Showing papers on "Quintic function published in 2011"

Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in H1(T3)

Abstract: A refined trilinear Strichartz estimate for solutions to the Schrodinger equation on the flat rational torus T3 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 Hs(T3) for all s≥1. This is the first energy-critical global well-posedness result in the setting of compact manifolds.

Equations for secant varieties of Veronese and other varieties

Abstract: New classes of modules of equations for secant varieties of Veronese varieties are defined using representation theory and geometry. Some old modules of equations (catalecticant minors) are revisited to determine when they are sufficient to give scheme-theoretic defining equations. An algorithm to decompose a general ternary quintic as the sum of seven fifth powers is given as an illustration of our methods. Our new equations and results about them are put into a larger context by introducing vector bundle techniques for finding equations of secant varieties in general. We include a few homogeneous examples of this method.

Resonant dynamics for the quintic non linear Schr\"odinger equation

Abstract: We consider the quintic nonlinear Schr\"odinger equation (NLS) on the circle. We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set, which have a non trivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomena does not depend on the choice of the resonant set. The dynamical result is obtained by calculating a resonant normal form up to order 10 of the Hamiltonian of the quintic NLS and then by isolating an effective term of order 6. Notice that this phenomena can not occur in the cubic NLS case for which the amplitudes of the Fourier modes are almost actions, i.e. they are almost constant.

Analytical solitary-wave solutions of the generalized nonautonomous cubic-quintic nonlinear Schrödinger equation with different external potentials

Abstract: A large family of analytical solitary wave solutions to the generalized nonautonomous cubic-quintic nonlinear Schr\"odinger equation with time- and space-dependent distributed coefficients and external potentials are obtained by using a similarity transformation technique. We use the cubic nonlinearity as an independent parameter function, where a simple procedure is established to obtain different classes of potentials and solutions. The solutions exist under certain conditions and impose constraints on the coefficients depicting dispersion, cubic and quintic nonlinearities, and gain (or loss). We investigate the space-quadratic potential, optical lattice potential, flying bird potential, and potential barrier (well). Some interesting periodic solitary wave solutions corresponding to these potentials are then studied. Also, properties of a few solutions and physical applications of interest to the field are discussed. Finally, the stability of the solitary wave solutions under slight disturbance of the constraint conditions and initial perturbation of white noise is discussed numerically; the results reveal that the solitary waves can propagate in a stable way under slight disturbance of the constraint conditions and the initial perturbation of white noise.

Design of rational rotation–minimizing rigid body motions by Hermite interpolation

Abstract: The construction of space curves with rational rotation-minimizing frames (RRMF curves) by the interpolation of G1 Hermite data, i.e., initial/final points pi and pf and frames (ti, ui, vi) and (tf , uf , vf ), is addressed. Noting that the RRMF quintics form a proper subset of the spatial Pythagorean–hodograph (PH) quintics, characterized by a vector constraint on their quaternion coefficients, and that C1 spatial PH quintic Hermite interpolants possess two free scalar parameters, sufficient degrees of freedom for satisfying the RRMF condition and interpolating the end points and frames can be obtained by relaxing the Hermite data from C1 to G1. It is shown that, after satisfaction of the RRMF condition, interpolation of the end frames can always be achieved by solving a quadratic equation with a positive discriminant. Three scalar freedoms then remain for interpolation of the end–point displacement pf −pi, and this can be reduced to computing the real roots of a degree 6 univariate polynomial. The nonlinear dependence of the polynomial coefficients on the prescribed data precludes simple a priori guarantees for the existence of solutions in all cases, although existence is demonstrated for the asymptotic case of densely–sampled data from a smooth curve. Modulation of the hodograph by a scalar polynomial is proposed as a means of introducing additional degrees of freedom, in cases where solutions to the end–point interpolation problem are not found. The methods proposed herein are expected to find important applications in exactly specifying rigid–body motions along curved paths, with minimized rotation, for animation, robotics, spatial path planning, and geometric sweeping operations.

Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear

Abstract: Using a variational approximation we study discrete solitons of a nonlinear Schrodinger lattice with a cubic-quintic nonlinearity Using an ansatz with six parameters we are able to approximate bifurcations of asymmetric solutions connecting site-centered and bond-centered solutions and resulting in the exchange of their stability We show that the numerical and variational approximations are quite close for solitons of small powers 1 Introduction The variational approximation (VA) has long been used as a semi-analytic technique to approximate solitary wave solutions of nonlinear evo- lution equations with an underlying Hamiltonian structure (13) There have been a number of papers exploring the VA with four parameters as a relevant approx- imation of localized modes in discrete nonlinear Schrodinger (DNLS) equations (6, 14, 19) Kaup (10) extended the variational approximation with six parameters that allowed him to construct not only site-centered solutions (also called on-site solitons) from (14) but also the bond-centered solutions (solitons centered at a mid- point between two adjacent sites also known as inter-site solitons) Site-centered and bond-centered solitons were recently considered in the context of the DNLS equations with competing cubic focusing and quintic defocusing non- linearities both in the space of one (3) and two (4) lattice dimensions It was found that the two branches exchange their stability while continued with respect to the underlying parameters A salient feature of this stability exchange is that the two branches of site-centered and bond-centered solitions do not intersect directly but are connected by an intermediate branch of asymmetric solitons It was argued in (4) that the discrete solitons have enhanced mobility near the regimes of stability inversion These properties were originally discovered in the DNLS equations with a saturable nonlinearity both in the space of one (8) and two (22) dimensions as 2000 Mathematics Subject Classication Primary: 35Q55, 37K60, 35B32; Secondary: 58E30

Global well-posedness of the Gross--Pitaevskii and cubic-quintic nonlinear Schr\"odinger equations with non-vanishing boundary conditions

Abstract: We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schr\"odinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.

Classical ladder operators, polynomial Poisson algebras and classification of superintegrable systems

Abstract: We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems respectively with a third and a fourth order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformed Lissajous's figures.

Spatial solitons with the odd and even symmetries in (2+1)-dimensional spatially inhomogeneous cubic-quintic nonlinear media with transverse W-shaped modulation

Abstract: We firstly investigate analytical spatial soliton solutions with the odd and even symmetries in (2+1)-dimensional spatially inhomogeneous cubic-quintic nonlinear media considering transverse W-shaped modulation. The power of localized states increases one by one along the line y = x when the soliton order number n increases. The stability analysis and numerical calculations show that stable fundamental solitons exist while higher order solitons are unstable in three nonlinear media, i.e. the defocusing cubic and focusing quintic nonlinear medium, focusing cubic and defocusing quintic nonlinear medium, and focusing cubic and focusing quintic nonlinear medium. While all solitons (even fundamental solitons) are unstable and ultimately decay into noise in the defocusing cubic and defocusing quintic nonlinear medium.

Spin curves and Scorza quartics

Abstract: In the paper (Takagi and Zucconi in “On blow-ups of the quintic del Pezzo 3-fold and varieties of power sums of quartic hypersurfaces”, pp 1–40, preprint, submitted, 2008), we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces. In this paper, we show that these quartics coincide with the Scorza quartics of general pairs of trigonal curves and ineffective theta characteristics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartics for any general pairs of curves and ineffective theta characteristics. We also give descriptions of the moduli spaces of trigonal even spin curves.

The construction of high-order b-spline wavelets and their decomposition relations for fault detection and localisation in composite beams

Abstract: B-spline scaling functions and wavelets have found wide applicability in many scientific and practical problems thanks to their unique properties. They show considerably better results in comparison to other wavelets, and they are used as well in mathematical approximations, signal processing, image compression, etc. But only the first four wavelets from this family were mathematically formulated. In this work, the author formulates the quartic, quintic and sextic B-spline wavelets and their decomposition relations in explicit form. This allows for the improvement of the sensitivity of fault detection and localisation in composite beams using discrete wavelet transform with decomposition.

High order instabilities of the Poincaré solution for precessionally driven flow

Abstract: Sloudsky in 1895 and Poincare in 1910 were the first to derive solutions for the flow driven in the Earth's fluid core by the luni-solar precession. In 1993, Kerswell investigated the stability of this so-called “Poincare flow” by applying a method devised in 1992 by Ponomarev and Gledzer to study the instability of flows with elliptical streamlines. They represented the components of the perturbed flow by sums of polynomials. Kerswell restricted attention to the linear and quadratic cases. Here cubic, quartic, quintic, and sextic generalizations are developed. Instabilities are located in new areas of parameter space, including some that verge on the small oblateness of the Earth's core

Invariant theory for the elliptic normal quintic, I. Twists of X(5)

Abstract: A genus one curve of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our algorithm for computing the invariants [12] and to extend our method in [14] for computing equations for visible elements of order 5 in the Tate-Shafarevich group of an elliptic curve. As a special case of the latter we find a formula for the family of elliptic curves 5-congruent to a given elliptic curve in the case the 5-congruence does not respect the Weil pairing. We also give an algorithm for doubling elements in the 5-Selmer group of an elliptic curve, and make a conjecture about the matrices representing the invariant differential on a genus one normal curve of arbitrary degree.

