Dirac operator invertibility proven for specific manifolds.
problem Invertibility of twisted Dirac operator on manifolds.
method Closed connected spin manifold with non-negative scalar curvature, flat Hilbert module bundle.
result Dirac operator is invertible under given conditions.
For a ring R, we denote by R[L] the free R-module spanned by the isotopy classes of singular links in S3. Given two invertible elements x,t∈R, the HOMFLY-PT skein module of singular links in S3 (relative to the triple (R,t,x)) is the quotient of R[L] by local rela…
Paper proposes a predictive maintenance system for solar plants using big data.
problem Fault prediction in photovoltaic plants to reduce downtime and maintenance costs.
method Data-driven approach with unsupervised clustering and Pattern Recognition Neural Network.
result Effective prediction of both generic and specific faults, up to 7 days in advance.
It is known, since works of Burde and de Rham, that one can detect the roots of the Alexander polynomial of a knot by the study of the representations of the knot group into the group of the invertible upper triangular 2x2 matrices. In this work, we propose to generalize this result by considering the representations…
We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, …
We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-a…
Paper investigates Lipschitz constants of self-attention modules in neural networks.
problem Lipschitz constants of self-attention modules in neural networks.
method Proved standard dot-product self-attention is not Lipschitz for unbounded input domain. Proposed L2 self-attention that is Lipschitz. Derived upper bound on L2 self-attention's Lipschitz constant.
result Proved standard self-attention is not Lipschitz for unbounded input domain and proposed an alternative L2 self-attention that is Lipschitz.
We define invariants of unoriented knots and links by enhancing the integral kei counting invariant Phi_X^Z (K) for a finite kei X using representations of the kei algebra, Z_K[X], a quotient of the quandle algebra Z[X] defined by Andruskiewitsch and Grana. We give an example that demonstrates that the enhanced invaria…
Neural spline flows enhance flow models with rational-quadratic splines.
problem Improving flexibility and density estimation in flow models.
method Proposes a new differentiable module based on monotonic rational-quadratic splines.
result Demonstrates improved performance in density estimation, variational inference, and generative modeling of images.
Let k be an integral domain containing the invertible elements α, s and \frac{1}{s-s^{-1}}. If M is an oriented 3-manifold, let K(M) denote the Kauffman skein module of M over k. Based on the work on Birman-Murakami-Wenzl algebra by Beliakova and Blanchet, we give an ``idempotent-like'' basis for the Kauffman skein mod…
We construct the Calderon projection on the space of Cauchy datas for a twisted Dirac operator in the Mischenko--Fomenko pseudodifferential calculus for operators acting on bundles of finitely generated C∗--Hilbert modules on a compact manifold with boundary. In particular an invertible double is constructed general…
Let k be a subring of the field of rational functions in x,v,s which contains x±1,v±1,s±1. If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the…
A commuting n-tuple (T1,…,Tn) of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module H over C[z1,…,zn] in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h…
In this paper we push forward results on the invariant F-module of a virtual knot investigated by the first named author where F is the algebra with two invertible generators A,B and one relation A−1B−1AB−B−1AB=BA−1B−1A−A. For flat knots and links the two sides of the relation equa…
Let A,B be invertible, non-commuting elements of a ring R. Suppose that A−1 is also invertible and that the equation [B,(A−1)(A,B)]=0 called the fundamental equation is satisfied. Then an invariant R-module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case hav…
Usually bundle gerbes are considered as objects of a 2-groupoid, whose 1-morphisms, called stable isomorphisms, are all invertible. I introduce new 1-morphisms which include stable isomorphisms, trivializations and bundle gerbe modules. They fit into the structure of a 2-category of bundle gerbes, and lead to natural d…
Constructs noncommutative spaces for D-branes on complex algebraic spaces.
problem Mathematical model for D-branes on noncommutative spaces.
method Toric geometry, Azumaya schemes, invertible sheaves.
result Embeds algebraic Calabi-Yau spaces into soft noncommutative schemes.
