The paper constructs minimal realizations of signed Gauss paragraphs using graph theory.
problem Constructing minimal realizations of signed Gauss paragraphs on surfaces.
method Theory of embedded graphs on oriented and compact PL-surfaces, intersection pairing of immersed PL-normal curves.
result The genus of the ambient surface can be a function of the maximum number of Carter's circles.
A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. W…
A Gauss paragraph is a combinatorial formulation of a generic closed curve with multiple components on some surface. A virtual string is a collection of circles with arrows that represent the crossings of such a curve. Every closed curve has an underlying virtual string and every virtual string has an underlying Gauss …
Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.
problem Understanding the relationship between Gaussian curvature and singularities of Gauss maps of cuspidal edges.
method Analyzes geometric invariants and types of singularities of Gauss maps to define and characterize positivity/negativity of cusps.
result Defines and characterizes positivity/negativity of cusps of Gauss maps by geometric invariants of cuspidal edges, and shows relation between sign of cusps and Gaussian curvature.
The paper proves properties of strain tensors on surfaces with changing Gauss curvature.
problem Regularity of solutions to strain tensor equations on surfaces with variable Gauss curvature.
method Proof of regularity, density property, and matching property.
result Established matching property and density of smooth infinitesimal isometries.
Bayesian approach improves paragraph vector entropy and uncertainty in text analysis.
problem Capturing semantic relationships in text of varying lengths.
method Probabilistic generative model for paragraph vectors with Bayesian inference.
result Entropy of paragraph vectors decreases with document length and uncertainty improves performance in text analysis.
New formula for spherical polygon area via prequantization.
problem Traditional area formula for spherical polygons requires measuring angles.
method Uses prequantization to create a new formula that doesn't require angle measurement.
result New formula applicable to a wider range of degenerate curves and polygons.
A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by coun…
New method learns more diverse topics from text documents considering paragraph structure.
problem Classic Topic Models ignore word position and use symmetric priors, limiting topic diversity.
method Exploits paragraph structure to distinguish between general and specific topics.
result Shows improved topic diversity and relevance in structured documents.
When the signed weighted resolution set was defined as an invariant of pseudoknots, it was unknown whether this invariant was complete. Using the Gauss-diagrammatic invariants of pseudoknots introduced by Dorais et al, we show that the signed were-set cannot distinguish all non-equivalent pseudoknots. This goal is achi…
Paper develops a framework for generating coherent image captions using visual features and hierarchical topics.
problem Generating semantically coherent paragraphs to describe image content.
method Plug-and-play hierarchical-topic-guided image paragraph generation framework integrating visual extractor and deep topic model.
result Proposed models can distill interpretable multi-layer semantic topics and generate diverse and coherent captions.
The paper identifies patterns in language model weights used for memorizing paragraphs.
problem Locating the specific mechanisms and weights used by language models to memorize paragraphs.
method Examined gradients and attention patterns in language models to identify memorized paragraphs.
result Gradients of memorized paragraphs have a distinguishable spatial pattern, and localized attention heads are involved in paragraph memorization.
This work speeds up unsupervised sentence learning using paragraph coherence.
problem Training fast unsupervised sentence encoders.
method Discourse-based objective function for neural network training.
result Models trained with this method are faster and perform well.
The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.
problem Computing curvature limits in affine and Minkowski groups.
method Analyzing Euclidean C2-smooth surfaces and curves in affine and Minkowski groups. result Gauss-Bonnet theorems in affine and Minkowski groups are proven.
New curvature K(x) measures manifold properties without integrals.
problem Understanding curvature on compact Riemannian manifolds.
method Developed index expectation curvature K(x) for 2D manifolds, constructed as a product of sectional index expectation curvatures.
result For small 2D manifolds with boundary, definite sign index expectation curvature K(x) exists and satisfies Gauss-Bonnet relation.
The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
problem Computing curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
method Sub-Riemannian limits of Gaussian curvature, Schouten-Van Kampen affine connections, and adapted connections.
result Gauss-Bonnet theorems associated with Schouten-Van Kampen affine connections in the Heisenberg group.
The Heisenberg group's curvature and Gauss-Bonnet theorem are explored using Riemannian approximation.
problem Defining curvature in the Heisenberg group for smooth surfaces and curves.
method Using a Riemannian approximation scheme to define sub-Riemannian Gaussian and signed geodesic curvatures.
result Proved a Heisenberg version of the Gauss-Bonnet theorem.
