Let Ng be the connected closed nonorientable surface of genus g >= 5 and Mod(Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod(Ng) is either trivial or Z if g is odd, and injects into the mapping class group of sphere with four holes if g is even.
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This paper was withdrawn by the author. The appearance of an author-written addendum [3] to the paper [2] made our correction note [1] to that paper superfluous and hence it is no longer available here. [1] Dror Bar-Natan and Ofer Ron, A Correction to "Groups of Ribbon Knots" by Ka Yi Ng, no longer available. [2] Ka Yi…
A framework connects Taylor methods with Fisher-efficient NG.
problem Combining Taylor-based methods with Fisher-efficient NG.
method Constructs a theoretical framework linking Taylor approximation and NG.
result Mathematical justification for combining higher order methods with NG.
Bayesian model learns cancer subtypes from diverse NGS data.
problem Overdispersed NGS count data and limited samples for specific cancer types.
method Bayesian Multi-Domain Learning (BMDL) model using hierarchical negative binomial factorization.
result BMDL achieves reproducible cancer subtyping without negative transfer effects.
Study forecasts Turkish residential NGD using JITL-GPR, reducing errors.
problem Accurately predict future monthly NGD for Turkey's import contracts.
method Used historical NG consumption data, applied various time series models, and introduced JITL-GPR.
result JITL-GPR reduces forecast errors compared to traditional methods.
A new batch optimisation framework using Natural Gradient improves DNN acoustic models.
problem Optimizing DNN acoustic models for better word error rate approximation.
method Proposes a Natural Gradient (NG) approach to sequence training, correcting the gradient based on local curvature of KL-divergence.
result The NG method converges more quickly and can be applied to any sequence discriminative training criterion.
Differential Expression Analysis of Dynamical Sequencing Count Data with a Gamma Markov Chainstat.AP
Develops GMNB model for better NGS data analysis.
problem Lack of tools for analyzing temporal NGS data.
method Integrates gamma Markov chain into negative binomial distribution model.
result GMNB outperforms existing methods in differential expression analysis.
Torus knots and provide counterexamples to a conjecture about knot contact homology.
problem Ng's conjecture about isomorphism between knot contact homology and character variety fails for certain torus knots.
method Demonstrated through the construction of ghost characters for specific torus knots.
result Counterexamples to Ng's conjecture for and torus knots.
We propose flow-based likelihoods to accurately capture non-Gaussian data.
problem Bypassing the Gaussian assumption in scientific analyses.
method Use optimization targets of flow-based generative models to reconstruct likelihoods.
result Flow-based likelihoods can accurately capture non-Gaussian data, improving parameter constraints.
NG+ method improves deep learning efficiency and accuracy.
problem Efficiency and accuracy in deep learning models.
method Proposes NG+ method using matrix-product natural gradient approach.
result Established global convergence and provided regret bound.
Paper tackles few-shot class-incremental learning with a neural gas network.
problem Incrementally learn new classes from very few labelled samples without forgetting old classes.
method Proposes TOPIC framework using a neural gas network to preserve class topology and adapt to new samples.
result Significantly outperforms other methods on CIFAR100, miniImageNet, and CUB200 datasets.
Unified product Lie groups and their quotient spaces are analyzed for dynamics.
problem Analyzing dynamics over homogeneous spaces using Lie group theory.
method Reduction and extension of Lie group structures to quotient spaces, formulation of Euler-Lagrange, Hamilton, and Euler-Poincaré equations.
result Unified product Lie groups and their quotient spaces provide a framework for formulating dynamics equations.
This paper has been withdrawn by the authors, due to a mistake pointed out by Lenny Ng and Josh Sabloff.
We generalize Ng's two-variable algebraic/combinatorial -th framed knot contact homology for framed oriented knots in to knots in , and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially co…
Paper introduces a new optimisation method combining NG and Hessian Free for sequence training.
problem Overfitting and inefficiency in training DNNs with sequence criteria.
method Combines Natural Gradient and Hessian Free methods for better path traversal on parameter manifold.
result Achieves larger reductions in WER and lower WER compared to NG and HF methods.
In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called -invar…
Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial level. We use the 1-jet of the classifying space of G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a diff…
MethaneMapper detects methane emissions with high accuracy and reduced model size.
problem Challenges in detecting and quantifying methane emissions from AVIRIS-NG data.
method Spectral absorption wavelength aware transformer network, introducing two novel modules.
result Achieves 0.63 mAP in detection and reduces model size by 5x.
A new method speeds up and stabilizes deep CNN training.
problem Training deep convolutional neural networks (CNNs) is slow and unstable.
method Nonlinearity generation (NG) method modifies activation functions to stabilize training.
result The method speeds up and stabilizes training of deep CNNs.
Study knot contact homology and character varieties to prove a conjecture about knots.
problem Relate knot contact homology to character varieties of branched covers.
method Study trace-free characters of knot groups and their connection to character varieties.
result Prove Ng's conjecture for 2-bridge and 3-bridge knots using ghost characters.
Advances of modern sensing and sequencing technologies generate a deluge of high dimensional space-temporal physiological and next-generation sequencing (NGS) data. Physiological traits are observed either as continuous random functions, or on a dense grid and referred to as function-valued traits. Both physiological a…
Classification of Legendrian knots of topological type with maximal Thurston--Bennequin numbermath.GT
Classifies Legendrian knots with maximal Thurston-Bennequin number.
problem Classifying Legendrian knots of a specific type.
method Using Thurston-Bennequin number as a criterion.
result Confirms conjectures about Legendrian knots of type .