New connections found for quantum flag manifolds modules.
problem Unique connections for relative line modules over quantum flag manifolds.
method Applied general results on quantum principal bundles to Heckenberger-Kolb calculi.
result Found bimodule connections with invertible maps.
Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of σ-connections on finitely generated projective modules. This ma…
Generalizes van Est map to sheaves of sections taking values in G-modules.
problem Classifying geometric structures involving Lie groupoids and stacks.
method Infinitesimal description of G-modules and generalized van Est map. result Definition and study of the generalized van Est map.
The classification of high-dimensional mu-component boundary links motivates decomposition theorems for the algebraic K-groups of the group ring A[F_mu] and the noncommutative Cohn localization Sigma^{-1}A[F_mu], for any mu>0 and an arbitrary ring A, with F_mu the free group on mu generators and Sigma the set of matric…
Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander module…
Develops graphical calculus for monoidal categories with twisted pivotal structures.
problem Constructing modules for surfaces with Morse functions or foliations.
method Graphical calculus and string nets for monoidal categories with twisted pivotal structures.
result Twisted string net modules assemble in an oriented categorified 2-TQFT.
New approach learns causally disentangled latent structures in generative models.
problem Fundamental tension between expressivity and structure in latent structure learning.
method Added a context module to an arbitrarily complex model to learn causally disentangled concepts.
result Causally disentangled representations can be composed for out-of-distribution generation.
Researchers compare two methods for handlebody constructions, finding they are related with a 'background charge'.
problem Comparing two methods for handlebody constructions in finite ribbon categories.
method Admissible skein module construction vs. ansular functor construction.
result An isomorphism between the two constructions is proven, with a 'background charge' that becomes trivial in the unimodular case.
In this paper we give a new basis, Λ, for the Homflypt skein module of the solid torus, S(ST), which was predicted by Jozef Przytycki, using topological interpretation. The basis Λ is different from the basis Λ′, discovered independently by Hoste--Kidwell \cite{HK} and Turaev \cite{Tu} w…
Let G --> G' be an embedding of semisimple complex Lie groups, let B and B' be a pair of nested Borel subgroups, and let f:G/B --> G'/B' be the associated equivariant embedding of flag manifolds. We study the pullbacks of cohomologies of invertible sheaves on G'/B' along the embedding f. Let O' be a G'-equivariant inve…
New bases for Kauffman bracket skein module of genus 2 handlebody.
problem Finding new bases for Kauffman bracket skein module of genus 2 handlebody.
method Using parting technique to convert elements in the Przytycki-basis to open braid form, defining an ordering relation, and relating the bases via matrix relations.
result Introducing BH2 as a more natural basis for KBSM(H2) and suitable for computing modules of 3-manifolds obtained from H2 by surgery. Let Γ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar curvature sequence for BΓ. The lower bounds are formulated in terms of the part …
Study on estimating invertible functions with minimax analysis.
problem Minimizing risk of estimating invertible functions on a plane.
method Introduce two types of L2-risks, derive lower and upper rates for minimax values, develop an asymptotically almost everywhere invertible estimator. result Invertibility does not reduce the complexity of the estimation problem in terms of the rate.
Local invertibility of higher order tensor transforms on compact manifolds.
problem Invertibility of higher order tensor transforms on compact manifolds.
method Local invertibility of transverse and mixed ray transforms of tensors on compact Riemannian manifolds.
result Local invertibility of transverse and mixed ray transforms of tensors for specific dimensions.
Study of strongly invertible Legendrian links in contact 3-space.
problem Characterizing and understanding strongly invertible Legendrian links.
method Equivariant analogs of basic results for strongly invertible and Legendrian links.
result Existence of maximal equivariant Thurston-Bennequin number for strongly invertible links.
Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.
problem Understanding gaps in genus between strongly invertible knots and their invariant Seifert surfaces.
method Analysis of invariant Seifert surfaces and proof of genus gaps, with variants of Edmonds' theorem.
result Gap between equivariant genus and usual genus can be arbitrarily large for strongly invertible knots.