Framework learns sentence order from paragraphs using attention and transformer networks.
problem Learning to order sentences from a paragraph.
method Bidirectional sentence encoder and self-attention transformer network for ranking.
result Framework outperforms state-of-the-art methods on sentence ordering and discrimination tasks.
Let (M,g) be a two dimensional compact Riemannian manifold of genus g(M)>1. Let f be a smooth function on M such that f≥0,f≡0,minMf=0. Let p1,…,pn be any set of points at which f(pi)=0 and D2f(pi) is non-singular. We prove that for all sufficiently small $λ>…
The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
problem Computing curvature and geodesic curvature for surfaces and curves in affine and rigid motions groups.
method Defined deformed Schouten-Van Kampen connections, computed Gaussian curvature limits, and signed geodesic curvature.
result Derived Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
The paper proves existence of horo-convex hypersurfaces in hyperbolic space with specific curvature conditions.
problem Existence of horo-convex hypersurfaces with prescribed shifted Gauss curvatures in hyperbolic space.
method Existence result obtained via standard degree theory based on a prior estimates for solutions to the prescribed shifted Gauss curvature equations.
result Existence of horo-convex hypersurfaces in hyperbolic space under certain conditions.
This study compares feature extraction methods using Neural Networks and Latent Dirichlet Allocation for movie synopses.
problem Extracting meaningful features from movie synopses for pattern detection and recommendation.
method Employed Latent Dirichlet Allocation for topic modeling and Neural Networks for distributed paragraph representations.
result Latent Dirichlet Allocation can provide meaningful features for movie synopses, comparable to Neural Networks.
A method of computing a basis for the second Yang-Baxter cohomology of a finite biquandle with coefficients in Q and Z_p from a matrix presentation of the finite biquandle is described. We also describe a method for computing the Yang-Baxter cocycle invariants of an oriented knot or link represented as a signed Gauss c…
Improved text embeddings enhance retrieval from a knowledge base.
problem Efficiently retrieving relevant paragraphs from a large knowledge base.
method Used Stanford Question Answering Dataset (SQuAD) for open-domain question answering. Compared various text-embedding methods and trained deep residual neural models for retrieval.
result Training deep residual neural models for retrieval purposes significantly improves paragraph recall.
Study on curvature functions for compact manifolds with boundary.
problem Understanding curvature functions on compact manifolds with boundary.
method Proves necessary and sufficient conditions for geodesic and Gaussian curvature, solves problems in the pointwise conformal case.
result New existence and nonexistence results for metrics with prescribed curvature, depending on Euler characteristic.
By adding or removing appropriate structures to Gauss diagram, one can create useful objects related to virtual links. In this paper few objects of this kind are studied: twisted virtual links generalizing virtual links; signed chord diagrams staying halfway between twisted virtual links and Kauffman bracket / Khovanov…
Convolutional autoencoding improves long text reconstruction.
problem Text reconstruction quality decreases with text length.
method Sequence-to-sequence, purely convolutional and deconvolutional autoencoding.
result Better at reconstructing and correcting long paragraphs.
Study on surfaces with constant anisotropic mean curvature in 3D space.
problem Characterizing surfaces with constant anisotropic mean curvature.
method Analyzing uniformly elliptic anisotropic functionals and proving properties of surfaces.
result Characterization of surfaces with constant anisotropic mean curvature.
The main purpose of the paper is twofold: First, to extend a well known theorem of Ruh-Vilms in the Euclidean space to symmetric spaces and, secondly, to apply this result to extend Hoffman-Osserman-Schoen Theorem (HOS Theorem) to 3-dimensional symmetric spaces. Precisely, it is defined a Gauss map of a hypersurface M^…
Study of rotation angles in a rotating disc model.
problem Understanding geometric phase in rotating systems.
method Analyzes a simple kinematic model of rotating discs.
result Explicit form of geometric phase Δg found using Baumkuchen lemma. Study calculates the renormalized area of catenoids in hyperbolic spaces.
problem Calculating the renormalized area of catenoids in hyperbolic spaces.
method Variational characterization and Chern--Gauss--Bonnet formulas for locally conformally flat manifolds.
result Renormalized area of catenoids varies continuously from negative infinity to twice the area of totally geodesic hypersurfaces.
