arXiv research

GONs improve predictions of maximizers from noisy black-box functions.

Geometrically classifies total stability spaces for Dynkin diagrams.

The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic geometry as well. In this paper, we generalize the discrete isoperimetric inequality t…

Classifies periodic points on regular and double n-gon surfaces.

In this short article, we find an explicit formula for Maslov index of Whitney n-gons joining intersections points of n half-dimensional tori in the symmetric product of a surface. The method also yields a formula for the intersection number of such an n-gon with the fat diagonal in the symmetric product.

We study 3-valent maps $M_n(p,q)$ consisting of a ring of $n$ $q$-gons whose the inner and outer domains are filled by $p$-gons, for $p,q \ge 3$. We describe a domain in the space of parameters $p$, $q$, and $n$, for which such a map may exist. With four infinite sequences of maps - prisms $M_p(p \ge 3,4)$, $M_4(4,q \g…

The paper shows that knot projections without triple chords can be simplified.

Study the space of simple polygons and their moduli.

We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely integrable systems on these moduli spaces by bending n-gons along their diagonals. Th…

We define an infinite series of translation coverings of Veech's double-n-gon for odd n greater or equal to 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a regular n-gon for even n greater or equal to 8. These families give rise to expl…

A regular $n$-gon inscribing a knot is a sequence of $n$ points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular $n$-gon for any $n$.

The reductivity of a spherical curve represents how reduced the spherical curve is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four by considering 4-gons.

In this paper, we construct new examples of Veech groups by extending Schmithusen's method for calculating Veech groups of origamis to Veech groups of unramified finite coverings of regular 2n-gons. We calculate the Veech groups of certain Abelian coverings of regular 2n-gons by using an algebraic method.

The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configuratio…

Unique Teichmüller curve found in complex geometry.

Researchers calculate complexity of billiard paths in regular polygons.

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $L/k$, where $L$ is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show that every doubled regular $n$-gon admits a $1/2n$-geodesic. For the doubled regu…

Inverse spectral theory reveals shapes from sound.

The equitangent locus of a convex plane curve consists of the points from which the two tangent segments to the curve have equal length. The equitangent problem concerns the relation between the curve and its equitangent locus. An equitangent n-gon of a convex curve is a circumscribed n-gon whose vertices belong to the…

We prove that every spherical football (also known as a spherical soccer ball) is a branched cover, branched only in the vertices, of the standard football made up of 12 pentagons and 20 hexagons. We also give examples showing that the corresponding result is not true for footballs of higher genera. Moreover, we classi…

The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.

We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in SU(2), denoted $C_r^n$, by the diagonal conjugation action …

Study calculates slope gaps on polygon surfaces, finding non-unimodal distributions.

Minimal surfaces in 3-sphere created by reflections from polygons, with new examples based on pentagons.

This paper proves that for large n, the regular polygon minimizes the first eigenvalue of the Laplacian.

In this paper we are interested in some Bonnesen-type isoperimetric inequalities for plane n-gons in relation with the two conjectures proposed by P. Levy and X.M. Zhang.

The space of polygons up to similarity is studied using the Schwarz-Christoffel formula.

We give a few simple methods to geometically describe some polygon and chain-spaces in R^d. They are strong enough to give tables of m-gons and m-chains when m <= 6.

Study of polygon degeneration to segments in complex space.

Fixed angles of convex polygons lead to combinatorially rich polytopes.

Study automorphism groups of quandles up to order 10.

The paper explores different perspectives on rhombile tilings.

We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in $\mathbb{Z}/a\mathbb{Z}$. We introduce a congruence level definition and a property of a primitive t…

Study on folded ribbon knots and their minimum length.

We prove that real projective space RP^{n-3} is homeomorphic to the space of all isometry classes of n-gons in the plane with one side of length n-2 and all other sides of length 1. This makes the topological complexity of real projective space more relevant to robotics.

Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. A related question is to estimate the number of connected components of the (super) level sets of a Neumann eigenfunction $u$. Indeed, in…

Study finds central points of double heptagon surface are not connection points.

Short proof for ideal polygons with near optimal orthogeodesic decomposition.

New set class preserves Fourier series terms for planar ovals, leading to isoperimetric inequalities.

We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute …

Maximizing energy on flexible curves yields regular or convex polygons.

Investigates conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.

A short proof for curve lengths on hyperbolic surfaces.

We describe the set of possible vector valued side lengths of n-gons in thick Euclidean buildings of rank 2. This set is determined by a finite set of homogeneous linear inequalities, which we call the generalized triangle inequalities. These inequalities are given in terms of the combinatorics of the spherical Coxeter…

Atiyah's conjecture concerning configurations of N points in the Euclidean three-space is verified for the following nonplanar configurations: The first m points lie on a line L and the remaining n=N-m (>2) points are the vertices of a regular n-gon whose plane is perpendicular to L and whose centroid is on L.

Call {\em i-hedrite} any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only and $p_2+p_3=i$. The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its {\em central circuits}, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge oppo…

We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($n\geq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with the conformal extremal metric on the regular $2n$-gon. The hexagon was considered…

New measures of asymmetry for triangles help evaluate electric power quality.