Quintic surfaces with maximum and other Picard numbers

Abstract: This paper investigates the Picard numbers of quintic surfaces. We give the first example of a complex quintic surface in P3 with maximum Picard number ρ = 45. We also investigate its arithmetic and determine the zeta function. Similar techniques are applied to produce quintic surfaces with several other Picard numbers that have not been achieved before.

Charles Hermite’s stroll through the Galois fields

Abstract: — Although everything seems to oppose the two mathematicians, Charles Hermite’s role was crucial in the study and diffusion of Évariste Galois’s results in France during the second half of the nineteenth century. The present article examines that part of Hermite’s work explicitly linked to Galois, the reduction of modular equations in particular. It shows how Hermite’s mathematical convictions—concerning effectiveness or the unity of algebra, analysis and arithmetic—shaped his interpretation of Galois and of the paths of development Galois opened. Reciprocally, Hermite inserted Galois’s results in a vast synthesis based on invariant theory and elliptic functions, the memory of which is in great part missing in current Galois theory. At the end of the article, we discuss some methodological issues this raises in the interpretation of Galois’s works and their posterity. Texte reçu le 14 juin 2011, accepté le 29 juin 2011. C. Goldstein, Histoire des sciences mathématiques, Institut de mathématiques de Jussieu, Case 247, UPMC-4, place Jussieu, F-75252 Paris Cedex (France). Courrier électronique : cgolds@math.jussieu.fr Url : http://people.math.jussieu.fr/~cgolds/ 2000 Mathematics Subject Classification : 01A55, 01A85; 11-03, 11A55, 11F03, 12-03, 13-03, 20-03.

Global classification of a class of cubic vector fields whose canonical regions are period annuli

Abstract: We study cubic vector fields with inverse radial symmetry, i.e. of the form ẋ = δx - y + ax2 + bxy + cy2 + σ(dx - y)(x2 + y2), ẏ = x + δy + ex2 + fxy + gy2 + σ(x + dy) (x2 + y2), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form. Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular, there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincare disc. Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular, we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover, we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.

An Efficient Numerical Method for the Quintic Complex Swift-Hohenberg Equation

Abstract: In this paper, we present an efficient time-splitting Fourier spectral method for the quintic complex Swift-Hohenberg equation. Using the Strang time-splitting tech- nique, we split the equation into linear part and nonlinear part. The linear part is solved with Fourier Pseudospectral method; the nonlinear part is solved analytically. We show that the method is easy to be applied and second-order in time and spectrally accurate in space. We apply the method to investigate soliton propagation, soliton interaction, and generation of stable moving pulses in one dimension and stable vortex solitons in two dimensions. AMS subject classifications: 65M70, 65Z05

Solution of Stochastic Cubic and Quintic Nonlinear Diffusion Equation Using WHEP, Pickard and HPM Methods

Abstract: In this paper, the cubic and quintic diffusion equation under stochastic non homogeneity is solved using Wiener- Hermite expansion and perturbation (WHEP) technique, Homotopy perturbation method (HPM) and Pickard approximation technique. The analytic solution of the linear case is obtained using Eigenfunction expansion .The Picard approximation method is used to introduce the first and second order approximate solution for the non linear case. The WHEP technique is also used to obtain approximate solution under different orders and different corrections. The Homotopy perturbation method (HPM) is also used to obtain some approximation orders for mean and variance. Using mathematica-5, the methods of solution are illustrated through figures, comparisons among different methods and some parametric studies.