Invertible networks help explain decisions and identify important features.
problem Interpreting and explaining the decisions of black-box neural networks.
method Two-stage approach: invertible transformation to feature space and linear classifier. Determining decision boundaries and feature importance using local linear models.
result Ability to explain decisions and identify important features in neural networks.
Table of symmetric diagrams for knots up to 10 crossings.
problem Finding symmetric diagrams for strongly invertible knots.
method Compilation of symmetric diagrams for knots up to 10 crossings.
result Similarity of transversal diagrams to symmetric union diagrams for strongly invertible knots.
Global invertibility proven for orientation-preserving maps without homeomorphic extension.
problem Global invertibility of orientation-preserving Sobolev maps.
method Avoiding homeomorphic extension, study of strictly orientation-preserving maps.
result Global invertibility can be achieved without homeomorphic extension.
Invertible neural networks with masked convolutions improve classification and generative models.
problem Building robust invertible neural networks for better model interpretability and generative tasks.
method Combining masked convolutions and iterative inversion methods to create invertible architectures.
result Invertible neural networks achieve competitive performance in classification and generative tasks.
HINT improves invertible neural networks for better density estimation and Bayesian inference.
problem Sparse Jacobians limit expressiveness in invertible neural architectures.
method Recursive hierarchical coupling within subsets of variables leads to dense, triangular Jacobian.
result HINT allows efficient sampling from joint and posterior distributions using a single network.
ISR creates analytical relationships from data via invertible maps.
problem Creating analytical relationships from datasets.
method Combines INNs and EQL, using invertible maps and sparsity promoting regularization.
result ISR can serve as a normalizing flow for density estimation and solve inverse problems.
Local invertibility of ray transforms on convex manifolds.
problem Invertibility of ray transforms on compact Riemannian manifolds with strictly convex boundary.
method Local invertibility results for transverse and mixed ray transforms of 1 and 1+1 tensors.
result Local invertibility of ray transforms near boundary points, leading to global results.
Deep invertible networks decode EEG signals better than chance.
problem Decoding brain signals from EEG data.
method Deep invertible networks for generating and classifying brain signals.
result Deep invertible networks generate realistic EEG signals and classify novel signals above chance.
This work tackles exploding inverses in INNs, revealing and mitigating their numerical non-invertibility.
problem Exploding inverses in INNs cause numerical non-invertibility, leading to failures in various tasks.
method Derived bi-Lipschitz properties of INN building blocks, proposed regularizers for local invertibility, and stable INN designs for global invertibility.
result Bi-Lipschitz properties and stable INN designs are crucial for addressing numerical non-invertibility.
Neural ODEs and i-ResNets can't approximate all continuous invertible functions.
problem Neural ODEs and i-ResNets' limitations in approximating continuous invertible functions.
method Proving the approximation capabilities of Neural ODEs and i-ResNets.
result Neural ODEs and i-ResNets can approximate homeomorphisms on a p-dimensional Euclidean space.
CF-INNs can approximate any invertible function, resolving a long-standing problem.
problem Whether CF-INNs can approximate any invertible function.
method Demonstrated CF-INNs are universal approximators for invertible functions by showing a convenient criterion.
result CF-INNs are universal approximators for invertible functions.
This work presents a novel approach to train invertible linear layers by adding rank-one perturbations.
problem Training invertible linear layers during optimization with gradient-based methods is challenging.
method Train rank-one perturbations and add them to weight matrices infrequently, keeping track of inverses and determinants.
result Invertible linear layers improve mixing and mode separation in normalizing flows.
Spaces of metrics with invertible Dirac operator are homotopy equivalent for cobordant manifolds.
problem Proving homotopy equivalence of spaces of metrics with invertible Dirac operator.
method Using cobordism theory and properties of Dirac operators.
result Spaces of metrics with invertible Dirac operator are homotopy equivalent for cobordant manifolds.
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
New findings on knot genera using advanced techniques.
problem Understanding the 4-genus of knots, especially strongly invertible and periodic ones.
method Innovative concordance group invariants, Donaldson's theorem, and g-signature.
result Many new examples showing the equivariant 4-genus is larger than the 4-genus.