Showing papers on "Quintic function published in 2006"

The Equation that Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry (APS Colloquium, 2006)

Abstract: For thousands of years, mathematicians solved progressively more difficult algebraic equations, from the simple quadractic to the more complex quartic equation, yielding important insights along the way. Then they were stumped by the quintic equation, which resisted solutions for three centuries, until two great prodigies independently proved that quaintic equations cannot be solved by simple formula. These geniuses, a young Norwegian named Niels Henrik Abel and an even younger Frenchman named Evariste Galois, both died tragically. Galois' work gave rise to group theory, the "language" that defines symmetry. Group theory explains much about the aesthetics of our world, from the choosing of mates to Rubik's cube, Bach's musical compositions, the physics of subatomic particles, and the popularity of Anna Kournikova. Some of the mysteries surrounding Galois' death, which have lingered for more than 170 years, are finally resolved in The Equation that Couldn't Be Solved. Livio will discuss this first popular-level book to explore group theory, not through abstract formulas but in a dramatic account of the lives and work of some of the greatest mathematicians in history.

Explicit lifts of quintic Jacobi sums and period polynomials for Fq

Abstract: In this paper, we construct explicit lifts of quintic Jacobi sums for finite fields via integer solutions of Dickson's system Namely we give a procedure to compute quintic Jacobi sums for extended field FP s+t by using quintic Jacobi sums for F Ps and for F pt . We also have the multiplication formula from F ps to F p ns as a special case. By the quintuplication formula, we obtain the explicit factorization of the quintic period polynomials for finite fields.

Disk enumeration on the quintic 3-fold

Abstract: Holomorphic disk invariants with boundary in the real Lagrangian of a quintic 3-fold are calculated by localization and proven mirror transforms. A careful discussion of the underlying virtual intersection theory is included. The generating function for the disk invariants is shown to satisfy an extension of the Picard-Fuchs differential equations associated to the mirror quintic. The Ooguri-Vafa multiple cover formula is used to define virtually enumerative disk invariants. The results may also be viewed as providing a virtual enumeration of real rational curves on the quintic.

Bifurcations and competing coherent structures in the cubic-quintic Ginzburg–Landau equation I: Plane wave (CW) solutions

Abstract: Singularity theory is used to comprehensively investigate the bifurcations of the steady-states of the traveling wave ODEs of the cubic-quintic Ginzburg–Landau equation (CGLE). These correspond to plane waves of the PDE. In addition to the most general situation, we also derive the degeneracy conditions on the eight coefficients of the CGLE under which the equation for the steady states assumes each of the possible quartic (the quartic fold and an unnamed form), cubic (the pitchfork and the winged cusp), and quadratic (four possible cases) normal forms for singularities of codimension up to three. Since the actual governing equations are employed, all results are globally valid, and not just of local applicability. In each case, the recognition problem for the unfolded singularity is treated. The transition varieties, i.e. the hysteresis, isola, and double limit curves are presented for each normal form. For both the most general case, as well as for various combinations of coefficients relevant to the particular cases, the bifurcation curves are mapped out in the various regions of parameter space delimited by these varieties. The multiplicities and interactions of the plane wave solutions are then comprehensively deduced from the bifurcation plots in each regime, and include features such as regimes of hysteresis among co-existing states, domains featuring more than one interval of hysteresis, and isola behavior featuring dynamics unrelated to the primary solution branch in limited ranges of parameter space.

Traveling wavetrains in the complex cubic–quintic Ginzburg–Landau equation

Abstract: In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate the periodic solutions of the complex cubic–quintic Ginzburg–Landau equation. The primary tools used here are Hopf bifurcation theory and perturbation theory. Explicit results are obtained for the post-bifurcation periodic orbits and their stability. Generalized and degenerate Hopf bifurcations are also briefly considered to track the emergence of global structures such as homoclinic orbits.

Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic

Abstract: We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P^2. In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.

Limit cycle analysis on a cubic Hamiltonian system with quintic perturbed terms

Abstract: This paper intends to explore bifurcation behavior of limit cycles for a cubic Hamiltonian system with quintic perturbed terms using both qualitative analysis and numerical exploration. To obtain the maximum number of limit cycles, a quintic perturbed function with the form of R(x, y, λ )= S(x, y, λ )= mx 2 + ny 2 + ky 4 − λ is added to a cubic Hamiltonian system, where m, n, k and λ are all variable. The investigation is based on detection functions which are particularly effective for the perturbed cubic Hamiltonian system. The study reveals that, for the Hamiltonian system [equation (1.5) in the introduction] with the perturbed terms mentioned above, there are 15 limit cycles if 15.1149 <λ< 15.1249; and 11 limit cycles if 15.1102 <λ< 15.1149. As numerical illustration, we numerically predict the detection curves and display graphically the distribution of limit cycles for the proposed perturbed Hamiltonian system.

