arXiv research

The paper studies stability of discrete planar curves using variational methods.

New method simplifies ideal curve flow with length constraint.

Critical trajectories in a sphere are found for a specific bending functional.

The paper studies how curves evolve under area constraints and converges to a critical point.

We prove the analyticity of smooth critical points for O'Hara's knot energies $\mathcal{E}^{α,p}$, with $p=1$ and $2<α< 3$, subject to a fixed length constraint. This implies, together with the main result in \cite{BR13}, that bounded energy critical points of $\mathcal{E}^{α,1}$ subject to a fixed length constraint ar…

The paper provides a theoretical justification for using stable SSM blocks in deep sequential models.

Algorithm minimizes loss and constraint violations in online convex optimization with smooth penalties.

Study of closed trajectories in hyperbolic plane with specific curvature constraints.

For curves of prescribed length embedded into the unit disc in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds $2π$ and in the large length limit. In the small excess length case, we prove convergence to a fourth order obstacle type problem with integral constraint on…

This paper constructs PH spline curves with prescribed arc lengths.

New algorithm reduces costs and latency for large language model inference.

Study finds knots with ideal length need not have smallest volume.

After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail.

Study shows curvature constraints force submanifolds to have specific topology or geometry.

Bayesian model predicts crack evolution on rails with uncertainties.

Computes volumes of metric maps on surfaces, linking to Weil-Petersson volumes.

Study on elastic curves with variable stiffness, derived from bending energy.

Paper approximates fractional harmonic maps with numerical methods.

The paper establishes bounds on the lengths of geodesics on manifolds with curvature constraints.

New algorithms optimize actions under time-varying constraints without projecting.

In this short article, we extend the cosine formula for the Möbius energy to generalized O'Hara energies. The newly derived formula gives us a condition for which the right circle minimizes the energy under the length-constraint. Furthermore, it shows us how far the energy is from the Möbius invariant property.

Random groups prove length constraints on product of conjugates.

Non-negative matrix factorization (NMF) is a dimensionality reduction technique which tends to produce a sparse representation of data. Commonly, the error between the actual and recreated matrices is used as an objective function, but this method may not produce the type of representation we desire as it allows for th…

New unknots with geometric constraints exist, proving a long-standing conjecture.

The ropelength problem asks for the minimum-length configuration of a knotted diameter-one tube embedded in Euclidean three-space. The core curve of such a tube is called a tight knot, and its length is a knot invariant measuring complexity. In terms of the core curve, the thickness constraint has two parts: an upper b…

Representation learning of pedestrian trajectories transforms variable-length timestamp-coordinate tuples of a trajectory into a fixed-length vector representation that summarizes spatiotemporal characteristics. It is a crucial technique to connect feature-based data mining with trajectory data. Trajectory representati…

Symmetric TSP is structurally equivalent to a constrained Group Steiner Tree Problem.

Exploiting a relationship between closed geodesics on a generic closed hyperbolic surface S and a certain unipotent flow on the product space T_1(S) x T_1(S), we obtain a local asymptotic equidistribution result for long closed geodesics on S. Applications include asymptotic estimates for the number of pants immersions…

Study explains Zipf's law using geometric mechanisms from a finite alphabet.

Estimates KVol on surfaces with geometric constraints.

Critical nets in $\mathbb{R}^k$ (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices (leaves) have a fixed position. In contrast to what happens on generic manifolds, w…

Curve diffusion flow straightens curves with endpoints on intersecting lines.

We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their kno…

The width $w$ of a curve $γ$ in Euclidean space $R^n$ is the infimum of the distances between all pairs of parallel hyperplanes which bound $γ$, while its inradius $r$ is the supremum of the radii of all spheres which are contained in the convex hull of $γ$ and are disjoint from $γ$. We use a mixture of topological and…

Dual decomposition provides a tractable framework for designing algorithms for finding the most probable (MAP) configuration in graphical models. However, for many real-world inference problems, the typical decomposition has a large integrality gap, due to frustrated cycles. One way to tighten the relaxation is to intr…

Recurrent Neural Networks (RNNS) are now widely used on sequence generation tasks due to their ability to learn long-range dependencies and to generate sequences of arbitrary length. However, their left-to-right generation procedure only allows a limited control from a potential user which makes them unsuitable for int…

New algorithm tackles dynamic assortment optimization with knapsack constraints.

Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes

Paper improves COCO problem, reducing constraint violation at the cost of slightly more regret.

Optimal control problems on Riemannian manifolds are solved by penalizing constraint violations.

Fixed angles of convex polygons lead to combinatorially rich polytopes.

We study the problem of finding strain-minimising stream surfaces in a divergence-free vector field. These surfaces are generated by motions of seed curves that propagate through the field in a strain minimising manner, i.e., they move without stretching or shrinking, preserving the length of their arbitrary arc. In ge…

Stability of branched immersions with energy constraints.

BOSS optimizes string inputs using string kernels and genetic algorithms.

In general relativity, spatial light rays of static spherically symmetric spacetimes are geodesics of surfaces in Riemannian optical geometry. In this paper, we apply results on the isoperimetric problem to show that length-minimizing curves subject to an area constraint are circles, and discuss implications for the ph…

The paper improves transformer generalization bounds using rank-dependent covering number bounds.

The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.

We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area constraint gives rise to equilibria with remarkable geometrical properties: not on…