Showing papers on "Quintic function published in 2014"

The observational status of Galileon gravity after Planck

Abstract: We use the latest CMB data from Planck, together with BAO measurements, to constrain the full parameter space of Galileon gravity. We constrain separately the three main branches of the theory known as the Cubic, Quartic and Quintic models, and find that all yield a very good fit to these data. Unlike in $\Lambda{\rm CDM}$, the Galileon model constraints are compatible with local determinations of the Hubble parameter and predict nonzero neutrino masses at over $5\sigma$ significance. We also identify that the low-$l$ part of the CMB lensing spectrum may be able to distinguish between $\Lambda{\rm CDM}$ and Galileon models. In the Cubic model, the lensing potential deepens at late times on sub-horizon scales, which is at odds with the current observational suggestion of a positive ISW effect. Compared to $\Lambda$CDM, the Quartic and Quintic models predict less ISW power in the low-$l$ region of the CMB temperature spectrum, and as such are slightly preferred by the Planck data. We illustrate that residual local modifications to gravity in the Quartic and Quintic models may render the Cubic model as the only branch of Galileon gravity that passes Solar System tests.

The observational status of Galileon gravity after Planck

Abstract: We use the latest CMB data from Planck, together with BAO measurements, to constrain the full parameter space of Galileon gravity. We constrain separately the three main branches of the theory known as the Cubic, Quartic and Quintic models, and find that all yield a very good fit to these data. Unlike in ΛCDM, the Galileon model constraints are compatible with local determinations of the Hubble parameter and predict nonzero neutrino masses at over 5σ significance. We also identify that the low l part of the CMB lensing spectrum may be able to distinguish between ΛCDM and Galileon models. In the Cubic model, the lensing potential deepens at late times on sub-horizon scales, which is at odds with the current observational suggestion of a positive ISW effect. Compared to ΛCDM, the Quartic and Quintic models predict less ISW power in the low l region of the CMB temperature spectrum, and as such are slightly preferred by the Planck data. We illustrate that residual local modifications to gravity in the Quartic and Quintic models may render the Cubic model as the only branch of Galileon gravity that passes Solar System tests.

On Scattering for the Quintic Defocusing Nonlinear Schrödinger Equation on R × T2

Abstract: We consider the problem of large-data scattering for the quintic nonlinear Schrodinger equation on R × T2. This equation is critical both at the level of energy and mass. Most notably, we exhibit a new type of profile (a “large-scale profile”) that controls the asymptotic behavior of the solutions. © 2014 Wiley Periodicals, Inc.

Interactions of nonlocal dark solitons under competing cubic–quintic nonlinearities

Abstract: We investigate analytically and numerically the interactions of dark solitons under competing nonlocal cubic and local quintic nonlinearities. It is shown that the self-defocusing quintic nonlinearity will strengthen the attractive interaction and decrease the relative distance between solitons, whereas the self-focusing quintic nonlinearity will enhance the repulsive interaction and increase soliton separation. We demonstrate these results by approximate variational approach and direct numerical simulation.

Conservation laws and Darboux transformation for the coupled cubic–quintic nonlinear Schrödinger equations with variable coefficients in nonlinear optics

Abstract: In this paper, by Darboux transformation and symbolic computation we investigate the coupled cubic–quintic nonlinear Schrodinger equations with variable coefficients, which come from twin-core nonlinear optical fibers and waveguides, describing the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the non-Kerr media. Lax pair of the equations is obtained, and the corresponding Darboux transformation is constructed. One-soliton solutions are derived; some physical quantities such as the amplitude, velocity, width, initial phases, and energy are, respectively, analyzed; and finally an infinite number of conservation laws are also derived. These results might be of some value for the ultrashort optical pulse propagation in the non-Kerr media.

Spin-5 Casimir operator its three-point functions with two scalars

Abstract: By calculating the second-order pole in the operator product expansion (OPE) between the spin-3 Casimir operator and the spin-4 Casimir operator known previously, the spin-5 Casimir operator is obtained in the coset model based on $ \left( {A_{N-1}^{(1)}\oplus A_{N-1}^{(1) },\ A_{N-1}^{(1) }} \right) $ at level (k, 1). This spin-5 Casimir operator consisted of the quintic, quartic (with one derivative) and cubic (with two derivatives) WZW currents contracted with SU(N) invariant tensors. The three-point functions with two scalars for all values of ’t Hooft coupling in the large N limit were obtained by analyzing the zero-mode eigenvalue equations carefully. These three-point functions were dual to those in AdS 3 higher spin gravity theory with matter. Furthermore, the exact three-point functions that hold for any finite N and k are obtained. The zero mode eigenvalue equations for the spin-5 current in CFT coincided with those of the spin-5 field in asymptotic symmetry algebra of the higher spin theory on the AdS 3. This paper also describes the structure constant appearing in the spin-4 Casimir operator from the OPE between the spin-3 Casimir operator and itself for N = 4, 5 in the more general coset minimal model with two arbitrary levels (k 1 , k 2).

Cubic fourfolds containing a plane and a quintic del Pezzo surface

Abstract: We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Cliord algebra over the K3 surface S of degree 2 arising from X. Specically, we show that in the moduli space of cubic fourfolds, the intersection of divisorsC8\C14 has ve irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaan and has nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov’s derived categorical conjecture on the rationality of cubic fourfolds.

Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation

Abstract: In this article we define a new class of Pythagorean-Hodograph curves built-upon a six-dimensional mixed algebraic-trigonometric space, we show their fundamental properties and compare them with their well-known quintic polynomial counterpart. A complex representation for these curves is introduced and constructive approaches are provided to solve different application problems, such as interpolating C 1 Hermite data and constructing spirals as G 2 transition elements between a line segment and a circle, as well as between a pair of external circles.

C2 interpolation of spatial data subject to arc-length constraints using Pythagorean-hodograph quintic splines

Abstract: In order to reconstruct spatial curves from discrete electronic sensor data, two alternative C^2 Pythagorean-hodograph (PH) quintic spline formulations are proposed, interpolating given spatial data subject to prescribed constraints on the arc length of each spline segment. The first approach is concerned with the interpolation of a sequence of points, while the second addresses the interpolation of derivatives only (without spatial localization). The special structure of PH curves allows the arc-length conditions to be expressed as algebraic constraints on the curve coefficients. The C^2 PH quintic splines are thus defined through minimization of a quadratic function subject to quadratic constraints, and a close starting approximation to the desired solution is identified in order to facilitate efficient construction by iterative methods. The C^2 PH spline constructions are illustrated by several computed examples.

Log canonical thresholds of certain Fano hypersurfaces

Abstract: We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kahler–Einstein metrics if they are general.

Growth of Sobolev norms for the quintic NLS on $\mathbb T^2$

Abstract: We study the quintic Non Linear Schrodinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis.

Smooth attractors for the quintic wave equations with fractional damping.

Abstract: Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existence of a smooth global and exponential attractors of finite Hausdorff and fractal dimension is verified.

Center of planar quintic quasi--homogeneous polynomial differential systems

Abstract: In this paper we first characterize all quasi--homogeneous but non--homogeneous planar polynomial differential systems of degree five, and then among which we classify all the ones having a center at the origin. Finally we present the topological phase portrait of the systems having a center at the origin.

Interpolation by G 2 Quintic Pythagorean-Hodograph Curves

Abstract: In this paper, the G interpolation by Pythagorean-hodograph (PH) quintic curves in R, d ≥ 2, is considered. The obtained results turn out as a useful tool in practical applications. Independently of the dimension d, they supply a G quintic PH spline that locally interpolates two points, two tangent directions and two curvature vectors at these points. The interpolation problem considered is reduced to a system of two polynomial equations involving only tangent lengths of the interpolating curve as unknowns. Although several solutions might exist, the way to obtain the most promising one is suggested based on a thorough asymptotic analysis of the smooth data case. The numerical algorithm traces this solution from a particular set of data to the general case by a homotopy continuation method. Numerical examples confirm the efficiency of the proposed method. AMS subject classifications: 65D05, 65D17

A Selection of Lower Bounds for Arithmetic Circuits

Abstract: Polynomials originated in classical mathematical studies concerning geometry and solutions to systems of equations. They feature in many classical results in algebra, number theory and geometry e.g. in Galois and Abel’s resolution of the solvability via radicals of a quintic, Lagrange’s theorem on expressing every natural number as a sum of four squares and the impossibility of trisecting an angle (using ruler and compass). In modern times, computer scientists began to investigate as to what functions can be (efficiently) computed. Polynomials being a natural class of functions, one is naturally lead to the following question:

Quintic B-spline method for function reconstruction from integral values of successive subintervals

Abstract: In this paper, we study a new method for function reconstruction from given integral values of successive subintervals by using quintic B-splines. The new method does not need any additional data and it is easy to implement. We are able to reconstruct the original function and its first-order to fourth-order derivatives. The reconstruction errors are well studied. Numerical results show that our method is very effective.

Usage of Higher Order B-splines in Numerical Solution of Fisher's Equation

Abstract: In this paper, the collocation method is performed with quintic B-spline functions on a uniform mesh to obtain the numerical solutions of Fisher's equation. Crank-Nicolson method is used for time dis- cretization. Von Neumann stability analysis shows that the given method is conditionally stable. In order to observe the effects of reaction and diffusion, four test problems related to pulse disturbance, step disturbance, super-speed wave and strong reaction are studied. A comparison between the obtained results and some earlier studies is presented.

Point compression for the trace zero subgroup over a small degree extension field

Abstract: Using Semaev's summation polynomials, we derive a new equation for the $\mathbb{F}_q$-rational points of the trace zero variety of an elliptic curve defined over $\mathbb{F}_q$. Using this equation, we produce an optimal-size representation for such points. Our representation is compatible with scalar multiplication. We give a point compression algorithm to compute the representation and a decompression algorithm to recover the original point (up to some small ambiguity). The algorithms are efficient for trace zero varieties coming from small degree extension fields. We give explicit equations and discuss in detail the practically relevant cases of cubic and quintic field extensions.

Distribution of orders in number fields

Abstract: In this paper we study the distribution of orders of bounded discriminants in number fields. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field.

A mirror theorem for the mirror quintic

Abstract: The celebrated Mirror theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of its mirror dual orbifold. In this article, we establish a mirror-dual statement. Namely, the B model of the Fermat quintic threefold is shown to be equivalent to the A model of its mirror, and hence establishes the mirror symmetry as a true duality. 14N35; 53D45

Quintic threefolds and Fano elevenfolds

Abstract: The derived category of coherent sheaves on a general quintic threefold is a central object in mirror symmetry. We show that it can be embedded into the derived category of a certain Fano elevenfold. Our proof also generates related examples in different dimensions.