Establishing the minimal index in a parametric family of bicyclic biquadratic fields

Let c ≥ 3 be a positive integer such that c, 4c + 1, c − 1 are square-free integers relatively prime in pairs. In this paper we find the minimal index and determine all elements with minimal index in the bicyclic biquadratic field K = ℚ(√(4c + 1)c, √(c − 1)c). The author was supported by the Ministry of Science, Education and Sports, Republic of Croatia, grant 037-0372781-2821.     

Complete solutions of a family of quartic Thue and index form equations

Continuing the recent work of the second author, we prove that the diophantine equation

for has exactly 12 solutions except when , when it has 16 solutions. If denotes one of the zeros of , then for we also find all with .

     

The cardinality of the set of all clones containing a given minimal clone on three elements

All minimal clones on three elements were found by B. Csákány. In this paper, for each minimal clone the cardinality of the set of all clones containing this clone is found.     

On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let m?−1 be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations

X³−mX²Y−(m+3)XY²−Y³=λ     

Countable infinite sequence of attractors'' families for the simplest known equivariant chaotic flow

More than 30 new families of periodic and strange attractors for the simplest known equivariant chaotic jerk equation are studied. Inside each family, both periodic and chaotic attractors result from the subharmonic cascade of a limit cycle born by saddle-node bifurcation. Our numerical results provide clear evidence for the following conjecture: the 35 observed families are the first members of a countable infinite sequence. Furthermore, our simulations point out four power laws relating to the initial infinite sequence of saddle-node bifurcations.     

On the solutions of a family of quartic Thue equations

In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for and , with square-free, the Thue equation

has no integral solution except the trivial ones: .

     

On the approximation of unstable parametric minimal surfaces

We prove optimal convergence results for discrete approximations to (possibly unstable) minimal surfaces. This appears to be the first class of results of this type for geometric objects solving a highly non-linear geometric variational problem. We introduce a number of new techniques which we expect will be of use in other geometric problems. The theoretical approximation results are confirmed by numerical test computations.     

Fundamental systems of units in biquadratic parametric number fields

Let D, d be integers with D > 0, d dividing D and d square free and let α and β be the two roots of the quadratic equation x² ? Dx ?d = 0. Suppose (α ? β)² = D² + 4d > 1 and α² + β² = D² + 2d be square free. Introduce $\text{e =}\text{(ω + α)}\text{β}$ and $\text{u =}\text{(ω + β)}\text{α}$, where $\text{ω =}\text{a}{}^2\text{+ β}{}^2$. Then it is proved that e, u and θ form a fundamental system of units in the field $\text{Q}\text{D}{}^2\text{+ 4d}\text{,}\text{D}{}^2\text{+ 2d}$, where

$\text{θ=}\text{α}\text{β}\text{if}\text{d ≠ ± 1}\text{and}\text{(D,d) ≠ (5,?5)}$

$\text{θ=}\text{α}\text{β}\text{=}\text{1+}\text{5}\text{2}\text{if}\text{(D,d)=(5,?5)}$

$\text{θ=α}\text{if}\text{d = ± 1}\text{and}\text{(D,d) ≠ (3,?1)}$

$\text{θ=}\text{α=}\text{1+}\text{5}\text{2}\text{if}\text{(D,d)=(3,?1)}$

     

On the order of elements in long minimal zero-sum sequences

Let G be a finite abelian group and S = a minimal zero-sum sequence in G of maximal length |S| = l. We study the order of the elements      

On the branch point index of minimal surfaces

Let Γ be a closed, sufficiently smooth Jordan curve in and denote by the class of disk-type surfaces which map ∂B continuously and monotonically onto Γ. Then any minimal surface possesses only finitely many branch points in , and the order of any such point is well-defined, and also the index of an interior branch point is defined in a natural way if X is nonplanar. We show that also the index of boundary branch points can be defined if the curvature κ and the torsion τ of Γ are strictly nonzero. Secondly we derive upper bounds for the index of any branch point in terms of the total curvature of Γ or of its cut number. Dedicated to Professor Heinz K?nig on the occasion of his 80^th birthday