The study models commuting distances based on income and finds a power-law distribution.
problem Understanding the relation between income and commuting distances.
method Data from Denmark, UK, and US; power-law distribution; alternative job search model.
result The commuting distance distribution decays as 1/r^3 and is independent of job quality.
There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into ac…
Estimates manifold distances using graph Laplacian, proving consistency.
problem Estimating distances in compact Riemannian manifolds.
method Graph Laplacian estimates of the Laplace-Beltrami operator, bounding errors.
result Proof of consistency for manifold distances.
Survey of distance computations in noncommutative geometry, linking to optimal transport.
problem Computing distances in noncommutative geometry.
method Review of explicit computations in various noncommutative geometries.
result Connes distance as a noncommutative version of Monge-Kantorovich metric.
We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savaré, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativi…
New approach predicts commuters' flow with 90.4% accuracy.
problem Limited ability to predict and reconstruct commuters' networks.
method Machine learning and 22 urban indicators.
result Predictions with 90.4% accuracy and 77.6% variance explained.
Unified proof of knot unknotting bounds using Ma-Qiu index.
problem Finding bounds on the number of moves to unknot knots.
method Using the Ma-Qiu index to bound presentation distances and Gordian distances.
result Unified proof of various unknotting number bounds.
Graph curvature measured by inverse resistance distance.
problem Defining and analyzing curvature in graphs.
method Defining curvature via inverse resistance distance and proving properties.
result Graphs with positive curvature have controlled diameter and spectral properties.
We propose a novel method to embed a functional magnetic resonance imaging (fMRI) dataset in a low-dimensional space. The embedding optimally preserves the local functional coupling between fMRI time series and provides a low-dimensional coordinate system for detecting activated voxels. To compute the embedding, we bui…
In this paper we address the problem of understanding the success of algorithms that organize patches according to graph-based metrics. Algorithms that analyze patches extracted from images or time series have led to state-of-the art techniques for classification, denoising, and the study of nonlinear dynamics. The mai…
Study extends biholomorphisms between convex domains in complex space without boundary constraints.
problem Extending biholomorphisms between convex domains without boundary regularity.
method Combining coarse geometry techniques with dynamical properties of maps in Gromov hyperbolic spaces.
result Proves extensions for biholomorphisms and quasi-isometries between convex domains.
New distance metric for neural architecture search reduces search space complexity.
problem Reducing the complexity of neural architecture search.
method Fisher task distance for measuring task similarity and online neural architecture search.
result Reduced search space complexity for task-specific architectures.
A new method for functional data clustering using varying smoothing parameters.
problem Determining dissimilarity between subjects in functional data.
method Measuring dissimilarity based on varying curve estimates with commutation of smoothing parameters pair-by-pair.
result The method effectively clusters subjects and has practical advantages.
Study on geodesic distances on SE(3)/SO(2) in machine learning.
problem Investigating the efficiency of computationally efficient sections in selecting geodesic distances.
method Analyzing geodesic distances on reductive homogeneous spaces, proving the efficiency of minimal distance sections.
result Minimal distance sections are not always geodesic minimizers, but minimal horizontal geodesics are.
Let F be Cayley's ruled cubic surface in a projective three-space over any commutative field K. We determine all collineations fixing F, as a set, and all cubic forms defining F. For both problems the cases ∣K∣=2,3 turn out to be exceptional. On the other hand, if ∣K∣≥4 then the set of simple points of …
New algebraic structures on manifolds generalize supergeometry concepts.
problem Developing algebraic structures for non-commutative manifolds.
method Introducing ρ-commutative manifolds, Q-manifolds, and modular classes. result Generalized modular classes for non-commutative spaces.
Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We …
Doodles link to commutator identities in a 2-sphere.
problem Understanding commutator identities in free groups via doodles.
method Analyzing doodles with proper noose systems and establishing bijections.
result A bijection between doodles and commutator identities.
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
Examining singularities of commuting vector fields on submanifolds.
problem Understanding singularities of commuting vector fields on submanifolds.
method Analyzing real submanifolds within Kähler manifolds.
result Characterized singularities of commuting vector fields.
The log3 Theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3-space is moved a distance at least log3 by one of the non-commuting isometries ξ or η provided that ξ and η generate a torsion-free, discrete group which is not co-compact and contains no parabolic. This theorem l…
Two commutators generate mapping class groups of surfaces with genus ≥5.
problem Generating mapping class groups efficiently.
method Analyzing commutators in mapping class groups of surfaces.
result Mapping class groups of surfaces with genus ≥5 are generated by two commutators.
Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.
problem Understanding stable commutator length on infinite-type surfaces.
method Analyzing mapping class groups of infinite-type surfaces, showing continuity and openness of commutator subgroups.
result Stable commutator length defines a continuous function on commutator subgroups of infinite-type mapping class groups.
New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
Study on commutator subgroups of welded braid groups, proving their finiteness and perfection.
problem Investigating the structure of commutator subgroups in welded braid groups.
method Proved finiteness and Hopfian property, showed perfection for n≥5, computed finite presentations. result Commutator subgroups of welded braid groups are finitely generated, Hopfian, and perfect for n≥5. Formulae for non-symmetric connections derived from covariant derivatives.
problem Deriving commutation formulae for non-symmetric affine connections.
method Covariant derivatives of tensors with respect to symmetric and non-symmetric affine connections.
result Formulae for non-symmetric connections derived from covariant derivatives.
