Showing papers on "Quintic function published in 2015"

Hypersurfaces that are not stably rational

Abstract: A fundamental problem of algebraic geometry is to determine which varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. In particular, we want to know which smooth hypersurfaces in projective space are rational. An easy case is that smooth complex hypersurfaces of degree at least n + 2 in P are not covered by rational curves and hence are not rational. By far the most general result on rationality of hypersurfaces is Kollar’s theorem that for d at least 2d(n + 3)/3e, a very general complex hypersurface of degree d in P is not ruled and therefore not rational [13, Theorem V.5.14]. Very little is known about rationality in lower degrees, except for cubic 3-folds and quintic 4-folds [4], [19, Chapter 3]. A rational variety is also stably rational, meaning that some product of the variety with projective space is rational. Many techniques for proving non-rationality give no information about stable rationality. Voisin made a breakthrough in 2013 by showing that a very general quartic double solid (a double cover of P ramified over a quartic surface) is not stably rational [23]. These Fano 3-folds were known to be non-rational over the complex numbers, but stable rationality was an open question. Voisin’s method was to show that the Chow group of zero-cycles is not universally trivial (that is, the Chow group becomes nontrivial over some extension of the base field), by degenerating the variety to a nodal 3-fold which has a resolution of singularities X with nonzero torsion in H3(X,Z). Colliot-Thelene and Pirutka simplified and generalized Voisin’s degeneration method. They deduced that very general quartic 3-folds are not stably rational [6]. This was striking, in that non-rationality of smooth quartic 3-folds was the original triumph of Iskovskikh-Manin’s work on birational rigidity, while stable rationality of these varieties was unknown [11]. Beauville applied the method to prove that very general sextic double solids, quartic double 4-folds, and quartic double 5-folds are not stably rational [1, 2]. In this paper, we show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d ≥ 2d(n + 2)/3e and n ≥ 3, a very general complex hypersurface of degree d in P is not stably rational (Theorem 2.1). The theorem covers all the degrees in which Kollar proved non-rationality. In fact, we get a bit more, since Kollar assumed d ≥ 2d(n + 3)/3e. For example, very general quartic 4-folds are not stably rational, whereas it was not even known whether these varieties are rational. The method applies to some smooth hypersurfaces over Q in each even degree. Section 3 gives some examples over Q which are not stably rational over C. The idea is that the most powerful results are obtained by degenerating a smooth complex variety to a singular variety in positive characteristic, rather than to a sin-

Forces and stress in second order Møller-Plesset perturbation theory for condensed phase systems within the resolution-of-identity Gaussian and plane waves approach

Abstract: The forces acting on the atoms as well as the stress tensor are crucial ingredients for calculating the structural and dynamical properties of systems in the condensed phase. Here, these derivatives of the total energy are evaluated for the second-order Moller-Plesset perturbation energy (MP2) in the framework of the resolution of identity Gaussian and plane waves method, in a way that is fully consistent with how the total energy is computed. This consistency is non-trivial, given the different ways employed to compute Coulomb, exchange, and canonical four center integrals, and allows, for example, for energy conserving dynamics in various ensembles. Based on this formalism, a massively parallel algorithm has been developed for finite and extended system. The designed parallel algorithm displays, with respect to the system size, cubic, quartic, and quintic requirements, respectively, for the memory, communication, and computation. All these requirements are reduced with an increasing number of processes, and the measured performance shows excellent parallel scalability and efficiency up to thousands of nodes. Additionally, the computationally more demanding quintic scaling steps can be accelerated by employing graphics processing units (GPU's) showing, for large systems, a gain of almost a factor two compared to the standard central processing unit-only case. In this way, the evaluation of the derivatives of the RI-MP2 energy can be performed within a few minutes for systems containing hundreds of atoms and thousands of basis functions. With good time to solution, the implementation thus opens the possibility to perform molecular dynamics (MD) simulations in various ensembles (microcanonical ensemble and isobaric-isothermal ensemble) at the MP2 level of theory. Geometry optimization, full cell relaxation, and energy conserving MD simulations have been performed for a variety of molecular crystals including NH3, CO2, formic acid, and benzene.

