This book is a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and ...the most important partial results related to these problems.
Let $K$ and $L$ be two convex bodies in $\mathbb R^5$ with countably many diameters, such that their projections onto all 4 dimensional subspaces containing one fixed diameter are directly congruent. ...We show that if these projections have no rotational symmetries, and the projections of $K,L$ on certain 3 dimensional subspaces have no symmetries, then $K=\pm L$ up to a translation. We also prove the corresponding result for sections of star bodies.
This volume contains the proceedings of the 2016 Summer School on Fractal Geometry and Complex Dimensions, in celebration of Michel L. Lapidus's 60th birthday, held from June 21-29, 2016, at ...California Polytechnic State University, San Luis Obispo, California. The theme of the contributions is fractals and dynamics and content is split into four parts, centered around the following themes: Dimension gaps and the mass transfer principle, fractal strings and complex dimensions, Laplacians on fractal domains and SDEs with fractal noise, and aperiodic order (Delone sets and tilings).
In 2013, while studying a relevant class of polyominoes that tile the plane by translation, i.e., double square polyominoes, Blondin Massé et al. found that their boundary words, encoded by the ...Freeman chain coding on a four letters alphabet, have specific interesting properties that involve notions of combinatorics on words such as palindromicity, periodicity and symmetry. Furthermore, they defined a notion of reducibility on double squares using homologous morphisms, so leading to a set of irreducible tile elements called prime double squares. The authors, by inspecting the boundary words of the smallest prime double squares, conjectured the strong property that no runs of two (or more) consecutive equal letters are present there. In this paper, we prove such a conjecture using combinatorics on words’ tools, and setting the path to the definition of a fast generation algorithm and to the possibility of enumerating the elements of this class w.r.t. standard parameters, as perimeter and area.
“Compactness”, or the use of shape as a proxy for fairness, has been a long-running theme in the scrutiny of electoral districts; badly-shaped districts are often flagged as examples of the abuse of ...power known as gerrymandering. The most popular compactness metrics in the redistricting literature belong to a class of scores that we call contour-based, making heavy use of area and perimeter. This entire class of district scores has some common drawbacks, outlined here. We make the case for discrete shape scores and offer two promising examples: a cut score and a spanning tree score.
No shape metric can work alone as a seal of fairness, but we argue that discrete metrics are better aligned both with the grounding of the redistricting problem in geography and with the computational tools that have recently gained significant traction in the courtroom.
Distinct angles in general position Fleischmann, Henry L.; Konyagin, Sergei V.; Miller, Steven J. ...
Discrete mathematics,
April 2023, 2023-04-00, Letnik:
346, Številka:
4
Journal Article
Recenzirano
Odprti dostop
The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is the Erdős distinct angle problem, the problem of finding the minimum number of distinct ...angles between n non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by analogous questions in the distance setting.
In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by n points in general position from O(nlog2(7)) to O(n2). We consider a point-set to be in general position if no three points lie on a common line and no four lie on a common circle. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we introduce a construction employing the geometric properties of a logarithmic spiral, sidestepping the need for a projection.
We also apply this configuration to reduce the upper bound on the largest integer such that any set of n points in general position has a subset of that size with all distinct angles. This bound is decreased from O(nlog2(7)/3) to O(n1/2).
This monograph leads the reader to the frontiers of the very latest research developments in what is regarded as the central zone of discrete geometry. It is constructed around four classic problems ...in the subject, including the Kneser-Poulsen Conjecture.
Based on the classic augmented Lagrangian multiplier method, we propose, analyze and test an algorithm for solving a class of equality-constrained non-smooth optimization problems (chiefly but not ...necessarily convex programs) with a particular structure. The algorithm effectively combines an alternating direction technique with a nonmonotone line search to minimize the augmented Lagrangian function at each iteration. We establish convergence for this algorithm, and apply it to solving problems in image reconstruction with total variation regularization. We present numerical results showing that the resulting solver, called TVAL3, is competitive with, and often outperforms, other state-of-the-art solvers in the field.
PDF
