Machine learning generates fragility curves for concrete bridges.
problem Estimating fragility curves for concrete bridges with uncertainty.
method Stripe-based approach using random forests.
result Fragility curves generated with reduced computational effort.
Meta-learning approach improves CNN architectures for concrete defect classification.
problem Challenging task of recognizing defects in concrete infrastructure.
method Two reinforcement learning based meta-learning approaches (MetaQNN and NAS) for finding suitable CNN architectures.
result Learned architectures have fewer parameters and better multi-target accuracy.
Introduces Schubert normal form for 3-bridge links and presents their groups.
problem Representing and understanding 3-bridge link groups.
method Develops Schubert normal form and uses it to present 3-bridge link groups.
result Two presentations of 3-bridge link groups based on Schubert normal form.
The study counts ideal points in 2-bridge knot complements using knot diagrams.
problem Counting ideal points in 2-bridge knot complements.
method Using knot diagrams, the structure of Serre trees for essential surfaces is determined, leading to a formula for ideal points.
result A formula for the number of ideal points associated with each incompressible surface in 2-bridge knot complements.
Method determines trace-free SL(2,C) reps for arborescent links.
problem Classifying trace-free representations of arborescent links.
method Completely determines trace-free mSL(2,C)-representations for arborescent links. result Concrete computations for a class of 3-bridge arborescent links.
FKEE estimates expectations without samples, using diffusion bridges and PINNs.
problem Estimating expectations without large sample sizes.
method Diffusion bridge models and Feynman-Kac operator approximation using PINNs.
result Significantly reduces variance and improves efficiency.
We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate t…
Study uses machine learning to predict optimal bridge types based on NBI data.
problem Improving bridge design selection through machine learning.
method Supervised learning on modified NBI dataset, feature selection, cross-validation, state-specific models.
result Supervised learning models predict optimal bridge types with high accuracy.
Paper proposes a holistic optimization for civil structures considering uncertainties.
problem Designing civil structures requires integration of material and structural design.
method Holistic optimization combining concrete mixture design and structural simulations.
result Inverted workflow allows consideration of new mixtures and uncertainties.
Study on higher dimensional Alexander polynomials for metabelian representations of knots.
problem Analyzing the asymptotic behavior of Alexander polynomials for metabelian representations of knots.
method Investigating the leading coefficients of the asymptotics of twisted Alexander polynomials for SL(n, C)-representations induced from irreducible metabelian SL(2, C)-representations.
result Concrete computations for all genus one two-bridge knots presented, including limits of leading coefficients in the asymptotics of twisted Alexander polynomials.
This is a survey paper, starting from the general notion of coordinate bundle taken from Steenrod. Its aim is to provide a motivation for the introduction of cyclic homology (and the closely related noncommutative de Rham cohomology) by Connes, Tsygan and the author. The bridge is made through a generalization of Chern…
Calculates volumes of knot cone-manifolds using Schläfli formula.
problem Calculating volumes of specific knot cone-manifolds.
method Schläfli formula, Hilden-Lozano-Montesinos-Amilibia instructions, Ham-Mednykh-Petrov methods, Hoste-Shanahan presentation.
result Concrete and explicit formula for volumes of C(2n,3) cone-manifolds. We study the twisted knot module for the universal deformation of an SL2-representation of a knot group, and introduce an associated L-function, which may be seen as an analogue of the algebraic p-adic L-function associated to the Selmer module for the universal deformation of a Galois representation. We…
While statistics focusses on hypothesis testing and on estimating (properties of) the true sampling distribution, in machine learning the performance of learning algorithms on future data is the primary issue. In this paper we bridge the gap with a general principle (PHI) that identifies hypotheses with best predictive…
New connection found between cluster algebras and knot theory.
problem Connecting cluster algebras to knot theory and Jones polynomials.
method Using continued fractions and snake graphs.
result Direct formula for the Jones polynomial of 2-bridge links.
