Related papers: Curious Pivot Points of Least-Squares Regression
After surveying some known properties of compact convex sets in the plane, we give a two rigorous proofs of the general feeling that supporting lines can be slide-turned slowly and continuously. Targeting a wide readership, our treatment is…
We construct a point set in the Euclidean plane that elucidates the relationship between the fine-scale statistics of the fractional parts of $\sqrt n$ and directional statistics for a shifted lattice. We show that the randomly rotated, and…
We study a family of maps from $S_n \to S_n$ we call fixed point homing shuffles. These maps generalize a few known problems such as Conway's Topswops, and a card shuffling process studied by Gweneth McKinley. We show that the iterates of…
In this paper the problem of retrospective change-point detection and estimation in multivariate linear models is considered. The lower bounds for the error of change-point estimation are proved in different cases (one change-point:…
Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but…
Let S be a set of 2n+1 points in the plane such that no three are collinear and no four are concyclic. A circle will be called point-splitting if it has 3 points of S on its circumference, n-1 points in its interior and n-1 in its exterior.…
In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface.
We study sets of bounded remainder for the two-dimensional continuous irrational rotation $(\{x_1+t\}, \{x_2+t\alpha \})_{t \geq 0}$ in the unit square. In particular, we show that for almost all $\alpha$ and every starting point $(x_1,…
The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail…
In this paper, we construct rotating frames for curves, including plane curves, space curves and curves on surfaces. Hence, the behaviour of an arbitrary moving point on a curve can be seen as the composite of linear motion and rotation.…
This paper proposes the capped least squares regression with an adaptive resistance parameter, hence the name, adaptive capped least squares regression. The key observation is, by taking the resistant parameter to be data dependent, the…
It is a well-known fact that genetic sequences may contain sections with repeated units, called repeats, that differ in length over a population, with a length distribution of geometric type. A simple class of recombination models with…
To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century when Carl Friedrich Gauss developed the…
The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to…
Fitting linear regression models can be computationally very expensive in large-scale data analysis tasks if the sample size and the number of variables are very large. Random projections are extensively used as a dimension reduction tool…
A nearly unstable sequence of stationary spatial autoregressive processes is investigated, when the sum of the absolute values of the autoregressive coefficients tends to one. It is shown that after an appropriate norming the least squares…
A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the…
Let S be a set of distinct points in general position in the Euclidean plane. A plane Hamiltonian path on S is a crossing-free geometric path such that every point of S is a vertex of the path. It is known that, if S is sufficiently large,…
We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled, and noisy observations. We focus on the low SNR regime, and show that a signal in $\mathbb{R}^M$ is uniquely…
Correlations between random variables play an important role in applications, e.g.\ in financial analysis. More precisely, accurate estimates of the correlation between financial returns are crucial in portfolio management. In particular,…