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Percolation on a plane is usually associated with clusters spanning two opposite sides of a rectangular system. Here we investigate three-leg clusters generated on a square lattice and spanning the three sides of equilateral triangles. If…
The problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container so as to minimize a measure of overlap between ellipsoids is considered. A bilevel optimization formulation is given, together with an algorithm…
An arrangement of circles in which circles intersect only in angles of $\pi/2$ is called an \emph{arrangement of orthogonal circles}. We show that in the case that no two circles are nested, the intersection graph of such an arrangement is…
We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can…
A loop structure was predicted to exist in the Bloch bands of Bose-Einstein condensates in optical lattices recently in [{\it Wu and Niu, Phys. Rev. A {\bf 61}, 023402 (2000)}]. We discuss how to detect experimentally the looped band with…
An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry.…
We propose a precise ellipsometric method for the investigation of coherent light with a small ellipticity. The main feature of this method is the use of compensators with phase delays providing the maximum accuracy of measurements for the…
An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base…
In this article, a posteriori error analysis of the elliptic obstacle problem is addressed using hybrid high-order methods. The method involve cell unknowns represented by degree-$r$ polynomials and face unknowns represented by degree-$s$…
Given two curves in $\PP^3$, either implicitly or by a parameterization, we want to check if they intersect. For that purpose, we present and further develop generalized resultant techniques. Our aim is to provide a closed formula in the…
Four-dimensional surgery is used to show that a two component link with Alexander polynomial one is topologically concordant to the Hopf link.
By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.
We prove an intersection formula for two plane branches in terms of their semigroups and key polynomials. Then we provide a strong version of Bayer's theorem on the set of intersection numbers of two branches and apply it to the logarithmic…
We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows us to connect the resulting Selmer group…
In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the…
A systematic construction of higher order splines using two hierarchies of polynomials is presented. Explicit instructions on how to implement one of these hierarchies are given. The results are limited to interpolations on regular,…
We present a novel interferometric technique for performing ellipsometric measurements. This technique relies on the use of a non-classical optical source, namely, polarization-entangled twin photons generated by spontaneous parametric…
How can one detect entanglement between multiple optical paths sharing a single photon? We address this question by proposing a scalable protocol, which only uses local measurements where single photon detection is combined with small…
We show how to detect entanglement with criteria built from simple two-body correlation terms. Since many natural Hamiltonians are sums of such correlation terms, our ideas can be used to detect entanglement by energy measurement. Our…
A classical two-stranded rope can be made by twisting two identical strands together under strain. Despite being conceptually simple, the contact-equations for helically twisted identical strands have only been solved within the last 20…