Cremona Groups and the Icosahedron

Abstract: Introduction Conjugacy in Cremona groups Three-dimensional projective space Other rational Fano threefolds Statement of the main result Outline of the book Preliminaries Singularities of pairs Canonical and log canonical singularities Log pairs with mobile boundaries Multiplier ideal sheaves Centers of log canonical singularities Corti's inequality Noether-Fano inequalities Birational rigidity Fano varieties and elliptic fibrations Applications to birational rigidity Halphen pencils Auxiliary results Zero-dimensional subschemes Atiyah flops One-dimensional linear systems Miscellanea Icosahedral Group Basic properties Action on points and curves Representation theory Invariant theory Curves of low genera SL2(C) and PSL2(C) Binary icosahedral group Symmetric group Dihedral group Surfaces with icosahedral symmetry Projective plane Quintic del Pezzo surface Clebsch cubic surface Two-dimensional quadric Hirzebruch surfaces Icosahedral subgroups of Cr2(C) K3 surfaces Quintic del Pezzo Threefold Quintic del Pezzo threefold Construction and basic properties PSL2(C)-invariant anticanonical surface Small orbits Lines Orbit of length five Five hyperplane sections Projection from a line Conics Anticanonical linear system Invariant anticanonical surfaces Singularities of invariant anticanonical surfaces Curves in invariant anticanonical surfaces Combinatorics of lines and conics Lines Conics Special invariant curves Irreducible curves Preliminary classification of low degree curves Two Sarkisov links Anticanonical divisors through the curve L6 Rational map to P4 A remarkable sextic curve Two Sarkisov links Action on the Picard group Invariant Subvarieties Invariant cubic hypersurface Linear system of cubics Curves in the invariant cubic Bring's curve in the invariant cubic Intersecting invariant quadrics and cubic A remarkable rational surface Curves of low degree Curves of degree 16 Six twisted cubics Irreducible curves of degree 18 A singular curve of degree 18 Bring's curve Classification Orbits of small length Orbits of length 20 Ten conics Orbits of length 30 Fifteen twisted cubics Further properties of the invariant cubic Intersections with low degree curves Singularities of the invariant cubic Projection to Clebsch cubic surface Picard group Summary of orbits, curves, and surfaces Orbits vs. curves Orbits vs. surfaces Curves vs. surfaces Curves vs. curves Singularities of Linear Systems Base loci of invariant linear systems Orbits of length 10 Linear system Q3 Isolation of orbits in S Isolation of arbitrary orbits Isolation of the curve L15 Proof of the main result Singularities of linear systems Restricting divisors to invariant quadrics Exclusion of points and curves different from L15 Exclusion of the curve L15 Alternative approach to exclusion of points Alternative approach to the exclusion of L15 Halphen pencils and elliptic fibrations Statement of results Exclusion of points Exclusion of curves Description of non-terminal pairs Completing the proof

Growth of Sobolev norms for the quintic NLS on 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 discussed in [15] for the case of the cubic NLS. This requires an accurate combinatorial analysis.

Mixed-Spin-P fields of Fermat quintic polynomials

Abstract: This is the first part of the project toward an effective algorithm to evaluate all genus Gromov-Witten invariants of quintic Calabi-Yau threefolds. In this paper, we introduce the notion of Mixed-Spin-P fields, construct their moduli spaces, and construct the virtual cycles of these moduli spaces.

Approximation of the KdVB equation by the quintic B-spline differential quadrature method

Abstract: In this paper, the Korteweg-de Vries-Burgers’ (KdVB) equation is solved numerically by anew differential quadrature method based on quintic B-spline functions. The weightingcoefficients are obtained by semi-explicit algorithm including an algebraic system with fivebandcoefficient matrix. The L2 and L∞ error norms and lowest three invariants 1 2 I , I and3 I have computed to compare with some earlier studies. Stability analysis of the method isalso given. The obtained numerical results show that the present method performs better thanthe most of the methods available in the literature.

