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We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

In this study, fractional differential equations having quintic nonlinearity are considered by proposing an accurate numerical method based on the matching polynomial and matrix-collocation system. This method provides an integration between matrix and fractional derivative, which makes it fast and efficient. A hybrid computer program is designed by making use of the fast algorithmic structure of the method. An error analysis technique consisting of the fractional-based residual function is constructed to scrutinize the precision of the method. Some error tests are also performed. Figures and tables present the consistency of the approximate solutions of highly stiff model problems. All results point out that the method is effective, simple, and eligible. © 2019 John Wiley & Sons, Ltd.

We prove the genus-one restriction of the all-genus Landau-Ginzburg/Calabi-Yau conjecture of Chiodo and Ruan, stated in terms of the geometric quantization of an explicit symplectomorphism determined by genus-zero invariants. This provides the first evidence supporting the higher-genus Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold, and exhibits the first instance of the "genus zero controls higher genus" principle, in the sense of Givental's quantization formalism, for non-semisimple cohomological field theories., 27 pages, comments welcome!

The triple conformal geometric algebra (TCGA) for the Euclidean R2-plane extends CGA as the product of 3 orthogonal CGAs and thereby the representation of geometric entities to general cubic plane curves and certain cyclidic (or roulette) quartic, quintic, and sextic plane curves. The plane curve entities are 3-vectors that linearize the representation of nonlinear curves, and the entities are inner product null spaces with respect to all points on the represented curves. Each inner product null space entity also has a dual geometric outer product null space form. Orthogonal or conformal (angle preserving) operations (as versors) are valid on all TCGA entities for inversions in circles, reflections in lines, and by compositions thereof, isotropic dilations from a given center point, translations, and rotations around arbitrary points in the plane. A further dimensional extension of TCGA also provides a method for anisotropic dilations. Intersections of any TCGA entity with a point, point pair, line, or circle are possible. The TCGA defines commutator-based differential operators in the coordinate directions that can be combined to yield a general n-directional derivative.

We describe a computational method to compute spectra and slowly decaying eigenfunctions of linearizations of the cubic–quintic complex Ginzburg–Landau equation about numerically determined stationary solutions. We compare the results of the method to a formula for an edge bifurcation obtained using the small dissipation perturbation theory of Kapitula and Sandstede. This comparison highlights the importance for analytical studies of perturbed nonlinear wave equations of using a pulse ansatz in which the phase is not constant, but rather depends on the perturbation parameter. In the presence of large dissipative effects, we discover variations in the structure of the spectrum as the dispersion crosses zero that are not predicted by the small dissipation theory. In particular, in the normal dispersion regime we observe a jump in the number of discrete eigenvalues when a pair of real eigenvalues merges with the intersection point of the two branches of the continuous spectrum. Finally, we contrast the method to computational Evans function methods.

For $n=3$, 4, and 5, we prove that, when $S_n$-number fields of degree $n$ are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices., 12 pages

If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$, $q(X)=0$, $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$, so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$, $16\mathsf{A}_{2}$, $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$.

This classical Korteweg capillarity system is here encapsulated in a quintic derivative nonlinear Schrodinger equation for a model Karman-Tsien type capillarity law. An integrable subsystem is isolated and invariants of motion are use to construct novel exact solutions. The latter involve parameters introduced via a gauge and reciprocal transformation.

SUMMARY This paper introduces a time-marching algorithm for linear systems based on quintic Hermite polynomial displacement interpolation. This fully implicit, single-step collocation method enforces acceleration and jerk constraints at a point in time that is selected by the user. The collocation point, then, is the parameter that determines accuracy, stability, and high-frequency damping behavior. Numerical experiments indicate that the method converges at a rate of four, even for problems involving harmonic external forcing and viscous damping. Significantly, the convergence rate also does not deteriorate when algorithmic damping is employed. Period error is extremely small, on the order of a few percent at moderate frequencies even with algorithmic damping. The undamped algorithm is shown to have the highest accuracy and to be symplectic, but it is also conditionally stable and nondissipative—the exact stability limits are calculated. A wide range of parameter settings provide an unconditionally stable algorithm, which preserves low modes while rapidly dissipating high modes. Three finite element analysis numerical examples demonstrate the capabilities of the algorithm on one-dimensional linear dynamic problems. This article is a US Government work and is in the public domain in the USA.

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.

