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The Mordell–Weil theorem states that for an abelian variety A over a number field k, the group A(k) of k-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group. The case with A an elliptic curve E and k the rational number field Q is Mordell’s theorem, answering a question apparently posed by Poincare around 1908; it was proved by Louis Mordell in 1922. The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(Q)=2E(Q) which forms a major step in the... 

In elliptic curves equation, if the number of solution is infinite, complication of calculation increase quickly. Even the simplest solutioncan be large. y2 + y = x35115523309x − 140826120488927 The x-coordinate number of smallest solution has 5454 digits. The theorem of Mordell and Weil states that “The set E(Q) of rational solutions has the structure of an infinitely generated abelian group.” E(Q) = (Z)rankE(Q) ⊕T Can the rank be arbitrary large? The current record is rank(E)=28.Manjul Behargava has recently made progress on the study of the average rank for all elliptic curves with rational coefficient. Every such curve has a unique equation of the form y2 = x3 +Ax+B, where A and B are... 

In this thesis, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than 1 are simple. Here we consider the multiplicities of the other (non- integral) Laplacian eigenvalues.We provide an upper bound and determine the trees of order n that have a multiplicity that is close to the upper bound (n-3)/2 , and emphasize the particular role of the algebraic connectivity.In continuation, let G be a graph and I be an interval. We present bounds for the number m_G I of Laplacian eigenvalues in I in terms of structural parameters of G. In particular, we show that m_G (n-α(G),n]≤ n-α(G) and m_G (n-d(G)+3,n]≤ n-d(G)-1, where...