No GANs can learn disconnected manifolds precisely.
problem Learning disconnected manifolds is challenging due to unimodal latent distributions.
method Formalized a no free lunch theorem and derived a rejection sampling method.
result Upper bound on the precision of the targeted disconnected manifold distribution.
GANs struggle with disconnected manifolds, our method improves their performance.
problem GANs fail to model distributions on disconnected manifolds.
method Using a collection of generators and a novel prior learning approach.
result Our method addresses the limitations of GANs on disconnected manifolds.
We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restricted holonomy groups of such manifolds and provide a construction method for manifolds with disconnected holonomy which starts from a Riemannian manifold …
We study the space of positive scalar curvature (psc) metrics on a 4-manifold, and give examples of simply connected manifolds for which it is disconnected. These examples imply that concordance of psc metrics does not imply isotopy of such metrics. This is demonstrated using a modification of the 1-parameter Seiberg-W…
Study shows disconnect between pseudo-Anosov homeomorphisms and hyperbolic 3-manifolds.
problem Disconnection between pseudo-Anosov homeomorphisms and hyperbolic 3-manifolds.
method Examined a family of braids and their associated homeomorphisms and 3-manifolds.
result Found that homeomorphisms converge but 3-manifolds do not, showing disconnect.
We derive an obstruction to representing a homology class of a symplectic 4-manifold by an embedded, possibly disconnected, symplectic surface.
We give examples of compact symplectic manifolds with disconnected contact type boundary in dimension 4n for any n≥1. The example is given by a subset of the tangent bundle of a compact quotient of the complex hyperbolic space endowed with the canonical symplectic form plus a generalized magnetic field and its …
The paper extends the isoperimetric inequality to disconnected regions.
problem Generalizing the isoperimetric inequality to regions that can split area.
method Analyzes the inequality in Euclidean, spherical, and hyperbolic geometries, providing conditions for multiple regions.
result Necessary and sufficient conditions for the inequality to hold for multiple regions in different geometries.
The study proves splitting theorems for manifolds with specific curvature and boundary conditions.
problem Proving splitting theorems for manifolds with specific curvature and boundary conditions.
method Warped product splitting theorem in manifolds with Ricci curvature bounded from below, requiring parabolic and convex boundary.
result Established splitting results for various manifolds with specific curvature and boundary conditions.
Edge augmentation connects disconnected graphs by elevating eigenvalues.
problem Connecting disconnected subgraphs in graphs with zero eigenvalues.
method Elevating zero eigenvalues of graph's spectrum to connect subgraphs.
result The algorithm consistently connects graph components, achieving >50% inter-community edges.
Proof of Hilbert-Smith Conjecture in 2D.
problem Hilbert-Smith Conjecture in 2D
method Proof in dimension two
result Proved Hilbert-Smith Conjecture in 2D
A flow of a low entropy hypersurface in 4D does not split.
problem Preventing splitting of hypersurfaces in 4D with low entropy.
method Level set flow analysis for hypersurfaces in R4 with entropy λ(Σ)≤λ(S2imesR). result The level set flow of a hypersurface in R4 of low entropy never disconnects. Symplectic and Poisson structures proved for information geometry's Frobenius manifold.
problem Connecting disconnected theories in information geometry.
method Proving symplectic and Poisson structures on the Frobenius manifold.
result Established a bridge between Vinberg, Souriau, and Koszul's theories.
Study on stable partitions in convex domains with three phases, finding disconnected phase stability.
problem Stability of partitions in convex domains with multiple phases.
method Careful derivation of the second variation of area, proving existence of stable partitions involving disconnected phases.
result Existence of stable partitions involving a disconnected phase in three phase problem.
New Coxeter groups have unique boundary structures.
problem Understanding boundaries of Coxeter groups.
method Recursive construction and amalgamation of CAT(0) groups.
result Totally disconnected Morse boundaries for new Coxeter groups.