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A- or B-adequacy), we prove that the growth of the…
Abstract: Linking field theory to Floer theory via regularization.
problem Finding periodic solutions of Hamilton's equation.
method Regularization scheme for polysymplectic formalism linking Euclidean field theory to hyperkähler Floer theory.
result Proved a cuplength estimate.
The paper introduces new measures for quantifying uncertainty in machine learning.
problem Uncertainty representation and quantification in machine learning.
method Proper scoring rules for aleatoric and epistemic uncertainty quantification.
result Established a natural bridge between credal set and second-order distribution representations of uncertainty.
We study optimization algorithms based on variance reduction for stochastic gradient descent (SGD). Remarkable recent progress has been made in this direction through development of algorithms like SAG, SVRG, SAGA. These algorithms have been shown to outperform SGD, both theoretically and empirically. However, asynchro…
Concrete distribution properties examined on simplex.
problem Properties of Concrete distribution on simplex.
method Reflection and location-scale transformation of uniform distribution; explicit parameterization to Poincaré half-space.
result Fisher information and information metric are hyperbolic space; Fisher-Rao geodesic distance computed.
Constructing exponential families from statistical manifolds.
problem The central problem of constructing exponential families from statistical manifolds.
method Constructive approach proving every compact statistical manifold admits a foliation of Hessian manifolds.
result Compact orientable leaves are either finite quotients of flat torus or mapping torus with periodic monodromy.
Concrete autoencoder selects key features for efficient data reconstruction.
problem Efficiently identifying and selecting important features for data reconstruction.
method Concrete selector layer with temperature-controlled selection during training, followed by reconstruction using a standard neural network.
result Concrete autoencoder selects a small subset of genes that can reconstruct the remaining gene expression levels, improving on existing methods.
New method detects essential tori in mixed singularity links.
problem Detecting essential tori in mixed singularity link complements.
method Analyzing properties of defining mixed polynomials.
result Explicit criteria for essential tori existence.
Concrete distribution relaxes discrete variables for gradient-based optimization.
problem Gradient-based optimization of discrete random variables.
method Concrete distribution as a continuous relaxation of discrete variables, enabling reparameterization and gradient computation.
result Concrete distribution allows for low-variance biased gradients in discrete stochastic nodes.
Study bridge spectra of 2-bridge knots and their cables.
problem Computing bridge spectra for 2-bridge knots and their cables.
method Computed bridge spectra of cables of 2-bridge knots.
result Results on bridge spectra and distance of Montesinos knots.
Paper proposes an algorithm for lifelong learning with shared structure.
problem Lifelong learning with shared structure in an online setting.
method Proposes a simple algorithm using multi-task empirical risk minimization.
result Establishes a sample complexity bound based on task-eluder dimension.
Paper introduces Floer theory for field theories, proving periodic solutions for particle-field systems.
problem Defining Hamiltonian Floer theory for covariant field theories, especially those with degenerate action functionals.
method Regularization procedure to handle degeneracy, leading to Floer curves that converge to periodic solutions.
result Existence of Floer curves and space-time periodic solutions for coupled particle-field systems.
Alternative proof for 2-bridge knots' bridge positions.
problem Proving genus one 1-bridge positions for 2-bridge knots.
method Alternative proof using disk complexes.
result Every genus one 1-bridge position is a stabilization.
For every bridge number, critical spheres exist for links.
problem Finding critical bridge spheres for links with arbitrary bridge numbers.
method Constructing infinite families of links with square bridge spheres.
result Existence of critical bridge spheres for links with arbitrary bridge numbers.
Study on knot diagrams showing bridge number can differ from crossing number.
problem Incompatibility between crossing number and bridge number for knot diagrams.
method Defined and compared various bridge number computations for knot diagrams, studied minimizing diagrams, and constructed families of minimal crossing diagrams.
result Found examples where bridge number differs from crossing number, and demonstrated this difference can grow infinitely.
