Some Notes on the Positive Definite Problem of a Binary Quartic Form

In this paper, we discuss the positive definite problem of a binary quartic form and obtain a necessary and sufficient condition. In addition we give two examples to show that there are some errors in the paper [1].

FIXED POINTS AND STABILITY FOR QUARTIC MAPPINGS IN β-NORMED LEFT BANACH MODULES ON BANACH ALGEBRAS

The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equationin β-Banach modules on Banach algebras. The concept of Ulam-Hyers-Rassias stability （briefly, HUR-stability） originated from Th. M. Rassias stability theorem that appeared in his paper： On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 （1978）, 297-300.

The Center and the Fine Focus for a Class of Quartic Polynomial Poincare Equations

We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fine focus at the origin for this class of equations is at most 6. Key words quartic polynomial Poincare equation - center - fine focus - order CLC number O 175. 12 Foundation item: Supported by the National Natural Science Foundation of China (19531070)Biography: TIAN De-sheng (1966-), male, Ph. D candidate, research direction: qualitative theory of differential equation.

MODIFIABLE QUARTIC AND QUINTIC CURVES WITH SHAPE-PARAMETERS

This paper presents a method for creating modificable quartic and quintic curves with shape parameters. The curves can achieve C 2 even C 3 continuity and unify both interpolation and approximation to the control points without solving a system of equations or inserting additional control points. They have the local properties like the cubic B spline. Besides, the quintic curve would be able globally to tend the control polygon.

A Remark on the Positive Definite Problem of a Binary Quartic Form

In this paper,we discussed the positive definite problem of binary quartic forms and obtained a necessary and sufficient conditions.

Pull-in instability analyses for NEMS actuators with quartic shape approximation

The pull-in instability of a cantilever nano-actuator model incorporating the effects of the surface, the fringing field, and the Casimir attraction force is investigated. A new quartic polynomial is proposed as the shape function of the beam during the deflection, satisfying all of the four boundary values. The Gaussian quadrature rule is used to treat the involved integrations, and the design parameters are preserved in the evaluated formulas. The analytic expressions are derived for the tip deflection and pull-in parameters of the cantilever beam. The micro-electromechanical system （MEMS） cantilever actuators and freestanding nano-actuators are considered as two special cases. It is proved that the proposed method is convenient for the analyses of the effects of the surface, the Casimir force, and the fringing field on the pull-in parameters.

Ground-state vortex structures of a rotating binary dipolar Bose–Einstein condensate confined in harmonic plus quartic potential

We consider a binary dipolar Bose–Einstein condensate confined in a rotating harmonic plus quartic potential trap.The ground-state vortex structures are numerically obtained as a function of the contact interactions and the dipole–dipole interaction in both slow and rapid rotation cases. The results show that the vortex configurations depend strongly on the strength of the contact interactions, the relative strength between dipolar and contact interactions, as well as on the orientation of the dipoles. A variety of exotic ground-state vortex structures, such as pentagonal and hexagon vortex lattice,square vortex lattice with a central vortex, annular vortex lines, and straight vortex lines, are observed by turning such controllable parameters. Our results deepen the understanding of effects of dipole–dipole interaction on the topological defects.

Quartic coupling and its effect on wetting behaviors in nematic liquid crystals

Based on the fact that rubbed groove patterns also affect the anchoring of liquid crystals at substrates,a quartic coupling is included in constructing the surface energy for a liquid crystal cell.The phase diagram and the wetting behaviors of the liquid crystal cell,bounded by surfactant-laden interfaces in a magnetic field perpendicular to the substrate are discussed by taking the quartic coupling into account.The nematic order increases at the surface while it decreases in the bulk as a result of the introduction of quartic substrate-liquid crystal coupling,indicating that the groove anchoring makes the liquid crystal molecules align more orderly near the substrate than away from it.This causes a different wetting behavior：complete wetting.

