Related papers: The 2-dimensional jacobian conjecture via Klein's …
We study Lie subalgebras $L$ of the vector fields $\mathrm{Vec}^{c}({\mathbb A}^{2})$ of affine 2-space ${\mathbb A}^{2}$ of constant divergence, and we classify those $L$ which are isomorphic to the Lie algebra $\mathfrak{aff}_{2}$ of the…
An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and…
We apply simple linear programming methods and an LP solver to refute a number of open conjectures in extremal combinatorics.
We describe an algorithm that computes possible corners of hypothetical counterexamples to the Jacobian Conjecture up to a given bound. Using this algorithm we compute the possible families corresponding to $\gcd(deg(P),deg(Q))\le 35$, and…
We show that the iterated images of a Jacobian pair stabilize; that is, the k-th iterates of a polynomial map of complex two-space to itself with a nonzero constant Jacobian determinant all have the same image for sufficiently large k. More…
We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus $3$, $4$ or $5$. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the…
A two-variable generalization of the Big $-1$ Jacobi polynomials is introduced and characterized. These bivariate polynomials are constructed as a coupled product of two univariate Big $-1$ Jacobi polynomials. Their orthogonality measure is…
An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.
We establish an equivalent condition to the validity of the Collatz conjecture, using elementary methods. We derive some conclusions and show several examples of our results. We also offer a variety of exercises, problems and conjectures.
We discuss various recent advances on weak forms of the Twin Prime Conjecture.
In this paper we obtain a partial answer to a conjecture on the solvabilty of linear difference equations in quasianalytic Carleman classes.
We determine the isomorphism classes of Jordan algebras in dimension two over the field of real numbers. Using techniques of non-standard analysis we study the properties of the variety of Jordan algebras, and also the contractions among…
In this paper we generalize previous work on decomposition in three-dimensional orbifolds by 2-groups realized as analogues of central extensions, to orbifolds by more general 2-groups. We describe the computation of such orbifolds in…
A method based on the symbolic methods of the classical invariant theory is developed for a representation of elements of kernel of Weitzenb\"ok derivations.
We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use…
Minimal length of a two-dimensional Klein-Gordon oscillator is investigated and illustrates the wave functions in the momentum space. The energy eigenvalues are found and the corresponding wave functions are calculated in terms of…
We have already conjectured 2 important guesses regarding Hypo-Lie algebra and modular simple Lie algebra. We would like to attach 2 important guesses more to this conjecture. Such new guesses are related to the Steinberg module.
The quadratic rank two Jacobi algebra is identified from the relations obeyed by the bispectral operators of the two variable Jacobi polynomials orthogonal on the triangle. It is seen to admit as subalgebras Racah and Jacobi algebras of…
The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools…
The well known Klein paradox for the relativistic Dirac wave equation consists in the computation of possible ``negative probabilities'' induced by certain potentials in some regimes of energy. The paradox may be resolved employing the…