Approximates cycles in planar and bounded-genus graphs.
problem Finding many disjoint cycles in planar and bounded-genus graphs.
method Constant-factor approximation algorithms for vertex-disjoint and edge-disjoint cycles.
result First algorithms for vertex-disjoint paths in fully planar and bounded-genus instances.
New algorithm for learning causal structures with disjoint cycles in linear non-Gaussian models.
problem Learning causal structures with cycles in linear non-Gaussian models.
method Characterizing when graphs determine the same model, using quadratic and cubic polynomial relations, and a strategy of decorrelating cycles and multivariate regression.
result Consistent and computationally efficient algorithm for learning causal structures with disjoint cycles.
Lin-Lu-Yau introduced an interesting notion of Ricci curvature for graphs and obtained a complete characterization for all Ricci-flat graphs with girth at least five [1]. In this paper, we propose a concrete approach to construct an infinite family of distinct Ricci-flat graphs of girth four with edge-disjoint 4-cycles…
The paper explores linked cycles in graphs and their properties.
problem Understanding the structure of linked cycles in graphs.
method Analyzing the set of all pairs of disjoint cycles in graphs and showing conditions for minimally linked sets.
result A minimally linked set of cycles in a complete graph Kp+q has at most eighteen elements. New proof shows no flat embedding for Petersen family graphs.
problem Proving Petersen family graphs have no flat embeddings.
method Applying Böhme's Lemma and the Jordan-Brouwer Separation Theorem.
result Every Petersen family graph has no flat embedding.
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.
A contractible simplicial complex is constructed that parametrizes different ways of representing a fixed one-dimensional homology class in a closed orientable surface by isotopy classes of systems of disjoint oriented simple closed curves. This is a variant on an earlier construction of Bestvina-Bux-Margalit.
We extend the edge version of the classical Menger's Theorem for undirected graphs to n-dimensional simplicial complexes with chains over the field F2. The classical Menger's Theorem states that two different vertices in an undirected graph can be connected by k pairwise edge-disjoint paths if, and only…
Hardness proven for embedding simplicial complexes in R^d, especially for k-dimensional ones.
problem Recognizing almost embeddability of k-dimensional complexes in R^d.
method NP-hardness proof using configuration spaces and preimage cycle properties.
result Embedding obstruction is incomplete for k-dimensional complexes in R^d.
A book representation of a graph is a particular way of embedding a graph in three dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to K6, t…
Characterizes weakly linked pairs of complete graphs in 3D space.
problem Identifying pairs of complete graphs that are weakly linked.
method Algebraic characterisation and geometric analysis of linking cycles.
result Characterization of weakly linked pairs of complete graphs.
Solves a triangulation problem by showing minimum tetrahedra equals minimum integral 3-chain.
problem Finding the minimum number of tetrahedra to extend a triangulation of a 2-sphere to a 3-ball.
method Relates the minimum number of tetrahedra to the minimum integral 3-chain norm, proving them equal and showing how to achieve the minimum.
result The minimum number of tetrahedra needed to extend a triangulation of a 2-sphere to a 3-ball equals the minimum integral 3-chain norm.
Analyzes how financial network dependencies can lead to multiple equilibrium outcomes and optimal bailout strategies.
problem Multiple equilibrium outcomes in financial networks due to dependency cycles.
method Characterized necessary and sufficient conditions for bank solvency, and provided upper bounds on optimal bailout payments.
result Minimum bailout payments needed to ensure systemic solvency and prevent cascading defaults.
The paper explores winding numbers of almost embeddings of a 4-vertex graph in the plane.
problem Understanding the winding numbers of almost embeddings of a 4-vertex graph in the plane.
method Constructing examples to show the only relation between the winding numbers of cycles in the graph.
result The sum of winding numbers is odd, and this is the only relation between them.
The paper studies the structure of a specific homology group related to mapping class groups.
problem Understanding the structure of a specific homology group of the Johnson kernel.
method Constructing and analyzing abelian cycles to describe the module structure.
result Described the structure of the subgroup of the homology group generated by simplest abelian cycles and found relations between them.
