Extremal families of cubic Thue equations

We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form $F(x,y)=1$ with at least $5$ such solutions. Our approach combines elementary arguments, with lower bounds for linear forms in logarithms and lattice-basis reduction

Multiples of integral points on Mordell curves

Let $P$ be a non torsion integral point on the minimal Mordell curve $E_B:y^2=x^3+B$. In this paper, we study integral multiples $[n]P$ of $P$. Among other results, we show that $P$ has at most three integral multiples with $n>1$. This result is sharp in the sense that there are points $P$ with exactly three integral multiples $[n]P$ and $n>1$. As an application, we discuss the number of integral points on minimal Mordell curves of rank 1

Thue equations

In this dissertation, we are mainly interested in effective methods to solve parametrized Thue equations. After briefly talking about the different effective methods, two parametrized families of cubic Thue equations are completely solved by using Pad é approximation and linear forms in logarithms. The Thue inequality ∣x³ + pxy² + qy³ ∣≤ k, is studied by using Bombieri's method. We find all solutions under some conditions on k, p and q. As an application of Thue equations, we find the integral points on the Mordell curves Y² = X³ + k for all nonzero integers k with ∣k∣ ≤ 10⁷ . Our approach uses a classical connection between these equations and cubic Thue equations.Science, Faculty ofMathematics, Department ofGraduat

Morphological and molecular analysis of the freshwater bivalve Anodonta anatina in Iran and Finland

Duck mussel, Anodonta anatina is a habitat generalist inhabiting both lentic and lotic aquatic ecosystems. Due to high morphological similarity and phenotypic plasticity, A. anatina has sometimes been misidentified as A. cygnea. Here, morphological and molecular studies were conducted on Anodonta mussels inhabiting North Iran and Finland. The individuals were collected from Anzali Wetland, Tajan River (North Iran) and Jyväsjärvi Lake (Finland). The COI sequence analysis showed the existence of A. anatina in the sampling areas. The Iranian and Finland specimens showed three and two haplotypes, respectively. The Iranian haplotypes were placed in a single clade, while the Finland haplotypes were clustered with those of Central Europe. The mean P-distance between these two clades was 2.4. The median-joining network showed that the Iranian haplotypes were lumped into a single haplogroup, while the Finland ones were in the same haplogroup as those from Central Europe. The Mediterranean haplotypes were the most divergent haplogroup from both Iranian and Central European haplogroups. In morphological characteristics, the shell pattern of all individuals from both Iranian and Finland specimens was stretched and slightly compact with light/dark brown periostracum. The mean length of the specimens from Anzali Wetland was significantly higher than those of Tajan and Jyväsjärvi. No significant difference was observed in morphometric characteristics between Tajan and Jyväsjärvi populations. The results did not indicate significant variation in shell morphology in the studied groups. In this regard, the conventional linear measurements can be supplemented using more complex geometric morphology in further studies