arXiv research

Study on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are dif…

The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or…

New approach to $Z$-stability and critical metrics on Kähler manifolds.

Rigidity of critical eigensections on spheres proven.

We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute …

We describe several configurations of clasped ropes which are balanced and thus critical for the Gehring ropelength problem of arXiv:math.DG/0402212.

We describe the configuration space $\mathbf{S}$ of polygons with prescribed edge slopes, and study the perimeter $\mathcal{P}$ as a Morse function on $\mathbf{S}$. We characterize critical points of $\mathcal{P}$ (these are \textit{tangential} polygons) and compute their Morse indices. This setup is motivated by a num…

In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring's problem to deal with links in a fixed link-homotopy …

The oriented area function $A$ is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the Morse index of a critical point. However, for planar polygons, the function $A$ i…

We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are co…

We study polygon spaces arising from planar configurations of necklaces with some of the beads fixed and some of the beads sliding freely. These spaces include configuration spaces of flexible polygons and some other natural polygon spaces. We characterise critical points of the oriented area function in geometric term…

Study spider mechanism configuration spaces using squared distance function.

In the present paper we introduce Mobius energy for the embedded graphs and formulate its main properties. This energy is invariant under the action of the group generated by all inversions in three-dimensional real space. We study critical configurations for the angles at vertices of degree less than five, and discuss…

In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter…

Study on surface configurations with curvature and elasticity.

We find that Koschorke's $β$-invariant and the triple $μ$-invariant of link maps in the critical dimension can be computed as degrees of certain maps of configuration spaces - just like the linking number. Both formulas admit geometric interpretations in terms of Vassiliev's ornaments via new operations akin to the Jin…

A new sampler tackles critical phenomena by leveraging scale invariance.

Morse inequalities for noncompact manifolds with group action.

Using results on the topology of moduli space of polygons [Jaggi, 92; Kapovich and Millson, 94], it can be shown that for a planar robot arm with $n$ segments there are some values of the base-length, $z$, at which the configuration space of the constrained arm (arm with its end effector fixed) has two disconnected com…

Polytopes $A(P)$ for posets $P$ compactify spaces of order-preserving maps.

Cellular network configuration plays a critical role in network performance. In current practice, network configuration depends heavily on field experience of engineers and often remains static for a long period of time. This practice is far from optimal. To address this limitation, online-learning-based approaches hav…

Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.

We develop a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on $\mathbb{CP}^2$ and the product metric on $S^2 \times S^2$. Using these met…

Assemblies of modular subsystems are being pressed into service to perform sensing, reasoning, and decision making in high-stakes, time-critical tasks in such areas as transportation, healthcare, and industrial automation. We address the opportunity to maximize the utility of an overall computing system by employing re…

Characterizes solutions to Z-critical equations on surfaces using effective conditions.

Under-parameterized networks can either copy or average teacher weights, leading to universal optimal solutions.

Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-secti…

Recurrent neural networks are a powerful tool, but they are very sensitive to their hyper-parameter configuration. Moreover, training properly a recurrent neural network is a tough task, therefore selecting an appropriate configuration is critical. Varied strategies have been proposed to tackle this issue. However, mos…

Study on the Euler-Plateau energy with elastic modulus, focusing on minimizers and critical surfaces.

New method uses RBM flows to find critical temperatures in Ising models.

The paper studies critical points and flows of a $G_2$-Hilbert functional on manifolds with circle actions.

We compare two combinatorial models for the moduli space of two-dimensional cobordisms: Bödigheimer's radial slit configurations and Godin's admissible fat graphs, producing an explicit homotopy equivalence using a "critical graph" map. We also discuss natural compactifications of these two models, the unilevel harmoni…

Performance of machine learning algorithms depends critically on identifying a good set of hyperparameters. While recent approaches use Bayesian optimization to adaptively select configurations, we focus on speeding up random search through adaptive resource allocation and early-stopping. We formulate hyperparameter op…

USAC balances pessimism and optimism in actor-critic training for better exploration and performance.

Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-\infty, a]$ of a Morse function $f: M \to R$, where $M$ is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets $f^{-1}(a)$ and give conditions under which the topology of $f^{-1}(a)$ changes when passing a cri…

The study connects K-stability and large complex structure limits in mirror symmetry.

Meta-active learning optimizes control of safety-critical systems by efficiently learning dynamics and configurations.

Acute kidney injury (AKI) in critically ill patients is associated with significant morbidity and mortality. Development of novel methods to identify patients with AKI earlier will allow for testing of novel strategies to prevent or reduce the complications of AKI. We developed data-driven prediction models to estimate…

The study proves the finiteness of moments for Gaussian field zeros and critical points.

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…

Theoretical understanding of how deep neural network (DNN) extracts features from input images is still unclear, but it is widely believed that the extraction is performed hierarchically through a process of coarse-graining. It reminds us of the basic concept of renormalization group (RG) in statistical physics. In ord…

Automated HPO design using Bayesian optimization and benchmarking.

Macromolecular and biomolecular folding landscapes typically contain high free energy barriers that impede efficient sampling of configurational space by standard molecular dynamics simulation. Biased sampling can artificially drive the simulation along pre-specified collective variables (CVs), but success depends crit…

The paper studies Einstein-Hilbert functional and its relation to K-semistability.

Paper develops an efficient approach to reduce HPO time.

Study finds minimal length networks connecting three points in Heisenberg group.