Geometric Searching over the Rationals

Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q

Let E be an elliptic curve over the rationals. A crucial step in determining a Mordell-Weil basis for E is to exhibit some positive lower bound λ > 0 for the canonical height ĥ on non-torsion points. We give a new method for determining such a lower bound, which does not involve any searching for points.

The Role of the Scholars` Tradition in Formulating Moral Propositions, with an Emphasis on Muhaqqiq Isfahani`s View

The issue of the source of ethical value is one of the most important issues in the field of metaethics. Moral philosophers have come up with different perspectives on this issue. Some Islamic thinkers, including Muhaqqiqi Isfahani, have considered the rationals` tradition as the formulating cause for moral propositions. This research is intended to emphasize on the viewpoint of Muhaqqiqi Isfah...

A Spectral Approach to Lower Bounds with Applications to Geometric Searching

We establish a nonlinear lower bound for halfplane range searching over a group. Specifically, we show that summing up the weights of n (weighted) points within n halfplanes requires Ω(n logn) additions and subtractions. This is the first nontrivial lower bound for range searching over a group. By contrast, range searching over a semigroup (which forbids subtractions) is almost completely under...

Newman Polynomials, Reducibility, and Roots on the Unit Circle

A length k Newman polynomial is any polynomial of the form za1 + · · ·+zak (where a1 < · · · < ak). Some Newman polynomials are reducible over the rationals, and some are not. Some Newman polynomials have roots on the unit circle, and some do not. Defining, in a natural way, what we mean by the “proportion” of length k Newman polynomials with a given property, we prove that • 1/4 of length 3 Ne...

Geometric Range Searching and Its Relatives Geometric Range Searching and Its Relatives