Study on trunk knot invariant and its relation to satellite and companion knots.
problem Understanding the trunk invariant of satellite knots and their companions.
method Analyzing the trunk invariant of satellite knots and their companions, deriving inequalities.
result Established inequalities relating the trunk of a satellite knot to its companion's trunk.
Backpropagation-free trunk training improves model performance on various benchmarks.
problem Memory inefficiency and noisy gradient estimates in deep network training.
method Split Forward Gradient (Split-FG) method that splits network into trunk and head, estimating only trunk gradient.
result Split-FG achieves better performance than pure forward-gradient training and backpropagation on various benchmarks.
Satellite knots have a higher trunk number than their base knots.
problem Understanding the relationship between satellite knots and their base knots.
method Using the Thurston norm and properties of satellite patterns.
result The trunk number of satellite knots is strictly greater than the product of the Thurston norm and the trunk number of their base knots.
The paper explores knots' height, trunk, and representativity, finding gaps and bounds.
problem Investigating the properties of knots and their invariants.
method Analyzing conjectures, defining new invariants, and comparing knot positions.
result Found gaps between height and minimal height, and bounds on representativity.
The trunk of a knot in S3, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots K1 and K2, we have tr(K1#K2)=max{tr(K1),tr(K2)}, confirming a conjecture of Ozawa. Another conjecture of Ozawa asserts that any…
We introduce two numerical invariants, the waist and the trunk of knots. The waist of a closed incompressible surface in the complement of a knot is defined as the minimal intersection number of all compressing disks for the surface in the 3-sphere and the knot. Then the waist of a knot is defined as the maximal waist …
Machine learning assesses balance outside clinics, improving therapy efficiency.
problem Lack of feedback from PTs in home balance training.
method Trunk sway data analysis with multi-class SVM.
result ML model achieved 82% accuracy in assessing balance.
Random sampling improves DeepONet training efficiency without sacrificing accuracy.
problem Training DeepONet models with high computational and memory costs.
method Random sampling of inputs in the trunk network of DeepONet.
result Significant reduction in training time with comparable accuracy.
We construct a new invariant-the trunkenness-for volume-perserving vector fields on S^3 up to volume-preserving diffeomorphism. We prove that the trunkenness is independent from the helicity and that it is the limit of a knot invariant (called the trunk) computed on long pieces of orbits.
A new training method improves stability and generalization of DeepONets.
problem Training deep operator networks (DeepONets) is challenging due to nonconvex and nonlinear nature.
method Two-step training method: first train trunk network, then branch network. Introduced Gram-Schmidt orthonormalization.
result Generalization error estimate and numerical examples demonstrating effectiveness.
Improved DeepONet variants using Transformer cross-conditioning enhance PDE solution efficiency.
problem Solving partial differential equations efficiently and accurately.
method Transformer-inspired DeepONet variants with bidirectional cross-conditioning.
result Improved efficiency and accuracy compared to modified DeepONet, with variant effectiveness tied to PDE characteristics.
DeepONet learns operators for PDEs with varying parameters and initial conditions.
problem Learning operators for partial differential equations with different parameters or initial conditions.
method DeepONet uses a Branch net and Trunk net to minimize error between evaluated and expected outputs, incorporating a scalar auxiliary variable approach for energy dissipation.
result DeepONet can accurately approximate operators for PDEs with varying parameters or initial conditions.
Defines combinatorial minimal surfaces in pseudomanifolds.
problem Finding minimal surfaces in complex geometric structures.
method Adapting thin position technique from pseudomanifolds.
result Combinatorial minimal surfaces always exist under mild conditions.
Flattenings of knotted surfaces help define new invariants.
problem Understanding and quantifying knotted surfaces in 4-sphere.
method Using hyperbolic decompositions and projections onto 2-sphere.
result Introduced layering, trunk, and partition number invariants.
