Binary Cubics Research Articles

Geometric properties of special orthogonal representations associated to exceptional Lie superalgebras

From an octonion algebra [Formula: see text] over a field [Formula: see text] of characteristic not two or three, we show that the fundamental representation [Formula: see text] of the derivation algebra [Formula: see text] and the spinor representation [Formula: see text] of [Formula: see text] are special orthogonal representations. They have particular geometric properties coming from their similarities with binary cubics and we show that the covariants of these representations and their Mathews identities are related to the Fano plane and the affine space [Formula: see text]. This also permits to give constructions of exceptional Lie superalgebras.

The Kostant invariant and special ϵ-orthogonal representations for ϵ-quadratic colour Lie algebras

Let k be a field of characteristic not two or three, let g be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of g. In this article we give various ways of constructing a colour Lie algebra g˜ whose bracket in some sense extends both the bracket of g and the action of g on V. Colour Lie algebras, originally introduced by R. Ree ([18]), generalise both Lie algebras and Lie superalgebras, and in those cases our results imply many known results ([13], [14], [5], [23]). For a class of representations arising in this context we show there are covariants satisfying identities analogous to Mathews identities for binary cubics.

Irreducible binary cubics and the generalised superelliptic equation over number fields

For a large class (heuristically most) of irreducible binary cubic forms $F(x,y) \in \mathbb Z[x,y]$, Bennett and Dahmen proved that the generalized superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,y \in \mathbb Z$ coprime, $z \in \mathbb Z$ and exponent $l \in \mathbb Z_{\geq 4} $. Their proof uses, among other ingredients, modularity of certain mod $l$ Galois representations and Ribet's level lowering theorem. The aim of this paper is to treat the same problem for binary cubics with coefficients in $\mathcal O_K$, the ring of integers of an arbitrary number field $K$, using by now well-documented modularity conjectures.

A Note on Generic Clifford Algebras of Binary Cubic Forms

We study the representation theoretic results of the binary cubic generic Clifford algebra $\mathcal C$, which is an Artin-Schelter regular algebra of global dimension five. In particular, we show that $\mathcal C$ is a PI algebra of PI degree three and compute its point variety and discriminant ideals. As a consequence, we give a necessary and sufficient condition on a binary cubic form $f$ for the associated Clifford algebra $\mathcal C_f$ to be an Azumaya algebra.

Polynomial invariants and moduli of generic two-dimensional commutative algebras

ABSTRACTLet V be a two-dimensional vector space over a field of characteristic not 2 or 3. We show there is a surjection ν from the set of ‘generic’ commutative algebra structures on V modulo the action of onto . In these coordinates (quotients of invariant quartic polynomials) many properties of the algebra are described by polynomial equations. The map ν is a bijection except over a degenerate elliptic curve Γ, and we give a parametrization of , also characterized as those algebras admitting non-trivial automorphisms, in terms of Galois extensions of . We show ν lifts to a map from the corresponding -orbit space into a hypersurface in a four-dimensional vector space whose equation is essentially the classical Eisenstein equation for the covariants of a binary cubic. This map is the restriction of a surjection from the -orbit space of ‘stable’ commutative algebras onto .

Interaction of punches on orthotropic half-space

An integral equation of the three-dimensional contact problem for an orthotropic half-space (9 independent elastic parameters in Hooke’s law) is obtained where its kernel does not include integrals, but it depends on the solution of a characteristic binary cubic. The interaction between two identical symmetrically embedded punches is considered for the case of the elliptic paraboloids. Galanov’s method of nonlinear boundary integral equations is used for solving the problem with an unknown contact domain that makes it possible to determine simultaneously the contact domain and the contact pressure. The exact solution to one elliptical punch is used for debugging the computer program. Contact pressures, contact zones and pressing forces are calculated for various orthotropic materials at the specified settlement, base forms of the punches, and relative distances between the punches. The orthotropic body model is applicable for describing lots of materials which are in-demand in the machinery and industry: sulfur, Rochelle salt, wolframite, barite, and various wood species.

Transport Protocol Throughput Fairness

Interest continues to grow in alternative transport protocols to the Transmission Control Protocol (TCP). These alternatives include protocols designed to give greater efficiency in high-speed, high-delay environments (so-called high-speed TCP variants), and protocols that provide congestion control without reliability. For the former category, along with the deployed base of ‘vanilla’ TCP – TCP NewReno – the TCP variants BIC and CUBIC are widely used within Linux: for the latter category, the Datagram Congestion Control Protocol (DCCP) is currently on the IETF Standards Track. It is clear that future traffic patterns will consist of a mix of flows from these protocols (and others). So, it is important for users and network operators to be aware of the impact that these protocols may have on users. We show the measurement of fairness in throughput performance of DCCP Congestion Control ID 2 (CCID2) relative to TCP NewReno, and variants Binary Increase Congestion control (BIC), CUBIC and Compound, all in “out-of-the box” configurations. We use a testbed and endto- end measurements to assess overall throughput, and also to assess fairness – how well these protocols might respond to each other when operating over the same end-to-end network path. We find that, in our testbed, DCCP CCID2 shows good fairness with NewReno, while BIC, CUBIC and Compound show unfairness above round-trip times of 25ms.