Neurally-Guided Structure Inference combines search and data-driven methods for efficient, robust structure inference.
problem Combining the advantages of exhaustive search and data-driven methods for structure inference.
method Neurally-Guided Structure Inference (NG-SI) uses a neural network to guide hierarchical search over structures.
result NG-SI outperforms search-based and data-driven methods on probabilistic matrix decomposition and symbolic program parsing.
We use the spanning tree model for Khovanov homology to study Legendrian links. This leads to an alternative proof for Ng's Khovanov bound for the Thurston-Bennequin number and to both a necessary and a sufficient condition for this bound to be sharp.
New invariant for surface-knots in 4D from marked graphs.
problem Invariants for surface-knots in 4D.
method Marked graph diagrams for surface-knots.
result Invariant constructed for surface-knots.
New invariant for surface-knots in 4D space.
problem Invariants for surface-knots in 4D space.
method Diagrammatic approach in 3D space.
result Invariant constructed for surface-knots.
The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing. This paper suggests the Theoretical Spectral Curvature Clustering (TSCC) algori…
We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.
Enhances GPLVM for multi-view data with scalable latent representation learning.
problem Limited kernel expressiveness and computational inefficiency in multi-view GPLVM.
method Introduces a new duality between spectral density and kernel function, uses NG-SM kernel, and applies random Fourier feature approximation for scalability.
result Consistently outperforms state-of-the-art models in learning meaningful latent representations across diverse datasets.
Researchers propose a non-monotone quantum natural gradient for quantum systems.
problem Applying natural gradient methods to quantum systems without monotonicity.
method Introducing a non-monotone quantum natural gradient (QNG) and demonstrating its superiority over conventional QNG.
result Non-monotone QNG outperforms conventional QNG in terms of convergence speed.
The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram has a ruling, therefore its Thurston-Bennequin number is maximal among Legendrian …
Redefined cord algebra using Morse Theory for knot invariants.
problem Defining a knot invariant using algebraic methods.
method Using Morse Theory to redefine the cord algebra.
result Proved the cord algebra is a knot invariant.
In this paper, which is mostly a research announcement, we give a new algebraic construction of knot contact homology in the sense of L. Ng [Ng05a]. For a link in , we define a differential graded (DG) -category with finitely many objects, whose quasi-equivalence class is …
We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic…
Defines singular grid diagrams for various types of links.
problem Unified description of singular links and related objects.
method Definition of singular grid diagrams and classification of equivalence relations.
result Unified description of singular links and related objects.
New method confirms conjectures about specific Legendrian knots.
problem Classifying Legendrian knots of specific topological types.
method New method of comparing Legendrian knots with nontrivial symmetry group.
result Conjectures about specific Legendrian knots are confirmed.
The (4,5)-torus knot has a unique ghost character with significant implications.
problem Understanding the character variety and fundamental variety of 2-fold branched covers of knots.
method Analyzing the (4,5)-torus knot's ghost character to derive new insights into knot representations and character varieties.
result The (4,5)-torus knot is the simplest example where certain representations cannot be realized by trace-free representations and provides a counterexample to Ng's conjecture.
Paper extends Willmore inequality to manifolds with negative Ricci curvature.
problem Establishing a Willmore-type inequality for hypersurfaces in manifolds with negative Ricci curvature.
method Using techniques from Riemannian geometry, the authors extend a classic result to manifolds with negative curvature.
result Constructed a Willmore-type inequality for hypersurfaces in hyperbolic space and characterized geodesic spheres.
A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form where is an exact symplectic manifold is established. The class of such contact manifolds include 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth)…
When a Lie group has a central -extension, there is a cocycle in the simplicial de Rham complex which represents the Dixmier-Douady class. Mickelsson and Brylinski, McLaughlin constructed a central -extension whose Dixmier-Douady class in is…
For any Legendrian link, L, in (\R^3, \ker(dz-y\,dx)) we define invariants, Aug_m(L,q), as normalized counts of augmentations from the Legendrian contact homology DGA of L into a finite field of order q where the parameter m is a divisor of twice the rotation number of L. Generalizing a result of Ng and Sabloff for the…
Examples are given of prime Legendrian knots in the standard contact 3-space that have arbitrarily many distinct Chekanov polynomials, refuting a conjecture of Lenny Ng. These are constructed using a new `Legendrian tangle replacement' technique. This technique is then used to show that the phenomenon of multiple Cheka…
Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.
problem Understanding transverse link invariants in the annular setting.
method Constructs a stable homotopy type for annular links and defines a map to the Khovanov skein spectrum.
result At extreme gradings, the map from the Khovanov spectrum to the Khovanov skein spectrum recovers the cohomotopy transverse invariant.
In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic m…
The paper proves there are many Lagrangian fillings for Legendrian links of affine type.
problem Proving the existence of many Lagrangian fillings for Legendrian links of affine type.
method Using cluster structures and Coxeter mutation to prove the existence of fillings.
result There are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type.
We define an algebraic/combinatorial object on the front projection of a Legendrian knot called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashb…
We show that there exist infinitely many pairs of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in and its congruence quotients, the classification of SOL (polycyclic) 3-manifold groups and an elementa…
Study of spacetimes in cosmology without symmetry assumptions.
problem Proving timelike incompleteness and properties of time functions.
method Rigorous mathematical proofs and analysis of spacetime properties.
result Timelike incompleteness for specific spacetimes with mean curvature constraints.