This paper classifies semi-equivelar maps on torus and Klein bottle with few vertices.
problem Classifying semi-equivelar maps on non-spherical surfaces.
method Analyzing known tilings of the plane to suggest possible types, then classifying maps with few vertices.
result Classification of semi-equivelar maps on torus and Klein bottle with few vertices.
The article improves and extends a heuristic for realizing surfaces with few vertices.
problem Realizing triangulated orientable and non-orientable surfaces with minimal vertices.
method Polyhedral embeddings and immersions of triangulated 2-manifolds with few vertices.
result Obtained numerous symmetric realizations of non-orientable surfaces with minimal vertices.
Characterizes simplicial complexes embedding into spheres with few vertices.
problem Characterizing simplicial complexes that embed into spheres with few vertices.
method Simple characterization using non-face families and analogy with Fáry's theorem.
result Recovery of van Kampen--Flores theorem and Erd\H os--Ko--Rado theorem.
Survey on spatial graphs with few vertices and edges.
problem Topology of spatial graphs with limited vertices and edges.
method Survey and focus on Brunnian θ-graphs.
result Survey reveals insights into spatial graphs.
In this survey on combinatorial properties of triangulated manifolds we discuss various lower bounds on the number of vertices of simplicial and combinatorial manifolds. Moreover, we give a list of all known examples of vertex-minimal triangulations.
Study of maps with -2 Euler characteristic and up to 12 vertices.
problem Enumerating and classifying semi-equivelar maps with specific Euler characteristics.
method Comprehensive enumeration and classification of semi-equivelar maps on a surface with χ=-2, up to 12 vertices.
result Determination of maps' vertex-transitivity and non-transitivity.
P. Arnoux and A. Marin showed that any triangulation of RPn contains more than 2(n+1)(n+2) vertices if n≥3. We construct some natural triangulation of RPn with 2n(n+5)−1 vertices for all n≥3. Previously, it was known that RPn has Z2n-e…
We describe an algorithm for the enumeration of (candidates of) vertex-transitive combinatorial d-manifolds. With an implementation of our algorithm, we determine, up to combinatorial equivalence, all combinatorial manifolds with a vertex-transitive automorphism group on n≤13 vertices. With the exception of act…
We study the localization of a cluster of activated vertices in a graph, from adaptively designed compressive measurements. We propose a hierarchical partitioning of the graph that groups the activated vertices into few partitions, so that a top-down sensing procedure can identify these partitions, and hence the activa…
We explicitly construct small triangulations for a number of well-known 3-dimensional manifolds and give a brief outline of some aspects of the underlying theory of 3-manifolds and its historical development.
New bounds on triangulations of manifolds with non-free fundamental groups.
problem Finding lower bounds on the number of vertices in PL-triangulations of manifolds.
method Using fundamental group structure and Lusternik-Schnirelmann category theory.
result Every PL-triangulation of a d-dimensional manifold with non-free fundamental group has at least 3d+1 vertices. New algorithm finds corrupted vertices in graphs with few queries.
problem Adversarial tampering of graph edges and vertices.
method Active learning algorithm with polynomial query complexity.
result Efficiently recovers corrupted vertices with small query complexity.
When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equat…
We show existence of centrally symmetric maps on surfaces all of whose faces are quadrangles and pentagons for each orientable genus g≥0. We also show existence of centrally symmetric maps on surfaces all of whose faces are hexagons for each orientable genus g=2k−1, k∈N. We enumerate centrally …
PriDE preserves differential privacy in vertically-partitioned datasets.
problem Privacy issues in distributed machine learning with vertically-partitioned data.
method PriDE uses (ε,δ)-distributed differential privacy to ensure privacy while allowing statistical estimation. result PriDE achieves bounded estimation error compared to non-private methods in distributed settings.
The paper finds and visualizes unique geometric polyhedra and tori with few vertices.
problem Finding and visualizing geometric polyhedra and tori with specific vertex configurations.
method Using Schlegel diagrams and geometric realization in 3D and 4D space.
result Identifies and visualizes 12 triangulations of the 2-torus and 12 triangulations of the 2D projective plane.
