A rope is a non-singular embedding of a closed interval into R^3, which sends the ends of the interval to some fixed points A and B such that |AB|=1. A rope is short if its length is less than 3. The main result of the paper is that the fundamental group of the space of short ropes is naturally isomorphic to the group …
Study confirms ropes and long knots have similar topological properties.
problem Classifying space of long knots and its equivalence to short ropes.
method Used techniques from Galatius and Randal-Williams to model classifying space and constructed a map.
result Space of short ropes is weakly homotopy equivalent to the classifying space of long knots.
What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others e…
In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantu…
RoPE framework calibrates misspecified simulators for reliable inference.
problem Misspecification compromises reliability of simulation-based inference.
method Data-driven calibration using optimal transport and a small calibration set.
result RoPE framework improves inference accuracy and uncertainty calibration.
Bayesian model compares ML algorithms on various datasets.
problem Comparing multiple machine learning algorithms across multiple datasets.
method Bayesian Bradley-Terry model, defining regions of practical equivalence (ROPE).
result Allows nuanced statements and ROPE definitions for algorithm comparison.
We describe several configurations of clasped ropes which are balanced and thus critical for the Gehring ropelength problem of arXiv:math.DG/0402212.
Unified framework analyzes and compares RFF and RoPE PEs for music generation.
problem Efficiently modeling music generation with positional encodings.
method Kernel methods to analyze and compare RFF and RoPE PEs.
result RoPEPool outperforms other methods in melody harmonization.
New examples of gordian unlinks show different rope geometries.
problem Tackling the existence of non-trivial unknots with prescribed length and thickness.
method Provided the first examples of gordian unlinks with thickness in [1,2).
result Thinner normal tubes lead to different rope geometries.
Misspecification-Aware Simulation-Based Inference via Side-Channel Guidance
problem Simulation-based inference (SBI) of latent parameters is hindered by simulator misspecification.
method Misspecification-Aware Simulation-Based Inference (MA-SBI) turns side-channel text into a posterior correction.
result MA-SBI matches the oracle posterior across 10 seeds and two backbones.
This work learns visual representations for deformable objects using contrastive estimation.
problem Challenges in learning plannable visual representations for deformable objects.
method Jointly optimizes visual representation and dynamics models using contrastive estimation.
result Substantial improvements in performance over standard model-based learning techniques.
New interdisciplinary branch in math physics tackles topological interactions of polymer-like objects.
problem Analyzing topological interactions in fluctuating non-phantom rope-like objects.
method Review of conceptual steps in statistical topology.
result Emergence of statistical topology as a new interdisciplinary field.
What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of Litherland, Simon et al. We prove a basic inequality between these thickness measu…
Thomas Rot discusses a topology theorem at a math conference.
problem Topology theorem presentation at a math conference.
method No specific method mentioned; presentation of a theorem.
result No specific key result mentioned; presentation of a theorem.
Generative model learns goal-directed visual plans from raw data.
problem Learning goal-directed visual plans from high-dimensional observations.
method Combines representation learning and planning, maximizing mutual information.
result Generative model learns low-dimensional representation explaining causal data.
Explains Conway's tangle trick and its mathematical origins.
problem Understanding the relationship between braids and elliptic curves.
method Discusses the tangle trick, its mathematical underpinnings, and historical context.
result Establishes the connection between braids and elliptic curves.
STRING improves 2D and 3D position encodings for better performance.
problem Efficient and accurate position encoding for 2D and 3D applications.
method STRING extends Rotary Position Encodings with a unifying theoretical framework, maintaining translation invariance and low computational cost.
result STRING shows substantial gains in open-vocabulary object detection and robotics.
The paper uses permutation representations to visualize group extensions and subgroups.
problem Visualizing and understanding group extensions and subgroups.
method Developing metaphoric rope-thread diagrams to represent semi-direct products and their constituents.
result Injective homomorphisms into semi-direct products are established.
Study of Milnor invariants and ropelength of spherical links.
problem Understanding the relationship between the thickness of spherical links and their Milnor invariants.
method Generalized Massey products and Milnor invariants to spherical links, finding optimal asymptotic bounds.
result Optimal asymptotic bounds on Milnor invariants in terms of thickness, revealing a polynomial vs exponential regime.
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 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…
Proposes a method to learn dynamic models for systems with variable number of objects.
problem Efficiently modeling systems with a variable number of objects.
method Uses graph neural networks and block-wise linear transition matrices to learn compositional Koopman operators.
result The method adapts to new environments and produces better control signals.
Zero-shot imitation learns to imitate tasks without expert demonstrations.
problem Learning to imitate tasks without expert supervision.
method Explore world without supervision, distill experience into goal-conditioned policy, use forward consistency loss.
result Zero-shot imitation policy can mimic expert tasks after seeing just goal demonstrations.
Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length $\mbox{Len}(K)$ of a knot K i…
Few-shot goal learning for robot tasks from a few examples.
problem Learning objectives for robot tasks from limited examples.
method Meta-learning a classifier to recognize new goals from a few examples.
result Meta-learning approach can be applied to both reinforcement learning and visual planning.
A graph model improves short text classification by integrating sentence relationships.
problem Sparse features in short text classification due to limited text length.
method PathWalk model combining graph networks and short sentences.
result PathWalk achieves state-of-the-art results on four datasets.
Study compares short vs long strategies for equity factors, finds short strategy better.
problem Determining the best market-neutral implementation of equity factors.
method Revisited the relative predictability of short and long legs, diversification, and costs.
result Long-Short implementation yields superior risk-adjusted returns compared to Hedged Long-Only.