Ensemble of GANs improves performance on disconnected data.
problem Disconnected datasets in computer vision cannot be represented by continuous GANs.
method Construct an optimization problem to relate single GAN, ensemble of GANs, conditional GANs, and Gaussian Mixture GANs.
result Ensemble of GANs outperforms single GANs with fewer parameters.
Let N be a hyperbolic 3-manifold and B a component of the interior of AH(π1(N)), the space of marked hyperbolic 3-manifolds homotopy equivalent to N. We will give topological conditions on N sufficient to give ρ∈Bˉ such that for every small neighborhood V of ρ, V∩B is disconnected. This …
Unbounded convex domains have zero mean curvature on disconnected boundaries.
problem Understanding mean curvature in unbounded convex domains.
method Analyzing mean curvature on disconnected boundary components.
result Mean curvature is zero on disconnected boundary components of unbounded mean convex domains.
In this paper we show that there exists a family of simply connected, symplectic 4-manifolds such that the (Poincare dual of the) canonical class admits both connected and disconnected symplectic representatives. This answers a question raised by Fintushel and Stern.
Study shows augmented deformation space of rational maps is disconnected.
problem Understanding the structure of rational maps with specific dynamics.
method Examined the augmented deformation space of a family of quadratic rational maps.
result The closure of the deformation space in the augmented space is also disconnected.
New method captures multimodal disconnectivity in schizophrenia.
problem Misinterpretation of single modality data in schizophrenia research.
method Gaussian graphical model and modularity-based approach on multimodal data.
result Identifies missing links in schizophrenia's default mode network.
Paper proposes a method to train neural networks incrementally using cloud computing despite disconnections and resource outages.
problem Frequent disconnections and resource outages in cloud computing and local machines hinder deep learning model training.
method Introduces an incremental learning framework that allows continuous training of neural networks even with interruptions.
result Demonstrates that incremental learning can maintain progress and train neural networks effectively despite interruptions.
Two proofs show that removing a loop from a plane circuit splits the plane.
problem Proving the Weak Jordan Theorem about plane circuits.
method Detailed presentation of Thomassen's and Filippov's proofs.
result The complement of any loop in a plane circuit is disconnected.
Expanding on subgroup dimension, this paper covers new dimensions and compactness.
problem Understanding subgroup dimensions and their properties.
method Generalization of dimension datum, study of disconnected subgroups, and isospectral sets.
result New insights into subgroup dimensions and compactness of isospectral sets.
New G_2-manifolds created from quotients of twisted connected sums.
problem Creating new G_2-manifolds with specific topological properties.
method Generalizing Kovalev's twisted connected sums by taking quotients of pieces before gluing.
result Realization of a wider range of topological types of G_2-manifolds.
Finite actions of lattices on manifolds proven for certain groups.
problem Finite actions of lattices on compact manifolds.
method Uses machinery from Brown, Fisher, and Hurtado.
result Finite actions proven for lattices in p-adic and S-arithmetic groups. Study sharp geometric estimates for critical metrics on compact manifolds.
problem Investigating critical metrics of the volume functional on compact manifolds.
method Establishing sharp estimates for mean curvature and area of boundary components.
result Sharp estimates for mean curvature and area of boundary components of critical metrics.
Proposes SCD-split for CP to balance interpretability and efficiency.
problem Difficult interpretation of disconnected subintervals in CP prediction sets.
method Incorporates smoothing operations into CP framework.
result SCD-split balances interval length and subinterval number, theoretically provable.
We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of orientation-reversing geodesics. Restricted to orientable surfaces, this result reduces…
Study on quasi-Einstein manifolds with boundary estimates and inequalities.
problem Understanding the geometry of compact quasi-Einstein manifolds with boundary.
method Sharp boundary estimates and characterization theorems for quasi-Einstein manifolds.
result New geometric inequalities and boundary estimates for quasi-Einstein manifolds.
