Veech surfaces and simple closed curves

We study the SL(2, ?)-infimal lengths of simple closed curves on halftranslation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths.We also revisit the “no small virtual triangles” theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spectrum of a half-translation surface either has a gap above zero or is dense in a neighborhood of zero.These results make use of the auxiliary polygon associated to a curve on a half-translation surface, as introduced by Tang and Webb.     

Hot spots conjecture for a class of acute triangles

We show that the hot spots conjecture of J. Rauch holds for acute triangles if one of the angles is not larger than \(\pi /6\). More precisely, we show that the second Neumann eigenfunction on those acute triangles has no maximum or minimum inside the domain. We first simplify the problem by showing that absence of critical points on two sides implies no critical points inside a triangle. This result applies to any acute triangle and might help prove the conjecture for arbitrary acute triangles. Then we show that there are no critical points on two sides assuming one small angle. We also establish simplicity for the smallest positive Neumann eigenvalue for all non-equilateral acute triangles. This result was already known for obtuse triangles, and it fails for the equilateral case.     

How to best fold a triangle

We fold a triangle once along a straight line and study how small the area of the folded figure can be. It can always be as small as the fraction \(2-\sqrt{2}\) of the area of the original triangle.This is best possible: For every positive number \(\varepsilon\) there are triangles that cannot be folded better than \(2-\sqrt{2}-\varepsilon\).     

The Smallest Convex Cover for Triangles of Perimeter Two

We find the unique smallest convex region in the plane that contains a congruent copy of every triangle of perimeter two. It is the triangle ABC with AB=2/3, B=60°, and BC1.00285.     

Minimal regular polygons serving as universal covers in R 2

A universal cover is a set K with the property that each set of unit diameter is a subset of a congruent copy of K. It is shown that the smallest regular n-gon, for fixed n 4, which serves as an universal cover in R ² is the smallest regular n-gon covering a Reuleaux triangle of unit width.     

Flatness of Gaussian curvature and area of ideal triangles

Let M be aC ^k ,k 4, compact surface of genus greater than two whose curvature is negative in all points but along a simple closed geodesic (t) where the curvature is zero at every point. We show that the area of ideal triangles having a lifting of as an edge is infinite. This provides a family of surfaces having ideal triangles of infinite area whose geodesic flows are equivalent to Anosov flows, in contrast with the well-known examples of surfaces with flat strips which also have ideal triangles of infinite area. By the CAT-comparison theory we can deduce, using these surfaces as models, that aC compact surface of non-positive curvature having one geodesic along which the curvature is zero has ideal triangles of infinite area.Partially supported by CNPq of Brazilian Government     

Packing similar triangles into a triangle

There exists a triangle T and a number \frac{1}{2}$$ " align="middle" border="0"> such that any sequence of triangles similar to T with total area not greater than times the area of T can be packed into T.     

Right angle free subsets in the plane

In this paper we investigate the complexity of finding maximum right angle free subsets of a given set of points in the plane. For a set of rational pointsP in the plane, theright angle number (P) (respectivelyrectilinear right angle number _R (P)) ofP is the cardinality of a maximum subset ofP, no three members of which form a right angle triangle (respectively a right angle triangle with its side or base parallel to thex-axis). It is shown that both parameters areNP-hard to compute. The latter problem is also shown to be equivalent to finding a minimum dominating set in a bipartite graph. This is used to show that there is a polynomial algorithm for computing _R (P) whenP is a horizontally-convex subset of the lattice × (P ishorizontally-convex if for any pair of points inP which lie on a horizontal line, every lattice point between them is also inP). We then show that this algorithm yields a 1/2-approximate algorithm for the right angle number of a convex subregion of the lattice.     

Random Triangle Theory with Geometry and Applications

What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to \((0,0)\) or reformulation as a \(2\times 2\) random matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage others to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the “square root ellipticity statistic” of random matrix theory. Another proof connects the Hopf map to the SVD of \(2\times 2\) matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight into both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed, and applications are proposed.     

Algorithms for optimal area triangulations of a convex polygon

Given a convex polygon with n vertices in the plane, we are interested in triangulations of its interior, i.e., maximal sets of non-intersecting diagonals that subdivide the interior of the polygon into triangles. The MaxMin area triangulation is the triangulation of the polygon that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. We present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in O(n²logn) time and O(n²) space. The algorithms use dynamic programming and a number of geometric properties that are established within the paper.     

Percolation of even sites for random sequential adsorption

Consider random sequential adsorption on a red/blue chequerboard lattice with arrivals at rate 1 on the red squares and rate λ$\lambda$ on the blue squares. We prove that the critical value of λ$\lambda$, above which we get an infinite blue component, is finite and strictly greater than 1.     

Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness property

We consider a natural class of composite finite elements that provide the mth-order smoothness of the resulting piecewise polynomial function on a triangulated domain and do not require any information on neighboring elements. It is known that, to provide a required convergence rate in the finite element method, the “smallest angle condition” must be often imposed on the triangulation of the initial domain; i.e., the smallest possible values of the smallest angles of the triangles must be lower bounded. On the other hand, the negative role of the smallest angle can be weakened (but not eliminated completely) by choosing appropriate interpolation conditions. As shown earlier, for a large number of methods of choosing interpolation conditions in the construction of simple (noncomposite) finite elements, including traditional conditions, the influence of the smallest angle of the triangle on the error of approximation of derivatives of a function by derivatives of the interpolation polynomial is essential for a number of derivatives of order 2 and higher for m ≥ 1. In the present paper, a similar result is proved for some class of composite finite elements.     

Triangles and groups via cevians

For a given triangle T and a real number ρ we define Ceva’s triangle ${\mathcal{C}_{\rho}(T)}$ to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ: (1 – ρ). We identify the smallest interval ${\mathbb{M}_T \subset \mathbb{R}}$ such that the family ${\mathcal{C}_{\rho}(T), \rho \in \mathbb{M}_T}$ , contains all Ceva’s triangles up to similarity. We prove that the composition of operators ${\mathcal{C}_\rho, \rho \in \mathbb{R}}$ , acting on triangles is governed by a certain group structure on ${\mathbb{R}}$ . We use this structure to prove that two triangles have the same Brocard angle if and only if a congruent copy of one of them can be recovered by sufficiently many iterations of two operators ${\mathcal{C}_\rho}$ and ${\mathcal{C}_\xi}$ acting on the other triangle.     

Congruence kernels in Ockham algebras

We consider, in the context of an Ockham algebra \({{\mathcal{L} = (L; f)}}\), the ideals I of L that are kernels of congruences on \({\mathcal{L}}\). We describe the smallest and the largest congruences having a given kernel ideal, and show that every congruence kernel \({I \neq L}\) is the intersection of the prime ideals P such that \({I \subseteq P}\), \({P \cap f(I) = \emptyset}\), and \({f^{2}(I) \subseteq P}\). The congruence kernels form a complete lattice which in general is not modular. For every non-empty subset X of L, we also describe the smallest congruence kernel to contain X, in terms of which we obtain necessary and sufficient conditions for modularity and distributivity. The case where L is of finite length is highlighted.     

Sequences of integers with missing quotients and dense points without neighbors

Let $A$ be a pre-defined set of rational numbers. We say that a set of natural numbers $S$ is an $A$-quotient-free set if no ratio of two elements in $S$ belongs to $A$. We find the maximal asymptotic density and the maximal upper asymptotic density of $A$-quotient-free sets when $A$ belongs to a particular class.It is known that in the case $A=\{p,q\}$, where $p$, $q$ are coprime integers greater than 1, the latter problem is reduced to the evaluation of the largest number of non-adjacent lattice points in a triangle whose legs lie on the coordinate axes. We prove that this number is achieved by choosing points of the same color in the checkerboard coloring.     

Belyi Uniformization of Elliptic Curves

Belyi's Theorem implies that a Riemann surface X representsa curve defined over a number field if and only if it can beexpressed as U/, where U is simply-connected and is a subgroupof finite index in a triangle group. We consider the case whenX has genus 1, and ask for which curves and number fields canbe chosen to be a lattice. As an application, we give examplesof Galois actions on Grothendieck dessins. 1991 MathematicsSubject Classification 30F10, 11G05.     

Packing coloring of Sierpiński-type graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\), \(i\in \{1,\ldots ,k\}\), where each \(V_i\) is an i-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by 8 in the family of Sierpiński graphs with base 3, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than 3. On the other hand, we prove that the packing chromatic number in the family of Sierpiński triangle graphs \(ST^n_3\) is bounded from above by 31. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpiński graphs \(S^n_G\) with respect to all connected graphs G of order 4.     

Arithmetic properties arising from Ramanujan’s theta functions

We prove some interesting arithmetic properties of theta function identities that are analogous to q-series identities obtained by Michael D. Hirschhorn. In addition, we find infinite family of congruences modulo powers of 2 for representations of a non-negative integer n as \(\triangle _1+4\triangle _2\) and \(\triangle +k\square \).     

An axiomatic look at the Erdős-Trost problem

The Erd?s-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC—with X, Y, and Z on the sides BC, CA, and AB, respectively—then one of the areas of the triangles BXZ, CXY , AY Z is less than or equal to the area of the triangle XY Z.” There are many different solutions for this problem. In this note we take up a very elementary proof (due to Szekeres) and deduce that the class of ordered translation planes is the level in the hierarchy of affine planes where the Erd?s-Trost statement still holds true. We also look at the conditions an absolute plane needs to satisfy for the validity of the Erd?s-Trost statement.