Sech-type and Gaussian-type light bullet solutions to the generalized (3$$$$1)-dimensional cubic-quintic Schrödinger equation in $$\varvec{\mathcal {PT}}$$PT-symmetric potentials

Abstract: We study two kinds of the generalized (3 $$+$$ 1)-dimensional cubic-quintic Schrodinger equation in $$\mathcal {PT}$$ -symmetric potentials and obtain two families (sech-type and Gaussian-type) and four kinds of analytical light bullet (LB) solutions. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulation. Results imply that sech-type LB solutions are unstable for all parameters only in the extended Rosen–Morse potentials. Sech-type and Gaussian-type LB solutions are both stable below some thresholds for the imaginary part of other $$\mathcal {PT}$$ -symmetric potentials in the defocusing cubic and focusing quintic medium, while they are always unstable for all parameters in other media. Moreover, we discuss the broadened and compressed behaviors of LBs in inhomogeneous hyperbolic system and periodic amplification system.

Quintic B-spline collocation approach for solving generalized Black-Scholes equation governing option pricing

Abstract: We develop a numerical algorithm for solving generalized Black-Scholes partial differential equation, which arises in European option pricing. The method comprises the horizontal method of lines for time integration and ? -method, ? ? 1 / 2 , 1 ] ( ? = 1 corresponds to the back-ward Euler method and ? = 1 / 2 corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic B-spline collocation method in uniform spatial direction. The convergence analysis and stability of proposed method are discussed in detail, it is justifying that the approximate solution converges to the exact solution of orders O ( k + h 3 ) for the back-ward Euler method and O ( k 2 + h 3 ) for the Crank-Nicolson method, where k and h are mesh sizes in the time and space directions, respectively. The proposed method is also shown to be unconditionally stable. This scheme applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence. Results shown by this method are found to be in good agreement with the known exact solutions. The produced results are also seen to be more accurate than some available results given in the literature.

Exact solutions of the (2+1)-dimensional quintic nonlinear Schrödinger equation with variable coefficients

Abstract: Using the self-similarity transformation, we find analytical spatial bright and dark self-similar solitons, i.e., the similaritons, of the generalized (2+1)-dimensional quintic nonlinear Schrodinger equation with varying diffraction, nonlinearity, and gain. Characteristic examples with physically relevant behavior of these similaritons are studied, and the stability of these solutions is verified with numerical integration.

Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities.

Abstract: The existence and stability of optical solitons in the semi-infinite gap of parity-time (PT)-symmetric optical lattices with competing cubic and quintic nonlinearities are investigated numerically. The fundamental and dipole solitons can exist only with focusing quintic nonlinearity; however, they are always linearly unstable. With the competing effect between cubic and quintic nonlinearities, the strength of the quintic nonlinearity should be larger than a threshold for the solitons' existence when the strength of the focusing cubic nonlinearity is fixed. The stability of both fundamental and dipole solitons is studied in detail. When the strength of the focusing quintic nonlinearity is fixed, solitons can exist at the whole interval of the strength of the cubic nonlinearity, but only a small part of the fundamental solitons are stable. We also study numerically nonlinear evolution of stable and unstable PT solitons under perturbation.

Modular-type functions attached to mirror quintic Calabi–Yau varieties

Abstract: In this article we study a differential algebra of modular-type functions attached to the periods of a one-parameter family of Calabi–Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is generated by seven functions satisfying functional and differential equations in parallel to the modular functional equations of classical Eisenstein series and the Ramanujan differential equation. Our result is the first example of automorphic-type functions attached to varieties whose period domain is not Hermitian symmetric. It is a reformulation and realization of a problem of Griffiths from the seventies on the existence of automorphic functions for the moduli of polarized Hodge structures.

A novel time integration method for structural dynamics utilizing uniform quintic B-spline functions

Abstract: The authors have proposed a new integration method for structural dynamics by utilizing uniform quintic B-spline polynomial interpolation. In this way, with two adjustable parameters, the proposed method is successfully formulated for solving of the differential equation of motion governing a SDOF system and later generalized for a MDOF system. In the proposed method, the straightforward recurrence formulas were derived based on quintic B-spline interpolation approximation and collocation method, and the calculation process for MDOF systems was also provided. Stability analysis shows that the proposed method can attain both conditional and unconditional stability. The validity of the proposed method is verified with three numerical simulations. Compared with the latest Bathe and Noh–Bathe methods, the proposed method not only has higher computation efficiency, but also possesses better numerical dissipation characteristics.