The paper studies Riemannian metrics on Lie groups with specific commutator subgroups.
problem Investigating Riemannian metrics on Lie groups with commutator subgroups of dimensions 1 and 2.
method Explicitly provided Levi-Civita connection, sectional curvature, and Ricci curvature; computed necessary and sufficient conditions for Ricci solitons; characterized Ricci solitons on Lie groups with one-dimensional commutator subgroups; examined indecomposable Lie groups with two-dimensional commutator subgroups.
result Characterization of all Ricci solitons on Lie groups with one-dimensional commutator subgroups and examination of indecomposable Lie groups with two-dimensional commutator subgroups.
Analytic torsion form constructed for non-commutative spaces.
problem Constructing analytic torsion form on non-commutative spaces.
method Using fiber bundles, flat vector bundles, and crossed product algebras, we define a non-commutative deRham differential form.
result The constructed torsion form appears in a transgression formula and leads to an index formula.
Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.
problem Conditions for homotopy commutativity in quasitoric manifolds.
method Analyzing characteristic matrices and polytope structures.
result Homotopy commutativity is determined by specific polytope and matrix conditions.
Study introduces dynamical ideals for non-commutative rings and classifies knots and links.
problem Classifying surface knots and links in smooth 4-manifolds.
method Introduced dynamical analog of prime ideals for non-commutative rings and proved a factorization theorem.
result Classified surface knots and links in smooth 4-manifolds.
Covariance is shown as a commutator in random variable calculus.
problem Expressing covariance as a commutator of operators.
method Demonstrated through commutator identities involving expectations and products of functions.
result Revealed the underlying Lie algebraic structure in efficient influence curve calculus.
Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.
problem Understanding the structure of homeomorphisms in Euclidean space.
method Proving every orientation-preserving homeomorphism can be written as a commutator of two such homeomorphisms.
result Every orientation-preserving homeomorphism of Euclidean space is a commutator of two homeomorphisms.
The study of commutator subgroups of virtual and welded braid groups.
problem Characterizing and generating commutator subgroups of virtual and welded braid groups.
method Obtained generators and relations for commutator subgroups, proved finiteness and perfection.
result Finitely generated commutator subgroups for VBn and WBn for specific n values. We show that solutions of Thurston equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston equation and a commutative ring associated to triangulated 3-manifolds.
Study classifies hypersurfaces in complex hyperbolic spaces with specific operator conditions.
problem Classifying real hypersurfaces with commuting structure Jacobi operators.
method Used simultaneous diagonalization for commuting symmetric operators to classify hypersurfaces.
result Complete classification of real hypersurfaces with commuting condition.
New bounds on twist commutators for separating curves on surfaces.
problem Estimating stable commutator lengths of Dehn twists.
method Using O(1/g) bounds for twists along separating curves. result Upper bounds on stable commutator lengths are O(1/g). The braid group's commutator subgroup is generated by two elements for n ≥ 7.
problem Generating the smallest possible generating sets for the commutator subgroup of braid groups.
method Analyzing specific cases of braid groups (n=4, 6, 5, 7+) to find generating sets of minimal size.
result For n ≥ 7, the commutator subgroup of the braid group is generated by two elements.
We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating cur…
Survey on invariant quasimorphisms and their relation to stable commutator length.
problem Understanding the relationship between invariant quasimorphisms and stable commutator length.
method Review of existing methods and examples in invariant quasimorphisms and their relation to stable commutator length.
result The existence of non-extendable invariant quasimorphisms is closely related to the behavior of stable mixed commutator length.
Paper proves non-compact inaudibility of symmetry and commutativity.
problem Proving inaudibility of symmetry and commutativity in non-compact settings.
method Using isospectral pairs of generalized Heisenberg groups.
result Proved inaudibility of weak symmetry and commutativity.
The paper finds a criterion for generating commuting pairs of structures.
problem Defining commuting pairs of generalized structures on product spaces.
method Proves a theorem for generating commuting pairs of generalized almost complex structures.
result Simple criterion for generating commuting pairs of generalized structures.
The paper proves norms on braid group commutator subgroup are unbounded.
problem Understanding norms on the commutator subgroup of infinite braid groups.
method Constructed stably unbounded norms and showed equivalence to biinvariant word norms.
result Found norms on commutator subgroup are equivalent to biinvariant word norms and are stably unbounded.
We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2 . It is shown that the commuting Ricci tensor gives that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a complete classifi…
The paper defines constraints for commuting endomorphisms in generalized tangent bundles.
problem Identifying constraints for commuting endomorphisms in generalized tangent bundles.
method Using Gröbner basis techniques to construct and study tensors forming ideals.
result Explicit construction and study of tensors forming ideals of commuting endomorphisms.
An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almo…
For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of…
The paper finds commutator formulas for Gradient Ricci Shrinker metrics and applies them to linear stability.
problem Linear stability of Gradient Ricci Shrinker metrics.
method Found commutator formulas and generalized a stability theorem.
result Generalized a necessary condition for linear stability.
The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations, using Nambu structures, i.e. integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures, and show some basic results about commuting Nambu struc…