Showing papers on "Quintic function published in 2007"

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. We hypothesize that 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 further check the conjecture by reproducing the first few instanton numbers by a localization computation directly in the A-model, and verifying Ooguri-Vafa integrality. This is the first exact result on open string mirror symmetry for a compact Calabi-Yau manifold.

Polynomial Structure of the (Open) Topological String Partition Function

Abstract: In this paper we show that the polynomial structure of the topological string partition function found by Yamaguchi and Yau for the quintic holds for an arbitrary Calabi-Yau manifold with any number of moduli. Furthermore, we generalize these results to the open topological string partition function as discussed recently by Walcher and reproduce his results for the real quintic.

Polynomial structure of the (open) topological string partition function

Abstract: In this paper we show that the polynomial structure of the topological string partition function found by Yamaguchi and Yau for the quintic holds for an arbitrary Calabi-Yau manifold with any number of moduli. Furthermore, we generalize these results to the open topological string partition function as discussed recently by Walcher and reproduce his results for the real quintic.

Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential

Abstract: Properties of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schr\"odinger equation (ICQNLSE) with an external potential are studied. When it is associated with the homogeneous CQNLSE, a general condition exists linking the external potential and inhomogeneous cubic and quintic (ICQ) nonlinearities. Besides for the nonpresence of an external potential, two classes of Jacobian elliptic periodic potentials are discussed in detail, and the corresponding ICQ nonlinearities are found to be either periodic or localized. Exact analytical soliton solutions in these cases are presented, such as the bright, dark, kink, and periodic solitons, etc. An appealing aspect is that the ICQNLSE can support bound states with any number of solitons when the ICQ nonlinearities are localized and an external potential is either applied or not.

D-brane superpotentials and RG flows on the quintic

Abstract: The behaviour of D2-branes on the quintic under complex structure deformations is analysed by combining Landau-Ginzburg techniques with methods from conformal field theory. It is shown that the boundary renormalisation group flow induced by the bulk deformations is realised as a gradient flow of the effective space time superpotential which is calculated explicitly to all orders in the boundary coupling constant.

D-branes and Normal Functions

Abstract: We explain the B-model origin of extended Picard-Fuchs equations satisfied by the D-brane superpotential on compact Calabi-Yau threefolds. Via the Abel-Jacobi map, the domainwall tension is identified with a Poincare normal function--a transversal holomorphic section of the Griffiths intermediate Jacobian. Within this formalism, we derive the extended Picard-Fuchs equation associated with the mirror of the real quintic.

D-brane superpotentials and RG flows on the quintic

Abstract: The behaviour of D2-branes on the quintic under complex structure deformations is analysed by combining Landau-Ginzburg techniques with methods from conformal field theory. It is shown that the boundary renormalisation group flow induced by the bulk deformations is realised as a gradient flow of the effective space time superpotential which is calculated explicitly to all orders in the boundary coupling constant.

Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method

Abstract: By using the extended hyperbolic auxiliary equation method, we present explicit exact solutions of the high-order nonlinear Schrodinger equation with the third-order and fourth-order dispersion and the cubic-quintic nonlinear terms, describing the propagation of extremely short pulses. These solutions include trigonometric function type and exact solitary wave solutions of hyperbolic function type. Among these solutions, some are found for the first time.

Log canonical thresholds of certain Fano hypersurfaces

Abstract: We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general.

The Brauer-Manin obstruction and Sha[2]

Abstract: We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in magma. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of Sha[2], for families of curves of genus 2 of the form y^2 = f(x), where f(x) is a quintic containing an irreducible cubic factor.

Exact solutions for the cubic–quintic nonlinear Schrödinger equation

Abstract: In this paper, the cubic–quintic nonlinear Schrodinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.

