Showing papers on "Quintic function published in 2010"

Generalization of the Fierz-Pauli action

Abstract: We consider the Lagrangian of gravity covariantly amended by the mass and polynomial interaction terms with arbitrary coefficients and reinvestigate the consistency of such a theory in the decoupling limit, up to the fifth order in the nonlinearities. We calculate explicitly the self-interactions of the helicity-0 mode, as well as the nonlinear mixing between the helicity-0 and -2 modes. We show that ghostlike pathologies in these interactions disappear for special choices of the polynomial interactions and argue that this result remains true to all orders in the decoupling limit. Moreover, we show that the linear and some of the nonlinear mixing terms between the helicity-0 and -2 modes can be absorbed by a local change of variables, which then naturally generates the cubic, quartic, and quintic Galileon interactions, introduced in a different context. We also point out that the mixing between the helicity-0 and -2 modes can be at most quartic in the decoupling limit. Finally, we discuss the implications of our findings for the consistency of the effective field theory away from the decoupling limit, and for the Boulware-Deser problem.

Multifield Galileons and higher codimension branes

Abstract: In the decoupling limit, the Dvali-Gabadadze-Porrati model reduces to the theory of a scalar field $\ensuremath{\pi}$, with interactions including a specific cubic self-interaction---the Galileon term. This term, and its quartic and quintic generalizations, can be thought of as arising from a probe 3-brane in a five-dimensional bulk with Lovelock terms on the brane and in the bulk. We study multifield generalizations of the Galileon and extend this probe-brane view to higher codimensions. We derive an extremely restrictive theory of multiple Galileon fields, interacting through a quartic term controlled by a single coupling, and trace its origin to the induced brane terms coming from Lovelock invariants in the higher codimension bulk. We explore some properties of this theory, finding de Sitter like self-accelerating solutions. These solutions have ghosts if and only if the flat space theory does not have ghosts. Finally, we prove a general nonrenormalization theorem: multifield Galileons are not renormalized quantum mechanically to any loop in perturbation theory.

The density of discriminants of quintic rings and fields

Abstract: We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic rings and fields.

Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations

Abstract: We compute the recently introduced Fan–Jarvis–Ruan–Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov–Witten genus-zero theory of the quintic three-fold via a symplectic transformation. More specifically, we show that the J-function encoding the Fan–Jarvis–Ruan–Witten theory on the A-side equals via a mirror map the I-function embodying the period integrals at the Gepner point on the B-side. This identification inscribes the physical Landau–Ginzburg/Calabi–Yau correspondence within the enumerative geometry of moduli of curves, matches the genus-zero invariants computed by the physicists Huang, Klemm, and Quackenbush at the Gepner point, and yields via Givental’s quantization a prediction on the relation between the full higher genus potential of the quintic three-fold and that of Fan–Jarvis–Ruan–Witten theory.

Superintegrability and higher order polynomial algebras

Abstract: We present a method to obtain higher order integrals and polynomial algebras for two-dimensional quantum superintegrable systems separable in Cartesian coordinates from ladder operators. All systems with a second- and a third-order integral of motion separable in Cartesian coordinates were studied. The integrals of motion of two of them do not generate a cubic algebra. We construct for these Hamiltonians a higher order polynomial algebra from their ladder operators. We obtain quintic and seventh-order polynomial algebras. We also give for the polynomial algebras of order 7 realizations in terms of deformed oscillator algebras. These realizations and finite-dimensional unitary representations allow us to obtain the energy spectrum. We also apply the construction to the caged anisotropic harmonic oscillator and a system involving the fourth Painleve transcendent.

Global well-posedness of the energy critical Nonlinear Schr\"odinger equation with small initial data in H^1(T^3)

Abstract: A refined trilinear Strichartz estimate for solutions to the Schrodinger equation on the flat rational torus T^3 is derived. By a suitable modification of critical function space theory this is applied to prove a small data global well-posedness result for the quintic Nonlinear Schrodinger Equation in H^s(T^3) for all s \geq 1. This is the first energy-critical global well-posedness result in the setting of compact manifolds.

A Fixed Point Approach to the Stability of Quintic and Sextic Functional Equations in Quasi- -Normed Spaces

Abstract: We achieve the general solution of the quintic functional equation and the sextic functional equation . Moreover, we prove the stability of the quintic and sextic functional equations in quasi- -normed spaces via fixed point method.

Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves

Abstract: Simple criteria for the existence of rational rotation-minimizing frames (RRMFs) on quintic space curves are determined, in terms of both the quaternion and Hopf map representations for Pythagorean-hodograph (PH) curves in ?3. In both cases, these criteria amount to satisfaction of three scalar constraints that are quadratic in the curve coefficients, and are thus much simpler than previous criteria. In quaternion form, RRMF quintics can be characterized by just a single quadratic (vector) constraint on the three quaternion coefficients. In the Hopf map form, the characterization is in terms of one real and one complex quadratic constraint on the six complex coefficients. The identification of these constraints is based on introducing a "canonical form" for spatial PH curves and judicious transformations between the quaternion and Hopf map descriptions. The simplicity of these new characterizations for the RRMF quintics should help facilitate the development of algorithms for their construction, analysis, and practical use in applications such as animation, spatial motion planning, and swept surface constructions.

