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We investigate the number of straight lines contained in a K3 quartic surface \(X\) defined over an algebraically closed field of characteristic 3. We prove that if \(X\) contains 112 lines, then \(X\) is projectively equivalent to the Fermat quartic surface; otherwise, \(X\) contains at most 67 lines. We improve this bound to 58 if \(X\) contains a star (ie four distinct lines intersecting at a smooth point of \(X\)). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided., Comment: One example removed. Some examples and some proofs now with more details

The singular structure of lower vibrational states of the difluorosilylene molecule (up to four total excitation quanta) is studied by expansion of energies of each state into series of the high-order Rayleigh–Schrodinger perturbation theory and analysis of their implicit properties of multiple-valuedness using Pade–Hermite fourth-order approximants. The quartic surface of potential energy in dimensionless normal coordinates is calculated quantum-mechanically at the MP2/cc-pVTZ level. It is shown that one of values of multiple-valued approximants reproduces the variational solution with a high accuracy, while other values (beginning with the fourth polyad) in many cases coincide with energies of other states of the same polyad. The Fermi and Darling–Dennison resonances are analyzed based on facts of coincidence of singular complex branch points of the approximants for mutually interacting states inside a unit circle or near it on the complex plane. It is found that a pair of states can have several coinciding branch points of the solutions (in particular, inside the unit circle). A conclusion is made that this approach is an effective method for the determination of the polyad structure of vibrational states. Calculation parameters necessary for the reproducibility of key results are selected. The calculations are carried out by means of a Fortran-based software package using an arithmetic calculation package with a long mantissa of real numbers (200 digits).

We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if the characteristic equals 3. If the quartic is not supersingular, the number of lines is at most 60 in both cases. We also give a complete classification of large configurations of lines., Comment: Final version accepted for publication. Theorem 1.1 had to be stated in a weaker version

In this paper, we prove that, over an algebraically closed field of odd characteristic p ( p ≠ 5 ), every supersingular K3 surface is isomorphic to a quartic surface in the three dimensional projective space .

Mustaţa has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in P 3 this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.

We construct an example of a smooth spatial quartic surface that contains 800 irreducible conics., Comment: The paper reflects the considerable development (explicit equations by X. Roulleau and B. Naskrecki) that followed the original version

We study the surface of Gauss double points associated to a very general quartic surface and the natural morphisms associated to it., 20 pages, 7 figures

Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Compos. 61(203) (1993) 117–130] investigated the Diophantine problem of representing integers [Formula: see text] in the form [Formula: see text] for rationals [Formula: see text]. For fixed [Formula: see text], the equation represents an elliptic curve, and the existence of solutions depends upon the rank of the curve being positive. They observed that the corresponding equation in four variables, the title equation here (representing a surface), has infinitely many solutions for each [Formula: see text], and remarked that it seemed plausible that there were always solutions with positive [Formula: see text] when [Formula: see text]. This is false, and the situation is quite subtle. We show that there cannot exist such positive solutions when [Formula: see text] is of the form [Formula: see text], [Formula: see text], where [Formula: see text]. Computations within our range seem to indicate that solutions exist for all other values of [Formula: see text].

With $\Fq$ the finite field of $q$ elements, we investigate the following question. If $\gamma$ generates $\Fqn$ over $\Fq$ and $\beta$ is a non-zero element of $\Fqn$, is there always an $a \in \Fq$ such that $\beta(\gamma + a)$ is a primitive element? We resolve this case when $n=3$, thereby proving a conjecture by Cohen. We also improve substantially on what is known when $n=4$., Comment: To appear in Math. Comp

We study a family of primary affine variety codes defined from the Klein quartic. The duals of these codes have previously been treated in Kolluru et al., (Appl. Algebra Engrg. Comm. Comput. 10(6):433–464, 2000, Ex. 3.2). Among the codes that we construct almost all have parameters as good as the best known codes according to Grassl (2007) and in the remaining few cases the parameters are almost as good. To establish the code parameters we apply the footprint bound (Geil and Høholdt, IEEE Trans. Inform. Theory 46(2), 635–641, 2000 and Høholdt 1998) from Gröbner basis theory and for this purpose we develop a new method where we inspired by Buchberger’s algorithm perform a series of symbolic computations.

