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…
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48 results for “defect formulae”
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.
Study of defects in gauge theories connects quantum field theory to classical integrability.
problem Vacuum expectation values of half-BPS surface defects in gauge theories.
method Analysis of Fuchsian systems, isomonodromic deformations, and blowup formulas.
result Establishes a relation between supersymmetric gauge theory and classical integrability.
Researchers lift spectral ζ-invariants to coverings using defect formulae.
problem Lifting spectral ζ-invariants to coverings of closed manifolds.
method Using lifted defect formulae and Wodzicki residues.
result Derived integrals of Pontryagin and Chern forms for certain lifted spectral ζ-invariants.
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.
We analyze the probabilistic variance of a solution of Liouville's equation for curvature, given suitable bounds on the Gaussian curvature. The related systolic geometry was recently studied by Horowitz, Katz, and Katz, where we obtained a strengthening of Loewner's torus inequality containing a "defect term", similar …
Topological recursion of the Weil-Petersson volumes of hyperbolic surfaces with tight boundariesmath-ph
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.
For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula co…
Differential Expansion for antiparallel triple pretzels: the way the factorization is deformedhep-th
Factorization of DE coefficients is violated in antiparallel triple pretzels, but described elegantly.
problem Understanding the origins of factorization in double braids and its extension to antiparallel triple pretzels.
method Defect-preserving deformation from trefoil to antiparallel triple pretzels, analysis of DE coefficients.
result Factorization of DE coefficients is violated but described by an elegant formula for symmetric representations.
Formula calculates mass using cube faces and edges.
problem Measuring mass of 3-manifolds.
method Cube faces and edges, mean curvature, dihedral angle, geodesic curvature, angle defect.
result Mass formula connects to Gromov's theory and Gauss-Bonnet theorem.
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.
Study on defect topology in smectic liquid crystals.
problem Topological classification and combination rules of smectic defects.
method Topological analysis of point defects and disclination lines in 2D and 3D smectic liquid crystals.
result Combination rules for smectic point defects in 2D and 3D, showing path dependence.
Extends RT TQFT to include surface defects and line defects.
problem Distinguishing non-isotopic embeddings of surfaces in 3D bordisms.
method Uses modular tensor categories, symmetric Frobenius algebras, and equivariant multi-modules.
result Invariant distinguishes non-isotopic embeddings of 2-tori but not 2-spheres.
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.
Bayesian detector improves defect detection on rough surfaces.
problem Defect detection on motion blurred images with rough texture.
method Developed Bayesian detector using reflected non-local prior distributions.
result Significantly enhanced accuracy in identifying defects.
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.
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 of Alexander polynomials.
result Proved Alexander polynomial degree correlates with knot defect, especially for defect zero.
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.
Two-dimensional Yang-Mills theory defects and orbifolds studied.
problem Discrete symmetries and their defects in Yang-Mills theory.
method Path integral over twisted G-bundles, orbifold construction, defect network approach.
result Exact computation of gauge theory partition function with defects.
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.
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.
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.
Metric Description of Defects in Amorphous Materialscond-mat.mtrl-sci
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 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.…
Parabolic skein modulesmath.QA
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 -polynomial, concrete computation method.
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.
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.
Research shows asymptotic non-defectivity of Segre-Veronese varieties.
problem Defectivity of Segre-Veronese varieties.
method Analyzing the asymptotic behavior of Segre-Veronese varieties using line bundles and embedding.
result Asymptotically, Segre-Veronese varieties are not h-defective for a certain range of h.
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…
Unified method for multi-defect microscopy image restoration with limited training data.
problem Challenges in applying deep learning methods due to limited training data for multi-defect microscopy images.
method Two-stage approach: data augmentation with GAN and conditional GAN training.
result Proposed method gives comparable or superior results to existing methods in image quality restoration.
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.
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.
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.
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…
A general state-sum construction of 2-dimensional topological quantum field theories with defectsmath.QA
This research extends TQFTs to include defects using state-sum construction.
problem Extending topological quantum field theories to include defects.
method General state-sum construction with extended Pachner moves, requiring flag-likeness of triangulations.
result Derivation of equations and string diagrams for 2D TQFTs with defects.
Study non-integrated defect relations for complete minimal surfaces in higher dimensions.
problem Understanding defect relations for Gauss maps of minimal surfaces.
method Extending previous work to higher dimensions, focusing on surfaces with finite total curvature.
result Non-integrated defect relations for Gauss maps of complete minimal surfaces in are established.
On structured surfaces with defects: geometry, strain incompatibility, internal stress, and natural shapescond-mat.mtrl-sci
The paper analyzes defects on structured surfaces and calculates stress and shape.
problem Analyzing defects on structured surfaces and their effects on stress and shape.
method Classified and quantified defects, derived strain incompatibility relations, and applied to shells.
result Determined internal stress field and deformed shape for shells with defects.
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 study improves bounds on non-secant defectivity of projective varieties.
problem Bounding non-secant defectivity of projective varieties.
method Using osculating projections and bounds on linear projections.
result Asymptotic non-defectivity of Grassmannians for .
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.