Showing papers on "Quintic function published in 1998"

Rational curves on hypersurfaces [after A. Givental]

Abstract: This article accompanies my June 1998 seminaire Bourbaki talk on Givental's work. After a quick review of descendent integrals in Gromov-Witten theory, I discuss Givental's formalism relating hypergeometric series to solutions of quantum differential equations arising from hypersurfaces in projective space. A particular case of this relationship is a proof of the Mirror prediction for the numbers of rational curves on the Calabi-Yau quintic 3-fold. The approach taken here is entirely algebro-geometric and relies upon a localization formula on the moduli space of stable genus 0 maps to projective space. A different proof of the quintic Mirror prediction may be found in the work of Lian, Liu, and Yau.

The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold

Abstract: The Abel-Jacobi map of the family of elliptic quintics lying on a general cubic threefold is studied. It is proved that it factors through a moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c_1=0, c_2=2, whose general point represents a vector bundle obtained by Serre's construction from an elliptic quintic. The elliptic quintics mapped to a point of the moduli space vary in a 5-dimensional projective space inside the Hilbert scheme of curves, and the map from the moduli space to the intermediate Jacobian is etale. As auxiliary results, the irreducibility of families of elliptic normal quintics and of rational normal quartics on a general cubic threefold is proved. This implies the uniqueness of the moduli component under consideration. The techniques of Clemens-Griffiths and Welters are used for the calculation of the infinitesimal Abel-Jacobi map.

The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold

Abstract: The Abel-Jacobi map of the family of elliptic quintics lying on a general cubic threefold is studied. It is proved that it factors through a moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c_1=0, c_2=2, whose general point represents a vector bundle obtained by Serre's construction from an elliptic quintic. The elliptic quintics mapped to a point of the moduli space vary in a 5-dimensional projective space inside the Hilbert scheme of curves, and the map from the moduli space to the intermediate Jacobian is etale. As auxiliary results, the irreducibility of families of elliptic normal quintics and of rational normal quartics on a general cubic threefold is proved. This implies the uniqueness of the moduli component under consideration. The techniques of Clemens-Griffiths and Welters are used for the calculation of the infinitesimal Abel-Jacobi map.

The mirror formula for quintic threefolds

Abstract: Our earlier proof of mirror formulas for genus 0 Gromov -- Witten invariants of Fano and Calabi -- Yau toric complete intersections is illustrated in the example of quintic 3-folds.

Bound states of dark solitons in the quintic Ginzburg-Landau equation

Abstract: We report results of systematic simulations of interactions between dark solitons in the complex quintic Ginzburg-Landau equation. Bound states of the solitons are found. The bound states (which are not possible in the cubic equation) exist in a wide range of parameters and are highly stable, providing an example of a stable bound state of solitary pulses in a generalized Ginzburg-Landau equation.

Solving Equations, an Elegant Legacy

Abstract: Solving equations-one of the primary themes in mathematics. In discussing this my secondary theme is that classical and modern mathematics are tightly intertwined, that contemporary mathematics contributes real insight and techniques to understand traditional problems. In the long second section I discuss some procedures that help to solve equations. There the discussion of symmetry is extensive because it is treated so inadequately as a fundamental thread throughout mathematics courses. The third section gives two different techniques to prove that equations have solutions. Most of the sections are independent; thus you can skip to examples that are more appealing. One ingredient in solving equations that I have not emphasized adequately is the basic role of inequalities. They are lurking here and there: the Euclidean algorithm and the application of the Brouwer fixed point theorem, to name two less obvious instances. To give inequalities their due would have changed the character of this article, which is drawn from the longer version [10]. 1. INTRODUCTION. Although by 1535 mathematicians had a formula to solve the cubic p(x) = X3 + bx2 + cx + d = 0, even without the formula it is easy to show that if the roots are x1, X2, X3, then expanding p(x) = (x - x)(x- x2) (x-X3) we get for instance x1+X2+X3=-b. (1) An immediate consequence is that if the coefficients in a cubic polynomial are rational and if two of the roots are rational, then so is the third root. This result introduces one thread in our story: even without a formula for the solution of an equation it may be possible to obtain useful information. Shortly after the cubic, the general quartic polynomial was solved. The next challenge was the quintic. If the coefficients of are real, for all large positive x we have p(x) > 0, while for all large negative x we have p(x) < 0. Thus if you graph the polynomial y = p(x), it is geometrically evident that it crosses the x-axis at least once and hence there is at least one real root x1 of p(x) = 0. The polynomial q(x) =p(x)/(x - x1) is then a quartic

Method for finding roots of quartic equation with application to RS codes

Abstract: A new method for finding the roots of a quartic equation over GF(2/sup m/) is described. Based on successive transformation, using the relationship between any two roots of the equation and looking up a simple table, the method is simpler to implement and faster than existing search techniques.

Congruence of Ankeny-Artin-Chowla type modulo p2

Abstract: In this paper the results of (4) and (5) on the congruence of Ankeny-Artin-Chowla type modulo p2 for real subfields of the Q(Cp) of a prime degree I > 2 are simplified to explicit forms (2) and (3) of the Theorem 1. The congruence is then used to calculations of class numbers in special quintic fields and to some calculations in cubic fields.

Soliton propagation and stability in a system with nonlinear amplifiers

Abstract: Soliton propagation in a system with linear and nonlinear amplifiers and spectral filtering is explored. We discuss different types of solutions of the cubic and the quintic complex Ginzburg-Landau equation (CGLE), namely solutions with fixed amplitude and solutions with arbitrary amplitude. The conditions to achieve a stable soliton propagation are analyzed within the domain of validity of the soliton perturbation theory. We obtain also a boundary for the region in the parameter space at which stable pulselike solutions of the quintic CGLE exist. In addition, an expression for the minimum value of the peak amplitude of these solutions is found, which depends uniquely on the quotient between the linear excess gain and the quintic saturating gain term.

The Modular Form of the Barth-Nieto Quintic

Abstract: Barth and Nieto have found a remarkable quintic threefold which parametrizes Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a (1,3)-polarization and a lecel 2 structure. A double cover of this quintic, which is also a Calabi-Yau variety, is birationally equivalent to the moduli space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2 structure. As a consequence the corresponding paramodular group \Gamma_3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is \Delta_1^3. The form \Delta_1 is a remarkable weight 1 cusp form with a character with respect to the paramodular group \Gamma_3. It has several interesting properties. One is that it admits an infinite product representation, the other is that it vanishes of order 1 along the diagonal in Siegel space. In fact \Delta_1 is an element of a short series of modular forms with this last property. Using the fact that \Delta_1 is a weight 3 cusp form with respect to the group \Gamma_3(2) we give an independent construction of a smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2).