Study of knotted defects in smectic liquid crystals using topological knot theory.
problem Understanding the topological structure of knotted defects in smectic liquid crystals.
method Investigation of screw and edge dislocations, focusing on their radial surface structure and knot fibration.
result Established a connection between smectic defects and knot theory, revealing the topological knotting of defects.
Study shortest geodesics on flat cone spheres with conical singularities.
problem Understanding the distribution of shortest geodesics on flat cone spheres.
method Proved a recurrent relation on the distribution of the length of shortest geodesics with respect to Thurston's volume form.
result Proved a recurrent relation on the distribution of the length of shortest geodesics.
This research extends topological recursion to hyperbolic surfaces with tight boundaries and conical defects.
problem Calculating volumes of hyperbolic surfaces with special boundaries.
method Generalized topological recursion to handle tight boundaries and conical defects.
result Weil-Petersson volumes are polynomial in boundary lengths for hyperbolic surfaces with tight boundaries and conical defects.
Bounds on saddle connections on flat spheres with conical singularities.
problem Counting saddle connections on flat spheres with conical singularities.
method Geometry of immersed disks and explicit upper bounds.
result Explicit upper bounds on the number and lengths of saddle connections.
Novel threefold partitioning of L2-spinor space on cone links.
problem Understanding equivariant Riemann-Roch defects in complex spaces with conic singularities.
method Investigates supertraces over local cohomology groups and spectral asymmetry.
result Novel complex equivariant ξT and ηT invariants defined. The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
Study compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
problem Compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
method Introduced a locally convex topology, extended compactness theorem, studied pseudo-differential operators, and applied to microlocal defect measures.
result Extended microlocal defect measures and compensated compactness theorem to Sobolev wave front set spaces.
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
Defines state sum models with defects in 3-manifolds.
problem Detecting and characterizing defects in 3-manifolds.
method Turaev-Viro-Barrett-Westbury state sum models with defects labeled by bimodule categories and functors.
result State sums are triangulation-independent and can be computed using polygon diagrams.
The study counts triangulations of a projective plane with specific vertex valencies.
problem Counting triangulations of a projective plane with unique vertex valencies.
method Analyzes the growth of triangulations with no more than n triangles, using complex mathematical functions and series.
result The number of triangulations grows as C·n^2 + O(n^3/2) with C ≈ 0.2087432125056015.
The study examines K-polystability on Fano 4-folds with specific Lefschetz defects.
problem Investigating K-polystability on Fano 4-folds with Lefschetz defect at least 2.
method Examining 19 families of Fano 4-folds with Lefschetz defect 3 and 175 families with Lefschetz defect 2, proving K-polystability and instability.
result Exactly 5 out of 19 families of Fano 4-folds with Lefschetz defect 3 are K-polystable, and 5 out of 175 Casagrande-Druel Fano 4-folds with Lefschetz defect 2 are K-polystable.
In this paper, we survey recent results on index defects of elliptic operators on manifolds with boundary. Index defects are similar to the Hirzebruch signature defects in topology, where the defects appear as the correction terms to the signature formula on manifolds with boundary. For some natural classes of elliptic…
We study the topology of smectic defects in two and three dimensions. We give a topological classification of smectic point defects and disclination lines in three dimensions. In addition we describe the combination rules for smectic point defects in two and three dimensions, showing how the broken translational symmet…
We define the sigma-model action for world-sheets with embedded defect networks in the presence of a three-form field strength. We derive the defect gluing condition for the sigma-model fields and their derivatives, and use it to distinguish between conformal and topological defects. As an example, we treat the WZW mod…
Alexander polynomial degree correlates with knot defect, proving conjecture for defect zero.
problem Characterizing knot polynomials and their defects.
method Analyzing differential expansions and degree in q±2 of Alexander polynomials. result Proved Alexander polynomial degree correlates with knot defect, especially for defect zero.