New examples of knots with special bridge positions found.
problem Understanding special types of bridge positions for knots.
method Derived examples of knots with unperturbed weakly reducible non-minimal bridge positions.
result Connected sum of unperturbed bridge positions is unperturbed (a new conjecture).
Reduction of knots in bridge position results in a lower bridge number if certain conditions are met.
problem Understanding the reduction of knots in bridge position and its implications.
method Analyzing the reduction of knots along bridge disks and considering the existence of cancelling pairs.
result If a reduction of a knot yields an (n-1)-bridge position, the knot bounds a disk containing the bridge disk and intersecting the sphere in n arcs.
The paper provides examples of keen weakly reducible bridge spheres for links in b-bridge position.
problem Characterizing and finding examples of keen weakly reducible bridge spheres.
method Analyzing bridge spheres and their properties in terms of compressing disks and width complex.
result Infinitely many examples of keen weakly reducible bridge spheres for links in b-bridge position.
Concrete proof that SO(n,1) is not a T-group.
problem Proving SO(n,1) is not a T-group.
method Constructed a concrete model for hyperbolic space and measure, proving hyperbolic distance equals measure up to a constant.
result Concrete proof that SO(n,1) is not a T-group.
Any 2-bridge knot in the 3-sphere has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.
Lower bounds on knot distortion related to bridge distance and number.
problem Finding lower bounds on knot distortion in 3D space.
method Extending Pardon's techniques to relate distortion to bridge distance and number.
result Lower bound on knot distortion proportional to minimum of bridge distance and bridge number.
Paper finds knots with stair-step bridge spectra but are not high distance.
problem Understanding the relationship between bridge spectra and knot distance.
method Computed bridge spectra and distances of generalized Montesinos knots, including pretzel and Montesinos knots.
result First example of knots with stair-step bridge spectra but not high distance.
New method finds infinitely many surface knots with specific bridge numbers.
problem Finding numerical invariants for surface links.
method Colorings of surface links by keis to prove bridge number existence.
result Existence of infinitely many surface knots with bridge number n for n ≥ 4.
Researchers found infinite links with specific bridge positions.
problem Finding links with minimal bridge positions.
method Applying Takao et al.'s criterion to create links with locally minimal n-bridge and globally minimal m-bridge positions. result Provided an infinite family of links with specific bridge positions.
Character varieties of 2-bridge knots and links explained using Farey recursion.
problem Understanding character varieties of 2-bridge knots and links.
method Using Farey recursion to define polynomials for character varieties.
result Character varieties described in terms of polynomials defined by Farey recursion.
New method speeds up deep learning optimization.
problem Scalable second-order optimization for deep learning.
method Second-order optimization with algorithmic and numerical improvements.
result Significant convergence and wall-clock time improvements.
We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degene…
We show that if K is a knot in S3 and Σ is a bridge sphere for K with high distance and 2n punctures, the number of perturbations of K required to interchange the two balls bounded by Σ via an isotopy is n. We also construct a knot with two different bridge spheres with 2n and 2n−1 bridges respecti…
Two-bridge ribbon knots have symmetric union presentations.
problem Characterizing two-bridge ribbon knots.
method Symmetric union presentations and partial knot analysis.
result Symmetric union presentations for various two-bridge ribbon knots.
Paper explores link and plat presentations, showing equivalence under bridge isotopy.
problem Relationship between links in bridge position and plat presentations.
method Established equivalence between Hilden double coset classes of plat presentations and bridge positions up to isotopy.
result Revealed equivalence of Hilden double coset classes for n-bridge unknot and torus knots in plat position.
We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.
We give a complete characterization of those essential simple loops on 2-bridge spheres of 2-bridge links which are null-homotopic in the link complements. By using this result, we describe all upper-meridian-pair-preserving epimorphisms between 2-bridge link groups.
Explains how Higgs bundles help study Fuchsian representations on surfaces.
problem Understanding components of character varieties for Fuchsian representations.
method Theory of Higgs bundles, focusing on real groups and their subtleties.
result Characterizes deformation spaces of Fuchsian representations using Higgs bundles.