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We first prove a one-to-one correspondence between finding Hamiltonian cycles in a cubic planar graphs and finding trees with specific properties in dual graphs. Using this information, we construct an exact algorithm for finding…
We automatically verify the crucial steps in the original proof of correctness of an algorithm which, given a geometric graph satisfying certain additional properties removes edges in a systematic way for producing a connected graph in…
Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of…
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…
We propose an algebraic study of the simple graph isomorphism problem. We define a Hopf algebra from an explicit realization of its elements as formal power series. We show that these series can be evaluated on graphs and count occurrences…
In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is…
We classify transverse Hopf links in the standard contact 3-space up to transverse isotopy in terms of their components' self-linking number.
The method of constructing trigonometric Hermite splines, which interpolate the values of some periodic function and its derivatives in the nodes of a uniform grid, is considered. The proposed method is based on the periodicity properties…
This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other…
Whether a system is to be considered complex or not depends on how one searches for correlations. We propose a general scheme for calculation of entropies in lattice systems that has high flexibility in how correlations are successively…
This paper is devoted to presenting a new approach to determine the intersection of two quadrics based on the detailed analysis of its projection in the plane (the so called cutcurve) allowing to perform the corresponding lifting correctly.…
Flip graphs of non-crossing configurations in the plane are widely studied objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian cycles, and perfect matchings. Typically, it is an easy exercise to prove connectivity of a…
Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, we define elliptic loops as the points of $\mathbb{P}^2(R)$ projecting to $E$ under the canonical modulo-$\mathfrak{m}$ reduction, endowed with an operation…
We present a protocol for increasing the entanglement between two atomic ensembles based on applying an approximate atom-light beamsplitter transformation to both ensembles. The effective asymmetric atom-light beamsplitter is created via a…
We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field $k$, this enrichment counts the number of lines meeting four lines defined…
We introduce a new method for predicting the formation of links in real-world networks, which we refer to as the method of effective transitions. This method relies on the theory of isospectral matrix reductions to compute the probability…
A framework to systematically decouple high order elliptic equations into combination of Poisson-type and Stokes-type equations is developed. The key is to systematically construct the underling commutative diagrams involving the complexes…
We propose a method called `coherence swapping' which enables us to create superposition of a particle in two distinct paths, which is fed with initially incoherent, independent radiations. This phenomenon is also present for the charged…
We explore exchange coupling of a pair of spins in a double dot and in an optical lattice. Our algorithm uses the frequency of exchanges in a bosonic path integral, evaluated with Monte Carlo. This algorithm is simple enough to be a "black…
Collision detection is a critical functionality for robotics. The degree to which objects collide cannot be represented as a continuously differentiable function for any shapes other than spheres. This paper proposes a framework for…