We introduce a Bayesian defect detector to facilitate the defect detection on the motion blurred images on rough texture surfaces. To enhance the accuracy of Bayesian detection on removing non-defect pixels, we develop a class of reflected non-local prior distributions, which is constructed by using the mode of a distr…
Automates defect detection using autoencoders on normal images only.
problem Manual defect detection is slow, tedious, and prone to human biases.
method Convolutional auto-encoder trained on normal images only, detects defects in residual masks.
result Achieved an impressive average F1 score of 0.885 on test images.
Framework for efficient defect classification and inspection.
problem Adaptive defect classification and inspection from high volume data.
method Continual learning framework for dynamic classifier updates.
result Efficient storage and computational needs reduction.
Graph-based ML improves defect prediction in software development.
problem Challenges in predicting defect-prone changes in complex software development.
method Building contribution graphs from developers and source files, using graph-based ML for defect prediction.
result Graph-based ML leads to significantly better defect prediction (F1 score up to 77.55%, MCC up to 53.16%).
Simplified 3D Dijkgraaf-Witten theory with defects explained geometrically.
problem Constructing 3D Dijkgraaf-Witten theory with defects.
method Symmetric monoidal functor from defect cobordism category to vector spaces, using geometric and homotopy theoretic methods.
result Explicit construction of 3D untwisted Dijkgraaf-Witten theory with defects.
The angle defect, which is the standard way to measure curvature at the vertices of polyhedral surfaces, goes back at least as far as Descartes. Although the angle defect has been widely studied, there does not appear to be in the literature an axiomatic characterization of the angle defect. We give a characterization …
Spin TFTs created by gauging line defects in 3D.
problem Creating spin TFTs from oriented TFTs with framed line defects.
method Constructing a spin TFT from an oriented TFT with framed line defects and a commutative Frobenius algebra.
result Spin TFTs extend earlier classifications and reproduce abelian spin Chern-Simons theories.
Defect of knot polynomials remains invariant under certain braid substitutions.
problem Invariance of knot polynomial defects under specific transformations.
method Investigation of defect invariants under antiparallel and parallel braid substitutions.
result Defect remains unchanged under antiparallel braid substitutions and changes by half the added length under parallel braid substitutions.
Extends knotted defect classification to bounded domains using handlebodies.
problem Classifying knotted defects in bounded domains.
method Using continuous maps and monodromies around meridional loops, global defects are described in terms of planar diagrams.
result Classification scheme for defects in handlebodies.
New theorem disproves Angle Defect for super triangles.
problem Angle Defect Theorem for N=1 super hyperbolic geometry.
method Action of OSp(1|2) on real super Minkowski space and brute-force computation.
result Disproves Angle Defect Theorem and provides novel additive function.
Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with a configuration of a body in classical elasticity is the sum of local contributions …
A modular tensor category C gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded C-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. …
A topological defect separating a pair of two-dimensional CFTs is a codimension one interface along which all components of the stress-energy tensor glue continuously. We study topological defects of the bosonic, (0,1)- and (0,2)-supersymmetric sigma models in two dimensions. We find a geometric classification of such …
We present a homogenization theorem for isotropically-distributed point defects, by considering a sequence of manifolds with increasingly dense point defects. The loci of the defects are chosen randomly according to a weighted Poisson point process, making it a continuous version of the first passage percolation model.…
Develops skein theory for 3-manifolds with defects, extending quantum character stacks.
problem Quantum character stacks and their applications in 3-manifolds with surface defects.
method Parabolic induction/restriction for quantum groups, quantum decorated character stacks, ideal triangulations, gluing equations.
result Knot invariants related to quantum A-polynomial, concrete computation method. We describe discrete symmetries of two-dimensional Yang-Mills theory with gauge group G associated to outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles, and calculate it ex…
A novel method classifies wafer defects using topological data analysis.
problem Classifying defect patterns on semiconductor wafers for maintenance and yield management.
method Representing defect patterns as vectors using topological features from persistent homology.
result The method outperforms CNN in accuracy and efficiency, especially with limited data.