Quintic parametric polynomial minimal surfaces and their properties

Abstract: In this paper, quintic parametric polynomial minimal surface and their properties are discussed. We first propose the sufficient condition of quintic harmonic polynomial parametric surface being a minimal surface. Then several new models of minimal surfaces with shape parameters are derived from this condition. We also study the properties of new minimal surfaces, such as symmetry, self-intersection on symmetric planes and containing straight lines. Two one-parameter families of isometric minimal surfaces are also constructed by specifying some proper shape parameters.

Self-similar optical pulses in competing cubic-quintic nonlinear media with distributed coefficients

Abstract: We present a systematic analysis of the self-similar propagation of optical pulses within the framework of the generalized cubic-quintic nonlinear Schroedinger equation with distributed coefficients. By appropriately choosing the relations between the distributed coefficients, we not only retrieve the exact self-similar solitonic solutions, but also find both the approximate self-similar Gaussian-Hermite solutions and compact solutions. Our analytical and numerical considerations reveal that proper choices of the distributed coefficients could make the unstable solitons stable and could restrict the nonlinear interaction between the neighboring solitons.

On the energy critical Schrödinger equation in 3D non-trapping domains

Abstract: We prove that the quintic Schrodinger equation with Dirichlet boundary conditions is locally well posed for H 0 1 ( Ω ) data on any smooth, non-trapping domain Ω ⊂ R 3 . The key ingredient is a smoothing effect in L x 5 ( L t 2 ) for the linear equation. We also derive scattering results for the whole range of defocusing sub quintic Schrodinger equations outside a star-shaped domain.

A Class of Solutions to the 3D Cubic Nonlinear Schrödinger Equation that Blows Up on a Circle

Abstract: We consider the 3D cubic focusing nonlinear Schrodinger (NLS) equation i∂ t u + Δu + |u| 2 u = 0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in of initial data, that blow up in finite time with a singular circle in the xy-plane. Our construction is modeled on Raphael’s [33] construction of a family of solutions to the 2D quintic focusing NLS, i∂ t u + Δu + |u| 4 u = 0, that blows up on a circle.

Zeros of p-adic forms

Abstract: A variant of Brauer's induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms. © 2009 London Mathematical Society.

Quintic b-spline collocation method for numerical solution of the extended fisher-kolmogorov equation

Abstract: In this paper, numerical solutions of the extended Fisher-Kolmogorov equation are obtained by using the quintic B-spline collocation scheme. The scheme is based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The accuracy of the proposed method is demonstrated by three test problems. The scheme has second order convergence.

Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equation with an External Potential

Abstract: An improved homogeneous balance principle and an F-expansion technique are used to construct exact self-similar solutions to the cubic-quintic nonlinear Schrodinger equation. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and the external potential. Some simple self-similar waves are presented.

Quartic rings associated to binary quartic forms

Abstract: We give a bijection between binary quartic forms and quartic rings with a monogenic cubic resolvent ring, relating the rings associated to binary quartic forms with Bhargava's cubic resolvent rings. This gives a parametrization of quartic rings with monogenic cubic resolvents. We also give a geometric interpretation of this parametrization.

Equations for secant varieties via vector bundles

Abstract: We introduce vector bundle techniques for finding equations of secant varieties. A test is established that determines when a secant variety is an irreducible component of the zero set of the equations found. We also prove an induction theorem for varieties that are not weakly defective, that allows one to conclude that the zero set of the equations found for s_{r-1}(X) have s_{r-1}(X) as an irreducible component when s_r(X) is an irreducible component of the equations found for it. The techniques are illustrated with examples of homogeneous varieties. We give an algorithm to decompose a general ternary quintic as the sum of seven fifth powers.

An improved lower bound on the number of limit cycles bifurcating from a quintic hamiltonian planar vector field under quintic perturbation

Abstract: The limit cycle bifurcation of a Z2 equivariant quintic planar Hamiltonian vector field under Z2 equivariant quintic perturbation is studied. We prove that the given system can have at least 27 lim...

Stability Problems of Quintic Mappings in Quasi--Normed Spaces

Abstract: We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi- -normed spaces and then the stability by using a subadditive function for the quintic function such that , for all .

Classifying brumer's quintic polynomials by weak mordellweil groups

Abstract: We develop a general classification theory for Brumer's dihedral quintic polynomials by means of Kummer theory arising from certain elliptic curves. We also give a similar theory for cubic polynomials.

On the field intersection problem of solvable quintic generic polynomials

Abstract: We study a general method of the field intersection problem of generic polynomials over an arbitrary field k via formal Tschirnhausen transformation. In the case of solvable quintic, we give an explicit answer to the problem by using multi-resolvent polynomials.