Resolution of algebraic equations by theta constants

Abstract: The history of algebraic equations is very long. The necessity and the trial of solving algebraic equations existed already in the ancient civilizations. The Babylonians solved equations of degree 2 around 2000 B.C. as well as the Indians and the Chinese. In the 16th century, the Italians discovered the resolutions of the equations of degree 3 and 4 by radicals known as Cardano’s formula and Ferrari’s formula. However in 1826, Abel [1] (independently about the same epoch Galois [7]) proved the impossibility of solving general equations of degree ≥ 5 by radicals. This is one of the most remarkable event in the history of algebraic equations. Was there nothing to do in this branch of mathematics after the work of Abel and Galois? Yes, in 1858 Hermite [8] and Kronecker [15] proved that we can solve the algebraic equation of degree 5 by using an elliptic modular function. Since \( \sqrt[n]{a} = \exp \left( {\left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern- ulldelimiterspace} n}} \right)\log a} \right) \) which is also written as exp((1/n) ∫ 1 a (1/x)dx), to allow only the extractions of radicals is to use only the exponential. Hence under this restriction, as we learn in the Galois theory, we can construct only compositions of cyclic extensions, namely solvable extentions. The idea of Hermite and Kronecker is as follows; if we use another transcendental function than the exponential, we can solve the algebraic equation of degree 5. In fact their result is analogous to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) . In the quintic equation they replace the exponential by an elliptic modular function and the integral ∫(1/x)dx by elliptic integrals. Kronecker [15] thought the resolution of the equation of degree 5 by an elliptic modular function would be a special case of a more general theorem which might exist. Kronecker’s idea was realized in few cases by Klein [11], [13]. Jordan [10] showed that we can solve any algebraic equation of higher degree by modular functions. Jordan’s idea is clarified by Thomae’s formula, 8 Chap, m (cf. Lindemann [16]). In this appendix, we show how we can deduce from Thomae’s formula the resolution of algebraic equations by a Siegel modular function which is explicitely expressed by theta constants (Theorem 2). Therefore Kronecker’s idea is completely realized. Our resolution of higher algebraic equations is also similar to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) In our resolution the exponential is replaced by tne Siegel modular function and the integral ∫(1/x)dx is replaced by hyperelliptic integrals. The existance of such resolution shows that the theta function is useful not only for non-linear differential equations but also for algebraic equations.

A quintic polynomial differential system with eleven limit cycles at the infinity

Abstract: In this article, a recursion formula for computing the singular point quantities of the infinity in a class of quintic polynomial systems is given. The first eleven singular point quantities are computed with the computer algebra system Mathematica. The conditions for the infinity to be a center are derived as well. Finally, a system that allows the appearance of eleven limit cycles in a small enough neighborhood of the infinity is constructed.

Normal form for spatial dynamics in the swift-hohenberg equation

Abstract: The reversible Hopf bifurcation with 1:1 resonance holds the key to the presence of spatially localized steady states in many partial differential equations on the real line. Two different techniques for computing the normal form for this bifurcation are described and applied to the Swift-Hohenberg equation with cubic/quintic and quadratic/cubic nonlinearities.

Collisions of counter-propagating pulses in coupled complex cubic-quintic Ginzburg-Landau equations

Abstract: We discuss the results of the interaction of counter-propagating pulses for two coupled complex cubic-quintic Ginzburg–Landau equations as they arise near the onset of a weakly inverted Hopf bifurcation. As a result of the interaction of the pulses we find in 1D for periodic boundary conditions (corresponding to an annular geometry) many different possible outcomes. These are summarized in two phase diagrams using the approach velocity, v, and the real part of the cubic cross-coupling, cr, of the counter-propagating waves as variables while keeping all other parameters fixed. The novel phase diagram in the limit v ↦0, cr ↦0 turns out to be particularly rich and includes bound pairs of 2 π holes as well as zigzag bound pairs of pulses.

FPGA Gaussian random number generator based on quintic hermite interpolation inversion

Abstract: In this work we present a very accurate floating point FPGA implementation of a Gaussian random number generator (GRNG) based on the inversion method. The inverse Gaussian cumulative distribution function (GCDF-1) is approximated using a quintic degree segment interpolation with Hermite coefficients and an accuracy-adaptative segmentation which divides the GCDF-1 into several non-uniform segments. Our architecture generates simple floating point samples of 32 bits with an accuracy of 20 bits of mantissa, achieving a 185 MHz speed and a throughput of one sample per cycle on a Xilinx Virtex-II FPGA.