We construct three elements in the kernel of the tame symbol on families of quartic curves. We show that these elements are integral under certain conditions on the parameters. Moreover, we prove that these elements are in general linearly independent by calculating the limit of the regulator.

We derive the quartic interaction vertex of pure Yang-Mills theory by demanding closure of the light-cone Poincar\'e algebra in four-dimensional Minkowski spacetime. This calculation explicitly shows why structure constants must satisfy the Jacobi identity. We then prove that corrections to the spin generator, for spin one at this order, vanish. We comment briefly on higher spin fields in this context., Comment: 10 pages. v2: minor changes, references added

The aim of this paper is first to reformulate the fixed point theorem [4, Theorem 1] in 2-Banach spaces, after it, we introduce and solve the radical quartic functional equation f ( x 4 + y 4 4 ) = f ( x ) + f ( y ) . We also show that the fixed point methods allow to investigate Ulam's type stability of radical quartic functional equation in 2-Banach spaces.

The previous approximation curves of conic section by quartic Bzier curves interpolate the conic section at the specified parameter values. In this paper, by solving the minimax problem, we present the optimal parameter values for approximating conic sections by the quartic Bzier curves. The upper bound on the Hausdorff distance between the conic section and the approximation curves is minimized. The method of solving the minimax problem is changed to solve a quartic algebraic equation by delicate reasoning.

In this paper, we get three integral representations for the linear independent solutions of differential equation y‴+xy=0 in terms of the quartic Levy stable functions. We also present the product...

Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Gruji\'c and Kalisch., Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1706.04659

Let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G, which possible groups G, In this new version we fix some errors in Table 5

Schmidt [‘Integer points on curves of genus 1’, Compos. Math. 81 (1992), 33–59] conjectured that the number of integer points on the elliptic curve defined by the equation $y^{2}=x^{3}+ax^{2}+bx+c$, with $a,b,c\in \mathbb{Z}$, is $O_{\unicode[STIX]{x1D716}}(\max \{1,|a|,|b|,|c|\}^{\unicode[STIX]{x1D716}})$ for any $\unicode[STIX]{x1D716}>0$. On the other hand, Duke [‘Bounds for arithmetic multiplicities’, Proc. Int. Congress Mathematicians, Vol. II (1998), 163–172] conjectured that the number of algebraic number fields of given degree and discriminant $D$ is $O_{\unicode[STIX]{x1D716}}(|D|^{\unicode[STIX]{x1D716}})$. In this note, we prove that Duke’s conjecture for quartic number fields implies Schmidt’s conjecture. We also give a short unconditional proof of Schmidt’s conjecture for the elliptic curve $y^{2}=x^{3}+ax$.

In this paper, we introduce a set-valued cubic functional equation and a set-valued quartic functional equation and prove the Hyers-Ulam stability of the set-valued cubic functional equation and the set-valued quartic functional equation by using the fixed point method.

A new method for approximation of conic section by quartic Bezier curve is presented, based on the quartic Bezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G2 continuous spline approximation of conic section when using the subdivision scheme, and the effectiveness of this method is demonstrated by some numerical examples.

Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{{\rm sa}}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential $(2,0)$-form on $Z$. The variety $Z$ is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of $Z$. One of these, denoted by $W$, is a kind of Igusa compactification such that the boundary $\partial W$ is a strictly normal crossing divisor. The main part of the L-function is described by some elliptic newform $g$ of weight 3. Then we construct an automorphic representation $\Pi$ of ${\rm GSp}_2(\A)$ related to $g$ and an explicit vector $E_Z$ sits inside $\Pi$ which creates a vector valued (non-cuspidal) Siegel modular form of weight $(3,1)$ so that $F_Z$ coincides with $E_Z$ in $H^{2,0}(\partial W)$ under the Poincar\'e residue map and various identifications of cohomologies.

The paper deals with the construction of bivariate quartic splines on Powell–Sabin triangulations. In particular, it provides a spline space that is C 2 everywhere except across some edges of the refined triangulation. The splines that belong to this space can be described uniquely with interpolation values and derivatives of order at most two. Moreover, they can be represented in a locally supported basis that forms a convex partition of unity. As an application of this result, quasi-interpolants that reproduce polynomials of degree four are derived.

