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We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are…
Special bases of orthogonal polynomials are defined, that are suited to expansions of density and potential perturbations under strict particle number conservation. Particle-hole expansions of the density response to an arbitrary…
Many parametrization and mapping-related problems in geometry processing can be viewed as metric optimization problems, i.e., computing a metric minimizing a functional and satisfying a set of constraints, such as flatness. Penner…
An end-grafted flexible polymer chain in 3d space between two pistons undergoes an abrupt transition from a confined coil to a flower-like conformation when the number of monomers in the chain, N, reaches a critical value. In 2d geometry,…
We propose a fast and scalable algorithm to project a given density on a set of structured measures defined over a compact 2D domain. The measures can be discrete or supported on curves for instance. The proposed principle and algorithm are…
Cages ($r$-regular graphs of girth $g$ and minimum order) and their variants have been studied for over seventy years. Here we propose a new variant, "weighted cages". We characterize their existence; for cases $g=3,4$ we determine their…
The theory of modular deformations is generalized for the category of complex analytic polyhedra which includes germs of complex space as well as any compact complex analytic space. The objective of the theory is a construction of fine…
We introduce constrained polynomial zonotopes, a novel non-convex set representation that is closed under linear map, Minkowski sum, Cartesian product, convex hull, intersection, union, and quadratic as well as higher-order maps. We show…
Modular invariance imposes rigid constrains on the partition functions of two-dimensional conformal field theories. Many fundamental results follow strictly from modular invariance, giving rise to the numerical modular bootstrap program.…
The two-matrix model can be solved by introducing bi-orthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of bi-orthogonal polynomials (called "windows") satisfy polynomial ODEs as well as…
This is a review of two-dimensional conformal field theory including some of the relations to integrable models. An effort is made to develop the basic formalism in a way which is as elementary and flexible as possible at the same time.…
Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes.…
The classical modular polynomial for $j$-invariants describes the relation between two elliptic curves connected by isogenies. This polynomial has been applied to various algorithms in computational number theory, such as point counting on…
In conformal differential geometry, there are some distinguished curves, often known as 'conformal circles,' since, on the round sphere, they are the round circles (and these are conformally invariant). But on the two-sphere, the curves of…
In this paper, we introduce multiple skew-orthogonal polynomials and investigate their connections with classical integrable systems. By using Pfaffian techniques, we show that multiple skew-orthogonal polynomials can be expressed by…
Two-component second and third-order Burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries. New symmetry integrable systems with their master symmetries are obtained. Some third…
We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by…
We consider harmonic functions of polynomial growth of some order $d$ on Cayley graphs of groups of polynomial volume growth of order $D$ w.r.t. the word metric and prove the optimal estimate for the dimension of the space of such harmonic…
Aiming to generate easy-to-handle assembly sequences for robotic assembly, this study tackles assembly sequence generation by considering two tradeoff objectives: (1) insertion conditions and (2) degrees of constraints among assembled…
We present a first exact study on higher-dimensional packing problems with order constraints. Problems of this type occur naturally in applications such as logistics or computer architecture and can be interpreted as higher-dimensional…