In order to investigate questions concerning changes in the atmosphere and other components of the Earth system, it is desirable to have a seamless modelling system that operates across a wide range of time-scales and horizontal resolutions. The development of a low-resolution configuration of the model would greatly assist in addressing these scientific questions whose long time-scales or complexity are computationally demanding—a cost that cannot be sustained by higher resolutions. However, in order to be confident that results remain robust to changes in resolution, it is essential that such models include the same underlying fundamental processes as those at higher resolution. Some resolution-dependent biases are found to grow (notably when horizontal resolution decreases below ∼150 km) as a result of the lack of transient eddy kinetic energy (TEKE). The advection scheme used by many of the current general circulation models follows the semi-Lagrangian (SL) dynamics approach. Because of the need for interpolation to the departure point, the scheme is highly diffusive when low-order interpolation schemes are used. We analyse how the impact of the high diffusivity of SL schemes leads to a lack of TEKE, which inhibits the development of mid-latitude variability phenomena such as synoptic cyclones or blocking events. Conversely, some examples where the performance of the low resolution is acceptable are provided. We investigate two different traceable solutions to partially offset the misrepresentation of dynamical features in the low-resolution version: a change of the interpolation scheme from quasi-cubic to quintic interpolation; and vorticity confinement: a parametrization designed to preserve vorticity features in dissipative numerical schemes. Both solutions are found to increase TEKE and improve many of the mid-latitude metrics considered. Copyright © 2012 British Crown copyright, the Met Office. Published by John Wiley & Sons Ltd.

In this paper looking for traveling wave solutions, we find that when the polynomial of velocity is quintic the generalized Ostrovsky equation (GOE) has abundant exact solutions that can be expressed in terms of the Jacobi elliptic functions. Hence, the GOE has a plenty of periodic waves, solitary waves, compactons, etc. These solutions are derived from the solutions of a simple non-linear ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd.

In order to find analytically the travelling waves of partially integrable autonomous nonlinear partial differential equations, many methods have been proposed over the ages: "projective Riccati method", "tanh-method", "exponential method", "Jacobi expansion method", "new ...", etc. The common default to all these "truncation methods" is to only provide some solutions, not all of them. By implementing three classical results of Briot, Bouquet and Poincare', we present an algorithm able to provide in closed form \textit{all} those travellingz waves which are elliptic or degenerate elliptic, i.e. rational in one exponential or rational. Our examples here include the Kuramoto-Sivashinsky equation and the cubic and quintic complex Ginzburg-Landau equations., Comment: 17 pages, to appear, Studies in Applied Mathematics

This paper introduces a quasi-interpolation method for reconstruction of data sampled on the Body Centered Cubic (BCC) lattice. The reconstructions based on this quasi-interpolation achieve the optimal approximation order offered by the shifts of the quintic box spline on the BCC lattice. We also present a local FIR filter that is used to filter the data for quasi-interpolation. We document the improved quality and fidelity of reconstructions after employing the introduced quasi-interpolation method. Finally the resulting quasi-interpolation on the BCC sampled data are compared to the corresponding quasi-interpolation method on the Cartesian sampled data.

We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in MAGMA. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of III [2], for families of curves of genus 2 of the form y2 = f(x), where f(x) is a quintic containing an irreducible cubic factor.

In this paper we review our experimental and theoretical investigations of ionization waves in a low-pressure discharge of neon. The main results are connected with the transition between regular and irregular wave motions. The experimental observed stability boundaries can well be understood within the hydrodynamic model. The point is the explanation of the experimentally observed mode transitions by means of a quintic amplitude equation of the Ginzburg-Landau type. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The implementation of two global surface descriptions in the physical-optics synthesis of shaped reflector antennas is compared with a view toward reducing the computational effort involved in the minimization of an appropriate objective function. The versatile behavior of quintic pseudosplines, recently explored in geometrical-optics syntheses, is now confronted with the well-established adequacy of Jacobi polynomial-sinusoidal expansions within the framework of a single offset reflector design for satellite communications. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 25: 412–416, 2000.

A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over the set of rational numbers. Remarks are given concerning the extension of these results to forms defined over number fields.

We study the weakly nonlinear evolution of acoustic instability of a plane- parallel polytrope with thermal dissipation in the form of Newton's law of cooling. The most unstable horizontal wavenumbers form a band around zero and this permits the development of a nonlinear pattern theory leading to a complex Ginzburg-Landau equation (CGLE). Numerical solutions for a subcritical, quintic CGLE produce vertically oscillating, localized structures that resemble the oscillons observed in recent experiments of vibrated granular material., 12 Latex pages using aasms4.sty, 2 postscript figures, submitted to the proceedings of the Florida Workshop in Nonlinear Astrophysics and Physics

