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Full-Text Articles in Algebraic Geometry
Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv
Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv
MSU Graduate Theses
Finding rational points that satisfy functions known as elliptic curves induces a finitely-generated abelian group. Such functions are powerful tools that were used to solve Fermat's Last Theorem and are used in cryptography to send private keys over public systems. Elliptic curves are also useful in factoring and determining primality.
On Elliptic Curves, Montana S. Miller
On Elliptic Curves, Montana S. Miller
MSU Graduate Theses
An elliptic curve over the rational numbers is given by the equation y2 = x3+Ax+B. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.
The Average Measure Of A K-Dimensional Simplex In An N-Cube, John A. Carter
The Average Measure Of A K-Dimensional Simplex In An N-Cube, John A. Carter
MSU Graduate Theses
Within an n-dimensional unit cube, a number of k-dimensional simplices can be formed whose vertices are the vertices of the n-cube. In this thesis, we analyze the average measure of a k-simplex in the n-cube. We develop exact equations for the average measure when k = 1, 2, and 3. Then we generate data for these cases and conjecture that their averages appear to approach nk/2 times some constant. Using the convergence of Bernstein polynomials and a k-simplex Bernstein generalization, we prove the conjecture is true for the 1-simplex and 2-simplex cases. We then develop a generalized formula for …