Paper proves ML-based vertex nomination is consistent and scalable.
problem Ordering non-interesting vertices to highlight interesting ones in graphs.
method Maximum likelihood estimation and vertex nomination scheme.
result ML-based scheme asymptotically matches Bayes optimal scheme performance.
New research shows graph embeddings fail to capture key network properties.
problem Graph embeddings fail to capture salient properties of complex networks.
method Mathematical proof and empirical study of various embedding techniques.
result Any successful graph embedding must have a rank nearly linear in the number of vertices.
Researchers calculate the Ray-Singer Torsion for S1 bundles.
problem Few explicit evaluations of path integrals in higher dimensions.
method Algebraic choice of gauge leading to factorization of path integral.
result Explicit calculation of Ray-Singer Torsion for S1 bundles. New algorithm detects cores in graphs with community structure, improving vertex selection for better clustering.
problem Understanding and detecting core-periphery structures in graphs with community structure.
method Introduces relative centrality to detect cores in graphs with community and core-periphery structures.
result Relative centrality solves bias issues in core detection, leading to better vertex selection and improved clustering performance.
A new method uses matrix sketches for efficient graph clustering in dynamic environments.
problem Efficiently clustering large, dynamic graphs in distributed memory systems.
method Inspired by spectral clustering, the approach uses random dimension-reducing projections to derive matrix sketches.
result The method produces embeddings that yield performant clustering results in a fully-dynamic stochastic block model stream.
A geodesic net with 4 boundary vertices and 25 balanced vertices is constructed.
problem Constructing geodesic nets with specific vertex types and properties.
method Novel approach to increase the number of balanced vertices from 16 to 25.
result First net with four boundary vertices and 25 balanced vertices, including non-symmetric balanced vertices.
New approach connects Finsler geometry's metric and connections.
problem Deriving Finsler geometry's metric and connections from compatibility axioms.
method Compatibility axioms between metric and Finsler connection.
result Metrical formulation of Finsler geometry for field theory.
The article explores symmetric maps on surfaces, focusing on semi-equivelar maps.
problem Identifying and classifying semi-equivelar maps on surfaces with specific Euler characteristics.
method Analyzing automorphisms and symmetry groups of maps on higher genus surfaces.
result There are at least 39 types of semi-equivelar maps on surfaces with Euler characteristic -2m, m ≥ 2, with symmetry groups isomorphic to dihedral or cyclic groups.
We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the eighties by S. Lins for closed 3-manifolds, which …
New schemes improve vertex nomination in stochastic block models.
problem Ordering vertices with unknown labels in a network.
method Canonical sampling and extended spectral nomination schemes.
result Improved precision and scalability of vertex nomination schemes.
We introduce Clique Matrices as an alternative representation of undirected graphs, being a generalisation of the incidence matrix representation. Here we use clique matrices to decompose a graph into a set of possibly overlapping clusters, de ned as well-connected subsets of vertices. The decomposition is based on a s…
String vertices proven in hyperbolic geometry, unique up to transformations.
problem Existence and uniqueness of string vertices in string field theory.
method Homological proof using hyperbolic metrics and geodesic boundaries.
result String vertices are sets of surfaces with systole greater than or equal to L. Solved Dudeney's 100-year-old puzzle about triangle to square dissection.
problem How to dissect an equilateral triangle into the fewest pieces to form a square.
method Reduced the problem to analyzing graph structures representing piece correspondences.
result Proved that four is the minimum number of pieces for an equilateral triangle to square dissection.
Origamis' orbits are non-planar except for a few specific cases.
problem Determining the planarity of origamis' orbits under SL(2,Z) action.
method Analyzing 4-valent graphs from SL(2,Z) action on origamis in H(2).
result Most origamis' orbits are non-planar, with specific exceptions.
First example of geodesic net with 4 boundary vertices, not a tree.
problem Constructing geodesic nets with specific properties.
method Constructing a geodesic net with 4 unbalanced vertices and 16 balanced vertices.
result First example of irreducible geodesic net with 4 boundary vertices, not a tree.
Geodesic nets with three vertices have at most one balanced vertex.
problem Characterizing geodesic nets with specific vertex configurations.
method Analyzing geodesic nets on non-positively curved planes.
result Geodesic nets with three boundary vertices have at most one balanced vertex.
