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane-closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane, the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one-half.

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24

Lercier, Reynald, Christophe Ritzenthaler, Florent Rovetta, and Jeroen Sijsling. "Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields." LMS Journal of Computation and Mathematics 17, A (2014): 128–47. http://dx.doi.org/10.1112/s146115701400031x.

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AbstractWe study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

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25

Bannai, Shinzo. "A note on splitting curves of plane quartics and multi-sections of rational elliptic surfaces." Topology and its Applications 202 (April 2016): 428–39. http://dx.doi.org/10.1016/j.topol.2016.02.005.

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26

Meidow, J. "ON THE VISUALIZATION OF POSITIONAL PRECISION." ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences V-4-2021 (June 17, 2021): 75–81. http://dx.doi.org/10.5194/isprs-annals-v-4-2021-75-2021.

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Abstract. Tasks such as image registration or pose estimation require the determination of transformations based on uncertain observations. Hence, the position of any geometric object transformed according to this estimate is also uncertain, at least in terms of precision. Often the knowledge of uncertainty changes the judgment of individuals. Thus, the visualization of this information is crucial whenever a human decision-maker is involved. In the absence of error-free reference data, we consider the estimated precision as the probably most important quantity characterizing the uncertainty. This contribution focuses on the visualization of positional precision as provided by estimated covariance matrices. Basic design principles such as coloration and contouring in 2D and 3D are presented and discussed in the context of practical applications, e.g., the superimposition of distance information as seen nowadays in sports broadcasts. As a novel contribution, we propose quartic plane curves to represent the confidence regions of the loci of conic sections.

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27

Wall, C. T. C. "Geometry of quartic curves." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 3 (May 1995): 415–23. http://dx.doi.org/10.1017/s0305004100073266.

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In recent work [5] which involved enumeration of singularity types of highly singular quintic curves, it was necessary to use rather detailed information on the geometry of quartic curves (for the case when the quintic consists of the quartic and a line). The present paper was written to supply this background. The cases of primary interest for this purpose are the rational quartics, and we concentrate on these.

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28

C.Dayanithi, C. Dayanithi. "Combination of Cubic and Quartic Plane Curve." IOSR Journal of Mathematics 6, no. 2 (2013): 43–53. http://dx.doi.org/10.9790/5728-0624353.

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29

Wall, C. T. C. "Quartic curves in characteristic 2." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 3 (May 1995): 393–414. http://dx.doi.org/10.1017/s0305004100073254.

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Simple singularities in positive characteristicSimple singularities in positive characteristic have been discussed by many authors, and the article [5] in particular establishes the subject on a firm footing. In it a simple, or ‘ADE’ singularity is defined by a list of normal forms and it is shown that the following conditions on a singularity are equivalent: (i) it is simple, (ii) it has finite deformation type, (iii) it has finite Cohen-Macaulay module type. Moreover, the normal forms for surface singularities coincide with the earlier list of Artin [1] and those for curves with the list of [9]: in those papers further characterizations were obtained.

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30

Plaumann, Daniel, Bernd Sturmfels, and Cynthia Vinzant. "Quartic curves and their bitangents." Journal of Symbolic Computation 46, no. 6 (June 2011): 712–33. http://dx.doi.org/10.1016/j.jsc.2011.01.007.

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31

Bauer, Thomas, Grzegorz Malara, Tomasz Szemberg, and Justyna Szpond. "Quartic unexpected curves and surfaces." manuscripta mathematica 161, no. 3-4 (November 22, 2018): 283–92. http://dx.doi.org/10.1007/s00229-018-1091-3.

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32

Movshovitz-Hadar, Nitsa, and Alla Shmukler. "Infinitely Many Different Quartic Polynomial Curves." College Mathematics Journal 23, no. 3 (May 1992): 186. http://dx.doi.org/10.2307/2686295.

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33

Movshovitz-Hadar, Nitsa, and Alla Shmukler. "Infinitely Many Different Quartic Polynomial Curves." College Mathematics Journal 23, no. 3 (May 1992): 186–95. http://dx.doi.org/10.1080/07468342.1992.11973455.

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34

Cassels, J. W. S. "The arithmetic of certain quartic curves." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 100, no. 3-4 (1985): 201–18. http://dx.doi.org/10.1017/s0308210500013779.

