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The representation of a graph as an abstract intersection graph of sets can be used to construct more concrete geometric intersection representations of the same graph. In particular, if a graph has intersection number k {\displaystyle k} , it can be represented as an intersection graph of k {\displaystyle k} -dimensional unit hyperspheres (its ...
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:
For the curve y 2 = x 3 + ax 2 + bx + c (the general form of an elliptic curve with characteristic 3), the formulas are similar, with s = x P 2 + x P x Q + x Q 2 + ax P + ax Q + b / y P + y Q and x R = s 2 − a − x P − x Q. For a general cubic curve not in Weierstrass normal form, we can still define a group structure by ...
In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis , following earlier results that every planar graph could be represented as the intersection graph of a set of simple curves in the plane (Ehrlich, Even & Tarjan 1976).
We can take a rotation of the axes where the b axis is the line y=x oriented northeast and the a axis is the line y=−x oriented southeast. These new axes are related to the original x-y axes by x=(b+a)/ √ 2 and y=(b−a)/ √ 2. We obtain, after substitution into (4) and expansion and simplification,
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The values of u, v, w such that g(u, v, w) = 0 are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such that w is not zero. An example is the Fermat curve u n + v n = w n, which has an affine form x n + y n = 1. A similar process of homogenization may ...