Elliptic solutions of the quintic complex one-dimensional Ginzburg Landau equation

Abstract: The Conte–Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a prior restriction is to simplify calculations by means of the use of a few Laurent-series solutions instead of one and the use of the residue theorem. The application of this approach to the quintic complex one-dimensional Ginzburg–Landau equation (CGLE5) allows us to find elliptic solutions in the wave form. Restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions for the CGLE5, have been found as well. We demonstrate that to find elliptic solutions the analysis of a system of differential equations is preferable to the analysis of the equivalent single differential equation.

Efficient computation of root numbers and class numbers of parametrized families of real abelian number fields

Abstract: Let {K m } be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials P m (x) = x3 - mx2 - (m + 3)x + 1 and P m (x) = x 4 - mx 3 - 6x 2 + mx + 1. We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest number fields. We deduce a method for computing the exact values of these Gaussian sums. These values are then used to efficiently compute class numbers of simplest fields. Finally, such class number computations yield many examples of real cyclotomic fields Q(ζ p )+ of prime conductors p > 3 and class numbers h + p greater than or equal to p. However, in accordance with Vandiver's conjecture, we found no example of p for which p divides h + p .

Spatial solitons of desired intensity and width and their self-tapering/uptapering in cubic quintic nonlinear medium

Abstract: In this paper, we have investigated the propagation behavior of a Gaussian beam in cubic quintic nonlinear medium with and without absorption or gain. A governing differential equation for the evolution of beam width with the distance of propagation has been derived using the standard parabolic equation approach. By solving the governing equation numerically for different sets of parameters, we have shown that spatial solitons of fixed width and desired intensity and of fixed intensity and desired width are possible. Such liberty does not exist in other saturable media. We have also investigated self-tapering and self-uptapering of spatial solitons in the presence of absorption or gain and showed that the rate of self-tapering/uptapering is not only controlled by the magnitude of absorption or gain but also by the values of cubic and quintic terms. It is revealed that by self-tapering, the smallest achievable soliton width decreases/increases by increasing the magnitude of the cubic/quintic term. It is also revealed that the smallest achievable soliton width by self-tapering, is smaller for a larger initial width.

Rational curves of degree 11 on a general quintic threefold

Abstract: We prove that the incidence scheme of rational curves of degree 11 on quintic threefolds is irreducible. This implies a strong form of the Clemens conjecture in degree 11. Namely, on a general quintic threefold $F$ in $\mathbb{P}^4$, there are only finitely many smooth rational curves of degree 11, and each curve $C$ is embedded in $F$ with normal bundle $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$. Moreover, in degree 11, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on $F$.

Jacobian elliptic function solutions of the discrete cubic–quintic nonlinear Schrödinger equation

Abstract: The study of solitary wave solutions is of prime significance for the nonlinear Schrodinger equation with higher order dispersion and/or higher degree nonlinearities in nonlinear physical systems. We derive the discrete cubic–quintic nonlinear Schrodinger equation from a Hamiltonian using different Poisson brackets. By using the extended Jacobian elliptic function approach, we investigate the abundant exact stationary solitons and periodic waves solution of this equation. These solutions include, Jacobian periodic solutions, alternating phase Jacobi periodic solution, kink and bubble soliton solutions, alternating phase kink soliton solution and alternating phase bubble soliton solution, provided that coefficients are bound by special relation. And then with the aid of symbolic computation, we present in explicit form these solutions. The stability of bubble and kink soliton as well as alternating kink and alternating bubble soliton are also investigated.

Real hessian curves

Abstract: We give some real polynomials in two variables of degrees 4, 5, and 6 whose hessian curves have more connected components than had been known previously. In particular, we give a quartic polynomial whose hessian curve has 4 compact connected components (ovals), a quintic whose hessian curve has 8 ovals, and a sextic whose hessian curve has 11 ovals.