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We present explicit descriptions of the decompositions of vertices of a hypercube graph with respect to its distinguished symmetric cycle.
We theoretically show how two impurity defects in a crystalline structure can be entangled through coupling with the crystal. We demonstrate this with a harmonic chain of trapped ions in which two ions of a different species are embedded.…
We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix…
We introduce a concept of an embedding of a quadratic space in an associative algebra. The general properties of such embeddings are analyzed by linking it to the Clifford algebra. Conversely, there isa simple description of the standard…
We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a…
We develop a circle of ideas involving pairs of lines in the plane, intersections of hyperbolically rotated elliptical cones and the locus of the centers of rectangles inscribed in lines in the plane.
In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number…
A bisection line divides a convex planar curve into two parts with equal areas. It is natural to study the envelope of these lines, which in general present singularities. The polygonal case is particularly inte\-resting, since there are…
In this paper, we study the embedded topology of smooth plane quartics and its bitangent lines via two-graphs and apply it to construct interesting examples for Zariski $m$-ple.
The theorem of Greenberg-Kazhdan-Drinfeld describes the formal neighborhood of a closed arc. After giving a complete proof with examples, two possible versions for the relative case of the theorem are discussed. Each one is shown to hold…
In this follow-up article to Symplectification of Circular Arcs and Arc Splines, biarc geometry is examined from a purely geometric point of view. Two given points together with their associated tangent vectors in the plane are sufficient…
In this paper we study how to distinguish two embeddings of a finite collection of disjoint circles into the plane up to planar isotopy. We adopt the spirit of the approach by V. Turaev, Operator Invariants of Tangles, Math. USSR-Izv. 35…
Partitionings (or segmentations) divide a given domain into disjoint connected regions whose union forms again the entire domain. Multi-dimensional partitionings occur, for example, when analyzing parameter spaces of simulation models,…
The task of approximating points with circular arcs is performed in many applications, such as polyline compression, noise filtering, and feature recognition. However, the development of algorithms that perform a significant amount of…
We consider the space of embeddings of finitely many circles that bound disks in non-positively curved surfaces. We index the connected components of this space with finite rooted trees and show that the connected components are classifying…
For certain bordered submanifolds $M\subset\CC^2$ we show that $M$ can be embedded properly and holomorphically into $\CC^2$. An application is that any subset of a torus with two boundary components can be embedded properly into $\CC^2$.
Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then…
Rationally convex topological embeddings of compact surfaces (closed or with boundary) into $\mathbb{C}^2$ are constructed.
We introduce an orbifold induction procedure which provides a systematic construction of cyclic orbifolds, including their twisted sectors. The procedure gives counterparts in the orbifold theory of all the current-algebraic constructions…
In this paper we show how, under surprisingly weak assumptions, one can split a planar curve into three arcs and rearrange them (matching tangent directions) to obtain a closed curve. We also generalize this construction to curves split…