Jacobi Quartic曲线上GLV/GLS标量乘算法

目前GLV/GLS (Gallant,Lambert,Vanstone/Galbraith, Lin, Scott)标量乘算法的研究主要集中在Weierstrass曲线上,尝试寻找和构造更多或者更高次数的可有效计算的自同态.本文主要研究了Jacobi Quartic曲线上GLV/GLS标量乘算法.首先利用曲线之间的双有理等价,给出了该类曲线在素域上可有效计算自同态的具体构造,得到2维GLV方法 .然后考虑椭圆曲线的二次扭曲线,利用曲线之间双有理等价和Frobenius映射,给出了该类曲线在二次扩域上可有效计算自同态的具体构造,得到2维GLS方法 .将上述GLV和GLS方法结合起来,同时利用曲线在二次扩域上的两个不同的自同态,得到4维GLV方法 .最后针对j不变量为0或1728两类特殊形式的椭圆曲线,利用更高次的扭曲线,得到4维GLV方法 .实验结果表明:对于Jacobi Quartic曲线,2维GLV方法和4维GLV方法比5-NAF方法分别提速37.2%和109.4%以上.同时,在三种不同的实现方式下,Jacobi Quartic曲线上标量乘效率都优于Weierstrass曲线.

A note on Pythagorean hodograph quartic spiral

By using the geometric constraints on the control polygon of a Pythagorean hodo- graph （PH） quartic curve, we propose a sufficient condition for this curve to have monotone curvature and provide the detailed proof. Based on the results, we discuss the construction of spiral PH quartic curves between two given points and formulate the transition curve of a G2 contact between two circles with one circle inside another circle. In particular, we deduce an attainable range of the distance between the centers of the two circles and summarize the algorithm for implementation. Compared with the construction of a PH quintic curve, the complexity of the solution of the equation for obtaining the transition curves is reduced.

Bioconvection Mangnetohydrodynamic Tangent Hyperbolic Nanofluid Flow with Quartic Chemical Reaction Past a Paraboloid Surface

In this numerical study,the effect of quartic autocatalysis type of chemical reaction,buoyancy force and thermal radiation phenomenon and magnetic effect on tangent hyperbolic nanofluid past an upper horizontal surface of a paraboloid has been studied.By considering the Buongiorno model approach,a diffusion of unequal coefficients in the presence of gyrotactic microorganism is discussed.Implementation of microorganism’s idea is used to stabilize the nanoparticles through bioconvection.The modeled PDEs of the problems are converted into nonlinear ODEs with the assistant of the similarity transformations.To tackle nonlinear ODEs,MATLAB package bvp4c is used.In addition,a hallmark of the Matlab code with the reported results in the literature is achieved by benchmarking.The variations in motion,concentration,temperature,and motile density due to sundry parameters have been analyzed in-depth via graphs.Our analysis shows that the density profile of motile of microorganism is hiked with an increment in the bioconvection Rayleigh number but decreases for higher thermal Grashof number.

An Unconditionally Monotone <i>C</i>²Quartic Spline Method with Nonoscillation Derivatives

A one-dimensional monotone interpolation method based on interface reconstruction with partial volumes in the slope-space utilizing the Hermite cubic-spline, is proposed. The new method is only quartic, however is C2 and unconditionally monotone. A set of control points is employed to constrain the curvature of the interpolation function and to eliminate possible nonphysical oscillations in the slope space. An extension of this method in two-dimensions is also discussed.

Green’s Function for the Quartic Oscillator

In this paper, a quantum mechanical Green’s function for the quartic oscillator is presented. This result is built upon two previous papers: first [1], detailing the linearization of the quartic oscillator (qo) to the harmonic oscillator (ho);second [2], the integration of the classical action function for the quartic oscillator. Here an equivalent form for the quartic oscillator action function in terms of harmonic oscillator variables is derived in order to facilitate the derivation of the quartic oscillator Green’s Function, namely in fixing its amplitude.

ERROR BOUNDS IN PERIODIC QUARTIC SPLINE INTERPOLATION

In this paper we develop periodic quartic spline interpolation theory which,in general,gives better fus to continuous functions than does the existing quintic spline interpolation theory.The main theorem of the paper is to establish that r=0,1,2,3.Also,the nanperiodic cases cannot be constructed empoly-ing the methodology of this paper because that will involve several other end conditions entirely different than(1,10).