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SynopsisLet F(X, Y, Z) be a non-singular quadratic form with rational coefficients. The curve EF(x2, y2, z2) = 0 is of genus 3. A procedure is described for deciding whether there is an effective divisor on E of degree 3 defined over the rationals. There is such a divisor if and only if there is a point on E defined over some algebraic number field of odd degree. An example is constructed for which there is no such divisor although (i) there are points on E defined over all p-adic fields and over the reals and (ii) there are infinitely many rational points on each of the three curves F(X, y2, z2) = 0, F(x2, Y, z2) = 0 and F(x2, y2, Z) = 0.

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35

NAJMAN, FILIP. "EXCEPTIONAL ELLIPTIC CURVES OVER QUARTIC FIELDS." International Journal of Number Theory 08, no. 05 (July 6, 2012): 1231–46. http://dx.doi.org/10.1142/s1793042112500716.

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We study the number of elliptic curves, up to isomorphism, over a fixed quartic field K having a prescribed torsion group T as a subgroup. Let T = ℤ/mℤ⊕ℤ/nℤ, where m|n, be a torsion group such that the modular curve X1(m, n) is an elliptic curve. Let K be a number field such that there is a positive and finite number of elliptic curves E over K having T as a subgroup. We call such pairs (T, K)exceptional. It is known that there are only finitely many exceptional pairs when K varies through all quadratic or cubic fields. We prove that when K varies through all quartic fields, there exist infinitely many exceptional pairs when T = ℤ/14ℤ or ℤ/15ℤ and finitely many otherwise.

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36

Dong Quan, Nguyen Ngoc. "The arithmetic of certain quartic curves." Monatshefte für Mathematik 168, no. 2 (February 21, 2012): 191–214. http://dx.doi.org/10.1007/s00605-012-0387-8.

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37

Rieken, Sander. "Moduli of real pointed quartic curves." Geometriae Dedicata 185, no. 1 (June 4, 2016): 171–203. http://dx.doi.org/10.1007/s10711-016-0174-0.

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38

DUBE, MRIDULA, and REENU SHARMA. "PIECEWISE QUARTIC TRIGONOMETRIC POLYNOMIAL B-SPLINE CURVES WITH TWO SHAPE PARAMETERS." International Journal of Image and Graphics 12, no. 04 (October 2012): 1250028. http://dx.doi.org/10.1142/s0219467812500283.

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Analogous to the quartic B-splines curve, a piecewise quartic trigonometric polynomial B-spline curve with two shape parameters is presented in this paper. Each curve segment is generated by three consecutive control points. The given curve posses many properties of the B-spline curve. These curves are closer to the control polygon than the different other curves considered in this paper, for different values of shape parameters for each curve. With the increase of the value of shape parameters, the curve approach to the control polygon. For nonuniform and uniform knot vector the given curves have C0, G3; C1, G3; C1, G7; and C3 continuity for different choice of shape parameters. A quartic trigonometric Bézier curves are also introduced as a special case of the given trigonometric spline curves. A comparison of quartic trigonometric polynomial curve is made with different other curves. In the last, quartic trigonometric spline surfaces with two shape parameters are constructed. They have most properties of the corresponding curves.

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39

Aguglia, Angela, Gábor Korchmáros, and Fernando Torres. "Plane maximal curves." Acta Arithmetica 98, no. 2 (2001): 165–79. http://dx.doi.org/10.4064/aa98-2-7.

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40

Assi, Abdallah. "Meromorphic plane curves." Mathematische Zeitschrift 230, no. 1 (January 1999): 165–83. http://dx.doi.org/10.1007/pl00004682.

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41

Georgiev, Georgi Hristov, Radostina Petrova Encheva, and Cvetelina Lachezarova Dinkova. "Geometry of cylindrical curves over plane curves." Applied Mathematical Sciences 9 (2015): 5637–49. http://dx.doi.org/10.12988/ams.2015.56456.

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42

Tzanov, Vassil V., Bernd Krauskopf, and Simon A. Neild. "Vibration Dynamics of an Inclined Cable Excited Near Its Second Natural Frequency." International Journal of Bifurcation and Chaos 24, no. 09 (September 2014): 1430024. http://dx.doi.org/10.1142/s0218127414300249.