Algorithm 954: An Accurate and Efficient Cubic and Quartic Equation Solver for Physical Applications

Abstract: We report on an accurate and efficient algorithm for obtaining all roots of general real cubic and quartic polynomials. Both the cubic and quartic solvers give highly accurate roots and place no restrictions on the magnitude of the polynomial coefficients. The key to the algorithm is a proper rescaling of both polynomials. This puts upper bounds on the magnitude of the roots and is very useful in stabilizing the root finding process. The cubic solver is based on dividing the cubic polynomial into six classes. By analyzing the root surface for each class, a fast convergent Newton-Raphson starting point for a real root is obtained at a cost no higher than three additions and four multiplications. The quartic solver uses the cubic solver in getting information about stationary points and, when the quartic has real roots, stable Newton-Raphson iterations give one of the extreme real roots. The remaining roots follow by composite deflation to a cubic. If the quartic has only complex roots, the present article shows that a stable Newton-Raphson iteration on a derived symmetric sixth degree polynomial can be formulated for the real parts of the complex roots. The imaginary parts follow by solving suitable quadratics.

Comparative Solution of Nonlinear Quintic Cubic Oscillator Using Modified Homotopy Perturbation Method

Abstract: We use modified homotopy perturbation method to find out the solution of nonlinear cubic quintic equation. Besides this method solution of the problem with the following methods is discussed, Energy Balance Method and He’s Frequency Formulation method and then compare the results with each other and Global Error Method. The results show that these three methods are effective as global error method for nonlinear cubic quintic oscillator equation with multiple nonlinear terms and have different effects on the solution. In particular, the homotopy perturbation solution is quite surprising. A cubic quintic nonlinear oscillator is used as an example to compare the results.

Algorithm 952: PHquintic: A Library of Basic Functions for the Construction and Analysis of Planar Quintic Pythagorean-Hodograph Curves

Abstract: The implementation of a library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph (PH) curves is presented using the complex representation. The special algebraic structure of PH curves permits exact algorithms for the computation of key properties, such as arc length, elastic bending energy, and offset (parallel) curves. Single planar PH quintic segments are constructed as interpolants to first-order Hermite data (end points and derivatives), and this construction is then extended to open or closed C2 PH quintic spline curves interpolating a sequence of points in the plane. The nonlinear nature of PH curves incurs a multiplicity of formal solutions to such interpolation problems, and a key aspect of the algorithms is to efficiently single out the unique “good” interpolant among them.

Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $\mathbb{R}^3$

Abstract: We prove almost sure global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $\mathbb{R}^3$ with random initial data in $ H^s(\mathbb{R}^3) \times H^{s-1}(\mathbb{R}^3)$ for $s > \frac 12$. The main new ingredient is a uniform probabilistic energy bound for approximating random solutions.

Improvements to the Deformation Method for Counting Points on Smooth Projective Hypersurfaces

Abstract: We present various improvements to the deformation method for computing the zeta function of smooth projective hypersurfaces over finite fields using $$p$$p-adic cohomology. This includes new bounds for the $$p$$p-adic and $$t$$t-adic precisions required to obtain provably correct results and gains in the efficiency of the individual steps of the method. The algorithm that we thus obtain has lower time and space complexities than existing methods. Moreover, our implementation is more practical and can be applied more generally, which we illustrate with examples of generic quintic curves and quartic surfaces.

Universal deformations of the finite quotients of the braid group on 3 strands

Abstract: We prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively, have finite rank. This is a special case of a conjecture by Broue, Malle and Rouquier for the generic Hecke algebra of an arbitrary complex reflection group. Exploring the consequences of this case, we will prove that we can determine completely the irreducible representations of this braid group for dimension at most 5, thus reproving a classification of Tuba and Wenzl in a more general framework.

Free vibration of symmetric angle ply truncated conical shells under different boundary conditions using spline method

Abstract: Free vibration of symmetric angle-ply laminated truncated conical shell is analyzed to determine the effects of frequency parameter and angular frequencies under different boundary condition, ply angles, different material properties and other parameters The governing equations of motion for truncated conical shell are obtained in terms of displacement functions The displacement functions are approximated by cubic and quintic splines resulting into a generalized eigenvalue problem The parametric studies have been made and discussed

The Craighero-Gattazzo surface is simply-connected

Abstract: We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero-Gattazzo surface is the only explicit example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod p technique and deformation theory.