Theorem regarding Orthogonal Conics

On a Theorem of Intersecting Conics

Given two conics over an infinite field that intersect at the origin, a line through the origin will, in general intersect both conic sections once more each, at points C and D. As the line varies we find that the midpoint of C and D traces out a curve, which is typically a quartic. Intuitively, this locus is the “average” of the two conics from the perspective of an observer at the origin. We ...

Some Consequences of a Well Known Theorem on Conics

Orthogonal Ham-Sandwich Theorem in R

The ham-sandwich theorem states that, given d ≥ 2 measures in R, it is possible to divide all of them in half with a single (d − 1)-dimensional hyperplane. We study an orthogonal version of the ham-sandwich theorem and define an orthogonal cut using at most d hyperplanes orthogonal to coordinate axes. For example, a hyperplane orthogonal to a coordinate axis and the boundary of an orthant are o...

Structure theorem of square complex orthogonal design

Square COD (complex orthogonal design) with size [n, n, k] is an n×n matrix Oz, where each entry is a complex linear combination of zi and their conjugations z∗ i , i = 1, . . . , k, such that O H z Oz = (|z1| 2 + . . . + |zk| 2)In. Closely following the work of Hottinen and Tirkkonen, which proved an upper bound of k/n by making a crucial observation between square COD and group representation...

Blumenthal's Theorem for Laurent Orthogonal Polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form Bn(x) = (x − βn)Bn−1(x) − αnxBn−2(x), with positive recurrence coefficients αn+1, βn (n = 1, 2, . . .). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. ...