We give a complete enumeration of all combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product S2×S1 and 615 triangulations of the twisted sphere product $S^2_\times_S^1$. All the 3-spheres…
We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in {\it Bulletin of the AMS}, 29(1):77--84, 1993. Its ends in the quotient are asymptotic to one full turn of the …
We prove that any complete surface with constant mean curvature in a homogeneous space E(κ,τ) which is transversal to the vertical Killing vector field is, in fact, a vertical graph. As a consequence we get that any orientable, parabolic, complete, immersed surface with constant mean curvature H in E(κ,τ) (different fr…
A method for matching vertices in large networks using seeds.
problem Matching vertices in large, overlapping networks.
method Identify seeds in local neighborhoods, match induced subgraphs, rank matches.
result Principled approach for large networks, demonstrated through simulations and real data.
Develops a method to disaggregate aerosol optical depth into vertical extinction profiles.
problem Uncertainty in measuring aerosol vertical distributions due to limited observations.
method Bayesian nonparametric Gaussian process modeling using meteorological predictors.
result Model reconstructs realistic extinction profiles with well-calibrated uncertainty, outperforming idealized baselines.
SplitNN-driven Vertical Partitioning enables distributed learning from diverse data sources.
problem Learning from vertically distributed features across institutions.
method A configuration of SplitNN that does not share raw data or model details.
result Flexibility in merging split model outputs and resource efficiency.
Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
problem Classifying 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
method Classifies vertical 3-manifolds as preimages of arcs on the plane for simplified (2,0)-trisection maps.
result Each 6-tuple of vertical 3-manifolds determines the source 4-manifold uniquely up to orientation reversing diffeomorphisms.
Improved bounds on ideal vertices in right-angled hyperbolic polyhedra.
problem Finding bounds on ideal vertices in hyperbolic polyhedra.
method Improved Nikulin's inequality and Nonaka's lower bound.
result Shorter proofs and improved bounds on ideal vertices.
The paper examines vertical curves and fibers in the Heisenberg group, proving properties and constructing counterexamples.
problem Characterizing and measuring vertical curves and fibers in the Heisenberg group.
method Metric analysis of vertical curves and fibers of maps from the Heisenberg group to the plane.
result Vertical curves in the Heisenberg group can have Hausdorff dimensions strictly larger or smaller than 2, unlike intrinsic Lipschitz graphs.
In this paper we study some problems related to a vertical Liouville distribution (called vertical Liouville-Hamilton distribution) on the cotangent bundle of a Cartan space. We study the existence of some linear connections of Vrănceanu type on Cartan spaces related to some foliated structures. Also, we identify a cer…
Paper shows minimum 10 vertices for hyperbolic origami 2-torus.
problem Finding minimum vertices for hyperbolic origami 2-torus.
method Geodesic triangulation and isometric polyhedral embedding.
result 10 vertices are the minimum required for a hyperbolic origami 2-torus.
New Lie group structure on vertical bisections of Lie groupoids.
problem Constructing Lie group structure on vertical bisections of Lie groupoids.
method Construct Lie group structure on the group of vertical bisections of a regular Lie groupoid, identify Lie algebra, discuss regularity properties.
result Established Lie theoretic properties of vertical bisections of Lie groupoids over non-compact bases.
Only vertical planes are asymptotic to other planes in 3D space.
problem Characterizing asymptotic planes in 3D space.
method Proof of uniqueness for complete translators with finite topology.
result Vertical planes are the only asymptotic planes in 3D space.
Asynchronous federated learning for vertically partitioned data improves efficiency and privacy.
problem Efficiently train models on vertically partitioned data without a trusted third party.
method Proposed AFSGD-VP and its SVRG and SAGA variants for asynchronous federated learning.
result AFSGD-VP and its variants achieve higher efficiency than synchronous algorithms.
Simplified approach to categorify knot polynomials using matrix models.
problem Evaluate quantum dimensions of knot polynomials in a simplified Chern-Simons theory.
method Use matrix model to reformulate quantum dimensions in terms of fat graphs and derive recursions.
result Explicit expressions for quantum dimensions of virtual knots, including negative ones.
An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic in a h…