Ternary Cubics Research Articles

Tackling Ternary Cubic Diophantine Equation

Abstract: A few interesting characteristics among the solutions and the patterns of non-zero integral solutions to the nonhomogeneous cubic equation with three unknowns represented by the Diophantine equation 2 2 3 4(   )  7( )  64 are examined.

Curvature loci of 3‐manifolds

AbstractWe refine the affine classification of real nets of quadrics in order to obtain generic curvature loci of regular 3‐manifolds in and singular corank one 3‐manifolds in . For this, we characterize the type of the curvature locus by the number and type of solutions of a system of equations given by four ternary cubics (which is a determinantal variety in some cases). We also study how singularities of the curvature locus of a regular 3‐manifold can go to infinity when the manifold is projected orthogonally in a tangent direction.

An Analysis on the Ternary Cubic Diophantine Equation

Abstract: In this article, we concentrate on identifying all the non-zero, infinitely many integral solutions to the ternary cubic equation

Integral Solutions of the Ternary Cubic Equation

Abstract: The non-homogenous cubic equation with three unknowns represented by the Diophantine equation ૜(࢞ ࢟ + ૛ ૛ ) − ࢠ૛૛૞ = (૚ + ࢟ + ࢞)૛ + (࢟࢞)૝ ૜ is analyzed for its patterns of stringfy integral solutions. A few interesting properties among the solutions and some special polygonal numbers are presented.

Properties of the Ternary Cubic Equation 5x2-3y2=z3

Abstract: To identify its different integral non-zero solutions, the ternary cubic equation 2 2 3 5x 3y  z is taken into consideration. Different integral solution patterns to the ternary cubic equation under consideration are obtained in each pattern by using the linear transformation and the method of factorization; interesting relationships between the solutions and some polygonal numbers, such as pyramidal and central pyramidal numbers, are also displayed.

Observations on the Homogeneous Ternary Cubic Equation with Four Unknowns 3(x3+y3)=2zw2

Observations on the Homogeneous Ternary Cubic Equation with Four Unknowns 3(x3+y3)=2zw2

On real typical ranks

We study typical ranks with respect to a real variety X. Examples of such are tensor rank (X is the Segre variety) and symmetric tensor rank (X is the Veronese variety). We show that any rank between the minimal typical rank and the maximal typical rank is also typical. We investigate typical ranks of n-variate symmetric tensors of order d, or equivalently homogeneous polynomials of degree d in n variables, for small values of n and d. We show that 4 is the unique typical rank of real ternary cubics, and quaternary cubics have typical ranks 5 and 6 only. For ternary quartics we show that 6 and 7 are typical ranks and that all typical ranks are between 6 and 8. For ternary quintics we show that the typical ranks are between 7 and 13.

Fundamental invariants of orbit closures

For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental SL-invariant function that relates the coordinate ring of the orbit with the coordinate ring of its closure. For the power sums we can write down this fundamental invariant explicitly in most cases. Our constructions generalize the two Aronhold invariants on ternary cubics. For the other objects we identify the invariant function conditional on intriguing combinatorial problems much like the well-known Alon–Tarsi conjecture on Latin squares. We provide computer calculations in small dimensions for these cases. As a main tool for our analysis, we determine the stabilizers, and we establish the polystability of all the mentioned forms and tensors (including the generic ones).

Associated forms and hypersurface singularities: The binary case

Abstract In the recent articles [5, 1], it was conjectured that all rational GL n {\mathrm{GL}_{n}} -invariant functions of forms of degree d ≥ 3 {d\geq 3} on ℂ n {\mathbb{C}^{n}} can be extracted, in a canonical way, from those of forms of degree n ⁢ ( d - 2 ) {n(d-2)} by means of assigning to every form with nonvanishing discriminant the so-called associated form. The conjecture was confirmed in [5] for binary forms of degree d ≤ 6 {d\leq 6} as well as for ternary cubics. Furthermore, a weaker version of it was settled in [1] for arbitrary n and d. In the present paper, we focus on the case n = 2 {n=2} and establish the conjecture, in a rather explicit way, for binary forms of an arbitrary degree.