Transformer models improve query-document retrieval efficiency and accuracy.
problem Efficiently retrieve relevant documents from large corpora for query matching.
method Designed paragraph-level pre-training tasks to optimize embedding-based Transformer models.
result Transformer models significantly outperform BM-25 and non-Transformer embedding models.
A new system learns entity representations to improve local entity disambiguation.
problem Local entity disambiguation in text.
method Entity-ELMo (E-ELMo) approach for contextual entity representation.
result Outperforms state-of-the-art models by 0.5% on AIDA test-b.
The paper evaluates different vector space models for text similarity.
problem Measuring semantic text similarity in natural language processing.
method Comparison of TFIDF, topic models, and neural models for patent-to-patent similarity.
result TFIDF performs well for longer, technical texts or finer distinctions.
Project aims to improve coherence in language generation models.
problem Models often generate inconsistent text that diverges from the prompt.
method Trained a sentence pair coherence classifier and co-trained GPT-2 with this coherence objective.
result Fine-tuned model generates coherent paragraphs without diverging.
An unsupervised method clusters patient incident reports for content analysis.
problem Lack of methods to extract interpretable content from electronic healthcare records.
method Combines text-embedding with paragraph vectors and graph-theoretical multiscale community detection.
result Extracts high-intrinsic-consistency groups of patient incident reports.
Gestalt combines two models to improve SQuAD2.0 performance.
problem Improving the accuracy of answering questions in context paragraphs.
method A stacking ensemble of ALBERT and RoBERTa models, combined with a CNN-based meta-model.
result Best ensemble achieved 87.117 EM and 90.306 F1 scores, improving baseline by 0.55% and 0.61% respectively.
The second H. Weyl curvature invariant of a Riemannian manifold, denoted h4, is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of h4 is that it is nonnegative for Einstein manifolds, hence it p…
The paper solves conditions for prescribing scalar and Gauss curvatures on manifolds with zero first eigenvalue.
problem Conditions for prescribing scalar and Gauss curvatures on manifolds with zero first eigenvalue.
method Local variational methods, local Yamabe-type equations, and monotone iteration scheme.
result The necessary and sufficient conditions for prescribing scalar and Gauss curvatures are established.
Local solubility of Bao--Ratiu equations proven for surfaces with specific curvature conditions.
problem Existence of asymptotic directions for volume-preserving diffeomorphisms on surfaces.
method Analysis of degenerate Monge--Ampère equation following Han's work.
result Asymptotic directions always exist locally about a point on surfaces with specific curvature conditions.
AutoML-GPT uses GPT to automate AI model training.
problem Manual model selection and tuning requires significant human effort.
method Develops task-oriented prompts and utilizes LLMs for automated training.
result Achieves remarkable results in various AI tasks.
We prove that an isometric immersion of a simply connected Lorentzian surface in R2,2 is equivalent to a normalised spinor field solution of a Dirac equation on the surface. Using the quaternions and the Lorentz numbers, we also obtain an explicit representation formula of the immersion in terms of the sp…
Gauss codes help uniquely identify virtual doodles.
problem Identifying virtual doodles uniquely.
method Introduced left canonical Gauss codes.
result Oriented virtual doodles uniquely presented by left canonical Gauss codes.
Better spectral partitioning of signed graphs using standard Laplacian.
problem Meaningless partitioning using signed Laplacian eigenvectors.
method Use standard graph Laplacian for spectral partitioning.
result Fiedler vector of standard Laplacian is easier to compute and more beneficial.
Gauss diagrams' properties can change with Hamiltonian cycle choice.
problem The impact of Hamiltonian cycle choice on Gauss diagrams.
method Examined realizable and unrealizable Gauss diagrams, and proved preservation of realizability under certain Hamiltonian cycle changes.
result Properties of Gauss diagrams can vary with Hamiltonian cycle choice.
Study on signed graphs with random signs, focusing on community detection.
problem Community detection in signed stochastic block models.
method Strong concentration inequalities for adjacency and Laplacian matrices, applied to signed Laplacian matrix.
result The sign of the first eigenvector of the Laplacian matrix defines a weakly consistent estimator for balanced community detection.
By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, …
SELO model predicts link signs better than SDGNN using subgraph encoding and linear optimization.
problem Inferring the sign of links in signed networks with limited sign data.
method Subgraph Encoding via Linear Optimization (SELO) approach to learn edge embeddings.
result SELO model outperforms state-of-the-art methods on multiple real-world signed networks.