A new deep metric learning method for defect classification in threaded pipe connections.
problem Defect classification in threaded pipe connections with limited and imbalanced multichannel functional data.
method COMPILED approach based on deep metric learning for imbalanced, multichannel, and partially observed functional data.
result Superior accuracy compared to existing benchmarks in a real-world case study.
The goal of this paper is twofold. First we prove a rigidity estimate, which generalises the theorem on geometric rigidity of Friesecke, James and Müller to 1-forms with non-vanishing exterior derivative. Second we use this estimate to prove a kind of spontaneous breaking of rotational symmetry for some models of cryst…
Theory of packing diabolic domains in liquid crystals.
problem Understanding the packing of diabolic domains in liquid crystals.
method Lorentz transformations and geometric analysis.
result Diabolic domains can lower the elastic energy of the system.
New conditions ensure points can be uniquely represented by combinations of variety elements.
problem Ensuring points can be uniquely represented by combinations of variety elements.
method Conditions on contact locus of general linear spaces.
result Conditions ensuring non tangential weak defectiveness of projective varieties.
We show that Bonnesen's isoperimetic defect has a systolic analog for Loewner's torus inequality. The isosystolic defect is expressed in terms of the probabilistic variance of the conformal factor of the metric g with respect to the flat metric of unit area in the conformal class of g.
Graph coloring is explained using a topological field theory with defects.
problem Graph coloring as a combinatorial problem is quantum in nature.
method Topological field theory with defects to interpret graph coloring.
result Graph coloring is related to sections of a certain bundle.
Recognition of defects in concrete infrastructure, especially in bridges, is a costly and time consuming crucial first step in the assessment of the structural integrity. Large variation in appearance of the concrete material, changing illumination and weather conditions, a variety of possible surface markings as well …
In this paper, using the method of moving frames, we generalise some of Terracini's results on varieties with tangent defect. In particular, we characterise varieties with higher order osculating defect in terms of Jacobians of higher fundamental forms and moreover we characterise varieties with "small" higher fundamen…
Deep learning methods are becoming widely used for restoration of defects associated with fluorescence microscopy imaging. One of the major challenges in application of such methods is the availability of training data. In this work, we propose a unified method for reconstruction of multi-defect fluorescence microscopy…
Extends a formula for the homomorphism defect of a signature map to coloured braids.
problem Evaluate the homomorphism defect of a signature map for coloured braids.
method Uses a 4-dimensional interpretation of the signature and new 4D tools like the Maslov index and isotropic functor.
result Generalizes the formula of Gambaudo and Ghys to coloured braids and tangles.
In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the de…
The paper characterizes coverings over the projective plane with minimal defect.
problem Characterizing minimal defect branched coverings over the projective plane.
method Characterization through properties of decomposable and indecomposable coverings.
result Extended family of realizations and generalized results on primitive permutation groups.
Refines knot defect measurement in 3D and 4D.
problem Measuring how far knots are from being alternating.
method Extends spanning surface defect to 4-ball, making comparisons and proving formulas.
result Connected sum formula proven.
Paper estimates manifold reach using convexity defect function.
problem Estimating the reach of submanifolds from point clouds.
method Relates reach to convexity defect function, uses stability properties, and combines with recent estimators.
result Uniform expected loss bound and minimax rate lower bounds for reach estimation are provided.
The study analyzes group testing algorithms for identifying defective items with high confidence.
problem Identifying defective items from a population using group testing with high confidence.
method Formulated as a function learning problem using the PAC framework, analyzed three algorithms: column matching, combinatorial basis pursuit, and definite defectives.
result Derived bounds on the number of tests needed for approximate set identification, comparing with existing bounds and simulating performance.
Sharp mass bounds for ALE and ALF toric 4-manifolds.
problem Establishing lower bounds for the mass of ALE and ALF toric 4-manifolds.
method Using gravitational instantons and conical angle defects, the mass is bounded below by a sum of the mass of the corresponding instanton and an expression determined by conical angle defects.
result The mass of an ALE or ALF toric 4-manifold is not less than the mass of the corresponding gravitational instanton.