In this paper, we present an optimal polynomial controller for reducing the peak response quantities of seismically excited non-linear or hysteretic building systems. A performance index, that is quadratic in control and polynomial of any order in non-linear states, is considered. The performance index is minimized based on the Hamilton-Jacobi-Bellman equation using a polynomial function of non-linear states, which satisfies all the properties of a Lyapunov function. The resulting optimal controller is a summation of polynomials in non-linear states, i.e. linear, cubic, quintic, etc. Gain matrices for different parts of the controller are determined from Riccati and Lyapunov matrix equations. Numerical simulation results indicate that the percentage of reduction for the selected peak response quantity increases with the increase of the earthquake intensity. Such load adaptive properties are very desirable, since the intensity of the earthquake ground acceleration is stochastic in nature. The proposed optimal polynomial controller is an effective and viable control method for non-linear or hysteretic civil engineering structures. It is an addition to available control methods in the literature.

then the canonical model C of C isthe base locus of a net of quadric hypersurfaces in P£. The discriminant curve of sucha net is a plane quintic which is the union of a non-singular quartic F and a line L.Moreover, F is endowed (in a natural way) with a non-effective theta characteristicn (that is, an invertibl rje shea on F sucf h that rj

A recent study demonstrated the superior stability of quintic pseudosplines, when compared with usual oscillatory series expansions, for global interpolation of numerically defined reflector surfaces presenting critical behavior of their principal curvatures. Although interpolating functions with continuous derivatives up to second order are necessary for ray-optical field evaluations, C-class surface descriptions are sufficient when the antenna secondary field is calculated by integration of physical-optics-induced currents. In this connection, the applicability of cubic pseudosplines is explored via numerical comparison with previously obtained results from the implementation of the quintic ones. © 1995 John Wiley & Sons. Inc.

We review recent results on pattern selection in the one- or two-dimensional reaction-diffusion system xt - Δx = f(x, y, λ), y t = eg(x, y), subject to global (〈x〉 = x 0) or long-range interaction; the source functions may be realistic kinetic functions or simple cubic or quintic f(x) functions for which the system admits inversion symmetry. This review discusses: (i) physical sources of such interactions and experimental observations in catalytic and electrochemical systems; (ii) the main emerging patterns and their classification according to their symmetry; (iii) the bifurcation between patterns; (iv) patterns when f(x) = 0 is tristable and can sustain several fronts.

Composite boundary elements are formed for plate-bending problems by attaching a circular sector to a triangle. Elements utilizing cubic and quintic interpolation are developed for convex and concave boundaries. Smaller error and less numerical effort result when triangular elements are replaced by the composite boundary elements on clamped and most simply supported convex, boundaries. For domains with concave boundaries, greater accuracy is obtained when the quintic composite boundary element replaces the triangular element in fine mesh configurations.

In this paper a curved finite element for the solution of shells of revolution is presented. The shell geometry in general, is approximated by a third order interpolation function. The displacement field has four components—three displacements and the rotation of the meridional plane of the of the shell. The displacement field is interpolated with cubic and quintic polynomials. The linear system involving the nodal unknowns is obtained from the principle of virtual work, together with the colateral condition relating the rotation to the other components of the displacement field. It is shown that broken generating lines for the shell surface do not limit the convergence of the solution. Finally, for numerical applications for static and dynamical problems presented.

A family of high-order, conforming triangles for the finite element analysis of plate bending problems is developed, using the displacement method of formulation. The development proceeds in two steps. A family of non-conforming triangles, based on polynomial expansions of order >4 is first generated, and its performance found to be unacceptable. Nodal variables for the non-conforming family are selected from the set w, wx, wy, wn. The elements are then made to conform, by adding a twist nodal variable wnt at the mid-side nodes. Rational functions define the three additional degrees-of-freedom. Simple and very accurate procedures solve the problems associated with the use of such functions. Numerical results are presented for the conforming triangles of order 4 and 7, following a brief description of the two elements. A listing of the explicit shape functions for the quartic triangle is included. The numerical results are very good. In particular, they are superior per degree-of-freedom to the results predicted by the various non-conforming and conforming, cubic and related elements at present used in computer packages. Together with a similar performance by the quintic member10 of the family, they validate the proposed theoretical solution.

Two explicit representation of a C1 quintic interpolant over triangles are given. These representations are derived by generalization of Coons' methods and Bernstein-Bezier methods, respectively.

Quintic designs, a class of PBIB designs having five associate classes, are defined. Properties of NN', where N is the incidence matrix, are examined to present necessary conditions for the existence of quintic designs. The variances of elementary treatment comparisons are found. The block structure of two types of quintic design is studied. Nous definissons les plans ‘quintic’, une classe de blocs imcomplets partiellement equilibres ayant cing classes associees. Nous etudions les proprietes de NN' ou N est la matrice d'incidence, dnecessaires pour que les conditions d'existence de tels plans soient satisfaites. Nous derivons lea variances des comparaisons de traitement elementaire, et etudions la structure de bloc de deux types de plans ‘quintic’.

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