Two complexes share a common covering but not a finite one.
problem Common covering of complexes with specific properties.
method Constructing CW-complexes K and L with a common covering.
result K is homeomorphic to a complex with a single 2-cell.
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48 results for “single 2-cell”
Structures of optimal discrete gradient vector fields on surface with one or two critical cellsmath.DS
The study identifies all possible vector field structures on specific 2D shapes.
problem Optimal discrete gradient vector fields on surfaces with 1-2 critical cells.
method Analysis of discrete vector fields on 2D shapes with minimal critical cells.
result All possible structures of discrete Morse functions on specified shapes.
Researchers found all embeddings of Kuratowski graphs on a double torus.
problem Characterizing embeddings of Kuratowski graphs and on the double torus.
method Constructive approach using Burnside's Lemma and automorphism groups.
result 14 orientable and 17 non-orientable 2-cell embeddings of on the double torus.
Characterizes Forman curvature bounds and proves curvature equivalence.
problem Characterize Forman curvature bounds and prove curvature equivalence.
method Contractivity of the Hodge Laplacian semigroup, translation between 2-cells and transport plans.
result Ollivier and Forman curvature coincide on edges when maximizing Forman curvature.
In this brief sequel to a previous article, we recall the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by cou…
A finite subset S of a closed hyperbolic surface F canonically determines a "centered dual decomposition" of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection …
A zigzag in a map (a -cell embedding of a connected graph in a connected closed -dimensional surface) is a cyclic sequence of edges satisfying the following conditions: 1) any two consecutive edges lie on the same face and have a common vertex, 2) for any three consecutive edges the first and the third edges are …
We use Klyachko's methods [A funny property of sphere and equations over groups, Comm. in Alg. 21 (1993) 2555--2575] (see also Fenn-Rourke, L'Enseignment Math. 42 (1996) 49--74 and math.GR/9810184 and Cohen-Rourke, math.GR/0009101) to prove that, if a 1-cell and a 2-cell are added to a complex with torsion-free fundame…
In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected -complex with a linear homological isoperimetric inequality, a bound on the length of attaching maps of -cells and finitely many -cells adjacent to any edge must …
We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the 1-cells with elem…
Symplectic Rational Blow-upmath.SG
Fintushel and Stern defined the rational blow-down construction [FS] for smooth 4-manifolds, where a linear plumbing configuration of spheres is replaced with a rational homology ball , . Subsequently, Symington [Sy] defined this procedure in the symplectic category, where a symplectic (given…
Nonpositive towers property in 3-manifolds spines.
problem Properties of 3-manifold spines.
method Analysis of 2-dimensional spines in aspherical 3-manifolds and 3-ball.
result Nonpositive towers property in 2-dimensional spines of aspherical 3-manifolds.
Study embeddability of 2-complexes in 4-space, proving Heawood family's excluded minors.
problem Whether a 2-dimensional CW complex embeds in .
method Operations preserving embeddability, constructions of non-preserving transformations, study of 4-flat graphs.
result Prove 78 graphs of Heawood family are excluded minors for 4-flat graphs.
We construct examples of free-by-cyclic hyperbolic groups which fiber in infinitely many ways over Z. The construction involves adding a specialized square 2-cell to a non-positively curved, squared 2-complex defined by labeled oriented graphs. The fundamental groups of the resulting complexes are hyperbolic, free-by-c…
We deal with the symmetries of a (2-term) graded vector space or bundle. Our first theorem shows that they define a (strict) Lie 2-groupoid in a natural way. Our second theorem explores the construction of nerves for Lie 2-categories, showing that it yields simplicial manifolds if the 2-cells are invertible. Finally, o…
Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston's conjecture by proving that such a manifold admits a piecewise Euclidean metric of non-positive curvature a…
New framework for sutured manifolds using handleslides and Heegaard invariants.
problem Naturality results for Floer homology theories in sutured manifolds.
method Definition of tight Heegaard invariants and their extensions to strong Heegaard invariants.
result Unique extensions of Heegaard invariants provide new frameworks for naturality results.