An accurate and precise polynomial model of angular interrogation surface plasmon resonance data

Abstract: We present a simple, statistically based method of fitting waveguide-coupled surface plasmon resonance (WCSPR) angular interrogation experiment data in the vicinity of the resonance angle using an appropriate polynomial model. This method allows one to determine the resonance angle to within precision of as little as 2% of the sampling step size, with mean results averaging about 8% of the step size, better than an order of magnitude improvement over no regression, achieved with little effort. In testing this method, we use theoretical and experimental WCSPR data. We have compared the statistical significance of using additional terms in a given polynomial representation. F -Ratio tests based on the “extra sum of squares” principle indicate that, in the vicinity of the resonance, approximately 20 millidegrees about the minimum, the addition of quintic or higher order terms to the quartic polynomial representation is not statistically significant. We have found that both cubic and quartic models produce estimates of the position of the minimum in which the confidence interval is both accurate and precise with an error of less than one tenth of a millidegree. In addition, a similar analysis of theoretical calculations suggests that this polynomial method, which is generally applicable to the determination of extrema in any spectrum, is capable of very high accuracy and precision.

Approximately Quintic and Sextic Mappings Form -Divisible Groups into Ŝerstnev Probabilistic Banach Spaces: Fixed Point Method

Abstract: Using the fixed point method, we investigate the stability of the systems of quadratic-cubic and additive-quadratic-cubic functional equations with constant coefficients form r-divisible groups into Ŝerstnev probabilistic Banach spaces.

Traveling Wave Evolutions of a Cosh-Gaussian Laser Beam in Both Kerr and Cubic Quintic Nonlinear Media Based on Mathematica

Abstract: With the aid of Mathematica, three auxiliary equations, i.e. the Riccati equation, the Lenard equation and the Hyperbolic equation, are employed to investigate traveling wave solutions of a cosh-Gaussian laser beam in both Kerr and cubic quintic nonlinear media. As a result, many traveling wave solutions are obtained, including soliton-like solutions, hyperbolic function solutions and trigonometric function solutions.

The Bring-Jerrard quintic equation, its solutions and a formula for the universal gravitational constant

Abstract: In this research the Bring-Jerrard quintic polynomial equation is investigated for a formula. Firstly, an explanation given as to why finding a formula and the equation being unsolvable by radicals may appear contradictory when read out of context. Secondly, the reason why some mathematical software programs may fail to render a conclusive test of the formula, and how that can be corrected is explained. As an application, this formula is used to determine another formula that expresses the gravitational constant in terms of other known physical constants. It is also explained why up to now it has been impossible to determine this expression using the current underlying theoretical basis.

Higher-Order Approximation of Cubic-Quintic Duffing Model

Abstract: We apply an Artificial Parameter Lindstedt-Poincar é Method (APL-PM) to find improved approximate solutions for strongly nonline ar Duffing oscillators with cubic–quintic nonlinear restoring force. This approach yields sim ple linear algebraic equations instead of nonlinear algebraic equations without analytical so luti n which makes it a unique solution. It is demonstrated that this method works very well for t he whole range of parameters in the case of the cubic-quintic oscillator, and excellent agreeme nt of the approximate frequencies with the exact one has been observed and discussed. Moreover , it is not limited to the small parameter such as in the classical perturbation method. Inter es ingly, This study revealed that the relative error percentage in the second-order approximate an alytical period is less than 0.042% for the whole parameter values. In addition, we compared th is analytical solution with the Newton– Harmonic Balancing Approach. Results indicate that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems. Utter simplicity of the solution procedure confirms that this method can be easily extended to other kinds of nonlinear evolution equations.

Derived resolution property for stacks, Euler classes and applications

Abstract: By resolving an arbitrary perfect derived object over a Deligne-Mumford stack, we define its Euler class. We then apply it to define the Euler numbers for a smooth Calabi-Yau threefold in the 4-dimensional projective space. These numbers are conjectured to be the reduced Gromov-Witten invariants and to determine the usual Gromov-Witten numbers of the smooth quintic as speculated by J. Li and A. Zinger.

Homological mirror symmetry for the quintic 3-fold

Abstract: We prove homological mirror symmetry for the quintic Calabi-Yau 3-fold. The proof follows that for the quartic surface by Seidel (arXiv:math/0310414) closely, and uses a result of Sheridan (arXiv:1012.3238). In contrast to Sheridan's approach (arXiv:1111.0632), our proof gives the compatibility of homological mirror symmetry for the projective space and its Calabi-Yau hypersurface.