Wide neural networks ensure disconnected decision regions.
problem Ensuring neural networks produce disconnected decision regions.
method Analyzing conditions for connectedness of decision regions in neural networks with specific activation functions.
result For leaky ReLU and pyramidal structures, wide hidden layers are necessary for disconnected decision regions.
New findings suggest no ensemble averaging for certain black hole observables.
problem Mystery in AdS/CFT correspondence regarding ensemble averaging of black hole amplitudes.
method Exploring sub-threshold observables in D=3 and proving novel results about hyperbolic geometry. result Connected solutions of Einstein's equations with disconnected boundary never contribute to sub-threshold observables.
Study finds Hilbert square of real surfaces can be maximal even when the surface has disconnected real locus.
problem Exploring conditions for maximality of Hilbert square of real surfaces.
method Analyzing Hilbert square of maximal real surfaces and examining specific examples.
result Hilbert square can be maximal even for surfaces with disconnected real locus.
For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five),w…
New examples show nonnegatively curved metrics with same soul but different moduli space components.
problem Finding nonnegatively curved metrics with same soul but different moduli space components.
method Examples of open manifolds with infinitely many metrics of nonnegative sectional curvature.
result Found open manifolds with souls of codimensions 3 and 2, showing different moduli space components.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…
Let a torus T act effectively on a compact connected cooriented contact manifold, and let Psi be the natural momentum map on the symplectization. We prove that, if dim T > 2, the union of the origin with the image of Psi is a convex polyhedral cone, the non-zero level sets of Psi are connected (while the zero level set…
We construct a graph TQFT for the minus flavor of Heegaard Floer homology. Our graph TQFT extends Ozsváth and Szabó's TQFT for closed and connected 3-manifolds, and allows for cobordisms with disconnected ends. As an application, we give an explicit formula for the chain homotopy type of the π1-action on Heegaard Fl…
In this paper, we study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds w…
Let Y be a closed 3-manifold such that all flat SU(2)-connections on Y are non-degenerate. In this article, we prove a Uhlenbeck-type compactness theorem on Y for stable flat SL(2,C) connections satisfying an L2-bound for the real curvature. Combining the compactness theorem and a previous…
This note proves properties of surface-links with trivial components.
problem Properties of surface-links with trivial components.
method Analyzes closed oriented disconnected surfaces and their links.
result For certain surface-links, trivial components result in ribbon links.
Researchers found different G_2 metrics on similar manifolds.
problem Identifying distinct G_2 metrics on manifolds with similar topology.
method Defined U(3)-structures invariants and constructed coboundary for G_2 manifolds.
result Discovered different connected components of G_2 moduli space.
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
A novel density regularizer improves data interpolation on non-simply-connected manifolds.
problem Topological differences between model-defined simply-connected manifolds and dataset's non-simply-connected regions.
method Density regularizer to circumvent low-probability-density regions (holes).
result Consistently better interpolation results on real-world image datasets.
New findings show exotic non-leaves in 4-manifolds with infinitely many ends.
problem Understanding non-leaves in smooth manifolds with low regularity.
method Developed new criteria for non-leaves in C1,0 and C2 regularity. result Found exotic non-leaves in S4 punctured by a Cantor set. Proposes a method to accelerate safe sequential learning using offline data.
problem Limited exploration due to disconnected safe regions and slow task learning.
method Safe transfer sequential learning using Gaussian processes and offline data.
result Enhances global exploration across multiple disjoint safe regions with lower data consumption.
The paper examines properties of self-affine Sierpiński sponges using metric invariants.
problem Investigating properties of self-affine Sierpiński sponges using metric invariants.
method Examined through maximal power law property and perfectly disconnectedness.
result Characterized self-affine Sierpiński sponges by their metric properties.
We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed stable symplectic hypersurface V in a symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders (in addition to insertions on X\V) for stable V. We obtain invariants of the deformation cl…