Consider a planar, bounded, -connected region , and let $\bordΩ$ be its boundary. Let be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair where is a genus …
We exhibit a set of edges (moves) and 2-cells (relations) making the complex of pant decompositions on a surface a simply connected complex. Our construction, unlike the previous ones, keeps the arguments concerning the structural transformations independent from those deriving from the action of the mapping class grou…
New geometric object for polynomials simplifies complex data.
problem Understanding the combinatorial and geometric properties of polynomials.
method Introducing a compact planar 2-complex for polynomials with distinct roots.
result Extracts combinatorial data from a geometric structure of polynomials.
Two simple homotopy equivalent 2-complexes K2 and L2 are related by an algebraic criterion of their corresponding presentations as stated in [HoMeSier]. Frank Quinn set it into a topological context (see [Qu1]) and call these 2-complexes related by an s-move. Using elementary 3-expansions, K2 extends to 3-cells in K3 r…
This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows o…
A dessin is a 2-cell embedding of a connected bipartite graph into an orientable closed surface. An automorphism of a dessin is a permutation of the edges of the underlying graph which preserves the colouring of the vertices and extends to an orientation-preserving self-homeomorphism of the supporting surface. A dessin…
2D complexes can be almost-embedded in 4D space without self-intersections.
problem Understanding the embedding properties of 2-dimensional complexes in 4-dimensional space.
method Analyzing specific 2-complexes constructed by Freedman-Krushkal-Teichner and showing they can be PL immersed in without self-intersections.
result Many 2-complexes can be PL almost-embedded in with singularities only as self-intersections of some 2-cells.
In this paper we continue the study started in part I (posted). We consider a planar, bounded, -connected region , and let $\bordΩ$ be its boundary. Let be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function…
Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph K(v,v). We investigate and mostly determine the set of triples (p,v,g) for which there exists a uniform lattice Γ in Aut(I(…
We consider collections of disjoint simple closed curves in a compact orientable surface which decompose the surface into pairs of pants. The isotopy classes of such curve systems form the vertices of a 2-complex, whose edges correspond to certain simple moves in which only one curve changes, and whose 2-cells correspo…
Bundle 2-gerbesmath.DG
We make the category BGrb_M of bundle gerbes on a manifold M into a 2-category by providing 2-cells in the form of transformations of bundle gerbe morphisms. This description of BGrb_M as a 2-category is used to define the notion of a bundle 2-gerbe. To every bundle 2-gerbe on M is associated a class in H^4(M;Z). We de…
New perspective on Heegaard splittings using square complexes and combinatorial measurements.
problem Measuring obstructions to Heegaard splittings in 3-manifolds.
method Square complexes and Guirardel's core, augmented Heegaard diagrams.
result Augmented Heegaard diagrams provide a new way to describe Heegaard splittings with desirable properties.
A new method for analyzing shapes and forms using additive models on manifolds.
problem Analyzing shapes and forms under geometric transformations.
method Extending generalized additive regression to models for shapes/forms using squared geodesic distance and Riemannian -Boosting algorithm.
result Automated model selection and intuitive visualization of covariate effects in shape/form space.
Random groups prove length constraints on product of conjugates.
problem Quantify products of conjugates in random groups.
method Sharp van Kampen diagram argument and boundary block-counting.
result Prove a sharp inequality for products of conjugates in random groups.
In a recent paper, {\it Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (II)}, we discussed two algorithms for deforming and contracting a simply connected discrete closed manifold into a discrete sphere. The first algorithm was a continuation of work that began in {\it Algorithms fo…
This paper extends the idea of Universum learning [1, 2] to single-class learning problems. We propose Single Class Universum-SVM setting that incorporates a priori knowledge (in the form of additional data samples) into the single class estimation problem. These additional data samples or Universum belong to the same …
ES-Single uses ES to estimate gradients in unrolled graphs, reducing variance and improving performance.
problem Estimating gradients in unrolled computation graphs with low variance and stability.
method Evolution strategies (ES) applied to unrolled graphs, with a single perturbation per particle.
result ES-Single reduces variance compared to PES, leading to better performance in various tasks.