Probabilistic galois theory for quartic polynomials

Abstract: We prove that there are only $O(H^{3+\\epsilon})$ quartic integer polynomials with height at most $H$ and a Galois group which is a proper subgroup of $S_4$. This improves in the special case of degree four a bound by Gallagher that yielded $O(H^{7/2} \\log H)$.

On the Solution of Quartic and Cubic Equations

Abstract: A method of solving a quartic equation, which does not require extracting the roots of complex numbers is explained in details. In the process, the solution of a cubic equation has also been presented, with the same degree of simplicity.

Opening Mirror Symmetry on the Quintic

Abstract: Aided by mirror symmetry, we determine the number of holomorphic disks ending on the real Lagrangian in the quintic threefold. The tension of the domainwall between the two vacua on the brane, which is the generating function for the open Gromov-Witten invariants, satisfies a certain extension of the Picard-Fuchs differential equation governing periods of the mirror quintic. We verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space. We reproduce the first few instanton numbers by a localization computation directly in the A-model, and check Ooguri-Vafa integrality. This is the first exact result on open string mirror symmetry for a compact Calabi-Yau manifold.

Improvement construction for planar g2 transition curve between two separated circles

Abstract: In this paper, we use the undetermined coefficient method to find a desirable pair of cubic Bezier spirals and a desirable pair of quintic PH spirals to generate planar G2 transition curve between two separated circles. The G2 transition curve can be gotten by the rooting formula, which simplifies the computation, and the ratio of two radii has no restriction, which extends the application area.

Elliptic functions to the quintic base

Abstract: We study two elliptic functions to the quintic base and find two nonlinear second order differential equations satisfied by them. We then derive two recurrence relations involving certain Eisenstein series associated with the group G0(5). These recurrence relations allow us to derive infinite families of identities involving the Eisenstein series and Dedekind ?-products. An imaginary transformation for one of the elliptic functions is also derived.

Cyclic polynomials arising from kummer theory of norm algebraic tori

Abstract: We study cyclic extensions arising from Kummer theory of norm algebraic tori. In particular, we compute quintic cyclic polynomials defining ‘Kummer extension’. The polynomials do not only give all the quintic cyclic extensions over the rationals by choosing the parameters but also classify all such extensions. Some arithmetic properties of the polynomials are also derived.

Counting the hyperplane sections with fixed invariants of a plane quintic

Abstract: We use three different methods to count the number of lines in the plane whose intersection with a fixed general quintic has fixed cross-ratios. We compare and contrast these methods, shedding light on some classical ideas which underly modern techniques.

A collective variable approach and stabilization for dispersion-managed optical solitons in the quintic complex Ginzburg–Landau equation as perturbations of the nonlinear Schrödinger equation

Abstract: With the help of the one-dimensional quintic complex Ginzburg?Landau equation (CGLE) as perturbations of the nonlinear Schr?dinger equation (NLSE), we derive the equations of motion of pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fibre optic links. The equations obtained are investigated numerically in order to view the evolution of pulse parameters along the propagation distance, and also to analyse effects of initial amplitude and width on the propagating pulse. Nonlinear gain is shown to be beneficial in stabilizing DM solitons. A fully numerical simulation of the one-dimensional quintic CGLE as perturbations of NLSE finally tests the results of the CV theory. A good agreement is observed between both methods.

Solitary Waves for Cubic-Quintic Nonlinear Schroedinger Equation with Variable Coefficients

Abstract: cubic-quintic 非线性的 Schrodinger 方程(CQNLS ) 在光纤维和原子水动力学起重要作用。由为 cubic-quintic 使用同类的平衡原则，钟类型，纽结类型，代数学的独居的波浪，和三角法的旅行波浪，有可变系数(vCQNLS ) 的非线性的 Schrodinger 方程在一套辅助高顺序的平常的微分方程的帮助下被导出(亚方程为短) 。在这篇论文使用的方法可能帮助一个为 otherhigh 顺序导出准确答案非线性的进化方程，和表演同类的平衡原则的新申请。

Adomian Decomposition Method for Approximating the Solution of the High-Order Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

Abstract: In this paper, the decomposition method is implemented for solving the high-order dispersive cubic-quintic nonlinear Schrodinger equation. By means of Maple the Adomian polynomials of obtained series solution have been calculated. The results reported in this article provide further evidence of the usefulness of Adomain decomposition for obtaining solutions of nonlinear problems.