On a cubic variational problem

An extremal curve of the simplest variational problem is a continuously differentiable function. Hilbert’s differentiability theorem provides a sufficient condition for the existence of the second derivative of an extremal curve. It is desirable to have a simple example in which the condition of Hilbert’s theorem is violated and an extremal curve is not twice differentiable.In this paper, a cubic variational problem with the following properties is analyzed. The functional of the problem is bounded neither above nor below. There exists an extremal curve for this problem which is obtained by sewing together two different extremal curves and not twice differentiable at the sewing point. Despite this unfavorable situation, an attempt to apply the method of steepest descent (in the form proposed by V.F. Dem’yanov) to this problem is made. It turns out that the method converges to a stationary curve provided that a suitable step size rule is chosen.     

On a certain non-split cubic surface

Science China Mathematics - This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of...     

On a wave equation with a cubic convolution

We study the well-posedness of the Cauchy problem and the asymptotic behavior of solutions of the nonlinear wave equation $\text{u}{}_{\mathrm{tt}}\text{? Δu + m}{}^2\text{+ u(V 1 u}{}^2\text{) = 0}$ in Euclidean space.     

On cubic surfaces with a rational line

We report on our project to construct non-singular cubic surfaces over \mathbbQ{\mathbb{Q}} with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.     

On cubic polynomials

It is shown that a system ofr homogeneous cubic equations with rational coefficients has a nontrivial solution in rational integers if the number of variables is at least (10r)⁵. For most such systems, an asymptotic formula holds for the numberz _P of solutions whose components have modulus <P.     

On lifting singular curves

The aim of this note is to lift singular curves of a certain type to characteristic zero. The liftings are obtained as suitable pushouts and the corresponding relative jacobians are identified as rigidifed Picard functors.     

On the existence of a certain cubic spline

We propose a constructive proof of the existence and uniqueness of a defect-1 parabolic interpolation non-periodic spline that does not require additional boundary-value conditions. An estimate of the interpolation error is obtained. Bibliography: 3 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 124–129.     

On the stability of a cubic functional equation

In this paper, we will find out the general solution and investigate the generalized Hyers-Ulam-Rassias stability problem for the following cubic functional equation
2f（x ＋ 2y） ＋ f（2x - y） = 5f（x ＋ y） ＋ 5f（x - y）＋ 15f（y）
in the spirit of Hyers, Ulam, Rassias and Gavruta.     

On a Frank-Wolfe type theorem in cubic optimization

ABSTRACT

A classical result due to Frank and Wolfe [An algorithm for quadratic programming. Naval Res Log Quart. 1956;3:95–110] says that a quadratic function f attains its supremum on a nonempty polyhedron M if f is bounded from above on M. In this note, we present a stringent proof of the extension of this result to cubic optimization (known from Andronov, Belousov and Shironin [On solvability of the problem of polynomial programming (In Russian). Izvestija Akadem. Nauk SSSR, Tekhnicheskaja Kibernetika. 1982;4:194–197. Translation appeared in News of the Academy of Science of USSR, Dept. of Technical Sciences, Technical Cybernetics.]). Further, we discuss related results. In particular, we bring back to attention Kummer's [Globale Stabilität quadratischer Optimierungsprobleme. Wissenschaftliche Zeitschrift der Humboldt- Universität zu Berlin, Math-Nat R. 1977;XXVI(5):565–569] generalization of the Frank-Wolfe theorem to the case that f is quadratic, but M is the Minkowski sum of a compact set and a polyhedral cone.     

On diagonal cubic surfaces

We show that the affine surfacesx ³+y ³+c z ³=c, c Q, in the casesc2,c=2, contain precisely 2, respectively 4, polynomial parametric solutions corresponding to curves of arithmetic genus 0 on the surface.However, these surfaces contain infinitely many polynomial parametric solutions corresponding to curves of arithmetic genus greater than 0.The author wishes to acknowledge the receipt of a Summer Support Grant for the College of Liberal Arts, Arizona State University, while this note was being written.     

On abelian cubic arcs

Work partially supported by a NATO Grant.     

On Manin's conjecture for a certain singular cubic surface

This paper contains a proof of the Manin conjecture for the singular cubic surface S⊂P³ that is defined by the equation . In fact if U⊂S is the Zariski open subset obtained by deleting the unique line from S, and H is the usual exponential height on P³(Q), then the height zeta function _∑x∈U(Q)H(x)^−s is analytically continued to the half-plane .     

On cubic Galois field extensions

We study Morton's characterization of cubic Galois extensions F/K by a generic polynomial depending on a single parameter s∈K. We show how such an s can be calculated with the coefficients of an arbitrary cubic polynomial over K the roots of which generate F. For K=Q we classify the parameters s and cubic Galois polynomials over Z, respectively, according to the discriminant of the extension field, and we present a simple criterion to decide if two fields given by two s-parameters or defining polynomials are equal or not.     

On subgroups lifting modulo central commutant

We consider a finitely generated group G with the commutant of odd order located at the center and prove that there exists a decomposition of G/G′ into the direct product of indecomposable cyclic groups such that each factor except at most n _l + ... + n _s factors lifts modulo commutant. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 742–745, May, 1998.     

On cubic étale algebras

We describe explicitly a group structure on isomorphism classes of cubic étale algebras with prescribed discriminant. The case of trivial discriminant corresponds to the group of cubic cyclic algebras.