Homogeneous instanton bundles on [formula omitted] for the action of [formula omitted

We classify k -instanton bundles on P C 3 which are homogeneous for the group SL ( 2 ) , acting linearly on P 3 with an open orbit. Besides the classical special instantons, we find isolated examples for SL ( 2 ) acting by the representation of binary cubics. We show that these examples are unique and that they exist only for k = a ( a − 1 ) / 2 , for some a ≥ 2 . We also compute their minimal free resolution in terms of homogeneous equivariant matrices.

Primes Represented by Binary Cubic Forms

Let $f(x, y)$ be a binary cubic form with integral rational coefficients, and suppose that the polynomial $f(x, y)$ is irreducible in $\mathbb{Q}[x, y]$ and no prime divides all the coefficients of $f$. We prove that the set $f(\mathbb{Z}^{2})$ contains infinitely many primes unless $f(a, b)$ is even for each $(a, b)$ in $\mathbb{Z}^{2}$, in which case the set $\frac{1}{2}f(\mathbb{Z}^{2})$ contains infinitely many primes. 2000 Mathematical Subject Classification: primary 11N32; secondary 11N36, 11R44.

On the Representation of 1 by Binary Cubic Forms of Positive Discriminant

We looked for types of irreducible cubic forms with integer coefficients and of positive discriminant for which the Thue equation f( x, y) = 1 has many or large solutions. We give a way to design even non-reversible forms which have at least the four different solutions (1, 0), ( x 1, y 0 ), ( x 2, y 0 ), ( x 3, y 0 ). Performance and results of a computer-search by which the solutions ( x, y) up to a given bound on | y| of some 85.000 of these equations had been determined will be described.

On classical invariant theory and binary cubics

Soit G un groupe algébrique complexe réductif et C[mV] G l’algèbre des polynômes G-invariants sur la somme directe de m copies de l’espace de représentation V de G. Il existe un nombre entier n=n(V) minimal tel que les générateurs et relations de C[mv] G puissent s’obtenir à partir de ceux de C[nv] G par polarisation et restitution pour chaque m>n. On borne n et les degrés des générateurs et relations de C[nV] G , en étendant des résultats de Vust. Ces techniques sont alors appliquées au calcul des invariants de plusieurs formes binaires cubiques.

Pencils of real binary cubics

A complete and satisfying account of the classification of pencils of binary cubics over an algebraically closed field was given by Newstead (2). Extending these results to the real case is not a matter of mere routine since new questions arise, for example the separation of roots of the cubics in a pencil (as well as their reality).

Sur certaines algebras de clifford

Clifford algebras of polynomial forms of degree n > 2, introduced by N. Roby are infinite dimensional when the module has rank bigger than I. However, in the case of diagonalizable forms and of binary cubics, we can show that these algebras have finite dimensional quotients. When forms are nondegenerate in a natural sense, these quotients are separable algebras. In the second case, we can also give a precise description of the Clifford algebra itself.

Covariants of pencils of binary cubics

SynopsisThe invariants of pencils of binary cubics were described in an earlier paper by the author. In this note, we give a complete set of covariants for such pencils, and show how each type of pencil may be denned by the identical vanishing of covariants.

Invariants of pencils of binary cubics

Let U denote the variety of pencils of binary cubics without base point defined over an algebraically closed field k whose characteristic is not equal to 2 or 3. (For full details of the terminology, see § 2.) There is a natural action of SL(2) on U; our main object is to show that U possesses a good quotient for this action in the sense of ((7), Definition 1·5) or ((4), p. 70), and to identify this quotient with the affine line over k. In fact, we proveTheorem 1. There exists a morphism φ: such that (, φ) is a good quotient of U by SL(2). Moreover, all but one of the fibres of φ are orbits; the exceptional fibre consists of two orbits corresponding to pencils with one or two triple points.

A Geometrical Interpretation of the Invariant System of Two Binary Cubics

A Geometrical Interpretation of the Invariant System of Two Binary Cubics

On Numbers Represented by Binary Cubic Forms

On Numbers Represented by Binary Cubic Forms

Investigation of Lagrange's Tangent Method on Diophantine Binary Cubics

Investigation of Lagrange's Tangent Method on Diophantine Binary Cubics

A Waring Problem for Binary Cubic Forms

Journal of the London Mathematical SocietyVolume s1-10, Issue 2 p. 84-88 Notes and Papers A Waring Problem for Binary Cubic Forms R. D. James, R. D. James University of California U.S.A. †National Research Fellow.Search for more papers by this author R. D. James, R. D. James University of California U.S.A. †National Research Fellow.Search for more papers by this author First published: April 1935 https://doi.org/10.1112/jlms/s1-10.1.84 AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volumes1-10, Issue2April 1935Pages 84-88 RelatedInformation

A Waring Problem for Binary Cubic Forms

A Waring Problem for Binary Cubic Forms