New Approach to Approximate Circular Arc by Quartic Bezier Curve

This paper presents a result of approximation an arc circles by using a quartic Bezier curve. Based on the barycentric coordinates of two and three combination of control points, the interior control points are determined by forcing the curvature at median point as similar as the given curvature at end points. Hausdorff distance is used to investigate the order of accuracy compare to the actual arc circles through central angle of . We found that the optimal approximation order is eight which is somewhat similar to preceding methods in the literatures.

Direct Fisher Inference of the Quartic Oscillator’s Eigenvalues

It is well known that a suggestive connection links Schr?dinger’s equation (SE) and the information-optimizing principle based on Fisher’s information measure (FIM). It has been shown that this entails the existence of a Legendre transform structure underlying the SE. Such a structure leads to a first order partial differential equation (PDE) for the SE’s eigenvalues from which a complete solution for them can be obtained. We test this theory with regards to anharmonic oscillators (AHO). AHO pose a long-standing problem and received intense attention motivated by problems in quantum field theory and molecular physics. By appeal to the Cramer Rao bound we are able to Fisher-infer the energy eigenvalues without explicitly solving Schr?dinger’s equation. Remarkably enough, and in contrast with standard variational approaches, our present procedure does not involve free fitting parameters.

On singular Lüroth quartics

Plane quartics containing the ten vertices of a complete pentalateral and limits of them are called Lüroth quartics.The locus of singular Lüroth quartics has two irreducible components,both of codimension two in P14.We compute the degree of them and discuss the consequences of this computation on the explicit form of the Lüroth invariant.One important tool is the Cremona hexahedral equations of the cubic surface.We also compute the class in M 3 of the closure of the locus of nonsingular Lüroth quartics.

TEN FORMULAE OF TYPE ANKENY-ARTINCHOWLA FOR CLASS NUMBERS OF GENERAL CYCLIC QUARTIC FIELDS

Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) with the prime number p = r^2+s^2 and s is even, then C_1h^-≡B_((p-1)/_4)B_(3(p-1)/4) (mod p) for p≡1 (mod 8); and C_2h^-≡E_((p-5)/8)E_((3p-7)/8)(mod p) for p≡5 (mod 8)where B_n and E_n are the Bernoulli and the Euler numbers. If the real K = Q((v(5+2(5^(1/2))))^(1/2),then C_3h^-≡h(Q((-v)^(1/2))) h (Q((-5v)^(1/2))) (mod 5). If 3 ramifies in K = Q(θ^(1/2)), then C_4h(K)≡h(K~*) (mod 3) with K~* = Q((-3θ^(1/2))). All the above C_i are explicitly given constants.Some relations between the factors of class numbers h^- are also obtained. These results forcyclic quartic fields are an extension of the results for quadratic fields obtained by Ankeny-Artin-Chowla, Kiselev, Carlitz and Lu Hong-wen from 1948 to 1983.

Certifying the Global Optimality of Quartic Minimization over the Sphere

The quartic minimization over the sphere is an NP-hard problem in the general case.There exist various methods for computing an approximate solution for any given instance.In practice,it is quite often that a global optimal solution was found but without a certification.We will present in this article two classes of methods which are able to certify the global optimality,i.e.,algebraic methods and semidefinite program(SDP)relaxation methods.Several advances on these topics are summarized,accompanied with some emerged new results.We want to emphasize that for mediumor large-scaled instances,the problem is still a challenging one,due to an apparent limitation on the current force for solving SDP problems and the intrinsic one on the approximation techniques for the problem.

Constraints on anomalous quartic gauge couplings via Wγjj production at the LHC

Vector boson scattering at the Large Hadron Collider(LHC)is sensitive to anomalous quartic gauge couplings(aQGCs).In this study,we investigate the aQGC contribution to Wγjj production at the LHC with√s=13 TeV in the context of an effective field theory(EFT).The unitarity bound is applied as a cut on the energy scale of this production process,which is found to have significant suppressive effects on signals.To enhance the statistical significance,we analyze the kinematic and polarization features of the aQGC signals in detail.We find that the polarization effects induced by aQGCs are unique and can discriminate the signals from the SM backgrounds well.With the proposed event selection strategy,we obtain the constraints on the coefficients of dimension-8 operators with current luminosity.The results indicate that the process pp→Wγjj is powerful for searching for the OM2,3,4,5 and OT5,6,7 operators.