On The Ternary Cubic Diophantine Equation 5(x2 + y2)

On The Ternary Cubic Diophantine Equation 5(x2 + y2)

ASSOCIATED FORMS OF BINARY QUARTICS AND TERNARY CUBICS

Let \( {\mathcal{Q}}_n^d \) be the vector space of forms of degree d ≥ 3 on ℂn, with n ≥ 2. The object of our study is the map Φ, introduced in earlier articles by M. Eastwood and the first two authors, that assigns every nondegenerate form in \( {\mathcal{Q}}_n^d \) the so-called associated form, which is an element of \( {{\mathcal{Q}}_n^d}^{\left(d-2\right)*} \). We focus on two cases: those of binary quartics (n = 2, d = 4) and ternary cubics (n = 3, d = 3). In these situations the map Φ induces a rational equivariant involution on the projective space ℙ\( \left({\mathcal{Q}}_n^d\right) \), which is in fact the only nontrivial rational equivariant involution on ℙ\( \left({\mathcal{Q}}_n^d\right) \). In particular, there exists an equivariant involution on the space of elliptic curves with nonvanishing j-invariant. In the present paper, we give a simple interpretation of this involution in terms of projective duality. Furthermore, we express it via classical contravariants.

Associated forms in classical invariant theory and their applications to hypersurface singularities

It was conjectured in the recent article by Eastwood and Isaev that all absolute classical invariants of forms of degree \(m\ge 3\) on \({\mathbb C}^n\) can be extracted, in a canonical way, from those of forms of degree \(n(m-2)\) by means of assigning every form with non-vanishing discriminant the so-called associated form. This surprising conjecture was confirmed for binary forms of degree \(m \le 6\) and ternary cubics. In the present paper, we settle the conjecture in full generality. In addition, we propose a stronger version of this statement and obtain evidence supporting it.

Extracting invariants of isolated hypersurface singularities from their moduli algebras

We use classical invariant theory to construct invariants of complex graded Gorenstein algebras of finite vector space dimension. As a consequence, we obtain a way of extracting certain numerical invariants of quasi-homogeneous isolated hypersurface singularities from their moduli algebras, which extends an earlier result due to the first author. Furthermore, we conjecture that the invariants so constructed solve the biholomorphic equivalence problem in the homogeneous case. The conjecture is easily verified for binary quartics and ternary cubics. We show that it also holds for binary quintics and sextics. In the latter cases the proofs are much more involved. In particular, we provide a complete list of canonical forms of binary sextics, which is a result of independent interest.

Self-Assembly of Ternary Cubic, Hexagonal, and Lamellar Mesophases Using the Lattice-Boltzmann Kinetic Method

We use a kinetic lattice-Boltzmann method to simulate the self-assembly of the cubic primitive (P), diamond (D), and gyroid (G) mesophases from an initial quench composed of oil, water, and amphiphilic particles. Here, we also report the self-assembly of the noncubic hexagonal phase and two lamellar phases, one with periodic convolutions. The periodic mesophase structures are emergent from the underlying conservation laws and quasi-molecular interactions of the lattice-Boltzmann model. We locate regions of the model's parameter space where the sequence of appearance of mesophases lamellar --> primitive --> hexagonal is in agreement with pressure jump experiments and the sequence cubic --> lamellar is in agreement with compositional variations reported in the literature. The ability of our lattice-Boltzmann model to simulate self-assembly of cubic and noncubic phases in a unified and consistent manner opens the way for further investigations into the transition pathways and kinetics of the phase transitions between these states as well as of the rheology of these phases.

Deformation and Defects in Laves Phases

ABSTRACTPrototype Laves Phases MgCu2 and MgZn2 have been deformed in compression at various temperatures and strain rates, and dislocations produced by deformation have been characterized by TEM. Ternary cubic Laves phases Mg(Cu,M)2, where M-Al, Ni, Si, and Zn, have also been deformed, and substantial solution hardening was observed

Geometry related to the key del Pezzo surface and the associated mapping of plane cubics

SynopsisWhen the points of a projective space [9] map ternary cubics the maps of perfect cubes are the points of a del Pezzo surface F. Several manifolds linked with F are listed and the cubics mapped on these are noted. A new approach leads to the first of Aronhold's two invariants.

On the Minimum of a Ternary Cubic Form

On the Minimum of a Ternary Cubic Form

Bildungsenthalpien ternärer Lavesphasen vom MgCu 2-Typ im System MgCu 2MgZn 2

The heats of formation of the ternary cubic Laves phases of the pseudobinary system MgCu 2MgZn 2 were determined by solution calorimetry methods. Replacing copper atoms with zinc atoms increases the negative heat of formation. This change is mainly due to the change of interatomic interactions within the copper sublattice. Relative minima appearing in the ΔH B vs composition-curve are interpreted in terms of ordering of copper and zinc atoms on the copper sublattice.

Compositions involving ternary cubics

Compositions involving ternary cubics

On the Minimum of a Ternary Cubic Form

On the Minimum of a Ternary Cubic Form