Study shows how many crossings arise in curves on surfaces.
problem Understanding crossings in curves on surfaces.
method Proved a minimum crossing number growth rate for m curves. result Effective bounds with optimal growth rates on every orientable surface.
Complex surfaces show non-simply connected diffeomorphism groups with non-homotopic loops.
problem Complex surfaces with non-simply connected diffeomorphism groups and non-homotopic loops.
method Exhibited examples of complex surfaces.
result Diffeomorphism groups of complex surfaces are not simply-connected and contain non-homotopic loops.
33 curves on a 3-genus surface, all intersecting at most once.
problem Finding a saturated system of curves on a surface of genus 3.
method Constructing 33 essential curves pairwise non-homotopic and intersecting at most once.
result The constructed system is saturated, not properly contained in any other system.
We prove that on a closed surface of genus g, the cardinality of a set of simple closed curves in which any two are non-homotopic and intersect at most once is ≲g2log(g). This bound matches the largest known constructions to within a logarithmic factor. The proof uses a probabilistic argument in graph th…
In this article we show that for any given Riemann surface Σ of genus g, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to Σ in terms of the genus and the combined extremal lengths on Σ of (g−1) disjoint, non-homotopic, simple closed comp…
Let S be an n-punctured sphere, with n≥3. We prove that (3n) is the maximum size of a family of pairwise non-homotopic simple arcs on S joining a fixed pair of distinct punctures of S and pairwise intersecting at most twice. On the way, we show that a square annular diagram A has a corner on …
We prove that on a punctured oriented surface with Euler characteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a cl…
Let Sg denote the genus g closed orientable surface. For k∈N, a k-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than k times. Juvan-Malnič-Mohar \cite{Ju-Mal-Mo} showed that there exists a k-system on Sg whose size is on the order o…
New surfaces in a 4-manifold are found that are not isotopic but homotopic.
problem Finding non-isotopic surfaces that are homotopic in a specific 4-manifold.
method Applying the Norman trick to a fixed immersed surface, using non-homotopic tubing arcs.
result Infinitely many embedded tori in T4#(S2imesS2) are non-isotopic but homotopic. New bounds on curves on torus with few intersections.
problem Finding the maximum number of non-homotopic curves on a torus with limited intersections.
method Analyzing the maximum size of sets of curves with at most k intersections, using combinatorial optimization techniques.
result The maximum size of such a set is k+6 for all k, and k+4 for large k.
New groups formed by twists on surfaces are free.
problem Understanding groups formed by Dehn twists on surfaces.
method Finding specific fillings that generate free groups.
result Groups generated by twists are free for certain configurations.
The study counts 23 maximal 1-systems on a torus with 2 punctures.
problem Counting maximal 1-systems on a torus with punctures.
method Defined and analyzed 1-systems, generalized results to surfaces with boundary.
result There are exactly 23 maximal 1-systems on a torus with 2 punctures.
Maximizes filling systems on surfaces with given boundary components.
problem Finding the maximum size of filling systems on surfaces with specific boundary conditions.
method Analyzing the structure of filling systems and their complements.
result The maximum size of a filling system on a surface of genus g with 1 ≤ b ≤ 2g-2 boundary components is 2g + b - 1.
Let M be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let cl(Σiness(1)) be the closure of the union of all singular knots in M with exactly one ordinary double point and such that the two resolutions repres…
The purpose of this article is two-fold: We first give a more elementary proof of a recent theorem of Korkmaz, Monden, and the author, which states that the commutator length of the n-th power of a Dehn twist along a boundary parallel curve on a surface with boundary S of genus g at least two is the floor of (|n|+3)/2 …
Paper detects anomalous edges in social networks using edge exchangeability.
problem Detecting anomalous edges in directed social networks.
method Exploits edge exchangeability and uses conformal prediction theory.
result Proposed anomaly detector has a guaranteed upper bound for false positives.
We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometri…
OL4EL optimizes edge learning on resource-constrained servers.
problem Resource constraints on edge servers hinder effective distributed machine learning.
method Online Learning for EL (OL4EL) framework using budget-limited multi-armed bandit model.
result OL4EL significantly improves learning performance while conserving resources.
New GPs model edge functions on complex networks, capturing divergence and curl.
problem Modeling flow data on networks with independent learning of Hodge components.
method Developed Hodge-compositional edge GPs using Hodge decomposition.
result Hodge-compositional edge GPs can represent any edge function and capture flow relevance.
In L^3, cuspidal edges can have bounded mean curvature under specific conditions.
problem Understanding cuspidal edges with bounded mean curvature in Lorentz-Minkowski 3-space.
method Investigated cuspidal edges and generalized cuspidal edges, analyzing their singular points and principal curvatures.
result Cuspidal edges with bounded mean curvature in L^3 occur only when the singular set is a light-like curve.
Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field ν is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of the cuspidal edge (as a regul…
Bundling of graph edges (node-to-node connections) is a common technique to enhance visibility of overall trends in the edge structure of a large graph layout, and a large variety of bundling algorithms have been proposed. However, with strong bundling, it becomes hard to identify origins and destinations of individual…
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.