We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free geometric presentation: A real arrangement L, whose graph of multiple points is a union…
Study on second homology group of genus 3 hyperelliptic Torelli group.
problem Understanding the structure of second homology group of genus 3 hyperelliptic Torelli group.
method Analyzing abelian cycles associated with disjoint separating curves and their algebraic properties.
result Simple abelian cycles are linearly independent in the second homology group.
Current multi-reference style transfer models for Text-to-Speech (TTS) perform sub-optimally on disjoints datasets, where one dataset contains only a single style class for one of the style dimensions. These models generally fail to produce style transfer for the dimension that is underrepresented in the dataset. In th…
A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators x1,...,xn and the cyclic relations: xikxik−1...xi1=xik−1...xi1xik=...=xi1xik...xi2 with no conjugations on the generators. We have alre…
Study abelian cycles in Torelli group homology, proving new results in stable rational homology.
problem Understanding the structure of Torelli group homology and its quotients.
method Analyzing the Johnson homomorphism and its induced map on rational homology groups.
result Proves new results about the stable rational homology of Torelli groups and their quotients.
Study on linking numbers in random book embeddings of complete graphs.
problem Distribution and mean of linking numbers in random book embeddings of complete graphs.
method Analyzes a family of two-component links arising from random embeddings of complete graphs, using Eulerian numbers and linear growth in mean linking number.
result Mean of squared linking number over all random embeddings is $ rac{i}{6}$, where i is the number of interior edges. The paper provides a converse to linking theorems for graphs in 3-space and higher dimensions.
problem Linking properties of graphs in 3-space and higher dimensions.
method Proves a converse to specific linking theorems for graphs in 3-space and higher dimensions.
result Proves a higher-dimensional analogue of a converse to a lemma by Segal-Spież.
For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of …
The paper extends Johnson's result on Torelli group homology.
problem Understanding the homology of the Torelli group and its subgroups.
method Analyzing the pushforward homomorphism on higher homology groups induced by Dehn twists.
result The pushforward homomorphism is injective for certain subgroups of higher homology groups.
Study on torsion in homology of Torelli group for surfaces.
problem Torsion in homology of Torelli group for surfaces.
method Analysis of abelian cycles and Dehn twists.
result Subgroup of homology generated by abelian cycles is a finite-dimensional Z/2Z-vector space for k=2 and g≥4. We present a short exposition of the following results by S. Parsa. Let L be a graph such that the join L∗{1,2,3} (i.e. the union of three cones over L along their common bases) piecewise linearly (PL) embeds into R4. Then L admits a PL embedding into R3 such that any two disjoint cycles…
A graph's winding numbers around two non-adjacent vertices differ by ±1.
problem Understanding the winding numbers of a specific graph configuration in the plane.
method Analyzing continuous maps from a graph to the plane, focusing on the winding numbers of specific cycles.
result The difference in winding numbers of a cycle around two non-adjacent vertices is ±1.
Study shows stability of tangent bundle through conifold transitions.
problem Stability of tangent bundle through conifold transitions.
method Hermitian-Yang-Mills metric and conformally balanced metrics.
result Tangent bundle T1,0Xt admits a Hermitian-Yang-Mills metric Ht near vanishing cycles of Xt. A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked if and only if it is not outer-planar. A graph is outer-flat if it can be embed…
We study Lagrangian embeddings of a class of two-dimensional cell complexes Lp,q into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type p21(pq−1,1) (Wahl singularities). We show that …
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…
A Heegaard splitting of a closed, orientable three-manifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the disjoint curve proper…
The paper defines plat closures for spherical braids and shows links in RP3 can be realized this way.
problem Defining and analyzing plat closures for spherical braids in RP3. method Defining plat closures, associating residual permutations, and presenting moves on spherical braids.
result The number of components of the plat closure link of a spherical braid is equal to the number of disjoint cycles in its residual permutation.