Proposes a new method to learn operators for stochastic problems using DeepONet with autoencoder.
problem Efficiently solve forward and inverse stochastic problems with limited data.
method MultiAuto-DeepONet, a multi-resolution autoencoder DeepONet model.
result The model effectively handles high-dimensional stochastic inputs and reduces the number of trainable parameters.
AMORE uses neural operators to efficiently predict multiple thermochemical states in stiff chemical kinetics.
problem Efficiently integrating stiff chemical kinetics systems to reduce computational cost.
method Developed AMORE, a framework of adaptive multi-output operator network with two adaptive loss functions.
result Demonstrated improved accuracy and efficiency in predicting thermochemical states from initial conditions.
The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in [Trunks and classifying spaces, Applied Categorical Structures, 3 (1995) 321--356]) and the classifying bundle is the first James bundle (defined in "James bu…
New invariants defined for volume-preserving flows on 3-manifolds.
problem Defining invariants for volume-preserving flows.
method Extending wrapping number and trunk to define invariants of links and flows.
result Wrappingness and trunkenness are not functions of helicity.
Innovative rack theory applied to Legendrian links.
problem Classifying and distinguishing Legendrian links.
method Purely rack-theoretic approach, Legendrian Reidemeister moves, cusps, homogeneous representations, modules.
result Invariant distinguishes infinitely many Legendrian unknots and trefoils.
Deep learning framework predicts surface texture parameters and their uncertainties.
problem Predicting surface texture parameters and their uncertainties from multi-instrument datasets.
method Reproducible deep learning framework using multi-instrument dataset, quantile and heteroscedastic heads for uncertainty modeling, and post-hoc conformal calibration.
result High fidelity predictions (R2: Ra 0.9824, Rz 0.9847, RONt 0.9918) and well-modelled uncertainty targets (Ra_uncert 0.9899, Rz_uncert 0.9955).
Study approximates operators on labelled conditional distributions for non-exchangeable systems.
problem Approximating operators on constrained probability measures for non-exchangeable systems.
method Combines cylindrical approximations and DeepONet-type neural architecture for finite-dimensional representations.
result Establishes a universal approximation theorem for continuous operators on Mλ. DeepONets improve surrogate modeling for engineering systems.
problem Accurately modeling complex PDEs for engineering systems.
method DeepONets specialize in approximating mathematical operators for PDEs.
result DeepONets achieve high prediction accuracy and zero-shot capability.
DeepONets enhance spatial-temporal surrogates for structural dynamics.
problem Creating full spatial-temporal surrogates for dynamical systems under uncertainty.
method Proposed Full-Field Extended DeepONet (FExD) to learn full solution operator across multiple degrees of freedom.
result FExD achieves superior accuracy and computational efficiency compared to other models.
Enhanced DeepONet framework with uncertainty quantification for complex operators.
problem Learning complex operators with uncertainty quantification.
method Generalised variational inference (GVI) using Rényi's α-divergence.
result Superior predictive accuracy and uncertainty quantification.
Deep RNN detects FoG episodes in Parkinson's disease with high accuracy.
problem Detecting freezing episodes in Parkinson's disease patients.
method Deep Recurrent Neural Network (RNN) with Long Short-Term Memory cells on 3D-accelerometer measurements.
result Frequency domain features from trunk sensor achieve an AUC score of 93% in subject-independent method.
Study examines how body segments respond to random vibrations.
problem Understanding human body responses to random vibrations.
method 35 participants were tested with random noise signals. Multiple linear regression models were created to determine influential predictors of peak translational gains.
result Multiple predictors, including motion direction and body segment, significantly influence peak translational gains.
The paper analyzes geometric densities and compression radii for knot types.
problem Optimizing geometric quantities associated with knot types.
method Develops a factorization framework for scale-covariant size functionals.
result Different minimizing sequences for density, compression, packing, and ropelength problems.
DeepONet learns nonlinear operators from data to identify differential equations.
problem Learning nonlinear operators from data to identify differential equations.
method DeepONet architecture with branch and trunk nets.
result DeepONet significantly reduces generalization error compared to fully-connected networks.