Review of Single-cell RNA-seq Data Clustering for Cell Type Identification and Characterizationq-bio.GN
In recent years, the advances in single-cell RNA-seq techniques have enabled us to perform large-scale transcriptomic profiling at single-cell resolution in a high-throughput manner. Unsupervised learning such as data clustering has become the central component to identify and characterize novel cell types and gene exp…
Single-head attention approximates any function under various norms.
problem Universal approximation of functions using attention mechanisms.
method Interpreting attention as partitioning and summing linear transformations.
result Single-head attention can approximate any continuous function under -norm and Lebesgue integrable functions under -norm.
AugurOne trains single image generators without GANs using image warps.
problem Training end-to-end single image generators without GANs.
method Non-affine augmentations of single input images for training an upscaling neural network.
result End-to-end training yields state-of-the-art performance on conditional generation tasks.
Improved GPLVM model for single-cell RNA-seq data.
problem Lack of effective scalable models for clustering cell types in large-scale single-cell RNA-seq data.
method Introduces amortized stochastic variational Bayesian GPLVM (BGPLVM) tailored for single-cell RNA-seq.
result Matches the performance of scVI on synthetic and real-world datasets and reveals more interpretable latent structures.
Kernel testing compares cell states in single-cell data.
problem Comparing non-linear cell states in single-cell data.
method Kernel-based testing framework for non-linear distribution comparison.
result Identifies subtle population variations in cell states.
New analysis shows actor-critic method converges efficiently in practical settings.
problem Understanding finite-time convergence of single-timescale actor-critic methods.
method Investigated online single-timescale actor-critic algorithm with linear function approximation and Markovian sampling.
result Proved convergence to ε-approximate stationary point with sample complexity of O(ε^(-2)).
Paper tackles continual learning with single-index models, proving regret bounds.
problem Continual learning with single-index models across multiple tasks.
method Proposes a randomized strategy to learn a common single-index and task-specific link functions.
result Proves regret bounds for the proposed strategy under various loss function assumptions.
SMAI framework tests and integrates single-cell data alignability.
problem Lack of a rigorous statistical test for alignability and distortion during alignment.
method Spectral manifold alignment and inference (SMAI) framework.
result SMAI outperforms existing methods in alignability testing and integration.
With ongoing developments and innovations in single-cell RNA sequencing methods, advancements in sequencing performance could empower significant discoveries as well as new emerging possibilities to address biological and medical investigations. In the study, we will be using the dataset collected by the authors of Sys…
Formula found for neural network error with fixed weights.
problem Understanding error in neural networks with fixed weights.
method Provided an explicit formula for approximation error.
result Explicit formula for neural network error with fixed weights.
Study SGD dynamics in sequence models, revealing training phases and influence of sequence length.
problem Understanding SGD in sequence models like attention networks.
method Derived closed-form population loss and analyzed SGD dynamics for SSI models.
result Two distinct training phases: escape from uninformative initialization and alignment with target subspace.
Single twist can unknot certain knots, study shows.
problem Can a knot be unknotted with a single twist?
method Classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory.
result Obstructions to unknotting with a single twist exist.
SGD shows distinct phases in learning single-index models, achieving optimal sample complexity and regret.
problem Learning single-index models with SGD in adaptive data settings.
method Stochastic gradient descent (SGD) with an optimal learning rate schedule.
result SGD achieves near-optimal sample complexity and regret guarantees across both burn-in and learning phases.
StrokeCoder uses Transformers to generate images from single examples.
problem Creating diverse images from a single example.
method Transformer Neural Network learns from a single path-based example to generate a set of images.
result The model can generate a large set of deviated images that still represent the original image's style and concept.