We prove several results about chordal graphs and weighted chordal graphs by focusing on exposed edges. These are edges that are properly contained in a single maximal complete subgraph. This leads to a characterization of chordal graphs via deletions of a sequence of exposed edges from a complete graph. Most interesti…
Under what conditions is an edge present in a social network at time t likely to decay or persist by some future time t + Delta(t)? Previous research addressing this issue suggests that the network range of the people involved in the edge, the extent to which the edge is embedded in a surrounding structure, and the age…
New maximally linkless graphs found with fewer edges.
problem Finding graphs without any links in 3D space.
method Demonstrated new maximally linkless graphs with improved edge count.
result Found maximally linkless graphs with m≤514n edges. In the emerging advancement in the branch of autonomous robotics, the ability of a robot to efficiently localize and construct maps of its surrounding is crucial. This paper deals with utilizing thermal-infrared cameras, as opposed to conventional cameras as the primary sensor to capture images of the robot's surroundi…
Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
Method certifies edge predictions with cloud-level reliability.
problem Ensuring reliability of edge intelligence models.
method Conformal alignment-based cascading mechanism.
result Certifies conditional coverage with user control over risk level.
Defense against user shilling attacks in collaborative filtering using edge reweighting.
problem Vulnerability of collaborative filtering to profile injection attacks.
method Adversarial robustness based edge reweighting to attenuate non-robust edges.
result Effective defense against various types of attacks demonstrated through experiments.
A hybrid neural network optimizes AI deployment on edge and cloud for energy efficiency.
problem Energy and resource constraints in edge devices for deep learning models.
method Conditionally deep hybrid neural network with quantized layers at edge and full-precision layers at cloud.
result Early classification at the edge reduces energy consumption by 5.5x on CIFAR-10 dataset.
This work surveys attacks and defenses on edge neural networks.
problem Security challenges of edge neural networks due to their compute and memory intensity, data-independence, and privacy risks.
method Taxonomy of attacks and defenses on edge-deployed neural networks.
result New security considerations and approaches are needed for edge DNNs.
CoMGNN models heterogeneous graphs with evolving nodes and edges.
problem Modeling complex, evolving graphs with diverse information.
method Meta graph attention on co-evolving heterogeneous graphs.
result Significant improvement over state-of-the-art methods.
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.
This paper proposes a method to learn graph representations by partitioning edges into communities.
problem Graph neural networks ignore how edges are formed, leading to suboptimal representation learning.
method Introduces a generative model to partition edges into community-specific weighted edges, then uses these for GNN-based inference and classification.
result The method learns discriminative representations for both node-level and graph-level classification tasks.
New research finds six bipartite intrinsically knotted graphs with 23 edges.
problem Identifying intrinsically knotted bipartite graphs with 23 edges.
method Analyzing embeddings and graph minors to find minimal intrinsically knotted graphs.
result No minor minimal intrinsically knotted bipartite graph exists with 23 edges.
Paper presents a lightweight, unobtrusive method to protect edge device data privacy.
problem Protecting inference data privacy in IoT edge devices with limited compute power.
method A lightweight neural network at edge devices to obfuscate inference data without indicating obfuscation.
result Effectively protects inference data confidentiality while preserving backend accuracy.
Graph neural networks improve with edge similarity constraints in RNA structure analysis.
problem Lack of edge similarity constraints in graph neural networks.
method Introduced a graph neural network layer that leverages prior information about edge similarities.
result Edge similarity constraints do not enhance performance in graph neural networks.
Johnson, Kidwell, and Michael showed that intrinsically knotted graphs have at least 21 edges. Also it is known that K7 and the thirteen graphs obtained from K7 by rY moves are intrinsically knotted graphs with 21 edges. We prove that these 14 graphs are the only intrinsically knotted graphs with 21 edges.
This paper optimizes AI inference on edge devices with reduced communication and computation costs.
problem Efficiently performing AI inference on resource-constrained edge devices with reduced communication and computation costs.
method A three-step framework for effective inference: model split point selection, communication-aware model compression, and task-oriented encoding of intermediate features.
result Our proposed framework achieves a better trade-off and significantly reduces inference latency compared to baseline methods.
A framework for real-time edge intelligence using federated meta-learning.
problem Real-time intelligent decisions at edge devices with limited resources and data.
method Federated meta-learning approach for rapid adaptation of learned models.
result Effective framework demonstrated on various datasets.
On-device federated learning updates edge models by exchanging trained results.
problem Limited training data at edge devices due to model drift.
method OS-ELM for sequential training and autoencoder for anomaly detection, combined with federated learning.
result The proposed approach produces a merged model as accurately as traditional methods with lower costs.
For non-homotopic maps u,v∈C∞(M,N) between closed Riemannian manifolds, we consider the smallest energy level γp(u,v) for which there exist paths ut∈W1,p(M,N) connecting u0=u to u1=v with ∥dut∥Lpp≤γp(u,v). When u and v are (k−2)-homotopic, work of Hang and Lin shows t…
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
Study on planar graph braid groups' second homology.
problem Characterize the second homology of planar graph braid groups.
method Analyzing configuration spaces of planar graphs under specific operations.
result The second homology is generated by three specific graphs.
Previous work in network analysis has focused on modeling the mixed-memberships of node roles in the graph, but not the roles of edges. We introduce the edge role discovery problem and present a generalizable framework for learning and extracting edge roles from arbitrary graphs automatically. Furthermore, while existi…
The paper provides conditions for realizing graphs and polytopes with specified edge lengths.
problem Proving the existence of planar embeddings or polyhedra with specified edge lengths.
method Practical sufficient conditions and software verification for non-self-intersecting perturbations of initial realizations.
result Existence of planar embeddings and polyhedra with specified edge lengths.
Edge features contain important information about graphs. However, current state-of-the-art neural network models designed for graph learning, e.g. graph convolutional networks (GCN) and graph attention networks (GAT), adequately utilize edge features, especially multi-dimensional edge features. In this paper, we build…