Let $$F\subseteq {\mathbb {P}^{3}}$$ be a smooth quartic surface and let $${\mathcal {O}}_F(h):={\mathcal {O}}_{{\mathbb {P}^{3}}}(1)\otimes {\mathcal {O}}_F$$ . In the present paper we classify locally free sheaves $${\mathcal {E}}$$ of rank 2 on F such that $$c_1({\mathcal {E}})={\mathcal {O}}_F(2h), c_2({\mathcal {E}})=8$$ and $$h^1\big (F,{\mathcal {E}}(th)\big )=0$$ for $$t\in \mathbb {Z}$$ . We also deal with their stability.

We show that there cannot be more than 64 lines on a quartic surface admitting isolated rational double points over an algebraically closed field of characteristic $p \neq 2,\,3$, thus extending Segre--Rams--Sch\"utt theorem. Our proof offers a deeper insight into the triangle-free case and takes advantage of a special configuration of lines, thereby avoiding the technique of the flecnodal divisor. We provide several examples of non-smooth K3 quartic surfaces with many lines., Comment: 31 pages, 3 tables, 1 figure

We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi-Yau threefolds, including their Hodge numbers and deformation theory., Comment: v2: Significant changes at the request of the referee. Section 3 has been rearranged to accommodate a revised proof of Proposition 3.5 (formerly 3.2). Section 5 has been removed completely, it will instead appear as part of Section 5 in arxiv:1601.08110

In this paper, we apply quartic B-splines properly to study a new approximation method for numerical solutions and numerical derivatives for a class of non-linear fifth-order boundary value problems. Their analytic solutions and any-order derivatives are well approximated with $${O(h^{6})}$$ errors. Numerical tests are performed and numerical results show that our new method is very practical and effective.

In this paper, we present a C 3 quartic B-spline approximation of circular arcs. The Hausdorff distance between the C 3 quartic B-spline curve and the circular arc is obtained in closed form. Using this error analysis, we show that the approximation order of our approximation method is six. For a given circular arc and error tolerance we find the C 3 quartic B-spline curve having the minimum number of control points within the tolerance. The algorithm yielding the C 3 quartic B-spline approximation of a circular arc is also presented.

We study exact multiplicity and bifurcation curves of positive solutions of the boundary value problem \begin{eqnarray} &u"(x)+\lambda (-u^4+\sigma u^3-\tau u^2+\rho u)=0, -1 < x < 1, \\ &u(-1)=u(1)=0, \end{eqnarray} where $\sigma, \tau \in \mathbb{R}$, $\rho \geq 0,$ and $\lambda >0$ is a bifurcation parameter. Then on the $(\lambda, \|u\|_\infty)$-plane, we give a classification of four qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, a $\subset$-shaped curve and a monotone increasing curve.

We provide explicit evaluations of the Chowla-Selberg formula for quartic abelian CM fields due to Barquero-Sanchez and Masri. These identities relate values of a Hilbert modular function at CM points to values of Euler’s gamma function Γ \Gamma and an analogous function Γ 2 \Gamma _2 at rational numbers. Our work consists of two main parts. First, we implement an algorithm in SageMath to compute the CM points. Second, we exhibit families of quartic abelian CM fields for which the part of the formula involving values of Γ \Gamma and Γ 2 \Gamma _2 takes a particularly simple form.

The aim of this article is to extend the applications of a fixed point method to provide the intuitionistic fuzzy versions of Hyers-Ulam stability for the generalized forms of cubic and quartic functional equations. This way shows that the concept of stability is related to some fixed point of a suitable operator.

There are seven time independent, integrable, Henon-Heiles systems: three with cubic and four with quartic potential. The cubic and one of the quartic cases have been separated in the last decades. The other three cases 1:6:1, 1:6:8 and 1:12:16 have resisted several attempts in the last years. In this paper we focus our attention on the 1:12:16 case whose equations of motion have been separated only in the degenerate case ab = 0. We give here the separation coordinates for the generic case using a method introduced by Franco Magri in 2005 under the name of Kowalevski’s Conditions.