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.
We study the stability of partitions involving two or more phases in convex domains under the assumption of at most two-phase contact, thus excluding in particular triple junctions. We present a detailed derivation of the second variation formula with particular attention to the boundary terms, and then study the sign …
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.
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.
This thesis revises phase space concepts in physics, incorporating physical dimensions.
problem Disconnection between theoretical models and units of measurement.
method Introducing unit-free manifolds and dimensioned algebraic structures.
result Reinterpretation of Jacobi manifolds as unit-free analogues of Poisson manifolds.
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.
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. 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.
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.
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.
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.
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 …
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.
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…
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…
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.
M-theory can be defined on closed manifolds as well as on manifolds with boundary. As an extension, we show that manifolds with corners appear naturally in M-theory. We illustrate this with four situations: The lift to bounding twelve dimensions of M-theory on Anti de Sitter spaces, ten-dimensional heterotic string the…
Adversarial dropout improves RNNs' performance on sequential data tasks.
problem Improving generalization performance of RNNs for sequential data.
method Adversarial dropout technique for RNNs using intentionally generated dropout masks.
result Adversarial dropout improves RNNs' effectiveness on sequential tasks.
This work reveals a new scaling law for optimal design of multirotor aerial vehicles.
problem Designing optimal configurations for fully-actuated multirotor aerial vehicles.
method Formulated on the product manifold of Projective Lines \RP^2^N, minimizing a coordinate-invariant Log-Volume isotropy metric.
result The topology of the global optima is governed by the symmetry of the chassis, leading to a N-5 Scaling Law.
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.
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 …
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…
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.
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.
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.
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
We prove that under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines which have disconnected realization spaces.
We generalize Hagopian's theorem characterizing solenoids to higher dimensions by showing that any homogeneous continuum admitting a fiber bundle projection onto a torus with totally disconnected fibers admits a compatible abelian topological group structure. The higher dimensional exponent group is then introduced.
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.
Real world systems typically feature a variety of different dependency types and topologies that complicate model selection for probabilistic graphical models. We introduce the ensemble-of-forests model, a generalization of the ensemble-of-trees model. Our model enables structure learning of Markov random fields (MRF) …
Improved pruning method using iterative sensitivity ranking before training.
problem Improper sensitivity propagation in existing pruning methods.
method Iterative application of SNIP criterion before training.
result State-of-the-art sparsity-performance trade-offs achieved.
The paper characterizes potential functions whose level sets are orbits in mechanical systems.
problem Characterizing smooth potential energy functions on the plane with specific level set properties.
method Analyzing inverse curvature flow and properties of level sets.
result Analytic or functions with totally path-disconnected critical sets must be radial, while every compact convex set is a critical set of a Levi potential.
Study on stable translating solitons without complex cycles.
problem Understanding the topology of stable translating solitons.
method Analyzing complete f-stable translating solitons to show absence of specific cycles. result Two-dimensional complete f-stable translating solitons have genus zero. 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.
We construct maps on hat Heegaard Floer homology for cobordisms decorated with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our construction uses Juhász's sutured Floer TQFT. We compute the maps for several elementary graph cobordisms. As an application, we compute the action of the fundamental …
We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples found in previous work by the authors from being Zariski pairs. Robustness is show…
We show that tree almost automorphism groups, including Neretin groups, satisfy the analogue of the F∞-finiteness condition in the world of totally disconnected groups: They possess a cellular action on a contractible cellular complex such that the stabilizers are open and compact and the restriction of the act…
We apply topological methods to study the smallest non-zero number λ1 in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set {S∈M2:λ1(S)>1/4} is unbounded and disconnects the moduli space M2.
HiSS sampling overcomes local mode traps in rugged discrete spaces.
problem Sampling multimodal discrete distributions with gradient-based methods.
method Integrates Metropolis-within-Gibbs framework with logistic convolution.
result HiSS outperforms alternatives on various tasks, including Ising models and binary neural networks.
Surface-links with specific groups are always trivial.
problem Understanding when surface-links are trivial.
method Analyzing surface-links with meridian-based free fundamental groups.
result Disconnected surface-links with meridian-based free fundamental groups are trivial.