We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note …
We present a new property, the Disjoint Path Concordances Property, of an ENR homology manifold X which precisely characterizes when X times R has the Disjoint Disks Property. As a consequence, X times R is a manifold if and only if X is resolvable and it possesses this Disjoint Path Concordances Property.
A new classification method using disjoint centroids and normalized distance.
problem Improving classification accuracy and feature selection.
method Nearest disjoint centroid classifier with normalized distance.
result Our method outperforms other classifiers in terms of misclassification rates and feature usage.
Odd crossing numbers and even rotation numbers for cycles in plane immersions.
problem Analyzing crossing and rotation numbers of cycles in plane immersions of graphs.
method Generic immersions and Legendrian embeddings of graphs, focusing on cycles of specific lengths.
result Sum of rotation numbers of all 5-cycles is even, and sum of crossing numbers is odd.
Proves inequality for 1-dimensional cycles.
problem Proving the Parametric Coarea Inequality for 1-cycles.
method Analytical proof based on conjecture by Guth and Liokumovich.
result Proved the Parametric Coarea Inequality for 1-cycles.
Two triples of triangles having pairwise disjoint outlines in 3-space are called combinatorially isotopic if one triple can be obtained from the other by a continuous motion during which the outlines of the triangles remain pairwise disjoint. We conjecture that it can be algorithmically checked if an (ordered or unorde…
This work introduces novel methods to identify and compare cycles across topological objects.
problem Identifying and comparing topological features, particularly cycles, across different topological objects.
method Two complementary approaches: dendrogram-based merge-tree algorithms and Stratified Gradient Sampling.
result Transformed cycle matching into hierarchical clustering and topological optimization framework.
This article introduces proximal planar vortex 1-cycles, resembling the structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867 and recent work on the proximity of sets that overlap either spatially or descriptively. Vortex cycles resemble Thomson's model of a vortex atom, inspired by P.G. Tait's …
Uniform bounds found for Sierpinski carpet hyperbolic components.
problem Bounding hyperbolic components of Sierpinski carpet type.
method Establishing uniform a priori bounds and analyzing quadratic-like restrictions.
result Sierpinski carpet hyperbolic components of disjoint type are bounded.
This paper identifies the unique efficient cycle for most hyperbolic manifolds but not for the figure-8 knot complement.
problem Identifying the unique efficient cycle for hyperbolic manifolds.
method Analyzing the limit of fundamental cycles and their ℓ1-norm convergence. result The uniqueness of the efficient cycle is proven for most hyperbolic manifolds but not for the figure-8 knot complement.
Study Agol cycles for pseudo-Anosov 3-braids.
problem Conditions for equivalent Agol cycles of pseudo-Anosov 3-braids.
method Investigate necessary and sufficient conditions.
result Necessary and sufficient conditions for equivalent Agol cycles of pseudo-Anosov 3-braids.
Study shows credit expansion in mortgage markets influenced U.S. business cycle.
problem Lack of causal evidence in cross-country business cycle studies.
method Unique research design combining cross-metropolitan U.S. data.
result Credit expansion caused stronger booms and busts in house-related industries.
Researchers compute Connes-Chamseddine cycle on 6D manifolds using noncommutative integral.
problem Computing the Connes-Chamseddine cycle for 6D manifolds.
method Using noncommutative integral on 6D manifolds, they compute the cycle.
result The Connes-Chamseddine cycle on 6D manifolds is computed.
Gauss diagrams' properties can change with Hamiltonian cycle choice.
problem The impact of Hamiltonian cycle choice on Gauss diagrams.
method Examined realizable and unrealizable Gauss diagrams, and proved preservation of realizability under certain Hamiltonian cycle changes.
result Properties of Gauss diagrams can vary with Hamiltonian cycle choice.
Proximal algorithms applied to current deformation into cycles.
problem Deformation of de Rham currents into cycles.
method Proximal algorithms, total variation denoising for differential forms.
result Calibrated cycles constructed in calibrated manifolds.