Short selling is key to exploiting arbitrage opportunities in financial markets.
problem Theoretical basis for differences in financial service regulations.
method Analyzing semimartingales to show arbitrage opportunities require short selling.
result Arbitrage opportunities can only be exploited through short selling.
Approximates bond option volatilities using affine short-rate models.
problem Calculating implied volatilities for bond options.
method Derive asymptotic approximation for bond option volatilities under affine short-rate dynamics.
result Accuracy of approximation validated through numerical experiments.
Modeling short selling risks to quantify losses.
problem Short selling constraints and associated risks.
method Optimal stopping model with margin risk and recall risk.
result Realistic short selling constraints lead to significant value loss.
Robust OPE framework uses human inputs to improve policy evaluation in changing environments.
problem Inaccurate policy evaluations due to shifts in environment properties.
method Adapts OPE methods to shifts on user-inputted covariates, providing more realistic utility estimates.
result Robust OPE framework yields less pessimistic policy evaluations and captures realistic dataset shifts.
Chronicals boosts LLM fine-tuning speed by 3.51x over Unsloth.
problem Memory bottleneck in fine-tuning large language models.
method Four optimizations: fused kernels, cross-entropy reduction, LoRA+, and sequence packing.
result 3.51x speedup on A100-40GB with Qwen2.5-0.5B.
Model uses RNN and CNN for sequence-based short-text classification.
problem Lack of sequence consideration in short-text classification.
method Recurrent and Convolutional Neural Networks for sequence data.
result Achieves state-of-the-art results on three datasets.
A new deep neural network improves short-speech recognition.
problem Improving speaker recognition for short-duration speech segments.
method Proposes UtterIdNet, a novel deep neural network architecture.
result Consistent and stable performance for short segments, especially sub-second durations.
We provide a combinatorial condition characterizing curves that are short along a Teichmueller geodesic. This condition is closely related to the condition provided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that short curves in a hyperbolic manifold homeomorphic to S x R are also short in the…
Investigates ropelength of complex knots and links.
problem Establishing ropelength for knots and links beyond tested crossing numbers.
method Investigated torus knots up to 1023 crossings, satellite knots up to 42 crossings, and used a model of repeated Hopf links.
result Found power-law scaling in T(p,p+1) and derived formulae for predicting crossing-ropelength relationships.
The paper explores bond pricing in short rate models using numerical and analytical methods.
problem Bond pricing in short rate convergence models of interest rates.
method Numerical and analytical methods for obtaining approximate solutions to partial differential equations.
result Approximations of bond prices in short rate convergence models.
Model combines long-term and short-term memory using conceptors.
problem Transfer between long-term and short-term memory.
method Recurrent neural network with gated reservoir for short-term memory and conceptors for long-term memory.
result Standard operations on conceptors allow combining long-term memories and describing their effect on short-term memory.
Margin trading and short selling boost green tech innovation in China.
problem Encouraging green technology innovation in Chinese companies.
method Quasi-experimental research using panel data of Chinese listed companies, double difference model.
result Margin trading and short selling increase green tech innovation significantly.
Short proof for ideal polygons with near optimal orthogeodesic decomposition.
problem Decomposing ideal polygons into orthogeodesics.
method Short proof with orthogeodesic decomposition of length at most 2log(n). result Optimal orthogeodesic decomposition of ideal polygons with length 2log(n). The paper introduces a new short rate model with memory components.
problem Modeling short rate dynamics with past values.
method Integrates memory (delay) components into Merton or Vasiček models.
result Analytical solutions for bond prices and forward rates.
Minimal number of caps covering an n-sphere is n+2.
problem Covering an n-sphere with minimal number of caps.
method Analyzing short closed sets and caps, proving theorems about their intersections and coverings.
result The minimal number of short closed sets (caps) covering an n-sphere is n+2.
Theory explains market crashes due to deleveraging and short sale constraints.
problem Market crashes caused by deleveraging and short sale restrictions.
method Developed a theoretical model of market dynamics involving two investor types.
result Short sellers can stabilize markets by providing supply or catching price drops.
A short proof for curve lengths on hyperbolic surfaces.
problem Proving a theorem about curve lengths on hyperbolic surfaces.
method Presented a concise proof for the theorem.
result A pair of curves has length at least half the perimeter of a specific polygon.
The study finds at least two short, simple geodesic chords on a disk with convex boundary.
problem Existence of short, simple geodesic chords on a 2-disk with convex boundary.
method Proof of existence using Riemannian geometry and bounds on lengths.
result Existence of at least two short, simple orthogonal geodesic chords on a 2-disk with convex boundary.
Two short proofs show intersection homology's stability.
problem Topological invariance of intersection homology.
method Two proofs: one short requiring support and cosupport axioms, the other longer adaptable to more perversities.
result Intersection homology's stability proven without axioms.
Study short maturity Asian options under CEV model, presenting an analytical approximation.
problem Analyzing short maturity behavior of Asian options in CEV model.
method Presented an analytical approximation for Asian options prices under CEV model.
result Good numerical agreement with Monte Carlo simulations and benchmark test cases.
RNNs excel at both short-term and long-term user interaction prediction.
problem Modeling user preferences over short and long time horizons.
method Evaluating RNN-based models on both short-term and long-term recommendation tasks.
result RNNs can predict immediate and distant user interactions.