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Inclined cables are essential structural elements that are used most prominently in cable-stayed bridges. When the bridge deck oscillates due to an external force, such as passing traffic, cable vibrations can arise not only in the plane of excitation, but also in the perpendicular plane. This undesirable phenomenon can be modeled as an auto-parametric resonance between the in-plane and out-of-plane modes of vibration of the cable. In this paper, we consider a three-mode model, capturing the second in-plane, and first and second out-of-plane modes, and use it to study the response of an inclined cable that is vertically excited at its lower (deck) support at a frequency close to the second natural frequency of the cable. Averaging is applied to the model and then the solutions and bifurcations of the resulting averaged differential equations are investigated and mapped out with numerical continuation. In this way, we present a detailed bifurcation study of the different possible responses of the cable. We first consider the equilibria of the averaged model, of which there are four types that are distinguished by whether each of the two out-of-plane modes is present or not in the cable response. Each type of equilibrium is computed and represented as a surface over the plane of amplitude and frequency of the forcing. The stability of the equilibria changes and different surfaces meet along curves of bifurcations, which are continued directly. Overall, we present a comprehensive geometric picture of the two-parameter bifurcation diagram of the constant-amplitude coupled-mode response of the cable. We then focused on bifurcating periodic orbits, which correspond to cable dynamics with varying amplitudes of the participating second in-plane and second out-of-plane modes. The range of excitation amplitude and frequency is determined where such whirling cable motion can occur. Further bifurcations — period-doubling cascades and a Shilnikov homoclinic bifurcation — are found that lead to a chaotic cable response. Whirling and chaotic cable dynamics are confirmed by time-step simulations of the full three-mode model. The different cable responses are characterized, and can be distinguished clearly, by their motion at the quarter-span and by their frequency spectra.

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43

Kunyavskiĭ, Boris È., Louis H. Rowen, Sergey V. Tikhonov, and Vyacheslav I. Yanchevskiĭ. "Division algebras that ramify only on a plane quartic curve." Proceedings of the American Mathematical Society 134, no. 4 (July 19, 2005): 921–29. http://dx.doi.org/10.1090/s0002-9939-05-08106-2.

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44

Duquesne, Sylvain. "Elliptic curves associated with simplest quartic fields." Journal de Théorie des Nombres de Bordeaux 19, no. 1 (2007): 81–100. http://dx.doi.org/10.5802/jtnb.575.

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45

Chien, Mao-Ting, and Hiroshi Nakazato. "Inverse Numerical Range and Determinantal Quartic Curves." Mathematics 8, no. 12 (November 26, 2020): 2119. http://dx.doi.org/10.3390/math8122119.

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A hyperbolic ternary form, according to the Helton–Vinnikov theorem, admits a determinantal representation of a linear symmetric matrix pencil. A kernel vector function of the linear symmetric matrix pencil is a solution to the inverse numerical range problem of a matrix. We show that the kernel vector function associated to an irreducible hyperbolic elliptic curve is related to the elliptic group structure of the theta functions used in the Helton–Vinnikov theorem.

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46

Korchagin, Anatoly B., and David A. Weinberg. "The Isotopy Classification of Affine Quartic Curves." Rocky Mountain Journal of Mathematics 32, no. 1 (March 2002): 255–347. http://dx.doi.org/10.1216/rmjm/1030539619.

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47

Arrondo, Enrique, and Carlo G. Madonna. "CURVES AND VECTOR BUNDLES ON QUARTIC THREEFOLDS." Journal of the Korean Mathematical Society 46, no. 3 (May 1, 2009): 589–607. http://dx.doi.org/10.4134/jkms.2009.46.3.589.

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48

Harui, Takeshi, Takao Kato, Jiryo Komeda, and Akira Ohbuchi. "Quotient curves of smooth plane curves with automorphisms." Kodai Mathematical Journal 33, no. 1 (March 2010): 164–72. http://dx.doi.org/10.2996/kmj/1270559164.

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49

Dimca, Alexandru, and Gabriel Sticlaru. "Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves." Geometriae Dedicata 207, no. 1 (October 9, 2019): 29–49. http://dx.doi.org/10.1007/s10711-019-00485-7.

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50

Vainsencher, Israel. "Counting rational plane curves." Anais da Academia Brasileira de Ciências 72, no. 4 (December 2000): 610. http://dx.doi.org/10.1590/s0001-37652000000400035.

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Journal articles on the topic 'Curves, Plane. Curves, Quartic'

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